224 Model Atmospheres where n>0 is an appropriate weight factor, and consider the transfer problem only at the continuum head, v = v0 [see also (195)]. If we define c = C/[4nw0(x0/hv0)B0 + C] then S0 = (1 - r)[i\ - s)J0 + fiB0] + rBo ~ 0 - m0 + zB0 (7-132) where & = £ + and the transfer equation in the Eddington approximation is 1 -(d2J0/dr0z) = e(j0 - S0) (7-133) Assuming p. is constant, and that the atmosphere is nearly isothermal so that B0 is roughly constant, the solution of equation (7-133) can be written immediately [cf. equation (6-9)] as JQ = B0{& + 1 - exp[-V^To]}/a + s= (7-134) which shows that (a) the solution thennalizes only at depths > l/e'% and (b) the departure coefficient at the surface is b^O) « Because & « 1, the departure from LTE is large, and persists to great depths. The actual value of s is a bit difficult to estimate if collisions dominate (i.e., e. > r), for the density, and hence &, increases exponentially into the atmosphere; on the other hand, the parameter r is almost depth-independent, and often dominates. For example, at T ~ 25,000°K, r ~ 1CT2 [see (633, 193)] and b1 ~ 10, which is in agreement with calculation. In summary we see that it is important to use the information inherent in the statistical equilibrium equations directly in the transfer equation; further, given the dominant scattering term, it is obvious that one must use care in satisfying the constraint of radiative equilibrium. We now shall consider a partial linearization method that is fairly general, and provides a satisfactory means of solving the non-LTE continuum-formation problem, with energy balance; the method fails, however, when lines are included, in which case one must use the complete linearization method described in the next subsection. The basic idea in the method [cf. (40; 41)] is to incorporate the statistical equilibrium equations into the transfer equations by manipulating them analytically to obtain explicit expressions for source functions, and to incorporate the constraint of radiative equilibrium into the transfer problem by linearization. Then, in principle, one solves all three sets of equations: transfer, statistical and radiative equilibrium, simultaneously. The transfer equation is (7-135) 7-5 Non-LTE Radiative-Equilibrium Models 225 where (omitting bound-bound transitions) * = 2>" ^""vA7K» + ZWUV, T)(\ - c-W) + nec and nv = (2hv3/c2)e~hvikT y>fa,.» + X nenAJy, T) (7-136) (1 _ „-hv/kr = k*Bv (7-137) We now divide the spectrum into a series of characteristic frequency ranges {vi0 < v ^ vti) upon which we assume the opacity and emissivity of a particular continuum i -> k is much larger than all others. For example we might assume that the Lyman continuum dominates for all frequencies such that X < 912 A, the Balmer continuum dominates for 912 A ^ A ^ 3650 A, the tree-tree continuum dominates for X > Xu the threshold of the last bound level, etc. We then factor out the dominant terms from both the emissivity and the opacity, and write, for (vi0 ^ v s£ vn), (nM = [nffln, - ti?*-HkTmivBv = SMibi - e-"^T) (7-138) where the three quantities Su ^ - nfe'hv/kT)/nf](Kt/Xx) (7-139) /v. and tv are all evaluated using current values of the temperature and level populations, whereas the term {bt ~ e~'"'!kT) in equation (7-138) is to be replaced with an analytical expression, involving the radiation field, to be found from the statistical equilibrium equations. The statistical equilibrium equation for level i is [cf. equation (7-126)] M + £ c.. = rkj + CiK + y bjCiJ (7-140) where rik and rki are defined, as usual, by equations (5-66) and (5-67) Solving for />, - e-hy>kT we then ha- jj- < i Ki - e-'"!ktrih (7-341) 226 Model A tmospheres 7-5 Non-LTE Radiative-Equilibrium Models 227 so that, defining (v = neajxv (again using current values), the source function in equation (7-135) on the range (vi0 ^ v < vfl) can be written as where (Div = 4jra,-K(v)//7v 7,-v ^ BJDiy (7-143) and £>,v is the term in square brackets in the denominator of equation (7-141). In the evaluation of yiv and t:iv we again use current values for the b/s of other levels and for the terms in Jv and T; it is thus clear that a large amount of information is lagged in this technique. Note that eiv contains an overlap integral from those frequency ranges outside the presumed range of dominance (v;0, vn) of the transition i -» rc; usually this integral is small compared to the term displayed explicitly in the scattering integral. Equation (7-142) accounts for the most important terms directly, but because the information involving other levels and in the nonlinear terms is lagged, iteration will still be required. Finally, discretizing, we have equations of the form d\fkJk)/dzk2 = j, - U (yik | w|<&h j/ + s;a) - :kJk (7-144) V l=km J on the range (/vi0 ^ k ^ kn) delineated for the 7th transition. We must now solve the system (7-144) subject to the constraint of radiative equilibrium which requires that k i *n I*" yJ-h - sv) dv = I wkXkjk -II wfc& JO k=1 i= 1 fc = k,-0 l = ki0 = 0 (7-145) We suppose that B„ the value of the Planck function that does yields radiative equilibrium, can be written in terms of the current value Bv° as Bv = Bv -i (cBv°ldT) AT. Substituting into equation (7-145) we find AT = I w^kJk - EL (7-146) where V=l ft = fciO // and £2 = £ I w^t^/fiTl Thus equation (7-144) becomes (7-148) (7-149) d2(fM/drk2 = (1 - CJJ, f wpuJt + £ik(dB°/0T)k I wk4k.Jk. (7-150) which is completed with boundary conditions given by equations (7-37a) and (7-37c) (the latter specifies the flux explicitly). Equation (7-150) has the following important properties, (a) It incorporates both the equations of statistical equilibrium and radiative equilibrium in terms of the new (i.e.. as yet undetermined) radiation field; thus a solution of the system satisfies both of these constraints automatically, (b) Cancellation of large scattering terms is done analytically in the coefficients [cf. espcially equation (7-147)] of the radiation field before the transfer equation is solved; hence good control of the residual thermal terms is obtained, (c) The equations are of the standard Feautrier form [equation (6-31)] with the frequency coupling spread over the entire spectrum, and can be solved using the usual elimination scheme of equations (6-39) through (6-41). (Why is the Feautrier method more efficient than the Rybicki scheme in this problem, even though the number of frequencies is large?) The whole solution proceeds iteratively. After solving equation (7-150) for Jv, the temperature distribution is revised via equation (7-146), the new radiative rates are used to solve the statistical equilibrium equations for new occupation numbers, new Eddington factors are then calculated from the revised values of ;/v and y_v, and the procedure starting with equation (7-139) is repeated. Methods of the type described above (which is related to the equivalent-two-leuel-atom approach for line-formation problems; see $12-2) have been developed by Feautrier (2(1) and Auer and Mihalas (40; 41). Feautrier's 228 lOg T F1GURH 7-19 Non-LTE departure coefficients in the first two states of hydrogen for a model with Torf = 15,000"K and log g = 4. Dashed curves: Lyman, Balmer and free-free cotUinua only; full curves: La line included. The optical depth scale in the abscissa is measured just longward of the Balmer jump. From (40). by permission. formalism differs significantly in appearance from that displayed above, but the physical content is about the same. These methods have been used successfully to construct continuum models for B-stars, including the Lyman continuum in cases where lambda iteration would be utterly hopeless. Characteristically the ground state of hydrogen becomes strongly overpopu-lated in the outer layers; see Figure 7-19. This has important consequences for the temperature structure and emergent spectrum of the atmosphere, as will be discussed below. Attempts have been made (40; 41) to apply the techniques described above to cases including line transitions. It is essential to include the lines because (a) they drastically affect both occupation numbers and energy balance in 7-5 Non-LTE Radial be-Equilib t'ium Models 229 the atmosphere, and (b) we obtain reliable level-populations that can be used to compute line profiles (to be compared with observation) only if line transitions have been taken into account in the solution of the statistical equilibrium equations. The importance of these effects is illustrated in Figures 7-19 and 7-20. In Figure 7-19 we see that the effect of including La is to drain the n = 2 level, which goes from being slightly overpopulated to markedly underpopulated. The drop in bx is caused by the drop in temperature; actual occupation numbers ni do not change much [see (40)]. In Figure 7-20 we see a similar effect for b2 and 63 when Ha is taken into account. If we had computed the Ha profile with the continuum-only values (which are similar to those of Figure 7-18) we would have found a spurious emission core in the line (because n — 3 would have been too overpopulated relative log T I'-IGURIi 7-20 Non-LTE departure coefficients in the second and third level of hydrogen in a model with Tefj = 15,000"K and log g = 4. Dotted curves: Lyman, Balmer, Paschen, and free-free continua only; full curves: tix included. Optical depth scale as in Figure 7-19. From (41), by permission. 230 Model Atmospheres to n = 2); inclusion of the hnc from the outset drains n = 3 into « - 2 and produces a strong absorption line (we shall compare with observations in §12-5). The above method does not work well for determining the temperature structure when lines are included; thus in (40), the solution for the temperature structure including La tended to stabilize rather than converge, and in (41) a solution including Ha (which is formed at about the same depths as the Lyman continuum) tended to become strongly unstable. The reasons for the failure can be traced to basic inadequacies of the method. (J) Many of the terms in the equations are lagged, and interactions among transitions are treated only iterative!}' rather than collectively from the outset. (2) The temperature dependence of many variables (e.g., the opacity) is ignored in the linearization procedure. (3) Most important, it is presumed that the temperature is, in some sense, "the fundamental variable" of the problem. Such a presumption is adequate for treating continua that are coupled strongly to the temperature via radiative recombination but, as noted before, is totally unsatisfactory for lines, where both emission and absorption rates are decoupled from the temperature structure. Let us now consider a method of great generality and power that overcomes these difficulties entirely. THE COMPLETE LINEARIZATION METHOD From a physical point of view, the solution of the non-LTE stellar atmospheres problem entails the specification, at each point in the medium, of the distribution of the radiation field as a function of frequency, the temperature and density of the material, and the distribution of atoms and ions over all bound states. These distribution functions are to be determined in such a way that the constraints of energy balance, momentum balance (i.e., hydrostatic equilibrium in the present work), steady-state statistical equilibrium, and charge and number conservation are all satisfied rigorously. To achieve the desired result, it is important to recognize two basic physical points, and to construct a generalized method in the light of them. First. no one variable is more "fundamental" than any other, for they all internet. Thus we must regard the solution at a given depth-point m(J to consist of the vector ^ = (i1)...,JK,WsT,«r,n1,...>nJr, (d = 1.....£>) (7-151) where D is the number of depth-points, K is the number of frequencies, and L is the total number of bound levels (of all kinds) considered. (Note that there is, in principle, redundant information in as written; e.g., given all the J's, the n-s follow from the rate equations. However, all of the information is of interest m practice, and hence will be retained.) The coupling among ail 7-5 Non-LTE Radiative-Equilibrium Models J31 these variables is intricate and essentially complete. For example, we have already seen in §5-5 that a change in the radiation field at any frequency implies a change in the occupation numbers of all bound states [cf. equation (5-108)]; further, any change in a radiation field implies a change in the local temperature and density, and so on. Second, the variables interact globally throughout the atmosphere, and a change in any variable at a given point implies changes in all other variables at all other points. Thus, if we alter occupation numbers locally, this changes the emissivity and opacity of the material, and hence, via the transfer equation, the radiation field throughout the atmosphere; when scattering terms dominate the source functions, information propagates over large distances in the material, and the global nature of the interaction is accentuated markedly. A truly adequate method must, therefore, put all variables on an equal footing, allow fully for all possible couplings among the physical variables resulting from the imposed physical constraints, and for the interaction of each variable with all others at every point in the atmosphere via transfer equations and requirements of energy and momentum balance. To determine the solutions \J/rf we require a total of X + L + 3 equations; we may choose these to be K transfer equations (Jt, k = 1, . .. , K); the equation of hydrostatic equilibrium (N); the equation of radiative equilibrium (7"); a total particle-number conservation condition {ne\\ and L statistical equilibrium equations specifying level-populations, total abundances of chemical species, and total charge conservation (;tl5 . . ., nL), exactly as written in §5-4 [see equations (5-91) through (5-93)]. The essential difficulty to be faced now is that these equations are nonlinear and must, in general, be solved by some kind of iteration procedure. In particular, we suppose that our desired (but as yet unknown) solution can be written in terms of the current (but imperfect) solution \\td° as t|/d = \fyd° + &\{td. We then choose Sv]/d so as to satisfy all constraints more closely; i.e., if fd{\\id) = 0 represents the entire system of constraints, we demand that f^f* + Svj/j) = 0, and solve for 5\j/rf by linearing the entire system: I S\jfd 0, the corrections 6\|/d -> 0. We have already stressed the physical implications of the complete linearization. Mathematically, the equations are internally self-consistent, and are equivalent to a generalized Newton-Raphson procedure; convergence, if obtained, should therefore be quadratic. This is not achieved in practice because, after each set of corrections 8\|/ have been determined and applied to the solution, it is necessary to recompute the Eddington factors (which were assumed fixed under linearization) from a formal solution using current source functions, as was done in the LTE case as well. Nevertheless, an order-of-magnitu.de reduction in the five's is often obtained from successive iterations, and on the whole the procedure is stable and efficient, and it easily handles physical problems that defeat the other techniques described above. It appears that at present the complete linearization method is the best technique available for solving non-LTE stellar-atmosphere problems. The method can handle multilevel, multiline, multispecies problems of great generality, and treats all the physical constraints because the full interaction 234 Model Atmospheres 235 among all variables is allowed, and a priori assumptions about the dependence of quantities upon only a restricted set of variables (e.g., T) are avoided. The complete linearization method has been used with good success to construct models in the temperature range from A-stars (42; 43; 368; 224) through O-stars (436 ;45). The early work, carried out before the introduction of the variable Eddington factor technique (44) made use of the Eddington approximation (i.e., one angle-quadrature point) and has been supplanted by the later calculations. An extensive grid of models for O- and B-stars is available (430;432). In this work, departures from LTE are taken into account for the first five levels of hydrogen, the first two levels of He I and He II, and for an "average light ion" (which represents C, N, and O) consisting of five stages of ionization, each with a ground-state only. These models typically allow for six hydrogen-line transitions: La, Lp, Ly. Ha, Hfi, and Pa for the O-stars, and Ha, Hfi, Hy, Pa, Pfi, and Ba for the B-stars (for which the Lyman lines may be set in detailed balance), and yield results for the physical structure of the atmosphere, continuum parameters, and H-line profiles. non-lte effects on energy distributions Departures from LTE affect both the continuum and the line-spectrum from a stellar atmosphere; the discussion in this section focuses on the continuum only. Enough results are available for models of early-type stars to allow us to delineate the regions of the H-R diagram where departures from LTE have important effects and where they can be ignored. At the present time the non-LTE models do not include line-blanketing effects; thus we compare LTE and non-LTE unblanketed models to determine differential effects (which then, presumably, can be applied as corrections to values of parameters obtained from blanketed LTE models). Extensive sets of results for continuum jumps and Stromgren-system colors are given in (430; 516, 241), and complete energy distributions are given in (432); we shall summarize some of the principal results here, and we suggest the reader examine the literature cited for further detail. For the B-stars (15,000°K ^ Teff ^ 30,000°K) the effects of departures from LTE in the visible spectral regions are generally found to be negligible for main-sequence stars, but become important for giants and supergiants (low gravities). Non-LTE effects oil colors are shown in Figure 7-21. There we see that results for both LTE and non-LTE models at a given gravity lie along the same curves, but the position of models with given Teff is somewhat different. Thus if we ignore departures from LTE, we introduce a systematic error into estimates of 7efr. For main-sequence (log y = 4) stars the errors can be neglected, whereas for log g = 3 the errors are about 200f'K and at log q = 2.5 they are about 500CK; these errors are small but systematic and 15 ° -0.11 -0.10 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 (b - y) FIGURE 7-2] Theoretical Stromgren-system colors for LTE and non-LTE models. Ordinate: [it - b); abscissa: (h - y). LTE values are represented by open circles and dashed curves; non-LTE values by solid dots and curves. Curves are labeled with log g, and individual models are labeled with 7 eff/103. could have significant consequences in certain applications. The effects of departures from LTE upon the continuum jumps at the Lyman, Balmer, and Paschen edges (DL, DBi and DP) are fairly substantial for B-stars. The Lyman jump is increased by non-LTE effects for most B-stars because the ground-state is overpopulated (b1 » 1) and hence the flux for A < 912 A is reduced; these changes are significant in making estimates of the far-ultraviolet energy output of these stars. Balmer jumps are generally decreased by non-LTE effects in B-stars because b2 < 1 while 63 > 1, so the opacity contrast across 236 Model Atmospheres 237 the Balmer edge is reduced by departures from LTE. The Paschen jump is only slightly affected because both /?3 and b4 > 1. and the opacity ratio stays about the same. In a plot of DB vs (h - y) (or some other flux ratio), the LTE and non-LTE models at a given gravity lie on the same curve, but slightly shifted (recall Figure 7-21). Tf one assigns values of TelT to stars using such a diagram, the departures from LTE again imply systematic errors in estimates of Teff; these errors are negligible for main-sequence stars, are about 350r'K at log g = 3, and are about 500' K at log g = 2. From the point of view of obtaining direct observational evidence for departures from LTE in the continuum, the results mentioned above are not helpful because there is not a clear discrimination between the two cases. Strom and Kalkofen (611) pointed out that the parameter = DP/DB provides a sensitive observational indicator of non-LTE effects, as may be seen in Figures 7-22 and 7-23. The LTE models predict values of 4> of about 0.16 to 0.17, independent of gravity, whereas the non-LTE models predict much larger values of increasing with decreasing gravity. This effect was found observationally (583); thesupergiants have systematically larger values of

= £V//->B; abscissa; Balmer jump DB. Solid dors and curves: non-LTE values: open dots and dashed curves: LTE values. Curves are labeled with log ij. 0.25 0.24 -0.23 -0.22 -0.21 -0.20 0.19 -0.18 -0.17 - 0.16 J__L 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 FKiIJRH 7-2? Same as Figure 7-22 for middle B-type .stars. Shaded area contains predictions of LTE models at various values of g: curves (labeled with surface-gravity g) show non-LTE values. new Vega calibration. Further, departures from LTE have major effects on the ultraviolet continua of A-stars (587; 588) for which it is found that the ground-states of CI and Sil are strongly underpopulated, and the flux obtained from LTE calculations is too low by very large factors. Only when the non-LTE effects are taken into account is a satisfactory agreement with space observations obtained for these stars. For O-stars (Tefr > 30,000°K) the effects of departures from LTE on visible continuum parameters and colors are much more important. For these stars, as described earlier, the ground state hydrogen becomes underpopulated, hence the flux below the Lyman limit increases, and the Lyman jump decreases. In contrast, at the X227 A ground-state edge of He II, the flux is decreased by non-LTE effects because n = 1 of He+ is overpopulated [see (45; 432)]. These changes are of importance in estimating the energy output from O-stars into, say, nebulae or the interstellar medium. The most significant change in the visible region is that the Balmer jump predicted by the non-LTE models is about 0.07 mag larger (for log g = 4) than that from LTE models, and shows but little variation with gravity, whereas the LTE —I_L___1__-_L___ 30 35 40 45 50 FIGUKt 7-24 Theoretic;]] Balmer jumps for O-stars. Ordinate: DB in magnitudes: abscissa; Teff/103. Solid symbols and curves: non-LTE models; open symbols and dashed curves: LTE models. Cur ves are labeled with log g. From (45). by permission. models predict emission edges at low gravities (see Figure 7-24). The absolute sizes of the non-LTE Balmer jumps are in excellent agreement with observations of O-stars if the Hayes calibration for Vega is adopted (as now seems correct). Further, the differential behavior of low-gravity versus high-gravity non-LTE Balmer jumps agrees with observation while the LTE results do not (405), and the same conclusion holds true for the differential behavior of O-star versus BO-star Z>,/s (45); both of these results are independent of the calibration and support strongly the non-LTE calculations. The fact that the Balmer jump remains about constant in strength for the non-LTE models but weakens markedly in the LTE models implies very large differences in the computed colors [say (u — 6)and(6 — y)] of the two sets of models(516.241). At the highest temperatures, the discrepancy in TelT for a given value of (u - b) is some 15,000°K. Comparisons of T[ff derived from colors with those derived from helium-line strengths supports the non-LTE results. The comparison could be much refined yet by using a large body of data, and comparing the O-stars differentially with B-stars (thus eliminating calibration problems) [see also discussion in (516, 241)]. In summary, departures from LTE have significant effects on the visible continuum parameters of O-stars and B-type giants and supergiants, but are negligible for B-type main-sequence stars. Deviations from LTE produce 7-5 Non-LTE Radialive-Equiiibrium Models 239 very large changes in the ultraviolet flux below the hydrogen Lyman edge, and in the resonance continua of certain light ions such as CI and Sil. For later-type stars one must examine departures from LTE in the H~ ion. With present estimates of the relevant reaction rates (556; 188), the predicted effects (607) are negligibly small for G and K main-sequence stars and giants [see also (491; 492)]. Virtually nothing is known about the possible importance of departures from LTE in other types of stars (e.g., M giants or super-giants), and much work remains to be done to evaluate these effects. Finally, the problem of accounting for non-LTE effects in line-blanketing remains to be attacked. TEMPERATURE STRUCTURE1 THE CAYREL MECHANISM AND LINE Ei-'l'ECTS As we have seen in Chapter 3, the temperature in a grey atmosphere decreases uniformly outward to a limiting value T0/Teff = 0.811. Further, for a nongrey atmosphere that has a large opacity jump or strong lines, under the assumption of LTE the boundary temperature falls below the grey value (exceptions: scattering lines leave the boundary temperature unchanged and, in certain special cases, "new" absorbers can cause small temperature rises). In sum, the basic prediction for LTE radiative-equilibrium atmospheres is that the temperature distribution is a monotone decreasing function outward. Often the temperature structure exhibits plateaus where the then-dominant transition (e.g., the Balmer continuum) has become optically thin while other transitions (e.g., the Lyman continuum and Lyman lines) are completely opague. followed by a series of drops as successively more opaque transitions become optically thin in turn. For non-LTE atmospheres the situation is quite different, and typically the temperature distribution passes through a minimum and then shows a rise outward. The basic reason this occurs was pointed out by Cayrel (142; 283, 169), who called attention to the similarity between the physical conditions in the outer layers of a stellar atmosphere and those in a nebula surrounding a star. Suppose the energy balance is completely determined by absorption and emission from a single state (e.g., the sole bound state of H~, or the ground slate of hydrogen). Then, in LTE, the condition of radiative equilibrium can be written n* f" avBv(T0) dv = n* f"' avJv dv = n*W f~ glvBs.(Tr) dv (7-160) JVO JVij JVf) where the last equality introduces a parametric representation of Jv in terms of a dilution factor W < 1 and a radiation temperature TR. As W < 1, it is clear that T0 < TR; in particular, if W = \ and TR = Tcff, T0 has essentially the grey value. The result just obtained follows because we have forced 240 Model A imospheves LTE. But now suppose that we allow departures from LTE; then n\ f* avBv(T0) dv = MT M„ rfv = b,n%W f" avBv(TR) dv (7-161) In the limit of low density the departure coefficient is determined by the photoionization and recombination rates [cf. equation (5-95)], and (7-162) bl = (avBjhv)dv (ajjhv)dv Hence combining (7-161) and (7-162) we have (*'" zvBv(TR) dvi \"yavBv(TR)v-1 dv Jvo / Jvo (7-163) which shows that T0 = TR, independent of the value of W\ Thus we expect typically the temperature will decline from T = Teff near T m 1, to a value Tmin) approximately equal to the grey-body boundary-temperature, and then rise to a value T0, with Tmil, < T0 < Tcff. These results can also be viewed in terms of the "quantity" and "quality" of the radiation field (239). That is, in LTE, it is the energy density (i.e., quantity) of the radiation that fixes T0; in the non-LTE case, the energy density may be lower than its equilibrium value, but each photon has an energy characteristic of TR and hence can still ionize the material and deposit an excess energy, per ionization, which is again characteristic of TR [see also (240)]. For the sun, Tef1 = 5900°K, and Cayrel estimated T0 - TR a 5600°K compared to Tmin i 4800°K (grey value). From a detailed calculation of a solar radiative equilibrium model, allowing for departures from LTE in the H~ ion, Feautrier obtained (211) Tmin » 4700°KandT0 =s 5200°K. Similar effects are found for O- and B-stars in which the main source of surface heating is the Lyman continuum. For example, in Figure 7-17 we saw that the LTE temperature structure of an atmosphere with Terf = 15,000°K, log g = 4, had a plateau at 10,400'JK, where the Lyman continuum was optically thick and all others were transparent, followed by a drop to T0 = 9400°K when the Lyman continuum became transparent. In the non-LTE model, the temperature for 1CT4 ■£ r 5 10"2 lies below the LTE value (because b2 < 1 in the Balmer continuum and therefore the efficiency of healing is reduced). The non-LTE temperature distribution shows a minimum near 10,100°K and then a rise outward to T0 « 10,350°K, which lies 1000°K above the LTE surface value. The agreement of this value of T0 with the LTE Balmer plateau temperature is probably fortuitous. In fact, it is hard 7-5 Non-LTE Radiative-Equilibrium Models 241 to make an a priori estimate of an appropriate value of TR in the Lyman continuum, for it, unlike H~ which is nearly grey, shows a sharp fall-off in opacity with increasing frequency, which implies that the surface radiation at high frequencies emerged from deeper, hotter layers and is characterized by larger values of TR. The final result is established by some kind of average of TR over frequency. Similar results are obtained at other effective temperatures (40; 211); detailed discussions of how particular transitions are affected may be found in the references. Let us now consider the effects of lines on energy balance in the non-LTE case; here again it will repay the reader's effort to reread this material after Chapter 11 has been studied. A qualitative feeling for the results to be expected follows immediately from the form of the non-LTE source function for a collision-dominated line (cf. §11-2), namely (7-164) where J denotes the average of Jv against the line-profile 0V. By an argument (625) exactly analogous to that leading to equation (7-86), we find a result that differs only trivially from that obtained there, even though the line-source function has a noncoherent rather than coherent scattering term. We therefore reach the same conclusion as before: for LTE (e = 1) there is a large drop in boundary temperature, but when e « 1, the lines have practically no effect on the boundary temperature. The argument is verified by a detailed picket-fence calculation (448) that allows the continuum to adjust self-consistcntly. In this work the line-strength is assumed to remain unchanged, which would be valid, say, for the resonance lines of the dominant ion of a given species (e.g., Ca 11 in the solar atmosphere). A very detailed calculation of non-LTE line-blanketing effects in the solar atmosphere (17) leads to the conclusion that a self-consistent model yields Tm-1B ~ 4330:,K, and that the outward temperature rise driven by the Cayrel mechanism in the continuum is strongly resisted by the lines. In early-type stars the effects of lines upon the energy balance can be studied in non-LTE models constructed with the complete-linearization technique (references given above); several interesting results emerge. For example, in Figure 7-17 we see the effect of including La (only) in addition to the Lyman continuum. In LTE the boundary temperature drops about 1600°K, from 9400°K to 7800°K; but in the non-LTE case the boundary temperature had risen to T0 « 10,350°K (owing to heating in the Lyman continuum), and inclusion of La produces a temperature drop of only about one-third that obtained in LTE, to 9800°K. This result is not surprising, for £ is « 1. The final temperature structure is relatively complex in the non-LTE case. An even more interesting example is shown in Figure 7-25, in which we see the importance of the coupling between lines and continua. Here the model hydrogen atom had three levels, and the formation of the first three 242 243 12 NLTE, no lines LTE, no lines NLTE. Hy. LTE, li'i / i / / FiGURi: 7-25 Temperature distribution for LTE and non-LTE models with Tlui curve: non-LTE model including lines; dashed curve: non-LTE continuum-only model; dolled curve: LTE model. Cross-bars on curves mark optical depth unity in most opaque region of transition indicated (those with ion designations are ground-state continua). From (436), by permission. -7 -6 -5 -4 -3 -2 -1 log in lem of non-LTE line-blanketing in stellar atmospheres has barely been approached, and a large amount of work remains to be done. 7-6 Extended Atmospheres For all of the models discussed thus far, it has been assumed that the atmosphere is stratified in plane-parallel layers; this is an excellent approximation when the density scale-height in the atmosphere is small compared to the radius of the star. However, many stars, in particular the supergiants and Wolf-Rayet stars, have extended atmospheres whose thicknesses are an appreciable fraction of a stellar radius; to a first approximation we suppose that these atmospheres are spherically symmetric. Atmospheric extension has important physical and observational implications. Thus stars with extended envelopes show a continuum energy distribution that has an anomalously low radiation temperature in comparison with the excitation temperature inferred from spectral lines. Equivalently, the energy distributions are "flatter", as a function of frequency, than those of main-sequence stars (which have compact, planar atmospheres) of the same spectral type, and show excess emission in the infrared and a deficiency in the ultraviolet. Almost always there are indications of rapid atmospheric expansion in stars with extended envelopes, so that one should consider dynamical models that include the effects of flow; we shall examine such models in Chapter 15, but for the 244 Mode! Atmospheres 7-6 Extended Atmospheres 245 moment will deal with the more limited problem of solving the transfer equation in static extended envelopes. In an extended, tenuous atmosphere, the radiation field at large distances from the underlying stellar disk becomes very dilute, and is confined primarily to a narrow solid angle (that subtended by the disk) around the radial direction. These facts imply that the temperature structure of the atmosphere must be quite different from that in a planar model, and that mathematical complications introduced by the angular peaking of the intensity will be encountered. The equation of transfer to be solved (cf. §2-3) is n(?lJ0r) + f-'d - fi2)(dljd(i) = n, - xJ* with moments (cf. §2-4) r-2[d(r2Hv)/dr-] = ih - lvJv (dKJdr) + r_1[3Kv - Jv) - -%VHV and or (7-165) (7-166) (7-167a) (7-167b) It is obvious that these equations are more complicated and more difficult to handle than their planar counterparts. For example, equation (7-165) is a partial differential equation involving explicitly two independent variables. Similarly the moment equations do not yield a simple form if, for example, we eliminate Hv between equations (7-166) and (7-167) (though a transformation will be described later that does allow a reduction of the two equations to a single combined moment equation of an attractive form). The moment equations could be dealt with, at least approximately, if we could relate Kv and Jv accurately, as is possible in the planar case. But now, even though./^. -> \ at great depth where the radiation field is homogeneous, fv -» 1 near the surface [recall Exercise (1-12)], and a direct application of the Eddington approximation, which gives remarkably good results in the planar case, will not be even roughly correct near the surface. As was recognized by Mc Crea (413) and Chandrasekhar (148), it is possible to construct a consistent approximation scheme in a tenuous atmosphere that envelopes a parent star of radius r^, by considering averages of the radiation field computed on the range //„. < /i < 1 [where = (1 — r^/r2)"*] separately from those computed on the range — 1 < /x ^ /i+ [see also (403)]. But such a method is suitable only if can be chosen unambiguously; for extended photospheres of appreciable density and nonnegligible optical depth this method breaks down [see, however, (635)]. It is thus not surprising that effective and general methods for treating transfer problems in spherical geometry have been slow to develop. A general and flexible numerical technique giving a direct solution of equations (7-165) through (7-167) will be presented later in this section. To treat the grey problem, an approximate solution will be obtained merely by patching together asymptotic results for the two limiting regimes mentioned above; this provides a good starting point for estimation of extension effects on the emitted energy distribution SPHERICAL GREY ATMOSPHERES The problem of grey spherical atmospheres in LTE and radiative equilibrium was analyzed approximately by Kosirev (360) and Chandrasekhar (148); numerical results of high precision have recently been obtained by Hummer and Rybicki (323). If we assume /v = ■/„ and integrate equations (7-166) and (7-167) over all frequency, omitting the subscript v to denote integrated quantities, we obtain r-2[d{rH)jdr~] ^ 0 (7-168) and \_c\fJ)/cr'] + r"l(3/ - l)J = ~XH (7-169) where in equation (7-168) we have demanded radiative equilibrium Equation (7-168) yields an integral for the total flux, r2H = H0 = L/\6n2 (7-171) where L is the luminosity of the star. Define the optical depth, measured radially from an arbitrarily large outer radius R, to be (7-172) Deep within the atmosphere (i.e., r « R, T » l), we expect the radiation held to become isotropic and / -> i; in this limit equation (7-169) becomes (cJIdr) = -y,H = ~3yj-~2H0 which yields the integral jw = ffo(3j0v2rfT' + C (7-173) (7-174) If the usual Eddington-type boundary condition at x = 0 could be applied then J(0) = 2ff(0) = 2H0/R2 so that ' J(r) = R~2H( 3Jo(K2/r2)dT' + 2 (7-175) 246 Model Atmospheres a result obtained by Chandrasekhar (148). If, further, we impose the-assump-tion of LTE, then rjv = xyBv = yBv and from equation (7-170) it follows that J(r) = B(t) = aT^jn, which assigns a temperature structure to the atmosphere. Equations (7-174) and (7-175) can be valid only at great depth. Near the surface, the free-flow regime occurs and / -> 1, in which case equation (7-169) becomes [d(r2J)/dr-] = -yr2H = -yH0 (7-176) which yields J(t) = r2H0(r + C) (7-177) In the limit / = 1, J(0) = H(0) = HJR2, so equation (7-177) becomes J(t) = r2H0(z + 1) (7-178) a result that is expected to be valid only when r « I, r ^ R. Considerable progress can now be made if we adopt a power-law opacity (i.e.. x " Cnr~"), as was done in the earliest work (360; 148). As noted by Kosirev, there is strong physical motivation for this choice in an expanding atmosphere, for the equation of continuity (see §15-1) demands thai pvr2 = constant, where p is the density, and v is the velocity of expansion; in the limit of very rapid expansion, when v > vCfc4(,e (actually observed), the material moves at practically constant velocity, so that p ~- r~2. The opacity of the material can be expected to vary as some power of the density (e.g.. linearly for electron scattering or as the square for free-free) and hence as some power of 1/r. Substituting the above expression for / into equation (7-172) and adopting, for simplicity, R = oc, we have T(r) = O"'"-"/^ - 1) (7-179) Using equation (7-179), the limiting form for J(t) las r -> 0, t » 1) is J -> [3(/i - l)/[n + l)]HQr-2x (7-180) while equation (7-177), valid for x < 1, remains unchanged. It is therefore attractive (387) to interpolate between these two extremes with an expression of the form M^)N(^t)] ™ Comparison with precise calculations demonstrates that equation (7-181) is quite accurate (323). Using the identity of J and B, and writing T1 for the temperature at t = 1. we can rewrite equation (7-181) as T(T) = T,t*" " + (n+ l)/3(n - (n + l)/3(« - I)]}* (7-182) 7-6 Extended Atmospheres 247 Equation (7-182) reveals an important characteristic difference between the planar and spherical cases, namely T -> 0 as r 0 in an extended atmosphere, rather than approaching a finite value. From this result we can see that the contribution from the outer coo! layers, which occupy a large volume, will enhance the flux observed at longer wavelengths and lead to the distinctively flatter energy distribution mentioned above. We may calculate the flux received by an observer at a great distance D from the center of the star by using the (p, z) coordinate system shown in Figure 7-27. The impact parameter p is the perpendicular distance of a ray from a parallel ray passing through the center of the star; z is the distance along the ray. measured from the plane through the center of the star perpendicular to the central ray. We shall take z to be positive towards the observer and formally place the observer al z = oo for purposes of calculating integrals. The coordinates (p, z) are related to polar coordinates [r, 6) as follows: r = rcos 0, p = r sin 0, and r = (p2 + z2)K If we choose a value of p, then the equation of transfer along the ray in the direction of increasing z is (dljdz) = ,7l. - yjv (7-183) which follows from first principles. The formal dilution of equation (7-183) for the intensity emerging at z = cc along the ray with impact parameter p may be written immediately as lv(p, go) = 5v[T(p, z)] exp[-t(p, z)]Z(p, z) dz (7-184) where zip, z) is the optical depth measured fromz = cc, inward along the ray. ^> Observer (z = co) F1GUKL 7-27 Coordinate systems for solution of transfer equation in spherical geometry. 248 Model A tmospheres Exercise 7-15: Calculate the operator (c/dz)p and show thai equation (7-183) is exactly equivalent to equation (7-165). In the (/>, z) coordinate system, the lines p = constant arc the characteristic rays of the partial differential equation, which reduces to an ordinary differential equation along these particular curves. The total flux received by the observer per unit receiver area is /v = 2nD'2 Jo" Iv(p, oc-)p dp (7-185) Following Kosirev (360) [see also (61, 165)] we change variables in equation (7-1841 to 0 = cos" \z/r) and write r(p, z) = x{p, 0) which, using a power-law opacity, becomes z(p, 0) = CHp~{n~1 V«(^) where Then (7-186) Iv(p, go) = Or'""11 £ 8,[T(p, 0)] exp[-T(p, f?)] sin""2 0 d0 (7-187) Substituting equation (7-187) into equation (7-185), and introducing successive transformations from p = r sin 0 to r{r), the radial optical depth given by equation (7-179), we obtain /v = %{RJDf | 5v[T(t)]t ^-^n{z)dx (7-188) where rx is chosen as the radius at which x = 1. and €>„(?) = 2 £ exp[-(n - ljx esc""1 0 oo implies planar models; n small yields large extent). Observationally, color temperatures Tc are assigned, at a wavelength Xc, by measurements of colors in a filter-system, or of the spectrophotometry gradient; both of these approaches, in effect, measure the slope of the continuum. Van Blerkom (61, 165) has calculated Tc at lc = 5000 A for various models with T1 -- 50,000°K. He finds that, atn - "oo, Tc/104 = 5, and at n = (10, 5, 3, and 2), Te/104 = (4.3, 3.5, 2.2, and 1.2) respectively, which shows that increasing atmospheric extent produces effects that simulate lower atmospheric temperatures. Indeed, a model with Tl = 50,000°K and n = 3 has a flux distribution nearly identical to 7-6 Extended Atmospheres 249 one with 7, - 30,000CK and n - 5. On the basis of these results we can understand why supergiants and WR stars have lower color temperatures than main-sequence stars of the same spectral type; it is also evident that atmospheric extension effects introduce ambiguities in the choice of a structural model for a star because of the trade-off between temperature and envelope-size in fitting the data. The frequency-variation of the flux from an extended grey atmosphere is shown in Figure 7-28, where it is compared to that from a Planck function at Tc, the color temperature of the flux at Xc = 5000 A, and to a Planck function at T(t = f) (the distribution that would emerge from a planar atmosphere). It is clear that the flux from an extended atmosphere shows a pronounced ultraviolet deficiency and an -1.0 - -0.5 - n 1.0 - FIGURE 7-28 Solid curve: emergent flux Fv from spherical grey atmosphere with 7 i =5 x 104 "K and n = 2. Dashed curve: black-body curve at color temperature T, that matches slope of Fv at /l5000 A: note ultraviolet and infrared excesses of Fv relative to £T(7 J. Dotted cun £■: black-body curve at T = Th = 2/3). the value characteristic of a planar atmosphere. Abscissa: [//., where a is in microns. From (61. 165], by permission. 250 Model Atmospheres infrared excess relative to that from a planar atmosphere of the same characteristic temperature. Relative to a Planck function at Tc, the temperature that would be assigned observationally, the flux distribution shows both an infrared and ultraviolet excess. These results emphasize the rather inhomo-geneous nature of the radiation field from an extended atmosphere, arising from the greater variation of the temperature through the envelope. We shall consider more realistic nongrey and non-LTE models after developing a general method for solving the transfer equation. solution of the transfer equation in spherical geometry A variety of methods have been employed to solve the transfer equation in spherical geometry. Both Kosirev (360) and Chandrasekhar (148) used variants of the Eddington approximation, which, however, is not accurate near the surface. Subsequent work (154; 679; 680) approximated the angular behavior of the radiation by an expansion in spherical harmonics, but only to second order. As demonstrated convincingly by Chapman (163), these methods cannot be adequate because the radiation becomes very sharply peaked in the outward direction {fK -»■ 1) at the surface [see (163, Fig. 1) or (323. Fig. 6)]. Methods that employ direct differencing of both the angular and spatial derivatives in equation (7-165). [related to the 5,,-method (127) of reactor physics] have been suggested (130; 255; 507), but these fail when the discrete spherical shells become optically thick, and hence must be supplemented by special techniques to be useful for stellar atmospheres work. The differential-equation technique to be described below, which is general, stable, and efficient, uses a Feautrier solution along individual impact parameters (tangent to discrete shells) to generate the angular information required to evaluate variable Eddington factors (323) from an estimated source function. Then, with known Eddington factors, it employs a Feautrier scheme to solve a combined moment equation obtained by using the elegant transformation introduced by Auer (32). Alternatively, a direct solution can be obtained with a Rybicki-typc method, if the scattering integral in the source function is frequency-independent (442); an equivalent integral-equation method has also been developed (558). Consider first the moment equations; regard all variables as functions of r and v, and introduce the radial optical depth scale dzv = —%v dr. Then the moment equations are and [3(fYJv)/drv-] - (3f. - \)JJ{yj-) = Bv (7-190) (7-191) 7-6 Extended Atmospheres 251 where Sv is assumed to have a general form Sv = av JR(r; v', v)Jv. dv' + fiv (7-192) For a pure continuum problem, we would have only a coherent scattering term-and Sv would be simpler. There are two essential difficulties with equations (7-190) and (7-191). (a) Direct elimination of H does not yield a simple equation, but rather a complicated one involving both first and second derivatives, (b) The term in equation (7-191) in (xvr)-1 tends to diverge strongly at the surface (recall xv is opacity per unit volume and hence varies with the particle number density over several orders of magnitude). This term destabilizes the system. Both of these difficulties can be eliminated entirely by the introduction (32) of a sphericality factor qv, defined by ln(r2qy) = £ [(3/v - l)/(r;/;,)] clr' + In r2 (7-193) where rc is a "core radius" corresponding to the deepest point in the atmosphere considered in the solution. It is obvious that qv is known if/" is known. The factor qv allows equation (7-191) to be rewritten as d(fqvr2Jy)/dTv = qvr2Hy (7-194) which, when substituted into equation (7-190), yields a combined moment equation '1 d(fvqvr2Jv) qv dzv d ox. = H(JV - S, (7-195) or, introducing a new variable dXv = —qvxv dr = qv dxv, d2(f,qy-jv)/ex/ = 4v~v2(Jv - sv) (7-196) Exercise 7-17: Verify that equation (7-193) allows the reduction of equations (7-190) and (7-191) to (7-195). To obtain an upper boundary condition we define K = ^I{R, ^v)pdftj^ I(R, fi, v)dfi so that from (7-194) (7-197) [5(/¥«vrVv)/aXv]P=li = hv(r2JvU, (7-198) 252 Model Atmospheres while at the lower boundary we apply a planar diffusion approximation to write flvl'v) =\ilv'l\^ldv\)re (7-199) and fix the gradient by demanding that the integral of Hv(t\.) over all frequencies equal the correct integrated flux Hc = Lj{\6n2r2). Then id(Uhr2JvydXvl^ = re2Hc BJdT) |J; X-\dBJdT)dv (7-200) The diffusion approximation, and hence equations (7-199) and (7-200), will be valid when the photon mean free path /v"1 « sR where e is some small number; this criterion can always be met by choosing re sufficiently deep in the atmosphere. For other physical situations (e.g., in a nebula), alternative inner boundary conditions can be posed (374). With the introduction of a discrete radius mesh {rd}, (d = 1,..., D), where R = i\ > r2 > ■ ■ ■ >rD = rc, and a frequency mesh {v„}, (n = 1,... ,JV), we may replace derivatives with finite differences [perhaps using splines (374; 442) or Hermite formulae (34)] and the frequency integral (if any) in the source function with a quadrature formula. Equations (7-196), (7-198), and (7-2001 are then of the standard tridiagonal form of equation (6-31) and can be solved with the usual Feautrier elimination scheme; the computing time scales as TM = c DN3, which is particularly economical if the source function is purely thermal or has only a coherent scattering term (N = 1). In the calculation just described, the scattering integral in the source function appears explicitly, hence the correct global thermalization properties of the solution are obtained. To carry out the computations described above, we require knowledge of the Eddington factors these can be found if we know the angular behavior of the radiation field at each depth. To obtain the required information we perform, at each frequency, a ray-by-ray solution along a grid of impact parameters {/;;}, chosen to be tangent to each discrete radial shell, augmented by an additional set of C impact parameters, which are chosen to intersect the core, and which include^ the central ray. The geometry of the situation is shown in Figure 7-29. The impact parameters are labelled with an index /, (z* = 1____, /), where I = D + C; p1 denotes the central ray; pc the last ray inside the core (i.e., pc. < pc + 1 = >\.\ and p{ = R. Each ray pt intersects all shells with rd > p,. and these intersections define a mesh of z-points {zdi}, (d = 1,..., Df). Here Dt = D + C + 1 - / for i > C, D, - D for / < C, and zdi = (r/ - p,2)*. It is seen from the figure that the ray p,- intersects the radial shell rd at an angle whose cosine is 253 Pi = 0 A Observer V [z = 00) Discrete (;>, z\ mesh used in solution of spherical transfer equation. The impact parameters [pi] are chosen to be parallel to the central ray, and tangent to spherical shells chosen to describe the depth-variation of physical properties of the envelope. The intersections of the rays with the radial shells define a z-mesh along each ray. i = p{r,i> Pi) = (>'/ — p,2)V'V = -ji/rd- Hence if we first construct the solution along all rays {pt}, and choose a particular value rdi then knowledge of the variation of Iv{zdi, pt) (i = 1,. . ., /,,) is equivalent to knowledge of the u-oariation of Iv (rd> p.) on the mesh {pdi}, (i = 1, . . . , Id\ which spans the interval 1 ^ p ^ 0. Here Id = I + 1 — d. It is therefore clear that the ray-by-ray solution described above will allow determination of the requisite Eddington factors. The astute geometrical trick employed here to synthesize the angular information from ray solutions is actually nontrivial and makes direct use of the symmetry of the problem, which allows us to treat all points on a given shell as equivalent; without strict spherical symmetry, the problem is much more complex. Consider now the ray specified by pt. The transfer equation along the ray is ± [dl+'iz, ph v)/dz] = n(r, v) - /\r, v)I±(z, p;, v) (7-201) where the + and — signs refer, respectively, to radiation flowing toward and away from the external observer, and we have written r as the space variable in >] and '/ will1 the understanding that r = r(z, pj = (p2 + z2f\ Defining the optical depth along the ray dx{z,phv) = —yfr, v) t/z, setting S(r, v) = n{i\ r)/yfr, v) (assumed to be known values), and introducing the 254 Model Atmospheres mean-intensity-like and (lux-like variables u{z, Pi, v) = i [/+(z, p„ v) + I'(z, Pi, v)] (7-202) and v(z, Pi, v) = ^ [J + (z, Pi, v) - /"(z, Pi, v)] (7-203) we obtain the second-order system [62u(z, Pi, v)/di2(z, Pi, v)] = «(z, pf, v) - S[r(z, pf), v] (7-204) with an upper boundary condition du(z,phv)/dz(z,Pi,y%...... = u(zmPhv) (7-205) where zmax = (R2 - p,-2)* The inner boundary condition depends upon whether the ray intersects the core, rDi = rc, or misses the core and intersects the central plane at z = 0. In the former case we apply the diffusion approximation as was done to obtain equation (7-200); in the latter, symmetry considerations show that u(0, Ph v) = 0, hence du(z, Pi, v)/ot(z, Pi, v)|s_0 = 0 (7-206) Equations (7-204) through (7-206), when written as difference equations, yield (with known S) a single tridiagonai system of the standard Feautrier form, and can be solved by the usual algorithm. The computing time for N frequencies, C core-rays, and D radial depth-points scales as j = cjv[D ■ C + £ D;] % c'ND2 for D » C. Having calculated the complete solution Mdili = u{zdyPi,\\) we construct the moments, as described above, Jan = Z wg>«dili (7-207) t= i and Kdl, = £ w<>di„ (7-208) ;= i and thus the Eddington factor fdll = KJJdn. Here the w's are appropriate quadrature weights, obtained analytically by integration of moments of a piecewise polynomial representation of u(rd, pi) on the mesh {^,-}, generated by the intersection of the rays {Pi} with the radial shell rd. Using the new Eddington factors, the moment equations are re-solved and the process is iterated to convergence; experience shows the convergence to be very rapid. Exercise 7-18: (a) Write difference approximations for equations (7-204) through (7-206). (b) Derive a second-order accurate lower boundary condition by ex- 7-6 Extended A imospheres 255 tending the difference equation for equation (7-204) beyond the central plane -d; = 0, and using symmetry of udin about this plane. In the case that the source function contains a single scattering integral involving J (or 7 in a line), rather than a frequency-dependent scattering term involving partial redistribution, we may avoid the iteration procedure between ray equations and moment equations, and develop a direct solution by a Rybicki-type scheme (442). Along each ray, for each frequency, we have a tridiagonai system of the form T,-„u,-„ - Uifl J + WiB {' = \ ■ ■ " ^ (7-209) (n = l.....N) where J describes the depth-variation of J [cf. equation (6-46)] on the range (d = 1,. . ., Dj), and u;„ the variation of u(z, Pi, v„) along the ray. This system can be solved for = C;„J + D,„, and this solution can be substituted into the equation defining J [cf. equation (7-207)] to yield a final system for J; the computing time for this solution scales as TD = cN Di for D » C. Details of this procedure, which is general, stable, and economical, can be found in the reference cited. extended models for early-type stars Nongrey spherical model atmospheres in LTE have been constructed for the central stars of planetary nebulae (130; 131; 376) and for O and B supergiants (136; 325; 376; 441; 442; 559; 516, 241), using a variety of techniques. Non-LTE models for both classes of objects are given in (376; 441; 442; 516, 241). All of these models assume hydrostatic equilibrium, and the extension effects are produced by a near-cancellation of gravity by radiation forces on the material. In actuality, there is strong evidence that atmospheric extension is almost always associated with large-scale expansion, and static models can be expected to yield, at best, only qualitative information; wc shall discuss dynamical models, which are more difficult to construct, in §15-4. In an extended atmosphere, it is necessary to account for the variation of the gravitational force with radius, and if J4 is the mass of the star, the equation of hydrostatic equilibrium becomes (dpjdr) = -(pGJf/r2) + (4rc/c) J*QX XvHv dv (7-210) Introducing the Rosseland mean opacity %K, the integrated flux H = L/(l6n2r2), and the parameter y = (xRH)-1 £zvHvdv (7-211) T 256 Mode! Atmospheres we have [dpjd(l/r)] = p\GJ4 - (yLzR/4ncp)-] (7-212) As was true for planar atmospheres, radiation forces lead to an increase in the scale-height (and hence the extent) of the atmosphere. The parameter T = y/RLI{4ncG-/Mp) measures the ratio of radiation to gravity forces. In the limit of a pure electron-scattering opacity we find an upper bound on the critical ratio for {LjjM) at which radiation forces just balance gravity forces, namely (L/J?)cri} = 3.8 x 104 {LJMQ). (cf. Exercise 7-1). We shall denote the value of f obtained from pure electron scattering as TL,. For an extended envelope, the meaning of the stellar "radius" (and hence of the effective temperature) becomes ambiguous; typically one uses the value rh at which Xr - ^ as a characteristic radius. The effects of radiation forces are quite important in the planetary nebula stars, as estimates of L and Ji for these objects (279) lead immediately to values of T on the range 0.8 < Ft, < 0.93. Further, it can be shown (442) that the absolute thickness of the atmosphere, Ar, is proportional g-f = [(1 - T)G.yM/R2]~'1, which means that for a given gt(f, the relative thickness of the atmosphere, Ar/R, will be larger for stars of small radius, such as the planetary nebula stars. Thus several of the models in (130; 131) have atmospheric thicknesses comparable to, or even larger than, the radius of the "core" star (depths below t = 10)! The effect of extension in these stars greatly reduces the color temperature relative to the effective temperature; at Xc = 5000 A, Tc/TcI{ for some of the models is as small as 0.3. The complete energy distribution shows the characteristic flattening described above for grey models [indeed in the visible, the nongrey energy distribution can be fitted quite accurately by grey models with n = 3 to 4 (61, 165)]. The flatter energy distributions predicted by these models strongly resembles those of WR stars (though the mode! parameters L, -JI, and R are not really appropriate for such stars). For O-type supergiants and Of-slars, it is clear, from the failure of planar models to match the energy distribution [see (376)] and to produce the observed emission lines, that the atmosphere must be extended. On the other hand, stellar evolution tracks at plausible masses all yield values of Fe < 0.5, which, taking into account the large radii of these stars, implies insignificant atmospheric extension (136). But absorption in spectrum lines can greatly increase the total force of the radiation on the material (102,404) and indeed it is precisely this mechanism that is thought to drive the winds from O-stars (see §15-4). In several calculations (376; 441; 442; 516, 241), attempts were made to simulate the effects of enhanced radiation forces by adopting ad hoc values for the radiation-force multiplier y in equation (7-212), chosen so as to raise the maximum value of F to values approaching unity (the most extreme case considered had rmM % 0.995). Although such models become very difficult to compute because of numerical instabilities, which directly reflect the physical instability of the atmosphere, a number of solutions were 7-6 Extended Atmospheres 257 obtained using a complete linearization procedure, both assuming LTE and taking into account departures from LTE. It should be stressed that, for such extreme values of F, hydrostatic equilibrium is very unlikely to be possible, and dynamical models are necessary; models such as these merely demonstrate the effects of a greatly increased scale-height for the envelope. Extension effects produce major changes in observable parameters such as colors, see Figure 7-30, which show a strong reddening with increasing atmospheric size. The colors for LTE and non-LTE models differ markedly because in LTE a spurious emission edge occurs at the Balmer jump (caused by the 1.7 -0.14 - -0.14-0.13-0.12-0.11-0.10-0.09-0.08-0.07-0.06-0.05-0.04-0.03-0.02 [b - y) FIGURE 7-30 Slromgren-syslem (u - b) and (b - v) colors for models with Jf = 60 JfG Open symbols: LTE models: (Hied symbols: non-LTE models. Models with equal values of TeU are joined on curves. Curves are labeled with parameters describing radiation-force multiplier Circles: ■■ = labels give value of?,; triangles: y = 1 + ••, exp(-tR), labels give value of From (376), by permission. 258 Model A tmospheres Schuster mechanism—see §10-2); detailed discussion of the overall properties of the energy distributions can be found in the references cited above. . 7-7 Semiempirical Solar Models All of the discussion presented thus far in this book has concerned the development of methods that allow the analysis of a stellar spectrum, and the inference of physical properties of the atmosphere, by a model-fitting technique. Our information about stars, which we perceive as mere points of light, is quite limited in quantity, quality, and scope, primarily because of the low intensity of the radiation received. But for one star, the sun, we have unparalleled opportunities to obtain information at very high spatial, temporal, and spectral resolution, over an enormous range of energies, from X-ray. ultraviolet, visible, infrared, and radio wavelengths. While stars are unresolved, structures of the order of 150 km in size can be distinguished in the solar atmosphere, and a wide variety of small-scale features can be seen. Further, measurements of velocity fields and magnetic fields can be made. The sun thus provides a unique testing ground for our theoretical methods, and, because so much information can be obtained by semiempirical analyses that rely only weakly upon theory, it offers guidance in seeing how to extend the theory in those places where it is oversimplified. It is fair to say that most (though not all) major advances in stellar atmospheres theory were motivated by attempts to understand solar phenomena; time and again, the refined confrontations possible between solar observations and the theory have forced radical changes in our interpretive picture. And many enigmas remain, so we may expect much further development yet. There is no hope of summarizing adequately, in a few pages, the vast amount of information known about the solar atmosphere, and the reader should consult the many excellent references available, particularly (244; 694; 11, Chaps. 9 and 10) for general information, (628; 19; 20) for detailed information about the chromosphere, and (94; 20) for material about the corona. The goal of the discussion below will be only to present a brief summary of some of the primary structural properties of the solar atmosphere, in order to provide background and orientation for the development of more realistic pictures of stars in general. In terms of basic morphology, the solar atmosphere can be divided into four major parts: (a) the photosphere, the opaque disk seen visually; (b) the chromosphere, a region extending some 2500 km above the limb, showing the characteristic emission spectrum of hydrogen; (c) the corona, a tenuous, faint outer envelope first seen at eclipses; and (d) the wind, a supersonically expanding region that streams past the earth. The overall temperature structure of the atmosphere is shown in Figure 7-31. In the photosphere the temperature decreases outward to a minimum of about 4200°K. At the relevant 260 Model Atmospheres temperature (T - 5 x 103 °K) and gravity (ť/ - 3 x 104) the pressure scale-height is about 120 km. This length subtends an angle of 0.15", which lies well below the typical limit of resolution set by seeing effects in the earth's atmosphere; as a result, the sun appears to have a crisply defined edge, which occurs at height h — 0 km on the scale of Figure 7-31. In the low and middle chromosphere the temperature rises to about 8000~'K and then to a plateau of around 30,000°K. There is then a very thin Transition region in which the temperature suddenly rises to coronal values of about 1.5 x 106 °K. In the corona the scale-height is about 50,000 km, which is a significant fraction of a solar radius, so the corona extends to large distances from the sun; indeed, even if the corona were static, it would envelope the earth with a substantial density of particles. The temperature structure described above has been inferred mainly from analysis of the solar spectrum, which is strongly affected by the nature of the temperature distribution. The features in each spectral region arise from some characteristic range of height (see Figure 7-32) and, since it has become possible to survey a wide range of wavelengths from space vehicles, our knowledge of the temperature structure has been greatly improved. In the spectral range from 1685 A < X < 350 p the continuum originates in the photosphere, and the lines have unit optical depth at heights that extend from the photosphere (in the line wings) into the chromosphere (at line center). Observed on the disk, lines with wavelengths X > 1900 A are domi-nantly in absorption, and a transition to emission lines occurs in the range 1700 ÁáK 1900 A. The continuum on the range 1525 Á < X < 16S5 A comes from the photosphere-chromosphere transition region, and for X < 1525 A is dominantly chromospheric. On the range 504 Á < 1 ž 912 Á the Lyman continuum is in emission; for X á 504 Á the emission is from the continuum of He I. Chromospheric emission lines extend at least down to 288 Á, through the resonance series of He II. Coronal emission lines begin to appear prominently at about 800 A, and extend down into the X-ray region. In the infrared, one sees most deeply into the atmosphere near 1.6 p., the minimum in the H~ opacity. At longer wavelengths, the temperature minimum region is encountered near 300 p; and at radio wavelengths of 100 cm and beyond, the continuum emission is entirely coronal. (The corona is completely transparent in the continuum at centimeter wavelengths and shortward.) Off the disk, the chromosphere and corona can be seen in strong emission lines, by using special instrumentation, or in special circumstances such as eclipses, from which an enormous wealth of data has been derived. A variety of analytical techniques have been used to make estimates of physical properties of the solar atmosphere from the data described above. A very powerful tool is the analysis of limb-darkening data. As we saw in §3-3, the Eddington-Barbier relation implies that the dominant contribution to IJQ, li) comes from Sv(tv ä pi); hence by scanning from center to limb, 262 Mode! Atmospheres 0 $ /i 1. we sample depths 0 < xv < 1. Suppose we assume'that the source function is the Planck function, and that we can write £v(tv)/Jv((X 1) -bv{xv). Then „ can be computed in terms of the expansion coefficients of the formula. By fitting this calculated function to the observations of V, one finds the coefficients and hence bv. For example, if we take then we find 0» = £fl*k!^ (7-214) (7-215) Thus a fit of equation (7-215) to the data yields the %'s, hence bv and, ultimately, if we know Jv(0,1) we obtain j5v(tv), and hence T(tv). A very large body of limb-darkening data exists, extending from ultraviolet through radio wavelengths. The procedure outlined above encounters many practical difficulties. Thus even large-amplitude fluctuations in the physical properties do not affect the observations if they occur in regions whose line-of-sight optical thickness is «1; we thus remain in ignorance of any such inhomogeneities. Similar remarks apply to horizontal structures that lie below the limit of resolution set by the observing procedure. Moreover, we clearly obtain but little information from depths with rv > 1, while a limit is also set at small depths by seeing effects at the limb; a typical resolution of 1" implies a blur of about 0.001 solar radius, which corresponds to p: « 0.05. Thus at any given wavelength we are confined to 0.05 < tv < 1. Further, with data oflimited quality the number of expansion coefficients that yield real information in equation (7-214) or (7-215) is restricted. For example, for data with an accuracy of + 1%, at most three coefficients can be obtained (98); attempts to use more coefficients yield results that fit the observed values of O,.(/.() but show wild oscillations in /\(tv). Numerical methods (such as the Prony algorithm) exist that automatically limit the coefficients to only those justified by the accuracy of the data (669). One may choose a variety of fitting functions in equation (7-214), each of which fits the observed iP,,(/i) equally well; in general these yield different values of bv(zv) and hence of T(xv). Among three-term solutions, discrepancies in the temperature of the order of + 200"K are found; these can be important in certain contexts. For example, in the solar atmosphere the dominant ionization state of iron atoms is Fe+. Using the Saha equation it is easy to show that an error of ±200'JK introduces an uncertainty in the 7-/ Semiempiriea! Solar Models 263 occupation numbers of low-lying states of Fe 1 (7.9 eV below the ionization limit) of about A log n ^ +0.25, or about +60",,; thus even assuming that every other step in the analysis of the spectrum were perfectly accurate, significant errors are introduced into estimates of element abundances from uncertainties in the model alone. The range of depths available to analysis using limb-darkening data may be extended by using different wavelengths. One then needs to map the curves of T(zJ from one frequency to another, which can be done if we know the frequency variation of the absorption coefficient—i.e., /{v)/x(vsld). Alternatively, we can demand that the different T(zv) curves correspond to a unique T[h) variation, and turn the procedure around to infer the frequency profile of the absorption coefficient. Studies of this kind (e.g., 517) have shown that the empirically-determined absorption coefficient is consistent with the hypothesis that H " is the primary opacity source (though additional sources, mainly lines, are needed in the ultraviolet). Another method that can be used to determine the temperature structure is the analysis of disk-center absolute intensities as a function of frequency. With the advent of precise absolute intensity measurements in the ultraviolet from space vehicles, this method has surpassed limb-darkening analyses in importance. .An extensive compilation and discussion of existing data is given in (646). The basic procedure employed is to construct models by (1) assuming a temperature distribution, (2| integrating the equation of hydrostatic equilibrium, and (3) solving the coupled transfer and statistical equilibrium equations for the relevant atoms (e.g., H, He, C. Si) to calculate absolute emergent intensities. The assumed T(h) relation is then adjusted to yield an optimum fit to all available data. A sequence of successively more refined models have been constructed in this way, starting with the Utrecht Reference Photosphere (283,239), which was soon displaced by the Bilderberg Continuum Atmosphere (BCA) (248). Introduction of ultraviolet space observations led to the highly-successful Harvard- Smithsonian Reference Atmosphere (HSRA) (summarized in Table 7-1) (249); subsequent work (645;646) has produced a new model ("Model M") that fits a very large body of data extremely well. Improvements in these models are constantly being made, and important advances will result when eclipse data are taken into account and spectrum-line synthesis is pushed to its utmost. It should be recognized that there are still many fundamental difficulties to be faced by these semi-empirical models. One of the most worrisome is that observation reveals that the layers under analysis contain a great deal of small-scale structure and are far from homogeneous. Some difficult questions remain essentially unanswered: To what extend does the value assigned to a physical variable in any layer of a one-dimensional chromospheric model represent the average property of that layer? How large is the fluctuation of any physical property about its mean value? Are fluctuations in various properties correlated? It 264 TABLL 7-1 Harvard-Smithsonian Reference Atmosphere" ft (km) t(k) 1.00 - 8 2.00 - 8 3.16 - 8 6.31 - 8 1.00 - 7 2.00 - 7 3.16 - 7 6.31 - 7 1.00 - 6 2.00 - 6 3.16 - 6 6.31 - 6 1.00 - 5 2.00 - 5 3.16 - 5 6.31 - 5 1.00 - 4 2.00 - 4 3.16 - 4 6.31 - 4 1.00 - 3 2.00 - 3 3.16 - 3 6.31 - 3 1.00 - 2 2.00 - 2 3.16 - 2 6.31 - 2 1.00 - 1 2.00 - 1 3.16 - 1 6.31 - 1 1.00 2.00 3.16 6.31 10.00 20.00 1860 1850 1840 1830 1820 1790 1760 1690 1620 1430 1230 947 840 720 654 588 557 515 487 447 420 379 352 311 283 241 212 168 138 92.6 63.1 22.6 0.0 -25.3 -37.1 -51.4 -60.8 -76.7 8930 8750 8630 8450 8320 8090 7910 7630 7360 6720 6180 5590 5300 4910 4660 4280 4170 4205 4250 4330 4380 4460 4525 4600 4660 4750 4840 5010 5160 5430 5650 6035 6390 7140 7750 8520 8880 9390 * The notation a ■ na - b denotes ti - an x Adapted from (249), by permission. ^(dynes cm 2) pe(dynes cm 2) np/[np + d(H)] 1.52 - 1 4.82 - 2 5.11 - 1 1.54 - 1 4.84 - 2 5.05 - 1 1.56 - 1 4.92 - 2 5.06 - 1 1.62 - 1 5.10 - 2 5.05 - 1 1.69 - 1 5.34 - 2 5.09 - 1 1.88 - 1 5.88 - 2 5.02 - 1 2.10 - 1 6.44 - 2 4.86 - 1 2.76 - 1 7.50 - 2 4.09 - 1 3.76 - 1 7.85 - 2 2.89 - 1 1.00 + 0 6.85 - 2 7.99 - 2 3.79 + 0 6.00 - 2 1.72 - 2 2.92 + 1 6.80 - 2 2.26 - 3 6.79 + 1 6.77 - 2 9.13 - 4 1.87 + 2 4.43 - 2 1.41 - 4 3.37 + 2 3.96 - 2 2.46 - 5 6.34 + 2 5.11 - 2 7.65 - 1 8.68 + 2 6.12 - 2 2.26 - 7 1.34 + 3 9.09 - - 2 2.13-7 1.77 + 3 1.20 - - 1 2.47 - 7 2.65 + 3 1.81 - - 1 3.40 - 7 3.46 + 3 2.37 - - 1 4.06 - 7 5.12 + 3 3.53 - - 1 5.43 - 7 6.65 + 3 4.65 - - 1 7.11 - 7 9.81 4- 3 6.86 - - 1 8.87 - 7 1.27 + 4 8.95 - 1 1.09 - 6 1.87 + 4 1.33 f 0 1.46 - 6 2.41 + 4 1.78 f 0 2.13 - 6 3.54 + 4 2.81 +■ o 4.42 - 6 4.56 + 4 3.95 4- 0 8.49 - 6 6.61 + 4 7.04 + 0 2.47 - 5 8.31 + 4 1.13 + 1 5.29 - 5 1.12 + 5 2.65 + 1 1.58 - 4 1.31 + 5 5.64 + 1 3.65 - 4 1.54 + 5 2.35 + 2 1.55 - 3 1.65 + 5 6.26 + 2 4.04 - 3 1.78 + 5 1.81 + 3 1.11 -2 1.86 + 5 2.61 + 3 1.67 - 2 2.00 + 6 5.00 + 3 2.79 - 2 7-7 Semiempirical Solar Models 265 is imperative that we face these queries, for one-dimensional models are the best we can hope to obtain for stellar chromospheres, and the validity of such models will be supported or undermined, depending on the answers to the questions posed above for the solar chromosphere. In the corona, which is optically thin, the density structure can be inferred from the brightness of the light scattered by free electrons, after allowing for the effects of integration along the line of sight [see, e.g., (94)]. Coronal density distributions have been tabulated for the equatorial and polar directions, for varying conditions during the solar cycle (477); a typical electron density near the base of the corona is of the order of 10s cm"3. That the temperature of the corona must be very high was first recognized by Grotrian, who suggested that two broad, faint absorption features seen in the light scattered by the corona were the very strong Ca II H- and K-lines of the solar spectrum, "washed out" by electron scattering at very high temperatures. This idea was further supported by the identification, by Elden, of coronal emission lines with transitions in highly ionized atoms—e.g., Fe X (;.6374) and [Fe XIV] (;„5303). Modern determinations of the temperature in the corona are in agreement with one another and give a typical temperature of about 1.5 x 106 K, based on (a) thermal line-widths, (b) ionization equilibria (including dielectronic recombination!), and (c) radio-wavelength noise temperatures. X-ray observations reveal intensely hot regions (several millions of degrees) near active regions. For more detailed discussion see (20) and (94). The temperature distributions shown in Figures 7-31 and 7-32 obviously bear little resemblance to the classical predictions of radiative-convective equilibrium models, and indicate a major breakdown in this approach in the outer layers of the atmosphere. The classical models are adequate to predict the structure of the photospheric layers where the visual continuum and line wings (and very weak lines) are formed; but once the temperature rises outward we must introduce new phenomena and an appropriate heating mechanism (clearly the Cayrel mechanism is not the cause of the rise, for Te becomes »Tcff!). It was suggested by Biermann (89) and Sch-warzschild (564) that acoustic waves would be generated in the solar convection zone, propagate outward, steepen into shocks, and deposit energy in the material, heating it to high temperatures. A specific mechanism for the production of soundwaves was suggested by Lighthill (394). Subsequent work has also called attention to the role of magnetohydrodynamic waves, for the outer solar atmosphere has a significant magnetic field. In addition, the discovery (393) that large regions on the solar surface oscillate with a period of about 300s provided another source of nonradiative energy that can be tapped to produce heating. The problems of wave generation, propagation, and dissipation are complex and difficult, and a tremendous 266 Model A imospheres 7-7 Sem[empirical Solar Models 267 amount of work has been done on the subject; an excellent review of the field has been given by Stein and Leibacher (602) [see also (20)]. No fully consistent picture of the heating has yet emerged, but current work indicates that the 5-minute oscillations can indeed heat the upper chromosphere and corona, while shorter periods are required to heat the lower chromosphere. Estimates of the actual amount of mechanical energy input can be made by comparing the semiempirical temperature distributions with radiative-equilibrium models [e.g., (17) or (380)] and calculating the energy required to produce the heating. Difficulties here are that at the temperature minimum (1) T in the empirical models lies below T in the radiative models, and (2) small errors in T (say ±100°K) imply unacceptable errors in the energy input estimates. Much work remains to be done to establish both the empirical and theoretical temperatures to the required accuracy. The presence of temperature plateaus followed by large jumps in temperature can be understood in terms of thermal stability of the gas. In essence, the gas is heated by mechanical input and cooled by radiative losses, which establish a balance [see (628) and (20)]. The losses are largest just as some atom is becoming nearly completely ionized, and this tends to provide a thermostatic action that holds the gas near a definite temperature. Thus, hydrogen provides the dominant cooling in the low chromosphere and keeps the gas at about 7000°K to 8000°K; after hydrogen becomes too strongly ionized to be effective, losses from He I and He II dominate, and the temperature jumps to about 20,000°K to 30.000°K; and, finally, after helium is strongly ionized, losses from ions of C, N, O, Ne, Mg, and Si dominate, yielding strong cooling rates at temperatures above 105 nK. In regions of steep gradients, conductive energy transfer also becomes extremely important and the final temperature profile is a result of all these mechanisms operating together. Given that the sun is a typical G-dwarf, it is clear that we must conclude that the chromosphere-corona-wind phenomenon must be a basic property of stars in general. There is ample evidence that this is true. Thus, most stars in which convection zones exist show chromospheric emission features in the Ca II H and K lines (cf.ss12-2), with many stars that are younger than the sun showing very active chromospheres. An extreme case is provided by the T-Tauri stars where most of the prominent spectral features arise in a "super-chromosphere" and very dense wind. Mass loss in stellar winds, particularly in the early-type supergiants and WR stars (estimated rates up to 10"6 —10_ 5 .,£Q/yea.r) is well established. These winds make the solar wind (mass-loss rate a 10"14 ./#0/year) seem puny by comparison. The solar atmosphere provides a kind of Rosetta stone that helps us first understand a rich '"literature" containing many dramatic stellar phenomena; these phenomena, in turn, extend our knowledge over a broad range ol physical conditions. Further, by studying large numbers of stars, we can hope to infer, from the behavior of the ensemble, the time-evolution of the phenomena for an individual such as the sun, on otherwise inacessible time-scales (billions of years). Work at the interface between solar and stellar atmospheres is both active and richly rewarding [see, e.g., (344)], and will unquestionably be synergistic in the development of our concepts of stellar atmospheres. (V/ Observational Quantities 269 The Line Spectrum: An Overview Superimposed upon the continuum of a star, we observe discrete spectrum lines, either in absorption or emission. These lines arise from transitions between bound states of atoms and ions in the star's atmosphere. An extremely wide variety of lines is found, from a wide range of atomic and ionic states, leading to very different-looking spectra for different classes of stars. A panoramic view of this variety is best obtained by inspection of actual stellar spectra, particularly as shown in (5) and (465) [see also (261, Chap. 14; 330, Chap. 1)]. One finds that lines in stellar spectra show enormous ranges of strength and striking variations in profile. A close examination shows that the spectra can be arranged into a two-dimensional scheme reflecting primarily the effective temperature and luminosity of the star. It would take us too far afield to describe here the details of this procedure-developed to a high state of refinement by Morgan and his collaborators—or the full implications of the results; therefore the references cited above should be studied carefully. Suffice it to say, the spectrum lines contain a wealth of information concerning the run of physical variables in the star and therefore provide important diagnostic tools for inferring the state of the atmosphere. Spectral lines are much more opaque in the core than in the wing, and hence can provide a sampling of a wide range of atmospheric depths, from very high layers (seen in the core) to the deepest points observable (the depth of continuum formation). Further, lines are narrow in frequency-width, and hence are sensitive to the effects of velocity fields; they thus provide the means by which motions of the material in stellar atmospheres can be studied. Moreover, it is clear that the strength of a given line must contain information about the number of atoms absorbing photons along the line of sight, and hence about the abundance of that chemical species in the atmosphere. Thus, by a suitable interpretation, the line spectrum offers the opportunity to perform a quantitative chemical analysis of the material of which stars are composed. This information, in turn, provides valuable clues when we attempt to construct a coherent picture of the structure and evolution of stars, the Galaxy, and the Universe as a whole. It is, therefore, of prime importance to develop a theoretical framework within which line profiles can be predicted and the desired physical information can be inferred. A great deal of effort has been devoted by many astronomers to this end, and considerable progress has been made. The second part of this book will describe the theoretical techniques that now exist to treat the problem of line-formation. In this chapter a few of the basic aspects of the problem, and a summary of the kinds of information required to solve it will be pointed out to orient our later work. 8-1 Observational Quantities A line in a stellar spectrum is most completely characterized by its profile, which is the observed distribution of energy as a function of frequency. For all stars except the sun, we can observe only the flux integrated over the entire disk of the star. We measure Fv (the flux in the line) relative to Fc (the flux in the continuum) and describe the profile in terms of its absorption depth A, = 1 - (FJFe) (8-1) or residual flux R, = (FJFC) =l-Av (8-2) In the case of the sun, the frequency distribution of the radiation can be observed at each point on the disk. We can then describe the profile in terms of 7„(0, fi), the emergent specific intensity, in units of the nearby continuum intensity 7,(0, p.), and write av(p) = 1 - [Jv(0, ju)/7e(0, p)] (8-3) 270 The Line Spectrum: An Overview 8-2 The Physical Ingredients of Line-Formation 111 or rM s= /v(0, ^/'/c(0,/i) = 1 - av(/i) ■ (8-4) Information about the center-to-limb variation of a profile is extremely valuable because it provides (via the Eddmgton-Barbier relation) an additional depth-resolution otherwise unavailable, and places important constraints on the theory. Such information is, of course, available only for the solar spectrum, and is one of the reasons why the solar spectrum provides an ideal testing-ground for proposed theories of line formation. Often, because of the low light levels involved, it is not possible to measure a stellar spectrum with sufficient resolution to determine a line-profile in detail, and one must then substitute the integrated line strength—the equivalent width—in place of this more detailed information. For stars, where we measure the ./fax from the disk, the equivalent width is defined as WY = ^Avdv (8-5) in frequency units (hz), or, more usually, as W?^^' AvdX (8-6) in wavelength units (A or mA). For the sun, where angular information is available, we can define, in addition, W;(ft) = f*av(ft)dv (8-7) or Wx{u) ~ ^ ajfj) dk (8-8) The equivalent width is clearly the width of a perfectly black line with the same area under the continuum level as the line under study (hence the name); obviously W gives a direct measure of the total energy in the continuum removed by the line (assuming that it is in absorption). Ideally, one attempts to obtain profiles rather than equivalent widths, for they contain far more information. In particular, it is apparent that there is an infinite number of radically different profiles (each with distinct implications for the structure of the atmosphere) that will produce a given equivalent width. An interpretation based on an equivalent width alone can be misleading (the same remark holds even for profiles!). Nevertheless, approaches exist that use equivalent width information from many lines simultaneously (the curve of growth), and these can yield important and reasonably unambiguous results. The actual measurement of the data requires refined instrumental techniques; we shall not discuss these techniques here as they lie beyond the scope of this book, but excellent discussions exist elsewhere [see, e.g., (300, Chaps. 2, 4, and 13)] 8-2 The Physical Ingredients of Line-Formation As in the case of the continuum, the calculation of the flux in a line requires the solution of a transfer problem, for the observed radiation originates from a wide range of depths, over which the physical properties of the material may vary more or less strongly. Let us inquire here what information is needed to formulate and solve the transfer equation in a spectrum line. Consider a frequency v, and suppose that we know the continuous absorption and scattering coefficients, kv and rrv, and the line absorption coefficient /j(j>v as a function of depth. We could then construct the line optical depth scale tv = ^(kv + av + %l(fv)dz' (8-9) and continuum optical depth scale xc = + oy)dz' (8-10) Then in addition, we knew the run of the source function 5V, we could immediately calculate Fv = 2j0 Sv(tv)£2(tv) rfrv (8-11) Fc = 2 J* 5c(t,)£2(tc) dxc (8-12) and hence AY from equation (8-1), and Wx from equation (8-6) if desired. Of course, in practice, it is the source function that must he determined. Only in the trivial case of LTE is Sv known beforehand (namely Sv = Bv). As was described in Chapters 2 and 7, in the general (non-LTE) situation, the source function and optical depth depend explicitly upon the occupation numbers of the particular levels involved, but these, in turn, depend upon the radiation field, and hence, ultimately, upon the source function. Thus what is required, as was the case for the continuum, is a simultaneous, self-consistent solution of the coupled transfer and statistical equilibrium equations. Before we attempt to do this, however, some insight into the kinds of information that will be required to attack the problem can be gained from the following phenomenological arguments. Consider the propagation of photons in a line with overlapping continuous absorption and scattering. Some photons will interact with the continuum, others with the line opacity. Some of the photons absorbed in the line will be scattered, and in general will suffer redistribution in frequency and angle according to the redistribution function r(v', n'; v, n). Others may be destroyed by collisional de-excitations or transitions to other levels. Photons may be introduced into the line by collisional excitations or by transitions into the upper level from other levels, with subsequent cascade to the lower level. A transfer equation 272 The Line Spectrum: An Overview accounting for these processes will be of the form Hidljdz) = -(rcv + <7V + j_i4>v)K + k*Bv + ovJv + f/j § {dto/Aa) J** <*v7(v\ n')i?(v', n'; v, n) + ij^JB9 (8-13) Here the coupling coefficients y and e describe, respectively, the fraction of the photons scattered, and those emitted by other processes at a characteristic temperature TR (not equal, in general, to the local electron temperature). At this point we can see four important ingredients needed to compute line profiles theoretically. (a) We must be able to calculate the absorption profile v. This will be treated in Chapter 9. (b) We must be able to specify the coupling between photons and the material via the parameters y and e. In general, the expressions for these quantities may be very complicated and may contain both radiative and collisional rates from the levels giving rise to the line, as well as terms coupling to other levels. The specification of y and s follows from the statistical equilibrium equations for the atom as a whole. We shall examine this aspect of the problem in Chapters 11 and 12. (c) We must be able to calculate the redistribution function R(v', n'; v, n), and to determine the effects of the details of photon scattering upon the line profile. We shall consider these problems in Chapter 13. (d) We must be able to solve the resulting transfer problem. Here we will simply refer back to the difference-equation methods discussed in Chapter 6. In all four of the areas listed above, great strides forward have been made recently. The greatest improvement in our understanding has come in regards to points (b), (c), and (d). In early treatments of line-formation, the question of redistribution was often evaded, and line-scattering was treated as coherent. We know today that this is a poor approximation, and that, in fact, a much better approximation is the opposite extreme assumption of complete redistribution over the line. In the specification of the parameters y and e, the classical approach was sketchy, and led to some serious misconceptions. More modern treatments have brought to light the importance of a clear understanding of these coefficients. In the area of actually solving the transfer equation, important advances have been made possible by application of high-speed, large-capacity computers, using the recently-developed, powerful, numerical techniques. Finally, it, it should be noted that in equation (8-13) it has tacitly been assumed that the atmosphere is static. The important effects of velocity fields upon line-formation are discussed in Chapter 14 where several different techniques for solving the transfer equation in moving media are described. The role of lines in establishing the dynamical state of the atmosphere is discussed in Chapter 15 in the sections on radiatively-driven stellar winds. I \ The Line Absorption Profile The profiles of lines in stellar spectra contain information about both the physical conditions and the abundances of chemical elements in the stellar atmosphere. Therefore they provide extremely valuable diagnostic tools and must be exploited as fully as possible. To carry out an analysis of observed line profiles one needs to know how the distribution of opacity with frequency in the line—the line absorption profile—depends upon local conditions of density, temperature, etc. For an isolated atom with levels having essentially infinite lifetimes, the spectral lines would be almost perfectly sharp; but in reality there are several different mechanisms that produce an indefiniteness in the energy levels of real atoms in a plasma, and thereby lead to line broadening. The first fine-broadening mechanism to be considered below is natural (or radiation) damping, which refers to the line width produced by the finite lifetime of the atomic levels set by their decay via the radiation process itself. Natural damping occurs even for a solitary, isolated atom. If the atom is imbedded in a plasma, then there will be an additional pressure broadening of the line caused by perturbations of the radiated wavetrain through collisions with other atoms, or charged particles, in the gas. r 274 The Line Absorption Profile Classically, pressure broadening is described in terms of two. limiting approximate theories. The first of these is known as impact theory. Here the radiating atom is considered to be an oscillator that suffers a collision that occurs essentially instantaneously, and that interrupts the radiation wave-train with a sudden phase shift, or by inducing a transition. These collisions thus cause the radiator to "start" and "stop" in intervals of finite duration, leading thereby to a frequency spread in the radiated wavetrain, and to a shift of the line away from its unperturbed frequency. The alternative approach is the statistical theory in which we consider the atom to be radiating in a field produced by an ensemble of perturbers. This field will fluctuate statistically about some mean value as a result of motions of the perturbers. At a given value of the field, the energy levels of the radiating atom are shifted slightly, and, correspondingly, the frequency of the line is altered. The intensity of the radiation at any specified frequency shift is taken to be proportional to the probability with which a perturbation of the appropriate field strength occurs. The basic limits on the classical theories of pressure broadening are set by their inability to account for the actual structure of the radiating atom, or for transitions produced by the collisions with the perturbing particles. Both of these defects are overcome in the quantum theory of pressure broadening, which yields results in good agreement with experimental determinations. Finally, we must account for the fact that in a stellar atmosphere we observe an ensemble of atoms moving with a velocity distribution along the line of sight. The profile for each atom is Doppler shifted according to its line-of-sight velocity, and the profile seen from the entire ensemble is a superposition of these shifted atomic profiles, calculated by a convolution with the velocity distribution. The theory of line broadening has progressed enormously in the past decade, and reliable calculations now exist for many profiles of fundamental astrophysical interest. The quantum theory has become very refined, but also quite complicated. As excellent treatises exist on the general subject of line broadening [see particularly (264; 268; 629)], only a brief summary of the most important results will be presented in this chapter; for further details one should consult the books just cited or the research literature, for which references are compiled in (228; 229 ;232). 9-1 The Natural Damping Profile ENERGY SPECTRA, POWER SPECTRA, AND THE AUTOCORRELATION FUNCTION We first derive some basic relations that will be needed in later discussions. Consider a time-varying oscillation of amplitude f(t). The Fourier 9-1 The Natural Damping Profile 275 transform F(vj) is defined to be F(oj) = j^/We"'*" dt which satisfies the fundamental reciprocity relation f(t) = (271)"1 f" F(co)eimt d(o J — CO The quantity E((ü) - [2n)-lF*(aj)F(yo) (9-1) (9-2) (9-3) is called the energy spectrum of the oscillator. This designation derives from the fact that f°° E(aj)da) = (2n)-1 f" F*(co)F(co) dm = f" f*{t)f(t)dt (9-4) J 'X J — CO J — CO ' ' which may be verified by direct calculation, using equation (9-1). If f(t) were the voltage across a one-ohm resistor, then f*(t)f(i) would be the instantaneous power delivered to the resistor, and the integral over all time gives the total energy. Thus, E(oj) is a direct measure of the energy in the wavetrain at frequency co. In many cases, the energy spectrum itself is not used, but rather the energy delivered per unit time—i.e., the power spectrum I{w), which is defined as I(co) = lim (271.7)" (9-5) However, for oscillations of finite duration (or with, say, an exponential decay), the power spectrum will be zero because on the average over an infinite time interval, the finite total energy emitted yields zero power. In these cases, which are of practical interest, we use the energy spectrum itself, assuming that we observe an ensemble of oscillators created at a constant rate with random phases; a finite power then results, with a frequency distribution proportional to the energy spectrum of an individual oscillator. In certain situations it is not possible to calculate the power spectrum directly, using equation (9-5). It is then valuable to make use of the autocorrelation function ®2 (9-H) which is an exponentially damped oscillation. Calculating the Fourier transform, assuming the oscillation starts abruptly at t = 0, we find *"(«>) = *o J7 e-^'-^e-^2 dt = x0/[i(fi) - co0) + iy] (9-12) The energy spectrum of the oscillator is then E(co) = (x02/2n) [foj - o>0)2 + (Ar)-]"' (9-13) The power spectrum of an ensemble of such oscillators created continuously with random phases is proportional to £(cu), hence the profile, normalized such that 7M tfw - 1 (9-14) is /(to) = (y/2n) [(© - 0)2 + (ir)2]-1 (9-19) From equation (9-17) we see that T is to be interpreted as the reciprocal of the mean lifetime of the upper state. If several transitions out of the upper state U are possible, then Tv= X AVi (9-20) is the appropriate width of the state. Suppose the line under consideration arises from a transition between two excited states, so that the lower state L also has a width TL given by an equation analogous to (9-20). The line profile must, in general, reflect the width of both states. We assume that the distribution of substates of each level around its nominal energy is given by a Lorentz profile with the appropriate T. Let 3 = F/2 for either level, andletx = (E — E0)/h be the frequency displacementofaparticularsubstate from the nominal energy E0. We assume that the probability of ending in a particular substate x' of the lower level is independent of the substate x of the upper level from which the transition starts; then the joint probability of starting in substate x and ending in substate x' is p{x.x') = (W2)[(*2 + v)(*'2 + (v)]"1 (9-21) 278 The Line Absorption Profile If we restrict attention to transitions producing radiation of a specific frequency co, then x and x' must be related by co0 +- x — x' = tu, or writing x0 = tu - w0, *' = x - x0 (9-22) The total intensity at co is obtained by summing over all upper substates x, with x' fixed by equation (9-22), that is i(co) = j*^ p(x, x - x0) * cos xy dy = e~x^ (9-37) from winch wc see that H(a, v) = «-* Jo" e-[fl-x+(A'2/4)l cos ox dx (9-38) Assuming that a « l, we can replace e~"x by its power series, and integrating term by term we find H(a, v) = V a"Hn(v) (9-39) where HH(v) = [(-I)"/**"'.] e'^V cos vx dx (9-40) From equation (9-37) it follows immediately that H0(v) = e~l,i (9-41) Integrating by parts, the first odd-order term can be written as Hi(v) = (-2/71*) [\ - v J* e-*2'* sin vx dx^j (9-42) and using the sine transform of the Gaussian J* e~yi sin 2vy dy = e~"2 £' ev" dy = F(v) (9-43) we have Ht{v) = (2/^)[2uF(y) - 1] (9-44) 9-i Collision Broadening: Classical Impaci Theory 281 The function F(v) is known as Dawson's integral: efficient techniques for computing F(v) are given in (170). The functions H„(v) are tabulated in (281) and (11,325) for/? ^ 4. Exercise 9-2: (a) Show that the Voigt function has the normalization (b) Prove the relation H„{v) = -[d2tf„_2(i>)/dz>2]/[n(rc - 1)] from which higher-order terms can be generated by recursion from H0(v) and H^u). (c) Write explicit formulae for H2(v'h H5(r). and HAy), expressing H3{v) in terms of F(v), (d) Show that for (a2 + v2) » 1. H(a, v) x (tt"*o)/(q2 + v2). In view of the result of Exercise 9-2(d). we see that for n » 1, H(a, v) a; <|/(jrry2). Thus a schematic representation of the Voigt function is H{a,v) ~ e~v2 + a/(7iV) (9-45) where the first term applies in the line core, v ^ v*, and the second in the line wing, v > u*, the quantity being chosen such that the two terms are equal. The line core is clearly dominated by Doppler broadening, while the line wings are dominated by the damping profile. 9-3 Collision Broadening: Classical Impact Theory the weisskopf approximation The simplest classical impact theory has its origins in an analysis by Lorentz, who considered the atom to be a radiating oscillator that suffers changes in phase during encounters with perturbing particles. It is assumed that the collisions occur between the radiating atom and a single perturber, one at a time. The collisions are assumed to occur essentially instantaneously, so that the wavetrain suffers an instantaneous phase dislocation that, in effect, terminates it. During the time between collisions the atom is assumed to be unperturbed. Thus suppose that the time between two successive collisions is T, and that in this interval the radiator emits a monochromatic wavetrain /(f) = exp(jo.»0f). The Fourier transform of this finite wavetrain is F{0}, T) = r di = gpfigj^n -' (9.46) J0 i(a> — OJ0) The energy spectrum E(w, T) of this wavetrain is given by substitution of F{o), T) into equation (9-3). In general there is not a unique time interval between collisions; rather, these intervals are distributed probabilistically about some mean value. If r i 282 The Line Absorption Profile the collisions occur as a result of a random-walk process, and the mean time between collisions is x, then the probability that the interval between two successive collisions lies on the range (T, T + dT) is W(T)dT = e-TlT(dT/t) (9-47) Hence averaging over all collision times T, we obtain a mean energy spectrum E{(o) = <£(w, T)}T = (Iny1 j*Q" F*{(o, T)F(co> T)W{T) dT (9-48) Computation of the integral, with normalization, yields (1/™) {T/2n) E[m) (co - to0)2 + (1A)2 (fi) - «0)2 + (T/2)2 Exercise 9-3: Derive equation (9-49). (9-49) The collision broadening theory described above again yields a Lorentz profile (a result of assuming that all the collisions are distinct), with a damping parameter T = 2/t. To complete the theory we must obtain an estimate of x. As was done for the radiation-damped oscillator, we take the profile of an ensemble of randomly phased oscillators, continuously created, to be proportional to the energy spectrum of a single oscillator [averaged over all times, as given in equation (9-49)]. If both radiation and collision damping occur, with widths TR and Tc respectively, and are assumed to be completely uncorrelated, then the profile is a convolution of the two Lorentz profiles. By an analysis similar to that leading to equation (9-24), one may readily show that the combined profile is again Lorentzian with a total width r = TR + Fc. The effects of Doppler broadening can be taken into account as in §9-2, by using a Voigt profile with the appropriate total damping width. We must now calculate the mean collision time x. If the radiating atoms and perturbing particles have atomic weights A,, and Ap respectively, and both have a Maxwellian velocity distribution at a temperature 77, then their average relative velocity is v = /yyi = \_(UTl7imH){A,rx + Ap-l)f (9-50) Assuming that the effective impact parameter of the collisions responsible for the broadening is p0, we then have r"1 - np02Nv (9-51) and r = 2np02Nv (9-52) where N is the perturber density. We must now determine p0. 9-3 Collision Broadening: Classical Impact Theory 283 Following Weisskopf (661) we assume that (a) the perturber is a classical particle; (b) the perturber moves with constant velocity past the atom on a straight-line path with impact parameter p; (c) the interaction between atom and perturber is described approximately by Aw = Cp/rp (9-53) where r(t) = (p2 + v2tTf, t = 0 occurring at the point of closest approach; and (d) no transitions in the atom are produced by the action of the perturber. The validity of these assumptions will be considered later. The form of the interaction in equation (9-53) is only approximate but holds over a fairly-wide range of distances. The value of the exponent p depends upon the nature of the interaction. Values of as trophy sical interest and the interaction they represent areas follows: p = 2, linear Stark effect (hydrogen + charged particle); p = 3, resonance broadening (atom A + atom A); p = 4, quadratic Stark effect (nonhydrogenic atom + charged particle); p = 6,van der Waals force (atom A + atom B). The interaction constant Cp must be calculated from quantum theory or measured by experiment. The phase shift induced by the perturbation is n(t) = (*'^ AcMt') dt' = Cp j'_ ^ (p2 + vH'2)-"12 df The total phase shift n(p) = i\(t = co) is found directly to be where % = **r[£(p - i)]/r&>) (9-54) (9-55) (9-56) Here T denotes the usual gamma function; for p — (2,3,4,6) one finds (£). Then we write .AO = exp ico0t + i Ato(r') dt' — f7iW+»!(()] (9-59) where r\{t) is the instantaneous phase of the oscillator. To obtain the line profile, we calculate the autocorrelation function {s) be defined by (p(s) = e~i(0oS®(s), which eliminates the unperturbed oscillation. Then from equation (9-6), (j)(s) = lim T-1 f*1/2 f ";",pi(?"'fj(£, s)]>r Further, writing d{s) = t (9-61) (9-62) (9-6?) where if denotes the change in phase occurring in the time-interval ds as a result of collisions that take place in that interval. The phase change cannot be correlated with the current value of the phase if the collisions occur at random. Thus the average of the product can be replaced by the product of 9-3 Collision Broadening: Classical Impact Theory 285 the averages, i.e., d7. = 0(s)T (9-64) If we can calculate the average ofe'Y, we obtain a differential equation for , = 2kNv ds rV^ - 1],, dp (9-65) The integral in equation (9-65) has both a real and imaginary part, so we write (0) = 1 we find ttlie i«tegrand of aK rapidly drops to zero [see (629,16) or (638.305)], and the dominant contribution to the line broadening comes from (strong) collisions inside the Weisskopf radius—-Le., (p/pw) < 1. In contrast, for ffj, the integrand for (p/pw) < 1 fluctuates in sign, and averages to nearly zero. Thus the dominant contribution to the line shift comes from (weak) collisions outside the Weisskopf radius. It is easy to understand physically how the shift arises. The very weak collisions fa « 1, p » pw) are extremely numerous and occur at an essentially constant rate, yielding an average phase change per unit time of (9-73) tj = 2?zNv * n(f>)p dp where p* is chosen to assure that rj{p*) « I. But as can be seen from equation (9-59), the rate of change of phase is by definition a change Aoj in the oscillator's frequency. SPECIFIC CASKS Lindholm theory has been most widely applied in astrophysical work for the cases p = 3,4, and 6. For these cases the integrals in equations (9-67) and (9-68) can be evaluated explicitly to yield the values listed in Table 9-1 [see (629, 14) for details]. The last line of the table gives the value of r\0 that, when inserted into the Weisskopf formula [equation (9-58)], gives the Lindholm V. As t]0 is always less than unity, it can be seen that the Weisskopf formula always leads to too small a value of T. TA1ÍÍ.F 9-1 Results of Lindholm Theory p 3 4 6 r Acu0 2ti2C3N 11.37} ; , 9.85] 2.94] 0 1.16 2.75 'Í0 0.64 0.64 0.61 Resonance broadening, p = 3. is of importance mainly for collisions of hydrogen atoms with one another. As the atmosphere must be hot enough that hydrogen is excited to the n = 1 state (to produce the observable Balmer Sines) but cool enough that it is not dominantly ionized, resonance broadening effects are of interest for solar-type stars. The interaction constant C3 in equation (9-53) for level n is C3 = {e2j\J2mo^n) (9-74) [see (662; 112, 231)]. A quantum mechanical calculation gives a F slightly different from the Lindholm value, namely T3 = (16*niC3/3) = (*nAe2flnßmvln) (9-75) For the hydrogen lines there is no shift Aa)0 because individual Stark components split symmetrically about line center (see §9-4) and the shift is identically zero. Resonance broadening is most significant for the lowest members of a series where Stark broadening is the smallest. The effects of resonance broadening have been shown to be important in the solar Ha line but negligible for higher series members (146). Quadratic Stark effect, p = 4, is important for the broadening of lines of nonhydrogenic atoms and ions by impacts with charged particles (electrons), and is the dominant pressure-broadening mechanism for these lines in the atmospheres of early-type stars. In applications of the classical Lindholm theory the interaction constant C4 was typically estimated from experimental measures of line shifts in electric fields or from time-independent perturbation theory for the quadratic Stark effect [see (11, 319-320) or (638, 326-328) for examples of this procedure]. The resulting damping widths are usually much too small, however, because the Lindholm approximation assumes the collisions are adiabatic (i.e., do not cause transitions in the radiating atom); this assumption is frequently poor, and accurate quantum mechanical calculations including nonadiabatic effects (cf. §9-5) yield much larger line-widths. Van der Waals interactions (p = 6) of non-hydrogenic atoms in collisions with neutral hydrogen atoms is the major source of pressure broadening in solar-type stars. The usual classical treatment accounts for the dipole-dipole interaction term in the potential, and yields [cf. (629, 91 -97; 638, 331 -334)] Am = C6/r6 = e2afl02[RH2 - Ä(2]/(fir' (9-76) where a is the polarizability of hydrogen and R2 is the mean square radius of the two levels. Quantum mechanical results are sometimes available for .R2; if not, hydrogenic estimates are used. Using C6 determined in this manner, F can be computed from Lindholm theory. When this is done (e.g., for lines of Fe I), it is found that the predicted values are much too small, by factors of 5 to 30 (382). Quantum mechanical calculations that again 2SS The Line Absorption Profile employ only the dipole-dipole term do not lead to large increases in T [e.g., (264; 98; 86; 599)], which points to the breakdown of the dipoie-dipole approximation rather than other theoretical problems [see also (301; 302; 541)]. Calculations using the more realistic Lennard-Jones potential (303) lead to significantly larger widths. Attempts have been made to include more terms in the expansion of the interaction potential (233); this increases T, but still falls short of the observed values. The expansion technique converges slowly, and an alternate approach, applied to the lines of Fe I, evaluated the exact expression for the interaction using scaled hydrogenic wave functions (116) and showed reasonable agreement between predicted and measured F's. Yet another method is based on the proposal that the dominant interaction leading to the line broadening is between the perturber and the valence electron (539; 540). One can then use a Smirnov potential (578) and obtain an expression for the damping parameter. This approach has been used to produce extensive tables (193) giving the parameters a and jj in the formula T = NaTp, as functions of the effective quantum numbers n* of the lower and upper levels, for s-p, p-d, and ^/'transitions. VALIDITY CRiTHRIA (1) Ail effective impact time ts can be defined such that is times the peak value of Aco for a collision at the effective impact parameter, namely CppQ~p, yields the total phase shift given by equation (9-55). This gives t, = WPPoM (9-77) For impact theory to be valid, we demand that only one collision at a time occur, so that ts < t = l/(N7ip02v), Writing JV = 3/(47tr03) where r0 is the mean interparticle distance, we find (ts/t) = l\j/p(p0/r0)3. Thus impact theory will be valid only when the density of particles is so low that the Weisskopf radius is small compared to the interparticle distance. (2) It is clear that, as p ^ go, the effective impact time ts becomes larger and larger, and eventually exceeds t, the mean time between collisions, so that the collisions overlap. Thus from the very weak collisions there is an essentially continuous perturbation of the atom, and here we expect statistical theory to begin to be valid. Indeed, we saw earlier that these weak collisions produce the line shift, just as would be given by the application of a steady perturbation. Even though Lindholm theory (an impact theory) treats the weak collisions, the calculation is not strictly logically consistent. (3) Impact theory fails for sufficiently large frequency displacements Aoj from line center, and statistical theory becomes valid. In impact theory, it follows from the general properties of Fourier transforms that the characteristic interruption time x corresponding to a frequency displacement Aoj is i ~ 1/Aco. For sufficiently large Aco's we will eventually have t « xs, and impact theory breaks down. These values of Aw correspond to large phase 9-4 Collision Broadening: Statistical Theory 289 shifts (i.e., Aons » 1) and hence to the strong collisions that occur inside the Weisskopf radius. It is difficult to construct a theory that makes the transition from impact to statistical theory. A useful conceptualization is to suppose that there is a "boundary" frequency Aws inside of which impact theory holds and beyond which statistical theory is valid. To a fair approximation [see (637; 306)] Acog m Acow where Aa)w denotes the frequency shift produced by a perturber at the Weisskopf radius pw \ i.e., Aojw = (v"/C^/y*o-^ (9-78) Note that Aa>w corresponds to a phase shift of unity. We shall see in §9-4 that equation (9-78) implies that broadening of the hydrogen lines by ions follows the statistical theory, while electron broadening is given by impact theory. (4) Classical impact theory assumes that the collisions are adiabatic—i.e., transitions are not induced in the atom. A collision occurring in an impact time ts will have Fourier components of frequencies up to cos ~ 1/ts. To guarantee that the collision is adiabatic, cos should be much smaller than any characteristic transition frequency w^; i.e., ct)s = 1/ts « a)ij = |£( - Ej\/h (9-79) For nondegenerate levels, the energy separation is often large enough that the condition stated above will be met. But for degenerate levels (e.g., for hydrogen), the energy separation between levels will be proportional to the perturbing field itself; i.e., \Et - E}\ & hCp/pp - h Ato(p). Then the condition for adiabaticity implies that Ato(p)xs = n{p) » 1; that is, only collisions inside the Weisskopf radius will be adiabatic. In the case of hydrogen, Acow for ions is very small, and for virtually the entire profile the statistical theory is valid, and the collisions causing the broadening occur inside the Weisskopf radius. Thus the ion broadening will be adiabatic. Precisely the opposite is true for electrons. Here Acow will be large, and almost the whole profile is described by impact theory with the relevant collisions lying outside the Weisskopf radius. The electron broadening is strongly nonadiabatic (and hence must be described by quantum theory). When the adiabatic assumption breaks down, much larger damping parameters than those predicted by classical theory are found; for this reason the modern quantum mechanical results are often drastically different from earlier classical work. 9-4 Collision Broadening: Statistical Theory The basic picture in this theory is that the atom finds itself radiating in a statistically fluctuating field produced by randomly distributed perturbers. The motions of the perturbers is ignored; this is known as the quasi-static approximation. (As we shall see later, this approximation is good for the 290 The Line Absorption Profile slow-moving ions—e.g., protons—in the plasma.) A specific distribution of perturbers produces a definite field; the relative probability of fields of different strengths is thus determined by the statistical frequency with which particle distributions producing the appropriate strengths are realized. For a given value of the field, the oscillation frequency of the radiator is shifted by a definite Aco. The intensity of the radiation at this Aco is assumed to be proportional to the statistical frequency of occurrence of the appropriate field. Thus the central problem is to determine the probability distribution of the perturbing fields. Once this is known, line profiles can be computed. The applications in this section will be restricted to quasi-static broadening of hydrogen lines by linear Stark-effect interactions with protons (though the theory is relevant in other contexts as well). THE NEAREST-NEIGHBOR APPROXIMATION As a first approximation, we assume that the main effect on the radiator results from the strongest perturbation acting at any given instant—namely, that from the nearest neighbor—and that the effects of all other particles are nedected. Then if W(r) dr is the probability that the nearest neighbor is located on the range {>; r + dr) from the radiator, the frequency spectrum is /(Aco) d{Aoy) x W(r)[dr/d(Aco)] co, W(p) f/T hence statistical theory predicts that, in the wings of a line broadened by linear Stark effect, the profile falls off as Aw-* in contrast with the prediction Aco-2 given by impact theory. The basic failing of this theory is that the profile is, of course, the result of perturbations by all particles, not just the nearest neighbor. To obtain accurate results a more elaborate theory must be constructed. HOLTSMARK THEORY The effect of an ensemble of particles upon a radiator was determined by Holtsmark (305), who calculated the net vector field, at the position of the radiating atom, from the superposition of the field vectors of all perturbers. An elegant treatment of the problem was given by Chandrasekhar (15T); this paper should be consulted for the derivation of the results quoted here. For an interaction of the form F = Cp;rp, the analysis yields W(p) = (2/J/tt) J" exp(-v3")j> sin fly dy (9-i Here (i = F/F0, the normal field strength f 0 now being defined as F0 = ycyv"3 (9-89) where y = {(2n2p)/[3(p + 3)T(3/p) sin(37i/2p)]}"/3 (9-90) In particular, for linear Stark effect, p = 2, Cp - e, and y = 2.6031 so that F0 = 2.6031eiV}, which differs only inconsequentially from the normal field strength given by nearest-neighbor theory. 292 The Line Absorption Profile 9-4 Collision Broadening: Statistical Theory 293 The integral in equation (9-88) cannot be evaluated exactly for p = 2, and W(ß) must be expressed in a series expansion. For small ß while for ß 1 one finds an asymptotic expansion W(ß) = 1.496/3-^(1 + 5.107/T* + 14.43/T3 (9-92) the leading term of which is essentially the same as given by nearest-neighbor theory. Tabulations of W(fi) are given in (151) and (629, 28). DEBV13 SHIELDING AND LOWERING OF THE IONIZATION POTENTIAL In deriving the probability distributions described above, interactions among the perturbers were ignored. In reality, the probability that a particle is found in a volume dV is not just N dV, but depends also upon the electrostatic potential (f> in dV. For example, if at some point > 0, electrons will tend to migrate toward it while ions will tend to migrate away, and vice versa if < 0. Following Ecker (203; 204; 205; 206), we may account for these effects schematically by introducing a Boltzmann factor depending on i// = (e/D2) where D = (kTI4ne2f + 1^ (9-99) is the Debye length. Solving for we find

D. One then finds W(ß, S) - (2ß ßy) dy where 3 A fx, g(y) = - y* J (1 - z sin z)z J dz (9-102) (9-103) 294 The Line Absorption Profile 9-4 Collision Broadening: Statistical Theory 295 and S = %7tD*N, the number of particles contained in the Debye sphere. As 5 -> oo, one expects to recover the Holtsmark distribution; i.e., W{fi, co) = WH{fi). From equation (9-102) one can show that for 8 co, W{[i, S) = Wj{f$) + S^^Fifi) where F(0) is a bounded definite integral. Recovery of the Holtsmark distribution for large and S{a) for numerous hydrogen lines are given in (634). The largest Ck for a line of upper quantum number n increases as n2, and recalling diat for j8 » 1, W{fi) cc /T!,we see from equation (9-108) or (9-110) that the Stark widths of lines rise rapidly up a series, roughly as n3. Let us now consider when the quasi-static profiles are applicable. Let % = X hnktL ^ tne sum being taken over p°sit've values only- Tnen' writing Aco = C2/r2 and F = e/r2, it follows from equation (9-107) that C2 = (3hnk/4nm) = 1.7387ft (9-112) with nk ~ £n(n ~ 1) for n » 1. From equation (9-78) the wavelength shift delimiting the transition between the impact and statistical theories is (for p = 2) AXiV = (v2X2/2n3cC2) (9-113) Notice that AXW cr v2, and thus AV(electron) ~ 103 A;.„,(proton). Using equations (9-112) and (9-50), we obtain the results listed in Table 9-2. It is obvious that the ion broadening is very well described by the quasi-static theory (especially when we note that Doppler motions will dominate in the core). The electrons, however, are in the impact-broadening regime except TABLF. 9-2 transition Wurelcngth &/.w(A\ between Statistical and Impact Broadening for Hydrogen Lines T Line Perturber 2.5 x 10* °K 104|2 (9-114) 298 The Line Absorption Profile where d is the atomic dipole moment. The total emission, summed over all possible substates contributing to a line is P{a>) = (4|2 (9-115) i. 3 Here p} is the probability that an atom is in the upper state j; in thermodynamic equilibrium Pj = = vLp[-Ej/kT)/U(T) (9-116) where (7(77) is the partition function. To calculate the broadening of lines emitted by an atom in a plasma, we consider the radiating system to consist of atom plus perturbers, and generalize the meaning of states |i> and jj> to include perturbers also. The profile of the line is then written as m = ZPj^tffl - a>£W\D\2 (9-117) j. j Now Pj refers to the probability of a particular state of atom and perturber. If the plasma is in thermal equilibrium, p, is proportional to e-H/kT = e-(HA + HP + V)lkT (9-118) where H is the total Hamiltonian of the system, HA and HP are the Hamil-tonians of the atom and perturber alone, respectively, and Kis the interaction Hamiltonian. It is easiest to account for collisions using the Fourier transform <£(t) = f_°^ I(co)eiM dw = £ p/ffl^||2 (9-119) which is analogous to the classical autocorrelation function [see (72, 498)]. The effects of collisions at a specific impact parameter upon the autocorrelation function (and statistical averages over all possible perturber paths) can be calculated directly. The intensity profile then follows from the inverse Fourier transform [cf. equation (9-7)]. The problem at hand is to obtain an expression for and [;> under the effects of perturbing collisions in terms of a time-development operator T(t, 0). This operator is defined such that a state of the system at time t is related to the state at time t = 0 by |h, t} - T(t, 0)|a, 0> (9-120) 9-5 Quantum Theory of Line Broadening 299 Now \a, t} satisfies the Schrodinger equation H\a, 1} = ih(d\a, t'yjdt) (9-121) Substituting equation (9-120) into (9-121) we can derive a Schrodinger equation for T(t, 0). Noting that \a, 0> is fixed in time, we find HT(t, 0) = ih[dT{l, 0)/dt] (9-122) which has the solution T(t,0) = e~iHllh (9-123) where the exponential is to be understood as an operator. We may now rewrite equation (9-119) in terms of time-development operators, including perturbations; we have m = itpJ^Ei~Ein<¥\j>\2 = ZpAjWye-^o^jy^ i. j = Zpj(i\dr\f) (9-124) t. j But the expansion rule, relative to a complete set of states |y), is (a\py = £ (9-125) 7 so we see that equation (9-124) can be rewritten as 0(0 - Z pjOldrdr'|y> = tr(p drdr^) (9-126) The trace includes both atomic and perturber states. This expression is quite general; the detailed form of depends upon the form of 77. THE CLASSICAL PATH APPROXIMATION Let us now consider the calculation of T in more detail. Let the wave function \j/ describing the solution of the complete system of atom plus perturber be the solution of the equation ih(d\}//dt) = (HA + Hv + V)\j/ (9-127) Note that HA is independent of perturber coordinates, HP is independent of atomic coordinates, while V depends on both. Further, because the atom 300 The Line Absorption Profile 9-5 Quantum Theory of Line Broadening 301 is in a static ionic field of strength F, HA = HA(F). For brevity, F will be suppressed here but included explicitly in the final result. To make progress, we now assume that the perturber follows its classical pař/i—i.e., a straight line past a neutral atom and a hyperbola around an ion. This assumption will be valid when the deBroglie wavelength is small compared to the impact parameter for those collisions that dominate the broadening. That is, we must have p » /. = (h/mv) or mvp » h. But mvp is just the angular momentum of the perturber ( — In); the criterion just stated is equivalent to the requirement that the quantum number I be much larger than unity. Under such circumstances the classical-particle picture would be expected to be valid on the basis of the correspondence principle. The validity of the approximation must always be checked in line-broadening calculations; in most cases of astrophysical interest it is found to hold. Some perturbers that violate the condition can always be expected, but the approximation remains useful if they do not dominate the broadening [see also (73, 498 ff.)]. A theory that does not use the classical path approximation is given in (64). The wave function for the system of atom plus perturber is assumed to be separable-Ä.Q., \jj{t) = a(t)it(t), where «(() and %{t) are the atomic and perturber wave functions. We further suppose the perturber path to be fixed and to be independent of the state of the atom with which the interaction takes place. In this way the effect of the perturber on the atom is taken into account, but the back-reaction of the atom on the perturber is ignored. This will be valid if the energy gained or lost by the perturber (of order fir, where T is the linewidth) is much less than its kinetic energy (kT), a condition that is almost always satisfied. A few collisions will always occur in which large energy exchanges take place, but again these will not invalidate the assumption if they do not dominate the broadening. Under the assumptions made above, n{t) is the solution of ih\cin{t)l = 0, = 0, and (|^> (9-135) Here we have neglected the variation of pa among the upper substates and have noted that only the time-development operators can depend upon the statistical averages. 302 The Line Absorption Profile It is convenient to replace the complete time-development operator Tby V the time-development operator in the interaction representation, defined as U{t,0) = eiH*""T{t,0) (9-136) This definition factors out the time-behavior of an eigenstate and isolates the perturbation effects; for an unperturbed eigenstate U{t, 0) = 1. By substitution into equation (9-135) we have 0(0 = pa £ <ŕ1d|«0|ŕ'> (9-137) Substituting equation (9-136) into (9-132) we obtain a Schrodinger equation for V(t, 0), ih[dU(t,Q)ídt] = eiH^hVcMe-iH^'U(t,0) ~ V'Jt)U(U0) (9-138) We solve this equation by iteration, starting with fj(t, 0) = 1 as a first approximation, to obtain a solution of the form rj(f, 0) = l + (ifi)"1 £ KM dt, + (ifi)"2 ^dt^'Mi) dnvjt,) + - ■ ■ (9-139) Now consider the calculation of the statistical average {UbU*}. We proceed in a way analogous to that used in Lindholm theory. That is, we write the change in {UbU*}, caused by a specific collision (treated as an impact), in some time At as A{UbU*} = {Ub{t + At, t)U*(t + At, t) - l}{UbU*}, where again we argue that the changes in (ŕ, t + At) are statistically independent of the current values, so that we can replace the average of the product by the product of the averages. Then using equation (9-139) on the interval (r, t + At) one obtains an explicit expression for {Ub{i + Ai, £) U*{t + At, t) - 1} [see, e.g., (264, 70; 268, 37)] of the form {Uh(t + M)U*{t + At) - 1} = ^-^(^AOe-^-11^ (9-140) We thus obtain a differential equation for {UbU*}, namely A{UhU*} = ^"-""W'^-^-^^'i^C/?} At (9-141) whose solution is \Ub(t, 0)U*(U 0)} = e^"-^1' exp[-i(H, - Ha)tjh + Oflbi] (9-142) Now, substituting into equation (9-137), we obtain 0(0 = Pa S lŕ'> (9-143) 9-5 Quantum Theory of Line Broadening 303 Then performing the inverse Fourier transformation, and reintroducing the quasi-static ion field F, we obtain the intensity profile Koj, F) = n V a, a', ft, ji' x -®ab(F)--[Ha(F)-Hb(F)-] '>|/3'> (9-144) where use has been made of the fact that O^F) is found to have a negative real part. If W{F) is the probability of an ion field of strength F, the final profile, averaged over all ion fields is K>|/0 )dF Uco) - W{F)tf< ( £ <)8'|d|a'> <«|(lft is straightforward in principle, it is complicated in practice, for it entails cutoffs both at small and large impact parameters, the former arising from strong collisions inside the Weisskopf radius (which are no longer correctly described by the iterative series development for U, the time-development operator) and the latter to account for Debye shielding effects. Further cutoff procedures are required to allow for the transition of the electrons from the impact- to statistical-broadening regimes. This work culminated in the publication of extensive tables (353; 356) for the first four members of the Lyman a;. (ä, 0.1 0.3 0.5 l.Ü 3.0 5.0 10.0 30.0 50.0 10.0 S(a) 0.01 - 0.001 - 0.01 0.03 0.05 0.1 0.3 0.5 3.0 5.0 10.0 Stirk profiles for Hoc at nr = 3.16 x 1014 cm"\ T = 104 'K. Anfrrf«/n*: Holtsmark profile for ion broadening only (634). Dashed curve: Profile for quasi-static .on broadening and impact broadening by electrons (650). Solid curve: Electron and ion-broadened profile convolved with thermal Doppler distribution of atoms. Notice that the line-core is dominated by Doppler effects, and that the absorption cross-section in the line wing is substantially larger when electron broadening is included. V-5 Quantum Theory of Line Broadening 305 and Balmer series lines. These tables give 5(a), analogous to the profile described by equations (9-109) through (9-111), but including the effects of electron impacts. An alternative approach developed by Cooper, Smith, and Vidal (581; 647; 648; 649) uses a unified theory that accounts automatically for the transition of electrons from the impact to quasi-static regimes. Extensive tables of results based on this theory have been published (650) for the first four members of the Lyman and Balmer series in temperature-density ranges appropriate to stellar atmospheres. These tables include the convolution of the Stark profile with a thermal velocity distribution of the hydrogen atoms. Figure 9-2 compares the profile of HS from the unified theory (including both electron and ion broadening) with the quasi-static theory for ions only. For higher series members one may use an approximate theory (262), after corrections to some of the matrix elements are made (265) [see also (45)]. In astrophysical applications, the hydrogen lines are significantly affected by broadening mechanisms other than the Stark effect. The core of the line is dominated by Doppler broadening. The effects of radiation and resonance damping may be important in the wings at low electron densities. Assuming these mechanisms are all uncorrected, we may account for their combined effects by a convolution procedure. Folding the Doppler profile with the Lorentz profiles from radiation and resonance damping gives a Voigt profile, H{a, v\ where a = {TnA + rres)/47i AvD,andu = (Av/AvD). This Voigt profile is then convolved (numerically) with the Stark profile S(a), yielding the cross-section av(Av) - (n*e2/mc)f J" _ S*(Av -I- v AvD)H(a, r) dc where 5* is S(a) converted to frequency units. (9-146) HYDROGENIC IONS Hydrogenic ions (of charge Z) have Stark patterns identical to those of hydrogen, though the energies are, of course, different. The profile from ion-broadenins alone is S(y.) = Z5S0S(Z5x) (9-147) where SQS is the quasi-static hydrogen profile given in (634). Note that on a wavelength scale the ion lines are narrower by a factor of 1/Z5. Indeed, because the Stark widths decrease for ions while the radiative transition probabilities increase, there comes a point where Stark broadening can be neglected compared to Tr;ili. The effects of electron broadening for hydrogenic ions are similar to those for hydrogen, though the expression for ($>ab changes because now the 306 The Line Absorption Profile 9-5 Quantum Theory of Line Broadening 307 perturber moves on a hyperbola around the positively charged ion instead of a straight-line path. Early calculations of the broadening of the lines He II A3203, Á4686 (prominent in O-star spectra) are given in (272), and much-improved calculations for He II U256, 304, 1085, 1216, 1640,4686, and 3203 are given in (354; 355). Unified theory (258; 260) calculations for He II ^304 are given in (259). Unfortunately, precise calculations for the astrophysically important Pickering series lines (4 -» n) (e.g., ^10124, 5412. 4542,4200, etc.) are not yet available, and only an approximate theory exists (262; 265: 45). NHUTRAL HELIUM LINES The lines of He I are prominent in the spectra of B-stars. Here the electron impact broadening and quasi-static ion broadening act by quadratic Stark effect, and for isolated lines yield profiles of the form W{F)dF (Aw - d - C+F2/e2)2 + w~ (9-148): where W(F) is the probability of a field of strength F, w is the electron-impact width of the line, and d is the line shift. Note that because F enters as F2, the ionic fields always skew the line components in one direction and thus lead to an asymmetric profile. Explicit expressions for w and d are given in (269; 264, 81 -86; 268, $II.3cce; 180) for isolated lines; tables allowing the calculation of the profile in terms of convenient dimensionless units are given in (269). Detailed numerical results for w and d for several lines are given in (269), and improved results are given in (180) and (67). A much more interesting (and difficult) case is presented by the diffuse series lines (23P-n3D), (21F-n1D) for n Js 4. As was; first recognized by Struvc (614:615), the (23P-43D) He I /.4471 line shows al "forbidden" component (23_P-43F) at /.4470. The other diffuse series lines also show these components which arise from the mixing of the (3D, 3F) or f1!), 1F) states m the presence of the electric fields in the plasma. As the., diffuse-series lines are among the best observed in stellar spectra, a reliable theory for them is of great interest. The first attempts at constructing such broadening theories (66; 245; 266) were not too successful, for they gave "forbidden" components that were too narrow and too intense. Compared; to observed stellar profiles the theoretical predictions showed too much contrast between the "forbidden" absorption component and the gap between it and the allowed component; the theory also disagreed with experimental-measurements (122; 123; 124). A comprehensive new theory was then developed for the He I A4471 (68) and A4921 (21P-41D) lines (69); predictions based on this theory are in excellent agreement with observed stellar spectra (438; 439). OTHER LIGHT ELEMENTS Electron collisions broaden the lines of other elements observed in stellar spectra. Widths and shifts for these lines may be calculated using techniques similar to those employed for He I [see (268, §§II.3c, il.3d)]. Extensive results for neutral atoms are given in (264,454-527; 268, Appendix IV; 84). For charged ions, the Coulomb interaction between radiator and perturber implies a hyperbolic path for the latter (110; 111), and line-widths calculated allowing for this are substantially larger than those calculated with a straight-line path. Extensive results of detailed calculations for ions are given in (268, Appendix V; 111; 548: 167; 549; 550). Convenient approximate formulae for estimating Stark-widths are given in (267; 183; 549). 10-1 Characterization of the Problem 309 Classical Treatments of Line Transfer In this chapter we discuss some of the early approaches to the line-formation problem; these provide background for the more modern treatments to be presented in following chapters. A fuller discussion of these older methods and of their application to stellar spectra can be found in (684, Chap. 7; 638. Chaps. 15-17; 15, Chaps. 12-16; 11, Chap. 8; 256, Chaps. 14-16). It is essential to be familiar with such methods because of the large body of literature based upon them. More important, one must understand the physical basis of the classical treatments in order to evaluate the reliability of spectroscopic diagnostics derived from them, and to realize the conceptual differences inherent in recent work. 10-1 Characterization of the Problem In the usual classical approach one notes, at the outset, the existence of two different line-formation processes: scattering and absorption. We have discussed these two categories in Chapter 2, and described there the physical differences between them. It is usually supposed that a fraction (1 - c) of the photons absorbed are simply scattered so that the excited electron returns directly to its original lower level, and normally it is assumed that the scattering is isotropic and coherent (actually complete redistribution is a much better approximation). Then the scattering contribution to the emission coefficient is n' = (1 - e)Zi/v' = ezrf&iT) (10-3) In the limit of strict LTE, e = 1, and all the emission is thermal. In addition, there are contributions to the opacity and emissivity from continuum thermal processes and electron scattering. Thus the transfer equation is p(dljdz) -- ~(kc + a + Xi4>v)f + kcBv + aJv + ext(f>vBv + (1 - e)x,0vJv (-10-4) We have ignored the v-dependence of kc and a because most lines are so narrow that these coefficients vary only negligibly over the line in comparison with the swift variation of v) dz, and let p = oj(kc + a), and A'v = Xi4>J{*c + ff) (10-5) we have p{dljdxv) = Jv - {[(1 - p) + ejSv]5v + [p + (1 ~ 6)0V]JV}/(1 + Pv) (10-6) Or. defining ;,. = [(1 - p) + e/Sv]/(l + 0V) (10-7) the transfer equation becomes H(dlv/6xv) = f~ XVBV - (1 - AVUV (10-8) E. A. Milne and A. S. Eddington developed equation (10-8) as an approximation to the line-transfer problem, and customarily it bears their name. 310 Classical Treatments of Line Transfer Excellent discussions of the physical implications of this equation have been given by Milne (416, p. 169ff) and Stromgren (613). From a physical point of view, equation (10-8) provides a rather severe idealization of line formation and may be criticized on several counts. (1) Line-scattering is not. in fact, coherent. This inadequacy is overcome if we write, instead of equation (10-8), JR{v', v)Jv-dv' (10-9) where R(v'„ v) is an appropriate redistribution function (cf. §2-1 and Chap. 13). (2) To solve equations (10-8) or (10-9), both the parameter e and the occupation numbers n,- and n} must be known. In classical treatments it is often assumed that LTE holds, so that c = 1 and n; - nf, n} = nf. It must be stressed, however, that this merely an assumption and, as will be seen in Chapters 11 and 12, the assumption is often unjustified and may yield results seriously in error. In many treatments, an ad hoc value is chosen for e, but the occupation numbers are still assumed to have their LTE values. Such an approach is internally inconsistent, for when scattering occurs in a line, the level-populations depend upon the radiation field via the equations of statistical equilibrium. (3) An equation of the form of (10-8) was derived by Milne for a strict two-level atom, and his analysis yields a unique (and correct) value for the parameter c (416, pp. 172-178). However, as mentioned above, coherent scattering is not an accurate approximation. More important, analysis of the equations of statistical equilibrium for general (i.e.. multilevel) atomic models shows (cf. Chapters 11 and 12) that other kinds of terms appear in the source function; these may depend upon the radiation fields in other transitions (continua and lines), and thus in principle couple all the lines in the spectrum together. In short, both equations (10-8) and (10-9) are seriously incomplete from a physical point of view, and this should be borne in mind during the discussion that follows in this chapter. 10-2 The Milne-Eddington Model DEFINITION Let us now consider the Milne-Eddington equation [equation (10-8)] under the simplifying assumptions that lv, e, and p are all constant with depth, and that the Planck function Bv is a linear function on the continuum optical depth scale r: i.e., Bv = a + bx = a + [bzj{l + p\)] = a + pyiv (10-10) 10-2 The Milne-Eddington Model 311 Under these conditions an exact solution may be obtained (158) but. this differs only slightly from the approximate solution derived below. Taking the zero-order moment of equation (10-8) we find (dHv/d*cv) = Jv - (1 - av)Jv - KBV = Wv ~ Bv) (10-11) Aside from the definition of Av, equation (10-11) is the same as equation (6-4), and from the analysis in §6-1 we know that the solution, in the Eddington approximation, is Jv = a + pvxv + (pv - V3a)exp[-(3AVK]/L73 + (3A,)*] (10-12) and the emergent flux is HJO) = (3)-*Jw(0) = -} [py + (3;.v)*fl]/{l + av-) (10-13) Equation (10-12) shows that thermalization (Jv -»■ Bv) occurs only at depths of the order of Xv~i. Recalling the definition of xv, wc see that this depth is (1 - p)~* in the continuum (/jv = 0), and e~* in a strong line (p\, -> co). In both cases thermalization occurs at a depth where p is the probability that a photon is destroyed and converted to thermal energy each time it interacts with the material; these results are compatible with the random-walk arguments given in Chapter 6. Note that these results apply only for coherent scattering (cf. Chapter 11). Equation (10-13) may be used to compute the profile of a line in a stellar atmosphere. In the continuum, j8v - 0;thenxv - (1 - p), and the continuum flux is Hc(0) = i \b + aVKT- P)]/[1 + (1 - pf] (10-14) Thus the residual flux in the line is, by equation (S-2). R. - P, + (3>lv)V 1 + xJ" 1 + (1 - P)* b + a^j3(\ (10-15) There are four important results that can be derived from this classical theory, which provided the conceptual orientation of much of the early work on line formation. Let us now examine these briefly. SCATTERING LINES Consider the case where p = 0. so that there is no scattering in the continuum. Further, suppose that e = 0, so that there is pure scattering in the 312 Classical Treatments of Line Transfer lines. Then /v = (1 + ft)-1 and the residual flux in the line is R„ = 2 1 + ft V + ßy ßa + b) (10-16) If we consider the case of a very strong line and take the limit ft. -»• oo, we obtain .R,, = Hv(0)/Hcf0) = 0, which shows that the core of a very strong line formed by scattering can be completely dark; this result is in contrast to the case of a line formed by absorption, as is shown below. Exercise W-l: Writing/?,. = jS0H(a, t1), where if denotes the usual Voigt function, plot residual flux profiles from equation (10-16) for scattering lines with /J0 = 1,10, 100, 1000, 104, and ib/a) = 1, 2, and 3, assuming a = HT3. Exercise 10-2: In the Schuster-Schwarzschiid model, the lines are assumed to be confined to a finite layer (the "reversing layer") of thickness ru, illuminated from below by an incident intensity I0. In the reversing layer the continuum opacity is zero, and the lines are pure scatterers. Using the two-stream approximation (/ = /+ for 0 =S p =S 1; / = J~ for ~ 1 < p < 0; and p = ±^ in the transfer equation) show that (a) Hv = - /v") = constant = £/0/(l + tv)- and (b)J,u,) = Hv(l + 2rv), 0 tv ^ v ABSORPTION LINES Again assume p = 0, but now set c = 1 (LTE in the line); then Xv = l,and Rv = [V'3a + 6(1 + ft)"1]/^ + b) (10-17) In this case, as ft -> oo, the flux in the line does not go to zero, but approaches a finite value Ro = = 1, ft - oo) = [1 + (b/yfa)] (10-18) This is to be expected, for as ft -*■ oo. only the surface layers of the star are seen, and the emergent flux is then determined by the surface value of the Planck function, which will be nonzero. In contrast, in the scattering case, photons are constantly diverted out of the pencil of radiation, and in the limit ft -> oo, none survive to emerge at the surface. Exercise W-3: Repeat exercise 10-1 for an absorption line, using equation (10-17). It is convenient to re-express equations (10-17) and (10-18) in terms of the Planck function and its gradient. Assume that on a mean optical depth scale, Bv(z) = BV(T0) + (dBJdx)0x = B0 + Bxz (10-19) 10-2 The Milne-Eddington Model 313 Using the Eddington-approximation result for the grey temperature distribution, namely T4 = 7704(1 + fx), it is easy to show that #i - ö ^o^o (10-20) where X0 = u0/(l - e""0) (10-21) and u(, = (hv/kT()). Thus the parameters in equation (10-10) are: a = BQ) and b = - XMk/k) P* = g X0Bq{k/k)/(1 + ft) (10-22) ;30-23) where k and k are the mean opacity, and the monochromatic continuum opacity at the line wavelength, respectively. Then equation (10-18) becomes R0 - (1 + [V'3X0(kA)/8]}-' Exercise 10-4: Derive equations (10-20) through (10-23). (10-24) For the sun, T0 ~ 4S00°K from a grey model, and if we choose X = 5000 A, then u0 -~ 6, X0 — 6, and k « ;7, so that equation (10-24) predicts Kn = 1 4 '3 1 = 0.44 (10-25) This value is in fair agreement with the depths of many of the stronger lines observed in that region of the solar spectrum. Some lines, however, are much deeper, particularly resonance lines such as the sodium D-lines; this fact led to the conceptual identification of resonance lines as "scattering" lines and subordinate lines (e.g., Ha) as "absorption" lines. It was believed that the central intensities of the latter group reflected information about the surface temperature of the atmosphere. Such a division of lines into two groups is intuitively not unreasonable, for we expect that in a resonance line the most probable route of exit from the upper state is, in fact, a direct decay to the lower state. In contrast, for subordinate lines a large number of possibilities will, in general, exist, and the photons may effectively be removed from the line and destroyed. It must be emphasized, however, that this characterization is only schematic and often does violence to the actual physics of line-formation. For example, we shall find in Chapter 11 that the boundary value 314 Classical Treatments of Line Transfer 315 of the source function of the Ha line (a subordinate line) has essentially nothing to do with the temperature of the outer atmospheric layers. CFNTER-TO-UMB VARIATION The specific intensity emergent at frequency v, on the disk at an angle 0 = cos"1 p from disk center, is given by = J* [B„ + (1 - ly)(Jy - Bv)] exp(-r»/^ dTv (10-26) where the form of Sv has been taken from equation (10-8). Substituting from equation (10-12) we find iv(0, p) = (a + pjt) + (pv - V3a)(l - K) v'3(l + V^v)(l + V^). In the continuum, p\, = 0, and again taking p = 0, /c(0, p) =-- a + bp (10-27) (10-28) The residual intensity t\(p) = /v(0, ,u)//c(0, /x) and the absorption depth av{p) = 1 — rv(p) follow immediately from equations (10-27) and (10-28). Consider first a pure absorption line, for which e = 1 and Xv = 1. Then rfp) = [1 + {b/a)fi/(l + ßv)]/[l + (b/a)fi] (10-29) Here we see that as the limb is approached (i.e., as p ^ 0), Jv -> Ic, and the line vanishes. This is compatible with the physical picture sketched above, for as the limb is reached, only the very surface can be seen at any frequency. For r. = 1, the same source function (the surface value of the Planck function) is observed throughout the profile, and the contrast between line and continuum disappears. On the other hand, if £ = 0 (a pure scattering line), Xv = (1 + /ivT\ and taking the limit as [f. -» co. equation (10-27) reduces to lv{0,p) = 0 for all ju's; thus the cores of pure scattering lines always remain dark, even at the limb, and there is a clear distinction between the center-to-limb behavior of absorption and scattering lines. A few representative values of the variation of line-depth with disk-position are given in Table 10-1 for several values of a and p\.. In this table (b/a) = 3; negative values imply emission, and are an artifact of the approximations made in solving the transfer equation. TABLE 10-1 Ccnwr-ii,-L)))ih Vuriaiion in the Milne-Edclington Model «vV.3£''<. 1 + ft U/U + ft; 1 + 1 + /3V (10-30) First, consider the case with c = 0; then Kv - 1/(1 + ft,), and the line is a pure absorption feature. This occurs because both the lines and continuum are formed by scattering, and the scattering length in the line is larger. On the other hand, suppose c = 1; then it is obvious from equation (10-30) that in the line core, as ft. -> oo, jRv -> (-feajlh). Thus the line can appear in absorption or emission depending on the ratio (a/b); the more shallow the temperature gradient (i.e., the smaller the value of b), the brighter is the line compared to the continuum. The reason for this is that because f. = 1, the source function in the line is everywhere equal to the thermal value while, in the continuum, scattering causes Jv (and thus Sv) to drop below Bv. If (a/b) is greater than the critical value (afb) = 2/-/3, the line is in emission for all values of ft. (i.e., at all frequencies in the profile). If [a/b) just equals the critical value, then both the extreme wing (ft. -> 0) and the very core of the line (ft. -> oo) lie at the level of the continuum, and all other points in the profile are in emission. If [a/b) is less than the critical value the line may have weak emission wings and a central reversal into an absorption core. If {a/b) < 1A/3, the line is everywhere in absorption. This interplay of scattering and absorption, which gives rise to either absorption or emission lines depending on the temperature gradient, is known as the Schuster mechanism; a thorough discussion of the various possible cases was given by Schuster himself (562). From time to time it has been suggested that this mechanism could be responsible for emission lines observed in certain early-type spectra, but on the basis of recent critical discussions (238; 280t this seems unlikely. 10-3 The Theoretical Curve of Growth Using equation (10-15) one could, in principle, compute the profile of a spectrum line and, by integration over frequency, determine its equivalent width. Such a procedure is laborious, however, and requires the use of a computer. It is instructive, therefore, to consider a simple model that allows the equivalent width to be computed analytically. In this way we can construct what is known as a curve of growth, which gives the equivalent width directly in terms of the number of absorbing atoms that produce the line. 10-3 The Theoretical Curve of Growth 317 To begin with, we assume that the line formation occurs in a layer to which we can assign a unique temperature and electron pressure. This assumption is normally valid in laboratory work, but in the stellar-atmospheres situation there are usually strong gradients in both temperature and pressure. Thus the choice of an appropriate temperature and pressure is difficult and, at best, can refer only to some ill-defined mean value in the line-forming region. Having made this choice, we can compute {assuming LTE) the populations of the atomic levels and, hence, the continuous opacity kc and the line opacity in a transition (/ Xij(v) = (ne2/mc)flAm ~ ^*"ru")0v = x,A (10-31) The line profile is assumed to be given by the Voigt function [equation(9-34)] H(a, v) = [a/x) e~y\(v - yf + a2]'' dy where v = Av/AvD, a ~ r/(47i AvD), and Avp = vi;()/c; here Co is the most probable velocity of the atoms in the material. We then write Xifv) = XqH{o, v) where %0 = Avd). Wc assume that both AvD and the param- eter a are fixed in the region of line-formation, and employ the Milne-Eddington model in which the ratio ft ~ &y(v)/Kf is independent of depth. (The assumption of depth-independent ft, is actually a fairly good approximation for some spectrum lines; e.g.. Mg II A4481 and Si II /U412S, 4131. Indeed for these lines both %0 and kc may vary over orders of magnitude in the atmosphere while their ratio remains nearly constant.) Finally, we assume the line is formed by absorption processes in LTE; the flux is then easily computed as fv = 2 L By[T(z)]E2 (1 + ft) d.z (10-32) As before, we adopt 5v[T(x)] = B0 + B^, where r is the continu optical depth. Then, having taken ft to be constant with depth, we h um ave = 2 J;; (B0 + B,r)£2[(l + ft,T](l + ft),/r = B0 + ^ BJ(l + ft) (10-33) From equation (10-33) the continuum flux is clearly Fc = B0 + |-B. so that the line-depth in the flux profile is Ar = DM1 + if:)] ] + 2^0/^) (10-34) 318 Classical Treatments of Line Transfer It is convenient to work in terms of the parameter A0, the centra] depth of an infinitely opaque line. Taking the limit of equation (10-34) as ßv -> r& we find 1 + l(B0/Bi) 2 (10-35) which is essentially the result stated in equation (10-18). Then equation (10-34) may be rewritten Av = AMU + /Jv) (10-36) The equivalent width may now be computed by integration of the line-depth over frequency to obtain ; = J"* Av dv = 2A0 AvD J7' jS(c)[l + im]'1 äv (10-37) W, where it has been assumed that the line is symmetrical about its center. But fi(v) = (xo/Kc)H(a' v) = ^0H(as v), and if we define the reduced equivalent width W* = W/{2A0 AvD), we have W*(a, ß0) = * ßom, v)[l + ßoH(a, i;)]"1 dv (10-38) Before we attempt to evaluate this integral, let us consider qualitatively how the line develops as more and more atoms absorb. At the start, with only a few absorbers, each will be able to remove photons from the radiation field, and the line strength should be proportional to the number of absorbing atoms. Only the Doppler core (where the opacity is highest) will contribute to the line strength; the line wings will be transparent and will not reduce the emitted flux. As yet more atoms absorb in the line, the core, at some point, becomes completely opaque, and the intensity there reaches its limiting value given by equation (10-35). Now, essentially all the photons that can be absorbed in the core already have been, and so long as the line-wings remain transparent, the addition of more absorbers does little to increase the equivalent width of the line, which is said to be saturated. Finally, when enough absorbers are present, the opacity in the wings becomes appreciable, and the equivalent width again increases as the contribution of the Sine wings grows. These different behaviors are shown in Figure 10-1. On the basis of the above discussion, we see that there are three basically distinct regimes in the curve of growth; these wilt be treated individually. The Voigt profile can be represented schematically [equation (9-45)] as H{a, v) ~ e~"2 + n~^av~2; here it is assumed that the first term applies only in the core for v < e*, and the second only in the wings for v 3s v*, where r* is chosen as the transition point where the two terms are equal. 319 MGURL" 10-I Development of a spectrum line with increasing number of atoms along the line of sight. The line is assumed to be formed in pure absorption. For fiQ < 1, the line strength is directly proportional to the number of absorbers. For 30 < fj0 < \0i the line is saturated, but ihe wings have not yet begun to develop. For /J0 > Kr1 the line wings are strong and contribute most of the equivalent width. Consider now the contribution from the core only, and write f](v) = p0e " v\ assuming that /J0 < 1. Then equation (10-38) becomes W = ß0 J7' e""2(l - ßoe-"2 + ■■■) dv (10-39) or W* = - tc±0o[1 - (ß0/^2) + (j30V3) - ■ ■ ■] (10-40) For small values of f}0 (weak lines) the linear term dominates, and the equivalent width of the line is directly proportional to the number of absorbers present. This is known as the linear part of the curve of growth. Note that /?0 varies as AvD"\ as does 147*, hence the equivalent width IF is independent of AvD on the linear part of the curve. In the saturation part of the curve of growth, f]Q is so large that the line core has reached its limiting depth, but not yet large enough that the line wings contribute to the equivalent width. Again fj(v) = fi0e~v\ and if we 320 Classical Treatments of Line Transfer 321 let u = v2, then dv = ju 1 du, and equation (10-38) becomes = ^£'(1 + eu-*)-xu-± du (10-41) where we have set /i0 = e\ This integral may be rewritten [see (160, 389)] as * [a(l + t)fdt r* [a(l - t)f dt 2W* = JV*rf« 1 1 (10-42) Following Sommerfeld, one may replace the upper limit in the third integral by yi> (because a » 1) and expand [«(1 + t)]* as a power series in t around r = 0. The results may be written in closed form in terms of the Riemann zeta function, and one obtains finally the asymptotic expression W* « - [7i2/24(ln /So)2] - [77i4/384(ln /S0)4] - ■ ■ ■} (10-43) The above expansion is only semiconvergent and must always be truncated after a finite number of terms; in practice, the series is useful for p"0 Z 55. From equation (10-43) we can see clearly that on the saturation, or fiat, part of the curve of growth, the equivalent width grows extremely slowly with increasing numbers of absorbers, namely W* x VhipV The weak dependence of W* on f}0 implies that W on this part of the curve is essentially proportional to AvD. It is easy to understand why this should be so: the depth of the line profile is fixed at A0l hence the integrated absorption must be proportional to the linewidth; i.e., to AvD (cf. Figure 10-1). Similarly the dependence of W on fj0 can be understood by recognizing that to produce a significant depression in the continuum, the optical depth in the line must exceed unity before the continuum reaches unit optical depth. This occurs at frequencies v < v0 where pV-""* ~ 1: clearly u0, which measures the width of the dark core (and hence determines IF*), varies as *jhi /30. Finally, for very large numbers of absorbers, the line wings become opaque enough to provide the dominant contribution to the equivalent width. Here we adopt H(a, v) (10-38) a/(Ti*v2), and writing C = fi0an 1 we find from equation W l l + ü70_1 dv = 27IV'"C = 2(naP (10-44) Thus W cc pV, giving rise to what is called the damping or square-root part of the curve of growth. Again, note that both a and fi0 each contain a factor of AvD; hence IF is independent of Avj, on this part of the curve. The entire curve of growth is shown in Figure 10-2. Notice that the larger the value of the damping parameter a, the sooner the wings dominate W, log ft, ["[(JURE 10-2 Curves of growth for pure absorption lines. Note that the larger the value of a, the sooner the square-root part of the curve rises away from the flat part. and hence the sooner the damping part of the curve rises away from the fiat part. Numerous curves of growth have been computed by various authors. A particularly useful set has been published by Wrubel for a wide range of the temperature-distribution parameters B0 and Bu and under different assumptions concerning the atmospheric model (i.e., Milne-Eddington or Schuster-Schwarzschild) and the transfer problem (i.e., absorption or scattering lines); see (687; 688; 689). 10-4 The Empirical Curve of Growth The curve of growth has long been one of the astronomer's favorite tools for performing an analysis of a stellar atmosphere, and the literature of the various applications of this approach is enormous. The reasons for the popularity of the curve of growth are that it provides estimates of several key parameters with speed and ease, and that it makes use of equivalent widths alone, which can be well-determined observationally even for faint stars where profiles would be impossible to measure accurately. As described in §10-3, the theoretical curve of growth gives log(lFv/AvD) = log(IFA/AAD) = log(lFAc/;^0) as a function of log pV Here £0 is the total random velocity of the atoms forming the line, and nle2f \ n?M - e ßo ~ — \ . 11/1 k„ \mc AvDJ kfl e-h*lkT} mc J \ 11;;, (10-45) 322 Classical Treatments of Line Transfer Here kc, the continuum opacity uncorrected for stimulated emissions, has been introduced for convenience (and continuum scattering has been ignored). The population nfjk, of excitation-state /, of ionization-stage j, of chemical species k, is assumed to be given by the Boltzmann-Saha equations as described in §5-1. In a stellar spectrum one can usually observe lines from several multiplets of a given ion, and one or more ionization stages of a given element, for each of several elements. Let us first consider only lines from a definite ion of a single element. Because of the factor exp( — xijk/kT) in the Boltzmann excitation formula, each multiplet has its own curve of growth. In view of equation (5-8) we may write log p,> = \og(gmfX) - 6zUk + log Cjk (10-46) where 9 = 5040/ T, % is expressed in eV, and Cjk = [JVJ-k/^k(r)](3tV/mc)(«r0/cc)-1 (10-47) Clearly the value of /30 will be affected by the choice for the temperature (6) in the atmosphere, and will be different for each multiplet (because of differences in %ijk), and for each line (because of differences in /-values). To construct an empirical curve of growth, we plot the value of \og(W;JX) versus \og(gf 1) for each line. Now if we assume that there is a unique relation between /i0 and Wk, we attempt to force all points for the different lines to define a single curve as nearly as possible. To do this, we adjust 6, and choose the value that minimizes the scatter around a mean curve. This value is called the excitation temperature, 6exc, and is considered to be the characteristic temperature of the line-forming region. It should be realized that it may not always be possible to derive a meaningful excitation temperature, for different lines will actually be formed in different regions of the atmosphere. For example, one would expect that lines from levels with high excitation potentials will be formed deeper in the atmosphere where temperatures are higher. Similarly, we would expect the average excitation temperature to be higher for higher stages of ionization. Such refinements can be taken into account by more elaborate calculations that use model atmospheres, but they are normally ignored in curve-of-growth work. There are other complications that arise in practice: there are errors both in the values of Wx and f and these introduce scatter into the curve. Also, lines arising from only a limited range of excitation potentials may be observable, and 6CXC may not be very well determined over this short baseline. Once the empirical curve, corrected for excitation effects, has been established, it may be compared with theoretical curves. To superimpose the two curves, a shift (both in abscissa and ordinate) of the empirical curve relative 10-4 The Empirical Curve of Growth 323 to the theoretical is normally required. From this fitting procedure we can deduce three essential bits of information. (1) The ordinate of the empirical curve is Xog^WJk), while that of the theoretical curve is log(W^/A^D) = \o%(WJX) - log(^0/c). Thus when the two curves are superimposed, the difference in the ordinates yields log(f0/c), and thus the velocity parameter £0. The value derived can be compared with the most probable thermal velocity at the excitation temperature 77exc, namely £therm = (2fcTexc/.4mH)* where A is the atomic weight of the element. It is usually found that c0 as inferred from the curve of growth exceeds £lhermJ sometimes by a large factor. To explain this difference, it has been customary to postulate the existence of additional nonthermal motions of the stellar material, which are usually referred to as microlurbutence. It is assumed that these motions occur on scales that are small compared to a photon mean-free-path and hence constitute, in effect, an additional source of line-broadening. If these small-scale mass motions have a Gaussian distribution around some most probable speed £t2urb, then £0 = [(2kTeJAmH) + i;lTb-]i (10-48) 'Turbulent" velocities have been derived for many stars by this method of analysis; the most dramatic results are obtained for supergiants where velocities in excess of the sound speed in the gas have been obtained. It should be realized, however, that such diagnoses are not on entirely firm ground because the introduction of a velocity field drastically affects the details of line formation (cf. Chapter 14). Indeed, with the high velocities sometimes derived, one must inquire whether the excitation state of the gas is affected by interchange of energy between mass motions and the internal energy of the material. Very little work has been done on this difficult problem, and our knowledge about small-scale stellar velocity fields remains rudimentary, almost to the point of merely recognizing their existence. (2) The difference between the abscissae of the empirical and theoretical curves of growth yields logC = log(3Q - [log(f/./7) - O^x] (10-49) To proceed further, we need an estimate of the electron density. This is derived from a theoretical model of the atmosphere. Once we know ne, we may compute kc, and as we already know £0, we find Njk, the number of atoms of chemical species k in ionization stage; directly from C, via equation (10-47). Then by use of the Saha ionization formula we can convert Njk to Nk, the number of atoms of chemical species k in all ionization states. More precisely, we obtain the abundance ak — (Nk/Nu), as kc is usually dominated by hydrogen or H", and thus is proportional to NH. In short, the horizontal 324 Classical Treatments of Line Transfer shift between the two curves of growth yields the abundance of the -element. This method of analysis has been applied to a wide variety of stars, and has shown that certain stars (e.g., Population II stars and peculiar stars) have abundances that differ markedly from solar values, which, in turn, are fairly typical of Population 1 stars. The uncertainties in a curve of growth abundance analysis should be clear: it has been assumed that the Saha-Bol tzmann relations are valid; depth variations of the relevant parameters (called stratification effects) have been ignored; and a schematic solution of the transfer problem has been employed. The accuracy of the results is, therefore, limited. (3) The horizontal and vertical shifts described above make use of the linear and flat parts of the curve respectively. For a given set of theoretical curves, a comparison between the observed and computed damping parts determines the value of a = F/(4ti Avd}, and hence T; this in turn may be compared with the value predicted by line-broadening theory. Let us now turn to a brief discussion of a few typical results. The solar spectrum has been analyzed extensively with curve-of-growth techniques; one of the most outstanding early treatments was by H. N. Russell (542), who derived element abundances (from eye-estimates of line-strengths!) that are in remarkably good agreement with current estimates. A very interesting study using curves of growth was carried out by K. O. Wright (686) who analyzed the Sun and three other solar-type stars. This study is by no means the most recent available [see. e.g.. (182)], but it is a classic example of the procedure, so we shall consider it here. From an extensive set of equivalent-width measurements and laboratory /-values, an empirical curve of growth was constructed using lines of Fe I and Ti I. In all, some 75 lines of Fe I arising from states with 0 s£ x ^ J -6 eV, and 137 lines of Ti I with 0 s£ % < 2.5 eV, were used to obtain the curve shown in Figure 10-3. Slightly different excitation temperatures were found for the two atoms, namely Tcxc = 4850°K + 150°K for Fe I and Texc = 4550JK ± 150°K for Ti I. These values are about what we would expect for a radiative-equilibrium model atmosphere. As can be seen, the curve is well defined, though it is true that the linear part is defined mainly by Ti 1 lines and the damping part by Fe I lines. It would be more satisfactory if lines of each atom were found in both the linear and damping portions of the curve. The vertical shift of the empirical curve, relative to a theoretical curve by Menzei, yields a velocity parameter £0 = 1.6 km s"1. As the thermal velocity for these atoms in the solar atmosphere is about 1.2 km s~J, this implies a microturbulent velocity of about 1.0 km s^1. By a fit to the damping part of the theoretical curves, log a = —1.4 was obtained. Adopting an average wavelength of 4500 A for the lines, this yields r = 1.7 x 109 s"1, which is very nearly a factor of 10 larger than the classical damping constant rc. It is clear that the main source of the line broadening must be collisions, 326 Classical Treatments of Line Transfer and because hydrogen is mostly un-ionized in the solar photosphere, the most likely process is van der Waals interactions between hydrogen and the radiating atom. An approximate result for rfi was obtained previously [equation (9-71)], namely F0 » 8.1 C6VNH where C6 * [13.6/0jion - yf)]2 x 10~32 [equation (9-76)]. For a typical value (y_ion - yf) = 6 eV, C6 « 5.2 x 10~32; typical values for v and ArH are 106 cm s_I and 1017 cm-3, so T6 «s 109, which is in basic agreement with the empirical value. The damping part of the empirical curve is found to have a slope (on a logarithmic scale) closer to 0.6 than 0.5. The likely explanation is that lines of widely differing strengths are formed in different layers, so that the parameters describing them (e.g., damping widths) are not identical, as was assumed in constructing the curve of growth. Many other analyses of the solar spectrum have been made. An improvement over the basic curve of growth method is obtained by using a saturation function (505) that accounts for the fact that the line cores and wings are actually formed in different layers of the atmosphere. This is conveniently done in the method of weighting Junctions, which is described in detail in (261, Chap. 4), and which has been applied in one of the classic abundance analyses of the solar spectrum (252). An important application of the solar curve of growth is in differential abundance analyses of stars with respect to the Sun. Excellent examples of this approach for G-type subdwarfs (extremely metal-deficient stars) and for Population II K-giants can be found in (13) and (294) respectively. An advantage of this approach is that oscillator strengths, which are often poorly known, cancel out of the analysis to first order. Also, because the temperatures of these stars are close to the solar value, one might hope that their atmospheric structures are at least roughly similar to the Sun's. If this is the case, then other effects, such as line-blending, stratification, departures from LTE, etc. might also cancel out to first order. The fundamental assumption made in a differential analysis is that the stellar curve of growth, log(vrw^0)* versus log is identical to the solar curve, ]og{Wc/AcQ)° versus log /ig. In practice we do not know [i% or and thus we cannot construct the stellar curve directly. What is known for each line is log(W/Xf and the value of log /?J for that line from its observed solar equivalent width. From the definition of [f, [see equation (10-45)], we have iog (I)-iog (I)+-e)+,og (US) (io-5o) If we define [A] = log(X°/X*) for any quantity A', then equation (10-50) may be rewritten as IW = [AW + Xtjk AO - Ko] - [AJ - [U] (10-51) 10-4 The Empirical Curve of Growth 327 Thus, if we plot \og(W/J)* for lines of a given ion, say Fe 1, versus log fi§ instead of log lines of different excitation potentials will scatter around a mean curve if there is a difference in excitation temperature between the star and the sun. Therefore, we plot iogiW/l)* versus (log j8o - Xm A0) and choose AO to minimize the scatter. For the G-subdwarfs it is found (13) that AO < 0.05, while for Population II K-giants A6 is about 0.25 to 0.35 (294). After the effect of AO is eliminated, the empirical curve for the star is superimposed upon the solar curve, in which log(Wc/A£0)° is plotted versus log ft®. The vertical shift gives directly [£0] while the horizontal shift yields the average value of S = \_Njk] — [_£0kcU] for the ion under consideration. For the G-subdwarfs c0 is found to correspond closely to a pure thermal value (no turbulence), while for the K-giants there is appreciable turbulence. As the temperatures of these stars are fairly close to the solar value, the partition-function ratio is usually set to unity, and the value of d depends primarily upon abundances, and upon differences in ionization and continuous opacity. Because H~ is the main contributor to kc in the temperature range under consideration, the value of [A'f] will essentially equal [n(,]. We can determine [ne] from an analysis of the ionization equilibrium using information from two stages of ionization of the same element k. If we have observations for two stages of ionization (say "0" and "1"), we may write 5jft = [AV] - [QoKl (./ - 0,1) (10-52) Using these two values of 5 we can derive A = 8ik - 30k = [Nlk] - [JV0k] = log{Nfk/Ng) - logCJVft/ATSk) (10-53) As logiNfJNg) is known, the observed value of A yields log{Nlk/N$k). But if we assume that the temperature is specified by 0*c (which is known), then Saha's equation gives log{r\"?knf/N'f)k); hence we may determine log n* and [nj. Estimates of [nj can be obtained from several different elements, and a mean value taken. Knowledge of [«e] allows us to evaluate [/.,;.] and log(Njfc/A/,*). Hence finally we can calculate the ratio of the abundance of the element in the star to its abundance in the Sun as log(JVfc°/JVk*J = log(A',?/./V|) + log(A^//V*) + \og(Nfk/N%) = \og(N?/N%) + log(N*/iV*) + 5 + Bo^t/] (10-54) because all four terms on the righthand side of the second equality are now known. The results of the two analyses mentioned above lead to the striking conclusion that the abundances of the heavy elements in Population II stars are lower than their abundances in the Sun by a factor of the order 328 Classical Treatments of Line Transfer of 102! This is a fact of tremendous significance for the construction of a picture of the evolution of the Galaxy, and it implies that the heavy elements in Population 1 stars such as the Sun are the result of nucleosynthesis in earlier generations of stars. The errors in abundances derived from curve-of-growth studies are typically of the order of a factor-of-two uncertainty; although such errors are serious, and motivate the development of more precise methods, they do not change the qualitative results for Population II stars mentioned above. 10-5 LTE Spectrum Synthesis with Model Atmospheres The curve of growth approach introduces a large number of simplifying approximations, which seriously limit the accuracy of the results obtained Considerable improvement is achieved if the physical assumption of LTE is retained, but now a model atmosphere is employed to represent the depth-variation of all physical parameters. Given such a model, one may compute both kc and yjd>v as functions of depth, allowing fully for the variation of the temperature, ionization-excitation equilibrium, Doppler widths, damping parameters, etc. It is then possible to compute the optical depth t¥(z) = j*;""" [kc(z') + xMW)~\ & (10-55) and hence the emergent flux in a line at frequency v Fv = 2joaoB¥[T(Tv)]E2(T¥)dTv (10-56) by direct numerical quadrature, with as much mathematical accuracy as desired. The first step in this method is to select the model atmosphere that most closely resembles the stellar atmosphere to be analyzed. This choice is made by comparing observed and computed values of certain key features in the spectrum. Typicallv the comparison is made for (1) continuum features such as the overall energy distribution, the Balmer jump DB, or a color such as (fc _ y): (2) line profiles of the hydrogen lines (which are density-sensitive); (3) ratios of line-strengths for lines of two ionization stages of a given element. For example, in B-stars, the continuum parameters determine mainly the effective temperature Teff; the hydrogen lines determine mainly log g; the ratio of, say, Si III A4552 to Si II A4128, 31 is a function of both TelT and log g (the ratio is insensitive to the element abundance). Examples of fits of computed and observed energy distributions are shown in Figures 7-4 through 7-6 and in Figure 7-8. Figure 10-4 shows a fit to the 329 0 10 20 30 40 a;. FIGURE 10-4 Hydrogen Balmer-line profiles in Vega. Solid dots: observed profiles from (509). Crosses: observed profiles from (71). Abscissa: a/, in A. Curves are computed profiles from a model with 7"err = 9650°K and log g = 4.05. From (555), by permission, observed hydrogen line profiles in the standard star Vega (555). Often a plot is made, in a diagram of Telf versus log g, of the loci of (7cfF, g) values for which the computed value of a given parameter [say DB or W{Hyj] equals the observed value. The loci for different criteria will intersect, and thus define an optimum value of (Tcff. g). Ideally the curves would intersect at a single point; in practice they will intersect within a small area, which introduces an uncertainty into both Teff and g. Examples of this procedure for the normal B-star y Peg and the He star HD 184927 are given in (508) and (299), respectively. Having chosen a model atmosphere, it becomes possible to carry out an abundance analysis. One now replaces the curve of growth by a computation of the equivalent width, of each line under consideration, as a function of element-abundance relative to hydrogen. Knowledge of the observed equivalent width of each line thus leads to an estimate of the element abundance; these estimates may be suitably weighted and averaged over all lines to yield a final abundance estimate. Although the choice of the model atmosphere used in the abundance analysis is based on continuum and hydrogen-line criteria, there is usually some ambiguity in (Teff> log g). Often the results of the abundance analyses for several elements can be used to narrow the range of uncertainty in 330 Classical Treatments of Line Transfer (Teff, log g), and to choose a more refined model, much in the same way as atmospheric properties may be inferred from a curve-of-growth analysis. For example, after the analysis is completed, the derived abundances for many lines may be examined to see if they show a correlation with excitation potential; such a correlation would be expected if the temperatures in the atmosphere are incorrectly chosen. The elimination of any such correlations may, therefore, lead to an improved choice of Te!f for the model. Similarly, for a given element, identical abundances should be derived from all ionization stages. Differences that are found between ionization stages contain information about errors in the choice of temperatures and pressures, and thus about Teff and log g for the model. Finally, one can examine the derived abundances to see if they show a correlation with equivalent width. If, for example, stronger lines systematically lead to larger abundances, the velocity parameter may have been underestimated, and thus information about turbulent velocities to be included in the model can be derived. (As we shall see in Chapter 11, however, departures from LTE must also be considered when a determination of the velocity parameter from strong lines is attempted.) Naturally, errors in the observed equivalent widths, in /-values, and in other atomic parameters will introduce scatter; hence the procedure outlined above may not be completely unambiguous. Nevertheless, examination of the correlations just mentioned can often lead to a significantly better choice for the model. Finally, if the derived abundances are very different from those used in construction of the model (as they might be for, say, Ap stars), it may be necessary to recompute models with appropriate abundances, re-derive Te{i and log g, and perform the analysis over. An example of the model-atmosphere method of abundance analysis is that for the two bright A-stars Vega and Sirius in (609). The results of this work show that Vega has very nearly solar abundances of the heavy elements, while Sirius has abundances a factor of 4 to 10 higher. In many respects Sirius mildly resembles the group of A-stars called metallic-line A-stars (Am stars) because of the strength of the metal lines in their spectra. The present literature of abundance analyses (and estimates of atmospheric parameters such as Tefr and log g) by both curve-of-growth and model-atmosphere techniques is vast; extensive lists of references to research papers on the subject for both normal and peculiar stars can be found in (12; 70, 57-204; 144; 450, 157-237; 493; 523; 552; 560; 658). The work thus far described makes use primarily of equivalent widths. For some stars, for which very high-quality spectroscopic data are available, a detailed point-by-point synthesis of the spectrum is possible. An example of such a synthesis for the Sun (538) is shown in Figure 10-5. Analyses of this kind can lead to accurate abundances and to the identification of hitherto unnoticed weak blends in the spectrum; applications of spectrum synthesis to the calibration of photometric systems were described in §7-4. 331 3984.0 3984.5 MGUTSE 10-5 Spectrum synthesis for solar Fe, Cr, and Ni lines. Curve: observed intensity; open circles: computed intensity. Ordinate: emergent specific intensity at disk-center relative to continuum; abscissa: wavelength in A. From (538) by permission. All of the methods described in this chapter make the assumption of LTE. As we shall see in Chapters 11 and 12. however, this assumption is often a poor description of the physics of line-formation, and can lead to results containing serious systematic errors. Let us now, therefore, turn to the problem of solving the combined equations of radiative transfer and atomic statistical equilibrium fully consistently. 11-1 Diffusion, Destruction, Escape, and Thermalization 333 Non-LTE Line Transfer: The Two-Level Atom Much of the progress in understanding the physics of line-formation has come from a study of solutions of the combined transfer and statistical equilibrium equations: the so-called non-LTE approach. In this chapter we shall consider some schematic line-formation problems that are simple enough to be solved readily, but which, nevertheless, provide a good description of the physically important processes—and yield considerable insight. It will be obvious that some of the assumptions made are oversimplifications, and are not valid in actual stellar atmospheres, for which elaborate numerical calculations are required to yield accurate results. On the other hand, the real goal is to understand the answers, not merely obtain them. This can be done only with a clear grasp of the prototype cases discussed in this chapter, which provide a conceptual framework of great usefulness for the interpretation of results from computations with complicated model atoms and detailed model atmospheres. 11-1 Diffusion, Destruction, Escape, and Thermalization The most important difference between the LTE and non-LTE treatments of line-formation is the way in which the coupling between the gas and radiation is handled. In the LTE approach, it is assumed that the local values of two thermodynamic variables (T and N) are sufficient to determine completely the excitation and ionization state of the gas (and hence transfer quantities such as xv, an^ Sv), independently of the state of the gas at other points. (This remark is strictly true only when the temperature structure is given; if it is determined by the condition of radiative equilibrium, then there is a coupling among different points in the atmosphere—see §7-2—but not in the sense we are concerned with here.) It has been emphasized in previous chapters that the excitation and ionization state of the gas is, in fact, strongly influenced by the radiation held, which, in turn, is determined by the state of the gas throughout large volumes of the atmosphere, via the transfer equation. In actuality the two problems of radiative transfer and statistical equilibrium are inextricably coupled, and must be considered simultaneously. Much of the basic physics of line-formation can be understood by consideration of the characteristic lengths for photon diffusion, destruction, and thermalization, which are related to photon escape and destruction probabilities. For simplicity we shall suppose that the atom consists of a ground state (denoted /) and a single excited state (denoted u). The distance over which a line photon will, on the average, move in the atmosphere between successive interactions with the material (absorptions) can be represented by the average mean-free-path I. The mean-free-path for a photon emitted at a given frequency is the distance interval whose optical thickness at that frequency is about unity—i.e., /t. x l/xv = (xiu4>v + Xc)~l where %lu and Xc are the line and overlapping continuum opacities, and 0V is the line profile. For complete redistribution, the probability of emission at frequency v is d>v, so / = — j lv(pv dv. It is obvious that photons emitted in the core travel relatively small distances, while those in the wings can travel much greater distances, up to a distance corresponding to unit optical thickness in the continuum. When a photon is absorbed in the line and excites an atom to the upper state, it is usually re-emitted in a radiative de-excitation and travels another mean-free-path. This process may occur again and again before the photon is ultimately destroyed, and has its energy deposited into the thermal pool, either by a collisional de-excitation or by an absorption in the overlapping continuum. Thus there exists a characteristic length L, the destruction length, over which a photon may travel before it is destroyed. The destruction length has a more basic physical significance than the mean-free-path, for it measures the distance over which a photon emitted at a given point retains its identity and hence can "communicate" information about conditions at that point to another. Thus L determines an interaction region: the volume containing those points that can influence one another via photon exchange. The relative sizes of L and / depend upon the photon destruction probability Pd, which gives the average probability that the photon is destroyed 334 Non-LTE Line Transfer: The Two-Level Atom I}-] Diffusion, Destruction, Escape, and Thermalization 335 when it next interacts with the material. The probability for photon destruction by collisions following absorption in the line is CJ(Ani + CHl) where Cul is the rate of collisional de-excitation and Au! is the spontaneous emission rate; now Cu! '/i nt„ so that it is clear that the contribution by collisions to the total destruction probability becomes large (i.e., approaches unity) in the deeper, denser regions of a stellar atmosphere, and can be quite small in the uppermost layers. If we assume that all photons absorbed in the continuum are destroyed thermally, the contribution of continuum processes to Pd is the average of yj(%hl(pv + %c) over the line profile. The continuum sets an upper bound on both / and L because a photon cannot travel more than a unit optical depth in the continuum before it is absorbed—and when it is, it is also destroyed. At great depths in the atmosphere, Pd approaches unity because of large densities (and hence collision rates) and because of the strength of the overlapping continuum. Then L -> /, for the photon is almost surely destroyed when it is next absorbed. Assuming that the continuum is already optically thick, it is clear that the radiation field becomes very strongly coupled to local conditions and thermalizes to its local thermodynamic equilibrium value. In contrast, when the destruction probability is very small, L » /, and the interaction region may become enormous compared to the volume over which a photon can diffuse in a single flight. In this case, the radiation field is dominated by nonlocal influences, and represents the result of physical conditions that may be quite inhomogeneous. For example, within the volume there may be large variations of the kinetic temperature that imply strong variations of thermodynamic properties and of the Planck function. The radiation field may then depart markedly from its local equilibrium value, and this departure will extend throughout the entire interaction region. The importance of these notions becomes manifest when we consider a sequence of test points approaching the surface of the atmosphere. At the deepest points we obtain equilibrium. But as we approach the surface, the distance L grows, and eventually the interaction volume contains the boundary itself (more precisely, extends above the depth at which % = 1 anywhere in the line). Then a new phenomenon enters the picture: photons escape from the medium into space. It follows that the radiation field, at test points whose interaction regions extend into space beyond the surface, must be depressed below the value it would have had if there were no boundary, for no radiation is emitted in the region void of material, and photon escapes are therefore uncompensated. This effect leads to a deficiency in the radiation throughout an entire interaction volume, and hence extends to depths at least of the order of L. But of course the radiation field at these points influences that at points that lie yet another destruction length L deeper, and therefore there is a "compounding" of the effect, which leads to a departure of the radiation field from its equilibrium value at positions far below the surface. The depth at which the radiation field (or source function) ultimately approaches closely its equilibrium value is the thermalization depth A; this concept, introduced by Jefferies (333), has proven to be extremely fruitful. To obtain a quantitative estimate of A we define the escape probability Pe(x) as the probability, averaged over the line, that a photon emitted at line optical depth t escapes from the medium before being absorbed along its path of flight. The mechanism competing with photon escape is photon destruction; the former induces a departure from equilibrium while the latter leads to coupling to the local thermal pool. We therefore compare Pe with Pd. Deep within the atmosphere where Pe(x) « Pd, photons are surely thermalized before they can escape, hence S -» B: at the surface where Pt,(i) » Prf, photons escape freely before thermalization, hence S will depart from B. It is therefore reasonable, on physical grounds, to identify the thermalization depth A as that point at which P^A) = P,,. Defined in this way, A is essentially the greatest depth from which photons have a significant chance to escape before being destroyed. Now the escape probability (and hence A) depends sensitively (a) upon the nature of photon redistribution over the line profile upon emission, and (b) upon the amount of background continuum absorption. The probability for photon absorption is highest at line center. If photons are emitted coherently, then those absorbed at line center will be re-emitted there, and hence will tend to be trapped by the large line-core optical depth up to the very uppermost layers, where the line finally becomes optically thin; this will tend to prevent serious departures of the line-core radiation field (which contributes heavily to the total photoexcitation rate) from its equilibrium value up to the shallowest layers of the atmosphere. In contrast, if photons are completely redistributed over the line profile, then there is a significant chance that after a number of scatterings a photon absorbed at line-center will be emitted in the line-wing where the opacity is low. and the probability of escape is high. Photons that would have been trapped, if emitted coherently, now freely escape from the atmosphere, depressing the line-core intensity (and photoexcitation rate) in much deeper layers in the atmosphere. The radiation field in the line as a whole now responds to the fact that the boundary lies within a mean-free-path at some frequencies, even if not at others. The role of the continuum is obviously important in the case of complete redistribution, for it sets a lower bound on the total opacity, and hence an upper bound on the depth from which escapes become possible at any frequency, no matter how far out in the line-wing. It is thus clear that we should expect the thermalization depth to be much larger, and the surface departures from equilibrium larger, when the scattering process is noncoherent instead of coherent; further, we expect the magnitude of these 336 Non-LTE Line Transfer: The Two-Level Atom 11-2 The Two-Level Atom without Continuum 337 effects to be larger, the more highly developed the line-wings are relative to the core. We shall find these expectations are fulfilled by the detailed analysis given below. It has been possible to extract practically all of the physical flavor of the problem using the qualitative arguments given above. Let us now turn to a mathematical discussion to obtain quantitative results, and to extend the analysis to cases where such heuristic discussion becomes less effective. 11-2 The Two-Level Atom without Continuum THE SOURCE FUNCTION Consider a schematic atomic model consisting of only two levels, I and u, between which radiative and collisional transitions can occur. This model is obviously very incomplete, but it nevertheless provides a fairly good description of the real situation for some lines. In particular, resonance lines arising from the ground state are well described by this model when the coupling of the lower and upper levels to the continuum (and of the upper level to other levels) is weak. For the present, assume that the onlv sources of opacity and emission at the line frequency are from the line itself; then the transfer equation is p(dljdz) = \_-jhBJv + ntl(Aul + BJv)]JJv - S,) f 11-3) where S, = nuAJ(niBhl - nuBJ = (2hv3/c?)[(n1gjn«gt) - l]"1 (11-4) the second equality following from application of the Einstein relations [cf. equations (4-8) and (4-9)]. As the factor v3 varies only negligiblv over the sharply peaked profile (pv, St is called the frequency-independent source function; when the emission profile differs from the absorption profile, S, becomes explicitly frequency-dependent (see §2-1 and §13-4). Equation (11-4) is an implicit form for the source function because the level-populations depend upon the radiation field; this dependence can be displayed explicitly by incorporating information from the equations of statistical equilibrium that determine n, and nu. The statistical equilibrium equation for level / is h, (b;„ j*■ BAT), and LTE is recovered. However, in virtually all situations of astro physical interest, s' « 1 in regions of line-formation, and, in general, the source function cannot be expected to have a value close to the Planck function. This state of affairs was partially recognized in the classical theory by the division of lines into the categories of "absorption" and "scattering" lines; this division, however, was largely ad hoc, and the thermal coupling parameter had to be guessed from heuristic arguments. In the present analysis the coupling parameter follows directly, and uniquely, from the statistical equilibrium equation. Further, in the classical treatments of "pure" scattering lines, it was sometimes incorrectly argued that the small thermal terms could be discarded. The important point to bear in mind is that, even if the thermal term r!B is small compared to the scattering term J locally, when integrated over the entire interaction region it accumulates to a value of importance; 338 Non-LTE Line Transfer: The Two-Level Atom moreover, at depths greater than the therm alization depth the in tensity must ultimately couple to the thermal pool. Indeed, if the thermal terms are discarded, the transfer equation becomes homogeneous in the radiation field and the scale of the solution is unknown; this scale is, in fact, fixed by the (small) thermal terms at the point of thermalization. SOLUTION 0¥ THE TRANSFER EQUATION Having obtained an expression for the source function, let us now consider the solution of the transfer equation. It is convenient to work with the dimensionless frequency-variable x, measured from line center in units of Doppler widths (or damping widths for Lorentz profiles). In terms of this variable we shall write Doppler profiles as (x) = n~ie~xl (ll-9a) Voigt profiles as (x) = an~:i e-y\{x - y)2 + a2]-1 dy (ll-9b) and Lorentz profiles as {z, x)[/(t, x) - S,(T)] (11-12) We now introduce discrete angle and frequency meshes {pm} and {v„}, and replace integrals with quadrature sums: fco N (11-13a) 11-2 The Two-Level Atom without Continuum 339 and m (ll-13b) where the points are chosen symmetrically—i.e.,x_„ = —x„and^_m= — pm (also «_„ = a„ and b-m = bm). Then the source function can be written in the same discretized form as was used in Chapter 6 [cf. equation (6-24a)], $d = (1 - fid) Z Lln X bnMdmn + bdBd (11-14) n m and the transfer equation can be reduced to the standard second-order differential equation form: &n2{d2u„Jdx2) = 2(umn - S) (11-15) where uv = j{Iv( + pt) + Iv(-p)]. Equation (11-15) can be discretized on a mesh {xd}, and solved numerically using either the Feautrier or Rybicki schemes discussed in §6-3. All of the relevant parameters (s, B, and 0) may be depth-dependent without causing any difficulty in the calculation. Alternatively, the transfer equation can be written in integral equation form. The formal solution of equation (11-12) is 0(ť, x) dť (pit, x) dt (11-16) For a depth-dependent profile, the argument of the El function will depend upon both x and t; this greatly complicates the analysis, so we shall consider only the case of a depth-independent profile, in which case only the displacement (t — x) enters: J(t, x) = Í0(x) J* Sl(t)E1 |(t - t)0(x)| (11-17) Substituting this expression for J into equation (11-11) yields an integral equation for 5,, S((t) = [1 - e(x)] J* Sl(t)K1 \t -x\dt + 8(x)B(x) (11-18) where the kernel function X, is given by K&) - \ £i(x2 ^ (11-19) 340 Non-LTE Line Transfer: The Two-Level Atom Exercise 11-2: Fill in the missing steps in the derivation of equation (11-18). Exercise 11-3: (a) Show that the normalization of K^s) is Jo' K^s) ds = \. (b) Using the result in (a), show that, for t co, where S,(t) must be a slowly-varying function and hence can be removed from the integral, S,(t) -» B. Numerical methods for solving equations of the form of (11-18) are discussed in detail in (52) and (18, Chap. 8); in most respects these resemble the Rybicki method of solving the differential equations, though Rybicki's method is simpler to apply when the profile is depth-dependent. In essence the solution is obtained by introducing a functional representation of S(x) on a discrete grid {t,,}; these functions are integrated analytically against the kernel to construct a final matrix system of the form S - KS + L (11-20) where S = (Sl5. . ., Sd>. . ., SD)T represents the depth-variation of the source-function. The system is then solved by standard numerical techniques. One of the advantages of the integral-equation formulation is that it displays explicitly the intimate dependence of the source function upon the mathematical behavior of the kernel function, and considerable insight can be gained by analytical study of the kernel. In particular, the asymptotic form of X,(s), for s » 1, shows that a characteristic feature of line-formation problems with noncoherent scattering is an extremely long-range interaction of one part of the atmosphere with another. Recall that for coherent scattering the appropriate kernel function in the formal solution is Et \t - x\ [cf. equation (2-57) and Exercise 2-10]. This kernel decays very rapidly, falling off as exp(— \t - x\)/\t - x\, thereby limiting severely the range of depth-points that are directly coupled together in the scattering process. In contrast, the asymptotic behavior of K^s) is found to be (53, Appendix I) K^s) ~ l/{4s2[!tt(s/7r±)]*} for Doppler profiles, K^s) ~ a*/{6s*) for Voigt profiles, and Ki{s) ~ l/(6s3) for Lorentz profiles. Clearly the range of these kernel functions is very large compared to that for coherent scattering, and this implies that the radiation fields at widely separated points in the medium become mutually interdependent. The physical reason for the long range of the noncoherent kernel functions is, of course, the redistribution of photons into the transparent line-wings. In the coherent case, any photon absorbed in the core will be re-emitted there, and hence always encounters high opacity and remains trapped. In the noncoherent case it has a probability of being emitted in the wings where, because of the lower opacity, it can travel over a large geometrical distance, and a correspondingly large integrated line optical depth, before it is again absorbed. 11-2 The Two-Level Atom without Continuum 341 the thermalization depth Let us now consider the behavior of the source function in an atmosphere in which e and B are constant with depth. Naturally this is a very schematic case, but it provides useful insight. One of the most important characteristics of the problem to be determined is the thermalization depth at which the line source function approaches closely the Planck function 73. Expressions for the thermalization depth can be derived by a wide variety of methods including (a) analysis of the asymptotic behavior of the solution of the integral equation (11-18) (53); (b) calculation of the distribution of the distances photons can travel from their points of creation to their places of destruction (545; 322); and (c) calculations of the probability distributions for photon escape from a given point of origin in the medium (214; 215; 216). One of the simplest and most appealing derivations can be made on the basis of the physical arguments given in §11-1: that the thermalization depth A must be near that point in the atmosphere where the probability of photon escape, PJA), following a given scattering event is equal to Pd, the probability of photon destruction (28). For the strict two-level atom with no overlapping continuum opacity, the probability of destruction per scattering event is just Pd = CJ(Aul + Cul) = e (11-21) where e is as defined by equations (11-7) and (11-8) (ignoring stimulated emission for simplicity). The escape probability can be calculated by summing exp( — z x1. Physically this states that photons are trapped in the core, where x 1, but in the wing where x !, the thermalizalion depth varies as A - f:~'f, where /? = a/(a - 1). The striking feature of the results given by equations (11-26) is that the thermahzation depth for a line with small e is enormous. Recall that for coherent scattering the thermahzation depth ~e~* (cf. §6-1 and 10-2). It is clear that the effects of noncoherence, with their attendant increase of photon diffusion in the line wings, greatly increase the depth in the atmosphere over which the source function can depart from the local Planck function. In fact, intercomparison of equations (11-26a) through (ll-26c) shows immediately that an increase in the relative importance of the line-wing within the profile leads directly to an increase in the thermahzation depth. It is worth stressing, however, that the above results do explicitly depend upon the assumptions of (a) no background opacity and (b) complete redistribution. We shall see later in this section that the effect of a background opacity can greatly reduce A. 11-2 The Two-Level Atom without Continuum 343 In Chapter 13 we shall find that in the case of resonance lines, the scattering in strong radiation-damping wings is nearly coherent, and A is better approximated by equation (11-26a) than (ll-26b). From a mathematical point of view, equations (11-26) show that any iterative solution of the transfer equation starting from LTE is futile for small e (much more so than the cases already discussed in §6-1!), and only direct solutions of the transfer equation will yield correct values. BOUNDARY-VALUE AND DEPTH-VARIATION OF THE SOURCE FUNCTION Having shown above that the source function may be decoupled from the Planck function over great depths in the atmosphere, let us now inquire what value the source function attains near the boundary of a semi-infinite atmosphere. If e and B are taken to be constant with depth, the transfer equation (11-12) is easily solved by the method of discrete ordinates (53; 476). Writing ln = l/0(x„) we have the coupled system of equations n m KnJdlnJdx) =Ima- -(1 - e) X an- £ bmJm,n> - sB (11-27) Considering first the homogeneous equation (B = 0) and seeking a solution of the form = g„m exp( — kx) one finds that lmn - c(l + Un/0_1e-kt (11-28) where the constants k are roots of the characteristic equation n m 2(1 - *) £ 4. I Ml - WO-1 - 1 (11-29) a = 1 m = 1 Exercise 11-5: (a) Verify equation (11-28). (b) Show that equation (11-29) is the appropriate characteristic equation for the system (11-27). For e = 0, the root k2 = 0 is a solution. For e > 0, k2 must be greater than zero. As in our earlier use of the discrete-ordinate method (§3-3, §7-4) we may delimit the roots by the poles at k2 = i/X2p.m2, and if we label these poles in the order of decreasing values of (A„fim) we may write 0 < k2 < (lix),-1 x + r) = (1 - k)JX «M*' dx1 + c,5 B (11-43) (11-44) It is easy to see that qx is the total destruction probability for a photon at frequency x, for it is the sum of (a) the probability that a photon is absorbed in the line, x SMEi \(t ~ i)(0, + r)\dt+l- rB J* E, |(f - t)(<^ + r)\dt (11-45) The solution given by equation (11-45) could be substituted into (11-43) to produce an integral equation for the total source function Sx{x\ but the resulting equation would be two-dimensional (x and x), and would be impractical to solve. Alternatively we can write an equation for S,(t); if the line source function is known, we can compute SJx) = [^xS,(x) + rB~\/((f>x + r) as desired. To obtain the equation for S,(t), the integral j Jx dx is calculated from equation (11-45) and the result substituted into equation (11-11) to give S,(x) = (1 - s)(l - S) J"St{t)K,.,|t - x\dt + (1 - 6)5B J* LUr\t -x\dt + sB (11-46) $ = '• [' f oM>, + r)~l dx 1 where and 1 Cx> L1>I.(s)s-r5-1J_oo£1[(^ + r)S]^ dx (H-47) (11-48) (11-49) The particular choice of coefficients in front of the integrals defining the two kernel functions was made to assure normalization of the kernels. Exercise 11-7: (a) Derive equation (11-46). (b) Show that lim,._>0 (5/r) = x\ limr_0 (1 - 5) = 1, and hence lim,._0 LUr(j) - 0 and lim,_>0 KUr{x) = K,(t) where K^x) is defined by equation (11-19). (c) Show that Ki ,. and LK, are normalized such that Kliriz)dz = i and Llr(z) dr = 4-, (d) lu view of (c), show that, for x -> oC', where St(x) is slowly varying and hence can be removed from the integral, St(x) -» B. The physical significance of equation (11-46) becomes clear if we introduce the average destruction probability q = J- £x dx = 6 + 8(1 — 3) (11-50) 352 Non-LTE Line Transfer: The Two-Level Atom for then (1 - I) = (1 - s)(l - <5) and we can rewrite equation (11-46) as S,(t) = (1 - £) J* S,(t)Xlj(. |r - t| dt + {I - e)B ^ L1>r |t - t| + «5 = (1 - ?) |oC S,(/)K1>r \t - x\ dt + ?5eff(t) (11-51) In the first form of equation (11-51), the first term obviously represents scattering in the line, with a total photon destruction probability given by J,. The second term represents photons fed into the line from the continuum, and the third term represents photons created by collisional processes. As in the pure-line case, the depth-variation of the source function will be determined by the properties of the kernel functions, and thermalization properties of St can be derived from a mathematical analysis of the asymptotic forms of Klt r and LY >f (315). On the other hand, the second form of equation (11-51) shows that the problem with a continuum is of the same form as the pure-line case, except that the coupling constant between the radiation and the thermal pool is J, not e.. Indeed, exploiting the same line of reasoning as before, but using c, for Pd, one finds that the thermalization depth is again given by equations (11-26), but with e replaced by 7;. Values of £ and A, for fi = 10"6 and for various values of r and a, are presented in Table 11-1. The striking result seen there is that £ can greatly exceed e, even when r ^ e, with a consequent dramatic decrease in A. These results show that by inhibiting photon transfer in the line wings, the continuum can dominate line thermalization. Note also that whenever r ^ s, then £ is already sufficiently large to assure that A < r~ \ as it must be, because we know from physical considerations than over distances greater than r"1 thermalization TAULli 11-1 Average Photon Destruction Probability I and Thermalization Depth A for a Line with Overlapping Continuum (r. = 10~hj a 0 10"3 SO"2 r A c A c A 0 10" fi 1 x 106 10" 0 10° 10 6 101U 10" 1 1.79 x 10" 6 5.6 x 10" 1.87 x 10" s 2.9 x 106 5.71 x 10" s 3.1 x 107 10" 6 8.26 x 10" ft 1.2 x !05 5.72 x 10" 5 2.1 x 105 1.78 x 10" 4 3.2 x 105 10" 5 6.69 x 10" 1.5 x 10* 1.85 x 10" 4 2.9 x 104 5.63 x 10" 4 3.2 x 104 10' 4 5.85 x 10" 4 1.7 x 103 7.84 x 10" 4 1.6 x 103 1.83 x 10" i 3.0 x 103 10" 3 4.98 x 10 3 2.0 x 102 5.23 x 10" 3 3.7 x 101 7.25 x 10" 3 1.9 x 102 Sourck: (315), by permission. /1-2 The Two-Level Atom without Continuum 353 in the continuum will occur; equations (11-26) with e replaced by ^ are thus consistent. To estimate the source function at the surface we can again use the similarity of equation (11-51) to (11-18) to infer that there exists a relation of the form 5,(0) = ?*Berr (11-52) where Befr is an appropriate average of Beff(x) over a thermalization length. It is apparent from the discussion given above that whenever 7; exceeds s (either because r > s or because extensive line wings assure that J, > e even though r < e), then both the destruction and creation terms (i.e., i?eff) are dominated by the continuum, and the line is said to be continuum controlled. This implies that S,(0), in particular, is set by the continuum even though the line opacity and line source function in the line core exceed the continuum terms. This is yet another manifestation of the fact, noted earlier, that the surface value of the source function is controlled by the line wings. Numerical solutions for S((t), in a semi-infinite atmosphere with B = 1 and v = \0~6, obtained using the method of discrete ordinates (315), are shown in Figures 11-5 and 11-6. It is easily seen that even modest values of 0 log S;(t) -2 - I \ \ t +T _ r = io-y / r =vrx / _r = \o-y / r = lO^^jy r=ffpf/ r = 0 i i i 1 1 -2-101234567 log t ! KiiJRE 11-5 Line source functions in a semi-infinite atmosphere for a two-level atom with r, = 10"6 and n pure Dopplcr profile (a — 0), with an overlapping continuum having various values of r. From (315), by permission. 354 11-2 The Two-Level Atom without Continuum 355 l0gT FIGURE 11-6 Line source functions in a semi-infinite atmosphere for a two-level atom with e = 10"6 and an overlapping continuum with various values of r, showing effects of different line profiles. From (315), by permission. r produce major changes in the source function; the effects are even more striking in finite slabs (315). The results in Figure 11-5 pertain to a pure Doppler profile; the vertical arrows indicate the predicted thermalization depths listed in Table 11-1, and the horizontal arrows show C% the value of S,(0) if Befr = 1- For Voigt profiles (Figure 11-6) a fairly good estimate for Sj(0) is obtained if one adopts Beff MA/a)"1 Jo(A/8)Bcff(T)dT (11-53) with a chosen to be a little larger than 3; this choice allows for the strong decrease in the true weighting function with depth. For finite slabs, a first approximation to Smsix at slab center is given by equations (11-40) with * replaced by 1\this accounts for photons emitted by the continuum. In real stellar atmospheres the effects of a background continuum on line-formation and thermalization are usually major. For early-type stars only the strongest lines have values of r as small as 10~4 or 10~3. and continuum terms therefore are generally dominant. On the other hand, the H and K lines of Ca II in the solar spectrum have r 10"9, and hence their thermalization is essentially independent of the overlapping continuum. EFFECTS OF DEPTH-VARIABLE THERMALIZATION PARAMETERS AND LINE PROFILES In the idealized problems considered thus far in this section it has been assumed, for the sake of simplicity, that all of the quantities B, i\ e, and v are independent of depth, whereas in real stellar atmospheres they may all be strong functions of depth. These variations can, of course, markedly affect the line-formation process, the behavior of the source function, and computed line profiles. To account for these complications in attempts to model accurately real stellar spectra, recourse must be had to direct numerical methods such as those described earlier in this chapter. But physical insight can be gained from a study of simple parameterizations of the variations of some of the quantities mentioned above. A discussion of the extremely important effects of Planck function variations will be deferred to §11-3, where line-formation in the presence of a chromospheric temperature rise will be described. Variations in the parameter r can be important, but are not easy to summarize in a few words; see (18, Chap. 3) for further discussion. We shall concentrate here on the effects of depth-variations of f. and (pv. The collisional de-excitation parameter e is proportional to the electron density, and hence must reflect the rise of the density with depth in the atmosphere. If the atmosphere is supposed to be essentially isothermal, then hydrostatic equilibrium implies that the total density increases linearly with m, the column-mass; if further we suppose that (yjp) is essentially constant, then t is also proportional to m. Then in an early-type star where the material is appreciably ionized, we expect ne to be proportional to z; in a later-type star the material may be neutral in the outer layers and may then abruptly ionize at some depth, in which case a much more sudden rise in ne (and s) may occur. Even though an actual numerical solution is required to determine 5,(t) with precision when e is variable, it is reasonable on physical grounds that we should still expect thermalization to occur at a depth A where Pe(A) =s PJAY To illustrate the usefulness of this idea, suppose a rapid rise in f. occurs, from some value s± to higher value e2, at a particular depth t0 in an atmosphere with r, B, and cp all constant. Then if r0 > l/£j (for a Doppler profile) it is clear that the line will already have thermalized, and hence the rise in c has no effect. In this case S,(0) will equal f.^B. Conversely, suppose that t0 < 3/e2- Then the line source function thermalization will have been delayed by the low value of s at the surface to a predicted depth of 1/c]; but at that depth e already equals e2 > e,. Hence the line will 356 357 -3 log r. - 4 - s (a) i r~ i ■ [ - i -|—-—r——r--1 b - A tl - 'iii..... 2 -1 0 12 3 4 5 6 7 8 log t i i i i i I I I i log S, -2 -1 lb) log t figure 11-7 (a) Variation of s wilh depth: curves are labeled with log i0, and e(t) = I0"J[1 - 0.99expf-t/t0)]. (b) Line source function in a semi-infinite atmospliere with B = 1, for a line with a pure Doppler profile and j:(t) as shown in (a). Curves labeled "a"" and "b" correspond to constant values oic of 10"5 and 10 "? respectively. From (284, 101). behave as if s were constant at £ = s2, and S,(0) will equal e2iB. These qualitative expectations are confirmed by detailed calculations (284, 101) such as those shown in Figure 11-7 where e is assumed to have the form e(t) = 10"3[1 - 0.99 exp(-t/'t0)]. Notice that, when t0 > 105, St behaves as if e = 10"5, whereas if t0 < 102, St behaves as if e = I0~3. A change in the form of the line profile alters the way photons get redistributed into the line wings, and their probability of escape. Numerical studies (320; 18, 51) have revealed a number of interesting effects of profile depth-variations upon the line source function. Suppose, for example, the Doppler width rises sharply near the surface in an atmosphere with B = log(S,/B) f1gurl I 1-K Line source functions in a semi-infinite atmosphere with B = 1, for a line withe = 10"4, r = 10"6, a = 10"3, and a depth-variable Doppler width, (a) Doppler width rising toward surface, Avd(t) = 1 + 2exp( —«tc). (b) Doppler width decreasing toward surface. AvcU) = 3 - 2 exp( -*r£). Curves arc labeled with parameter a. Curves labeled "0" have constant Doppler width. Ordinnies: Left-hand scale and solid curves give \og{S,/B); right-hand scale and dashed curves give Avn. From (18, 52), by permission. constant (see Figure 11-Sa). Then we find that the surface value S,(0) is increased markedly because the broader line-wings can now intercept radiation from deeper layers. As the position of the rise in AvD moves deeper in the atmosphere, the surface effect diminishes, basically because the surface layer becomes more and more opaque in the wings, and the bright underlying r 358 Non-LTE Line Transfer: The Two-Level Atom continuum radiation becomes more attenuated. At the same time, the source function at great depth also increases, basically because the higher opacity in the line-wings in the surface layer impedes the escape of radiation from below; this is sometimes referred to as the reflector effect. The effect at depth is greater, the thicker the upper layer (until, of course, the rise lies below the point of thermalization). Notice in Figure 1 l-8a that when a - 106 the upper layer is optically thin, and thus there is essentially no reflector effect even though the surface rise is largest. When the Doppler width decreases sharply at the surface, the value of Sj(0) drops because the narrower profile no longer intercepts bright radiation in its wings. Radiation trapping is also reduced in lower layers, and hence S; decreases there as well. At very great depth, however, the effect is the opposite, and S, actually rises a little above its value for constant AvD, because now the decreased bandwidth available to the lines implies a decreased escape probability. It must also be noted that, in addition to changing S,(tJ, variations in the absorption profile change the run of optical depth, at a given frequency, with physical depth. In view of the Eddington-Barbier relation, it is obvious that this implies a change in the "mapping" of St into /v(0), and hence in the emergent intensity distribution in the fine profile, and its center-to-limb behavior [see. e.g., (26)]. 11-3 The Two-Level Atom with Continuum THE SOURCE FUNCTION The discussion given in the preceding sections of this chapter is based on a very schematic and admittedly restrictive atomic model. Naturally the true situation for any real atom is more complicated. To introduce some of the physically important effects while retaining analytical simplicity, let us now consider an atomic model consisting of two bound states and a continuum. This model provides at least a rough representation of reality for resonance lines and also for subordinate lines when the resonance lines are in radiative detailed balance. The addition of the continuum greatly increases the number of processes that may take place. In the strict two-level case, the only processes that can occur are photoexcitations or collisional excitations from the lower to upper state, and their inverses. Now there are. in addition, photoionizations and collisional ionizations from the bound levels to the continuum, and radiative and collisional three-body recombinations to each bound level. It is clear that this is a much more general model, and we shall find that the additional physics has major implications for the line-formation process. 11-3 The Two-Level Atom with Continuum 359 As before, assume complete redistribution, so that 5, is given by equation 1-4). The statistical equilibrium equations now are ni[Blu jvJvtlv + Clu + RlK + ClK Bul j>v/v dv + C,„J + nf(RK, + ClK) (11-54) and nH (A,d + Bul j(J>vJv dv + Cu! + R,IK + Cu = n„ A = >h [BlH jvJv dv + sBv{T) + 0 'I v) where r. is given by equation (11-7), (RUK + CBJn(*{/?k., + ClK) - 9l{Rlh + ClK)n*{RKU + C^/g,, and 6 = 2hv3\/ gt C,K) + n*{RKU + CHJ] c2 JVlJj [nf{RKl + Clfc) + ri*(RKU + C1(J] (11-56) (11-57) (11-58) Equation (11-56) was first derived by Thomas (622) as an extension of earlier work by Milne (416. 159 -164) and Stromgren (613), and was studied extensively in an important series of papers by Jefferies and Thomas (335; 336; 623; 337). Exercise J1-8: Derive equations (11-56) through (11-58) in detail. Despite the apparent complexity of these expressions, each term admits a simple interpretation. Consider the numerator of equation (11-56). The first term again represents the scattering reservoir. The second term is the thermal source, giving the rate at which photons are created by collisional excitation; note that this term depends upon the local value of Te, the electron kinetic temperature. The third term is proportional to the total rate at which electrons are ionized from the ground state to the continuum, 360 Non-LTE Line Transfer: The Two-Level Atom 11-3 The Two-Level Atom with Continuum 361 times the fraction that recombines to the upper state and thus becomes available for emission into the line by radiative decay to the ground state. Similarly, in the denominator, the second term accounts for photons destroyed by collisional de-excitation of the upper state. The third term is a new sink term that is proportional to the total ionization rate from the upper state times the fractional recombination rate to the lower state; this term clearly accounts for the destruction of potential line photons via the continuum processes. We thus find all of the basic physics of the situation represented clearly in the source function. Further insight follows from rewriting 0 as nB*, where B* is found to be B* 2hv <9i '{Ruk + C!tK)(RK, + CJ (RlK + ClK)(RKU + C,J - 1 (11-59) Exercise 11-9: Verify equation (11-59). It is clear that B* bears a formal resemblance to the Planck function, and, in fact, we can think of it as B* = Bv{Tr), where TT is a characteristic radiation temperature whose value is set by the photoionization and recombination rates in the two bound-free continua. This radiation temperature can, in general, be quite different from Te. At great depth in the atmosphere where the continua are optically thick, Jv -> Bv(Te); then RlK -»■ RfK - R*„ and KK -> = R*u, so that B* -> Bv(Te), as expected, and St -> Bv(Te). But near the surface, the continua may become transparent (while the line remains optically thick) and the continuum radiative rates become essentially fixed. Depending on the relation of the radiation temperature (which characterizes Jv emerging from zc ~ 1) to the local electron temperature, quite distinct situations emerge. For example, suppose that collisions are negligible, and that RlK > RKt while RK!I > RUK (i.e., suppose that a relatively cool medium is irradiated by a "hot" radiation field in the ground-state continuum). Then it is clear that B* > Bv(Te), and that S, will be larger than the value it would have had by coupling to the thermal pool; the excess emission comes from a preferential depopulation of the lower state into the upper via the continuum, followed by radiative de-excitation in the line. If the line is an absorption line, an increase in 5, tends to weaken it; taken to the extreme limit, this is the mechanism that produces the photoionization-recombination-cascade emission spectrum in a nebula. If the inequalities posed above are reversed, then the upper state is selectively depopulated into the lower, and 5; decreases; if the line is an absorption line, it will strengthen. The two essential points that have emerged from the above analysis are the following, (a) The source function for a given line contains terms in other transitions (the two continua for the present model). This result is quite generally true and carries over to the multilevel case (see §12-1). (b) The new terms do not couple S; to the local thermal pool, characterized by Te, as do collisions, but rather to a radiation temperature that may be markedly hotter or cooler than the local kinetic temperature. CLASSIFICATION OF LINES In view of the fundamentally differing natures of the various source and sink terms that appear, we may expect that line-transfer problems may show substantially different characteristics depending on which terms provide the dominant contributions to St. Thomas has suggested (622; 626, 174) a classification of lines into broad categories by consideration of which terms are the largest for different atoms in typical stellar-atmosphere regimes. For example, if s > r\ and sBv > 77B*, the line is called collision dominated; here S, couples to Te. On the other hand, if n > & and tjB* > eB, the line is photoionization dominated, and the line couples to a characteristic T, # Te. Intermediate cases where, say, e > rj but nB* > eB, or vice versa, are called mixed domination lines. The recognition of these classes of lines represents a considerable advance over the rather ill-defined classical division of lines into "absorption" and "scattering" categories, and has led to important insights about line-formation. In particular, the emergent profiles for lines in these classes are quite different when a chromospheric temperature rise occurs in the outer layers of the atmosphere. The category to which any particular line belongs depends (via the atomic cross-sections involved) upon the structure of the ion from which it arises, and upon the structure of the atmosphere (because of the dependence of the relevant rates upon atmospheric parameters such as temperature, density, and incident radiation fields). Different lines of the same ion will, in general, fall into different classes, and a comprehensive a priori classification is not possible; rather, each case must be examined in turn. Broad groupings of lines in a solar-type atmosphere have been suggested by Thomas; these are displayed in Table 11-2, and can be understood qualitatively as follows. The resonance lines of the singly ionized metals are collision dominated because the excitation energy is only a few electron volts (compared to a thermal TAI5LL 1 1-2 Categories of Line Source Functions in a Solar-Type Atmosphere Collision dominated Photoionization dominated Resonance lines of singly ionized Resonance lines of neutral metals (Mg + , Ca + , Sr+, etc.) metals Resonance lines of H and other Hydrogen Balmcr lines nonmetals (C, N, O, etc.) Source; Adapted from (623; 626. 174). 362 Non-LTE Line Transfer: The Two-Level Atom energy of about 0.5-1 eV), while the ionization energies are 11-15 eV; recalling that both the collision and photoionization rates scale as exp(— E0/kT), it is plausible that the collisions should prevail. Even for hydrogen, where E0 in La is 10 eV, the collision rates dominate because the solar u.v. radiation is weak and the photoionization rate is small. In contrast, for the hydrogen Balmer lines and the neutral metals, the relevant photoionization edges lie in regions where the solar radiation field is intense, and the photoionization rates prevail over collision rates. The sodium D-lines are an exception because of an unusually large collision cross-section (284, 333; 284, 347); this example serves as a warning that the broad classes listed are illustrative, and detailed analysis is needed in each case. In a higher temperature regime (e.g., in an O-star) some of the remarks just made are no longer valid, and lines switch categories. For example, the radiation held in the Lyman continuum becomes extremely intense (the hydrogen is virtually completely ionized), and the hydrogen Lyman lines become photoionization dominated. The higher subordinate lines now have energies £0 < kT (~3 eV at O-star temperatures), and are relatively weak, hence are formed in deeper, denser layers; this tends to lead to collision domination of the subordinate series. LINE-FORMATION IN THE PRESENCE OF A CHROMOSPHERE The great physical importance of the division of lines into the two broad categories described above becomes strikingly apparent when we consider the nature of the source function variation, and emergent line profile, in an atmosphere with a chromospheric temperature rise outward. The basic point is that collision-dominated lines are linked to the local electron kinetic temperature, while photoionization-dominated lines are not. The latter couple instead to a radiation temperature characteristic of the energy distribution (emitted at some other point in the atmosphere) in the continua of the upper and lower levels. We therefore expect collision-dominated lines to be at least partly responsive to the local temperature, and to exhibit, in their profiles, features attributable to the outward temperature rise. On the other hand, we expect photoionization-dominated line-profiles to be insensitive to variations in local parameters (in particular, to variations in Tc). The strong dichotomy of behavior just described was clearly demonstrated in the pioneering work of Jefferies and Thomas (336). They showed that these considerations explained the observed presence of emission cores in lines such as the H and K lines of Ca+ (and the analogous lines of Mg+), and their simultaneous absence in the hydrogen Balmer lines. In their work Jefferies and Thomas adopted a schematic continuum source function of the form Sc(t) = BJTJ = 5,(1 + azc + ße~™) (11-60) 11-3 The Two-Level Atom with Continuum 363 which provides an approximate representation of a temperature distribution that has: a uniform gradient in the photosphere; a plateau at a minimum temperature at some characteristic depth (they used Tmin ss 4000' K at zc % 10"2); and then a steep outward gradient mimicking the sharp chromospheric rise. The transfer equation was solved in the Eddington approximation: \{d2JJck2) = (0V + r)\Jv - (0V + r)-\rSc + 0^)] (11-61) using the method of discrete ordinates, for a Doppler profile and for typical values of r, c, n, and B*. Results for a collision-dominated line (n = 0) with a — 10 -4 r = 10" and various choices for the continuum source function are shown in Figure 11-9. Note that at great depth S, thermalizes to Sc, but as the wings begin to become transparent (tc 5 1), St drops below Sc. Proceeding outward, Sc rises very steeply at xc x 10~2. The line source function tries to follow this rise through collisional coupling, and actually does increase outward, but ultimately the effects of scattering dominate, and, at the surface, S, lies about three orders of magnitude below Bv(Te). It follows from the Eddington-Barbier relation that the depth-variation of S,(t) should be reflected in the frequency-variation of the emergent intensity. The computed emergent \nS/Sl log ic FIGURE 11-9 Line source functions for a collision-dominated line in a semi-infinite atmosphere with a chromospheric temperature rise. Upper curves show continuum source function Sc, and lower curves show corresponding line source functions S,. In all cases £ = 10-4" and r = 10"4. From (336) by permission. 364 (It - IY)/S, 0 12 3 Av/AvD FKiUKE 11-10 Line profiles for a collision-dominated line in a semi-infinite atmosphere with a chromospheric temperature rise. Curves are coded to correspond to those in Figure 11-9. From (336), by permission. intensity is shown in Figure 11-10; only half of the profile (which is symmetric about line-center) is shown. Note that the lines show an emission peak near + 1.5 Doppler widths, with a central absorption reversal; lines of this form are called doubly reversed. The overall variation of intensity within the line is in good qualitative agreement with the observed behavior in the solar Ca II H- and X-lines and the corresponding lines of Mg II. Numerous other calculations, with a chromospheric temperature rise, for other choices (sometimes including depth-variations) of e, a, r, A. where A = or a'i~2 for Doppler and Voigt profiles, respectively, and q is the total destruction probability given by equation (11-50)], then St saturates to /? at depth and falls by a factor of Jf at the surface. If the slab is optically thin, then S, just equals the local creation term ^/?. If the slab is effectively thin, the maximum value of St is given by equations (11-40) with T = (yr)-1. Note that the quoted results for the Voigt profile presume that a(yr)"1 > 1; if a is so small that this is not true, the line wings are negligible and the results listed for a Doppler profile apply instead. The surface value of 5, can be derived by recognizing that if Smax a: £fl(n} (where gives the mean number of scatterings needed to escape), then from random-walk arguments we expect S,(0) oc £/J(iV>* Note that the effectively-thin results merge smoothly with the effectively-thick results when (yr)'1 = A, and with the optically-thin results when (yr)"1 = 1. From the results of Table 11-3 we see that, with a given chromospheric structure, some collision dominated lines will respond strongly to the temperature rise, while others may not, depending on the values of s, r, and a appropriate to them. For example, for the Ca II lines in the solar chromosphere we have (to order of magnitude only!): a — 10~3, r ~ 10~9, e - 3 x 10~4, and y - 106; therefore A - 104, while (yr)"1 - 103, so the lines are effectively thin. Furthermore a(yr)"1 — 1, so (5mav//J) &(yra)~'s ~ 0.3, and thus the coupling of the Ca II lines to the temperature rise is relatively inefficient. In contrast, for the Mg II resonance lines, r is about a factor of 10 smaller (because Mg is about 10 times more abundant than Ca), and the other parameters are about the same; in this case we obtain a much stronger coupling to the rise and much brighter emission, as observed. It should be noted that the arguments given about are meant only to be gualitative, for the depth-variation of the parameters is actually fairly complicated, and not well-represented everywhere by the values chosen; in particular, equation (11-60) provides a relatively poor fit to the actual run of Bv(t). Detailed analyses of Ca II line-formation in stellar chromospheres have been performed using realistic atomic and atmospheric models (56; 58), and a physically plausible explanation of the Wilson-Bappu effect has begun to emerge (57). 11-4 Static Extended Atmospheres As was described in §7-6, many stars have extended atmospheres whose thicknesses are comparable to the radius of the star. Wc shall suppose that, to a first approximation, these atmospheres are spherically symmetric. For the purposes of the present discussion we shall consider the stellar radius r% to be that of the surface on which tr as f. We imagine that this "photospheric core" is surrounded by an envelope of large size, within which unit optical depth in the most opaque spectral regions (i.e.. line centers) is encountered at radii R » r^. In reality, virtually all stars with very extended atmospheres also have large-scale velocity fields (usually overall atmospheric expansion) that strongly affect—indeed dominate—transfer in the lines, so that the assumption of a static atmosphere is physically less useful in an analysis of line-formation than it was for the continuum. Nevertheless, there are some extremely important effects of a fundamentally geometric origin that enter; it is worthwhile to examine these here, and to defer a discussion of velocity-field effects until Chapter 14. As seen by an outside observer, the size of the emitting surface where tv = 1 at the more opaque frequencies, in particular the cores of spectral lines, can be much larger than that of the continuum. Then the line has a larger effective emitting area, and if we assume LTE and suppose that the envelope is essentially isothermal, it is clear that the line will appear in emission relative to the continuum. This behavior contrasts with the result for an isothermal planar atmosphere where the line is neither in absorption or emission. In fact, the basic geometric effect just described is actually the primary mechanism that produces the extremely intense emission in very opaque spectral lines observed—e.g., in Wolf-Rayet spectra. The assumption 368 Non-LTE Line Transfer: The Two-Level Atom that the atmosphere is essentially isothermal implies a nonradiative source of energy input, for we have seen in §7-6 that in radiative equilibrium (in LTE), J ~ B ~ r~2 in an extended atmosphere; thus the situation just described may be a bit extreme (though we have no strong physical reason to prefer radiative equilibrium—recall the solar corona!). Also, if the line source function has a scattering term, we expect S,(t) to decrease outward (in fact the drop is enhanced by extension effects; see below). However, it is clear that, for any given S,(t), the change in effective emitting area from line core to wing will always tend to increase the core emission, relative to the continuum, compared to the value it would have had in a planar atmosphere. Absorption lines will tend to weaken, and emission lines will tend to grow brighter, as atmospheric size increases. A second effect of atmospheric extension is a systematic increase of the escape probability at a given (radial) optical depth. It is obvious from elementary geometric considerations that along all rays, except the single ray at p = +1, the optical path-length from the test point to the surface is smaller in a spherically symmetric atmosphere than in a plane-parallel semi-infinite atmosphere of identical properties (i.e., same run of physical variables along the radial direction in the two cases). Thus, in view of equation (11-22), Pe{x) increases, and we expect S,(t) to decrease accordingly. A third effect, which also increases the effective probability of escape, is the systematic bias toward larger radii in the scattering. Suppose that the scattering process is isotropic and that the material is homogeneous, so that the photon mean-free-path is the same in all directions, and that / = 1 defines a spherical volume around the test point. Then, from the basic geometry of the situation, it is easy to see that, in a spherical atmosphere, more photons end their flights at larger radii (hence closer to the surface); in contrast, in a planar atmosphere the probabilities that the photon ends its flight at a greater or smaller depth (by a given amount) are equal. If the material has an opacity that decreases strongly outward, the bias is enormously enhanced [see Table II and Fig. 2 of (374)]. This bias implies that the net chance of escape is enhanced even further. The transfer equation for the two-level-atom line-formation problem [i.e., Si given by an equation of the form of (11-56)] in spherical geometry is easily solved by the methods described in §7-6 [see equations (7-190) through (7-208) and related discussion]. Calculations for idealized models (374) similar to those employed earlier in this chapter provide illustrations of the concepts outlined above. We characterize the atmosphere by its outer radius R (in units of the core radius rc = 1), a total line optical depth Th a continuum optical depth Tc, and opacities /, oc r~2 and %c cc r~2. We set s = constant, andB = 1. Source functions for an envelope with no background continuum, surrounding an empty core (i.e., a nebula) are shown in Figure 11-12. The two 369 log ^ i i i _i_I -10 12 3 l0gT figure 11-12 Line source functions in a spherical atmosphere of outer radius R (in units ofrc = 1), B = \, s = constant, T, = 0, T, = 10s, and opacity x, -K r 2. Curves are labeled with R, and results for two values oft;, corresponding to effectively thick and effectively thin media, are shown. From (374], by permission. values of e chosen yield effectively thick and effectively thin media. The major effects of increased photon escape in decreasing St are readily apparent. Note that, for the effectively thick medium, the thermalization depth (A « 102) is scarcely affected by sphericity. Emergent flux profiles for these source functions are shown in Figure 11-13. There we see that the central reversal, which is a prominent feature in the planar limit (note log scale), vanishes for very 370 371 log Fx - I 1 "—= 1 _ ^\ \\\io\ _ e - 10"2 - 30 >v\ \\ - 100' \ \\\ - 300\ \\ i i \ - - \ \ "^\\ \3 \ - 6 = 10"4 - \^ 30\\^ 300\ - i i 0 12 3 4 x figure 11-13 Emergent flux profiles from spherical atmospheres with same properties as used in Figure 11-12. From (374), by permission. extended atmospheres; this is a manifestation of the greater effective emitting area in the line core. Analogous results for atmospheres with Tc = 2 and s = 2 x 10~3, and otherwise identical to those mentioned above, are shown in Figures 14-11 (source functions; note only curves with umax = 0) and 11-14 (line profiles). Again, the source functions are dramatically reduced by increased escapes, and the line profile transforms from a pure absorption line to an emission line with a central reversal (the reversal results only because 5, has decreased so much at x = 1; if S, remained unchanged, the whole line would come into emission). 0.6 - i 1 l - - \ /30 / - 1 i --•--R - 3 i i i - -3 _ i _ 0 1 2 3 X figure 11-14 Emergent flux profiles from spherical atmospheres with B — 1. r. = 2x 10"-\ Tc - 2. T, = 10\ and X h Bi„ f vjYdv + c„, - n„ = n HKi + ClK)+ X nfAnZn + »,.C,^ (12-2) and for the upper level we have „ [A,,, + Bul UXJV dv + Cul + X AläZui + X CujYuj + Rm + Cu uvJvdv + {s' + 0)Bv(Te) (1 + c' + n) (12-4) where s' is defined as in equation (11-7), while n = [a2«3 - (gA>ifl4]/lA((«2 + (12-5) and 0 = [nffllfl4(l - e-"^T)']/l^Alll(a2 + aA]\ (12-6) where, in turn, «i = + ClK + I AUZU + 1 Crfj (12-7) a2 = nf(RKl + ClK) + £ njAjfZj, + £ w,Ca*i (12-8) l< ji=u i<-\ a, = Rm + CUK+ £ AuiZld + Y CujYUJ (12-9) U>i±t uv)Iv + IcSc + xdA (12-11) where Sc denotes the continuum source function (which is not necessarily the Planck function). Writing div = + xM dz, and substituting for S, from equation (12-4), equation (12-11) can be rewritten as p(dljdxv) = Iv - ay j*0vJv dv - bv (12-12) which can be recast into a second-order form, and solved by the standard Feautrier or Rybicki methods described in Chapter 6. A calculation of this type is carried out for each line of the entire transition array considered. Note that to compute Xi, and hence tv, a,, and bv in equation (12-12), it is necessary to have actual values for n, and /?„; as these quantities are known only approximately at any stage of the calculation, it is clear that an iteration procedure is required, in which successively more accurate values of the occupation numbers are employed. This iteration may be carried out simultaneously with the more basic iteration required to obtain estimates of the radiative rates in all lines. 12-1 The Equivalent-Two-Level-Atom Approach 379 In a calculation designed to simulate conditions in a real atmosphere, it is, in general, necessary to account for the variation in the ionization state of the particular atomic species under consideration (perhaps over a sequence of several different ionization stages). Although continua arising from excited states of '"'impurity" species will generally be transparent (compared to the dominant absorption and emission terms from H and He), so that the radiation field may be considered to be fixed, this situation will not usually be true for ground state continua, which normally will be opaque enough to determine their own radiation fields. Thus it is necessary to supplement the line transfer equations (12-12) with corresponding continuum equations, as described in the discussion of the formation of the Lyman continuum in §7-5 [see equations (7-127) through (7-131) and (7-135) through (7-144)]. The complete set of rate equations for a particular impurity species will be composed of as many equations as there are bound levels (over all stages of ionization taken into account), plus a final equation setting the sum of all occupation numbers equal to nalom, the total number density of the species. For a given abundance of the element, relative to hydrogen, natom is a pre-specified function of depth. In overall form these equations are similar to the first MHc lines of the matrix displayed, in §5-4, between equations (5-91) and (5-92). The iteration procedure required to obtain the solution (which consists of the run of the occupation numbers of all the levels with depth) is fairly straightforward in principle, though often complicated in practice. If one starts with an initial estimate of n^z) for all atomic levels i, say from LTE relations, then provisional optical depth scales can be constructed. By regarding xv and nv as known [via equations (7-1) and (7-2)], an estimate of the radiation field in each transition may be obtained by a formal solution of the transfer equation (i.e., Sl is regarded as given). One may use this estimate to compute continuum photoionization and recombination rates. A transfer equation of the form (12-12) for each line (and opaque continuum) is then solved in sequence, supposing, initially, that all the net brackets Zjt and Yij appearing in equations (12-7) through (12-10), and their continuum analogues, may be set identically to zero. This sequence of calculations produces radiative rates Ru in all lines and opaque continua. The rate equations stfn = are then re-solved, at each depth, to obtain an improved estimate of all the occupation numbers n. All Z^'s and Y^s, and hence n and 6 for each line can now be evaluated via equations (12-5) through (12-10) because all of the required rates and level-populations are known. Equations (12-12) are then re-solved, using the new values for tv, n, and 6 and one obtains, thereby, new estimates of Jlu and Slu in every line (and continuum). The rate equations are again re-solved, and the process is iterated to convergence. Once a converged solution is obtained, all the line source functions are known, and line profiles may be calculated for each line. 380 Non-LTE Line Transfer: The Multilevel Atom As mentioned above, the practical implementation of a successful iteration procedure using the equivalent-two-lcvel-atom formalism is often quite complicated, for the rate of convergence (or lack thereof!) may be strongly affected by technical details, such as the way the net rates are computed, source functions are evaluated, and many others. Careful discussions of these points can be found in (18, Chap. 4; 23, 27-63; 23, 113-132; 187), as well as in the references cited therein. A more fundamental difficulty with this overall approach is that the iteration procedure may fail to converge but, instead, stabilizes on a solution that is inconsistent: see the discussion in (18, §4.2) and (23,27-63). This failure, which is not too common, and which can sometimes be overcome with special procedures, is, nevertheless, not surprising: as we shall see in the next section, there are many physical situations in which the radiation fields in different lines are very strongly interdependent, in contradiction to the basic assumption of the equivalent-two-level-atom approach. The inconsistency problem is overcome wholly by the complete-linearization method presented in §12.3. application The equivalent-two-level-atom approach has been extensively applied to calculations of a wide variety of spectra, particularly for the solar atmosphere and for solar-type stars. For example, for the sun, analyses have been made of the Ca II H- and X-lines (400; 401; 569), O I and C II lines (168), the Mg I 6-lines and Na I D-lines (21), and the spectrum of Fe I (22), using sophisticated multilevel atoms and elaborate atmospheric models. Very complete spectrum syntheses, using this general approach, of both continuum and line intensities have led to refined photospheric-chromospheric models (645; 646). Further, similar analyses have been made of the Ca II H- and X-lines in solar-type stars (56; 58). Space does not permit a discussion of all of these results, and the reader should examine the references cited. It is, nevertheless, of interest to quote a few results (401) for the solar Ca II lines, which play a central role in studies of the chromosphere. The H and K lines arise from the transitions 4s 2Sj_ -* 4p 2i\ 4 (see Figure 12-1); the two upper states are coupled by collisions, and may also decay to the metastable M 2Di± states in the infrared triplet ;.;.8498, 8542, 8662. As the next levels lie well above the 4p level (recall kT - 0.5 eV in the solar photosphere), the five levels mentioned above, plus the continuum, suffice to provide an accurate description of the physics of line-formation. A solution of the transfer and statistical equilibrium equations for this 5-leveI Ca+ atom yields source functions in all 5 lines; the frequency-independent X-line source function (obtained assuming complete redistribution) is shown in Figure 12-2, along with the Planck function Continuum 11.87 eV 7.22 eV 3.14 eV - 1.70 eV - OeV - I-'KiURE 12-i Energy-level diagram for lower states of Ca + . The five levels 4rS_:, 4p'P_...... and 3(/2IX , suffice to describe the line-formation process for the Ca 11 /-/ and K lines at ;J3968 and 3933, and the infrared-triplet A/.8498, 8542, 8662. Nolc that the next levels lie fairly high in energy, and hence can be ignored, though interactions with the continuum are included. From (401), by permission. (Courtesy of the Publications of the Astronomical Society of the Pacific.) 382 10-6 1 1 1 - 1 1 1 i i i i i - 10"1 1 1 1 1 1 1 1 10 1 1 1 102 103 104 io5 to5 -X 1 1 [ 1 1 J 1 1 ! 1 1 1 1 1 2500 2000 1500 1000 500 0 h (km) FIGURE 12-2 Ca II X-line source function S, and corresponding Planck-function B, in a model solar atmosphere. Source function is in absolute units: ergs cm-2 sec"1 hz"1 sr-1. Optical depth scale is at line center and dashed lines indicate computed intensities at Kls K2, and K$ (sec Fig. 12-3). From (401), by permission. (Courtesy of the Publications of the Astronomical Society of the Pacific] corresponding to the adopted run of temperature Te(h). The emergent intensity profiles, at \i = 1, for the H- and X-lines and the infrared triplet, are compared with observation in Figures 12-3 and 12-4, respectively. The agreement between theory and observation is quite satisfying. However, when the computed variation of the H- and X-line profiles from center to limb is compared with observation, significant disagreements are found; as we shall see in §13-4, these are removed when the partial coherency of the scattering process, and the resulting frequency-dependence of the source function, are taken into account. Further, the calculations shown in Figures 12-3 and 12-4 represent only the average quiet chromosphere. To match plage profiles (569), a different atmospheric model is required; and to match the detailed, variation of the X-line profile, as observed from point to point on the disk, will require a full treatment of the three-dimensional fine-structure and the velocity fields in the chromosphere. Such calculations have not yet been undertaken in a completely satisfactory fashion, but are needed, both to diagnose details of chromospheric structure, and to answer the question "To what extent I(AA)//, 0.021-1-1-'-—'-1-1— 0 0.2 0.4 0.6 AA FlGUKli 12-3 Computed H- and K-line profiles at ji = 1 (dashed lines] compared with observed profiles (solid lines). From (401], by permission. (Courtesy of the Publications of the Astronomical Society of the Pacific.) 1 1 1 0.6 - 8498 Calculated^ Observed 0.4 1 8542 Calculated'^/ Observed —' — 0.2 "T^^ \ Observed **" 8662 Calculated i 1 i 1 0 0.2 0.4 0.6 0. AA J-IGURii 12-4 Computed Ca II infrared-triplet profiles at fi = 1 compared with observed profiles. From (40!), by permission. (Courtesy of the Publications of the Astronomical Society of the Pacific.) 384 Non-LTE Line Transfer: The Multilevel Atom (if any) is it possible to replace the complicated multidimensional structure of the chromosphere with a meaningful horizontally-averaged 'mean chromosphere'?" 12-2 Effects of Level Coupling: Source Function Equality in Multiplets As was emphasized in the preceding section, the equivalent-two-level-atom approach works best when the coupling terms from two particular levels to all other levels are small, compared to those between the levels themselves and from these levels to the continuum. There are, however, many important physical situations for which this will not be the case, and it is important to understand the effects of strong coupling among several levels. These effects have important implications that affect both our perception of line-formation, and our choice of numerical methods used to obtain solutions of multilevel transfer problems. Most of the fundamental physical concepts were outlined by Jefferies (333; 334, Chap. 8), and much of the subsequent theoretical work [e.g.. (567)] has been based on his ideas. Numerous instructive calculations, using idealized multilevel atomic models and parameterized model atmospheres, have been carried out; see, e.g., (54; 18, Chap. 4; 187: 213; 217; 218. Rather than attempt to summarize these diverse investigations here, we shall, instead, confine attention to two concrete examples, which illustrate the nature of interlocking effects in a particularly illuminating way. First we shall consider, briefly, the case of three levels coupled together by two lines of a resonance series, and one subordinate line. The primary result of interest here is photon degradation from the higher resonance line into the lower resonance and subordinate lines; this phenomenon is related to fluorescence in nebulae and to the Rosseland cycles discussed in §5-5. Next, we shall devote the remainder of the section to the interlocking of lines within a multiplet; here photon conversion may occur as collisions shuffle electrons among the fine-structure levels of the spectroscopic terms involved. As a result, the source functions in the different lines become dependent upon one another, and in a certain limit become equal at each depth in the atmosphere. Source-function equality in multiplets is of great theoretical importance, for it implies that we may replace several lines within a multiplet by a single representative line, and thereby reduce greatly the amount of computation required. Further, source function equality, when it occurs, allows application of a non-LTE method of analysis of line-profile data, by means of which the physical properties of an atmosphere may be inferred directly with a minimum of theoretical interposition. In keeping with the remainder of the book, in which theoretical prediction rather than analysis of observations has been emphasized, this analytical method will 12-2 Effects of Level Coupling: Source Function Equality in Multiplets 385 not be discussed here, and the reader is encouraged to refer to the complete discussion by Jefferies (334, Chap. 9). PHOTON DEGRADATION AND CONVERSION Consider, first, a three-level atom with two resonance lines 1 <-*■ 2 and 1 <-> 3, and a subordinate line 2 3. As an example, in hydrogen we would have the La, L/J, and Ha lines; in fact, for expository convenience, the three lines will be referred to with these names, even though we may not be dealing with hydrogen. For the present purposes, interactions with the continuum will be ignored, and attention focussed entirely upon the lines. Generally, we can expect the ground-state population n, to be vastly larger than the upper-state populations n2 and n3; hence the resonance lines will be much more opaque, and will have much larger optical depths, at a given physical depth, than the subordinate lines. Suppose now that all three lines were somehow uncoupled, and were formed in distinct two-level atoms; each would then thermalize at a characteristic depth Au % l/c;/, where is the collisional destruction probability in transition i <-*/'. Because /rv23 « hvl2 (or /rv13) we will generally find t:23 » fi12 (or s13); hence the subordinate line will thermalize in a smaller number of optical depths, measured on its own scale, than the resonance lines. But this will be overshadowed by the much greater opacity in the resonance lines, and the subordinate-line thermalization will occur at much greater geometric depth in the atmosphere. Put another way, the subordinate line is so much more transparent than the resonance lines, that photon-escapes (and hence departures of S from B) first occur in this transition, as one proceeds outward from great depth toward the surface of the atmosphere. There will thus be a certain range of depths over which the resonance lines, were they not coupled to the subordinate line, would by themselves be thermalized, but within which the subordinate line has become transparent enough to permit escapes. In this critical range, some of the electrons photoexcited by Lfi from level 1 to level 3 will decay into Ha, and the Ha photons will escape; there will thus be a systematic degradation of Lfi photons into Ha and La photons, at a rate determined by the branching ratio ■^32/04.31 + ^32)- Note that the inverse process of conversion of Ha photons into Lf$ is essentially ineffectual. If an Ha photon is absorbed, leading to an excitation 2 -> 3, most of the subsequent emissions from state 3 will be in the 3 -> 1 transition; because, however, r13 is so large and cl3 is so small, these photons are trapped, and merely scatter until a 3 -» 2 transition is finally made, and the photon escapes. In this atmospheric region, La will adjust in such a way as to remain almost in detailed balance (with its source function near to the Planck function), while L$ excitation is drained, and its source function is depressed. 3S6 Non-LTE Line Transfer: The Multilevel Atom Ignoring stimulated emissions, we can express "reduced" source functions as (S21/B21j - {b2/bt\ (S31/B3S) = (M>i), and (S32/B32) = (b3/b2), where Bjj denotes the Planck function B{vu. 7"), and bi = nfnf. When La adjusts to detailed balance, b2 = bY (recall the discussion pertaining to the ground-state population of H in B-stars given in §7-5), and thus (S32/B32) - (S31/B31), so that the "reduced" L/? and Ha. source functions become equal below the point of La thermalization. As we proceed to the outermost layers of the atmosphere, the L/? line becomes more and more transparent and, ultimately, the probability of direct escape exceeds the branching ratio into Ha., with its consequent escape; at this point the L/J and Ha source functions become uncoupled. Results from a calculation (218) for a three-level atom with the parameters 1,^/1,2 = 10, = e, = 10" r'23 = 9 x 10" (Ti2/Ti s) = 6.2, and U23A13) = 1-6 * 10"4 [(S3JB3l)/(S32/B32)] are shown in Figure 12-5. ThereweseethatLathermalizesatT13 x 2 x 102, which implies t, 2 - 103. as expected from e12.AXt13 % 10, the probability of L/i escape is comparable to the probability of branching into Ha; therefore for r13 < 10 the "reduced" Lfj and Ha source functions diverge. But below this depth, they rapidly figure 12-5 Source functions for 3-level atom in which both resonance lines, (1 <-> 3) and (1 <-» 2), and the subordinate line (2 <-> 3) are radialively permitted, Abscissa: optical depth in the 1 —► 3 line; onliiuite: "reduced" source functions [S^tO/S^oo)]; ij-,(cc) is equal to the Planck function at the appropriate frequency. From (218), by permission. 12-2 Effects of Level Coupling: Source Function Equality in Multiplets 387 approach equality, at a value well below that which the Lfi source function would have attained if degradation did not occur. Thermalization in Ha (and, along with it, Lfi) occurs at about t13 « 10s, which implies t23 - 10, as would be expected from Further discussion and interpretation of the results may be found in the paper cited, but it is clear from what has been said here that there are major effects of interlocking on the source function variations in this problem. See also (18, §4.8). Let us now turn to the case of a three-level atom consisting of a ground state, and two close-lying upper states. We assume that both radiative and collisional transitions can take place between states 1 and 2, and between 1 and 3, but that only collisional transitions occur between states 2 and 3. This model simulates the actual physical situation for a resonance doublet, where the upper states arc separated by fine-structure splitting. For example, the sodium D-lines arise from transitions from the 3s 2S4 ground state (stale 1) to the upper states 3p zPi and 3p 2P4 (states 2 and 3). Consider first the limit in which the 2 *-> 3 collision rate is zero (i.e., the uncoupled case). The transitions 1 <-> 2 and 1 <-> 3 can then take place without reference to one another, and the two lines are formed independently. Each line will have a source function that falls below the local Planck function near the surface, and equilibrates to the Planck function at thermalization depths given by A12 « (a21 + C21)/C21 (12-13a) and A,3 ~(-<3, + C3i)/C2 (12-13b) Here we have made the simplifying assumptions of Doppler profiles and an absence of strong gradients. In the limit considered here, the run through the atmosphere of the source functions for the two lines will, generally, be different, and at any given depth they will not usually have the same value. On the other hand, suppose that collisions occur very rapidly between levels 2 and 3. In this case, excited electrons are shuffled from one level to the other; we then say that photons have been converted back and forth between the two lines. In the transfer problem the two upper states become effectively a single state, and in the extreme limit, the occupation numbers of the two upper states will be proportional to their statistical weights. The source functions for the two lines (which depend on the ratio n^/Vy?,) then become equal at each point in the atmosphere. (Here we ignore the inconsequential differences that may arise because v12 is not exactly equal to v13). In general, the actual situation will be intermediate between the two extremes described above, and we expect that the source functions will be equal only over a limited range in the atmosphere. Proceeding outward from the deepest layers of the atmosphere, we will find that the source functions in the lines begin to drop below the Planck 388 Non-LTE Line Transfer: The Multilevel Atom function at depths smaller than their thermalization depths; this occurs (cf. §11-2) because the escape probability begins to exceed the destruction probability. But if the collision rates C23 and C32 are nonzero, there is now a finite chance that the photon will be switched from one line to another, rather than be destroyed or escape, and there is now a conversion length (333; 284, 177) over which the source functions in the two lines are still coupled together. The lines become uncoupled only when the escape probability exceeds both the destruction and the conversion probabilities—i.e., when P^i2) * (C21 + C23)/(A21 + C21 + C23) (12-14a) and 13; > (C31 + C32)/(A31 + C 31 c32) (12-14b) From equations (12-14), we see that the two lines could thermalize independently over the entire range predicted by equations (12-13) only if, everywhere on that range, C3i > C32 and C21 > C23 (by a fair margin); if either of these conditions is violated, photon conversion plays an essential role. Again assuming Doppler profiles, we can estimate the optical depths from which photons in each line can retain their strict identity, and can emerge, without being either collisionally destroyed or converted, as and A*2 {A2i + C21 + C23)/(C21 + C23) A?3 * {A31 + C31 + C32)/(C31 + C32) (12-15a) (12-15b) in the 1 <->■ 2 and 1 <-> 3 transitions respectively. Let z%2 and zf3 be the geometric depths (from the surface) corresponding to Af2 and Af3. Then the two groups of photons may propagate independently only over a depth z* - min(z?2, zf3). From a mathematical point of view, whenever conversion is competetive with thermalization, the usefulness of the iteration procedure employed by the equivalent-two-level-atom approach becomes less clear. It then appears more attractive to consider the two (or more) lines, and their effects upon one another, simultaneously, and to develop a different computational technique. OBSERVATIONAL INDICATIONS OF SOURCE FUNCTION EQUALITY To motivate further the theoretical development, let us consider some of the observational evidence that source function equality in multiplets actually occurs. An excellent example is provided by an extensive set of precise observations of the sodium D-lines at various positions on the disk of the Sun (655). The emergent intensity at a specified frequency and 12-2 Effects of Level Coupling: Source Function Equality in Multiplets 389 disk-position is given by Jv(0, p) = p-1 J* [Sc(tc) + jJvS,(tc)] exp -p-1 pQc (1 + )f) dtc (12-16) where xc is the continuum optical depth scale, and />„ = Zi4>vllc- In the core of the line p\, » 1, so that, with good accuracy, we may write 7V(0, p) = p-1 J7 S,(t) exp(-T0v/A#v dx (12-17) where x is now the line optical depth scale. For the D-lines one has (fn/fn) — 2; thus we may write '12(0, p, v) = Jj* S12(t12) exp(-0vt12//i)((£v/ji) dx12 (12-18a) and /13(0, p, v) = S13(x12) exp(-2(hvxl2/p)(2 (C31iA431) and (C23iA421) > (C21/A21) (conditions that are met for the solar D-lines) to obtain a very near (though not exact) equality of the source functions. These conditions are sufficient to produce profiles that look identical to within the accuracy of the observations. / 2-2 Effects of Level Coupling: Source Function Equality in Multiplets 391 SOLUTION OF THE TRANSFER EQUATION IN MLJLTIPLETS Let us now consider how the coupled transfer problems for lines in a multiple!, arising from a single lower level to two (or more) upper levels that interact via collisions, are formulated and solved. For simplicity, stimulated emissions and coupling to the continuum will be neglected, and the differences in line frequencies ignored; these may, of course, be taken into account, but the algebra becomes, much more complicated without a compensating enhancement of the physical content. The source function in the (1 -> j) transition is Su = (2/7vV)fe1V'^iJ (12-19) The required population ratios are obtained from rate equations; for the case of only two upper levels these are: n2(A21 + C2l + C23) = nx{Bl2Jl2 + C12) + n3C32 (12-20a) and n3(A3l + C31 + C32) = n^B^J^ + C13) + n2C23 (12-20b) where 1 p. 2 dv /12 = C2i/A2U c]3 = CS]//431, and >/13 = C32//431, we may rewrite equations (12-20), in terms of source functions of the form of (12-19), as and 512 = (J Ii + ßi2ßv + fli2S13)/(l + cl2 + nn) (12-22a) 513 = (^i3 + fiia^v + Ti^/U + Ei 3 + r]13) (12-22b) These expressions display transparently the linear dependence between S12 and Su.Wesee that if 7712 = ni3 = 0, the source functions are uncoupled. But if either n12 » 1 or n13 » 1, then S12 -» S13. We see further, that even if the strong inequalities just mentioned are not satisfied, merely having '7ij » Kij and rjijSik » r.ljBv will imply a near-equality of 512 and S13, for then the physical source-sink terms in each source function will be dominated by the other line. The nlj terms are thus expected to influence the solution in a way reminiscent of the collisional versus photo ionization domination found in the two-level case (i.e., the lines in a multiplet may be "conversion-dominated1'). In view of the linear dependence discussed above, it is evident that a simultaneous solution for both source functions is mandatory. Let us 392 Non-LTE Line Transfer: The Multilevel Atom consider a single ground state and an arbitrary number of upper levels. Equations (12-22) can be replaced by the general form su = Vij + EfrfcSi* + 7j (12-23) k where the sum extends over all upper levels. This system can be solved (at each depth-point in the medium) to obtain an expression of the form Si/r) = EM^ikW + W (l2-24) k Exercise 12-2: Write explicit analytical expressions, of the form of equation (12-24), for the case of only two upper levels, starting from equations (12-22). To write the transfer equation we now adopt the optical depth xlt in some particular line (1 -> /) as standard, and write yj = (dxljfckli) = {fijfn). Then for each line (1 ->./') we have ji[rf/u(T, u, v)/dt] = v)[/:j-(r, u, v) - SJt)] (12-25) We introduce depth, angle, and frequency discretizations {zd}, {/;,„}, and {v„}, and appropriate quadrature sums required to perform the integrations indicated in equation (12-21). Then defining "j.dmn = "ljfo, /V V«) = \ Ulj(*d, + t*m> VJ + hjfrd, - fan, O] (12-26) and rewriting equation (12-21) as Jj.j - Jlj{*d0j, J2 ^j,dnw - ajk,dJj,a ~ bjy^j (12-28) Here <£/)rf„ = <£j(Tj, v„), a,-^ = a/k(td), bjd = b}{z4), and d = yj[Td) the sum extends over all L lines (or upper levels) considered. An equation of the form of (12-28) can be written for each line. To solve the system we may use the Rybicki method. Define Uj. ton — (Uj, lmm • ■ ■ ■> Uj,m„, to generate a final system of the form £ PifcJ; = Q;, 0 = 2,...,/.+ 1) (12-33) Here each Pjk is a (/J x D) matrix, so the whole system is of order LD x LD. The system is then solved by standard numerical methods, and yields the full depth-variation in all lines simultaneously. When stimulated emission terms are included, the system becomes nonlinear and requires an iterative solution. A linearization method for the problem is described in (23, 1); we shall not discuss this particular procedure, but instead will describe, in §12-3, a more general method that handles this problem and many others from the outset. The sodium D-lines have been studied (51), using schematic atomic and atmospheric models, with an integral-equation method essentially equivalent to the system described above. For these particular lines the following relations among atomic parameters exist: g1 = 2, g2 = 2, g3 = 4, A3l = A21, B13 = 2B12, C21 - C31; hence e12 - el3 and n12 = 2nl3. Adopt the 1 <-> 3 transition as standard so that y2 = \ and y3 = 1. Solutions were obtained for a range of typical values of the parameters s13 and n13, and deoth- variations of the Planck function. Results for cases with Bv = 1, b = 10~4, and n = r/u - 0, 10"4, 10"3, 10"2. 10"'. and 1, are shown in Figure 12-7. We see that, while r/, 3 would have to be larger than 1 to guarantee strict source function equality to the very surface, much smaller values of '?i3 yield equality for t £ 1, which is the significant range for determining the emergent intensity. The emergent specific intensity for p = 1.0, 0.8, and 0.6 is shown in Figure 12-8. Clearly the profiles agree very closely when n = 10~3, and are indistinguishable when n = 10~z. These calculations show that, for all practical purposes, source function equality can occur for if « 1; indeed this result is not surprising, for we would, in fact, expect I 394 395 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 ■v x FIGURE 12-8 Line profiles comparing Il2(0, i/i, v) with /13{u, ft, v) for various values of/;, at pi = 1, 0.8, and 0.6. Solid curves give /13, dashed curves give 712. Abscissa: displacement from line center in units of Doppler widths. From (51), by permission. r 396 Non-LTE Line Transfer: The Multilevel Atom conversion to dominate individual line-thermalization as soon as n » s. The effects of a schematic temperature gradient were also studied, and it was found that, when Bv(t) increases inward, the value of r\ required for source function equality decreases (compared to the Bv = 1 case), and increases when Bv(t) decreases inward. These results, however, may depend upon the specific forms chosen for Bv(t), and may not be general. 12-3 The Complete Linearization Method In the light of the discussion presented thus far in this chapter, it can be appreciated that the effects of interlocking in the multilevel line-formation problem are important, complex, and lead to systems of equations that may be ill-conditioned. Neither of the methods discussed thus far is entirely satisfactory for multilevel problems. The equivalent-two-level-atom formulation is ill-posed when there is strong coupling among lines. The method developed to treat multiplets is too specialized. What is needed is a method that is general, flexible, and computationally robust, and that handles the physical complexities of the problem. These requirements are met by the complete linearization method (42). In this method, the full set of rate equations is incorporated into the transfer equation, from the outset, by linearizing in terms of all occupation numbers and their dependences upon the radiation field. The rate equations themselves constitute the physical prescription of how the photons interact with the material, are created, destroyed, converted, degraded, and interlock within the collective photon pool envisioned by Jefferies. The transfer equations determine how this information is propagated from one depth to another. By means of the linearization procedure, a set of equations is developed that describes fully consistently (to first order) the response of the material, at every point in the atmosphere, to the radiation field at any frequency and any depth-point and, reciprocally, the response of the radiation field at all frequencies and all points in the medium to a change in material properties at any point. When this system is solved (iteratively), one has a simultaneous solution of the full set of rate equations and transfer equations that reflects both the global nature of the transfer problem (inherently large destruction/conversion lengths interplaying with an open boundary surface) and the intricate infrastructure of the statistical equilibrium equations. In practice, this method has provided a means of treating extremely complex model atoms with a very high degree of realism. We shall examine here the statistical equilibrium problem for a multilevel "impurity" atom that has no effect upon the structure of the atmosphere. We regard the model atmosphere as given—i.e., the temperature (T), electron density {nci mass density (p), and the total number density of the species under consideration (natom) are all specified, fixed, functions of depth. The atom is assumed to have L discrete levels, distributed, perhaps, over several 12-3 The Complete Linearization Method 397 ionization stages. The rate equations are formulated to describe all interactions among all levels, and to fix the total number density at the specified value naIom. These equations will be of the general form of the first A/Hc lines of the matrix displayed in §5-4, supplemented by an equation of the form Y,u,j)nij = nau.m- For ease of exposition we make the simplifying assumption that, at any chosen frequency v,„ there is only one transition (say / *-+ u) of the atom that can interact (bound-bound or bound-free) with the radiation field, and that all overlapping sources of opacity and emission are fixed. The transfer equation to be solved at each frequency v is (12-34) From equations (7-1) and (7-2) we can write the opacity and emissivity in a bound-bound transition as Xv = «Mln, - (0,/0„)nB] + Xv (12-35a) //,. = (2/n'V)ai>W0.X + Ev (12-35b) and that in a bound-free transition as Zv = Mv)[», - ns(«i/«J*c-*1''kr] + Xv (12-36a) l1v = (2hvyc2)aJv)nM/nure-^kT + Ev (12-36b) where Xv and Ev are fixed. To unify the notation, write Zv = aiH(v)[h, - GJvK] + Xv (12-37a) nv = (2/wV)««u(v)GlB(vK + Ev (l2-37b) where G,u(v) = (gjgj for bound-bound transitions, and Glu(v) = nc<$,(T) exp( - hvjkT) for bound-free transitions; here ,(T) is the Saha-Boltzmann factor for level / [cf equation (5-14)]. The system is discretized in depth \md}, (d — 1, ..., £>), and frequency {v,J, (fc = 1,. . ., K). We then can write a transfer equation of the form of equation (7-37) at each frequency vk; these equations are linearized to obtain an equation of the form of (7-39). where now §Xdk ant3 dVdk can De expressed simply as hdk = a/M(vjt)[<5rai, d ~ Glu(vk) dnu> rf] fytk = (2H3/c2)alH(vfc)Giu(vfc)^nH>(( (12-3 8a) (12-38b) The tin's can be expressed in terms of changes in the radiation field only, as T and ne are assumed fixed, and from equation (5-102) or (7-157) we can write K t?n k = i VlJJ kfd 12-39) 3l)s Non-LTE Line Transfer: The Multilevel Atom where analytical expressions for (5n/dJk) are given by equation (5-108). By means of equations (12-38) and (12-39), it is now easy to eliminate all the t>j?'s in the linearized transfer equation analytically, in terms of the 5J's only (33), and to obtain equations in the standard Feautrier form -A, + B„ 5J, - Cd 5Jd + 1 = L, (12-40) where SJ, = (5Jdl,SJdk,. . . , SJdK)T (12-41) and At = CD = 0. Exercise 12-3: Write explicit expressions for the elements of the matrices Ad, Bd, and Cd, and the vector Lj in equation (12-40). Tn the formulation of the linearization procedure described in §7-5, both the SJ's and dn"s appeared explicitly. By eliminating the drfs the system size has been reduced from [(K + L) x (K + L)] to (K x K), which saves computer storage; but now A and C are full, whereas in the earlier formulation [cf equation (7-159)] both A and C were diagonal (omitting the constraint of hydrostatic equilibrium which is not relevant in the present context). All of the operations in the elimination scheme given by equations (6-40) and (6-41) now involve full matrices, and the computing time therefore increases; the overall scaling is T = c DK3, compared to the older scaling T = d D(K + Lf, but c is much larger than d. The solution of equations (12-40) is carried out iteratively, as described below. The great power of the method is that equations (12-40) account for the full depth-to-depth and frequency-to-frequency coupling in the problem. That is, the effect of dJdk at vt) upon Jd,k. at all other (rd., vk.) is given consistently, and thus photon propagation within the collective pool is taken fully into account. In actual application of the method, we must now consider how to obtain a starting solution, and how to treat a large number of lines, as required by a realistic model atom. These questions are related, and the latter is of particular importance. For example, suppose the model atom has, say, 20 levels. On combinatorial grounds there could be of the order of 200 transitions; if we require, say, 10 frequencies per line profile, the problem becomes unmanageable. But of course spectroscopic selection rules severely limit the number of transitions that are actually possible in the spectrum, and among 20 levels one would typically have about 30 permitted lines. Of these, only a particular subset will directly influence those levels of primary interest in the treatment of definite limited set of lines in the spectrum, and usually it is possible to divide the transition array into a set of "primary" lines for which a strictly self-consistent treatment is necessary and a set of "secondary" lines for which a less precise treatment is adequate. This division is exploited in the initialization procedure. 12-3 The Complete Linearization Method 399 To initialize the problem, the model atmosphere is used to determine background opacities and emissivities, and the radiation field is computed, assuming LTE populations for the atom under study. This radiation field is used to calculate photoionization rates, and the statistical equilibrium equations are solved assuming radiative detailed balance in the lines, but not in the continua; in this way one obtains the correct asymptotic solution at depth. It is then possible to do an equivalent-two-level-atom solution to find Jlu in each line. The resulting j's allow the radiative rates in the lines to be computed, and the complete statistical equilibrium equations can then be solved to obtain improved level-populations. This process may be iterated, and may be extended to include the analogue of the equivalent-two-level-atom solution for the opaque continua as described in §7-5 [cf. equations (7-135) through (7-144)]. The process outlined here could, in principle, be iterated to consistency, and one would then have the method of §12-1; however, in the present context, only one or two iterations are performed, and the results are adopted as initial estimates of the n\ jv's, and Eddington factors required to start the linearization procedure. At this point, and henceforth, the radiative rates in the "secondary" lines are held fixed at their equivalent-two-level-atom values, and only the "primary" lines are treated explicitly in the linearization. This will provide an adequate treatment of the secondary lines if they have been chosen astutely, and if they (a) are very weak, and therefore dominated by overlapping absorption-emission processes, or (b) are, in fact, "isolated", and therefore accurately described by a two-level atom, or (c) are only eery weakly coupled to the lines of primary interest, even though they are members of, say, a mulliplet, and the two-level-atom approximation is poor. Starting from the initial solution just described, the linearized equations (12-40) are now solved for SJ at all depths. The resulting SJ's are used to obtain a more accurate estimate of the radiation field; the radiative rates are then updated with this revised field, and the statistical equilibrium equations are re-solved for new level populations. With the new occupation numbers, opacities and emissivities may be computed, and a formal solution (A-iteration) is then performed to obtain the radiation field and Eddington factors (this step also provides a smoothing of the solution). The convergence of the method is ordinarily very swift, yielding ||<$JV/JV|| ;£ 10~5 in 4 or 5 iterations, typically with order-of-magnitude improvement between successive iterations. This scheme is the one used to carry out most of the work discussed in §12-4. A detailed description of a particular version of the method, including a program adapted for calculations in the solar atmosphere, is given in (36). A difficulty with the Feautrier-type approach developed above is that it becomes costly for computations involving large numbers of frequencies. i 400 Non-LTE Line Transfer: The Multilevel Atom The problem can be overcome, however, with a new method (37) that uses a Rybicki-type elimination. Again we assume that at frequency vk there is only one transition (r), arising from levels / and u. At this particular frequency, the linearized transfer equation (7-39) can be written, in view of equations (12-38), as T,; 8J, + Lk 8n; + Uk Sn„ = R, (12-42) where each of the 5-vectors contains the depth variation of the quantity—e.g., SJ, =..........W (12-43) and the matrices Tft, Lfc5 and Vk are of dimension (D x D) and are tridiagonal. Equation (12-42) may be solved to obtain an equation of the form SJ, + &k Sn; + «Uk Sn„ = &k (12-44) wnere eg anc\ qf are now full matrices. Now the basic radiation-dependent quantities entering the rate equations are radiative rates integrated over the transitions in question. We therefore introduce the variations in the net rates, defined as (§Z()d = }\ddRlu_d - n^d8RuUd - X [4nwkaJvk)/hvk~][nLd - Glu(vk)nu^~\ 3Jdk (12-45) k where the sum extends only over those frequencies contained within transition t. If we substitute equations of the form of (12-44) into (12-45), and perform the indicated summations, we obtain finally 5Zr + At 8n, + B, 5n„ = Q (12-46) where A, and B, are full matrices. Exercise 12-4: (a) Write explicit expressions for the elements of the matrices TJ(, Lk, and Uk, and the vector Rk in equation (12-42). (b) Assuming that the matrices ££\, °lik, and the vector fflk are known, write explicit expressions for the elements of A„ B„ and C, in equation (12-46). From equations (5-108) and (12-39) it follows that one can write Snm = VD^SZ, (12-47) r where D„„ is a diagonal matrix with elements (D,^^ = (dnJdZt)d = {^d)mjl — (^)miS where is the unperturbed rate matrix at depth-point 3r3, which for, say, T ~ 20 and D — 50 becomes impractically large. The system is therefore solved by iteration, using a successive over-relaxation (SOR) method (526, 438). In this approach there are two basic iteration cycles: (a) the SOR iteration, to obtain a definite set of SZ/s (t = 1,.... T) within a given stage of linearization, and (b) the overall linearization procedure, where successive sets of 8Z/s are used to update rates, and the full statistical equilibrium equations are then re-solved. The SOR procedure is started by computing the solutions of the systems E, 8ZJ01 = C, (t = 1,..., T); this initial solution requires c D3T operations, and the resolved systems (equivalent to E,"1) are saved. Then with any set of current estimates of the SZ's, the righthand side of equation (12-48) can be evaluated for each transition in turn (note that only vector multiplications are involved, so the procedure is very fast); this yields a single vector of known value on the righthand side, and, using the previously resolved E„ a new value of SZ, is obtained. Each cycle in the SOR procedure requires c D2T2 operations, so if I iterations are necessary, the overall computing time scales as TSOR = c DiT + c'l D2T2, which is clearly favorable compared to Tn if I < DT. By actual tests it is found (37) that this method works well, even though the SOR iterations resemble the equivalent-two-level-atom approach, in that only one transition at a time is treated. The reason is that this part of the calculation is required only to determine the SZ's, which are merely one step of the overall linearization procedure (in itself designed to handle the interlocking problem self-consistently). Inasmuch as further steps in the linearization are presumed, the 8Z's at any given stage need not be known perfectly, but only with sufficient accuracy that the error in the current estimate of 8Zf is smaller than the full size of the 8Z£'s of the next linearization step. In practice, the requirement \dZ{'] — t>ZJ'-1)| < s^Zfj (where i denotes the SOR iteration number) works well with « set at about 10~3. Both of the methods described in this section have proven to be very effective in a wide variety of physical problems, for different atoms, in stellar atmospheres of various types; we now turn to a discussion of some of the results obtained for early-type stars. 12-4 Light-Element Spectra in Early-Type Stars From our study of the combined equations of transfer and statistical equilibrium, we have gained deep insight into the physics of spectral line-formation in stellar atmospheres. But it has also emerged that, when departures from LTE are taken into account, the equations to be solved become extremely intricate, and require special methods (which are computationally expensive). 402 Non-LTE Line Transfer: The Multilevel Atom It is therefore of considerable interest to answer the questions: "Do departures from LTE actually have a sensible effect in real stellar spectra?" "Are serious errors made in diagnostics of element abundances, or physical properties of the atmosphere, when non-LTE effects are ignored?" "Are there regions of the spectrum, portions of the H-R diagram, or classes of atoms and ions, for which we can conclude categorically that the use of an LTE approximation is safe, or is unsound?" From the practical point of view, it is important to know when it is adequate to assume LTE in the analysis of a stellar spectrum, for then the amount of computational work required is greatly reduced. Equally important is knowledge of when the results obtained from such analyses are not to be trusted. Answers to the questions posed above have come forth only quite recently as it became possible to obtain accurate numerical solutions of the non-LTE transfer equations, for elaborate multilevel atomic models in realistic model atmospheres, by applying on high-speed computers the techniques discussed in this chapter. It is now known that non-LTE effects are indeed of great consequence in several situations of basic interest and importance, and that a number of serious discrepancies between theory and observation are removed when departures from LTE are taken into account. A few of the results obtained thus far will be described below; further details can be found in the papers cited as well as reviews given in (433; 434: 435). It is not an exaggeration to state that work in this essential area has just begun, and much is yet to be learned from further efforts. In many respects, the most fundamental spectrum in early-type stellar atmospheres is that of hydrogen. Not only do the hydrogen-line profiles serve as effective tools to measure basic stellar parameters, such as the effective temperature and surface gravity, but also, when one recognizes that hydrogen dominates the transfer of radiation through the major part of the spectrum in early-type stars, one realizes that a serious discrepancy between theory and observation for these lines has grave implications for the overall properties of the radiation field. Although very good agreement is usually obtained between predicted and observed H-line profiles for B-and A-type stars (cf. §10-5), strong disagreement has been found for the O-stars (Tcff > 30,000°K). The basic problem is that the H-hnes are observed to have nearly-constant strength in the O-stars over the spectral range 09-05 [as can be seen by inspection of the plates in (465) or (5)], while the LTE equivalent widths decrease markedly over the corresponding temperature range (30,000DK ^ Tcii < 50,000°K), owing to increased hydrogen ionization. This decrease can be offset somewhat by assuming a higher surface gravity in the models, but then inconsistencies arise. Typically it is found (420) that, if a match is made between the observed H-line equivalent widths and those calculated from LTE models, then the surface gravities required to obtain the fit are too large (by about a factor of 3) compared to those deduced from fundamental measurements of stellar masses and radii, or 12-4 Light-Element Spectra in Early-Type Stars 403 from stellar-structure computations. Similarly, when detailed comparisons are made with line profiles (510), it is found that, if a fit is made to the line-wings, the line-core predicted by LTE is much too weak. These discrepancies vanish when departures from LTE are taken into account (436; 45). and it is found that the non-LTE profiles are invariably much stronger than the LTE profiles, as shown in Figure 12-9 for a typical model. A comparison between the observed equivalent widths of Hj8 and both LTE and non-LTE calculations is shown in Figure 12-10. It can easily be seen there that the non-LTE results are in much better agreement with observation, and that the LTE equivalent widths are systematically too small by factors of 3 to 5 in the extreme cases [see also (174)]. Further, an excellent fit can be obtained to observed profiles by using the non-LTE computations; this can be seen by comparing the fits displayed in (510) and (45), both of which use the same observational data. In sum, by taking into account departures from LTE, major improvements are obtained in matching observed H-line profiles for the O-stars. For the B-stars, the hydrogen lines are much less strongly affected by deviations from LTE, and for a line such as Hy, LTE actually provides a very good approximation. Some important effects are found for Ha (511; 430), where departures from LTE produce a deeper line-core, and a shallower wing than given by LTE. The latter effect arises because at the depths where the Ha-wing is formed, b3 > h2 (cf. §7-5) and therefore S23 > B(v22, T); the predicted changes have been confirmed by observation. After hydrogen, helium is the next most important element in stellar atmospheres, and is represented by the He I spectrum in B-stars, and by both the He I and He II spectra in O-stars. These lines may be used to 0.5 1—i—J—i—1_i_I_t_L_j_1_i_I ■ I 0 2 4 6 8 10 12 14 Ak figure 12-9 Ha profiles for O-slar models with 7cff = 45.000' K; curves are labeled with .surface gravity g. Abscissa gives displacement from line-center in A; ordinate gives residual flux. Solid curves: non-LTE profiles; dashed curves: LTE profiles. Note marked strengthening of line by non-LTE effects. From (45), by permission. 3T 404 BO 09.5 09 0807 06 05 30 35 40 45 50 Teff/103 1'IGURE 12-10 Hf> equivalent widths in O-stars. Solid curves: non-LTE calculations; dashed curves: LTE calculations. Dots: observed values (693). Ordinate gives equivalent width in A; abscissae give (TefF/103) and spectral types. estimate the helium abundance N(He)/A7(H), and in the O-stars, the ratio of strengths of the He II lines to He I lines provides a sensitive temperature-indicator. LTE calculations of He I lines in B-stars have produced excellent agreement with the observations for the blue-violet lines [see, e.g. (390; 480; 481)], especially when the most accurate line-broadening theories available are used (438; 439). However, lines in the yellow-red regions of the spectrum [e.g., A5876 (2p 3P-3d 3D) and 26678 (2p lP-sd lD)~] are always found to be stronger in B-star spectra than given by LTE computations, using the same model and abundance that fit the blue-violet lines. Formally these lines indicate abundances almost three times larger than that given by the blue-violet lines. When a detailed non-LTE calculation is made (46), using a realistic atomic model (see Figure 12-11), it is found that the lines mentioned above are strengthened dramatically, and that good agreement with observation is obtained, as shown in Figure 12-12 for A6678. It is interesting that the largest non-LTE effects are found for the lines with the smallest values of hv, for, as mentioned in §5-3, these lines, with relatively small values of hv/kT, have collisional rates that are comparable to the radiative rates, and the classical argument is that these lines should therefore be in LTE. The reason for the large effects can be seen by examining the source function in the limit 5 = hv/kT < 1. If we write (b,/Z>„) = (1 + /?), then (SJBV) = (e>™lkT - mhlK)e^kT - 1] = (/ - + jg) - 1] * «5/(/J + 8) = (I + 405 'S 1P° yD *F° 3S 3P° 3D 3F° FIGURE 12-11 Model He I atom used in non-LTE calculation of He I spectrum. Ordinate gives wavenumber in cm"1 times 10"i. All the transitions shown were included in the calculation explicitly as "primary" transitions in the sense defined in §12-3; all other dipole-permitted lines up through n = 10 were treated as "secondary" transitions, with rates fixed by an equivalcnt-two-levcl-atom computation. The singlets and triplets are treated separately. From (46), by permission. which shows that a given fractional departure fi of the level-population ratio is amplified by a factor 8 ~1 when 8 < 1. The occupation numbers, and hence the ratio (b}/bu) will, in general, be determined by processes other than those in the line. In the O- and B-stars the level populations are set by photo-ionizations and recombinations, and thus /? will generally be nonzero despite a large collisional rate in a line. Once /? ^ 0, large departures of 5, from Bv become possible; the same argument, by the way, explains why even tiny departures from LTE have very large effects in the radio-frequency recombination lines observed in nebulae (199). In the O-stars, LTE predicts too-weak lines for both He I and He 11. If one attempts to match the observed equivalent widths, without regard to line-profiles (420), the derived helium abundances are about a factor of two higher than the value obtained from B-stars, or from the nebulae in the interstellar medium from which the O-stars have just formed. If a fit is made to the wings of the line profiles (510), it is again found that the line cores are much too weak. On the other hand, a non-LTE calculation (45) yields excellent agreement with observed line strengths and profiles at the "standard" abundance JV(He)/iV(H) = 0.1 [see also (174; 175)]. Departures from LTE on occasion introduce large errors into abundance estimates based on LTE calculations. For example, LTE determinations of T 406 AX fk; url 12-12 Comparison of observed (390) He 1 ,16678 profiles (dots) wilh non-LTE (solid curves) and LTE (dashed curves) calculations for three B-stars. Ordinate: residual flux; abscissa: displacement from line-cenlcr in A. From (46), by permission. the abundance of Mg in O-stars are in error by a factor of 10 or more (431), and only when a non-LTE calculation is made, is a fit obtained to the observations, with an abundance near the solar (and accepted "cosmic") value, as shown in Figure 12-13 [see also (384; 586)]. Similarly, LTE analyses of the spectrum of Ne I in B-stars have routinely given a neon abundance of 5 x 10~4 relative to hydrogen, in disagreement with the value 10~"4 obtained from nebulae, the solar corona, the solar wind, and cosmic rays (both galactic and solar-produced). Again the discrepancy is removed when a non-LTE calculation is done (47) and it is found that the Ne I spectrum, consisting primarily of lines in the red, is affected by essentially the same mechanism as that described above for He I. Finally, the spectra of Si III and Si IV in B- and O-stars show significant non-LTE effects (349). Departures from LTE increase the computed line-strengths by 50 to 70 percent, and line-core intensities decrease by a factor of 0.6 relative tq LTE. To 407 T 15 17.5 20 22.5 25 27.5 30 32.5 35 37.5 40 l1 It jure 12-13 Strength of Mg II ;.4481 in B- and O-siars. Ordinate: equivalent width in A; abscissa: (Teff/103). Squares: observed values at the indicated spectral types. Dotted curves: LTE calculations at a solar or ten-times solar Mg abundance, with zero microturbulence. Dashed curve: solar Mg abundance, and 4 km s"! microturbulence. Solid curves: non-LTE calculation at solar Mg abundance, and zero (open circles) or 4 km s"' (filled circles) microturbulence. Note that for the O-slars the choice of microturbulent velocity does not influence the result, and thi.it the LTE predictions yield much too small a line-strength, while the non-LTE results agree well with the observations. produce lines of similar strength in LTE, an abundance 5 to 7 times the solar value would be required. In short, there is ample evidence that, at least for early-type stars, deviations from LTE may produce large and important changes in abundance estimates. Yet another context in which non-LTE effects enter preeminently in early-type stellar spectra is in the production of emission lines. For example, in the Of stars, the lines of He II A4686, C III 25696, and N IN A/4634-41 are seen in emission, while other lines of the same ionic spectra are in absorption. In the case of He II /4686, studies have been made of selective excitation mechanisms (45) and the effects of atmospheric extension in static models (376); while weak emission lines have been obtained in such work, it is clear that a quantitative fit to the observed line-strengths will be obtained only when large-scale atmospheric expansion is taken into account (see Chapters 14 and 15). The C III line has not yet been subjected to analysis. For the N III lines there exists a subgroup of the Of stars, designated 0((f)), in which the N III lines are weakly in emission and the He II line is in absorption (657); these appear to be near-main-sequence objects in which the effects of atmospheric extension and expansion, if present at all, may be ignored. 408 Non-LTE Line Transfer: The Multilevel Atom A detailed non-LTE study of the N III doublet spectrum (429; 440: 443; 433) gives a plausible explanation of how the A/.4634-41 lines from the 3p 2p _^ 2d 2D transition can be in emission, while the next lower multiplet, .U4097, 4103 (3s 2S -»■ 3p 2P) is in absorption. The two essential ingredients required to produce this result are (a) a mechanism for populating the 3c/ state with enough electrons to induce the 3d -*■ 3p emission, and (b) a way of depopulating the 3p state so that emission does not appear in the 3p 3s transition. Both the 3s and 3c/ states can be populated directly by transitions from the ground state (see Figure 12-14) and, when the ultraviolet radiation field at A374 and X452 is intense, these states will have large occupation numbers. In static atmospheres, however, the resonance lines come into detailed balance, and therefore do not lead to a very large overpopulation of the 3d state. Rather, the overpopulation results from dielectronic recombination from the 2s2p{1P)3d state—which, for the N+ + ion, just happens to lie barely above the ionization limit. Stabilization of this state occurs when the 2p electron drops to the 2s level, and proceeds at a very rapid rate, feeding electrons directly into the 3d state. These electrons then decay 3d -» 3p, producing the emission. Further, it turns out that, because of the particular structure of the N++ ion, the most probable route of exit from 3p is not 3p -> 3s, but rather via "two-electron jumps'1 of the form 2s23p -»■ 2s2p2 (see Figure 12-14): the latter process occurs so efficiently that it drains the 3p state, leaving the 3s -> 3p line in absorption. Inasmuch as the /U4634- E(cm_1)/103 FIGURIi 12-14 Simplified term diagram for the lower stales of N+ "\ as well as the autoionizing state 2s2p(1P)3t/ from which dielectronic recombinations can occur directly into the 2sz3d stale, leading to emission at /./.4f>34-40. 12-4 Light-Element Spectra in Early-Type Stars 409 41 emission lines can be produced in planar, static atmospheres, solely as a result of the atomic structure of the ion responsible, they must be regarded as intrinsic emission lines. In expanding atmospheres, direct pumping of the 3s and 3d states will occur in the 2p -+ 3s and 2p 3d transitions, because the resonance lines will be Doppler shifted into the bright adjacent continuum and will not be in detailed balance. This greatly enhances the 3d -> 3p emission (thus explaining the very bright emission seen in the Of stars, which are known to have expanding envelopes and stellar winds), while the increased 3s population, coupled with the drain from 3p, assures that the 3s -» 3p lines remain in absorption. The calculations show (see Figure 12-15) that the -U4634-40 lines make the transition from absorption to emission -0.5 1—1--'-1—--j-1_i__._i_l_ 32.5 35 37.5 40 42.5 45 47.5 50 Teff/103 ficurl 12-15 N III linc-strcngths for the 3s - ip (/.4097) and 3p - 3d {U4634- 40) lines in planar, static, model atmospheres. Ordinate: equivalent width (negative values denote emission); abscissa: (Ttfl-/103). Nole that the A4097 line remains in absorption while /./.4634-40 make a transition from absorption to emission al a spectral type near 06. From (443) by permission. 410 N on-LTE Line Transfer: The Multilevel Atom at about spectral type 06 near the main sequence, in agreement with observation (443). Many more analyses, for a wide variety of atoms and stellar atmospheres, need yet to be done (particularly for later spectral types) before the questions posed at the beginning of this section can be answered fully, and efforts in this direction will be richly rewarded with interesting results. Line Formation with Partial Frequency Redistribution In the process of scattering in spectral line-formation, an atom is excited from one bound level to another by the absorption of a photon, and then decays radiatively back to the original state, with the emission of a photon. In our work on line-transfer we have, thus far, assumed that scattering is either strictly coherent, or that the photons are completely redistributed over the line profile. Neither of these limits is achieved exactly in stellar atmospheres, and it is necessary to consider the redistribution of photons, in angle and frequency, in some detail, and to calculate redistribution functions, which describe the scattering process precisely. This calculation proceeds in two steps. We first consider a single atom in its own frame of reference, and compute the form of any redistribution that occurs within the substructure of the bound states. Then, recognizing that what is actually observed in a stellar atmosphere is an entire ensemble of atoms moving with a thermal velocity distribution, we take into account the Doppler redistribution in frequency produced by the atoms' motion. Doppler redistribution arises because the incident and emergent photons travel, in general, in different directions; in this event the projection of the atom's velocity vector along the propagation vectors will be different for the two photons, and a differential 412 Line Formation with Partial Frequency Redistribution 13-1 Redistribution in the Atom's Frame 413 Doppler shift must occur. The final redistribution function is obtained by averaging over all possible velocities. Once the redistribution function is known, a transfer calculation can be carried out, taking into full account any correlations (or lack thereof) between incoming and outgoing photon frequencies. In this chapter the notation and methods of Hummer (313) will be adopted; the reader is referred to this paper for further details and other results. An interesting analysis of the problem from a rather different conceptual point of view can be found in (295) and (296). 13-1 Redistribution in the Atom's Frame Let us first consider the nature of redistribution in the rest frame of the atom. Let the frequency displacement from line center of the incoming photon, measured in the atoms frame, be denoted by tf its direction by n', and the frequency displacement and direction of the outgoing photon by Č, and n. Assume that the material has no preferred directions on an atomic scale so that the atomic absorption profile/(£') is isotropic;/ is normalized such that 1*«, fičf) dč,' - 1. Further, suppose that Vac frequency redistribution function p( Č) gives the probability that a photon absorbed in the frequency range ;->■ /. Further, by inspection of the denominators in equation (13-5), it is easy to see that for a given value of £, there are two relative maxima, one with = £ and the other with = c"0- These can be understood physically as follows. (1) Most absorptions will arise from the center of the lower level; when these excite a particular substate of the upper state, the decay occurs most often back to the center of the lower level. It is clear that £ will then equal 4', and that this process occurs with a relatively high probability. (2) Alternatively, there is a very high probability that an atom will be excited from the center of the lower state to the center of the upper by photons with % = £0; the upper state may then decay to any arbitrary substate of the lower, in particular, yielding the frequency q. The inverse also occurs: a transition from an arbitrary substate of the lower level has high probability of exciting the center substate of the upper level, and the most probable decay is back to the center of the lower level. Thus there will be a peak in emission at line center (i.e., at £ = <;0), and indeed one sees that there is such a peak in equation (13-5). The laboratory-frame redistribution function corresponding to equation (13-5) has not appeared in the literature [owing to use of the erroneous result of Heitler], though it could be derived straightaway (the calculation would be tedious, however). Moreover, the formulation of a correct transfer equation for subordinate-line redistribution is quite complicated; we shall not, therefore, consider Case IV further in what follows. Of the four cases defined above, Cases II and III are of the greatest astro-physical importance. In fact, neither of these extreme limits is attained, and 13-2 Doppler-Shifi Redistribution 415 the typical situation of interest for resonance-line formation is one in which the upper state is broadened by both radiation damping and elastic collisions, with damping-widths 5R and Sc respectively. In this case, one would expect the line profile to be given by equation (13-3) with 6 = 3R + <5C. Of the atoms in the upper state, a fraction y = 3R/(5R + 3C) would be expected to decay radiatively. and hence to emit coherently in the atom's frame (recall the lower state is perfectly sharp). The remaining fraction (1 — y) = 8c/(dR + Sc) would have suffered collisions, and would be expected to have been completely redistributed. Thus we could now write pit', c) = y > n) d% d^(doj'/4n)(dco/4n). Transforming this expression to the laboratory frame via equations (13-7) we can write R0{v\ ri; v, n) = f(v' - v0v ■ rijc)p{v' - v0v ■ n'/c, v - v0v ■ n/c)g(ri, n) (13-8) where the subscript v implies that the redistribution is produced by an atom of velocity v. To find the net result for the entire ensemble of atoms, we must average over the velocity distribution, which is assumed to be Maxwellian. To perform this average, introduce an orthogonal triad of reference axes (nl5 n2, n3), with nL and n2 chosen to be coplanar with n' and n, and with nx bisecting the angle 0 between them (see Figure 13-1). Then we may write and an1 jSn2 ßn2 (13-9) (13-10) figure 13-1 Coordinate axes used in calculation of redistribution function. The vectors n1. n2, n, and n' are coplanar. The vector n, bisects the angle 0, (0 < 0 n), between n' and n. J3-2 DopplerShift Redistribution 417 Resolve v along these axes, and write vt = v ■ n,; for convenience express velocities in dimensionless thermal units u = v/u, = {mJlkTfy (13-11) where mA is the mass of an atom, and introduce the Doppler width w = (v0/c)(2kTfmA)> = v0dwrmJc) (13-12) Then the Maxwellian velocity distribution [equation (5-1)] is P(wl5 u2, u3) du± du2 du3 = exp[ — (u12 + u22 + u32)~] dux du2 du3 (13-13) We now average equation (13-8) against the velocity distribution of equation (13-13) to obtain R(v\ ri; v, n) = du. du2 I du3 P{uu w2, u3)jR„(v', n'; v, n) = n !a(n', n) du, e Wl du2 J - --r, j - oc x e~U2lf\y' - wfai*! + pu2j] x p[y' - w(au} + fiu2\ v - w{ocui - pu2)] (13-14) where the integration over uz has been carried out explicitly. An alternative form for R, which will prove useful, can be derived by choosing n: to lie along n'. Then v ri = vu while v ■ n = i\ cos 0 + v2 sin 0 = yJv1 + fi'v2 (13-15) We then find R(v', ri: v, n) = n~1g{n\ n) duy e~"l2f(v' — wu^ x j" du2 e~"22p[y' — wuu v — w(a'u1 + P'u2j] (13-16) With the aid of these general formulae, we can now compute the redistribution functions for the various cases defined in §13-1. Note first, however, that in both Cases 1 and IT the scattering is coherent in the atom's frame, so that p(C', £) = ö{c' -0 = 8{v - V + 2wßu2 (13-17) In these cases h5 no longer appears explicitly in the expression for p, and the integrations written above are simplified. Substituting equation (13-17) into (13-14), the integration over u2 can now be performed explicitly (transforming to the variable z = 2wfiu2 in order to preserve normalization of the Dirac 418 Line Formation with Partial Frequency Redistribution S-function). We then obtain, for /? ^ 0, ^■(v'n;v-,,) = ^7exp (V - V')2" 4ß2wz x j"^ e""2/ü(v + v') - waw] du (13-18) while if ß = 0 we find Rc(v', n'; v, n) - 7i"^/(n', n) <5(v' - v) j*^ e~"2f(v' - wu) du (13-19) Here the subscript c denotes coherence in the atom's frame. Exercise 13-1: Verify equations (13-18) and (13-19). Finally, we note that the results can be written in a concise and convenient form by expressing frequency displacements from line center in Doppler units: x = (v - v0)/w (13-20) A' = V — Vn) W (13-21) and and writing R(x', ri; x, n) = R(v\ n'; v, n)(dv'/dx')(dvfd.x) = w2R(v', ri; v, n) (13-22) to yield normalization of R(x', ri; x, n) when integrated over x' and x. RESULTS FOR SPECIFIC CASES (a) Case I. If we substitute /(£') = - v0) into equation (13-18), we obtain immediately R,(v', n';v, n) = -—exp -(v - v'Y 4ß2w2 e.\p -(v + v' - 2v0)} 4a2w2 (13-23) Now noting that laß = 2 sin(|0) cos(|0) = sin 0, that a2 + ß2 = 1, and transforming to dimensionless units via equations (13-20) through (13-22), we have Rj(x', ri; x, n) = [g(ri, ii)/jt sin 0] exp[-x2 - (x' - x cos 0)2 esc2 0] (13-24) Exercise 13-2: Fill in the steps required to obtain equation (13-24). 13-2 Doppler-Shifi Redistribution 419 From this result, we immediately notice that, even though the scattering in the atom's frame was presumed to he strictly coherent, there is significant redistribution in the laboratory frame. This is a point of very great physical significance, as was stressed by Thomas (622). He examined the frequency dependence of scattered radiation [cf. equation (2-9)] using several simple analytical expressions for the incident intensity, and showed that in the Doppler core (x < 3), over which the absorption profile varies by a factor of 104, the emission profile departs from the absorption profile by less than a factor of four! Thus, within the Doppler core, the assumption of complete redistribution of the scattered radiation provides an excellent approximation (it will be easy to see this using the angle-averaged redistribution functions derived in §13-2; see also Figure 13-2). This result provided much of the motivation for assuming complete noncoherence in the treatment of line formation, as was done in Chapters 11 and 12. Outside the line core, complete redistribution is not achieved as closely, and one must use the correct redistribution function. For example, in the case considered here, a simple physical argument (597) shows that the radiation in the far line-wings is about two-thirds noncoherent, and one-third coherent. On the other hand, outside the Doppler core the line-wings will often be swamped by the continuum (unless the profile has a large damping parameter), in which event the details of redistribution become less significant. (b) Case II. Here we substitute equation (13-3) into (13-18) to obtain n / / / \ ft"'' Ru(v, n ; v, n = —exp ZKCf.ßw -(v - V')' 4ßzw2 / + v — 2v0 2") exp[-4-(.v - x1)2 esc2 £©] x'J sec ^0] n sin 0 x H[a sec yO, j(x (13-26) where a = (5/w). Although this result is relatively complicated, efficient methods to evaluate H(a, v) exist, and Rn can be calculated fairly easily. As will emerge in §13-3, Ru simulates nearly complete redistribution in the line core (x < 2.5), but becomes nearly coherent in the line wings. Exercise 13-3 : Fill in the missing steps leading to equation (13-26). 420 Line Formation with Partial Frequency Redistribution (c) Case 111. Here, as was noted in equation (13-4). p(c', is independent of c'. In this case equation (13-16) is particularly useful, and substituting from equations (13-3) and (13-4) we find Rmiv', n'; v, n) = g{n', n) S2"l - 1 du2 e~»-2 (?) {[v - w(u, cos 0 + u2 sin 0) _ v0]2 + 62}-1 (13-27) Converting to dimensionless units, and again using the definition of the Voigt function, we find Rm(x'. n'; x, n) = g{n\ nl e "2H{(7, x esc 0 — u ctn 0) (x' - u)1 + a2 du (13-28) where a = a esc 0 and a = {S/w). This result is no longer expressible in terms of simple functions, and must be evaluated by numerical integrations (529). Exercise 13-4: Fill in the steps required to obtain equation (13-28). Exercise 13-5: In the limit that Compton scattering effects arc negligible (i.e., hv/mc2 « 1 where m is the mass of an electron), the scattering of radiation by electrons is coherent, p(£, <) = HC - c), and grey, /(# = — n' changes the sign of a and /? in the expressions for incoming photons, hence if g(n\ n) is an even function of ri ■ n, R{~x\ -n';x, n) = w2%~ lg(n', n) j**^ dut e'1"2 duz x c~"2J[w(-x' + + lhi2)]p\_w{-x' + auj + ft<2), w(x - aui + ft/2V] (13-36) which will reduce to (13-29) in certain cases. In particular, the integrals will be equal if /(£') = 8(?) or if J(?)P(Z\ Z) = f{-Op{~£, C)- The former is true for Case I, while the latter is true for Case III where /(<:') = f {-£,') 422 Line Formation with Partial Frequency Redistribution 13-3 Angle-Averaged Redistribution Functions 423 and píč', č) = fič). Hence we conclude that /?,.(-x', -n';x,n) = R;(-x',n'; x, -n) = R^x', n': x, n), (i = I, III) (13-37) where the second identity follows from equation (13-33). applications The solution of the transfer equation allowing for full angle and frequency redistribution is fairly difficult, and requires techniques more powerful than those discussed in this book; both Monte Carlo methods (31; 50) and difference-equation methods (460) have been developed to handle the problem. There are a number of situations where both angular and frequency effects may become important. For example, the escape of La photons from a very thick nebula depends sensitively upon the details of the scattering process, and a complete, careful treatment is required (31). In addition, when the material has a macroscopic velocity, both the absorption and emission coefficients become angle-dependent in the observer's rest frame (cf §2-]). There then results an inextricable coupling between angles and frequencies in the transfer problem (see §14-1), and the details of the redistribution process may be of consequence; in this book, however, we shall consider only the case of complete redistribution in moving media, and will not pursue this point further. Finally, when observable angle-dependent information is available (e.g., center-to-limb variations as measured in the solar atmosphere), the possible importance of angular effects in the redistribution process must be examined. The investigations of this question conducted thus far (640; 641; 460) show that, at least for semi-infinite plane-parallel atmospheres with homogeneous layers, the differences between results obtained using angle-averaged redistribution functions and those obtained using the full angle-dependent functions are negligible. On the other hand, for optically thin media or small-scale structures (e.g., chromospheric hue-structure, spicules, etc.) the angular effects are often quite important (186), and should be considered in detail. However, these questions lie beyond the scope of this book, and will not be discussed here; rather, we now turn to the derivation of angle-averaged redistribution functions, and then to their use in radiative transfer calculations. 13-3 Angle-Averaged Redistribution Functions As was noted in §2-1, if we assume that the radiation at any particular point in the atmosphere is essentially isotropic, then the contribution to the emissivity from scattering processes can be written in the form ns(r, v) - ff(r) R(v', v)Jv, dv' (13-3S) where Jv is the mean intensity and R(v', v) is the angle-averaged redistribution function R{v', v) = (4k)-1 j) R(V, n'; v, n) dm' (13-39) The rationale for this approach is essentially as follows. The radiation field will depart significantly from isotropy at any given frequency only at points whose optical depths tv from the surface are of order unity or less; at depths zv > 1, the radiation field is essentially isotropic. Now a basic characteristic of non-LTE line-formation that emerged clearly in Chapter 11 is that the surface value of the source function is determined by photons contributed from over an entire destruction length, and over virtually all of this region the radiation field will, in fact, be isotropic. We may expect, therefore, that even at the surface where I(u, v) shows departures from isotropy, Sv will still have a value already fixed by processes occurring at depths where the anisotropy is negligible; hence the value of J(tv = 0, v) computed from this Sv should be quite accurate. On the other hand, use of the angle-averaged functions accounts completely for the frequency-reshuffling of scattered photons, the action of which (as we saw in Chapter 11) affects crucially the photon escape-probability, and hence the thermal-ization process. In short, in this approach we account for the critical aspects of the redistribution process, and sacrifice information only in an area of secondary importance. The functions defined by equation (13-39) are normalized such that (aj dv f " dv' R{v', v) = 1 (13-40) J - oc j -vj Integration over all emitted photons yields the absorption profile dv' R(v\ v) dv = (v')ö[v - v') R[v, v) = in equation (13-46) is trivial, we have Ru(v', v) = ^ dp' f(v' - wp'u) j*^ dpg{p\ p)p(y' - wp'u, v - wpu) (13-48) For simplicity, in what follows only isotropic scattering in the atom's frame will be considered: in the case y(p, p) = \ [formulae for a dipole phase function are given in (313)]. Applying this restriction we have R-a, x(v'> v) = i J.11 dl1' f(v' ~ wß'u) 1 dp p(v' - wp'u, v - wpu) (13-49) Although equation (13-49) is general, it is not convenient in the case of coherent scattering in the atom's frame because of complications that arise in setting the limits of integration; let us, therefore, derive a more refined formula for this case. If the scattering is coherent in the atomic frame, p(y' — wp'u, v — wpu) = <5[v' — v — \vu(p! — pf\ (13-50) Because the range of integration for p and p is only (—1, 1), it is clear that for a given value of u, the singularity of the <5-tunction will be outside the range of integration for sufficiently large values of |v' — v|, and RA,u(y\ v) will, accordingly, be zero. Physically this corresponds to the fact that an atom moving with velocity u can change a photon's frequency by no more than 2uw, this maximum shift occurring if the propagation vectors of the incoming and outgoing photons lie along the velocity vector and are oppositely directed. We substitute equation (13-50) into (13-49), and consider first the integration over p. Let y = wpu, and write I = (wu)"1 Jn<>[v - (v - V + wup')] dy (13-51) The integral will equal (1/ww) if — wu ^ v — v' -f wup' ^ wu, and will be zero otherwise. Define A(x) such that A = 1 if — l^x^l, and A = 0 426 Line Formation with Partial Frequency Redistribution 13-3 Angle-Averaged Redistribution Functions All otherwise. Then equation (13-49) can be rewritten, using equation (13-51), as Ra,u{v\ v) = (4wm)"1 j\ /(V - wupt')A[pi' + (wuyHv - v')] dpi' (13-52) If u is sufficiently small, then |(v — v')/wu\ > 1 and A will vanish for all values of pi'. Thus there is a minimum speed umin, whose value we must determine, for which scattering from V to v can actually occur. Define v = max(v', v)andv = minfv', v). First suppose that v > v'; then the requirement that the argument of the A-function fall in the range ( — 1,1) implies that [(v - v')/wu] - 1 = [(v - v)/ww] -1^1; this inequality yields «™in = (v - v)/2w = |v - v'|/2w (13-53) But the same result is obtained by similar reasoning if we assume v' > v; therefore equation (13-53) is general. For u < wmin, Ru will be zero. For u > umi!l, a contribution to Ru will come from part of the range of integration over pi'. To determine this range, we suppose, for definiteness, that v > v'. Then a contribution is obtained when -1 ^ pi' ^ 1 - [(v - V)/wu] = 1 - [(v - v)/wu] which implies that v — wu ^ y — wupi' ^ v + wu; recalling that v = v' by hypothesis, the result just stated can be rewritten as v — wu ^ v' — wupi' ^ v + wu (13-54) If we assume instead that v' > v, we again obtain equation (13-54), which is, therefore, general. Now introducing the Heaviside function (x, x0), defined such that O — 1 when x > x0 and 0 = 0 when x < x0, making the substitution y = v' — wupi', and using equations (13-53) and (13-54), equation (13-52) may finally be written as Ra.u« v) = (4w2a2)-10(M - |v - v'|/2w, 0) £*™ f(y) dy (13-55) Finally, the results expressed in equations (13-49) and (13-55) must be averaged over the Maxwellian velocity distribution P{u) du = 7i" V"2(4™2) du (13-56) From equation (13-55) we then have for coherence in the atom's frame (Cases I and II) RA(v',v) = {niw2y1 f°° due-"2 Q__W\f(y)dy (13-57) From equation (13-49) we have, for noncoherence in the atom's frame (e.g., Case III), RA(y', v) = 7r_i ^ du u2e~"2 dpi' f(v' - wpi'u) x dpip(v' — wpi'u, v — wpiu) (13-58) Finally it is convenient to use Doppler units [cf. equations (13-12), (13-20), and (13-21)] and write RA(x', x) = RA(V, v)(dv'/dx')(dv/dx) = w2RA(V, v) (13-59) RESULTS FOR SPECIFIC CASES (a) Case I. Here/(y) = d(y - v0); hence the integral over y in equation (13-57) is nonzero only if v + wu > v0 > v - wu. This implies that wmin now becomes effectively u'min = max(jx'[, |x|), which clearly satisfies the inequality Wmin 5s umin as given by equation (13-53). Then from equations (13-57) and (13-59) Ri,a&> x) = Xl e~u2 du = \ erfc(wmin) (13-60) where the complimentary error function is defined as erfc(x) = In"" £° e~z2 dz (13-61) Substituting for u'min, Ri,a(x',x) = \ erfc[max(|4 |x'|)] (13-62) This redistribution function is easy to compute from well-known approximation formulae for erfc(x) (4, 299); asymptotic formulae and results for dipole scattering are given in (313). A plot of Ri!A(x', x)j(p{x') is shown in Figure 13-2; the curves are labeled with the incoming photon frequency x', and give the probability for emission at frequency x, per absorption, as a function of x. We see that a photon absorbed at frequency x' will be emitted with equal probability at all x such that - |x'| < x ^ |x'|, and with exponentially decreasing probability beyond this range. It is easy to understand this result. The absorption can occur only when c;' = 0 in the atom's frame. Therefore atoms absorbing at frequency x' in the laboratory frame have a velocity of at least x' Doppler units. As the emissions occur with equal probability in all directions, the photons can be redistributed with equal probability over the entire range ±x'. 428 429 1.0 ■3- 0.5 ~y. 0 -4 1 1 1 1 1 [ 1 1 ! 1 1 1 1 A x' = 0 1 1 - / V1 - / / \ \ jc' = 2 / / \ \ \ x' - 3 Ji yy/\ , i , i^x>^^\ iV, -3 -2 -1 figure 13-2 Probability of emission at frequency x, per absorption at frequency x', for pure Doppler redistribution. Ordinate: R^x', x)fcb(x'); abscissa: emission frequency x. Curves are labeled with frequency x' of absorbed photon. (b) Case II. Substituting equation (13-3) into (13-57), we have *n,>', v) = (wVr1 ö £n du e-"2 dy [(y - v0)2 + ci2]"1 (13-63) Converting to Doppler units, writing z = (y — v0)/5 = x/a, and writing umin = j\x — x'\ from equation (13-53) we have R-ii, a(x', x) = ti 3 due-"2 [lx-+HV"dz{l + z2)~X = 71 x'-x\/2 tan" x + u — tan" x — u du (13-64) where x ^ max(|x|, |x'|) and x = min(jx|, jx'|). The asymptotic behavior of Rn, and results for dipole scattering, are given in (313); an accurate method for the evaluation of Rn is given in (6). The redistribution function Ru is of great interest, for it describes the important case of scattering by a resonance line that is broadened by radiation damping, and it has been extensively studied. A plot of Ru A(x', x)/(p(x') for a = 10~3 is shown in Figure 13-3; again, the curves are labeled with the incoming photon frequency x', and give the probability, per absorption, of a subsequent emission at frequency x. Here we see that for small x' (x' < 3), the curves resemble those for Rh because most of the figure 13-3 Probability of emission at frequency x, per absorption at frequency x', for Doppler redistribution from a coherently scattering (in the atom's frame) Lorcntz profile with a ^ 1()~3. Ordinate: Rj[(x', x)/tp(x'); abscissa: emission frequency x. Curves are labeled with frequency x' of absorbed photon. emissions are from atoms absorbing at line center and moving with velocities near x'. For large x' there will be very few atoms with high-enough velocity to absorb at line center, and most emissions then come from atoms moving with low velocities and absorbing in the line-wing. As the scattering process is coherent in the atom's frame, and the appropriate atoms are nearly at rest in the laboratory frame, the scattering will be nearly coherent in the laboratory frame as well. Thus in the line-core there is Doppler redistribution and strong noncoherence, while in the wing the scattering is more nearly coherent; this dichotomy has important implications for the line-transfer problem, as we shall see in §13-4. In early work by Jefferies and White (338), it was suggested that a simple approximation for Rn could be written in the form RlltA(x', x) w [1 - a(x)]{x) + a{x)(p(x) d(x - x') (13-65) where a(x) is nearly zero for x < 3 and approximately unity for x > 3. The original suggestion for the form of a(x) is not adequate, however, and fails to meet requirements of normalization, symmetry, and wing-coherence. Nevertheless it is easy to define an appropriate function (358), and an approach of this type does simplify the calculation somewhat, although in the most precise work the correct form for Ru [equation (13-64)] should be used. 430 Line Formation with Partial Frequency Redistribution (c) Case III. In this case we no longer have coherence in the atom?s frame, and equation (10-55) may be used without complication to give 1 ri {& (5/n) dpi 1 n (Öfn) dp1 4 J-i (v' - wp'u — v0)2 + 52 J-i (v — wpu — v0)2 + S2 1 4n w W tan" tan' x' + u — tan' x — u X + u — tan' X — u (13-66) Averaging over a Maxwellian velocity distribution and converting to Doppler units we have -Km, J* tan"1 . fx + U\ , /x — u tan"1 - - tan"1 x + u tan - j x — u du (13-67) Asymptotic formulae, results for dipole scattering, and computational methods for evaluation of Rm are given in (313,212; 529). A plot of Rm(x', x)/0(x') for a — 10"3 is shown in Figure 13-4. For small x', most absorptions are at line center by atoms moving with velocities near x', hence redistribution occurs with equal probability over the range — x' < x < x', as was true for Rf and Ru. Again, for large x' most absorptions occur in the line wings of nearly stationary atoms, but now the emitted photons are completely redistributed over the absorption profile in the atom's frame, hence Rm(x', x)j (x) for x' » 1. It has, on occasion, been argued on intuitive grounds that if the redistribution process is completely noncoherent in the atom's frame, and if this is combined with random Doppler motions, then the redistribution should be completely noncoherent in the observer's frame as well; this conclusion is false, however, as is clearly shown by Figure 13-5. In fact, Doppler motions introduce a correlation between incoming and outgoing frequencies near the line core, and the deviations from complete frequency redistribution in the laboratory frame can be large. Despite these deviations, it turns out (see §13-4) that the assumption R(x\ x) — (f>(x')(f>(x) produces line profiles quite similar to those obtained from the exact Rm{x\ x), and in practice, the case of complete noncoherence in the atom's frame may be treated as complete noncoherence in the laboratory frame also, without serious errors. 13-4 Radiative Transfer with Partial Redistribution 433 1000 figure 13-5 Ratio of actual redistribution function, for complete redistribution in the atom's frame and Doppler redistribution in the laboratoiy frame. Rm(x', x), to the limiting case of complete noncoherence in the laboratory frame, ^(i). The absorption profile ip is a Voigt profile with a = 10" From (212), by permission. Exercise 13-6: Show lhat the angle-averaged redistribution function for scattering by electrons (under the same assumptions as employed in Exercise 13-5) is Re A(v',v) = w'1 ierfc|(i>' - v)/2w| where icrfc(x) = erfc(z) dz = tT^-1' - x erfc(x) and w denotes the electron Doppler width, w = vü{2kT/me^/c, which is about 43/4* times as large as the Doppler width of an atom of atomic weight A. See also (318) and (39). SYMMETRY PROPERTIES From equation (13-31) we see that, after angle-averaging, we must have R(-x\ -x) = R(x,x) (13-68) which is general so long as equation (13-30) holds. Further, by a direct extension of the arguments leading to equation (13-35) one finds Ri{x\x) = R^x'X (/ = T, II, III) (13-69) The validity of this result can also be seen by inspection of equations (13-62), (13-64), and (13-67). From equation (13-69) it follows that the natural-excitation emission profile [see equation (13-42)] W(x) EE faix), (i = I, II, III) (13-70) Exercise 13-7: Verify the result stated in equation (13-70) for Case I by direct integration of Rt A{x\ x) over x'. 13-4 Radiative Transfer with Partial Redistribution Consider now the problem of accounting for the effects of angle-averaged partial redistribution in spectral line formation. For simplicity we shall confine attention to a two-level atom with continuum, and develop a general method that gives a rigorous solution of the problem. (This method can be generalized easily to more complicated atomic models as well.) Next, a somewhat simpler, but less general, approach will be developed, which still treats the troublesome stimulated emission term correctly. Then a still simpler method, which treats the stimulated emission terms only approximately, will be described, and results from the application of this method in highly idealized models will be discussed. Finally, results from calculations of resonance-line profiles, using the general method, for realistic model atoms and stellar atmospheres, will be presented. FORMULATION FOR A TWO-LEVEL ATOM Let us examine the formation of a resonance line connecting a perfectly sharp lower level to an upper state that is broadened into a distribution of substates by both radiation damping and collisions. Denote the population of the lower level by nu the total population of the upper level (summed over all substates) by n2, and the number of ions by nK. The distribution of the atoms over the upper state is specified in terms of the observer s-frame emission profile >pv, defined as the fraction of all atoms in the upper state that, if they decay radiatively, emit photons of frequency v as seen in the laboratory frame. This method of counting atoms in the upper state is well-posed physically, for it describes the distribution in terms of observables: mathematically, ij/v is a complicated, but unique, one-to-one mapping from the distribution of atoms over their own rest-frame frequencies c, and velocities v. the transformation being specified by the redistribution function 434 Line Formation with Partial Frequency Redistribution 13-4 Radiative Transfer with Partial Redistribution 435 (456). Clearly i/\ ls normalized, and n2 = n2 j \pv dv. We shall find it convenient to define an auxilliary variable n2 such that n2(v) = n2tj/v = n2{v)v gives the number of upper-state atoms in substates that can emit photons of frequency v; the dominant variation of the upper-state distribution is factored out this way, and n2 reflects only the departure from natural excitation. The substate occupation number n2{v) (or, equivalently, i/rv) is specified by a rate equation of the form n2(v)(A2i + B21Jr + C21 + R2K + C2K) BL1 JR(v', v)Jv. dv' + C12y* + nttf(RK2 + C2K) (13-71) This equation has a simple physical interpretation [see also (456)]. The term on the lefthand side is the population of the substate, times the total rate of exit (a) to the ground state by spontaneous emissions, stimulated emissions and collisions, and (b) to the continuum by photoionizations and collisions. The only noteworthy term is that for stimulated emission which is strictly coherent in both the laboratory and the atom's frame. [This follows from the fundamental quantum-mechanical characteristics of the process (197, §62; 293, §17). in which a photon, incident upon the atom, induces an emission in such a way as to create yet another photon of exactly the same properties: energy (hence frequency), momentum (hence direction of propagation), and polarization. Thus a photon (v', n') in the laboratory frame undergoes a definite transformation (depending on n' and v) to (£', n') in the atom's frame, and creates another identical photon; both photons undergo exactly the same inverse transformation back into the laboratory frame, returning two photons (v', n') into the radiation field.] The terms on the righthand side describe processes of excitation to the substate nz(v), from the ground state and by recombinations from the continuum. The latter produce atoms in level 2 distributed according to the natural-excitation profile ; nf denotes the LTE particle density nf = nKnev [see equation (13-70)]. From the ground state, atoms excited by collisions are also distributed according to t//*. The radiative excitations are given by the number of absorptions of photons of frequency v', namely i^B^J^, times the joint probability R{v', v) of absorbing at v' and emitting at v, summed over all v'. In the scattering process we assume that a fraction y of all excited atoms emit coherently, and the remainder are completely redistributed in the atom's frame by elastic collisions [so that />(£', c) is given by equation (13-6)]. The correct laboratory frame redistribution function in this case would be RW, v) = i*u(v', v) + (1 - y)Rm{v', v) (13-72) but in practice it is adequate (and much easier) to use R(v', v) = yRu(v', v) + (1 - y)d>v4v (13-73) which assumes that complete redistribution in the atom's frame leads to complete redistribution in the observer's frame. The rate equation for the ground state is , (Byljv + A'v and nv - {2hv3/c2)a12{gjg2)fl2(v)fi>v + Ev and in the ground-state continuum X, = «i»K - nfe-W) + Xv (13-79) and nv = (2hv5/c2)e'hv/kToclK(v)nf + Ev (13-80) Here Xv and Ev represent (fixed) background opacity and emissivity sources, while a12 = (B12hY)/4n. The solution of the transfer equation is now quite complicated because the emission profile \j/v is not known a priori, but follows from the statistical equilibrium equations (13-71) through (13-75). Unlike the case of complete redistribution, where only the ratio (n2//7j) is required to specify the source function [cf. equation (11-4)] and hence only one statistical equilibrium equation is needed, we must now compute \j/v, and this introduces as many equations of the form of (13-71) as are required to define this function to 436 Line Formation with Partial Frequency Redistribution 13-4 Radiative Transfer with Partial Redistribution 437 the desired precision. Indeed, the situation at hand strikingly resembles that of a multiline "multiplet" problem with very strong interlocking effects, each frequency within the line playing the role of a separate transition, and the line as a whole acting as the collective photon pool. This analogy suggests that the solution can be obtained effectively by means of a complete linearization technique. METHODS OF SOLUTION A general and powerful method of solving equations (13-71) through (13-75), simultaneously with the transfer equations (13-76) through (13-80), is to use the complete linearization technique. We introduce a discrete set of upper-state substates specified by frequencies {v/,, measured relative to line center. The substate populations can then be written as n2{vj) = ft2jjf and the rate equations discretized as Bl2 £ WjtpjJj + Cl2 + RlK + ClKJ - X wfi2jHA2i + BuJj + C21) -nMnMT)(RKL + ClK)-]=0 (13-81) L J + c124>i + n2j(Pj(A21 + B21Jj + C21 + R2K + C2K) - nK[n^2(T)(RK2 + C2lt)] = 0 (13-82) and «i + Z W/\A + nK = "atom (13-83) j Here 0ft> is a discrete representation of Kn(v;, v,-) in an appropriate quadrature. Writing n = (nlt ft21, «22,. . ., n2j,. . ., n2J, nK)T, where J denotes the total number of substates, equations (13-81) through (13-83) are of the form Suppose we have an estimate of the occupation numbers n for all depths; then current values of Xv, and Jv may be computed at all frequencies in the line and continuum. We may then linearize the transfer equation in terms of SJr, 8xv, and 5}]v, and express the latter two quantities in terms of .5«!, the dn2j% and SnK. In turn, them's can be written as 8n = £ (dn/v - "2^v 1 ~ (91*2/92"l)<*>v Here, and in what follows, we assume that a current estimate of a\. is known. From equation (13-71), omitting the continuum terms, _ A^-'iR(v',v)Jv.dv' + C2I(1 - e-hvlkT)Bv(T) (2hvi/c~)(gln24/v/g2n1(f>v) = and from equation (13-74) ^21 + B2lJv + C21 (13-85) r, < > 1 A2i + B21{Je - a)vJa)+ C21(l - (Ove'hvlkT) [1 " foin2/02niK] - \ i~7%---" (1-3-86) where Ja = f vJv dv and Je = f ij/vJv dv - \vwvJvdv. Defining f: = C21(l - e-hvlkT)/A2l we have S,(v) = cy 0v_1 (R(v'.v)J,, dv' + h'Bv ; 13-87) 438 Line Formation with Partial Frequency Redistribi 13-4 Radiative Transfer with Partial Redistribution 439 where _JV^i:Jf_g1iJg + Cy)l(A21 + B21JV + C21)] 1 + (ß2iM2i)(^ - >j>* + (13-90) the transfer equation now reduces to the general form d2{fvJv)/dxv2 = Jy - av JR(v', v)Jv t/v' + bv (13-91) which may be solved straightaway by the standard Feautrier technique. In this method we use current, estimates of cov and Jv to calculate £v in equation (13-88), and of the level populations and tuv to determine -/A{v) in equations (13-89) and (13-90). The transfer equation (13-91) is then solved for improved values of Jv. These Jv's are then used in equations (13-81) through (13-83) to update n^ n2{v), and nK, and from these, and thus cov. The process is then iterated to convergence. One would expect this method to work well whenever (B2lJ/A2i) « 1; when stimulated emissions are very important one must use the full linearization technique described above. RESULTS FROM IDEALIZED MODELS A great deal of insight into the nature of partial redistribution effects can be obtained from studies similar in spirit to those described in Chapter 11, using idealized model atmospheres and atoms. To simplify the problem as much as possible, it is customary to make the additional physical assumption that the stimulated emission profile is given by not i//v, and that the stimulated emission rate in equation (13-71) can be written as n2{y)BztJa rather than n2{v)B2lJv. Then the parameter cov defined above is identically unity, 7e = Ja, and £v = (1 + e')"1* and equation (13-87) reduces to 5,(v) = (1 - £)/1 JR(v', v)Jv. dv' + eBv (13-92) where s = c'/{l + e'). With this source function, the transfer equation can be solved in a single step without iteration. This approach has been used to estimate the differences between the frequency-dependent S,(v), obtained when partial redistribution effects are taken into account, and the frequency-independent Sf* = (1 - s) 14>VJV dv + e.Bv (13-93) obtained from complete redistribution, for Rh Ru, and Rm in constant-property atmospheres (53; 212; 316). The source functions 5,(v) and S,CR may also be used to compute line-profiles; the errors made in calculated profiles if complete redistribution is assumed in place of an accurate redistribution function can then be assessed. An intermediate approximation is the iterated source function SRv) = ee (1 -— e)4>v_1 JR(v\ y)j£K dv' + sBv (13-94) which is evaluated using the mean intensity obtained from the complete redistribution solution based on equation (13-93). It has been found that S\i}(v) in isothermal media is nearly equal to S,(v) (316), and obviously it is much simpler to compute. Let us first consider the case of redistribution by Doppler shifts only—i.e., R(v', v) = Ri{v\ v). A number of solutions for S^v) have been obtained (316) for both finite and semi-infinite isothermal media, assuming zero continuum opacity, with e = lO'^and 10~6. Results for variation of the source function with frequency and depth are shown in Figure 13-6 for e = 10"4. The vertical arrows designate the frequency at which the monochromatic optical depth tx = 1. It is clear that, at all frequencies where tx ^ 1, 5,(x) essentially equals S,CB; large deviations occur when xx < 1, but are of little consequence because these optically thin regions do not contribute significantly to the intensity in a line profile. In fact, the line profiles computed from S,CR and from Si(x) are virtually identical; hence we conclude that for Case I the scattering process is, for all practical purposes, adequately described by the simpler assumption of complete redistribution. A similar conclusion, is reached (212) for the case of complete redistribution in the atom's frame—i.e., R(v\ v) — Rlu(v', v). Here the iterated source function, equation (13-94), was evaluated in an isothermal semi-infinite atmosphere, for lines with a = 10"3, and s = 10"6 and 6 x 10"3. Again it is found that S'^Xx) departs from S,CR only for %x < 1, and that emergent line profiles computed from are almost identical to those computed from the frequency-dependent function. Thus the assumption of complete redistribution provides a very useful and accurate approximation for Case III as well. The situation for Case II (coherent scattering in a broadened profile in the atom's frame with Doppler redistribution in the laboratory frame) is quite different. Results from an isothermal atmosphere of total thickness T = 10s, for a line with a = 10"3 and e = 10"4, given in (316), are shown in Figure 13-7. Here we see that at line center, S,(v) is near S,CR for shallow optical depths, but rises above S,CR at great depth, and thermalizes to the Planck function sooner than the complete-redistribution source function. This result is obtained because the coherent nature of the scattering process in the line-wings inhibits photon escape from the line-core, and forces more 440 0 -1 \og(S/B) ~2 I________i_I_i_I__i_L 0 12 3 x I 1GURE 13-6 Source functions in an isothermal, serai-infinite atmosphere for a line with e, = 10' 4, assuming pure Doppler redistribution (case I). The vertical arrows show the frequency at which the monochromatic optical depth xx = 1. Dashed curves: frequency-independent SjCR obtained assuming complete redistribution. Solid curves: frequency-dependent S^x) obtained using correct redistribution function. Abscissa; displacement from line center in Doppler units. From (316), by permission. i = 0 0 12 3 4 x FIGURE 13-7 Source functions in an isothermal atmosphere of total thickness T = 106, for a line with e = 1()"4 and a = 10~-\ assuming coherent scattering in a radiation-damping profile in the atom's frame, and Doppler redistribution in the laboratory frame. Dashed curves: frequency independent SfCR obtained assuming complete redistribution. Solid curves: frequency-dependent S{(x) obtained using correct redistribution function. Abscissa: frequency displacement from line-center in Doppler units. From (316), by permission. 442 Line Formation with Partial Frequency Redistribution 13-4 Radiative Transfer with Partial Redistribution 443 rapid thermalization in the core by reducing the net escape probability for such photons. In contrast, in the line-wings, the frequency-dependent source function lies substantially below SfR, because photons are no longer being fed into the wings from the core as efficiently. These effects are even more pronounced for finite atmospheres where S,CR may exceed S,(x) by orders of magnitude in the line wing! The emergent line profiles faithfully reflect the discrepancies between S,(x) and 5,CR. Profiles for the correct source function have intensities that lie well below those for complete redistribution in the wings, and that nearly agree with the intensities predicted by coherent scattering. Although these effects will be diminished when there is an overlying continuum, we may, nevertheless, expect partial redistribution effects to be important for resonance lines formed in the outer layers of stars where densities, and thus collision rates, are low. application to solar and stellar resonance lines The resonance lines of many ions [e.g., those of hydrogen (particularly La), Ca+, and Mg+] are formed in chromospheric layers of low density, and hence are rather well characterized by the redistribution function of equation (13-73) with y almost unity. Several calculations have now been made for solar lines, using the HSRA (249) or related models, and for solar-type giants (where densities are even lower than in the sun) using realistic model atmospheres. One of the first examples of the importance of partial redistribution effects in resonance-line formation arose from attempts to fit the observed solar chromospheric La profile (645). It was found that when the line profile was computed under the assumption of complete redistribution, using models that provided accurate fits to the continuum data formed in the same atmospheric layers as the La line-wing, the intensity in the calculated profile wing was much larger (by a factor of 5 to 6) than observed. From the first study (645) it emerged that a much better fit to the profile is obtained if the scattering is assumed to be about 93 percent coherent, and only 7 percent completely redistributed. Subsequent work showed (456) that this parameterization is equivalent to using equation (13-73) with realistic values of y (determined from the known collisional and radiative rates), and finally, that when the full depth-dependence of the profile and redistribution functions, atomic rates, level populations, and background opacity are taken into consideration, an excellent fit to the observations is achieved (457). An even more interesting example is provided by the solar Ca II H- and K-lines, for which the earlier work (e.g., 401) assuming complete redistribution gives a good fit to the disk-center profiles, but fails to fit the observed center-to-limb variation. Calculations using a five-level atom similar to that shown in Figure 12-1 were made (570) [see also (642)] for three model atmospheres: (1) the HSRA with a depth-independent microturbulence of 4.5 km s_1; (2) the HSRA with the microturbulent velocity distribution given by (401); and (3) the same as model 1, but with the temperature structure modified to be T3 = max (THSRA, 4450°K), which raises the temperature minimum by about 300 0. while both models 1 and 3 yield an excellent fit to the data. Model 2 does not do as well, though it provides a better fit to certain data for K2 (not discussed here). In part (b) of the figure we see that CR fails badly to fit the limb-darkening of K1 intensity, while PR (with any of the three models) yields the correct center-to-limb variation (notice the logarithmic scale). Note that the absolute intensity of the Kl feature is reduced by PR, relative to CR, for a given model (compare the open and filled circles at a = 1). This results from the essentially coherent nature of the scattering process in the Kx feature, and is consistent with the results for Rn redistribution in the idealized models discussed above. One way to obtain a fit to the absolute intensity is to raise the temperature minimum by 300!)K to 400°K over the HSRA value, as was done for model 3. While this change seemingly is small, it should be recalled that a change of about 100°K near Tmin changes the energy content of that region by an amount comparable to the energy content of the entire corona (cf. §7-7.). One hesitates to conclude on the basis of one line that T„,in must be higher than the HSRA model value; however, similar results are also obtained from an analysis of the Mg II h~ and k-lines (59), and it may, in fact, be necessary to raise the empirical value of Tmin to about 4400°K. Such an adjustment would bring the semiempirical result into harmony with the estimates of Tmin from radiative equilibrium models, and would eliminate the difficulty described in §7-7. In any case it must be emphasized that the drop in intensity of PR relative to CR is a differentia! effect for a given model, and therefore will occur no matter what model is used. Thus if Tm[n is estimated from a lit to the K1 intensity (as is sometimes done in stellar work), one must use a PR description of the scattering process, especially for giants (571), or systematic errors will be made. Finally, it is found that the PR calculation accurately reproduces the relative behavior of the H- and X-lines while CR does not. The marked differences in the CR and PR predictions of limb-darkening and wavelength-position of Kt can be easily understood in terms of the differences in the depth-variation of the source function 5(CR, which is << < S 2 x 10 (b) P- figuri! 13-9 (a) Wavelength position of minimum intensity outside emission core, K:, as a function of fi, cosine of angle from disk center. (b) Limb-darkening at ; note logarithmic units of intensity. Open circles: complete redistribution results, model 2 (see text). Filled circles: partial redistribution, model 2. Open triangles: partial redistribution, model 1 (see text). Crosses: partial redistribution, model 3 (see text). From (57(1), by permission. frequency-independent, and the frequency-dependent PR source functions 5,(v) shown in Figure 13-10. There we see that CR yields a unique source function that has a single absolute minimum. As one observes from center to limb, the slant-length optical depth at a particular frequency increases; if S,CR is used, this implies that the intensity at AAK|(/i = 1) must rise as \i decreases. The minimum S,CR will manifest itself only at some larger AX where the material is more transparent and thus reflects the source function 446 1(T71-1-1--I-1-J-1 I_I 1CT7 1(T6 ltT5 10"4 10"3 10"2 10"1 1 10 m (gm cm ~ 2) FIGURF. 13-10 Depth-variation of K-line source function (for a five-level atom) and Planck function in model 1. Dashed curve: Planck function Bv. Dotted curve: complete-redistribution source-function Sj (see also Figure 12-2). The other curves show the partial-redistribution source-functions at line-center (K3), the emission peak (K2), and the profile minimum outside the peak (iCJ. Abscissa: column density gm/cm2. From (570), by permission. deeper in, near the minimum. This explains both the rapid increase in AAfXj) as a function of/*, and the near-constancy of IKi predicted by CR. In contrast, the PR source function at line-center (K3) lies above the CR value (as a result of photon trapping as discussed previously). But at Kl the source function has an essentially coherent-scattering character, is decoupled from other frequencies, and shows a monotonic decrease outward in the atmosphere [similar results were obtained in (642)]. For this behavior of S,, changing the point of observation from center to limb merely samples S^Kf) higher in the atmosphere, where it has (for PR) a lower value, and hence produces a decrease in IKi, as desired, at about the same Al In short, taking partial redistribution into account yields a substantial improvement in the degree of agreement between theory and observation for the solar Ca II H- and X-lines. Similar calculations, yielding similar results, have also been made for the Mg II h and k (3s-3p) lines for the solar atmosphere (458; 59) and for solar-type stars (455). It now appears that it is mandatory to account for partial redistribution effects in the interpretation of strong chromosphere resonance lines with strong radiation-damping wings, and further efforts h this direction will undoubtedly be amply rewarded. Radiative Transfer in Moving Atmospheres The existence of macroscopic motions (i.e., nonthermal velocities that are coherent over distances much larger than a particle mean-free-path) in stellar atmospheres is well-documented by a wealth of observational evidence. These motions appear to be present on all scales, from "eddies" whose sizes are small compared to a photon mean-free-path, up to expansion of the atmosphere as a whole. Although velocity fields have but little effect on radiative transfer in the continuum, they strongly influence line-formation because even a small (Doppler) frequency shift of a line produces a major change in its absorptivity as seen by a stationary observer. In the analysis of spectra of supergiants, Struve and Elvey (617) discovered that the Doppler widths inferred from the position of the flat part of the curve oj growth (cf. §§10-3 and 10-4) were far in excess of the thermal value. They attributed this broadening to nonthermal "turbulent" velocities, presumed to have a Gaussian distribution. The geometric scale of these motions (which are called "microturbulence") is supposed to be so small that they act as additional line-broadening agents, and thus enhance the line-strength. The inferred velocities often approach or exceed the speed of sound in the material, and it is clear that astrophysical "microturbulent" velocities are not to be identified 448 Radiative Transfer in Moving Atmospheres with turbulence in the strict fluid-dynamical sense, but rather with unresolved motions. The picture is clarified when we examine the solar spectrum. From curves of growth, or from spectra of low spatial, spectral, and temporal resolution, one again infers a significant microturbulent velocity. But high-resolution spectra have a characteristic "wiggly-line" appearance [see Figure II1-2 of (20) for an excellent example], showing distinct Doppler shifts and asymmetries that fluctuate rapidly along the length of the slit, and in time. Probably a large part, or even all, of the velocity field is composed of wave motions of various types, scales, and periods, which (when superposed) give rise to a pattern that appears chaotic and "turbulent". However, the diagnostics are still in a primitive state, and the precise nature of the velocity field is not at all well known. Evidence for velocity patterns on a large scale {"macroturbulen.ee") was provided by Struve's observation (616) that the widths of line profiles in certain stars exceeded the Doppler widths obtained from the curve of growth of their spectra; here the line strength is unchanged, and one envisions areas on the stellar surface so large as to be practically independent "atmospheres" moving systematically along the line of sight. A discussion of the observations, with numerous references, can be found in (261, Chap. 8). Further, periodic Doppler shifts of the lines in some stellar spectra reveal that they are from pulsating stars. Beyond this, objects such as the WR stars, P-Cygni stars, and early-type supergiants all show characteristic line profiles, with blue-shifted absorption components and red-shifted emission components, indicative of large-scale expansion. It is clear that a characterization of stellar velocity fields by the two extremes of "micro"- and "macro"-turbulence is an oversimplification and, in the end, we wish to know distribution functions describing the amplitudes and scales of the velocity patterns. Further, one would like to relate observed parameters to more fundamental quantities such as the velocities of convec-tive motions. Finally, a truly consistent theory of stellar atmospheres will require a dynamical theory of the interaction of material velocities, the thermodynamic state of the matter, and the radiation field; only then will we be able fully to understand stellar chromospheres and coronae. At the present time, however, we are far from having such a complete theoretical structure. In this chapter we shall focus almost entirely on the "kinematics" of radiative transfer in moving media; i.e., given the velocity field and the model atmosphere, compute the emergent spectrum. A variety of techniques exist to attack the problem just posed. Observer's-frame methods can handle complicated velocity fields and multidimensional structures (although we shall confine attention to one-dimensional problems only), but are generally restricted to velocities of the order of a few Doppler widths, and hence are not well suited to handle rapid atmospheric expansion. Sobolev's method, on the other hand, provides an approximate solution in the case of 14-1 The Transfer Equation in the Observer's Frame 449 rapid flow with large velocity gradients. Comoving-frame methods span the two extremes and provide general solutions applicable in both limits. Special techniques have been developed to treat random or stochastic velocity fields. 14-1 The Transfer Equation in the Observer's Frame FORMULATION AND SOLUTION OF THE TRANSFER EQUATION When the material in the atmosphere moves with velocity v(r) relative to an external observer at rest, there is a Doppler shift of photon frequencies between the observer's frame, and the frame of the atoms of which the material is composed. If the frequency in the observer's frame is v, then in the atom's frame the frequency at which a photon traveling in direction n was emitted, or can be absorbed, is v' = v - v0(n ■ y/c) (14-1) Thus, the opacity and emissivity of the material, as seen by a stationary observer, become angle-dependent. The transfer equations for a time-independent moving medium in planar geometry is then p[dl(z, p, v)/dz] = n(z, p, v) - yfz, p, v)J.{z, p, v) (14-2) It is convenient to measure frequency displacements from line center in units of a fiducial Doppler width Avg = v0yt|/c, where v?h is a thermal velocity parameter, and to measure velocities in the same units, V = vjvfh. Then the transformation between observer's frame and atom's frame frequencies is x' = x - pV (14-3) where x = (v — v0)/Avd and x' is defined similarly. The effects of Doppler shifts are inconsequential for continuum terms, which do not vary much over the frequency range implied by velocity shifts, so we account only for changes in line terms and write yfz, p, x) = Xc{z) + yjz)(z,p,x) = (f>{z;x - fiV) (14-6) For example, for a Doppler profile,