450      Radiative Transfer in Moving Atmospheres
where S measures the local Doppler width in units of the fiducial value; i.e., t>(z) = AvD(2)/Avj$. We now define line and continuum source functions S,(z) = and Sc(z) = /?,(z)/&{z); a total source function
S(z, pt, -v) e        pi, x)St(z) + f(r)S(fz)]/[^(3,    x) + >1z)] (14-8)
where /-(z) = /jr(-V;M2); ar>d arj optical depth scale measure along a ray specified by pi,
x{z,pi,x) = jU~5  J"1'" X{z,l*,x)dz
(14-9)
where z.
denotes the upper surface o( the. atmosphere. Then the transfer equation becomes
[af(z, pi, x)/dx{z, u. x)] = /(z,    x) - S(z, pt, x) (14-10)
The formal solution of equation (14-10) can be written immediately as /(zmax, ^ x) = 7(0, pi, x)<r«°-*-*> + Jf3'*"*' S{z, pi, X)e-^"-^ dx(z, p., x) = 7(0, pi, x)c-z{0^'x)
+ ^ ii-l[<j>{z.pi,x)S1\z) + r[z)Sc(z)\e-^-"-x,yJz)dz
(14-11)
from which we can compute the emergent intensity for given source functions. (For example in LTE, S£ = S, = 73; or we might use the values of 5, found for the line in a static atmosphere, an approximation that often is surprisingly accurate as we shall see below.) Tn equation (14-11) we have taken the atmosphere to be a finite .slab with intensity incident at z = 0; for a semi-infinite atmosphere, we set r(0, pi, x) = to and omit the term in 7(0, pi, x). The formal solution allows a direct evaluation of the effects of velocity fields on profiles by accounting for the velocity-induced shifts in the opacity and emissivity of the material.
The line source function will, in general, contain a scattering term, and therefore depend on the radiation field; thus the source function can be strongly affected by material motions. For example, an expansion at the upper surface of an atmosphere can displace a Sine away from a dark absorption feature, at the rest position, into the bright nearby continuum, thus raising J (and St) dramatically. If we assume that photons scattered by the line are completely redistributed, then the source function for a two-level atom becomes
S,(z)
1 (i - ft) §\ dx J't dpi Hz, pi, x)4>{z, pi, x) + &{=) (14-12)
14-1   T/ie Transfer Equation In the Observer's Frame      451
where e, is the usual thermalization parameter. Note that in the scattering integral we can no longer replace / with J because (f> is angle-dependent; note also that the intensity can no longer be assumed to be symmetric around the line center, hence the full profile must be considered. The approximation of complete redistribution becomes questionable for moving media, as the conditions that help validate it in static media no longer occur; a good discussion of this point is contained in (273,87) (this is a superb paper that is highly recommended to the reader). Recently some work has been devoted to the problem of partial redistribution in moving atmospheres; it has been shown that to treat the problem in the observer's frame, the full angle-frequency dependent redistribution function must be employed. In contrast, in a comoving-frame method (see §14-3), one can employ static redistribution functions in the fluid frame, and angle-averaging again yields accurate results.
Accurate calculation of the scattering integral in equation (14-12) with a quadrature sum poses a fundamental difficulty in an observer-frame solution for two reasons. (1) The line-profile <f>(x — piV) is clearly shifted by an amount 2F in frequency as pi varies from —1 to 1. Thus, in the frequency quadrature, an amount equal to twice the maximum macroscopic flow velocity must be added to the bandwidth required to describe the static line-profile. This requirement is not severe in studies of, say, wave motions in the solar atmosphere, but becomes prohibitive for atmospheres in supersonic expansion where v/c 0.01, or 2{v0v/c)/Av'^, w 200. (2) The angle-quadrature scheme must employ a large number of angles. Because the argument of the profile function is (x —piV), there is an inextricable coupling between the angular and frequency variations of the intensity. Thus if some maximum frequency increment Axmax(«^) is required to obtain sufficient precision in the frequency quadrature, the maximum tolerable angle increment will be Ajumax = Axmat.T< which is quite stringent! These difficulties are ameliorated by transforming to the comoving frame.
Equation (14-10) may be cast into second-order form. If the line profile is symmetric about line center, then <fi(—x + piV) — d>(x — piV), which suggests that we group the two pencils 7(z, pi, x) and I(z, —pi, —x) together, for dx(z, pi, x) = c/t(z, —pi, —x) and 5(2, pi, x) = 5(z, —pi, —x). Thus, defining
and
u(z, pi, x) = - [7(z, pi, x) + J(z, -pi, -x)]
v(z, pi,x) = - [7(z, pi, x) - l(z, -pi, -x)]
we obtain    [d2u(z, pi, x)/dz(z, pi, x)2] = u(z, pi, x) — S(z, pi, x) At the upper boundary there is no incoming radiation, hence [du(z, pi, x)/dx{z, pi, .v)]_^ = «(zmax, pi, x)
(14-13)
(14-14) (14-15)
(14-16)
452      Radiative Transfer in Moving Atmospheres
1'4-1   The Transfer Equation in the Observer's Frame 453
At the lower boundary we assume either that the incident radiation is specified, in which case
[du(z, p, x)/dr(z, pi, x)]z = 0 - 1(0, pi, x) - u{0, pi, x) (14-17)
or that, in a semi-infinite atmosphere, the lower boundary is chosen to be so deep that the diffusion approximation is valid [which demands that the velocity gradient be small enough that y~1(dV/dz) « 1; i.e., there is a negligible change in the velocity over a photon mean-free-path], in which case
ouyz, jl, xj
dz{z, pi, x)
ß föBA \dT j{z, ß, x)\dTJ \ dz
(14-18)
As in the static case, we introduce a discrete depth-mesh {zd\, angle-mesh {/(,„}, and frequency-mesh {xn}, and combine angles and frequencies into a single quadrature set {/j,, x,} = (pi,,,, x„) where / = m + (n — 1) m; the angle-points are distributed on the interval [0,1], while frequencies must now span a range [.\mill, xmaJ, .xmin < 0 and xmdX > 0. large enough to contain both halves of the line profile and to allow for Doppler shifts ±2VnVtlK. We then replace equations (14-15) through (14-18) with difference equations and write
Sdl = S(:^„xt) = ocdl Ja + j8dI (14-19) where « and f> are the appropriate combinations of rd, (pdh and ed5 and
(14-20)
where (pdl = 4>(za- xi ~ t^iKt)- The resulting system is then of the standard Rybicki form [see equation (6-47)] and is solved for J as described in §6-3. An analogous integral-equation solution can also be constructed (273, 120), but in application the differential-equation method is easier to use. The whole procedure is stable and general, and quite efficient, as the computing time TR = cL D2 -f c' is only linear in L, the number of angles and frequencies. The depth-mesh must be chosen sufficiently line to assure that only modest changes in V(zd), say ^ 4-, occur between successive depth-points; otherwise the profile function (j>dt may change radically with depth and lead to inaccuracies in the optical depth increments. Except for supersonic winds, this is not a stringent requirement. Note also that the same methods can be used to construct the formal solution, when 5 is given, by solving a single tridiagonal system (at each angle-frequency point desired) of the form T,u, = S,; here the computing time required is only Ts = cL D, which is minimal.
LINL-FORMATlON WITH SYSTEMATIC MACROSCOPIC VELOCITIES IN PLANAR ATMOSPHERES
The effects of velocity fields on line-formation in planar atmospheres have been studied by a number of authors; we shall review some typical results here. Basic insight into effects of motions can be gained by using just the formal solution. Consider, for example, a semi-infinite atmosphere with a line formed in LTE, with r = constant, and SL = Sc = Bv = B0{\ + at,), where t, denotes the static line optical depth. Choose a velocity held of the form v(z,) = d0/[1 4- (VTo)j\ taken to be positive toward the observer (i.e., toward increasing z). Then it is easy to calculate the emergent intensity
x) over the surface of the star, and to construct the flux F [x)by integrating over pi; results for models with a = 3 x 10~2, r = 10~2, v0 = (0, 1, 3, 5, .10), and t(, = (1, 10, 100) are displayed in Figure 14-1. There we see that the line flux profile shows an asymmetry toward the blue. Similar asymmetries result even if there is no velocity gradient and the atmosphere is assumed to expand with constant velocity, because of the way velocities and intensities are weighted in the flux integral (Cf. Exercises 14-1 and 14-2).
Exercise 14-1:   (a) For a linear limb-darkening law ^(p) = = I + fipt,
show that the function normalized to give unit flux is 4>* (/<) = (1 + fiu)l(k + jID-lb) Assume that a weak line is formed on the surface of an atmosphere expanding with velocity d0, and that the line depth, as a fraction of the continuum, does not vary with fi. Derive an expression for the radial velocity measured from observations of the flux in a spectrogram. In particular, show that for the grey-body limb-darkening law, in the Eddington approximation, i;oh5 = (17/24)%. What is the ratio of vohJv0 for fi = 0; for fi = cc?
Exercise 14-2: Calculate the flux, profile from a line, idealized as a delta-function of constant depth, on a stellar surface expanding with constant velocity i.e., take Hji.x) = (1 + /Jjf)[l - a05{x - /iV0)]/(t + */')■ Derive an explicit expression for F(x;«(,, fi, V0), and plot the profile in the limiting cases fi = 0,fi ~ cc. (b) Extend the analysis to a line with a Gaussian profile, and compute numerically a typical flux profile.
The formal solution can be used to evaluate proposed velocity-diagnostic techniques by computing profiles for given velocity fields, subjecting these profiles to the diagnostic analysis, and comparing the inferred with the originally-assumed fields. For example, the "'bisector shift" technique has been examined (373) for a variety of cases. This method supposes thai the displacement d.x, from the static line-center, of the position of the point mulway between two pomU ol equal intensity in the line-profile, gives the Doppler shift caused by velocities in a layer at unit optical depth for a (static) line frequency Ax, where 2 Ax is the full distance between the two points on the profile. It is found that the inferred velocities are in fair agreement with
454
14-1   The Transfer Equation in the Observer's Frame 455
F(x)/Fc
Hx)/Fc 0.6-
F(x)/F
-4-2        0 2
Flux profiles from an expanding atmosphere, for a line formed m LTE with B <J= B0{1 +       and       = »0/[l + (V*o)].   Ordinate: flux m umt of fon nuum    AbjLa: frequency displacement from fine center m Doppler umts. Par—.'arc B0 =1,   «=3x10-   ^ 10- for all curves. Each curve ts abeled with v0. Panels (a), (b), and (c) have x0 = 1, 10, 100, respectively.
the input velocities for measurements in the line-core, but that spurious velocities at great depths are inferred from the wings. It is easy to see why this is so. Suppose that the atmosphere moves with velocity v0 on the range 0 ^ t, 5$ t1s and is at rest for i, > t1. It is clear that the shift of the line in the upper layer forces the line-wings to be asymmetric, because opacity from the upper layer intrudes into the wing and absorbs radiation from below. By assuming that radiation at frequency displacement Ax from line-center arises from depths t, -~ 1/0(Ax), velocities will automatically be ascribed to these depths even if i} > tx. This example shows that care must be taken in inferring velocity fields!
A further example of such problems is shown in (321) where a calculation is made [using the Riccati method (544)] of the non-LTE source function of a line with e = 10~3 and r = 0, in a differentially expanding finite slab of total (static) optical depth tmax = 50. The expansion is taken to be symmetric about the midpoint of the slab (assumed to be at rest) with a linear velocity law of the form V{x) — V0 + I^t,; the medium is effectively thin, and simulates an expanding nebula. Line profiles are shown in Figure 14-2. There we see that the usual central reversal arising in the material nearest the observer is blue-shifted, and thus obliterates the blue emission peak, while the red peak is enhanced because photons more easily emerge from below; the line as a whole appears red-shifted even though the average velocity of the material is zero! One cannot, therefore, immediately conclude that a small observed red shift implies a receding emitter.
To evaluate the effect of velocity fields on the source function one must solve the transfer equation self-consistently. Some of the early work on this
lAn = 1)
0.03
0.02
0.01
1	i l	i i
	Ao\	
- 2/7		/ \
		V li
		
-3-2-1      0       1       2 3
x
figure 14-2
Normally-emergent intensity from differentially expanding slab with total (static) thickness timx = 50, for a line with e = 10"3 and r = 0.   Abscissa: x = Av/AvD. Curves are labeled with c0, velocity of expansion at surface; velocity law is linear in i and gives zero velocity at slab center.   From (23, 215).
456     Radiative Transfer in Moving Atmospheres
problem {370; 371) treated the case of an isothermal atmosphere-with a velocity jump A at a depth t1; with constant velocities above and below. The transfer equation was solved in the Eddington approximation, for a two-level atom, using a discrete-ordinate method. When A = 0, the static solution for the appropriate values oft: and r is obtained. As A becomes greater than about 4, the lines (which have Doppler profiles) in the upper and lower regions are strongly shifted with respect to one another and no longer interact. The atmosphere then acts as if it consists of two independent parts: (a) a finite upper layer of optical thickness t1s and (b) an underlying semi-infinite atmosphere in which t = 0 at the depth, ri, of the velocity jump. In this limit, the source functions in the two layers both achieve their respective static limits. Thus for A = OandA -> co, one recovers profiles identical to those computed from static source functions, the major effect just being the Doppler shift of the line-center in the formal solution. This result is strengthened when there is an appreciable background continuum [see also (372)]. For A on the range of 2 to 3, the two layers interact strongly and a full solution must be found.
A more realistic problem (273, 120) is presented by an atmosphere with a "chromospheric" Planck-function rise at the surface (see Figure 14-3a) and with velocity laws of the form F(t,) = 10/[1 + (t,/t0)], where t, is the static optical depth in the line. The resulting source functions for a line with r = ]0~4andt: = 10~2 and various values of t0 are shown in Figure 14-3a, while emergent intensity profiles are shown in Figure 14-3b, and flux profiles in Figure 14-3c. The striking result seen in Figure 14-3a is that the line source function is only weakly affected by the velocity field, even though the profiles show drastic changes. The basic reason for this result is that photon-escape through the outer layers is increased in the red wing, but decreased in the violet wing—and, to a large measure, these effects nearly cancel [see also (18, 53)]. On the whole, the photon-escape probability is slightly enhanced by the atmospheric expansion, which explains why St tends to lie below its static value for 1 ^ x, 5 102. In the case t0 = 10, the value of S, increases for Xj < i0 because the line intercepts underlying continuum radiation while it is optically fairly thin, which leads to an increase in J; for larger values of t0, the line becomes optically thick above the velocity rise, and the effect vanishes. When r0 <: 10\ the point of the velocity rise already lies below the thermalization depth of the line: line-formation in the upper (effectively thick) layer then proceeds as if the atmosphere were static, and the static value of 5; is recovered very closely. The flux profiles for x0 = 102 to 103 show "P-Cygni" features with red emission components and violet-shifted absorption. Here the emission, however, arises from the assumed temperature rise, and not from the geometrical effects that occur in extended atmospheres. To a high degree of approximation, one would find the same profiles from a velocity-dependent formal solution using static source functions.
0.15 r
Up = i.x)
1.25
U)0
F(x) 0.75
of/	10	Ao	10"
-		^C^^o3	
-	0 V /	10 i       i       , i	i
\	0	4 8	—  [-_j_-- 12
0.25
-4
(c| x
figure 14-3
(a) Planck function and source function in an expanding atmosphere, for a line with e = 10" and r = 10"4    Abscissa: static line optical depth,   (b) Normally-emergent intensity. Abscissa: x = Av/ArD.   (e) Flux profiles. Curves are labeled with value of r0 in velocity law vix) = 1D./[1 + (t,/t0)].   From (273, 120).
458      Radiative Transfer in Moving Atmospheres
14-1   The Transfer Equation in the Observer's Frame 459
The results described above apply lo expansion, where an increased escape probability in one wing can be compensated by a decrease in the other. In fluctuating velocity fields, however, the coherence of shifts from one point in the atmosphere to another is lost, and effects similar to a marked depth-variation of the line profile are produced. The effects of fluctuating mesoscale velocity fields on line-formation have been studied (568) for sinusoidal waves with V(i,t) = /JsinpTrU"1 log10t -I- f)], and for "sawtooth" waves; the latter simulate steepened shock-like structures. Using the Rybicki-type method described earlier, the transfer equation was solved at times spaced equally over a period, and time-averaged profiles were found for various values of ji and X. The limit a -» 0 corresponds to a "microturbulent" regime, while I -> oo yields a "macroturbulent" limit. For a given source function (e.g., S, = Bv), the line profiles for finite X invariably lie between those given by the two extremes X — 0 and X = cc. The lines computed with a given /? and X = 0 are always stronger than those for X - go, as expected. When non-LTE cases are considered, the source functions are modified by the velocity field; characteristically, S, shows ripples as a function of t, and the departure of the results for finite X from the microscopic limit (X = 0) are larger at smaller values of e. The primary result found for non-LTE isothermal atmospheres is a significant rise in the core intensity of the time-averaged profile for finite X\ the profile lies between the limiting (X = 0,
- oof profiles only in the wings, and is much brighter than both in the core (by a factor of 2.5 for /? = 2.5 and X = 4). The same behavior of the line core is also found in computations using the HSRA, and substantial changes in the source function occur. In particular, even though a collision-dominated source function is used, S, rises above Bv (as it can for photo-ionization-dominated lines), because velocity shifts allow the line to intercept bright continuum radiation. The brightening of the line core appears to improve greatly the agreement between the observed and computed solar Na I D-lines at disk center, without recourse to unusually high densities as hitherto required; this result may also offer an explanation for similar discrepancies observed in the solar Ca I and Fe I lines. Further work in this area will be quite rewarding.
The results discussed above take no account of possible effects of the velocity field on the state of the gas. Recently a study has been made (291) of effects of acoustic pulses on formation of the solar Ca Tl lines, allowing for the temperature and density changes the pulses produce in the gas. These computations show that, if changes in the physical variables are ignored, the velocity field alone produces little change in the source function, and the correct profile is predicted using the static source function in a velocity-dependent formal solution; a similar conclusion was reached in (185). In contrast, the changes in T and N produced by the pulses have major effects on Sj and hence on the profiles. In particular, both T and ne increase together,
and increase the local coupling of St to Bv; this, in turn, leads to substantial increases first in the violet, and then the red, emission peaks of the doubly-reversed profile. These results again point to the need for a dynamical theory for the velocity fields.
spherical atmospheres: low-velocity regime
Observer's frame calculations in the low-velocity regime have been carried out for radially expanding spherical atmospheres (375). Such an approach can be useful in studying line formation in the deeper layers of expanding atmospheres, but for the large-velocity regime a comoving-frame formulation is preferable. The method is similar to that described in §7-6 for static atmospheres, and carries out a ray-by-ray solution in the same (p, z) coordinate system. The transfer equation along the rav is
±[_vl±{z, p, x)loz\ ~ n{z, p, a) - x(z, p, x)I^{z, p, x) (14-21)
where x(z, p. x) = ic{r) + yd(r)<t>(z, p, x), and a similar expression defines rf{z, p, x); we use the relations r(z. p) = (z2 + />2)*and p.(z,p) = z/(z2 + p2ý. The profile is defined as <p(z, p, x) = (p\r{z. p); x - p(z, p)V(r)], The velocity V(r) is positive in the direction of increasing r. Introducing the optical depth along the ray
z(z, p, x) = j*™ x(z\ p, x) dz (14-22) and defining      u(z, p, x) = ^ [7 + (z, p, x) + ľ(z, p, -x)] (14-23)
and y(z, p. x) = ^ [7+(z, p, x) ~ I~(z, p, -x)] (14-24)
we may rewrite equation (14.21) in second-order form:
[ô2u(z, p, x)/ôt(z, p, x)2] = u{z, p, x) - S(z, p, x) (14-25)
where S{z, p, x) = f/(2, p, x)/x{z, p, x) has the general form S(z, p, x) = y.(z, p, x)J[ŕ'(z, p)] + p(z, p, x). Here a and p contain combinations of the line parameters s and r0 = xdXh an(l the profile function <j)(z, p, x\ while
J(r) = dx Jo ^ ^ x - f-lV(r)]u[z(r, p), p(r, p), x] (14-26)
In formulating the boundary conditions a difficulty arises. On the axis z = 0 we can no longer write u(0, p, x) = 0 because two frequencies ( + x) of radiation are involved. We may circumvent this problem by following the ray for its entire length—i.e., we consider the whole interval [ - zmax, zmjI(]. The lower
460      Radiative Transfer in Moving Atmospheres
14-1   The Transfer Equation in the Observer's Frame 461
and upper boundary conditions for rays that do not intersect the core then become
[du{z,p,x)/di;{z,p,x)]z=±z^ = ±u(z, p, x% = ±,mji (14-27)
J{rd) *   I   w„ £ adi4>ird-xn - p(rd, Pl)V{rd)-]udiH (14-29)
But, from the spherical symmetry of the problem, I±(z, p, x) = p, x),
and thus u(z, p, -x) = u(-z, p, x), and u(j, p, -x) = -u(-z, p. x); these relations allow elimination of the values of u at negative x and positive z, in equation (14-29), in terms of u at positive x and negative z. Thus
N I,,
J = X w» Z '^Wfe** " WhKi]"j,-k + + /idi^]^'(„j (14-30)
it=l i=l
where tf = D; + 1 - d. Equation (14-30), when used in the Rybicki method, yields V-matrices that are rectangular chevron matrices. Using equations (14-28) for all values of i and n in equation (14-30), we obtain a final system for J, which is then solved. The computing time required for the solution scales as TR - cN D3 + c' D3; this is less favorable than the result for the planar case, because now there are about as many angles (i.e., impact parameters) as there are depths. The method is stable and easy to use for small velocities (i.e., a few times thermal). For larger velocities, the number of depth-points required to resolve the velocity field becomes large, and the
computing time is prohibitive; in this case one may use a comoving-frame method. An advantage of the observer-frame method is that it can be used for arbitrary variations in the velocity field (e.g., nonmonotone flows), which is not true for comoving-frame methods, as currently formulated.
Exercise 14-3: (a) Verify the symmetry relations quoted above for ]±, u, and v, and the reduction of (14-29) to (14-30). (b) Sketch the form of the rectangular chevron matrices U,n and Vin in the Rybick i scheme, and show that the dimensions of the matrices mesh in the correct way to allow a solution.
The method described above has been applied (375) to highly idealized spherical atmospheres with power-law opacities and linear velocity laws of the form V[r) = V[R)(r - rc)/(R - Calculations were made in extended isothermal models with R/rc = 30, for a line with r. = 10"2 and 10~4 and zero background continuum, with V(R) = 0, 1, and 2. The results include the following, (a) The source function is more strongly alfected by extension than by small velocity fields; to a first approximation, one may use the static spherical source-function in a velocity-dependent profile computation, (b) The effects of extension and velocities are more significant for smaller values of s. (c) The dominant effect of the velocity field is to reduce photon trapping, and hence to increase the photon escape probability, (d) The line profiles become skewed lo the red as the central absorption feature obliterates the violet emission peak [true also in planar geometry, cf. Figure (14-2)]. In a second sequence of models, the maximum velocity was held fixed, V(R) = 2, and R(rc was chosen to be 3, 10, and 30; B = r~2, e = 10"4, xdXi = 10~4, and yd = Cr~2. The effects of velocities on the source function are small compared to the changes produced by spherical geometry, and the relative departure from static results increases as R decreases, probably because the velocity gradient increases. The emergent profiles all show a strong P-Cygni character. A final calculation worthy of mention is a case in which the velocity is constant throughout the atmosphere; i.e., V(r) = V0. For a planar medium the source function would, of course, be unchanged (though the flux profile computed by averaging over the surface of a star changes; recall Exercise 14-1). For a spherical medium however, the radii diverge from the center, leading to a transverse velocity gradient that decreases photon trapping; as a result, the escape probability increases, and the source function decreases. Even a modest constant velocity of expansion has quite dramatic effects on emergent flux profiles.
HFFI-CTS OF LINES ON ENERGY BALANCE TN MOVING MEDIA
The energy balance in the outer layers of an atmosphere can be dominated by spectral-line contributions. Hence Doppler shifts, which may move a line away from its rest position and allow it to interact with the continuum (in
For rays that intersect the core, p si rc, we either (a) apply a diffusion approximation at an opaque core (stellar surface), which yields r(zmill, p, x) directly, or (b) for a hollow core (nebular case), apply equation (14-25) at zmin (forcing the points at ±zmh, to be identical), and equation (14-27) at the ends of the ray.
To solve the system we introduce the same discrete meshes [rd] and {pj used in §7-6 to solve the static problem. The frequency mesh now includes the whole profile {x,,}, n = ± 1, .. ., ± N. with x_„ = -x„; we shall, however, be able to eliminate half of these (see below). We again obtain equations of the form of (6-27) and (6-48), and hence can apply the Rybicki method to obtain J. Because J(r„) need be defined only for \rd),   1 ^ d ^ D, while
udin    u(zd, ph xj is defined on a mesh {zd.},   4=1,----Dh which runs
the whole length of the ray, it now turns out that, while the tridiagonal T-matrix is square, the U-matrix is rectangular, and is a chevron matrix. Solution of these systems for each choice of (i, n) yields an expression of the form
u„ = A,.„J + B,„ (14-28) Equation (14-26) defining J can be written in the discrete form
jv Id
462
Radiative Transfer in Moving Atmospheres
14-1   The Transfer Equation in the Observer's Frame 463
which the intensity can be markedly different), can significantly alter the temperature distribution. Qualitatively, we can expect three effects to be present, over and above the usual boundary-temperature change and back-warming that occur in static media. (1) Lines Doppler-shifted away from their rest frequencies can intercept continuum photons from deeper layers; we may call this the irradiation effect. The absorbed continuum flux provides additional energy input to the gas and, because the color temperature of the flux exceeds the ambient temperature locally, irradiation will lead to a net heating of the outer layer. The effectiveness of the input is determined by the strength of the coupling of the lines to the thermal pool; thus heating will be greatest when e = 1, and should be negligible for e 0. (2) As the lines shift away from their rest positions, photons that would have been trapped in the deeper layers by overlying line absorption now encounter only continuum opacity, and hence may diffuse freely to the surface and escape; we may refer to this effect as escape-enhancement. In general, an increase in photon escapes will lead to a cooling of deeper layers. (3) A velocity gradient in the atmosphere smears the lines over a larger bandwidth, thus impeding the free flow of photons; we may call this the bandwidth-constriction effect. At depth in the atmosphere, where the diffusion approximation is valid, a velocity gradient causes little, if any, change in the temperature structure when the scale of the velocity variation is large compared to a photon mean-free-path. But if a velocity shift of the order of a Doppler width occurs in a mean-free-path, then bandwidth constriction leads to a decrease in the effective radiation-diffusion coefficient, and hence to increased backwarming. In the extreme limit of an abrupt velocity step near the surface, photons emerging from lower layers in formerly-open continuum bands encounter opaque material, and have their escape impeded; this might more appropriately be termed a "backscattering" or "reflector" effect.
Studies of the influence of line-shifts on energy balance have been made for highly schematic picket-fence models, in planar geometry with an abrupt velocity jump (428), and in spherical expanding atmospheres (445). The velocity-step can be regarded as a caricature of a shock front. The escape-enhancement and irradiation effects show very clearly for the velocity-jump, where the layers above thejump heat markedly (one case gives AT — 1100°K for Teff = 10,000°K), and a cooling of a few hundred degrees occurs immediately below. For the expanding spherical atmospheres one obtains large irradiation-effect rises at the surface, and substantial backwarming below. The irradiation-effect temperature rise is larger for extended models than for nearly-planar models; this is because the discrepancy between the color temperature of the flux, and the ambient temperature characteristic of the local energy density, becomes larger as atmospheric size increases. For one extreme case, the velocity field produces a change AB/B » 3, which implies AT IT a 0.33, or AT - 10,000°K for an O-star.
Even though the models upon which the results quoted above are based are very schematic, it is clear that velocity-field effects on line-absorption can lead to very substantial changes in the energy balance of the outer layers of stellar atmospheres. These changes could, in principle, influence the hydrodynamics of the flow. Thus, in a pulsating atmosphere, energy deposition in the lines produces a kind of radiative precursor that could affect shock propagation; for expanding atmospheres, significant energy deposition could occur in the transsonic flow region, which might alter the nature of the stellar wind. It is also possible that velocity-dependent line-absorption contributions to energy balance could affect the flow-dynamics of novae, supernovae, and mass-exchange in binaries. Much further work remains to be done on this subject.
line-formation in turbulent atmospheres
As mentioned earlier, the dichotomy of velocity-field effects on spectrum line-formation into the "microturbulent" and "macroturbulent" limits is obviously oversimplified. In these two extreme limits, the effects of the velocity field upon line strengths and profiles may be predicted from simple phenom-enological arguments. To study the influence of velocity fields that have a scale which is neither zero, nor infinite, with respect to a photon mean-free-path, a detailed computation must be made. One could, in principle, specify a particular run of the velocity, solve the equation of transfer, and average over as many realizations of the velocity field as are necessary to embrace the possible ranges of its inherent degrees of freedom. Such an approach, however, would be costly, and would not yield direct insight into the problem. An attractive alternative (motivated by the expectation that in stellar atmospheres the velocity field is chaotic, and possibly even turbulent in the hydro-dynamic sense) is to assume that the velocity is a random variable, described locally by a probability distribution for the amplitude, and nonlocally by a characteristic correlation length.
Considerable progress in solving the transfer equation in turbulent media has recently been made with two distinct approaches. One formulation, developed by the Heidelberg group (70, 325; 234; 235; 236; 557), employs the joint probability P(z; v, I) that, at point z, the velocity lies in the range (v, v + dv\ and the intensity in the range (7,1 + dl\ for Markov-process variations of v and /. P, or some other suitable distribution function derived from P, is found by solving a Fokker-Planck equation. This method is powerful and general, and allows the treatment of velocity fields that are continuous functions of depth. However, the resulting partial differential equations are difficult to solve, and a succinct description of the method would presume considerable familiarity of the reader with the mathematical methods for treating Markov processes. A rather different formulation has
464      Radiative Transfer in Moving Atmospheres
been developed by the Nice group (49; 226; 227). The flow is conceived as consisting of turbulent eddies or cells. The velocity is taken to be uniform within each cell, and to jump discontinuouslyd\ sharp boundaries that separate the cell from neighboring cells with uncorrelated velocities; this description is called a Kubo-Anderson process. Although the discontinuity of the velocity structure is unphysical and introduces some artifical high-order correlations in the field (49; 70, 325), this approach nevertheless has the advantages of yielding exact analytical results in certain limits, and of expository simplicity. We shall therefore describe the Nice method here, but quote results from both bodies of work.
We assume that the cell boundaries, at which the velocity changes, are located at random continuum optical depths {t„}, distributed according to a Poisson law characterized by an eddy density n(z), which gives the reciprocal of the correlation length i (in continuum optical depth units) of the velocity field. The probability that no jump has occurred on the interval (t', t) is given by exp[ — n{z") dx"~\. Let vh(z) denote the hydrodynamic velocity of the cell at t ; these velocities are independently distributed according to a probability distribution function P(vh), which we shall take to be Gaussian. Let vth{z) denote the thermal velocity of the line-forming atoms at z, and measure all velocities in thermal units, uh(z) = vh(x)/vth(x), and frequencies from line center in units of the corresponding Doppler widths, x = Av/Avd(t). Characterize the turbulent field by a dispersion q (in thermal units); then
P(uh) = exp(-Wh2/f V(ti^)
(14-31)
The transfer equation to be solved in
fi[dl{z, & x)/5z] = + yj<px)I(z, ft, x) + XA + XiSi (14-32)
where tpjr) = (p[x — piuh(rj]. Equation (14-32) is a stochastic equation—i.e., the coefficients in the equation are random variables. We now assume that the turbulent velocity field influences only the line absorption coefficient via Doppler shifts; fluctuations in the continuous opacity, source functions, and occupation numbers are expected to be of secondary importance, and are ignored here. We adopt LTE and ignore scattering, so that St = Sc - Bfz), and employ a Voigt line-profile so that
buWM/ZcW = MH[a(x),x - mi(z)] (14-33)
where       flr) = (n<e2jmclf^if - cTftv'kT)]/|>f(T) Avd(t)] (14-34)
and a(z) = TjAn Avd(t) (14-35)
Here T is the damping width, and n; denotes the occupation number of the lower level of the transition.
14-1   The Transfer Equation in the Observer's F)
<-ame
465
We may obtain an analytical solution of the problem if we introduce the following additional assumptions: (1) set Bv(z) = B0(\ + ax); (2) use a Milne-Eddingionmodel—i.e.,/3 - constant,a = constant,AvD = constant; (3) adopt a constant eddy density n(z) = n. All of these assumptions may be relaxed if the problem is solved numerically (49; 226). To simplify the treatment still further, we consider the intensity emergent at disk-center, and set /t = 1. The transfer equation thus becomes
[dl(z,x)/dx] = [1 + /5if,(T)][/(i,x) - B0(\ + az)] (14-36)
where Hx(z) denotes H[a, \ — «h(t)] and is constant between successive jump-points. If we define
ijx(z) = exp j-J"J [1 + /jtf(T')] f/T'j
then the emergent intensities in the continuum and the line are
Ic = ^' B0{\ +xz)e-'d-and /(0, x) = JoM B0{\ + ax)qx[z)\_\ + dz
(14-37)
(14-38) (14-39)
It is easy to show that /, = B0(l + «), and that the intensity in a line of infinite strength (fj -> cc) is B0; the absorption depth of such a line will be A0 = a/(l + a). Then, in general, the line absorption depth ax = [Jc - 1(0, x)]//c can be written
(ax/A0) = I - J* ^(T)
dz
(14-40)
The ensemble average of the line profile over all possible realizations of the velocity field is thus
(a
(14-41)
while the average of the reduced equivalent width is
<H/*> = (Wv/A0 AvD> = A0~l f" <ax) dx (14-42)
Thus the crux of the problem is. the calculation of (qx(%))>.
The function \qx{r)/ can be found from the solution of an integral equation, which is obtained by the following arguments. First, at a given point t, the probability that no jump has occurred on (0. t) is exp( — nz); the corresponding contribution to <<ja(t)> is then expf - nxKqx(z)ys, where the
466      Radiative Transfer in Moving Atmospheres
14-1   The Transfer Equation in the Observer's Frame 467
static average is
<cM>s = <exp[-(l + £H,)t]>
=       exp{-[l + m(a, x - uh)-]x}P(uh) duh (14-43)
Here uh (and hence Hx) is constant over the entire interval (0, t). On the other hand, suppose that one or more jumps have occurred on the interval (0, t), and let x'{<x) denote the last jump point. On the interval (%', t), Hx will be constant, hence
qjx) = exp[-(t - t')(1 + ^)]<yV) (14-44)
The probability that the last jump, at occurs between %' and x' + dx' is exp[-n(i - t')]« dx'; thus averaging (14-44), and summing over all x', we obtain the contribution to (qx(x)} from jumps on (0, x), namely
<<L^)>jumP = /J <cxp[-(t -        + PHJ]qJ?')> exp[-n(r - x')~\ndx'
(14-45)
Now all steps in HJx") on the interval 0 < x" < x' are independent of the value of     on the interval (x', x), hence
<exp[-(t - t')(1 + mx)]gx(x')} = <exp[-(r - t')(1 + ^j]><^')>
= - t')>s<^(t')> (14-46)
Adding the static and jump contributions we have, finally,
<fte(T)> = e'"\qM>s + £ <qxWXqx(i ~ t'))^""'1"0" dx> (14-47)
Because we have taken n to be depth-independent, the integral in equation (14_47) is a convolution integral, and we may apply a Laplace transformation to obtain the solution. Thus let
Qx(s)^ ^e-%qx(x)>dx (14-48)
and
Sx{s) = j7 e's\qM)s dx =       dub P(uh) J" exp[-(l + mx)^'n dx
= <[s + (1 + iSffJ]-1) (14-49) Then from equation (14-47) we find
QJs) = Sx(s + n)/[l - nS,(s + «)] (14-50)
Exercise 14-4:   Derive equation (14-50).
We do not require the inverse transform of Qx because, as can be seen by comparison of equations (14-48) and (14-41), the residual intensity can be expressed entirely in terms of Qx{0). Thus we have the general result that
(axy/AQ = i - <(„ + i + mxrl>\i - «<(n + 1 + m.r'yy1
(14-51)
We can recover the macroturbuleni limit by taking the correlation length / = co, or n = 0, so that the entire atmosphere along the ray moves with constant velocity. From equation (14-51) we find
<flv($>macro = A0^HJ{1 + £HJ> (14-52)
This result agrees with our intuitive expectations when we recognize that the argument in the bracket is just the emergent residual intensity of a line in the Milne-Eddington model (cf. §10-3), shifted bodily by a velocity uh, and then averaged over the probability distribution of uh. We may express the general result in terms of the macroturbulent limit as
(ajfi)) = (n + l)<flje[jS/(n + 1)]>™„o[1 + ^o~V*[M" + l)])™™]"1
(14-53)
To obtain the microturbulent limit, we let n -> oo. First, note that for /? -> 0, equation (14-52) can be expanded as <fl;c(i9)>macro * A0(P(H} - P2(H2}), which yields
4o-I<a*(/3)>micro = fl<H{a, x)>/[l + p<H{a, x)>] (14-54) where    <H(a, x)> = (tt^)"1 J__^ H(a, x - uh) exp(-wh2/f2) duh (14-55)
Because the Voigt function is a convolution of a Lorentz profile with a Gaussian [cf. equation (9-34)], we obtain, from an interchange of the order of integration,
<H(a,x)> = (1 + £2r*H[fl(l + fl-i)X(l + a-*] (14-56)
That is, in the microturbulent limit the Doppler width AvD is increased by a factor of (1 + £2)*
In the limit of zero turbulence the standard Milne-Eddington curve of growth is given by [cf. equation (10-38)]
Wofa $ - jZ0 $H(a>       + £H(fl> x)]_1 dx (14-51)
Now, substituting equation (14-52) into (14-42) and interchanging the order of integration, we find W%acro(a, fi) = (W%(a, fi)} = W%(a, /?); i.e., in the macroturbulent limit the curve of growth is unchanged—which, of course,
468      Radiative Transfer in Moving Atmosphere.
14-1   The Transfer Equation in the Observer's Frame 469
is the expected result. In the microturbulent limit, by substitution of equations (14-54) and (14-56) into (14-42), we obtain immediately
inic>,/i) = a + c¥*no/u + a* mi + en
This shows that the linear and damping parts of the curve are unaffected, while the flat part rises by a factor of (1 + ^'2)i. Between these limits the curve of growth is found by numerical integration of equation (14-42), using equation (14-53). In addition to expressions for <ax> and PI7*, it is possible to derive an expression for gx, the rms fluctuation in the residual intensity that would be seen along the slit in a spectrogram of perfect resolution; i.e.,tf, = /IcT'KV) - [see (49)].
Results for the average depth, and its dispersion, of a strong line in an atmosphere with the turbulent-velocity parameter I equal to the thermal velocity, and a density of 10 "eddies" per unit continuum optical depth, are shown in Figure 14-4. Curves of growth for £ = 1 are shown in Figure 14-5. There we see that the dependence of the theoretical curve upon the eddy density, n, implies that a comparison of an observed curve to a theoretical curve with c = 0 cannot lead to a unique value for q. In general, the value oft; deduced in the microturbulent limit will be a lower bound on the actual value. Note that this effect is opposite to that produced by departures from LTE— which, typically, raise the flat part of the curve of growth even when no velocities are present (cf. §11-4).
Detailed calculations, using a realistic solar atmosphere, have been made for the O I AA7771, 7774, 7775 lines, and the Fe I /U5576, 5934, 6200 lines (235); an excellent fit to observed profiles is achieved. In the fitting procedure, the loci of points in the (c, /) plane [/ = 1/n (km)] that match the observed
figure 14-4
Absorption depth in line with 0 = 100, in a turbulent atmosphere with turbulent velocity I = 1, and eddy-density n = 10. Abscissa: x = Av/Avn.   Solid curve: average profile from all realizations of velocity distribution.   Dashed curves: average profile ± the rms fluctuation seen by spectrograph of perfect resolution.   From (49), by permission.
log W*
figure 14-5
Curves of growth in turbulent atmospheres with c = 1. Ordinal?: Logarithm of reduced equivalent width W* = (\\\M0 Avn). Abscissa: Logarithm of line-strength fi = XuXi- Curves are labeled with depth-independent eddy density. From (49), by permission.
470
Radiariue Transfer in Moving Atmospheres
intensities al several values of A/, from line center, as well as the equivalent width, intersect at almost a unique point. Thus it proves possible to determine both u ;md / — 1/n (km) uniquely; numerically c * 2.2 km s"1, and / s: I50km. The velocity is larger than (hat usually adopted as a microturbulent velocity; the correlation length is of the same order as the photo-spheric scale height. It would be of interest to ascertain whether the apparent near-equality of the correlation length with the scale height has hydro-dynamic significance, or is mere coincidence.
Allowance for a finite scale in the velocity structure also has important implications for the interpretation of the center-to-limb variation of solar line profiles. A long-standing problem has been that the profiles at disk center have a characteristic V-shaped appearance, suggestive of macro-turbulence, while those at the limb have a U-shaped appearance, suggestive of microturbulence. Moreover, the microturbulent velocities derived from limb profiles are typically larger than those derived from the same lines at disk center.
This result has been advanced as evidence for anisotropic turbulence (a situation difficult to understand hydrodynamically). Both of these effects can be understood, at least qualitatively, in terms of a finite eddy density for the velocity field (226). If n0(x) gives the eddy density at fi = 1, then the appropriate density at other values of fi is nQ{x)f/i; this follows from the requirement that the turbulence be isotropic, so that we encounter the same eddy density per unit path-length along the ray. If the density is depth-independent. equation (14-53) is unaltered except that we replace n with n0, and now A0 = v.pi(\ 4- a/0~ 1 ■ In general, hp(t) must vary with x. Suppose we demand that the eddies have constant geometrical size L, then nQ(x) = [^A1)^]"1-For the HSRA %c(x) cc t, hence n0(x) cc x~1. Thus the eddy density pertinent to observations near the limb must be higher than at disk center, and this result provides at least a qualitative explanation of the center-to-limb effects mentioned above. An analysis of the Mg I 24571 line has been performed (226) using the HSRA. A good fit to the profiles is obtained with isotropic turbulence characterized by 't — 1.2 km s-1 and \ a 70 km. The smaller value for / quoted here, compared to the results for O I and Fc I mentioned earlier, may reflect the fact that this analysis is based on a Kubo-Anderson discontinuous velocity field while the other uses a continuous field. It turns out that for a given a given line-strength is always achieved at a smaller value of / in the discontinuous case (237).
Non-LTE effects produce line profiles that are deeper in the core, and approach the LTE profile in the wing (227; 236). The deviations of the non-LTE profiles from their LTE counterparts become larger as the turbulent velocity parameter £ increases, and as the correlation length / decreases. Much useful information about the nature of velocity fields in stellar atmo-
14-2   Sobolev Theory      47 \
spheres will undoubtedly emerge from further detailed application of the methods described above to the analysis of the solar spectrum, and to stellar spectra where possible.
14-2   Sobolev Theory
The existence of large-scale, rapid (sometimes violent) expansion in stellar atmospheres is well established observationally. Probably the first objects in which such motions were unequivocally recognized were the novae (and later the supernovae). In their spectra, after the explosive increase in the star's luminosity, one observes absorption lines strongly violet-shifted from their rest positions, indicating material flowing rapidly toward the observer. These lines are accompanied by extensive red-shifted emission features, resulting in characteristic P-Cygni profiles resembling those in Figure 14-6. In nova spectra these features are transients, and indicate episodes of violent ejection of the outer layers of the star. In other objects [the classical P-Cygni stars (79:366)], these lines, though variable, are more-or-less permanently present in the spectrum, and indicate persistent outflow of material. Beals first recognized (75; 77) that the great breadths of lines in WR spectra (indicating velocities of the order of 3000 km s-1) could be interpreted in terms of rapid outflow of material. He suggested that the flow was driven by radiation pressure, a conclusion supported by current dynamical models. Similar conclusions can be reached for the Of stars. We know today that in the WR and Of stars, and in many early-type supergiants. there are transsonic stellar
2.5 2.0 1.5
F(M}/Fe 1.0 1.0 1.0 0.5
		A	
HD1	J0603		
	Jill		k
	Hfi		
	Hy		
			
figure 14-6
P-Cygni profiles of the hydrogen lines in the spectrum of HD 190GO3 as observed by Beals (79).   Ordinate: observed flux in units of the continuum (successive profiles are displaced for clarity), -ibscissa: displacement from line center in velocity unils—i.e., v = c AX/Z.
-200      0 +200 v (km sec~')
472      Radiative Transfer hi Moving Atmospheres
winds (cf. §15-4). that have vanishingly-small outward velocities in the deeper layers, and a large outward acceleration producing very large velocities (v/c a; 0.01) at great distances from the star.
The solution of the transfer equation in a spherical expanding medium is difficult, and in the early work by Beals (75; 76; 77; 78), Chandrasekhar (149), Gerasimovic (243), and Wilson (674) it was assumed that the material was optically thin so that transfer effects could be ignored. This approach, while obviously oversimplified, has nevertheless contributed a good deal to our basic picture of the physical situation. A major advance occurred with the brilliant realization by Sobolev (590; 591; 15, Chap. 28) that the presence of the velocity gradient in an expanding medium actually simplifies line-transfer problems, for it dominates the photon escape and thermalization process, and implies a geometric localization of the source function not present in static problems. In Sobolev's theory the solution of the transfer problem is, in effect, replaced by the calculation of escape probabilities; the basic theory has been refined and extended by Castor (134), and has been applied to fairly realistic calculations of spectra from multilevel atoms in WR envelopes (139; 140).
surfaces of constant radial velocity
Consider a spherically-symmetric, radially-expanding envelope surrounding a star with a fairly well-defined photospheric surface, as sketched in Figure 14-7. With this basic model we can explain qualitatively the main features of P~Cygni profiles such as those in Figure 14-6. For the discussion in this subsection, we shall assume that the envelope is essentially transparent, so that every photon emitted towards an external observer can be received. This approach yields insight, and results that will be useful later. Throughout the discussion we make use of the fact that most of the line emission (or absorption) occurs at line center; thus radiation from a given region, as received by an external observer, appears mainly at the line-center frequency, Doppler-shifted by an amount corresponding to the velocity of the material along the line of sight.
The material behind the stellar disk is in an occulted, region, and cannot be seen by an external observer. The matter projected on the stellar disk can either (a) simply emit radiation without significant reabsorption as occurs in, e.g., a forbidden line in a nebula or in a thermally excited medium where Te » Tc (the color temperature of the radiation from the underlying photosphere), or (b) absorb the incident photospheric radiation and scatter it out of the line of sight. From this material, in case (a) we would obtain a violet-shifted emission feature, while in (b) we obtain a violet-shifted absorption dip characteristic of P-Cygni profiles. From the matter in the emission
473
Observcr
FIGliRF. 14-7
Schematic diagram of expanding envelope surrounding a stellar surface. The material in the occulted region is blocked from view by the stellar disk, and cannot be seen by an external observer.
lobes to the sides of the disk, we receive photons either emitted thermally or scattered from both the stellar and diffuse (from the envelope itself) radiation fields. The velocities along the line of sight in the emission-lobe material range from positive through negative values, and produce a symmetric emission feature extending from a wavelength to the violet of the rest-wavelength, to redder wavelengths. Because material is occulted by the star, the maximum redshifts that could be produced will not be observed, and, in general, we expect to derive information about the maximum flow
474      Radiative Transfer in Moving Atmospheres
14-2   Sobolev Theory 475
velocities from the position of the blueward edge of the absorption feature (or emission feature if no absorption is present). The volume of the emitting region can be enormous compared to that of the star, and the integrated contributions to the observed emission line from this volume may far outweigh the amount of energy received from the stellar disk. As a result, the peak intensities in strong emission lines may be several times the background continuum value (see Figures 14-6 and 14-9). Also, when the size of the stellar disk is much smaller than that of the emission region, occulta lion effects become unimportant. Finally, some lines are much more opaque than others and may, therefore, have a larger effective volume for emission; thus in Figure 14-6 we see a transition from quite strong emission at Ha, to practically no emission at Hy, where we see essentially the photospheric Hy absorption-line.
To make these notions more quantitative, we can compute the energy received at frequency v by an external observer as
£v = J^0-,v)<*ai- (14-58)
where the integration is carried out over the entire unocculted volume. In performing the integration we may use, as was done in our earlier work in spherical geometry, either (r, 8) coordinates with the axis of symmetry from the center of the star through the observer, or ip, z) coordinates (see Figure 7-27). Equation (14-58) can be made more explicit if we write n{r, v) -j](r)(j)[y — v0(l + px>Jc\\, which accounts for the shift of line center, as seen by an external observer, to v0(l + vjc), where u. = pi\ is the velocity along the line of sight resulting from the expansion velocity i;,.. If we suppose that rj(r) -- j]0{p/paf, (where reasonable values for a lie in the range 0 < a ^ 2) and, further, assume v = o0(r/r0)", then from the requirement of continuity, ()vr2 = p0v0r02, we obtain finally rj{r) = t]0(r/r0)~ln + 2la. Choose units such that r0 = y0 = 1 (these quantities referring to values at the photospheric surface), and measure frequency displacements from line center in units x = (v - v0)/AvD) where AvD = v0v0/c. Then
E* = no jv 4>\* ~ Mr)>-(B + 2)a J3r (14-59)
In principle V refers to the entire unocculted volume; in actuality the volume of integration can be defined more precisely.
Most of the emission observed at frequency x will arise from regions where the line center frequency, after Doppler shifting, is at the observed value. The observed flow velocities (up to 3000 kms"1) in WR and Of atmospheres vastly exceed the thermal velocity (~30 km s~!). Therefore the geometrical region from which the emission at any one frequency arises must be a very
thin zone, centered on a surface of constant radial velocity such that v. = uvr = x. (Note: here the term "radial velocity" has the usual astronomical meaning of velocity along the line of sight, not the more fundamental meaning of the velocity vr measured from the center of the star.) In the idealized limit that the width of the line profile is negligible (because t\h(.rmal « r,)pw), the zones degenerate to the radial velocity surfaces themselves, which therefore play a basic role in the theory.
The shape of these surfaces depends upon the nature of the velocity field—which, ultimately, must be obtained from dynamical calculations. VVc can gain insight, however, by consideration of some simple velocity laws of the form v = r" (in units vQ = r0 = 1). (a) Suppose vt = constant. This law could apply to a thin spherical shell (e.g., a planetary nebula at large distance from the star), or in high-velocity flows nearing terminal velocity [see (c) below], (b) Suppose u = r. This law could apply in the case of an explosive ejection that started at some time t0 such that (t — t0) = r/v; here the faster-moving particles outrun the slower ones, giving the linear relation of v with r. (c) Suppose the gas leaves the star with velocities greater than escape velocity. We then can write v = v<yr>(l — rc/r)^, which provides a crude simulation of atranssonic wind; the flow accelerates everywhere on the range rc ^ r ^ cc. (d) If the material is ejected with just the escape velocity and is decelerated by gravity, we may take v = r~*. Each of these laws has a distinctive set of constant radial-velocity surfaces.
Exercise 14-5: Show thai the surfaces vs = constant for cases (a) and (b) are the cones 0 ~ cos-1 p, — constant, and the planes z = constant, respectively, in the usual (r, 0) and (p. z) coordinate systems.
The constant-radial-velocity surfaces for laws of the form (c) and (d) are shown in Figure 14-8 (a and b).
From what has been said above, some far-reaching conclusions can be drawn. The fact that the surfaces of constant radial velocity extend over large regions (infinite if the flow is not decelerating) implies a complete breakdown of the Eddington-Barbier relation for expanding atmospheres. We can no longer associate a given frequency in the line profile with a specific position r in the envelope, but only with a wide range of values (r, r + A/-). From the viewpoint of an outside observer, geometric localization occurs only if variations in total particle density and ionization-excitation equilibria confine the region of high emissivity. What is worse, this conclusion is independent of the intrinsic line-strength {317). So we can no longer, in principle, obtain a depth-analysis of atmospheric structure, with a precision betier than the characteristic Ar defined above, by examining weak and strong lines. Clearly these considerations imply severe modeling and diagnostic
(b) Observer
fig URL 14-8
SurLices of constant radial velocity, r, = constant.
(a) (-{;■) = r . (I - i/r)':. Curves are labeled with r./i;
(b) v(r) = r~±.   From (366), by permission.
14-2   Sobolev Theory 411
problems for expanding atmospheres. The problems are even more severe in the case of decelerating flows where, as can be seen in Figure 14-8b, a particular line of sight may intersect a surface of constant radial velocity at two distinct points; hence two regions, which may have vastly different physical properties, contribute to the information received by the observer. Furthermore, in this case these two distinct regions can also interact radia-tively, and the Sobolev method to be described below requires reformulation.
Taking into account the geometry of the surfaces of constant radial velocity, equation (14-59) can be applied to calculate observable line profiles. For instance, as Beals first showed (75; 76; 77; 78), if we can ignore occulta-tion, then the profile from an optically thin shell expanding with v = constant is fiat-topped; an example of such a profile appears in Figure 14-9, where the X5696 line of C III and the A5808 line of C IV in the spectrum of the hot WR star HD 165763 are shown. The rounded profile for the C IV line indicates it is optically thick (see below). Beals's result can be seen by inspection of equation (14-59) using an (r, 0) coordinate system. We note that radiation at each value of vz (and hence of x in the line profile) is contributed by a conical volume element centered on u. = cos 9 = x, in a range dp; the volumes of all such elements are manifestly identical. Results for other velocity laws may easily be derived so long as the line is optically thin; this approximation will not be true in general, and the need for it is overcome by Sobolev's method.
5700 5800
i'igukl 14-9
Observed profiles of C ITT /5696 and C iv in ihc
WC5 star HD 165763. Notice the flat-topped profile for the transparent /.56% line, and the rounded profile for the optically-thick /.5S08 line.   From (369), by permission.
T
478      Rculuiiive Transfer in Moving Atmospheres
Exercise 14-6: (a) Suppose thai the intrinsic line profile is <p(x) = 8(x). Consider an envelope for which v = vQ = 1. Ignoring occupation, show that Ex = EQ == constant for — 1 ^ x ^ 1, and Ex = 0 for |x| > 1. Derive expressions for £n from equation (14-59). and show thai different results arc obtained for a < 1.5, a. = 1.5, and a > 1.5. Show that for a. ^ 1.5 the envelope must be bounded, r sc J?, but that no restriction is required for 2 > 1.5. Accounting for occupation, show that Ex = 0 for x < ,\'imri, where xmill = —(1 - R~2t, and write an expression for Ex on the range xmjll ^ x ^ 0. (b) Perform a similar analysis tor 0 = r; determine £0 for appropriate ranges of a, and derive analytical expressions for the profiles o£Ex/E0, including occupation effects.
escape and thermauzation in an expanding medium
Consider now the formation of a line in an optically-thick expanding envelope surrounding an opaque core of radius rc; assume that the effects of a background continuum can be ignored. If we now transform to a coordinate system at rest with respect to a particular fluid element, and inquire what happens as we look along a given ray, it is clear that (as a result of the velocity gradient) there will be a differential Doppler shift of each successive sample-point along the ray relative to the test-point. Eventually this shift becomes so large that no line photon emitted within the effective bounds of the line profile (assumed to be limited by some ± xmax) can interact with the line profile at the test point. The velocity field has introduced an intrinsic escape mechanism for photons; beyond the interaction limit (measured from the point of emission) they no longer can be absorbed by the material (even if it is of infinite extent!) but escape freely to infinity. Thus there is a definite limit to the size of the region within which photons emitted, or scattered, can have any effect upon the intensity within the line at the test-point. In the limit of large velocity gradients the interaction region will be small, and may, therefore, be presumed to be nearly homogeneous in its physical properties (temperature, density, ionization state, etc.). The theory then can be formulated in terms of local quantities, and a parameter (1 that gives the probability of photon escape summed over all directions and line-frequencies. In the limit of negligible transfer effects, we can therefore write
J(r) = (1 - jS)S(r) + M (14-60)
The first term is derived from the value that J would have in the limit of no escapes, namely J = S, corrected for velocity-induced escapes. The parameter f}c measures the probability of penetration (summed over frequency and angle) of the specific intensity /c, emitted from the core, to the test-point. We must now calculate /i and
14-2   Soboteu Theory 479
As was done earlier, we measure velocities in units of a thermal velocity [i.e., V{r) = o(r)/vth] and frequency displacements from line center in Doppler units, [i.e.,x = (v - v0)/Avu, where AvD = v0vjc]. The optical depth along a ray to an observer at infinity can then be written [cf. equation (14-22)]
t(z, a x) = J* x(z', p, x) dz' =      yj(r')(p(x') dz' (14-61)
where /■' = (z'2 + p2)\   ^ = {z!/r% and
x' - x'(z\ p, x) = x - Vz{z) - x - pi'V(r') (14-62)
The main contribution to the integral in equation (14-61) must come from the region where x' - 0—i.e., from z' = z0(p,x), where z0 is chosen such that ^o~lV(r0) = x; here r0 = (z02 + p2^. The surface z0(p, x) is, of course, just a surface of constant radial velocity. To a good approximation, we can then replace ya{r') with yd{r0), and remove this factor from the integral. We now change the variable of integration from z' to x'\ in view of equation (14-62) the transformation is
-(dx'/cz)p = (3Vz/dz)p = (t>(z,p)K[r(z, pj]}/dz)p
= fi2(dV/dr) + (f - n2){Vlr) == Q(r,fi) (14-63)
where ji and r are again understood to be functions of z and p. If the interaction region is small the transformation coefficient written above may be assumed to be essentially constant and may be evaluated at the resonance point z = zQ(p, x). Then if we define
<D(x) =       ${£,') d£ (14-64)
where clearly 0(- co) = 0, and 0>(co) - 1, we can rewrite equation (14-61) as
t(z, p, x) = t(- cc. /). x)0[x'(z, p, x)] (14-65)
where x'(z, p, x) is defined by equation (14-62) and
T(-co,p,x) = Zi(f0)/Q(rOs/i0) = T0(r0)/[1 + [i2[(dln V/dhi r) - 1]}0
(14-66)
Here Xi(i-o) = (net/mcWjln&o) ~ (.y;/^)^(r0)J/AvD (14-67)
aild T0(r0) = Xi{ro)/{V/r)0 (14-68)
4&0      Radiative Transfer in Moving Atmospheres
In equations (14-66) through (14-68) it should be borne in mind that r0 and (xQ are functions of p and x, i.e., ri} = rQ\_z0{p, x), p~\ and p0 = Po[zo(P> XX p]-Let us now choose a fixed value of rand calculate /?(r); because of spherical symmetry, integration over p. can be effected by using the above results for various values of p. The escape probability along any ray is just exp( —Atot) where At^ denotes the optical path length from the test-point to infinity. Thus, summing over angle and frequency, we have
/j(r} = l      dp       dx <P[x'(z. p, x)] exp{ -t[z(r, p.), p(r, p), x]j (14-69)
Here we have assumed that photons that hit the opaque core are absorbed and hence lost. To evaluate equation (14-69), we use equations (14-64) through (14-67), and assume that the material in the interaction region is sufriciently homogeneous that the distinction between r0 and r may be ignored. Then
jg(r) = i J_\ dp JJ rf* exp[-Zl(r)0/Q(r, //)]
= Xt'K'-) £ (1 " exp[-2le)/e(r, ^jQfr, ju) <fc (14-70)
For the special case that V = /<r, <2(r, p.) = /c, and equation (14-70) reduces considerably to
jg(r) = [1 - exp[-t0(r)]}/To(r) (14-71)
where now x0(r) = k~~1 Xl{r). The same result is obtained if the angle-dependent terms in equation (14-63) are merely ignored.
To calculate /?c, assume that the test point is relatively far from the core (i.e., that the surface of the core is at — go). Then, from its physical meaning, fic can be written
l r-*
= Zi_1(r)^ J,',*1 _^PL-xiW/etr^Bje^/o^ d4-72)
where ~ [1 — (rc/V)2]*. Again, for the special case of a linear velocity law we obtain a considerable reduction, namely jSc(r) = Wf3(r), where is the usual dilution factor given by equation (5-36). The result just quoted is what would be expected physically, because W is the fraction of the full sphere contained in the solid angle subtended by the disk, while /? measures the probability of penetration from the disk to the test point.
14-2   Soboleu Theory
481
Note that both ji and fic are defined essentially in terms of local quantities: the local opacity and velocity gradient. Given these values, one can compute J from equation (14-60) without actually solving a transfer equation; thus we see the enormous simplification that has been achieved. For the particular case of a two-level atom, where the source function (assuming complete redistribution) is given by S = (1 — e)J + we may use equation (14-60) to write
S = [(1 - e)PJe + eB]/[(I - e)/? + e] (14-73)
which shows that knowledge of/] and /ic is sufficient to determine S. Further, jf we ignore the continuum contribution (j}cIc = 0), equation (14-60) allows us to write the net radiative bracket for each line of a multilevel atom immediately, namely Zjt = we shall exploit this result in the discussion of multilevel atoms.
It is very instructive to consider a uniformly expanding plane-parallel atmosphere, for then we can obtain expressions that show the effects of the velocity gradient on the thermalization of the source function in a particularly transparent way [see (273, 87) and (406)]. Let x denote the integrated line optical depth defined for a medium at rest; assume the velocity gradient 7 = cVjoi is everywhere constant. The specific intensity at a test point z in direction p is
i(T,/i..v) = S(t')exp
■p.     I       <!>(x + ypt)dt
<p[x + yp(z' - t)] dx'/p (14-74)
Thus the source function for a two-level atom is given by the integral equation S(T) = (1 - c)J(t) + cB{z) = (1 - e) j*^ Kp |t' - t| S(t') dx' + nB(x)
where the kernel function
Kp(s) = ^       dx £ dV M 1 <£(*)</>(■* + yps) exp
(14-75)
-p'1 \cb(x + ypt) dt
(14-76)
It is easy to show that, unlike the static case where the kernel is normalized to unity, in the present case the effects of escapes lead to
P K kl dx = 1 - B
J - (JO 11
(14-77)
482      Radiative Transfer in Moving A imospheres
where jí is the planar-atmosphere escape probability that follows from equations (14-70) and (14-63) in the limit that 1/r -> 0, namely
0 = \y\ £ (1 - exp[- \/(\y\ dp (14-78)
Exercise 14-7:   Verify equations (14-77) and (14-78).
Equation (14-75) may be cast into the standard form for a two-level atom by renormalizing the kernel to K*(t) = Ke(r)/{1 - jj), and defining 1 — e* = (1 —       - e) and B*(t) = bB{t)j&*. Then
S(t) = (1 - £*) j"^ X* |r' - t| S(f) df + e*B*{t) (14-79)
When thermalization is achieved, S varies slowly and may be removed from under the integral to yield S(z) = B*(t) = eB{x)/(& + /? - ejS). For e » /?, S(t) -* B(z), as expected. But for p1 » a, escapes dominate and S(t) &B(f)ifi, showing that S decreases to the local creation rate e,B as /J -> 1, which is reasonable on physical grounds. If the medium has a boundary surface and B is constant, then [cf. (406)]
5(0) = (e*)*B* - cB/(e*)* (14-80)
Thus when e » /Í we recover the usual static result S(0) = e*B, while for ft » £, we find S(0) = cB./p1* — p1*^, where denotes the asymptotic value for S at depth.
LINE PROFll.tS
Let us now derive expressions for the line profiles seen by an external observer. The flux emergent at frequency x is proportional to
Fx = 2k Jj0 /(go, p, x)p dp
= 2n j^SffoK 1 - exp[-i(-ao,p, x)]\pdp
+ 2k j'Qc S{r0){\ - exp[-t(-     />, x)0>(x,)]}p dp
+ 2nlc jrQl txp[-z{-co, p,x)<t>{xc)~]p tip (14-81)
where, as above, r0 denotes the value of r at the surface of constant radial velocity specified by x, and xc is the value of x' given by equation (14-62) at ť = rc and p' = [1 — (p/''c)2]*- The first term gives the emission from the part of the envelope seen outside the disk (i.e., p > rc). The second term
14-2   Sobokv Theory 483
gives the emission from the part of the envelope superposed on the core, the factor <S>(xr) correcting for occultation of material by the core. Note that, for an expanding atmosphere, <X>(xe) equals zero for x < 0 and will be essentially unity for x > 0, showing immediately the effect of core occultation on the red wing of the profile. The last term gives the continuum contribution from the core; in view of the properties of 0(xc) just mentioned, we see that it is uuattenuated in the red wing, and more or less heavily-attenuated in the blue wing of the line. The flux in the continuum outside the line is proportional to
Fe = 2nlcfcpdp = nrc2Ie (14-82)
Transforming the variable of integration from p to r on surfaces (z/r) V{r) = x, equations (14-81) and (34-82) can be combined to yield an expression for the line profile Rx = (Fx — Fc)/Fc, namely
Rx = 2(re2/erl f°°    ^MToWM-co.ftxail - exp[-t(-co,p,jc)]]r<fr
*! ' hull (a 1
- 2{ri2l[)-,      S{rc){Qxp[-i(-n,p, x)0(xe)]
- exp[-r(-co.p, xj]}pdp
- 2rc-2|0re{l -exp[-t(~Go,p,x)4>(xc)]}pdp (14-83)
where rmin(x) is the radius at which V(r) = x, and p is regarded as p(i\ x). Note the change in sign convention [relative to equation (8-2)] that has been made to give positive numbers for emission lines. Each term in equation (14-83) can be interpreted in parallel with terms in equation (14-81).
If we ignore the last two terms, from the core, in equation (14-83), we consider a two-level atom for which the envelope is so thick that the source function achieves its asymptotic value S = sB/fi, and we replace t(— cc, p, x) with t0, then, in view of equation (14-71), we may write
R ^ <eBT0> p  bBt0  2r dr = ,4<eBTQ) "v ~    h     *r° <eBr0> i\2 Ic
where <sBt0> is a typical value of sBxQ; here the quantity A denotes the effective emitting area measured in core units. For sufficiently large effective emitting areas, the line can become quite bright relative to the continuum. In fact, most strong emission lines result largely from this geometrical effect.
Another interesting result follows easily from equation (14-81) [see, e.g., (15, Chap. 28)]. Consider an envelope with constant velocity of expansion V; then Q(r, pi) = (V/r) sin2 0, and the surfaces of constant radial velocity are given by cos 0 = (x/V) = constant. The transformation from p to r is
4
3
F/Fc 2
1.0 0.8 0.6 0.4 0.2 0 -0.2-0.4-0.6-0.8 -1.0 (a) {c Av/voi;^)
1.51--r     i i—' i      I   ~J     "i-   i i
1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 (b) (c Av/v0O
io I--1—  i      i-1-T~—r~~1     1      1 ~~
1.0    0.8    0.6    0.4    0.2     0  -0.2 -0.4 -0.6 -0.8 -1.0 <c) ((■■ Av/voO
figurl 14-10
Calculated lint profiles in expanding spherical atmospheres.
(a) e = 0.0092 and t0(max) m 15.   (b) c = 0.002 and :0fmax) ss 0.5.
(c) £ = 0.021 and t0(max) ~ 2-   From <I34>- b^ Permissio11-
14-2   Sobolev Theory 485
p = r sin 0, and in the limit that we can neglect the contribution from the core,
Fx = 2tc sin2 6 J* S(r){l - exp[-Xl(r)r/(V sin2 0)]} r dr (14-85)
If the envelope is opaque, the exponential term vanishes, and the integral becomes a constant, so that Fx = C sin2 6 = C[l — (x/T7)2]; the line profile in this case is rounded (specifically, it is parabolic). This conclusion is of importance because it shows that rounded profiles occur naturally, as a result of optical-depth effects, even if the velocity is constant. In contrast, an interpretation based upon an analysis that assumes the lines are optically thin would necessarily have involved an accelerating, or decelerating, velocity field (which would, of course, have quite different dynamical implications). We thus see that the spectroscopic diagnostic procedure must be carried out with care, and to a high degree of consistency, if physically meaningful results are to be derived.
Detailed calculations of flux profiles, using equation (14-83), have been made (134) using the two-level-atom source function of equation (14-73), along with assumed distributions of V(r), t0(r), and the constants e and B/Ic (the latter chosen always to have the numerical value 5). The velocity law was taken to be of the form V(r) — 1^,(1 — rjrf. The adopted distributions of x0(r) all are characterized by a maximum on the range 1.1 < (''Ac) ^ 4. If monotone decreasing distributions are used, very asymmetric profiles (not observed) result. Presumably this indicates that the lines observed in real stars arise in shell-like zones, produced by variations in the ionization equilibrium that yield a dominance of a particular ion in a definite range of radii. A wide variety of profiles can be produced by suitable choices of the parameters. Three characteristic types of profiles, similar to those observed in WR stars, are shown in Figure 14-10: (a) rounded emission with violet absorption, such as observed in the C III 24650 and N III 26438 lines; (b) flat-topped emission with violet absorption, such as seen in the He I lines; (c) very intense rounded emission with no absorption, such as observed in the He II lines. In each case the intensity of the emission is proportional to A(_EBT0y/Ic as expected from equation (14-84); note that the flat-topped profile results from an optically thin line.
multilevel atoms: application to wolf—r a yet stars
In the spectra of Wolf-Rayet stars, extensive series of extremely strong emission lines can be observed. The spectra fall into two broad classes: WC, in which lines of C and O are prominent while those of N seem to be practically absent; and WN, which have prominent lines of N and essentially no lines of C. The He II Pickering series (n = 4 -> n') is very strong and, by
486     Radiative Transfer in Moving A tmospheres
14-2   Soboleo Theory 487
comparison of lines with odd n (not overlapped by a hydrogen line) and even ri (blended with a hydrogen Balmer line), it is found that the hydrogen emission is weak, and hence we conclude that the hydrogen to helium ratio must be significantly less than unity. To derive quantitative information about these interesting abundance anomalies, as well as about the physical structure of the envelope, it is necessary to carry out a complete multilevel analysis of the spectrum. At present it is not possible to specify the atmospheric structure in detail, and studies (139: 140) have been carried out in the spirit of a coarse analysis (making a fair number of approximations), with the goal of obtaining estimates of the physical properties at a single typical point in the envelope.
' The statistical equilibrium equations are of the form 0tt + f€-t = 0 where 01- and are, respectively, the net rates at which level i is populated by radiative and collisional processes. We have one such equation for each level of the ion under consideration, plus one additional equation specifying a total abundance for the chemical species. The net collision rate can be written (cf. §5-4)
(14-86)
while the net radiative rate is
+ nfAn J* ail£(v)(/iv)-1Bu(Te)[l - exp(-fev/feTB)] dv
- nAn J" ailc(v)(fcv)-Vv[l - V exp(-/iv//cTe)] dv (14-87)
Here Te denotes the envelope temperature, and nf = nKne<biK(Te) [cf. equation (5-14)] where nK is the actual ion density. The expression for is useful as written, for a given value of Te and ne, but that for m{ must be rewritten. To simplify the bound-bound rates we use equation (14-60) to obtain
nfiAji + BjJij) - «(-Bl7Jo- = ~ " njB^WB^TMj i14^
where is given by equation (14-71), and %u (called t0 there) is given by equations (14-67) and (14-68); in equation (14-88) we have used the approximate result fjc « Wfi, and have parameterized lc in terms of a radiation temperature T,. The bound-free terms are somewhat harder to reduce, because the Doppler shifts produced by expansion scarcely affect continuum formation, and no essential simplification is introduced by intrinsic escapes, as it is for lines; in fact, a solution of the static transfer problem is, in principle,
required. To circumvent the need for a detailed solution, write Jv = Jf +Jvd, where Jf represents the radiation emitted by the stellar core, and J/ is the diffuse field from the envelope. Allowing for absorption, we adopt
Jve ~ WBv{Te)e~^ (14-89)
where a representative optical depth between the test-point (R) and the core (rc) is taken to be
Tv =        v)[l - b~: exp(-hv/kTe)](R - rc) (14-90)
Further, if we assume that (a) the envelope is homogeneous, and (b) the optical depth from the test point to the boundary in any direction is tv> we then may adopt
J/ « SJv)(l - e~^) (14-91)
where SiK(v) = (2/iv3/c2)[Z>,- exp{hv/kTe) - 1]"l. With these approximations we have rate equations of the form s//n = & where ,o/ and ffl contain rate coefficients, line escape probabilities, and analogous continuum quantities. To solve the system, the parameters that must be specified are T„ rc, R, v(R), Te, nc. and the total number density »alom of the species under consideration. Most of these quantities can be specified from independent considerations, and typically only Te, ne, and ?;atom are free parameters to be determined from model-fitting. The system of statistical equilibrium equations is nonlinear because the optical depths tip xv, and the departure coefficients in the continuum terms, depend upon the solution; it is necessary, therefore, to solve the system iteratively. The iteration may be effected using a Newton-Raphson procedure, which yields swift convergence.
To compute line-strengths we use equation (14-83) and, in the spirit of coarse analysis, we assume that Si} and tu are constant for rc < r < R, and ignore angular factors. Then the three contributions to the line profile are constant, and are equal to (a)R(, = (£>/)[S;/Bv(7;)](l - e"^)from the emission component; (b) RQ = -[5y/Bv(rc)](I - e"r,J) from the occulted material; and (c)Ra = -(1 - e~Zii) from the absorption component. The blue half and red half of the line profile each has a width, in wavelength units, of AX = lv(R)/c. Thus the equivalent width (taken positive for emission) is
Wk = (Xv/c)(l ~ e-«'){[(2R2/re2) - l][Sy/B¥(rc)] - 1] (14-92)
If occultation and absorption effects are ignored, the two terms involving " -1" in the above formula are suppressed. Equation (14-92) is only approximate, because of the assumption of homogeneity of the envelope, and could be in error by as much as a factor of two. The above results can be used to
488      Radiative Transfer in Moving Atmospheres
14-2   Sobvlev Theory 489
compute intensities and line strengths if Su and xt} (i.e., the level-populations) are known, or, alternatively, may be used in diagnostics to determine these quantities.
The methodology developed in this section has been applied (140) in a thorough analysis of the He TI spectrum of two WN6 stars; HD 192163 and HD 191765. As a first step, the total line intensities (which show neither absorption components nor occultation effects) are used to establish the level populations empirically. For the line (w -> /) the total intensity is
Iul cc JAulhvulnu(r)i3Jr) dV cc §nul{r)Pul(r) dV
oc jsul(r)[v0v(r)/c]{l - exp[-t;u(r)]} dV (14-93)
where equation (14-71) was used for /?„,, and equations (14-67) and (14-68) for xlu. Therefore, assuming homogeneity,
iWv*,4) = K{\- exp[-t^K)]}/[(^/^nJ - 1] (14-94)
where K is the same for all lines. For any line that is optically thick, we can set the exponential term to zero; this is expected to be true for A4686 (tt — 3 _>. n = 4). From the observed intensities, the "reduced intensities" (relative to /A6S6)Jrul = {1'Jv„,*)/{/43/v434) can be formed. From equation (14-94),
,ful = [1 - exp(-tju)][(g4n,/^3«4) - l]/[(ftA/ff,nJ - 1] (14-95)
where we have set exp(-t34) = 0. Further, we can write
x!u = Alfnjgt) - (njgjjigjtuiyinjgj (14-96)
where ,4 = {Ke2/mc\(nJg^)\_R/LiR)'] (14-97)
If we now assume that both the A3203 (5 3) and the A10124 (5 -> 4) lines are also opaque, we may solve for the numerical value of (g^jg^nf) = JsA\y — Jr53)/-ir53, and S43 in each star. The three lines considered thus far are, in fact, opaque, but this need not be true for higher series members which have much smaller/values. Suppose, however, we presume that all Pickering-series line are opaque; then from equation (14-95) we can obtain empirical values for (g^njgju). At reasonable values of nc and Te, the upper levels of the He+ ion should become dominated by collisions, and their occupation numbers should have the LTE values
(njgj = nionne ffi/lxmkTrf expfo/fcTJ (14-98)
for u > 10. Because '/JkTe « 1 for u » 1, {njgu) should approach a constant value. This, however, is not found for the empirical values just obtained; but rather (n,jgu) cc (/2)4u, as expected for optically thin lines because of the factor [1 - exp( - t4J] k x4u cc (/).)Au for xAu « 1. We thus conclude that the upper Pickering-series lines are optically thin; this implies an upper bound on the parameter ,4 defined in equation (14-97). By imposing the physically reasonable requirements that (njgj) (a) be a monotone decreasing function of u (i.e., no population inversions), and (b) become constant for u > 10, we may set limits on A; the results are 3 < A < 6 for HD 192163, and 2 < A < 8 for HD 191765.
As a next step, one may use the known absolute magnitude Mv to obtain, from standard relations, the absolute continuum flux at A5500 A; if we adopt Tc x 40,000°K, as indicated observationally, we can then deduce rc =t 13RQ. Using the observed ratio FC(;.4686)/FC(>15500), we find Fc(a46S6(; then using the observed value of the emission line strength J?e(/,4686) we can find R2SAi. This in turn yields R as 70 RQ because S43 is already known. The ratio (R/re) as 5-6 is in agreement with direct interferometer measures for the WC star y2 Vel, and also explains the absence of any occultation effects in the profiles. If we adopt v(R) = 1000 km s"1 from the observed linewidths, and use the known values of A and R in equation (14-97), we deduce a numerical value for (njgui and hence for all in,, </„). from the empirically determined ratios (g^n^/g,,!!^). If these values are used in the Saha equation (14-98) for u > 10. and if we adopt Te - 10s ftK and set ne = 2n,- (i.e., a completely ionized atmosphere of helium), we find ne « 5 x 10" cm3, which implies that the optical depth of the envelope in electron scattering is xe neaeR = 1.5. Finally, knowing that the upper lines are optically thin, the observed excess emission in Pi 14 (which is blended with H 7) relative to Pi \ 5 can be used to estimate n(H + )/n(He + + ) n(H)/n(He) < 0.5; in these stars the helium-to-hydrogen ratio is thus enhanced by a factor of 20!
Having fixed rc, Tc, R, v(R), and the rate equations can be solved for various assumed values of Tt, and rc(He); the results yield theoretical values of A and(n,/Vi). Calculations (140) were done for a30-lcvel atom. Good agreement with the empirical results are obtained for n(He) % 2.5 x 1011 cm-3 and Te % lO""' "K (the fit is not unique). The calculations show that the upper levels are indeed collision dominated, and that bu -» 1 for u » 1. Further, because Te > Tc, it appears that a nonradiative source of energy input is required to maintain the excitation of the envelope.
A second application of the methods described above has been made in an analysis of the C III line strengths (in the ultraviolet, visible, and infrared) observed in the WC 8 star y2 Vel (139). In this work, independent estimates could be made of the parameters rc. Tc, R, andu(R). The values of Te, nc, and /i(C++) were determined by fitting the observed equivalent widths for
490   Radiative Transfer in Moving Atmospheres
14-3   The Transfer Equation in the Fluid Frame 491
10 lines to theoretical widths obtained from equation (14-92), using occupation numbers from a model atom consisting of the lowest 14 terms of C+ +. Transitions to higher levels and to the continuum were ignored; this compromises somewhat the accuracy of the upper-state populations. Good agreement with observations was obtained for
Te = 22,000°K,   ne = 4 x 1011 cm"3,   and   «(C++) = 1 x 109 cm"3
[having adopted Tc = 30,000°K, (R/rc) = 3.6, and v(R) = 900 km s"1]. The lowest four excited states are found to have a Boltzmannian distribution relative to the ground state, while the higher levels are underpopulated. Assuming that all the carbon is in the form C+ H", and that all the electrons come from hydrogen, a lower bound is found for the ratio «(C)/n(H)—-namely, 2.5 x 10~3. This value exceeds the normal cosmic abundance by a factor of 8, and suggests that the carbon abundance in WC stars is indeed enhanced, and that the prominence of the carbon lines in the spectrum is a result of this enhancement. Allowing for other ions of carbon and for electrons from other sources (e.g., He) would raise the lower limit mentioned above. For y2 Vel, Ts < Tc, which does not present a compelling argument for nonradiative energy deposition, and even offers the possibility that the envelope may be in radiative equilibrium.
14-3   The Transfer Equation in the Fluid Frame
The formulations of the transfer equation discussed in the preceeding sections of this chapter are all based in the stationary frame of the observer, who views the stellar material as moving. As we have seen, the complication in this approach is that the opacity and emissivity of the material become angle-dependent, owing to the effects of Doppler shifts and aberration of light. There results an inextricable coupling between angle and frequency that presents severe difficulties in the calculation of scattering terms with a discrete quadrature sum. It then becomes attractive to treat the transfer problem in a frame comoving with the fluid.
There are two strong motivations for working in the moving frame of the material. (1) From the point of view of the transfer equation itself, there is the advantage that both the opacity and the emissivity are isotropic in the comoving frame. Further, in problems involving partial redistribution effects, we may use the standard static redistribution functions. In addition, in the calculation of scattering integrals we need consider only a frequency bandwidth broad enough to contain fully the line profile; this bandwidth is independent of the fluid velocity. Finally, the angle quadrature may be chosen on the basis of the angular distribution of the radiation alone.   (2) Dynamical
calculations in spherical flows (e.g., pulsation, expansion) can be handled accurately in a Lagrangian coordinate system (i.e., the comoving frame). The Lagrangian equations of gas dynamics are easy to formulate, and offer many physical and computational advantages. Obviously it is desirable to be able to treat the radiation field in a closely parallel way. On the other hand, a disadvantage of the comoving-frame formulation is that present methods of solution work only for relatively simple velocity fields; otherwise it becomes too difficult to pose boundary conditions on the problem (cf. Exercise 14-12 below).
In the comoving frame we shall develop both (1) the monochromatic equation of transfer, used for calculating, e.g., line profiles, and (2) frequency-integrated moment equations, which specify the total radiative contributions to the energy, momentum, and pressure of the gas-plus-radiation fluid.
To obtain expressions describing how the relevant physical variables change between the rest and the comoving frames, Lorentz transformations are applied. Here we encounter a problem: strictly speaking, a Lorentz transformation applies only when the velocity v of one frame relative to the other is uniform and constant. But in stellar atmospheres we are concerned with situations where v = v(r, t), and hence the fluid frame is not an inertial frame. We must then imagine transformations taking place from uniformly-moving frames that instantaneously coincide with the moving fluid. Actually, the difficulty just mentioned introduces considerable complication into the analysis. It is easy to show that the form of the transfer equation is the same in two uniformly moving frames (i.e., the transfer equation is covariant), providing we account for the effects of Doppler shifts and aberration of photons when we calculate atomic properties. Further, it is straightforward to derive the behavior under transformation of the atomic properties themselves. But for unsteady or steady differential flows, new terms appear in the equations that account, in effect, for changes in the Lorentz transformation from one point in the medium to another; these terms can be derived by application of the differential operator (c~x ajdt + n ■ V) in the transfer equation to the transformation coefficient of the specific intensity.
the local frequency transformation
Before discussing the details of transformation of the physical variables in the transfer equation, it is worthwhile to extract the essential physical flavor of the problem in simplest possible terms. A velocity field produces a Doppler shift and aberration of photons, and gives rise to advection terms describing the "sweeping up" of radiation by the moving fluid. Formally, these terms are all of order (v/c). However, in the case of line-profiles, the effect of a frequency shift Av becomes important not when Av/v = v/c is significant,
492
Radiative Transfer in Moving Atmospheres
but rather when Av/AvD = (r/r!h) is significant (i.e., of order unity); in essence, velocity-field Doppler effects arc amplified a factor of (c/vlh) by the swift variation of the line profile with frequency. To a first approximation, then, it is sufficient to consider only Doppler shifts and to ignore aberration and advection.
If v is a frequency seen in the observer's frame, then v0 = v(l — pv/c) is the corresponding comoving-frame frequency. The differentia! operator p(d/dz) in the observer's frame (for a time-independent, planar atmosphere) is evaluated at constant frequency v; but if we move a distance Az, holding v fixed, v0 = v0(v, z) will change because v changes. Thus [0/0z)v (d/8z)VQ + {Ov0/0z)v(8/dv0)2lj. Clearly (cv0/cz)v = -(vQp0/c)(dv/cz), to order (v/c). Substituting into the transfer equation we thus obtain
p0[dl°(z, p0, voydz] - [(po2^/c)(dv/dz)][d!0(z, tlQ, v0)/r;v0]
-Az,v0) -X°(z,v0)nz,p0,v0) (14-99)
The corresponding result for spherical geometry is d!°(r, p0, v0) , (1 - Po2) dl°{r, p0, v0)    /v0«
or
dp0
cr
(1 - p02) + Po'
d In v dhir
csV\r, p0, v0)
= '/V.v0) - /(r,v0)i°(r,/to.v0) (14-100)
Exercise 14-8:  Derive equation (14-100).
Several points should be noticed about equations (14-99) and (14-100). (1) The variables /, p, and v are now all in the comoving frame, and in that frame % and r\ are isotropic. (2) It is clear that any scattering terms need be evaluated only on a (small) definite range of v0. (3) The transfer equation is now a partial differential equation. The frequency derivative accounts for the change, with position, of a given photon frequency v0 in the comoving frame, as seen hy an external observer—or, equivalent^, for the frequency shift of photons as seen in the comoving frame. In particular, suppose the atmosphere is expanding so that dv/dz (or dv/Or) is greater than zero; we then see that photons arc always systematically redshifted as they transfer from one point in the atmosphere to another, as expected intuitively. Note that in planar geometry only velocity gradients matter; in spherical geometry a net effect occurs even when v(r) = constant, because divergence of the rays still implies a transverse velocity gradient in this case. (4) From a mathematical point of view, equations (14-99) and (] 4-100) yield a hyperbolic system of equations [cf. (181. Chap. 5; 530, Chaps. 9 and 12; 462, Chap. 4)], and pose an initial-boundary-value problem requiring boundary conditions in the spatial coordinate and initial conditions in frequency to effect their
14-3   The Transfer Equation in the Fluid Frame 493
solution. We shall discuss a numerical method for solving these equations below.
Equations (14-99) and (14-100) contain all the essential physics of the transfer problem, but there are additional angle-dependent terms that have been omitted by our neglect of aberration and advection; to obtain these terms (which are important for the fluid equations), we must now develop the transformation properties of the relevant variables.
lorentz transformation of the transfer equation
We consider here transformations between the rest system, specified by the four coordinates (x1, x2, x3, x4) = (x, y, z, ict), and the fluid system (x0, >"0, z0, ictQ), moving relative to the rest system with a constant velocity v in the z-direction. This choice for v is the one most physically important to our work, and simplifies the calculation; generalizations for an arbitrary orientation of v are given in (621). Changes from one system to the other are effected by means of a Lorentz transformation, which corresponds to a proper rotation in four-dimensional space-time. Physically the Lorentz transformation is chosen in such a way that the equation for the wavefront of a light wave is of the same form (covariant) in both systems (i.e., such that the velocity of light is always = c in both reference frames).
The mathematical form of the transformation isx0a - L/x^' (a = 1,..., 4) where the Einstein convention of summing over repeated indices is employed. The transformation can be represented [cf., e.g., (392, 29; 253, 191)] by the matrix
L =
11	0	0	0
0	1	0	0
0	0	y	m
|o	0	-ifly	y
(14-ior
where y = (1 - v2/c2)~i and f} = (v/c). Note that L is Hermitian; i.e., L = L' where "t" denotes the adjoint (i.e., conjugate transpose). Note also that L ~1 = LT where "T" denotes the ordinary transpose. In matrix notation, x0 = Lx, where \0 and x are column vectors. Clearly x = L_1x0 = LTx0; equivalently, xa = (L~l)pxx0li.
The Lorentz transformation can be applied to arbitrary four-vectors and to four-tensors of rank two. The transformation rules of tensor analysis assure that these quantities are covariant under the Lorentz transformation (because it is a proper rotation in four space); hence physical laws written in terms of four-vectors and four-tensors are automatically covariant. The transformation of an arbitrary contravariant four-vector A [e.g., a space-time increment (Ax, Ay, Az, i'cAf)T] is defined such that A0* = {rx0a/dxf*)Als = LpaAp, or, in matrix notation, A0 - LA, A = L_1A0. The transformation
494      Radiative Transfer in Moving Atmospheres
14-3   The Transfer Equation in the Fluid Frame 495
of a covariant four-veclor B (e.g., the gradient operator [djdx, d/dy, d/dz, (/c)"1 r/ctf) is defined such that (B0)a = {dxfi/dx0a)Be = (L~ which, in matrix notation, is equivalent to B0T = BTL 1 or, transposing, B0 = (L_1)rB = LB;also,B = L-1B0. Finally, the transformation ofCa", acon-Iravariant tensor of rank two, is defined such that C0xli - L*L/Cy6, which (in matrix notation) is equivalent to C0 = LCLT = LCL"1 (i.e., the transformation is a similarity transformation); also we have C - LTC0L = L" 1C0L. (Note: The word covariant describing the four-vector B is used in a different sense from that used two sentences earlier in relation to physical laws, where it meant "of the same form"; this double meaning, however deplorable, is standard usage.)
Equipped with the transformation rules given above, we can now derive a number of important results. First, applying the transformation to the coordinates themselves it is easy to show that the measurement of intervals perpendicular to the z-axis is unaffected by the relative motion of the two frames—i.e., Ax0 = Ax and Ay0 = Ay. But an object of length As at rest in the fixed frame will be measured by an observer in the moving frame to have a length
Az0 = y-1 Az (J4-102a)
This result is the celebrated Lorentz- Fitzgerald contraction effect. Tikewise a time-interval At measured in the fixed frame will be measured by an observer in the moving frame to be
At0 = yAt (14-102b)
This is the time-dilation effect. From these results we conclude that a space-time volume element is invariant; i.e..
dVdt = dVn dt
0 Mt0
(14-103)
a result we shall use repeatedly below. Next, applying the Lorentz transformation to the four-gradient (a covariant vector), we obtain
3 d d 1 d ox" dy' cz' ic dt
oo to
ß±) i(JL-.cßJL
c dtQr ic \dt0 dz0
(14-104)
Let us now turn to the transformation of the radiation field and transfer-related variables [further discussion can be found in (521; 551; 621)]. For any particle, the four-momentum is Pa = (px, py, pz. iE/c)T, where p} is the ;th component of the ordinary momentum, and E is the total energy of the particle. If the particle has rest mass m0, then p2c2 + (m0c2)2 - £2, where p2 = p2 + py2 + p,1. Photons have m0 = 0and£ = /iv,hencep = /iv/c,and
= (hv/c)(nx, riy, nzy if = (/iv/c)(n, i?
(14-105)
Here n is the direction of photon propagation, and (nXi ny, nz) - (sin $ cos <p, sin 6 sin 0, cos 6*). Applying a Lorentz transformation to equation (14-105) we find
O'o"*0, v0ny°, vQnz°, iv0) = [ynx, vny, vy{nz - ivy{\ - njj] (14-106) which is equivalent to
= [>; y-\\ - /r)V(l - /xjS); {ft - R)/(l - rf)- vy(l - pR)] (.14-107) The inverse transformation gives
[(1 - /i2)i;/t;v] = [y-l[l - p02)±/(l + p0j3);(p0 + fi)/(\ + {i0fi); v0yfJ + /^)]
(14-108)
Equations (14-106) through (14-108) contain the familiar results for Doppler-shift and aberration; the classical results can be derived by keeping only terms of 0(v/c), (i.e., by setting y - 1). From equations (14-107) and (14-108), it is easily seen that dvQ = (v0/v) rfv and dp0 = (v/vQ)2 dp,; hence, using do) = sin 6 dO d<p = dp d(j), we find
v dv d.O) = vn dvn dcOr
(14-109)
To establish the transformation properties of the specific intensity we calculate (621) the number of photons passing through an area dS oriented perpendicular to the z-axis, in frequency interval dv, into solid angle dco, propagating at angle 0 = cos"1 p to the z-axis, in a time dt. We suppose that dS is stationary in the rest frame; then N = [I(p, v)//iv] do} dv dS cos 6 dt. This must be the same number that would be counted passing through the same surface element by an observer in the comoving frame, namely
N0 = \_lQ{p0,vQ)ihvQ\do)Qdv0 x (dS cos 00 dt<
dS v dt0); here the first
term gives the number that would have been counted if dS had been stationary in the comoving frame, and the second gives the density of photons (I°/hv0c) times the volume (dS v dt0) swept out by dS in a time dt0. Using equations (14-102). (14-108), and (14-109), we find
l(p, v) = (»'/v0)3J<W v
(14-110)
Next, consider the emissivity. The number of photons emitted from a given volume, into a specified solid angle and frequency-interval, in a definite time interval, must be the same in both frames, hence
(nv da) dv dV dt)/hv = (nvQ do)0 d.v0 dVQ dl0)/hv0>
which implies that
n(p,v) = (v/v0)V(v0) (14-111)
496      Radiative Transfer in Moving Atmospheres
14-3   The Transfer Equation in the Fluid Frame 497
(where we have noted explicitly that r\ is isotropic in the fluid frame). To be able to establish energy balance in any frame, we must be able to balance emission losses with absorptions; hence, from equations (14-110) and (14-111), we conclude that
X{tu v) = (v0/v)z°(v0)
Finally, between two frames moving uniformly with respect to one another, the differential operator in the transfer equation transforms, using equation (14-104) and (14-106), to
c-fdldt) + (n • V) = (v0/v)[c'\o/dt0) + (n° ■ V0)] (14-113)
We can now show that the transfer equation is, in fact, covariant, for we see from equations (14-110) through (14-113) that
(dljdt) + (n- V)7t = ;?v - X,f-
(14-114a)
transforms to
(voMlc-'id/oto) + (n° ■ V°)][(v/v0)3/%uo> v0)]
= (v/V0)2[^(v0)-/(v0)/Vo-V0)]
which, if (v/v0) is a constant—as it will be if the two frames are in uniform motion with respect to one another (and only then)—can be written as
[c-\d/dt0) + (n° ■ V°)]/Vo, v0) = t/°(v0) - f{vQ)I0{Ho, v0)   (14-114b)
Obviously equations (14-114a) and (14-114b) are of the same form. Two points must be stressed, (a) Despite the similar form of the two equations, equation (14-114b) (at rest relative to the fluid) is actually much simpler because of the isotropy of n°{vQ) and jc°(v0). (b) The reduction of equation (14-114a) to equation (14-114b) is not valid if the two frames do not move uniformly with respect to each other; i.e., this equation does not apply in, say, an expanding or pulsating atmosphere.
One approach to coping with the shortcoming mentioned above is to leave the streaming terms described by the differential operator, and the radiation field itself, in the observer's frame, but to use first-order expansions to write the source-sink terms in the comoving frame (521, Chap. 6). Thus, using equations (14-111), (14-112), and (14-107) we can write
[c-\d(dt) + (n ■ VJ]/(jh, v) = rj°{v) - /'(v)l(fi7 v)
+ (n ■ v/c){V(v) - v(dif/dv) + [/'(v) + vidf/dvflKii, v)} (14-115)
This approach may be adequate for continua. where the frequency derivatives are well defined and nearly constant, but will not be accurate enough for lines
where the changes in rjy and %y are so swift that a first-order expansion is not sufficient. A complication in application of equation (14-115) is that both even and odd terms in p appear on the righlhand side, so that the usual reduction to a second-order transfer equation is not possible; a satisfactory numerical scheme can, nevertheless, be devised using a staggered mesh [see, e.g., (460)].
transformation of moments of the radiation field
The total energy density, energy flux, and radiation pressure in the field are specified by the frequency-integrated quantities [cf. equations (1-7) (1-19) (1-28)]
ER(r, t) = c-'Jj dv <j) doj /(r, n, v, i) (14-U6a) ^(r, 0 =      dv j> dm /(r, n, v, t)n (14-116b)
P(r, r) = c"1 |oJ' d.v (J) doj I(r, n, v, f)nn (14-116c)
These are related by the frequency-integrated moment equations [cf. equations (2-75) and (2-68)]
and
c *W
and
W/dt) + V ■ P = c"1 J* dv j) dcolnir, v. t) - yfr. v, t)/(r, v, ,)]n
(14-117a)
(dER/dt) + V ■ & = Jo" dv 03 do[)j(T. v, f) ~ X(r. v, t)l{r, v. t)]   (14-117b)
As written, equations (14-117) are already covariant. One way this may easily be seen is to exploit the results of electromagnetic theory in which one finds [see, e.g., (331, Chap. 12; 386, Chap. 4; or 494. Chap. 21)] that the momentum-energy conservation laws of electrodynamics can be written in the covariant form
dT#/dyf = -r (14-118)
Here T"" is the stress-energy tensor of the electromagnetic field, and /* is a four-vector giving, per unit volume, the three components of the Lorentz forces of the field on charged matter, and (i/c) times the rate of work done by the field on charges and currents. The stress-energy tensor is a contra-variant four-tensor of rank two, namely
T =
— TM icG' tcG    - W
(14-119)
498      Radiative Transfer in Moving Atmospheres
14-3   The Transfer Equation in the Fluid Frame 499
where TM is the Maxwell stress tensor (cf. §1-5), G is the momentum density, and W is the energy density of the held.
We have seen in Chapter 1 that a one-to-one correspondence between electromagnetic field quantities and moments of the radiation field can be made; specifically. TM = -P. G = c~2iF, and W = ER. We thus conclude that the stress-energy tensor of the radiation field must be
R =
ic~-9f -Er ,
dv j) dco Inn i jdv j) day In i jdv j dco In   — jdv<ji dco /
(14-120)
We can see by inspection that R is actually a four-tensor merely by noting that it is formed from the outer product of the four-vector (vn. iv) [recall equation (14-105)] with itself, times the invariant [see equations (14-109) and (14-110)] (/ civ doj/v2), integrated over all angles and frequencies. Similarly, the Lorentz force can be identified with the rate at which momentum is transferred from radiation to matter, and the rate of work with the rate of energy input of the radiation into the matter. Thus we conclude that we can write
jdv (j dco (;/_! - n)n, i jdv j) doj (yl - if)
(14-121)
as a four-vector. This can again be seen to be the case, for gfi is composed of the four-vector (vn, iv) times the invariants (yl dv dafj/v or (n dv dco)/v. The four-divergence of a four-tensor is automatically a four-vector, hence
3ä">/Ry* = -gR<
(14-122)
is, in fact, a covariant representation of the conservation relations for the radiation field. One can see immediately that equation (14-122) produces equations (14-117a) and (14-117b), when the latter is multiplied by {-ic\ and that these equations are, therefore, covariant between frames moving uniformly with respect to one another; for nonuniform motion, additional terms will appear in the comoving-frame moment equations.
In working with the equations of radiation hydrodynamics (§15-3), it will be useful to have transformations accurate to 0(v/c) of ER, .W, and P (or, equivalently, J, H, and K) between the fixed and comoving frames. These are most easily obtained by using equations (14-109), (14-110), and (14-107), and expanding to first order in (v/c). Thus, setting y = 1, we find readily that /v° dv0 dco0 = (v0/v)2iv dv doj « (1 - 2/?/i)/v dv dco, from which it follows
that
J° = J — 2ßH
(14-123a)
Similarly, Iy°pQ dv0do)0 = {p-ß){\ - pß)Ivdvdco a; [>-/?(! + pl)]f.dv do),
and //Vo2 dvu da>0 = [p - ß) Iv dv dm
{p2 - 2ßß)Iv dv do), from which
it follows that
H° = H - ft(J + K) (14-123b)
and                                 K° = K - 2[JH (14-123c)
The inverse transformations yield
(J, H, K) - [J° + 2/?tf° H° + fi(J° + K°), K° + 2^H°] (14-124)
Notice that these transformations are valid only for the frequency-integrated moments.
Exercise 14-9: Derive equations (14-123) and (14-124) by applying a Lorentz transformation to the stress-energy tensor [equation (14-120)] and expanding the result to Oin/c).
the comoving-frame equation of transfer
The full transformation of the equation of transfer (including the differential operator) for a nonuniform velocity field can be done rigorously using covariant differentiation (399; 135; 278); however this approach requires considerable familiarity with tensor calculus. We shall instead use a simple first-order expansion method that yields results correct to 0(v/c). The equations of transfer thai we consider are (in planar and spherical geometry, respectively)
[c-\0/dt) + M(^)][(v/vo)3/0(ro^o,v0.r0)]
= (v/v0)2[f/°(v0) - z°(vo)J°('o, p0, v0, r0j] (14-125a) and   [c-\d/3t) + ii(d/8r) + r^U - A*2)(5/^)][(v/v0)3/°(r0, pQ, v0, i0)]
= (v/v0)2[/y°(v0) - x0(vo)l°(ro-/<o, v0, f„)] (14-125b)
We consider one-dimensional flows, and apply a local Lorentz transformation to a frame that instantaneously coincides with the moving fluid. We shall ignore terms of 0(v2/c2), and set y = 1. We then have
rn = r
ct0(r, t) = ct - c 1     v(r', t) dr'
(14-126a) (14-126b)
Equation (14-126a) states that space increments will be judged to be the same by observers in both frames (i.e., no Lorentz contraction). Equation (14-126b) accounts for the finite propagation velocity of light and accounts for the classical retardation effect. [The significance of this term can be understood more fully by the following thought-experiment. Suppose clocks at a number of stations around some point i\ in the rest frame are initially synchronized.
500      Radiative Transfer in Moving Atmospheres
34-3   The Transfer Equation in the Fluid Frame 50)
Then suppose that at some time f, all these stations simultaneously emit pulses of light received by (a) an observer at rest at rx, and (b) an observer at yx moving with velocity v. Observer fa) will receive the pulses synchronously, while observer (b) receives pulses from directions r > t\ sooner than from r < ry. He thus concludes that the emission times (r0) were earlier for r > rt than for r < as is shown by equation (14-126b) (for simplicity consider the case of v = constant).] The retardation effect is, in one sense, not a relativistic term, but merely acknowledges a finite light-propagation speed; its significance from a relativistic viewpoint is, of course, profound.
To evaluate the derivatives in equations (14-125) the chain rule is applied.
ftVt
cr
dv,
or
d
dp/rvl
öy0
d — +	(tVo
t &0	\c7
\ d_	
J IM &V0	+ t
d ___^	fdp^
t ^0	\Cp/
Wo
dr)m fro
(14-127a)
	) —		d
	rvt fr'o	+ U^J	rvt Of0
	d __i_	fop^\	d
\fr)n,V	Sr0 +	\ ot )^	dpa
♦(£	) -		1 I7JV <• (
(14-127b)
— (14-1270
We use the first-order expressions (v/v0) = 1 + fipQ, (v0/v) = 1 - ftp, fk] = (fl - - fjp)-\ and p = {;/0 + /i)(l + M~\ and make the additional approximation that fluid accelerations (which are identically zero for steady flow) are so small that the change in any velocity during the time of a photon flight, over a mcan-free-path, is negligible compared to the velocity itself. We then neglect idv/dt) and derivatives of the form (dxjdt) fovx0 = r0,/t0, or v0, and retain only (dtJOt) = 1. The remaining coefficients in equations (14-127), to 0(v/c), are easily derived from equations (14-126) and the first-order relations written above. One finds
or
0
(r0,Ko.v0, tQ) = [1, ' Wv/dr0), -c-'povotfvßro), -/*/c]
II vt
dp
(14-128a) (14-128b)
Starting with equation (14-125a), rewritten with the approximation mentioned above as
{{v/vQ)[c-\c/dt) + p(c/vr)-} + 3pld(v/Y0)/dr])I°(r0,p0,v0,t0)
substituting from equations (14-127) and (14-128) and the first-order expressions for (v/v0), etc., and retaining only terms of first-order in (v/c), one obtains:
"1 c
c dt0
cj or
VoMo2 I fr\ d
dr0J opo c   \ dr0J dv0       c o
dv c   \ dr
I (r0, pQ, v0, t0) (14-129)
For spherical geometry we have the additional term
(vo/vjVHl - ;i2)(?/3//)[(v/v0)3i°]
= iv/VfS)r\l - p2)(cl°/cp) + 3r~Hl ~ /i2)[3(v/v0)/3/i]/°
Again expanding to first order in (v/c), and using the result from equation (14-107) that v2(l - p2) = v02(l - /i02), and also [d(v/v0)/dp] = (v/c), we obtain for these additional terms
r0~\l - *>2){[U + Md/dpo) - J?v0(3/i3v0)] + 3$}!°
Thus the comoving-frame transfer equation to order O(vfc) in spherical geometry is
v\ o cfdt-
(I - /V
PqV
c
VqV
c''o 3u
cr
1 - lh2 ( 1 -1 - Po2 ( 1 ~
d In v		
d In r0/_		dp0
d In v \	d	
d In r0y		
d In v V d In r()J	J i°{r0,	
- '?°(v0) - X>o)A''o^o^0,t0) Exercise 14-10:   Verify equations (14-129) and (14-130) in detail.
(14-130)
502
Radiative Transfer in Moving Atmospheres
Equations (14-129) and (14-130) are the comoving-frame equations including all terms of 0(v/c), and were first derived consistently by Castor (135). The time-derivative written in these equations is, in essence, still in the fixed frame (though it allows for retardation); the Lagrangian time derivative, which follows the motion of a fluid element (cf. §15-1) consists of the two terms (D/Dt) = (d/dt) + {v/c){8/8r\ the second term being the advection term. For steady flows, the terms in (d/dt) are identically zero. The equations just derived are obviously fairly complicated. As we shall see, it is important to retain all of the terms in the moment equations. However, for a solution of the transfer equation itself, it is sufficient for most astrophysical flows [as verified by a detailed calculation (446)] to retain only the frequency-derivative terms because of the effective amplification of these terms by (c/vth), as discussed earlier. In this limit, equation (14-129) for steady How reduces to equation (14-99), while equation (14-130) reduces to (14-100). We shall describe a numerical method for solving equation (14-100) in the next subsection.
For problems of radiative transfer involving partial redistribution, it is useful to derive frequency-dependent moment equations from equations (14-129) and (14-130). For spherical geometry we obtain
18JV° v_oJl_ 1 d(r2H°) c  at      c or J
Vi-
er
(3/v(
8
c or 1 (2v    8v\ d(v0Kv
.er I ov0
„0C,   ~! „,0
~ [vo(3K,° - Jv0)] - - - +
0\
for the zero-order moment, and for the first-order moment
1 ^ + v_8Hl + OKI + V ,o _ 0 +       + ^Ho
c   dt      c  or —       " ' -
or) ov0 (14-131a)
or
c \r dr
I) f-- *.o)] - I p + 3 M = _ xW
er! ovq c \ r     or J cv0
where
1 ri
Ny° = 2.)-i 7V,/Wo, t)p03 duo
;i4-131b) (14-132)
and, for brevity, we have written Jv° = J°(r, v0, t\ etc., and suppressed the suffix "0" on r and t. Again, for practical transfer calculations with steady
14-3   The Transfer Equation in the Fluid Frame 503
flow, it is sufficient to replace equations (14-13la) and (14-131 b) with
r2[d(r2H°)/dr-] - a[_c(Jv° - K»)/8vQ + b(dKv°/8v0)} = n°(v0) - Z°(v0)Jv°
(14-133a)
and   (8Kv°/8r) + r"i(3Kv° - Jv°) - a[8(Hv° - Nv°)/8v0 + b{dN°/dv0)-]
= ~X>o)Hv° (14-133b)
where a = (v0v/cr) and b = (d In v/d In r) [see (447)].
For problems of radiation hydrodynamics we require the frequency-integrated moment equations, which follow immediately from equations (14-131):
(dER°/dt) + v(8ER°/8r) + r-2[5(r2^°)/G>]
+ (v/r){3ER° - pR°) + {8v/dr)(ER° + pR°)
= 4tt Jo ' [)?°(v0) - x>o)J°l dv0 (14-134a) c-2(dSF°/dt) + (v/c2)(8^°/8r) + (dpR°/dr) + {3pR° - ER°)/r
+ (2v/c2r)[l + (d In v/d In r)]^° = -c"1 Jq" %0{v0W° dvQ   (14-134b)
where ER° = (4n/c) j J°(v0) dv0 = {4n/c)J° and pR° = (4n/c)K°. In planar geometry, equations completely analogous to (14-131) through (14-134) can also be written down. Equations (14-134) contain additional velocity-dependent terms on the lefthand side compared to their counterparts [cf. equations (14-117)] in the fixed frame. But this is more than adequately compensated by the tremendous simplification of the righthand side, where the isotropy of the opacity and emissivity in the comoving frame allows the integrals to be expressed in terms of the moments themselves, instead of as double integrals over the specific intensity. The only question remaining to be faced is: "How do we actually solve the comoving-frame transfer equation for 7°(r, ju0, v0) and its moments?"
solution for spfierically symmetric flows
Let us now consider methods for solving the comoving-frame transfer equations in the limit that only Doppler shifts are taken into account—i.e., equations of the form of (14-99) and (14-100). As was mentioned earlier, the transfer equation in the comoving frame is a partial differential equation, which was first obtained by McCrea and Mitra (414). Chandrasekhar (156; 157) obtained solutions of these equations for planar geometry, with a linear
504      Radiative Transfer in Moving Atmospheres
velocity law, a two-stream description of the radiation field, and strictly coherent scattering; generalizations of this approach are given in (1; 2; 3; 276, 199). Lucy (403) solved the equations in planar geometry, with coherent scattering, in the high-velocity limit, by ignoring the spatial derivative and treating the equation as an ordinary differential equation in frequency alone. An integral-equation method for planar geometries has also been devised (574); this method is restricted to linear velocity laws and hence is not useful for realistic models. A general and flexible numerical method developed by Noerdlinger and Rybicki (478) solves the equations in planar geometry using a Feautrier-type elimination scheme; this method can treat problems involving partial redistribution. In a ray-by-ray solution for spherical geometry, the number of angles must be of the same order as the number of depth-points, and a Feautrier-typc solution becomes costly. If complete redistribution is assumed, we can construct an efficient Rybicki-type solution (444), as will be described here; for partial redistribution we use moment equations [e.g., equations (14-133)], thereby eliminating the angle-variable, in which case a Feautrier-typc solution again becomes practical (447).
In spherical geometry we adopt the (p, z) coordinate system introduced in §7-6. Then, along a ray specified by constant p, equation (14-100) becomes
±{_dl±(z, p, v)jdz] - y(z, pildlHz, p, v)/5v] = n(r, v) - x(r, v)I±(z, p, v)
(14-135)
where y(z, p) = [vy(r)/cr][l - p2(d In v/d In /-)] (14-136)
and r = (p2 + z2'f, p. = (z/r). In equations (14-135) and (14-136) we have suppressed the suffix "0" for notational simplicity (and continue to do so henceforth in this chapter), but it is to be stressed that all quantities are evaluated in the comoving frame. Now introducing the optical depth along the ray, c/t(z, p, v) = -yfz, p, v) dz, and the variables
P, v) = \_ [Hz, P, v) + 7-(z, p, v)] (14-137)
and v(z, p, v) = \ [/+(z, p, v) - I~(z, p, v)] (14-138)
we can obtain from equation (14-135) the system
[cu(z, p, v)/dx(z, p, v)] + y{z, p, v)[dv(z, p, v)/dv] = v(z, p, v)   (14-139)
and   [dv(z, p, v)/3t(z, p, v)] + y(z, p, v)[du{z, p, v)/flv] = u(z, p, v) - S(z, p, v)
(14-140)
14-3   The Transfer Equation in the Fluid Frame 505
where y(z, p, v) = y(z, p)/x(z, p, v), and the source function is assumed to have the form for an equivalent-two-level-atom with complete redistribution—i.e., S(z, p, v) = S[r{z, p), v] = a(r, v)J(r) + p». The coefficients a and p1 contain the thermalization parameter g, the opacity ratio yjyd, and the profile function, while
J(r) =        dv <p(v) JoX dp. u[z(r, p), p(r, p), v] (14-141)
In equation (14-141), vmin and vmax are chosen to contain the whole line profile as seen in the comoving frame. Note particularly in equations (14-137) and (14-138) that, because we are working in the comoving frame, we can now average I+ and /" at a given value of v, in contrast to the situation in an observer's-frame formulation [cf. equations (14-23) and (14-24)].
Spatial boundary conditions are now required. At the outer radius r = R I~ = 0; therefore u = v, hence
[du(z, p, v)/5t(z, p, v)]Zm:ix + y(zmax, p, v)[oM(zmax5 p, v)/5v] = u(zmax, p, v)
(14-142)
At the plane of symmetry z - 0, we can now write u(0, p, v) = 0, hence for rays that do not intersect the core,
■i i
[du(z, p, v)/£t(z, p, v)]z = 0 = 0 (14-143)
For rays that intersect the core (i.e., p < rc) we (a) apply the diffusion approximation for an opaque core (stellar surface), which specifies v, or (b) set v = 0 (by symmetry) for a hollow core (nebular case).
In addition, an initial condition in frequency is required. For an expanding atmosphere [i.e., one in which v > 0 and (dv/dr) > 0], it is obvious that the high-frequency edge of the line profile (in the comoving frame) cannot intercept line photons from any other point in the atmosphere, because they will all be systematically red-shifted; any photon incident at the high-frequency edge must be a continuum photon. To specify the required initial condition we may therefore either (a) solve equations (14-139) and (14-140) in the continuum, omitting the frequency-derivative terms (which yields the standard second-order system) to obtain u(z, p, vmax) - wconlinuum, or (b) fix the derivative (8u/dv)VimK to any prespecified value given by the slope of the .! continuum; in particular, the choice (du/dv) = 0 leads to equations identical
to option (a) just mentioned.
The system is now discretized using the same grids {rd}, {pj, {zd.} as were employed in §7-6 and §14-1. We now choose the frequency grid'{v„} (n = 1,.. ., N) in order of decreasing values (vx > v2 > ■ ■ ■ > vN) because the initial condition is posed at the highest frequency. We replace equation
500      Radiative Transfer in Moving Atmospheres (14-14)} with a quadrature sum
JO;) = I w. I cidi<P(r<h vju[z(rd, Pi)>Pt, vj (14-144)
11= L i=l
Equations (14-139) and (14-140) are replaced with difference approximations
(H-J+I.jjj — ii,|ijiV'^TJ + -i-. in ~  J'J + -i. j*i + bd + -l,i,n~±{Vd+\-.jn ~~ V<i + i, i, 11 - l)
(14-145)
and ,„ - f1j-+,l-„)/Ar(,i,l =       - Sdill + V «•-*("«<». - "<«.■-1)
(14-146)
where u is presumed to be defined on the mesh-points zd = z(rd, />,), and udin = w(jdf /v v„), while i; is presumed to be defined on the interstices zd±y = \(zd + zd±1) and ^±,_J(1 = ';(z</+r Pi, v„). Further, we have defined
and
(14-147) (14-148) (14-149) (14-150)
Similar difference equations may be written to represent the boundary conditions (444). In equations (14-145) and (14-146), an implicit frequency differencing is used Jo assure stability (444; 462; 530).
Equation (14-145) can be solved analytically for vd + i._ in to yield
Vd + i, in ~ {[_{UJ± 1, in ~~ udin)/&xd+i, in J + ^d + f, i, n-i^d+i, i. n - 1 }/( ' + $d h- 4- i, " ~ +■'
(14-151)
Organizing the solulion into vectors that specify the depth-variation along a particular ray at a given frequency—i.e.,
and vin - (vit. .. , i„)T
equation (14-151) can be written in the form
v,-,, = GínUí„ + Hf„v/in_j
(14-152a) (14-152b)
(14-153)
14-3   The Transfer Equation in the Fluid Frame 507
where G is bidiagonal and H is diagonal. Equations (14-151) can be used, to eliminate ŕ!li±-L,ŕ„ from (14-146); we then obtain a set of second-order equations for uriin, namely
{U'((+i.iH - "diJ/[ATd++.f)l(l + ^ + *.i,„-*)]
= (1 + ^di.ii-i)lidin ~~ Srfiii ~ $di,n-\ udi,n~i
+ [v*,.-.«-±(i + v-.-.^rv-
- <W;;(1 + t>rf+i.„_i)"1 í'ri+i>ŕ>„_ J/At^
(14-154)
Adding the boundary conditions to equation (14-154) we obtain the system
Tl(lulfl + U^u,-H-i + V,„v,,^( + \V„,J = X,,, (14-155)
where T,„ is tridiagonal, U;„ and W,-„ are diagonal, Vjn is bidiagonal, and Xin is a vector.
Exercise 14-11: Verify equations (14-151) and (14-í 54), and sketch the forms of the C, U, and V matrices.
To solve the complete system, we choose a definite ray, specified by a given p„ and carry out a frequency-by-frequency integration procedure, with n ranging from 1 to N. This is effected by noting that the initial condition in frequency implies that Uŕl, Vrl and HfI are all exactly zero: thus we can obtain expressions of the form u,, = A,-, — BrlJ and va = C(1 — DnJ, where An — T^X^ is a vector, Brl = T^Wn is a matrix, Crl = GŕlAŕl, and Dn = GnBn. Similar substitutions are carried out for successive values of n, to yield
and where
ll;„  =  A;„  - B,„J
v!j1 G;(1 D;,,J
and
B,n = Tr^Vto - U^B^.i - V^D,,^, G;,, = G,'(,Ai(, + H,-nC/iH_i Di„ = Gr(IB,„ + H,,,D, „_!
(14-156) (14-157) (14-158) (14-159) (14-160) (14-161)
508      Radiative Transfer in Moving Atmospheres
Each result, of the form of equation (14-156), for every frequency yn, along every ray p,-, is substituted into equation (14-144) to obtain a final system for J of the form
(l+ £F„,bJ)j = £F/A„ (14-162)
\ i, 11 / l. K
where the Fs contain the quadrature weights. The solution of this final system yields j, and hence Sir. v), and u(z, p, v) and v(z, p, v) from equations (14-156) and (14-157). Knowledge of u(r, p, v) implies knowledge of u(r, pi. v), so it is clear that we can calculate J°(r, v) and K°(r, v) in the comoving frame; similarly we can calculate the flux H\r, vj from v(r, p, v). Thus we obtain a complete solution for the radiation field and its moments in the comoving frame.
The number of operations required to obtain A,„ and Bin in equation (14-156) is proportional to D2, so summing over all frequencies on all rays one obtains Ts = cN Di + c' D3 as an estimate of the computing time required. Note that this time is linear in the number of frequencies. A similar formulation can be written down for planar geometry; in this case we dispense with the rays and use M fixed angles {u,}. In planar geometry the computing time required for the solution is TP = cNM D2 -f- c D3. For partial redistribution in spherical geometry one would use a Feautrier solution of the moment equations (447), obtaining the Eddington factors from a ray-by-ray formal solution with a given estimate of the source function. For partial redistribution in planar geometry the method of Noerdlinger and Rybicki (478) is applicable.
Exercise 14-12: Consider flows with monotonia velocity fields [i.e., (dv/dr) everywhere ^0 or everywhere ^0]. (a) In planar geometry show that the choice of the initial condition in frequency is unique and depends only on the sign of (dv/dr) [cf. (451) for a discussion of non-monotonic velocity fields], (b) In spherical geometry show that unique conditions can be found only if [i; > 0, (dv/dr) > 0] or [v < 0, (dv/dr) < 0], and that, owing to projection effects along a ray at the plane of symmetry (z = 0), velocity distributions of the form [v > 0, (dv/dr) < 0] or [v < 0, (dv/dr) > 0], though monotone in the radial direction, produce nonmonotonic fields along tangent rays.
The method outlined above has been used to calculate source functions and line profiles in idealized model atmospheres (444). Each atmosphere is characterized by an outer radius R (in units of rc — 1), a continuum optical depth Tc (at r = rA, a static line optical depth T,, opacities %L cc r~2 and r/c cc r~2, and constant Planck function B = 1. The two-level-atom thermal-ization parameter was set to c = 2/T,. The velocity field was chosen to be either (a) dv/dr = constant, or (b) v(r) = y0 [tan" \ar + b) — tan"1 (a + b)], which gives a sharp rise at the point i\, =■ — (hja), and has constant terminal
-2,0 -1,0 o t.o 2.0 3.0
log T
FfOUKE 14-1 [
Line source functions in expanding atmospheres for various values of outer radius R (in units of rr\ and terminal velocity rinax (in thermal velocity units). For all models Tc =2,   T, = 103,   5=1, and c = 2 x I0"3.   Abscissa: log of static line optical depth.   The dashed line gives the mean intensity in ihc continuum. The curves are labeled with rmay = r(R).   From (444). by permission.
velocity at large r. Form (b) is a caricature of stellar-wind solutions. Results for the source function in several models are shown in Figure 14-11; for thesemodelsT,=: 103, Tc = 2, rv = (R + l)/2, B = l,and£ =2 x 10~3. We see that the basic effect of the velocity gradient at depth is to increase the escape probability; hence the source function drops below its static value and, for large values of <;inax, S, approaches Jc, the mean intensity
510
F   0.9 -
figure 14-12
Line profiles (emergent fluxes integrated over the disk) from expanding spherical atmospheres.   Ordinate: flux relative to continuum flux,   abscissa: Ar/AvD.   Tor all models, R = 300, 1\ =2.   B = 1,   r. = 2/TL. and (dvkh^ = - )00/T,_. From (444), by permission.
in the continuum (dashed line). Near the surface, the enhanced escape probability competes with interception of continuum radiation by the Doppler-shifted line. In planar geometries the latter effect dominates, and produces an increase in S;; in very extended atmospheres the former effect dominates, and S; decreases. Flux profiles, as seen by an external observer, are shown in Figure 14-12 for models with parameters specified in the figure caption. Characteristic P-Cygni profiles are obtained with both the emission intensity and the absorption depth increasing with increasing optical depth.
The solution of the comoving-frame transfer equation by the method described above is convenient, efficient, and easily generalized to realistic stellar models. The method can also be extended to apply to multilevel atoms, and should permit the computation of line spectra for realistic model atoms in expanding atmospheres. Accurate theoretical spectra, when compared with observation, should assist in the determination of the physical structure of the atmospheres of stars with expanding envelopes and stellar winds.
Stellar Winds
The outermost atmospheric layers of many stars are in a state of continuous rapid expansion, and the material lost from a star in such a flow is called a stellar wind. These winds have a wide range of properties. At one extreme are very massive flows (mass-loss rate —lO-' J40/ye'dr) that are optically thick in spectral lines (and even in some continua) and produce emission lines and P-Cygni profiles; at the other are relatively tenuous flows such as that of the Sun, which is optically thin and inconsequential in terms of mass-
loss (10"
0/year), but still of great importance to the solar angular
momentum balance. In Chapter 14 we discussed the problem of spectrum formation in a given flow; here we shall examine the dynamics of the wind and analyze questions of momentum and energy balance. We shall find that there are two primary mechanisms for producing stellar winds. (1) In stars with hydrogen convection zones (such as the Sun), the outer atmosphere is a mechanically-heated corona of very high temperature. Here we find that the corona cannot establish a static pressure balance with the interstellar medium, but must inevitably expand supersonically, driving the flow by tapping the thermal energy of the gas. (2) In early-type high-luminosity stars, the radiation field is so intense thai momentum imparted to the gas by photons drives
512      Stellar Winds
the material in a transsonic flow. (There is also some observational evidence that there may be a corona near the surface of these stars, and our theoretical models for their winds are only preliminary.)
Stellar winds have important implications for many astrophysical problems. In some cases, the mass-loss rate is so large as to produce a significant change in the star's mass on a thermonuclear-evolution time-scale, and hence directly to affect the star's evolution track. In other cases, the possibility of noncatastrophic mass-loss over its entire lifetime may permit a star to evolve to a white-dwarf configuration without becoming a supernova. Stellar winds act as brakes on stellar rotation, and hence strongly influence the angular momentum content of stars. Further, stellar winds represent important sources of mass- and energy-input into the interstellar medium, and thus help to determine its composition and thermodynamic state. In this book we shall consider only winds from single, isolated, stars. Stellar winds can also occur in binaries, where they may induce rapid mass-exchange that radically alters the course of stellar evolution, or, in some cases, produce exotic objects such as X-ray sources; although space does not permit a discussion of these phenomena, the material presented here is basic to a study of the more complex cases just mentioned, and provides a background for an approach to the literature.
15-1   The Equations of Hydrodynamics for an Ideal Compressible Fluid
In this section we shall develop briefly the equations of hydrodynamics for an ideal (nonviscous) compressible fluid, which is taken to be a perfect gas. We shall ignore ionization effects, and assume that the material is already essentially completely ionized. No attempt will be made to discuss the equations in great depth, as numerous excellent texts and monographs on the subject of hydrodynamics are readily available (385; 692; 104; 490). For expository convenience, the equations will be derived in Cartesian coordinates when explicit reference to a coordinate system is required, then restated in vector-tensor notation, and finally rewritten in spherical coordinates (assuming spherical symmetry) for application to the stellar wind problem.
kinematics
Let us first consider some of the basic kinematic properties of the fluid. We consider the gas to consist of a mixture of particles of different species (e.g., protons, electrons, heavy ions). Each species k has a mass mk, and a space and velocity distribution function fk(r, V, t\ defined such that fk dxt
15-1   The Equations of Hydrodynamics 513
d.x2 dx3 dV\ dV2 dV3 gives the number of particles of type k in the volume element (r, r + dr) = [(x:, x: + dxy\ (x2, x2 + dx2). (x3, x3 + dx3f\. with velocities on the range
(V, V + dV) = [(K(, Vl + d\\\(V2, V2 + dV^iV,, V3 + dl;)]
at time t. The plasma is assumed to be chemically homogeneous, so that the relative numbers of particles of different species are the same throughout the gas. The velocity distribution is characterized physically in terms of macroscopic flow velocities and a microscopic thermal distribution; the former describe the average motions of particles (and hence the bulk fluid motion) as seen in a fixed laboratory frame, while the latter gives the random individual particle motions relative to the average. We assume that the Coulomb collision rate in the plasma is so large that (a) there is no drift of any species relative to any other, and (b) on the microscopic level there is perfect equipartition of energy at each point, so that all species have the same thermal distrubtion specified by a unique temperature T(r). Further, the microscopic velocity distribution is assumed to be isotropic.
The number density (cm-3) of particles of species k is given by
nk(r,t)= \"dVl \K dV2 f°° dV3fk(r,V,l) (15-1)
The mass density of species k is mknk(r, t), and the total density (gm cm-3) is
p(r,t) = 5>knk(r,t) (15-2)
The average velocity (i.e.. the fluid-flow velocity) in the z'tb direction is
nkO'uk =       dVx       dV2       dV3 fk(r, V, W (15-3)
As mentioned above, <K>k is taken to be the same for all species, hence the subscript may be omitted. The full velocity component V, of any particular particle may now be written as Vs = <!•■> + V\. where V\ is the random thermal velocity in the /th direction. Clearly <k';> - 0. The complete fluid velocity is
v(r, t) = <ki>i + <F2>j + <l/3>k
= vfi, t)i + ii2(r, t)j + u3(r, Ok (15-4)
The motion of the fluid results in mass transport and the mass Jlitx is given 2>k«ft«Pi>i + <Vi>i + <V3>k) = (j^m^y = py (15-5)
514      Stellar Winds
15-1   The Equations of Hydrodynamics 515
Similarly, the particles carry momentum, and the rate of transport of the ith component of momentum, across a surface oriented perpendicular to the yth direction, by particles of species k, is
nf/(r, t) = mk       dVt       <IV2       dV% ./,(r. V, t)^.
= mk J*^ dVt       dV2       dV, fk(t, V, t)(v, + V'divj + V))
= mknk(r, t)(vtVj + rr-<F}>k + u/KJX + <^K}>k)
= mtnk(r, f)to + O^'})*' (15-6)
The quantity (y'iV'f)k is the average of the ith and jth components of the random thermal velocity, and because the thermal distribution is isotropic and the individual components are uncorrected.
Hence
<W>k = <(^r>k5ťJ- = (fcr/mk)5řJ-
n;/ = mknkviv} + {nkkT) 6i} = m^v-Oj + pk ötj
(15-7)
(15-8)
where pk is the partial pressure from species k. Summing over all species, the total momentum flux tensor is given by
(15-9)
where p is the total gas pressure.
Finally, there are two useful (and conceptually rather different) schemes for describing the changes that occur in the fluid as a result of material motions. As an external observer views the fluid, the natural description of its properties will be to write a(xl5 x2, x3, t) for any property a. The variation of a, as a function of time and position, is described in terms of a time derivative [c/rt) computed at fixed (xi.x2,x3), and space derivatives (d/dXf) evaluated at & fixed t. This system is called the Eulerian description. Alternatively, imagine that we follow the motion of a fluid element, consisting of a definite sample of material; this system is called the Lagrangian description. The time variation of the properties of a Lagrangian fluid element is described in terms of"the fluid-frame time derivative (D/Dt) (also known as the "Lagrangian", "total", or "substantial" derivative). The relationship of the Lagrangian derivative to derivatives in the Eulerian frame may be obtained as follows. The derivative (Da/Dt) is defined as the limit, as At -> 0, of [a(f + At) - a(í)]/Aí, where a is measured in the fluid frame at times t and t + Ai, at (in general) two different positions; r, and r + Ar = r + v Ai. Allowing for the change in
position of the fluid clement we have
a(t + At) = a(r, t) + At
= a(r, t) + A*[(da/t/)r + (v ■ V)a] It thus follows that for any a (scalar or vector)
(Da/Dt) = (da/dt) + (v • V)a
115-10)
The Lagrangian system is particularly advantageous for time-dependent one-dimensional (planar or spherically symmetric) flows, while the Eulerian system has advantages in complicated geometrices. For steady flows [i.e., all properties constant in time as viewed by an external observer, so that (d/dt) = 0], one usually employs the Eulerian system.
the equation of continuity
Consider a fixed volume element dx in the fluid. If we demand overall mass conservation, then the rate of decrease of the mass in dx is given by the net mass flux out of the element through the surface S enclosing the element. That is,
^-jp dx = -(j) (pv) ■ dS - - jV • (pv)   dr (15-11)
where the second equality follows from the divergence theorem. The equation written above is true for an arbitrary element dx, and hence the integrands must be equal; therefore we have the equation of continuity
(öp/öt) + V ■ (pv) = 0 or, in view of equation (15-10),
(Dp/Dt) + pV ■ v = 0
(Í5-12)
.15-13)
Equation (15-12) was also derived in §5-4 from the non-LTE rate equations [cf. equation (5-50)]. For steady flow, (d/dt) = 0, hence
V ■ (pv) = 0 (15-14)
For a one-dimensional spherical flow, equation (15-12) becomes
(dp/dt) + r-2[d(r2pr)/dr] = 0 (15-15)
516      Stellar Winds
which, for steady flow implies
4nrzpv = constant = J4
(15-16)
Here J4 is the mass-loss rate through an entire spherical surface and v denotes the velocity in the radial direction.
Exercise 15-1: Show that for any physical variable a (scalar or vector) the equation of continuity implies that
p{DajDt) = c{pa)lct + V • (pav)
(15-17)
momentum equations
The momentum density (i.e., momentum per unit volume) in the flow is pv. If we again consider a fixed volume element dx, we may equate the time rate of change of the momentum in dx to the sum of the losses from the momentum flux through the surface S enclosing dx, and the gains from the force per unit volume f acting on the material in dx. That is,
= J(f- V-Tl)dx (15-18)
where the divergence theorem has again been employed. As the element dx is arbitrary, we conclude that
8(p\)/dt = - v ■ n + f Substituting for n from equation (15-9) we have
[d(py)/dt] + V-(pvv) = -Vp + f
;i5-19)
(15-20)
In view of equation (15-17), we can rewrite equation (15-20) to obtain the equations of motion
p{Dv/Dt) - p[(dv/dt) + (v ■ V)v] = - Vp + f (15-23)
For a spherically symmetric one-dimensional flow with a gravity force f. = — (G.M pjr2\ equation (15-21) becomes
p(ovjdt) + pv{dv/dr) - -(dp/or) - {GJ4pjr2\ For steady flow the first term of equation (15-22) vanishes.
(15-22)
15-1   The Equations of Hydrodynamics 517
energy equation
The statement ofenergy conservation for an element of gas is conveniently given by the first law of thermodynamics, which equates the change in the specific (i.e., per unit mass) internal energy, de, plus the work done by the gas pressure p when the specific volume changes by an amount d.(l/p), to the amount of heat per unit mass added to the element, dq. That is,
de + p d(l/p) =
(15-23)
Imagine that these changes occur as we follow the motion of a Lagrangian fluid element for a time At, and assume that heat exchange with the surroundings occurs by means of conduction only (other sources and sinks—e.g., radiation and mechanical energy dissipation—being neglected) with a conductive flux qc, which yields a rate of energy loss per unit mass of (1/p) V • qc. Then equation (15-23) yields the gas energy equation
p{{De/Dt) + p[D{l/p)/Dt]} = -V-qc
(15-24)
Normally the conductive flux is written as qc = —k(\T) where k is the thermal conductivity of the material.
From the equation of continuity, one sees that p[D(l/p)/Df] — V ■ v; hence equation (15-24) can be written in the alternative form
p(De/Dt) + p(V-v) = -V qc
(15-25)
Now, from the momentum equation, a statement of mechanical energy conservation can be derived by taking the dot product of equation (15-21) with v, to find
p[D&v2)IDt] + (v ■ V)p = v ■ f (15-26)
Adding equations (15-25) and (15-26), and noting that V ■ (ah) = (b ■ V)a + a(y ■ b), we find a total energy equation
P\P&2 + e)/Di] + V ■ (pv) = v ■ f - V ■ qc (15-27)
Applying equation (15-17) we may write instead
[Sftpo2 + pe)/di] + V ■ \ikpv2 + pe + p)v + qj = v ■ f (15-28)
For one-dimensional spherically-symmetric flow equation (15-28) becomes
dt
{{pv2 + pe)
1 d
r2pv[±v2
e + ~ P
df
or
GJdpv
; 15-29)
518      Stellar Winds
For steady spherically symmetric flow
rpiih2 + h) - v2k
17
or
(r1pv)(GJ?/r2) = 0 (15-30)
where h = e + (p/p) is the specific enthalpy. Recalling that {r2pv) = constant, we may integrate equation (15-30) to obtain the total energy integral
(4m-» [{v2 + h - {G,Mfr)\ - {4nr2)K(dT/dr) = C (15-31)
Equation (15-31) states that the total energy flux through a spherical surface (consisting of the flux of kinetic energy, gas enthalpy, potential energy, and heat conduction) is a constant in a steady flow.
sound waves
When a compressible fluid suffers a small impulsive disturbance, an oscillatory motion of small amplitude (consisting of alternating compressions and rarefactions) is generated and propagates through the medium. These oscillations are sound waves, and their characteristic speed is the speed of sound. Suppose the medium is planar and homogeneous, and originally at rest with an ambient density p0, and pressure p0. Write the perturbed values of density and pressure as p = p0 + p1 and p = p0 + pl5 respectively, where it is assumed that px « p0 and p1 « p0; further, d, the velocity of the disturbance, is to be regarded as a small quantity. Then substituting these expressions into the equation of continuity (15-12) and the momentum equation (15-21) (with the external force / = 0), expanding, and retaining only terms of the first order in the perturbations, we find
and
(dpjdt) + Po(dv/dx) = 0 Po(dv/vi) + (dpt/dx) = 0
(15-32) (15-33)
For an ideal fluid (i.e., no viscosity or conductivity), there is no thermal energy exchange from one fluid element to another, and the material behaves adiahatically. We may therefore write
ifpjdx) = (dpfdp)s{dpjdx) = a\dpjdx)
where (8p/dp)s is the isentropic derivative of the pressure with respect to density. Now, differentiating equation (15-32) with respect to t, and (15-33) with respect to x. and eliminating the cross derivative [d2v/dt dx), we obtain
(d2PJdt2) - a2(82pjdx2) = 0
(15-34)
15-1   The Equations of Hydrodynamics 519
Equation (15-34) is the wave equation, in which the wave velocity is a. We thus identify the adiahatic speed of sound as
= [{?pfip)sy
(15-35)
For a perfect gas undergoing adiabatic changes, (p/p0) = (p/po)y, where y is the ratio of specific heats at constant pressure and constant volume (y = § for an ideal monatomic gas); thus
a> = (yp/ft)- = iykT/pmup
:i5-36)
where p is the number of atomic units per free particle. In certain situations there can be a very free exchange of energy between the wave and its surroundings via conduction or radiation; in this case the temperature fluctuations in the sound wave are destroyed (the wave is thereby damped), and the wave propagation proceeds isothermally. In this event p - p(kT0/piuH) and the isothermal sound speed is
a - (fc70//imH)i
(15-37)
the rankine-hugoniot relations for stationary shocks
The speed of sound determines the rate at which a disturbance propagates naturally in a compressible fluid. Jf the velocity of the flow exceeds the speed of sound, then the adjustments of the fluid state by sound waves cannot proceed quickly enough, and situations occur where the properties of the flow may change markedly over a rather small distance. This gives rise to almost discontinuous changes at a narrow interface called a shock front. Shocks may occur in a wide variety of circumstances. For example, in a pulsating star, a wave initiated at great depth will propagate with larger and larger velocities as it progresses upward in the atmosphere, because of the decrease in density; eventually it runs through the material at supersonic velocities, producing a shock front that travels outward into the upper layers of the atmosphere. Such shocks produce interesting spectroscopic phenomena in RR Lyrae and Cepheid variables, but we shall not be able to pursue this subject here. Alternatively, in a steady flow, a stationary shock front may be produced where the gas, in effect, encounters an obstacle (e.g., a constriction in a nozzle or flow channel or at the point where a stellar wind runs into the interstellar medium). For the purposes of this chapter, only stationary shocks will be considered.
The shock front itself has a structure whose properties are determined by dissipative processes involving viscosity, heat conduction, and radiation; furthermore, the downstream material may be forced out of equilibrium in
520      Stellar Winds
15-2   Coronal Winds 521
one or more degrees of freedom, depending on how long are the characteristic relaxation times that restore the equilibrium [see (692) for a detailed discussion of these phenomena]. These effects will be ignored here, and it will be assumed that the fluid is a perfect monatomic gas in equilibrium throughout, and that the shock is a discontinuity at an infinitely sharp interface. If we consider the flow of particles across an interface, it is clear that the flux of mass, momentum, and energy must be conserved. Using subscript 1 to denote upstream quantities (i.e., material flowing into the shock), and subscript 2 to denote downstream quantities (i.e., post-shock material), the conservation requirements just mentioned become (in planar geometry)
Pi*>i = Pzvz (15-38)
Pi + Pi»i2 = Pz + i'2^22 (15-39)
and   p1o1[el + (p,,Vi) + 2'^] = Pi^[e2 + (.P2/P2) + T^a] (15-40)
The appropriate fluxes have been identified from equations (15-5), (15-9), and (15-28). In view of equation (15-38), and the results for a perfect gas that e = (p/p)/\y - 1) so that h = e + (p/p) = y{p/p)/(y - 1), equation (15-40) can be rewritten as
[yp.A; - Dpi] + W = [ypJir - Dp2] + W U5-41)
If we eliminate v2 by means of equation (15-38), use the definition of the sound speed in equation (15-36), and define the Much number Mi = v\/a„ then equations (15-39) and (15-41) may be rewritten as
(P2/Pi) = 1 + yMftl - (Pi/p2)] (15-42) and 2[(PiP2/p2p1) - 1] = (y - OM^l - (Pl/p2)2] (15-43)
which are known as the Rankine-Hugoniot relations. Equation (15-42) gives p2 as a function of p2 for given (p.. pv). In principle, two solutions exist: (a) p2 > Pi and p2 > pt and (b) p2 < pt and p2 < p,. However, it is found [see (385. §§82-84) or (692, §17)] that case (b) is excluded on the basis of the second law of thermodynamics and stability considerations, and that only compression shocks are physically possible.
Equations (15-42) and (15-43) can be solved simultaneously for (p2/Pi) and (p2/Pi)> yielding
{Pi/Pi) = <vM = (T + l)Mi7[(7 - 1)AV + 2] (15-44)
and U'2/Pi) = [2yM:2 - (y - 1 )]/(>■ + 1) (15-45)
Applying the perfect gas law we also may write
(7VT,) = [2;-Af,2 - (y - D][(y - DM,2 + 2]/M12(y + l)2 (15-46)
Exercise 15-2; (a) Verify equations (15-42) and (15-43). (b) Show that equation (15-43) can be rewritten in the customary forms
e2- ex= iCpr1 - p2~1)(Pi + Pi)
h2 - hL = Kp,"1 + p2_1)(P2 ~ Pi)
Exercise 15-3: (a) Verify equations (15-44) through (15-46). (b) Show that the downstream Mach number is
M22 = [(y - l)Mf + 2]/[2yM12 - (y - 1)]
and that this implies the downstream flow is always subsonic.
15-2   Coronal Winds
The outermost layers of the solar atmosphere comprise the corona, a tenuous (characteristic electron density ~4 x 108 cm-3), high-temperature (77 ~ 1.5 x 106 °K) envelope which can be observed, at eclipses, to extend to several solar radii. As was described in §7-7, the high coronal temperature is a result of energy input from wave dissipation. The ultimate source of the waves providing this mechanical heating is the hydrogen convection zone, and we believe that all stars that have extensive convection zones should also have coronae.
For many years the corona was regarded as an essentially static envelope bound to the Sun, and although it was realized that particle bombardment of the Earth from energetic events (e.g., flares) on the Sun produced auroral and geomagnetic effects (164; 165), it was Biermann (91; 92; 93) who first advanced the idea of continuous particle emission ("corpuscular radiation") from the Sun. Realizing the importance, at coronal temperatures, of energy transport by conduction, Chapman (166) showed that the corona extended far out into interplanetary space and, in fact, enveloped the Earth in a low-density high-temperature medium. Subsequently Parker showed (496; 498) that any reasonable hydrostatic model of the corona, starting from known conditions near the Sun, led to such high pressures at large distances, as to preclude the possibility of containment by the pressure of the interstellar medium. Thus static models are internally inconsistent, and large-scale coronal expansion must occur. This expansion provides the source of the particle emission proposed by Biermann.
Parker developed a theoretical model indicating a flow with low velocities near the Sun, rising to very large supersonic values at large distances; he called this transsonic flow the solar wind, and made predictions of typical flow velocities, densities, and particle fluxes at the Earth's orbit. These predictions were supported by many bits of evidence [see the summaries of the fascinating historical growth of our conceptions of the solar wind in (107, Chap. 1) and (324, Chap. 1)], but an alternative model was proposed
r
522      Stellar Winds
by Chamberlain (147), which was everywhere subsonic and produced only a low-velocity solar breeze at the Earth's orbit. The ensuing debate in the literature, based almost entirely on theoretical considerations, led to a sharpening of the formulation of the theory (and makes interesting reading today because of the high standard of scientific argumentation it contains!). But the question of which picture really applied remained undecided until direct measurements from space vehicles provided conclusive evidence in favor of a high-speed wind.
Detailed measurements of solar-wind properties show that there is considerable fluctuation of all the physical variables; many of the variations correlate with a solar rotation period and hence reflect changes in the initial conditions of the flow at the corona, while.others are the result of violent events such as flare-generated blast waves. Despite these variations, it is useful to consider the background low-speed "quiet" wind, whose characteristics must result from the conditions prevalent in the corona as a whole. A summary of quiet solar-wind properties measured at the Earth's orbit is given in Table 15-1. Although it must be emphasized that these data define only a high-order abstraction of the real solar wind, they nevertheless provide typical values against which theoretical results can be compared. From the data given, we see that the wind is a highly supersonic (Much number a 8), nearly-radial flow. It is easy to show (Exercise 15-4) that the effects of the mass-loss in the wind on the evolution of the Sun as a star are negligible. Moreover, the corona-wind ensemble is optically thin, Thus at first sight the flow seems to be of little significance from a stellar-astro physical point of view. We shall see, however, that even this rather weak (see below) flow has implications of great consequence to the question of angular-momentum loss by the Sun.
TABLE 15-1
Properlies of the Quid Solar Wind at the Earth's Orbis
Radial component of flow velocity	300-325 km s"'
Nonradial component of flow \docity	S km s"1
Proton (or electron) density	9 cm "3
Average electron temperature	1.5 x 10s "K
Average proton temperature	4 x lu4 -'K
Magnetic field intensity	5 x 10"J gauss
Proton flux	2.4 x 10s cm-2 s-'
Kinetic energy flux	2.2 x 10"' erg cm"2 s"'
Enthalpy flux	8 x 10"-' erg cm'"2 s"1
Gravitational flux	4 x lO-'ergcm-'s"1
Electron heat-conduction flux	7 y 10"3 erg cm "2 s"1
SouKCr: Adapted from (324, 44—45), by permission
15-2   Coronal Winds 523
Exercise 15-4: Using the particle flux given in Table 15-1. show that the rate of mass-losi, M = 4nr2ni;ui. where n is the particle density and m is a mean particle mass, is - 2 x 10" 14 „//0/year. Compare this with the thermonuclear mass-loss rate required to yield the solar luminosity.
As mentioned previously, we believe there should be coronae for all stars that have hydrogen convection zones (i.e., spectral types F and later); these stars should also have stellar winds. There is, in fact, direct observational evidence (e.g., P-Cygni line-profiles) for massive coronal winds in many late-type supergiants and giants. Thus estimates of mass-loss rates [sec, e.g., (624, 238; 70, 246)] range from 2 x 10"10 ,//0/year for K-giants to 10"8 ,/#Q/year for M-giants; and from 10"7 .^0/year for G- and K-supergiants to 10"6 or 10"5 JfQ/yeav for M-supergiants. All of these are coronal winds resulting mainly from the high temperature of the stellar corona (except possibly for the M-stars, where radiation pressure on grains may be important). Another class of massive winds (10"6-10~5 „#0/year) is found for early-type stars (OB supergiants and WR stars); these flows are driven by radiation pressure acting on the material (cf. §15.4).
In this section we shall examine the basic physics of coronal winds, using simplified and idealized descriptions, both of the material properties and of the nature of the flow. Our goal will be to gain insight into the overall nature of these winds, rather than to attempt detailed modeling. We shall repeatedly refer to the solar wind as an example, and as an indicator of what other processes need to be considered in an "ultimate" stellar-wind theory.
expansion of the solar corona
Suppose the corona is taken to be a fully-ionized pure-hydrogen plasma with T0 % 1.5 x 106 °K and h0^4x 108 cnT3, where n0 denotes the density of protons or electrons (np = ne for charge neutrality). At these high temperatures the electrons have large velocities, and can conduct energy efficiently, producing an energy flux qc = -kVT, where the conductivity k is given (598, 86) by k = k0T* with k0 & 8 x 10"7 ergs cm-1 s"J deg-i'. At 106 °K, the conductivity of the plasma far exceeds ordinary laboratory conductors! If we assume that conduction is the dominant energy transport mechanism in the corona, then for equilibrium we must have V ■ qc = 0, which implies that
r-2d[r2K0T*{dT/dr)~]/dr = 0 (15-47)
This equation can be integrated twice, and demanding that T -> 0 as r -> go,
W6find T{r) = T0(r0/,f (15-48)
where r0 denotes a suitable reference-level in the corona (which is of the order of RQ = 7 x 1010 cm). The temperature fall-off implied by equation
524      Stellar Which
15-2   Coronal Winds 525
(15-48) is extremely slow, and at the earth's orbit (r@ = 1.5 x 1013 cm) >we find T a 3.3 x 105 °K for T0 = 1.5 x 106 °K; thus the earth is enveloped by high-temperature coronal material.
If the corona were in hydrostatic equilibrium, we would have
(dp/dr) = ~(GJ/oP/r2)
(15-49)
The Debye length in the corona at the temperature and density quoted above is only about 0.3 cm [cf. equation (9-101)]. Hence the plasma is neutral except on the smallest scales, and we may write ne = np = n and p — n[pi + rne) ~ nm, where m denotes the mass of a hydrogen atom, and p = 2nkT. Then the hydrostatic equation becomes
d(nT)/dr = -(GJf Qmßk){nlr
(15-50)
If T were constant, and we were to consider only distances r a= r0, we could approximate equation (15-50) with the usual planar limiting form d(\n n)jdr = -(\/H), where the scale-height H = (GJi Qm^kT^2)'1 ar 105 km. As shown in Exercise 15-5, this scale-height is so large that the predicted particle density near the earth's orbit is quite large. We would then conclude that the Earth is immersed in a hot, fairly dense, extension of the corona. Actually the assumption that T = T0 is too crude, and produces somewhat of an overestimate of n(r0). If. instead, we make use of equation (15-48), we may rewrite equation (15-50) as
which yields the solution
d[(r0/rfii]/d{r/r0) = -(r0/r)2n/H (15-51)
n[r) = n0(r/r0)*exp{-7r0[l - 0-o//f ]/5f/} (15-52)
from which we deduce n ~ 9 x 104 near the earth for n„ = 4 x 10g.
Exercise 15-5: Assume the corona is isothermal, T = TQ, but allow for the variation of the gravitational force to obtain an expression for n(r). With n0 4 x 108 cm3, show that the particle density near the earth's orbit is about 4 x 105 cm "3. Exercise 15-6: Show that equation (15-52) yields a minimum for n at (r/r0) = (7/4)(r0/7i)s and that n(r) increases outward beyond that point (formally this configuration would be unstable against a Rayleigh- Taylor instability).
Combining equations (15-48) and (15-52), we find that the pressure is
p(r) = p0exp(-7r0[l - (r0/rf]/5H} (15-53)
Instead of vanishing at (r/r0) = oo, the pressure approaches a finite value. Adopting p0 ~ 0.2 dynes cm"2, we find px x 10"5 dynes cm"2. This pressure is to be compared with the pressure expected from the material and
magnetic fields in the interstellar medium. The average density of the medium is of the order of 1 cm "3; for an H1 region, T - 102 °K, while in a typical HII region T ~ 104 "K, so that we expect pressures to range from 1.4 x 10"1A to 3 x JO"12 dynes cm"2. The magnetic field in the medium is of the order of 10""5 gauss; hence the magnetic pressure (B2/&n)k of the order of 4 x 10"12 dynes cm'2. The pressure attributable to cosmic rays is estimated to be about 2 x 10"12 dynes cm"2; hence a reasonable upper bound of 10"1! dynes cm'2 can be assigned as the total interstellar pressure. This value is a factor of 10° smaller than that estimated for a hydrostatic corona, and we conclude that the corona cannot be contained, but must undergo a steady-expansion. We therefore abandon the assumption of hydrostatic equilibrium and inquire into the hydrodynamics of coronal expansion.
one-fluid models of steady, spherically symmetric coronal winds
To simplify the treatment of the hydrodynamics of a coronal wind, the following assumptions are made: (1) the How is taken to be steady; (2) the wind is spherically symmetric; (3) the gas is an ideal compressible fluid. Clearly all of these assumptions are idealizations, and they restrict the applicability of the results for, say, the real solar wind; nevertheless they provide a good basic physical model for general study.
The equations describing a steady flow are the equation of continuity (mass conservation)
the momentum equation
ld(r2pv)/dr = 0
pu(dv/dr) = -(dp/dr) - P(GJiQ/r2)
(15-54)
;i5-55)
and the energy equation (including conductivity)
l_d_ 7Jr
rpv   jv- + e
P
= -pv
For a fully ionized gas of pure hydrogen, p = 2nkT and p = nm, where n = np = ne and m is the mass of a hydrogen atom. The specific internal energy of the gas is 2{\nkT)/(nm) = (ikT/m), and (pip) = {2nkT)j(mn) = (2kT/m), so the specific enthalpy is h = e + (p/p) = (5/cT/m). Note that equation (15-56) does not include any terms representing radiative losses or mechanical energy deposition from dissipation of waves. The assumption made is that these processes occur in a relatively thin layer at the base of the corona, and that above some reference level only the kinetic, potential, thermal (enthalpy), and conductive terms need be taken into account. The continuity equation can be integrated to give
4nr2nv = F = constant
;i5-57)
526     Stellar Winds
15-2   Coronal Winds 527
where F denotes the particle flux, or
Jl = mF = 4nr2nnw ~ constant (15-58)
where.// denotes the rate of mass-loss. The energy equation can be integrated to give
(4%r2nv)[{mv2 + 5kT - (G,M0mjr)\ - (4nr2)ic(dT/dr) = E = constant
(15-59)
which states that the total energy flux through a spherical surface is a constant of the flow. Equations (15-57), (15-55), and (15-59) comprise a nonlinear system of two first-order differential equations, containing two constants of integration; two more constants will result from the integration of the system; hence a total of four conditions (boundary conditions or specifications of the nature of the solution) must, in general, be imposed. Before discussing this general problem, however, it is extremely instructive to consider the simple case of an isothermal corona.
If we demand that the flow be strictly isothermal, we need to specify only n and y, and hence we may omit the energy equation and employ only the momentum and continuity equations. [The ad hoc assumption T = constant is equivalent to invoking some unspecified mechanism to heal or cool the gas in just the right way as to yield the desired temperature.] Equation (15-55} becomes
nmv(clvfdr) = -2kT0(dn/dr) - nm(GJ/Jr2) (15-60)
and using equation (15-57) to eliminate n we find
i[l - (2kT0/mo2)](dv2/dr) = -{G-^/r2)[l - (4AT0r/G.,#>m)] (15-61)
This equation yields several families of solutions that have substantially different mathematical behaviors and physical significance. For the solar corona, {4kT0rJGJ?om) * 0.3. Thus it is clear that the righthand side of equation (15-61) is negative for r < rc and positive for rc < r < co, where rc is the critical radius
rc = (GJ?om/4kT0) (15-62)
at which the righthand side is exactly zero. For the solar corona, {rjr0) « 3.5. At r = rc, the lefthand side of equation (15-61) must also be zero; this may occur in one of two ways. We may either have
(dvfd>% = 0 (15-63)
or v(rc) = (2/<T0/m); = vc (15-64)
The critical velocity vc defined by equation (15-64) is equal to the isothermal sound speed.
We now restrict attention to solutions for which both v and (dv/dr) are single-valued and continuous. First, suppose that equation (15-63) is satisfied. We may then construct solutions for which 1 - (2kT0/mv2) has the same sign for all r. If v(rc) < vc then v{y) will have a relative maximum at /;.; this yields solutions of type 1 shown in Figure 15-1. These solutions are everywhere subsonic. On the other hand, if v{re) > vc, then v(r) will have a relative
0 123
FIGURE 1 5-1
Schematic variation of velocity in units of the critical velocity l-„ as a function of radial distance in units of the critical radius rc, for stellar winds and breezes. Solutions of type 1 are the subsonic breezes. Solutions of type 2 ure everywhere supersonic. Solutions 3 and 4 are the transsonic critical solutions that pass through the critical point (heavy dot) continuously. Solutions of types 5 and 6 are double-valued, but are important for fitting shock transitions.
528      Stellar Winds
529
minimum at rc, and will be everywhere supersonic; these are the type-2 solutions shown in Figure 15-1.
Alternatively, if equation (15-64) is satisfied, we obtain a critical solution (i.e., v = vc at r = rc) that has definite slope at r = rr. Suppose that {dvjdr\.c > 0; then we obtain a unique transsonic solution that is monotone increasing from subsonic speeds (v < vc) for r < rc to supersonic velocities {v > vc) for r > rc. This unique solution is shown as type 3 in Figure 15-1. \i(dv/dr),.c < 0, then we obtain a unique solution in which v{r) is monotonically decreasing from supersonic speeds for r < rc to subsonic speeds for r > r,; this is shown as type 4 in Figure 15-1.
Finally, there exist two families of solutions defined by the restrictions r ^ r% > rc or r ^ < rc, both of which have v{r.^) = vc and {dv/dr)rii = co. These give rise to the double-valued solutions of types 5 and 6 shown in Figure 15-1. Initially we exclude consideration of these solutions because they are double-valued; we shall find later, however, that they may be used to provide part of a complete solution that includes a shock transition to match boundary conditions as r -* cc.
To decide which solution to choose, we must now invoke physical boundary conditions. First, both the entire family of type-2 solutions and the unique type-4 solution can be excluded because they predict velocities v > i\. ^ 170 km s_1 in the low corona, which is not observed. This leaves solutions of type 1 or the unique transsonic solution of type 3. We may make the choice on the basis of the behavior of the solution as r -» cc. Equation (15-61) may be integrated straightaway to yield
(v/vf - ln(v/vc)2 = 41n(r/rc) + 4(rjr) + C
(15-65)
Exercise 15-7: (a) Verify equation [15-65). (b) Evaluate C for the critical solutions.   |c) Evaluale C for solulions oftypes ! and 2.
Consider first solutions of type 1 asr -> co.Here(u/t;c)is < 1 and decreasing as r 00j hence on the lefthand side the dominant term will be - 2 ln(i>/rc), and on the righthand side it will be 4 ln(r/rL.). Thus as r -> oc, then v <r_ r~~, which implies (from the equation of continuity) that n remains finite. These solutions thus yield a pressure at r = co which greatly exceeds the ambient interstellar pressure, and therefore they can be rejected on the same grounds as the hydrostatic solution was. On the other hand, the critical solution has (v/vc) > 1 and increasing as r -> co, hence we find v ^ 2vc ln(r/rr)* In this case n cr_ r~2v"L -» 0 as r -> cc, and we conclude that this critical solution can indeed match the boundary condition at infinity.
Following a line of reasoning similar to that outlined above, Parker concluded (correctly!) that the solar corona must be undergoing a transsonic expansion into interplanetary and interstellar space (496), and that near
4	X	to6 °k ; [	
" f	X	106 °K |	-
	X	106 °k !	-
1.5	X	106 °K \	-
1	X	106 °k !	
10
x 106 °K
Earth
_L_L
140 160
FK.URE 15-2
Isothermal solar wind solutions; curves are labeled with coronal temperature.   Ordbuile: velocity in km s "1;   abscissa: heliocentric distance in 10" km. From (496), by permission.
the orbit of the Earth the solar wind has velocities of the order of a few hundred km s"1. Several wind solutions for isothermal coronae at different temperatures are shown in Figure 15-2. The wind solution given above is not entirely satisfactory because (v/vc) increases without limit as (r/rc) -* co. This behavior is an artifact of the assumption that the corona is strictly isothermal. The corona is, in fact, nearly isothermal near the Sun because conduction is effective and distributes the heat deposited by wave dissipation very efficiently, thus obliterating large temperature gradients. At large distances, however, expansion of the gas must ultimately force it to cool; if we insist that T remain fixed, we have, in effect, introduced a (spurious) energy source to maintain the temperature. This effectively results in a continuous deposit of energy into the enthalpy of the gas, and this energy is available to do work to continue to accelerate the gas without limit. The problem is easily overcome, however, and Parker was able to produce satisfactory wind solutions by using either of two assumptions, (a) The corona is isothermal for r0 < r ^ and expands adiabatically for r > 1%. In this case, for r > the pressure and density are related by a polytropic law of the form P = Poip/Po)7 witn)' = f (ideal gas), (b) The corona is everywhere polytropic with a polytropic exponent y < f. Both of these classes of solutions avoid an unphysical increase in v at large r, and one finds that v approaches a finite value vIx.. as /■ 00.
The isothermal solutions described above reveal the broad outlines of the nature of coronal expansion into a supersonic wind, but clearly are inadequate for a detailed description of the flow. In particular, in solving the
530      Stellar Winds
problem, we wish to determine the temperature distribution T(r). We m'ust, therefore, turn to the full set of equations including energy balance, namely equations (15-55), (15-57), and (15-59). As before, there are two basic classes of solutions to be considered: (1) subsonic solutions, called stellar breezes, which resemble those of type I shown in Figure 15-1; (2) transsonic critical solutions, stellar winds, resembling the unique solution of type 3 shown in Figure 15-1. These two classes are distinguished from one another by the value of the total energy flux E in equation (15-59). The breeze solutions all have E = 0, while the wind solutions have E > 0.
Although we are primarily interested in wind solutions in the solar context, the breeze solutions, which have been studied extensively (147; 532), played an important role in the development of the theory, and it is worthwhile to consider them briefly here. In the breeze solutions, as r -»■ co, then u 0, T 0, and p -» 0. There is a whole family of such solutions that are differentiated from one another by the limiting value oi(tnv2/kT) as r -* co (532). Because a match to the interstellar pressure can now be achieved, we cannot exclude these solutions from the outset, as we could for an isothermal wind; indeed. Chamberlain (147) advocated this type of solution for the solar corona, in which case the velocity of the material near the Earth's orbit would be only 20km s"'. He argued that the correct hydrodynamic solution should be consistent with an evaporative model of the corona, in which individual particle motions are calculated assuming a critical level above which the density is so low that fast-moving particles suffer no further collisions, but escape. He found that in the evaporative model the mean speed of ions near r$ would be about 10 km s"1, and concluded that the breeze (rather than wind) solution was correct. The question was laid aside when direct observation proved the validity of the wind solution. But subsequent work (108; 339) has shown that the basic premise that the hydrodynamic and evaporative pictures should agree is, indeed, correct, and that the difficulty lay in the calculation of the original evaporative model. In particular, when account is taken of the fact that particles moving at different speeds escape from different levels, not from a single "critical" level (the faster particles escape from deeper, denser layers), and when a correct dynamic calculation is made of the electric field in the plasma that couples the electron and proton flows together, then the evaporative solutions yield n x 10 cm"3, v « 300 km s"1, and T(proton) a; 5 x 104 'K, in good agreement with the observed values. We would now conclude that the correct evaporative model supports the solar wind solution.
Let us now examine the wind solutions in greater detail. In all of these solutions, as r -> cc, then v -+ (a nonzero value), n -> 0, and T -> 0. As mentioned above, four distinct conditions are required to specify the solution uniquely. Typically, these may be chosen to be values for the coronal density and temperature, and to meet the requirements that the solution pass
15-2   Coronal Winds 531
smoothly through the critical point, and that T(r) -»■ 0 as r -> co. The last of these conditions, however, is actually more complex than it seems, and it is now known that the nature of the variation of T with r depends upon the mechanism of heat transport at r = co [see (200; 201; 202; 532;324,47)]. Suppose first that the heat-conduction flux remains finite at r = co, with a value £e(oo). From equation (15-59) it is clear that we will then find vx = {2[E - £c(co)]/y//}+, and further, because [^THdTjdr)]^ = constant, the temperature obeys the asymptotic law T x r~* This is the same law found by Chapman, and used by Parker (499) in his wind solutions. Next we might suppose that both a conduction flux and an enthalpy flux persist to r = co, and that the ratio (5kTu)/[r2T*{dT/dr)] -» constant. In this case both fluxes approach zero at r = co, and vK = {lEjJlf. Further, the special condition imposed on the ratio of the fluxes implies that T co a solution first obtained by Whang and Chang (667). Finally, we might suppose that the conductive flux vanishes, as r -> co, more rapidly than the enthalpy flux. Again i-rSi = {lEjJf. In this case there is no energy exchange mechanism in the flow as r -> co, and the gas merely expands adiabatically. If the poly-tropic exponent is y, then T cc p11'"1'. But p cc r~2\ hence, as r -> co, then Tec (r"2)<*'or Tec r_t, for y equal to f (ideal gas).
The relationship of these solutions to one another was explained lucidly by Durney (200), who considered wind solutions that were all chosen to have the same value for T0, but different values for n0. For small values of nQ, the critical solutions yield large values for e(co) s= (E/F) which is the residual energy per particle at r = co, and they have T x r~* As n0 is increased, the conductive part of e(oo), say ee(co), becomes smaller, and at a particular value (ng), e,(co) = 0 and T cc r~* (i.e., this particular value gives the Whang and Chang solution). As n0 is increased further, ec(co) remains equal to zero, and now e(cc) decreases; here T x r~* Finally, when n0 increases still further, a limiting value is reached at which e(co) = 0, and a stellar breeze solution is obtained (see Figure 15-3). Physically, these results imply-that, as n0 increases, more and more of the conductive flux at infinity is consumed by the expansion (the flow is more and more massive), until at some point ec(co) vanishes. If we now add more material, the flow continues to be supersonic, but more and more thermal energy is consumed in the adiabatic expansion, and <?(oo) decreases. Eventually e(co) vanishes and the flow becomes subsonic.
In a systematic study of stellar wind solutions, it is very convenient to work with dimensionless variables. Chamberlain (147) suggested the transformations
t = (T/T0) (15-66a) it = (v2 pm/kT0) (15-66b) and a = (GJ/Q pm/kT0r) (15-66c)
532
15-2   Coronal Winds 533
T0
PIG U RE: 15-3
Regions of stellar wind and breeze solutions as a function of coronal temperature t0 (in units of CJ/Qiirn/kr()) and density a'„ [in units of 2icnk~ 1 (g, Shaded area contains breeze solutions. The
dotted curve marked S delimits the region (to the right) where the flow is already supersonic at the base of the corona (r = r0). The region P contains wind solutions with a Parker temperature law, T >x r~ :, as r -» q-j ; these solutions have a nonzero conductive flux at infinity. The region D contains Durney temperature law (7 <x r~";) winds, which become adiabatic as r -> x. The curve marked WC oives the locus of solutions with a Whang-and-Chang temperature law, T </_ r~i as /■ — m\ these particular solutions occur just when the conductive flux at infinity first drops to zero.   From (532).
where p is the number of atomic mass units per particle ( = £ for ionized hydrogen). In these units, equations (15-57), (15-55), and (15-59) become, respectively,
„X~2^ = (kTJttm)*F/l4nG2jei>) = C (15-67)
i[l - (t/i//)](#/^-) = 1 " 2(t/A) - (chidK and AzHdx/d).} = c«, -hji + X - |t
where = fiE/kT0F
(15-68) (15-69) (15-70)
is the residual energy per particle at infinity (in units of /<T0), and
A = 4itK0Gfi2mJ/oT0</{k2F) (15-71)
In these equations we need specify only and A to perform the integration (imposing, of course, the requirements that a critical solution be obtained, and that t -> 0 as X -» 0); by redimensionalization, one can obtain several solutions from a single dimensionless one. A further transformation (531) reduces the number of arbitrary constants to one (for winds only). Thus, write Tjj. = t/s^, i//^ = i}//er,, and = X/c.^. Then equation (15-68) has the same form in terms of the new variables, while equation (15-69) becomes
KziUhJdX*) = 1 +
15-72)
where K = s.^A. A large number of solutions for a wide range of the parameter K are given in (201), along with an example of how to recover a unique dimensional solution having specified, say, T0 and F. A detailed discussion of breeze solutions is given in (532).
Exercise 15-8: Carry out the transformation to dimensionless variables, and verify equations (15-67) through (15-72).
transition to the interstellar medium
The solutions derived above ail have pressures that vanish at r = ct. Actually the interstellar medium does have a finite (if small!) pressure Pi ^ 10~n dynes cm~2. When the high-speed gas in the wind encounters the interstellar medium, a stationary shock front is formed (624, 306; 665) in which the flow speed suddenly diminishes to a small value, and the density and temperature rise. The solution jumps discontinuously from the critical solution (curve 3 in Figure 15-1) to one of the subsonic solutions of type 6 shown in Figure 15-1. Beyond the shock the material finally cools and recombines, and the velocity drops to zero [see, e.g., (106)].
At large distances from the star, the wind is highly supersonic, and the gas pressure is negligible compared to the kinetic energy density. Thus, balance with the interstellar pressure is achieved through the impact pressure of the material in the wind. We then have
mnv
Pi
4 5-73)
From the equation of continuity and the assumption that v is approximately constant, we know that n(r) = >i©(f'©/J')2> and using this relation we may
estimate the radius of the shock front in the solar wind as
(rs/>©) = (mn^jlpif
(15-74)
534      Stellar Winds
15-2   Coronal Winds 535
Adopting ne x 10 cm-3, v & 300 km s_1, and the upper limit on p; given above, we find i\ > 30 a.u. (i.e., outside the orbit of Neptune). The physical conditions in the flow beyond the shock can be estimated by applying the Rankine-Hugoniot relations [equations (15-44) through (15-46)]. On the conservative assumption that the temperature falls as r""% the temperature in the preshock flow is of the order of 4 x 104 °K, which implies a sound speed in ionized hydrogen of about 25 km s-1, so the Mach number is about 12. In the large Mach-number limit, the Rankine-Hugoniot relations reduce to
v2 = Vl[y - l)/(y + 1) (15-75)
n2 = n.iy + i)/(y - 1) (15-76)
and p2 = Irt^nv2^/ + 1) (15-77)
where in deriving equation (15-77) we have written
PiM,z = {pja2)v2 = (pjy)v,2
Here the subscript 1 denotes preshock and 2 denotes postshock conditions. Combining equations (15-76) and (15-77) into a perfect gas law, we estimate the postshock temperature to be
T2 = (y - l)mv2/[{y + \)2k~\ (15-78)
where we have assumed that the material remains fully ionized, and that the proton and electron temperatures equalize. Assuming y = f, we see that v2 = ~ 75 km s"1 while T2 = 3mu2/(32fc) « 106 °K [which shows that the postshock material is indeed subsonic, as expected from Exercise 15-3(b) for Mj » 1].
The injection of hot, high-velocity material from stellar winds has significant implications for the energy balance in the interstellar medium. The picture developed here is purposely quite simplified; more elaborate calculations have been made allowing for the effect of an "interstellar wind" (arising from the Sun's peculiar velocity with respect to the interstellar medium), magnetic fields, and thermal conduction [see, e.g., (498, Chap. IX)].
the magnetic field and braking of stellar rotation
From direct observation it is known that the Sun has a magnetic field. Because coronal material is completely ionized, it has an extremely high electrical (as well as thermal) conductivity, and thus magnetic fields are "frozen" into the material (i.e., the charged particles cannot diffuse readily across field lines). The large-scale expansion of the corona thus implies that there will be a transport of the solar magnetic field into interplanetary space. If the Sun did not rotate, the field lines would be drawn out radially. The Sun
does rotate, however, and the field lines must be considered to be anchored at the solar surface. Field lines from a particular point on the surface will be drawn out along the streamlines of fluid elements as seen by an observer in the rotating frame of the Sun; this yields a spiral pattern for the interplanetary field (496:498, 137; 107, 67; 324, 1J).
When the magnetic field is taken into account, the physics of the fluid flow in the wind becomes much more complicated, and considerable mathematical complexity results; we shall not pursue this problem here but merely refer the interested reader to the literature [e.g., (463; 659)]. It is of considerable interest, however, to calculate the angular momentum carried away in the wind, for this loss has important implications for stellar rotation. The magnetic field must satisfy Maxwell's equation V ■ B = 0, from which it follows, in spherical coordinates, that B£r) = (r0/r)2Bf(rp). The azimuthal component in the spiral pattern, B^, can be computed in terms of Bry and hence the total field can be calculated. We can then estimate the ratio of the energy density in the flow to the magnetic energy density—i.e., a = (fmnu2)/{2?2/87i); one finds that a » 1 in the vicinity of the Earth's orbit, but a is small at the base of the corona. This implies that—although the fluid motions will dominate the field at large distances, and will drag it along with the material—deeper in the flow, the magnetic field will dominate and will be able to drag the material along with the solar rotation as seen in a fixed frame. Thus there is a region with r ^ rA in which the material is forced into corotation with the Sun, and a region r > rA where the material streams essentially radially. Inuitively we would expect the transition to occur at the radius rA where a = 1, and detailed analysis [(463; 659; 324, §111.15; 107, §3.7)] shows this conjecture to be correct. The speed of the hydromagnelic Alfven waves is vA = (B2/4np)i, hence a is nothing more than the square of the Alfvenic Mach-number a = (v/vA)2 = MA2; thus the flow corotates with the sun inside the Alfvenic critical point, and streams radially outside it. The Alfven speed can be directly measured in the vicinity of the earth's orbit; using the equation of continuity and the inverse-square radial dependence of B, and demanding that MA = 1 at the Alfvenic critical point, we find
(rA/re) - vA{r@)ilv(rA)v{^)f (15-79)
We may obtain a lower bound for rA by assuming v(rA) ~ v(r@), and substituting numerical values we find rA * 20 RQ. This region is large, and thus the effects of magnetic fields greatly increase the angular momentum contained in the material flowing in the wind.
Exercise J5-9:   Verify equation (15-79) and the numerical estimate of rA.
The angular momentum of a particle of unit mass at the equator of a star is /0 = wrA2\ where co is the angular frequency of stellar rotation. Hence the
536      Stellar Winds
15-2   Coronal Winds 537
total angular momentum lost from the stellar surface as a whole is
UlL/dt) = -{4nrnum)l0 J"'2 cos3 0 dO = -f J/l0 (15-80)
where we have accounted for the fact that (at latitude 0)}[()) = !0 cos2 tf. This angular-momentum loss into the wind implies a braking of the star's rotation. We may estimate the rotational-braking decay-time x by writing {dLjdt) ^ — L'r, and expressing L as L = ho, where 1 is the moment of inertia, from which we rind r -- UjUir/). For the Sun, / = 6 x iG°-1 gm cm2, and using Ji = 2 x 10-u.^o/year and rA « 20 R0,wefmdT ^ 1010 years, which is comparable to the thermonuclear-evolution time-scale for the Sun. That is, if the solar wind maintains its present properties, we expect the solar angular momentum to be significantly diminished during the main-sequence
lifetime of the Sun.
One of the striking features of the statistics of stellar rotation is the marked drop in the observed average rotation speed <y sin i>, as a function of spectral type, for stars of spectral types F and later. Although it is possible that this decrease is associated with the formation of planetary systems (in the solar system the Sun has only 2 percent of the angular momentum), it is also extremely suggestive that this is precisely the point where stars develop deep hydrogen convection zones, and hence presumably have coronae and winds. It is thus very attractive lo hypothesize (554) that all later-type stars lose their angular momentum via magnetic braking in stellar winds. Time-scales for such braking can be estimated by an examination of mean rotation speeds of stars in clusters whose ages can be determined from stellar-evolution considerations; when this is done, it is found that there is indeed a strong decrease with age (362). and that the braking times may be as short as 5 x 10s years. Although the stellar time-scale appears much shorter than that derived from the present solar wind, it is known that chromospheric activity (and presumably also the activity of the corona-wind complex) decreases with age (675; 677); hence it is possible that in the early history of the Sun the wind was stronger, and provided more efficient braking. Actually this discussion provides a good example of the fruitful exchange of ideas that occurs when solar properties are used to develop detailed physical models of a particular phenomenon, which are then applied in a stellar context, where results of evolutionary significance on long time-scales can be inferred.
detailed physics of the solar wind
The study of the solar wind by means of direct measurements from satellites has greatly deepened our understanding of the physics of coronal flows. Tn an effort to fit the observations, numerous theoretical refinements have been introduced to extend the basic model described above, and to achieve greater realism. Space does not permit anything more than mere
mention of the many physical problems that have been addressed: for further details see, e.g., (324) or (107) and the references cited therein.
First we might ask how well a one-fluid conductive model fits the data; taking the model by Whang and Chang (667) as typical, one finds/; = 8cm~\ v = 260 km s"1, and T = 1.6 x 103 °K at the radius of the earth's orbit. Comparing with the observed values listed in Table 15-1, we see that the model gives good agreement for the density, and the one-fluid temperature agrees well with the electron temperature, but is a factor of four larger than the proton temperature; the velocity is about 20 percent too low. The fact that the protons and electrons can have different temperatures is explained by the very low densities, and hence collision rates, in the solar wind near t(q ; these rates are so low that they cannot equilibrate the thermal energy in the two separate components (moreover, the thermal velocity distribution for both electrons and protons is observed to be anisotropic). Two-fluid models have been constructed (618; 282) in which electrons and protons are allowed to have distinct temperatures; an energy equation is written for each component, including an energy exchange term of the form jvk(Tp — Te), where v is a collision frequency calculated for Coulomb collisions of the charged particles. Typical models of this type yield n % 15 cm-3,   r ^ 250 km s_1, Te * 3.4 x I05 °K, and Tp % 4.4 x 103 UK at r = />. The densities are about a factor of two too large, the velocity is 20 percent loo low, the electron temperature is a factor of two too high, and the proton temperature is a factor of 10 too low. Although the basic concepts used in the two-fluid model do admit more physical realism than is inherent in a one fluid description, the quantitative results are not impressive; in particular it appears that the energy exchange between protons and electrons must proceed more efficiently than is predicted from Coulomb collisions alone. The exchange has been treated on the assumptions that the particle-velocity distributions are isotropic and Maxwellian, and that magnetic field effects may be ignored; possibly all of these assumptions are inadequate. However, most of the discrepancy seems to be removed when the effects of viscosity are taken into account.
Magnetic fields may play an important role in determining the nature of the solar-wind flow, through magnetic forces, by modification of transport coefficients such as the conductivity, and through energy transport and dissipation by hydromagnetic waves. Each of these effects leads to a significant increase in the complexity of the theory, and can appreciably alter the detailed results. Worse, allowing major sources of momentum of energy input into the wind throughout the flow (from any source, not just magnetic), changes the whole topology of permitted solutions of the equations, and introduces many possibilities beyond those shown in Figure 15-1 (307).
The models described above all omit the effects of viscosity. When viscous forces are included, the equations become of higher order, and the momentum
5?8      Stellar Winds
15-2   Coronal Winds 539
equation no longer possesses a singular or critical point (668). The earliest one-fluid viscous models (553; 668), using classical viscosity coefficients for a proton-electron plasma, were disappointing, for they yielded flow speeds and temperatures about a factor of two too small. A resolution of the difficulty is achieved when magnetic effects upon the viscous stress tensor are taken into account (660), and modern models including viscous terms (681) have the right velocities, and raise the proton temperature (from the spuriously low values found in the inviscid two-fluid models) to values close to those observed.
Even the most refined and accurate spherically symmetric models still represent only very high-order abstractions of the real solar wind, for it varies, on time-scales of days, over wide ranges of all the physical variables. Prominent individual features that appear are high-speed plasma streams; these often are observed to recur with a synodic solar-rotation period, and are, therefore, the result of specific conditions in localized regions of the corona. Further, there are energetic flare-produced shock waves that give rise to geomagnetic storms. These structures often interact in a complex way and, more and more, it seems that it is an oversimplification to view the solar wind as a smooth flow upon which "atypical" structural features are superposed; rather, these complex structures are, in some sense, the wind itself. Similarly we are coming to understand that the wind does not arise from a single smooth coronal condition, but that it may be produced in highly specific regions of markedly differing properties, and can be substantially modified by interaction with magnetic fields and by rapid (nonspherical) divergence from a limited initial volume. At some point the problem of the time-dependence of a full three-dimensional model must be solved; only for the solar wind do we have data that require (and, reciprocally, permit!) such solutions, but it is clear that the results will have important implications for stellar winds as well.
stellar coronal and winds
As mentioned above, it is reasonable to suppose that all stars having hydrogen convection zones should also have coronae and winds. Further, many stars clearly have much more massive flows than the Sun, for the winds are sufficiently optically thick in some spectral lines to give rise to displaced lines or P-Cygni features. Needless to say, we know less about stellar winds than about the solar wind, and much work remains to be done in this rapidly developing field; nevertheless, a number of interesting results emerge even from very simplified calculations.
Parker emphasized (498, Chap. XV) that the amount of energy consumed by coronal expansion rises very rapidly with increasing coronal temperature; for example, for the Sun, the energy consumption ranges from 1027 to 3 x
1030 ergs s 1 if T0 varies from 106 "K to 4 x 106 T<L Thus a coronal wind acts as a very effective thermostat for controlling the coronal temperature. Tf a wind is present, we demand that the temperature be compatible with the flow—i.e., T0 < [GJi\mj4kR^. For the Sun, the numerical factor of 4 is replaced by 10, so one might expect
T0 * 0.\{GJt^m/kR) (15-81)
In main-sequence stars, {.M.JR^) varies only by about a factor of two above or below the solar value, which suggests a similar restricted range for T0 (less than the variation of photospherictemperatures!). In contrast, the energy in the wind may vary widely (perhaps more so than the stellar luminosity). For a giant or supergiant, {JtJR*) is much smaller than in main-sequence stars and, accordingly, T0 should also be much smaller. For example, in an M-supergiant, equation (15-81) suggests T0 ^ 4 v 104 °K. At such low temperatures conduction becomes ineffective, and one must invoke some mechanism to heat the corona as a whole (665; 666); quite possibly, significant wave dissipation occurs throughout the corona. Similarly, the velocity of the winds ultimately approach some significant fraction of the escape velocity, and thus we can expect
i\„ > (15-82)
Again this implies little variation along the main sequence, and implies that supergiants have quite low wind velocities. The picture developed above depends on the assumption that the wind is subsonic at the coronal temperature maximum. If, however, the critical point in the wind lies below this maximum, the run of temperature with distance is no longer monotonic and the properties of the flow may change substantially.
To construct detailed models or apply existing computations [e.g., (201)], we need to specify the relevant coronal conditions, such as n0 and T0, from theoretical calculations—or to specify one of these quantities and a flow parameter such as the mass-loss rate, which can sometimes be determined observationaljy. The computation of coronal conditions is exceedingly difficult. The basic technique is to calculate the acoustic energy flux created in the convection zone [see, e.g., (394; 395; 525; 601)], compute the dissipation of this flux in the coronal material, and (taking into account losses by radiation and the energy transported by conduction) obtain a temperature-density structure in a model corona [see, e.g., (192; 377; 378; 630; 631; 632)]. Each of these steps requires use of an uncertain theory, and of necessity many approximations must be made, so a high degree of reliability cannot be assigned to the final results. In particular, a good fit is not obtained to observed solar coronal properties [see Fig. 21 of (192)] in all cases. But if we simply accept the calculations, however uncertain, at face value, it appears that for
540     Stellar Winds
15-3   Radialion Hydrodynamics 541
main-sequence stars the maximum acoustic flux is found for stars near spectral types F0; for these stars T0 % 4 x 106oK.andn0 % 3 x IO,0cnT3 (192). One would expect substantial coronal winds in these stars. If T0 and n0 are regarded as given, then the calculations cited previously can be used to derive wind models.
Consideration of the energetics of stellar coronae and winds is also quite instructive (290). A corona is heated by the mechanical flux Fm delivered to it, and loses energy via radiation and conduction, and in the kinetic energy of the wind. Analysis shows that, for a given coronal pressure, the rate of energy loss by mass-ejection and by conduction increases with rising temperature, while that by radiation decreases. Therefore, for a given pressure, a temperature exists at which the total energy loss from the corona is a minimum. Further, losses from all three processes increase with increasing pressure, so for minimum-flux coronae there is a monotone relation between coronal pressure and the energy flux required to maintain the corona and wind. This suggests that a given Fm determines a unique coronal temperature and pressure, and hence wind. (Actually it remains to be demonstrated that, if one has a given minimum-flux corona, and arbitrarily changes Fm, then the corona necessarily adjusts to a new minimum-flux configuration.) The competition among these processes establishes three basic classes of coronae. (1) For low values of F,„ the main losses are by conduction and radiation. An example is the solar corona-wind complex where only about 10 percent of the mechanical flux delivered to the corona goes into the wind. (If the wind arises in geometrically localized regions of the corona—e.g., coronal holes— the local conversion efficiency may be much higher.) (2) At intermediate values of Fm the losses are mainly from radiation and mass ejection. An example is the wind in the A2 supergiant a Cyg where losses into the wind are dominant, while those by radiation are significant, and those by conduction are negligible. (3) At very large values of Fm, mass loss totally dominates. For early-type stars the rate of mass ejection is as high as 10~6 to 10"5 J/0f year, and exceedingly high coronal temperatures (which appear to be excluded by the observations) would be required to drive the winds; here the winds are driven instead by radiation forces (see §15-4).
15-3   Radiation Hydrodynamics
In both the atmospheres and interiors of stars, there exist intense radiation i fields that can influence heavily the momentum and energy gains and losses ? of the material, and hence its motions. To study the dynamics of a flow occurring in such a situation, it is fruitful to consider the fluid as consisting of both material particles and photons, and to calculate the contributions of both types of particles to the equations of motion and of energy conserva-
tion. In this way we obtain the equations of radiation hydrodynamics, which describe the coupled flow of the gas and radiation. In the applications of interest here, it will be assumed that the flow velocities v « c, so that the material particles can be treated nonrelativistically; in certain other contexts (e.g., in supernovae or thermonuclear explosions), this assumption may not be valid. Despite the restriction to v « c, the fact that the photons have velocity c leads to subtleties in the way the radiation and material terms interact, and it is a nontrivial task to obtain equations that describe the interaction fully consistently. In the end it is simplest and most reliable to develop them in a relativistically covariant form from the outset (621; 524; 664; 135). These equations may subsequently be simplified to retain only terms of order (v/c), and to omil terms of 0(v2/c2) and higher. It must again be noted that, as was true in our earlier treatment of the comovmg-frame transfer equation, it is not sufficient to apply a unique Lorentz transformation, because the fluid velocity is, in general, a function of position and time. As before, we consider transformations from a set of uniformly-moving frames that instantaneously coincide with the moving fluid.
The radiative contributions to the equations can, in principle, be written in terms of either laboratory-frame or fluid-frame quantities. Although in some ways it is easier to write down the equations of radiation hydrodynamics in a stationary frame [see, e.g., (521; 551; 692)], they are actually of a simpler form when written in the comoving frame of the fluid, for the radiative terms are most easily evaluated in that frame (621; 135). In the past this approach was not widely employed, because one must be able to solve the comoving-frame transfer equation to calculate the necessary radiation-field quantities. As was shown in §14-3, the comovmg-frame transfer equation is easily solved with present-day techniques (and is, in some ways, simpler than the fixed-frame equation, which is complicated by anisotropics in the absorption and emission coefficients arising from Doppler shifts). We shall, therefore, concentrate primarily on a comoving-frame formulation, but it will also be shown that these equations are consistent (to order v/c) with the fixed-frame equations. Only one-dimensional spherically-symmetric flows will be treated in this book; an extensive collection of formulae for three-dimensional flows in various coordinate systems can be found in (521, Chap. 9).
the material stress-energy tensor and the radiating-fluid equations of motion
In the Cartesian coordinate system (x1, x2, x, x4) = (x, y, z, ict), a co-variant formulation of the equations of motion and of energy conservation for the material alone yields a system of equations of the form
{dT^/dx") = F\   (a = 1,... t 4) (15-83)
542      Stellar Winds
15-3   Radiation Hydrodynamics 543
where F* is a four-vector force (the Minkowski force) and T3" is a tensor describing the momentum flux (stress), momentum density, and energy density of the material. Similar equations were written in §14-3 for the radiation [cf. equations (14-118) through (14-122)], and by analogy with equation (14-120) we might expect Ta/f to be of the form
He
where IT, G, and E are suitable covariant generalizations of the momentum flux tensor, the momentum density vector, and the total energy density of the material, respectively. We place three demands upon T: (1) that it be written in terms of scalar invariants and four-vectors, so that it is co variant; (2) that it reduce to the correct limit in the frame at rest with respect to the fluid; and (3) that it yield the correct nonrelativistic limit in the laboratory frame.
First, we may define the proper time, a four-scalar (invariant) as
(dx)2 = -(dx, dxa)ic2 = dt2 - c-2(dx2 + dy2 + dz2) (15-85)
Clearly dx reduces to di as v -»■ 0. Then we define the contravariant four-velocity as
V = {dx*/dx) (15-86)
Noting from equation (15-85) that (dt/dx) = (1 - v2/c2yi = y, where v is the ordinary velocity v2 = [{dx/dt)1 + (dyfdt)2 + {dzjdt)2\ we have
V' = (dxj/dt)(dt/dx) = (1 - u2/c2)~WM)
= yv',   (i = 1,2,3) (15-87a) K4 = icidtjdx) = icy (15-87b)
Note in passing that V\V'- = —c2, a world scalar (which is invariant under any arbitrary coordinate transformation, as can be readily shown from the transformation properties of covariant and contravariant vectors). Next, if p0, e, and p are the invariant mass density (i.e., the particle number density times rest-mass per particle), the specific internal energy, and the pressure measured in the rest-frame of the fluid, then the equivalent total mass density is
Poo = PoU + e/c2) (15-88) Also, following Thomas (621) we define
Pooo = Poo + Vic2 = Pod + hfc2) (15-89)
Here h is the specific enthalpy of the gas, and (pQh/cz) gives the mass equiv-j alent (per unit volume) of the total energy contained in the microscopic
i motions of the gas.
Noting that in the nonrelativistic limit TTJ = pu'V + p 3ij, while G' =
i po\ we may hypothesize, by analogy, that
i
T3" = PoooV'V + pSxfi (15-90)
where <5a/j denotes the standard Kronecker symbol. It is obvious that equation (15-90) satisfies requirement (1) stated above. The individual components may be written more specifically as
Tij = PqqqVW + pSij = p1vivj + p3ij (15-91a) Tu = T4f = icyPw)oV- = icpili (15-91b)
and       T44 - -c2yVooo + p= -c2Pl + p (15-91c)
where i = 1, 2, 3, /' = 1, 2, 3, and p, = ?2Pooo- Notice, now, that in the frame of the fluid itself, where v = 0, T becomes diagonal, with T " = p = (FI'%, and T44 = — (p0c2 + p0e); this gives the correct static pressure and the correct, energy per unit volume of the fluid (including the material rest-energy). Thus requirement (2) stated above is satisfied. Next, let us examine the nonrelativistic limit (o/c « 1) of T. Note first that, if p0 is the mass density in the fluid frame, then [remembering that a volume element transforms as dV ~ y~l dVQ (Lorentz contraction)], the density seen in the laboratory frame will be p = yp0. Thus, expanding each element of T, and retaining only the leading terms, we see by inspection that T!J and T'4 (or r4i) reduce to the correct momentum-flux tensor and momentum density, while
T44 = -yz(p<,c2 + p0e + pv2/c2) — -y(pc2 + pe)
* ~(pc2+\pu2+pe) (15-92)
which is the nonrelativistic energy density (including the rest-mass energy). Thus T satisfies requirement (3) staled above.
The action of external forces upon the fluid are expressed in terms of a four-force (per unit volume) F" whose spatial components give the rate of increase of the momentum of the matter per unit volume and whose time component specifies the rate of increase of energy per unit volume. The expression FaFK is a scalar invariant, and hence may be evaluated in any frame. In particular, we may evaluate it in the frame comoving with the fluid. In this frame
K,F* = (1/4F4)0 = -cz(dPo0/dx) (15-93)
544      Stellar Winds
15-3  Radiation Hydrodynamics 545
In the absence of gencral-relativistic effects, the derivative on the righthand side will be zero unless there is a change in the proper mass density arising from a chemical energy release or from a conversion of matter to energy by, say, a thermonuclear reaction. Thus Vjr* = 0; note that this relation implies that in the comoving frame (F4)n = 0, a result that will be used below.
The equations of motion for the material alone are given by equation (15-83), where T*0 is defined by equation (15-90). If Rj/i denotes the radiation stress-energy tensor [cf. equation (14-120)], then the equations of motion for the complete fluid (matter plus radiation) are
or
8{Tyß + Raß)/öxß = Fa
{dTjß/dxß) = F* + cf
(15-94) (15-95)
where ga is given by equation (14-121). In addition, we may write an equation of continuity for the material particles alone (the photons are massless); demanding number conservation, we have
d(p0V*ydx* = 0
(15-96)
the fluid-frame energy equation
If we were to assume that the first law of thermodynamics remains valid for the combined material and radiation fluid in the comoving frame of the material, then the gas-energy equation [cf. equation (15-24)] would need to be modified only to account for gains and losses of energy (per unit volume per unit time) by the material from interaction with the radiation field. Thus we could write
pQ(De/Dt) - (p/p0)(Dp0/Dt) = 4te Jq (Zv°Jv° - ,U(>) dv (15-97)
where (D/Dt) denotes the Lagrangian time-derivative, following the motion of the material, and y^°, and Jf are evaluated in the comoving frame, which is al rest with respect to the gas. It will now be shown thai this a priori expectation is right, and that equation (15-97) is rigorously correct (621; 135). Forming the dot product of the equations of motion (15-95) with — Va [and using the explicit form for T*0 given by equation (15-90)] one finds
d\Pooo
Pqqq
oh?
~V"^7,= ~ V*F* ~ Klf U5-98)
But I,\J = -c\ so V7{dValdx*) = a(lKF*)/7V = 0 and, recalling that F„P = 0, equation (15-98) becomes
c2d{p000V*)/dx* - V"{dp/cx*) = -Vac?
(15-99)
Now, subtracting [cz + e + p/p0) times equation (15-96) from equation (15-99) and writing [D/Dt) = V%dfdx*), we obtain
PoiDeiDt) - (p/p0){Dp0/Dt) = - V^cf (15-100)
Exercise 15-/0:   Fill in (he missing steps between equations (15-99) and (15-100).
As the lefthaud side of equation (15-100) is evaluated following the fluid flow, the righthand side can be evaluated in the comoving frame. In this frame V1 =0 (i = 1,2, 3), and F4 = icinstantaneously, and from equation (14-121)
g4 = (4ti//c) j(Xv°Jv° - nv°) dv (15-101)
where advantage has been taken of the isotropy of the absorption and emission coefficients in the comoving frame. It is thus clear that equation (15-100) reduces to equation (15-97), which is then, in fact, the correct gas-energy equation for the material as it interacts with the radiation field.
the fluid-frame momentum equations
To obtain an equation of motion in the comoving fluid frame, one might argue that the forces exerted by the radiation field on the material should be added to the usual body forces exerted by, say, gravity. The net force exerted by the radiation, when calculated in the comoving frame, is given by [cf. equation (14-121)]
(15-102)
where again x,° and .^v° are evaluated in the comoving frame, and advantage has been taken of the isotropy of and i]v°. Thus from equation (15-21) we expect that
p{D\/Di.) = -Vp + F + c~l jyj-^f dv
(15-103)
To verify equation (15-103) we examine the z'th component of equation '15-95), which, using equations (15-91), may be written
[d(p,rV)/<V] + [d(PiVl)/dt] + (op/ox*) = F + g1 (15-104)
or
dv' dv' idipiu*)
ct ox' 0XJ
dp
7 = F + g' (15-105)
But the time-component of equation (15-95), multiplied by vl is
v'ldip^ydx1] + vild(pl - c-2p)/ct] = y'(F4 + g4)/(ic) (15-106)
546     Stellar Winds
15-3   Radiation Hydrodynamics 547
Hence by subtraction
dv1
övl dx~j
cp_ 8xl
? oř
— (F4 + cf) (15-107)
Writing (D/Dtf = V*(d/dx*) and /> = yp000, equation (15-107) is equivalent to
p(D\/Dt) = -Vp - {y/c2)(dp/ct) + F + g - v(F4 + g*)/ic (15-108)
Again, the lefthand side of the equation is evaluated following the fluid flow, so the righthand side can be evaluated in the comoving frame of the fluid (in which v is instantaneously zero). We then find equation (15-103). as expected, when g is evaluated in the comoving frame from using equation (14-121).
It is convenient to rewrite equation (15-108) with gy replaced by -(dR^/dx?) [cf. equation (14-120)] to obtain
1 v fdER
Dv
p- + Vp +
cl at
8t
+ v-F
(15-109)
where the rate of increase of material energy per unit volume has been written as F4 = -(v ¥)/ic. In equation (15-109) the radiation-field quantities arc now expressed in an aribitrary (laboratory) frame. Notice now that, if we are concerned with describing only a nonrelativistic fluid flow, the term from F4 is 0{v2/c2) with respect to F and hence may be omitted. Likewise, if we compute the fluid flow (as opposed to the time-evolution of the photons), the characteristic times involved will be At - Ax/v where v is the fluid velocity, and the term in (rp/dt) is clearly 0(v2/c2) compared to Vp and may be omitted. Finally, ER ~ pR; hence (v/c2){8ER/8t) is 0(u2/c2) compared to V ■ P and may also be omitted. All of these terms will be dropped henceforth.
the inertial-frame equations
Most treatments of radiation hydrodynamics formulate the equations in an inertial laboratory frame. Let us now show [working to O(rfc) only] that the inertial-frame and comoving-frame equations are consistent. This may be done by expressing the inertial-frame radiation-field quantities in terms of their comoving-frame counterparts via equation (14-124), which implies:
ER = EK° + 2<TV^° (15-110a) Fft°v + v P° (15-110b)
TO
P = P°
(15-HOc)
Consider first the momentum equation (15-109), which for a spherically symmetric flow with a gravity force becomes
Dv PDt+~~
típ 8r
1 vcF ~č2~8t~
8r +
(3Pii - F,
or
2&
(15-111)
where use has been made of equation (l-43b) for V ■ P. To obtain an equation correct to O(vfc) the full transformations of equations (15-110a) and (15-110c)
or cpR
o.
must be used. However, even in the free-llow limit, J5"' hence it is clear that all of the terms in ;¥ in equation (15-111) are already tit most only 0(v/c) (remembering that we consider At - Ax/v). Thus the additional terms in v in equation (15-110b) will introduce terms of 0{vz/c2) and can be omitted, so that it is sufficient to write fF = Substituting for the inertia]-frame radiation field as indicated, we thus obtain
P
Dv_ Dt
	"i ar0 v i		OPr
Or	c2  dt c2	or	- -— dr
	+ (3Pü° -r	Er°)	2#-° (8v c    \ or
G.
- (15-112)
Exercise 15-1! :   Verify equation (15-112).
In view of the comoving-frame transfer equation {14-134bt, all of the terms in the square brackets collapse to yield
piDvjDt) + (opfdr) = -(G./fp/r2) + (An/c) J7jHv° dv (15-113)
It is thus apparent that the comoving-frame transfer equation derived in Chapter 14 renders the inertial-frame and fluid-frame equations of motion fully consistent to order (o/c).
Let us now consider energy conservation (ignoring thermal conduction) as expressed by equation (15-28). The time derivative operates on the energy per unit volume, and to this term we should add ER; the divergence operates on the energy flux, and to this term we should add Hence
[d(pe + ipv2 + ER)/8t] + V ■ [(pe + \pv2 + p)v +     = v F
(15-114)
To obtain a gas-energy equation, we subtract out the mechanical work terms obtained by taking the dot product of v with the momentum equations (15-109). Note that the two terms in (r/c2) will then become 0(u2/c2), as will c~2\ ■ (c.^/ct) for time-intervals At ~ Ax/v: hence
p[D(h2)/Dtj + (v ■ V)p + v - (V ■ P) = v ■ F (15-115)
548      Stellar Winds
15-4   Radiatively Driven Winds 549
and by subtraction from equation (15-114) we find
[d(pe + ER)j8t] + V ■ {pev + &) + p(V ■ v) - v ■ (V ■ P) = 0 (15-116) In view of equations (15-17) and (15-13), equation (15-116) reduces to
p(De/Dt) - (p/p)(Dp/Dt) + [(dER/8t) + V ■ & - v ■ (V ■ P)] = 0 (15-117)
It is clear that, in transforming to comoving-frame quantities, we must now carry all three terms of equation (15-110b) for 3F; but manifestly the second terms of equations (15-110a) and (15-110c) will produce terms of 0(v2/c2) in time-intervals characteristic of the fluid flow; hence these terms may be omitted. Thus we calculate V ■ ,W = V ■ (^° + £R°v + v ■ P°), and note that, for spherically symmetric flow,
V-(£Ä°v) = -EÄ°(V-¥) + (vV)£Ät
(15-118a)
V-(fP°) = r~2 B(r2vpR°)/8r
= v(dpR°/dr) + pR°[(dv/8r) + 2(«/r)]  (15-118b)
Further, v ■ (V • P°) = v[(vpRQ/dr) + (3pR° - £R°)/r] (15-llSc)
so, expanding the term in square brackets in equation (15-117), we have
8t
8Et
8v 2v
or
8F u\ /f>F 0
+ v
ot  /       \  Or j     \ or (£/ + p/)| + (3£/-p/)(-
(15-119)
by virtue of equation (14-134a). Equation (15-117) is thus identical to the comoving-frame equation (15-97).
Finally, it is useful to combine the momentum and gas-energy equations into a total energy equation that displays the effects of radiative momentum and energy inputs in a transparent way. Let qR be the rate of net gain of energy density by the material from the radiation, and let iR be the force per unit volume exerted by radiation. Then, from equations (15-97) and (15-13),
p(De/Dt) + p(V -y) = qR (15-120)
and from equation (15-103), by forming the scalar product with v,
pWiv?)/Dt] + (v ■ V)p = v ■ (f + fK) (15-121)
By addition of equations (15-120) and (15-121), and use of equation (15-17),
[_B(ipv2 + pe)/dt] + V ■ [&pv2 + pe + p)v] = qR + v ■ (f + fK)
(15-122)
For a steady (8j8t = 0) spherically symmetric flow, integration of equation (15-122) with the usual gravity force gives
(4nr2 pv)[}v2
(pjp) - (GJt/r)-]
4%
(Ur + v ■ fÄ)r2 dr = E = Constant (15-123)
which is analogous to equation (15-59) with added radiative terms. For later work it will be convenient to denote the first set of terms by EQ and to write
E0(r) + 4% £° (qR + v • fR)r2 dr = E = Constant (15-124)
15-4   Radiatively Driven Winds
Observational evidence gathered over the last decade has demonstrated compellingly that rapid mass loss via transsonic winds is ubiquitous throughout the high-temperature, high-luminosity portions of the Hertzsprung-Russell diagram. The basic theoretical framework within which such flows are explained involves winds driven by momentum input into the gas from the intense radiation fields of these luminous stars. Although the basic outlines of the theory seem fairly well established at present, this is a very active and rapidly developing field, and a number of important questions remain open, awaiting future study. We shall, therefore, confine attention to admittedly idealized models that illustrate the basic physics; detailed fitting of computed models to observed data is only beginning, and the reader should turn to the current research literature to follow these developments.
550
Swl/w Winds
15-4   Radial irely Dri uen Winds 551
observational evidence for transsonic winds in early-type stars
A large number of characteristic features (e.g., P-Cygni profiles, emission lines, line asymmetries) in the visible spectra of many O and early B stars (particularly supergiants, Of, and WR stars) have long provided evidence that these objects have extensive envelopes, and that the material being observed in the lines is flowing outward from the stellar photosphere [see, e.g., (261, Chap. 10)]. The case for actual mass loss was not unequivocal, however, because the observed velocities measured from the short-wavelength edge of the absorption in P-Cygni features (typically 200-400 km s~ *) did not exceed the surface escape velocity
i\sc = 620(J?/J{o)HR;Ro)-> kms"1 (15-125)
which is of the order of 1000-1500 km s"1 for a main-sequence O-star, and 600-900 km s~: for an OB-supergiant. The data obtained from ground-based observations suffer from a fundamental limitation: all of the lines observed are subordinate transitions that arise from levels with high excitation potentials, and hence with small populations outside the regions of high temperature and density. The column density of absorbers in these lines is therefore quite small, and one observes only the innermost layers of the envelope, just outside the photosphere.
In contrast, complementary information is obtained from the ultraviolet region of the spectrum accessible to observation from space, which contains resonance lines arising from the ground states of the dominant ionization stages of abundant light elements. The column densities in these lines are so large that one may sample the outermost parts of the envelope. Decisive direct evidence of mass loss was thus first provided when Morton (467) discovered displacements corresponding to outflow velocities in the range 1500-3000 km s"1 in the P-Cygni profiles of the ultraviolet resonance lines of Si IV xl402.8 A and CIVA1549.5 A in spectra obtained from rockets [see also (468; 469; 470; 129; 600; 579; 580)]. Combining the ultraviolet and ground-based data, wc infer a transsonic flow in the expanding envelope. The measured resonance-line velocities are, in fact, only lower limits on the actual terminal velocity of the flow, because variations in the ionization equilibrium may cause the absorbing ion to vanish at some level in the envelope (or the material may just become optically thin and produce features below the detection threshold). Recently, results of great accuracy and sensitivity have been obtained for ultraviolet OB spectra from the orbiting observatory Copernicus (589).
Further evidence for mass loss is provided by infrared and radio continuum observations (of several OB and WR stars), which are most readily interpreted in terms of free-free emission from an extended, optically-thick
envelope having a density profile consistent with steady outflow of the material (171; 241; 277; 685).
The ground-based data are of great importance for they, in principle, provide information in the region where the flow passes through the sonic velocity. The hydrogen Ha line and the He II /4686 line are among the strongest lines in the visible spectrum of OB stars, and hence usually provide the first indications of atmospheric extension and expansion by coming into emission. Extensive observational surveys have been made of both lines. Ha is found (537) to show emission in luminous B-stars of all types from B0 to A3. There are well-defined lower limits to the luminosity at which Ha emission first becomes conspicuous, namely Mv —5.8 (Mbol ?t —8.8) near spectral type B0, and Mv ^ — 6.S (MboI % —7.3) near spectral type AO, and there is a definite relation between the net emission strength of Ha and the stellar luminosity. There is clear evidence for differential expansion of the atmosphere for stars about 0.5 mag less luminous than those that show-conspicuous Ha emission. For the O-stars (177) there is generally a close correlation between the emission strength of He II A46S6 and that of Ha, and emission at either line is an indicator of an extensive envelope around the star. Further, it is found that the luminosity at which Ha weakens, or comes into emission, is Mv % —6 (Mba] % —9). The Ha profiles typically exhibit P-Cygni characteristics, or show an extended blue ward absorption wing indicating expansion; typical velocity widths are of the order of ± 600 km s"1. As the expanding envelope of an OB star becomes more extended and dense, a definite series of other spectroscopic effects are manifested, and a useful qualitative assessment of the state of the atmosphere can be obtained by an examination of the particular effects that do appear (328).
Measurements of the radial velocities (in the usual spectroscopic sense) indicated by different lines of various ions in Of spectra (326; 327), show some interesting features. (1) There is a velocity progression with line-series (e.g., the Balmer lines), with the strongest lines showing the largest velocities of approach: this indicates an accelerating outward flow, for the outermost layers are seen in the strongest lines. By theoretical fitting of the observed profiles it is possible, in priniciple. to infer the variation of velocity with radial distance from the star. If such information can be obtained reliably, it is of great value, for it can provide important constraints on possible theoretical models in the crucial transsonic flow regions. For one star (HD 152236), it was found (327) that the velocity rises sharply outward, from a value near the sound speed (— 25 km s "1) to v ~ 300 km s ~1 in about 0.5 stellar radii above the photosphere; for two other stars this rise is much more gradual, with velocities of a few hundred km s"1 being attained at two or three stellar radii above the photosphere. We shall note the significance of these results later in this section. (2) There is a clear correlation of larger velocities with smaller lower-level excitation potentials for the lines. If the
552      Stellar Winds
excitation were presumed to be in LTE, this correlation would imply that the temperature decreases outward in the envelope (as would be expected if the envelope is in radiative equilibrium, or is expanding adiabatically). Of course, LTE is not likely, and probably the correlation reflects increasing dilution of the radiation field (hence lower radiative excitation rates) or decreasing densities (hence lower collisional excitation rates). Both sets of results mentioned here urgently need to be refined with trustworthy diagnostics, based on careful solutions of the coupled transfer and statistical equilibrium equations, for various flow models. Finally, it should be mentioned that broad, faint emission features have been observed (678) underlying the relatively narrow, bright, emission lines of He IIA4686, C III A5696, and N III /L4634-40 in some Of spectra. These features have total velocity widths up to 4000 km s"1, and if real (they are not seen by all observers) could possibly result from emission originating in the extensive, rapidly-expanding outer envelope of the star.
A large body of ultraviolet data has now been accumulated, thanks to the possibility of making long-term observations from the satellite Copernicus (589). Because the observed resonance lines are intrinsically strong and easily detected, they provide extremely sensitive indicators of stellar winds. From an examination of ultraviolet spectra of 47 O-, B-, and A-stars (see Figure 15-4 for an example), it is found that mass loss occurs over a much wider range of temperature and luminosity than indicated by effects seen in the spectrum visible from the ground. Essentially all stars with luminosities greater than 3 x 104 L0 (a reduction of a factor of 15 relative to the ground-based limit) are found to show mass loss. The observed terminal velocities
1160        1180        1200        1220        1240 1260
;.(A)
figure 15-4
Far-ultraviolet scan of the spectrum of C Pup (05f) obtained from the satellite Copernicus. Ordinate gives flux (in arbitrary units), and abscissa gives wavelength in A. The positions of the C III All 75.7, Si ITT 41206.5, H I ;.1216, and N V /1238.8, 1242.8 lines are marked with vertical arrows. The smooth upper curve is an estimated "continuum" level, drawn for illustrative purposes only; note the marked P-Cygni character of the C III and N V lines.   From (589), by permission.
15-4   Radiatively Driven Winds 553
are all clearly larger than the surface escape velocity (and hence much larger than <;esc at great distance from the star), and range from 300 km s"1 to 3500 km s_1. There docs not appear to be a significant correlation of va. with stellar temperature, luminosity, gravity, or rotation velocity.
A wide variety of ions are observed, ranging from Mg II and C II in the coolest stars, through C IV, Si IV, and N V in the hotter stars. Lines arising from excited states (e.g., N IV A1718.5 at 16.1 eV, or He II A1640 at 40.6 eV) show lower velocities (470), again indicating decreasing excitation with increasing distance from the star; as noted above, this probably reflects decreasing densities and radiation fields. In several stars the O VI A1032 and A1038 lines are observed (533; 589). These lines are unexpectedly strong, and indicate a higher density of 05+ than would be estimated from the color temperature of the stellar radiation field. If it is assumed that the ionization equilibrium is established by collisions (coronal case), the temperature derived is about 2 x 105 C'K, which is taken as evidence for coronal heating in these stars. On the other hand, an upper bound of about 3 x 105 °K on T follows from observations of lower ionization stages (e.g., C III and N III) in the flow, and from comparisons of upper limits of X-ray fluxes to Ha emission strengths (145). We shall see below that temperatures in this range are too low to drive the flow by a coronal-wind mechanism and produce terminal speeds in the observed range.
Mass-loss rates have been estimated from the observed line-strengths (468) and profiles (329) for the Orion supergiants <5, e, and C Ori, yielding values of about 1 to 2 x 10"6 ^/year. Detailed fits to line profiles for ( Pup (05f) yield a mass-loss rate of 7 (± 3) x 10"6 ^//0/year (383).
BASIC DYNAMICS OF RADIATION-DRIVEN WINDS
The first question that arises is whether the flows observed in early-type stars can result from coronal expansion of the kind considered in §15-2. As recognized by Lucy and Solomon (404), the answer to this question is probably negative. To begin, the OB stars are thought not to have extensive convection zones, and are not expected a priori to have coronae. But even if they did have coronae for some reason, then equations (15-59), (15-62), and (15-64) imply that the critical-point temperature Tc required to drive the flow must satisfy the relation 2kTc \mv,2. Adopting yx ~~ 3 x 103kms"1, we find Tc ~ 3 x 107 °K, a value that is completely excluded by (a) the absence of soft X-ray emission from the stars, and (b) the presence of lines from ions such as C IV, N V, and Si IV (observed to exist in the flow at velocities from ~ to vV3\ which would be destroyed by collisional ionization at temperatures greater than about 3 x 103 °K.
We must therefore seek an alternate mechanism to drive the wind. Lucy and Solomon (404) suggested that this mechanism is direct momentum input
554      Stellar Winds
into the gas through the absorption of radiation by the strong resonance lines observed in the ultraviolet spectrum. The momentum input results when photons are absorbed by certain ions from the stellar radiation field, which is strongly outward-directed, and then scattered isotropically. Because the emission process is isotropic, it produces zero net change in the momentum of the material; hence there is a net gain of outward momentum from the incident radiation field. The absorbing ions are thus accelerated radially, and they then suffer collisions with all other particles in the medium; in this way the momentum gained by the particular ions absorbing the radiation is shared with the other atoms in the gas, and accelerates the material as a whole. The outward acceleration experienced by the gas is then
Or = (4n/cp) J" j£w°J/v° dv (15-126)
where %v° denotes the absorption coefficient per unit volume from all sources (continua, electron scattering, and lines); Hv° is the incident flux.
The outward radiative acceleration is to be compared with the inward acceleration of gravity, g = {GJijr2); iff/ is everywhere greater than gR, then the atmosphere remains in hydrostatic equilibrium and does not expand. We therefore must investigate the circumstances under which gR will exceed g; for convenience we shall define
r = (gjg) (15-127)
In O-stars the continuous opacity is dominated by electron scattering in those spectral regions where most of the flux emerges. Pure Thomson scattering is frequency independent, and the resulting T can be written immediately as
Fe = (seL/47tcGy£) (15-128)
where se — (neae/p) is the electron-scattering coefficient per gram. Recalling the results of Exercise 7-1, Fe x 2.5 x 10"5 (L/L0)(J4Q/Jf); for an O-star (L/Lq) « 106, (,J//.yMQ) « 60,andre & 0.4. It is thus clear that continuum absorption alone cannot produce a farce that exceeds gravity. (We shall see below that it is essential for a transsonic wind that it does not!). We must therefore look to the spectral lines to produce the required force.
At great depth in the atmosphere the diffusion approximation is valid, and Hv cc Xv~1 [cf- equation (2-91)]; in this limit the product %VHV appearing in equation (15-126) is independent of the value of %v; i.e., in the diffusion limit the lines are no more effective than the continuum in delivering momentum to the gas. Thus at depth, T remains essentially equal to Fe as given by equation (15-128). On the other hand, at the surface of the atmosphere Hv may rise far above its diffusion-approximation value, because intense radiation emerges from the material below, and none is incident from above. To estimate
15-4   Radial ively Driven Winds 555
the maximum force that can result from a single line, we assume that the line intercepts unattenuated continuum radiation—i.e., Fv = Fc = Bv(Tcff). Then an upper limit to the acceleration of the material by a single line of an atom of chemical species k, in excitation state i, of ionization stage,;, is
gR° = (^/mc^faBJT^^ (15-129)
where niJk is the population of the particular level, Njk is the total number of atoms in all excitation states of ionization stage.;, Nk is the total number of all atoms and ions of species k, ak is the abundance of species k relative to hydrogen, and X is the mass fraction of the stellar material that is hydrogen. Lucy and Solomon considered the C IV line at xl548 A, and adopting,/' = 0.2, Terr - 25,000;1K (to maximize Bv). ac = 3 x 10"4, X = 1, and K-jc/^jc) = 1, they found
log(0/),1548 = 5-47 + log(iVjC/Ay (15-130)
For a typical O-supergiant, log g a 3; hence the upper limit on the force (obtained when NjC/Nc = 1) from even this one line exceeds the force of gravity by a factor of 300.
Of course, the estimate just derived is (purposely) a gross upper limit, because the carbon atoms in the photosphere of the star produce a dark absorption line in which Fv « Fc. To account for this, Lucy and Solomon solved the transfer equation approximately, and found that above a certain critical level in the atmosphere the radiation force, computed using equation (15-126) for the C IV A1548 line, still exceeded gravity. [Interestingly, similar results had also been found in standard plane-parallel, static, model-atmosphere calculations (7; 298). For early-type stars, the radiation force on a realistic line spectrum exceeded gravity at the surface of the model; but this was regarded as "nonphysical" for the purposes of model construction and was suppressed in the calculation!] In summary, one finds that, for O-stars, the forces obtained when the atmosphere is assumed to be static are incompatible with that assumption; hence hydrostatic equilibrium in the outermost layers is not possible, and an outflow of material must occur.
Once the uppermost layer begins to move, the lines will be Doppler-shifted away from their rest positions and will begin to intercept the intense flux in the adjacent continuum; this enhances the momentum input to the material and hence increases the acceleration. The underlying layers must expand to fill the rarefaction left by acceleration of the upper layers. Furthermore, the lines in these lower layers become unsaturated (because the absorption lines in the upper layer have been Doppler-shifted); hence these underlying layers also begin to experience a radiative force that exceeds gravity. In this manner, a flow can be initiated; it remains to be shown that (a) the amount of mass loss produced is significant, and   (b) the variation of the radiation force
556     Stellar Winds
with depth is consistent with transsonic flow. Let us consider the latter point first.
To obtain a transsonic flow, certain conditions at the sonic point must be met (408; 132), as is the case for coronal winds. For steady flow, mass conservation is expressed by equation (15-16), while the momentum equation (15-103) can be written, using equation (15-127), as
v(dv/dr) + p-l(dpjd.r) = -GJi(\ - Y)/r2 (15-131)
Here F is assumed to be a given function of the radius r. The pressure may be expressed in terms of the density and isothermal sound speed a [cf. equation (15-37)] as p = a2p; a is assumed to be a function of the radius r. Then using the equation of continuity (15-14), we find
p-\dpldr) = (da2/dr) - (2a2/r) - (a2/v)(dv/dr) (15-132)
Substituting equation (15-132) into equation (15-131), we obtain
|[1 - (a2/v2)](dv2/dr) = (2a2/?') - (da2/dr) - GJ4(\ - Y)/r2 (15-133)
For simplicity, assume the envelope is isothermal (a good approximation), so that the term in (da2jdr) may be omitted. Then it is clear that, if we are to obtain a smooth transition from subsonic flow at small r to supersonic flow at large r, the righthand side of equation (15-133) must (1) vanish at some critical radius r = rc, where v = a; (2) be negative for r < rc; and (3) be positive for r > rc. The condition for r < rc can be met only if T < 1 in that region; i.e., in the subsonic-flow region the radiation force must be less than that of gravity. If T is greater than unity everywhere (which implies that the whole star is unstable), transsonic flow becomes impossible, and one has either an initially subsonic flow that decelerates, or a supersonic flow that accelerates. For r > rc in transsonic flows, F may become arbitrarily large; indeed the larger it is, the greater is the momentum input into the gas, and the larger (dv/dr) will be. This is, of course, of great interest for early-type stellar winds, for we have seen that T can become very large in the supersonic-flow region, where the lines are sufficiently displaced from their rest frequencies that they absorb continuum radiation; indeed it is just these large values of Y that lead to the high values of vx. that are observed. Because the flow is already supersonic where Y exceeds unity, information about the momentum input cannot propagate back upstream, and the flow at the sonic point itself is essentially unaffected.
It is important to recognize that the radiation force on the continuum and on spectrum lines has precisely the right properties, as delineated above, to produce the desired transsonic wind. That is, Y is less than unity inside the star where the diffusion approximation is valid, approaches unity as the lines desaturate, becomes greater than one when the lines are optically thin, and
15-4   Radiatively Driven Winds 557
reaches very large values when the lines shift into the continuum. We shall find below that, to a good approximation, the force depends upon a power of the velocity gradient; this dependence allows the force, and the flow it produces, to accomodate to one another, so that a steady transsonic wind can be obtained.
Before leaving the question of how the flow is achieved, it is worthwhile to comment on some other points that have been made in the literature. Cassinelli and Castor (132) studied the problem of energy input into an optically thin flow with continuum opacity only. They reached several important conclusions, (a) A flow can be driven by thermal energy-input from radiation into the gas via true absorption processes near the critical point. Such a mechanism deposits energy in an extended region, analogous to the solar corona, with radiation now playing the role conduction has at coronal temperatures. It was noted that the term E0 in equation (15-124) is negative near the star, but must become positive at large distances for finite vx. It was argued that this change must be brought about by the integrated absorptive energy input (qR), and was concluded that true absorption is essential to effect the transition from subsonic to supersonic flow. This conclusion, however, is too restrictive (307), for it is clear from equation (15-124) that, although appropriate thermal energy input can drive the flow, the work done by radiative forces can serve equally well. In fact, it is precisely this work that drives the winds calculated by Lucy and Solomon (who assumed that the lines scatter the radiation conservatively, so that qR = 0), as well as those described later in this section, (b) In the event that the wind is driven by true absorption, then virtually every early-type star has a wind, but in most cases the sonic point is so far from the stellar surface that the mass-flux is miniscule (the density continues to fall with an essentially hydrostatic scale-height inside the sonic point). Only if unrealistically large values for the absorption coefficient and the parameter Y (assumed constant) are adopted do significant winds result. These difficulties are overcome entirely when a realistic representation of the radiative force on lines is used (see below), (c) The time required by the material to gain and lose energy radiatively is short compared to the time a fluid element requires to move through a density scale-height. Therefore, to a high degree of approximation, the energy balance is given by radiative equilibrium; this result is exploited in the models described later.
The first successful radiatively-driven wind models for O-stars were constructed by Lucy and Solomon (404). They assumed (1) planar geometry (adequate for the flow inside the sonic point); (2) constant temperature; (3) and ionization equilibrium fixed by equation (5-46) with W — \, and Tc and Te equal to 0.7Teff (the effective temperature of the model photosphere); and (4) that the radiation force is dominated by absorption in resonance lines of a few ions (C III, C IV, N III, N V, Si IV, S III, S IV, and
558      Stellar Winds
15-4   Radiatively Driven Winds 559
S VI). The momentum equation was written as
i[l - (a/vf](do2/dr) = (15-134)
where jyeff = g^ — gR,i; = g — (izFsJc), which allows for the radiation force from electron scattering; and gR (is the radiation force on the lines. To calculate gRh the lines were treated as pure scatterers, illuminated from below by a photospheric intensity
Iv{Qsfi) = oBv[Tc„)/[a + xi(v)] (15-135)
where a = neaei the electron scattering coefficient per unit volume. Equation (15-135) gives a rough representation of the emergent intensity in the profile of a pure scattering line formed in the photosphere. In calculating the intensity higher in the envelope, re-emissions were ignored because, on the average, they contribute nothing to the force exerted by radiation on the material. In this case Iv(tv) decays exponentially, and we can write /v(tv) = Iv(0) exp( — Tv/fi), where tv is the optical depth, at frequency v, from the base of the envelope to the test point, allowing for Doppler shifting of the line profile along the path. Then
gRtl = (2*/cp) £ £ dfi^ dvyjv)!^-)\ue-^ (15-136)
where the sum extends over all lines.
The density pQ at the base of the envelope is taken from a model atmosphere. Then a trial value is chosen for the velocity v0; this fixes the mass-flux J = PqVq. The mass-flux is an eigenvalue of the problem: if too large a value is chosen, then at the sonic point, where v = a, the radiation force will be too small [essentially because tv in equation (15-136) will be too large], and gefi will be > 0, making a continuous transition to supersonic flow impossible. Likewise, if J is chosen too small, gcfi at the sonic point will be <0. For precisely the right value of J, the condition gen = 0 will be met, and transsonic flow is possible. A large number of solutions, for a range of stellar parameters, were obtained in this way, and it was found that they gave mass-loss rates of the order of 10"8 ^#0/year (or less), which is about a factor of 100 smaller than the observed values, despite the fact that reasonable terminal velocities, i.^ ^ 3300 km s-1, were obtained.
Let us now inquire: "How large a mass-flux can be driven by radiation from a star?" Suppose that the spectrum contains a large number of lines, each of which totally removes the momentum from the radiation field at each frequency where it absorbs. Assume that the material is accelerated from v = 0 to v = va. in the process, so that a line at frequency v,- is spread over a frequency range Av^ = (VjU/r). Then the maximum mass-loss that can be driven is obtained by putting the material momentum-flux equal to the
radiative momentum absorbed in the lines (assuming perfect efficiency); i.e.,
,Mvr, - i4nr2/c) V F(v;) Av; (15-137) i
where the sum extends over all lines. Lucy and Solomon argued that the sum would be dominated by a single line located near the maximum of the flux distribution, and wrote = (4nr2/c) - F1„av(vm.]XL-tc/c), or
Ji = (4nr/c2)Fmaxvmax * (4nr2/c2)F = (L/c2),
where F denotes the integrated flux, and L is the stellar luminosity. This yields i = 7x 10" I4(L/L0X//o/year, or about 7 x 10"8 ^Q/year for L = 106Lo. Lucy and Solomon considered this result to be an upper limit to the mass loss; actually it is more nearly a lower limit (627), for in essence it is obtained by accelerating a single line across the entire spectrum, to drive the material (formally) to the speed of fight. A better estimate (132) is obtained by replacing (4nr2) £F(v;) Av^ in equation (15-137) with L; then
J. ^ (L/u^c) = 7 x lO-1-(L/L0)(3OOO/yJ^0/year (15-138)
where ux is expressed in km s_i. For L = 106Lo, 1 = 7 x 10~6 .MnJ year, which is the observed value for £ Pup. This result presumes that all the momentum carried by stellar photons is converted into mass-loss in a single scattering; in actuality only some fraction t: will be so converted, but observations indicate that e may be substantial, perhaps as large as 0.5. Thus it appears that adequate mass-loss rates can be obtained, if (and only if) enough lines are included in the calculation of gR [see also the discussion in (145)]. A still larger bound on Ji may be obtained when account is taken of the possibility that some photons may scatter several times in the envelope, passing back and forth to regions on opposite sides of the central star between successive scatterings. Quantitative estimates of the importance of this effect have yet to be made.
line-driven winds in Of stars
The most complete and internally consistent theory for radiatively driven stellar winds at present is that of Castor, Abott. and Klein (138; 145). The physical parameters employed make this theory applicable to Of stars. The flow is assumed to be time-independent and spherically symmetric; the gas is taken to be a single fluid; and conduction and viscosity are neglected. The gas is assumed to absorb momentum from the radiation field in spectral lines according to a particular force law discussed below.
The single-fluid approximation can be justified (145) by a comparison of the fluid-flow velocity to the drift velocity of the ions absorbing the radiative
560     Stellar Winds
15-4   Radiatively Driven Winds 561
momentum (relative to the remainder of the material with which they suffer Coulomb collisions). For a representative electron density of ne — 1011 and temperature T ~ 40,000°K, the drift velocity of C + 3 ions is found to be 0.7 km s"\ which is clearly negligible in a medium with a flow velocity of 1000 km s"1. Also the lines of all chemical species are observed to span the same velocity range, indicating that there is no systematic separation of the material, and supporting the single-fluid picture. To justify the neglect of viscosity, one computes the Reynolds number ?Jt = (vl/v), where I is a characteristic length-scale in the How at density p and velocity i\ for a fluid with kinematic viscosity v. The Reynolds number essentially gives the ratio of inertia! to viscous forces (490, 19; 385, 62); at very large values of £#. the fluid may be regarded as inviscid. In an Of wind the minimum calculated value of M is found (145) to be of the order of 101 °, which assures that viscosity can be neglected. Finally, because the temperature of the material is not extremely high, the thermal conductivity is low; and further, because the mass flux is large (10s times the solar wind), the conductive flux is found (145) to be about eight orders of magnitude smaller than heat transport by bulk motions, and hence it can be ignored.
Let us now consider how to calculate the force exerted by the radiation on the material; the essential point is to account for saturation of the lines, so that the transition between the optically thick and thin limits is handled correctly. This problem has been treated in detail by Castor (137); his analysis leads to a simple result, for which a heuristic argument that contains the essence of the physics will be presented here. Suppose that the absorbing lines are confined to a discrete layer overlying the photosphere. Let the continuum flux incident from below be %FC, and approximate the momentum absorbed (per unit mass) from the unattenuated continuum, by a line of opacity xi and width AvD, as gRA(0) = (tzFjj AvD/cp). To account for attenuation we note that (as only the net momentum input is needed) re-emissions, which are presumed to be isotropic, can be ignored. In this case the incident flux decays as e~t! where t, is the line optical depth computed allowing for Doppler shifts. Thus the average rate of momentum input into the layer is
Ti<Sju> = 9r,i(0) fj'c"1'^' (15-139) or - {nFc7j AvD/cp)(l - p-*>)/t, (15-140)
In their work, Castor, Abott, and Klein replace the term Tj_1(1 — e~t() with min(l, t;-1), which provides an adequate approximation.
At a given observer s-frame frequency, the effective optical thickness t, will be limited by (a) the amount of material in the line-layer if the medium is static, in which case
Tf = j*x,dr (15-141a)
where R is the photospheric radius, or (b) by the velocity gradient (which Doppler-shifts the line from its rest frequency) in a moving medium, in which case
h * WtoWdr)-1 (15-I41b)
where z>tll is the thermal velocity of the absorbing atoms. Equation (15-141b) is derived from considerations similar to those employed in development of the Sobolev approximation, and is the planar equivalent of equations (14-61) and (14-62).
It is more convenient to have a depth-scale that is independent of line-strength, so we introduce ^, = (yjfo-), and calculate an equivalent electron optica] depth scale t = t,//?,, which for an expanding atmosphere is defined to be
t = cv^idv/dr)-' (J5-142)
We shall use equation (15-142) throughout the wind, even though it becomes invalid in the stellar photosphere, for the radiation force on the lines becomes negligible there anyway. The total line force is obtained by summing equation (15-140) over all lines, and can be written
gRl = (nFa/cp)M(t) = (seL/4Kcr2)M{t) (15-143)
where M{t) =       £ Fc(v,) AvD,l min(ft, r1) (15-144)
i
is the radiation-force multiplier. The calculation of the radiation force is thus reduced to the evaluation of M(t), which is a function of only one parameter (t).
Castor, Abott, and Klein evaluated M{t) on the assumption that the line spectrum is the same as that of C III (for which an extensive set of /-values was available), and by assigning a total abundance of C+ + relative to hydrogen of 10~3 (which is the total abundance of C, N, and O taken together). The occupation numbers were computed using LTE. Although these assumptions are somewhat rough, the results are certainly qualitatively correct. The numerical values of M{t) are found to be well fitted by the formula M(t) = kt~a, with/c ^ ^janda = 0.7. In more recent work (145), all elements from H through Ni are included, and the coefficients k and a are allowed to depend on the relevant physical variables (but these refinements will not be considered further here). Using equations (15-142) and (15-143), and the formula for M(t), we find finally
(seLk\( 1  dv\a     C ( . dvY (1C ^
where the equation of continuity has been invoked, and the constant C is C = (stM/4nc)[4n/(sevlhJ/)]c' (15-146)
562      Stellar Winds
Using equation (15-145) for the line contribution to the radiation force, the equation of motion [cf. equation (15-133)] becomes
dv2 dr
2cf r
dr
c + 7
dv dr
(15-147)
Or, defining the new variables w = ^v2 and u = -r~\equation (15-147) is equivalent to
F(u, w, w') = (1 - \a2\\>- v)w' - h{u) - C(w'f = 0 (15-148)
where w' = (dw/du), and
h(u) = -GM{\ - T(J) - 2a2u-' - (da2/du)
(15-149)
Equation (15-147) [or equation (15-148)] has a singular point at which solutions terminate, have cusps, or show other discontinuities; this point is not the sonic point. At the sonic point, where r = a, it is easily seen that the lefthand side of equation (15-147) vanishes, and that the righthand side can be made to vanish as well with a suitable choice of (dv/dr\ which need not be infinite or discontinuous. This difference from standard coronal-wind theory results from the force depending on (dv/dr) instead of just r itself. The critical-point analysis is relatively complicated, compared to the theory described in §15-2, because the equation is nonlinear in (dv/dr). Detailed study (138) of the situation shows that the locus of singular points is specified by
[3F(u, w, w')/dw'~\ = (1 - $a2w~l) - adw'f-1 = 0 (15-150)
But not every point on this locus yields an acceptable solution; if W is to be continuous the additional condition
(oF/ou) + w'(dF/dw) = 0
(15-151)
must also be satisfied. Equations (15-148), (15-150), and (15-151) determine («, w, and w') if C is given, or (iv, w', and C) if u is given. Analytical expressions can then be obtained for the rate of mass-loss, the velocity law, and the acceleration (dv/dr). In the limit v » a these are quite simple:
„2 _
1 - aV1"*
(15-152)
\lGJi{\ - rja/(l - a)][(l/rs) - (1/r)] (15-153)
and
0-J'S) = 1 + {-jn + [In2 + 4 - 2n(n + 1)]
(15-154)
15-4   Radiatively Driven Winds 563
where r, is the sonic radius (essentially equal to R, the photospheric radius), and rc is the critical radius. Equation (15-154) is based on the assumption that T cc r'""; likely values for n lie between 0 (isothermal) and \ (radiative equilibrium), which implies that 1.5 < (rjrs) < 1.74.
The stellar model is parameterized by a choice of L, ,/M, and R, and by an assumed relation for T(r) [or a2(r)\. The mass-loss rate is fixed almost entirely by Jl and L (via TJ; the characteristic temperature Teff also enters into vlh. In practice, the model is determined by guessing a value for r(. from equation (15-154) with rs = R; equation (15-147) is then solved numerically, and the exact relation between radius and optical depth is computed. The value of re is then adjusted until a reasonable photospheric optical depth (^f) is attained at r = R. Having constructed a dynamical model, one may use the resulting density structure in a spherical model-atmosphere code to adjust the temperature structure in such a way as to satisfy the requirement of radiative equilibrium. The new temperature structure is then employed to reconstruct a new dynamical model, and the process is iterated. In practice the dynamics—and hence the velocity, density structure, and mass-loss rate— is insensitive to the temperature structure, and the iteration process converges rapidly.
Castor, Abbott, and Klein have published (138) a solution for parameters appropriate to an 05 star: J4 = 60JfQ, L = 9.7 x 105Lo, R = 9.6 x 1011 cm = 13.8Kf
ve. ,cff = 49,300°K, log g - 3.94, r, = 0.4. The resulting mass-loss rate is Ji = 6.6 x 10"6 ^#Q/year, a value appropriate to a star like £ Pup. The terminal velocity is v,^ = 1500kms_1,so^ « ^(L/t^c), which shows that about one half of the momentum originally carried by radiation has been transferred to the matter. Furthermore, about one-half of the continuum radiation field is blocked by the lines; this value appears to be in good agreement with observation (304). Standard stellar evolution theory (606) gives main-sequence lifetimes at this mass of about 3 x 106 years, which implies a total mass-loss of about j the original mass; it thus appears that the stellar winds of these stars will have very significant effects on their evolution.
The density and velocity structures of the model described above are shown in Figures 15-5 and 15-6; the letters P, S, and C designate the photosphere, sonic point, and critical point respectively. The striking characteristic of the solution is the "core-halo" nature of the density structure: inside the sonic point the model has a nearly hydrostatic density gradient, while outside the critical point, p cc r~2. The model inside rc is essentially planar. The velocity rise outside r, is abrupt and, in fact, the predicted gradient is much steeper than than those inferred observationally for some Of stars (326; 327); however, these empirical analyses are rather rough, and a much more accurate study is urgently needed. The velocity variation is, of course, fundamental to
564
565
10"
	1 1
	c
-	■s
_	p
-	
101
10";
101
101
1-1gure 15-5
Variation of velocity (in km s'1) with radius (in cm) in an Of stellar-wind model.   From (138), by permission.
M
FIGURE 15-7
Variation of radiation force multiplier M with continuum optical depth in an Of stellar-wind model. From (138), by permission.
10"7 -
10"9 -
10"
10"
10"
10"
FIGURE 15-6
Variation of density (in gm cm -1 with radius (in cm] in an Of stellar-wind model.   From (138), by permission.
the dynamics, and its empirical determination strongly merits any efforts necessary to yield accurate results.
The variation of the radiation-force multiplier is shown in Figure 15-7. There one sees that in the outer envelope M»5, which implies that the radiation force on the lines is about twice the force of gravity; adding the acceleration from electron scattering (and subtracting the force of gravity), we see that the material experiences a net outward acceleration of about 1.4 times gravity.
The emergent relative flux distribution from the wind model is almost identical to that from a planar static model with TeU a; 50,000°K. However, the outer envelope has an electron-scattering optical depth of about 0.16, and this layer (which scatters conservatively and hence does not affect the relative energy distribution) increases somewhat the observed radius of the star (measured interferometrically), and thus reduces the absolute flux. A critical discussion (304) of both the visible and ultraviolet data for £ Pup (allowing for interstellar reddening effects and line-blanketing) shows that, if both the observational data and the theoretical models are pushed to the allowable limits, agreement is obtained with the Castor-Abbott-Klein model. Earlier work had suggested the need to consider sphericity effects in an extended atmosphere to fit the flux distribution.
Finally, a calculation of the Ha and He II A4686 line profiles from the wind model yields profiles of shape similar to those observed, and with emission
566      Stellar Winds
15-4   Radiatively Driven Winds 567
strengths and redshifts in qualitative agreement with typical values measured for Of stars; these need to be refined with a more accurate calculation. On the whole, the Castor-Abbott-Klein model appears to give a fairly satisfactory picture of the basic dynamics of Of atmospheres; but many questions remain open.
frontiers
The theory of radiatively driven winds is at an early stage of its development, and many interesting (and challenging) problems remain to be attacked. Within the framework of the Castor-Abbott-Klein theory, refinements of the force law to include the complete spectrum, a more realistic ionization-excitation equilibrium, and more accurate treatment of the transfer (accounting for the fact that a photon scattered at one point in the envelope can subsequently interact with the material at some other point), should yield more precise and reliable results.
A more difficult problem is posed by the question of how to treat properly the energy equation and thereby determine the temperature structure of the flow. The observations (533) of O VI lines, described earlier in this section, have been interpreted {assuming collisional ionization only—an assumption that requires further consideration) as requiring temperatures of the order of 2 x 10s °K, which is much higher than can be produced radiatively. This suggests that there may be mechanical energy deposition that produces a (relatively cool) corona. Mechanisms for producing a mechanical flux (which presumably dissipates and heats the outer layers) have been proposed in (289; 290; 404), but these are not entirely satisfactory in their present form. If the flow were to become turbulent (see below), energy dissipation and heating could occur and, because pv2 » kT, even a rather low efficiency in the conversion of flow energy to heat could have a large effect on the temperature. Nevertheless, it must be stressed that, although current models do not convincingly determine the temperature structure, the dynamics of the flow will remain essentially unaltered for T < 3 x 107 nK (a value that seems completely excluded observationally), unless the energy deposition changes the topology of the solution [e.g., by the introduction of additional critical points (307)] and, thereby, even the qualitative nature of the solution. These possibilities all await further exploration.
One of the problems that will ultimately have to be faced by the "core-halo" models is that evidence has been presented for atmospheric extension effects in the continuous energy distribution of the most extreme Of stars (367; 466). Unless it can be shown that there are errors in the observations or in their reduction (e.g., in the allowance for interstellar reddening), it will be necessary to find ways to construct models showing a slower outward rise of the velocity. Such models seem essential for WR stars, where a critical analysis
(304) of the energy distribution of HD 50896 (WN5) seems absolutely to require an extended subsonic-flow region; some models of this type have been constructed (133). but only with ad hoc force laws, and much further work remains to be done before they can be regarded as satisfactory.
Although all theoretical models of radiatively driven winds assume that the flow is steady, there is ample evidence that the spectrum (and hence the wind) of Of stars is time-variable. A wide range of time-scales (176; 536) has been noted. It appears, in fact, that essentially all early-type supergiants that show emission lines are intrinsic spectrum variables (537). In a few cases, pathological spectra with transient inverse P-Cygm profiles (presumably indicating temporary inflow of the material) have been observed (173). If the time-scale of the variations is long compared to the time a fluid parcel requires to move from the photosphere to the critical point, then one can argue that the flow can be considered as a sequence of quasi-stationary states, each of which is well approximated by steady flow. The only problem then remaining would be to understand the mechanism leading to the variations. On the other hand, if very short time-scales are ever observed, then a fully time-dependent treatment might be required, and this would introduce staggering difficulties into the problem.
Although the assumption of spherical symmetry of the wind is a reasonable starting point, it may not adequately describe the flow for some stars. In particular, if the star is rapidly rotating, then centrifugal forces can appreciably lower the effective surface gravity, to the point where it barely exceeds even continuum radiation-forces. This may lead to enhanced mass loss from the equatorial regions of the star (409), in which case the flow becomes axisymmetric instead of spherically symmetric. The divergence of streamlines away from the equatorial plane again introduces the possibility of a radical alteration of the topology of the solution (307). Furthermore, rotation implies that the flow has nonzero vorticity and, in the presence of rotational shear, the flow may disintegrate and become turbulent, as suggested by the very large Reynolds numbers mentioned earlier in this section. In this event, gross inhomogeneities may develop in the wind, and again the theoretical complexity becomes overwhelming.
Finally, there is the question of whether magnetic fields play a significant role in winds from early-type stars. The O-stars are very young, and have only recently formed from the interstellar medium. Presumably any fields present in the medium could persist as weak fields in the atmospheres of these stars. There would then be the possibility of a region of forced corotation out to an Alfvenic point, with subsequent radial expansion. Could this give rise to the profiles with broad emission wings extending beyond the short-wavelength edge of the P-Cygni absorption feature, as observed in some stars? Could such fields produce structural inhomogeneities with flow-tube divergences such as are seen in the solar corona and wind (307)? Could they
56£
Stellar Winds
provide structures against which shear (as a result of rotation) w.ith consequent turbulence can develop? The present observational detection threshold for stellar magnetic fields is several hundred gauss; much weaker fields (a few tens of gauss) in the atmosphere could have major effects on the flow.
It should be clear from the points raised above that there remains much to be learned about the physics of stellar winds for early-type stars. Without doubt it is unrealistic to suppose that these issues can be decided on the basis of theoretical considerations alone. It is obvious that a thorough analysis of the spectroscopic data, at a high level of internal consistency, with the goal of diagnosing the physical conditions in the flow semiempirically, is required, and that such efforts will be immensely rewarding.
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Glossary of Physical Symbols
Physical symbols used in the text are listed below, along with a brief description of their meaning and the page number on which each first appears. Standard mathematical symbols, dummy variables and indices, and notations used only in one location are not included.
a           Ratio of damping width to Doppier width, T/4n AvD 279
a           Isothermal sound speed 519
at          Quadrature weight 65
Branching ratio j -> i in radiative decay of state j 142
aR         Radiation constant in Stefan's law 7
as          Adiabatic sound speed 519
av Macroscopic absorption coefficient uncorrected for
stimulated emission 78
av          Coefficient in linear expansion of Planck function 148
«0          Bohr radius 89
dj(t)       Coefficient of cigenstate ij/j in expansion of a general
state t/d0 85
afp)      Absorption-depth, relative to continuum, in spectrum-line intensity-profile at angle cos-1 a from disk-center,
1 - rv(n) 269
A          Vector potential 7
Rate matrix of statistical equilibrium equations 138
Aa         Autoionization transition probability 134
Aj Atomic weight of chemical species / 110 4^         Einstein spontaneous-emission probability for transition
j - ' 78 As         Stabilization transition probability in dielectronic
recombination process 135
Ar Absorption-depth, relative to continuum, in spectrum line
flux-profile, 1 - Rv 269
588      Glossary of Physical Symbols
A0 Central absorption depth of an infinitely opaque line 318 A((         Matrix coupling deplli-pomts d - 1 and d in Fcaulricr
difference-equation solution of transfer equation 155
b,-          Non-LTE departure coefficient (n;/nf) for level i 219
bv          Coefficient in linear expansion of Planck function 148
ba(x)       Planck function normalized to value at surface of
atmosphere Bv[r(t)]/£v(70) 76
bv(rv)      Planck function normalized to disk-center intensity
5v[T(tv)]/7v(0, 1) 262
B          Magnetic induction 7
30          Righthand-side vector of statistical equilibrium equations 138
B;j         Einstein absorption probability for transition i -> j 77
Bji Einstein induced-emission (or stimulated-emission)
probability for transition j -» i 78
B0         Planck function at T = T0 312
B1          (dBJdz) evaluated at t = 0 312
B* Equivalent Planck function for recombination emission in
non-LTE source function 360
Brf         Coupling matrix at depth-point d in Feautrier difference-equation solution of transfer equation 155
B(T)      Frequency-integrated Planck function 52
B^((t)     Effective thermal source in line with overlapping continuum 352
BV(T)      Planck function 7
B*[T(t)~]      Planck function for temperature distribution that gives
radiative equilibrium 179
c           Velocity of light 4
Collisional transition rate from level; lo level / 127
CiK         Collisional ionization rate of level i to continuum 123
Ck Interaction coefficient of /<th component of hydrogen Stark
pattern 295
Cp         Specific heal at constant pressure 186
Cp, Cjs C4, C6      Coefficient of power-law expression for perturber-radiator
interaction 283
C\.         Specific heat at constant volume 186
C0 Numerical constant in collision rate formula, na^iSk/nm)'1 133 Cd         Matrix coupling depth-points d and d + I in Feautrier
difference equation solution of transfer equation 155
'6,          Net collisional rate into level; 486
d           Electron-impact line shift 306
d           Electric dipole moment 86
dt Departure of ratio of non-LTE to LTE population of level i
from unity, (bL - 1) 219
	Glossary of Physical Symbols	589
d.....	Matrix element of dipole moment ((l)*,\&\4>n/	86
	Volume element	8
: D	Distance from star to observer	11
i D	Debyc length, Debye radius	122
1 D	Electric displacement	7
	Balmer jump (in magnitudes)	195
Dp	Paschen jump (in magnitudes)	195
j D,j	Auxiliary matrix used in developing Feautrier difference-	
	equation solution of transfer equation	156
i {D/Dt)	Fluid-frame or Lagrangian derivative	514
1 (D/Dt)con	Time rate of change of distribution function caused by	
	collisions	33
e	Electron charge	81
e	Specific internal energy of a fluid	517
e(co)	Residual energy per particle at infinite distance in stellar	
	wind, E/F	531
E	Total particle energy	494
' E	Total energy flux in stellar wind	526
E	Electric field	7
i E	Matrix giving definition of J in terms of iij's in Rybicki	
	difference-equation solution of transfer equation	159
S	Energy in radiation field	3
Ei	Energy of atomic state / relative to ground state	22
Ev	Contribution to emissivity from (fixed) overlapping	
	transitions	397
Ev	Energy received at frequency v from an expanding	
	atmosphere by an external observer	474
; £o	Magnitude of electric field	8
E0	Threshold energy of collision process	132
E%	Thermodynamic equilibrium value of radiation energy-	
	density	7
' E{co)	Energy spectrum of oscillator	275
£c(cc)	Heat-conduction flux at infinite distance in stellar wind	531
En{x)	Exponential integral of order n	40
Er(t, v, £),"]		
	Monochromatic energy density in radiation field	6
^R\>- 'h I-----------oj -------J-----------
ER(r, tf)
ER(r\ > Total (frequency-integrated) energy density in radiation field 6 E* j
f Force-per-unit-volumc acting on a fluid 516
590      Glossary of Physical Symbols
Glossary of Physical Symbols 591
ft
fc
fj /-/("', ")
/("', l'\n,l)
Buoyancy force
Continuum oscillator strength
Oscillator strength of transition i -> j
Monochromatic flux received by observer
Oscillator strength for transition between states with principal quantum numbers n and n
Oscillator strength for transition from substate I' of state ri to substate / of state n
Amplitude of time-variation of oscillator
Maxwellian velocity distribution
Particle distribution function
Variable Edditigton factor KJJV
Fraction of chemical species k in ionization stage j, NjkfNk
Kramers'-formuki oscillator strength for transition ri ~* n in hydrogen
Photon distribution function
Perlurber field-strength
Particle flux in stellar wind
Imposed external force
Frequency-integrated flux
Flux in continuum near spectrum line
Energy flux transported by convection
Energy flux transported by radiation
Four-force
Normal field strength
Radiative damping force on an oscillator
Frequency-integrated flux from blackbody
Spontaneous recapture probability for electrons of velocity v
Dawson's function
& Jn Monochromatic "astrophysical" flux of radiation field
Fourier transfrom
Fourier transfrom of wavetrain of duration T
Monochromatic flux of radiation field in direction normal to atmospheric layers
123 84
Monochromatic vector flux of radiation field
11	^bd(v) g	Flux from blackbody Surface gravity in planar atmosphere	12 170
88		Surface gravity at which radiation force exceeds	
88		gravitational force in aunosphere Net acceleration, gravitational minus radiative, of stellar	170
274		atmospheric material	256
110	y<	Statistical weight of atomic state i	77
		Statistical weight of excitation state i of ionization state./ of	
32		chemical species k	110
	Gnl	Statistical weight of substate I of state n	88
18	9r	Acceleration produced by radiation force	554
		Radiative acceleration produced by all spectrum lines	558
114	gR°	Acceleration produced by radiation force in a single	
		spectrum line	555
		Four-force of radiation on matter	498
90 4 291			
	g(n',p,<fr)\	Angular phase functions in scattering process	29
	yip', p)l		
525	g{in, ri)	Gaunt factor, for bound-bound transition ri -* rt in	
32		hydrogen	90
44 269	Quiri kl]	Gaunt factor for bound-free transition n -» k in hydrogen	99
188	ShA 0,1		
190		Gaunt factor for free-free transitions in hydrogen	101
541	fln.(v, T)j		
291	G	Newtonian gravitation constant	255
82	Gem	Momentum density in electromagnetic field	14
12	GR	Momentum density in radiation field	10
94	G(u)	Induced recapture probability for electrons of velocity v	94
280	G(v), )	Generalized statistical weigh! ratio in stimulated emission	
	G,„(v)j	correction	165
10	h	Planck's constant	4
	h	Reduced Planck's constant h/2%	85
275	h	Specific enthalpy of a fluid	518
281	K	Eddington factor giving ratio J7v(0)/,/v at surface of	
		atmosphere	157
JO	H	Frequency-integrated Eddington flux	54
	H	Pressure scale height	188
592      Glossary of Physical Symbols
1(t, n, v, 0,
IM
Iv
I (to)
fit, V):
fy, v),
j j J J
J. J
J°
7„
Magnetic field 7
Nominal Eddington flux, oRT*(?j4n 174
Hamiltonian for atom %$
Pcrturber Hamiltonian 298 Current value of Eddington flux in Avrett-Krook procedure 174
Magnitude of magnetic field g
Flux-constant in extended atmosphere, r2H = L/16n2 245
Voigt function 279
Hamiltonian operator 85
Monochromatic Eddington flux, £FT = j^JAtz 10
Limb-darkening function 70
Expansion function in power-series expression for Voigt
function 280
Unit vector in x-direction 3
Frequency-integrated specific intensity 54
Moment of inertia of a star 536
Specific intensity emitted from core in expanding, extended
atmosphere 478
Fractional intensity of kih component of hydrogen Stark
pattern 296
Total line intensity in transition u -> / 488
Ionization energy of hydrogen 133
Specific intensity of radiation 2
Power spectrum of oscillator 275
Emergent specific intensity along ray with impact parameter
p in extended atmosphere 247
Specific intensity traveling in + p. direction 36
Specific intensity traveling in —p direction 36
Unit vector in y-direction 3
Current density 7
Jacobian of transformation of coordinates 32
Total angular momentum of atom 92
Current value of mean intensity in Avrett-Krook procedure 174
Mean intensity averaged over line profile, j <{)VJV dv 129
Discrete representation of depth-variation of J{z) in
Rybicki difference-equation solution of transfer equation 159
Glossary of Physical Symbols 593
Jfr.r), 7 ./(r, v, 1), J(r, v), Jlz, v),
JM
J,
k
k
k
k
K K
K
jf
K U
m
Frequency-integrated mean intensity, \% Jv dv
Mean intensity
Boltzmanns constant 7
Wavenumber 8
Continuum-state quantum number for hydrogen 98
Unit vector in z-direction 3
Continuum opacity uncorrected for stimulated emission 321
Root of characteristic equation in discrete-ordinate method 66
Frequency-integrated second moment of radiation field 55
Numerical coefficient in hydrogen cross-section 99
Discrete representation of depth-variation of thermal source terms at angle-frequency point i in Rybicki difference-equation solution of transfer equation 159
Second angular moment of monochromatic radiation field 16
Line-formation kernel function for an expanding
atmosphere 481
Line-formation kernel function 339
Kernel for line-formation with overlapping continuum 351
Azimutbal quantum number 89
Continuum-state quantum number for hydrogen 100
Convective mixing length 188
Correlation length in turbulent velocity field 464
Photon mean-free-path 51
Stellar luminosity 49
Photon destruction length 333
Rotational angular momentum of a star 536
Total orbital angular momentum of atom 92
Lorentz transformation 493
Critical luminosity at which radiation force exceeds
gravitational force in atmosphere 171
Integration constant in discrete-ordinate method 67
Source-term in Feautrier difference-equation solution of
transfer equation 155
Continuum kernel function 351
Mass of electron 83
594     Glossurv of Physical Symbols
m Magnetic quantum number . 89
m          Column mass in atmosphere 170
mf          Mass of electron 89
mp          Mass of proton 89
mH         Mass of hydrogen atom 170
m0         Rest (proper) mass of a particle 494
m          Average mass per nucleus (atoms + ions) 170
M          Mach number 520
Ji         Stellar mass 171
Mrj         Integration constant in discrete ordinate method 67
Ji'         Mass-loss rate 516
M[t)       Radiation force multiplier 561
n           Index of refraction 4
n           Principal quantum number 89
n,   n'      Directions of radiation propagation 2
n Occupation-number solution-vector of statistical
equilibrium equations 138
iij Number density in doubly excited state of dielectronic
recombination process 135
ne          Number density of free electrons 94
n;          Number density of atoms in state i 78
nf          LTE value of number density of atoms in state i 79
niJk Number density of atoms in excitation state i of ionization
stage./ of chemical species k 110
nr          Proton number density 138
n0          Propagation vector of plane wave K
n(z)        Turbulent eddy density 464
nk(r, /.)     Particle density of particle species k in a gas 513
J7;(v)       Number density of atoms in state i capable of absorbing
radiation at frequency v 77
rt;(v)       Ratio of population in subs täte v of state i to line profile,
n,(v)/0(v) 436
N          Total particle density (all species) 115
JV          Number density of perturbers 282
JVj-fe        Number density of atoms in all excitation states of
ionization stage j of chemical species k 111
Nk Number density of atoms of chemical species k in all
excitation and ionization states 114
A\.         Number density of nuclei (atoms -h ions) 115
Afv         Third angular moment of monochromatic radiation field 502
.yV'km       Number of transitions k -» m 87
p           Total pressure 170
Glossary of Physical Symbols 595
p           Impact parameter of ray in extended atmosphere 247
p Exponent in power-law expression for perturber-radiator
interaction 283
p           Momentum of a particle 32
pe          Electron pressure 103
pQ          Tola! gas pressure 115
Pi          Generalized momentum coordinate 85
pt          Pressure in interstellar medium 533
Pji         Cascade probability of state j to state i 142
pk          Partial pressure of particle species k in a gas 514
pv          Probability of photo ionization at frequency v 94
pv Coefficient in linear expansion of Planck function on iv scale 310
p(x, x')     Joint probability of absorption from substate x and return
to substate x' in a line transition 277
p{£', £) Redistribution probability in atom's rest frame 412 Pr(t, v, f),l
pR{~. v, t), >    Monochromatic radiation pressure scalar 16 Pr(v) j
Pr(i> v> 0, {    Thermodynamic equilibrium value of monochromatic
p%iz, v, t) f       radiation pressure scalar 17
P           Radiation-pressure tensor 12
Pd         Photon destruction probability 333
Ptj         Component ij of radiation pressure tensor 12
P;j         Tola) transition rate from level i to level j 128
P"          Four-momentum 494
P          Mean radiation pressure 13
P(t)        Power radiated from accelerating charge 82
P(uh)      Probability distribution function for dtmensionless velocity
in turbulent atmosphere 464
<P(tu)>      Average power radiated at circular frequency o by
harmonic oscillator 82
Pfr)      Photon escape probability 334
P„((r)      Radial charge density 89
q           Heat delivered to a gas, per unit volume 517
q{          Generalized space coordinate 85
qv          Sphericality factor in extended atmosphere 251
q.          Conductive heat flux 517
q{z)         Hopf function 55
qtj{T)     Collision rate i    j, per atom in state /, per electron, averaged over Maxwellian velocity distribution at
temperature T 132
1
596     Glossary of Physical Symbols
qx(i)       Exponential absorption factor for a turbulent atmosphere   • 465
<q((t)>s      Static average of qjx) 466
Q          Integration constant in discreie-ordinate method 67
O Factor correcting for ionization and radiation pressure
effects on mean molecular weight of a gas 188
Qij          Collision cross-section in units of izaQ2 132
Q(r, p)     Derivative of radial velocity along a line of sight in Sobolev
method 479
Qx{s)      Laplace transform of qx{r) 466
r           Radial distance from center of a star 3
r           Distance between two test points 4
/■           Ratio of continuum to line opacity yJjA 36
/*           Opacity ratio in schematic Lyman continuum problem 222
r           Position in a stellar atmosphere 2
re          Core radius in extended atmosphere 251
rc          Critical radius in transsonic wind 526
rb          Sonic point radius 562
rA          Alfvenic radius 535
rQ          Mean interatomic distance 111
rQ Radial coordinate of surface of constant radial velocity,
(V + P2f 479
rf          Radius at Rosseland optical depth tr = § 256
r.          Stellar radius 11
f            Unit vector in radial direction 3
rv[ft)       Residual specific intensity, relative to continuum, in
spectrum line at angle cos"! \i from disk center, 1 — afp) 270
R          Stellar radius 49
M          Rydberg constant 89
$          Reynolds number 560
R           Stress-energy tensor of radiation field 498
Rllb        Dielectronic recombination rate d -> h 135
R;j         Radiative transition rate from state i to slate j 11
RiK         Photoionization rale of level i to continuum 123
Rjt Radiative de-excitation rate j -> i scaled to equilibrium
value Rn = nfR'jfnf 129
R'yt         Radiative de-excitation rate j ->■ i per atom in upper state 129
Rk! Radiative recombination rate k -*■ i scaled to equilibrium
value RK-t = n*R'KJnf 131
R'Ki Radiative recombination rate rc -> i per ion in ground state 130
Rv Residual flux, relative to continuum, in spectrum line
profile, 1 - Av 269
R0         Residual flux at center of infinitely opaque line 312
Glossary of Physical Symbols 597
?A{         Net radiative into level i 486
R(x',x)     Angle-averaged redistribution function in dimensionless
frequency units 427
R{x\ n'; .v, n)      Redistribution function in dimensionless frequency units 418
R{v\ v)     Angle-averaged redistribution function 28
R(v\ n'; v, n)     Redistribution function for scattering process 27
R£.(v', n'; v, n)     Redistribution function in laboratory frame for coherent
scattering in atom's frame 418
Rni(r)      Radial wavefunction 89
R„(/> v)     Angle-averaged redistribution function for atom moving
with (dimensionless) velocity u 425
R„(v\ n'; v, n)      Redistribution function for atom moving with velocity v 416
s           Path length 31
s            Spin quantum number 89
,s'e          Electron scattering coefficient per gram of stellar material 554
S           Surface area 3
S           Frequency-integrated source function 54
S           Poynting vector 11
S           Total spin angular momentum of atom 92
S,          Line source function 80
Sraax       Maximum source function in finite slab 347
S(; Discrete representation of source function at depth-point d
in Feautrier difference-eq nation solution of transfer
equation 157
S(i,j)      Line strength in transition / -+ / Si>
S(t, v),j
S(z, v), >    Source function, ny/xv 35 S, J
S(a)       Normalized Stark profile 296
S(—p)     Angular distribution of intensity emergent from grey
atmosphere 70
•S(£/f)     Strength of line within multiple! 92
£/'(J{)     Multiplet strength 92
Sx(s)       Laplace transform of (^.(t)).,- 466
t           Time 2
r           Current optical depth scale in Avrett-Krook procedure 174
t Equivalent electron-scattering optical depth in expanding
atmosphere 561
ic          Self-relaxation time for electrons in a plasma 122
lr           Average recombination time 122
T          Absolute thermodynamic temperature 7
T           Stress-energy tensor of electromagnetic field 497
1
598      Glossary of Physical Symbols
Glossary of Physical Symbols 599
	Color temperature	248	i\.	Critical velocity in transsonic wind	526
Tc	Radiation temperature of core in expanding atmosphere	486	i.:esc	Escape velocity From stellar surface	550
T,	Kinetic temperature of electrons	122 '	vt	Expansion velocity in radial direction	474
reff	Effective temperature	49	V.	Expansion velocity along a ray with impact parameter p	474
Tk	Kinetic temperature of atoms and ions	123	V0	Most probable speed	110
Tr	Radiation temperature	360		Terminal velocity in stellar wind	530
Tr, T	Total optical thickness of finite slab	36	<	Fiducial thermal velocity	449
To	Boundary temperature of atmosphere	61		Average speed of conveclive elements	188
T,	Temperature perturbation in Avrett-Krook procedure	174	V	Acceleration	81
T;	Tridiagonal matrix representing differential operator for		p{z, v, fl), vYli	Antisymmetric angle-average of specific intensity	
T(k2), T{X)
u
u(z, v, p), un uh{x)
U{t, 0)
n, I, m, s) VJk(T) v
v
V
angle-frequency point fin Rybicki difference-equation
solution of transfer equation 157
Maxwell stress tensor 14
Characteristic function in discrete ordinate method 66
Time development operator 298
Time-development operator for atom 301
Time-development operator for perturber 300
Radiation temperature 121
Current temperature distribution in Avrett-Krook
procedure 174
Velocity in units of thermal velocity (m/lkT)^ \ 417
Discrete representation of u(zdi v, p) in Feautrier difference-equation solution of transfer equation 155
Discrete representation of depth-variation of u(z, v;, p;) in
Rybicki difference-equation solution of transfer equation 157
Symmetric angle-average of specific intensity
£[/{v, +/() + /(v,-m)]                        ' 152
Hydrodynamic velocity in turbulent atmosphere in units of
local thermal velocity vh(x)/vlh(T) 464
Matrix giving depth-coupling to J at angle-frequency point i in Rybicki difference-equation solution of transfer
equation 159
Time-development operator in interaction representation 302
Electron orbital 91
Partition function of ionization stage; of chemical species k 111 Frequency displacement from line-center measured in
Doppler widths, (v - v0)/AvD 279
Average relative velocity of colliding particles 282
Velocity 7
v(r) Expansion velocity of atmosphere
<;,((t) Hydrodynamic velocity in turbulent atmosphere
V Volume
V Perturbation potential
V Velocity of atmosphere in fiducial thermal velocity units,
Vt The fth component of total particle velocity in a gas
V'i The ith component of thermal velocity of a particle in a gas
()Z> = v,     The ixh component of fluid (or mean flow) velocity in a gas Vm„        Matrix clement of perturbation potential <(/>*Jl'"j0„> V" Four-velocity
V; Matrix containing depth-variation of quadrature weights
and profile functions at angle-frequency point i in Rybicki difference-equation solution of transfer equation
Vcl(t)      Classical interaction potential
V'd(t)      Canonical transformation of classical interaction potential to interaction representation
w Electron-impact line width
vv Doppler width corresponding to thermal velocity
Ivo/cWkT/m)*-wk Quadrature weight
wl5   w2      Relative probabilities of line and continuum bands in picket-fence model
W Dilution factor
W Matrix in final system WJ = Q in Rybicki difference-
equation solution of transfer equation
We Electron distribution function
Wi Ton distribution function
W},   Wv Equivalent width of spectrum line
W* Reduced equivalent width, W/2AnAvn
449 464 6 86
449 513 513 513 86 542
159
301
302 306
417 144
207 120
160 292 292 270 318
600      Glossary of Physical Symbols
W{r)       Nearest-neighbor distribution function _ 290
W{[)       Energy density in electromagnetic field 8
W{fi)      Field-strength distribution function 291
W({3~ S)     Field-strength distribution function allowing for shielding
effects 293
W(^)      Distribution function for velocities along line-of-sight 279
WH{ji)     Holtsmark field-strength distribution function 294
W?(p),   WJp)     Equivalent width of spectrum line at angle cos"1 p from
disk-center 270
a-           Cartesian coordinate in horizontal direction 3
x Frequency displacement from line-center measured in
Dopplcr or damping widths 338
x,          \/ka, where ka is a characteristic root 69
x Minimum of absolute value of incident and scattered
photon frequencies measured in dimensionlcss units from
line-center 428
x Maximum of absolute value of incident and scattered
photon frequencies measured in dimensionless units from
line-center 428
Xx         xgz = l//<a2, where    is a characteristic root 69
Xv Generalized optical depth variable in spherical atmosphere 251
Xv Contribution to opacity from (fixed) overlapping transitions 397
A"0         «o/0 ~        where n0 = OivJkT) 313
XT[f(tj]      K-integral operator 41
v           Cartesian coordinate in horizontal direction 3
Y          Ratio of abundance of helium to hydrogen by number 138
Yfj         Net collision"^ bracket 132
Y"'{6,4>)      Spherical harmonic 89
z Cartesian coordinate in vertical direction (normal to
atmospheric layers) 3
z           Path-length along ray in extended atmosphere 247
z0(p, x).   z0     The r-coordinate of surface of constant radial velocity
corresponding to frequency shift x 479
Z          Total geometrical thickness of finite slab 36
Z          Charge number of atomic nucleus 91
Z,          Ionic charge 330
Zji         Net radiative bracket in transition j ->■ / 129
a           Stark shift in A per unit normal field strength, AX/F0 296
ak          Relative abundance of chemical species k 115
Energy absorption cross-section per atom 94
as          Stellar angular diameter 12
y,[t)        Atomic wave function 300
Glossary of Physical Symbols 601
otij(v)      Absorption cross-section at frequency v in bound-bound
transition /    j 128
aIK(v)      Absorption cross-section at frequency v in bound-free
transition / —- ic 130
^dr(P)     Dielectronic recombination coefficient 135
krr(T)     Radiative recombination coefficient 131
aKK(v)      Free-free absorption cross-section at frequency v 165
ß           Ratio of line to continuum opacity in picket-fence model 207
ß           Field strength in units of normal field strength, F/F0 291
j                     ß           Velocity in units of speed of light 493
ßc Probability of penetration of core radiation to test point in
expanding atmosphere 47S
ßv Fractional departure of monochromatic opacity from
mean value 74
ßv Ratio of line opacity to continuum opacity,
:                                     iMIXc = zAÄKc + o) 309 ß0          Ratio of line to continuum opacity for line with Voigt
profile. ßr = ßbH(a, v),   ß0 = x0/Xc 312
j                     ß{r)        Photon escape-probability in expanding atmosphere 478
I                       y            Classical damping constant 82
y           Ratio of specific heats for an ideal gas 186
y            Convectivc efficiency parameter 189
y Ratio of radiation force to its limiting value in diffusion
approximation 255
y Fraction of all emission that occurs coherently in atom's
rest-frame 415
• y Velocity gradient dVjcx in uniformly expanding atmosphere 481
7           Lorentz transformation factor (1 — t.'2/c2)~- 493
yv,  y     Ratio of monochromatic opacity to mean value or
continuum value 74
yv Generalized noncoherent scattering coefficient in non-LTE
source function 223
y(z, pj     Coefficient of frequency-derivative in comoving-frame
transfer equation using optical depth scale 504
y{z, p)     Coefficient of frequency-derivative in comoving-frame
transfer equation 504
F Ratio of specific heats for non-ideal gas (i.e., including
ionization and radiation pressure effects) 186
T          Ratio of radiation force to gravity force 256
F          Reciprocal lifetime of excited state 277
F Total damping width of a line 278 Ft          Ratio of radiation force from electron scattering only to
gravity 256
602     Glossary of Physical Symbols
Vc CoIIisional damping width • 282 T,,   r0      Reciprocal mean lifetime of lower and upper states of
transition L «-> U 211
TR         Radiative damping width 282
Tw         CoIIisional damping width in Weisskopf theory 283
T3         Resonance damping width 287
T6         Van der Waals damping width 326
YjfiT) Secondary temperature-dependent factor for collision rates 133 5, SL, Sv, d\, Sc     Reduced damping widths, T/2 in circular frequency units,
r/4n in ordinary frequency units 277
6 Number of perturbers in Debye sphere 294 5           Ratio of continuum to total opacity averaged over line
profile 351 tW.   5Na     Perturbation of total number density (at depth-point d) in
linearization procedure 118
<3T,  5Td     Perturbation of temperature (at depth-point d) in
linearization procedure 118
6N        Perturbation of total number density distribution in
linearization method 184
6T Perturbation of temperature distribution in linearization
method 184 (5,7 Kronecker (5-symbol 14
6ne,   dne j      Perturbation of electron density (at depth-point d) in
linearization procedure 117
on,,   (5rafi rf,"|    Perturbation of level-populations (at depth-point d) in
gnd   J       linearization procedure US
<57v,   5Jdn     Perturbation of mean intensity (at depth-point d) in
linearization procedure 143
6Jk        Perturbation of depth-variation of mean intensity
(at frequency vk) in linearization procedure 184 S(x) Dirac delta-function 8 5{z)        Ratio of Doppler width to fiducial Doppler width,
AvD(r)/Av* 449 op,  Spd     Perturbation of mass density (at depth-point d) in
linearization procedure 183
<5*?y>   orlan     Perturbation of emissivity (at depth </, frequency v„) in
linearization procedure 183
<>Zv,  &'/,dn     Perturbation of opacity (at depth U, frequency v„) in
linearization procedure 183
S%v        Change in opacity produced by departures from LTE 219
6\|/d        Perturbation of solution vector in complete linearization
method 231 AB(t)     Correction to integrated Planck function al depth t in
Unsold-Lucy procedure 63
Glossary of Physical Symbols 603
AH(t)
ATOn) At? Av
A*
A/,.
A/,w
AvD
Av£ Airf + i
Aro,,
Acow
Au)0 A«j0 Acu(t)
£
e, s'
no
n'
nit)
n\r, v, r), n!(y)
Error in integrated Eddington flux in Unsold-Lucy procedure
Temperature perturbation at depth (column mass) m Difference in tfeil. between Sun and a star Frequency shift arising from Doppler effect Lowering ofionization potential Classical damping width in wavelength units Wavelength shift produced by perturber located at
Weisskopf radius Doppler width in frequency units Fiducial Doppler width
Optical depth increment between mesh-points d and d + 1 in difference-equation solution of transfer equation
Frequency shift from line-center at boundary between impact and statistical-broadening regimes
Frequency shift produced by perturber located at Weisskopf radius
Line shift in Lindholm theory
Normal frequency shift
Instantaneous frequency shift induced by collision with
perturber Electric permittivity
Fraction of line emission that is thermal (classical theory) CoIIisional thermalization parameter in non-LTE source function
Generalized thermal emission parameter in non-LTE source function
Thermaliziilion parameter in schematic Lyman continuum problem
Non-LTE source function parameter Photoionizalion coupling parameter in non-LTE source function
Critical phase shift in Weisskopf theory Phase change in time-interval ds Instantaneous phase shift induced by collision with perturber
Change in phase in time interval (t, t + s) Emission coefficient
Total phase shift produced in collision with perturber at
impact parameter p Thermal emission coefficient
64 173 327 279 112 276
296 279 449
154
289
289 285
290
283 7 35
337
223
224 226
359
283
284
283
284
25
283 26
604     Glossary of Physical Symbols
f/5(r, v, t)..
(r, v, f), r\*{y 0 6
Ocff
0„„
K
A A,-A([/(0]
/<, /i'
P-
/' Pi
Pw v,
v
Scattering emission coefficient ■ 28
Thermodynamic equilibrium value of emission coefficient 26
5040/7* 322
Polar angle between direction of pencil of radiation and
normal to atmosphere layers 3
Recombination source term in non-LTE source function 359
5040/Terr 213
Excitation temperature parameter deduced from curve of
growth, 5040/7^ 322
Unit vector in direction of change in polar angle for
orthogonal spherical coordinate system 16
Polar angle of a point on a spherical surface 3
Angle between incident and scattered photons 416
Thermal conductivity 5 ] 7
Continuum absorption coefficient 205
Absorption-mean opacity 60
Planck mean opacity 59
Absorption ("true") coefficient 24
Thermodynamic equilibrium value of "true" absorption
coefficient 26
Wavelength 8
De Broglie wavelength 300
Ratio of true absorption to total opacity
kJ(kv + tr,) = (1 - pr) ]48
Fraction of total emission that is thermal in classical line-formation theory [(1 ~ p) + cp\]/(l + /?,) 309
Thermalizaiion depth 335
Discrete matrix representation of A-opcrator at (v;, p,) 160
Lambda (mean-intensity) operator 41
cos 0 (cosine of polar angle of pencil of radiation) 3
Magnetic permeability 7
Number of atomic mass units per free particle in a gas 519
Angle-point in discrete-ordinate method 65
Reduced mass of hydrogen atom 89
Frequency 2
Frequency associated with transition i -* j 22
Threshold frequency for ionization from nth state of
hydrogen 99
Glossary of Physical Symbols 605
v
v
vd
c
I"
t
71(0
n
p p p Pi
Pa Pp Pw
Pv
Pa PQ
Pou
Pooo
a ce
öiirt 0,
Threshold frequency for continuum absorption Line-center frequency
Minimum of incident and scattered photon frequencies Maximum of incident and scattered photon frequencies Auxiliary vector used in developing Feautrier difference-equation solution of transfer equation
Line-of-sight velocity
Photon frequency in atom's rest frame
Most probable line-of-sight thermal speed at temperature
Tcxc deduced from curve of growth Most probable line-of-sight speed of small-scale "turbulent'
mass motions in atmosphere Photon destruction probability at frequency x from line
center
Non-LTE source function parameter
Most probable line-of-sight velocity
Average photon destruction probability in a line
Perturber wave function
Gas pressure tensor
The i/th component of gas pressure tensor
The i/th component of partial pressure tensor for particle
species k in a gas Charge density Mass density
Impact parameter in collision Density matrix clement Density matrix for atomic states Density matrix for perturber states Weisskopf radius
Ratio of scattering coefficient to opacity,
aJ{kv + ov) = (1 - K) Effective impact parameter for collision-broadening Invariant mass density
Equivalent mass density of invariant mass density plus
internal energy of fluid Equivalent mass density of invariant mass density plus
enthalpy of fluid Continuum scattering coefficient Thomson scattering cross-section for free electrons Total scattering cross-section Imaginary part of collision integral Stefan-Boltzmann constant
123 ■278 426 426
156 279 412
323
323
350 225 279
351
300 516 514
514
7
170 283 298
301 301 283
43 283 542
542
542 309 106
84 285
12
606     Glossary of Physical Symbols
	Real part of collision integral	285 |		Phi (flux) operator
a{t, v, t),l			'L	Opacity in continuum
a(z, v, r),>	Scattering coefficient	24      ' |	Lj	Line opacity in transition i ->./,   x,(v) = yAj4>v
o				Excitation potential of state i, relative to ground state, of
v 1	Cross-section for transition i    / induced by collisions with	i		ionization stage j of chemical species k
ajv)				
	electrons of velocity v	132 I	/ion, Xl	Ionization potential
X	Mean time between collisions	282 ;	X	Mean opacity
i	Proper time	542	Xc	Chandrasekhar mean opacity
zc	Continuum optical depth	271 \	Ir-	Flux-weighted mean opacity
xe	Optical thickness of corrective element	189 1	is,	Rosseland mean opacity
T|	Static line optical depth	453 |	to	Line opacity assuming a Voigt profile, ^((v) = y.QHia, v). Xu = Xij!(.Kl&V0)
	Effective impact time	288 |		
Tjl	Rosseland mean optical depth	58 ;	Z(r, v, 0,1 l(z, VJ)\	Extinction coefficient, opacity, total absorption coefficient
Ti	Optical-depth perturbation in AvreLt-Krook method	174		
T	Mean optical depth	56	*v J	Line opacity
				
T(r. vj,	Monochromatic optical depth		ii/	Numerical factor in expression for total phase shift
r(z, v}\		34		Solution vector at depth-point d in complete linearization
% J				method
	Line-o("-sight optical depth through uniformly expanding		H*i> ■■■■> riv)	Wave function of JV-electron atom
	envelope in Sobolev theory	479		
0	Azimuthal angle of pencil of radiation around normal to			Line emission profile
	atmospheric layers	3	i/,,.(r, t)	Time-dependent wave function of atomic state 1
	Scalar potential	7		Natural-excitation line emission profile
	Ratio of Paschen jump to Balmer jump, Dp/DH	236	(0	Solid angle
	Unit vector in direction of change in azimuthal angle for		(0	Circular frequency
	orthogonal spherical coordinate system	16	10.....	Circular frequency associated with transition m -> n
	Reduced autocorrelation function	284	<ov	Ratio of line emission profile to absorption profile, \pJ4>y
4>(vX *v	Line absorption profile	27		Resonant frequency of an oscillator
	Time-independent wave function of atomic state i	85		Line-center frequency
	Azimuthal angle of a point on a spherical surface	3	V, VK	Logarithmic temperature-pressure gradient in ambient
	Matrix element describing broadening of spectrum line			atmosphere and convective elements
	a-*b	302		Adiabatic and radiative logarithmic temperature-pressure
0>v	Continuum photon-absorption rate coefficient, 4nv.Jhv	223		gradient
(t>(s)	Autocorrelation function	275		Gradient with respect to momentum coordinates
<D(xj	j'l. 4>{x)dx	479	©	Earth symbol
<vm	Saha-Boltzmanu factor of excitation state ? of ionization		0	Sun symbol
	stage./ of chemical species k relative to ground-state population of ionization stage.; + \,nfjk = ?7t.no..;-m, a^-ja	113		Integral over all solid angles
	Saha-Boltzmamt factor of excitation state / of ionization			
	stage j of chemical species k relative to total number density in ionization stage j + ], nfik = ncNjk<$ljk	113		
	Limb-darkening function	262		
Glossary of Physical Symbols 607
41
35
316
110
94
56 74
57
58
317
35 283
230
84 27
85 29
3 8
86 437
82 276
187
186 33 524 171 5
Index
Aberration of photons, 491-493, 495 Absolute intensities method (solar
temperature structure), 263-265 Absorption, bound-bound, 22 Einstein coefficients for, 77-93 excitation equilibrium, 126-127, 128-130
non-LTE radiative rates, 128-130 spectral line modeling, 167-169, 198-204, 316-331, 361-367, 371-373, 380-384, 401-410, 442-446, 485-490 Absorption, free-free, 22 vs. bound-free absorption, 102 by hydrogen, 100-102 Absorption, negative. See Emission,
stimulated Absorption, probability of (Einstein coefficient), 77-93 classical oscillator calculation, 81-84 net radiative bracket, 129-130 non-LTE radiative rates, 128-130 proportional to oscillator strength, 84
quantum mechanical calculation, 84-93
Absorption coefficient, 23 corrected for stimulated emission,
95-96 free-free, 101-102
See also Line absorption coefficient and Opacity Absorption depth (spectral line), 269-271, 316-321, 345-346, 362-365, 371-373, 402-407. 465-467 Absorption process, 20
bound-bound absorption, 22, 77-93 bound-free absorption, 22 free-free absorption, 22 photoexcitation, 22 photoionization, 22 See also Opacity Absorption profile, 27, 29. 78, 84, 273-307, 309, 338, 355-358, 423-424 See also Line profile Abundance, 115, 323-324, 372-373 carbon in WC stars, 489-490 differential curve-of-growth analyses,
326-327 helium, 403-405, 486, 489 light element, 405-407 LTE vs. non-LTE abundances,
372-373, 403^107 Population II vs. sun, 326-328 spectrum synthesis, 329-331,
402-407 See also Population Advection, 491-493
610 Index
Index 611
Angular phase function, 29-30, 412,
416-421, 424-425 Atmosphere, Bilderberg Continuum
(BCA), 263 Atmosphere, convection in, 185-192 Atmosphere, extended: combined moment equation,
250-251 early-type star models, 255-258 expanding spherical atmospheres, 459-462, 471-485, 502-510, 521-540, 549-568 line formation, 367-371 observational effects, 243, 247,
248-250, 256-258 Sobolev's method, 471-490 solutions of transfer equation, 250-255
transfer equation and moments, 244 Atmosphere, finite slab, 346-349, 354, 365-367
chromosphere, 346-347, 365-367 moving atmosphere, 449-450, 455 Atmosphere, grey, 53-76 boundary temperature comparable
to non-LTE temperature
ininimum, 240 Eddington approximation, 60-63 emergent flux, 73-74 extended spherical atmosphere,
245-250 grey and nongrey atmospheres
compared, 56 grey to nongrey atmospheres by
means of temperature
correction, 60 grey vs. nongrey surface temperatures, 205-207, 212-214 Hopf function, 55, 71-73, Table 3-2 mean opacities, 56-60 Milne's equation, 54 Milne's problem, 54 small departures from greyness,
74-76
temperature correction procedures, 62-64
Atmosphere, Harvard-Smithsonian Reference (HSRA), 263, 442-444, 458, 470, Table 7-1
Atmosphere, model. See Model atmospheres problem
Atmosphere, nongrey:
comparison to grey atmosphere, 56 constant frequency variation of opacity at all depths, 74-76, 207, 212-213 convection, 190-192 eigenvalue solution of transfer
equation, 150-151 energy balance, 170-181 mean opacities, 56-60, 211-212 nongrey transfer equation in LTE,
35, 75, 181 nongrey vs. grey surface temperature,
205-207, 212-214 picket-fence model, 207-212 transfer equation and moments. 44-46, 56 Atmosphere, plane-parallel. See Atmosphere, finite slab and Atmosphere, semi-infinite Atmosphere, semi-infinite:
boundedness condition for transfer
equation, 37 definition, 36
diffusion approximation, 49-52 discrete ordinates method, 64-69 moving atmosphere, 449-459 non-LTE line transfer, 345-358,
362-365, 374-410, 438-446 solution of transfer equation,
146-161
uniformly expanding plane-parallel atmosphere, 481^482 Atmosphere, turbulent, 463-471 Heidelberg approach, 463, 469-470 nice approach, 464-471 macroturbulent limit, 467 microturbulent limit, 467-468 Atomic transitions. See Transitions, atomic
Autocorrelation function, 275,
284-285, 298 Autoionization, 134-137 Autoionization transition probability,
135-136
Avrett-Krook procedure (temperature correction), 174-175, 220
Backscattering, 462
Backwarming effect, 167, 210, 212, 462
Balancing, detailed, 79, 94,-218-219, 278, 409 collisional, 132 ionization equilibrium, 125 radiative, 130
requirements for LTE, 119-120 two-level-atom model, 337, 358
Balancing, radiative detailed (of spectral line), 130, 218-219
Bands, spectral: color systems, 194, 204-205 non-LTE effects on colors, 234-235,
238, 257-258 photometric, 167
Bandwidth-constriction effect, 462
Bisector shift technique, 453. 455
Blocking effect, 167. See also Back-warming effect and Line-blanketing
Bohm-Vitense procedure (constraints), 176-178
Boltzmann equation, 32-33
Boltzmann law. See Excitation equation, Boltzmann
Boltzmann statistics, 79, 110-112
Boundary conditions (transfer
equation), 36-37, 51-52, 150-155, 181, 251-252, 254, 451-452, 459-460, 505, 507
Boundedness condition (transfer equation), 37. See also Atmosphere, semi-infinite
Bracket, net collisional, 132, 376-377, 379, 486
Bracket, net radiative, 129-130, 376-377, 379, 481,486
Braking, rotational, 534-536
Breeze solution (solar wind), 522, 530-533
Bremsstrahlung, 22
Broadening, collision. See Line-broadening, pressure
Calibration, fundamental, 193-195
flux of Vega, 196-198 Cayrel mechanism, 239-241, 265 Characteristic equation, 66-67, 69-70,
209, 215, 343 Characteristic function, 66-67,
69-70, 210-211
Characteristic rays (of partial differential equation), 248 Charge conservation. See Conservation
of charge Charge density of electronic wave
function, 89, 91-92 Chi operator, 41-42 Chromosphere, 258-260, 263-265, 266, 362-367, 372, 442-446, Table 11-3 Classical path approximation, 299-301 Cluster-expansion method, 295 Coefficient, absorption. See Absorption
coefficient Coefficient, departure, 219-223.
228-229, 235-236, 240 Coefficient, Einstein, 77-93
autoionization transition probability,
134-136 net radiative bracket, 129-130 stabilizing emission transition probability, 135-137 Coefficient, emission. See Emission
coefficient Coefficient, energy absorption. 78. 94-105. 123, 128-131, 27S-279 quantum defect method of calculation, 97-98 Coefficient, extinction. See Extinction
coefficient Coefficient, line absorption. See Line
absorption coefficient Coefficient, rate. See Coefficient,
Einstein Coefficient, recombination. See Recombination coefficient Coefficient, scattering. See Scattering
coefficient Collective photon pool, 375, 384-388,
396, 398, 436 Collisional de-excitation parameter.
See Coupling coefficient, thermal Collisional destruction probability.
See Coupling coefficient, thermal Collisions, three-body. See Recombination, collisional Colocation, spline, 155, 252 Color system. See Bands, spectral Column mass, 170 Comoving frame method (velocity fields), 449, 451, 461, 490-510
612 Index
Index 613
Comoving frame method {continued) frequency-integrated moment equations, 497-499, 503, 546-549
monochromatic equation of transfer, 491^97. 499-510 Complete linearization procedure (LTE), 182-185; (non-LTE), 230-234, 396-401,436-437 Complimentary error function, 427, 432
Computing time estimates, 157-158, 160-161, 185, 233, 252, 254-255, 398, 401, 452, 460-461, 508 Conductivity, radiative (diffusion
approximation), 52 Conductivity, thermal, 517-518, 523,
525-526, 531, 537, 559-560 Conservation of charge, 115, 138-139,
232, 292-293 Conservation of mechanical energy,
517, 547 Conservation of particles. 115,
138-139, 142, 232 Constraint procedures (Energy balance): Bohm-Vitense procedure, 176-178 complete linearization method,
180-185, 230-234 Feautrier's procedure, 178-179 partial linearization, 179-180, 226-227
Continuity, equation of, 128, 515-517,
525, 528, 544, 561 Continuum: absorption cross-sections, 96-105 blocking and backwarming, 167-169 departures from LTE. 220-222, 223-224, 228-230, 235-239 destruction probability. 334-335, 350-354
"effective depth" of formation, 62 Einstein-Milne relations, 94-96 electron velocity distribution,
121-123 emission coefficient, 25 emissivity, 96, 165-166 example of source function, 35-36 fitting LTE models to stellar spectra,
193-205 isotropic, coherent scattering
assumption, 30
Lyman continuum formation, 222-228
non-LTE solution by lambda-iteration, 218-222
opacity, 165-167
oscillator strength formula, 97, 99 partial linearization method for
continuum formation, 224-228 Saha ionization equation, 112-114 scattering cross-sections, 105-107 slope (in spectrum) indicative of
temperature, 248-250 solar, 260-263
two-level-atom model, 350-355, 358-365, 372-373 continuum controlled line, 353 See a/so Continuum, Lyman and Continuum, Paschen and Schuster mechanism Continuum, Balmer, 198-204, 214,
239-240, 242 Continuum, Lyman, 213-215,
222-228, 239-242, 260 Continuum, Paschen, 194-198
discrepancy with Balmer jump, 196 Convection, 185-192 Conversion, photon, 375, 384,
387-388, 393, 396 Coordinate system, Lagrangian, 491,
502,514-517, 544-549 Coordinate systems: conventions defined, 3 impact parameters, 247 Lorentz transformation of, 493-494 used in angle-averaged redistribution, 424
used in Doppler-shift redistribution, 416,417 Core, line. See Line core Core-halo models, 563, 566-567 Corona: classes of coronae, 540 collisional ionization dominant,
124, 141 conductive transport, 47, 521, 523-526, 531. 537, 539-540, 560
coronal holes, 540
dielectronic recombination, 136, 141
electron and radiation temperatures
for the sun, 124 rate equation, 141
Corona (continued)
solar corona, 258-260, 265-267,
293, 444,511,521-540 spherical coronal wind models, 525-540 breeze solutions, 522, 530, 532 isothermal corona, 526-529 two-fluid models, 537 Correction, bolometric, 198 Correction procedures, temperature. See Temperature correction procedures Coulomb approximation, 93, 97 Coupling, level. See Interlocking Coupling, Russell-Saunders, 92 Coupling coefficient, thermal. 35-36, 206-239, 215-216, 224, 241, 308-312, 314-316, 337, 341-353, 355-358, 377, 385-387, 391, 393-396, 456-457, 462, 481-482 depth variations of, 355-358 Coupling parameter, thermal, 148-150,
215-216, 308-312, 314-316 Critical point, Alfvenic, 535, 567 Critical radius, 526-529, 539. 553,
556, 558, 562-563, 566-567 Critical solution, 526-528, 562 Cross-sect ion, absorption, 23, 78-107 continuum absorption by helium,
104- 105
continuum absorption bv hydrogen, 98-102
continuum absorption by negative and other ions of hydrogen, 102-104
continuum absorption cross-sections, 96-105
continuum scattering cross-sections,
105- 107
Einstein coefficients, 77-93 classical oscillator calculations, 81-84
quantum mechanical calculation, 84-93
Einstein-Milne continuum relations, 94-96
Einstein relations, 78-79 hydrogen cross-sections, 98-102, 213-214
light element cross-sections, 97-98 Rayleigh scattering, 106-107 Thomson scattering, 106
Cross-section, collisional, 131-134 equations for rate estimation,
133-134 Thomson formula, 124, 127 Cross-section, excitation, 122, 131-134 Cross-section, recombination, 122,
130-131, 132 Cross-section, scattering. See
Oscillator, harmonic Curve of growth, 270, 373, 447-448, 467-469 empirical, 321-328
Population II vs. Population I abundances, 327-328 theoretical, 316-321 Cycles. Rosseland's theorem of. See Rosseland's theorem of cycles
Damping, collision. See Line-broadening, pressure
Damping, natural. See Damping, radiation
Damping, radiation, 273, 276-278, 282, 305, 343, 413, 415, 442-443, 446. See also Line-broadening
Damping, resonance. See Line-broadening, resonance
Damping constant, classical (harmonic oscillator), 82, 276
Damping parameter, Weisskopf, 283
Dawson's integral, 281
Decrement, Balmer, 142
De-excitation, collisional, 22, 333-334. See also Thermalization and Cross-section, collisional
Degradation, photon, 384-387. See also Rosseland's theorem of cycles
Density, radial charge, See Charge density of electronic wave function
Density structure, 170-171, 255-257, 264-265. See also Equilibrium, hydrostatic
Depth, absorption. See Absorption depth
Depth, thermalization, 119-120, 141, 149, 311, 334-335, 341-342, 345-347, 352-355, 387-388, Table 11-1
Destruction length (photon), 333-334, 350
614 Index
Destruction probability (photon),
333-335, 342, 350-352 Destruction probability, total.
350-352, 366, Table 11-1 Difference equations, 153-161, 171,
181-183, 192, 230-233, 252-255,
398-401,436-437, 451-452,
504-508
Diffusion approximation, 49-52, 153, 157, 168, 177, 189, 191, 252, 254, 451,460, 505, 554 consistent with assumptions of
Rosseland mean opacity, 58 grey atmosphere, 55 See also Diffusion limit Diffusion limit, 168, 462, 554 boundary condition for semi-infinite atmosphere, 37 form of the radiation field in, 17, 50-51
See also Diffusion approximation Dilution factor, 120-121, 123-125,
127, 239-240, 244 Discrete ordinates method: grey transfer problem, 64-71 non-LTE line transfer, 343-344 nongrey transfer equation, 150-151 picket-fence model, 208-211, 215-216 Distribution, temperature. See Temperature distribution Distribution function, macroscopic velocity, 442-443, 453-458, 461, 464, 469-471, 508, 563-564 Distribution function, opacity,
167-171, 193, 198, 205. See also Picket-fence model Distribution function, particle, 32 Distribution function, photon, 4, 6, 10, 13
Double-Gauss formula, 65-66 Doublet, resonance (spectral line), 387-390, 393-396
Eddington approximation, 60-63, 148-149, 224, 234, 244
Eddington-Barbier relation, 39, 206, 346, 358, 363. 475 Eddington approximation. 61-62
Eddington factor, variable. See Variable Eddington factor
Effective thickness (of atmosphere),
347-348, 366, 369 Eigenstates, 85-90 Eigenvalue solution (to transfer
equation), 150-151 Einstein-Milne continuum relations,
94-96
Einstein relations, 79. See also Einstein-Milne continuum relations Emission (thermal), 20 bremsstrahlung, 22 collisional recombination, 22 completely linearized line transfer, 397
free-bound transition, 22, 96 LTE radiative rates, 166 moving medium, 30-31. 449 non-LTE radiative rates, 129-131, 165
Planck mean opacity, 59 probability of spontaneous and
stimulated emissions, 78-80 radiative recombination, 22,
130-131
thermodynamic equilibrium, 25-26 See also Emission coefficient and Source function Emission, induced. See Emission,
stimulated Emission, spontaneous, probability of, 78-81, 88, 277, 336-337, 376-377, 385-388, 391, 393, 434-438, 486, 488 non-LTE radiative rates, 129, 130-131
Emission, spontaneous thermal. See
Emission (thermal) Emission, stimulated: correction for stimulated emission,
80, 95-96 definition, 24 Einstein coefficient, 78, 88 non-LTE radiative rates, 129, 130-131
partial redistribution, 434, 437-438 Emission, stimulated, correction for,
80, 95-96, 165-166 Emission, stimulated, probability of.
78, 88
non-LTE radiative rates, 129, 130-131
Emission coefficient (scattering)
27-30, 309-310, 423, 435-438 Emission coefficient (thermal): classical line transfer, 309 continuum emissivity, 96, 130-131,
165-166 definition, 25
in comoving frame for velocity
fields, 490, 492, 495-496 moving medium, 30-31, 449, 490 proportional to intensity in thermodynamic equilibrium, 26 probability of spontaneous and stimulated emissions, 78 Emission profile, 27-30, 78-80. 423-424 complete noncoherence, 29 complete redistribution, 29, 430^132 partial redistribution, 433-438 See also Excitation, natural, emission profile Emissivity. See Emission coefficient Emitting area, effective, 367, 483 Energy density (of radiation field): comoving frame method for velocity
fields, 497-498, 503, 546-548 definition from electromagnetic
theory, 8 fixes surface temperature in LTE, 240
monochromatic and total densities
defined, 6 proportional to mean intensity, 6 proportional to mean radiation
pressure, 13 specifies radiation pressure tensor in isotropic radiation field, 17 thermal equilibrium, 6-7 zero-order moment of transfer equation, 43-44 Energy distribution (emergent): LTE model atmospheres, 193-205 non-LTE atmospheres, 234-239 Energy equation, total, 517-518,
544-545, 549 Energy transport, 47-49, 163-164, 185, 188. See also Transfer equation and Convection Equilibrium, convective, 47-48, 187, 191-192
Equilibrium, excitation, 126-127, 137-139, 143-145
Index 615
Equilibrium, hydrostatic, 163, 170-171,
183-184, 232-233, 255-257, 263,
355, 521, 523-525, 554-555, 563 See also Density structure and Equilibrium, ionization Equilibrium, ionization, 123-126,
137-139, 140-142, 143-145 Equilibrium, local thermodynamic.
See Local thermodynamic
equilibrium Equilibrium, non-local thermodynamic.
See Non-local thermodynamic
equilibrium Equilibrium, radiative. See Radiative
equilibrium Equilibrium, thermodynamic. See
Thermodynamic equilibrium Equivalent-two-level-atom method
(for multilevel atom), 375-383,
505
Ergodic hypothesis. 285
Escape-enhancement, 462
Escape probability (photon), 333-335. 341-342, 347, 355-358, 368, 385-388, 478-482, 486-488, 509
Eulerian description (fluid flow), 514
Evaporative model (solar wind), 530
Excitation, natural: definition, 29
emission profile. 423-424. 433, 434 Excitation equation, Boltzmann, 79, 110-112, 322 extended by Saha ionization
equation, 112-114 LTE equation of state for ionizing material, 114-119 Exclusion principle, Pauli, 91 Expansion, atmospheric, 447-449, 451, 453-457
comoving frame method, 491-493,
499-510 energy balance, 461-463 infinite plane parallel medium,
481-482 Sobolev's method, 471-490 spherical atmospheres, 459-461, 478-481, 482-485, 503-510 stellar winds, 511-512, 521-540, 549-568 Exponential integral, 40-41 Extinction coefficient, 23-25. See Opacity
616 Index
Index 617
Extinction coefficient {continued) See also Absorption, Absorption processes, Absorption cross-section, Energy absorption coefficient, Scattering, and Scattering coefficient
Feautrier's procedure (constraint), 178-179
Feautrier's solution (of transfer
equation), 156-158, 161, 233, 252, 398, 436, 438, 504, 508 complete linearization method, 233 optimum for coherent scattering
problems, 158 partial linearization method for non-LTE continuum, 226-227 Field, radiation. See Radiation field Field strength, normal, 291, 296 Fluorescence, 22
Flux, absolute, 12, 194, 198-200, 489 Flux, astrophysical:
computation of, 161
definition, 10
grey atmosphere, 55, 73-74 line and continuum, 269, 271, 317, 328-329, 370-371,402-410 Flux, conductive, 517-518, 525,
530-532 Flux, Eddington: comoving frame method for velocity
fields, 497-499. 502-503 emergent flux from transfer
equation, 161 first moment of radiation field, 10 plane wave in extended stellar envelope, 17 Flux, emergent:
extended grey atmosphere, 247-250
grey atmosphere, 73-74
line in expanding atmosphere,
453-459, 482-485, 510 nongrey atmosphere, 193-205,
234-239, 256-257 spectrum line, 269, 271, 311, 317, 328-329, 370-371, 371-373, 402-410 See also Energy distribution
(emergent) and Flux, absolute Flux, energy, 518, 526, 531, 549 Flux, mass, 511, 516, 523, 525-526, 553, 558-559, 562-563
Flux, momentum. 516 Flux, radiation: comoving frame method for velocity
fields, 497-499, 502-503 constant in radiative equilibrium for
planar geometry, 48 definition, 9
first moment of radiation field, 9-10 interior point of semi-infinite
atmosphere, 41, 50-51 observational determination of
stellar flux, 11-12 Poynting vector, 11 proportional to momentum density,
10
Flux, residua], 269. 311-313, 316, 328-329, 370-371, 371-373, 403-405
Flux ratio (for model fitting), 193-194,
196-197, 234-239 Force, buoyant, 188 Force, gravitational, 170-171, 255-257,
516. 547 Force, Lorentz, 497 Force, Minkowski, 542 Force, radiation. See Radiation force Force, van der Waals, 283, 287-288,
326
Four-force, 497, 542, 544-546 Fourier transform. See Transform,
Fourier Four-momentum, 494
Gas energy equation, 517, 544-545,
547-549 Gaunt factor:
bound-bound, 90
bound-free, 99
thermal average of free-free, 101 Gaussian formula ("single-Gauss
formula"), 65-66, 69 Geometry, planar: boundary conditions for transfer
equation, 36-37 comoving frame equation of transfer, 492. 501
constant flux in radiative equilibrium, 48
conventions in, 3
examples of transfer equation for planar atmospheres, 37-38
Geometry, planar (continued) moments of transfer equation,
43- 47
transfer equation for planar
atmosphere, 32, 38 Geometry, spherical: boundary conditions for transfer
equation, 36-37 conditions for radiative equilibrium,
49
comoving frame equation of transfer, 492, 501-510
conventions in, 3
conventions for extended atmospheres, 243-244, 247, 252-254
differential element of solid angle. 3, 5
equation of continuity, 515-516 impact parameter, 247, 252-253 moments of transfer equation,
44- 46
transfer equation for axially
symmetric atmosphere, 34 transfer equation for spherically symmetric atmosphere, 33-34 Gravity, surface, 170-171, 193, 195, 204, 234-238, 255-256, 328-330, 402
H-R diagram, 234 Hamiltonian (operator), 85-86,
299-303 multi-electron atom. 91 Hartree-Fock method (transition
probability calculations), 91-93 Ffartree's self-consistent field method,
91
Heaviside function, 426
Helium: line-broadening, 305-306 non-LTE vs. LTE line transfer,
403-406 opacity, 104-105
Wolf-Rayet stars, 485-486, 488-489 Hermite integration, 155, 252 Holtsmark theory, 291-292, 294-295. 304
Hopf function, 55 Eddington approximation, grey
atmosphere, 61-63 exact solution, 71-73, Table 3-2
Hydrodvnamic equations, 512-521, 525-526
energy equation, 517-518, 525-526 equation of continuity, 515-516,
525-526 momentum equation, 516, 525 Hydrodynamics, radiation, 491, 540-549 fluid-frame: energy equation, 503, 544-545 momentum equation, 503, 545-546 inertial-frame, 546-548 stress-energy tensor, 497-499, 541-544 Hydrogen:
absorption cross-sections, 98-102 Balmer decrement, 142 calculation of wave functions and
oscillator strengths, 88-90 comparison of LTE and non-LTE treatments, 215, 218-230, 234-239, 241-243, 257, 402-404 convection in hydrogen ionization
zones, 187 departures from Maxwellian velocity
distribution by electrons, 123 effects of temperature on hydrogen
and metal atmosphere, 116 H~ for solar opacity, 76, 102-104. 260
H I region, 525 H II region, 525
impact theory for hvdrogen lines, 303-305
LTE ionization of pure hydrogen gas, 116
model hydrogen atom, 214-215. 220-221, 222-223, 228-230, 234
negative ion cross-sections, 102-104 other hydrogen ion cross-sections, 104
quasi-static ion broadening of
hydrogen lines, 295-297 Rayleigh scattering, 107 recombination with electrons, 122 sample equations of state, 138-139 Wolf-Rayet stars, 486, 489 Hydrostatic equation, 170-171,
183-184, 188, 232-233, 255-257, 524
618
Index
Index
619
Impact approximation, 288-289, 301-303
Impact parameter, 247, 252-253,
282-286, 288-289, 303 Impact theory of pressure broadening.
See Line-broadening, pressure Instability, Rayleigh-Taylor, 524 Integra-differential equation, 32, 36, 65 Intensity, mean: approximate closure for solving
transfer equations, 46-47 comoving frame method for velocity
fields, 498-499, 502-508 Eddington approximation, 61-63 interior point of semi-infinite
atmosphere, 40, 51-52 lambda-operator, 41-43 Peierls's equation, 41 plane wave in extended stellar
envelope, 17 proportional to energy density, 6 Schwarzschild-Milne equations, 40 zero-order moment of radiation field defined, 5 Intensity, specific: comoving frame, 495 determination of, from spatially
resolved source, 5 first-order moment of, 9-10 invariance property, 4 Laplace transform of source
function, 39 macroscopic definition, 2-3 microscopic definition, 4 moments of, 5, 9-10, 12, 43-46 proportional to electromagnetic
energy density, 9 second-order moment of, 12 zero-order moment of, 5 Interaction potential, classical, 283,
286-288, 301-302 Interaction region, 333-335, 371-373,
474-477, 478, 480-481 Interlocking (of line radiation fields),
375-378, 384-393, 396-401 Interstellar medium, 524-525, 533-534 Interstellar reddening, 194, 198 Ionization, collisional, 22 dominant in corona, 124, 141 ionization equilibrium, 124-127,
137-139, 140 rate of, 123-124, 132-134
Ionization equation, approximate, 125, 140-141
Ionization equation, non-LTE, 137-139, 140-141, 143-145 Ionization equation, Saha: curve of growth, 322, 327, 330 derivation, 112-114 equation of state for ionizing
material, 114-119 negative hydrogen ions, 103 Irradiation effect, 462
Jump, Balmer, 194-196, 213, 220, 235-238, 257, 328-329 discrepancy with Paschen continuum, 196, 220 Jump, Lyman, 213-215, 235, 237 Jump, opacity, 205-206, 212-214 Jump, Paschen, 194, 235-237
Kirchhoff-Planck relation, 26, 50-51,
80, 96, 166 Kirchhoffs law, 26 Klein-Nishina formula, 106 Kubo-Anderson process, 464, 470
Laguerre polynomials, 89 Lambda-iteration: difficulties with, 62-63, 126, 147-150,
172-173, 222-224, 343 LTE model atmosphere, 172-173 non-LTE model atmosphere,
218-220, 222-224, 227-228 temperature correction procedure,
62, 172-173 unsuited to non-LTE transfer
equation, 149-150, 222-224, 343 Lambda-operator: Bohm-Vitense constraint method,
176-178 definition, 41 Laplace transform, 39, 71, 280 Legendre functions, 65, 89 Length, conversion. See Conversion, photon
Length, correlation, 464, 467, 469-470 Length, Debye, 293-295, 524 Length, destruction (photon), 333-334, 350
Limb-darkening function, 70-71, 314-315
Limb-darkening law, 61, 63, 72,
120-121, 260, 262-263, 442-446, 453
Lindholm approximation, 284-288,
Table 9-1 Line, emission, 314, 316, 348-349,
362-367, 370-371, 372, 408-410,
448, 471-478, 483-485, 487-490,
510, 549-553 Line, forbidden, 37, 306 Line, resonance, 143, 167, 313, 358,
380-383, 384-387, 387-388, 413,
415, 428-438, 442-446, 550,
552-553, 555 Line, subordinate, 143, 313, 358 Line absorption coefficient, 77-78, 80,
317, 328, 333, 336 classical line transfer, 309, 317 moving atmosphere, 449-451, 464, 479
partial redistribution, 435, 438 Linearization: complete linearization method,
230-234, 396-401 Newton-Raphson method, 117-119,
233
solution of non-LTE equations of state, 143-145 Linearization method: LTE model atmosphere, 180-185 non-LTE model atmosphere,
230-234 with convection, 192 Linearization procedure, partial,
179-180, 224-230 Line-blanketing, 167-169, 193, 196-198, 205-212, 215-216, 241-243 effect of enhanced line opacity,
200-204 See also Picket-fence model and Opacity distribution function Line-broadening, 273-307. See also Damping, radiation and Line-broadening, pressure Line-broadening, Doppler, 274, 279-281, 286, 305, 338, 340, 342, 356-358 Line-broadening, pressure, 273-274, 278, 281-307, 324, 326, 328 classical impact theory, 281-289 quantum theory, 297-303
classical path approximation,
299-301 impact approximation, 301-303 statistical theory, 289-297 Debye shielding, 292-295 Holtsmark theory, 291-292,
294- 295, 304
nearest neighbor approximation,
290-291 quasi-static ion broadening,
295- 297, 303, 304, 305, 306 See also Line broadening
Line-broadening, quasi-static ion,
295-297, 303-306 Line-broadening, resonance, 283,
286-287, 305 Line-broadening, statistical theory of
pressure. See Line-broadening,
pressure
Line core, 29, 281, 296, 305, 312, 314, 318-320, 333, 335-336, 341, 348, 367-368, 383, 389-390, 403-406, 419, 429-430, 439, 442, 443_446, 453-455, 458, 470 emission cores, 362-367, 370-371, 372, 383, 443-446 Line profile, 276, 278, 285, 296, 303-307, 311-313, 328-331, 382-383, 404-406, 442-446, 448, 453-459, 461, 471, 482-485, 510, 550, 552 asymmetric, 306, 453-455, 461,
471-474, 482-485, 510 comoving frame for velocity fields,
508-510 doubly reversed, 364, 370-371,
383, 443-446, 458-459 expanding plane-parallel atmosphere, 453-457 flat-topped, 477, 484-485 rounded, 477, 483-485 self-reversal, 348, 365-367, 370-371,
372, 455 Sobolev theory, 482-485 surfaces of constant radial velocity,
472-477, 479 turbulent atmospheres, 464 See also Absorption profile Lines, spectral: absorption depth and residual
flux, 269 blocking effect, 167-169, 204-205
620 7m/e.v
Index 621
Lines, spectral (continued) center-to-limb variation, 314-315, 442-446
chromosphere, 362-367, 372-373 classical line transfer, 308-331 coherent scattering, 30 collision and photoionization
dominated lines, 361-362,
363-365, 372-373 comoving frame theory, 503-510 complete redistribution, 29 continuum controlled line, 353 curve of growth, 270, 316-328 effects on temperature structure
of non-LTE atmosphere, 241-243 Einstein coefficients, 77-80 emission with departures from LTE.
26, 78-80, 165 energy balance in moving media,
461-463 equivalent width, 114-115, 270,
316-328, 372-373, 402-404,
406-407, 409, 465-470,
487-490
example of source function, 35-36 expanding spherical atmospheres,
459-461,471-481,482-490,
503-510 formation in moving planar
atmospheres, 449-459, 481-482 helium lines, 93, 403-406, 488-489 hydrogen lines, 88-90, 100, 193,
228-230, 241-243, 328-329,
402-403 line absorption coefficient, 80 line broadening, 273-307 lines in the partial linearization
solution of non-LTE
continuum, 228-230 line source function, 80, 309-310,
336-337, 359-360, 376-378,
391-392, 404-405, 435, 437-438 LTE spectrum synthesis, 328-331 Lyman and Balmer lines, 228-230,
234, 241-243, 402-404, 442,
471. 474, 551 non-LTE line transfer with
continuum, 358-367, 376-380,
396-401 non-LTE line transfer without
continuum. 336-349 opacity, 167-169
opacity distribution function, 168-169
opacity sampling technique, 169 physics of non-LTE line formation, 332-336, 374-375, 384-388, 472-475, 478 picket-fence model, 207-212
scattering lines, 215-216 radiative detailed balance, 130 redistribution, line transfer with,
433-446, 504, 508 Sobolev theory, 471-490 turbulent atmosphere, 463-471 See also Absorption profile, Emission profile, Line-blanketing, Opacity and Transitions, atomic Lines, spectral (scattering): center-to-limb variation, 314-315 classical line transfer, 310, 311-312 no effect upon surface temperature,
207, 216, 241 picket-fence model, 215-216 Line shift, 284-288, 306-307, Table 9-1 normal frequency shift, 290-291 See also Shift, Doppler Line source function. See Source
function, line Line strength, 88, 92-93 Line wings, 281, 291, 303-305, 316, 318-320. 333. 335-336, 340, 343, 345, 352-353. 356-358, 403, 419, 429-430, 439-442. 446, 455, 456 Local thermodynamic equilibrium: assumptions of, 26-27, 108-110 Boltzmann excitation equation,
110-112 classical line transfer, 308-331 correction for stimulated emission,
80, 95-96 departures from LTE in continuum,
216-243 Eddington-Barbier relation, 39 effects of radiation field upon
opacities, 43, 108-109 equation of state for ionizing
material, 114-119, 123-127 excitation equilibrium, 110-112 extended grey atmosphere, 245-250 flaws in the theory. 119-121, 123-127, 332-335, 371-373
Local thermodynamic
equilibrium (continued) ionization equilibrium, 112-114 Maxwellian velocity distribution, 110 methods for radiative-equilibrium models, 164-185 construction of models, 170-185 determination of temperature structure, 171-180, 184-185, 205-216 opacity and emissivity, 166 results of models, 192-236 microscopic requirements of,
119-120 non-LTE vs. LTE line transfer,
371-373, 401-410 Saha ionization equation, 112-114 source function in, 80 Lorentz-Fitzgerald contraction effect, 494, 543
Lorentz transformation, 493-495, 541
of radiation field, 495-499 Luminosity, 49, 171, 245, 252, 256
Mach number, 520-521, 522, 534
Alfvenic Mach number, 535 Macroturbulence. See Velocity
fields and Turbulent atmospheres Magnetic field (of a star), 534-536, 537-538, 567 Alfven waves, 535 Markov process, 463 Mass loss, 511-512, 521-540, 549-568
See also Mass loss rate Mass loss rate, 511, 516, 523, 526. 531, 536, 540, 553, 558-559. 562-563
Mass motion, 323, 373, 447-510,
511-568. See also Microturbu-
lence, Atmosphere, turbulent, and
Velocity fields Maxwell's equations, 7-8 Mean-free-path, 23, 35, 51, 149, 323,
333-336, 350, 371, 447, 452, 462 See also Length, destruction Mean opacity. See Opacity Microturbulence, 323, 327, 373,
442-443, 447-448, 458, 463,
467-470 Milne-Eddington model (for line
transfer), 310-316, 317, 321, 465,
467, Table 30-1
Milne's equation, 54
Milne's problem, 54
Mixing-length theory, 48, 187-190
Model atmospheres problem: assumptions, 162-164 convective model atmospheres, 190-192
hydrostatic equilibrium, 170-173,
255-257
methods of LTE model construction, 164-185 non-LTE model construction, 216-234 complete linearization method
(non-LTE), 230-234 difficulties. 217
formation of Lyman continuum,
222- 228 lambda-iteration fails, 125-126,
223- 224
models for Early-type stars with extended atmospheres, 255-257
partial linearization method for continuum formation,
224- 227
solutions of spherical atmosphere transfer equations, 250-255 results of LTE models, 192-216 emergent energy distribution,
193-205 extended grey atmosphere,
248-250 temperature structure, 205-216 results of non-LTE models, 234-243 emergent energy distribution,
234-239 temperature structure, 239-243 temperature distribution with radiative equilibrium, 171-185, 205-216, 239-243, 246-247 treatment of spectral lines, 167-169 Moment, dipole, 86-88, 297-299, 301-303 free-free hydrogen opacity, 100 Moments of the radiation field: astrophysical flux, 10 comoving frame for velocity fields,
497^199, 502-503 Eddington flux, 10 flux defined (first moment), 9-10
622 Index
Index 623
Moments of the radiation field (continued) mean intensity denned (zero-order
moment), 5 radiation pressure (or stress) tensor
(second moment), 12-14 Schwarzchiid-Milne equations, 40-43
Momentum, angular, 522, 535-536 Momentum, particle: hydrodynamics, 514, 516, 542-544 Lorentz transformation of, 494 Momentum, photon, 494-495 Momentum density (of radiation field), 10, 497-498 first-order moment of transfer
equation, 44-46 fluid flow, 546 Monte Carlo technique, 294, 422 Multiplet, 88, 93, 161, 322, 375, 387-388
solution of transfer equation for, 391-393
source function equality, 384, 387, 388-390, 393-396
Nearest neighbor approximation, 290-292, 294. See Line-broadening, pressure Newton-Cotes formula, 65 Newton-Raphson method. See
Linearization Non-local thermodynamic equilibrium: compared to LTE line transfer,
371-373, 401-410 effects of radiation field upon
transfer equation, 43, 108-109, 217
line transfer, 336-371, 374-410, 433-446, 449-461, 471-490, 503-510 model atmosphere construction, 216-234 complete linearization method,
230-234 difficulties, 217
formation of Lyman continuum,
222- 228 lambda-iteration fails, 125-126,
223- 224
partial linearization method for
continuum formation, • 224-227 results of non-LTE models, 234-243
non-LTE equation of state, 140-145 opacity and emission rates, 80,
95-96, 165, 219, 232 rate equations, 127-145
Oblique rotator model, 200 Observer's frame method (velocity fields), 449-471 spherical atmospheres, 459-461 transfer equation, planar atmospheres, 449-452 velocity fields in planar atmospheres, 453^159 Occupation numbers (of atomic levels). See Population (of atomic levels) Opacity:
completely linearized line transfer,
397-398 constant frequency variation of
opacity at all depths, 74-76,
212-213 definition, 23-25 departures from LTE, 26, 78, 80,
108-109, 165, 219, 232 grey atmosphere, 53 helium opacity, 104-105 hydrogen opacity, 100-101, 102 LTE continuum opacity, 166 LTE radiative-equilibrium models,
165-169, 200-204 mean opacities, 56-60, 168 absorption mean opacity, 60, 174 Chandrasekhar mean opacity, 74-75
flux-weighted mean opacity,
56-57, 174 Planck mean opacity, 59-60,
168, 173-174, 211-212, 214 Rosseland mean opacity, 57-59,
168, 181,211-212, 214,
255-256
moving medium, 30-31, 449-450,
459, 495-497 negative hydrogen ion 76, 102-104 non-LTE opacity, 80, 95, 165, 219,
232
) Opacity (conlinued)
' optical depth scale, 34-35
power-law, 246, 461, 508 Rayleigh scattering, 106-107 spectral lines, 167-169 i Thomson scattering, 106
See also Absorption, Absorption processes, Cross-sections, absorption, Scattering, f Scattering coefficient, and
Source function Opacity, absorption mean, 60, 174 Opacity, Chandrasekhar mean, 74-75 Opacity, flux-weighted mean, 56-57, 174
Opacity, Planck mean, 59-60, 168,
173-174, 211-212, 214 optical depth scale, 173 Opacity, power-law, 246, 461, 508 Opacity, Rosseland mean, 57-59, 168,
181,211-212,214,255-256 Opacity distribution function, 167-169,
193, 198. See also Picket-fence
model
Opacity sampling technique, 169 Operator. See Chi-operator, Lambda-operator, and Phi-operator ; Optical depth scale:
definition, 34-35 definition of "very large t,"
140-141, 149. See also Ther-J malization depth
j Eddington-Barbier relation, 39
i lines and continuum, 271, 328, 368
' moving atmosphere, 450, 479,
560-561
Optical depth scale, Rosseland, 49, 58, 181, 191, 210, 367. See also Opacity, Rosseland mean Optical thickness, 36, 38, 346-349, 366 Orbitals, electron, 91-92 Oscillator, classical. See Oscillator,
harmonic Oscillator, harmonic, 81-84, 276 Oscillator strength (of atomic transi-1 tion), 84, 88, 90, 92-93, 133, 326
continuum, 97, 99, 101, 123
P-Cygni (line characteristics), 448, 456, 461, 471-475, 484-485, 510, 523, 538, 550-553, 567
Parasites, 150-151
Partial redistribution. See Redistribution
Particle conservation. See Conservation of particles
Particle distribution function, 32
Particle number conservation equation. See Conservation of particles
Partition function, 111, 298, 322, 326-327
Peierls's equation, 41
Phi-operator, 41-42, 161
Photoexcitation, 22, 126-127, 128-130
Photoionization, 22, 94-105 bound-free vs. free-free absorption, 102
departure from LTE, 125-126 ionization equilibrium, 123-126 non-LTE radiative rates, 130-131 rate of photoionization, 123, 130 Photoionization, probability of, 94-95 Photoionization edges, 99-100,
104-105, 194, 213-214, 235-239 Photon absorption. See Absorption Photon distribution function, 4, 6, 10, 12-13
Photon pool. See Collective photon pool
Photon scattering. See Scattering Photosphere, Utrecht Reference, 263 Pickering series, 488-489 Picket-fence model, 207-212, 215-216, 241, 462. See also Opacity distribution function Planck function, 7, 26, 78-79, 94, 96, 119-121, 309, 310, 312, 317, 337, 359-360, 362-365, 372 Poisson law, 282, 464 Poisson's equation, 292 Population (of atomic levels), 78-80, 94-96, 108-109, 220-222, 228-230, 235-237, 403-405, 408-409, 488-490 Boltzmann excitation equation,
110-112 equation of state for ionizing
material (LTE), 114-119 non-LTE effects on energy distributions, 234-239, 257, 263 non-LTE effects on spectrum fine profiles, 345-346, 348-349, 363-367, 369-373, 380-383,
624 Index
Index 625
Population (of atomic
levels) (continued) 393-396, 402-410, 442-446, 470, 482-485 non-LTE equation of state, 140-145 non-LTE rate equations, 127-139 Saha ionization equation, 112-114 Wolf-Rayet stars, 486, 488^90 Potential, ionization, 94, 112 Potential, Lennard-Jones, 288 Potential, Srairnov, 288 Poynting vector, 11, 14, 82 Pressure, impact, 533 Pressure, radiation. See Radiation
pressure Pressure tensor, radiation. See
Radiation pressure tensor Probabilities, cascade, 142 Probabilities, transition. See
Coefficient, Einstein Probability of absorption. See
Absorption, probability of Probability of spontaneous emission.
See Emission, spontaneous,
probability of Probability of stimulated emission.
See Emission, stimulated,
probability of Profile, absorption. See Absorption
profile
Profile, damping. See Profile, Lorentz Profile, Doppler. See Line-broadening,
Doppler-shift Profile, emission. See Emission profile Profile, line. See Line profile Profile, Lorentz, 84, 276-278, 282,
285, 305, 338, 340, 342, 346-
347, 413
Profile, Voigt. See Voigt function Prony algorithm, 262
Quadrature sum, 65-69, 144, 150, 158-161, 177-180, 192, 208-211, 226-227, 254, 338-339, 343, 392, 400, 436, 451-452, 460, 506
Quantum defect, 97-98
Quantum numbers, 89, 91-93 imaginary, 98, 100
Quantum theory of pressure
broadening. See Line-broadening, pressure
Radial velocity, surface of constant. See Velocity, surface of constant radial
Radial wave function, 89, 92-93 Radiation, solar: center-to-limb variation, 5, 61-63,
260, 262-263, 315, 442-446, 470 dielectronic recombination, 136 energy transport in photosphere, 47, 62
partial redistribution in lines,
442-446 radiation pressure, 170 residual flux of lines, 313 resonance broadening of Ha, 287 semi-empirical model atmospheres,
258-266 spectrum, 260, 265, 362-367,
372-373, 380-384, 389-390,
442-446, 458-459, 469-471 temperature and electron density of
outer layers, 124 See also Sun Radiation field: anisotropic and non-PIanckian in
stellar atmosphere, 121 dilution factor, 120-121, 123-125,
127, 239-240, 244, 480, 486-487 effect of absorption and scattering,
20-23
energy density defined, 6 energy equation, 44 flux defined, 9-10 interior point of semi-infinite
atmosphere, 38, 50-51 macroscopic description using
specific intensity, 3 mean radiation pressure, 13 microscopic description using
photon distribution function, 4 momentum density defined, 10 momentum equation, 45-46 observational determination of
stellar flux, 11-12 Poynting vector, 11 radiation pressure tensor, 12-17 radiation temperature defined, 121 thermal equilibrium, 6-7 Radiation field, second moment of: approximate closure for solving
transfer equations, 46-47
Radiation field, second moment of (continued) comoving frame method for
velocity fields, 498-499,
502-503, 508 interior point of semi-infinite
atmosphere, 41-42, 50-51 plane wave in extended stellar
envelope, 17 proportional to radiation pressure
(scalar), 16 proportional to variable Eddington
factor, 18
Schwarzschild-Milne equations, 40-43 See also Chi-operator Radiation force, 45-46, 57-58, 170-171, 255-257, 498, 544-547, 554-556, 558-561 Radiation-force multiplier, 561-562, 565
Radiation pressure, mean, 13 Radiation pressure (scalar):
comoving frame, 503, 547-548
definition, 16-17
equals mean radiation pressure in
isotropic radiation field, 13 flux-weighted mean opacity gives
correct radiation pressure in
transfer equation, 57 hydrostatic equilibrium, 170-171,
181, 255-257 non-isotropic radiation fields, 16-17 proportional to second moment of
radiation field, 16 proportional to variable Eddington
factor, 18 thermal equilibrium, 17 Radiation pressure tensor:
comoving frame, 497-499, 546-548 definition, 12
expressed by scalar radiation
pressure and energy density in one-dimensional atmosphere, 16-17
negative of Maxwell stress tensor, 14-15
relation to volume force, 13-14 See also Radiation field, second moment of Radiation stress tensor, 12. See Radiation pressure tensor
Radiative equilibrium, 47-49 complete linearization method for
LTE model, 180-185 complete linearization method for
non-LTE model, 230-234 conditions for, 47-48 extended grey atmosphere, 245-246 grey atmosphere problem, 54 in early- and late-type stars, 47, 62 partial linearization method for
continuum, 226-227 picket-fence model, 207-212,
215-216 temperature structure for LTE model atmosphere, 171-180, 184-185, 205-216, 239-240 temperature structure for non-LTE model atmosphere, 238-243 Radius, Bohr, 86, 89, 111, 132 Radius, critical. See Critical radius Radius, Debye, 122, 293-295, 524 Radius, Weisskopf, 283, 286, 288, 303 Random-walk process, 149, 282, 311,
333-336, 366 Rankine-Hugoniot relations, 519-521, 534
Rate equations, 23, 80, 108-109, 127-139, 219, 223, 391 autoionization and dielectronic
recombination, 134-137 collisional rates, 131-134 complete linearization method,
143-145, 233 completely linearized line transfer,
397-398, 400-401 complete non-LTE rate equations,
137-139 continuity equation, 128 equivalent-two-level-atom method,
376-377, 379-380 non-LTE continuum, 225 non-LTE rate equations, 127-139 partial redistribution, 433-438 radiative rates, 128-131 two-level-atom model, 337 two-level-atom model with
continuum, 359 Wolf-Rayet stars, 486-487 Ratios, branching, 142, 143, 385 Recombination, collisional, 22, 132, 141
626 Index
Index 627
Recombination, collisional (continued) collisional vs. radiative recombination, 124-125 ionization equilibrium, 125 net collisional bracket, 132 Recombination, dielectronic, 125,
134-137, 141,265, 408 Recombination, radiative, 22, 94-96, 122, 130-131, 141 induced recapture probability (of
electrons), 94 ionization equilibrium, 125 radiative vs. collisional recombination, 124-125 spontaneous recapture probability
(of electrons), 94 spontaneous recombination, 130. See also Saha ionization equation stimulated recombination, 131 See also Recombination, collisional and Recombination coefficient Recombination coefficient, 131, 141-142
Redistribution, angle-averaged, 28-29, 451, 508 vs. angle-dependent, 422 application to line transfer, 433-446 definition, 423 electron scattering, 432 general formulae, 424-427 specific cases, 427-433 symmetry properties, 432-433 Redistribution, complete, 29, 80, 127, 159, 309, 335, 411, 415, 419, 420, 424, 430, 435, 438-439, 450-451 in laboratory frame, 419, 424, 427, 429, 430, 435, 439-442 Redistribution, partial, in transfer equation, 310, 433-446, 490, 504, 508
Redistribution by Doppler shifts, 21, 411-412 angle-averaged, 421-433 general formulae, 424-427 specific cases, 427-433 electron scattering, 420 general formulae for full angular and frequency dependence, 415-418 applications, 422 specific cases, 418-422
partial redistribution in transfer
equation,433-446 symmetry properties, 420-422 Redistribution function, 21, 27-30, 310, 411-433. See also Absorption profile and Emission profile Redistribution in atom's frame, 21, 29, 412-415
coherent, 30, 413, 415, 417-418, 419,
421, 425-426, 427-429, 434, 439 noncoherent, 29, 413-415, 420, 424,
427, 430-432, 434-435 See also Redistribution, complete,
Redistribution function, and
Redistribution in the laboratory
frame
Redistribution in the laboratory frame, 21, 27-29, 415-446 angle-averaged redistribution functions, 28-29, 423-433 general formulae, 424-427 specific cases, 427-433 symmetry properties, 432-433 coherent, 30, 424, 429, 434, 442 Doppler-shift redistribution, 415-422 applications, 422 general formulae, 415-418 partial redistribution in transfer
equation, 433-446 specific cases, 418-422 symmetry properties, 420-422 partial redistribution in transfer equation, 310, 433-446, 490, 504, 508 application to solar and stellar
resonance lines, 442-446 idealized models, 438-442 methods of solution, 436-438 two-level-atom, 433-436 Reflector effect, 358, 462 Relaxation time, 122 Residue theorem, 278 Retardation effect, 499-500 Reversing layer, 312 Reynolds number, 560, 567 Riemann zeta-function, 7 Rosseland's theorem of cycles,
142-143, 384 Rybicki's solution (of transfer equation), 158-161, 184-185, 227, 233, 255, 339, 340, 378,
Rybicki's solution (of transfer equation) (continued) 392-393,400-401,451-452, 459-461, 504-508
Rydberg constant, 89, 91, 93, 98-101
Saha formula. See Ionization equation, Saha
Saturation function, 326
Scale height, pressure, 188, 256, 260,
470, 524 Scattering (of photons): absorption profile, 27 angle dependence, 25 causes lambda-iteration to fail,
147-150, 217, 224, 343 classical line transfer, 308-312,
314- 316, 346
by classical oscillator, 83-84 coherent scattering, 29-30 complete noncoherence, 29 complete redistribution, 29 Compton scattering, 21 continuum, 106-107, 148, 165-166,
315- 316 definition, 20-21 emission profile, 27-29 noncoherent scattering term in
Lyman continuum source function, 223, 226 Rayleigh scattering, 21, 106-107, 148 redistribution function, 27-30 scattering coefficient, 24, 27, 29-30 Thomson scattering, 21, 106, 148 Scattering coefficient, 24, 27, 29-30 Scattering, Compton, 21 Scattering, dipole, 30, 106, 412, 425, 428-430
Scattering, electron. See Scattering,
Thomson Scattering, isotropic, 30, 35, 106, 147,
412, 425
Scattering, Rayleigh, 21, 106-107, 148 Scattering, Thomson, 21, 106, 148, 554 Scattering efficiency. See Oscillator,
harmonic Schrodinger's equation, 85-87,
299-302
Schuster mechanism, 258, 315-316 Schuster-Schwarzchild model (line
transfer), 312 Schwarzchild-Milne equations, 40-43
Shielding, Debye, 292-295, 303 Shift, Doppler, 23-25, 411-412,
447-449, 449-461, 461-463,
463-471, 471-485, 490-493,
495-510, 550-553, 553-561,
567-568
differential Doppler shift, 453-455,
456-457, 478-485, 491-493, 499-503 See also Line-broadening, Doppler Shift, line. See Line shift Shift, phase (in pressure broadening of
lines), 281, 283-286, 289 Shock, stationary. See Shock front Shock front, 169, 462, 519-521, 528 radius of, for solar wind, 533-534 Simpson's rule, 65 Single-Gauss formula. See Gaussian
formula Slater determinant, 92 Snefi's law, 4
Sobolev's method (for velocity fields), 448, 471-490
application to WR spectra, 485-490
opaque envelope, 478-485
surfaces of constant radial velocity, 472-478 Sonic point, 558, 562-563
See also Critical radius Sound, speed of, 518-519, 527, 556 Sound waves. See Waves, acoustic Source function:
classical line, 309-310
collision-dominated line, 361-362
continuum, examples of, 35, 147, 219, 226
definition, 35
dependence upon radiation field, 43 Eddington-Barbier relation, 39 effects upon radiation field, 42 explicit form of, 81, 337, 359,
'376-378, 391-392 for partial linearization, non-LTE
continuum, 223, 226 frequency-independent, 336 implicit form of, 80, 336 line source function, 80, 336-337,
358-360, 376-378, 386-388,
391-392 LTE, 80
Lyman continuum, 222-226 moving atmosphere, 449-452, 481-482, 505
628 Index
Index 629
Source function {continued) photo ionization dominated line, 361-362
specific intensity is Laplace transform of, 39
transfer equation with sample
source functions, 35-36, 39, 42
See also Emission (thermal),
Opacity, and Source function, line
Source function, line, 80, 241
boundary value and depth variation, 343-348, 353-358, 362-367, 368-370, 382, 385-388, 393-396, 439-442, 442-446, 456-459, 461, 482, 508-510
in a chromosphere. 362-367, Table 11-3
collision dominated, 361-362, 363-365
equivalent-two-level-atom method, 376-380
explicit and implicit forms, 80-81.
336-337, 359 frequency-independent source
function, 336 integral equation form. 339-340 iterated source function. 439 moving atmosphere, 449-452,
459-460, 481-482, 505 multiplets, 387-388, 391-392 net radiative bracket. 129-130,
376-377, 379-380 in partial redistribution, 435-438 photoionization dominated,
361-362, 363-365 "reduced" source function, 386 reflector effect, 358 source function equality, 384-390,
393-396
thermalization depth, 341-343, 352,
Table 11-1 two-level-atom with continuum,
358-362
two-level-atom without continuum, 336-338
Spectrum, energy (of an oscillator),
275-276, 281-282 Spectrum, power (of an oscillator), 275 Spectrum lines. See Lines, spectral Spectrum synthesis (with LTE model atmosphere), 328-331
Spherical harmonic function, 89, 91 Sphericality factor (extended spherical
atmospheres), 251-252 Spline colocation, 155, 252 Stability criterion, Schwarzchild,
186-187. See also Mixing-length
theory
Stabilizing transition probability (for dielectronic recombination), 135-137
Standard stars. See Calibration,
fundamental Stark components, 295-297 Stark effect, 283, 287, 290-291,
295-297, 303-306 Stark pattern, 296, 303-305 Stars:
A-, 163, 166-167, 185, 187, 193, 194, 204, 234, 237, 402, 540, 552
Ap-, 200-203
B-, 166, 193, 195, 196, 198, 200, 220. 228, 234-236, 238. 240, 255, 306. 328-329, 402-406, 523, 550-553
early-type stars, 166, 185, 187, 192, 198, 218, 241-243, 287, 354-355, 540
F-, 185, 187, 191. 198. 536, 540
G-, 166, 204. 239, 266, 326-327, 523
K-, 166, 239, 326, 523
late-type stars, 166-167, 185, 187, 205
M-, 166, 187, 191, 192, 239, 523, 539
0-, 166, 170-171. 193, 194, 198, 220. 221, 234, 237-238. 240, 255-256, 306, 316, 362, 402-409. 471, 474, 523, 550-555, 557, 560, 563, 566, 567
P-Cygni. See P-Cygni (line characteristics)
planetary nebula stars, 255-256
Population II, 326-328
Procyon, 193, 196, 197
Sirius, 330
solar temperature and cooler stars, 166-167, 240-241, 258-267, 287,324, 331-332, 355, 361-362, 364-367, 446, 536
supergiants, 170, 192, 234-238. 243, 249, 255-256, 266, 323, 448. 471, 523, 539-540,553, 567
Stars (continued) T-Tauri, 266
Vega, 196,197, 198. 220, 237-238. 329,330
WC and WN stars, See Wolf-Rayet stars
WR-, 243, 249, 256, 266, 367, 448, 471, 472, 474, 477. 485-490, 523, 550 dc Leo, 200 a*CVn, 202-204 y Peg, 329 9 Aur, 200, 201 C Pup. 552-553, 559, 563, 565 State, equation of, 114-119. 140-145 Stationary states. See Eigenstates Statistical equilibrium equations. See
Rate equations Stefan-Boltzmann constant, 12 Stefan's law, 7 Stochastic equation, 464 Stress-energy tensor, 497-498, 542-543 Stress tensor, Maxwell. 14-15,497-498 Stress tensor, radiation. 12. See
Radiation pressure tensor Successive over-relaxation method (SOR), 401
Sun:
central temperature, 52 chromospheric temperature rise,
259-260, 264-266, 362-367 coronal wind, 258, 266-267, 521-540 curve of growth abundances.
324-326 differential abundance analyses,
326-328 ET opacity source. 76, 102-104,
260, 263
ionization of atmosphere, 115, 266,
Table 7-1 limb darkening, 260, 262-263 magnetic field, 534-535 photosphere, 258, 260, 263-265 radiation pressure, 170-171 solar chromosphere, 122, 258-260,
263-267, 348, 362-367, 372,
442-446
equivalent-two-level-atom method, 3S0-384
solar model atmospheres, 258-267 temperature structure, 241, 258-261, 263-266
solar wind, 258, 266-267, 521-540 temperature minimum, 241,
258-259. 266, 444 turbulence, 324, 469-470 See also Chromosphere, Corona, and Limb-darkening law Surface, constant radial velocity. See Velocity, surface of constant radial
Taylor's expansion, 155 Temperature: effects of lines on surface temperature, 205-212, 215-216 effects upon two-element gas, 116 extended grey atmosphere, 245-247 grey atmosphere vs. nongrey atmosphere surface temperatures, 205-206, 212-214 in LTE, 109-110
non-LTE model atmosphere, 239-243 Temperature, color. 248-250, 256 Temperature, effective, 48-49. 52. 59,
73, 172, 175, 181, 190-193,
195-206, 212-215, 234-243,
328-330, 402-409, 536 Temperature, electron, 121-123,
359-367, 371-372, 382, 442-444,
487-490, 521-522, 537, 553 Temperature, excitation (curve of
growth), 322-323, 326-327 Temperature, kinetic, 121-123, 359,
362
Temperature, radiation, 121, 123-125,
141, 239-241, 360-365 Temperature correction procedures: Avrett-Krook procedure, 174-175, 220
constraint methods, 176-180, 184-185
discrete ordinates method, 74-75 extends grey to nongrey solution. 60 lambda-iteralion, 62*-63, 172-173 in solving LTE radiative transfer
equation. 171-176 Unsold procedure, 63-64 Unsold-Lucy procedure, 174-175 See also Radiative equilibrium Temperature distribution:
complete linearization method
(LTE), 180-185, (non-LTE),
230-234
630 Index
Index 631
Temperature distribution (continued) construction of LTE model, 180-185 convective atmosphere, 190-191 extended grey atmosphere, 245-247 srey atmosphere, 54-55, 61-64. 68, 72
line strengths determine distribution.
206-212, 215-216 mean opacities with temperature
correction procedures, 60 non-LTE atmospheres, 239-243 partial linearization solution of
non-LTE continuum, 226-227 radiative equilibrium, 172 results from LTE model, 205-216 solar distribution, 240-241, 258-265, 380-382, 444 with Eddington approximation for
grey atmosphere, 61-63 with Rosseland mean opacity, 58 Temperature minimum (non-LTE
atmospheres), 239-241, 266, 444 Temperature rise (chromospheric), 259-260, 264-266, 362-367, 372-373, 382, 456 Thermaiization (of electrons), 121-122
Thermaiization (of photons), 20. 333-336 colHsional de-excitation, 22 See also Coupling parameter, thermal and Thermaiization depth
Thermaiization (of spectral lines), 130, 311, 334-336, 342-346, 353-356, 384-388 See also Thermaiization depth Thermaiization depth, 141, 149-150, 311, 334-335, 341-342, 352-355, 387-388, Table 11-1 Thermodynamic equilibrium: Boltzmann excitation equation, 110-112
dependence of temperature upon
thermal emission, 49 description of, 25-27 detailed balancing, 79, 94, 119-120 emission and absorption coefficients, 26 Kirchhoff-Planck relation, 26, 80, 96 Maxwellian velocity distribution, 110 microscopic requirements of, 139-127
occupation numbers of energy
levels, 79, 110-112 See also Local thermodynamic equilibrium and Non-local thermodynamic equilibrium Thickness, atmospheric, 256 Thickness, optical, 36, 38, 346-349, 366, 479^180, 489. 558, 560-561 Three-level-atom, 384-388 Time-development operator, 298-303 Time-dilation effect, 494 Transfer equation, 30-31 axially symmetric medium, 34 as a Boltzmann equation, 32-33 boundary conditions, 36-37, 153 classical line transfer, 309-310 closure problem, 46-47 comoving frame method, 490-510 solution for spherical symmetry,
503-510 transformation into moving frame. 491-493, 495-497, 499-503
complete linearization method for
LTE, 181-182 completely linearized line transfer,
396-401
with constraint of radiative equilibrium, 176-180
difference equation methods for numerical solution, 153-161, 503-510
Eigenvalue methods unsatisfactory, 150-151
with Einstein coefficients for a line,
92, 336, 376-378 equation for non-LTE continuum,
219, 225-226. 232-233 equation for spectral line, 272 equivalent-two-level-atom method,
376-380
examples of transfer equations for planar atmospheres, 37-38, 39-40
extended grey atmosphere, 245-246 extended nongrev atmosphere, 244, 250-255
Feautrier's solution, 156-158, 161, 233, 252, 398. 436, 438, 504, 508
formal solution of. 38-39 integrating factor for. 38
Transfer equation (continued) grey, 54-56
interlocking multiplets, 391-393 lambda iteration unsatisfactory, 147-150
lower boundary condition, 36-37, 51
nongrey transfer equation in LTE, 35-36, 75
non-LTE line transfer with continuum, 359-360
non-LTE line transfer without continuum, 336-340, 350-352
numerical solution, 153-161, 252-255, 392-393, 397-401, 435-438, 449-452, 459-461, 503-510
observer's frame method for velocity fields, 449-471 formal solution. 450 spherical expanding atmosphere,
459-461 turbulent atmosphere, 463-471 one-dimensional planar atmosphere
(time dependent), 32, 43 partial redistribution, 435-438 for photoionization, recombination, 95
picket fence, 208-209, 215-216,
462-463 Rybicki's solution. 158-161, 184-185, 227, 233. 255, 339, 340, 378, 392-393, 400-401, 451-452. 459-461,504-508 second-order form, 151-153 solution at interior point of semi-infinite atmosphere, 38, 50-51 solution by moments, 46-47 solution for emergent intensity, 38-39
solution in spherical geometry,
250-255 source function, 35 spherically symmetric medium,
33-34
in static medium, 30-32, 33-34 time dependent, 31 time independent equation, 32 two-point boundary value solution,
151-361 See also Transfer equation,
moments of Transfer equation, moments of, 43-47 comoving frame, 502-503
extended atmosphere, 244, 250-252, 254
combined moment equation, 251-252 first-order moment, 44-46 grey atmosphere, 54-56 to solve transfer equation, 46-47,
157-158 zero-order moment, 43-44 Transform, Fourier, 275, 278, 280, 298, 303
energy spectrum of an oscillator,
274-277 reciprocity relation, 275 Transformation, Lorentz, 493-495,
497-499, 542-544 Transitions, atomic
autoionization and dielectronic
recombination, 134-137 bound-bound, 22
dipole transition selection rules, 134 Einstein coefficients, 77-79 non-LTE collisional rates, 131-134 non-LTE radiative rates, 128-131 probability calculations, 81-93 Transitions, bound-free, 22. See Pliotoionization and Ionization, collisional Two-level-atom models, 214, 227 with continuum, 358-367, 371-373 without continuum, 336-358 moving atmosphere, 450, 459,
481, 505 partial redistribution, 433-438 See also Equivalent-two-level-atom method
Two-stream approximation (for
stellar radiation field), 61, 151, 312 definition, 18
Units. Doppler, 279-281, 317-318, 338 ff., 418, 427, 449-450, 479
Units, thermal. See Units, Doppler
Unsöld-Lucy procedure (temperature correction), 174-175
Unsold procedure for temperature correction, 63-64
Variable Eddington factor, 63, 153, 161, 244, 245-246, 397, 435, 508 approximate closure for solving transfer equations, 47
632 Index
Variable Eddington factor [continued) complete linearization method,
181-185, 233-234 definition, 18
differential equations technique for spherical atmospheres, 250-254 Feautrier's solution of transfer
equation, 157-158 interior point of semi-infinite atmosphere, 51 Velocity, critical, 526, 527-529 Velocity, surface of constant radial,
472-478, 479, 482-485 Velocity distribution, electron, 121-123 Velocity distribution, Maxwellian, 110,
119, 121-123, 279, 323 Velocity field: energy balance, 461^163 observer's frame method for lines
in planar atmospheres, 449-459 Sobolev's method for spherical expanding atmospheres, 471-479 spherical atmospheres, 459-461,
503-510 stellar winds, 526-534, 562-564 transfer equation in the fluid frame,
490-493, 499-503 turbulent atmospheres, 463-471 Velocity field, random. See Atmospheres, turbulent Velocity field, stochastic. See Atmospheres, turbulent Voigt function, 279-281, 286, 305, 317-318, 320, 338, 340, 342, 346, 347, 349, 354, 366, 419-420, 464-467
WC stars. See Wolf-Rayet stars WN stars. See Wolf-Rayet stars Wave equation, 518
Wave function, 84-85, 277, 297-303 approximate, for light elements, 91-92 approximations for helium, 105 calculation of continuum absorption
cross-sections, 97 for hydrogen, 89 Waves, acoustic, 265, 458-459, 519-521, 539-540 speed of sound, 518-519 wave equation, 518 Waves, Alfven, 535 Waves, magnetohydrodynamic, 265 Weighting functions, method of, 326 Weights, statistical, 77, 79-80, 88, 97, 110-114
Weisskopf approximation, 282-284, 286
Width, Doppler (of a line), 279, 305,
317, 323, 338, 356-358 Width, equivalent (of spectral line), 114-115, 270, 316-328, 372-373, 402-404, 406-407, 409 turbulent atmosphere, 465-470 Wolf-Rayet star, 482-485, 487-490 Wilson-Bappu effect, 365-367 Wind, radiation driven, 549-568 dynamics, 553-563 Of stars, 559-568 Wind, solar, 521-540, Table 15-1 radius of shock front, 533-534 rotational braking, 534-536 spherical coronal wind models, 525-533, 536-540 breeze solutions, 527, 530, 532 isothermal corona, 526-529 two-fluid models, 537 Wind, stellar, 447-448, 463, 471-472,
475, 511-568 Wings, line. See Line wings Wolf-Rayet stars, 243, 249, 256, 266, 367, 448, 471, 472, 474, 477, 485-490, 523, 550