is 8S = c_1J(r, n, v, t) da) dv dV; by integrating over all solid angles and over the entire volume, we find the total energy contained in V, namely:
@{x, v, t)(/v = (T1
jvdvj)do)I{r, n, v, f)
dv
(1-6)
But if we pass to the limit of infinitesimal V, I becomes independent of position in V, and the integrations can be carried out separately. The monochromatic energy density, ER{r, v, t) = S\r, v, t)/V is thus
ER(r, v, t) = c~1 (j) /(r, n, v, t) dw = (4n/c)J(r, v, i) (1-7)
ER has dimensions of ergs cm"3 hz"1. The total energy density (dimensions: ergs cm"3) is found by integrating over all frequencies:
£„(r, t) = EK(r. v, t) dv = (4n/c) j* J(r, v, t) dv = [4n/c)J(r% t) (1-8)
PHOTON PICTURE
It is easy to show that the results derived above are consistent with the photon picture of the radiation field. By definition,/fi(r, n, v, 0 is the number of photons, per unit volume, of energy hv propagating in direction n into intervals dv dco. The energy density clearly is just this number, multiplied by the energy per photon, summed over all solid angles: i.e.,
ER(r, v, i) = hv (j)_/R(r. n, v, í) doj
(1-9)
But from equation (1-2), hvfR = c 1I, hence equation (1-9) is seen to be identical with equation (1-7).
EQUILIBRIUM VALUE
In thermal equilibrium the radiation field inside an adiabatic enclosure is uniform, isotropic, time-independent, and has a frequency distribution
given by the Planck function 8,(7) = (2hv3/c2){ehv/kT - l)"1 [see (520). (392, 365)]. Thus, in thermal equilibrium the monochromatic energy density is E%(v) = (4n/c)Bv(T\ and the total energy density is given by Stefan's law:
E$ = (Znh/c3) ^
,hv/kT _
1)~V dv = aRTA
;i-10)
where aR = 87[3/<4/(i5c'3/i3). Here, as elsewhere in this book, we denote a quantity computed from thermodynamic equilibrium relations with an asterisk.
Exercise 1-3: Derive Stefan's law by substituting x = liv/kT, and expanding - I)-1 — e~x(l — e"A)_1 as a power scries in e"\ The sum obtained from the term-by-term integration is related to the Riemann zeta-function [see (4, 807)].
Stefan's law is valid in the interior of a star, and in the deeper layers of stellar atmospheres, where thermal gradients over a photon mean-free-path are extremely small, and the radiation becomes isotropic and thermalizes to its equilibrium value. At the surface, the radiation field becomes very anisotropic and has a markedly non-Planckian frequency distribution, as a result of steep temperature gradients and the existence of an open boundary through which photons escape into interstellar space; here Stefan's law becomes invalid.
ELECTROMAGNETIC DESCRIPTION
Electromagnetic theory provides an alternative description of the radiation field; we shall show how a one-to-one correspondence can be made between the macroscopic and electromagnetic descriptions of the radiation field. The electromagnetic held is specified by Maxwell's equations [see, e.g., (331, Chap. 6)] which, in Gaussian units, are
and
V ■ D = 4np
V -B = 0
(V x E) + c-1 (dB/dt) = 0 (V x H) - c-l(dB/dt) = (4jt,V)j
(Mia) (1-1 lb) (1-llc) (Mid)
The electric field E is related to the electric displacement D in terms of the permittivity s, namely D = eE. Similarly, the magnetic induction B can be expressed in terms of the magnetic field H and the permeability p. by the relation B - pH. For vacuum, e = p = 1. In equations (1-11), p is the charge density and j is the current density j = pv associated with charges moving with velocity v. The electric field and magnetic induction can be derived from a scalar potential d> and a vector potential A, which are defined
8 The Radiation Field
!l
such that ' i
B = Vx A (i-12a).
and E = -\4> - c-'(d\/dt) (l-12b)
Equation (l-12a) satisfies equation (1-1 lb), while (l-12b) satisfies (1-1 lc). Because B is defined as the curl of A, the divergence of A may be specified arbitrarily; one of the most convenient choices is to impose the Lorentz ;
condition \
V ■ A — -c-\d(j>/dt) (1-13)
With this choice, Maxwell's equations can be reduced to i
V> - c-2(d2 |r — r |
where, as indicated, p and v at r' are evaluated at the retarded time f = \ t - c-1|r' - r[ which takes into account the finite speed of propagation of electromagnetic waves.
One of the most important solutions of Maxwell's equations is that for monochromatic plane waves in vacuum, propagating in direction n0 with velocity c:
E(r, t) = E0 cos[27i(/cn0 ■ r - vt)] (l-16a)
and H(r, /) = H0 cos[27t(/m0 ■ r - vt)] (l-16b)
where k = l~x = c_1v. The vectors (E0, H0, n0) form an orthogonal triad withH0 = n0 x E0, so it follows that |H0| = |E0|. The result obtained from electromagnetic theory for the instantaneous energy density W(t) in the field is
W{t) = (ED + B-H)/8ti (1-17) ;
Averaging in time over a cycle introduces a factor of !>r = h and using the relations |£0| = \H0\ and /i = e = 1 (for vacuum), equation (1-17) 1 reduces to W = (W{t)yr = E02/8k. In terms of the macroscopic picture, a monochromatic plane wave propagating in direction n0 [specified by angles (60, <£0)] has a specific intensity /(/*, ) = I0 5{fi - pQ) 5( - 0) where 5
1-3 The Flux 9
denotes the usual Dirac function. Substitution into equation (1-7) yields the energy density ER = c-170, a result that is intuitively obvious for a plane wave propagating with velocity c. Therefore we obtain a correspondence between the two descriptions by making the identification
I0 = c£02/8tc (1-18)
It will be shown below that this choice yields consistent relations between the Poynting vector and Maxwell stress tensor and their macroscopic counterparts. The results derived here apply, strictly, only to a monochromatic plane wave, but are easily generalized to fields having arbitrary angle and frequency distributions by summing over suitably chosen elementary plane waves.
1-3 The Flux
MACROSCOPIC DESCRIPTION
We define the flux of radiation J^(r, v, t) as a vector quantity such that ^ ■ dS gives the net rate of radiant energy flow across the arbitrarily oriented surface dS per unit time and frequency interval. Noting that n dS = dS cos 6, where 9 is the angle between the direction of propagation 11 and the normal to dS, we immediately recognize that the flux can be derived from the specific intensity via equation (1-1), for 6$ as written there is, in fact, nothing more than the contribution of the pencil of radiation moving in direction n to the net energy flux. Thus we merely sum over all solid angles and obtain
^(r, v, t) = (j) 7(r, n, v, f)n dco (1-19)
The flux has dimensions: ergs cm"2 sec"1 hz"1. Note that ^ is the first moment of the radiation field with respect to angle. In cartesian coordinates we have
(1-20)
where dco = -dpi d(f>, nx = (1 - p2)* cos , ny = (1 - fi2)'2' sin , nz - fi. If the radiation field is symmetric with respect to an axis, it follows that there will be a ray-by-ray cancellation in the net energy transport across a surface oriented perpendicular to that axis, and that the net flux is identically zero across this surface. In particular, for a planar atmosphere homogeneous in x and y\ only J\ can be nonzero; we shall therefore require only this component of the flux, and shall refer to it as "the" flux, as if it were a scalar,
r
10 The Radiation Field
1-3 The Flux 11
and write
&{z, v, t) = 2tt I{2,n,v,t)fidn
(1-21)
Exercize 1-4: (a) Show that and J5",, vanish in an atmosphere with azimuthal (0) independence of 1. (b) Show that in a spherically symmetric atmosphere only #V is nonzero and is given by equation (1-21) with z replaced bv r. (c) Evaluate for ~ J»/f": show thai only the odd-order terms contribute to .J7.
In astrophysical work it is customary to absorb the factor of n appearing in equation (1-21), and to write the astrophysical flux as F(z, v, t) = n~ l,^{z, v, t). Further, regarding the flux as one of a sequence of moments with respect to ji, one may define the Eddington fiux
H(z, v. t) = (47r)"'.^(z, v, 0 = \ f1 Hz, t)n d\i (1-22)
2 J '
which is in a form similar to equation (1-5) for the mean intensity.
PHOTON ENERGY FLUX
The same results for the energy flux may be obtained from the description of the radiation field in terms of photons. The net number of photons passing, with velocity c, through a unit surface oriented at angle 0 to the beam, per unit time, is clearly
N[r, v, t.) = c ij) fR{r, n, v, t) cos 0 dio (1-23) Each photon has energy hv, so the net energy transport must be
Xr. v, t) -- [chv)j)fR{r, n, v, t)n dio
(1-24)
In view of equation (1-21 equation (1-24) is obviously identical to equation (1-19).
Furthermore, photons of energy hv propagating in direction n have momentum hvn/c. Thus il is clear that c"1^ ■ dS dt gives the net momentum transport across the surface dS in time dt, by particles moving with velocity c. It therefore follows that the momentum density associated with the radiation held is Gfi — c~2.^; we shall find further significance of this result in §2-3 and shall use it in §14-3.
Exercise 1-5: Verify the assertion that c~ check units for consistency.
2.t/' represents a momentum density;
THE ROYNT1NG VECTOR
In electromagnetic theory, the energy flux in the field is given by the Poynting vector
S = (c/4jt)1E x H) ■ (1-25)
Considering a plane wave as in §1-2. the average power over a cycle is r = cr/47r - (c7. Again, this result can be generalized to arbitrary angle and frequency distributions of the radiation field.
OBSERVATIONAL SIGNIFICANCE
The energy received from a star by a distant observer can be related directly to the flux J\ emitted at the stellar surface. Assume that the distance D between star and observer is very much larger than the stellar radius r^. so that all rays from star to observer may be considered to be parallel. The energy received, per unit area normal to the line of sight, from a differential area on the star is c//v = dto where dco is the solid angle subtended by the area, and lv is the specific intensity emergent at the stellar surface. Considering the geometry shown in Figure 1-3 we see that r = sin0 so that the area of
To observer
figurl 1-3
Geometry of measurement of stellar flux. The annulus on the surface of the star has an area dS = 2nr dr = Inl sin 0 cos 0 cW normal to the line of sight; thi.s area subtends a solid angle do = dS/D1 as seen by the observer.
12 The Radiation Field
1-4 The Radiation Pressure Tensor 13
a differential annul us on the disk is dS = 2nr dr = 2nr.i2pi dpi, and- dm = 2n(r.M/D)2pi dpi. The radiation emitted from this annulus in the direction of the observer emerged at angle 0 relative to the normal; hence the appropriate value of the specific intensity is I(r^, pi, v). Integrating over the disk, we find
A = MrJDf £ /(/V ^ % dpi = {rJD)\nr^ v) = l- a,2^(r„ v) (1-27)
where x% is the angular diameter of the star. [In the above calculation we have assumed there is no radiation incident upon the surface of the star; i.e., /{r^, — pi, v) = 0.] For unresolved objects (e.g., stars), we can measure only the flux. The energy received falls off as the inverse square of the distance (because the solid angle subtended by the disk varies as D~2). If the angular diameter is known, then the absolute energy flux measured at the earth can be converted to the absolute flux at the star.
Exercise 1-6: Show that the flux emergent from a small aperature in an adiabatic enclosure (blackbody) is y*m{v) = nBy{T). Show that the integrated flux is ^B — a*T4 where |
0) we must also specify the polarization of the wave via the angle i//0. Here t//0 measures the angle of rotation of E around S from the plane through n0 and k (the unit vector in the z-direction); see Figure 1-4. It is easy to see that
Ex = £0(sin ij/0 sin 0 — cos \(/Q cos T - -(£02/8ti) cos2 Q0 which is indeed — PS3; note that the final result is independent of ijj0.
Exercise 1-8: Calculate the remaining components of Tw and show that TM = — P, independent of ij/0.
The above results demonstrate that a complete correspondence exists between electromagnetic theory and the macroscopic or photon descriptions of (he radiation field; we shall exploit this correspondence in a useful way in §§14-3 and 15-3 where we will be able to use the known Lorentz-transformation properties of electromagnetic field quantities to establish those of their radiation-field-description counterparts.
16 The Radiation Field
1-4 The Radiation Pressure Tensor 17
LIMITING CASES! SYMMETRY, ISOTROPY, EQUILIBRIUM, PLANE WAVES
In a one-dimensional planar or spherically-symmetric atmosphere the radiation field is azimuthally invariant, hence the pressure tensor becomes diagonal
'Pr 0 0 Pír, v, t) = I 0 pR 0
where
and, in turn,
v0 0 pR
Pr(z, v, 0 = (4tz/c)K(z, v, t)
1 fi
3pR - ER 0 0^
0 3pR - ER 0 ] (1-40)
0 0 0,
K(z,v, r^-J^/fc^ v, t)p2 dp is the second moment of the radiation held in Eddington's notation.
(1-41) (1-42)
Exercise 1-9: (a) Derive equation (i-40) for the conditions stated, (b) Show that the same expression for P is obtained in spherical symmetry relative to the orthogonal triad (, 9, f). (c) Verify that the tensor shown in equation (1-40) is consistent with equation (1-32).
It is clear from equation (1-40) that, for a one-dimensional atmosphere, only two scalars (pR and ER) are sufficient to specify the full radiation pressure tensor. Further, for such atmospheres, derivatives with respect to (x, y) or (0, in the planar and spherical cases respectively, are identically zero, and the only nonvanishing components of the divergence of the radiation pressure tensor are
(V ■ P)z = dpR(z, v, t)/dz (l-43a)
in planar geometry—or, in spherical geometry,
(V ■ P)r = [dpR(r, v, t)/dr] + [3pK(r, v, /) - ER(r, v, l)]/r (l-43b)
In this book we shall confine attention strictly to one-dimensional problems, and with the exception of further formal development of the equations of radiation hydrodynamics in §15-3, the full tensor description of the radiation field will not be required; for ease of expression, we shall therefore refer to the single scalar pR as "the" radiation pressure.
Exercise J-10: Show that, for any diagonal tensor A, in spherical coordinates (V-A), = (cAJdr) + (2A„ - A00 - A^/r; use this result to derive equation (l-43b).
In general, pR as defined by equation (1-41) will not equal P defined by equation (1-32), but the two become equal if the radiation field is isotropic. For an isotropic field, / is independent of p, and equations (1-7) and (1-41) immediately yield pR = ^ER, so that equation (1-40) reduces to
P(r, v, t)
(1-44)
That is, when the radiation field is isotropic, the radiation pressure tensor is diagonal and isotropic, and may be replaced, for all purposes of computation, by a scalar hydrostatic pressure, or even eliminated entirely in terms of ER. If in addition to being isotropic the radiation field has its thermal equilibrium value, then the monochromatic radiation pressure is
pt(z, v, ř) = i E*{z, v, t) = (4n/3c)Bv(T)
and the total radiation pressure is
■ftz, i)=
(1-45)
;i-46)
a result first obtained by thermodynamic arguments (160, 55; 565, 123).
From equations (1-7) and (1-41) we see that pR is an average of I(p) weighted by p2 whereas ER is a straight average. If the radiation field becomes peaked in the direction of radiation flow out of the atmosphere, the larger values of u become more heavily weighted in pR (recall p = 1 for 6 = 0) and pR will exceed its isotropic limit of ^ER. The most extreme departure from isotropy occurs when the radiation flows in a plane wave. For a wave in the outward direction we can write I(z, v, p) = I(z, v) 8(p — 1); then K(z, v) = j(z. v) = H(z, v), and pR(z, v) = ER(z, v). This extreme limit is approached in the outermost layers of very extended stellar envelopes (or in nebulae) in which the radiation field originates from a stellar surface that occupies only a very small solid angle as seen from the point in question.
Exercise 1-11 \ (a) Show that for a plane wave moving along one of the coordinate axes, the radiation pressure tensor has only one non-zero component, (b) Show that the pressure tensor is isotropic if the angular dependence of the radiation held is given by I(p) = I0 + I^p. This result is important because the radiation held is accurately described by an expression of the stated form in the diffusion limit which obtains at great depth in the atmosphere (cf. §2-5). Exercise 1-12: Suppose an observer is at distance r from the center of a star of radius ri; which has a uniformly bright surface (i.e., / is independent of p). Derive analytical expressions for J, H, and K in terms of 0^ = %m~\rjr), that in the limit (77VJ -» 00, J = H = K.
18
The Radiation Field
VARIABLE EDDINGTON FACTORS
From the results derived above it follows that the ratio pR(r, v, i.)/ER(r, v, t) or K(r, v, t)/J(r. v, t) is a dimensionless number whose value is fixed by the degree of isotropy of the radiation field, and typically ranges between \ and 1. It will be shown later (§6-3) thai this ratio can be used in certain numerical methods to reduce the number of independent variables in the transfer problem; further, it may be used to effect a closure of the system of moment equations derived from the transfer equation. It is useful, therefore, to define the variable Eddington factor
The Equation of Transfer
f(r, v, t) = K(r, v, t)fj(r, v, f) or, in abbreviated notation fv = KJJV.
;i-47)
Exerci.se 1-13: (a) Consider an expansion of the form /(/*) = I0 + £„/„/i";show that / = j if the sum includes only odd powers n. (b) Suppose that l(p) ~ Z( for (0 ^ n ^ 1) and I{p) = Iz for (-1 ^ fi ^ 0); show again that / = |. This representation of / provides a rough description (the two-stream approximation) of a stellar radiation field, for we may let J,/^ ->■ 0 at the surface and I2!Ii -> 1 al depth, (c) Show that, for a slab of infinite extent in (x, y) and finite extent in z, / may drop below \.
As radiation passes through the gas composing a stellar atmosphere, it interacts with the material and is absorbed, emitted, and scattered repeatedly. These phenomena determine how radiative transfer occurs in the atmosphere. In this chapter, macroscopic quantities that define radiation-matter interactions are introduced (§2-1), and the equation of transfer (which describes the transport of radiation through the medium) is developed (§2-2). Using this equation, we can compute the emergent spectrum from a star, and calculate how the angle-frequency variation of the radiation field changes with depth in the atmosphere. The time-dependent equation of transfer will be derived in order to obtain moment equations (§2-3) that describe the dynamical behavior of the radiation field, but the discussion will then be restricted to static atmospheres in all subsequent work through Chapter 13. In Chapters 14 and 15 radiative transfer and its dynamical effects in steady (i.e., time-independent) flows will be considered.
20 The Equation of Transfer
2-1 The Interaction of Radial ion with Matter 21
2-1 The Interaction of Radiation with Matter
DISTINCTION BETWEEN SCATTERING AND ABSOKPTION-EMISS10N PKOCESSHS
In the interaction of radiation with matter, energy may be removed from, or delivered into, the radiation field by a wide variety of physical processes. For the present, it is adequate to characterize these processes by macroscopic coefficients; as will be seen in Chapters 4 and 7, these coefficients are specified by atomic cross-sections and occupation numbers of energy levels of the constituents of the stellar material. It is worthwhile, from the outset, to make a distinction between L'true" absorption and emission on one hand and the process of scattering on the other for, as we shall see repeatedly in the development of the theory, the physical nature of the interaction between the atmospheric material and radiation is quite different in these two cases. However, it is also important to realize that in spectral lines the dichotomy between these processes can be established uniquely only when we consider a transition between two specified atomic states, with no coupling to any other states allowed. As soon as sequences of transitions among several interacting states are considered, fundamental ambiguities arise, and it is no longer possible to describe a given line as an "absorption" or a "scattering" line in a rigorous way; nor would it be important or useful to do so. Nevertheless, if is fruitful to have at least an intuitive notion of the contrast between these two basic processes, obtained by consideration of some definite examples.
We may identify as scattering processes those in which a photon interacts with a scattering center (perhaps producing a change in the scatterer's internal excitation state) and emerges from the interaction in a new direction with (in general) a slightly altered energy. The essential point is that in this process, the energy of the photon is not converted into kinetic energy of particles in the gas. In contrast, we shall identify absorption processes as those in which the photon is destroyed by conversion of its energy (wholly or partly} into the thermal energy of the gas. In this process we say that the photon has been thermalized. The crucial physical point to note is that the local rate of energy emission in scattering processes depends mainly upon the radiation field (which may have originated at some other remotely situated point in the atmosphere) and has only a weak connection with the local values of the thermodynamic properties (e.g., temperature) of the gas. Absorption processes, on the other hand, feed photon energy directly into the thermal kinetic energy of the gas. and hence are more intimately coupled to local thermodynamic properties of the material. Conversely, the inverse of absorption, thermal emission, transfers energy from the thermal pool of the gas directly into the radiation field. Thermal absorption and emission processes thus tend to produce local equilibrium between the radiation and material; but scattering
processes allow photons to move from one part of the atmosphere to some other part without coupling to local conditions, and thus tend to delocalize the control of the gas-radiation equilibration process, and to introduce global properties of the atmosphere (e.g., the presence of boundaries) into the problem.
To illustrate the ideas developed above, let us consider the following as typical examples of scattering processes.
(a) The interaction of a photon with an atom in bound state a leading to the excitation of a higher bound state b (the photon's energy being converted to internal excitation energy of the atom), followed by a direct return to state a with the emission of a photon. In general the emitted photon will propagate in a different direction from that of the incident photon. Further, both the lower and upper states a and h of an atom in a radiating gas will not be perfectly sharp, but will have finite energy widths arising, for example, from the finite lifetime of each state produced by radiative decay, or from interactions of the atom with other particles of the plasma in which it is imbedded. Each of the bound states can, therefore, be considered to consist of a distribution ofsubstates, with radiative transitions possible from any subsfate of one level to any substate of the other. Thus if the decay of the upper level occurs to a different substate of the lower level than that from which the excitation occurred, or if there is a redistribution of the excited electron from the original excited substate to some other substate (because, say, the atom suffers an elastic collision with another particle) then the emitted photon's energy may be slightly different from the incident photon's. Similarly, motions of the scattering centers with respect to the fixed laboratory frame can change the emitted photon's energy from the incident energy if the projection of the scatterer's velocity along the direction of propagation is different for the two photons, for then a differential Doppler shift can occur. (Example; imagine the incident photon to be moving in the same direction as v of the scatterer and the emitted photon to move in the opposite direction. The emitted photon will be redshifted by an amount Av = —2v0v/'c relative to the incident photon). Changes in photon direction and frequency during scattering are described by redistribution functions (see below). Note that in this process no significant part of the photon energy is imparted to the material.
(b) Scattering of a photon by a free electron (Thomson or Compton scattering) or by an atom or molecule (Rayleigh scattering). Thomson scattering may be viewed as the result of the free charge oscillating in the electromagnetic field of the radiation. Compton scattering as a collision of a photon with a free charged particle, and Rayleigh scattering as a resonance of a permitted "oscillation" of the bound system with the field. The remarks made in (a) above concerning redistribution and lack of coupling of radiant energy to the thermal pool apply here as well.
22 The Equation of Transfer
2-1 The Interaction of Radiation with Matter 23
Similarly, we may consider the following to be examples of thermal absorption processes (and their inverses as thermal emission).
(a) A photon is absorbed by an atom in a bound state, and ionizes the bound electron, allowing it to escape with finite kinetic energy into the continuum. In this process of photoionization or bound-free absorption, the photon is destroyed and the excess of its energy over the electron's binding energy goes initially into the electron's kinetic energy, and ultimately into the general thermal pool after the electron suffers elastic collisions that establish a thermal velocity distribution for the particles. The inverse process, of a free electron dropping to a bound state with the creation of a photon whose energy equals the sum of the electron's kinetic and binding energies, is called direct radiative recombination. These processes clearly transfer energy back and forth between the radiation field and the thermal pool of the material.
(bj A photon is absorbed by a free electron moving in the field of an ion, resulting in an alteration of the electron's kinetic energy relative to the ion. The electron then, classically speaking, moves off on a different (hyperbolic) orbit around the ion. This process is known as free-free absorption because the electron is unbound both before and after absorbing the photon. The inverse process, leading to the emission of a photon, is referred to as bremsstrahhmg.
(c) A photon is absorbed by an atom, leading to a transition of an electron from one bound state to another; this process is called photoexcitation or bound-bound absorption. The atom is then de-excited by an inelastic collision with another particle. Energy is put into the motion of the atom and the collision partner and thereby ends up as part of the thermal pool. The photon is said to have been destroyed by a collisional de-excitation. The inverse process leads to the collisional creation of a photon at the expense of the thermal energy of the gas.
(d) Photoexcitation of an atom with subsequent collisional ionization of the excited atom into the continuum. Photon energy again contributes to the thermal energy of particles. The inverse processes is called (three-body) collisional recombination.
To illustrate the conceptual limitations of the kinds of arguments given above, let us now consider some ambiguous cases. Suppose an atom has three bound levels a, b, and c, in order of increasing energy, and a photoexcitation from a to c occurs. Then suppose that c decays radiatively to b, and b then decays radiatively to a; this process is called fluorescence. Here a single photon of energy hvac = Ec — Ea is degraded into two photons of energies hvab = Eh - Ea and hvbc - Ec - Eb. Was the original photon scattered or absorbed? By our original definition it has not been "scattered" and, moreover, the new photons may have vastly different properties (e.g., probability of escape through the boundary surface) from the original, so
that the nonlocal behavior of the radiation field has been altered. On the other hand, no contribution has been made to the thermal energy of the gas. Alternatively, consider the same process, but now with a collisional de-excitation c -> b followed by an emission b -» a. The original photon can be said to have been destroyed (absorbed); but is the emitted photon "thermally" emitted when most of the original energy was derived from the radiation field? Many other more complex and subtle cases may be constructed, which, taken together show the limits of usefulness of the absorption-vs-scatlcring description.
In fact, a truly consistent picture emerges only when we write down the full equations of statistical equilibrium (cf. Chapter 5). which describe all possible processes (both radiative and collisional) that couple an arbitrary state / to some other state j, and solve these together with the equations (transfer equations) that describe how the radiation is absorbed, emitted, and transported through the atmosphere. To do this is, in general, quite difficult, and formulation of successful methods of solution of the problem will occupy the bulk of this book. (The full import of these comments will emerge only when the student has studied the material through Chapter 12; nevertheless they should be borne in mind at all stages of subsequent development).
THE EXTINCTION COEFFICIENT
To describe the removal of energy from the radiation field by matter let us introduce a macroscopic coefficient #(r. v, t) called the extinction coefficient, or opacity, or sometimes (loosely) the rotal absorption coefficient. This coefficient is defined such that an element of material, of cross-section dS and length ds, removes from a beam with specific intensity I(r, n, v, t), incident normal to dS and propagating into a solid angle dco, an amount of energy
SE = /(r. n. v, t)I(r, n, v, r) dS ds dw dv dt (2-1)
within a frequency band dv in a time dt. The extinction coefficient is the product of an atomic absorption cross-section (cm2) and the number density of absorbers (cm-3) summed over all states that can interact with photons of frequency v. The dimensions of % are cm-1, and {\jy) gives a measure of the distance over which a photon can propagate before it is removed from the beam—i.e., a photon mean-free-path (cf. §2-2).
The frequency variation of x may be extremely complicated, and may include thousands or millions of transitions (bound-bound, bound-free, and free-free). For static media in which there are no preferred directions imposed on an atomic scale (e.g., by a magnetic field), the opacity is isotropic. For moving media, the opacity has an angular dependence introduced by the
24 The Equation of Transfer
2-1 The Interaction of Radiation with Matter
25
Doppler shift that radiation experiences in the fluid frame relative to its original frequency in the stationary laboratory frame; this Doppler shift obviously depends on the projection of the velocity vector onto the direction of the incident beam. In what follows we consider only static atmospheres.
As outlined earlier in this section, it is sometimes useful to distinguish between "absorption" and "scattering"; hence we introduce volume coefficients jc(r, v. t) andďfr, v, t) that describe [via equation (2-1)] the rate at which energy is removed from the beam by "true absorption" and "scattering," respectively. The total extinction is given by
v, t) = Kir, v, t) + ff(r, v, t) (2-2)
That is, both processes are assumed to occur independently and to add linearly. In actual practice % is sufficient to describe energy removed from the beam; the distinction between k and a is useful mainly in defining the emission coefficient.
In the calculation of % it is necessary to include a correction for stimulated emission (see §§4-1 and 4-3). This is a quantum process in which radiation induces a downward transition from the upper state at a rate proportional to the product of a cross-section, the upper-state population, and the specific intensity. Because the process is proportional to l(r, n. v, f) and effectively cancels out some of the opacity, it is convenient to include it in the definition of /. Stimulated emission occurs only when the emitting system exists in a definite upper state (whether bound or free). There is thus no stimulated emission in Thomson scattering (free electrons) or Rayleigh scattering (involves virtual states) but there is stimulated emission in spectrum lines, even if they are described with a "scattering" coefficient.
If we know the value of %(r, v, t) [or of k{t, v, t) and cr(r, v, tf], we have a complete macroscopic description of the rate at which material removes energy from a beam of radiation. But it is crucial to emphasize that the "completeness" of the description is illusory. The reason this unpleasant comment must be made is that the simple picture we obtain from equation (2-1) glosses over the fact that the level populations, which "determine" the rate of energy removal from the radiation field by their contribution to %, are, in turn, determined by the radiation field via photoexcitations, photo-ionizations, radiative emission, radiative recombination, and related processes. Thus, in reality, the interaction of the radiation field with the absorbing material is nonlinear. The problem just described still remains (though more subtly) even if it is assumed that we can calculate level populations by local application of thermodynamic equilibrium relations that depend only on the density and temperature. (This is the so-called local thermodynamic equilibrium or LTE approximation.) The reason is that the temperature is determined by overall balance between energy emitted and energy absorbed
by the material, and thus by the nature of the radiation field and its response to the global properties of the atmosphere (e.g., boundaries, scattering, gradients, etc.). The remarks made in this paragraph carry over with equal force in the macroscopic description of emission. Again, the full significance of these remarks will emerge only with considerable further development (see.particularly §§5-1 and 5-3).
THE EMISSION COEFFICIENT
To describe the emission of radiation from the stellar material, we introduce a macroscopic emission coefficient or emissivity r\{\, n, v, t) defined such that the amount of energy released from an element of material of cross-section dS and length ds, into a solid angle da), within a frequency band dv, in direction n in a time interval dt, is
SE = n(r, n, v, t) dS ds doj dv dt (2-3)
The dimensions of n are ergs cm-3 sr"1 hz"1 sec""1. As was true for the opacity, thermal emissivity is isotropic for static media (without imposed preferred directions) but is angle-dependent for moving material owing to Doppler-shift effects. For radiation emitted in scattering processes, there is normally an explicit angle-dependence, even for static media. The emissivity is calculated by summing products of upper-state populations and transition probabilities over all relevant processes that can release a photon at frequency v. In writing the transfer equation we shall usually use the unembellished symbol rj to denote the total emissitivity; if electron scattering terms appear explicitly in the same equation, r\ will then denote all other emission. Subscripts "c" and 'V may on occasion be used to denote continua and lines, respectively. Again, we must realize that the simplicity of this description is deceptive, for the reasons given above in the discussion of the extinction coefficient.
An important relation exists between the emission and absorption coefficients in the case of strict thermodynamic equilibrium (T.E.). If we consider an adiabatic enclosure in steady-state equilibrium containing a homogeneous medium, we know that the material will have the same temperature T throughout (otherwise it would be possible to devise processes to extract work from the temperature gradient, in violation of the second law of thermodynamics). Further, we may expect the radiation field to be isotropic and homogeneous throughout the enclosure (including at the surface of the walls), for if it were not. beams traveling in opposite directions would not be exactly similar and a directional transport of energy would result, from which work could be extracted, again in violation of the second taw of thermodynamics. Consideration of the energy absorbed and emitted in
26 The Equation of Transfer
2-1 The interaction of Radiation with Matter 27
angle-frequency ranges dco dv b> an element of material, in time dt, now shows that if a steady-state thermal equilibrium is 1o be achieved (no net gain or loss of energy by the matter), the thermal emission must be given by
}f(v) = K(v)7(n, v) (2-4)
which is known as Kirchhoffs law. For an enclosure in strict T.E. at temperature T the intensity of the radiation field is given by the Planck function BV{T), so that
/,*(v) = k*(v)Bv{T) (2-5)
which is the Kirchhoff-Planck relation. This result has been obtained without reference to the composition of the material and is valid (in T.E.) for all materials. [See the excellent discussion of the interaction of matter and radiation in T.E. in (160.199 -206) and in an article by Milne in (416,93-96).] Strictly speaking, the Kirchhoff-Planck law applies only in the case of a system in T.E. But if the material is subject only to small gradients over the mean free path a photon can travel before it is destroyed and thermalized by a collisional process (as is true, e.g., in the interior of a star), then we could expect equation (2-5) to be valid to a high degree of approximation at local values of the thermodynamic variables specifying the state of the material. In such a case we write
f(T. v, i) = K*(r, v, t)Bv[T(r, f)] (2-6)
The hypothesis of local thermodynamic equilibrium (or LTE) just mentioned makes the assumption that the occupation numbers of bound and free states of the material, the opacity, the emissivity—indeed all of the thermodynamic properties of the material—are the same as their T.E. values at the local values of T and density, throughout the entire atmosphere, out to the outermost regions. Only the radiation field is allowed to depart from its T.E. value of Bv[T(r)], and is obtained from a solution of the transfer equation. Such an approach is manifestly internally inconsistent, although LTE expressions remain valid for certain quantities even in the general case. For example, equation (2-6) is a valid expression for the continuum emission coefficient even in the presence of departures from LTE so long as the velocity distribution of recombining (or. for free-free emission, colliding) particles is Maxwellian; the equation is not valid for line emission, and further, the LTE formula for the opacity is not correct. The use of LTE is a computational expedient that simplifies the calculation of models of stellar atmospheres, and has been widely applied. (We shall employ it in places to provide a prototype with which we can introduce basic mathematical techniques of solving transfer problems, and will discuss models built assuming LTE in §§7-2 through 7-4. (But it must be stressed that stellar atmospheres are regions
I in which there are large gradients of material properties, and an open
boundary through which radiation freely escapes; the radiation field is : therefore highly anisotropic and has a markedly non-Planckian character.
[ One might argue for using LTE, even though the radiation field is obtained i from solving a nonlocal transfer equation, if one could show that some J mechanism, specifically collisions among the particles, enforced LTE popu-
•f lations. As already mentioned above, and as we shall see in detail in §5-3
; and in Chapters 7 and 11 -14, this will not, in general, be the case. Rather,
• the radiation field determines the state of the material, and hence equation - (2-6) becomes invalid; in the end we must carry through a general analysis
in which we specify the thermodynamic state of the gas and the distribution function of the radiation field simultaneously by solving the coupled equations of transfer and statistical equilibrium,
Let us now consider radiation scattered by the material. For simplicity of J notation, we suppress explicit reference to r, though all of the quantities may
| be time-dependent. As described earlier, in scattering processes both the
I direction and frequency of a photon may change. These changes are described
: by a redistribution function
I R(v'. n'; v, n) dv' dv(clco'/4ii){doj/47i)
'i which gives the joint probability that a photon will be scattered from direction
~! n in solid angle dio and frequency range (v', v' + dv') into solid angle (V) = (47T)-1 j) dm dv R(v', n'; v. n) (2-8)
which by virtue of equation (2-7) is normalized such that j cj)(v') dv' = 1. If | (v')J(r, v') dv'—i.e., that the scattering process is conservative.
Equation (2-9) gives the full angle-frequency dependence of the emission profile. It is usually difficult to treat radiative transfer problems in the degree of generality implied here, and useful simplifications of the problem can be made. For example, if we are primarily interested in redistribution in frequency and not in angle, we could assume that I(r, n, v) is nearly isotropic, and replace it in equation (2-9) with J(r, v). Then the emission into dv dm is
}f{r, v) = r/0(r)
R(v\ v)J{r, v') dv'
(2-10)
where the angle-averaged redistribution function
R(v',v) = (4ny1 R(v',n';v, ri) dm' = (47i)_1 j)R(v', n'; v, n) dm (2-11)
gives the redistribution probability from (v', v' + dv') to (v, v + dv) and is normalized such that
J7 dv' J7 dv R(v', v) = j;
dv' = 1
(2-12)
The function R(v', v) is rendered independent of angle as a result of integrating over either dm' or dm; this follows from the fact that (cf §13-2) R(v', n'; v, n) depends only on the angle between n' and n. Equation (2-10) provides an extremely useful approximation in line transfer problems because the crucial phenomenon there is the frequency diffusion of photons from the opaque line core (where they are trapped) to the more transparent line wings (whence they may escape from the atmosphere at depths where / is, in fact, very nearly isotropic). In the angle-averaged approximation the emission profile
i//(v) = ŕ?(r, v) I i/(r, v)dv
is given by
m = j7 R(v', v)J(r, V) dv'/fj0 tf>(vV(r, V) dv' (2-13)
2-1 The Interaction of Radiation with Matter 29
which shows that the distribution of emitted photons depends upon the frequency profile of the incoming radiation.
In the limiting case that the intensity is independent of frequency, we obtain natural excitation, with
(v')(f>(v). For complete redistribution, the emissivity is
ns(r, v) = ffo(r)0(v) j7 ^(v')^(r, V)
dv'
(2-15)
from which we see clearly that the emission and absorption profiles are identical. Complete redistribution is also a good approximation within the Doppler core of a spectrum line, and actually provides an excellent first approximation in line transfer problems. We shall, in fact, assume complete redistribution in our discussion of line formation until Chapter 13.
Another class of problems arises when we focus attention on the angular redistribution of the emitted radiation, but assume that the scattering is essentially coherent (i.e., v' = v). This is the situation of interest, e.g., in scattering of light by large particles in a planetary (including earth's) atmosphere. We can then write
jR(v', n'; v, n) = g(n', n)(v') <5(v —
(2-16)
where d is the Dirac function and g is an angular phase function normalized such that
(47i)-1^(n',n)^' = 1 (2-17)
30
The Equation of Transfer
31
Two important phase functions are those for isotropic scattering
g(n\ n) = 1
(2-18)
and the dipole phase function (which applies for Thomson and Rayleigh scattering)
g(a', n) = ^ (1 + cos2 = n' ■ n. The phase functions for scattering by large particles (i.e., whose size is comparable to a wavelength of light) are often extremely complicated and show large, rapid variations as a function of angle (312; 359, Chap. 4).
For coherent scattering, equation (2-9) reduces to
tj (r, n, v) =
- varies slowly over the range corresponding to a Doppler shift). On the other hand, for continuum scattering (e.g., by electrons) the frequency distribution of radiation is smooth and essentially constant over the typical frequency shifts occurring in the scattering process. For this reason continuum scattering processes are customarily treated as if they were coherent (though this may be inadequate near a spectrum line). Moreover, as the angular redistribution effects from a dipole phase function are usually very small in a stellar atmosphere, it is customary to assume that continuum scattering is also isotropic, and to write
tf(r, v) = ff(r, v)J(r, v)
(2-21)
2-2 The Transfer Equation
DERIVATION
Let us now consider the problem of radiative transport. Choose an in-ertial coordinate system and examine the flow of energy through a fixed volume element in a definite time interval. Let us assume that the radiation field is, in general, time-dependent. If we suppose the material to be at rest, then both % and r\ will be isotropic (unless we consider anisotropic scattering). In moving material one must account for changes in photon frequency and
lit + Ar, ii, v, / + Al)
/(r, n, v, t) figure 2-1
Element of absorbing and emitting material considered in derivation of transfer equation.
direction (Doppler shift and aberration) resulting from the transformation between the laboratory frame and the fluid frame. These effects depend upon n ■ v; hence both x and y\ will have an explicit angle-dependence in this case. Now calculate the energy in a frequency interval dv, passing in a time dt through a volume element of length ds and cross-section dS oriented normal to a ray traveling in direction n into solid angle dm (see Figure 2-1). The difference between the amount of energy that emerges (at position r + Ar at time t + At) and that incident (at r and t) must equal the amount created by emission from the material in the volume minus the amount absorbed. That is,
[/(r 4- Ar, n, v, t + At) - I(r, n, v, r)] dS dco dv dt
= [?j(r, n, v, t) - x(r, n, v, t)I(r, n; v, t)] ds dS dw dv dt (2-22)
Let s denote the path-length along the ray; then At = As/c, and
I(r + Ar, n, v, t + At) - /(r, n, v, t) + [c'^dl/dt) + {31/ds)] ds (2-23)
Substituting equation (2-23) into equation (2-22) we have the transfer equation
[c~ Hd/dt) + (d/3s)]I(r, n, v, t) = n(r, n, v, 0 - yfr, n, v, t)I(r, n, v, t) (2-24)
The derivative along the ray may be re-expressed in terms of an orthogonal coordinate system:
81
OS
OS 1 \ GX
dy\cl ds)\dy
OS I \vz I ox
dy
31
:d~z
(2-25)
where (nx, nyi nz) are the components of the unit vector n. We may thus rewrite equation (2-24) as
[c" Ho/ot) + (n ■ V)]J(rs n, v, t) = n(r, n, v, () - yjr, a, v, t)I(r, n, v, t) (2-26)
32
The Equation of Transfer
2-2 The Transfer Equation 33
For a one-dimensional planar atmosphere nz = (dz/ds) = cos 6' = /.(; further, the derivatives (5/dx) and {d/dy) are identically zero, and we obtain .
[c-Hd/dt) + fi(d/dz)]Hz> n. v, t) = rj(z, n, v, f) - yfz, n, v, t)I{z. n, v, t)
(2-27)
or, for the time-independent case,
pi[8I(z, n, v)/dz] = n(z, n, v) - yfz, n, v)I(z, n, v) (2-28)
!
Equation (2-2S) is the "standard" transfer equation for plane-parallel model-atmospheres calculations; the coordinate z increases upward in the atmo- i sphere (i.e., toward an external observer). For static media, the specification (z, n, v) in n and % may be reduced to (z, v) only. Note that if n and / are given, equation (2-28) is an ordinary differential equation, which may be solved for \ all relevant choices of \i and r. When r\ includes scattering terms, the transfer equation becomes an integro-differential equation containing angle and i frequency integrals of /.
i
THE TRANSFER EQUATION AS A BOLTZMANN EQUATION ;
The basic equation describing particle transport in kinetic theory is the Bottzmann equation; we shall now show that the transfer equation is just the Boltzmann equation for photons. Suppose we have a particle distribution function fit, p, t) that gives the number density of particles in the phase ; volume element (r, r + dr), (p, p + dp). We follow the evolution of /' within a particular phase-space element for a time interval dt, in which r r + v d.t and p -> p + F dt, where F denotes externally imposed forces acting on the particles. The phase-space element evolves from -
(d3rW3p)o - {d3r)(d*p) = J[(d2r)0(d"p)0-]
where J is the Jacobian of the transformation. >
Exercise 2-2: Show thai to first order in di the Jaeobian of the transformation
of a phase volume element is./ = 1. ;
In view of the result of Exercise 2-2 we see that the phase-space element is : deformed, but its phase volume is unchanged. If all external forces F are continuous, then the deformation of the phase-space element is continuous, t and all particles originally within the volume remain there; as the volume itself is unchanged, the particle density is unchanged. But if, in addition, collisions occur, individual particles may be reshuffled from one element of i phase space to another "discontinuously"; i.e., their neighbors may be totally ; unaffected during the same time interval. Therefore, the change in the particle
number density within a phase-space element must equal the net number introduced into the element by collisions; i.e.,
h (1)1) ♦(?)© * ©(i)
or. in more compact notation
(df/dt) + (v ■ V)/ + (F ■ yp)f = (Df/Dt)coU (2-30)
For a "gas" consisting of photons (with rest-mass zero), in the absence of general relativistic effects, F = 0, and photon propagation in an inertial frame occurs in straight lines with v — en, while the frequency remains constant. The distribution function fR can be written in terms of the specific intensity by means of equation (1-2). The analogues of "collisions" are photon interactions with the material, and the net number of photons introduced into the volume will be the energy emitted minus the energy absorbed, divided by the energy per photon. Thus for photons equation (2-30) becomes
(c/n-r1 [(£//&) + c(n ■ V)/] = (rj ~ ^/)/(fev) (2-31)
which is identical to the transfer equation (2-26). In effect the transfer equation is a Boltzmann equation for a fluid that is subject to no external forces but which suffers strong collisional effects. As will be seen in §2-3, the moments of the transfer equation yield dynamical equations for the radiation field, just as moments of the Boltzmann equation for a gas lead to equations of hydrodynamics.
SPHERICAL GEOMETRY
Tn a spherically symmetric medium, the specific intensity will be independent of the coordinates 0 and 0 of the triplet (r. 0. of the pair (0, (f>) which specifies the direction of the beam relative to the local outward normal f. Thus I(r, n, v, t) reduces to I(r, 0, v, t). In writing the transfer equation, starting from the general form of equation (2-24), we must now account for the variation of B along a displacement, and employ the general form ds = dr r 4- r dO 6. As is clear from the geometry of the situation (see Figure 2-2), dr = cos 0 ds, while /• dO = -sin 6 ds (note that d$ < 0 for any ds), so that
(d/ds) cos 0(d/dr) - r"1 sin 0(8/80) = (i(d/dr) + r_1(l - »2){d/dfi)
(2-32)
34
2-2 The Transfer Equation 35
6 + dO
ds
■dO
>dr
FIGURE 2-2
Geometric relations among variables used in derivation of transfer equation in a spherically symmetric medium.
O
where, as usual, ji = cos 0. Hence the transfer equation for a spherically symmetric atmosphere is
[c-1(d/dt) +fi(8/5r
L(l - ß2)(d/dfi)]I(r, pi, v, t)
= n(i\ v, t) - x(r, v, t)I(r, p, v, f) (2-33)
which simplifies in an obvious way in the time-independent case. Note that now, even with n and specified, equation (2-33) is a partial differential equation in r and p. However, this added complexity can be avoided by using the (straight-line) characteristic paths that reduce the spatial operator to a single derivative with respect to pathlength (see §7-6). In fact, equation (2-33) is not structurally different from equation (2-28) and can be solved almost as easily.
Exercise 2-3: (a) Consider an atmosphere which is axially symmetric but not spherically symmetric (e.g., a rotationally flattened star). Show that now /(r, n) = I(r,®,d,4>). (b) For the general case where J(r, n) = /(/-, 0, <&, 0, )
+ r
- ť)(dlfin) - (a cot G/r)(dI/e) = where y = cos sin 6 and o = sin 4> sin Ö.
OPTICAL DEPTH AND THE SOURCE FUNCTION
For the remainder of §2-2 let us confine attention to the time-independent planar transfer equation (2-28). Writing dx{z, v) = — x(z, v) dz, we define an
optical depth scale t(z, v) which gives the integrated absorptivity of the material along the line of sight as
t(z, v)
X(z', v) dz'
(2-34)
The negative sign is introduced so that the optical depth increases inward into the atmosphere from zero at the surface (where z = zmax), and thus provides a measure of how deeply an outside observer can see into the material [cf. equations (2-47) and (2-52)]. Recalling that y'1 is the photon mean-free-path, we recognize that t(z, v) is just the number of photon mean-free-paths at frequency v along the line of sight from zmax to z. In addition, we define the source function to be the ratio of the total emissivity to total opacity,
S(z, v) = n(zy v)/x(z, v) (2-35)
To simplify the notation we shall for the present suppress explicit reference to z and fi, and denote frequency dependence with a subscript v. The equation of transfer may then be written in its standard form
p{dIJK) = iy - Sv
(2-36)
From the discussion of §2-1 we can write prototype expressions for the source function which we use to study methods of solving equation (2-36). Suppose first we have strict LTE. Then from equation (2-6) we have
S„ = B,
(2-37)
If we have a contribution from thermal absorption and emission plus a contribution from a coherent, isotropic, continuum scattering term (say from Thomson scattering by free electrons or from Rayleigh' scattering) then we could write
Xv = kv + <7v (2-38)
and Sv = (kvBv + v (2-40)
where %c and %t denote the continuum and line opacities, respectively. If we assume that a fraction s of the line emission occurs from thermal processes and the remainder is given by angle-averaged complete redistribution [equation (2-15)], we can write
sBv + (1 - e) J>VJV
dv
(2-41)
36 The Equation of Transfer
2-2 The Transfer Equation 37
and
S„ =
r + Ec TT'i
"(1 - e)0v" r + fa
jvJvdv
(2-42)
where r = yjy,. (We may ignore the frequency variation of the continuum over a line-width.) In equations (2-39) and (2-42) we have examples of the explicit appearance of integrals of the radiation field over angle and frequency, which shows the integro-differential nature of the transfer equation. It must be stressed that the source functions in equations (2-37), (2-39), and (2-42) are only illustrative; they are based on essentially heuristic arguments, and a physically rigorous formulation can be provided only after the equations of statistical equilibrium (Chapter 5) are developed.
BOUNDARY CONDITIONS
Solution of the transfer equation requires the specification of boundary conditions. Two problems of fundamental astrophysical importance are (a) the finite slab (in planar geometry) or shell (spherical geometry), and (b) a medium (e.g., a stellar atmosphere) that has an open boundary on one side but is so optically thick that it can be imagined to extend to infinity on the other side—the semi-infinite atmosphere.
For the finite slab of total geometrical thickness Z and total optical depth Tv (defined to be zero on the side nearest the observer), a unique solution is obtained if the intensity incident on both faces of the slab is specified. Writing 8 for the angle between a ray and the normal directed toward the observer, and pi = cos 6, we must specify the two functions I+ and I~ such that
í(tv = 0,pi,v) = r(pi, v), {-l*Zti<0)
at the upper boundary, and
J(tv = Tv,pi,v) = r(^v), (0 < < 1)
(2-43)
(2-44)
at the lower boundary. For a shell of outer radius R and inner radius rc, equation (2-43) still applies at r = R.
Exercise 2-4: (a) At the inner boundary of a spherical shell, r = rc, show that the lower boundary condition is given by !(rc, + pi, v) ~ I(rc, —pi, v) if the central volume r ^ rc is void; this is Milne's "planetary nebula" boundary condition, (b) If the volume contains a point source (a star) of intensity /0, show that the result in (a) must be augmented by /{rr, +l,v) = I0{v)d{fi - I), (c) Extend results (a) and (b) to the case of the volume being partially filled by an opaque source on the
range 0 ^ r ^ r,.. (r^ < rc) with intensity J0(pi'. v) where pi' is the angle cosine at the opaque surface; this case simulates an envelope around a star, (d) Show in case (a) above thai the flux is identically zero at all points r ^ rc.
In the semi-infinite case (planar or spherical), the radiation field incident upon the upper boundary must be specified by an equation of the form of (2-43); for stellar atmospheres work, it is customary to assume /" = 0 (clearly this would not be done, e.g., in a binary system). For a lower boundary condition one may replace equation (2-44) by a boundedness condition in analytical work where the limit tv -> co is taken. Specifically we impose the requirement that
lim I(Tv,fi,v)e-^'1 = 0 (2-45)
T-> CO
The reasons for this particular choice will become clear in the discussion below. Alternatively, at great depth in the atmosphere we may write I(xv, pi, v) in terms of the local value of Sv and its gradient, or may specify the flux; these conditions follow naturally from physical considerations in the diffusion limit where the photon mean free path is much smaller than its optical depth from the surface (see §2-5).
SIMPLE EXAMPLES
Before writing the formal solution of the equation of transfer, it is instructive to consider a few simple examples in planar geometry.
(a) Suppose no material is present. Then= rpv = 0, and equation (2-28) reduces to (dlv/dz) = 0 or /v = constant. This result is consistent with the proof in §1-1 of the invariance of the specific intensity when no sources or sinks are present.
(b) Suppose that the material emits at frequency v, but cannot absorb. Then equation (2-28) is pi(dljdz) = nv, and for a finite slab the emergent intensity is given by
/(Z,/í, v) = p~l j*ti{z, v) dz + l+(0,pi,
(2-46)
The physical situation described above occurs in the formation of optically forbidden lines in nebulae. Atoms may be excited to metastable levels by collisions; because nebular densities are so low, the chances of a second collision leading to de-excitation are very small, so the atoms can remain unperturbed in these levels for long periods of time, and large numbers of atoms may accumulate in these states. Eventually some of the atoms decay via "forbidden" transitions, which have very small but nonzero transition probabilities, and emit photons. Because the line is forbidden, the probability of reabsorption is negligible, and the photon escapes. Thus photons are created
38 The Equation of Transfer
2-2 The Transfer Equation 39
at the expense of the energy in the thermal pool, and none are destroyed by absorption.
(c) Suppose that radiation is absorbed but not emitted by the material. Then p(dljcz) = -Xvh, and defining dxx = -%v dz, the emergent intensity from a finite slab of total optical thickness Tv is
/(Tv = 0, p, v) = r(Tv, p, v) exp(- TJp)
(2-47)
Equation (?-47) applies, for example, to radiation passing through a filter m which photons are absorbed and degraded into photons of some very difler-ent frequency (e.g., extreme far-infrared heat radiation) before being re-emitted, or are destroyed and converted into kinetic energy ot the particles iu the absorbing medium.
FORMAL SOLUTION
Let us now obtain a formal solution of the equation of transfer; we confine attention exclusively to planar geometry. Regarding S as given, equation (2-36) is a linear first-order differential equation with constant coefficients, and must therefore have an integrating factor. The integrating factor is easily shown to be exp( — xjg), so that
c[J¥ expl-rv//i)]/ 0) we can (in principle) infer information about Sv(tv) for 0 < tv < 1. For stars, we cannot observe the center-to-limb variation, but it is clear that if we observe at different frequencies (e.g., within a spectral line), we encounter unit optical depth in higher layers for frequencies with high opacity (e.g., line-center) and in deeper layers for frequencies with low opacity (e.g., line-wings). If we then know something about the frequency-variation of Sv, we can infer information about its depth-variation; for example, in LTE, Sv = Bv and the frequency-variation over a narrow line can be ignored, so that one can, in principle, infer the run of the temperature with depth. Although it is extremely useful conceptually, the Eddington-Barbier relation should not be applied indiscriminately and used literally to argue that 1(0, p. v) is identical to Sv(tv = pi because (a) there are always significant contributions to 7V{0) from other depths (i.e., there is an intrinsic "fuzziness" in the problem), and (b) the assumptions from which it follows may not be valid. A detailed critique of the limitations of the Eddington-Barbier relation can be found in (18, 121-130) and (20,20-30).
Exercise 2-5: Suppose the source function is to be represented by a power-scries expansion about the point ; i.e., S(x) ^ S(x%) + S'(t^)(t - x^) + jS"(x%)(x - t*)2. Calculate the emergent intensity and show that the choice = p is "optimum" in the sense that it eliminates the contribution of S' and minimizes the contribution ofS" to H0, p).
Another instructive example to consider is a finite slab of optical thickness T, within which S is constant, and upon which there is no incident radiation.
40 The Equation of Transfer
The normally emergent intensity is 1(0, 1) = S(i - c~T).ForT » 1,7 = S. This result is sensible physically, for the radiation that emerges consists of, those photons emitted over a mean-free-path from the surface; the emission rate is r\ and the mean-free-path is x~\ hence I = rj/yL = S. For T « 1, e~T ft: 1 — T,so/ a; ST. Again this result is physically reasonable, for in the optically thin case we can see through the entire volume; hence the energy emitted (per unit area) must be the emissivity per unit volume r\ times the total path-length Z, or / = r\Z = (n/yj(yZ) = ST. Note that this result is consistent with equation (2-46).
THE SCH WAR ZS CHILD—MILNE EQUATIONS
By integration of the formal solution for the specific intensity over angle, concise expressions for the moments of the radiation field may be derived. Thus by substitution of equations (2-50) and (2-51) into equation (1-4) we have for the mean intensity
dt
(2-54)
Equation (2-54) is reduced to a more useful form by interchanging the order of integration, and making the substitution w = ±l/p in the first and second integrals respectively. Then
1
dw
,-W(t-Tv)
vv
j>s.«J7
dw
w
(2-55)
The integrals over vv are of a standard form and are called the first exponential integral. In general, for integer-values of n, the nth exponential integral is defined as
EH(x) = Jl' r"e~xtdt = x" In terms of £A(x), equation (2-55) can be rewritten as
A.(?v) = j Jo sv(tv)£i|fv - tv|dtv
f2-
57)
Equation (2-57) was first derived by K. Schwarzschild and is named in his honor; Schwarzschild's paper (416, 35) is one of the foundation stones of radiative transfer theory, and merits careful reading. Because the integral appearing in equation (2-57) occurs so often in radiative transfer theory it has
2-2 The Transfer Equation 41
been abbreviated to an operator notation:
Exercise 2-6: (a) Show that equations (2-50) and (2-51) are equivalent to
J(i\ n) = ["J"iai n(x') cxp[- r(r, r')] d\r' - \\
where r'(.s-) = r - sa, t(i\ r) = fjj _t| ^[r'(-t)] ds, and .s'ma, is the distance along the ray to any boundary surface in the direction (-n); jinax = co for outward-directed rays in a semi-infinite medium, (b) Substitute the above result into the definition of J(r) [equation (1-3)] to derive Peterls^ equation:
J(t) = (47!)" 1 jy {';(r')exp[-r(r',r)]/[r' - r[2} rfV where V denotes the entire volume containing material.
By an analysis similar to that used to obtain equation (2-57) we can derive expressions for Fv and Kr first obtained by Milne (416. 77):
FvlTv) = 2 J" Sv(tY)E2(tv - xx)div
- 2 fj Sjtv)E2(zv - tv)dtv (2-59) and Kfxv) = I Sv(tv)£3[rv. - tv| dtv (2-60)
We also define the corresponding operators
Ot[/(ij] = 2J"f{T)E2(t - r)dt
- 2 j*f(t)E2(t ~ t)dt (2-61) and Xlf(t)] = 2 j"* f(t}E2\t - i| dt (2-62)
Exercise 2-7: Derive equations (2-59) and (2-60).
The mathematical properties of the exponential integrals are discussed in detail in (4, 228-231) and (161, Appendix I), and the properties of the A, 4>, and X operators are discussed in (361, Chap. 2). A few of the most important results are mentioned in the following exercise.
Exercise 2-8: (a) Differentiate equation (2-56) to prove E'u(x) = —£„_!(*}. (b)Integrateequation(2-56)bypartstoshowthat£„(.v) = [e~x - xEn. fx)~]/(n - 1) for» > 1. (c) Show that the asymptotic behavior (x » l)oiEn(x)isEfl(x) - e~*/x.
42 The Equation of Transfer
2-3 Moments of the Transfer Equation
Some interesting physical insight can be gained by considering a linear source function S(z) = a+ bt; with the results of Exercise 2-8, it is easy to show that
K(a + bt) = (a + bz)+- [6£3(t) - a£2(T)] (2-63) v('vUv(U] + \J[l - pv(tv)]BK.(K)}
where pv = oJ{kv + £R(r, t)/dt] + V ■ J^(r, t) = 4n [>?(r, v, t) - y{x, v, t)J(r. v, f)] dv (2-69)
For a time-independent radiation held in a one-dimensional planar static medium, equation (2-67) reduces to the "standard" result
[dH{z, v)/t?z] = ri{z, v) - X{z, v)J(z, v) (2-70)
or, abbreviating the notation and using equations (2-34) and (2-35),
{oHJdiv) = Jv - Sv (2-71)
For spherical geometry (in a time-independent static atmosphere), by use of the appropriate expression for the divergence, we find
'■[d(r2Hf)/dr] = nY - XvJv
(2-72)
Equations (2-70) through (2-72) will be employed repeatedly to obtain solutions of the transfer equation, and equation (2-69) will be used to develop the equations of radiation hydrodynamics (cf. §15-3).
Exercise 2-11: Derive equation (2-72) directly from equation (2-33).
The ftrsl-order-moment equation with respect to the ith coordinate axis is obtained by multiplying equation (2-66) by ??f and integrating over (dco/c), which yields
c-2[ajr.(r, v, t)/dt] + £ [rP0.(r, v, t)/dXj] j
= c~1 j) \\}{x, n, v, t) - %{r, n, v, t)J(r, H, v, t]\nt da> (2-73)
or, in vector notation,
c~2[d^(r,v,t)/dq + V-P(r, v, t)
= c~l ^ [7j(r, n, v, t) - z(x, n, v, ()/(r, n, v, f)]n dco (2-74)
Recalling that the momentum density in the radiation field is GK - c~2-^ (see §1-3), we see that equation (2-74) is analogous to the hydrodynamical equations of motion, and may be viewed as a dynamical equation for the momentum in the radiation field at frequency v. When we integrate over all frequencies, we obtain a total momentum equation for the radiation, which we shall use in the equations of radiation hydrodynamics (cf. Chapter 15):
c~2[&^(r, t)/8t] + V ■ P(r, t)
= c
Jo" dv (j) doj[n(r, n, v, t) - /(r, n, v, t)/(r, n, v, *)]n (2-75)
Equation (2-75) states that the time rate of change of the total momentum density in the radiation field is equal to the negative of the volume force exerted by radiation stresses (cf. §1-4) plus a term that must represent the net momentum gain (or loss) from interacting with the material [see further discussion following equation (2-76)]. As in equation (2-68), equation (2-75) allows for the possibility of material motions. If the medium is at rest, then the integral over n vanishes (which merely states that the net momentum loss from the material by isotropic emission is zero, as is physically obvious}, while the second term reduces to an integral over the flux:
c"2[t\^(r, t)/dt] + V ■ P(r, t)= - 1 [w y_(r, v, t).^{r, v, t) dv (2-76)
The physical significance of the integral on the righthand side of equation (2-76) can readily be seen by the following argument. Consider a beam of specific intensity /, entering an element of absorbing material of surface area dS, at an angle $ relative to the normal. The energy absorbed by material of opacity x from a solid angle dco and frequency band dv in time di is
dE = yj dS cos 0 ds doj dv dt
where ds = [dz/cos 0) is the slant-length of the ray through the element of thickness dz along the normal. The component of momentum deposited in the material along the direction of the normal is c"1 dE cos 0; hence the momentum deposition per unit volume per unit time is
(c_1 cos 0 dE)/{dz dS dt) = c"1^/ cos 0 do.) dv
46 The Equation of Transfer
2-4 The Condition of Radiative Equilibrium 47
If we sum over all angles and frequencies, we obtain precisely the integral in equation (2-76). We have thus shown that the integral is the radiation force, per unit volume, on the material; this interpretation is clearly compatible with the overall physical meaning of the equation.
For a time-independent radiation field in a one-dimensional static planar medium, equation (2-74) reduces to
[3pR(z.v)fdz] = -(4n/c)x(z,v)H(z,v) (2-77a)
or, integrating over all frequencies
\dpR{z)idz] = -(4te/c) x(z, v)H(z, v) dv (2-77b)
Alternatively we can write
[6K(z, vfdz] = -yfz, v)H(z, v) (2-78)
or idKv/dxv) = Hv (2-79)
In spherical geometry, under the same assumptions, equation (2-74) reduces, with the aid of the expression derived in Exercise 1-10 for (V ■ P),., to
(dKJdr) + r~l(3Kv - Jw) = -XvHv (2-80)
Exercise 2-/2: Derive equation (2-80) directly from equation (2-33).
Thus far we have examined the moment equations primarily from the standpoint of their dynamical significance; but in the time-independent case they may also be used as tools to solve the transfer equation. By the introduction of moments, the angle-variable is eliminated and the dimensionality of the system to be solved is reduced. As wc have seen in §2-2, the mean intensity can be determined from the solution of an integral equation (see Exercise 2-10). This gives the source function, from which the higher moments (e.g., the flux) can be determined by quadrature. The question now arises whether we can solve the moment equations as differential equations. Examination of equations (2-71) and (2-79) immediately reveals an essential difficulty: the moment equation of order n always involves the moment of order n + 1, hence there is always one more variable than there are equations to determine them! This difficulty is known as the closure problem: one additional relation among the moments must somehow be obtained to "close" the system. For solving transfer equations, a variety of methods exist that employ moments of arbitrarily high order, and introduce ad hoc closure relations [see, e.g., (361, 90-101; 365)]. However, in this book attention will be confined entirely to the moments Jv, Hr, and Kv (an exception appears in §14-3), and the system will be closed (see §6-3) by eliminating Kv in terms of Jv and
the variable Eddington factor f,—i.e., Kv = f,Jv. The factor/;, is obtained by iteration and allows us to effect an approximate closure of the exact system (if the iteration converges, the closure is also exact). Alternatively, it will be shown in §6-3 that the transfer equation can be rewritten in terms of angle-dependent mean-intensity-like and flux-like variables, and that exact closure of an angle-dependent equation that resembles a moment equation can be obtained. This equation is easily discretized and solved. In sum. the solution of the transfer equation in terms of moments or equivalent variables can be effected by differential-equation techniques of great generality and power.
2-4 The Condition of Radiative Equilibrium
Deep within the interior of a star, nuclear reactions release a flux of energy that diffuses outward, passes ultimately through the star's atmosphere, and emerges as observable radiation. In normal stars there is no creation of energy within the atmosphere itself; the atmosphere merely transports outward the total energy it receives. In a time-independent transport process, the frequency distribution of the radiation, or the partitioning of energy between radiative and nonradiative modes of transfer, may be altered; but the energy flux as a whole is rigorously conserved.
There are two basic modes of energy transport in those atmospheric layers in which spectrum-formation takes place: radiative and convective (or some other hydrodynamical mode). In these layers conduction is ineffective and can be ignored (it becomes important in coronae at temperatures of the order of 106 °K). When all of the energy is transported by radiation, we have what is called radiative equilibrium; conversely, pure convective transport is called convective equilibrium. Whether or not radiative transport prevails over convection is determined by the stability of the atmosphere against convective motions.
The criterion for the stability of radiative transport was first enunciated by K. Schwarzschild (416, 25) in another of the fundamental papers of radiative transfer theory. Schwarzschild was able to demonstrate convincingly that the dominant mode of energy transport in the photosphere of the sun is radiative. Since his work, a number of analyses of radiative stability have been carried out for a variety of stellar types; results are summarized in (638. 215; 11,449; 654, 432). The basic picture that emerges for a star like the sun is that radiative equilibrium obtains to continuum optical depths of order unity, and that below this depth the atmosphere becomes unstable against convection. Convection zones exist below the outer radiative zone in ah stars of spectral type later than about F5. For earlier spectral types, radiative equilibrium prevails throughout the entire outer envelope of a star. In this book primary concern will be given to the early-type stars, and accordingly
48 The Equation of Transfer
2-5 The Diffusion Approximation 49
emphasis will be given to the radiative equilibrium regime. The theory of convective transport is not, at present, in a fully satisfactory form, and only the mixing-length theory (a phenomenological approach that has been widely applied in astrophysics) will be described (§7-3).
Let us now consider some of the implications of radiative equilibrium; assume the medium is static and the radiation field is time-independent. From the discussion in §2-1, it is clear that the total energy removed from the beam is
j"" dv (j) dm yfr, v)/(r, n, v) = 4% J" %(r, v)J(r, v) dv (2-81)
where / denotes the total extinction coefficient. The total energy delivered by the material to the radiation field is
Jo" dv j) dm n(r, v) = 4n yfr, v)S(r, v) dv (2-82)
where equation (2-35) has been used. The condition of radiative equilibrium demands that the total energy absorbed by a given volume of material must equal the total energy emitted; thus at each point in the atmosphere
or 4ti x(r, v)[S(r, v) - J(r, v)] dv = 0 (2-83b)
Exercise 2-13: Suppose that Sv is given by equation (2-39). Show that, in the condition of radiative equilibrium, the scattering terms cancel out to yield
kA(T) dv = kvJv dv
Using equation (2-83) in equation (2-69) we have, alternatively,
V ■ & = 0 (2-84)
Hence in planar geometry the condition of radiative equilibrium is equivalent to the requirement that the depth derivative of the flux is zero—i.e., the flux is constant. Physically, equations (2-83) and (2-84) have the same meaning, but mathematically the requirement $F — constant is rather different from the expressions in equations (2-83); either form of the constraint may be used in constructing model atmospheres.
Because the total flux is constant in a planar atmosphere, it may be used as a parameter that describes the atmosphere; an equivalent quantity often employed is the effective temperature. From Exercise 1-6 we know that the integrated flux from a black body of absolute temperature T is J^BB = aR T4.
Although the radiation emerging from a star is by no means Planckian, it is nevertheless customary to define the effective temperature as the temperature a black body would have in order to emit the actual stellar flux—i.e.,
"A = J7 dv = 4k Jo" Hv dv = Lf(4izR2) (2-85)
Here L is the total luminosity and R is the radius of the star; the atmospheric thickness is assumed to be negligible compared to R. Although Teff has only an indirect physical significance, it is a convenient parameter with which to characterize the atmosphere, for typically the actual kinetic temperature Twill equal Tetf near the depth from which the continuum radiation emerges (i.e., unit optical depth at frequencies where the opacity is lowest). In spherical geometry equation (2-84) implies
r2& = constant = L/4n (2-86)
In an extended atmosphere it is no longer really possible to choose a unique radius R for the star and to define a unique value of Tcff; rather, L or the quantity r2^ should be regarded as fundamental. However the identification R = r(TR = f) is sometimes made, and a value of "Tcff" derived with this radius; here xR is the Rosseland optical depth scale (cf. §3-2).
Finally, it is important to return to the point raised in the discussion of the formal solution. Suppose that the opacity kv is independent of T; then for physically reasonable distributions of kv, the integral j kvBv(t) dv (which gives the total thermal emission) will be a monotone increasing function of T. Thus when we fix the total thermal emission at some value, we fix the local value of T. It is then clear from equation (2-83b) (and from the result of Exercise 2-13) that the local value of T is determined by the mean intensity, which depends upon the global properties of the atmosphere because it follows from a solution of the transfer equation. Thus the temperature at a given point in the atmosphere is to some extent determined by the temperature at all other points and, at the same time, helps to establish the temperature structure elsewhere. This nonlocalness in the problem is a result of radiative transfer, through which photons moving from one point in the medium to another lead to a fundamental coupling (i.e., interdependence) of the properties at those points.
2-5 The Diffusion Approximation
At great depths in a semi-infinite atmosphere the properties of the radiation field and the nature of the transfer equation become extremely simple. We can obtain in a straightforward way an asymptotic solution that applies
50 The Equation of Transfer
2-5 The Diffusion Approximation 51
throughout the interior of a star (except, of course, in convective zones), and that provides a lower boundary condition on the transfer problem in the stellar atmosphere. Consider first the properties of the radiation field. At depths in the medium much larger than a photon mean-free-path, the radiation is effectively trapped, becomes essentially isotropic, and (eventually) approaches thermal equilibrium so that SY -» Bv. Choose a reference point tv » 1, and expand Sv as a power-series:
SM = I [d?BJdxv"]{tv - ts)7«1
(2-87)
b = 0
Calculating the specific intensity for 0 < // < 1 from this source function with equation (2-50) we have
(2-88)
A similar result for -1 < ft ^ 0 follows from equation (2-51) and differs from equation (2-88) only by terms of order e~xl,i; in the limit of great depth the latter vanish and equation (2-88) applies on the full range - 1 ^ pi ^ 1. By substitution of equation (2-88) into the appropriate definitions we find for the moments
Adv)= t (2"+ irKd2"Bjdxv2»)
n = 0
Bv{xv) + ^{d2Bv/dx2)
Hv(xv) = X (2b + 3)-Hd2"+1BJdz2"+\
n = 0 1
(2-89a)
and
= -(dBJdxv) +
K„(tv) = t {In + 3)-\d2"BJdT2n)
(2-89b)
3
Bv(tv) + -{d2BJdx2]
(2-89c)
Note that only even-order terms survive in the even moments Jv and Kv and only odd-order terms in Hv.
We now inquire how rapidly these series converge. The derivatives can be approximated, at least to order-of-magnitude, by appropriate differences—
i.e., \d"Bv/drv"\ - BJxf. Then it is clear that the ratio of successive terms in the series is of order 0(1/tv2) or 0(1/<%V>2 Az2) where is the average opacity over the path-length Az. In terms of the photon mean-free-path /v ~ 1 /Xv> tr,e convergence factor is 0(/v2/Az2). It is clear that the convergence is quite rapid; indeed the estimate just given turns out to be conservative. Also, it is obvious that convergence will be most rapid at frequencies where the material is quite opaque. For a star as a whole one expects Az to be some significant fraction of a stellar radius, say Az ~ 10i0 cm, while — I (which implies a photon mean-free-path of 1 cm), so that the convergence factor of the series is of the order 10"20. It is clear that in the deep interior of the star only the leading terms are required.
In the limit of large depth we may therefore write
& Bv{Tv) + ft(dBv/dxv) Jv(tv) ^ Bv(xv)
Bv(xv) * {-{dBJdxf)
KjTv)^-By{xy)
(2-90a) (2-90b)
(2-90c) (2-90d)
In equation (2-90a) we have retained two terms so as to account for the nonzero flux [cf. equation (2-90c)]. Note that equations (2-90b) and (2-90d) show that lirn^ [Xv(tv)/Jv(tv)] = |, which is what we would expect for isotropic radiation; we shall show below that the ratio of the anisotropic to isotropic term in I(x, /i) becomes vanishingly small as i -> go, so that the limit just found is appropriate. Insofar as the specific intensity as given by equation (2-90a) was computed from the formal solution of the transfer equation, in effect using a source function Sv(tv) = Bv(tv) + (tv - xv)(dBv/dxv), equations (2-90) should obviate further use of the transfer equation. It is easy to verify that this is so, for by inspection one sees that substitution from equations (2-90) reduces both the transfer equation (2-36) and the zero-order moment equation (2-71) to the single requirement d2BJdx2 - 0 (already assumed), while the first-order moment equation (2-79) is identical to (2-90c). Thus, in effect, at great depth the transfer problem reduces to the single equation
1
Hv = -
dxv
(UK 3T
dT dz
(2-91)
It is clear that equations (2-90) and (2-91) can be used, as mentioned in §2-2, to set lower boundary conditions on the transfer equation in a semi-infinite atmosphere.
52
The Equation of Transfer
Equation (2-91) [and also equations (2-90)] is referred to as the diffusion approximation, primarily because of its formal similarity to other diffusion equations, which are of the form
(flux) = (diffusion coefficient) x (gradient of relevant physical variable)
e.g., d> = — k VT for heat conduction. The coefficient ^xv~1(^BJdT) is, in fact, sometimes called the radiative conductivity, a designation that is quite appropriate in view of the fact that %v~l = lv is the photon mean-free-path. Note that equation (2-91) exhibits the essential physical content of our earlier result that the flux computed by application of the O-operator to a linear source function depends only on the gradient of 5 [see discussion following equation (2-65)]. Also it shows that the mere fact that energy emerges from the star implies that the temperature must increase inward. Indeed, replacing H with (L/4nR2) and (dB/dz) with {aRTcAj%R\ and taking (yy ss 1, it is easy to show that the central temperature Tc of the sun must be of the order of 6 x 106 °K, a result consistent with our earlier statement that the ultimate energy source in a star is thermonuclear energy-release at the center.
In an intuitive picture of diffusion, one usually conceives of a slow leakage from a reservoir of large capacity by means of a seeping action. These ideas apply in the radiative diffusion limit as well. The diffusion approximation becomes valid at great optical depth (i.e., many photon mean-free-paths from the surface) whence many individual photon flights, with successive absorptions and emissions, are required before the photon finally trickles to the surface, and issues forth into interstellar space.
If we integrate equations (2-90a-c) over all frequencies, we obtain 7(t, ft) B(r) + 3>f.iH. The ratio of the anisotropic to isotropic terms gives a measure of the "drift" in the radiation flow; this ratio is
Anisotropic term 3H Isotropic term B
4
T
_cff
T
(2-92)
Clearly at great depth, where T becomes » TelT, the "leak" becomes ever smaller. The same result is found by a physical argument from a slightly different point of view. If kF is the energy flux carried from an element of material by photons of velocity c, the rate of energy flow per unit volume is (nF.'c): the energy content per unit volume is (4nJ/c) =k (4kB/c), so that (Rate of energy flowJ/(Energy content) = {F/4B) = £(Tet-r/T)4. Again, we see that diffusion, in the intuitive sense described above, occurs at great depth where T » Teff, while free flow of radiant energy occurs at the surface where
T a r,rf.
3
The Grey A tmosphere
The grey atmosphere problem provides an excellent introduction to the study of radiative transfer in stellar atmospheres. The nature of the defining assumptions is such that the problem becomes independent of the physical state of the material, and requires the solution of a relatively simple transfer equation. At the same time, the grey problem demonstrates how the constraint of radiative equilibrium can be satisfied, and the solution can be related to more general and more realistic physical situations. Furthermore, an exact solution of the problem can be obtained, and this provides a comparison standard against which we can evaluate the worth of various approximate numerical methods that can be applied in more complex cases.
3-1 Statement of the Problem
The problem is posed by making the simplifying assumption that the opacity of the material is independent of frequency; i.e., %v = This assumption is of course unrealistic in many cases. Yet as we shall see in later chapters, the opacity in some stars (e.g., the sun) is not too far from being grey and, in
54 The Grey Atmosphere
addition, it is possible partially to reduce the nongrey problem to the grey case by suitable choices olmean opacities. Thus the solution also provides a valuable starting approximation in the analysis of nongrey atmospheres.
If we assume yv = /, then the standard planar transfer equation (2-36) becomes
H{dlv/dz) = lv - Sv (3-1)
Then by integrating over frequency, and writing
and similarly for J, S, B, etc., we have
Hidl/cz) = I - S
(3-2)
(3-3)
If we impose the constraint of radiative equilibrium [equation 2-83b)], we require
Jo' ^ dv = Jo" *S> dv which, for grey material, reduces to J = S. Thus equation (3-3) becomes
p(dl/di) = 1 - J (3-5) which has the forma] solution [equation (2-57)]
J{x) = AT_S(t)] = Ajy(t)] = \ |o'"' J(t)Ei\t - r| di (3-6)
Equation (3-6) is a linear integral equation for J known as M lines equation; the grey problem itself is sometimes called Milne's problem. It is important to recognize that, when a solution of equation (3-6) is obtained, it satisfies simultaneously the transfer equation and the constraint of radiative equilibrium. The determination of such solutions in the nongrey case will occupy most of Chapter 7.
If we now introduce the additional hypothesis of LTE, then Sv = BV(T)—■ which, from the condition of radiative equilibrium, implies that
j(T) = S(t) = B[T(i)] = (<7RT*)/n
(3-7)
Thus, if we are given J(x\ the solution of the integral equation (3-6), then the additional premise of LTE allows us to associate a temperature with the radiative equilibrium radiation field via equation (3-7).
Several important results may be obtained from moments of equation (3-5). Taking the zero-order moment and imposing radiative equilibrium we have
(dH/dx) = J ~ S = J - J s 0
(3-8)
3-1 Statement of the Problem 55 which implies the flux is constant, while the first moment gives
(dK/ch) = H (3-9) which, because H is constant, yields the exact integral
K(x) = Ht
1 r
c = -Fx + c
(3-10)
To make further progress, we must relate J(x) to K(x). This is easily done on the basis of the discussion in §2-5, where we showed that at great depth the specific intensity is quite accurately represented by /(/() = 10 + /1(u, which produces a nonzero flux and also implies that, for t » 1, K(x) = ^J(t). Thus the fact that K{x) \Fx for x » 1 implies that at great depth
J(x)
- Fx (T » 1)
(3-11)
That is, asymptotically the mean intensity varies linearly with optical depth. On general grounds we expect the behavior of J{x) to depart most from linearity at the surface [note equation (2-63)], which suggests that a reasonable general expression for J(x) is
fix) = \f[t + q(x)-] = ~ (aA/n)lx + q{xj\ (3-12)
The function q{x), known as the Hopf junction, remains to be determined; from equation (3-6) it is clear that q{x) is a solution of the equation
z\ dt
(3-13)
Finally, we notice that because
lim
~ J(x) - K(x)
= -F lim [t 4- q(x) - t - c] ~ 0 (3-14)
we have c = qfxS) and hence can write equation (3-10) as
K(x) = -F[x + q(K)-]
(3-15)
The solution of the grey problem consists of the specification of q{z). Given q(x), the temperature distribution is obtained by combining equations (3-7) and (3-12) into the relation
(3-16)
56 The Grey Atmosphere
3-2 Relation to the Nongrey Problem: Mean Opacities 51
We shall derive approximate expressions for q(x) in §3-3 and describe the exact solution in §3-4. First, however, it is useful to delineate the nature, and extent, of the correspondence between the grey and nongrey problems.
3-2 Relation to the Nongrey Problem: Mean Opacities
The opacity in real stellar atmospheres usually exhibits strong frequency variations, at least when spectral lines are present. Although numerical methods now exist that allow a refined solution of nongrey transfer equations and an accurate determination of the temperature structure in a nongrey atmosphere, the calculations are, at best, laborious, and it is important to ask whether a significant connection exists between the grey and nongrey cases. We shall show in this section that such a connection, though limited in scope, does exist, and that, among other things, it permits the temperature distribution of the deepest atmospheric layers to be determined quite accurately from the grey solution.
Let us first compare side-by-side the grey and nongrey transfer equations. Starting with the transfer equation and calculating the zero and first-order moments we have, in the nongrey and grey cases respectively:
p(dljdz) = yJSv- lv) (3-17a) p(dl/dz) = y{J - I) (3-17b)
(dHJdz) = Xv{Sv - ivj (3-18a) (dH/dz) = 0 (3-lSb)
[cKJdz) = -XvHv (3-l9a) (dK/dz) = -yH (3-3 9b)
Here variables without frequency subscripts denote integrated quantities, as in equation (3-2). We now ask whether it is possible to define a mean opacity /. formed as a weighted average of the monochromatic opacity, in such a way that the monochromatic transfer equation, or one of its moments, when integrated over frequency, has exactly the same form as the grey equation. Several possible definitions have been suggested.
FLUX—WEIGHTED MEAN
Suppose we wish to define a mean opacity in such a way as to guarantee an exact correspondence between the integrated form of equation (3-19a) and the grey equation (3-19b). If such a mean can be constructed, then the relation K{x) = Hx + c will again be an exact integral, as it was in the grey case. Integrating equation (3-19a) over all frequencies we have
-(dK/dz) = Jj0 lvHv dv = yFH (3-20)
where the second equality produces the desired identification with equation (3-19b) if we define
Jf = H~l j» XvHvdv (3-21)
The opacity yF is called the flux-weighed mean. Note that this choice does not reduce the nongrey problem completely to the grey problem, for the monochromatic equation (3-18a) does not transform into equation (3-lSb) with this choice of y. Further, there is the practical problem that Hv is not known a priori, and therefore yF cannot actually be calculated until after the transfer equation is solved. This latter difficulty can be overcome by an iteration between construction of models and calculation of yF. Although the desired goal has not been fully attained, the fact that the flux-weighted mean preserves the K-integral is important, for it implies that the correct value is recovered for the radiation pressure [cf. equation (1-41)]. It also follows that the correct value of the radiation force, which is the gradient of the radiation pressure, is likewise obtained. Thus from equation (2-77) we have
(dpR/dx) = -xF"\dpR/dz) = [4n/cX,.) Jj° XvHv dv = (4nH/c) = (aT*f/c)
(3-22)
so that use of the flux-mean opacity produces a simple expression for the radiation pressure gradient. This is a result of practical value in the computation of model atmospheres for early-type stars, because in these objects radiation forces strongly affect the pressure (or density) structure of the atmosphere via the condition of hydrostatic equilibrium (or momentum balance in steady How).
ROSSELAND MEAN
Alternatively, suppose we wish to obtain the correct value for the integrated energy flux. From equations (3-19) it follows that this may be done if y is chosen such that
-J0" yv~l(dKv/cz) dv = jo" Hv dv = H~ -r\dK/dz) (3-23) or, equivaleutly,
Tl = J7 Xv-\dKJdz) dvjj* (BKJdz) dv (3-24)
Again we face the practical difficulty that Kv is not known a priori, and hence the indicated calculation cannot be performed until the transfer equation is
58 The Grey Atmosphere
3-2 Relation to the Nongrey Problem: Mean Opacities 59
solved. But the mean defined m equation (3-24) can be approximated' in the following way: at great depth in the atmosphere, Ky -> £jv while Jv -> Bv. Thus may write (dKJdz) a ^BJdT)(dT/dz). We then define the mean opacity y_R as
1
1r
cir
dz
ÖT
1 dB,
Jo v.. c
3\^;j°
Ay. ) I
j:
dB,
dv
or
X«-1 ^(7i/4aRT3)
^dBJdT) dv
(3-25)
(3-26)
The opacity y_R is called the Rossetand mean in honor of its originator. Note that the harmonic nature of the averaging process gives highest weight to those regions where the opacity is lowest, and, whereas a result, the greatest amount of radiation is transported—a very desirable feature. Again the use of yR or the mean defined by equation (3-24) does not permit a correspondence between equations (3-18a) and (348b), and hence does not allow the nongrey problem to be replaced by the grey problem. On the other hand, it is obvious that the approximations made to obtain equation (3-26) are precisely those introduced in the derivation of the diffusion approximation to the transfer equation (2-90; i-e.,
1 SBAtäT"
Hence use of %K is consistent with the diffusion approximation. Therefore on the Rosseland-mean optical-depth scale xR we must recover the correct asymptotic solution of the transfer equation and the correct flux transport at great depth. This implies that at great depth (TR » 1) the temperature structure is quite accurately given by the relation TA = f T*ff[tfl + '/(tr}]; see equation (3-16). It is therefore clear why Rosseland mean opacities are employed in studies of stellar interiors. Note also that so long as the diffusion approximation is valid, a simple expression can be written for the radiation force, namely
(dpR/dxR) = {\6KORT*ßcxR){-dTjdz)
(3-27)
Exercise 3-1: Derive equation (3-27).
Although the diffussion approximation is nearly exact at great depth, and provides the verv useful results just discussed, it must of course break down at the surface, and exact flux conservation is not guaranteed in the upper layers by use of the Rosseland mean, nor will it give the temperature structure
or the radiation force correctly in the outermost regions of the atmosphere. This point must be recognized clearly, for it is precisely these layers in which spectrum-formation occurs, and hence which are of primary interest in the analysis of stellar spectra.
planck and absorption means
Several other expressions for mean opacities may be chosen. For example, if we demand that the mean be defined to yield the correct integrated value of the thermal emission, then we require
Kp —
kvBv{T) dv
!B(T) = % Jo" KvBv(T) dvlaRT* (3-28)
Note that only the true absorption is used, and scattering is omitted. The opacity kf is known as the Planck mean; it has the advantage of being calculable without having to solve the transfer equation. On the other hand, kp does not permit a reduction of equation (348a) to (3-lSb) nor of (3-19a) to (349b), and therefore it lacks the desirable features possessed by yF or Xr. Nevertheless this mean does have additional significance.
In particular, near the surface of the star, the physical content of the condition of radiative equilibrium is expressed most directly by equation (3-4). In view of this constraint, a correspondence between equations (3-18a) and (348b) can be made near the surface if k satisfies the relation
J"* (kv - k)(Jv ~ Bv) dv = 0
(3-29)
Once the material becomes optically thin (i.e., tv < 1 at all frequencies), Jv becomes essentially fixed, and the integral above will be dominated by those frequencies where kv » k. If we represent Bv by a linear expansion on a i-seale, i.e.,
Bv(t) = Bv(x) + (dBy/dx)(l - t) w Bv(t) + (dBJdx)(KK)(t, - tv),
then by application of the A-operator we find [cf. equations (2-57), (2-58), and (2-63)],
-\bJt)E2(xy
(k/ky)(dBJdT)
(3-30)
In the limit x 0, E2(x) 1, and E3(t) - \, so the first term yields -£Bv(t) while the second becomes least important when kv » k [precisely the region of highest weight in equation (3-29)]. Thus to satisfy equation (3-29), k
60
The Grey A tmospherc
3-3 Approximate Solutions 61
should essentially fulfill the requirement
k,„B„ dv = K
Jo
B„ dv
(3-31)
which shows that the Planck mean is the choice most nearly consistent with the requirement of radiative equilibrium near the surface.
Alternatively, we might demand that the mean opacity yield the correct total for the amount of energy absorbed. We then obtain the absorption mean
^ = io
k„Ju dv / J
(3-32)
Again only the true absorption is included, and scattering is omitted. As was true for JF, we cannot calculate k, until a solution of the transfer equation has been obtained. Further, iCj does not permit a strict correspondence between the grey and nongrey forms of the transfer equation or any of its moments (as was also true for the Planck mean).
SUMMARY
We have seen that no one of the mean opacities described above allows, in itself, a complete correspondence of the nongrey problem to the grey problem. Yet mean opacities provide a useful first estimate of the temperature structure in a stellar atmosphere if we assume, as a starting approximation, T(zR) = Tgrey {zR), and then improve this estimate with a correction procedure that is designed to enforce radiative equilibrium for the nongrey radiation field. Indeed, the mean opacities %F, fcP, and k3 appear explicitly in some temperature-correction procedures.
From an historical point of view, it should be recognized that, before the advent of high-speed computers, the nongrey atmosphere problem required far too much calculation to permit a direct attack, and the use of jr and kf provided a practical method of approaching an otherwise intractable problem. In fact, the answers obtained in this way often do not compare too unfavorably with more recent results despite the apparent crudeness of the approximation. Only some of the more basic properties of mean opacities have been mentioned here; further information may be found in (419) and (361, §§34-35).
3-3 Approximate Solutions
THE EDDINGTON APPROXIMATION
In §2-5 it was shown that, at great depth in a stellar atmosphere, the relation / = 3K holds; further, in §1-4 (cf. Exercise 1-13), it was demonstrated
that this relation is also valid for a variety of other situations, including the two-stream approximation, which provides a rough representation of the radiation field near the boundary. In view of these results, Eddington made the simplifying assumption that J = 3K everywhere in the atmosphere. Then the exact integral K = \Fz + c implies that in the Eddington approximation JE(%) = |Ft + d. To evaluate the constant d we calculate the emergent flux and fit it to the desired value. Thus from equation (2-59) we have
t Ft + d ] E2(r) dx = 2c'£3(0) +-F
~ - 2£4(0) .i
(3-33)
so that, using the relation £„(0) = l/(n - 1) and demanding F(0) = F, we find d = \F. Thus
3
J At) = ~r F í t + —■
2
(3-34)
In Eddington's approximation q(x) = f. Imposing the constraint of radiative equilibrium and the assumption of LTE we have from equation (3-16)
(3-35)
Equation (3-35) predicts that the ratio of the boundary temperature to the
effective temperature is (F0/Tcff) =
= 0.841, which agrees fairly well
with the exact value (T0/Tcff) = (31/2/4)1/4 - 0.8114. Assuming S(z) = JE(z) we may calculate the angular dependence of the emergent radiation field in the Eddington approximation by substituting equation (3-34) into equation (2-52) to obtain
lE(0,p)^-F^(z
3
p-'expi-t/^dz = |fL + ^| (3-36)
which yields a very specific form for the Eddington-Barbier relation [cf. equation (2-53)]. The center of a star s disk, as seen by an external observer, corresponds to 0 = 0C', or p = 1; the limb is at 0 = 90", p = 0. The ratio 1(0, p)/1(0, 1), which gives the intensity at angle Ö = cos-1 p relative to disk-center, is referred to as the limb-darkening law. In the Eddington approximation the limb-darkening is
(3-37)
This result predicts the limb intensity to be 40 percent of the central intensity. Observations of the sun in the visual regions of the spectrum are actually in good agreement with this value and, in fact, it was precisely this agreement
62 The Grey A tmosphere
3-3 Approximate Solutions 63
that led K. Schwarzschild (416, 25) to propose the validity of radiative equilibrium in the outermost layers of the solar photosphere.
Equation (3-35) predicts that T = Tc[[ when x — §. This result has given rise to the useful conceptual notion that the "effective depth" of continuum formation is x % f; in fact, this is often a rather good estimate. In support of this idea we may note that a photon emitted outward from t = § has a chance of the order of e~0-67 x, 0.5 of emerging from the surface; this corresponds in a reasonable way with the place we would intuitively identify with the region of continuum formation.
Exercise 3-2: The Eddington-Barbier relation shows that the intensity 1(0, p.) is characteristic of S(t) at t{jx) ss p. Show then that the average depth that characterizes the fiux is = |.
Anticipating the exact solution given in Table 3-2, we can evaluate the accuracy of JE(r); one finds that the worst error occurs at the surface, where AJ/J = (JE - JwaciV^xaci = 0.155. Both the size of the error and the fact that it occurs at % = 0 are unsurprising when we recognize that the basic assumption upon which the derivation is based, namely J = 3K, is known to be inaccurate at the surface. We know that J(x) must satisfy the integral equation (3-6), and we know further that the A-operator has its largest effect at x = 0 [see equation (2-63) and associated discussion]. This suggests that an improved estimate of J(x) can be obtained from
jfHx) = Ae[JE(t)] = K
[~3 / A1 3 _
-Fit - i- - = ~*F x
_4 V V. 4
(3-38)
Recalling the properties of E„(i), it is clear that the largest difference between J{i\x) and JE(x) occurs at the surface, where we find J(P(0)/J^[0) — \. The new estimate of T0/Tetl is thus (-&ji/4 = 0.813 (exact value is 0.8114) and q(0) drops from f to = 0.583 (exact value is 1/^/3 = 0.577).
It is thus clear that an application of the A-operator has produced a dramatic improvement in the solution near the surface. Note, however, that there is no improvement in the solution at x -> go, where q remains at its original value §. In principle, successive applications of the A-operator should improve the solution, and, eventually, produce the exact solution. In fact, one can show that lim„^ A"(l) = 0 [see (684, 31)] so that an initial error s at any depth can ultimately be reduced to zero by repeated application of the A-operator. In practice, however, the convergence is too slow to be of value, for the effective range of the A-operator is of order At = 1, so errors at large depth are removed only "infinitely slowly." (We shall encounter this problem with A-iieration repeatedly in a wide variety of contexts! See, e.g.,
§6-1, §7-2.) Further, even a second application of the A-operator to J{E\x) introduces the functions A^E„(t)'}, which are cumbersome to compute [see (361), equations (14-50). (14-53). and (37-36) through (37-44)]. Therefore alternative methods for obtaining a solution must be developed.
Exercise 3-3: Using the results of Table 3-2, evaluate the percentage errors of J£(t) and Jjf'fr) and display them in a plot. The required values of Elt(r) can be found in (4, 245).
Exercise 3-4: Show that, although J t:{x) was derived assuming F = constant, the flux computed from Je{t) via equation (2-59) is not constant; make a plot of the error &F(z)/F.
Exercise 3-5: Apply the Z-operator [cf. equations (2-62) and (2-65)] to J E(%) to obtain K%\x) = t%F[It + | - f£4(T) + 2£5(t)]. Use this result to write an analytical expression for the variable Eddingfon factor fir) = X(t)/J(t). Show that y(T = cc) = £and/(r = 0) = $ = 0.405. Using the results of Table 3-2, evaluate the fractional error in /(t) [recall equation (3-15)] and plot it. Exercise 3-6: Show that the improved estimate of the emergent intensity obtained by using Jfii) is /<2,(0. fi) = |F('-,72 + 4// + lift + V) ln[(l + ft)/fi]}. Compare this result and IFA0,p) given by equation (3-36) with the exact result shown in Table 3-1, and plot a graph of their fractional errors.
ITERATION; THE UNSOLD PROCEDURE
The primary shortcoming of the A-iteration procedure is its failure to yield an improvement in the solution at great depth. Unsold (638, 141) proposed an ingenious alternative method that overcomes this inadequacy and can be generalized to the nongrey case. The basic idea is to start from an initial estimate for the source function Bít), and to derive a perturbation equation for a change AB(x) that more nearly satisfies the requirement of radiative equilibrium.
If we calculate the flux from the initial guess B(t), we will find that it is a function of depth, H{v), and not exactly constant unless B{x) happened to be the exact solution of the problem. From the first-order moment equation (3-9) we then have
K(x) = I H{x) ch'
C
(3-39)
If we make the Eddington approximation J(x) = 3X(t) and evaluate C by writing 7(0) = 2H(0) [cf. equation (3-34)] we obtain
J(x) x 3poH{ť)dx' + 2H(0) But from the zero-order moment equation (3-8) we have B(x) = J(x) - [dH(x)idx\
(3-40)
(3-41)
64 The Grey Atmosphere
so that
B(x) ft 3 £ H(t') dt' + 2fJ(0) - [dH(z)/dz] (3-42)
Equation (3-42) cannot be exact because of the approximations made in its derivation, but it can be used with sufficient accuracy to compute perturbations. In particular, suppose AB(x) is chosen just so the flux computed from B{x) + AB(z) is constant; thus
B(z) + AB[i) » 3 j*J H dz' + 2H (3-43)
and by subtraction of equation (3-42) from (3-43) we obtain an expression for AB(t), namely
AB(x) = 3 Jj AH(z') dz' + 2AH(0) - [d AH(x)/dT~] (3-44)
Thus if we know the flux errors AH{z) = H - H(x) we can compute the correction AB(z); this correction is then applied and new values of the flux are computed, which give new (smaller!) errors AH; the process is iterated until H becomes constant and AB -» 0 at all r. Equation (3-44) can be generalized for nongrey atmospheres: see equation (7-IS). Unsold's procedure is very powerful compared to A-iteration, for it provides a great improvement in the solution at depth as well as at the surface; this result is demonstrated in the following exercise.
Exercise 3-7: Assume a starting- solution Biz) = 3H(t +■ c); i.e., q{z) = c. (a) Show that AH{z) =e H - H(x) = jH[_EJx) - c£3(t)]. Obtain expressions for AH{(i) and d{AH)/dz. (b) Apply Unsold's- procedure and show that
AB(t) = 3H
~ll _ c - 1 cE2(z) + -n E3(x) + \ cEJ?) ~ \ E5ir) 24 2 2 2 i
(c) Show that, independent of the initial choice of c, the improved solution has qlO) = Yi = 0.583 (exact value 0.577| and q(o-_J = {i = 0.708 (exact value = 0.710). (d) Show that, in contrast, the A-opemtor acting on q = e gives yjD) = {c + 5, which agrees with Unsold's value only if c - f: and y(oo) = c, which shows no improvement whatever at depth.
THE METHOD OF DISCRETE ORDINATES
The method to be described now furnishes a means of obtaining both approximate solutions and the exact solution of the grey problem. More important, it introduces the fundamental mathematical scheme that provides the basis for practically all modern techniques of solving transfer equations. Introducing the definition of J [equation (1-4)] into equation (3-5), the
3-3 Approximate Solutions 65
transfer eqiim». i> d may be written in the form
p[di(z, p)jdz] = I(z, p)~l- f_s 1(t, p) dp (3-45)
which is classified as an integro-dijferential equation. The essential difficulty in obtaining the solution arises from the presence of the integral over angle. However, definite integrals such as that in equation (3-45) may be evaluated numerically by means of a quadrature sum consisting of the function evaluated at a finite set of points on the interval of integration times appropriate weights. Thus introducing {pj} on [ — 1, 1] we write, for any function f(p),
In. 1 "
j{p) dp & - aJXfij)
Z j=~n
2 J-i
(3-46)
The numbers {pj} are called the quadrature points, {o^} the quadrature weights, and {f(pj)} the discrete ordinates. Having chosen a definite quadrature formula, we replace the integro-differential transfer equation (3-45) with a coupled system of 2n ordinary differential equations:
pjoljdz) = - - x ajlj, (i = ±1,
±n)
(3-47)
where It denotes /(t. ^). The radiation field is no longer represented as a continuous function, but rather in terms of a set of pencils of radiation, each of which represents the value of I(p) over a definite interval. On physical grounds it is reasonable to expect the solution to become exact in the limit
n —> co.
The accuracy of the quadrature depends both upon the number of points, and upon their distribution on the interval. If the points are distributed uniformly on the interval we obtain a Newton-Cotes formula, of which Simpson's rule with points at {ju,-} = (—1,0, 1) is a familiar example. A better choice is to use a Gaussian formula, in which the 2n points on [ — 1, 1] are chosen to be the roots of the Legendre polynomial of order In. It would take us too far afield to discuss the construction and accuracy of quadrature formulae [see (161. Chap. 2)]; an important result that we shall merely state is this: an m-point Newton-Cotes formula gives exact results for polynomials of order m — 1 (for m even) or m (for m odd), but an m-point Gauss formula is exact for polynomials of order 2m — 1. For solving the transfer equation the double-Gauss formula is superior (619) to the ordinary (or "single") Gauss formula. Here one chooses two separate n-point quadratures on the ranges (— i ^ p ^ 0) and (0 < ^ < 1); on each range the n points are given by the roots of the Legendre polynomial of order n, scaled from [—1, 1] to
66
The Grey Atmosphere
3-3 Approximate Solutions 67
the appropriate range. This approach has the advantage that + p) and 7(t, -j-t) are approximated independently, and thus the integration formula can account, without difficulty, for the physical fact that l{-u) = Oatr = 0 while I{ + p) remains finite. In the single-Gauss formula, the discontinuity in I(p)alp = Owhent = 0 introduces significant errors. In all of these formulae the points are chosen symmetrically about zero so that ft_j = — ph while
We now wish to solve the system of equations (3-47). Observing that the system is linear and of the first order, we take a trial solution of the form 7; = fa exp(-kx), where g{ and k are to be specified. Substituting into equation (3-47) we find
(3-48)
so that gt = c/(l + kpt). If we use this specific form for gi and again substitute the trial solution into equation (3-47), we find
i t aj{l + kfij)-1 = 1
(3-49)
This is the characteristic equation, which is satisfied only by certain values of fc, called the characteristic roots (eigenvalues). Recalling that a_j = ai and - = —p^ equation (3-49) can be used to define the characteristic function T{k\
T(k2) = 1 - J o/l - fcV)"1
(3-50)
The roots of T [i.e., those values of k for which T{k2) = 0] are the desired characteristic roots. If we set f{p) = 1 in equation (3-46) we see that
- V a, = Y a-. = 1 -l j i—i i
zj=-» ;=l
hence it follows that k2 = 0 is a solution of the characteristic equation; i.e., T(k2 = 0) = 0. There are an additional (n - 1) nonzero roots, which may be seen as follows. Note that k2 = pf2 is a pole of T, which becomes infinite for these values of k2. For k2 = - e, T(k2) < 0, and by making e arbitrarily small, T(k2) -> - oo. Similarly, for k2 = 2 + e, T{k2) > 0, and as ?. 0, r -> +oo. It is thus clear that T must pass through zero somewhere on the interval between successive poles, hence the (n — 1) nonzero roots must satisfy the relations
Ml"2 < kl2 < 1*2
- 2
<
< k;_} < /(„"
where we have ordered {/V, such that pt > pi+l. Note that the largest ps must be less than unity, hence the smallest nonzero k2 must be greater than unity. In all there exist In - 2 nonzero values of the k\ in pairs of the form ±kt(i = l,...,n - I). The general solution of the system (3-47) is therefore of the form
(3-51)
We must now seek the special solution corresponding to the root k2 = 0. In view of equation (3-11) which shows that J(r) must become linear in t at depth, we examine trial solutions of the form 7- = b(x + qt). Substituting this expression into equation (3-47) we obtain
4i = fr + ^ x ajaJ
(3-52)
Now observing that if we set f[p) = p in equation (3-46) the quadrature sum £ ajfij is zero, we see that equation (3-52) is satisfied by the simple choice . To satisfy the latter constraint we set
= 0 (3-55)
and use the upper boundary condition to write the n equations
q - ,(i + "£ LJl - k^)-' = 0, (i = l,...,«)
a = l
(3-56)
Solution of equations (3-56) yields the » unknowns Q and I,. In addition, squire that the flux equal the nominal flux F. Thus we demand that
2 P I{p)pdpi = 2 I djfijlj = F J " i j = — H
Substituting equations (3-53) (with L_a ^ 0), we have
2b
j = - n j - " 3
(3-57)
(3-58)
In view of equation (3-46) the first sum is zero, the second equals f, and from the characteristic equation the fourth is found to be zero.
Exercise 3-8: Verify the statement made above about the values of the sums in equation (3-58).
Thus we find that b ~ |F, as would be expected from equation (3-11); note also that the quadrature calculation yields a constant flux automatically. Finally, the complete solution for the semi-infinite atmosphere may be written
H-l
(i = 1, ■ ■ ■ , «)
(3-59)
We may compute J(x) by substhutmg equations («^*^tUie formula; making use of the characteristic equation (3-49), we obtain
it - 1
=_F t + ß + I
4 \ a = 1
(3-60)
3-3 Approximate Solutions
69
and thus the discrete-ordinate representation of q(x) is
~ x * x\~2
j=i a — ßj j=i j=i A — P-j
v
ajftf
- X (3-62)
To clear T{X) of fractions, multiply through by Yl'}=i {ft/ - X); this yields P(X) = f[ (Hj2 ~ X)T{X) = t atp2 f[ W ~ X) (3-63)
J = l i=l ii-i
which is clearly a polynomial of order (n — 1) in X. But we know that T(X) has the (n - 1) roots A'j = 1/fc,2, . . . , .Y„_ 1 = 1/7^-,, so P(X) must have the form C(X - XX)---{X — A'„_ ]). To evaluate the constant, we note that the coefficient of the term in A"'7-1 in equation (3-63) is(— l)"~L Ya = i aifii2 = (—1 ^; this is simply C itself. We thus have
P(X)=l-(X1 -A)
-x)
70 The Grey Atmosphere and therefore
T(X) = -3
n ^ -
n (*2 - *>
(3-64)
From equation (3-62) we see that T(X - 0) = lf hence setting X = 0 in equation (3-64) we obtain the useful (see below) result that
(3-65)
Now consider the emergent intensity 2(0, p.). Define a function S(^) such that
2=1
The surface boundary conditions in equation (3-56) may then be written
We then generalize S(/*) to apply at all values of p and write
HO, p) = -FS(~u)
(3-67)
(3-68)
for u > 0 Note that, with this generalization, 1(0, - ji) is not sO, but will in general have nonzero values for By working with S(p) we can
obtain an expression for /(0, p) that does not involve the constants La and Q explicitly. Clear fractions from equation (3-66) to obtain
Sipi'li f1 - M = (q - ^) ri U ~ M + i ^ ii (1 ~ M (3-69)
The righthand expression is obviously a polynomial of order n in ^u. But S(p) has the it roots pu ..., /(„; hence this polynomial must be of the form C(p — jUi) ■ ■ • (ju — /:<„)■ To find C we note that the coefficient pn on the righthand side of equation (3-69) is (-1)%--■/<„_!, which is C itself. Therefore
{ß) n:-i a - m n;=i & -
which, when inserted into equation (3-68), yields the desired expression for 1(0, p). It is customary to define a limb-darkening function H(p) as
(3-71)
(3-70)
H(p) s /(0,/t)//(0,0)
3-4 Exact Solution 71
[note that unlike equation (3-37), the reference point here is the limb not disk-center]. In the discrete-ordinate approximation we find from equations (3-68) and (3-70)
H(p) =
Fl (1 + *~V)
n (i + m
(3-72)
By further analysis using S(p) it is possible to write explicit expressions for the Lfs and 0 m terms of the points {Mt} and the roots {ka} [see (161, 78-79; 684,25)]. ' 1
Before leaving the discrete ordinate method, let us show that ^(0) = I/J3 is the exact value. First, note that in the nth discrete approximation
while from equation (3-59) we find
(3-73)
(3-74)
independent of the order n. Thus we conclude that in the exact solution J(0) 1(0, 0). But from equations (3-68) and (3-70)
W0)=3-FS(O) = 3-Fp1---pnkl...ki,_1
(3-75)
Then, combining equations (3-65)., (3-75), and (3-73), we deduce that, independent of the order n, qH(0) = 1/^3; hence this result is exact.
3-4 Exact Solution
The exact solution for q(x) and H(p) can be obtained by taking the discrete-ordinate method to the limit n -> 00 (161, Chap. 5; 361, §27), by applying the principle of invariance (161, Chap. 4; 361, §28), or by direct Laplace transform methods (361. §29; 684. Chap 3). Only certain important results will be quoted here, and the reader should consult the references cited for details.
Several expressions for H(p) exist (361, 186-187); a form convenient for numerical computation is
H(ft) = (/' + 1)"1,2 exp
1
J
*/2#tan ^jiitanö)
71 JO
1 - 0 cot 0
dO
(3-76)
72 The Grey Atmosphere
table 3-1
Exitct Limb-Darkening Law for Gn>\> Atmosphere*
1(0, n)lF i< i(0, u) J-
0.0 0.43301 0.6 0.95009
o.i 0.54012 0.7 1.02796
0.2 Ü.62802 0.8 1.10535
0.3 0.71123 0.9 1.18238
0.4 0.79210 1.0 1.25912
0.5 0.87156
Evaluation of this integ 3-1.
ral (152; 518) yields the results summarized in Table
The value of t/(co) can be obtained by noting from equation (3-15) that K(0) = Hq(co), and thus, using equation (3-71),
£ H(/,)/i2 rfji I^H[n)iidM (3-77) From the known expressions for H(p) one then obtains
which yields q(co) = 0.71044609 (519). Finally, a closed-form expression can
(/(co) =
table 3-2
Tluj Exact Solution for q{:
t t
0.00 0.577351 0.8 0.693534
0.01 0.588236 1.0 0.698540
0.03 0.601242 1.5 0.705130
0.05 0.610758 2.0 0.707916
0.10 0.627919 2.5 0.70*191
0.20 0.649550 3.0 0.709806
0.30 0.663365 3.5 0.710120
0.40 0.673090 4.0 0.7 L0270
0.50 0.680240 5.0 0.710398
0.60 0.685S01 co 0.710446
3-5 Emergent Flux from a Grey Atmosphere 73 be written (407) for q{x\ namely
1 i*i e~xiudu
q{x) - q{rxf)---
2/3 Jo H(u)Z{u)
where H[u) is as defined above and Z(u) ~
1--u In —
2 \ 1 — u
2 1 2 *
(3-79)
(3-80)
Results obtained from a numerical evaluation of equation (3-79) are given in Table 3-2.
3-5 Emergent Flux from a Grey Atmosphere
The basic physical assumption made in the grey-atmosphere problem is that the opacity is independent of frequency. In this event, the constraint of radiative equilibrium reduces to S(x) = J(t), and the problem simplifies to that of obtaining the solution of equation (3-6). If, in addition, it is assumed that LTE prevails, then we may equate B(x) = {urT^/ti to J(x), and thus arrive at equation (3-16) for T(t). The radiation field has a dependence upon frequency because the source function, which we assume is Bv(x), depends upon frequency. Given the source function, the flux, also frequency-dependent, can be computed at any depth by means of equation (2-59), which now reads
Fv(t) = 2 J" Bv[T[tj]E2(t -x)dt~2 po Bv\T(t)\E2{x - t) dt (3-81)
The temperature appears in the Planck function only in the combination (hv/kT); further, the ratio T(r)/Teff is a unique function of t [cf. equation (3-16)]—say, l/p(i). We may therefore simplify the equation by introducing a parameter a = (hv/kTei(\ in terms of which we can write (hv/kT) = ap(x). Expressing the flux in the same units by writing Fa(x) = Fv(x){dv/da\ and using the relation F = (grE^/k, we may rewrite equation (3-81) as
KM F
4nkA h*c2aR
E2(t - x) dt
exp[ap(í)] - 1 J" exp[ap(í)] — 1
E2(x - t) dt
(3-82)
The expression in the brackets is a function of a and x only, and may be calculated once and for all. A tabulation of Ea(x)/F is given in (161, 295), and a plot of the function is displayed in Figure 3-1. The figure shows clearly the degradation of photon energies as they transfer from depth to the surface; for example, the most common photon energy at t = 0 is only about
74
3-6 Small Departures from Greyness 75
F
0.20
0.15
0.10
- t = y \ 1 1 1 1 -
/ Av'xv
_ / / /i.o -
/ / / X= 20 -
t = 0\\ \\t = 2,0 -
W x l i 1 0.4 —\< -
2 4 6 8 10 12
mgure 3-1
Frequency distribution of tlux at selected depths in a grey atmosphere. From (155), by permission,
75 percent of that at t = 1, This progressive reddening of photons in the outer layers results from the outward decrease in temperature produced by the requirement of radiative equilibrium.
3-6 Small Departures from Greyness
By use of an appropriate mean opacity, it is possible to account for small departures from greyness, at least approximately, and thus to extend greatly the usefulness of the results obtained for a grey atmosphere. Suppose that the frequency variation of the opacity is the same at all depths so that we can write
7M + (Q = Xcjy (3-83)
where
Xv
Xv
F{}} dv
(3-84)
lu equation (3-84), Fj,11 denotes the flux in a grey atmosphere. The mean opacity %c defined in equation (3-84) is known as the Chandrasekhar mean; as we shah" see in what follows, this mean is constructed in a way that makes optimum use of the information contained in the grey solution. Unlike the flux mean [equation (3-21)] the Chandrasekhar mean can be computed straightaway for any given frequency dependence (i.e., p\, or yv) of the opacity because F[l) is a known function.
Let us now consider how; we may solve the nongrey transfer problem using a method of successive approximations. The nongrey transfer equation (assuming LTE) is
p(eljef) = (7JyXIy ~ Bj = (i + /M/v " Bv) (3-85)
If we suppose that the departures p\ may be regarded as small, then a first approximation to the solution of equation (3-85) is obtained by setting fjf = 0 initially. The transfer equation then returns to the equation for the grey problem itself, namely
p(dl[y\!dx) = l\» - By (3-86) whose solution is already known. To obtain a second approximation we write
p(el\2)jdx) = /<2) - Bv + fijll" - Bv)
= /<2) - Bv 4- MdlWdf) (3-87)
by substitution from equation (3-86). If we demand that the radiation field resulting from this second approximation should satisfy the constraint of radiative equilibrium, then we must have (JFl2)fdz) = 0, where F<2) is the integrated flux; then from equation (3-87)
J,2) - B +
d
dx
= 0
(3-88)
(3-89)
(3-90)
J ff.F[J>dv
But note that equations (3-83) and (3-84) imply that
IF = Z j," 0 + /w dv = ycF + yc J* BVF[» dv
or that
J; /w ^ = o
Therefore the radiative equilibrium constraint for a nongrey atmosphere treated by the above approximation scheme collapses to Ji2) = B(x). This shows that the grey solution for T(x) on the Chandrasekhar-mean optical-depth scale will automatically satisfy the condition of radiative equilibrium in the first approximation to the nongrey atmosphere. The method for obtaining higher approximations is described in (161,296 ft.)
At this first level of approximation we may compute the nongrey emergent intensity as
1M l') = J7 BXT{t)] exp{-ytt/Ml(yv/;0 dx (3-91)
76 The Grey Atmosphere
and the flux as
F„(0) = 2 J7' £v[T(t)]£2(7vT)yv dr (3-92) If we introduce the parameter a = (hv/kTLli) as was done in §3-5, and write
~exp(a7cff/70) - 1
= Bv(T0)bJz) (3-93)
exp[ap(T)] — 1 equations (3-91) and (3-92) reduce to the parametric forms 1J(X p) = JBV| 70) J* ha(T) exp( - >\m(yM ) may be computed once and for all; tables of these functions are given in (161, 306-307).
The functions /J) and .^(v, normalized such that J 0V dv = 1. Thus if the total number of atoms in state i is nh the number capable of absorbing at frequency v is n.(v) = riifa. In making the transition from level i to level j, the atom absorbs photons of energy hvu = Ej - E,. Thus the rate at which energy is removed from an incident beam of radiation is
4-1 The Einstein Relations for Bound-Bound Transitions 79
ajv ~ n,-(ßrj/ivrj/47i)0v/v
(4-2)
where av denotes a macroscopic absorption coefficient (per unit volume), uncorrected for stimulated emission (see below).
For atoms returning from level j to level i, two processes are possible. The first of these is a spontaneous transition with the emission of a photon. Writing the probability of spontaneous emission per unit time as A}i, the rate of emission of energy is
//.(spontaneous) = fi;(J4jí/ivij/4ji)i/'ť
(4-3)
Here the emission profile \f/v specifies the number of atoms in the upper state that can emit photons on the frequency range (v, v + dv); it is normalized such that | \j/v dv = 1. The other possible return process is a transition induced by the radiation field (stimulated emission). The rate at which such emission occurs is assumed proportional to the intensity of the incident radiation field. The energy emitted may be written in terms of the Einstein coefficient Bjt as
^(induced) = nfBjfivJAn^J, (4-4)
In writing equation (4-4), use has been made of the result that the profile for induced emission is the same as that for spontaneous emission, as can be shown from general quantum mechanical considerations (197, §62). It should be noted that spontaneous emission takes place isotropically. Induced emission is proportional to and has the same angular distribution as Iv; because of this, induced emissions are sometimes considered to be negative absorptions, though this is not quite correct, for in general t//v will not be identical to ,,.
The coefficients A}i, B}i, and Bu are simply related, as can be shown by calculating rates of absorption and emission in thermodynamic equilibrium. In strict T.E., the radiation field is isotropic, and Iv = By, the Planck function. Furthermore, in T.E. the occupation numbers of levels i and j are related
by the Boltzmann law [cf. equation (5-5)]
("A)* = (gj/gdexpi-hvtj/kT)
(4-5)
Moreover, in T.E., \j/v = $
(4-18)
where 0 denotes the angle between the acceleration v and f, 6 and are unit vectors in a spherical system defined by v and f, and v is evaluated at a
82 Absorption Cross-Sections
lime r' = t ~ (rfc). The power radiated per cm2 is given by the Poynting vector [cf. equation (1-35)]
S = (c/4tt)(E x H) - (^/4«f V) sin2 ö f (4-19)
Now integrating over a sphere of radius r by forming S ■ dA where dA = (r2 dco)v = (r2 dp d)> = (e2x02o>4/3c3)
(4-21)
Then
Because the oscillator is radiating away energy, the oscillation will eventually decay. We may describe the decay in terms of a damping force that may be viewed as the force exerted on the moving particle by its own electromagnetic field. To calculate the effective damping force, we assume that the rate of work done by it accounts for the energy loss by the oscillator. Thus from equation (4-20) we write
Frad ■ v + {2eH*/3c3) = 0 (4-22)
J]'2 (FMiJ ■ v) dt + [2e2/3c3) ^ " v ■ v - J'2 V ■ v dtj = 0 (4-23)
Over a cycle the integrated term vanishes, therefore on the average
Frad = (2e2/3c*)\ (4-24)
To a good order of approximation we may calculate V from its value for the undamped osdlJalor, namely V = —(o02\, and thus we can write
(4-25) (4-26)
where
Frad = -myv
y = (2e2w02/3mc3)
The constant y is called the cluneal damping constant because of the formal resemblance of the radiation reaction term as expressed in equation (4-25) to a viscous damping term.
4-2 The Calculation of Transition Probabilities S3
We now can calculate the scattering coefficient for a classical oscillator in an imposed electromagnetic field. In the classical picture the interaction is a conservative scattering process; hence we can compute the energy scattered out of a beam by calculating the energy radiated by an oscillator driven by the electromagnetic field of the incident radiation. The equation of motion for an oscillator of mass m and charge e driven by a field of amplitude E0 and frequency u> is
mix
= eE0eiat - myx
(4-27)
Taking a trial solution for x that is proportional to exp(tor). we find for the steady-state solution
x = Re from which we derive
x = v = Re
ie/ni^e1
(or
0)(
iyoj
-(eco2/m)K0e'-
(4-28)
(4-29)
(CO2 ~ QJ02) + iyoj
Thus, substituting into equation (4-20) and averaging over a period, we have
' e^m
=
F 2
3m2c*)(co2 - co02)2 +
(4-30)
which is to be identified with the total energy scattered out of the beam To calcu ate the energy scattered, we suppose that the scattering cross-section
ha0oTod write %j *]r/o % ~ ^ ^ - ^ m"1-2 ™ fo-d
that to produce a correspondence between the macroscopic and electromagnetic descriptions of radiation, we have /0 = (cE02/Sn). Thus
'cE2
^d j^dpSip- p0)ö(
0o
(«)> = cr(o)) = 3.. (4-43)
A general state of the system at time t = 0 can be expanded in terms of the eigenstates (which form a complete set) by writing
J|y)(0)>. For a system in a general state with wavefunction .
86 Absorption Cross-Sections
If the atom is unperturbed (i.e., H ~ HA\ then the afs are constant. If, however, the atom is perturbed by some potential V, then the cCs will change with time; this is interpreted as the atom undergoing transitions from one state to another. An example of such a perturbation is that exerted by an external electromagnetic field upon the atomic electrons. In the lowest order of approximation we can assume the atom is in a uniform, time-varying electromagnetic field, E = (£0 cos wt)i. The assumption of uniformity is reasonable for light waves, which have wavelengths (X ~ 10"~5 cm) that are large compared to a typical atomic dimension characterized by the Bohr radius (a0 = 5 x 10"9 cm). The potential of the atomic electrons in the field is
V = e £ E ■ Xi = E ■ d = (£0 cos cot)(i ■ d) (4-46)
i = i
where d is the dipole moment of the atom. With a perturbing potential, Schrodinger's equation becomes
(HA + F)i// = ih(8if//dt) Substituting equation (4-45) for }j/ we have
In view of equation (4-39), equation (4-48) reduces to
(4-47)
(4-48)
(4-49)
We may isolate a particular coefficient d,„ by using the orthogonality of the t/)'s. Thus, multiply equation (4-49) by - E a»W exp[i(£m - £»)t/A]<^*Mtfn>
(4-50)
Now writing wm = (£m - En)/h and Fm„ = \4>'i\V\n>, a^d using equation (4-43), equation (4-50) reduces to
(i
For the perturbing potential given by equation (4-46), we see that Vmn = (E0 cos cot)i ■ <0*|d|0„> = (E0 cos «t)(T ■ dmn)
= 2hmi! cos tot
(4-52)
4-2 The Calculation of Transition Probabilities 87
The quantities d„,H are called the dipole moment matrix elements. Substituting equation (4-52) into equation (4-51) we have
djt) - (ihr^cUm^-'ie^
(4-53)
We now make the simplifying assumption that, at time t = 0, the atom is in a definite eigenstate k, and we consider a time interval T so short that this state is not appreciably depopulated. That is, at t = 0, we assume ak(0) = 1 anda„(0) = 0 for all n ^ k. Moreover, we choose Tsuch that ak(t) « 1 for all t ^ T. Then the sum in equation (4-53) may be replaced by a single term
djt) = (ihy'h^e^ie^ + e-***) Integrating equation (4-54) with respect to time we obtain
(4-54)
a (t) = — [cxp^^Mfc ~ co)t] - 1 + exp[i{comk + co)t] - lj ih \ {comk - of) {ojmk + of) j
As we are interested in absorption processes we choose Em > Ek, so that coink > 0. From the denominator of the first term in the braces, we see that the dominant contribution to am(t) will come when oj ~ comk (i.e., radiation near the line frequency is most effective in producing transitions). It is clear that the second term can be neglected in comparison with the first. Then, writing x = (oj — comk\ and forming \am\2 = a*am> we have
2 = 4ft"2/,
lF 2li ■ H \2x'
1
-xt
2
(4-56)
Equation (4-56) gives the number of k -> m transitions (per atom in the initial state k) produced in time t by radiation of frequency v = to/2-jr. To calculate the total number of transitions, we must sum over all frequencies that can contribute. Suppose that the line has a profile (pv that falls sharply to zero over some characteristic frequency interval Av, and that over this range (at least) the intensity of radiation (and hence £02) is constant with a value Jv. Then integrate over dv = doj/2ii = dxj2n, and define u = %xt, to obtain
= (£07«2)|i-dmJ2r
(4-57)
Now for thermal radiation, a characteristic frequency interval Aco over which the intensity will be constant is of the order (kT/Ii) ~ 1015, while transition times t ~ 10"8 sec; hence the limits + U on the integral may be extended formally to ± oo. The value of the integral is then found in standard tables to be 7i. Further, as was shown in §1-2, ER = (4nJv/c) = (E02/Sn), therefore
JVkn = (87i2/ft2c)|i-dmJ2Jv£
(4-58)
88 Absorption Cross-Sections
Now in terms of the Einstein coefficient Bkm,
km = BkmJvt
hence Bkm = (8n2/h2c)\\ ■ = dmk2 Q,4>) - RMWiO,® (4-71)
where Y"' is the spherical harmonic function expressible in terms of associated Lcgendre functions, and Rnl is the radial function, which can be expressed in terms of associated Laguerre polynomials and exponentials. These functions are normalized such that
and Jo ^ Jo $)\*Y, '"'(0, n). The statistical weight is
(4-74)
90 Absorption Cro.ys-Scciions
which follows from the fact that the allowed values of J are 0 ^ / ^ n - 1; those of m are — / ^ m < /; and each »im state has two possible spin orientations s = ±j.
Exercise -1-2: Derive equation (4-74).
Because the wave functions are known analytically, explicit expressions can be derived for the oscillator strengths:
1/1
fin', 0=3-^-^5
l\max(/, 0 l>;nJ}
1)
(4-75)
and
1 f1
fin, n) = -—,2 "
v -1
£ l'a2(n,l';nj' - 1) ' = 1
n'- 1
r = o
where
(4-76)
(4-77)
An explicit expression for a2 was first derived by Gordon (254) and an explicit KTfk n) was derived by Menzel and Pekens (417). Extensive tables ^"j ) can be found in (4)7) and (257). A very convenient form for hydrogen oscillator strengths is obtained by expressing them in terms of the semi-classical value derived by Kramers (363), namely
JkW- »1
32
W3 V"'
-3
n3n'5
(4-78)
which shows the principal dependences of / upon n' and 11. It is then customary to express the exact /-value in terms of Kramers' approximation fK by writing
f(n', it) = g,(n', n)fK(n', n) (4-79)
where gr{n', n) is called the Gaunt factor. The Gaunt factors are all numbers of order unity; an extensive tabulation of t/jfn', n) can be found in (60).
Exercise 4-3: Using the analytical expressions for hydrogen wave functions given in texts on quantum mechanics [e.g., (392, 183)], calculate the /values for La(n' = 1 -» n ~ 2) and Ha(n' = 2 -• n = 3). Obtain values for each f(n\ /'; n, I) and combine ihese lo find f(n', n). Compare your values with those given in tables [e.g.. (9, 70)].
4-2 The Calculation of Transition Probabilities 91 TRANSITION PROBABILITIES FOR LIGHT ELEMENTS
(a) Hartree -Fock Method. When the atom has more than one electron, the wave equation can no longer be solved in closed form, and approximations must be made. The actual Hamiltonian for an N-electron atom is
H= ~(h2/2m) Y V,2 - Z (2c2Ai)
(4-80)
all pairs U. J)
The first term represents the kinetic energy of the electrons, the second their electrostatic potential with the nucleus of charge Z, and the third their mutual Coulomb repulsion. It is the last term that causes the principal difficulties.
One of the most important methods of deriving approximate wave functions is Hartree's self-consistent field method. In this approach, the sum over electron pairs is replaced for each electron by its spherical average. An excellent description of how this average is computed is given in (576, Chaps. 3 and 9). Each electron then moves in a potential that depends only upon its distance from the nucleus, and we make the replacement
I
(4-81)
ill pairs (*". J)
This results in the approximation of the actual potential by a central field. With a central field potential, the angular factors in the Schrodinger equation can be separated out in exactly the same way as for hydrogen, and for each electron the wave function has the form
[/,.(/-. 0, 0; n, I, m, 5) = r~ xPJr)YT{B, 4>)X(s)
(4-82)
where the normalizations given in equation (4-72) still apply. The functions [/; are called electron orbitals. The radial equation for each orbital is of the form (r in units of a0, E in Rydbergs)
(d2PJdr2) + [En! + 2/-"1Zeff(r) - /(/ + ljr"2]^, = 0 (4-83)
Here Zeff(r) is the "effective nuclear charge" sensed by an electron after allowance is made for shielding by other electrons [using the central fields of equation (4-81)]. The atom is now considered to be made up of N such orbitals, and these are used to construct the wave function for the entire configuration.
Because of the Pauli exclusion principle, the set of four quantum numbers (n, I, m, s) for each orbital cannot be identical for any two orbitals. Also, the
92 Absorption Cross-Sections
wave function of the atom must be constructed so that it is antisymmetric under the interchange of the coordinates of any two electrons. In practice these conditions may be met by writing the wave function as a Slater determinant (576, Chap. 12),
(4-84)
where the numbers 1, 2,.... N denote the orbitals of electrons 1, 2, etc., while a, /i,..., v stand for the space and spin coordinates of elections
a,..., v, respectively.
The solution for the wave function is carried out Ueratively. Thus Zeif(r) depends in an involved way on integrals over the electron orbitals, but in turn it determines those orbitals. Therefore, we start with an approximate set of orbitals, compute Zcff, solve for the F„,"s, recompute Z£{f, and iterate until the procedure converges. The calculations are time-consuming and laborious, but are within the capabilities of modern computers, and a large number of wave functions for a wide variety of atomic configurations are now available.
A specific term in an atomic spectrum can be characterized by certain quantum numbers describing the atom as a whole. In light atoms these describe the total orbital angular momentum L (the vector sum of the individual 1,'s), the total spin angular momentum S (the vector sum of the s/s), and the total angular momentum J. which is the vector sum of L and S. This type of coupling of the individual momenta is called [L-S] or Russell-Saunders coupling. As a given L, S, and J may result from more than one arrangement of the individual W m's, and s's of the orbitals, the complete wave function will, in general, consist of a sum of Slater determinants, and thus may be very complicated.
In calculating transition probabilities, it is generally assumed that only one orbital is different between the initial and final state—i.e., only one electron undergoes a transition. In this case the matrix element ri} can be split into factors, one coming from the initial and final radial wave functions, and another depending on the angular and spin wave functions. It is customary, therefore, to write the expression for the line strength in the form
S(n', If, S\ J'\n._ L, S, J) = aQ2eza2{n', l ■ n, lYAJfyn^) (4-85)
Here
(4^-ir(j0"^iv*
(4-86)
4-2 The Calculation of Transition Probabilities 93
where /m.lx = max(/, /'). The factor y\Jf) is the strength of the multiplet, depending on nLS and n'L'S', and the factor y>\<£) is the strength of the line within the multiplet. Extensive tables of .9'{Ji) and .y(if) can be found in (11, Chap. 8 and Appendix; 9, §§26-28; 250; 251), and general formulae for computing these factors are given in (534; 535; 572, §§10.8-10.10; and 191). Generally, the most difficult part of the calculation is the determination of a2, but serious complications also occur when there are deviations from L-S coupling.
(b) The Coulomb Approximation. Because of the labor involved in obtaining ), He I A6678 Up 'P-3d lDl and He 1x4471 (2p 3P-4.f 3D) lines using equations (4-66) and (4-85) with Coulomb approximation values of o2 and multiplet- and tine-strengths from tables. Compare with standard values in (672).
94
Absorption Cross-Sections
4-3 The Einstein-Milne Relations for the Continuum
The Einstein relations were generalized to bound-free processes by Milne (461) in a paper of considerable interest and importance. We consider photoionization processes that start with atoms (or ions) in a definite bound state (not necessarily the ground state) and produce an ion in a definite state (perhaps excited) of the next highest ionization stage plus a free electron moving with velocity v. The inverse process is a recombination of a free electron by a collision with an ion (in the particular state mentioned above) to form an atom (in the proper state). The recombination process can occur spontaneously or can be induced by incident radiation. Let n0 be the number density of the atoms, nl the density of the ions, and ne the density of free electrons. The electrons have a Maxwellian velocity distribution, and we write njv) dv for the number with speeds on the range (i\ v 4- dv). Let pv be the probability of photoionization of an atom by a photon in the frequency range (v, v + dv); then the number of photoionizations in time dt on this frequency range is n0pvIv dv dt. The usual energy absorption coefficient Thus to reduce equation (4-90) to the Planck function we must have
F(v) = (2hv3/c2)G{v)
(4-94)
and
p, = ($nm2v2qJh2q0)G(,v) -= (4nc2m2v2giilr'gK)v3)F{v) (4-95)
These are the continuum analogues of equations (4-8) and (4-9). Again we recognize that, although these relations have been derived from thermodynamic equilibrium arguments, the quantities pv, F(v), and G{v) must really depend only on the properties of the atom; hence equations (4-94) and (4-95) are true in general.
The great importance of the results just derived becomes more clearly manifest when we write the transfer equation assuming that at the frequency under consideration only the particular photoionization and recombination processes considered above occur. The generalization to a multilevel, multi-atom case with several overlapping opacities and cmissivities is trivial because each term adds linearly and the conclusions we shah derive apply to the sum. The transfer equation is
pidljdz) = -n0pjivf. + iVi»[F0:) + G(i;)/v](/i2v/m) (4-96)
Neither n0 nor n: necessarily has its LTE value in the above equation. If we are to write the transfer equation in standard form, then it is clear that the absorption coefficient corrected for stimulated emission must be
kv = [«0 - nxne(v)[_hG{v)lmp^)(pyhv)
(4-97)
Using equations (4-88), (4-91), (4-92) and (4-95), and recalling that av - pvhv, we find
kv = (n(} - n%e-hv!kT)xv (4-98)
In equation (4-98), n% denotes the LTE value of n0 computed from equation (4-92) using the actual values of nx and ne (i.e., the LTE population relative
96
A bsorption Cross-Sections
to the actual ion density). In the particular case of LTE where n0 = ng,
= ngav(I - e
(4-99)
As was true for bound-bound transitions, the term (1 — c~'n'!kl) is usually called the correction factor for stimulated emission; but it is clear that this expression is correct only for LTE. Indeed we see from equation (4-98) that the stimulated emission always occurs at the LTE rate (if we understand Hq to have the meaning given above); this must be true because the recombination process is a collisional process involving particles with an equilibrium (i.e., Maxweihan) velocity distribution. Note the contrast here with the result given in equation 14-13) for bound-bound transitions, where in general the stimulated emission term does not have its equilibrium value. When departures from LTE affect the bound-free opacity, they change the direct absorption term involving n() (which in general will not equal r$) We shall use these results both in calculating the stimulated emission rates in the equations of statistical equilibrium [of. equation (5-63)] and in writing a general expression for the opacity [cf. equation (7-1)].
Returning to equation (4-96) and examining the term involving F{v), it is clear that the emissiviiy is
(4-100)
which, with the help of equations (4-88), (4-91), (4-92), and (4-95), can be written
nv = {2hvi/c2Have
-™*T = n3«v(l - e-hv,Kr)Bv = k*Bv(T) (4-101)
Thus the continuum emissivity always occurs at the LTE rate (if n% is defined as above), which is what we would expect, for the recombination process is collisional. Notice that this derivation recovers the Kirchhoff-Planck law. equation (2-6), and extends its validity somewhat. Again, notice the contrast with the bound-bound spontaneous emission where departures from LTE enter directly if nj is not identical to nf. These results will be exploited in calculating spontaneous emission rates in the statistical equilibrium equations [cf. equation (5-61)] and in writing a general expression for the emissivity [cf. equation (7-2)].
Exercise 4-5: Verify equations (4-93), (4-98), and (4-101).
4-4 Continuum Absorption Cross-Sections
Cross-sections for bound-free absorption can be calculated quantum mechanically by essentially the same methods as used in ^4-2 for bound-
4-4 Continuum Absorption Cross-Sections 97
bound transitions. Consider absorptions from a bound state n, of statistical weight g„, to the continuum in a frequency interval Av. The free states have wave functions characterized by E, the energy of the free electron, and are normalized such that
<£'|£> = S(E' - E) (4-102)
so that there are AE states in the energy interval AE. Thus by analogy with equation (4-65) and in view of (4-36) we can write
gn), in a series. Defining £,,, = — l<'v„i\ we can determine the behavior of p(rnl, I) versus Enl; in favorable cases p is a simple function of e (say, constant, or linear in e). It is then assumed that this variation of p with f can be extrapolated into the continuum (i.e., for r. > 0) to give ;j'(e). This establishes the properties of the
98
Absorption Cross-Sections
4-4 Continuum Absorption Cross-Sections 99
continuum wave functions. The radial matrix element can then be evaluated using hydrogenic wave functions, and the cross-section when the energy of the ejected electron is k2 == Zze (in Rydbergs) can be written
a(nik2) = 8.56 x l n) is
hvu-„ = <«[(l/n')2 - (1/n)2] (4-109)
If a free state has the imaginary quantum number ik, then by analogy
hvn,k = #[{W2 + U/M2] = {l
(»ln'2) + - mv2
(4-110)
where the first term clearly represents the ionization potential from bound state n', and the second the energy of the free electron. Note that k -> oo at the ionization limit and becomes small high in the continuum.
The formula for the continuum oscillator strength follows from a generalization of equations (4-78) and (4-79) to
1
3n^3j\n'^J{n
1 1\"3 J2+ Sil« k)
(4-n r
where gn is the bound-free Gaunt factor. Formulae for the Gaunt factor are given in (417) and an extensive numerical tabulation is given in (352); glt is a number of order unity at the ionization threshold, shows a slow rise to about 1.10 (in the limit as n' -» oo) at about 1 Rydbcrg above threshold, and then decreases to small values in the A'-ray region. The absorption cross-section can now be derived by substituting equation (4-111) into (4-105), noting from equation (4-110) that for n fixed,
(dk/dv) = ~(hk3/2@)
We then find
which, in view of equation (4-69). reduces to
*,, = /^WV 32
ri5k*J (hvjPlf
64n4 me11 3N/3 ch6
L •)
ft v
nv~
(4-112)
(4-11:
(4-114)
where :/C = 2.815 x iO29. Thus bound-free absorption from level n commences abruptly at the threshold frequency v„ = {M/hn1) and falls off at higher frequencies as v~3 (neglecting the weak variation of the Gaunt factor). The threshold cross-section is given by
a(vwn)— 7.91 x 10~1S nga{n, v„) cm2
The opacity per cm5 of the stellar material can be computed by multiplying the cross-section for level n by the number of hydrogen atoms (per cm3) in that level, and summing over all levels that can absorb at a given frequency v (i.e., all ft such that v„ ^ v). The bound-free opacity of hydrogen calculated in this way has a jagged character, as shown in Figure 4-1. Except for the hottest stars, most of the hydrogen is in the ground stale, and the absorption edge at A912A (one Rydberg) is extremely strong. For 912 A ^ X ^ 3647 A, absorptions from the ground state can no longer occur, and the dominant opacity source is photoionization from the n = 2 level (Balmer continuum). Similarly, for 3647 A s£ ;. ^ 8206 A, both n = 1 and n = 2 cannot absorb, and the dominant continuum is from n = 3 (Paschen continuum); and so on. Actually the opacity variation shown in Figure 4-1 is idealized in that there
100
-17 F\ » = 4
log kv
1/A
figure 4-1
Opacity from neutral hydrogen at T = 12.500 K. and T = 25.000' R, in LTC: photcionizatiori edges are labeled with the quantum number of stale from which they arise. Ordinal?: sum ofboii'id free and free-free opacily in cm: ''atom: itbsciw. \ jk where /. is in microns.
exists a series of lines converging on each photoionization threshold at the series limit. Near the limit the lines blend together smoothly and merge into the continuum. Bound-free absorption from hydrogen is the dominant continuum opacity source in stars of spectral types A and B.
Let us now consider the free-free opacity of hydrogen. In this process, a free electron passing near a proton causes a transitory dipole moment, and absorptions and emissions of photons (with a consequent change in the electron's energy) become possible. By analogy with the calculation of bound-free absorption, we introduce imaginary quantum numbers for both the initial and final states, say Ik and (7, such that, iff is the initial velocity of the free electron, and v is the frequency of the radiation absorbed, then
g$k 2 = ~ mv2
(4-115)
and
4-4 Continuum Absorption Cross-Sections 101 Mk~2 + = m~2 (4.116)
Assume that absorptions take place from a band of states dk into a band of states dl = (dl/dv) Av; then in equation (4-105) we replace fnk with fu dk, and Ak with dl to obtain
a(v, v) - (ne2/nic)fkl dk(dljdv)
(4-117)
as the absorption coefficient per ion and per electron moving with velocity v The appropriate generalization of equations (4-78) and (4-79) is
/« =
64 1/1
I2 ~/c3/3
(4-118)
where yk is the statistical weight of a free electron, given by quantum statistics as
fh = (2/T3)(47wrV Ju) = (16n&m2v/h3k3)dk (4-119)
the second equality following from equation (4-115). Substituting into equation (4-117) we find
a(\\ u) —
64
31,3
h>k
tic'
'' ~ " 11 " " I / i */nit"' v) iiu \ ™ >\3n^j{l6nMmh)[hv) ~kH3~\Jv^ {4A20}
ffm(v. t') /dl
and making use of the relation that for k (or r) fixed, (dl/dv) = (hi3/2<%) from equation (4-116), we obtain
ct(v, v) =
IPlhe2 \{fim(v,v)
(4-121)
The total absorption cross-section per ion and per electron is obtained by summing over all incident electron velocities, assuming a Maxwellian velocity distribution as given by equation (4-91). The result is
(4-122)
where use has been made of equation (4-69), and gm is the thermal average ot the Gaunt factor
0iii(v, T) ~ tjm(v, v)e-"du
where u = (mv2/2kT).
Exercise 4-6: Verify equations (4-122) and (4-123).
(4-123)
102
Absorption Cross-Sections
Inserting numerical values for the atomic constants, multiplying-by the electron and proton densities, and correcting for stimulated emission (notice that because the process is collisional it is always in LTE, at the actual electron and ion densities), we obtain the opacity coefficient
Kv(free-free) = 3.69 x 108 gin(v, T)v ^"^(l - e
) (4-124)
Formulae for Z;w are given in (417) and extensive tables can be found in (85) and (352). The free-free opacity plays an ever more important role at low frequencies compared to the bound-free, because of the decreasing number of photoiouization edges that contribute as v -> 0. Further, the free-free becomes more important at high temperatures, for as can be seen from equation (4-92), in the limit (kT/-/Uin) » 1, the bound state populations vary as ni a ivipT_t; hence the ratio of free-free to bound-free opacity rises -j. T in the high-temperature limit. The free-free process is the dominant true absorption mechanism in, e.g., the O-stars.
the negative hydrogen ion
Hydrogen, because of its large polarizability, can form a negative ion consisting of a proton and two electrons. This ion has a single bound state with a binding energy of0.754eV. Because of its low binding energy, H~ does not exist at high temperatures (it is destroyed by ionization) but is prevalent mainly in the atmospheres of solar-type and cooler stars. It was recognized by Pannekoek and Wifdt that H" could be an important opacity source in such stars. As it turns out. the abosprtion cross-section of H- is large and, although only a small fraction of the hydrogen exists in this form, the opacity from H " is the dominant one in the atmospheres of cooler stars.
The negative hydrogen ion can absorb and emit radiation via both bound-free and free-free processes; i.e.,
FT + hv "H + e{v)
where hnv1 = hv - 0.754 eV, and
(4-125)
(4-126)
where hnv'
H + e{v) + hv ^ Ft + e(v')
wuciu j"w — 2"'v + hv. In the free-free process, an electron passing near to a neutral hydrogen atom induces, by polarization, a temporary dipole moment that can interact with Ihe radiation field, leading to absorptions and emissions. The bound-free absorption process has its threshold at about 16500 A (1.65/if corresponding to the detachment energy. It reaches a maximum cross-section of about 4 x 10"17 cm2 at 8500 A and decreases toward shorter wavelengths. The free-free cross-section is about equal to the
103
10 12 14 16 if A(A)/1000
20 22 24 26 28
figure 4-2
Bound-free and free-free opacity from H " at T = 6300°K. Ordinate: cross-section (x 1026) per neutral H atom and per unit electron pressure p, = nekT; abscissa: a/1000 where I is in A.
bound free near 15000 A (1.5 p) and increases towards longer wavelengths. The summed absorption coefficient (see Figure 4-2) has a minimum at about 1.6 p; although other absorption processes act to wash out the minimum, the opacity for cool stars is smallest near this wavelength.
The determination of cross-sections for the two processes mentioned above is difficult and has been attempted both theoretically and experimentally. Very elaborate wave functions are required to give the desired accuracy. Pioneer calculations that gave fairly accurate values were carried out by Chandrasekhar and Breen (162). These were shown to be in accord with empirically deduced values for the absorption coefficient in the sun, and led to the firm identification of FT as the major opacity source in the solar atmosphere (see §3-6). More accurate values are now available for both the bound-free (242) and free-free (604) cross-sections; these are in good agreement with experimental values.
In LTE, the number of H " ions per cm3 is given by a Saha formula [see equation (5-14)] that is of the form «*(H~) = n(H)peQ>(T) where n(H) is the number of hydrogen atoms per cm5, pL, = nekT, and O(T) contains the temperature dependence of the ionization equilibrium. The LTE opacity can thus be written k*(H~) = ov(H~)h(H)pe0>(7)(l - e~hv'kT); departures from LTE may enter both in the calculation of n(H~) and in the stimulated emission
104
Absorption Cross-Sections
4-5 Continuum Scattering Cross-Sections 105
correction factor. Because /v*(H~) is proportional to pe, it is clear that it will be a more important opacity source in dwarfs than in giants. Also, because the electron density in G-type and cooler stars depends upon the abundance of the metals, H" will be a much weaker opacity source in Population II stars (which have low heavy-element abundances).
OTHER IONS Ol7 HYDROGEN
Hydrogen exists in two other forms thai can contribute significantly to the opacity in stellar atmospheres, namely H2 + and H2~. The positive ion H2+ consists of a single electron shared by two protons; absorption cross-sections are given in (72) and (117). As the number density of H2+ is proportional to o(H)npi H2 + contributes significant)) to the total opacity only for the temperature-pressure range where both neutral and ionized H-atoms exist simultaneously in appreciable numbers; i.e., where the hydrogen is about half-ionized. This range is characteristic of the A-stars, and H,+ makes about a 10 percent contribution to the opacity in the visible part of the spectrum (the H> + absorption peak at /A 100 A is swamped by the Balmer continuum of hydrogen).
The negative molecular ion H2 ~ exists only at relatively low temperatures, characteristic of the M-slars, and its free-free continuum makes a significant contribution at long wavelengths (the bound-free process is negligible). In this process, an electron passing near an H2 molecule temporarily induces a dipole moment by polarization effects, and this moment can interact with the radiation field. The H2"" continuum tends to fill in the opacity minimum of at 1.6/j. The free-free cross-section is given in (592).
HELIUM
Helium is observed in stellar spectra in both its neutral and singly-ionized states. Because the ionization potential of neutral helium is 24.58 eV. it persists to temperatures characteristic of the B-stars, where hydrogen is already strongly ionized; in the O-stars, He IT becomes a major opacity source. The threshold for abosprtion from the ground state of He I is at /.504 A: the ultraviolet spectrum for / < 504 A is dominated by He I absorption for stars of types B0 and cooler. The excited stales of helium fall into two groups, singlets and triplets, and each (n, /, 5) state has a different ionization energy. Roughly speaking, the ionization energies lie close to the hydrogenic value at the same n; thus helium contributes several absorption edges near each hydrogen edge (for n 5= 2). Because the excitation energy of even the lowest excited state is so large (19.72 eV), helium adds to the opacity in the visible regions of stellar spectra only in hot (B-type) stars. Generally, helium
is appreciably ionized before the excited states contribute to the opacity significantly. In a few stars the helium to hydrogen ratio is anomalous, and approaches or exceeds unity; here helium can dominate the opacity.
Helium is a three-body system, and exact wave functions cannot be obtained. A number of special methods can be applied to obtain accurate approximate wave functions [see (87, §§24-32; 577, §§18.1 — 18.3)]; variational techniques applied to the ground state have been refined to the point of yielding very precise wave functions. The ground-state absorption coefficient calculated from an accurate Hartree-Fock wave function is given in (603); this agrees well with experimental values (311). Absorption cross-sections from the235\ VP, 2lSy2{P levels have been calculated using accurate variational bound-state wave functions and close-coupling free-state wave functions (332), For higher excited states, precise cross-sections have not been published, and here one may use the quantum defect method.
Ionized helium is a hydrogenic ion with Z = 2. As energies in such ions scale as Z2, the frequencies of the ionization edges, v„, are larger by a factor of four, and the ground-state edge occurs at 1227 A. This edge dominates the far ultraviolet spectrum of O-stars, except at the very highest temperatures where the helium becomes doubly ionized. The n = 2 edge of He II coincides with the hydrogen Lyman continuum; higher edges from states with even quantum numbers coincide with hydrogen edges from states with n = »(He II)/2, while those from states with odd quantum numbers fall between the hydrogen edges.
Hydrogenic cross-sections can be used for He II. but the bound-free cross sections are a factor of Z4 larger, and the free-free are a factor of Z2 larger. The hydrogenic Gaunt factors apply if one evaluates them as functions of (v./vj. He II affects the visible spectrum only in stars of types BO and hotter.
Finally, helium can give rise to a free-free opacity in cool stars. Cross-sections for this process are given in (593; 340).
Exercise 4-7: Calculate the pUotoioni2,aUon crass-sections of Hel from the four a — 2 states by the quantum defect method and compare these results with the more accurate values cited above.
4-5 Continuum Scattering Cross-Sections
As mentioned in Chapter 2, continuum radiation may be scattered as well as absorbed. In the latter case, photons are destroyed, and their energy contributed at least partially to the thermal content of the gas. In a scattering event, the photon is not destroyed, but merely redistributed in angle, and perhaps shifted slight!)' in frequency. Cross-sections for the two most important scattering processes in stellar atmospheres are derived in this section.
106 Absorption Cross-Sections
THOMSON SCATTERING
The scattering of light by free electrons is referred to as Thomson scattering. The classical formula for this process can be obtained directly from equation (4-32) by noting that for an unbound electron both the resonant frequency ) - (STre^mV^oA'K.;2 - co2)2 = ffJiXA^j2 - to2)2 (4-128)
Far from the resonant frequency, a(oj) vanes as cm4 or A"4, which leads to a strong color dependence of the scattered radiation; a well-known example of this dependence is the blue color of the sky, resulting from sunlight scattered by molecules of air.
5-1 Local Thermodynamic Equilibrium 109
The Equations of Statistical Equilibrium
ionizations (and their inverses), and thus acts to help determine the occupation numbers of the gas; we shall show in fact, that radiative transitions dominate the state of the gas. In this case the occupation numbers must be determined from equations of statistical equilibrium, which specify all of the microprocesses that produce transitions from one atomic state to another.
The fact that the state of the material depends upon the radiation field introduces the essential difficulty of stellar atmospheres theory for, as we mentioned in Chapter 2, the radiation field, in turn, depends on the occupation numbers via the absorptivity and emissivity and their effects upon the transfer of radiation through the atmosphere. Thus what is required is a completely self-consistent simultaneous solution of both the radiative transfer and statistical equilibrium equations. This is a difficult problem in general, and its solution occupies the bulk of Chapters 7, 11, and 12 of this book. For the present we shall only show that there are strong expectations that the state of the material will depart from that predicted by LTE, which is therefore at best a computational expedient. If in any particular case the occupation numbers obtained from the general analysis happen to agree with those predicted by LTE, then one may legitimately use the LTE assumption; but for a wide range of problems (line-formation in particular), such agreement is not generally attained (nor can we accurately predict a priori when it will be for most cases of interest!).
Stellar atmospheres are regions of high temperature and low density. Therefore the gas consists mainly of single atoms, ions, and free electrons; in cooler stars molecules also form. Because of the low densities, the material always behaves as a perfect gas. The state of the gas is specified when we know the distribution of the particles over all available bound and free energy levels—i.e., when we know the occupation numbers of these levels. We then have the information required to compute the gas pressure, mass density, opacity, emissivity, etc. of the material.
To specify occupation numbers, we must deal with the phenomena of excitation and ionization of each chemical species in the gas. One approach is to assume that we may apply the equilibrium relations of statistical mechanics and thermodynamics at local values of the temperature and density; this is the local thermodynamic equilibrium (LTE) approach. As we shall see, LTE provides an extremely convenient method for computing the particle distribution functions. One of the fundamental properties of stellar atmospheres, however, is the presence of an intense radiation field whose character is very different from the equilibrium Planck distribution. This radiation field interacts strongly with the material via radiative excitations and photo-
5-1 Local Thermodynamic Equilibrium
In thermodynamic equilibrium, the state of the gas (i.e., the distribution of atoms over bound and free states) is specified uniquely by two thermodynamic variables (we shall choose the absolute temperature T and the total particle density N) via the well-known equilibrium relations of statistical mechanics. These relations will not all be derived in this chapter, as they are easily found in standard texts [see. e.g., (565, Chaps. 12, 14. and J5; 11, Chap. 3)] but will be summarized in forms useful for further developments in this book. The assumption of LTE asserts that we may employ these same relations in a stellar atmosphere at the local values T(r) and N(r) despite the gradients that exist in the atmosphere. This simple assumption is actually a very strong one, for it implies that we propose to calculate the above-mentioned distribution functions without reference to the physical ensemble in which the given element of material is found. Thus it is assumed that it is irrelevant whether the material is contained within an equilibrium cavity (the classical hohlraum), an atmosphere with a strong radiation field, or in the exhaust of a space vehicle, despite the obvious dissimilarities of these situations! In LTE, we have a purely local theory, which makes no allowance for coupling of the state of one element of gas with that of another,
110 The Equations of Statistical Equilibrium
say by radiative exchange (except as may be imposed by certain global constraints on the atmosphere—-e.g., hydrostatic or radiative equilibrium). Moreover, in LTE the absolute temperature T has a quite general significance. The same T applies in the calculation of the velocity distribution functions of atoms, ions, and electrons; the distribution of atoms and ions over all states (Boltzmann-Saha equations); and the distribution of thermal emission (Planck function). In short, the full implications of the LTE assumption are quite sweeping. It is this very fact which makes it so effective in reducing the complexity of the equations, and at the same time so difficult to justify physically and so vulnerable to error.
the maxwelliav velocity distribution
The probability, in thermodynamic equilibrium, that a particle of mass m at temperature T has a velocity on the range (v, v 4- dv) is given by the
Maxweltian velocity distribution
f(\) dvx dvy dvz =
m
2%kT
exp[-m(aJC2 4- v2 + fz2)/2kT] dvx dvy dvz (5-1)
or, in terms of speeds on the range {l\ v + dv)
f{v) dv = l~T^) exp(-mv2/2kT)4nv2 dv \2nkiJ
(5-2)
These distributions may be characterized in terms of the most probable speed
v0 = (2/cT/m)' = 12.85(X7lCrW km/sec (5-3)
where A is the atomic weight of the particle. Related parameters are the root-mean-square speed * = (IkT/m)*, and the root-mean-square velocity in one component (e.g., along the line of sight) = (fcT/m)*.
the boltzmann excitation equation
In thermodynamic equilibrium at temperature T, atoms are distributed over their bound levels according to the Boltzmann excitation equation. Let n,jk denote the number density of atoms in excited level i of ionization state j of chemical species k. Let; = 0 denote neutral atoms, / = 1 singly ionized atoms, etc. Measure the excitation energy yijk relative to the ground state of the atom. Let gijk denote the statistical weight assigned to the level to account for degenerate sublevels (e.g., the 2J + 1 m-states in the absence of a magnetic field). Then, according to the Boltzmann law the population of
5-1 Local Thermodynamic Equilibrium 111
any excited level is
(>Vno,-J* = (9ukhloik)^p(-y.uJkT)
(5-4)
where the subscript 0 denotes the ground level and the superscript * denotes LTE. For any two excited levels./ and m,
= (Qmjijfhjk) txp{~hvtJkT)
(5-5)
where !iv,m is the energy of a photon that equals the energy difference between the levels. In calculations of ionization equilibria, we typically wish to know the total number of atoms in a particular ionization stale, which can be written as
Njk = = ^lQjk/(lnjk)YdfJiik^P(-Xijk/kT)
i i
= (nU/g0!h)UJk(T) (5-6) where L'rt(T) = £ gijk exp(-y_ijk/kT) (5-7)
i
is called the partition function. A form of equation (5-4) customarily used in classical eurve-of-growth analyses of spectra (see sjijlO-3 and 10-4) is
("ijkl%k)* = gm^P(-Xtjk/l lik(v) to denote the number of ions in the ground level with accompanying free electron with speed in the range (v, v + dv), we may apply equation (5-4) to write
Ki.^KcJ* = W«)/»o,o.Jexp[-(^i0,k + hnv2)/kT] (5-9)
We identify tyeicclron with the number of phase space elements available to the free electron, which, according to quantum statistics, is
Qciceu™ = 2(dx dX (h ilPx dP>- dp*)/*1*
where the factor of 2 accounts for the two possible orientations of the electron spin. We choose the space-volume element to contain exactly one free electron, and make the substitution dx dy dz - nt~\ We rewrite the momentum volume element in terms of the electron's speed,
dpx dpy dpz = 4jip2 dp = Annr'v2 dv Then equation (5-9) becomes
[«0.1.AlrVn0.0,k]*
= 8wn3/T3to0. iVffo.o.kK"1 ™v[-('Ai.o,k + jmv2)/kT] v2 dv (5-10)
5-1 Local Thermodynamic Equilibrium 113
Now, summing over all final states by integrating over the electron velocity distribution we obtain
(no,i.ftnf/"o.o,fc)*
= 8mn3h-3(g^UkfgD 0tk){2kTjmf Qxp{~Xl0k/kT) e~x\2 dx (5-11) or, evaluating the integral,
«o,o, ft = ^i.^Ht^Trm/cT^^o^/^jJexpf^o^y^T) (5-12)
which is a basic form of Sahas equation. Note that in the derivation we made no explicit reference to the ionization state of the initial "atom," hence we may extend equation (5-12) to apply between any two successive stages of ionization
nojk = <,•+!.*»* W/2nmkT)*{g0ik/g0tJ+uk) expfa^/ZcT) (5-13)
If, further, we apply Boltzmann's formula, equation (5-4), we obtain an expression for the occupation number of any state of ion j in terms of the temperature, electron density, and ground state population of ion j + 1, namely
nM = no,j+ijMgijk/9o.j+i,k)CIT^^exp[(Xljk - Xijk)/kT]
= n0,j+iikne^iJk(T) (5-14)
Equation (5-14) is the most useful form of Saha's equation for the formalism we shall employ, and will be used to define LTE populations in the full non-LTE equations of statistical equilibrium (for this reason the superscript * on n0,j+i.ft and ne has been omitted). The constant has the value C7 = 2.07 x 10"16 in cgs units.
By applying equation (5-6), we may rewrite equation (5-14) as
nfjk = Nf+lf&e[g0iJ+1jUj+lik{T)]®lJk{T) s NJ+^nfi^T) (5-15)
Further, by summing over all bound levels of the lower ionization stage and again using equation (5-6), we obtain an equation for the ratio of the total number of atoms in successive stages of ionization;
(Njk/Nj+iJJ* = nlU^TyUj^JT^CJ-* cxp(Xljk/kT) = n^jk(T)
(5-16)
By recursive application of equation (5-15) between successive stages of ionization, we can obtain an expression for the fraction of atoms of chemical
114 The Equations of Slat is! teal Equilibrium
species k in ionization stage j relative to the total number of atoms of that species:
fM, T) = (NJk/JVk)*
(NJ-uk/NJk)*---(Njk/Nj+1:kr _
~ 1 + {Nj-uk/NJkr + tJVJ-1,t/JVJk)*(WJ_2it/JVJ_1,J1)* + ■ • ■
+ (JV^1>k/iVJfc)*---(JV0Jt/JVlfc)*
= Jn l«A(T)]/x 'n^nAtnj
= Pjk(ne> T)/St0\„ T), 0= l,...,Jk) (5-17)
where Jfc is the last ionization stage of species k considered. We observe the convention that the product term for / = Jk (which formally becomes void) is replaced by unity in both the numerator and denominator.
Consideration of the above results shows that, if we know (ne, T\ then we ; may determine, for any chemical species k, the fraction in any chosen ionization stage from equation (5-17), and in any particular excitation state from equation (5-15). If, in addition, we know the total number density of atoms of this species, we can obtain absolute occupation numbers nijk. In j practice this procedure is useful in LTE calculations of line spectra where \ we are given a model atmosphere that specifies nfz) and T(z). In computation !: of the model itself, however, we generally do not know ns{z\ but rather the total particle density N{z); we must then determine nei and as can be seen from equation (5-17), this implies we must solve a nonlinear set of equations. Let us therefore now consider methods of solving the nonlinear problem.
5-2 The LTE Equation of State for Ionizing Material
The Saha-Boltzmann relations allow a computation of the fraction of each chemical species in various stages of ionization, and the number of free electrons that each contributes to the plasma. Stellar atmospheres consist of a mixture of elements with widely differing ionization potentials; in general some of the species may be neutral while others are singly or multiply ionized. Usually the transition from one ionic stage to the next occurs fairly abruptly with increasing temperature, and normally a particular chemical species exists essentially entirely in two successive ionization stages. This provides a sensitive diagnostic tool to infer the temperature structure of a stellar atmosphere, for it implies that ratios of line strengths of two successive ionic spectra (e.g., He I and He IT, or Ca I and Ca II) will vary
5-2 The LTE Equation of State for Ionizinu Materia! 115
rapidly as a function of temperature. In fact, this was the basis upon which the first understanding of the spectral sequence as a temperature sequence was built by Sana (546; 547), Pannekoek (495), Cecilia Payne (501), and Fowler and Milne (222; 223). In normal stellar atmospheres hydrogen is by far the most abundant constituent, and helium is next most abundant with N(He)/JV(H) « 0.1. The heavier elements have much smaller abundances relative to hydrogen [see, e.g., (252) for element abundances in the solar atmosphere]. At typical temperatures m the solar atmosphere (6000eK) hydrogen is essentially neutral, and the electrons are contributed mainly by the "metals" such as Na, Mg, Al. Si, Ca, and Fe. At higher temperatures, characteristic of the A-stars H0,000°K), hydrogen ionizes and becomes the dominant source of electrons. At very high temperatures, characteristic of the O- and early B-stars, helium ionizes and makes an appreciable contribution of electrons.
CHARGE AND PARTICLE CONSERVATION
In calculations of stellar atmospheres we specify the gas pressure from the equation of hydrostatic equilibrium. Thus, given pfi and T, we know the total number density N from the relation
pq = NkT= (A>aIonis + A/,ons + ne)kT=(t\n + n,)kT (5-18)
Here NN denotes the density of "nuclei"; i.e., atoms and ions of all types. In equation (5-18) and subsequent equations of this section we suppress the L'*" that denotes LTE for notational simplicity. We define the abundance ak of chemical species k to be such that Nk = akNN where T,kak = 1. Then
Nk - ak(N - n.) (5-19)
summarises the constraint of particle conservation (i.e., I>Nk = A/N). In addition we require the plasma to be electrically neutral; then the number of free electrons equals the total ionic charge, and the condition of charge conservation reads
«* = z i JK* = i A', E jfJk{»,, T) = {N- ne) I ak £ jfJlc(ne. T)
k j= 1 k j= 1 k j= 1
(5-20)
As mentioned above, if we know (ne, T) we may calculate N and the fJk directly. But if we know (N, T), we must find ne from a nonlinear equation. Before the availability of electronic computers this problem was solved by constructing tables of log pg{T, log pL.) (here pe = nekT\ in which interpolations could be made to find log pJT, log />„). Examples of such tables are
116 The Equations of Statistical Equilibrium
5-2 The LTE Equation of State jor Ionizing Material 117
given in (11, 130) and (638, 104). We shall develop a different procedure, along lines suggested by L. H. Auer, that is better suited for machine computation, and that fits into the overall approach of Chapter 7 for the computation of model atmospheres. But first consider an instructive example that yields physical insight in limiting cases.
Suppose that the gas consists only of hydrogen {yH = 13.6 eV) and one metal "M" with a single ionization stage of much lower ionization potential (say — 4 or 5 eV) and an abundance relative to hydrogen aM « 1. At high temperatures where the hydrogen is appreciably ionized, it will contribute most of the electrons; at lower temperatures the hydrogen is neutral, and ne is determined by /M, the ionization fraction of the metal. The number of particles of all types is
N = hh|1 + fH) + txMnti(l + /M)
while the number of electrons is
Then
ne = np + nM+ = nH(/n + 1, and as aM « 1, {pjpa) -*■ i At intermediate temperatures, where aM « fK « 1, and at the same time fM « 1, (pJPg) ~ /h- At ,ow temperatures, fu ~* 0 while fjfu » 1, hence {pe/pg) -»■ «m/m- We thus see tbat at high temperatures the metals are essentially irrelevant to the determination of pjpg, while at low temperatures they play a crucial role. In particular, note that the metal abundance enters directly in fixing pe in cool stars; this is important because the dominant opacity source in cooler atmospheres is absorption by the H" ion, and n(H~)/n(H) is proportional to ne. Thus in these stars the metal abundance fixes the opacity as well.
For a pure hydrogen gas, equations (5-16), (5-19), and (5-20) may be solved analytically to obtain
n,(H) = ^[(JV^n + ^ ~ 0 (5-24) which shows that at low degrees of ionization, nL, ~ N- for a given T.
Exercise 5-1: Derive equation (5-24).
If only the metal in our two-component gas described above is ionized (/h « hm) then we have
n„(M) » M"
kMAf$M + -(1 + 2aM)
(5-25)
A fairly good estimate of ne can be obtained from equations (5-24) and (5-25) if aM « 1 and yu « yn by writing ne x «e(H) + ne[M).
solution by linearization
Let us now turn to the problem of determining ne for a given value of (N, T) by means of an iterative linearization procedure (generalized Newton-Raphson method). We shall describe the procedure m fair detail because it is a simple example of the approach we shall use in more complicated cases (e.g., the non-LTE rate equations and the transfer equation). The only equation to be solved (contrast this with the non-LTE case, cf. §5.5!) is equation (5-20) where fjk(ne> T) is given by equation (5-17). Suppose that we have an initial estimate of the electron density, nt,°; suppose also we find that using ne° to evaluate the nghthand side of equation (5-20) yields a density nel ne°. It is then clear that the true density differs from nj\ so we write ne = ne° + Sne where Sne is to be determined in such a way as to satisfy equation (5-20) exactly. Because the equation is nonlinear, we cannot determine this 6ne exactly, but on the supposition that SnJnJ* « 1, we can estimate 3nt by expanding all terms to first order and solving for bne. Then we have
n,° + 6ne * [(N - nf - bne) • f,(ne°, 7)] + (N ~ «,°)[5S(ne, T)/3«JB.0 3ne
(5-26)
or Sne x [(N - nf)Z - ne°][l + Z — (N - ne°)(dt/dns)] ~1
where
Z(ne, r)^^-^ T) YjPik(ne, T)
* J=i
(5-27) (5-28)
Note that we may rewrite the functions P(ne, T) and S(ne, T) as
i=j
and
SJA„ T)=Jf P,7i(n,, 7") = £ n»>-»{\ (T)
j=o j = 0 'k
(5-29) (5-30)
The value of 3ne given by equation (5-27) will not be exact, so we iterate the procedure by using a new estimate ne°(new) = ne0(old) 4- Snv to reevaluate Z and dZ/dne, and to compute yet another value of SnL,.
The convergence of this procedure is quadratic (if our original estimate lies within the range of convergence) so, if the first fractional error dnjne is c, subsequent iterations will produce corrections of order c2, e4, e8, etc.,
118 The Equations of Statistical Equilibrium
which implies that one can obtain the result to the desired accuracy quickly. It is also worth noting that the derivative ffLfdn. can be evaluated analytically:
{dl.fdne) = i
a.
(5-31)
where (dPjk/dne) and (dSk/dne) follow immediately from equations (5-29) and (5-30) and produce a compact expression for (5-31). In general, the derivatives appearing in linearization procedures can be estimated numerically; however, we shall usually be able to obtain analytical derivatives, and experience has shown that in this way we obtain better control of the calculation.
Finally, having obtained a satisfactory value for ne, and, as a byproduct the fjk, we may calculate any particular occupation number from equation (5-15)
nijk = Nj+,,knJf>iJk{T) = v.k(N - ne)tiefj+Uk{ne,T)^iik(T) (5-32)
this completes the computation of the LTE equation of state.
The procedure outlined above has a larger significance than indicated thus far. We have assumed that A' and T are given. But these quantities follow from constraints of pressure and energy balance, and in general are known only approximately at any particular stage of calculation of a model. As we shall see in Chapter 7, we may apply the linearization procedure to all the variables involved, and hence we shall need to evaluate the response of the occupation numbers to the perturbations SN and ST. Perturbing equation (5-20) we obtain
ne + Sne = (N + SN - ne - bne)*L
+ (N- nM&/dne)öne + (dl/ÖT) ST]
(5-33)
or, assuming that ne is a solution of equation (5-20) at the current values of (A', T),
K = [1 F 1 - (N - ne)(tl/Cne)]-l[Z SN + (N - ne){dtfdT)ST]
= (dnJdN)T SN + (cnJdT), ST (5.34)
where again oZ/?T may be evaluated analytically. Further, from equation (5-32) we may develop an expression for 6n!jk of the form SniJk = Aj SN + A2 ST + A3 Sne, which can be collapsed down by use of equation (5-34) to an expression of the form
On
(dnijk/dN)r ON
(dnijk/dT)N ST
(5-35)
5-3 The Microscopic Requirements of LTE 119
Exercise 5--: Obtain expressions for the coefficients in equation (5-35) in terms of Pjk, SK, and ,^, and their derivatives.
Equations (5-34) and (5-35) provide the information we shall require in §7-2 to find the response of the opacity and emissivity (S%, St]) to changes in the model structure (SN, ST).
Exercise >i: Show that oNJk., for the last ionization stage of element A. has a particularly simple form because fJk involves only Sk. Then show that expressions for 5Njk of lower ions can be evaluated recursively from equation (5-16), and that these lead from equation (5-15) to simple expressions of the form of equation (5-35] for oh;;,,.
5-3 The Microscopic Requirements of LTE
Before we develop the equations of statistical equilibrium, it is worthwhile to discuss qualitatively the microscopic requirements of LTE. An interesting commentary on these requirements by K. H. Böhm may be found in (261, Chap. 3); we shall summarize and discuss this analysis here along with other material of relevance.
DETAILED BALANCE
In thermodynamic equilibrium, the rate at which each process occurs is exactly balanced by the rate at which its inverse occurs, for all processes; i.e., each process is in detailed balance. This is a very strong requirement, and it proves to be very useful in constructing relations among rate coefficients (recall the use of this procedure in Chapter 4). We may classify processes that produce transitions from one state to another (bound or free) into two broad categories: radiative and collisional. Collisional processes are the processes invoked in statistical mechanics to establish equilibrium, and can be expected to be in detailed balance whenever the velocity distribution of the colliding particles is the equilibrium (i.e., Maxwellian) distribution. We shall show below that this can be expected to be the case in stellar atmospheres. Furthermore, we may make the same statement about processes which are essentially collisional in character, even though a photon is emitted (e.g.. free-bound radiative recombination and free-free emission); we can therefore use detailed balancing arguments to calculate the rates of these processes when convenient to do so. In contrast, radiative processes (e.g., photoexcitation, photoionization) depend directly upon the character of the radiation field, and will be in detailed balance only if the radiation field is isotropic and has a Planck distribution. We shall show below that this is not the case in stellar atmospheres.
t
120
The Equations of Statistical Equilibrium
If some processes are in detailed balance while others are not, the final occupation numbers will be determined by a competition among them and • may depart more or less strongly from an equilibrium distribution. LTE will be valid in the deepest layers of stellar atmospheres where densities are high and the collision rates become large, and the optical depth is so large that no photon escapes from the atmosphere before being thermalized, so that the radiation field approaches the Planck function. But in the observable lavers, precisely the opposite regime is found.
THE NATURE OF THE RADIATION FIELD
A stellar atmosphere is not in any sense a closed system in equilibrium at a uniform temperature. Indeed the opposite situation prevails: radiation flows freely from the surface layers of the star into essentially empty space, which implies that the radiation field is decidedly anisotropic, and that the atmosphere has a large temperature gradient. The radiation field at any point is the integrated result of emissions and absorptions over the entire (possibly large) volume within which a photon can travel from its point of emission to the test point. This volume may include the boundary surface and empty space beyond, with a consequent reduction of intensity, as well as layers of higher temperatures and densities from which intense radiation originates. The radiation field therefore is distinctly nonlocal in nature, and has an absolute intensity, directional distribution, and frequency spectrum that may have no resemblance whatever to the local equilibrium distribution BV(T). Radiative rates may therefore be far from their equilibrium values, and thus tend to drive the material away from LTE.
The radjation field is plainly anisotropic because the radiating surface subtends a solid angle less than 4n, and essentially no radiation enters from the surrounding void. We may describe this geometrical effect b) introducing a dilution jactor W defined to be 0)J4k where is the solid angle subtended by the stellar disk.
Exercise 5-4: Show thai
W = ~{l-[\ -(r*/r)2]^
(5-36)
where is the radius of the radiating surface and r denotes the position of the observer. Show that for rjr «1. W = i(rjr)2
As defined. W clearly measures the factor by which the energy density m the radiation field is reduced as the source of radiation moves to a large distance. At the '•surface" of a star, it is obvious that W = \ (actually a little
5-3 The Microscopic Requirements of LTE 121
less because of limb-darkening) but in an extended stellar envelope W « 1 (and in a planetary nebula, W - 10"i4). Thermodynamic equilibrium requires that W = L so it is clear that detailed balancing in radiative transitions cannot in general occur in a stellar atmosphere.
Jn addition to being dilute, the stellar radiation field has a markedly non-Planckian frequency distribution. As we know from the Eddington-Barbicr relation, the emergent specific intensity at frequency v is approximately equal to the source function Sv at tv = 1. Even if Sv were Bv, the fact that the material is vastly more opaque at some frequencies than at others (line to continuum ratios are often I03 and may reach much larger values) implies that the radiation will emerge from greatly differing depths at substantially different temperatures; the radiation field is therefore a composite of widely differing radiation temperatures. The effects of the temperature gradient become extreme when bv'kT > 1, for then the Planck function varies as exp( — hv/kT) and becomes very sensitive to small changes in T. If we were to parameterize the radiation field by introducing a radiation temperature TH[p, v) such that for p > 0, l{r^ /<, v) - WBv[TR(p, vj], we would find marked variations of T R with both v and p. For example, in the solar spectrum, TR ranges from 4800'K in the visible to "-25,00(FK in the ultraviolet in the ground-state continuum of He + . In sum, the radiation field displays an extremely complex behavior, and the conditions required to assure LTE a priori are simply not met.
THE ELECTRON VELOCITY DISTRIBUTION
In stellar atmospheres, the free electrons are produced by photo-ionization and collisional ionization. The inverse processes are radiative recombination and three-body collisions, which lead to recaptures of electrons into bound states. While in the continuum, an electron may undergo elastic collisions with other electrons and inelastic collisions (leading to excitation or ionization of bound electrons) with atoms and ions. The elastic collisions redistribute energy among the electrons and tend to lead to an equilibrium partitioning—-hence a Maxwellian velocity distribution. If a Maxwellian velocity distribution is in fact attained, we may define the local temperature to be the kinetic temperature of the electrons. On the other hand, inelastic collisions and recombinations disturb the achievement of a Maxwellian velocity distribution, for the inelastic collisions involve electrons only in certain velocity ranges and tend systematically to shift them to much lower velocities, while recombinations remove electrons from the continuum and prevent further elastic collisions. Whether or not the Maxwellian velocity distribution is established hinges upon how rapidly thermalization by elastic collisions occurs compared to the perturbing processes: if it occurs much more rapidly, the velocity distribution will be very nearly Maxwellian.
122 The Equations oj Statistical Equilibrium
5-3 The Microscopic Requirements of LTE 123
The thermalization rate can be measured in terms of the relaxation time j
of the system, which, for particles interacting with themselves, is J
tc = mH3kTp/[ll.9nceAZA ln(D//)0)] sec (5-37) |
(see 598, Chap. 5). Here D is the Dehye radius (see §9-4) D = {kTß%e2nef
and p0 = e2/mv2 is the impact parameter for a 90° collision. Now consider :
recombinations; if a is the average cross-section for the process then the i
mean time between recombinations is :|
tr = (NaCvyy1 = N-^-^nmßkTj* sec (5-38) |
where N is the density of the particles with which recombination occurs. J
Two astrophysically important processes are (a) H + e H" and(b)H+ + [
c _> h. At T ~ 6000"'K (a typical solar temperature) ou--3 x 10"22 J
cm2 and ne/Nti ~ iO"4 At T - 10000°K, au - 6 x 10"21 cm2 and |
njnp ~ 1. Substituting these values into equations (5-37) and (5-38), we :|
und tTjt£ - 105 for process (a) and tjtc ~ 107 for process (b). We conclude, |
therefore, that under representative conditions in stellar atmospheres a | free electron will undergo an enormous number of elastic scatterings between recombinations, and that the latter will not seriously hinder equilibration to a Maxwellian distribution.
Let us now consider inelastic collisions. Collisions of electrons with the most abundant element, hydrogen, occur frequently, but the excitation
energy of hydrogen is 10 eV while the thermal energy of the electrons j
is 1 eV. Thus only 3 x 10"5 of the electrons have sufficient energy to [
induce the excitation, and only a fraction of these will be effective. Using j
typical excitation cross-sections one finds that (at 10,000"K) the rate of \ inelastic excitations is of the same order as the recombination rate—i.e.,
very small compared to the elastic collision rate. One must also consider [
collisions with other elements, which may be grouped as follows: (a) the j
alkalis, which have large cross-sections but low abundances (10~ö); (b) Fe, j
which has numerous low-lying levels and moderate abundance (4 x 10"?); j
and (c) C, N, and O, which have small cross-sections but large abundance [ (10~3). Most of the levels for groups (b) and (c) are metastable, so that most
of the inelastic excitations are subsequently cancelled by collisional de- [ excitation; ignoring this effect we overestimate the number of inelastic
excitations. Taking the various factors into account and ignoring com pen- \ sating de-excitation, Böhm estimates (elastic collisions/inelastic collisions) -
103 and hence concludes that a Maxwellian velocity distribution that defines ;
Te is established. Recent work (573) suggests that departures from a Max- J wellian distribution in a pure hydrogen gas can occur in the high-energy
tail if (a) the ionization level is very low [nJnH % 0.01) and (b) the ground-slate t
population is far from its equilibrium value; these conditions can occur in the solar chromosphere.
Finally, one may ask if the atoms and ions in the atmosphere also have a Maxwellian velocity distribution, and if their kinetic temperature Tk — Tc. An analysis of this question (88) for a pure hydrogen atmosphere of atoms, ions, electrons, and radiation, demanding a steady-state solution, while allowing for energy exchange among the four components of the medium, shows that ifne > 1010 (a condition easily met in the bulk of the atmosphere) and 5 x 103 < T, < 10s, then \Tk - T\ ^ 10"3 Te. It thus appears safe to conclude that a unique local kinetic temperature applies to all the particles in most atmospheric regions.
THE IONIZATION EQUILIBRIUM
The degree of ionization of stellar material is determined by the balance of photoionizations and collisional ionizations against radiative recombinations and three-body collisional recombinations. Let us first examine the relative rates of photoionization and collisional ionization; it suffices to obtain only an order-of-magnitude estimate.
The energy absorbed by an atom in bound state i at frequency v in interval dv is AnJva-,{v) dv; each photon has energy hv, hence the total number of photoionizations is
niRiK = n^rdr1 j*' ^(vlJ,.!"1 dv (5-39) To estimate RiKi we adopt a hydrogenic cross-section
av = (tie2/mc)fc(2v02/v3)
where fc is the integrated oscillator strength for the continuum. Further, we write
jv = WBV(TR) = W{2hv*/c2) x exp(-nhv/kTR)
n= ]
Then RiK = (16n2e2v02/mc3)fcW £ E^nhvJkT^ (5-40)
(i=i
The rate of collisional ionizations can be computed from a{v\ the collisional ionization cross-section for electrons of velocity
"iC-K = «/«? v{v)f{v)v dv (5_4i)
124
table 5-1 . Ratio of Radiative to CoUisioiml Ionization
Rales
Star
= 8eV
= lcV
Sun O-star
10J 20
2
0.2
Sourct- From data by K. h. Böhm, in Stellar Atmospheres, cd. J. L. Grcenslein, Chicago: ScU of Ch.cago Pres., I960, by perm.ss.oi,.
To obtain an estimate, we adopt the semiclassical Thomson formula [cf. (684, 120)]
a(v) = y^E-'Khvo)-1 - (5-42) where E = \mv2 is the energy of the incident electron. Substituting equations (5-2) and (5-42) into (5-41) and integrating we obtain
ne[12nV/c/(2mfc3Te3)*]«o"1£2(«(
(5-43)
- hv IkT In the limit that hv0 » kTR and hv0 » kT we retain :^tot^f Suation (5-40), and use the asymptotic result that for x>>1, £2(x)-Et(x)-^7-^o obtain
4(2%*kfk\\
4 exp
/iVf
1
_L
(5-44)
For photospheric layers we could adopt ff ä 3, TR * t(,. Böhm calculates estimates of RiJCiK for representative cases of levels with ionization potentials of 1 eV and 8 eV for conditions characteristic of the outer layers (t ~ 0.05) of the sun and an O-star. In particular, for the sun he adopts ne ä 3 x 1012, and T « 5 x 10J °K while for the O-star he uses «e ä 3 x 1014, T ä 3.2 x 10"4" 'JK and finds the values for R!K/CiK listed in Table 5-1. It is clear that in stellar photospheres, the radiative rates dominate, except for high-lying levels at high temperature and densities. In fact, for O-stars the important levels have even larger values of yAon than those listed in Table 5-1 (e.g., the ground state of H at 13.6 eV and the ground state of He I at 24.5 eV), and are even more markedly radiatively dominated. Thus the ionization equilibrium is vulnerable to departures from LTE if Jt departs from By. Note in passing that in the corona of a star where Tc ~ 2 x 10fl °K and tr ~ 6 x 103 "K (for the sun), while the relevant values of hvQ are around 300 eV, the exponential factor in equation (5-44) becomes very small and collisional ionizations dominate.
5-3 The Microscopic Requirements of LTE 125
We may carry out similar estimates for the rates of radiative recombination and three-body collisional recombination; these processes are both essentially collisional and hence per ion occur at the LTE rate. We can then use detailed balancing arguments to compute the rates in terms of the equilibrium values of the upward rates. We may still use equation (5-44), except that for the radiative recombinations the appropriate temperature is now Te not TR, and W = 1. We then find that radiative recombinations always outweigh collisional recombination, both in the photosphere and corona (in the corona, yet another mechanism—dielectronic recombination—outweighs radiative recombination).
The ionization balance is thus determined by photoionizations and radiative recombination; to establish the equilibrium the numbers of ionizations and recombinations are equal :ntRiK = nKRKi = nfRfK where the last equality follows by a detailed-balance argument. Hence for the ground state,
.) = 4nW 1 (hv)-\,BfTR)dv 4n \ ' (hv)-la,Bv(Te) dv
= W
e x
1W r
/ Jln-JkTR)?exp(-hvJkT^
(5-45)
where we have again used hydrogenic cross-sections. If we substitute for n~Qj from the Saha equation (5-13) we may obtain the approximate ionization equation
(>W,'+l/»0,jJ
= W ■ (2g0J+i/g0^)(27imkTR/h2)' ■ [TJTR)* • Qxp(-Xlj/kTR) (5-46)
which has been extensively applied—e.g., in analyses of gaseous nebulae.
To analyze the ionization balance in stellar atmospheres we now must decide (a) how to choose Jv, and (b) which levels dominate. Böhm suggested comparing the values of 4n j' (hv)" 1kvJv dv with 4n §{hv)~ 1kvBv dv, where kv is the total opacity from all overlapping continua and Jv is the mean intensity obtained from LTE model-atmosphere calculations. If these numbers are equal the claim is made that LTE is self consistent. Böhm examines the Fcl <-> Fe II equilibrium in a model solar atmosphere and finds that the rates mentioned above have a ratio of 2.9 at t = 0.01, 1.3 at t = 0.05, and essentially unity at t > 0.1; from this one is tempted to conclude that the Saha ionization formula is valid below? = 0.1.
There are, however, flaws in this argument. First, it is clear that integrated rates summed over all continua of an atom may be subject to cancellations
126 The Equations of Statistical Equilibrium
127
and compensations, and it is not at all clear what a given departure between the two integrals implies for any particular level (i.e., some levels may be overpopulated and others underpopulated and the integrals could balance). Second, and far more important, the reasoning is circular if from the outset we use Bv as Sv to calculate Jv, for we know that
Jyi?*) = Atv[B„(tv)] = Bv(xv) + 0(,^)
i.e., Jv is forced to Bv at zv ,..The number of transitions produced by incident intensity /v in the frequency interval dv and solid angle doi is iijBij^Jv dv iho/4n or ni(cf.i]i'hv)x, away from vir We have written the downward rate in this particular way because it is then of exactly the same form as the downward rate in the continuum; moreover, the downward collision rates will also have a factor of (n,-//!,)* appearing explicitly. In the end we achieve nolational economy in the full rale equations by using equation (5-57) rather than the simpler Einstein probability form. Finally, it is sometimes useful to work with the net rate from level; to level i.
niBijJij = lljA ■,;'/.:;
(5-58)
130 The Equations of Statistical Equilibrium
where the term Zn is called the net radiative bracket (NRB). Net radiative brackets are useful notational devices that we shall employ in Chapters 11 and 12. Further, Zj7 can be rewritten as
Zfi=l- JifaBij - njBjMnjAji) = 1 - (Jy/Sy) (5-59)
where is the frequency-independent line source function. Because the NRB contains only the ratio of J to S, it is often true that it is known to much higher accuracy in an iterative procedure than cither S or J themselves. Under favorable conditions, use of NRB's can significantly enhance the convergence of certain types of solutions of multilevel line-formation problems. If a particular line i -> j is in radiative detailed balance, then Z}i = 0, and we may cancel the corresponding terms out of the rate equations analytically (i.e., omit Rtj and Rjf); this situation occurs when a particular line thermalizes, and the cancellation procedure is of great use in simplifying the rate equations (cf- §7-5).
(b) Bound-Free Transitions. Let us now calculate the radiative rates from a bound level i to the continuum k. Let c.iK(v) be the photoionization cross-section at frequency v; then the number of photoionizations is calculated by dividing the energy absorbed in interval dv by the appropriate photon energy hv, and summing over all frequencies. Thus the number of photoionizations is
HjRfe = n-M aiK(v)(hv)-\Jydv (5-60)
We may calculate the number of spontaneous recombinations by use of a detailed-balancing argument. In thermodynamic equilibrium, the number of spontaneous recombinations must equal the number of photoionizations calculated from equation (5-60) when (a) Jv has its equilibrium value (i.e., Bv) and (b) we correct for stimulated emissions at the T.E. value by multiplying by a factor of (1 — e~hv'ri) (cf. §4-3). Thus if nK denotes the ion density,
K^Xon - n?4n a;K(v)(/7v)-^(l - e~hv'kT) dv (5-61)
The recombination process is a collisional process involving electrons and ions, and therefore is proportional to nK ■ ne. For a given electron density and a given Te, which by definition describes the electron velocity distribution, the rate just calculated above must still apply per ion, even out of T.E. Hence to obtain the non-LTE spontaneous recombination rate we need correct equation (5-61) only by using the actual ion density nK. Then
WApon = nMKT ■ 4tt aJv^hvy'Bfl - e~hv'kT) dv
= uMKYAn f* a-Jv)(hv)-\2hv'ic2)e~^kT dv (5-62)
5-4 The Non-LTE Rale Equations 131
Recall from equation (5-14) that (ttj/nj* = new;0(T) (5-68)
where v0 is the velocity corresponding to £0, the threshold energy of the process-—i.e., jtnev02 = E0. The downward rate (j -> i) can be obtained immediately on the basis of detailed-balancing arguments, for the electron velocity distribution is the equilibrium (i.e., rvlaxwellian) function; thus we must have
"fCij = nfCSi (5-69)
from which it follows that the number of downward transitions is
= n^/UjfCj = *jlnJnj)*iw,jiT) (5-70)
As was the case for radiative transitions, it is sometimes useful to introduce the net collisional bracket Yn and write the net rate for collisions i -> j(Ef < Ej) as
nf'^u s w.-Cy - njC}i = n.-CyD - (nj/nj){nf/ni)'] (5-71)
The actual cross-sections required to compute rates are found either experimentally or by rather complicated quantum-mechanical calculations; it would take us too far afield to describe these methods here, so we merely refer the interested reader to (410). There exists a vast literature containing results (theoretical and experimental) for a variety of transitions of astro-physical interest; bibliographies of this literature are issued from time to time by the Information Center of the Joint Institute for Laboratory Astrophysics of the University of Colorado and the National Bureau of Standards. (This center also maintains current literature references in an on-line computer.) As indicated by equation (5-68) we are more directly interested in rates for a given cross-section, so let us examine qu in a bit more detail. Usually cross-sections are measured in units of na02, where a0 is the Bohr radius; i.e., we write c7u = na^Q^. Also, Qu is usually tabulated in terms of the energy of the exciting particle, so writing \nw2 --= E, and substituting
5-4 The Non-LTE Rate Equations 133
equation (5-2) into (5-68), we find
qtj{T) - C0T± j'" QulukT)ue-" du (5-72)
where u = E/kT, and C0 = na02(8kjm7z)± = 5.5 x 10"n. Writing x = {u - u0\ where ul} = E0/kT, we obtain
%{T) = C0T*txp(-E0/kT)ru(T) (5-73)
where TU(T) = ^ QSj(E0 + xkT)(x + u0)e-vdx
(5-74)
hxerciac 5-5:
Verify equations (5-72) through (5-74).
The advantage of writing the collision rate as in equation (5-73) is that the principal sensitivity to the temperature has been factored out in the product T* exp( — E0/kT) while T^IT) is a slowly-varying function of T.
Of course the main problem in application is to obtain reliable values of Qjj. A characteristic difficulty for astrophysical work is that for many transitions of interest, kT « EQ, so that the rate depends extremely sensitively upon values of CJi; near threshold. LInfortuuately, for E E0 a great computational effort is required to obtain accurate cross-sections because the simplifying approximations that are valid for E » £0 break down, and because complicated variations of Qtj result from resonances in the collision process. When values for Qu can be obtained, one typically fits them by numerical procedures to simple analytical approxtmants, against which the integration in equation (5-74) can be performed analytically.
For the astrophysically important spectra of H. He 1, and He II, accurate experimental cross-sections exist for excitation and ionization from the ground state. For transitions arising from excited states one must rely upon theoretical calculations. For many atoms and ions of interest there may be no detailed estimates whatever available, and one must have recourse to rough methods to estimate rates. A very useful (though quite approximate) expression for excitation rates in radiatively permitted transitions can be written (639) in terms of the oscillator strength namely
C0 = C0uX^XPh!F.o)2}uq exp(-Ho)re{«o) (5-75)
where w0 = E0/kT, lu is the ionization energy of hydrogen, and for ions
TP[u0) = max 0.276 cxp{u0)E1(u0j] (5-76)
The parameter g is about 0.7 for Transitions of the form nl -» nl', and about 0.2 for transitions of the form nl -> n'C, ri # n (95). For neutral atoms r,,(u0) has a different form [see (47)]. It is worth stressing that equations
134 The Equations of Statistical Equilibrium
(5-75) and (5-76) provide, at best, rough values and should be applied with caution. In particular, collisions are not restricted by the dipole transition selection rules A/ = ±1, and cross-sections for other values of Al may be as large as for A! = ± 1 despite fu being zero in the dipole approximation. For collisional ionizations there exists a semi-empirical formula (402)
oiK(E) = ™02[2.5C(/h/£o)-] HE/E0)[] - fcexp[-cf£ - E0)/Eo]}/{E/E0)
(5-77)
which yields a rate
CIK = CQneT^C{lH/E(])2]u0[EL(t2, respectively (here Z is the charge on the ion). The same caveats expressed about equation (5-75) apply to equations (5-78) and (5-79) as well.
AUTOIONIZATIOM AND DIELECTRONIC RECOMBINATION
In complex atoms with several electrons, the ionization potential is determined by the lowest energy to which a sequence of bound states with only one excited electron converge (to the ground state of the ion plus a free electron). If two electrons are excited within the atom, they will, in general, give rise to states with energies both below and above the ionization potential defined above. Subject to certain selection rules [(172, 371; 297, 173)] the states above the ionization limit may autoionize to the ground state of the ion plus a free electron. The inverse process is also possible and, if an ion in the ground state suffers a collision with an electron of sufficiently great energy, then a doubly excited state of the atom may be formed. In general, this process will be of little interest because the compound state will immediately autoionize again (typical autoionization transition probabilities Aa are in the range 1013-1014!), and its equilibrium population will be small. In some cases, however, a stabilizing transition occurs in which one of the two excited electrons (usually the one in the lower quantum level) decays radiatively to the lowest available quantum state, leaving a bound atom with a single excited electron. This process can provide an efficient recombination mechanism referred to as dielectronic recombination.
5-4 The Non-LTE Rate Equations 135
In particular, for an ion of chemical species X and charge Z, we consider processes of the type
XHZ\n, I) + e(E, I" + I) ^ X+(Z-v. / + l;n", I") (5-80a) followed by the stabilizing transition
X + (Z"ls(n', 1 + 1; n". /") -> X^-^fn, /; n", I") + hv (5-80b)
which leaves the ion (Z — 1) in a bound excited state. As an example, for He+ we might have
He+(ls) + e(E, I" + 1) -» He°(2p; «"/")
Hel"Y2p; n"l") -> He°(ls; n"l") + hv
If we denote the doubly excited state by d, the final bound state of the ion (Z - 1) by b, and the ground state of the ion Z as k, then the number of dielectronic recombinations to state h from d can be written as nKRdh = iijA^ where A, is the spontaneous transition probability for the stabilizing emission; to a good approximation (particularly for large n") /I, — A(n\ I + 1; h, /) for the Z ion. In the limit of low radiation fields, the reverse process in equation (5-S0b) can be ignored, and if Aa measures the transition probability for autoionization, nd can be written (73,258) in terms of its equilibrium population nf
Jirf = ri$AJ{Aa + /},) (5-81) where h* = nKne{gd/gK)CjT~* cxp(-^x/kT) = nKne so that nd is given by its equilibrium value (relative to actual ion densities), equation (5-82); furthermore one need sum over only a few states. On the other hand, these processes occur deep enough in the atmosphere that one must account for the inverse transitions
5-4 The A'on-L TE Rate Equations 137
produced by the radiation field in the stabilizing transition. If state d is characterized as {n1, T, L") and state b as (n. /, L) we have the total dielectronic recombination rate
nKRdb = ryj, V h Z l;ri, l')JnT
»\ r
where /; n\ I') = A*(n\ /'; «, tyg^/lhvhj,).
(5-86)
COMPLETE RATE EQUATIONS
Having examined all of the processes of interest, we may now assemble the individual rates into a single complete equation of the form of equation (5-52) for each bound state i of each ionization stage of each chemical species in the material. We shall fa) ignore explicit mention of dielectronic recombination because the rate has the same form as for radiative recombination, and we shall assume that both are included; and (b) assume that all ionizations from bound states of ion j go to the ground state only of ion j + 1 (generalization is easy but complicates the notation and discussion). We then may write
;x MUr, + Cri) + (n,,/n^iRiV. + Cri} + £
(Ru- + cu.
- I M»M*{RVi + Q,,) = 0 (5-87)
138 The Equations oj Statistical Equilibrium
where the radiative rates are defined by equations (5-66) and (5-67), the collision rates by equation (5-68), and the LTE population ratios by equations (5-5) and (5-14) for bound states, and bound and free states, respectively. One such equation may be written for each bound state. We have one more variable (the ion density nK) than we have equations. If we wrote down an ionization equation
Z nt(RiK + CiK) - nK x (niM*(RKl + CJ = 0
we would find it to be redundant with the set (5-87).
(5-1
Exercise 5-7: Show that equation (5-1 over all bound states.
results from summing equation (5-87)
We therefore invoke an additional physical constraint to complete the system. For an impurity species (i.e., afc/aH « 1), we close the system by demanding that the total number of atoms and ions (of all kinds) of the species equal the correct fraction of the number of all hydrogen atoms (including protons); i.e.,
(5-89)
Z «ü.fc - («*/<%) Z ni,u + np = 0
Alternatively, we can close the system by invoking charge conservation (saving the total number conservation for use elsewhere) and write
ZZ-^j* + iip =
(5-90)
where Njk = nijk. The final system, for all levels of all ions of all species is written in the general form
y/n = iM (5-91)
where ii denotes a vector that lists all occupation numbers (say . i'~ of them) while s/ is an (Jf x Jf') matrix and $ is a vector in which only one element is nonzero [from equation (5-89) or (5-90)].
To make these considerations more definite, let us consider a case that is simple enough to be manageable and complicated enough to be of general applicability. Suppose we have an atmosphere composed entirely of hydrogen and helium (of abundance >\ by number, relative to hydrogen). We consider the helium to consist of a ladder of three ionization stages, He0, He + , and He+ +, and we suppose that these ions have L0. L + , and 1 levels respectively. Further we write MHc = L0 + L+ + J; i.e., MHe is the total number of helium states of all kinds. Similarly we consider Ln bound states
5-4 The Non-ETE Rate Equations 139
of hydrogen, yielding AfH = LH + 1 states in all (including protons). Then, using A"s to denote nonzero elements, the rate matrix jtf has the form
Row Number
L0 + 1
X X X X
X X
1 1
EH
MHc + MH \0 1
-y -y
0
X x • X x -
X a -
0 I 1 1 ■ ■ ■ 1 j 2 Column Number
0 0
0
1
The first LQ rows correspond to equation (5-87) for He0, the next L_>. rows give equation (5-87) for He+, the MHeth row gives the abundance equation (5-89), the next Lu rows give equation (5-87) for H, and the last row gives charge conservation. The vector n consists of elements
n = [Hl(He°),.... nu(He°), n^He^), . . ., nL+(He+),
n(He+ + hnm,.-.-nLHmnpy (5-92)
m = (0,...,0,>iir)T (5-92)
For given values of ne, T, and the radiation field, equation (5-91) is a linear system in ii, and may be solved by standard numerical methods (526, Chapter 9).
and
140 The Equations oj Statistical Equilibrium
5-5 The Non-LTE Equation of State
From the results of the preceding sections, we see that in LTE each occupation number at a specific point in the atmosphere is a function of only two thermodynamic variables; i.e., n,- = n;(N, 7") where Tis the absolute temperature at that point. In contrast, in the non-LTE case, the full rate equations imply that n, = nt{N, T, Jv) where Jv denotes the frequency dependence of [he radiation field over (he entire spectrum and T is now a kinetic temperature describing only the particle velocity distribution function. We now have as many new (fundamental!) thermodynamic variables as are required to specify the distribution of radiation in frequency. [Note that (/ we could simplify the description of this distribution—e.g., if we could write Jv = WBy(T)—then the situation would be vastly simplified; but in genera) we may need to consider perhaps hundreds of new variables.] As was the case for the LTE equation of state, the uon-LTE statistical equilibrium equations are actually nonlinear in the electron density ne. and we shall require a linearization procedure to solve for the occupation numbers; but now we shall have to extend the linearization to include changes in the radiation field as well. We shall see in §7-5 that this approach provides a method for coupling the transfer equations and statistical equilibrium equations together, and allows us to determine the global response of the gas to the radiation field simultaneously with the reciprocal response of the radiation field to material properties.
Before developing the linearization procedure required in the general case, it is worthwhile to consider a few examples that illustrate clearly the essential physical content of the statistical equilibrium equations.
LIMITING CASES
Consider first an atom consisting of a single bound level that can ionize to its continuum. We then have one rate equation which states that (ignoring stimulated emissions)
(njnf) =
4nj*'(avBJh\)dv + neqlK i 4tu J* {zyJJhv)dv + nt,qlK
(5-94)
We note first that, as the electron density becomes very large, so that colli-sional rates exceed the radiative rales, then
lim (»,/"*) = lim (u,,qUJ\qlK) = 1
i.e., LTE is recovered. Further, at very large optical depth, Jv —► Bv and clearly n^jnf -> I; i.e., // the radiation field is perfectly Planckian we recover
5-5 The Non-LTE Equation of State 141
LTE. as expected. Two comments are necessary here, however, (a) To obtain LTE in a multilevel atom Jv must equal Bv in all transitions. If any transition is transparent, then LTE will not be obtained (unless densities are so high that collisions dominate), not only for the particular levels involved in the transition under consideration, but actually for all other levels as well because the radiation field in each transition influences the populations of all levels (see below), (b) We have left unanswered the question of how large is "'very large1' optical depth. As we have indicated earlier, tv <; 1 is not sufficient to guarantee Jv -» Bv. Rather tv must exceed a thermalizaiion depth, for which precise estimates will be given in Chapters 7 and 11. In the low-density limit (e.g., in a nebula), equation (5-94) reduces to
(njnf) =
f* (xvBv/hv)dvl f* fajjliv)
d\
(5-95)
which states that, if the recombination rate exceeds the photoiouization rate, the level is overpopulated; and it is underpopulated if the reverse is true. Equation (5-95) is, of course, equivalent to equation (5-46) which is often applied in nebular analyses. In the coronal case, we have 7~,.( 106 K) » TR{ -~6000°K)S which implies that collisional ionizations exceed radiative [see equation (5-44) and related discussion] while radiative plus dielectronic recombinations, both of which proceed at a rate specified by Tt.. exceed collisional recombinations. Then
so that
nynvqlK - 'V\,(aRR + aDR) K>i) = qiJ(aRR + y-UR) = f{T)
(5-96)
That is, the coronal ionization balance depends only on temperature and is independent of the electron density, a fact that vastly simplifies analysis of the corona. Both the coronal and nebular situations represent extreme departures from LTE.
Let us now consider some multilevel problems. Suppose we have a volume of pure hydrogen gas illuminated by a very dilute radiation field (i.e.. a nebula). We anticipate that virtually all of the hydrogen will be in its ground state, and we assume that all the resonance lines are completely opaque (and hence in detailed balance). Further, we assume that, after an atom is photo-ionized from the ground state, recombinations occur to all states, but the populations of the upper states are so small and the incident radiation field so diluted that (a) we can ignore photoiouization out of these states, and (b) electrons in any excited state cascade downward at rates determined by the Einstein coefficients Aj{ without reabsorption upward (i.e., the subordinate lines are transparent). We further assume densities are so low that we
142 The Equations of Statistical Equilibrium
may neglect collisions. Then we have an ionization equation
n,RlK-ne2 i«rb0-,T) (5-97a)
and a number conservation equation
(5-97b)
where nH is the (given) hydrogen density. / is the total number of bound states considered, and ne = np (for pure hydrogen). RlK is assumed given in terms of Jv = WBJTRl as in equation (5-40). For any subordinate state we can calculate the population in terms of the branching ratios an ~ AjJYjKj Aji, and the cascade probabilities p,7 which are defined recursively as pi+lif = ai+ui, and pjf = + S=i + i PAfor U = i + X ■ ■ ■, A Then for level i we find
- 0,
(5-98)
Exercise 5-8: (a) Verify the expressions for p;i given above and derive equation (5-98). (Hint: Start with level I and work downward.) (b) Show that equations (5-97) and (5-98) yield a quadratic equation in ne that allows the determination of nL,(iin, T), and hence all the h/s. (c) Show that f>n = 1 (J > \); inierpret ihis result physically.
From equation (5-98) we may estimate ratios of occupation numbers, and hence ratios ofline intensities along a series. For example we can compute the relative intensities of the Balmer lines (the Baimer decrement) as
/(Hft)/i(Hj) = inkAk2hvk2)/(njAi2hvj2)
and compare the theoretical results with observation. The approach outlined in equations (5-97) and (5-98) (with extensive elaboration and refinement!) forms the basis for the analysis of nebulae [see (15, Chaps, 23-25; 10, Chap. 4; 415, pp. 40-110; and 350, Chaps. 1-3)].
Finally, consider an atom that consists of Ihree stales (1, 2, 3) in order of increasing energy in a rarefied medium (neglect collisions) and a dilute radiation field. A famous result regarding such a system is Rosselamfs theorem of cycles, which states that the number of radiative transitions in the direction 1 3 -* 2 -> 1 exceeds the number in the inverse direction 1 -> 2 ->■ 3 -> 1. A consequence of this result is that energetic photons are systematically
5-5 The Non-LTE Equation oj State 143
degraded from high energies (say far ultraviolet) to low (visible and infrared); for exampie, in a nebula. Lyman continuum photons are degraded-—-e.g., into Balmer continuum photons plus La photons (state 1 = Is, state 2 = 2p, state 3 = continuum). We may calculate the ratio Rl^s^.2-\IR\-2->s->\ quite easily. The number of excitations 1 -> 3 is nlBliWB[yi2). Of theexcited atoms in state 3, a fraction A32/(A32 + A3i) decays to state 2, and of the atoms in state 2 a fraction -f21/[/t2l + B2iWB{y2i\\ decays to state 1 (here we have ignored stimulated emission). Thus
»1*1-3-
By similar reasoning
ntB13WB{v,3)A32A21 (A32 + A3l)[A2l + B23WB(v23)]
nR _ ihBi2WB(v12)B2,WB(v2,)A31
so that
[A2I + B2iWB(v23)](A32 + A31)
(5-99)
>-100)
*l-2-3-l/-*l ->3-2->l
= W/[5J2B(v12),M31][fl23B(v23)/.432][>l31/B13B(v13j] (5-101)
But using the relations among the Einstein coefficients and writing Bv in the Wien approximation [hv/kT » 1) one finds [/^(v;,-)//!,-,-] = (n^/n,)*, so equation (5-101) reduces to/?, ^2_,3_1/W[_3j:_.t = W < 1, which proves the theorem. The result clearly follows from the fact that in the cycle 1 -» 3 -> 2 -> 1 the dilution factor enters only once, while in the reverse process it enters twice. In stellar atmospheres, Rosseland's theorem is relevant because at certain depths one may have resonance lines that are opaque (i.e., W = \) exciting atoms to upper states, from which the subordinate lines are transparent; in such cases we anticipate a systematic photon degradation.
LINEARIZATION
As mentioned before, the general system c/n = can be solved as a linear system for n if ne, T, and Jv are all specified. But, in practice, we do not know exact values for these variables in the course of a model-atmosphere computation (recall the discussion for the LTE equation of state) but have only current estimates in an overall iterative process. We expect all of these variables to change by amounts Sne, ST. SJV, etc. to satisfy better the constraints of energy and pressure balance, and must evaluate the response of n to these changes, in the form
k
5n - (dn/dnJSnc + (en/3T) 5T + £ (vn/cVk)SJb (5-102)
144 The Equations of Statistical Equilibrium
Here Jk{k = ],..., K) is the mean intensity at discrete frequencies that sample the spectrum- These frequencies are chosen such that all integrals over frequency are replaced by quadrature sums—i.e.;
F(v) dv = £ vv(iF(vf
(5-103)
We obtain equation (5-102) by linearization of the original equations (5-91) [and also can find parallel linearized equations that give, in essence, bJk{6T, dne, 5n) from the transfer equations; cf. §7-5]. Equations of the form (5-102) are required in two contexts: (a) model atmosphere calculations where all variables may change in an iteration cycle, and (b) multilevel statistical equilibrium calculations for a given model (nc, T, and total particle density fixed). The procedure for case (b) will be deferred until Chapter 12. and we shall consider only case (a) here. [In case (b) one may use a special technique motivated by consideration of the computational methods of solving transfer equations, to be developed in Chapter 6.]
If x denotes any variable, then by linearization of equation (5-91) we have
g'x
~dx~
ds/J
■ n
ox
(5-104)
where we have assumed that n is the solution of the current system ,r/Jw = 38 (we might introduce a subscript zero, or some similar device, but it would become unwieldy). An extremely important feature of this approach is that every derivative in equations (5-102) and (5-104) can be written down analytically (though the inverse must be computed numerically); this produces a system of high accuracy and reliability. To illustrate the procedure, we shall write down some representative derivatives for the model atoms discussed at the end of §5-4; more comprehensive collections of formulae are given in (42) and (437). In what follows, we use the auxiliary \ector a = (c {) < i)
(5-105a)
I «y(vkj + S «J,(vfe)(Hj/HI.J*i?-^kr
(5-105b)
and (d^/dJk)ij= -[47rwJla(./vJt)/hvt](n1/wJ.)*e-"Vft/fcT) (./">/) (5-105c)
5-5 The Non-LTE Equation of State 145
from which we find
a; = [(„(/V - Bj]
If there were no scattering, pv = 0, then J„ could be calculated, as a quadrature, from Bv; when pv ± 0, we must solve an integral equation for Jv. One of the first methods that comes to mind to effect such a solution is iteration. As we know that Jv -> BY as tv oo. let us deal with (J„ — Bv). Suppose pv were everywhere zero; then (./,, - Bv) would equal (Bv - Bv) where Bv(tv) = Arv[Bv]. If pv is not zero, we could regard this value as a first approximation and write
tJv - Bvf = (Bv - *v) + AT„[pvU, - Bv)°] = (Bv - Bv) + A,£pM - Bv)~] = (Bv ~ Bv\ + A(1> (6-1)
Then by iteration, we find
(Jv - B,
B,
I A(">
(6-2)
where A(n) ~ Atv[pv Aln_1)]. In practice we continue the iteration until some convergence criterion—e.g., JjAl"V(/v — By)[n)\\ ^ c, where c « I—is satisfied. It is clear that, if \\pv\\ « 1, the iteration procedure of equation (6-2) can be expected to converge, for successive corrections A(m must be of order \\py\\" relative to (Jv - Bv), If, however, ||pv|j « 1 over a large depth of the atmosphere, the iteration method
148
Solution of the Transfer Equation
The circumstance just mentioned actually occurs in stellar atmospheres, and the thermal coupling parameter/,, = 1 — pv may be very small throughout a large part of the atmosphere. For example, in very hot stars the principal source of continuum opacity in the outer layers is electron scattering, and /,v may be of order 10"4 very deep into the atmosphere (until finally, as the density rises, free-free thermal absorption overwhelms the electron scattering). In cool stars of low metal abundance, the hydrogen is neutral in the upper atmosphere and free electrons are scarce, so Rayleigh scattering by h and FF dominates the Ft' opacity, and pv is nearly unity until great depth (at some point the hydrogen rather abruptly becomes excited and ionized, and ?,v suddenly rises to unity). For lines, the corresponding thermal parameters may be very small, Xv - 10"8 (see Chapter 11).
The symptomatic behavior of the iteration method in these cases is that the solution stabilizes, and although successive iterations differ fractionally only by some small value, the A's are monotonic, and are nearly equal in iteration after iteration. In such cases, although the fractional change per iteration is i: (c « I), there is no guarantee that, say, 1/s more iterations may not actually be required to reach the final solution. The discussion thus far has been couched in terms of integral equations using the A-operator, but it should be stressed that the same difficulties would arise with a similar iterative solution of the transfer equation as a differential equation (we shah, in fact, refer to either procedure as "A-iteration" even when we do not actually employ the A-operator). The failure of A-iteration to converge is a point of crucial importance whose physical significance must be understood completely; to this end we may consider the following simplified analysis.
Suppose that the depth-variation of the Planck fund ion can be represented with sufficient accuracy by a linear expansion
Bv(tv) = a, + byxv (6-3)
and that pv is constant with depth. The zero-order moment of the transfer equation can be written, using equations (2-71) and (2-39)
{dHjdxJ = Jv ~ Sv = Wv
(6-4)
while the first-order moment gives
{CKJCzv) = Hv
(6-5)
If we use the Eddington approximation Kv = %JV and substitute equation (6-5) for Hv into equation (6-4) we obtain
(6-6)
6-1 Iteration: The Scattering Problem
149
where the second equality follows from the form of Bv assumed in equation (6-3). The solution of equation (6-6) is
Jv - Bv = «v «p[-(3xr)*xj + ßv exp[ + (3Av)*Tv]
(6-7)
As we demand that Jv -*■ Bv as xv -» cc, we must have = 0. To evaluate av we make use of the boundary condition Jv(0) = sj3 HJO) = (dJv/dzv)0/^/3 [the second equality following from equation (6-5) in the Eddington approximation]. We thus find from equation (6-7)
J„(0) = av + a, = {dJJdx)QjJl = |>v - aJZXvf]js/3 (6-8) Hence we obtain finally
Jv(i,) = av + brxt. 4- (bv - y/3 av) exp[-(3;.v)*Tv]/[V'3 + (3xv)*] (6-9)
Equation (6-9) reveals the essential physics of the problem. First, it shows that J¥ may be markedly different from Bv at the surface. For simplicity, consider an isothermal atmosphere—i.e., l\. = 0and5v = av;thenatxv — 0, Jv(0) = + Kk') = ?JBJ(1 + a;-). Thus when « 1, then Jv is
much smaller than Bv at the boundary. Second, we see that this departure extends deep into the atmosphere because the slow decay of the exponential term implies that Jv(tv) -> Bv(xv) only at depths xv > (/.J-*; in view of the small values quoted above, these are very large depths indeed. When Jv has approached Bv arbitrarily closely, we say that the solution has thermalized; we therefore refer to iv_i as the thermalization depth (a concept that will be generalized in Chapters 7. 11, and 12).
We may obtain an intuitive understanding of the thermal ization depth from the following physical argument. The parameter = kvj{kv + ov) clearly is just the probability that a photon is destroyed (i.e., converted into thermal energy) per scattering event. To assure thermal destruction, the photon must be scattered about n = 1//.,, times. If the photon progresses through the atmosphere by a random-walk process, with mean free path At (which must be approximately unity), then the total optical thickness through which it may pass without destruction is At = AtAv~* a; a,,"1. Photons emitted at greater depths are unlikely to escape without being thermalized (hence Jv -> Bv\ while those emitted from shallower depths manage to escape and allow ,/v to fall below the thermal value (i.e., Bv).
We now can understand why A-iteralion fails when we adopt Jv = Bv as an initial estimate. Each successive iteration can propagate information about the departure of Jv from Bv only over an optical depth At ^ 1—i.e., a mean free path [recall that £i(At) falls off as e~ A'/Ax for At » 1]. Thus we must perform of order xv~ = iterations to allow the effects of the boundary to make themselves felt in the solution to a thermalization depth. When
150
Solution of the Transfer Equation
} « 1 such a procedure becomes computationally prohibitive, and we conclude that any useful method must account for the scattering terms in the source function from the outset and provide a direct solution for such terms.
6-2 Eigenvalue Methods
A characteristic mathematical difficulty that emerges in treating the transfer equation as a differential equation arises from the nature of the boundary conditions. Suppose we use the method of discrete ordinates, replace the angular integral for Jv by a quadrature sum, and attempt to integrate numerically the system
1 "
L j=
(1 - p)B, (i = ±1,. - ±n) (6-10)
To effect the integration, we require starting values for f, for all values of /; these are fixed by the boundary conditions. As described in Chapter 2, the boundary conditions fall into two groups, namely /;(0) = 0, (i = -1,..., — n), for incoming rays on the range — 1 ^ pt ^ 0 and ^Ow) = gift) [e-g., gift) = Bv~], d = I- ■ ■ * »)> for outgoing rays on the range 0 ^ pt ^ 1. Here zmax refers to the deepest point actually treated in a semi-infinite atmosphere. The problem is this: suppose we wish to start the integration at z = 0, and proceed step-by-step inward; we cannot, for we do not know the values of /;(0). Similarly at rmax we lack values for
Thus we face an eigenvalue problem of order n. We could, for example, guess a set of values for / ,( rmax) and use these to integrate toward the surface. When the integration reaches the surface, we would in general find I _ ;(0) ^ 0. In principle, we could then adjust the values of f_,(traax), and by successive trials find those values that forced 7-/(0) = 0. In practice, however, this method is strongly unstable and can work only if tmax is not very large. We can see this as follows. As we know from the grey problem, the discrete ordinate method leads to exponential solutions of the form expf ±kz) where the A's are of order 1/u. In cases where the coefficients (such as pv) are depth-variable, the solution no longer consists of pure exponentials, but, nevertheless, still has an exponential character, perhaps f(x) exp{±kz) where / is a weak function of z. In a semi-infinite atmosphere we must suppress the ascending exponentials. For the grey problem this can be done explicitly, for we have an analytical form with which to work. But in the nongrey variable-coefficient case, the solution is known only numerically, and unless exactly the right choice of starting values is made, it contains both the
6-3 The Transfer Equation as a Two-Point Boundary Value Problem
151
ascending and descending exponentials. Therefore, in general the terms in exp(/a) will be present; these are called parasites, and they increase at a rate of order exp(2fcr) relative to the true solution. Thus if our starting values are wrong by an error the parasite will be of order t: exp(2kzmax) ~ e i0JiC"'" compared to the true solution at the other boundary, and it is obvious that, unless our initial choice is very good (c « 1), the parasite will swamp the true solution, which will then be lost. In fact, to retain any vestige of the real solution, we must employ n ~ kzm.A^ significant figures. If several angle-quadrature points are used, some pt « 1 and hence some k » 1, so even with a moderate rmax x 10 we will lose the solution on typical computers. At t a 1 in the continuum, zm3X may be ~ 103 to 104 in the lines, which shows the hopelessness of this approach. In summary, the mathematical structure of the problem requires that we employ a method that accounts explicitly for the two-point nature of the boundary conditions from the outset. We now turn to a discussion of such methods.
Exercise 6-1-- (a) Solve equation (6-10) with p = 0. B ~ const, for I + with p± = +4. Show that d2Jjdx2 = 4(J — B) and write exact solutions for J, / + J and /_, calculating constants of integration from boundary conditions. Suppose one had chosen I<-fz,mi) = 7_(rmax) = B; evaluate the (false) .solution and show that the error s = B exp( — 2tmax) at the lower boundary amplifies to e = B at the surface, (b) Generalize the discussion to the case where p # 0 (but constant).
6-3 The Transfer Equation as a Two-Point Boundary Value Problem
In this section we shall derive two very general, flexible, and powerful approaches for solving transfer problems. These approaches result from writing the transfer equation as a second-order differential equation subject to two-point boundary conditions. Most of the basic ideas were presented in an important paper by Feautner (209). These methods have proven to be stable and easy to implement; each offers advantages in complementary ranges of the parameters that set the scale of the computational effort to solve a given problem.
SECOND-ORDER FORM OF THE EQUATION OF TRANSFER
In plane-parallel geometry we may write two equations governing the outgoing and incoming radiation field at ±p\
±fi[dl(z, ±p. v)/5z] = X(z, v)[S(z, v) - I(z, ±p, v)] (6-11)
152 Solution of the Transfer Equation
where we restrict p to the half-range 0 ^ /* < 1. We now define symmetric and antisymmetric averages
i[i(z,^v) + J(z, ~p,v)] (6-12)
6-3 The Transfer Equation as a Two-Point Boundary Value Problem
153
u(z, p., v)
v(z,p,v)^\\f^^)'1^-^
and - -- (6-13)
which have, respectively, a mean-intensity-like and a flux-like character. In terms of u and i: we can construct a system of two first-order equations by adding the two equations (6-11) to obtain
u{z, p, v)]
p[dv(z, /i, v)/3z] - z& v)[S(z, v
and subtracting them to obtain
M[5u(z,/(,v)/^]= -x(MM^v)
(6-H) (6-15)
r « i6 I ^ into (6-14) we can eliminate v and obtain Then substituting equation (6-15) into to l<*}
a single second-order system tor u:
h(z,^t)- S(z,v) (6-16)
X(z: v) vz
1 Ou^u^v) /(£> v)
oz
, , . * rfTfz v) = -y(z,v)dz and abbreviating the notation, or, defining dxv = ai^, VI ^ > '
p2(d%Jdx2) = «^
(6-17)
In writing equation (6-15) we have assumed that 5 is symmetric in //; this will be true for most of the source functions we shall consider—e.g., those of the form
Sv = av J t(>vJv. H + ft (6-18)
or Sv = av JR{v\ v)Jv. dv' + /iv (6-19)
but may not be true if the redistribution is angle-dependent [in which case other techniques are required, cf. (460)] or if there are motions in the atmosphere (see §14-1). In equations (6-18) and (6-19) the as essentially stand for scattering coefficients divided by the total opacity and the /Ts represent thermal terms. It must be stressed that these choices of Sv are purely illustrative, in the sense that we shall later (cf. §§7-2 and 7-5) find similar-looking terms that involve the radiation field over the entire spectrum (imposed by a radiative equilibrium constraint) or for the entire transition array for a multilevel model atom. The analysis given below still applies in such cases.
Note that in contrast to the moment equations, which do not close, equation (6-17) [first derived by Feautrier (209)] yields exact closure of the system in terms of the angle-dependent symmetric average u^.. We shall see below that it is sometimes advantageous to follow an intermediate course and to use an approximate closure of the moment equations in terms of variable Eddington factors.
boundary conditions
Equation (6-17) must be solved subject to boundary conditions at x = 0 and at x = rm.lx [which denotes the thickness (or ha If-thickness) of a finite slab, or a great depth where the diffusion approximation applies for a semi-infinite atmosphere]. At t = 0, 1(0, —p,v) = 0 which implies that ^v(0) = u!lv(0) so that
p(eujdx,)0 = «„(0) (6-20)
At t = xmax, we specify I(xmax, + p, v) = I+(p, v), and write v(rmax) = I+{p, v) - ivv(rmaJ so that
If the diffusion approximation is valid at xmax. then
JtW, I1' v) = #v(w) + M
\X v
-Hp;
ciz
(6-21)
(6-22)
so that ivXw) - #v(w), i^vCw) = MXv \SBjdz\
cm,
and
oxv
(I 3BV
= p - -—
oz
(6-23)
Exercise 6-2: (a) Generalize equation (6-20) when 7(0, ~m>v) # 0- (b) Show that for a symmetric slab (infinite in x and y\ of finite thickness (in z) zmax, the lower boundary condition can be written at x = ^rmai_ as [du^Jdx^) = 0. This implies that we need consider only half the slab: 0 ^ t ^ +tm.lx.
difference-equation representation
We now convert the differential equation (6-17) into a set of difference equations by discretization of all variables. Thus we choose a set of depth points [xd], (d = 1,. . ., D) with tx < x2 < • • ■ < rD; a set of angle points {/;,„}, (in = 1,. .. , M); and a set of frequency points {v,,}, (n = 1,. . . , N). For any variable g, we write g(zd, pm, v„) — gdm„. We replace integrals by
154 Solution of the Transfer Equation
quadrature sums-e.g., for equation (6-18) we write
Sd» = E Z bmitdmil + pin (6-243)
11= 1 »1 = 1
Further, we group angles and frequencies into a single serial set of values subscripted i such that v;) = (jxm, v„) at / = m + (n — 1)M, and hence reduce (6-24a) to
i
Sdi = adi X ^4dv^v + (iu H= 1.....I) (6-24b)
(■' = i
Similarly, for equation (6-19) we have
= a* £ 0* i-, A,i< + Ai (i = 1. ■ ■ -, I) (6-25)
£'=1
Note in passing that these source functions are independent of angle, and hence this description contains redundant information (which can be removed when we introduce variable Eddington factors). Equation (6-24b) has an additional redundancy because the scattering integral is independent of v (or of i); we shall exploit this later in Rybicki's method of solving the equations.
Further, we replace derivatives by difference formulae, and write, e.g.,
(dXjdx)d + i » [AXd + i/Axd + i) = (Xd + 1 - Xtl)/(xd+ ! - xd) (6-26)
and (dzX/dz2)d x [(dX/drh^ - (rfX/rfT)„_J
-(Ar(J+.
+ Axd-i
thus, defining
and
we rewrite equation (6-17) as
A%di = ~(Axd-hi + Axd+hi)
(6-27) (6-28)
(6-29)
^ATjj-a,At,j.,-
Mi
1 1
(; = 1,...,/) "d+i.f = ~ Sdi ^ = 2,. . ., Z) - 1)
(6-30)
6-i? 77ti? Transfer Equation as a Two-Point Boundary Value Problem 155
where Sdi has the form of equation (6-24) or (6-25), and indeed can be generalized still further (to include, e.g., the entire spectrum; cf. §§7-2 and 7-5). As indicated, there is one such equation for each angle-frequency point i, at each of D — 2 depth points.
If we now define the vector ud, of dimension V", to consist of the angle-frequency components at depth-point d—i.e., (ud),- = udi—then equation (6-30) can be written as a matrix equation
— A,u,
B„u, - Cu
du(! + l
(6-31)
The (/ x I) matrices Ad and Cd are diagonal and contain the finite-difference representation of the differential operator. Bd is a full matrix that has the differential operator down the diagonal plus off-diagonal terms that come from the quadrature sum representing the scattering integrals in equations (6-24) and (6-25). hd is a vector containing the thermal source terms. More accurate difference representations than equation (6-30) may be written using spline colocation (374). (442) or Hermite integration formulae (34), but these do not change the general form of equation (6-31) (though A,, and Cd may become full).
To complete the system, we use the boundary conditions. At the surface we could write
PkiMn - uu)/Axhi = Mlif (6-32)
which is only of first-order accuracy; second-order accuracy can be obtained (30) from the Taylor's expansion «2 = ux + Axi.(du/dx)1 + ^Ax±2(d2u/dx2)li which, using equations (6-17) and (6-20). yields
tt("2« - "uJ/At.,.,- = «u + f*ij(uu - Su) (6-33) or, in matrix form
Mi ~ Cii*2 = Li (6-34) Similarly, equation (6-21) at the lower boundary becomes
ji{{uDi - ua-ui)IAxD-hi - IDi + which, in matrix form is
-A^.j + BDuD = LD Note that AA = 0 and CD = 0.
*Di ~ ( ^ ^D-i, iJ Hi) (UDi ~ SDi)
(6-35) (6-36)
156 Solution of the Transfer Equation
Exercise 6-3: Derive equations (6-33) and (6-35), specialize the latter to the diffusion approximation using equation (6-23).
THE FEAUTR1ER SOLUTION
The set of equations (6-31), (6-34), and (6-36) have the overall structure
h \
U
-A2 B2
0 -A3
-c2
B3
-C
*D-1
^BD_5 ^.JIU^.I I^D-ll
— AD BD / w / \LD /
(6-37)
i
"2
Each element indicated is either an (/ x /) matrix or a vector of length I: the grand matrix has a block rridiagonal structure, and the solution proceeds by an efficient forward-elimination and back-substitution procedure (209). In this scheme we in effect express each ud in terms of ud + l and substitute into the following equation. Thus from equation (6-34) we can write
ux = B]_1C>2 + B1 xh} = D,u2 + V! (6-38)
Substituting equation (6-38) into equation (6-31) for d = 2 yields u2 D2u3 + v2 where D2 = (B2 - A2Dl)~LC2
and v2 = (B2 - A2D,)"1(L2 +■ A2vt)
We therefore have in general
where
(6-39) (6-40) (6-41)
D, = (Brf -
and vd ~ (Brf - A^-^'f1^ + Arfvl(_
for d = 1,..., D. Starting at d = 1, we compute successive values for D,, and \d through d = D - 1. At the last point, d — D, CD = 0. hence DD = 0, and uD = vb [which still follows from equation (6-41)]. Having found Up we then perform successive back-substitutions into equation (6-39) to find ud, [d = D — I,..., 2, 1). Having found udmtr we may then evaluate JAlt = £]f=1 hmudmir and the source function, which involves frequency integrals of Jf. e.g., Sdt, = y.dn £„- wn4in-Jd„- + filbv
The forward-backward sweep described above accounts explicitly for scattering terms and the two-point boundary conditions. Feautrier's method
6-3 The Transfer Equation as a Two-Point Boundary Value Problem 157
has proven itself to be very stable, and has many desirable properties. Note, for example, that at depth the system tends to become diagonal (the terms in I/At2 ~> 0) and hence j,, -> Sd, as expected; in fact, we find j -> S + fr(d2S/dz2), which recovers the diffusion approximation automatically. The depth-discretization is commonly taken to give equally-spaced steps in log t, usually with 5 or 6 steps per decade of t; such a choice has the advantage that at different frequencies with widely differing opacities (e.g., a line-core vs. nearby continuum) one has a reasonable distribution of depth-points.
We can estimate the amount of computing time required in a given problem by counting the number of multiplications needed to solve the system; the solution of a linear system of order n requires 0(rc3) operations, so the time required by Feautrier's method is TF = cDI3 = cDM3N3 where D is the number of depth-points, M the number of angle-points, and N the number of frequency points. It is clear that one pays a penalty for any unnecessary redundancy in the angle-frequency information, and that the representation of these variables must be economized as much as possible. If we have a coherent scattering problem, N = 1, M is generally small, and Feautrier's method is optimum. However, in other problems the number of frequencies can be large because we must satisfy the constraint of radiative equilibrium, or statistical equilibrium in several transitions; but the angular information is essentially unnecessary because only Jv, not u/iV, enters these constraints. We therefore eliminate the angular information by introducing variable Eddington factors f. = KJJV (44).
By integration of equation (6-17) over p we obtain
d2{fvJv)idxv2 = j¥ - sr
and the boundary conditions yield
[5(/vJv)/3tJ0 = Mv(0)
(6-42,
(6-43)
and
-[([
cnv ^"4 cz
(6-44)
where hv = ifv(0)/Jv(0). Equations (6-42) through (6-44) may be differenced in the same way as the angle-dependent equations, but the solution of this system requires a time of only Tv — cDN3, which represents a considerable saving. To solve these equations we must know the depth-variation of f. at all frequencies. We proceed as follows, (a) From any given Sv (e.g., Sv = Bv as a first estimate) we can solve equation (6-17) for u[lv one angle and frequency at a time. In matrix form we have T,-u(- = S; where T is tri-diagonal, and u, and S£ represent the depth-variation of udi and Sdi respectively. Solution of a single tridiagonal system of order n requires 0(n) operations, so the time required to evaluate the full angle-dependent radiation field for
15R Solution of the Transfer Equation
gtcen 5V is Tu = c'DMN. (b) Given udm„ we then calculate
fdn ~ £ kjii/^ii^'rimjr/X ^mUdmn m J
and = Z« ^wrtnl'iBin/Z','"ui'"'>- ^ote t'iat even ^ tne radiation field is known only with modest accuracy, the Eddmgton factor may be determined with substantially better precision (e.g., if u is in error by a scale-factor, / is still correct), (c) Now, given fda, we solve equations (6-42)-(6-44) for Jv using explicit expressions of the form of equations (6-24) and (6-25) for Sy (written in terms of Jv). We then re-evaluate Sv using the new values for Jv. (d) Because Sv found in step (c) differs from that used in step (a), we iterate steps (a)-(c) to convergence. If L is the number of iterations the total computing time is TE = L(cDN3 + c'DMN) « cDM^N* for moderate L. Experience with this method for a very wide variety of physical regimes in stellar atmospheres has always shown extremely rapid convergence (L usually is 3 or 4), and substantial economies (about a factor of ten) are achieved. Finally, we note that additional equations can be added to the transfer equations at each depth-point d; these arise from other physical constraints—e.g., statistical, hydrostatic, or radiative equilibrium (see §7-5). The basic form of equation (6-31) remains unaltered because these constraints involve information only at one or two depth-points at a time. Thus if we have C constraints the total computing time becomes TE ~ L[cD(N + C)3 + c'DMN]; this result bears on the question of whether it is advantageous to use Feautrier's solution or Rybickfs solution, which we shall discuss next.
Exercise 6-4: This exercise requires access to a digital computer (of smal] capacity), (a) Write a computer program to perform the formal solution of the transfer equation with a given Sv for uflv, one angle at a time as described above, and to evaluate the variable Eddmgton factors at all depths. Use equally-spaced steps in A log t starting at t = 10~3, up to r = 10 (5 or 6 steps per decade), and use a double-Gauss angle quadrature (4. 921); experiment with the number of angle-points M to examine the sensitivity of the Eddmgton factors to the quadrature, (b) Write a computer program to solve equations (6-42) through (6-44) with given Eddington factors, assuming coherent scattering—i.e., Sv = ajv + /?. Integrate the two programs and study the convergence of the iteration process in cases with a = (1 - k), />' = ;:. z « 1. starting with Jv s 1, for i: = 0.1, 001, 10~4.
THE RYB1CK1 SOLUTION
As we have seen above, the Feautrier solution organizes the calculation in such a way as to group all frequency information together at a given
6-3 The Transfer Equation as a Two-Point Boundary Value Problem 159
depth, and .to solve depth^by-depth; in that method we may treat a fully frequency-dependent source function [e.g., equation (6-25)] with partial redistribution, but the computing time scales as the cube of number of frequency points. In a beautiful paper. Rybicki (543) pointed out that, in the most commonly considered case of complete redistribution, much of this frequency-dependent information is redundant, for to specify the source function [equation (6-24)] we need only the single quantity J = \~3 (Note: these c's are not numerically equal to those in the formulae for TF, TE, etc.). Unlike the Feautner system, in which the computing time scales as the cube of the number of angle-frequency points (i.e., MiN3), Rybicki's method is linear in MA'. It is obvious that Rybicki's method is vastly more economical than Feautrier's (even with variable Eddington factors) when a large number of frequency-points is required. Recall, however, that Rybicki's method works only if Sv can be written in terms of a single quantity J in the scattering integral, while Feautrier's method works for general scattering integrals. In principle one could use variable Eddington factors with Rybicki's method, but the advantage gained would likely be small (if any) because iterations would then be required. It should also be emphasized that Rybicki's method is exactly equivalent to the integral equation approach in which one writes u; = A,J + M;, where the A matrix is generated by analytical integration of the kernel function against a set of basis functions representing J. In fact, T(-'J is the A,- matrix, and inversion
6-3 The Transfer Equation as a Two-Point Boundary Value Problem 161
of T; is markedly less costly than any other approach for generating A (34); put another way. one may use integral-equation techniques if one wishes, but one should do so by means of Rybicki's method for generating Af.
Exercise 6-5: Using a digital computer, write a program to solve the transfer equation by Rybicki's method for a coherent scatteringsource function Sv = aJv + (i for the same values of r. as were used in Exercise 6-4. Note that the Rybickt method docs not show its advantage here because only one frequency-point is involved.
Finally, let us mention the effects of constraints in Rybicki's solution. For each constraint that introduces essentially new information into the problem, one requires an additional new variable similar to J, along with its defining equations. For example, in a multiplet problem (see §12-3) one requires a J for each independent transition, and in problems where one has introduced the full set of statistical equilibrium equations by linearization a new variable is required for each level of the model atom or every line in the transition array (see §12-4). If we have a total of C variables describing the constraints, then each U matrix must consist of C diagonal (D x D) matrices side by side while each V matrix consists of C diagonal (D x D) matrices stacked into a column, and E becomes a matrix of dimension (CD x CD). In this case the computing time for a direct solution becomes
TR = c(Dz ■ M-N-C) + c'(DC)?
for O » 1 this value exceeds the corresponding value for Tr, and at first sight Feautrier's method looks more attractive for dealing with systems involving many constraints (which is why we shall apply it in §7-5 for non-LTE model construction). Nevertheless, for statistical equilibrium calculations, Rybicki's method has been applied successfully even for large values of C by using an iterative solution of the overall system (cf. §12-4).
computation of the flux
To compare with observations, we must calculate the emergent flux. This may be done in a variety of ways. If Feautrier's method is used with variable Eddington factors, hv is available, and hence HV(Q) = hvJr(Q) can be calculated directly. If Rybicki's method or the angle-dependent Feautrier equations are used, we can calculate Hv(0) = £,„ bmpmu(Q, p,„, v). Alternatively, having 5v(rv) we can use the O-operator [equation (2-61)] to find /\(0) = (I)0[S(tv)]; in practice this operation is done using a quadrature sum, for which several choices are available [see, e.g., (141; 246; 8, 33)]. If the flux is required at points internal to the atmosphere one may apply the operator 0T to Sv. or one may compute v{zd±±, pm, v„) from equation (6-15) and find ffj+.i „ = o^u.^di-t,,,,,,, (note that this defines the flux at midpoints of the depth mesh).
7-1 The Classical Model-Atmospheres Problem 163
Model Atmospheres
7-1 The Classical Model-Atmospheres Problem: Assumptions and Restrictions
The model-atmospheres problem refers to the construction of mathematical models that provide a description of the physical structure of a stellar atmosphere and of its emergent spectrum. In its greatest generality, the problem is one of enormous complexity, and presents both physical and mathematical difficulties thai are beyond solution at the present time. It is therefore necessary to make a number of simplifications, and to deal with idealized models that are rather high-order abstractions from reality. Such abstractions are useful inasmuch as they enhance our insight without overwhelming us with detail; yet it is important to state, at the outset, some of the restrictions we have imposed, not only because this helps to define the problem, but also as a reminder of the almost limitless numbers of fascinating research questions left to explore. The assumptions used in our work tall into several broad categories:
(a) Geometry. We assume that the atmosphere is composed of homogeneous plane-parallel layers when the thickness of the atmosphere is small
compared to the radius of the star, or (in 7-6} homogeneous spherical shells when the thickness is an appreciable fraction of the radius. The assumption of homogeneity makes the problem one-dimensional and thus greatly simplifies the analysis; but it excludes many interesting phenomena involved in small-scale structures seen in the solar atmosphere. For the stars we have almost no information about the homogeneity of the atmosphere [see, however, (261, Chap. ID] and we can only hope that one-dimensional models yield some kind of "average" (in an ill-defined sense) information. However, because the "averaging" process is nonlinear, the question is really an open one, and it is not at all clear whether such models always do yield meaningful averages, (e.g., in chromospheres), although they may be satisfactory for some cases. In particular, in the solar atmosphere many of the inhomo-geneities arise from hydrodynamic phenomena driven, ultimately, by the convection zone; for stars without strong convection zones, the atmospheres may indeed be homogeneous. (Counterexample: the Ap stars, which show gross variations of physical properties over their surface, presumably associated with the existence of strong magnetic fields).
(b) Steady state. We shall assume that the atmosphere is in a steady state, and shall avoid discussion of all time-dependent phenomena—e.g., stellar pulsations, shocks, transient expanding envelopes (novae, supernovae), heating by a binary companion, variable magnetic fields, etc. In this chapter we consider only static atmospheres; in Chapters 14 and 15 we extend the theory to steady flows (expanding atmospheres). We shall assume that the transfer equation is time-independent, and that level-populations are constant in time and are specified by statistical equilibrium equations (a special case being LTE) that equate the number of atoms leaving a level by all microprocesses to the number that return.
(c) Momentum balance. Having specified a steady state, we shall consider either hydrostatic equilibrium in which the static gas pressure distribution just balances gravitational forces, or one-dimensional, laminar, steady flows. Here we are ignoring the possibly large effects of magnetic forces: both large-scale (as in the Ap stars) and small-scale (e.g., in sunspots or in the concentrated knots of the general solar magnetic field). We further ignore the effects of small-scale motions such as waves, and larger scales such as super-granulation flows, convective cells, etc.. as well as major tidal distortions in close binaries.
(d) Energy Balance. Usually we shall assume that the atmosphere is in radiative equilibrium, which again implies that it is static; in §7-3 we shall consider the effects of convection, but only in the roughest terms. In Chapter 15 we shall generalize to steady flow and include one-dimensional hydro-dynamic work terms. The existence of complicated motions in the solar atmosphere is well documented observationally [see, e.g.. (694, Chaps. 9 and 10) or (244, Chap. 5)] and, although data for stars are less complete,
164
Model Atmospheres
there is little doubt that complex mass motions play an important role in the atmospheres of many stars (e.g., supergiants). But in its present state the theory is unequipped to handle with full consistency the details of energy exchange between the radiation field and hydrodynamic motions. Turbulent dissipation in convection; wave generation, propagation, and dissipation; effects of shear in rotating atmospheres; magnetic field effects; and a variety of other phenomena are all essentially overlooked [ These are vital phenomena, for without them we cannot account for chromospheres and coronae (in this book we shall approach these regions from a semiempiricai diagnostic view because we do not have an ab initio theoretical method). U remains true that important limits on our understanding of stellar atmospheres are imposed by our inability to handle the intricate interchange of energy between radiative and nonradiative modes, and that development of a satisfactory theory to handle such interactions is probably the most vital research frontier in this field of astrophysics.
It should be said, however (lest the reader receive an unduly gloomy picture of our efforts to date), that progress has been rapid, and continues at an accelerating rale, so that we may reasonably expect at least some of the inadequacies of the present-day theory to be ameliorated in the near future. Moreover, the framework imposed above does appear to yield many successful predictions of continuum features and line profiles for many (perhaps most) stars.
7-2 LTE Radiative-Equilibrium Models
Ju this section we develop the methods that can be used to construct planar, static, radiative-equilibrium models assuming LTE; the results of such calculations will be described in §7-4. As was discussed in Chapter 5, the assumption of LTE vastly simplifies the computation (as one can see by comparing the methods of this section with those of §7-5). We criticized the use of LTE because it does not give an accurate description of the interactions of radiation and matter in stellar atmospheres, and is totally deficient in many important conceptual points (especially regarding line-formation). But on the pragmatic side, LTE models allows treatment of many effects (e.g., line-blanketing) that are of importance in the application of the results of stellar atmospheres computations to the interpretation of photometric indices, stellar temperatures and luminosities, etc., but that still lie beyond the present capabilities of a non-LTE calculation. In a sense, then, the two approaches are complementary: the non-LTE theory provides deep physical insight while LTE allows a preliminary assessment of complexities in the models. Of course the end goal will be to have non-LTE models that are as "refined" as any LTE model can be.
7-2 LTE Radiative-Equilibrium Models 165
THE OPACITY AND EMISSIV1TY: CONT1NUA AND CINE-BLANKET ("NO
The frequency variation of the opacity and emissivity in stellar atmospheres plays a key sole in determining the nature of the emergent spectrum. For example, the sharp decrease in flux shortward of about /.3650 A in A-stars can be explained by the huge jump in the opacity caused by photoionizations from the n = 2 state of hydrogen. Because the material becomes more opaque, we see less deeply into the atmosphere, and therefore receive energy only from the outer, cooler, layers. We have already seen (Chapter 3), that we cannot reduce the problem of an atmosphere with a nongrey opacity to the grey problem by any choice of average opacity, and we must, therefore, make allowance for the detailed frequency-dependence of the absorption coefficient from the outset. At the very minimum we must treat the opacity variation in the continuum, which accounts for the gross features of the energy distribution in the emergent spectrum; in more refined work we must also include the effects of lines.
The opacity at any given frequency contains contributions from all possible transitions (bound-bound, bound-free, free-free) of all chemical species that can absorb photons at that frequency. From equations (5-53) and (5-60) we see that the direct absorption coefficient for process (i -» /) from level i is /!;«!_,-(v). Stimulated emissions return energy to the beam at a rate proportional to Iv; hence (assuming identity of the emission and absorption profiles) we correct the opacity by subtracting stimulated emissions from the absorptivity. In view of equations (5-54) and (5-64), the correction is H;K;/(v)G(v) where G(v) = g-Jg- or G(v) = {njnff exp( — hv/kT) for bound-bound or bound-free processes respectively. Let nf denote the LTE population of state i computed from the usual Saha-Boltzmarm formulae [equation (5-14)] using the actual ion density. Then summing over all levels and processes we have the non-LTE opacity
I I [»,■ - (yMn^jiy) + 2>,. - flf^ftv/kr)««(v)
+ E'WUv, 7-)(l - + l(p(Tt,
(7-1)
where the four terms represent, respectively, the contributions of bound-bound, bound-free, and free-free absorptions, and of electron scattering (other scattering terms—e.g., Rayleigh scattering—may also be added). To calculate the spontaneous thermal emission (non-LTE) we use the rates derived in equations (5-55) and (5-62) to write
I I njiffi/g^v) + J nfaiK(v)e-hvlkr
L < .;><■ i
t]v = (2/lv3/c2)
(7-2)
166 Model A tmospheres
The three terms again describe bound-bound, bound-free, and free-free processes. Emission from continuum scattering terms will be written separately in the transfer equations. Equations (7-1) and (7-2) apply in the non-LTE case; if we assume LTE they simplify to
x (1 __ e-hvlkr) + neae
and
<=(2hv3/c2>"',WkT
11 «f^ some chosen value Xh and plot a graph of this fraction against X;. The result is a smooth curve that can be well approximated by a small number of subintervals (possibly of differing widths) containing constant opacities appropriate to the curve. This procedure may be carried out for a mesh of temperatures and densities to produce a description of the variation of the line opacity through the atmosphere. A critical study of this approach (126) shows that opacity distribution functions yield excellent results, and reproduce both the emergent fluxes and physical atmospheric structure given by detailed direct calculations to satisfactory accuracy.
V
FIGURE 7-2
Schematic opacity distribution function of the spectrum in Figure 7-1, A relatively small number of representative opacities suffice to describe this smooth distribution.
The main limitation of the opacity distribution function approach is that it implicitly assumes that the positions of the lines (in frequency) do not change markedly as a function of depth, measured in units of a photon mean-free-path (i.e., unit optical depth in the continuum). It is crucial to the transfer problem whether a line in one layer of the atmosphere coincides in frequency with a line or with a continuum band in an overlying layer, for photons might freely escape in the latter case, but not the former. Marked variations in the line spectrum, which invalidate the opacity distribution function approach, can occur in a number of situations—for example, the following, (a) Molecular bands of two species may overlap; one species may show a rapid decrease or increase with depth relative to the other. Even though the total opacity of the two bands together might not change, the positions of the two sets of lines could be radically different, (b) A strong shock in the atmosphere might produce an abrupt change in the excitation-ionization state of the gas over a small distance. The line spectra through the shock front might change radically, (c) Velocity shifts in expanding atmospheres systematically move lines away from their rest positions; this strongly affects momentum and energy balance in the material (cf. ^14-1 and 15-4). In such cases one must employ either the direct approach, or a generalization of the statistical approach that in some way allows for the changes in the frequency positions of the lines. An alternative approach, called the opacity sampling technique (based on a random-sampling procedure) has recently been suggested (585); although this method appears computationally more costly than the opacity distribution function method, it also appears that it may not suffer from the limitations just described, and should be tested further.
170 Model A tmospheres
7-2 LTE Radiative-Equilibrium Models 171
hydrostatic equilibrium
In a static atmosphere, the weight of the overlying layers is supported by the total pressure, and it is this balance, in essence, that determines the density structure of the medium. Thus
Vp = pg
(7-5)
where the total pressure p = pg + pR (dynes cm"2); the gas pressure p - NkT; the radiation pressure pR = (4n/c) jKvdv; g is the surface gravity (regarded as a fundamental parameter describing the atmosphere). Here p is the mass density (gm cm"3) which, using the notation of 5-2, can be written
p = (]V - nJmH £ ctkAk = (N - ne)m (7-6)
where mH is the mass of a hydrogen atom, and Ak is the atomic weight of chemical species k with fractional abundance afc. If we define the column mass m (gm cm""2) measured from the outer surface inward as our new in-
dependent variable—i.e.,
dm = ~ p dz
(7-7)
then we may rewrite equation (7-5) as dp/dm = cj, which yields an exact integral p(m) = gm + c. It is obviously advantageous to be able to write such a result, so we shall use m as the independent variable henceforth; the choice of m instead of z has no significant effect on the transfer equation. Using equation (2-77b) for the radiation pressure gradient we can rewrite equation (7-5) in another useful form:
(dpJam) = g - {4ii!c) J* {xJpWv
dv
(7-8)
which shows that radiation forces tend to cancel gravitational forces, and lead to a smaller pressure gradient in the atmosphere. Put another way, the material tends to "float" upon the radiation field. As was shown in Chapter 1, the radiation force is related to the flux through the atmosphere, and we can thus see that for a given Tcll there will be some lower bound on g below which radiation forces exceed gravity and blow the material away. Specifically, Underfill] (633) showed that gravity forces will exceed radiation forces only if g > 65 (7cfl/104)4 cm sec-2. Clearly radiation pressure forces are negligible for the sun (Tcfl important for an O-star (Teff a g is quite low (and indeed approaches gcrii). In fact, as we shall show in Chap-
6 x 103, g « 3 x 104} but become very x 104, g ä 104j and in supergiants where
ter 15. for some O-stars the radiation forces on spectrum lines in stellar winds exceed g and accelerate the material to very large velocities (~ 3000 km sec"l).
Exercise 7-1: Consider fully ionized stellar material of hydrogen and helium (abundance Y). (a) Show that nvoQjp, which provides a lower bound on the opacity, is (7,.(1 + 2Y)/mH(\ + 4Y). (b) Take advantage of the grey nature of electron scattering to show that, if gravity is to exceed radiation forces, then g must be ^gCIil where gctll = aE{{ + 2Y)(oltT^(S)/[cmu{l + 4Y)J. (c) Re-express this result to show that (he luminosity L of the star must be ^ Leta = 3.8 x 104 (.,///.//e)Ls.
For computational purposes we can rewrite (7-5) as a difference equation connecting the depths specified by column masses md and mll + 1, namely
jV
NjcT, - Nd^1kTa^i + (4tt/c) £ w„(fdnJd„ - fd-Ul!J d-Un) = g(md - m,^)
n= 1
(7-9)
Here Kv is expressed in terms of the mean intensity and a variable Eddington factor; i.e., Kv = fvJv. We can obtain a starting value from equation (7-8) by assuming that the radiation force remains constant from the boundary surface upward, and thus
A/j/cTj - ml
g - (4%/c) £ wn(%uJP\)hJ\,.
(7-10)
Equations (7-5) through (7-10) are valid for both LTE and non-LTE atmospheres.
Notice that, if we knew the temperature structure T(m), and could either (a) ignore radiation forces or (b) estimate them, using equation (7-8), as (x/p){vRTtff/c) where x is a suitable mean opacity, then we immediately could derive the density structure N(m). From this we could calculate */*{N, T), n*(N, 77), solve the transfer equation, and thus determine all model properties of interest. Of course in general we do not know the temperature structure, and we must now address the issue of how it is to be determined.
radiative equilibrium: temperature-correction procedures
For a given temperature distribution, the equation of hydrostatic equilibrium can be integrated as described above, and opacities and emissivities
172 Model Atmospheres
derived. The radiation field then follows from a solution of the LTE transfer equation
d2(j;jv)/dTv2 = Jv - ft? + Wv)/X? = 0 - fWjtfVv - «/Zv*)Bv
(7-H)
at all frequencies and depths using the techniques described in Chapter 6. For an atmosphere in radiative equilibrium the total energy absorbed by the material must equal that emitted, hence in LTE
471 So ['J* - (** - n°a*)J^ Ch = 471 So K*(Bv - Jv) dv - 0 (7-12) or, in discrete form, (and allowing for departures from LTE),
^Iw^-^-n^Jj^O (7-13)
In radiative equilibrium the total flux AnH - o~RT%f = constant, and we may choose it (or Teff) as another fundamental parameter characterizing the model. Now in general we do not know the temperature distribution that produces radiative equilibrium, and using our present estimate of T(m) we will normally find that the radiation field does not satisfy equation (7-12) or (7-13). Tt is therefore necessary to adjust T(m) iteratively in such a way that the radiation field does ultimately satisfy the requirement of energy balance. The determination of T(m) is, in fact, the very heart of the problem of constructing LTE models. There are basically two strategies we may use: (a) temperature correction procedures, and (b) solution of the transfer equation subject to a constraint of radiative equilibrium. In temperature correction procedures one attempts to use information about the radiation field calculated from a given X(m) in an a posteriori fashion to estimate a change AT(m) that will cancel out the errors found in the flux and in the flux derivative [equivalent to equations (7-12) and (7-13); see equation (2-71)]. In the second approach, one attempts from the outset to formulate the transfer equation in such a way that the resulting radiation field will automatically satisfy radiative equilibrium. The first approach (corrections) was historically the one originally used to solve the nongrey atmospheres problem, and the methods are often quite ingeniously constructed. The second approach (constraints) is more subtle and powerful, and overcomes inadequacies fatal to "correction" procedures in the non-LTE case, thus allowing a deep penetration into problems of considerable complexity. Ironically, the roots of the idea of using "constraints" are to be found in the methods used to solve the grey problem. We first consider temperature correction procedures.
The first, and most obvious, method is the so-called lambda-iteration procedure. Here we suppose that from a given run of T0(m) we have, in effect,
7-2 LTE Radiative-Equilibrium Models 173
computed Jv = A^[B,(T0f], hence the name of the method, and that equation (7-12) is not satisfied. We then assume that the run of T(m) that does satisfy the condition of radiative equilibrium is T(m) = T0(m) + AT(m), and require that
Jo" k*Bv(T0 + AT) dv = J"* K*JV dv (7-14) Expanding BJT(t + AT) x BV(T0) + AT(dBJdT) we find
AT * J; k?[Jv - BV(T0)] dvj$* K*{dBJdT)To dv (7-15)
It must be emphasized that Jv in equation (7-15) denotes the value already computed from BJT0). If one carries through the process and recomputes a new model with the new temperature distribution, some improvement in satisfying equation (7-12) usually will be found. However, the procedure sulfers from several severe defects.
(a) Because Jv = Atv[Bv(T0)] = BV{T0) + 0(e~Tv), it is clear that at depth the temperature correction goes rapidly to zero, no matter how bad the solution actually is at those points. We found a similar result in the grey problem.
(b) If the frequency variation of k* is such that the opacity is much larger (say several orders of magnitude) at some frequencies than at others, the method again fails. The reason is that in the opaque frequency bands the contribution to the numerator vanishes as tv -» 1 while the contribution to the denominator swamps that of all other bands. In effect the A-iteration procedure is effective only over Atv - 1 for the most opaque frequencies.
(c) Equation (7-12) places a condition only on the flux derivatives; hence we have no way of specifying the actual value of the flux to which the solution converges (if it does).
(d) The real failure of the A-iteration procedure is that it ignores the effect that AT, computed at some depth x, has on Jv(z') at all other depths (i.e., Jv is presumed to be fixed). This oversight necessarily leads to spurious values of AT. Actually Jfx'v) = At.[Bv(T + AT)], which means we should, in reality, solve an integral equation for AT; we shall return to this point later.
When the reasons for the failure of the A-iteration procedure were understood, it was realized that methods were needed that made use of information about errors in the flux itself (which gives direct information about the temperature gradient at depth) as well as in the flux derivative. One method of doing this was suggested by Lucy (283, 93), who generalized the method devised by Unsold for the grey problem (sec §3-3) to the nongrey case. If we use a Planck mean [equation (3-23)] optical depth scale dx = ~k$ dz, then exact frequency integrals of the moment equations (using quantities
174 Model A tmosplieres
without subscripts to denote frequency-integrated variables) are (dH/dz) = [KjIk$)J - B [dKjdz) = {yJ/>4)H
(7-16)
and {dKjdz) = {Zr/Kr)" f 7-3 7>
where icj is the absorption mean [equation (3-32)] and yji is the flux mean [equation (3-21)]; note that the scattering coefficient is included in yf but not in the other means. Then relating K to J with the Eddington approximation, equations (7-16) and (7-17) are combined to give an expression for B(r), which, treated as a perturbation equation for a correction AB(T) = 4f7RX3 AT/n gives finally
AB{z) = -d{AH)jdx + {kJ/}4)
3 £ 0$/Kp) AH(r') rfr' + 2 AH(0)
(7-18)
Here A7/(r) = H - H(z). The first term on the righthand side of equation (7-18) is the correction predicted by the A-iteration procedure; the other terms introduce new information that gives nonnegligible values of AB at depth and produces a response to Mux errors at the surface. Experience has shown the Unsold-Lucy procedure to be quite effective in constructing LTE radiative-equilibrium models (but it has no obvious generalization to the non-LTE case).
Exercise 7-2: Derive equations (7-16) and (7-17) and, applying reasoning similar to that yielding equation (3-44), derive equation (7-J 8).
Another very clever and quite useful method of calculating temperature corrections was suggested by Avrett and Krook (55), who introduce perturbations to both the temperature and the optical depth scale. That is, we suppose that the current temperature distribution Tu(t) is related to the desired temperature distribution T{x) (which yields radiative equilibrium) by a pair of relations: T = T(] 4- Tl and x = t + tv The transfer equation is then expanded to first order in the perturbations x1 and T\, and by taking moments of the resulting first-order equation of the perturbation expansion, equations are derived for tl and 1\. These equations [extended to allow for scattering terms (421) and with an improved closure relation (351)] are
= (1 - jr/H*) - 3-* /; x,V° - B,(T„)] rfv/j; ,,.H° dv (7-19)
and
7-2 LTE Radiative-Equilibrium Models 175
r:=j(l+t,) )~yJ]-pv)[J»-Bv(T0)-]dv
(7-20)
where the prime denotes the derivative d/dt: quantities with superscript or subscript zero denote current values; H° = \$ dv; ^denotes the nominal flux (...,£>)
(7-23)
7-2
LTE Radiative-Equilibrium Models 177
where the Md v represent the contribution of the interval (td oo) to the integral. Substituting equation (7-23) into (7-21), and (a) assuming ;c* is unchanged by AT, (b) writing BfT + AT) » Bv(t) + WBJdT) AT, and (c) introducing a frequency quadrature {v,,}, (n = 1,..., N), we find a set of linear equations for the values of ATrf;
I
I ^UdBldT)d.R(8dd. - Add,n)
ATd,
iV = 1
, (d= 1,, . ., D) (7-24)
Exercise 7-3: Verify equation (7-24).
The solution of this system yields the change in the temperature consistent with the global properties of the radiation field. Because our expansion of Bv is only linear, the system has to be iterated to convergence by using the new temperatures to recalculate Br, (dBJdT), k*, etc.. and re-solving the system; if assumption (a) is valid, we would expect quadratic convergence. There are some defects to this approach. (1) The computation of the A matrix is cumbersome and costly, and (because rc* really is a function of 7") must be done again for each iteration. (2) It is possible to calculate the response of the A matrix to changes in tv (arising from changes in k* caused by the temperature change) but this again is extremely cumbersome and costly (also, there are problems of stability) [see, e.g., (347; 575)]. The method described in the final subsection of this section overcomes these difficulties. (3) As originally formulated, and as described thus far, the method does not force convergence to a prespecified flux. This may be done by applying the diffusion approximation at the lower boundary and demanding the correct flux transport (32). Thus, we write for tv > td. v
Bv(tv) = Bv(Tn)
1_ 0BV
dT dz
k* dT
dT dz
(7-25)
(7-26)
Integrating against p, and over all frequencies, we find
3\Jo K*
J dB,
k* dT
dv
dT
dz
(7-27)
which fixes dT/dz in equation (7-25), and introduces the flux into the quantities Mdi v; note the similarity of this device to that used in the grey problem.
178 Model Atmospheres
Exercise 7-4: (a) Evaluate the elements Add. by assuming a piecewise linear interpolation for BJLxJ on a discrete grid; i.e., on
[xj,rd + 1].selB(T) = [BAm " T) + fi
, for v > v0; parame-
lor an Opacity si^j iu.ivlw. . .vv - v,,
terize the problem in terms of the value of =■ hvo/kT^, which specifies the frequency of the step. Noiiee that the A matrix and M vector are independent of frequency (though different) in the two ranges (v ^ vg}, (v > v0); hence the integrals over frequency of Bv and 8Bv/dT be done analytically and expressed in terms of elementary functions [by using the known result for the complete interval (0, co) and appropriate expansions for (0, v0) when ft « 1 and for (v0, ocj when ji » 1] or in terms of Debye integrals (4,998). Solve the problem for several values of a and /?, starting from the grey temperature distribution (on the Rosseland mean .scale); compare your results with those in (128: 605; 381
A second constraint procedure was suggested by Feautrier (283,108; 210). Noting that radiative equilibrium implies that
ii ii
at all depths d, he solved the transfer problem [equation (6-30) or (6-42)] with the source function
Sdn = Bdll ■ V w,,Kd*tt.Jdtt.l1Zwn.Kt.Bdtt.
(7-29)
where the Ts are regarded as unknowns. Note the conceptual similarity between this approach and that used to solve the grey problem! In contrast to the integral operator approach described above, Feautiier's method is very easy to formulate and solve using the difference-equation procedures described in Chapter 6. In this method the ''scattering" integral in equation (6-24) now involves the entire frequency spectrum. This shows explicitly the physically important fact that the radiation field at any frequency actually depends upon the field at all other frequencies. Using current estimate of Bv and k*, equations (7-29) and the discretized form of (6-42) are solved for Jv at all depths. These values are used in equation (7-28), which is solved for the new temperature that satisfies it (linearizing, in principle, both Bv and in terms of AT and iterating). Because opacities, etc., will be altered as a result of the changes in T, the whole process must be iterated to convergence.
In his original analysis Feautner did not introduce the desired flux explicitly into the problem; one may do so easily, however, by using equation (7-27) to fix \dTjdz\ in the lower boundary condition [equation (6-44)]. If one uses Feautrier's method to solve the system, the cost is high because N, the
7-2 LTE Radiative-Equilibrium Models 179
number of frequencies, must be large (the angles can be eliminated in terms of variable Eddington factors); if radiative equilibrium is the only constraint involved, it is cheaper to use RybickTs method, letting 7 denote the term in the numerator of (7-28); this equation in effect, replaces equation (6-4S). Feautrier applied his scheme with good results for both LTE and non-LTE continuum models. The basic drawback of the method is that it is not clear how to generalize it, as it focuses entirely on the temperature correction (which is not sufficient in general).
Exercise 7-6: With the help of a computer, use the method just described to solve (a) the grey problem for q(x); start with q{x) = C and try several values for C; (b) the opacity-step problem of Exercise 7-5 [cf. (128; 605; 38)]. In part (a) use a quadrature for the frequency integral (not the exact results—-this would make the problem trivial) and Rybicki's method for solving ihe final system.
An alternative method that also uses second-order difference equations for the transfer equation was proposed by Auer andMihalas (38); this method is very easily generalized to extremely complicated problems, and forms the basis of the methods described in the final subsection of this section and in §7-5 for non-LTE problems. If the temperature structure T*(m) of the atmosphere were precisely that which produced radiative equilibrium, and 5f the corresponding Planck function, then the solution of the transfer equation
oUJv)/dxv2 = Jv - B* with lower boundary condition [using equation (7-27)]
8(fJv) = {H_dB?
' tfl + ATd+ n)
(7-38b)
First-order boundary conditions have been written for simplicity it is easy to include second-order terms.
182 Model Atmospheres
If we solved equations (7-37) we would find that the constraint of radiative equilibrium, equation (7-13), is not satisfied; we must therefore change the temperature Tim) in such a way as to more accurately satisfy the conditions of radiative equilibrium, and iterate. There are two difficulties: (a) the problem is nonlinear, and (b) the coupling is global. That is, any change STd implies a change 5Nd (from hydrostatic equilibrium) and therefore b%d, 5r}d, and hence (5Jd.„ at all d' and n throughout the atmosphere. To handle these problems, we linearize the equations, replacing each variable x by x0 + Sx, and retain only first-order terms in the <5's. The power of this method is that (a) it may be applied to a wide variety of constraints, and (b) it produces systems that allow for the effects of a change in a variable at a given point in the atmosphere upon all other variables at all other points. In particular, the linearized transfer equations describe fully how a change in material properties or radiation field at any point propagates and affects the solution at every other point. We may use the linearized transfer equations frequency-by-frequency to eliminate the 5f$ from the constraint equations (radiative and hydrostatic equilibrium), yielding a final system for the perturbations of the "fundamental" variables SN and ST. Thus, linearizing the transfer equation (assuming the Eddington factors remain unchanged) we have, away from the boundaries, and at each frequency v,„
Id- l.ii <5Jj-1, „
1
1
Jdii_
+ 1 -
Xd„
6J,
- {Vdn + "e, d<*Jj)-2 + - + — Sih; .
where
Xdn Xdn XAu
= ftln + Jdn ~ (ne,dae^dn + ^did/Xdn
&dv, = (fdn^dn ~ Jd- 1, n^d-i, nlK^d-n ^dn)
7dn = (jdu^dn ~~ Jd+t,nJd+l,n)A&l:d + i,n£LTdn)
+ adH5u}d-u„ A- bdndo}dn + cdH8cad+liH
a,,„ s
and
Ddn adn + Cdii
Wd„ = Xdn/Pd
(7-39)
(7-40) (7-41) (7-42)
(7-43)
(7-44)
(7-45) (7-46)
7-2 LTE Radiative-Equilibrium Models 183
Note that equations (7-39) through (7-46) apply both for LTE and non-LTE cases.
Exercise 7-9: (aj Derive equation (7-39). (b) Derive linearized expressions for the upper and lower boundary conditions, equations (7-37a) and (7-37c), See also (437).
In equation (7-39), assuming LTE, all material variations are expressed in terms of 5N and ST. Thus, from equations (7-6) and (7-3)
3pd = m(5Nd - Sned) (7-47) ht = (Oy^'dT),, STd + {dXticine)a Sne (y - \)/y = (d In T/d In p)A = VA (7-62)
In stellar atmospheres the gas is not perfect because of the effects of ionization and radiation pressure; we may account for this by generalizing y to F (160, 57) and writing VA = (F — 1)/F where T will not, in general, equal its value for a perfect monatomic gas, namely y — {Cp/Cu) ~ |. Convenient formulae for the calculation of T, allowing for radiation pressure and ionization, have been given by several authors [see, e.g., (638, §56; 643; 364)]. These effects can be of major importance, and may drastically lower VA,
7-3 Convection and Models for Late-Type Stars 187
and hence the critical value of VR at which convection occurs. Thus for a perfect monatomic gas VA = f/f = 0.4. while for pure radiation pressure f = f, so VA — 0.25, and for conditions where hydrogen is ionizing f may be only 1.1 so VA drops to 0.1! These results clearly suggest that we may expect convection to occur in hydrogen ionization zones. This expectation is strengthened by noting that in the limit of the diffusion approximation (- dTjdr) = (3nFfR)/0 6 VA (7-63)
Consider now a rising element of material. If<377is the temperature difference between the element and its surroundings, the excess energy delivered per unit volume when the element merges into the surroundings is pCp ST. The temperature difference arises from the difference between the gradients of the element and the surroundings. Thus for elements traveling over a distance Ar with an average speed v, the energy flux transported is
K^conv = pCpvbT = pCpv[\~dTjd.r) - [~dTJdi%] Ar (7-64)
At a given level in the atmosphere we will find elements distributed at random over their paths of travel; averaging over all elements, we set Ar = 1/2 where I is the mixing length. Further, using the hydrostatic equation (dp/dr) — - pg, and introducing the pressure scale height H = (- d In p/dr)"1 = p/Ujp) we can rewrite (7-64) as
rc/w = h>CpvT(V - Vfi)(//H) (7-65)
To estimate v, we calculate the work done by buoyant forces on an element and equate this to its kinetic ene2"gy. If Sp is the density difference between the element and its surroundings, then the buoyant force is fb = — g dp. The equation of state yields In p = In p — In T + In p, where u is now considered to be variable to allow for effects of ionization and radiation pressure. Thus we may write d(\n p) = d{\n p) - Qd[]n T). where 0 = I — (3 In p/d In T)p, and, demanding pressure equilibrium {bp = 0), we have bp = -Op ST/T, so that
fb = (gQp/T)&T = (gQp/T)[(-dT/dr) - (-rfT/dr^] Ar (7-66)
The buoyancy force is thus linear in the displacement; integrating over a total displacement A, and setting A ~ 1/2 to account for the average over all elements passing the point under consideration, we obtain the average work done on the elements
w = £ fh(Ar) d{Ar) = (gQPH/$){V - VE)(//H)2 (7-67)
We now suppose that about one-half of this work will be lost to "friction" in pushing aside other turbulent elements and the other half wili provide the kinetic energy of the element (i.e., \p\r ^ |tv) from which we find
v = igOHjmy - VE)Hl/H) (7-68) and. therefore, from equation (7-65),
nFcon* = (gQH/32r\pCpT)(V - VE)*(//H )2 (7-69)
One of the uncertainties of this approach lies in the question of how to specify the mixing-length I; usually it is assumed that / is simply some multiple of H, say 1 or 2.
To complete the theory, we need another relation that will allow us to express V and VE in terms of VR and VA; this may be done, following Unsold, by considering the efficiency of the convective transport. As an element rises, its temperature exceeds that of the surroundings (which accounts for the energy transport); the temperature excess implies that it will lose some energy to its surroundings by radiation. This energy loss will diminish the excess energy content of the element and therefore decrease the energy yield when the element "dissolves" at the end of its mixing length. We therefore define the efficiency parameter as
excess energy content at time of dissolution ^ ^
' energy tost by radiation during lifetime of element
The excess energy content of the element is proportional to (V — VE) [cf. equation (7-65)]; had the element moved adiabatically, the energy content would have been proportional to (V — VA). Therefore the loss by radiation is proportional to (V - VA) - (V - VE) - (VE - VA) so that
r = (V - VE)/(VE - VA) (7-71)
Alternatively, we may calculate the quantities in the numerator and denominator of equation (7-70) in terms of local variables. Thus for an element of volume V, with excess temperature ST, the excess energy content is pCpV ST. The radiative loss depends on whether the element is optically thin or thick. In the thin limit the rate of energy loss will be 4nxR AB, from a volume V, over a lifetime (l/v). Assuming an average excess of (<5X/2) over this lifetime, we have
?\hin - (pCpVdT)/lM4aRT3/k){ST/2mV)(l/v)]
where xe denotes the optical thickness of the characteristic element size I, tc = xRi Equation (7-72) applies when xe « 1. At the opposite extreme, ie » 1, we apply the diffusion approximation to determine the radiative flux lost by an element of characteristic size /. with fluctuation ST, by writing i — dT/dr) 35 (<5T/7). Assuming a surface area A and the same lifetime as before, we now have
>W = (pCp('<5r)/[(f6frKr3/3lwl^770/I(/^)] = (pCpv/16cRT3)3xR(V/A)
(7-73)
190 Model Atmospheres
The choice of (V/A) is ambiguous and introduces another source of uncertainty into the theory; if the elements are presumed to be spherical, (V/A) = 1/3 and
Tthicic = hÁpCp)/{^RT3) (7-74)
We interpolate between the two extreme cases by writing
7 = [{pCpm*°*T3)] ■ [(I + k.2)/rj (7-75)
Combining equations (7-71) and (7-75), and substituting equation (7-68) for V we derive finally
V^-Va = _J6^2aRT^^J^- s B (7.76) (V - VE)* pCp(gQH)Hl/H) (1 + W)
The final requirement we place upon the theory is that the correct total flux be transported by radiation and convection together—i.e.,
%F = 7iFrad + 7iFconv = oRT% (7-77)
The mixing-length theory described above is the simplest (and most widely used!) convection theory in astrophysics. Numerous refinements have been proposed, attempting to introduce nonlocal information into the theory; it would take us too far afield to attempt to describe these here, and the interested reader should examine the literature. [See, e.g., (594; 595; 450, 237; 479) and the references cited therein.]
CONVECTIVE MODEL ATMOSPHERES
The computation of connective model atmospheres is more complicated than for radiative models (even assuming the mixing-length theory) because there are two transport mechanisms that must be brought into a final balance to satisfy equation (7-77). We may proceed as follows. Suppose we assume some specification of the temperature distribution—e.g., the grey distribution on a Rosseland-mean optical-depth scale. We then carry out a step-by-step integration of equations (7-35) and (7-36), as before. At each point we may-calculate VR = VX(T, p, pj and VA = VA(T, p, p„). If at some point we find that the instability criterion is satisfied, we must determine the true gradient V, VR V 3s VA, which satisfies equation (7-77). If the instability occurs deep enough for the diffusion approximation to be valid, then {Frj(l/F) = (V/VR). and equations (7-77) and (7-69) reduce to
^4(V - VE)* = VR - V (7-78)
| 7-3 Convection and Models for Late-Type Stars 191
j where A depends only on local variables. Adding (V - VB) -t- (VE - VA) to
I both sides of equation (7-78) and using equation (7-76) to eliminate (VE - VA),
| we find a cubic equation for (V - VE)* = x, namely
A(V - VE)^ + (V - VE) + B(V - VE)* = (VK - VA) (7-79)
j which may be solved by standard methods for the root jc0. We thus obtain
j the true gradient V = VA + Bx0 + x02, and proceed with the integration,
j now regarding T as a function of p. If, at some point, convection ceases,
[ we revert to the original T(fR) relation (adjusted to match the current values
| of T and tr) and continue the integration into a radiative zone,
j In the case that the material is presumed grey [or, for nongrey material,
[ the convection zone is really deep enough that the diffusion approximation
1 is correct, and the true nongrey temperature distribution is known near
j the surface] the treatment described above is essentially exact. Using this
approach for grey atmospheres, Vitense (653) performed computations for a wide range of effective temperatures and gravities; this work nicely delineates the role of convection in stellar atmospheres over much of the H-R diagram. In a general way, the outermost layers can always be expected to be in radiative equilibrium because densities and opacities are small and radiative transport is more efficient than convective. In deeper lavers, the opacity and density rise, ionization may occur, and convection may begin. Convection will have its largest effects in stars of low effective temperatures (in which the hydrogen is essentially neutral in the outer layers) and high gravities (which imply large densities and heat capacity, hence efficient thermal transport). When convection is efficient, it will transport practically all the flux, and V will be close to VA; indeed, in stellar interiors, convection (which it occurs) is so efficient that one may set V = VA and dispense with the mixing-length theory entirely. When convection is inefficient, the true gradient V will lie close to VK, and a substantial part of the flux may be carried by radiation; in this regime the uncertainties of the mixing-length theory make themselves felt fully.
When the convection zone lies close enough, to the surface that the diffusion approximation used to derive equation (7-78) is invalid, it is then necessary to calculate Frad from the solution of the nongrey transfer equation, and employ an iterative temperature-correction procedure. In any such procedure it is essential to account for changes in both Frad and Fconv induced by the proposed alteration of the temperature structure. Methods for constructing convective models based on a generalization of the Avrett-Krook procedure have been used to study F-type main-sequence stars (422), middle-type supcrgiants (500), and M-stars (dwarfs through supergtants) (48). A detailed description of a computer code that treats convection is given in (379). An extensive grid of nongrey models (4000°K ^ Tt{f ^ 50,000°K, 2 log g ^
192
Model A tmospheres
5), including convection effects where appropriate and making allowance for line-blanketing, is available (247. 377). More limited grids of blanketed convective models may be found in (512; 513; 514), and extensive computations for M giants and supergiants, allowing for molecular line-blanketing, are given in (341; 342). The solar convection zone has been studied with both the mixing-length approximation (652) [see also (479)] and more detailed hydrodynamical theories (99; 100). Recently, methods for computing convective models using a linearization procedure similar to that described in §7-2 have been developed (274; 275; 479). The basic change in the formulation is to use equation (7-77) as the energy balance equation; introducing a discrete representation of the flux [cf equations (6-15) and (6-26)]. On an angle-frequency mesh {(.ti7 vt f we can write
— G it Tff r
(7-80)
The convective flux can be regarded as FcmJp, pg, T, V) [given these variables, VE follows from equation (7-76) and Fconv from equation (7-69)]. The radiative term may be linearized as before. In linearizing the convective term, the total pressure is fixed, and the derivatives appearing in the expression
fconv = + (dFe/dpt) 5pa + [oFJdT) 3T + (0FJ0V) SV (7-81)
are computed numerically. Several approximations are introduced (274) to reduce this to an expression in 57Tonly, and practical procedures for handling numerical problems have been developed (274; 275). Improvements in convergence might be obtained by including terms in SN as well as ST, but, as described earlier, this is inherently more costly.
At the present time, convection theory as applied in stellar atmospheres analysis is only heuristic; improvements to the physical theory are being actively pursued and, when more accurate treatments of convection become available, our understanding of the atmospheres of late-type stars will be improved significantly.
7-4 Results of LTE Mod el-Atmosphere Calculations for Early-Type Stars
The largest group of reliable model atmospheres available pertains to solar and earlier spectra! types; therefore we shall confine attention primarily to these stars. For later types, many difficult problems related to molecular line-blanketing and the hydrodynamic structure of the atmosphere must be overcome. There is now a very large literature concerning LTE, plane-parallel, model stellar atmospheres, which cannot be described fully here;
7-4 Results of LTE Model-Atmosphere Calculations 193
we shall merely give a few typical references and invite the reader to examine these papers and the references cited therein. A comprehensive list, through 1965, can be found in (506); many of the models in that list use a grey temperature distribution on a mean optical-depth scale. Extensive grids of unblanketed, nongrey, radiative-equilibrium models can be found in (421; 608); models Including hydrogen-hue blanketing by the "direct approach" have been calculated for A- and B-type main-sequence stars and giants (423; 357), and for white dwarfs (620; 412). Models for O- and B-stars, allowing for blanketing by hydrogen lines and strong lines of abundant light ions, by the direct approach, are given in (449; 7; 298; 105; 471). Major improvements in the simulation of real atmospheres have been achieved by including the blanketing of thousands to millions of lines, using various types of opacity distribution functions. A preliminary model of Procyon (F5IV) allowed for about 30,000 lines (612); extensive grids including hundreds of thousands of lines semtempirically have been published in (247; 512; 513; 514); and recently these efforts have culminated in the publication of models (331;516, 271) allowing for 1,760.000 lines on the range
8000JK ^ Tel, ^ SO.OOO'K, 2 ^ log y £ 5
(as well as a solar model). A few illustrative results horn these calculations will be described below.
EMERGENT ENKRGY DISTRIBUTION
The ultimate goal of stellar atmospheres analyses is the construction of mathematical models that describe the physical properties of the outer layers of stars. Having computed detailed models on the basis of the theoretical principles described in this chapter, one then compares predicted and observed values for the distribution of radiation within the spectrum, and attempts to associate a real star with a definite model. In this way values of the parameters that describe the model, (Teff. log g, chemical composition), can be assigned to the star. We shall concentrate here on the comparison of observed and computed values ox continuum features, deferring a discussion of lines to the second half of this book. In early-type stars, spectroscopic information about gravities comes mainly from profiles of the hydrogen lines (for which the broadening mechanisms are density-sensitive) and about abundances from an analysis of line-strengths; we shall therefore focus mainly on the determination of Teff and related parameters—e.g., the bolo-metric correction. In fitting the continuum we may follow several approaches.
(a) A fit can be made to the entire spectrum. This assumes that a complete energy distribution (perhaps including spectral regions inaccessible to ground-based observations) is available. In most cases the comparison is based on the relative distribution of energy in the spectrum—i.e., Fv/FVo, where r0
194 Model Atmospheres
denotes some prechosen frequency. In a few cases it is possible to make the comparison in absolute energy units using fluxes in ergs cm""2 sec-1 hz"1 for both the star and model; here we obtain an enormously important check on the validity of the whole theory.
(b) More limited information concerning a few outstanding features in the flux distribution may be used. For example, in A- through O-stars the slope of the Paschen continuum (3650 A ^ / < 8205 A) is useful; the name is derived from the fact that the dominant opacity source on this wavelength range in early-type stars is from photoionizations of the n = 3 level of hydrogen. Two other important features are the Buhner jump,
DB ~ 2.5 Iog[Fv(A3650+)/Fv(A3650")]
and the Paschen jump, DP = 2.5 log[Fv(/l8205+)/Fl,(i.8205")]. These parameters give measures of the effects of the onset of photoionization edges near the wavelengths stipulated.
In particular, at the Balmer jump, towards shorter wavelengths the opacity is large, owing to photoionizations from the n = 2 level of hydrogen, hence we receive radiation only from the upper, cooler layers; whereas towards longer wavelengths, the material is much more transparent and we see deeper, hotter, layers from which the flux is larger. The result is a fairly abrupt drop in the flux across these frequency boundaries (actually the drop is not sharp because of the opacity of overlapping lines of the series converging on the continuum). The continuum slope can be observed and computed unambiguously, but one must be able to correct the observed values for interstellar reddening effects, and must have a reliable absolute energy distribution standard (see below). The ''jumps'* are not as strongly affected by reddening or calibration problems because they are defined over a very limited frequency range. However, although the flux ratio is obtained easily from unblanketed models, this abstract quantity is not actually measureable, owing to the confluence of lines near the series limit; hence one must use blanketed models, and apply the same operational process to both observed and computed distributions to obtain meaningful comparisons.
(c) Finally, we may employ colors measured with filters that isolate specified bands within the spectrum. Colors can be obtained easily and accurately by standarized observational techniques, and such measures can be extended to very faint stars by use of broad-baud filters. On the other hand, it is easier to calibrate colors against theoretical models for narrow bandwidths, for then one can account more accurately for line-blanketing effects in the model. In practice a compromise must be struck, and a large number of color systems with various properties exist, many of them measuring parameters that are specially designed to characterize the properties of particular groups of stars [see, e.g., the systems described in (516)]. A widely used system that has been well-calibrated in terms of models is the Stromgren uoby system.
7-4 Results of LTE Model-Atmosphere Calculations 195
AH comparison between models and observations rest, in the end, on the fundamental calibration of the energy distribution of a standard star (or stars) in the sky. and it is impossible to overemphasize the importance of this basic connection between theory and observation [see also (516, 241)]. Because it is, in practice, impossible to make an a priori determination of the absolute efficiency of the telescope-spectrometer-receiver system, one proceeds by comparing a star to a standard blackbody source of known emissivity, using the same observational apparatus. It would take us too far afield to describe the details of this procedure here; it is worth the reader's effort to study the literature on the subject [e.g., (261, Chap. 2; 484; 485 ; 486; 285; 487; 286; 287;288) and references cited therein]! For main-sequence B-stars, both the slope of the Paschen continuum and the Balmer jump depend strongly on FcfF and are insensitive to surface gravity (see Figure 7-3); hence both may be used to infer Tcff.
FIG URL 7-:
Balmer jumps computed from LTE model atmosphere* a« a function of effective temperature and gravity. Ordinate-Balmer jump in magnitudes; abscissa:
Unblanketed and blanketed energy distributions for models with Ttff = 6500°K and log g = 4, compared with obsei vat ions of Procyon. Ordinate: relative flux in magnitude units; abscissa: wavelength 2 in A. From (612), by permission.
198 Model Atmospheres
energy distribution quite well, whereas the unblanketed model is much too bright. Blanketing effects are minor in the visible for B- and O-type stars but become large in the ultraviolet; fits made to models ignoring u.v. line-blanketing will be systematically in error (see below").
An entirely different approach to the derivation of effective temperatures can be made using absolute fluxes. From the fundamental calibration one can determine the actual energy output of a star at a particular wavelength; specifically, for Vega [a Lyr), which is the standard star, the average of the Palomar and Mt. Hopkins work vields (287) a flux, at the earth, of /„ = 3.50 x 10"20 ergs cm"2 sec-1 hz"1 at/.5556 A. For any other star we use the magnitude difference Am of the star relative to Vega (at this wavelength) to scale the flux quoted above by 10~°'*Am. As was discussed in §1-4, we can convert fluxes measured at the earth to fluxes at the stellar surface if we know the angular diameter of the star. Angular diameters have been measured (113) for 32 stars on the spectral-type range 05 to F8; these may be used to construct an effective temperature scale. One could, for example, deduce the absolute stellar flux at some particular wavelength, and choose the model that yields the same flux to assign Teff. By comparing the total energy emission with that observable in the visible, one can then obtain the bolometric correction.
Such an approach is vulnerable, however, to serious systematic errors if inadequate allowance is made for line-blanketing (190), and will tend to assign too-high values of 7cff and bolometric corrections. The nature of the ultraviolet line-blanketing is illustrated in Figure 7-7. The blanketed model (449) allows for the strong lines of H, He, C, N, O, Si, CI, Fe, etc. on the range 912 A ^ A ^ 1600 A by the "direct" approach. The effects of the blanketing are quite dramatic. The integrated flux of the blanketed model corresponds to Te!f = 21,900°K, but so much flux has been removed from the ultraviolet, and redistributed to longer wavelengths, that the energy distribution there most closely resembles an unblanketed model with Teff = 24,000°K. Had we used unblanketed models to fit the visible energy distribution (whether absolute flux values or the Paschen continuum slope), the derived effective temperature would have been systematically too high by 2100°K! In fact, "direct-approach" models provide, at best, a lower bound on the total amount of blanketing, and only the recent calculations (381) allowing for millions of lines with opacity distribution functions provide reliable estimates of these effects.
In the face of these difficulties it is preferable to avoid direct reference to the models, and use known angular diameters, visible energy distributions, and recent space observations in the ultraviolet to construct complete absolute energy distributions empirically (516, 221; 169). In this procedure there are nontrivial problems of ultraviolet calibrations and interstellar reddening effects but, with care, these can be overcome (96). From the integrated flux,
200
u s r—
1000
2000
5000 6000
3000 4000 A(A)
figure 7-S
Comparison of empirical absolute energy distribution of a Leo (B7V) (516. 221) with a line-blanketed model (.181) of the same effective temperature (12,200"K). The agreement is excellent and lends strong support to the model techniques. Ordinate: absolute flux 10y f}_ in ergs cm"2 sec"1 A"1 at the earth; abscissa: wavelength ), in A.
the actual effective temperature is obtained; this value is essentially independent o( any model atmosphere. A comparison of the empirical absolute energy distribution with that from a model that has the same (i.e., the empirical) Tcfr is therefore extremely significant, for it provides a test of both the absolute and relative flux predictions of the model. Such a confrontation is shown in Figure 7-8 for the B7V star a Leo (516, 221) and a blanketed model (381) of the same effective temperature. The agreement is excellent, and lends strong support to the validity of the new models.
As an example of an extreme case of line-blanketing effects, it is interesting to consider the ultraviolet flux distribution in the Ap stars as observed by OAO-2. The Ap stars are objects with anomalous abundances of certain elements (e.g., Si, Mn, Cr, Eu. Sr) that are enhanced by factors of 102 to 103. These stars have strong magnetic fields and show spectral variations with time; the observed variation of the field is well explained by an oblique rotator model in which the magnetic axis is inclined to the rotation axis of the star, while the spectral variations indicate concentrations of the elements into definite zones or patches on the stellar surface [see, e.g., (522; 125; 194)]. The greatly enhanced heavy-element abundances produce strong additional blanketing in the ultraviolet, over and above that found in normal stars. The effect is nicely illustrated in the peculiar (Si 3995) star 0 Aur (see Figure 7-9), whose energy distribution in the visible matches a normal star of the same color, but in the ultraviolet (391) fits that of a cooler star. The effect of enhancing the line opacities in models is shown in Figure 7-10, which reproduces the behavior seen in Figure 7-9 at least semiquantitative^. Note that
-1.8 -
-1.6 -
-1.4
-1.2 -
-1.0
-0.8 -
-0.6 ~
_. -0.4
M
_o -0.2
in
(N 0.0
1 + 0.2
+ 0.4
+ 0.6
+ 0.8
+ 1.0 1
0 Aur, A0p, B — v = -0.08
— 134 Tau, B9.5 V, B - V = -0.07
— 7 UMa, AO V, B - V - 0.00
______ i__i_i_L__I
1000 1500 2000 2500 3000 3500 4000 4500 5000 5500
Wavelength (A)
figukl' 7-9
Comparison of relative energy distribution of the peculiar (Si 3995) A-slar 6 Aur with those of 134 Tau (B9.5V) and y U Ma (AOVj (391 j. Because of the enhanced line blanketing arising from the greater heavy-element abundances in the peculiar star, the energy distribution of 0 Aur matches neither of the normal stars, but resembles the coole it a i' in the ultraviolet and the hotter star in the visible. Ordinate, relative flux in magnitude units; abscissa: wavelength in A. From (391), b> permission.
14.000u, Normal opacities 13,090°, 100 x Normal opacities 11,000", Normal opacities
1000 1500 2000 2500
3000 3500 4000 Wavelength (Ä)
4500 5000 5500
6000
figure 7-10
Lme-blanketed models (391) showing effects of inn ma i.
-i
202
203
3317 A
0.4 0.6 Phase
i'IGURO 7-11
Variation of the peculiar fSi-Cr -Eu| A-star y1 CVn in UBV and al A3317 A as measured from OAO-2. From (464], by permission.
the peculiar star has a lower Tcff than a normal star of the same visible color for energy distributionJ, and a total energy distribution that is distinct from a normal star of the same Tcff. A further effect is shown in the Ap (Si-Cr-Eu) spectrum variable a2CVn. The light variations in the visible are shown in Figure 7-11 along with a near-ultraviolet band observed from OAO-2 (464); the far ultraviolet behavior is shown in Figure 7-12 where we see an antiphase variation. These results are easily interpreted in terms of much-increased ultraviolet line-blanketing at phase 0.0, which depresses the ultraviolet flux and redistributes the energy into longer wavelength bands, thus forcing a brightening in the visible; this interpretation is consistent with the fact that the rare-earth lines reach maximum strength at this phase. In contrast, at phase 0.5 we observe regions of the atmosphere where the rare-earth lines are at a minimum, hence the ultraviolet blanketing is lowest; at these phases flux emerges more freely in the ultraviolet (leading to a brightening there) and is
a. ©
a
x
3317
2985 2945 2462
2386
1913 1554
1430
1332
0.0
0.2
0.4
0.6
0.i
0.0
Phase
figure 7-12
Variation of or CVn in ultraviolet as measured from OAO-2. Curves are labeled with wavelengths (A) of filters. Note antiphase variation of far ultraviolet flux relative to visible! From (4641. by permission.
204 Model A tmospheres
7-4 Results of LTE Model-Atmosphere Calculations 205
not redistributed into the visible regions, which, accordingly,-decrease in brightness. The existence of a "null wavelength" near A2960, which shows no variation, supports the differential hne-blanketing interpretation and argues against others involving, e.g., gross geometrical deformations of the stellar surface.
For most stars we do not have detailed energy distributions, but only much more limited information such as colors measured in a photometric system. By suitable choices of filter combinations, colors can be obtained that are sensitive to effective temperature, gravity, and metal abundance and allow a determination of the amount of interstellar reddening. For example, in the Stromgren uvby system for, say, A-G stars, the index (b — y) is a good temperature indicator, cy = (u — v) — (v — b) is gravity-sensitive, while mt = (v — b) — (b — y) is sensitive to metal abundance. To recover the information available in these data the system must be calibrated against model atmospheres. A first step in the procedure is the determination of normalizations between observed colors and those computed from the known filter transmissions. If T^?.) denotes the filter transmission in color i, then
we have
5500
(b - y)
FIGURE 7-13
Comparison of observed (dots) Stromgren b — y) indices for main-sequence stars (which have log v0. Then the atmospliere will equilibrate to some new surface temperature T0 given by
kc /J0 BY{T'0) dv + y,cc J* BV{T0) dv = kc Jj" J, dv + yKf £ Jv dv (7-84)
Assuming that for v ^ v0, J,, Jv° (i.e., neglecting backwarming), and noting that for v > v0, the surface value of Jv % iB(T'0), equation (7-84) can be rewritten as
kc J*Bv(T'0)rfv « kf Jo' Jv° dv
- k, I (y -11 f" BV(T'Q) dv + f' 7V° v0 while tv < 1 for v < v0. Then the mean intensities in the square bracket of the second equality of equation (7-85) saturate to the local Planck function and the whole bracket vanishes, and T'(tv) equals
T0 for the grey case; i.e., the surface temperature drops only in (hose layers
where the opacity jump has become transparent.
Suppose now there are spectrum lines at frequencies {v^ that add to the
opacity, xv = kc + a + l-t4>v, and make both a thermal and scattering
contribution to the emissivity, nv = kcBv + oJv + ^],/i^v[£(B, + (1 — £,)./,,]■
Then the condition of radiative equilibrium reduces to
kc§°J BJTQ)dv = Kcj*j»dv
X WBVi.(T'0) - J,-] + kc X J" (Jv° - Jv) dv
(7-86)
7-4 Results of LTE Model-Atmosphere Ccdculations 207
where 7; denotes j 0), then (just as is the case for continuum scattering!) they have no effect upon the energy balance and the boundary temperature is not changed markedly. We shall see that this conclusion is also supported by the detailed analysis to which we now turn.
The qualitative results obtained above can be put on a quantitative footing by consideration of the illuminating treatment of line-blanketing offered by the picket-fence model proposed by Chandraskhar (150) and further developed by Munch (474). In this model we assume (a) the continuum opacity is frequency-independent, (i.e., rcv = ;c); (b) the lines have square profiles of constant width and a constant opacity ratio /? — 1/k relative to the continuum; and (c) the lines are distributed at random uniformly throughout the spectrum, such that within a given frequency band a fraction w1 contains pure continuum, and a fraction w2 = 1 — wt contains continuum plus lines. (Alternatively, the probability of finding line opacity at a specified frequency is vv2.) A pictorial representation of the problem is given in Figure 7-14, which shows why the name "picket-fence" is appropriate. (A slightly different interpretation of \v\ and w2 allows treatment of an opacity step; see below.) Adopting the continuum as the standard optical depth scale, we have for
09 + 1)K h
F1GURL 7-14
Picket-fence model. Lines are assumed to be a factor of /i more opaque than continuum, and to occur with a probability w2 = 1 — iv j in any frequency band.
208 Mode! Atmospheres
frequencies in the continuum,
pidl^/di) = /<" - Bv (7-87a)
and, in the line
lt(dl{2}/dx) = (1 + $1™ - (1 - e)pJ™ - (1 + e/J)B„ (7-87b)
Integrating over all frequencies (radiation quantities without a subscript v denote integrated quantities) and accounting for the relative probabilities that the band is covered by line or continuum, we find
p(dl{i)/dz) = Tl) - \vxB (7-88a)
and p{dli2)jdx) = (1 + 0)/l2) - (1 - £)/3J,Z) - (1 + s£)w2B (7-88b)
These equations are to be solved simultaneously with a constraint of radiative equilibrium, which is obtained by integrating over angle and demanding that p + (1 + ef>)J'2) = K + vv2(I + m\B (7-89)
Consider now the case of LTE (i.e., e = 1). Let y\ - 1 and yz = 1 + /i. Then equations (7-88) become
p(dlw/dx) = v,U{,) - wtB), (/ = 1,2) (7-90) where, from equation (7-89),
(7-91)
To solve this system we use the discrete-ordinate approach, and choose {/t(.],(? = +1, . . ., +n), such that
Then, substituting equations (7-91) and (7-92) into (7-90), we have
(/ = 1, 2)
(7-92)
vv,
If we now assume a solution of the form
Jf = CWle-fa/(l + WTi)
(7-93) (7-94)
7-4 Results of LTE Model-Atmosphere Calculations 209 we find that k satisfies the characteristic equation
l 2 «
X = I w„ym X cij/il - k2nj2/ym2) (7-95)
m= 1
This equation yields In - 1 nonzero roots for k2 (bounded by poles at l/fh2, ■•■> 1/vV and y2jpx2,y2jp2) and hence 4n - 2 values for k of the form ±kr In addition, we see by inspection that k2 - 0 is a root of the characteristic equation; this root yields a particular solution of the form
/|!) = bwt(r + Q + pjyi)
(7-96)
which may be verified by direct substitution into equation (7-93). The general solution (7-93) is thus of the form
(II r ~kaT 2k— 1 r k z
-h M 1 + feaW/ri B^ 1 - A-aJUl./y(
(/= 1,2)
0' = ±1,-.., +»)
(7-97)
Demanding that the solution not diverge exponentially as t ^ oo. we set L_a = 0 for all a. Requiring that the total flux
F =21 I
(=1 J=-b
we find
6 = Uf)/j>*
-1
(7-98)
(7-99)
The constant 0 and the Lfs, are determined from the surface boundary conditions 7(i?,.(0) = 0, which yield a linear system of In equations in In unknowns:
Q - (I'M + Y Lj{\ - kapM = o, (/ " l'2)
{i =]....,n)
Using equations (7-99), (7-97), and (7-92), we find
and, from equation (7-91),
M i - KW/yi
3
2n~l
m= 1
(7-100)
(7-101)
(7-102)
210
Model Atmospheres
We shall see below that, for the picket-fence model, (k/kr) = Z wm7m~\ so from equation (7-102) we see that the asymptotic solution for B(x) is f Fth, as would be expected. The Rosseland mean scale xK exceeds t; in the limit of infinitely strong lines (y, -* oc), xR(x) = z/\\\ and we see from equation (7-102) that the temperatures must be larger at depth. This is the backwarming effect, and clearly depends mainly upon the bandwidth available for continuum flux transport.
Exercise 7-11: (a) Verify that equation (7-96) is a particular solution of the transfer equation, (b) Verify equations (7-99), (7-101), and (7-102).
As was true for the grey problem, we may calculate the value of 5(0) explicitly. Define the function
S{x) = Q-x+ Z La/(1 - fcax)
(7-103)
The boundary conditions (7-100) show that S(x) = 0 at the In values x = pLilfi. But if we clear equation (7-103) of fractions by multiplying through by a function composed of the product of the 2// — 1 denominators [i.e., R(x) = n*^1 ^ ~~ Kx)l*tfien trie product R[x)S(x) is clearly a polynomial of order 2n in x. But we know In zeros of S(x). hence the polynomial must be of the form R(x)S{x) = C{x — pf) • • • [x — p„)(x — pjy) ■ - ■ {x — pn/y). If we equate the coefficients of the two terms in x" on the two sides of this equation we can evaluate C as C = kik1 ■ • • /<2„-1, and hence obtain finally
which implies that
n n (x- m
1=1 i = l
i2n - i
Flo - M) (7-104)
« = 1
s(0) = k1k2---k2H-lliL2lii2---tin2/y*
"Now consider the characteristic function
(7-105)
T(X) = Z w„x
m= 1 2
= Z wmy*
m= 1 2
1 - X flj/(l - ii/lyJX j=t
1 - X V flj/(X tfjyj)
(7-106)
where A' = l/k2. We clear equation (7-106) of fractions by multiplying through by the product of the 2n denominators. The resulting function is a
7-4 Results of LTE Model-Atmosphere Calculations 211 polynomial of order 2n - 1 in X; now we know that T{X) has 2n - 1
IZ^yT5 Xm = 1/V S° thC PoI>'nomiaI must be of the form i - ',^(A, ~ Xl-"-^- To evaluate C we equate the coefficients of
the terms m X2" 1 to find
ml m
C = (-Ip-'tZ^^XZ^/) = (-l)2"-1 x l(£ Thus we have
3 v»=i 2 t = i / U=i j=i J
(7-107)
From the middle equality of equation (7-106), we have 7\0.l = Z wmym; and from equation (7-107), we have
TiO) = I (V «'1Mym-1)/[(/^2 ■ " ■ H»% ■ " ■ OA'"]2 Combining these two results we then find, from equation (7-105),
3 Z wmy,
(7-108)
(7-109)
S(0)-( Z "W^1
vii = 1 / \ in — 1
But comparison of equations (7-102) and (7-103) shows that
Hence we conclude that
[S(0)/F] = (V'3/4)[(X wmym)(£ wmym-l)]-i (7-110)
This result may be restated in a form that reveals its physical content. The Planck mean opacity is
= B~ 1 j"*' kvB^dv = B~h<(wxB + \v2yB) = k(w, + w2y) (7-1U) while the Rosseland mean opacity is
(lcRyx =-- (dBjdT)'1 J*' kv-\dBjdT)dv
= (dB/dT)"' + vv2/y)u/ß/WF) (7-112)
or fcfi = K(vv't + w2/y)_1 (7-113)
213
212
Grey
¥
oToA^ToT^A 0.5 0.6 0.7 0.8
FKitlRL 7-15
Depth-variation of integrated Planck function in picket-fence models. Solid curve: grey solution, ji = 1; solid dots; y = 10, w, = 0.8, ic2 = 0.2, c = 1 (LTE); triangles: y = 10, w, = 0.8, w, = 0.2, c = 0 {pure scattering). Note back warming effect in both blanketed models, the large surface-temperature drop in the LTE model, and the absence of a surface effect in the scattering model. From (474), by permission.
Thus equation (7-110) reduces to
[B(0)/F] = U/3/4)(ka/Kp)*
or
(T0/Teff)
U/3/4)*(«
(7-114) (7-115)
\JI LIZ.
Now in the limit as y -> oo, the Rosseland mean (being a reciprocal mean) simply saturates at a value k/wj (which, in effect, shows the decrease of bandwidth available for flux transport) while the Planck mean increases without bound; thus the effect of opaque lines in LTE is to lower the boundary temperature (in principle to very low values). An example is shown in Figure 7-15, where B{x)/F is plotted for the grey case and for one of Munch's solutions withe = 1, Wj = 0.8, w2 = 0.2, and y = 10; in this case B(0)/F decreases from the grey value 0A330 to 0.286; i.e., T0/Tcff drops from 0.811 to 0.721.
The analysis just described can also be applied to an opacity jump at a critical frequency, v0, beyond which the opacity increases by a factor of y. We now apply the two versions of equation (7-90) on the ranges (v ^ v0) and (v ^ v0), and define w\B = \v0° By dv and \v2B = Bv dv. We must then assume that vv, and w2 are constant with depth; for example we may
0.8 h
FIGURF. 7-16
Ratio of boundary temperature Tu to effective temperature TaU as a function of 0cff = 5040/Teff. The break near 0eff = 0.25 results from inclusion of the Lyman continuum in the high-temperature models. Upper line gives value of T0/Teff for a grey atmosphere.
choose the values appropriate at T = Teff. As pointed out by Munch (261, 38), this assumption is crude; but we employ it because it simplifies the analysis while retaining the essential physical content. Consider the results shown in Figure 7-16 for the ratio T0/Teff from nongrey LTE model-atmospheres calculations. For all 0eff > 0.25, the Lyman continuum has been omitted. For the coolest models T0/Teff is near its grey value; this is not surprising because the dominant opacity source is H~, which is only weakly frequency-dependent. At higher temperatures the effects of the Balmer jump become important and T0 drops below its grey value. At 0eff = 0.23 the curve shows a sharp break caused by the effects of the Lyman continuum, which is first included at that temperature. At higher values of 7^ff, hydrogen becomes more strongly ionized, the size of the Lyman jump diminishes, and the flux maximum moves beyond the jump, so TQ/Ts{f rises toward the grey value again. At still higher temperatures T0/Teff drops again as a result of the He I edge at A504 A and the He II edge at ?226 A.
We can estimate the drop in the boundary temperature caused by the Lyman jump by applying equation (7-115). Assume that bound-free and free-free absorption by hydrogen are the only sources of opacity. Using equation (4-124) for the free-free contribution, summing nfav> t(b — f) over all bound levels with un - n~2{%-IQJkT) ^ u = (hv/kT), using equations (4-114) and (5-14), correcting for stimulated emission, and setting all Gaunt factors to unity, we may write the opacity coefficient in the form
k* = Cw"3(l - e~u)
l + Y, 2utn 3 QXp(uJn2)
(7-116)
214
Model Atmospheres
where the first term in the square bracket accounts for free-free and the second for bound-free absorption. Because the Rosseland mean is a reciprocal mean, it will be essentially unaltered whether the Lyman continuum is included or not. Thus we need calculate only the Planck mean with and without the Lyman continuum, and use these values to estimate the ratio of T0 for these two cases. We take the limits of the integral in equation (7-111) to be 0 and u0, where uv = ut when the Lyman continuum is excluded, and w0 = gc when it is included. Writing Bv = CVV""(1 — e'")~1,
kp(u0) = C"\l - e~"« + £2u,/i"3[l - exp(-«0 + T2Hi)]} (7-117)
I
Now for 0eff - 0.23, ul = 2.3 x 0.23 x 13.6 = 7.2; if u0 = oo, the exponential term is zero identically, while if u0 = ut it can be neglected unless n = 1 because u1 » 1. Thus
(7-118)
KP(co)/Kp(wi) = |i + 2ux y; n^jjy1 + 2«i I n
= (1 + 2.4«t)Al + 0.4Ul)
For «j = 7.2 we thus find kp(co)/k>(hi) = 4.7, hence
T0(Lyman cont.)/T0(no Lyman com.) = (4.7)" = 0.825 (7-119)
In Figure 7-16, extrapolation of the results without Lyman continuum to (?cff = 0.23 yields to T0/Tei( & 0.65, while TJTefI % 0.56 when the Lyman continuum is included, which gives a boundary temperature ratio of 0.865, in good agreement with equation (7-119) (considering all the approximations that have been made). It should be noted that this temperature drop occurs only at very shallow depths where the Lyman continuum becomes transparent. At optical depths of even 10"4 in the visible, the Lyman continuum is opaque, and temperatures in models with and without the Lyman continuum are practically the same.
A further illustration of the cooling effects of LTE continua and lines is given in Figure 7-17, which shows the temperature structure in a model atmosphere with Trff = 15,000°K, log ^ = 4, consisting of hydrogen schematized as a two-level atom plus continuum (40). The transitions allowed in this atom are La, the Lyman and Balmer continua, and the free-free continuum. The temperature plateau at T x. 10,300'3Kfor — 4 < logr ^ —2 occurs where the Balmer continuum is optically thin but the Lyman continuum is thick; this temperature gives a "T0"/T£{i % 0.68, in fair agreement with Figure 7-16 where the Lyman continuum is omitted. Including the Lyman continuum drops T0 to 9400°K; adding the Lcc line (alone) produces
215
_i. jo frit)-1
F X
- NLTE, no lines 1 -- LTE, no lines -■ NLTE, /.«
- LTE, Ly.
1 i f , 3
FKiURH 7-17
Temperature distribution for LTE and non-LTE models with = 15,00!TK and log y = 4. The atmosphere is composed of hydrogen, which is represented by a schematic model atom with two bound levels and continuum. This model atom accounts for the Lyman. Balmer. and free free continua, and the Lyman-a line. From (40), by permission.
-X -7 -6 -5 -4 -3 -2 -1 logi
a further drop lo 7800°K. Further lines wo uld produce yet additional cooling; the non-LTE results will be discussed in §7-5.
If we now consider scattering lines (e ^ 1), the results obtained above change radically. Define ), ~ (1 + ey?)andn- = (Wl + Xw^1. Then equation (7-89) becomes B = a{Jtu -f- )J<2)) and equations (7-88) become
uidl^idx) = /(1> - wlff(J -f AJi2i) (7-88a') and p(dl^ldz) - y/(2> - - vv2ffAJ(1) (7-
where
and
i= - II
1 "
(7-121a)
(7-121b)
Equation (7-120) has 2n - 1 positive root, /- ti
positive roots /v The solution for 5(T) is
3
(7-122)
216 Model A tmospheres
where M, - cL^GJA - Gafx + {X/y)Ha{Ha - l)"1] and where, in turn, the 2n - 1 constants L3 and the constant Q are determined from the boundary conditions (0) = 0, which imply
2n-l a= 1
(7-123a)
and
Q + (yw2)
a= 1
(7-123b)
jExerei.se 7-/2: Verify equations (7-120) through (7-123).
A solution obtained by Miinch for wi = 0.8, w2 = 0.2. y = 10, and e = 0 is shown in Figure 7-15. Here one finds that the boundary temperature lies only slightly below its grey value, with B(0)/F = 0.4308 compared to the grey result 0.4330. Thus lines, when formed by scattering, have almost no influence upon the boundary temperature; that is, the effect of lines upon the boundary temperature depends sensitively upon the mechanism of \\ne-formation. The backwarming effect is, of course, still present because the frequency band for free-flowing radiation has been restricted. In fact, the backwarming effect is almost identical in the two cases, which shows that backwarming is determined mainly by the frequency bandwidth blocked by the lines, and but little by details of the line-formation process. It is important to realize that LTE line-blanketing cools the surface layers (and produces darker lines); but scattering lines are also dark (cf. §10-2) even when there is no cooling at the boundary. It is not valid to argue for low values of T0 in a stellar atmosphere just because observed lines have dark cores for, in general, the lines may be decoupled from the local temperature distribution, and their central depths may have nothing to do with T0. We shall return to this point again in our work on line-formation. Finally, it is interesting to note that under certain circumstances the abrupt introduction of an opacity edge can cause local heating in the atmosphere [cf. (198)].
7-5 Non-LTE Radiative-Equilibrium Models for Early-Type Stars
The methods and results described thus far in this chapter have been based on the simplifying assumption of LTE. We now turn to the more general
7-5 Non-LTE Radiative-Equilibrium Models 217
problem of constructing models in which the populations of the atomic levels and the radiation field are computed by self-consistent solutions of the equations of transfer and of statistical equilibrium. To understand fully some of the difficulties inherent in this problem, the student should, ideally, already have mastered the material in Chapters 11 and 12; on the other hand, some of the material presented here provides background for those chapters. It is recommended, therefore, that this section be read again after Chapters JI and 12 are studied. In this section we shall follow a somewhat "historical" approach in developing methods that treat, first, continuum-formation alone, and then, a final method that treats both continuum and lines. We shall not describe the line spectrum here (cf. §12-4), but will discuss the effects of lines mainly from the point of view of energy balance.
The fundamental difficulty in the solution of the non-LTE model-atmospheres problem is that the occupation numbers in the outer layers of the atmosphere are determined mainly by radiative rates. Thus the state of the material is only weakly coupled to local conditions (e.g., temperature and density) and is dominated by nonlocal information contained in the radiation field, which responds to global properties of the atmosphere, including boundary conditions. We shall see that the mathematical manifestation of this physical circumstance is that the source functions implied by the equations of statistical equilibrium contain dominant (noncoherent) scattering terms. We have already seen (§6-1) that these terms introduce mathematical difficulties into the solution of the transfer problem.
The first approaches to the non-LTE model atmospheres problem used an iteration procedure, which was successful only for eontinua in which the scattering terms were not large. Subsequent approaches attempted to solve the transfer equations simultaneously with the rate equations by introducing information from the latter explicitly into analytical expressions for the source functions used in the transfer equations. Scattering terms reduce the degree of the coupling of the material to the local thermal pool and hence also tend to introduce nonlocal information into the energy-balance criterion. Thus it becomes difficult to satisfy the requirement of radiative equilibrium. As we noted earlier in our discussion of temperature correction procedures (cf. &7-2), even small errors in energy balance may severely affect the solution of the statistical equilibrium equations. It is thus necessary to find methods that apply the constraint of radiative equilibrium in addition to the simultaneous solution of the transfer and rate equations. Initially this was done by a linearization procedure for the temperature alone; this procedure works when there is fairly direct coupling to the temperature structure (as there is for eontinua via radiative recombinations) but fails for lines where neither the emission nor absorption rates depend directly upon temperature. For models including lines it becomes necessary to make a sweeping generalization to a complete linearization procedure that places all physical variables
218
Model Atmospheres
of interest on an equal fooling and accounts for the global interactions of all variables throughout the atmosphere.
solution bv iteration: detailed balance in the lines
The first attempts to construct non-LTE model atmospheres employed an iteration procedure in which one (a) starts with estimated occupation numbers (say from LTE, (b) uses these to compute the radiation field, and then (c) uses this radiation field to compute radiative rates in the statistical equilibrium equations, which are then solved for a new estimate of the level populations. In practice it was found that this lambda iteration procedure failed (283. 217) when lines were included. The lines are very weakly coupled to local conditions (see Chapter 12) and are very opaque; therefore the severe problem of radiative control of the populations over a very large range of optical depths is encountered. Just as described in §6-1 for the archetype scattering problem of the transfer equation, the iterative process then stabilizes to a spurious value without converging, and successive iterations differ but slightly, even though the current estimate is far from the true solution.
It is therefore of interest to inquire whether it is possible to treat only the continua and to ignore, or at least defer, treatment of the lines. The continua are basically simpler because (a) they are strongly coupled to local thermal conditions (via recombinations), and (b) they are relatively transparent down to depths where densities are high enough to assure domination by collisions (and hence recovery of LTE). This means that the self-consistency problem occurs in regions that are not optically thick, and hence that the iteration procedure has a chance of working (these remarks do not apply in the Lyman continuum, which is as difficult to handle as the lines). An affirmative answer to the question posed above was given by Kalkofen [(283, 175; 345; 346); see also (424)], who showed that, for early-type stars, the Lyman and Balmer lines are so opaque that, at depths where the visible continuum is formed, they can be expected to be in radiative detailed balance. In this case the bound-bound radiative rates upward and downward essentially cancel each other. In particular, for Tef[ - 104 °K, it is found that the detailed balance criterion is met for continuum optical depths r50Q0 > 10 "4, which implies that the continuum is already formed (i.e., is optical!) thin) before the lines go out of detailed balance; thus the continuum-formation problem can be treated essentially independently (except for the Lyman continuum, which is about as opaque as the lines). This result is valuable, for it offers an opportunity to assess the importance of departures from LTE from continuum observations alone.
Mathematically, radiative detailed balance implies that m the rate equations (5-87) we may analytically cancel out (or, equivalents, omit) all pairs
7-5
Non-LTE Radiative-Equilibrium Models 219
of terms of the form [n,^ — nj(nfnjfRji~\; we thus eliminate the most troublesome terms from the equations at the outset. Physically, the approximation proposed here recognizes that photons first "see" the surface in the most transparent regions of the spectrum (i.e., in the continuum), and that free escape out of the atmosphere in these bands leads to departures from LTE at the greatest geometrical depths in the star. The simplified continuum-only problem leads, therefore, to the correct asymptotic behavior at depth, and provides a starting point for the solution of problems that include the line terms.
In practice, the iteration procedure treats the departures from LTE as a perturbation away from the LTE state. Comparison of equations (7-2) and (7-4) shows that with the line-terms omitted, departures from LTE do not affect the expression for the emissivity [if by nf, the LTE population of level /, we mean the value calculated from the Saha-Boltzmann equation (5-14) using the actual (non-LTE) electron and ion densities]. Comparison of equations (7-1) and (7-3) shows that (again omitting lines) we can write 7.v ~ 7* + &7.v ^ we define bt = nfnf, then &%v = ££ (r) where
[fi = b-, ~ 1. Now suppose that at any stage of the calculation we regard as given both T(m) and either (a) the values of bi for all bound levels or (b) the values of all radiative continuum rates. We may then integrate the hydrostatic equation in the usual way. and solve for the electron and ion densities using either essentially the same formalism as in §5-2, but with n* replaced with bp* throughout, or the linearization method in §5-5 with all terms in <5T and bjk set to zero. The latter method yields a consistent current value for ne and «i011; the former does not, for it ignores the non-linearity in J?(, in the collision rates. In the work cited below where this iteration method was employed, the former alternative was used, and the whole process iterated to convergence. We next solve the transfer equation
p(dljdz) = + 3X)f + k*Bv + ne: 10"; curves are labeled with quantum number of the level of the model atom. Note thai level 2 is underpopulated while higher levels are overpopulated. Levels 1 and 2 are locked together by the assumption of detailed balance in the Lyman continuum.
On the other hand, for n 2= 3. hv/kT < 1, and the dilution factor of \ in /v outweighs the effects of the temperature gradient, so Jv < Sv and the levels are overpopulated. In Figure 7-18 we have dt = d2 because it was assumed that the Lyman continuum was also in radiative detailed balance; in this case the radiative rates in equation (7-126) for n = 1 cancel analytically, and we are left with dl Yj = i £\j ~ Z;=2 d.jC\j which implies d{ % d2 because CI2 » C{j for j > 2. Thus, collisional coupling of n = 1 to n = 2 allows the upper level to drive the same departure into the ground state population. Closer to the surface, where the Lyman continuum comes out of detailed balance, the n = 1 level becomes overpopulated (see below). At higher temperatures, characteristic of the O-stars (i.e., Te[[ ^ 35,000:K), the situation is different, for now n — 2 becomes overpopulated, and the ground state n — 1 becomes underpopulated at depths where the Lyman continuum is formed. This would be expected on the basis of the scaling of hv0jkTc{[-
222 Model Atmospheres
mentioned above; these results may also be understood (346) in terms of the anticipated variation of the flux with depth in the various continua (recall that dHv/dxv ~ Jv - Sv). Finally, it should be emphasized that the departure coefficients obtained from the procedure described here cannot be used to calculate line profiles, for they lead to spurious results, as would be expected because the level-populations will be inconsistent with the values they would have in the presence of lines (43).
FORMATION OF THE LYMAN CONTINUUM
The calculations described above assume that the Lyman continuum is in radiative detailed balance. But at some point in the outer atmosphere, the Lyman continuum must begin to become transparent, and significant transfer effects occur that force ntRiK to depart from nKR'Kh and hence lead to an uncoupling of the n = 1 state from the n = 2 stale. This situation becomes most relevant at high values of Te[t, where the high degree of ionization of hydrogen implies that the Lyman continuum is weakened to the point of being only somewhat (rather than markedly) more opaque than the visible continuum. Application of the iteration method to the Lyman continuum fails, and we can gain some important physical understanding of the problem (and also a preview of the problems of line-formation) by analyzing why this is the case. We shall see that one must account for the information in the statistical equilibrium equations by introducing them directly into the transfer equations in such a way as to yield a simultaneous solution of the two sets of equations.
Consider the following simplified problem. Represent the model hydrogen atom by two bound states and continuum, and assume that departures from LTE occur only in the ground state. Let the Lyman continuum threshold frequency be v(>; consider only frequencies v > v0, and suppose that hvJkT » 1 so that stimulated emission can be neglected. Ignoring electron scattering and Gaunt factors, the ground-state, upper-state, and free-free opacities ail have the same "profile" cpv = (v{1/vJJ. Writing n1 = b^n'f, we then have x,. = Xo'K = (Di/.f + Zu)v> where the superscript * denotes LTE values, and the subscript it denotes the contribution of the upper-level and free-free continua. Similarly, i?v - (n* + ?j*)<£v - {"/.* + X*)Bvv. Let dvQ --— '/0dz be the optical depth at the continuum head. Then the transfer equation to be solved is
P(dl,/dx0) =