Boltzmann equation for a photon gas interacting with a plasma

ANNALS

OF PHYSICS:

Boltzmann

69, 321-348 (1972)

Equation

for

a Photon

Gas Interacting

with

a Plasma*

ROBERT J. GOULD Department of Physics, University of California, San Diego, La Jolla, California 92037 Received May 20, 1971

The dynamics of a photon gas is described in a Boltzmann-equation formulation for the rate of change of the photon occupation number. The interaction with the electron component of a hydrogen-helium plasma by Compton scattering is introduced as a Fokker-Planck operator. This operator, as well as the total energy exchange rate by Compton scattering, is derived by simple physical arguments from the limiting formula for the case of a zero-temperature nondegenerate electron gas; the effects of stimulated scattering are discussed. Photon production by various atomic processes (bremsstrahlung, direct radiative recombination, line emission, and 2+1s two-photon continuum emission) is described as source terms in the Boltzmann equation. The spontaneous emission by these atomic processes is conveniently described in terms of the basic bremsstrahlung term which dominates the other atomic processes for kT, 2 1 Ry; at lower temperature the other processes, which involve bound atomic states, dominate. Absorption and stimulated emission by the atomic processes are discussed briefly. The process of radiative Compton scattering (double Compton scattering) is evaluated in the limit of low photon energies for interaction with a zero-temperature electron gas; this process is shown to be negligible compared with bremsstrahlung for the important application to the problem of the primordial plasma and photon gas.

1. INTRODUCTION Because of its cosmological origin, there has been a great deal of interest lately in the 3°K microwave background radiation [l]. In particular, some investigations have been made into the problem of the expected deviations in this spectrum from blackbody form [2-61. This is essentially a problem of the nonequilibrium statistical mechanics of a photon gas interacting with a partially ionized gas of hydrogen and helium, and this paper is devoted essentially to these interaction processes and their formulation in a Boltzmann equation for the photon gas. The important epoch in the evolution of the cosmic blackbody radiation is the period during decoupling from the matter gas, since after this occurs the matter and radiation have different temperatures (cf. Ref. [7]). Deviations from a * Research supported by NASA through Grant NGL 05 005 004 and by an Alfred P. Sloan fellowship grant.

321 0 1972 by Academic Press, Inc.

322

GOULD

Maxwellian distribution in the matter gas are expected to be small [S] and we shall assume that its distribution is exactly Maxwellian at a temperature1 T, . There are many processes involving the matter gas which produce new photons and destroy others; there is also the important process of Compton scattering which changes the energy of a photon. These processes, if not exactly balanced by their reverse processes, will cause deviations in the photon spectrum from blackbody form; in the following sections they are considered, and an attempt has been made to simplify derivations of the relevant formulas whenever possible. First we treat Compton scattering, and here some simplification results because for nonrelativistic electrons interacting with low-energy photons the energy exchange in each collision is small. This allows a “Fokker-Plan& type formulation for the process (see Section 3). Processes such as bremsstrahlung, radiative recombination, and photoionization are of a different type and are considered together in Section 4 and 5 on “Atomic” processes. Section 6 treats “radiative Compton scattering” [9]; under some conditions this could be an important process because it is a mechanism for producing a new photon. All of the interactions considered in this paper involve nonrelativistic particles; synchrotron radiation, which usually involves relativistic electrons, will not be included, although it can produce low-energy photons. Application of the results derived here in a reinvestigation of the decoupling phenomenon of matter and radiation will be given in a subsequent paper. A critical discussion of previous work [2-61 will also be deferred to this later treatment. The enumeration of the various photon processes is given now in the present paper because of the many related applications of these processes.

2. GENERAL BOLTZMANN EQUATION

As in describing distributions of particles with mass by means of a Boltzmann equation, it is convenient to introduce as distribution function the number of photons per unit volume (d+ = dsr d8k) of phase space. Dividing this phase space volume by (277=Qs,the differential number of photons in d4 and in polarization state OLis then dN, = E,(r, p) d3r d3k/(2&Q3.

The distribution function E, is, of course, the dimensionless for the photon gas and in equilibrium A,(,,) = (e+= - 1)-l.

0) occupation number

(2)

1 Essentially, it is the thermal properties of the electron component of the gas that are important.

PHOTON GAS INTERACTION

323

WITH PLASMA

The Boltzmann equation for the photon distribution

function would be of the form

where the vai terms represent “collision integrals” (interaction terms) and the qwjterms represent sources and sinks of photons. The mean velocities of the photon gas in polarization state a. are (c,) and (k,) in position and momentum space, respectively. We shall, in fact, assume a uniform and isotropic photon gas where both of these velocities are zero.2 Essentially, our task will be an evaluation of the terms vmiand qaj on the right-hand side of (3). Generally we shall not be interested in the (two) photon polarization states and shall only consider an unpolarized gas. In the photon-interaction processes an average over initial polarizations and a sum over final polarization states are always assumed. Thus we write C ii, = 2ii, etc *, (Y

and the Boltzmann

(4)

equation simplifies to

an/at = c vi + C qj . z i We now evaluate the terms (vi and qj) for the various interaction

3. COMPTON

SCATTERING.

FOKKER-PLANCK

processes.

OPERATOR

Consider a collision between a photon and an electron in which the energies of each are appreciably less than mc2; then the electron before and after the collision is nonrelativistic. Designating the photon momentum before the collision by k and the electron momentum by p and the quantities for the scattered species by a subscript 1, the momentum and energy conservation relations are k - k, = PI - P,

(6)

k - kl = (p12 - p2)/2mc.

(7)

If II and n, are unit vectors in the directions of k and k1 and 8~ = (k, - k)c = e1- E a There are very few conditions under which a photon gas would have a finite &J; gas in a gravitational fieId could produce a small photon “acceleration.”

(8) a photon

324

GOULD

is the magnitude of the photon energy exchange in the collision, and (7) to find 8~ in terms of3 E, p, n, and n, , -SE

=

(+w2)(l

-

we can use (6)

n * a) + (+zc) p . (n - n,) + (SE)~/~~YZC~ - n * n,) - p * n,/mc

I + (+rc”)(l

(9)

This is an exact equation in the nonrelativistic limit, and could be solved for &. However, 6~ is always small and the term (S~)~/2rnc~ is negligible. It is interesting to note that for p = 0 (electron initially at rest) the formula for 8~ is identical to the exact relativistic formula if (8~)~/2mc~ is neglected. The case p = 0 corresponds to the zero-temperature electron gas, and since e/me2< 1 we have, in good approximation, SE(E,19,)= -(e”/mc”)(l

- cos 0,)

(for p = 0, l /mc2 < l),

(10)

where 8, is the scattering angle. Suppose we have a photon moving in such a zerotemperature electron gas of density n, . Then in each scattering the photon loses a small amount of energy determined by (10) and by the differential scattering cross section; the cross section is, in the Thomson limit [l 1,9], du/dQ, = Br()“(l + co? e,),

(11)

where r,, = e2/mc2 is the classical electron radius. The mean energy loss rate of the photon is (d~/At)~,c,,

= n,c

s

Sr[l + n(~,)] da.

(12)

In this expression the factor 1 + E(cl) is introduced to include the contribution from stimulated scattering4 [12]; c1 is the energy of the outgoing photon and is very nearly equal to E. Substituting (10) and (11) into (12) we then get (d~/dth-,=~

= -(u&m>

n,c2[1 + E(e)],

(13)

where uT = 87r r,-,2/3 is the Thomson cross section. In the integration (12) no contribution comes from the term cos til in (10) because da is symmetric around 0, = 42 and cos 8, is antisymmetric. The total energy exchange rate per unit volume for the photon gas is then computed from

s Kompaneet’s [lo] expression for this quantity contains a minor error in sign. * It is clear that there must be a factor 1 + ii to account for stimulated as well as spontaneous scattering since scattering can be regarded as absorption and reemission of a photon. It follows from the principle of detailed balance that the ratio of stimulated to spontaneous emission is just ii; thus, the factor 1 + R(E~ would be required by detailed balance arguments.

PHOTON GAS INTERACTION

325

WITH PLASMA

where dn,h/de is the spectral number density of photons and the integral in (14) is over this spectrum. Using (13), this integral can be readily evaluated for any photon spectrum. However, the most interesting case is that for a blackbody photon gas at a temperature T,a for which E would be given by (2). The integral (14) can be evaluated in terms of the total photon energy density Uph( cc Tph4) to give (&.th/&-,,O

=

-4(‘7,/mC)

n,kTphU,h

(15)

_

If we had not included stimulated Compton scattering we should have obtained a result smaller than (15) by about four percent.5 Stimulated effects are small here essentially because of the factor l 2 in (13) and another phase space factor c2 in the integrand of (14); thus, large E-values are weighted in the integrand and when E becomes of the order of a few times kTph the occupation number E(E) is small. However, although the stimulated correction is small, it is not negligible and this little problem is a very simple example of the occurrence of a photon-stimulated process. The formulas for the Compton-scattering energy exchange rate for finite electron temperature can, in fact, be found from some simple considerations of the result (15) for the zero-temperature case. Let us continue to take the positive direction of energy flow as that to the photon gas. Now we know that our general expression for d&h/dt (= --du,/dt) must go to zero as T, + Tr,h, corresponding to thermal equilibrium; moreover, the genera1 expression must reduce to (15) as T, ---f 0. Also we expect our general formula to have as a factor the product I1,Uph as in (15). In other words, the general formula must differ from (15) by, say, instead of a factor Tph , a function f (Tph , T,) which approaches Tph as T, + 0 and approaches zero as T, --f T,h . The obvious choice for this function is simply f = T,h - T, . For clearly, the problem is essentially one of kinematics and, in general, we could assume that f could be written as a power series in Tph and T, with general term of the form TLh Teq. However the magnitude of these terms compared with the linear term, Tph - T, would be of order some power of kTph/mc2 or kTJmc2 which is small. Thus the general formula for the energy exchange rate must be, for a blackbody photon gas, hph/dt

= --du,/dt

=

--4(u,/mC)

n,u,hk(T,h

-

T,);

(16)

this formula was derived first by Weymann [13]. We want now to evaluate the spectral change rate in the photon gas due to Compton scattering off a Maxwellian electron gas; in other words, we want to 5 Precisely, for the case of a dilute photon gas (W --, 0, so no stimulated scattering) the numerical factor in the result (15) would be smaller by a factor 5(5)/c(4) = 0.958, where the I’s are Riemann I-functions.

326

GOULD

evaluate the term yc = (ai#t), on the right-hand side of (5). To obtain the expression for vc we shall, in fact, make use of (16); however, the result will be valid for any arbitrary photon distribution interacting with a Maxwellian electron distribution. An integration over all photon energies can then give a more general expression for dz.&~t than (16). We consider an isotropic photon gas, so that the relationship between differential number density and occupation number is dn * (,rr2~3k3)-1 E(E) c2de = ICE(E)c2de.

(17)

Since in Compton scattering the number of photons is conserved, there must exist a photon conservation equation of the form6

where j, is the photon current density in energy space due to Compton scattering. Thus we can write the Compton-scattering term in the Boltzmann equation as vc = (&i/at),

= -(1/Ke2)(ajc/&).

(19)

Now, in a manner similar to that in Kompaneets’ paper [lo], we shall try to arrive at the form for j, by physical arguments. First we make a convenient change of variable, and express the photon energy in terms of the parameter kT, and a dimensionless variable x, x = E/kT, .

(20)

We are treating the process in the Fokker-Planck approximation whereby the photon energy change in each scattering is small. So we expect our expression for vc to contain derivatives like &/ax (“dynamical friction” terms) and a2E/ilax2 (“fluctuation” terms); this means that j, will contain only first-derivative terms Z/ax. A useful requirement for the form of j, is provided by the condition that in thermal equilibrium (at a temperature T,) when ii(x) -+ f&(x)

= (e” - 1)-l,

j, -+ 0.

(21)

Now since

aii,dax = -+ies(i + fies>, and since we know j, must contain a term X/ax,

(22)

the restriction (21) combined

6 We formulate this as an equation for dn/de rather than R(E) essentially because motion is in a three-dimensional space and the photon energy results from three components of tiomenta tc = (Pm8 + p*= + PSYCI.

PHOTON

GAS

INTERACTION

WITH

PLASMA

327

with the identity (22) tells us there must also be a term $1 + ?I) with the same coefficient. In other words, j,(x)

cc pi/ax

+ n(1 + n)].

(23)

The additional important factors in j, can be found readily. Clearly there must be a phase-space factor x2 [see Eqs. (17)-(19)]; there must also be another factor x2 which arises essentially from the kinematics of the Compton scattering. One might expect for this kinematic factor something similar to the mean photon energy change for scattering, since perhaps j, cc (6~). Then because of the result (9) we might be led to try something of the form Ax2 + 3x for this factor. However, the proposed term Bx associated with the term (+zc) p - (n - n,) in (3) should not be expected.’ For the kinematic factor must be some function of the photon momentum and, for an isotropic photon gas, an even function. Since for photons p2 = G/c2 = (xkT,l~)~, this would allow the term Ax2 but not Bx. One might ask why there could not be terms with higher even powers of x. These can be ruled out because they would give the wrong dependence on Tph for the computed expression for Au&At for a blackbody photon gas which must be identical to (16). In fact, in our expression [see Eqs. (19) and (20)]

vc-

_

an (xc=1

A --i a - K(kT,)3 x2 ax [x4 (s

we shall fix A by comparison any photon gas dz&dt

of the flz&dt

= j e(a/at)(dn/de)c

dc = -/

+ $1 + E))],

(24)

using (24) with the result (16). For

6(8j,/&)

de = K j e3vc de.

(25)

Now if T, # TDh , but fi is blackbody at Tph, i%/ax + ii(1 + ti) = e”(e’ - l)-” (1 - Te/Tph),

(26)

y = xT,/Tph .

(27)

where

Substitution

of (24), (26), and (27) into the last integral in (25) yields, on two simple

’ This is not because this term in (9) would average to zero. For this averaging must have a weighting factor ~(1 - j3 cos 0) which is the relative velocity of the electron and photon along the Iatter’s direction [14, 151. Then, for example, <(~/mc)p * n> = -(+nc)
328

GOULD

integrations by parts, a value for A when the result is compared with (16). Substituting back into (24) we get our final expression for vc , an

_~--a,n,kT, mc

vc = (4at c -

1 a ~2 ax [

x4 g (

+ E(1 + ii,)].

The dimensions of vc are reciprocal time and we might introduce a characteristic relaxation time 7c in terms of the coefficient in (28) and express it in the following way: rC = mc/a+,kT, = 3t,/(/3,2); (29) here tc = (n,ca,)-l is the time for a photon to traverse one Compton mean-free path (h, = l/n,ar) and (Be”) is the mean-squared electron velocity in terms of c.

4.

“ATOMIC”

PROCESSES.

SPONTANEOUS

EMISSION

The processes considered here involve essentially three particles: a photon, an electron, and a nucleus or ion. The electron may be either free or bound in an atomic state initially or finally in the process. The spontaneous processes are then bremsstrahlung, radiative recombination and line emission. The induced processes (absorption and induced emission) involve an additional external radiation field and are calculated from the basic atomic parameters associated with the spontaneous processes; how the induced processes are to be included is described in the following section (5). The effects of atomic processes on the cooling of an electron gas were treated in [16]; however, in this section and in this paper we are primarily interested in the photons that result from the processes. In the paper by Kompaneets [lo], aside from Compton scattering, only bremsstrahlung was included. However, as we shall see explicitly, radiative recombination and line emission dominate bremsstrahlung for kT 2 I,(= 1 Ry) and certainly must be included in a treatment of primordial matter during the evolutionary phase of decoupling from the radiation field.8 Thus, in this section, we shall often compare the results for these processes directly with those for bremsstrahlung. A. Bremsstrahlung For electrons incident on ions of charge ze possessing a pure Coulomb field the differential cross section for emitting a bremsstrahlung photon of energy within de is most conveniently written in the form [16, 171 da, Z = 243-3/2vxx3z2X,2(d+) g,,, , * This

was realized

by Peebles

[3] and Zeldovich

et

al. [4].

(30)

PHOTON GAS INTERACTION

WITH PLASMA

329

where X, = filmv, is the deBroglie wavelength of the incident electron, a: is the fine structure constant, and gs,%is the “Gaunt factor.” The Gaunt factor is essentially a constant and, except in the low photon-energy limit, is close to unity. We are interested in evaluating the bremsstrahlung source term qB,z on the right hand side of (5); this is given by [see Eq. (17)] qB.z

=

j$

1

dnB

&-

z

=

2

I

do, z dn, ,

V, --&-

(31)

where the integration is over a Maxwellian velocity distribution of the electrons [dn, = ~&JO,) dv,] and ~1, and n, are the ion and electron number densities. Performing this integration by taking a constant value for g,,, which we replace by an appropriate thermal average9 gB,z , we get qB

,r = 2g’23-3’2n5’2(e3h/m)2 (m/kT,)“”

n,n,Z2gg,zE-3e-E’kTE;

(32)

values for gB,Z may be taken from the paper by Karzas and Latter [18]. The result (32) is the same as that given by Syunyaev and Zeldovich [6]. We shall compare the expressions for v for two other continuum-photon producing processes with it; these processes are direct photon production in radiative recombination and twophoton emission in the 2s + 1s transition following radiative capture and cascade. As we shall see, these processes allow the emission of essentially “suprathermal” photons of energy E> kT, . The energy comes, of course, from the binding energy of the atom involved in the radiative transition. B. Direct Emission in Radiative Capture In the recombination of an electron of energy E, into a hydrogenic level of principal quantum number n, the energy of the emitted photon is

4%) = E, + z2&/n2,

(33)

where IH = 1 Ry = 13.6 eV. There results a term vR,z,n on the right-hand side of (5) due to recombination onto level n, (34) The recombination

cross section oz.%is closely related to the bremsstrahlung

DThis is sometimes written as gefI or instead of g.

cross

330

GOULD

section (30) (see [16]), and is written in terms of a recombination gR,Z,lz for level n which is averaged over Z-values,

Gaunt factor

CJz,n = 253-3~2m3a,,z[z4fH2/E8en(Ee)](gR,Z,n/n3).

(35)

We then readily find, in terms of q&z , (36) (37) where S is the step function

S(x)= 0 tx < 11, = 1 tx > 0,

(38)

and arises because of the minimum value E, = 0 in (33). The important factor in (36) and (37) is the exponential which can get very large at low temperatures and small n. In fact, this limit is the most important case since for a gas ionized thermally the condition of 50 % ionization usually occurs for I/kT N 10. Thus we see that recombination radiation can dominate bremsstrahlung and must be included in a Boltzmann-equation treatment of a photon gas. The relevant quantity for recombination is, say, the ratio (36), (37) summed over all levels it. Actually, the range of the summation is dependent on E; because of (38) we must have n > z(I&l/2 = n, . (39) Thus, for example, in the limit of low photon energy only large n contribute. Then the sum can be replaced by an integral and, neglecting the dependence of j&.&n on n,

(40) =

(~TR.J~B.&(~""~~

-

1)

w

(gR.J&,A

E/kT,

(% > 1).

Here gR,Z is an effective recombination Gaunt factor evaluated by an appropriate averaging procedurelO and is close to unity. In the limit of high temperature and lo The weighting factors in the averaging would be essentially l/n” in this case.

PHOTON GAS INTERACTION

331

WITH PLASMA

away from the low-energy limit the exponentials and the step functions can be replaced by unity and we geP pR/B

=

(gR.z/gB,z)(2Z21H/kTe)

5c3)

(n, < 1; zV,/kT,

< l),

(41)

where c(3) = 1.202 is the Riemann 5-function. While there is no dependenceon E in the result (41), in the low and intermediate temperature case (zzIH/kTe 2 1) and for general 12,there are the pronounced recombination edges.This is illustrated for the case where z21H/kT, = 1 in Fig. 1 where pxIB is plotted as a function of v = c/kT, . For lower temperatures the recombination edges are even more pronounced.

L

001

-1

?=E/hT

FIG. 1. Ratio of the spontaneous source terms for total radiative recombination and bremsstrahlung as a function of photon energy for a hydrogenic plasma at a temperature T, = z2ZH/k. Edges due to recombination to the levels of various principal quantum numbers are indicated. I1 If, however, we have a situation where the plasma is optically thick to photons with c > z2Z~, radiative captures to the ground state are balanced by a photoionization, thereby reversing the process. In this case captures to the ground state would not be included in the summation and C(3) in (41) would be replaced by t(3) - 1 M 0.202. Also the resulting ea.. would be a little larger (and quite close to unity) since the ground-state term would not be included in the average. 595/69/2-3

332

GOULD

A general and simple criterion as to whether recombination radiation or bremsstrahlung is more important can be found by computing, say, the total rates for direct emission of energy in photons by recombination and bremsstrahlung. Using the cross sections (30) and (35) multiplied by photon energy we get, averaging over a Maxwellian distribution, (~E,/&&,d

=

(gR,z/~~,z)(~z”I~/rcT,)

c(3).

(42)

This formula,12J3 which holds for all temperatures, is identical to (41) and shows that recombination dominates when T, < z21H/k = 1.6~~ x lo5 “K. All of these results are for hydrogenic species and have been known for a long time. For a hydrogen-helium plasma they would apply for all processes except the direct radiative recombination to the ground state of He.14 For this process we take uR.He(ground) M +“t(1He/mcv,)2, (43) where ut = 7.4 x IO-l8 cm2 is the threshold photoionization cross section of He in the ground state. Formula (43) follows from an application of detailed balance, taking an E-~ dependence for the photoionization cross section [16]. We then find for the ratio of the ground-state recombination contribution to the e-He+ bremsstrahlung contribution to q qR,He(grOund) qB.He+

(44)

C. 2~1s Emission Continuum

Another photon-production process which can dominate bremsstrahlung at low temperatures is the two-photon decay of the hydrogenic 2s state following radiative capture to levels with n 3 2 and cascade to the 2s state. In this decay process the sum of the two photon energies equals the energy difference between the levels. El + E2 = 2221, = Emax. (45) The probability

distribution

for the emission of photon energies between 0 and

I2 Note that this ratio is a little different from &s/& in [13]. ER in (42) results from the total photon energy resulting from the electron kinetic energy plusthe binding energy. la The remarks in Footnote 11 apply here, as well. I4 Excited states of He are approximately hydrogenic. Also, since bremsstrahlung is closely related to recombinations involving high-n states we expect the e-He+ bremsstrahhmg cross section to be approximately the same as that for e-H+.

333

PHOTON GAS 1NTERACTION WITH PLASMA

emaxis symmetric around &c,, and has been computed by Spiker and Greenstein [19] and is plotted in Fig. 2. With the normalization “max

W(E)de = 2,

(46)

s 0

w max FIG. 2. Distribution

function for emission in the 2s-1s two-photon

transition.

wrnax= 1.592~-~ x lOllerg-l. For atomic hydrogen the reciprocal lifetime of the 2s state for decay by this process is 8.227 set-l [19]. The production rate of these photons is computed by calculating the radiative capture rate to states with n 3 2 multiplied by the probability of resulting cascadeto the 2s state. As long as the plasma density is lessthan about log cm-3, the atom will then decay by two-photon emission rather than undergo a collision-induced transition. This rate has been computed by Pengelly [20] for multiples of 2 in 7’,/104“K between l/S and 8. However, it is convenient to express the results for 2s-1s continuum in terms of the recombination coefficient (II,(2)for radiative capture to stateswith n > 2. That is, if the rate of these radiative captures per unit volume into hydrogenic speciesz is nezz, n 01~~)the number of 2s-1s photons produced per cm3 per set per unit photon energy will be (dn/dt dE)2s-ls= ~2scpn,n,w’(E);

(47)

334

GOULD

here tzS is the probability that these captures lead to the 2s state. Equation (47) is a convenient way of expressing the two-photon production rate because tS9 is a very weak function of temperature. The recombination coefficient ac2) is given by [16] aF)(TJ = 263-3’2~3~a,2(2k7’~/~~)1’2y[$(y) = 1.308 x lo-‘“T,““y[+(y)

- y$(y)] &

(48)

- $l(y)] &),

where y = .z21,JkT, , 4 and 41 are slowly varying transcendental functions of y and gg’ is the effective recombination Gaunt factor. Now these quantities are easily computed and are given in Table I. The quantities + and +1 were computed

Recombination

Te(“K)

4

(l/8) x 10” (l/4) x 1W (l/2) x 104 104 2 x 104 4 x 104 8 x 104 16 x 104

3.287 2.942 2.596 2.253 1.912 1.575 1.250 0.946

TABLE I Parameters (z = 1)’

&, (CaseA)

2R

0.992 0.984 0.970 0.943 0.897 0.824 0.684 0.594

0.893 0.891 0.890 0.882 0.881 0.881 0.887 0.894

0.938 0.943 0.948 0.946 0.947 0.947 0.966 0.981

0.080 0.091 0.105 0.124 0.148 0.180 0.200 (0.210)

6%(CaseB) 0.243 0.267 0.295 0.327 0.361 0.399 0.403 (0.406)

a See text for discussion.

as outlined in [16, 21, 221; & represents, of course, the ground-state contribution which must be substracted out. If we were interested in the total recombination coefficient, the term in brackets in (48) would be simply 4 and instead of g$) a slightly different gR would be used; these effective Gaunt factors were evaluated by the method in [22] using the Glasco and Zirin tables15[23]. The essentialquantity [29 was computed essentially from Eq. (47) and Pengelly’s tables [20]. Pengelly gives results for a “Case A” and “Case B” corresponding to the situation where the Lyman lines are optically thin or thick. That is, for example, in CaseB the transitions to the ground state are exactly balanced by an absorption (the reverse process) so that this cascadechannel is essentially ignored (in CaseB). CaseB is, in general, I5 For T, = l/S x lo* “K the Glasco-Zirin tables could not be used and the slightly more inaccurate temperature averaging procedure of Delmer, Gould and Ramsay [19] was used to compute gR and &). Also, for T. = 8 x IO4 “K, yllz w 1.4 so an additional weighting factor of 0.4 for II = 2 was used to compute .#Rby the method in [19].

PHOTON GAS INTERACTION

335

WITH PLASMA

the more realistic condition in astrophysical plasmas. The parameter tZs is given for both casesin Table I. We seethat (especially for Case B) tZ9 is very slowly dependent on T, . This allowed an extrapolation for the value at T, = 16 x lo4 “K; Pengelly’s calculations did not go to this temperature and the extrapolated values are in parenthesesin Table I. The values given in Table I are in essentialagreement with those computed by Brown and Mathews [24] by a similar method over the narrower temperature range 400&20,000 “K. For hydrogenic species with z # 1 the appropriate tZs at a temperature T, ten be gotten from the values in Table I for z = 1. Since recombination rates to individual hydrogenic states of various z-species are equal for equal values of Te/z2, values for tZs (and all other quantities involved in recombination) for z # 1 at a temperature T, can be identified with the values for z = 1 at a temperature16 T,/2. For 2s-1s photon production the source term in the Boltzmann equation (5) can be expressedconcisely in terms of the bremsstrahlung term (32). We find (49) -

FIG. 3. Ratio of the spontaneous source terms for 2tils emission and bremsstrahlung function of photon energy for Case B [E,,, = (3/4) z2Zw ; y = YZ,JkT,].

as a

I6 Note the remark to this effect in the Appendix of [19] which corrects the suggestion in [23].

336

GOULD

This ratio is plotted in Fig. 3 for Case B and y = z21H/kT, = 1, 2, and 4. We see that for the largest E, the 2.9-1s continuum emission can dominate bremsstrahlung, The total photon energy emitted in the 2s-1s continuum can also be computed easily and also shows explicitly that the radiation dominates bremsstrahlung at low temperatures. The result is [cf. Eq. (42)]

D. Line Emission A process which is closely associated with the others considered in this section is the production of a line-emission spectrum from atomic transitions between discrete levels as a result of radiative capture into excited levels followed by cascade. Like the other processes involving bound levels, line emission dominates bremsstrahlung at low temperatures (T, 5 z21H/k). Because of the many radiative cascade paths from excited states, the calculation of the intensities of these lines presents a computational problem. The most accurate calculations of this type are those of Pengelly [20] for hydrogenic systems and his results will be used here. The emission lines from low-density plasmas have thermal Doppler widths characteristic of the temperature of the plasma. The Doppler width [Am/w - v/c N (kT/Mc2)1/2] is, of course, quite small and for some radiation problems the line shape could be replaced by a a-function. For example, we could write the line-emission source term in the Boltzmann equation (5) as 4L .z

-

nenz

C

nn’

yz.d8(~

-

~,,d),

(51)

where Ez,12121 is the energy of the photon emitted in the transition n + n’ in species z and yzsnnt is a coefficient which depends on T, and determines the intensity of the line. Unfortunately, we cannot derive simple analytical expressions for these intensity coefficients as in bremsstrahlung and radiative recombination. However, one can make semitheoretical fits to Pengelly’s results to arrive at some simple, though approximate, expressions giving the temperature dependence of these intensity coefficients. A reasonable assumption that one can make is that the temperature dependence Of Y.&m ’ is the same as that of the recombination coefficient @ for captures into states with principal quantum number >n. We have essentially already made such a formulation in (47) in the discussion of the 23-1s continuum. Let us introduce a dimensionless probability factor .$,,,,# which is the probability that the total

PHOTON

GAS

INTERACTION

WITH

331

PLASMA

number of recombinations to levels >n give rise to line y1-+ n’. Then, as with t2s 9 Lm ’ will be a weak function of temperature.l’ Then by (17) and (19) we have Yn,nn’ = f z,nn’%(n)lw,nd2.

(52)

This coefficient can also be related to a parameter which is in more common use by astronomers. They measure the integrated line intensity which is related to the line-energy production rate per unit volume I,,,,, (in erg/cm3 set). This rate is proportional to nenz , and in terms of it we can write Yz,nn’ = (z,,.,,ln,n,)lK(E,,,,,)3.

(53)

Simple expressions were suggested for, for example, the Balmer lines of hydrogen by Delmer, Gould, and Ramsay [22]. Based essentially on the result (A.6) quoted in the Appendix, their formulas can be generalized slightly to the form Lnnhnz

= A,.,,, T~1’2[ln(zzIH/kT,)

- c,,,,,],

(54)

where the constants A,,,,, and c,,,, ’ can be fixed by fitting to Pengelly’s results [20] in the temperature regions of interest. For the case of hydrogen (z = 1) one can fit to Pengelly’s results for T, = 0.5 x lo4 and 1 x lo4 “K to give18 c1,32 = 0.68 for the Ha line (n = 3 + 2). For the H/3 line one finds thatlg c1,42= 0 (almost identically). Surprisingly enough, the higher Balmer lines can also be fit quite well to the value cl,,, = 0. The values Al,n2 can then be fixed by fitting to, say, Pengelly’s result at T, = lo4 “K and are given in [22] for IZ > 4; for the Ha line the revised value for the fit (54) is 1.710 x 10-23 cgs units. The form (54) is appropriate in the low-temperature limit and would apply as long as y 2 3, say. At higher temperatures, and in particular for transitions starting at high n, analytical expressions for line intensities can be inferred from formula (A.8). Taking just the first term in (A.8) and using (52), (53) and (48) we expect

Y~,~,c= L,,

0~TL3’*Mn2/v) + 1 - rl

(r/n” Q 1).

(55)

On this basis we expect, for example, the ratio of line intensities starting at n = 20 for T, = 8 x lo4 “K to that at 4 x lo4 “K to be 0.40; this compares with Pengelly’s ratios of 0.49 and 0.43 for the Balmer and Paschen lines respectively. Perhaps the I7 This is more nearlycorrectfor higher n, since for large n the atom tends to cascade “in the classical manner,” that is, by dn = - 1; then once the level n is reached, the subsequent decay probabilities are purely atomic parameters. Recently this type of procedure was used to estimate the production rate of HP photons in high-temperature plasmas 1251. I8 This is for “Case B” where the Lyman lines are optically thick. Is Also we might, at this point, give a typical value for 5 in (52); for the HP line at T, = lo4 “K, 6 m 0.22.

338

GOULD

discrepancies result from the I-value restrictions placed on the specific choices of final states for which I= 0,l and 0, 1, 2, respectively; this would explain why the Paschen-series value is closer to the semitheoretical prediction. The Balmer-series lines give the bulk of the hydrogen-line energy radiated from astrophysical plasmas, and in fact the lower Balmer lines give the main contribution. The overall production rate of Lyman lines, in particular the Ly 01line, is much smaller since the photons are almost immediately annihilated in absorption. The total energy in discrete lines is essentially the energy difference between the initial level n onto which the recombination took place and the n = 2 level. In other words, if the initial recombination is to level IZ, the subsequent energy radiated in lines is E z.n,lines

=

~21d1/4

-

lln2)*

(56)

Here the assumption that is made is that when the atom in its cascade lands on the n = 2 level, no new line photons are produced. If the 2p state is reached, the resonance photon that is subsequently emitted is essentially immediately reabsorbed. If the 2s state is reached, a two-photon continuum is emitted which, of course, is not reabsorbed; we have already taken this process into account. The amount of line radiation emitted per unit volume per unit time is then (57)

where 01,,, is given by (48) with 4 - +1 = c,U~)replaced by I,& [see in Appendix Eq. (A.l)]. As we did for recombination and 2s-1s emission, we can compute the ratio of this power radiated per unit volume to that for bremsstrahlung [see (42), (50)] and we find

the functions #“)(v) and In@ are defined in (A.2) and (A.10). This ratio (58) can become larger than unity only at quite low temperatures, essentially because the middle factor in (58) is always small. For example, for y = z21H/kTe = 1, &U3’ - ~‘3~)= 0.04. Th us, although the energy emitted in lines can be larger than the bremsstrahlung emission at low temperatures, at all temperatures the lineemission energy is appreciably less than the energy emitted in direct recombinations and in two-photon emission [cf. Eqs. (42) and (50)]. While line emission from radiative capture and cascade is not of major importance, electron collision-induced lines can comprise a large fraction of the integrated photon-production spectrum. This is evident from, for example, the work of Tucker and Gould [26], Gould and Ramsay [27], and Gould and Thakur [16] who calculated the cooling rates by various processes in a low-density plasma.

PHOTON

GAS

INTERACTION

WITH

339

PLASMA

We can write the collision induced term in the Boltzmann

equation (5) as [see (51)] (59)

where n, is the number density of the atom (or ion) in which the transition place. Also, we can write [see (17), (19) (52)]

Ya,nn’(coll) = [~(%,nn~)21-1 c ~~,,d(coll)~)

~n.“,,,~

;

takes

(60)

II”

here CJa,n”Ccoll) is the cross section for excitation of state n” and X,q,,,, is the probability that in the resulting radiative cascade the line n -+ IZ’ is produced. Usually, several lines comprise the main contribution to (59) (see [13]) and for these lines X,-,,,, = or - 1. We might compare in order of magnitude the ratio of the contribution for a particular line from inelastic collisions to that from radiative capture and cascade; typically, 4L(COll) 4LomC)

%

-N-

4

<~a.n@OllP) Cy(n)

.

*

(61)

Now the ratio of the collisional excitation rate constant to the radiative recombination coefficient is essentially the ratio of the cross sections involved; the inelastic cross section is of order 7ruo2 and the recombination cross section is of order [see (35)l a3~uo2. However, in the collision-induced process a threshold energy (E,) is involved, so there is a factor exp(-E&T,) in the associated rate. Then

4L(COlU n, exp(--EnlkTe) --013 4L~casc) n, and the ratio is essentially equilibrium

(62)

determined by the atom to ion ratio n&r, . In thermal

where X&= fi/pJ is the electron de Broglie wavelength thermally averaged and I is the ionization energy of the atomic species [7]. On the other hand, for an ionization steady state based on collisional ionization and radiative recombination [7, 161, Wnh~

ion

rad

ret

-

2e1’kr~,

(64)

which would give for the ratio (62) simply exp[(l - E,J/kT,] which can become appreciably greater than unity. Since this latter condition is perhaps the one with more common applications, we see how the production of collision-induced lines

340

GOULD

can be quite important. In general, however, the process depends on the (nonequilibrium) ionization conditions and cannot readily be compared with bremsstrahlung, etc. Given the ionization conditions, the line production rates may be computed from the atomic parameters summarized in [16].

5. STIMULATED ATOMIC PROCESSES

For each spontaneous atomic process involving the production of a new photon there is an associated process corresponding to stimulated emission and its inverse (absorption). Then the total source term in the Boltzmann equation is, for each basic photon-producing process, 4 =

Moreover,

~spon

+

qstim

em +

qabs

.

(65)

we have the basic relation qstim

em

=

nqspon

.

(66)

The expression for qabs can also be written down quite readily; it must be negative and, of course, proportional to fi. Also it must be proportional to the relative number of atomic species in the lower state. Since it is the exact inverse of stimulated emission, then 4as b /4 smem=t’

NtINu 3

(67)

where N,/N, is the ratio of the number of interacting atomic species in the lower and upper states. When the species is in equilibrium at a temperature T, , (NJNJes = exp[(& - EJ/KT,] = exp(E/kT,). We could express the nonequiIibrium population in terms of this equilibrium distribution by introducing a correction factor 7, NJN,

= qecfkTe = -qe”.

(68)

Then 4 = (1 + ii - -qn e”) qspon ,

(69)

and under conditions of complete thermal equilibrium [q = 1; T, = Tph; 1)-l] q = 0. For bremsstrahlung the electrons involved are in the continuum and have a Maxwellian distribution; thus the total q would be given by (69) with r] = 1 and general E(C). For radiative recombination, however, we may have to consider ji = jieq z.z (eE -

PHOTON

GAS

INTERACTION

WITH

PLASMA

341

the nonequilibrium parts of the electron distribution, namely, the bound electrons. In a low-density plasma the atomic levels will not be maintained in a Boltzmann distribution because of the tendency of excited levels to depopulate by emitting photons; as a result, excited levels tend to be underpopulated with respect to the ground state and the continuum. This would affect the overall radiative capture rate to excited states, since the reverse process (absorption) would be reduced. In other words, 7 in (69) would be small and we would have, for the total effective source term for photons emitted in recombination to excited states,

where the spontaneous term is given by (36). The stimulated correction to the recombination rate is also of interest; for example, it affects the production rate of recombination-cascade lines. The appropriate form for this correction can be found readily for recombination to hydrogenic systems, since the recombination cross section is of the form [see Eqs. (33) and (35)]

u z,n = Cz4/c(e - En) n3,

(71)

where E, = z21H/n2. The recombination coefficient including stimulated recombination is then computed by integrating over a Maxwellian distribution and summing over n,

The variable of integration can be changed from v to E by (33), and we see that the total recombination coefficient can be written as wot = (1 + 6))

%pon ,

(73)

where olspon is given by an expression like (48) with the term in brackets replaced by $ (or $(m) if the recombination only to excited states is counted). The appropriate averaged (ti) in (73) is

if recombination to all states is considered, just #W(y)/y (see Appendix); the numerator

m = 1. The denominator

in (74) is

must be evaluated for the particular

342

GOULD

photon spectrum. For the case of a blackbody photon spectrum at a temperature Tph # T, , the integral in the numerator of (74) can be written as (7 E Te/Tph) z = z, (5)

= j;,,

e-ye’u - 1>-’ u-1 du, (75) (1 + w Y 37 n2

=gK[

where K(x) is the exponential would come from I = 1 and

integral (A.3). For e7Y> 1, the main contribution

(6) w [l/+‘“‘(y)]

i 5 ev’la*K [ (T +,el’ ‘1, n=m

(76)

which can be evaluated easily. A formula of the type (73) can also be used to correct, in an approximate way, expressions for the production rate for recombination-cascade photons. For the stimulated enhancement of these photons, including the 2~1s continuum, is due essentially to the stimulated enhancement of recombination to the higher levels. The cascade-path relative probabilities are changed slightly but the main effect of stimulation on the cascade is to speed it up. Then we can (approximately) correct formulas such as (52) by simply multiplying by a factor 1 + (E) with (fi) again given by (74) with m set equal to the value of IZ corresponding to the upper level for the particular line. For collision-induced lines the intensities are not changed appreciably due to stimulation effects. The only change is due to the slightly perturbed cascade path probabilities. This would not be a large effect for collision-induced lines, since usually for a particular line the main contribution comes from direct production of the upper level in a collision (that is, without cascade). Perhaps it should be emphasized again that, if stimulated emission is to be included, absorption must be as well. However, for bound electrons, absorption usually takes place only from the ground state. The associated source (or sink) term in the Boltzmann equation would be given by (77)

where n, is the number density of atomic species s and us(e) is the associated atomic absorption cross section. One source of absorption is photoionization for which the cross section can usually be written in the form “,-photo(E)

=

“,,t(f~/E)~

‘%E/Js)>

(78)

PHOTON GAS INTERACTION

WITH PLASMA

343

where u~,~ is the threshold photoionization cross section for species s (with ionization energy I,), and the step function S is defined in Eq. (38). The important atomic speciesin the primordial plasma are H, He+, and He; for H and He+, 4 w 3 and the threshold cross sections are the (exact) theoretical values (cf. [17]) 6.304 x lo-l8 cm2 and 1.783 x 10-l’ cmz, respectively; for He, q = 2 and the (experimental) threshold cross section is 7.4 x 10-l* cm2 (see Section 4-B). The other source of absorption corresponds to transitions to higher-bound levels of the species(line absorption) for which we can write the cross section as, summing over lines,

The integrated absorption cross section can be expressedin terms of the oscillator strength and other well-known atomic factors,

here fs,l is the absorption oscillator strength for the particular line and may be found in various atomic data tables (cf. [17, 281). This line-absorption process can, of course, lead to production of other lines in the atomic cascadefollowing absorption.

6. RADIATIVE

COMPTON

SCATTERING

The possibleimportance of this process,which is more commonly called “double Compton scattering,” was mentioned by Weymann [ 131 as a mechanism which changes the total number of photons. Radiative or double Compton scattering is essentially ordinary Compton scattering with the emissionof an additional photon; thus it is a higher-order processand the associatedcross section has an additional factor CLThe cross section has been calculated (cf. [9]) and displays the typical infrared divergence associatedwith higher-order processes.The magnitude of the cross section is comparable to the first radiative correction to the ordinary (radiationless) Compton cross section. Bremsstrahlung is another processby which the number of photons is changed. The bremsstrahlung production rate is proportional to the square of the electron density while the radiative Compton production rate is proportional to the product of the electron and photon densities. Therefore, one might expect that radiative Compton scattering might dominate bremsstrahlung if the photon density is very high. However, it turns out that for the cosmic primordial plasma bremsstrahlung dominates. Although radiative Compton scattering is negligible then, it is instruc-

344

GOULD

tive to give a brief quantitative treatment of the problem, if only to demonstrate that the process is indeed negligible. As with ordinary Compton scattering, it is convenient to formulate the problem for a zero-temperature electron gas; it is also convenient to consider only the limit of low energy for the emitted photon. We shall compute the source term due to radiative Compton (rC) scattering, q&e) = (KG-l

dn,,/dt de.

(81)

Further, we can write dnrC/dt de = (dn,,/dt)(dw,/dt),

where dn,,/dt is the number of Compton and20

(82)

scatterings per unit volume per second

dw, = (201/37~)(~lv,/c)~ de/e

(83)

is the probability that in the scattering in which the electron’s velocity changes by Av, a soft photon is emitted with energy within de. We assume that this photon energy (e) is small compared with that of the photon that scatters. We denote the incident energy and energy change of this latter photon by E’ and AL. Then, for an electron initially at rest [see Eq. (lo)], (Av,/c)~ = 2(~‘/rnc~)~(l - cos 13,‘).

(84)

The average in (78) is over the scattering angle 8,’ and in this averaging (cos (9,‘) = 0. The spontaneous differential scattering rate is given by k%@>

sp,,n= n,ca,(dn,,Jde’) de’

and we eventually integrate over this photon spectrum. Inserting a factor 1 + n(e’) to account for stimulated scattering we then have, by Eqs. (81)-(84), 401 necoT 1 W(E) = -3rr (mc”)” Kc3 s d2[ 1 + n(e’)](dn,iJde’) de’,

which is the expression for spontaneous emission in (spontaneous plus induced) Compton scattering. The integral in (85) has already been encountered in (14) and is, for a blackbody photon gas, 4kT$,uph , where Uph is the photon energy density (UT:,; a = 7.568 x lo-l5 cgs units). Then setting uT = (8~/3) ro2 we get, finally, qrc = (128/9)[a~02c/(mc2)2](kTDh/K~3)n&ph

.

(86)

2. This formula is valid when the charged particle that undergoes a change in velocity is nonrelativistic. For a derivation of this useful formula see [ll], 1151, or [29].

PHOTON GAS INTERACTION

345

WITH PLASMA

Now this result might be compared with that for bremsstrahlung in the lowfrequency limitzl [(Eq. (32)], taking some value for the electron temperature. One finds,22 taking a temperature T, = Tph = 3000 “K, corresponding to the epoch of decoupling between matter and radiation [7], that qrc is smaller than the bremsstrahlung term by a factor of about 1O-s,

APPENDIX:

RADIATIVE

RECOMBINATION

AND LINE

EMISSION

FORMULAS

In the radiative recombination formula (48) the transcendental function f$ is given by [16, 21, 221 KY) = f

b(u)

= f

often the notion #*)(y)

(y/n”) eY@W+%

?L=l

?l=l

64.1)

is introduced where P(Y)

= 2 vL(Y>n=n%

64.2)

K(x) = J,” u-‘e-% du,

(A.3)

In (A. l), K is the exponential integral

which has the asymptotic expansions23 m C-x)’ C r!

K(x) = - y - In x -

(small x),

(A.4)

(large x).

(A.3

T=l +

[e-"/x)

f.

r!

(-

$j'

By means of the Euler-Maclaurin (E-M) sum formula [30, 311 the summation (A.l) was performed [21] yielding the simple formula +(y> = W.735 + Iny + (6y>-9

(‘4.6)

which holds to surprising accuracy for y 2 0.5. For small y (high temperature) one can also derive a useful power seriesin y [26]; however, the most important 21A logarithmic expression for the bremsstrahlung Gaunt factor should then be used [26]. 22Of course, the expression for qrc for finite T, would be somewhat different from (82), although the magnitude would be about the same. z3 Only (A.5) is, strictly speaking, an asymptotic expansion; the series diverges and must be broken off at some value (p) of Y. Note the misprints in [21] as noticed in [16].

346

GOULD

domain is that of large and intermediate y where (A.@ is valid. The functions $trn) can usually be found easily from 4 by simply subtracting +I + C& + .*v + c$,,+~ . One could also use the E-M sum formula to evalute +crn). The corresponding integral (times 2) is [21] 2 SW (y/t3) eYit2K(y/t2) dt = s:‘“” 111

e”K(z) dz, (A-7)

= eY@“K(y/m2)

+ y + ln(y/m2).

In this case, although the correction terms in the E-M formula involving the Bernoulli numbers may be small, their evaluation may be difficult using (A.4) and (A.5) if y/m” N 1. For y/m” < 1, (A.4) can be used to evaluate these terms as well as to expand (A.7) to give d’“‘(Y)

= &

+ &

(In $-

+ 1 - y) + &

(3 In $

(ln -$

- 2 - 3~) + 0 (-$)

- y) + 0 (-+$).

(A.8)

In (A.8), the first term comes from the integral (A.7), the second term is just bum, the third term comes from the correction term involving the Bernoulli B, , the term of order y/ma would come from B4 , and the term of order y2/m4 comes from the higher terms in the expansion (A.4). Another transcendental function of interest (seeSection 4-D) is 8(y) = f

y

= fl

4(Y)

(A.9)

?I=1

as well as the more general function pqy)

= i k(Y). rl=m

The series (A.9) and (A.lO) converge very rapidly and as a result an evaluation by means of the E-M formula is not as accurate as, say, the result (A.6). The integral in the E-M formula can be evaluated since (as can be readily shown) m m s (YP)

eYlt2K(y/P) dt

= (2~)~’ s:‘“’ ze”K(z) dz, = (2y)-l [y/m” - (1 - y/m”) e’@K(y/m2) - y - 1n(y/m2)].

(A.ll)

PHOTON

GAS

INTERACTION

WITH

347

PLASMA

However for the evaluation of 0 (that is, the case m = 1) the cutting off of the asymptotic series involving the Bernoulli-number correction terms is quite critical (and somewhat ambiguous)24 and no useful formula for 8 is obtained. It is better to simply substitute the expression (A.l) or, if applicable, the approximations (A.4) and (A.5) and (A.lO) and try to sum directly. In fact, from (A.4) and (A.5),

&z(Y)-+ (l/W

- VY)

+ W~5>bW/r) - rl and a simple approximate

u/n” > 1,

(A.12)

< 1,

(A.13)

YP

formula for e(y) for large y is obtained from

ecy) IV g -$ = C(3) - Jallr $

= C(3) - $.

II

(A.14)

A somewhat better formula is, also for large y, -

In Y 7’

(A.15)

(Y Q 11,

(A.16)

When y < 1, (A.13) can be summed to give e(Y) = ~[2c,(5) - 5Wb

+ 741

where C,(5) = f

%

= 0.02858;

l(5) = 1.0369.

n=2

For y/9 < 1, a more accurate expression for t?@)(y) is obtained by subtracting out the n = 1 and 2 terms from (A.16), ey y) = ~(0.0322 - 0.0057 In y) + 0(y2/27). In the low-temperature

(A.17)

limit where y/9 > 1 we have [see Eqs. (A.12), (A.15)] (A.18)

*4 See [21] regarding this cut-off criterion.

59dW2-4

348

GOULD REFERENCES

1. P. THADDEUS, Annu. Rev. Astron. Astrophys. 10 (1972), in press. 2. R. WEYMANN, Astrophys. J. 145 (1966), 560. 3. P. J. E. PEEBLES, Astrophys. J. 153 (1968), 1. 4. YA. B. ZELDOVICH, V. G. KURT AND R. A. SYUNYAEV, Zh. Eksp. Teor. Fiz. 55 (1968), 278 [Sov. Phys.-JETP 28 (1969), 1461. 5. YA. B. ZELDOV~CH AND R. A. SYUNYAEV, Astrophys. Space Sci. 4 (1969), 301. 6. R. A. SUNYAEV AND YA. B. ZELDOMCH, Astrophys. Space Sci. 7 (1970), 20. 7. R. J. GOULD, Annu. Rev. Astron. Astrophys. 6 (1968), 195. 8. R. J. GOULD AND R. K. THAKUR, Phys. Fluids 14 (1971), 1701. 9. J. M. JAUCH AND F. ROHRLICH, “Theory of Photons and Electrons,” p. 235, Addison-Wesley Pub. Co., Reading, MA, 1955. 10. A. S. KOMPANEETS, Zh. Eksp. Teor. Fiz. 31 (1956), 876 [Sov. Phys.-JETP 4 (1957), 7301. 11. J. D. JACKSON, “Classical Electrodynamics,” p. 489, John Wiley and Sons, New York, 1962. 12. H. DREICER, Phys. Fluids 7 (1964), 735. 13. R. WEYMANN, Phys. Fluids 8 (1965), 2112. 14. R. J. GOULD, Am. J. Phys. 39 (1971), 911. 15. G. R. BLUMENTHAL AND R. J. GOULD, Rev. Mod. Phys. 42 (1970), 237. 16. R. J. GOULD AND R. K. THAKIJR, Ann. Phys. (N.Y.) 61 (1970), 351. 17. H. A. BETHE AND E. E. SALPETER, “Quantum Mechanics of One- and Two-Electron Atoms,” p. 322, Academic Press, New York, 1957. 18. W. J. KARZAS AND R. LAITER, Astrophys. J. Suppl. Ser. 6 (1961), 167. 19. L. SPITZER, JR., AND J. L. GREENSTEIN, Astrophys. J. 114 (1951), 409. 20. R. M. PENGELLY, Mon. Notic. Roy. Astron. Sot. 127 (1964), 145. 21. G. R. BURBIDGE, R. J. GOULD AND S. R. POITASCH, Astrophys. J. 138 (1963), 945. 22. T. N. DELMER, R. J. GOULD, AND W. RAMSAY, Astrophys. J. 149 (1967), 495. 23. H. P. GLA~CO AND H. ZIRIN, Astrophys. J. Suppl. 9 (1964), 193. 24. R. L. BROWN AND W. G. MATHEWS, Astrophys. J. 160 (1970), 939. 25. R. J. GOULD, Astrophys. Left. 8 (1971), 129. 26. W. H. TUCKER AND R. J. GOULD, Astrophys. J. 144 (1966), 244. 27. R. J. GOCJLD ANLI W. RAMSAY, Astrophys. J. 144 (1966), 587. 28. H. H. LANDOLT AND R. BERNSTEIN, “Zahlenwert und Funktionen,” Band 1, Teil 1, p. 260, Springer-Verlag, Berlin, 1950. 29. R. J. GOULD, Amer. J. Phys. 38 (1970), 189. 30. E. MADELUNG, “Die mathematischen Hilfsmittel des Physikers,” p. 21, Springer-Verlag, Berlin, 1936, Dover Pub., New York, 1943. 31. E. T. WHITTAKER AND G. N. WATSON, “Modem Analysis,” p. 127, Cambridge University Press, Cambridge, 1927.