M A R C H 1 5 , 1 9 4 0 P H Y S I C A L R E V I E W V O L U M E 5 7

Transformations of Matter-Radiation Gas Mixtures

ROBERT M. WHITMER

Purdue University, Lafayette, Indiana

(Received December 16, 1939)

Emden found that for a perfect material gas, the transformations along the isochor, the adiabat, the isotherm and the isobar are all forms of the general polytropic transformation ^ = dQ/dT=const. An alternative definition of a polytrope is r = — (dp/p)/(dv/v)= const. When radiation energy density and pressure are considered, it is found that X is a con­stant only on the adiabat, and r only on the isobar. A discussion follows concerning the relation of these results to certain polytropic distributions of interest in astrophysics.

I. INTRODUCTION

EMDEN in his book Gaskugeln1 has found that for a perfect material gas the trans­

formations along the isochor, adiabat, isotherm (which in this case is coincident with the isen-ergic) and the isobar are all particular forms of the general polytropic transformation

dQ/dT = a, constant, (la)

where Q is the heat added to the system of the gas and T the temperature. Emden applied his results to certain problems in stellar structure. Later Eddington2 included considerations of radiation energy density and pressure. By plausible qualita­tive reasoning he showed that the ratio of the partial pressures of matter and radiation probably remains essentially constant throughout a star. From this it follows that3

pvT = a, constant, (ib)

where T is a constant, p the total pressure and v the volume. This is an alternative way of defining a polytropic transformation.

It is the purpose of this note to show that when the radiation is considered—as of course it must be when one is dealing with stellar interiors—the only one of Emden's transformations which re­mains a polytrope is the adiabat, and even this does not fulfill condition (lb). We will also show the relation of Eddington's results to the generalization of Emden's transformations. |r' Emden also found the algebraic signs of the elementary changes in heat, entropy, internal

1 R. Emden, Gaskugeln (Leipzig, 1907). 2 A. Eddington, The Internal Constitution of the Stars

(Cambridge University Press, 1926). 3 Eddington, reference 2, § 84.

energy and mechanical work for the various transformations. We will show that except for a minor effect due to the splitting of the isenergic and isotherm these signs and that of the ele­mentary temperature change remain unaltered by the generalization.

We will then show the relation between the conditions (la), (lb) and a polytropic distribu­tion suggested by Milne which may be realized under somewhat more general conditions than those assumed by Eddington.

In the following we consider a unit mass ofthe material gas, and we assume that the tempera­ture changes are not large enough to affect the average molecular weight.4 Table I contains the notation to be used, together with that of Eddington and Emden.

We assume that the material gas obeys the perfect gas law, so that

pp = RT/v.

Our fundamental equation is

dQ = de+pdv, (2)

where e — CvT-\-avTA. (3)

The v appears in the second term on the right of (3) because we are dealing with a unit mass of the material gas, rather than unit volume. We see that

de=(Cv+±avT*)dT+aT4dv} (4)

4 It should be remembered that under stellar conditions no polyatomic molecules can exist, and all atoms are highly or completely ionized. Our "molecules" are thus free elec­trons and nearly bare nuclei. Changes in the degree of ionization of such a gas will not have much effect on the average molecular weight.

516

M A T T E R - R A D I A T I O N G A S M I X T U R E S 517

whence

dQ=(Cv+4:avr3)dT+(p+ar*)dv. (5)

The total pressure is

p = (RT/v)+$oT*. (6)

It will be of interest to find the differential equation of a transformation in the p—v plane. We have by definition dQ/dT=\\ putting this and (6) into (5) we find

/RT 4 \ \dT=(Cv+4:avT*)dT+l—+-aT*\dv

or if we let

or

(7) dT /RT 4 \

T(Cv-\+4avT3) H •—+-aT i )dv = 0. T \ v 3 /

The coefficient of dT/T may be written as

T(Cv-X+4avT') = v\(Cv-\)—+4aT*\

[/ \\CvBp "I

=^ |——/S+12(1- /3)1 , (8)

where A = X/C„. The coefficient of dv becomes

(RT/v) + (4aT4/3)=p(4-3B). (9)

From (5),

dp /R 4 \ RT

dT dv

whence

dT dv = p{4-3B) Pj>-,

T v

dT dp/p+Bdv/v

T 4-3/3

(10)

(11)

r=-(4-SBy

l - A 1

7 - 1 -0+12( l - |8 )

then dp/p + Tdv/v = 0.

(13)

(14)

Under conditions such that V is a constant this equation may be integrated at once to give pvv — constant. This is correct for Emden's con­ditions (j3 = l, X a constant) and it seems to be true to a good approximation for Eddington's conditions. However, it is not generally valid.

Two other relations which will be useful later are obtained from the definition of /3. We have

whence

Pp = RT/v and (l-p)p = &T\

(3 = 3R/(3R+avT*)

= l-aT*/3p.

(15)

(16)

All transformations will be referred to the p — v plane, and they will be taken in such a direction that dp^O, dv^O.

Note: One may observe that if in (13) we set j3 = 0 then r = 4/3, apparently for all values of X and A. However, it must be remembered that all quantities are referred to unit mass of the material gas. When 0 is allowed to approach zero, we are actually allowing the system to expand to

Putting (8), (9) and (11) into (7) and simplifying, we find

dp -+•

(4-3/3)2

l - A

IT-I -0+12(1 -0 )

dv -0 (12)

TABLE I. Notation used

QUANTITY

Heat added to the system Entropy Internal energy Mechanical work Total pressure Specific volume Specific heat at constant pressure,

material gas alone Specific heat at constant volume,

material gas alone Ratio of specific heats, material

gas alone Gas constant Coefficient of dv/v in dp/p-\-Tdv/v

= 0 Absolute temperature Ratio dQ/dT Ratio X/Cv Fractional pressure of material gas Coefficient in Stefan's law Order of polytrope= l / ( r - l )

in this

THIS PAPER

Q •n e

w p V

cv

cv y R

r T X A 0 a n

paper.

EDDINGTON

Q s Ev

P UP

r K/n

T T

0 a n '

EMDEN

0 V e

w p V

cp

Cv

K

AH

k T y

n

518 R O B E R T M . W H I T M E R

an infinite volume. This means that the energy given by (3) becomes infinite. For each of the transformations except the adiabat, X then becomes infinite and the term (1— A) 18/(7 — 1) appearing in T becomes an indeterminate form. The limits of T for /3 = 0 which are found for the various transformations may then be looked upon as the results of the evaluation of these indeterminate forms.

II. ISOCHOR

Here dv = 0, so that

dQ = de=(Cv+4avT*)dT. (1)

From I, (6) we see that

R 4 dp

(K 4 \ = ( — + - a T * \ d T > 0 , .'. dT>0.

Then dQ = de>0 and drj>0. Also since dW== — pdv, dW=0. (With this sign convention, W represents the mechanical work done on the gas.)

Since for a constant volume the temperature increases, we see in I, (15) that p decreases for this transformation. From (1),

\ = dQ/dT=Cv+4:avTz (2) •

and it is obvious that X is positive and increases. With this expression for X, together with I, (15) for P it may be shown that the denominator of the first term of T, (I, (13)) vanishes, so that r = <*>. We see from I, (14) that it must approach + °° if we approach the isochor from the left.

Summarizing, for the isochor, we have

dQ>0 dv>0 de>0

dT>0 dW=0

III.

/3 decreases X increases r = 00

ADIABAT

Here of course dQ = dr] = 0 and X is also zero. Thus the transformation is polytropic according to Emden's definition I, (la). We now have

de+pdv = 0, (1)

by I, (2), so that

Likewise dW>0. From I, (4) we see that when dv<0 and de>0 it must be that dT>0.

In the expression I, (15) for /3 the fact that volume and temperature change in opposite directions makes it impossible to determine the direction of change of p. However, let us form

d(l/$) = (a/3R)(T*dv+3vT2dT) = (aT2/3R)(Tdv/dT+3v)dT. (2)

We already know the sign of dT\ it remains to find the sign of the quantity,

y=Tdv/dT-\-3v. (3)

Referring back to I, (5) we find that for the adiabat

hence

dv Cv+4:avTz

dT p+aT*

CvT+4avT*

p+aT4

3CVT - (7 -4 /3 ) . p+aT*

(4)

Thus the sign of y depends upon the magnitude of the ratio of specific heats of the material gas, 7. In dealing with stellar interiors we may take 7 > 4 / 3 . Then y>0, d(l/0) > 0 and finally p is a decreasing function as we move along the adiabat.

We now consider V. For X = 0 = A,

r=-(4 -3 /3 ) 2 ( T - l )

P*+I2(l-P)(y-1) + ,

This may be rearranged to the form

/ 5 2 ( 4 - 3 7 ) + 4 ( 4 - 3 £ ) ( 7 - l ) r=- j8(13-127) + 1 2 ( 7 - l )

(5)

de=-pdv>0.

The behavior of this function depends markedly upon the value of 7. However, it is difficult to conceive of conditions under which 7 would depart appreciably from 5/3, the value for a monatomic gas, unless at the same time the tem­perature were so low that we could take £ = 1 . With such conditions T = 7. In the opposite extreme we have no material gas present, and 0 = 0. Then T = 4/3, the well-known "ratio of specific heats" for a pure radiation gas.

M A T T E R - R A D I A T I O N G A S M I X T U R E S 519

In problems of stellar structure we may take 7 = 5/3; then

Then

T = (^+8/3-32/3)7(7/3-8). (6)

This is then a monotonic increasing function of /3 such that 4 / 3 < T < 5 / 3 . Since we have found above that /3 is a decreasing function for the adiabatic transformation, T also decreases for this transformation.

Summary for the adiabat:

dQ = drj=^0 /3 decreases de>0 X=0

dT>0 T decreases for conditions hold-dW>0 ing in stellar interiors.

IV. ISENERGIC

For the isenergic we have from I, (4),

de=(Cv+4:avTz)dT+aT4dv = 0. (1)

Since dv<0 we still have dT>0; this indicates that in the p — v plane the isenergic lies above the isotherm, and this will be borne out by the value of T to be found below. It is also evident from (1) that when radiation is absent the isenergic and isotherm coincide. By I, (2) dQ<0 and dri<0. Also dW>0.

We may write /3 as

P = 3R/(3R+avT*) = 3R/(3R-Cv+e/T). (2)

As we move along the isenergic T increases, therefore e/T diminishes and /3 increases.

As to X, we find from (1) and from I, (2),

\ = dQ/dT= -p(Cv+4avT*)/dT4>

_ 0 + 1 2 ( 7 - 1 X 1 - 0 ) . Ks V

3/3(1-/3) (3)

This function goes to — <*> for /3 = 0, 1, and has a maximum somewhere between these limits. It is impractical to determine the value of /3 for the maximum except when y = 5/3. Then Xmax occurs at/3 = 2(4-v2) /7 = 0.74.

We now consider T. From (3) above

A = — / 3 + 1 2 ( T - l ) ( l - 0 )

3 Y - l ) l - « ( 4 + / 3 ) + / 3 2

r = (4) 1 2 ( 7 - l ) ( l - j 8 ) + 0

This function is unity for /3 = 0, 1 and does not change rapidly between these limits regardless of 7. For a more detailed discussion we again take 7 = 5/3; then T reduces to

r = (^+6/3-8) / (7/3-8) . (5)

Numerical values of this function are:

P /*. 0.0 0.2 0.4

1.000 1.024 1.045

0.6 0.8 1.0

1.062 1.065 1.000

Since T changes so slowly, the polytropic con­dition I, (lb) is very nearly satisfied.

Summary for the isenergic:

dQ<0 drj<0 de = 0

dW>0 dT>0

Here

/3 increases X negative, maximum at /3 = 0.74 and

-00 for 0 = 0, 1. T nearly constant, ^ 1 .

V. ISOTHERM

dQ=(p+aTA)dv<0

and dr}<0, dW>0; de = aT*dv<0.

p=t3R/(3R+avT*)

is an increasing function. Since dT>0 and dQ<0 as we approach the

isotherm from the isenergic, X—->— 00 on the isenergic side, X—>+ 00 on the isobar side of the isotherm. By I, (13) as X—•Too, r—>/3, and hence T increases.

Summary for the isotherm:

dQ<0 drj<0 de<0

dW>0 dT=0

/3 increases X = =Foo

r = /3, increases; 0 < T < 1

3/3(1-/3)

We note here that for a pure radiation gas, the pressure depends only on the temperature, and the isotherm and isobar coincide. Thus T = 0 for /3 = 0 is consistent with-the results of the next section.

520 R O B E R T M . W H I T M E R

VI. ISOBAR

By I, (9),

dT dv ( 4 - 3 / 3 ) — = £ — < 0 ; . \ dT<0. (1)

T v

By I, (4), de<0; then by I, (2) dQ<0 and drj<0. Also, of course dW>0.

From I, (16) we see t h a t since the temperature diminishes for a constant pressure, /3 increases.

From (1) and I, (5) we find

dQ pv — = \=Cv+4avT*+—(4-3/3)2, (2) dT $T

bu t by I, (15) we may express avTz in terms of 0. Also, from the definition of 0,

we see tha t Pp = RT/v,

pv/T=R/p.

When these results are substi tuted into (2) we obtain

/ 1 6 12 \

q 3). \B2 3 /

(3)

This is a monotonic and decreasing function of fi. Since £ is an increasing function for this t rans­formation, X is a decreasing function whose value lies between the limits of infinity and Cv.

From (3),

and

X / 1 6 12 \ > = — = 1 + ( Y - 1 ) ( - 3 )

C% \B2 B /

1 - A 12 16

= 3+ 7 - 1 P &2

3^2 + 1 2 ^ - 1 6

Put t ing this into the expression for T we find tha t

r=o, (4) Summary for the isobar:

dQ<0 dv<0 de<0

dT<0 dW>0

13 increases X decreases

r=o.

VII . R E L A T I O N TO EDDINGTON'S R E S U L T S

Following Eddington3 we observe t h a t if we eliminate the temperature between the two equations

pp = RT/v and {\-&)p = \aTA

we arrive a t the expression

-3# 4 ( l - /3) - |3 pv 4/3

nR\l-(3)V (1)

Now from considerations of radiative equilib­rium Eddington6 concludes t h a t in stellar in­teriors j8 is probably a t worst a very slowly varying quant i ty . This makes the right-hand member of (1) a constant , and satisfies the con­dition stated in I, ( lb) for a poly tropic t rans­formation. If then we differentiate (1) we find tha t

dp/p=(4/3)dv/v = 0. (2)

Now this must be identical with I, (12), which is generally true, so t ha t

4 - . (3) U±-W)2/(——0+12(l-0))l+/3=-.

Rearranging (3) we find t ha t j3 disappears and

A = X / a = 3 ( 4 / 3 - 7 ) . (4)

Thus condition I, ( la) is also satisfied. Eddington 6 also considers the poly trope for

which n= & ( r = 1) and which he assumes to be an isothermal distribution. From our discussions of the isenergic and isotherm it is apparent t h a t this polytrope is much nearer to the former than to the latter. The way in which the temperature changes may be seen by integrating I, (12) with T = 1; this gives pv = const. From this it is obvious t h a t RT+avT*/3 = const., or in terms of pressure RT-\-aTA/$p = const.

VII I . R E L A T I O N TO E M D E N ' S POLYTROPES

We have shown t h a t except for the adiabat none of the ordinary transformations remain polytropes when the radiation gas is considered. In fact, even for the adiabat Y is not strictly a constant, although its range is small, so t h a t the

5 Eddington, reference 2, § § 81, 83. 6 Reference 2, § 63.

M A T T E R - R A D I A T I O N G A S M I X T U R E S 521

alternative polytropic condition I, (lb), pvv

= constant is not satified. On the other hand, the isenergic very nearly satisfies this last condition but except over a narrow range of values of £, X is not even approximately constant. On the isobar T is constant (zero) but the isobar is of no astrophysical interest.

As was to have been expected Emden's ele­mentary energy inequalities are unaffected by the generalization. It is of more importance to find that the order in which the transformations occur in thep — v plane is also unaffected. This order is determined by the relative values of I\ and is shown in the table below. The only change from the pure material gas is that now the isotherm is split off from the isenergic, and in the limit of a pure radiation gas it coincides with the isobar.

Transformation path r

Isochor oo Adiabat | > r > | Stellar polytrope -f Isenergic ^ 1 Isotherm 1 > T > 0 Isobar zero

It is possible to show in another way that when the ratio of the partial pressures is constant, the transformation is polytropic with r = 4/3. We have from I, (5),

X= Cv+4:avT*+(p+aT4)dv/dT, (1)

and we see in I, (15) that if fi is constant then

vT* = constant. (2)

Differentiating this last, we find

dv/dT=-3v/T. (3)

Substituting (3) into (1) we obtain

X = a ( 4 - 3 7 ) or

A = X / a = 4 - 3 7 . (4)

If we put this value of A into the general expres­sion for T, I, (13) we find that

r = 4 / 3 . (5)

Thus each of the alternative conditions for a polytrope is satisfied.

IX. MILNE'S POLYTROPE

. Milne7 shows that if we allow ft to vary with the temperature according to the relation

then r3R*-sl-l3c 1 iW<*-o

^(4 - * ) / (3 - S ) = # (^

L a /3C4~* TV J

Here s is a constant, probably between zero and one. /3C and Tc are the values of /3 and T which occur at the center of the star. Since these are fixed for a given star, (1) is a polytrope according to I, (lb), with T = (4 -<0 / (3 - s ) . If we put this value of T into I, (13) we obtain

1-A 3£2-120+165/3

7 - 1 P2(3-s)-p(4:-s)

Except for s = 0 this is not independent of 0, hence A is not constant and Emden's polytropic condition is not satisfied.

This is not a serious difficulty, however. As Milne points out in his Handbuch article, the reason astrophysicists search for polytropic dis­tributions is the following: For his perfect ma­terial gas Emden derived the condition I, (lb) from his definition I, (la). Then he found a certain differential equation connecting the density of the stellar material with the gravita­tional potential and, indirectly, the temperature. The quantity T appeared in this equation as a parameter. Emden then found numerical solu­tions of this differential equation for many numerical values of r . (Actually of n, where T — l + l/n.) So now, if reasons may be estab­lished for believing that the material in a star obeys the polytropic condition I, (lb) then even though the condition I, (la) does not hold still Emden's numerical results may be applied.

7 Milne, Handbuch der Astrophysik, Vol. 3, No. 1, p. 215.