80 
Gaseous Stars 
[ch. Ill 
1 „ i /1 1 0 2 1 3 4 \ ( 0 9 0 2 \(G\ /(TO ox 
= +5(5 + 7SP+9Si + -) r 3 + 53^ + -)(x) - (73 ' 3) > 
, d Id 
where = — j- 5 -. 
cv «/? or 
Besides failing owing to the neglect of terms in G, equations (72*2) and 
(723) fall further into error near the surface of the star, owing to there being 
no true approximation to thermodynamical equilibrium between matter and 
radiation in these regions. The extent of this error has been discussed by 
the present writer*, Milnef, Freundlich, Hopf and WegnerJ, and others. 
The Configurations of a Star in Radiative Equilibrium. 
The General Equations. 
74. We have seen how radiation completely outstrips the material carriers 
in the transport of energy to the star’s surface. As a consequence the ordinary 
coefficient of conduction of heat is of no importance, and the build of a star 
is entirely determined by the values of k, the coefficient of opacity in its 
interior. If this coefficient is everywhere zero, the star is entirely transparent, 
and so cannot retain any heat; we have a star of zero temperature and 
therefore of infinite extent. If, on the other hand, the coefficient of opacity is 
everywhere infinite, the star is completely opaque, so that all radiation 
accumulates where it is generated until the star’s temperature becomes 
infinite, and we have a star of infinite temperature and consequently of 
infinitesimal radius. Naturally, only the intermediate values are of any 
practical interest, but the two extreme cases just mentioned shew how the 
whole build of a star depends on the value of the opacity coefficient k. 
So much is this the case that attempts to investigate the build of stars 
before the value of this coefficient was known can only be regarded as 
speculation ; much of it was, moreover, .unfortunate speculation, since the 
results obtained were mostly at variance with the results subsequently 
obtained by using the true value of the coefficient of opacity. 
75. We have already noticed that the problem of determining the equi 
librium configurations of a sphere of gas only becomes definite when some 
relation, outside the dynamical equations, is introduced to fix the temperature 
of the gas at every point. Emden’s solutions assumed the adiabatic relation ; 
each element of a star was supposed to have the temperature which it would 
assume if the stellar material was being continually stirred up. It is now 
clear that the proper relation to take is one expressing that the temperature 
of each element of the star is that determined by the flow of radiation. This 
flow of radiation is in turn determined by the rate at which energy is being 
generated in the star’s interior. The outward flow of energy across a sphere 
* M.N. Lxxvii. (1917), p. 32. f Ibid, lxxxi. (1921), p. 361. 
J Ibid. Lxxxvxn. (1927), p. 139.