SOLUTION OF THE EQUATIONS 
115 
unknown law of liberation of subatomic energy, but it may be expected 
to be approximately the same for all stars. 
Substituting in (81-4) we have 
82. We are about to introduce an approximation with regard to the 
behaviour of rjk; but before doing so we note that the exact equation 
(81-7) enables us to set an upper limit to the opacity k in any star (perfect 
gas or not) for which L and M are known by observation. 
The temperature must in any case increase inwards and it seems 
extremely unlikely that the density can diminish inwards. If both 
temperature and density increase the material pressure p G must increase. 
Hence for an inward step dp a is positive, so that by (81-3) 
Similarly for Sirius we find k < 630, and for the sun k < 13,200. Note 
that k cannot rise above these values in any part of the star unless there 
is a reversal of the density gradient. 
The physical explanation of these upper limits is that the radiation 
observed to be emitted must work its way through the star, and if there 
were too much obstruction it would blow up the star. 
The upper limits found for Capella and Sirius are sufficiently low to 
narrow the field of speculation. Absorption coefficients higher than these 
have been measured in laboratory experiments. The upper limits in fact 
are only 4 or 5 times greater than the definitive values of k found later. 
83. We shall now work out the case in which rjk is constant throughout 
the star. This requires that the absorption coefficient should be nearly 
constant, decreasing a little towards the centre to counterbalance the 
increase of rj. Reasons will be given in due course for believing that the 
absorption coefficient does behave in this way, and that rjk = const, is 
a very close approximation. One simplification resulting from the as 
sumption that rjk is constant is that radiation pressure and gas pressure 
are in the same ratio throughout the star. 
Accordingly let rjk = const. = k 0 (83-1), 
so that k 0 is in a sense the boundary value of k. But it must be understood 
thgt the value of k in the photosphere may be widely different from k 0 . 
(81*7). 
dpR < dP. 
Then by (81-7) 
and, since y > 1, 
k < 4 TrcGMjL 
< 25100 M/L. 
For Capella (§ 13), M = 8-3.10 33 , L = 4-8.10 35 . Hence 
k < 435 c.g.s. units. 
8-2 
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