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
f