SOLUTION OF THE EQUATIONS
133
/
in this ratio. By Table 9 the value of 1 — /3 for a perfect gas is -05*, so
we must decrease it to -0031 for the sun.
By (84-3) k is proportional to (1 — /3)^//?*; hence we find that the
reduction in log 10 k is 0-4280. The value of log 10 k for a perfect gas and
for the sun’s mass is given in the last line of Table 13 as 14-5739; the
reduced value 14-1459 is seen to correspond very nearly to the third line
of the Table. By a slight interpolation the corresponding value of p m /p 0
is 0-18. Since p m for the sun is 1-41 it follows that p 0 = 7-8.
The result is that if the limiting density of stellar matter under extreme
pressure is 7-8 the sun’s brightness will fall 3 magnitudes below that
predicted for a perfect gas owing to the purely mechanical effect on the
equilibrium.
But in addition there will be a further reduction of brightness owing
to reduced transparency, which we can roughly estimate.
Failure of the gas laws would somewhat modify the distribution of
density in the sun, but this effect is of minor importance since the mean
density is prescribed. The main effect is a decrease of internal temperature;
with reduced compressibility a lower temperature is sufficient to withstand
the compressing force of gravity. The outward stream of radiation pro
portional to the gradient of T i is accordingly reduced, and it is this effect
which has been calculated above. But according to our absorption law
if the temperature is decreased without changing the density the opacity
is increased proportionately to T~'-. The factor hindering the outflow is
increased very nearly as much as the factor causing the outflow is reduced.
The full reduction of brightness is thus about double that stated above,
and if p 0 = 7-8 the sun will be about 6 magnitudes fainter than a perfectly
gaseous star of the same mass.
97. Further illustrations of the use of Table 13 will be found in the
author’s earlier papersf where curves are traced showing the rise to a maxi
mum of the effective temperature and subsequent fall as a star of constant
mass contracts. The values of h and p 0 (both assumed constant) were fixed
by observational data for the sun and for a typical giant star. An oft-quoted
result that a star of mass less than | that of the sun cannot rise to the
temperature of type M, must now be regretfully consigned to oblivion.
In a somewhat later paperf in which the variation of Jc with temperature
and density was taken into account, it was seen that the permissible
* The calculation is here made for ¡x = 2-1.
f Monthly Notices, 77, p. 605; Zeits. fur Physik, 7, p. 377.
f Monthly Notices, 83, p. 98. See especially Table 2, p. 104, where p 0 =13 was
selected as corresponding to Eggert’s value /la = 3-3 which was then current; but it
was noted that p 0 was very sensitive for changes of p, and the same table gives
p 0 =83 for the modern value p =2-2. Moreover it now becomes unnecessary to adopt
a different p for Sirius.