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.