120 
SOLUTION OF THE EQUATIONS 
To these may be added 
which is contained in (84-1). This shows that T 2 /p, which is constant 
throughout any one star, is also constant for all stars of the same mass. 
(As usual we neglect possible small differences of /x.) 
Hence in stars of the same mass the temperature at homologous points 
in the interior varies as the cube root of the mean density. 
The effective temperature follows a different law. The levels where the 
temperature is equal to T e (somewhere in the photosphere) are not at 
homologous points; in fact as the density and temperature of the star 
increase, the photosphere comes relatively nearer to the surface. By (31T) 
the black-body radiation of matter at temperature T is 
since this gives the temperature of the black body giving the same amount 
of radiation as the star. 
The mean density of the star is 
by (87-2). The central density being a constant multiple of p m we have 
We may take the effective temperature for type M to be 3000°, and 
for type A 10,500°. Hence if L is constant (as the observations appear 
to indicate) the range of central temperature is 12 : 1 . The range of mean 
density should be the cube of this—about 2000 : 1 —and this is in accord 
ance with our general knowledge of the densities of these types. 
For the reasons already stated it is difficult to judge how closely the 
rule that L is constant for stars of the same mass is supported by observa 
tion. As bias may enter into our estimates it may be best to quote an 
opinion formed by the writer before he had arrived at any theory as to 
what the variations of k 0 ought to be*. He then concluded from the 
observations that k 0 might vary as rapidly as T l - or but was not 
likely to pass beyond these limits. This corresponds to a range of 3-5 : 1 
per sq. cm. per sec. 
hence from a sphere of radius R the radiation per second is 
L = 4:ttR 2 . \acT 4 = 7 T acR 2 T i . 
Accordingly the effective temperature of a star is defined by 
L = nacR 2 Tf 
(87-2), 
so that by (87-1) we have for stars of fixed mass 
T C *L-*T' 
(87-4). 
* Zeits. fiir Physik, 7, p. 368.