122 
SOLUTION OF THE EQUATIONS 
which when applied to the different parts of a single star (or to stars of 
the same mass) reduces by (87-1) to 
k oc (89-2). 
This coincides with one of the limits suggested in § 87 as being consistent 
with the observed magnitudes of the giant series. At present the physical 
theory of absorption is not definitive, and we must not lay undue stress 
on the odd half power of the temperature. But (89*1) is the law which 
appears most likely according to our limited knowledge and it is appropriate 
to develop its consequences fully for comparison with astronomical 
observation. 
The function 77 depends on the relative distribution in different parts 
of the star of the source of stellar energy—a distribution at present 
unknown. What makes progress possible is that 77 is comparatively in 
sensitive to very great changes in the assumed distribution of the source. 
The general effect of such changes can be seen in the following way. The 
temperature gradient in a star achieves two purposes : it gives a pressure 
gradient great enough to distend the mass M of the star against gravity 
and it drives the radiation L through the obstructing opacity to the surface. 
Now L comes from varying depths according to the distribution of the 
source. If we concentrate the source of L to the centre a greater pro 
portion of L comes from the deeper parts and more temperature gradient 
is required near the centre to drive it through. Or, if the temperature drop 
at our disposal is limited by the known mass and distension of the star 
(i.e. to the amount required to achieve the first purpose) we must be 
content with less L. Hence in general L/M will be decreased by concentrat 
ing the source to the centre. What concerns us is the average optical 
depth of the source of energy below the surface. To evaluate this precisely 
we must know the law of distribution of the source. But it is clear that, 
whether the source is spread uniformly over the mass or whether it is 
strongly concentrated to the centre, the difference in average depth will 
be a factor of the order 2 or 3 of no great consequence for a first 
approximation. 
90. Let e be the rate of liberation of energy per gram. We can represent 
various degrees of concentration of the source to the centre by taking e to 
vary as T s within a single star. 
First consider the law e <x T . Then the average value of e within a 
sphere of radius r will be proportional to the average temperature within 
the sphere. Comparing r = 0 and r = R, we have 
Ve _ 
1 T m ’ 
since the ratio of the 77 ’s for two spheres is the ratio of the average values