CHAPTER VI 
SOLUTION OF THE EQUATIONS 
81. The fundamental equations of the theory of the interior of a star 
are the hydrostatic equation (54-3) and the equation of radiative equi 
librium (71*1), viz. 
dP 
dr < 8M >> 
(81-2). 
dr c 
The whole pressure P is made up of gas pressure p G and radiation pressure 
p R , so that 
P = Vo + Vr (81-3). 
From (81-1) and (81-2) 
dp R = ^dP (81-4). 
In a steady state the amount of radiation 4nr 2 H flowing per second 
outwards across a sphere of radius r must be equal to the amount of energy 
liberated within the sphere, probably from subatomic sources. Denoting 
by L r the liberation of energy per second within the sphere, we have 
H = L r /47Tr 2 , g = GM r /r 2 , 
so that - = -Ll± (81-5). 
g 477 Cr M r 
The quantity L r fM r is the average rate of liberation of energy per 
gram for the region interior to r. Presumably this liberation is greater 
at the hot dense centre than in the outer parts, and L r IM r will decrease 
as r increases and the successively cooler layers are brought into the average. 
But we do not anticipate that the decrease will be rapid. The rate of 
generation of energy may decrease rapidly with temperature, but the 
change of L r fM r will be much less marked since it is toned down by the 
averaging. 
Let M be the mass of the star, and L the total emission of energy per 
second from its surface. (The observed bolometric magnitude is deter 
mined by L.) Then L/M is the boundary value of L r /M r . We set 
k ■ M < 81 - 6 >’ 
so that rj increases from 1 at the boundary to some unknown but not 
very large value at the centre. The form of the function 77 depends on the