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