[. Ill 
73-75] j Radiative Equilibrium 81 
3), 
) and 
>eing 
■ and 
d by 
triers 
in ary 
i star 
n its 
irent, 
! and 
ity is 
ation 
tomes 
ly of 
any 
v the 
nt k. 
stars 
id as 
e the 
Lently 
equi- 
some 
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ition; 
would 
¡ now 
•ature 
This 
being 
phere 
of radius r must be equal to the rate at which energy is being generated 
inside the sphere of radius r ; this leads to the equation 
. AaT'CdT [r 
- W -§*T * = Jo * 7 r f ,r 0dr (75 1)1 
where G is the rate of generation of energy per unit mass at a distance r 
from the centre of the star. As we have seen in § 73, terms in G ought to be 
added to the left-hand member of this equation, but a numerical discussion 
shews that in the present problem they are inappreciable in comparison with 
the term on the right, and so may legitimately be disregarded. 
The approximate equation (751) may accordingly be taken to be the 
equation of radiative equilibrium. The physical meaning of this equation is 
that the temperature gradient dTjdr at every point must be just that required 
to discharge the stream of radiation generated at all interior points. 
To examine the build of a star in radiative equilibrium we combine this 
equation with the dynamical equation which expresses the condition that the 
star shall be in dynamical equilibrium, namely (§ 58), 
^ ^ ¡ r 4nrpr*dr (75-2), 
dr r 2 Jo 
in which p must now be taken to represent the total pressure, comprising 
both gas-pressure and pressure of radiation. The physical meaning of this 
equation is of course that the total pressure p is just adequate to support the 
weight of the whole column of gas standing above it. 
It can be shewn that the pressure of radiation is, with sufficient accuracy 
for the present problem, equal to its value when there is a state of thermo 
dynamical equilibrium, namely £ aT 4 . If we still assume the gas-pressure to 
be given by the ordinary laws of Boyle and Charles for an ideal gas, the total 
pressure p is 
p=£^ T+ i aTi < 7 «- 3 >- 
Writing G for the mean value of G throughout a sphere of radius r, the 
equation of radiative equilibrium (751) assumes the form 
rr 2 Cl d C r 
-V £ 0^0 ~«J, (75-4), 
or, transformed by the use of equation (75’2), 
<”■*>■ 
If we write p Q , p R , for the gas-pressure and the pressure of radiation 
(£ aT 4 ) respectively, this may be put in the form 
«»>. 
This and the dynamical equation (75'2) determine the configurations of 
equilibrium of the gaseous star. 
J 
6