88 
POLYTROPIC GAS SPHERES 
have no boundary; the “star” extends to infinity and the mass is infinite. 
These distributions are of no interest to us here. 
In the critical case n = 5, the differential equation (55*8) is soluble, 
the solution being 
u = (1 + \z 2 ) 
The distribution thus extends to infinity and R is infinite. The whole 
mass, however, is finite, since (— z 2 du/dz) z =«> is equal to y/3. 
Another simple solution occurs when n = 1, viz. 
u = sin z/z. 
The boundary of the star is then at 2 = tt. We have 
R' = TT, M' — TT, p 0 /p m = TT 2 /3. 
The solution for n = 0 is also simple since it corresponds to a uniform 
sphere. 
62. If the material is a perfect gas and jS/x is constant the mean 
temperature of the star can be calculated. The mean temperature T m 
averaged with respect to mass is given by 
rM 
T m = TdM r -r- M. 
J 0 
Hence since 
t/To — 4 > l c t > o> 
T* 
T n 
M 
1 
Wo 
(n + 1) 
3M(f> 0 
cf)dM r 
by (60-22) 
n + 1 GM 2 
5 — 71 R 
n + 1 M’ 
5 — n R' 
1 
'Wo 
by (57-3) 
(62-1). 
Tor n = 3 this gives T m /T 0 = 0-584. 
Substituting for T 0 from (58-4) 
rp 1 G 
m = 5^9? ~R 
(62-2). 
It is of considerable interest to have obtained at this early stage a 
simple formula for the mean temperature of the material of a star. If we 
are definitely assured that the temperature is some millions of degrees 
we know the kind of conditions with which the more detailed theory will 
have to deal. At present the chief point left in suspense is the value of 
n which will ultimately be found from the theory of radiative equilibrium. 
Equation (62-2) shows to what extent the mean temperature is affected 
by this determination. By changing the value of n we obtain a series of