96 
POLYTROPIC GAS SPHERES 
A wide range of conditions will be covered if we take the material 
beyond to conform to a new polytropic law P = Kl p (1+1 / s) , where s 
differs from n, and k x from k. The gravitational force in this region can 
be set equal to GMJr 2 since its own mass is very small compared with 
that of the rest of the star. Then, as in (55-42), we shall have 
P = GM 1 /1 _ 1 
p s + 1 R)’ 
since <f> is now GM x fr — GMJR. Hence, setting r = R 1} 
1 _ 1 
R\ ~ R 
(s+1 )Pi 
= 9-90.10- 14 x (s + 1) 
GM lPl 
so that R can be found. We find for the added mass beyond R 1 
.(67-2), 
1 /1 
^ ( 1 
A M = 4:7r Pl R 3 I (- — 1) a 2 da -f- ( — — 1 
where % = RJR. 
From (67-2) and (67-3) the following results are obtained- 
.(67-3), 
s 
R 
A M 
0 
7-54.10 11 
•0479.10 33 
1 
8-15 
•0508 
3 
9-72 
•0534 
5 
12-04 
•0549 
7 
15-81 
•0561 
Whereas in the range s = 0 to 7 the radius of the star is doubled, the 
consequent change in mass (M 1 + AM) is not much more than 1 part in 
1000 . 
The mass is thus very insensitive to conditions in the low temperature 
part of the star, and our procedure is amply justified so far as the mass is 
concerned. We must be prepared to admit an uncertainty of, say, 30 per 
cent, in the radius. It should be remembered that when there is wide 
diffusion of the outer material, e.g. with the law s = 7, the radius of the 
photosphere may be considerably smaller than the radius R representing 
the extreme limit of the stellar atmosphere; by taking higher values of 
s, R could be increased without limit, but I do not think that the photo 
sphere (which in practice is regarded as the surface of the star) would be 
much increased.