POLYTROPIC GAS SPHERES 
89 
nfinite. 
soluble, 
whole 
mean 
ire T m 
models with different degrees of concentration of mass to the centre. The 
following results (derived from Emden’s tables) show the progression— 
n = 0 
1 
Ц 
2 
2Ì 3 
4 
Ц 
4-9 
Pol Pm = 1 
1-84 
3-29 
6-00 
11-40 
24-08 54-36 
623-4 
6378 
934800 
5/(5 - n) — 1 
Ml 
1-25 
1-43 
1-66 
2-00 2-50 
5-00 
10-0 
50-0 
The last line shows how the mean temperature increases 
with 
the con- 
centration (the mass, radius and molecular weight being fixed). 
Unless the density decreases inwards, there is a minimum value of 
the mean temperature given by the form n = 0. This is proved more 
generally in § 66 where the discussion is not limited to polytropic models. 
The actual mean temperature is not inordinately higher than the minimum 
unless we have extreme concentration of mass to the centre as shown in the 
second line of figures; but in that case there is practically no density in 
the outer part of the star so that \^e are virtually dealing with a star of 
smaller radius. 
The high temperatures inside the stars are often considered rather 
startling and it is well to realise that they are not dependent on the more 
advanced developments of the theory. 
• 1 ). 
• 2 ). 
stage a 
If we 
degrees 
згу will 
alue of 
ibrium. 
The Isothermal Gas Sphere. 
63. A mass of perfect gas at uniform temperature is the limit of the 
polytropic distribution for n = oo. Certain modifications of the analysis 
are necessary for this case. Although it has no direct application to actual 
stars a study of the isothermal distribution is useful for purposes of 
comparison. 
By (58-2) P = Kp, k — WT/fifi (63-1), 
and (54-4) gives d(f) = dP\p = t<d (log p). 
Hence integrating, p = p 0 ^ K (63-2), 
where p 0 is the density at cf> = 0. Since n > 5 the distribution extends to 
infinity and the mass is infinite. Previous conventions fixing the zero of 
<f> at the boundary of the star or at infinity therefore break down. For 
convenience we now take (f> to be zero at the centre, so that p 0 denotes the 
central density as before. 
Poisson’s equation becomes 
^M + ?# + 4 ^ 0 e«” = 0 (63-3). 
dr 2 r dr 
iffected 
eries of 
Write 
ф = KU, 
(63-4).