92 
POLYTROPIC GAS SPHERES 
the second to the third shell. Since p 2 > p z it will be possible to treat a 
small mass in this way without reversing the density gradient. It was 
shown in Problem I that this transfer reduces the pressure interior to the 
shell p 2 by a constant amount. Since SP = WpST/p,, the reduction of 
temperature will be inversely proportional to the density and therefore 
least at the centre and increasing (or stationary) outwards. Thus the 
temperature gradient interior to p 2 will not be reversed; moreover, the 
temperature distribution outside p z is unaffected. The temperatures in 
the two shells will change in a more complicated way but since T x > T 2 > T, 
there is a small margin for such changes without reversal of the temperature 
gradient. 
This shows that it will always be possible to reduce the central tempera 
ture without violating the conditions by making a small transfer at a 
point where both p x > p 2 > p z and T x > T 2 > T z . Hence when the central 
temperature is an actual minimum one of these inequalities must be 
replaced by equality (not necessarily the same one in all parts of the star). 
The central part of the star must be isothermal. For if a mass Sili is 
taken from the outer part and distributed uniformly through a small 
sphere of radius r x at the centre, the increase of pressure tends to infinity 
like SM/t-l whilst the increase of density is proportional to SM/r x 3 . Hence 
if r x is taken small enough the temperature falls. If, however, the central 
part of the star is isothermal, this transfer is ruled out because it would 
make the central temperature less than surrounding temperatures. The 
isothermal region must be limited because a completely isothermal star 
has no boundary. We have shown that where the star is not isothermal 
it must be of uniform density. 
Hence the star of minimum central temperature consists of a region 
of uniform temperature surrounded by a region of uniform density. It 
remains to determine the extent of these regions. 
Let T x be the temperature of the isothermal region, p x the uniform 
density in the outer region. Then the pressure P x at the transition is 
Let R x be the radius of the transition sphere and let p m be the mean 
density within this sphere. Let 
Pm Pi ^ 
(65-1). 
We have M = § 7 t Px R 3 + § 7 r { Pm - Px ) R 3 
= irrp x R 3 (\ + a 3 fi) 
At a point in the outer region 
g = %TrG Pl r + |t tG ( p m - p x ) R x 3 /r 2 , 
so that by (54-3) — dP — ±TrGp x 2 dr (r + /3a 3 R 3 /r 2 ). 
(65-2).