POLYTROPIC GAS SPHERES 
93 
treat a 
It was 
to the 
tion of 
lerefore 
hus the 
rer, the 
ures in 
T t >T z 
erature 
mpera- 
er at a 
central 
mst be 
star). 
8M is 
small 
nfinity 
Hence 
central 
would 
s. The 
al star 
hernial 
region 
ty. It 
niform 
s 
mean 
• 1 ). 
• 2 ). 
Hence integrating between aR and R 
P 1 = §7 tG Pi 2 R 2 (1 - a 2 + 2 Pa 2 (1 - a)} (65-3). 
By (65-2) and (65-3) 
< RT 1 P x GM (1 - a) (1 + a + 2£a 2 ) 
ir"* "is t+f (85 ' 4) - 
The quantities a and /3 are connected by the theory of the isothermal 
part of the star. With the notation of § 63 
Pilpo = &u , 
, 3 du 
PjPo - - - - 
Hence 
a 3du 
p = e~ u - 1 
z dz 
Also since aR = (k!^ttGp 0 )^ z and P x = Kp 0 e u , (65*3) gives 
6 _ (1 — a) (1 + a + 2/3a 2 ) 
z*e l 
.(65-51). 
.(65-52). 
Choosing an arbitrary value of z we find e u and du/dz from Table 7. 
Then /3 and a can be found from (65-51) and (65-52), and the factor 
S = (1 — a) (1 T a T 2/3a 2 )/(l + fia 3 ) 
is calculated. The following results are found— 
.(65-6) 
z 
/3 
a 
s ■ 
4 
•782 
•663 
•646 
4-5 
•929 
•673 
•641 
5 
1-073 
•680 
•640 
6 
1-348 
•687 
•645 
We see that S has a minimum value about -640 at about a = -676. Hence 
by (65-4) a star with an isothermal region extending to -676P surrounded 
by a region of uniform density has the minimum central temperature, viz. 
GfxM 
T x - 0-32 
91 R 
.(65-7). 
It may be noted that for a star of constant density throughout the factor 
is 0-5, and for the poly trope n = 3 (the model chiefly used in this book) 
the factor is 0-856. Thus the temperatures which we find in the detailed 
investigation are not greatly above the minimum values. 
66. Problem III. To find the minimum value of the mean temperature 
in a star of mass M and radius R of perfect gas of molecular weight p., subject 
only to the condition that the density does not decrease inwards. (Radiation 
pressure is neglected.) 
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