/ 
POLYTROPIC GAS SPHERES 
85 
ns of the 
(57-4). 
(57-5), 
Hence 
(57-6). 
îe tables, 
iensity to 
the mass 
the con- 
Similarly 
of a unit 
entry B' 
of perfect 
equation 
(58-1), 
molecular 
epresents 
to denote 
drawn in 
radiation 
sure p G to 
.(58-2). 
.(58-3). 
n terms of 
ìe constant 
s m = R/H, 
By (57-3) the central temperature is determined by 
B' G fiixM 
0 (w+lJJTSR B 
,(58-4). 
Provided that /3p, is constant through the star (which is actually a fair 
approximation) we have by (58-3) T oc cf>. Hence 
T/T 0 = u (58-5). 
59. To show the method of using these formulae and tables we give 
calculations for the bright component of Capella. For the assumed re 
lation between pressure and density we take Pxp* so that n = 3 and the 
appropriate table is Table 6. The mass and radius of Capella (§ 13) are 
M = 8-3 . 10 33 gm., B = 9-55 . 10 11 cm., 
whence the mean density is 
Pm = -00227. 
From Table 6, column 6, the ratio p 0 lp m is 54-36. Hence 
p 0 = -1234 gm. per c.c. 
Also from the last line of Table 6 
M' = 2-015, B' = 6-901. 
Hence by (57-3) 
, 6-66.10- 8 x 8-3.10 33 x 6-901 , _ 
2-015 x 9-55.10 11 1-982.10 , 
and by (55-42) 
P 0 = ipo&o = i x -1234 x 1-982.10 15 = 6-11.10 13 dynes per sq. cm. 
To determine the central temperature we must assume a molecular 
weight, and, if radiation pressure is not neglected, a factor )3. We take 
/xj8 = 2-0 which is probably fairly near the truth, then by (58-3) 
2-0 x 1-982.10 15 
= T20.10 7 degrees. 
u 4 x 8-26.10 7 
To find the conditions at another point in the star, take for example 
the line z = 3-5 in Table 6. This relates to a distance from the centre given 
by 
- = 2 = — - 50" 
B B' 6-901 ’ 
or a little more than half-way from the centre to the surface. Here 
T = -27629P 0 - 3-32 . 10 6 , 
p = -02109p 0 = -00260, 
P = -005828P 0 = 3-56 . 10 u . 
Also at this point 
1-8203 
2-0150 
: = 0-90,