SOLUTION OF THE EQUATIONS 123
/
of e in the two spheres. (Central values are denoted by the suffix c.) By
(62-1) T m /T c = 0-584. Hence
Vc = 1-7.
If we proceed from the centre to a distance including 95 per cent, of
the mass the temperature drops in the ratio 1 : 0-21 (Table 6). Hence by
(89-2) k increases in the ratio 1 : 2-2. Meanwhile 77 undergoes practically
its full decrease in the ratio 1-7 : 1. Evidently r]k keeps remarkably steady
in the main part of the mass of the star. A more detailed calculation for
intermediate points is given in Table 10.
The law e cc T represents a moderate concentration of the source of
energy to the centre*. If the source is subatomic we can well imagine that
a much stronger concentration occurs. It is desirable therefore to consider
e oc T 2 and e oc T 4 . Results obtained by rough calculation of 77 are given for
four different laws in Table 10. The star is divided into 10 shells of equal
mass, and the values of T , 77 and 77 T~- are given for the limit of each shell,
the central temperature being taken as unity|.
Table 10.
Test of constancy of yT K
M r /M
T
e = const.
e <x T
€ OC
rp2
€ OC
J '4
vT~-
V
V
7] T~b
V
7] T~*
0-0
1-00
1-00
1-70
1-70
2-57
2-57
4-71
4-71
0-1
0-88
1-07
1-57
1-68
2-18
2-32
3-38
3-60
0-2
0-80
1-12
1-49
1-67
1-97
2-20
2-81
3-14
0-3
0-73
1-17
1-41
1-65
1-81
2-12
2-40
2-81
0-4
0-66
1-23
1-36
1-67
1-67
2-06
2-07
2-55
0-5
0-60
1-29
1-31
1-69
1-54
1-99
1-80
2-33
0-6
0-53
1-37
1-25
1-72
1-41
1-94
1-58
2-17
0-7
0-46
1-47
1-19
1-76
1-31
1-93
1-40
2-06
0-8
0-38
1-62
1-13
1-84
1-20
1-95
1-24
2-02
0-9
0-28
1-89
1-07
2-02
M0
2-08
111
2-10
Mean
1-32
1-74
2-12
2-75
The constancy of 77 T - is best for e oc T, but it is also reasonably close
for the other laws. In Table 10 T c has been taken as unit, so that
T-* = */*,.
* The law e oc T also corresponds to the contraction theory of stellar energy if
the star passes through a series of homologous states.
f The solution based on the approximation -qk — const, is used to calculate these
values; so that, for example, the values of are only to be trusted if they turn
out to be reasonably constant. The table tests the first approximation; it would be
a dubious procedure to use it as a starting-point for a second approximation.