88—90J
Configurations of Equilibrium
97
ExM°(\ + iy *g*Tf
( 90 * 4 )
Table X. Solutions for various internal arrangements of stars.
K
n
*i
Pc
p
o
1 (M
B
£(0=1)
(n + ljti* \pj
1-4
2-5
5-42
24-1
6-60
0-235
1-02 xlO 69
255
1-33
30
6-90
54-4
6-04
0-285
1-27 XlO 69
319
1-31
3-25
8-00
88
5-84
0-322
1-43 X 10 69
358
1-29
3-5
9-52
152
5-67
0-374
1-60 XlO"
400
1-25
4-0
15-00
623
5-42
0-554
2-00 x 10 69
500
1-222
4-5
32-14
6378
5-20
1-124
2-46 X 10 69
614
1-204
4-9
169-47
934,800
5-20
5-524
3-03 X 10 69
758
1-200
5-0
00
00
5-20
00
3-19 x 10 69
799
1-14
7-0
00
00
5-20
oo
—
—
Further progress can only be made by assigning a definite value to l. As
the labour of continuing the calculations for all values of l would be excessive,
we may confine our attention to two values for l, namely l = ^ and 1 = 0,
trusting to interpolation or extrapolation to give an adequate idea of the
solutions for other values of 1.
Solutions when j = 0 and l = \ (Eddingtons model).
90. When l~\, equation (89'3) gives n = 3, independently of the value
of X c ; this, as we have already seen (§ 76), is the model discussed by Eddington.
For this value of n, Table X gives p c =54 , 4p, so that the central density is
always equal to 54’4 times the mean density, and 5 = 319.
Equation (891) now becomes
X c V 4 if 2 = 319 (1 + X c ) 3 (901),
which gives X c at once in terms of M and g, and we have already noticed
that X is constant throughout the star, as is also obvious from equation (82'7).
Equation ( 88 * 6 ) gives the central temperature in the form
r,= 19 ' 8 xl 0 ‘rTx( i r) (90-2)
The total emission of radiation E is in general equal to MG, where G is
measured at the surface of the star, and when all stars are supposed built
on the same special model (n = 3), this is proportional to MG where G is
measured at the star’s centre. Inserting the value of G from equation (81-3),
or from Eddington’s integral (761), we find that
E x
M J
.(90-3).
X (X "I - 1)
Using equations (901) and (90‘2), and also the relation E = ±Trar 2 T e \
where T e is the effective temperature of the star’s surface, we find that