85, 86] Configurations of Equilibrium 91 
For all values of n less than 5, the differential equation (65*5) gives the 
star an infinite radius; for values of n less than 5, the radius is finite. From 
equation (85*3), the critical value n = 5 corresponds to = 0*286 and, by 
equation (85*5), the corresponding value of \ c is 0*35. Even this improved 
approximation, however, fails for the limiting case of a star of infinite radius, 
for I have found, as the result of numerical calculations by quadratures, that 
the actual value of \ c for which the radius first becomes infinite is quite close 
to X—0'5. For this reason we revert to the simpler approximation again 
from § 88 onwards. 
Distribution of Density. 
86. Emden’s numerical solutions give the following values for the radius r x 
of masses of gas arranged in adiabatic equilibrium with different values of ft, 
and for p c /p, the ratio of the central density to the mean density: 
n = 3-25 4-0 4-5 4-9 >5-0 
r x - 8-00 15-00 32-14 169-47 oo 
p c /p=88 623 6378 934800 oo 
We see that gaseous stars of large mass have very high central condensation 
of mass; the greater part of their mass is condensed in a very small central 
region, while their outer layers are mere tenuous unsubstantial veils drawn 
round small and massive central cores. 
For instance, from Emden’s solution* for a star arranged according to the 
law n = 4*9, I have calculated the following values for M r /M, the ratio of the 
mass enclosed within a sphere of radius r to the total mass : 
rit = 0 1 2 3 4 5 10 50 169-5 
M r /M =0 0-125 0-436 0-657 0788 0-850 0'958 0*997 l'OOO 
The main part of the mass is concentrated very near to the centre of the 
star. Ninety-six per cent, of the whole is enclosed within a sphere whose 
radius is one-seventeenth of that of the star, the remaining space, which is 
99*98 per cent, of the total volume of the star, containing only 4 per cent, of 
the star’s mass. 
To make the example more concrete, let us imagine that Betelgeux, with 
a radius 300 times that of the sun, is arranged internally according to the 
law w = 4*9. The above figures shew that a sphere of radius equal to that of 
the sun (r x = 0*56) drawn round the centre of Betelgeux will contain over a 
twentieth part of the star’s mass, or a mass which is almost certainly greater 
than that of the sun itself. Thus to within a sun’s radius from its centre, 
Betelgeux would, if it had the structure we have imagined, be denser than the 
sun. The remaining 299 radii are comparatively devoid of matter and it is 
their inclusion in the star’s volume which reduces its mean density to the low 
value of 0 000002 . 
Gas Kugeln, p. 81.