86 
POLYTROPIC GAS SPHERES 
so that although only about one-eighth of the volume is within this radius 
it contains 90 per cent, of the mass*. 
In Fig. 1, we show a star divided into shells corresponding to (a) ten 
equal steps of temperature, (6) ten equal masses; that is to say, the circles 
on the left correspond to T/T 0 = 0-9, 0-8, 0-7, etc. and the circles on the 
right correspond to M r /M = 0-1, 0-2, 0-3, etc. These diagrams are for the 
polytrope n = 3, which is believed to correspond nearly to the actual 
conditions of the stars. 
Potential Energy. 
60. We can determine the negative potential energy of a poly tropic 
star, i.e. the work done by gravitation in drawing together the material 
from a state of infinite diffusion. The analysis is due to Emden. 
Consider two concentric shells of masses dM r and dM s . The inner 
shell attracts the outer but not vice versa. The mutual potential energy 
(negative) is thus 
G dM r dM s 
where r is the radius of the outer shell. 
The mutual energy of dM r and of all shells interior to it is accordingly 
GdM f Mr 
dM. 
r . o 
GM r dM r 
r 
* It will be understood that we are here merely illustrating the mode of using 
the formulae, and the results (although probably fairly near the actual values) are 
not intended to be our final conclusions as to the state of Capella (cf. § 13).