POLYTROPIC GAS SPHERES 
87 
his radius 
to (a) ten 
bhe circles 
les on the 
tre for the 
he actual 
■qual mass. 
Dolytropic 
3 material 
The inner 
ial energy 
icordingly 
de of using 
values) are 
3). 
Hence the negative potential energy of the whole star is 
n n I' M M r dM r 
Q = e l 0 r — (60-11) 
= iff rms 
Jo r 
M 2 i'R M 2 
- ie S + j 0 w *. 
by integration by parts. Or since GM r jr 2 = - dfi/dr 
Q = i<? Tf (60-12). 
Again since GM r = - r 2 d<f>/dr and dM r = 4tt r 2 pdr, 
(60-11) gives Q = - 477 I R r 3 p dr. 
Jo dr 
Writing p = \(j) n in accordance with (55-41) 
f<i> = o 
Q = -^1.L/ (W 
= f ^ n+1 by integration by parts 
3 r 
= I 4> • Tpr 2 dr (60-21) 
= ^Tlf^ dMr ( 60>22 ) 
= (60-3) 
by integration by parts. 
Comparing (60-12) and (60-3) 
Q,= iG — + ^ 1 O 
^ R + 6 
3 M 2 
Hence Q = - ¿r-p- (60-4). 
O — ??/ it 
We note for reference that 
Q = 3 I K P AjrrHr = 3 | Pdv (60-5), 
Jo J 
where dv is the element of volume. This follows from (60-21) and (55-42). 
Uniform density is given by n = 0 and (60-4) then gives the usual 
expression for a uniform sphere Q. = | GM 2 /R. For other distributions the 
numerical coefficient is greater, the mass being more concentrated to the 
centre. 61 
61. The result (60-4) shows that there must be some break-down in 
the analysis when n > 5. The failure occurs because such distributions 
/