312 The Evolution of Binary Systems [ch. xi 
certain extent by tidal friction, and this diminution will certainly delay, and 
may entirely prevent, the fission of the two constituents separately. 
Let us first disregard the action of tidal friction and treat the angular 
momentum of each component after fission has once occurred as a constant 
quantity. Let us further treat the masses as incompressible or nearly so. 
Let M be the mass of the parent star and M its angular momentum at 
the moment of fission, and after fission let one component have mass f m M and 
angular momentum / a M. When the rotations of two masses result in their 
having configurations of similar shape, although of different dimensions, 
(ù 2 / 27 r<yp is the same for both, so that their angular momenta are proportional 
to Let the angular momentum M be supposed to be 
M = a M*p~t, 
where a depends only on the shape of the mass. 
Apply this equation first to the parent star at the moment of fission, and 
next to the constituent star when, if ever, fission overtakes it. If p P , p G are 
the densities of the parent and constituent at their moments of respective 
fission, we find 
M = a M*p P ~l, 
f a -HL = <x{f m Mfp c -K 
From these two equations 
Pc f >» 10 
Pp fa 
Values of f a corresponding to different values of f m have already been 
calculated in § 264. From the table there given we deduce 
Value of f m 
0 
029 
0*33 
0-50 
066 
0-71 
1 
» fa 
0 
0-039 
0-046 
0077 
0135 
0-160 
1 
» Pc/Pp 
1040 
1740 
4760 
2590 
2460 
1 
00 
15 
17 
21 
16 
15 
1 
We see that a second fission cannot take place until the density has 
increased to some thousands of times the density at the first fission. The last 
line gives the values of ( Pc/fmPpf, this being the ratio of the dimensions of 
the orbit produced by the first fission to that produced by the second fission. 
The last line shews that when a triple system is formed in the way we 
have imagined, the separation of the two components of the main system 
will be from 10 to 20 times that of the components of the subsidiary system, 
and if still further subdivision takes place, a similar ratio must prevail. 
a* 
The relation P = 2tt . 
7 *(M + My 
now shews that the period as the main system must be at least about