Tidal Friction 
297 
263-265] 
and £ = 0. The whole momentum is orbital, and in this extreme case the 
constancy of M requires that l shall remain absolutely constant. We may 
reasonably suppose that all compressibility lessens the possible range of in 
crease in l, so that the increase of 90 per cent, just calculated for an incom 
pressible mass is the maximum possible, always provided the mass ratio does 
not exceed 2| : 1, and that the system remains free from external disturbance. 
265. Under these same assumptions there is a simple relation between 
the dimensions of the present orbit of a binary star generated by fission and 
those of the parent star out of which the system originated. 
Let r and p denote the mean radius and mean density of the parent star 
at the instant at which the pseudo-spheroidal form first becomes unstable, so 
that M + M' = %irpr\ and let 0 denote the value of <u 2 /27ty p at the same 
instant. Then the angular momentum at this instant is 
k 2 
M = (M+ M') k 2 a> = ryh(M+ M'f r* (f ef. 
When fission has taken place and the components have become thoroughly 
separated, the orbital momentum is 
MM ' . . . 
(M+M')* y + ^ 1 ’ 
Since this must always be less than M, it follows that at any stage of 
the star’s orbital motion 
i (M + M'Y k* 
r M 2 M' 2 r 4 * * 
If the parent star was wholly incompressible, the critical figure is a 
Maclaurin spheroid at its point of bifurcation, for which Ic 2 /r 2 — 0‘3838 and 
0 = 018712, so that the inequality becomes 
-<-04135 (if+ ^ 
r< M’M'* 
.(265-1). 
For a compressible mass the value of k 2 lr 2 is less, but 0 is greater. 
Calculation makes it clear that compressibility decreases k 2 /r 2 much more 
rapidly than it increases 0, so that compressibility lessens the numerical 
factor '04135—for instance, for Roche’s model it reduces to zero. Thus the 
inequality (265'1) remains true independently of the compressibility of the 
mass. 
If the two components are of very unequal mass 
(M+ M'Y/M 2 M' 2 
is very large, so that l may be very large compared with r —the components 
can separate to a distance large compared with the dimensions of the parent 
star. But for the normal binary in which M/M' < 2£, the value of 
(M + M'YJM 2 M' 2 is less than 24'01 and inequality (265'1) takes the form 
l< '9928r, 
T 9 
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