296 The Evolution of Binary Systems [ch. xi 
when the components are of approximately equal mass, as is the case with 
the majority of binary stars. 
264. In § 217 we obtained the total angular momentum M of a newly- 
formed binary system in which the components revolved in a circular orbit 
of radius R, in the form 
M = 
Mie + M'k'> + 
MM ' 
(M + M') 
7 i(l + Ç)i(M+M')i R-* ... (264-1). 
Consider first the extreme case in which the masses are supposed homo 
geneous and incompressible. With a view to obtaining some idea of the 
ratio of division of M into its rotational and orbital parts, I have calculated 
the ratios of the separate terms in M for Darwin’s figures of closest 
approach from the data already tabulated in § 218, with the following results: 
m 
JT = 
0 
0-4 
0-5 
10 
Rotational momentum of M' 
0 
•039 
•046 
077 
» » n 
1 
T60 
T35 
•077 
Orbital momentum 
0 
•801 
•819 
•846 
Total 
1-000 
1000 
1000 
1-000 
With very few exceptions all known binary stars have values of M'/M 
which lie between 0"4 and 1-0. Excluding the few systems for which M'/M 
is less than 0 - 4, it appears that the orbital momentum must initially be at 
least 80 per cent, of the whole if the components move in circular orbits; 
if they moved in stable elliptical orbits, it would of course be greater still. 
Thus no matter for how long tidal friction or other similar tendencies 
act, the orbital momentum cannot increase to more than T25 times its 
initial value throughout the whole course of a binary star’s history. In the 
subsequent motion of the star, the orbital momentum is 
----- - ryh (1 + £)i li (264-2), 
(M + Mf YK ' ' 
where l is the semi-latus-rectum of the elliptical orbit. It follows that, 
throughout the whole life of the star, (1 -f £) l cannot increase to more than 
(1*25) 2 or 1*56 times its initial value. For bodies at a considerable distance 
apart f=0; for two similar ellipsoids in contact £ = 0*22, which is the 
maximum value of £. Thus in the whole course of evolution the value of 
1 + ? cannot decrease more than in the ratio 1*22 :1. It follows that l cannot 
at the very most increase in a ratio greater than 1*56 x l - 22 or 190. 
These calculations have referred to a perfectly homogeneous mass. To 
study the effect of heterogeneity, let us pass to the extreme case of matter 
so compressible that Roche’s model (§ 228) may be supposed to give an 
approximation to the arrangement of density. We may now put k 2 = k" 2 = 0