298 
[CH. XI 
The Evolution of Binary Systems 
so that under no circumstances can the semi-latus-rectum of a binary system 
in which M/M' < 2\ exceed the mean radius of the parent star at the 
instant at which the spheroidal or pseudo-spheroidal form first became 
unstable. 
266. For a Centauri the present value of l is 2 5 x 10 14 cms., subtending 
an angle of 12-75". If the system was generated by rotational fission and has 
been under no influence beyond that of tidal friction, the mean radius of the 
pseudo-spheroid just before elongation commenced must have been at least 
2’5 x 10 14 cms., so that the major-axis must have been at least 6 x 10 14 cms., 
subtending (at its present distance) an angle of at least 30". The mass of the 
system being 3-8 x 10 33 grms., the mean density must have been less than 
6 x 10 ~ u . 
These figures are so remote from possibility as to dispel at once the 
supposition that the present long periods and large orbits of binaries can be 
ascribed to the workings of tidal friction on systems evolved by rotational 
fission, and we are compelled to turn elsewhere for a solution of the problem. 
Secular Decrease of Mass. 
267. The foregoing discussion has supposed the masses of the two com 
ponents of the binary to remain constant; we have seen that in actual fact 
they must undergo a slow secular decrease as a consequence of the emission 
of radiation from the star. In 1924* I investigated whether this secular 
change could produce the lengthening of the period and increase in the 
eccentricity which are observed to occur in binary systems. 
When a force F acts upon a body of mass m which is losing mass by 
radiation at a rate —dm/dt per unit time, the momentum of the mass mv 
experiences an increase at a rate F from the action of the force, and a loss 
at a rate ^ v\ from the momentum carried away by the emitted radiation. 
Thus the change of momentum is governed by the equation 
d , . dm 
- (mv) = F +1F v, 
or, simplifying, = F (267*1). 
The ordinary Newtonian equations are accordingly applicable without 
modification to a body whose mass is changing as a result of the emission 
of radiation. 
Motion of Particle about a Body of Decreasing Mass. 
268. Consider first the abstract problem of the motion of a particle of 
small mass about a gravitating body whose mass is decreasing as a result of 
the emission of radiation. 
* M.N. lxxxiv. (1924), pp. 2 and 912.