272-275] Motion subsequent to Fission 261 



Equipartition of Energy 



274. Let us examine how the problem stands when account is taken of 

 the forces from the other stars. We have seen that in the present state of 

 the universe the forces from neighbouring stars may be neglected in a 

 statistical discussion ; the forces acting on a binary system from outside may 

 be supposed to be those arising from the gravitational field of the universe as 

 a whole. 



Over the extent of any binary system, the potential of this gravitational 

 field may be represented by a single second harmonic term, say 



the centre of gravity of the binary system being taken for origin. We may 

 first examine whether the action of such a field is to increase the eccentricity 

 and period of the orbit of the binary system. 



The field may be regarded as the superposition of two fields, having 



potentials respectively 



- J fj, (# 2 + y 2 + z 2 ) and | yiu? 2 . 



The first of these fields gives rise merely to a force of repulsion pr acting 

 away from the centre of gravity of the star. So long as //. remains constant, 

 the apsidal distances of the orbit naturally remain fixed ; when jj, increases, 

 there is an increase in the dimensions and period of the orbit, but when p 

 decreases, there is a corresponding decrease. Thus any secular change in the 

 dimensions and period of the orbit can depend only on yu, 2 , which is so small 

 as to be negligible. 



The field of force of potential f [MOC* gives rise to a repulsive force 3/u? away 

 from the plane of yz. As the line of apses of the orbit must be supposed to 

 make all angles indifferently with the axis of x, it is readily seen that the 

 average effect of these forces- must be nil. Thus any secular change can be 

 proportional only to the negligible quantity //. 2 . 



It acccfrdingly appears that in the present state of the universe, the 

 general gravitational field can only have an infinitesimal effect upon the 

 orbits of binary stars. 



275. We have already had occasion to contemplate a past epoch in the 

 history of the universe, in which the stars were much closer together than 

 they now are. We have found reasons for supposing that at this time the 

 stars were close enough to affect one another's orbits in space, to an appreci- 

 able degree, and it is natural to inquire in what way the close encounters of 

 this epoch would influence the relative orbits of binary stars. 



The problem now before us has a close analogy in the Kinetic Theory of 

 Gases. In the present epoch the stars have been seen to behave like the 



