262 The Evolution of Binary and Multiple Stars [CH. xi 



molecules of a gas in which no collisions occur. In the past epoch that we 

 have under consideration, they behaved like the molecules of a gas in which 

 collisions occurred in the ordinary way. Single stars must have behaved like 

 monatomic molecules and binary stars like diatomic molecules. There is 

 however the welcome difference that we understand the dynamics of a binary 

 star, whereas we do not yet understand the dynamics of a diatomic molecule. 

 The tendency of encounters with other stars must have been towards estab- 

 lishing an internal distribution of energy such as would be in equilibrium 

 with the translational energy of the stars. It will, however, be best to state 

 the problem in a form which does not imply or presuppose any analogy with 

 the Kinetic Theory of Gases. 



276. Let M, M' be the masses of the two constituent stars of a binary 

 system of total mass M+ M'. Let u, v, w denote the components of the 

 velocity of the centre of gravity in space, and let r, 0, </> be polar coordinates 

 of M relatively to M'. Then the whole kinetic energy of the system is 



| (M+M')(u*+ y 2 + O + ijj^ fef /(^+ r *& + r'sin 2 Oft) ...(586). 



Following our view of the genesis of binary system, we suppose that when 

 a double-star first comes into being, the value of r 2 will be very small, while 

 the value of r 2 + r 2 sin 2 6ft, the square of the tangential velocity, bears no 

 relation at all to the translational velocity of the system as a whole. The 

 theorem of equipartition of energy shews that the tendency of stellar 

 encounters must be towards equalising the mean values of the different 

 terms in formula (586). In the final steady state which would be attained 

 after an infinite number of encounters, the mean square of the tangential 

 velocity would be equal to twice the mean square of the radial velocity ; it 

 would also be equal to f ( M + M'^jMM' times the mean square of the velocity 

 of translation in space. 



Consider the description of an orbit of eccentricity e. Without loss of 

 generality the plane of the orbit may be supposed to be < = 0, and the equation 

 of the orbit will be 



- = 1 -f e cos 6. 



The motion will have the usual integral of momentum r 2 # = h, so that 



dt=r 2 d0/h. 



Using a bar over a quantity to denote its mean value at all instants of 

 the description of a complete orbit we readily find that 



