238 The Evolution of Star-Clusters [CH. x 



instead of being measured in billions (10 12 ) of years they must be measured 

 only in millions. In a period which is, astronomically speaking, a short time, 

 each star may be expected to have been knocked about considerably both by 

 direct collisions and by close encounters with other stars. The only factor 

 which makes a complete calculation of the final result impossible is that we 

 do not know for how long the period in which the stars were close together 

 must be supposed to have lasted. 



244. If this period lasted indefinitely we know quite certainly what the 

 final result would be it would be a final steady state in which Maxwell's law 

 of distribution of velocities and the law of equipartition of energy would 

 hold. The final law of distribution would necessarily be of the general type 

 found to be necessary for every cluster, namely 



(575), 



but Maxwell's law fixes the form of the function /. The appropriate form 

 for a rotating system is known to be* 



f(E lt BT 8 )= Cfe-^PEi + fcwa] = (7g-/Wc*-2#F+2(^-^)] ...... (576), 



where (7, h and k are constants. 



The shape of the cluster is determined by making V satisfy equation (562). 

 It is clear that this demands that the shape and arrangement shall be the 

 same as that of a uniformly rotating mass of gas in isothermal equilibrium, 

 the stars of types M, M', . . . playing the same part in the clusters as molecules 

 of different kinds of gas. Now a uniformly rotating mass of isothermal gas 

 cannot form a figure of equilibrium at all its molecules merely fly off into 

 space. It follows that a star-cluster cannot ever finally attain to the equi- 

 partition law expressed by equation (576). 



245. We must therefore suppose our system to have moved part way on 

 the path towards equipartition but not to have attained it by the time that 

 its expansion had become so great that all hope of finally attaining it had 

 disappeared. We may suppose the law of distribution in our system to be 

 approximately of the general type (575) and we may further suppose the 

 equipartition law (576) to give a very rough approximation to the actual 

 form of this law. 



Let us examine some of the consequences of the law (576) with a view to 

 testing whether any of them are fulfilled, even approximately, in our system. 



The first property implied in the law of distribution (576) is one of 

 correlation between a star's mass and its velocity, this being of the kind 

 predicted by the well-known theorem of equipartition of energy. If c, c' 



* Jeans, Dynamical Theory of Gases (2nd Ed.), 107, 113. 



