244 The Evolution of Star-Clusters [CH. x 



collision once in 4 x 10 9 years ; assuming the masses to be equal to those in 

 our system, encounters producing a deflection of more than 1 would occur 

 about once in 10 8 years ; the cumulative effect of encounters feebler than this 

 produces a cross-velocity of 1 km. a sec. in about 4 x 10 7 years; the "time of 

 relaxation" (228) is of the order of 3 x 10 10 years. 



These estimates suggest that the typical globular star-cluster is hardly 

 likely to have attained fully to the final steady state of equipartition of 

 energy, but that this state is likely to be more closely approximated to than 

 in our own system. 



253. If we suppose the cluster to have formed out of a rotating nebula, 

 the law of distribution of density and velocities must, as in 240, be of the 

 general form 



/(^^H/ftOP + e'+Z')-!^] (578). 



Approximation to the final equipartition state will be shewn by the 

 function / approximating to the special form expressed by equation (576). 

 This law of distribution can never be fully attained ; as it is approached, the 

 stars having the highest total energy escape from the main cluster and form 

 runaway stars in space just as those molecules which are endowed with the 

 highest total energy may escape from a planetary atmosphere and describe 

 orbits in space. These stars carry with them an undue share both of energy 

 and of angular momentum, with the result that the cluster contracts and 

 rotates less rapidly ; the cluster must continually approach, but never quite 

 reach, a spherically symmetrical configuration. 



The investigations of Pease and Shapley to which reference has already 

 been made ( 6) suggest that the majority of star-clusters still shew evidence 

 of a flattened form, but the approximation to a globular configuration is 

 nevertheless tolerably close. The approach to a spherical form will be indicated 

 by the function / (E l , v? s ) depending less and less upon <7 3 . In the final 

 spherical form the law of distribution will reduce to the law/(j' 1 ) discussed 

 in 237. 



In this law the stars behave like the molecules of a gas ; different forms 

 of the funct ion f correspond to different relations between pressure and density 

 in this supposed gas. In the final equipartition law, f (E^ reduces to the 

 exponential form Ce~ 2hE >, and the corresponding law between pressure and 

 density is that of an isothermal gas. With this law the density at a great 

 distance from the centre falls off as I/?' 2 . Thus the total mass is infinite or 

 the central density infinitesimal ; the law is not one which can ever be 

 attained in an actual star-cluster. 



254. Let us simplify the problem by limiting the possible relations 

 between pressure and density in the supposed gas to those which correspond 

 to adiabatic equilibrium ; different laws are supposed (quite* arbitrarily) to 



