228 The Evolution of Star-Clusters [OH. x 



A comparison of these expressions shews that, whatever the values of 

 v l and v 2 , sin 2 /? must always be between J and 1. For rough numerical 

 estimates, which can at best only be accurate as regards order of magnitude, 

 we may take sin 2 /3 uniformly equal to f , giving 



........................... <> 



This value of v n * becomes very large if there are encounters for which V 

 is very small, but our original formula is not applicable to these. Moreover, 

 whatever the law of distribution of stellar velocities, the frequency of encounters 

 for which V is small must be proportional to V 3 d V, so that the influence of 

 such encounters on v n 2 is negligible. We may accordingly suppose V to be 

 replaced by its average value V. 



In an interval t, the number of encounters for which cr lies between a and 

 cr + da- may be taken to be 



so that the expectation of v n 2 after time t will be 



^t = -^L^--\og(^}t (551). 



In this evaluation of v n 2 we have assumed M' to be about the same for 

 all encountering stars ; if it is not we need only replace M /2 by its mean value. 



Further, as the limits indicate, we consider only encounters for which cr 

 lijs between two values <T I} <r 2 . We notice at once that the expression on 

 the right would become infinite both for a-^ = and for o- 2 = oo . We are not 

 entitled to put c^ = because by doing so we should be taking into account 

 the effect of violent encounters, for which our formula does not apply. We 

 have seen that encounters for which i/r > 1 will be of extremely rare occur- 

 rence. Reserving these for separate discussion, we may give to CT O the 

 minimum value for which t/r< 1, which we have seen to be about 10 15 ' 5 cms. 



The circumstance that expression (551) becomes infinite when cr 2 =oo 

 shews that encounters with very distant stars contribute greatly to the value 

 of v n \ We may, however, use our formula to find the effect of encounters 

 with fairly near stars, say stars within 20 parsecs, and to do this we put 



cr 2 = 20 parsecs = 10 19 ' 8 cms. 



With these values, we have 



Vi 104-3 = 9-9. 



Thus we find for the expectation of v^ produced by non- violent encounters 

 with stars within 20 parsecs, 



