222-225] 



General Stellar Dynamics 



225 



attraction appreciably influences their motion may conveniently be referred 

 to as an " encounter." It now becomes necessary to estimate the frequency 

 of stellar encounters. 



225. In fig. 42 let OPQ be a slightly curved orbit described by a star 

 of mass M about G the centre of gravity of a second star of mass M ' and itself. 



The distance of closest approach PGP' will be denoted by a. The velocity 

 of M relative to G will be M'V\(M + M'), where V is the relative velocity of 

 the pair of stars. 



When the star M is at R, the acceleration along GR is yM' cos 2 0/<r*, 

 y being the gravitation constant. Thus the rate of change of velocity along 

 PG is yM' cos 3 0/cr 2 , and the total change of velocity along PG will be 



(546). 



This total change of velocity must however be equal to 



M'V ' 



where -fy is the total deviation of either orbit ; thus 



.(547). 



It must be remembered that this formula may only be used when ^ is 

 small. 



A certain value of ^r will correspond to each value of closest approach a > 

 and conversely. The value of a which corresponds to a deviation i/r of 1 is 



.(548). 



Taking, as values appropriate to our present universe, M+M '= 6'8 x 1 38 gms., 

 V 4 x 10 6 cms., we find as the value of a- about 3'2 x 10 15 cms., or about seven 

 times the radius of Neptune's orbit. 



j. c. 15 



