225-228] 



General Stellar Dynamics 



227 



new velocity ZyM'/o-V along SP. Thus after encounter the components of 

 the velocity of M along and perpendicular to OP will be 



cos/3, 



sin /3 



.(549). 



Fig. 43. 



228. Let TP be the intersection of the plane POQ with the plane per- 

 pendicular to PQ in which the direction of closest approach must lie. Let 

 the angle OPQ be 0, and the angle TPS be <. Then cos ft = sin 6 cos cj>. 



In a series of encounters all directions in the plane TPS are equally likety 

 for the direction of closest approach PS, so that all values are equally likely 

 for </>. Thus, using a bar to denote mean values over a series of encounters, 



cos/3 = 



cos 2 /3 



sn 



sm 2 a 



Returning to formula (549) it is clear that the expectation of the com- 

 ponent along OP is simply v l ; the velocity along the original path remains 

 unaltered. The component of velocity perpendicular to this, say v n , can be 

 in any direction perpendicular to OP. After any number of encounters the 

 expectation of the value of v n 2 will be 



Now 



sin 2 3 = 1- 



sin 2 /3. 



sn 2 a 



and since F 2 = v? + v<? 20^ cos a, this can be expressed as 



87V 



152 



