317 
284, 285] 
The Ages of the Stars 
In forming such estimates our unit of time is virtually the interval 
between one stellar encounter and the next, so that we begin our investigation 
by considering the frequency of stellar encounters. 
285. When two stars come so close as to exert appreciable forces on one 
two. In fig. 52 let S be the centre of gravity of two stars of masses m, m, 
which are pulling each other appreciably out of their courses, let V 0 be the 
velocity of m' before the encounter began, and let V be its velocity at the 
moment of closest approach, both velocities being measured relative to the 
centre of gravity S. Let p be the perpendicular distance of the undeflected 
path from S , and let a be the distance at the instant of closest approach. 
The orbit described by m is that which would be described under a 
gravitational force 7 m 8 /(m + ra ') 2 r 2 directed towards S. Thus the principles 
of conservation of energy and momentum supply the relations 
and as total angle of deflection yjr of either orbit is equal to 2 cosec -1 e, we 
obtain 
This gives the relation between a and t/t ; to find the relation between p 
The Dynamics of Stellar Encounters. 
another each describes a hyperbolic orbit about the centre of gravity of the 
Fig. 52. 
(285-1), 
P V 0 = aV 
The elimination of V between these equations gives 
2 g , 2ym 3 a 
p2 ~ a2 + (m + my F 0 2 
(285-2). 
(285-3). 
The eccentricity of the orbit, e, is given by 
p A - 1 n 2 2 rs / m . 3 
and yjr we eliminate a between this and equation (285'3) and obtain 
(285-5).