226 The Evolution of Star-Clusters [CH. x 



Encounters for which <r is equal to, or less than, this value i.e. encounters 

 in which the deviation is 1 or more may be expected from formula (545) to 

 occur only about once in 10 11 years. It is true that formula (545) was 

 obtained only by neglecting the curvature of the stars' paths produced by 

 gravitational attraction, but it is quite legitimate to apply it to determine 

 the present critical value of a-, in which by hypothesis the total deflection of 

 path is only 1. 



A similar consideration also justifies our estimate of 4 x 10 12 years obtained 

 in 223 for the interval between collisions. In obtaining this estimate we 

 neglected the gravitational curvature of the path ; formula (547) now shews 

 that this curvature would only be about 7, of which only half, or 3^, would 

 occur before the collision took place. Thus the estimate will remain very 

 approximately true even when the full effects of gravitation are taken into 

 account. 



226. We have now, allowing fully for gravitation, obtained two estimates 

 applicable to a system of stars in the state in which our universe now is. 

 For the frequency of actual collisions, even assuming the stars to have a 

 diameter equal to that of Neptune's orbit, we have found 4 x 10 12 years, and 

 obviously for smaller stars collisions would be still more infrequent. For 

 encounters producing a total deflection of path of 1 or more, we have ob- 

 tained a frequency of one in 1*4 x 10 11 years. These intervals of time are 

 both so long in comparison with astronomical times that it is clear that, in 

 statistical calculations dealing with our universe as it now is, we may neglect 

 altogether the possibility of collisions and of encounters in which ty is as 

 large as 1, and confine our attention to encounters for which ^ is less than 

 1. For such encounters formulae (546) and (547) may be regarded as 

 accurate. 



227. We proceed now to study the cumulative effect of these feeble 

 encounters. 



It will be remembered that we have so far been concerned only with 

 motion relative to the centre of gravity of two stars. In fig. 43 let OP, OQ 

 represent the velocities in space of the two stars M, M' ; let these velocities 

 be denoted by v lt v 2 and be inclined at an angle a. Let G divide PQ in the 

 ratio M' : M, then OG will represent the velocity of the centre of gravity of 

 the two stars. 



The direction of the line of closest approach may be supposed to be SP. 

 This is necessarily perpendicular to the direction of relative velocity PQ. 

 Let the angle SPO be j3. 



The effect of the encounter is to superpose on to v 1} the velocity of M, a 



