228-230] General Stellar Dynamics 229 



With 'the numerical values already assumed (v 10~ 56 ' 54 ; M' = 10 33 ' 53 , 

 F= 4 x 10 6 ), this gives v n * = 7 x lQ- 9 t. 



Here t is measured in seconds. If t is measured in years, the formula 

 becomes, nearly enough, 



v n = ^t (552). 



Thus in one year the expectation of cross-velocity is J cm. a second ; in a 

 million years it is '005 km. a second. It only reaches 1 km. a second after 

 40,000 million years. Taking v = 25 kms. a sec., the time required for v n * to 

 become comparable with v 2 , i.e. for the direction of motion to be entirely 

 changed by encounters for which A|T < 1, is found to be of the order of 

 10 13 ' 5 years*. 



These numbers, it will be remembered, refer only to our universe as it 

 now is, and measure the cross-velocity to be expected from encounters with 

 neighbouring stars within 20 parsecs. It is accordingly clear that the velo- 

 city set up by encounters with neighbouring stars is quite negligible. Thus 

 we arrive at the very important result that the changes in stellar velocities 

 may be regarded as coming from the forces exerted by the main body of the 

 universe : near stars need not be taken into the calculation at all. 



229. It now appears that, for our present universe, the problem of stellar 

 dynamics is the same as the problem of the kinetic theory of gases with the 

 collisions left out. This being so, stellar dynamics is naturally very much 

 simpler than gas-dynamics. As Eddingtonf has remarked, it may, in virtue 

 of the result just obtained, be regarded as a quite different study from gas- 

 dynamics, or from that of the motion of any type of system that has yet been 

 investigated. For the action of contiguous units, which becomes gradually 

 simpler as we pass in succession through rigid dynamics, hydrodynamics and 

 gas-dynamics, disappears entirely when we come to stellar dynamics. 



230. Just as in gas-dynamics, the units in any small region of space may 

 be classified according to their velocities into a system of showers of parallel- 

 moving units. But there is the essential difference between the two cases, 

 that in stellar dynamics these showers retain their identity through very long 

 periods of time, whereas in gas-dynamics they do not. 



Suppose that in any small region of space dxdydz, the number of stars 

 which have velocities lying within a small range dudvdw surrounding the 

 values u, v, w at time t is 



f (u, v, w, x, y, z, t) dudvdivdxdydz (553). 



* This time corresponds to the "time of relaxation" in a gas. For this same time, 

 Prof. Charlier has obtained, by a somewhat different process, a value of the order of 10 1(! years 

 (Lund Meddel. u. No. 15). 



f Stellar Movements, p. 256. 



