236 The Evolution of Star-Clusters [CH. x 



Let us pass to cylindrical coordinates OT, 0, z and let the components of 

 velocity at any point in these directions be denoted by II, , Z. Then 

 'cr 3 = -C3-II, and the law of distribution (572) becomes 



/[Hn a + @ 2 + Z 2 )-F,<sr<H)] (574). 



The velocities at any point are again not distributed uniformly for all 

 directions in space, but the velocity diagram at any point will be a figure of 

 revolution having the direction of 6 increasing for axis. In other words star- 

 streaming will take place, the direction being everywhere along the circles 

 CT = cons., z = cons., which are circles coaxal with the axis of the whole 

 universe. 



This type of system, we have now seen, is the only type of system in a 

 steady state which can have originated out of a rotating system. Thus if we 

 assume, as we most reasonably may, that the cluster of stars generated out of 

 a rotating nebula will ultimately assume a steady state, then this state must 

 be one expressed by formula (574). In particular, if our universe is believed 

 to be in a steady state, the hypothesis that it has originated out of a rotating 

 1 nebula must fall unless the stellar motions are found to conform to a law of 

 the type of (574). 



STELLAR MOTIONS IN THE GALACTIC UNIVERSE 

 241. Let us examine the special problem presented by our own universe. 



Charlier, who has made a special study of tlie distribution of stellar 

 velocities, believes that the velocity surface is approximately an ellipsoid of 

 revolution ; in his opinion the axis is approximately, though not exactly, per- 

 pendicular to the radius vector to the centre of the system *. We have seen 

 that if the system, whatever its origin, were in a steady state, the axes of the 

 velocity surfaces would have to be either exactly radial or exactly perpen- 

 dicular to the radius vector at each point. Charlier's result accordingly 

 indicates that the system has not yet finally attained to a steady state, but 

 that it is approaching a steady state of the type indicated by the law of dis- 

 tribution (574). And this steady state is, as we have seen, the one to which it 

 would necessarily tend if it had originated, as we conjecture, out of a rotating 

 nebula. 



242. Let us try to estimate the length of time required for this final 

 state to be reached. We have already supposed our universe to have a mass 

 of 5 x 10 42 gms. and an equatorial radius of 2000 parsecs or 6 x 10 21 cms. The 

 period of a star describing a circular orbit round the equator would be about 

 160,000,000 years f, and the period of description of any orbit by any star 



* The Observatory, 40 (1917), p. 390, or Scientia, Aug. 1917. 



t Eddington, in his Stellar Movements, using somewhat different data, obtains periods of 

 300,000,000 years (p. 255) and of 0-5" a century or 259,200,000 years (p. 261). 



