The Great Nebulae 
354 
[CH. XIII 
equal to \n r, so that the time necessary for a sphere to shrink under gravity to 
any small fraction of its original radius r is approximately 
(7—/ 
If we put M = 4<x 10 42 and r — 285,000 parsecs = 9 x 10 23 cms. in this 
formula, we find that the time necessary for a condensation in the primaeval 
medium to shrink to the dimensions of a spiral nebula is of the order of 6 x 10 10 
years. This supposes that each particle falls solely under the attraction of the 
attractions of all the masses in the neighbourhood, and these various attrac 
tions would to a large extent neutralise one another, while as soon as the 
just calculated. 
324. Any currents or motion in the original medium would contribute angu 
lar momentum to the nascent nebulae, and as these shrank tonebulardimensions, 
the constancy of angular momentum would result in fairly rapid rotations of 
the shrunken masses; currents of well below a kilometre a second in the 
primaeval gas would be adequate to produce the observed rotations of the two 
nebulae whose rotations have been measured. 
The question arises as to what length of time must elapse before a mass 
whose angular momentum was ini tially scattered in the form of random currents 
can assume a state of approximately uniform rotation about a definite axis. 
We saw in § 243 that inequalities of rotation would be reduced to half-value 
across a distance r in a time of the order of magnitude of pr^/rj, where 77 is the 
coefficient of viscosity. Since a nebula is almost transparent, radiative viscosity 
is non-existent, so that 77 may be taken to be the coefficient of material viscosity 
\pcl, where c is the mean velocity and l is the mean free path, and the time in 
question becomes Sr 2 /cl. 
For a final nebula, let us putr= 4 x 10 20 , l = 10 14 , c = 10 7 , and we find that 
inequalities of rotation are halved over a distance of about a quarter of the 
radius of an average nebula in a period of 1*6 x 10 1S years. 
In an earlier stage of the shrinkage the time for equalisation over a 
corresponding fraction of the radius is less. When the nebula has n times its final 
dimensions, equalisation takes place n times as rapidly. In view of the slowness 
of the earlier stages of nebular contraction, it seems likely that a nebula may 
acquire fairly uniform rotation in the process of shrinkage, but we have seen 
that if this does not happen, the whole of astronomical time will barely suffice 
to smooth out such inequalities of rotation as remain. 
The calculation hardly suggests that a nebula will acquire absolutely 
uniform rotation, and neither does observation suggest that the rotations of 
the nebulae, are uniform. Photographs of different lengths of exposure may be 
condensation to which it belongs. Actually each particle would be under the 
motion had proceeded to any extent, it would be further retarded by mole 
cular collisions. Thus the actual time would be many times longer than that