216 The Evolution of Rotating Nebulae [CH. ix 



For instance (cf. 6) van Maanen has found the period of rotation of the 

 Ursa Major nebula M. 101 to be about 85,000 years for a number of points at 

 average distance of 5' from the centre. The angular velocity is not quite 

 uniform, increasing somewhat as the nucleus is approached, but if, for pur- 

 poses of a rough calculation, we assume the period of rotation of the nucleus 

 to be 85,000 years, we find that the mean density of the nucleus must be 

 about 3'8 x 10~ 17 grammes per cubic centimetre. At this density there will 

 be about a million atoms or molecules of atomic or molecular weight 20 per 

 cubic centimetre, the mean free path being of the order of two thousand 

 kilometres. 



215. There remain some characteristic features of spiral nebulae which 

 have not so far been predicted or explained by our theory, and which will 

 accordingly provide further tests of the tenability of this theory. In particular 

 may be mentioned the characteristic shape of the arms (cf. 3) and the con- 

 densations or nuclei in these arms. The determination of the shape of the 

 arms to be expected on our theory seems at present to be beyond the 

 reach of mathematical analysis, but the formation of condensations admits 

 of discussion. 



In examining the ejection of streams of gaseous or other compressible 

 matter under tidal forces ( 160) we found that a long stream of gas must 

 become longitudinally unstable and will tend to break up into condensations 

 or nuclei under its own gravitational attraction. Exactly similar effects are 

 naturally to be expected in the present problem, and these seem to provide a 

 very natural and satisfactory explanation of the nuclei observed in the arms 

 of spiral nebulae. 



In a stream of compressible matter of uniform density p, the distance 

 apart of successive nuclei was found to be (approximately) 



For a gas, or quasi-gas formed of dust or meteorites, p = Jp<7 2 , where C is 

 the molecular velocity, so that 



Here p may be taken to be the mean density in the nebular arms. The 

 mean density of the nucleus, which we have so far denoted by p, will probably 

 be somewhat greater : let us denote it by Op. Then equation (534) becomes 



&> 2 = 0'35 x 27rjBp = 26yp (say), 

 so that equation (539) gives 



e c 



