277-279] Motion subsequent to Fission 265 



The figure of the component under consideration will now be a Maclaurin 

 spheroid just about to give place to a Jacobian ellipsoid. The increase in 

 ft) 2 /27r/3 since fission took place is one of about 4*3 times, whence it follows 

 that the stars will now be separated by about 4 diameters and the tidal dis- 

 tortion of one on the other may legitimately be neglected. The increase of 

 density in the mass under consideration will be about (4'3) 3 or 79'5. A further 

 increase of density to 4'3 times this value is found to bring the figure to the 

 critical Jacobian ellipsoid, after which fission of the component takes place 

 and the star forms a triple system. The total increase in density since the 

 first fission occurred is 79*5 x 4*3 or 342 times, so that the linear dimensions 

 of the sub-system will be about one-seventh of those of the original system, 

 while their periods will be in a ratio of about 18 to 1. 



279. This calculation has neglected tidal friction altogether ; it is clear 

 that any action of tidal friction will postpone the formation of a sub-system 

 and so will increase the inequality of dimensions, density and period between 

 the two systems. The calculation has also supposed the masses to be incom- 

 pressible ; it is easily seen that compressibility will further increase the 

 inequalities, for the ratio of rotational to orbital momentum in the original 

 pair decreases with compressibility. 



Thus the inequality we have calculated is the minimum possible. When 

 triple systems form under natural conditions, the density at the second fission 

 must be more than 342 times that at the first, and so on*. 



With still further increase of density either component of the sub-system 

 may again sub-divide, but this cannot happen until the original density is 

 more than (34-2) 2 or 11,700 times that at the original fission, B 



while the period of the final system must be less than ^ times A 



that of the sub-system of which it is part and less than 

 3^2 times that of the main system. 



A typical multiple system of the kind predicted by the 

 rotational theory appears to be found in Polaris. This shews 

 spectroscopically periods of 4 days and 12 years, while 

 Courvoisier finds that the spectroscopic triple system is in 

 orbital motion with a fourth visible star, the period being 

 20,000 years. 



A typical visual system of the kind predicted by theory 

 is illustrated in fig. 45, this being the star 1502 in Jonck- c 



heere's Catalogue^. The figure is drawn to scale to repre- ? c 

 sent the projection of the system on the celestial sphere, Fig. 45. 



* Russell, to whom the first investigation of this question is due, gives 380 as the minimum 

 increase, the mass ratio in the first fission not being greater than 3 : 1 (Astrophys. Jo-urn. 31, 

 1910, p. 196). 



+ Memoirs R.A.S. Vol. 61 (1917). 



