186 Compressible and N on- Homogeneous Masses [OH. vn 



gas or other compressible matter in adiabatic equilibrium will break up by 

 fission if 7 is greater than 22 ; it will break up equatorially if 7 lies between 

 1'2 and 2'2. This latter range of course includes the values of 7 for all 

 gases whose density is so low that Boyle's law is approximately satisfied ; 

 for these 7 is always less than T66. 



Similarly as we pass along the chain C of generalised Roche's models, 

 the value of s, the ratio of the volume of the nucleus to that of the 

 atmosphere, varies from oo to 0. The critical point is found to occur at 

 about s = J. Thus when the atmosphere is less than a third of the volume 

 of the nucleus, the mass will break up by fission ; when the atmosphere is 

 greater than this the mass will break up equatorially. 



These various results may be exhibited diagram matically as in fig. 38. 



Region of 

 Fissional 



Break-up. 



Region of 



(0-1871 2) 



(0-36075) 



Fig. 38. Kotational break-up. [The figures in brackets denote the values of o> 2 /27r/5.] 



The Tidal Problem 



186. In the tidal problem we have found precisely similar results, the 

 incompressible model breaking up by a process very closely analogous to that 

 of fissional break-up in the rotational problem, and Roche's model breaking 

 up by a process which is at least suggestive of the equatorial break-up of a 

 rotating mass. 



Going further into detail, we have four 1 thnt the incompressible mass 

 will, under small tidal forces, have the shape of a prolate spheroid. As the 

 tidai forces increase, the elongation of this spheroid increases. When the 

 elongation reaches a stage such that the axes are approximately in the ratio 

 17:8:8, this spheroidal figure becomes unstable. Dynamical motion ensues, 

 the elongation at first increasing rapidly until finally furrows form on the 

 mass and it breaks up into several detached masses (cf. fig. 23, p. 127). 



Roche's model also will assume the shape of a prolate spheroid so long as 

 the tidal forces in action are small. As the tidal forces increase the boundary 

 of the figure departs from a true spheroidal form ; conical points form and 



