150 Compressible mid Non- Homogeneous Masses [OH. vn 



The volume of the critical equipotential determined by equation (401) is 

 easily found. Putting r z = ^ 2 + w 2 , the equation becomes 



If we put ta- = 2x3-0 sin 6, 



_ 1 - 4 sin 2 ^ 



so that the volume is 



4?r I ziffdis = 327ror ft 3 I -^ - cos 2 6d (cos 6} 



J z =v Jo 4cos 2 0-l 



I ' 

 = 327rw 3 x -0225466. 



The mass is equal to p times this, where p is the mean density, hence 



...(402). 



Thus the series of configurations possible for the mass in question form a 

 single linear series, starting from &> = and ending abruptly when 



ft) 2 /27rjo = -36075. 

 For greater values of o> there is no equilibrium configuration possible. 



As a homogeneous mass shrinks, keeping its moment of momentum con- 

 stant, it is easy to shew that ca?/p will continually increase. It cannot be 

 rigorously proved that the same is necessarily true for a non-homogeneous 

 mass, but obviously the normal event will be for shrinkage to be accom- 

 panied by an increase of o) 2 /p. 



We can imagine a mass shrinking and a> 2 /27r^ continually increasing until 

 it reaches the value '36075, at which the mass begins to break up. When 

 this stage is reached matter begins to escape at the sharp edge of the 

 boundary (A A' in fig. 28), and will escape at just such a rate that o) 2 /27rp 

 retains the critical value '36075 for the main mass. The subsequent mo- 

 tion, as well as certain complications that arise, will be considered in a later 

 chapter. 



153. We have already seen that two distinct mechanisms may come 

 into play to effect the break-up of a rotating mass, and that there are only 

 two such mechanisms possible. The two models we have studied, namely 

 the incompressible mass and Roche's model, have now been found to pro- 

 vide examples of these two methods of break-up, the incompressible model 

 breaking up by fission into two parts, and Roche's model breaking up by 



