154, 155J Roche's Model 153 



Thus it wili be seen that the value of the ratio v A /v N steadily decreases 

 from oo to about J as we pass along the Maclaurin series to the point of 

 bifurcation, and it is readily found that it decreases still further as we pass 

 along the Jacobian series. Its value at the pear-shaped point of bifurcation 

 is about ^. 



We can now describe the series of equilibrium configurations assumed by 

 this model as its angular momentum continually increases. Suppose first 

 that the ratio V A /V N is greater than J. 



For small values of co, the boundary of the nucleus and the atmosphere 

 will both be spheroids of small eccentricity. For larger values of o> the 

 boundary of the nucleus will remain spheroidal, while that of the atmo- 

 sphere will be a pseudo-spheroid coinciding with one of the external equi- 

 potentials. As &> still increases this pseudo-spheroid will develop a sharp 

 edge, this occurring when the critical volume V A ' is equal to V A . After 

 this, matter will be ejected from the sharp edge on the equatorial plane 

 of the mass. By the time the rotation is given by a) 2 /27rp = 18712, 

 the atmosphere is reduced to about ^V N in volume. Thus p = f /o , and 

 o) 2 /27rp = '2496. At this stage the figure loses its symmetry, being no longer 

 a figure of revolution. The nucleus becomes ellipsoidal, while the boundary 

 becomes a pseudo-ellipsoidal figure having two sharp pointed ends, and as 

 the rotation still increases, two streams of matter will be ejected from these 

 ends. Gradually the nucleus becomes -more elongated and the atmosphere 

 diminishes more and more, until the pear-shaped point of bifurcation is 

 reached. After this the nucleus will divide into detached masses, each of 

 which will be surrounded by a thin atmosphere. 



If the original atmosphere were of volume less than ^V N , the course 

 of events would be the same except that none of the atmosphere would 

 be thrown off until after the symmetry of revolution had been lost. In 

 this case the sequence of figures would be spheroids of small eccentricity, 

 pseudo-spheroids, pseudo-ellipsoids, pseudo-ellipsoids with pointed ends and 

 a stream of matter emerging from each, finally ending in detached masses 

 surrounded by thin atmospheres. 



The Tidal Problem 



155. In the tidal problem to vanishes but V T does not, so that equa- 

 tion (400) becomes 



where M, as before, is the mass of the primary, and its centre of gravity is 

 taken as origin. 



