184,185] Summary of Results 185 



The Rotational Problem 

 185. Let us consider the rotational problem first. 



For the incompressible model A, the mechanism of breaking up is very 

 fully known to us, thanks mainly to the investigations of Maclaurin, Jacobi, 

 Kelvin, Darwin and Poincare. For small rotations the mass will be spheroidal 

 in shape, but as soon as the angular velocity exceeds a value co given by 

 ft) 2 /27r/3 = 018712, the spheroidal form no longer remains stable, but gives 

 place to an ellipsoidal form. With still further increasing rotation*, the 

 mass elongates until a furrow begins to form across a section of the ellipsoid, 

 giving it a pear-shaped appearance. After this furrow has once started, the 

 motion is cataclysmal until the mass divides into two detached parts. 



For Roche's model B, the mechanism of break-up is also fully known. 

 As the rotation gradually increases, the equator of the mass bulges more 

 and more, until finally a sharp edge forms on the equator, so that the whole 

 figure becomes lens-shaped (see fig. 28, p. 149). Any further increase of 

 rotation now results in matter being thrown off from the equator in a 

 continuous stream, owing to centrifugal force outweighing gravity on the 

 equator. 



Thus models A and B both break up with increasing rotation, but they 

 break up in very different ways. We have been able to shew quite generally 

 that there are only these two distinct ways of breaking up ; the method of 

 breaking up of any other mass must be a variant of one or other of these 

 two. It will be convenient to refer to the first method of break-up, that of 

 the incompressible mass, as fissional break-up ; and to the second method of 

 break-up, that of Roche's model, as equatorial break-up. 



It follows that as we pass along either of the chains of models C and D 

 which connect A and B, or along any other chain of models connecting 

 A and B, there must be some point on each at which fissional break-up 

 gives place to rotational break-up. At such a point, the two methods of 

 break-up must be about to begin simultaneously with the same rotation. 

 Thus the condition determining such a point is that centrifugal force shall 

 be precisely equal to gravity on the equator of that configuration at which 

 the rotation reaches such a value that a figure of revolution is no longer 

 a stable form for the mass. 



We have determined this critical point on each of the two chains of 

 models C and D. Of these the adiabatic chain D is the more important. 

 As we pass along this chain from A to B, the value of 7 varies from x to 

 1*2 ; the critical point is approximately given by 7 = 2*2. Thus a mass of 



* The critical angular velocity is w 2 /27r/> = 0-14200, so that u[p^ has decreased, but the con- 

 stancy of angular momentum requires that /> shall have increased so much that w is found also 

 to have increased (cf. G. H. Darwin, Tides, p. 371). 



