210 The Evolution of Rotating Nebulae [CH. ix 



steadily. In the motion after the sharp edge has formed, both the angular 

 momentum and the mass decrease owing to loss from the sharp edge. The 

 lost particles are those for which the angular momentum is greatest, so that 

 a result of this loss is that the ratio of angular momentum to mass decreases. 

 The decrease has to be at just such a rate that w 2 /27rp remains constant ; 

 this condition determines the rate at which matter is thrown off from the 

 edge of the main mass. 



During the later stages of the motion it will not be legitimate to neglect 

 the gravitational field set up by the ejected matter. The ejected matter will 

 exert gravitational forces on the main mass, in directions such as to reinforce 

 centrifugal force and to neutralise the gravitational attraction of the main 

 mass. Thus each element of matter which is ejected provides in itself a force 

 tending to increase the rate of ejection of matter. It is at once clear that 

 the motion of ejection must ultimately become of a "cataclysmic" nature; 

 it cannot cease until either the main central mass has become entirely 

 disintegrated, or until the physical conditions of the problem change in 

 some way. 



The Theory of Laplace and Roche 



208. Laplace and Roche both imagined the mass to throw off matter for a 

 time and then to stop. The ejected matter was supposed to form an annulus 

 or ring out of which planets were ultimately formed. Roche went further 

 and imagined a continual series of alternations in the physical condition of 

 the mass, so that the mass threw off a succession of detached rings at distinct 

 intervals *. 



Clearly the ejection of matter at the sharp edge can cease only when the 

 condition for the existence of a sharp edge and the condition of the constancy 

 of angular momentum (without ejection of matter) become identical. So long 

 as the ideal gas laws are obeyed, the former condition requires that w 2 /P shall 

 remain constant, while the latter requires that o> 2 /p shall vary as (p)*. Thus 

 it is quite clear that the ejection of matter, when once it has started, can never 

 cease so long as the ideal gas laws are obeyed. When these laws are sub- 

 stantially departed from, the ejection of matter may cease, but so long as we 

 regard our nebulous mass as being approximately a perfect gas, the emission 

 of matter from the sharp edge will be a continuous process. 



209. Let us now examine the motion of the matter thrown off from the 

 sharp edge. Each particle or molecule, as it leaves the edge, will have a tan- 

 gential velocity in space equal to cor, arising from the rotation of the mass in 

 space, and superposed on to this velocity there may be a velocity of ejection 



* Cf. footnote to p. 209. 



