207-210] The Theory of Laplace and Roche 211 



arising from the motion of the matter inside the mass, and also a velocity of 

 molecular motion when the nebula is gaseous. 



As the problem is one of great complexity, it will be well to separate the 

 difficulties, and consider first the motion to be expected when the_velocity cor 

 exists alone or preponderates enormously over the other velocities. This 

 condition would be approximately satisfied if we could regard the matter of 

 the nebula as fluid, although it must be remembered that in actual fact a 

 sharp edge could not form on a fluid mass in rotation unless there was a 

 considerable central condensation of mass. 



Since the sharp edge is determined by the condition that centrifugal force 

 shall exactly balance gravity, it is clear that the tangential velocity cor will 

 be exactly that required for the description of a circular orbit. Thus at first 

 the ejected particles will form a chain of infinitesimal satellites in contact 

 with the main mass. 



As the main mass shrinks, this contact will of course be broken. More- 

 over, with the shrinkage of the main mass, the gravitational field will change, 

 so that the velocity of the satellites will no longer be that appropriate to the 

 description of circular orbits. Clearly a shrinkage of the main mass will 

 result in a lessening of the radial gravitational force at a fixed distance r, so 

 that the tangential velocity cor will become greater than the velocity for a 

 circular orbit at distance r, and the ring will begin to expand. Throughout 

 this motion the ring will remain circular, its angular velocity being deter- 

 mined by the constancy of its angular momentum ; cor 2 will always be equal 

 to the value at the instant of projection. 



The ring will not expand indefinitely. To a close approximation each 

 separate particle will describe an elliptic orbit, the loci of these particles at 

 each instant being a circle. Thus it appears that the ring may be expected 

 to expand and contract rhythmically. 



Immediately after the ejection of the first ring, a second ring may be 

 thought of as being ejected with approximately equal values of co and r, and 

 this will be followed by a continuous succession of other rings. In the 

 process of expansion and contraction these rings will collide and interfere with 

 one another's motion. 



210. To follow out the train of thought of Laplace and Roche, we should 

 have to imagine these rings to coalesce into a single ring of finite size which 

 would rotate with a uniform angular velocity co about the main mass. This 

 ring will be held together by its own gravitational cohesion but, like the 

 main mass, it will be subject to the disruptive effects of rotation. Clearly it 

 will only be matter of very considerable density that will possess sufficient 

 gravitational cohesion to form a definite ring. Poincare*, by a very simple 



* Lecons sur les Hypotheses Cosmogoniques, p. 22. 



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