136 Cataclysmic Motion [OH. vi 



break up into detached masses, much as it was seen to do in the tidal problem, 

 and of these the innermost will fall in towards the primary while the outermost 

 will recede from it. If we think of these fragments as ultimately describing 

 elliptic orbits, the point at which instability sets in will approximately coincide 

 with the aphelia of the inner pieces and with the perihelia of the outer ones. 



If no change of density takes place in the matter of the satellite, the 

 orbits of the inner fragments will all be within the radius of limiting stability, 

 so that for each fragment the same process must repeat itself indefinitely, a 

 limit only being reached when the fragments are so small that their chemical 

 cohesion is able to defy the disruptive effects of gravitation and rotation. 

 The outer fragments, on the other hand, will describe orbits which will all lie 

 outside the radius of limiting stability, and so they will not suffer further 

 disintegration at first. But the perihelia of these orbits are already very 

 close to the sphere of limiting stability, and if the agencies which drove the 

 original satellite inside this sphere are still operative, it may be expected that 

 before long the new satellites also will be driven in and broken up in turn. 



136. If the matter of the satellite is even slightly compressible, and 

 therefore liable to changes of density, an entirely new feature presents itself. 

 For the initial elongation of the satellite when the configuration of limiting 

 stability is reached will be accompanied by a rapid diminution of pressure in 

 the interior of the satellite, and therefore by a rapid diminution of average 

 density. The radius of the sphere of limiting stability is however a function 

 of the density p of the satellite (cf. equation (65)), its radius varying as 

 p~ . Thus the elongation of the satellite will be accompanied by a rapid 

 expansion of the sphere of limiting stability; when the satellite breaks into 

 fragments all these will be within the new sphere of limiting stability, and 

 the process of breaking up will repeat itself indefinitely. 



Whichever way we approach the problem, the final result of the motion 

 must be a ring of broken fragments, each fragment being so small that 

 its forces of cohesion can resist the mechanical tendency to disintegration. 

 Roche has suggested that Saturn's rings may have formed in this way, a 

 suggestion borne out by the following figures : 



Radius of Saturn's outermost ring = 2 '30 radii of Saturn. 



orbit of Saturn's innermost satellite = 3'07 



Jupiter's = 2'o5 Jupiter. 



Mars' =2-75 Mars. 



Roche's critical radius, it will be remembered, has been found equal to 

 2 '45 radii of the primary when the densities of primary and satellite are the 

 same. 



