286-289] The Rotational Theory 273 



extreme central condensation is impossible, have arrived, naturally enough, 

 at the conclusion that rotational break-up was also impossible. Such extreme 

 condensation as is demanded by the rotational theory will be admitted to 

 be highly improbable, but there seems to be no way of proying it to be 

 impossible. 



At the same time, as we shall now see, a slight change in the form of 

 the argument brings to light considerations which suggest very strongly that 

 Laplace's hypothesis musfc be abandoned, at any rate if we hold to the assump- 

 tion that the angular momentum of the system has remained constant since 

 its birth. 



288. Small values of & 2 /r 2 can only mean that the matter in the out- 

 lying parts of the nebula is of density low compared with the mean density p. 

 Let p e denote the density of matter near the edge. The interior matter may 

 then be supposed of density greater than p e . The moment of inertia is 



= O 2 + f)pdxdydz, 

 and, since p > p e except near the edge, this requires that 

 Mk* > p e (x 2 + f) dxdydz. 



For the figure corresponding to extreme central condensation ( 152), the 

 integral is easily evaluated, and found to be equal to 0'52313r 2 times the 

 volume of the figure. It follows that 



0-523 ^< -. 



P r a 



The table of 287 (opposite) now assigns upper limits to p e /p. We have 

 r = Radius of present sun pelp-< 0'137, 



T Q = orbit of earth p e /p < 0'009, 



r = Neptune p e /p< 0*0017. 



289. We have seen that the method of break-up, if this occurred at all, 

 must have been that of equatorial ejection, as imagined by Laplace and 

 Roche. They imagined the next stage of evolution to be the formation of a 

 ring. This, on account of the conservation of angular momentum, must have 

 rotated with angular velocity &> practically equal to that of the original mass, 

 and so given by o> 2 /27rjo = 0'36. The theorem of Poincare, quoted in 210, 

 now assigns a lower limit to the mean density p r of this ring; it must be 

 greater than &> 2 /27r, and therefore greater than 0'36/3. 



Thus for evolution to have taken place on the lines imagined by Laplace 

 and Roche, p r must have been much greater than p e ; the ejected matter must 

 have increased in density, and so contracted, before a ring could form. 

 j. c. 18 



