280 The Origin and Evolution of the Solar System [CH. xn 



296. Formula (593) shews that the frequency of encounters at a distance 

 less than R is proportional to R*. It follows that tidal break-up of our sun, 

 if ever it occurred, is likely to have been caused by an encounter in which the 

 value of R was not far below the maximum possible. In 294, assuming 

 certain definite numerical data, we found the greatest value of R at which 

 tidal break-up could possibly be effected to be about 1*27 x 10 15 cms., the 

 corresponding relative velocity at periastron being about 4 kms. a second. 

 But a relative velocity as small as this must be excluded on dynamical 

 grounds. The relative velocity at periastron must, except in the special case 

 of a triple encounter, be greater than that due to a fall from infinity, and 

 when the assumed masses (0 and 2 ) fall from infinity to a distance of 

 1*27 x 10 15 cms., they acquire a relative velocity of 7*9 kms. a sec. It now 

 follows that the relative velocity is not likely to have been much greater than 

 that due to a fall from infinity. 



In general this velocity is given by Rv* = 2(M + M') while in 130 we 

 saw that an encounter would be " transitory " if 2 M'/Rv* was small. Clearly 

 the encounter we are now considering is far from transitory and may perhaps, 

 with fair accuracy, be treated as slow. If so, the critical value for R is simply 

 proportional to r c , the sun's mean radius, and we see that the sun is most 

 likely to have been broken up when its density was very low. 



297. On the tidal theory, as we are now considering it, the planets must 

 have been formed as condensations in an arm of matter thrown out from the 

 sun towards a passing mass. In terms of the molecular velocity C and mean 

 density p of the matter in this arm, the mass of each planet ought to be of 

 the order of J C 3 y ~ * p ~ , if its birth occurred in the manner considered 

 in 217. 



The calculations already made have shewn that our system is a priori 

 most likely to have broken up when it was of low density and when our stellar 

 universe was in the earliest stages of its existence. Let us conjecturally 

 assume for the nebular arms a mean density 5'5 x 10~ 13 , this being one-tenth 

 of that of our sun spread through a sphere of radius equal to that of the orbit 

 of Neptune ; let us assume a molecular velocity of 4 x 10 4 , this being about 

 that of hydrogen or oxygen at their boiling points. The mass of the resulting 

 condensations is found to be about 10 :i grammes a mass intermediate between 

 those of Jupiter and Saturn. It is clear that if our system contained, beyond 

 the central sun, only planets of masses of the order of those of the two greatest 

 planets, the tidal theory would provide a highly satisfactory explanation of 

 the genesis of the system. 



298. The tidal theory can only inspire confidence if it proves able to 

 account for the small planets as well as for the large planets, and also if it can 

 account for the satellites of the planets in the same way in which it accounts 



