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



could break up a sun of density such that it just filled the present orbit of 

 Neptune. Moreover since the critical value of R only varies as p *, it is 

 clear that even grazing contact would not suffice to break up a primitive sun 

 of density less than this. 



We are hardly free to suppose the secondary to have had a mass much 

 greater than that already assumed, nor to suppose the primitive sun to have 

 had a radius much less than that of Neptune's orbit. For, as we shall see 

 immediately, either of these suppositions would result in encounters capable 

 of effecting tidal break-up becoming excessively rare events. We accordingly 

 retain the already assumed values Jlf' = 4xl0 33 grammes, p = 5'5 x 10~ 12 , 

 and examine the effect of assigning a smaller value to the relative velocity v. 



With a relative velocity of only 4 kms. a second, formula (592) gives 

 1*28 x 10 15 cms. as the limit for R, but encounters with this velocity are 

 no longer transitory; on taking v = 4 x 10 5 the calculation of 130 gives 

 1'6 x 1C 16 as the closest distance of transitory encounters. Indeed the 

 encounter is so far from transitory that we may expect the calculations 

 for slow encounters to give a better approximation. Taking M'/M = 2, the 

 critical value of R for a slow encounter is 2'78r for an incompressible mass 

 and 2*87r for Roche's model. Taking r = 4*5 x 10 14 cms. (the radius of Nep- 

 tune's orbit), these limits are found to be 1*25 x 10 15 and 1*29 x 10 15 cms. 

 respectively. 



Thus with a primitive sun filling Neptune's orbit our calculations give 

 the following critical distances for a mass double that of the sun, passing 

 with a velocity of 4 kms. a second : 



Incompressible mass, transitory R = T28 x 10 15 cms. 



slow R = 1-25 x 10 15 



Roche's model R = 1'29 x 10 15 



The encounter we are here considering (R= about T27 x 10 1S , v 4 kms. 

 a second) is neither slow nor transitory, and the actual sun is not likely either 

 to have been incompressible or to have conformed to Roche's model. But 

 the calculated values of R agree so closely among themselves that there is not 

 likely to be much error in taking the limiting value of R to be T27 x 10 15 cms., 

 or about 2'8 times the radius of Neptune's orbit. 



This value of R corresponds to a velocity of 4 kms. a second. Higher 

 velocities of course require closer approaches, a velocity of 40 kms. a second 

 requiring as we have seen an approach to a distance of 4 x 10 14 cms., which 

 represents grazing contact. On the other hand lower velocities do not permit 

 of larger values of R, for these lower velocities give rise to slow encounters 

 for which R is independent of the velocity. Thus the largest value of R for 

 which tidal break-up can occur in a sun of density such that it fills a sphere 

 of radius equal to Neptune's orbit is of the order of T27 x 10 15 cms. 



