269-272] Motion subsequent to Fission 259 



With the various maximum figures which have now been mentioned, the 

 greatest increase possible in P is found to be one of 13'6 times. This is a 

 maximum, and entirely abnormal increase ; for most binaries e is not greater 

 than J, so that (1 e 3 )- is not less than 0'65, and the maximum possible 

 increase in P is one of 4'4 times. 



271. Detailed calculations can be made for individual stars. For a Cen- 

 tauri, P = 81-18 years, M=M' (approximately) and e = 0'53. The parallax 

 is 0'76" and the semi-major-axis subtends an angle of I7'7l", whence 

 a = 3'5 x 10 14 cms. and I = 2'5 x 10 14 cms. Since M = M' it appears from the 

 figures on p. 257 that the maximum possible increase in /* (1 + f)* cannot be 

 greater than 1 : 0*846, so that Z(l + ?) cannot have increased by more than 

 40 per cent., and I cannot have increased by more than 71 per cent. Thus in 

 the very earliest stages of the star's history I cannot have been less than 

 59 per cent, of its present value, say 1*5 x 10 14 cms., and a cannot have been 

 less than the same amount. The period, which is now 81 '18 years, can never 

 have been less than 20'4 years. 



272. Still assuming that the binary system may be supposed to have 

 been free from external disturbances, a simple relation can be obtained 

 between the dimensions of the present orbit of a binary star evolved by 

 fission and those of the primitive nebula out of which the system originated. 



Consider the primitive nebula of mass M + M' at the instant at which the 

 pseudo-spherical form first became unstable. Let r and p denote its mean 

 radius and mean mass at this instant, so that M + M' = f Trjor 3 , and let 6 denote 

 the value of a> 2 /27rp. The angular momentum at this instant is 



M = (M + M') k*a> = (M + Jf ') r (f 0). 



After fission has taken place and the components have become thoroughly 

 separated, the orbital momentum will be 



MM' p 

 (If +#"')* 



Since this must always be less than M, it follows that at any stage of the 

 star's orbital motion 



I (M + MJK 



r M*M'* i- X *- 



Suppose first that the primitive nebula is wholly incompressible. The 

 figure is a Maclaurin spheroid at its point of bifurcation, so that & 2 /r 2 = 0'3838 

 and = 0-18712. Our inequality becomes 



172 



