260 The Evolution of Binary and Multiple Stars [OH. xi 



For a compressible mass the value of k' 2 /r 2 is less, but the value of is 

 greater. It is clear that k 2 /r z will decrease much more rapidly than 6, so that 

 compressibility lessens our calculated factor '04135 for instance for Roche's 

 model it reduces to zero. 



Thus the inequality (585) will be true independently of the compressibility 

 of the mass. 



For a binary in which the components are of very unequal mass 



will be very large, so that I may be very large compared with r the com- 

 ponents can separate to a distance large compared with the dimensions of 

 the primitive nebula. But for binaries in which M/M' < 2J, the value of 

 (M + M')'M*M'* is less than 24*01 and our inequality (585) becomes 



I < '9928r. 



Thus under no circumstances can the semi-latus rectum of a binary system 

 in which M/M' < 2J exceed the mean radius of the primitive nebula at the 

 instant at which the spheroidal form became unstable. 



For a Centauri the present value of I is 2 '5 x 10 14 cms., subtending an 

 angle of 12'75". If it was generated by fission, the mean radius of the 

 pseudo-spheroid just before elongation commenced must have been at least 

 2*5 x 10 14 cms., so that the major-axis must have been at least 6 x 10 14 cms., 

 subtending (at its present distance) an angle of at least 30". The mass of 

 the system being 3'8 x 10 33 gms., the mean density must have been less 

 than 6 x 10" 11 . 



273. The results just obtained dispose at once of the hypothetical inter- 

 pretation put upon Campbell's table on p. 255. So long as a binary star is 

 regarded as a self-contained system, we have seen that, for the great majority 

 of binaries, the linear dimensions and period of the orbit can only vary 

 slightly through the whole course of the star's life. A binary star can no 

 longer be supposed, as it grows older, to pass in turn through the different 

 columns of that table ; on the contrary, except in rare cases, it will remain 

 continually in the same column. Its spectral type and eccentricity will vary 

 as its evolution progresses, but the general order of magnitude of its period 

 must remain perpetually the same. If we suppose spectroscopic binaries to 

 have originated by fission, the problem of explaining why it is that short- 

 period binaries are generally of small eccentricity of orbit and of early spectral 

 type, the reverse being true of long-period binaries, admits of no answer so 

 long as we regard a binary star as a self-contained dynamical system. 



