110 THE TIDAL PROBLEM. 



With an initial density of 1.4 tlie final density comes out imaginary, indi- 

 cating that incompatible conditions have been imposed. That is, it is impos- 

 sible for a star having the density and twice the mass of the sun to divide 

 and be driven by tides into a binary system having a period of 100 years. 

 When (To = 4X10~^" the final density is infinite, another impossible result. 

 When the initial density is 10~*** we find /c=1.03, indicating an expansion 

 instead of a contraction. The only initial densities compatible with the 

 assumptions lie between 4X10~^" and 10~'". Consequently a double star 

 having two equal components with combined mass twice that of our sun and 

 a period of 100 years could not originate by fission except when it was in 



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the nebulous state with a mean density not exceeding -— ^ of that of our 



atmosphere at sea-level. On the other hand, if the close binaries revealed 

 by the spectroscope have originated by the fission of a single star, their 

 period can never become great through the effects of their mutual tides. 

 It follows from (92) that this relation between the initial period and final 

 period depends only upon the law of density and the amount of the con- 

 traction, and is entirely independent of the mass of the system. 



The question may be raised whether the results would not be less 

 unfavorable to the theory if the mass of the system were unequally divided 

 between the two stars. The discussion of section VII shows that the problem 



remains in a general way the same, 2771^ being replaced simply by w^ + f — j 



We must conclude from this discussion that approximately equal 

 binary stars with long periods can have originated by the fission process 

 only when the parent mass was yet in the nebulous state. In fact, it 

 removes the chief support of the belief that there is any such thing as fis- 

 sion among the stars simply because of rapid rotations. From other con- 

 siderations Jeans ^ has arrived at the same conclusion. 



iThe Astrophysical Journal, XXII (1905), p. 101. 



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