158 THE TIDAL PROBLEM. 



VIII. SUMMARY. 



The problem under consideration is that of the fission of celestial bodies 

 because of rapid rotation when they are not disturbed by important exter- 

 nal forces. The attack is made through well-known results concerning the 

 figures of equihbrium and conditions as to stability of rotating homogeneous 

 incompressible fluids. It is recalled that for slow rotation a nearly spherical 

 oblate spheroid is a stable form of equilibrium; that for greater rates of 

 rotation the corresponding figure is more oblate; that when the eccentricity 

 of a meridian section becomes 0.813 the figure loses its stability and at this 

 point a stable line of three axis ellipsoids branches; that when the longest 

 axis of the ellipsoid becomes about three times the axis of rotation a new 

 series, known as the pear-shaped figures (or better, perhaps, the cucumber- 

 shaped figures) branches, and that before this point is reached there is no 

 possibility of fission. We are almost entirely ignorant as to what may happen 

 after this point is passed, and it must be remembered that it has not been 

 proved that in any case fission into two stable bodies is possible. 



The celestial bodies differ from those just considered in two important 

 respects. In the first place their densities increase toward their centers. 

 For a given rate of rotation and mean density this central condensation 

 makes them more nearly spherical, as is shown both by theory and by com- 

 parison of the observed figures of the planets with the computed forms of 

 corresponding homogeneous masses. In the case of Saturn, for example, 

 the eccentricity computed on the hypothesis of homogeneity is 0.607 while 

 the observed value is only 0.409. It seems certain that this central conden- 

 sation tends toward stability. The second important difference between 

 the ideal homogeneous incompressible fluids and the celestial bodies is that 

 the latter are compressible. This latter factor, at least under certain cir- 

 cumstances, tends toward instability. 



The opposing quantitative effects of central density and compressibility 

 undoubtedly differ greatly in different masses and can not be easily deter- 

 mined in any case. However, if we may assume that they approximately 

 offset each other, we may reach some conclusion respecting the possibility 

 of the fission of the actual celestial bodies by discussing the corresponding 

 homogeneous incompressible body. This is the assumption adopted here, 

 but, because of its uncertainty, in the applications to the solar system, where 

 it turns out fission is impossible, all approximations are made so as to favor 

 fission, and it is assumed that in the actual bodies fission may be immanent 

 long before it is possible in the homogeneous ones. These safeguards and 

 simplifications are possible and easy because it is a negative result which is 

 reached. 



The actual problem is not one in which the rate of rotation changes 

 while the density remains constant, though this is the one heretofore treated 

 in the mathematical discussion. In the physical problem the rate of rota- 

 tion and the density change simultaneously with the shape in such a way 

 that the moment of momentum remains constant. Imposing this condition, 

 we arrive in the case of the spheroids and ellipsoids at relations between 

 the density and respective shapes, the coefficients depending upon the mass 



