150 THE TIDAL PROBLEM. 



VI. APPLICATIONS TO THE SOLAR SYSTEM. 



In applying the formulas above to the solar system we must remember 

 that they are strictly valid only when the masses are homogeneous. Now 

 the sun and planets are certainly not homogeneous, but we have seen reasons 

 for believing that nevertheless the formulas will give results which are not 

 remote from the truth. But, because of this uncertainty, in the applica- 

 tions which follow we shall not attempt to draw conclusions except where 

 the margin of safety is extremely great. 



Let us consider first the sun. We shall find its density for various de- 

 grees of oblateness, and its oblateness for a certain very high density.^ Re- 

 ferring to (6) we see that the greater the moment of momentum of a body 

 the less dense it will be for a given oblateness, and the more oblate it will be 

 for a given density. Consequently we shall be favoring the conclusion 

 that the sun will eventually suffer fission if we use too large a value of M. 

 The moment of momentum is most easily computed if we suppose the sun 

 is homogeneous, and the result obtained in this way will certainly be in 

 excess of the true value. 



Using the mean solar day, the mean distance from the earth to the sun, 

 and the mass of the sun, as the units of time, distance, and mass respec- 

 tively, we find that the density of water is 



<7water=l>567,500. 



Taking the sun's density as 1.41 on the water standard, its period of rota- 

 tion as 25.3 days, and its radius as 433,000 miles, we find for its moment of 

 momentum 216 



Now we may apply equation (6) to find how dense the sun will be before 

 the Jacobian ellipsoids branch off. It is hardly possible that the sun could 

 suffer fission before this point is reached, the shape of the spheroid being 

 given in figure 19. The computation shows that when the sun shall have 

 reached this degree of oblateness its density will be 



<7 = 307 X 10" on the water standard. 



This density corresponds to an equatorial radius of the sun of 11 miles. 

 Since this density is millions of times greater than it is supposed matter 

 ever attains under any circumstances, we must conclude that the oblate- 

 ness of the sun can never approach that for which the Jacobian figures of 

 equilibrium branch. Or, in brief, the sun can never contract so much that 

 its rotation will threaten it with disruption. 



Notwithstanding the extreme character of these figures, one might still 

 be so ultra-skeptical as to doubt the conclusion, since it is based on the com- 

 putation of the point of bifurcation for homogeneous masses. However 

 that may be, we must admit that the sun will be stable until its oblateness 

 reaches that of Saturn at present. We have seen that the eccentricity of a 



* Strictly speaking the computations are made for homogeneous bodies of the same 

 mass and having the same moment of momentum, but no confusion will result from this 

 mode of expression. 



