bracketleftBigg F \u03b3\u03b1 E \u03b3\u03b1 tan \u03b2 sin \u03b2 tan \u03b3 cos \u03b2 E \u03b3\u03b1 tan 3 \u03b3 cos 2 \u03b1 cos 2 \u03b2

Bracketleftbigg f γα e γα tan β sin β tan γ

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bracketleftBigg F ( γ,α ) E ( γ,α ) tan β sin β tan γ + cos β E ( γ,α ) tan 3 γ cos 2 α cos 2 β tan 2 γ cos 2 α bracketrightBigg , (3.140) respectively, where E ( γ,α ) = integraldisplay γ 0 (1 sin 2 α sin 2 θ ) 1 / 2 d θ, (3.141) F ( γ,α ) = integraldisplay γ 0 d θ (1 sin 2 α sin 2 θ ) 1 / 2 , (3.142) are special functions known as incomplete elliptic integrals . 4 The integral α 0 , defined in (3.107), transforms to α 0 = 2 (cos β cos γ ) 1 / 3 a 0 sin γ F ( γ,α ) . (3.143) Finally, making use of some of the analysis in the previous two sections, the normalized angular momentum, and normalized mechanical energy, of the planet can be written hatwide L = = 6 10 1 + cos 2 β (cos β cos γ ) 2 / 3 hatwide ω, (3.144) hatwide E = 3 5 (cos β cos γ ) 1 / 3 sin γ F ( γ,α ) + 3 20 1 + cos 2 β (cos β cos γ ) 2 / 3 hatwide ω 2 , (3.145) respectively. Now, the constraint (3.139) is obviously satisfied in the limit β 0, since this implies that α 0 and E ( γ,α ), F ( γ,α ) γ . Of course, this limit corresponds to the axisymmetric Maclaurin spheroids discussed in the previous sec- tion. Jacobi, in 1834, was the first researcher to obtain the very surprising result that (3.139) also has non-axisymmetric ellipsoidal solutions characterized by β > 0. These solutions are known as the Jacobi ellipsoids in his honor. The properties of the Jacobi ellipsoids, as determined from Equations (3.139), (3.140), (3.144), and (3.145), are set out in Table 3.2, and illustrated in Figures 3.5 and 3.6. It can be seen that the sequence of Jacobi ellipsoids bifurcates from the sequence of Maclaurin spheroids when e 13 = 0 . 81267. Moreover, there are no Jacobi ellipsoids with e 13 < 0 . 81267. However, as e 13 increases above this critical value, the eccentricity, e 12 , of the Jacobi ellipsoids in the x 1 - x 2 plane grows rapidly, approaching unity as e 13 approaches unity. Thus, in the limit e 13 1, in which a Maclaurin spheroid collapses to a disk in the x 1 - x 2 plane, a Jacobi ellipsoid collapses to a line running along the x 1 -axis. Note, from Figures 3.5 and 3.6, that, at fixed e 13 , the Jacobi ellipsoids have lower angular velocity and angular momentum than Maclaurin spheroids (with the same mass and volume). Furthermore, as is the case for a Maclaurin spheroid, there is a maximum angular velocity that a Jacobi ellipsoid can have ( i.e. , hatwide ω = 0 . 43257), but no maximum angular momentum. Figure 3.7 shows the mechanical energy of the Maclaurin spheroids and Jacobi ellipsoids plotted as a function of their angular momentum. It can be seen that the Jacobi ellipsoid with a given angular momentum has a lower energy 3 See On Jacobi’s Figure of Equilibrium for a Rotating Mass of Fluid , G.H. Darwin, Proc. Roy. Soc. London 41 , 319 (1886). 4 See Handbook of Mathematical Functions , M. Abramowitz, and I.A. Stegun (Dover, New York NY, 1965).
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Hydrostatics 51 e 12 e 13 hatwide ω hatwide L hatwide E e 12 e 13 hatwide ω hatwide L hatwide E 0.00 0.81267 0.43257 0.30375 0.50452 0.60 0.85585 0.42827 0.30984 0.50138 0.05 0.81293 0.43257 0.30375 0.50459 0.65 0.86480 0.42609 0.31296 0.49975 0.10 0.81372 0.43257 0.30375 0.50459 0.70 0.87510 0.42288 0.31760 0.49734 0.15 0.81504 0.43256 0.30377 0.50458 0.75 0.88705 0.41807 0.32462 0.49372 0.20 0.81691 0.43253 0.30380 0.50457 0.80 0.90102 0.41069 0.33562 0.48814 0.25 0.81934 0.43248 0.30388 0.50453 0.85 0.91761 0.39879 0.35390 0.47908 0.30 0.82237 0.43237 0.30402 0.50445 0.90 0.93778 0.37787 0.38783 0.46295 0.35 0.82603 0.43220 0.30427 0.50432 0.95 0.96340 0.33353 0.46860 0.42782 0.40 0.83037 0.43191 0.30468 0.50410 0.96 0.96950 0.31776 0.50078 0.41499 0.45 0.83544 0.43146 0.30532 0.50376 0.97 0.97605 0.29691 0.54672 0.39771 0.50 0.84131 0.43078 0.30628 0.50326 0.98 0.98317 0.26722 0.62003 0.37241 0.55 0.84808 0.42976 0.30772 0.50250 0.99 0.99101 0.21809 0.76872 0.32842 Table 3.2: Properties of the Jacobi ellipsoids.
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