Jacobi ellipsoid

A Jacobi ellipsoid is an oblate ellipsoid under equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gustav Jacob Jacobi.^[1]

History^[2]^[3]

Before Jacobi, the Maclaurin spheroid which was formulated in 1742, was considered as the only type of ellipsoid which is in equilibrium, even though Joseph-Louis Lagrange in 1811^[4] considered the possibility of an ellipsoid being in equilibrium, but in conclusion he inferred that the two axes of the ellipsoid must be equal, leading back to the solution of Maclaurin spheroid. But Jacobi realized that Lagrange's demonstration is a sufficiency condition, but not necessary. His remarks were "One would make a grave mistake if one supposed that the spheroids of revolution are the only admissible figures of equilibrium even under the restrictive assumption of second degree surfaces" and further adds that "In fact a simple consideration shows that ellipsoids with three unequal axes can very well be figures of equilibrium; and that one can assume an ellipse of arbitrary shape for the equatorial section and determine the third axis (which is also the least of the three axes) and the angular velocity of rotation such that the ellipsoid is a figure of equilibrium."^[5]

Jacobi formula

Haumea, a dwarf planet with triaxial ellipsoid shape.

For an ellipsoid with semi-principal axes $a,\ b,\ c$ , the angular velocity $\Omega$ about $z$ axis is given by

{\frac {\Omega ^{2}}{\pi G\rho }}=2abc\int _{0}^{\infty }{\frac {udu}{(a^{2}+u)(b^{2}+u)\Delta }}\ ,\quad \Delta ^{2}=(a^{2}+u)(b^{2}+u)(c^{2}+u)

where $\rho$ is the density and $G$ is the gravitational constant, subject to the condition

a^{2}b^{2}\int _{0}^{\infty }{\frac {du}{(a^{2}+u)(b^{2}+u)\Delta }}=c^{2}\int _{0}^{\infty }{\frac {du}{(c^{2}+u)\Delta }}

For fixed values of $a$ and $b$ , the above condition has solution for $c$ if

{\frac {1}{c^{2}}}>{\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}

The integrals can be expressed in terms of incomplete elliptic integrals.^[6] In terms of the Carlson symmetric form elliptic integral $R_{J}$ , the formula for the angular velocity becomes

{\frac {\Omega ^{2}}{\pi G\rho }}={\frac {4abc}{3(a^{2}-b^{2})}}(a^{2}R_{J}(a^{2},b^{2},c^{2},a^{2})-b^{2}R_{J}(a^{2},b^{2},c^{2},b^{2}))

and the condition on the relative size of the semi-principal axes $a,\ b,\ c$ is

{\frac {2}{3}}{\frac {a^{2}b^{2}}{b^{2}-a^{2}}}(R_{J}(a^{2},b^{2},c^{2},a^{2})-R_{J}(a^{2},b^{2},c^{2},b^{2}))={\frac {2}{3}}c^{2}R_{J}(a^{2},b^{2},c^{2},c^{2})

The angular momentum $L$ of the Jacobi ellipsoid is given by

{\frac {L}{(GM^{3}{\bar {a}})^{1/2}}}={\frac {\sqrt {3}}{10}}{\frac {a^{2}+b^{2}}{{\bar {a}}^{2}}}\Omega ,\quad {\bar {a}}=(abc)^{1/3}

where $M$ is the mass of the ellipsoid and ${\bar {a}}$ represents the radius of sphere of same mass $M$ as the ellipsoid.

References

↑ Jacobi, C. G. (1834). Ueber die figur des gleichgewichts. Annalen der Physik, 109(8–16), 229–233.
↑ Chandrasekhar, S. (1969). Ellipsoidal figures of equilibrium (Vol. 10, p. 253). New Haven: Yale University Press.
↑ Chandrasekhar, S. (1967). Ellipsoidal figures of equilibrium—an historical account. Communications on Pure and Applied Mathematics, 20(2), 251–265.
↑ Lagrange, J. L. (1811). Mécanique Analytique sect. IV 2 vol.
↑ Dirichlet, G. L. (1856). Gedächtnisrede auf Carl Gustav Jacob Jacobi. Journal für die reine und angewandte Mathematik, 52, 193–217.
↑ Darwin, G. H. (1886). On Jacobi's figure of equilibrium for a rotating mass of fluid. Proceedings of the Royal Society of London, 41(246–250), 319–336.

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Jacobi ellipsoid

History[2][3]

Jacobi formula

See also

References

History^[2]^[3]