Jacobi ellipsoid

Jacobi formula

For an ellipsoid with equatorial semi-principal axes ${\displaystyle a,b}$ and polar semi-principal axis ${\displaystyle c}$, the angular velocity ${\displaystyle \mathrm{\Omega }}$ about ${\displaystyle c}$ is given by

${\displaystyle \frac{{\mathrm{\Omega }}^2}{\pi G\rho }=2abc{\int }_0^{\mathrm{\infty }}\frac{udu}{(a^2+u)(b^2+u)\mathrm{\Delta }},{\mathrm{\Delta }}^2=(a^2+u)(b^2+u)(c^2+u),}$

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

${\displaystyle a^2{b}^2{\int }_0^{\mathrm{\infty }}\frac{du}{(a^2+u)(b^2+u)\mathrm{\Delta }}=c^2{\int }_0^{\mathrm{\infty }}\frac{du}{(c^2+u)\mathrm{\Delta }}.}$

For fixed values of ${\displaystyle a}$ and ${\displaystyle b}$, the above condition has solution for ${\displaystyle c}$ such that

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

The integrals can be expressed in terms of incomplete elliptic integrals. In terms of the Carlson symmetric form elliptic integral ${\displaystyle R_J}$, the formula for the angular velocity becomes

${\displaystyle \frac{{\mathrm{\Omega }}^2}{\pi G\rho }=\frac{4abc}{3(a^2-b^2)}[a^2{R}_J(a^2,b^2,c^2,a^2)-b^2{R}_J(a^2,b^2,c^2,b^2)]}$

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

${\displaystyle \frac{a^2{b}^2}{b^2-a^2}[R_J(a^2,b^2,c^2,a^2)-R_J(a^2,b^2,c^2,b^2)]=c^2{R}_J(a^2,b^2,c^2,c^2).}$

The angular momentum ${\displaystyle L}$ of the Jacobi ellipsoid is given by

${\displaystyle \frac{L}{\sqrt{G{M}^{3}r}}=\frac{\sqrt{3}}{10}\frac{a^2+b^2}{r^2}\sqrt{\frac{{\mathrm{\Omega }}^2}{\pi G\rho }},r=\sqrt[3]{abc},}$

where ${\displaystyle M}$ is the mass of the ellipsoid and ${\displaystyle r}$ is the mean radius, the radius of a sphere of the same volume as the ellipsoid.

Relationship with Dedekind ellipsoid

The Jacobi and Dedekind ellipsoids are both equilibrium figures for a body of rotating homogeneous self-gravitating fluid. However, while the Jacobi ellipsoid spins bodily, with no internal flow of the fluid in the rotating frame, the Dedekind ellipsoid maintains a fixed orientation, with the constituent fluid circulating within it. This is a direct consequence of Dedekind's theorem.

For any given Jacobi ellipsoid, there exists a Dedekind ellipsoid with the same semi-principal axes ${\displaystyle a,b,c}$ and same mass and with a flow velocity field of

${\displaystyle \mathbf{u}=\zeta \frac{-a^{2}y\hat{\mathbf{x}}+b^{2}x\hat{\mathbf{y}}}{a^2+b^2},}$

where ${\displaystyle x,y,z}$ are Cartesian coordinates on axes ${\displaystyle \hat{x},\hat{y},\hat{z}}$ aligned respectively with the ${\displaystyle a,b,c}$ axes of the ellipsoid. Here ${\displaystyle \zeta }$ is the vorticity, which is uniform throughout the spheroid (${\displaystyle \mathrm{\nabla }\times \mathbf{u}=\zeta \hat{\mathbf{z}}}$). The angular velocity ${\displaystyle \mathrm{\Omega }}$ of the Jacobi ellipsoid and vorticity of the corresponding Dedekind ellipsoid are related by

${\displaystyle \zeta =\left(\frac{a}{b}+\frac{b}{a}\right)\mathrm{\Omega }.}$

That is, each particle of the fluid of the Dedekind ellipsoid describes a similar elliptical circuit in the same period in which the Jacobi spheroid performs one rotation.

In the special case of ${\displaystyle a=b}$, the Jacobi and Dedekind ellipsoids (and the Maclaurin spheroid) become one and the same; bodily rotation and circular flow amount to the same thing. In this case ${\displaystyle \zeta =2\mathrm{\Omega }}$, as is always true for a rigidly rotating body.

In the general case, the Jacobi and Dedekind ellipsoids have the same energy, but the angular momentum of the Jacobi spheroid is the greater by a factor of

${\displaystyle \frac{L_{\mathrm{J}\mathrm{a}\mathrm{c}}}{L_{\mathrm{D}\mathrm{e}\mathrm{d}}}=\frac{1}{2}\left(\frac{a}{b}+\frac{b}{a}\right).}$