ThreeDimensionalConfigurations/Stability/RiemannEllipsoids

Lebovitz & Lifschitz (1996)

Lebovitz & Lifschitz (1996)

Here we review the work of 📚 N. R. Lebovitz, & A. Lifschitz (1996, ApJ, Vol. 458, pp. 699 - 713) titled, "New Global Instabilities of the Riemann Ellipsoids," and discuss various extensions that have been made to this work. We were prompted to tackle this review by an email received in December 2021 from Howard S. Cohl.  
 
 
 
 
 

Background

In Figure 1, the abscissa is the ratio $b/a$ of semiaxes in the equatorial plane, and the ordinate is the ratio $c/a$ of the vertical semiaxis to the larger of the equatorial semi axes. This diagram shows what 📚 Lebovitz & Lifschitz (1996) — hereafter, LL96 — refer to as "the horn-shaped region of existence of S-type ellipsoids and the Jacobi family;" it underpins all four panels of the LL96 Figure 2.

Figure 1: The Horn-Shaped Region of S-type Ellipsoids

• Jacobi sequence — the smooth curve that runs through the set of small, dark-blue, diamond-shaped markers; the data identifying the location of these markers have been drawn from §39, Table IV of [EFE]. The small red circular markers lie along this same sequence; their locations are taken from our own determinations, as detailed in Table 2 of our accompanying discussion of Jacobi ellipsoids. All of the models along this sequence have $f\equiv \zeta /{\mathrm{\Omega }}_f=0$ and are therefore solid-body rotators, that is, there is no internal motion when the configuration is viewed from a frame that is rotating with frequency, ${\mathrm{\Omega }}_f$.
• Dedekind sequence — a smooth curve that lies precisely on top of the Jacobi sequence. Each configuration along this sequence is adjoint to a model on the Jacobi sequence that shares its (b/a, c/a) axis-ratio pair. All ellipsoidal figures along this sequence have $1/f={\mathrm{\Omega }}_f/\zeta =0$ and are therefore stationary as viewed from the inertial frame; the angular momentum of each configuration is stored in its internal motion (vorticity).
• The X = -1 self-adjoint sequence — At every point along this sequence, the value of the key frequency ratio, $\zeta /{\mathrm{\Omega }}_f$, in the adjoint configuration $(f_{+})$ is identical to the value of the frequency ratio in the direct configuration $(f_{-})$; specifically, $f_{+}=f_{-}=-(a^2+b^2)/(ab)$. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [EFE].
• The X = +1 self-adjoint sequence — At every point along this sequence, the value of the key frequency ratio, $\zeta /{\mathrm{\Omega }}_f$, in the adjoint configuration $(f_{+})$ is identical to the value of the frequency ratio in the direct configuration $(f_{-})$; specifically, $f_{+}=f_{-}=+(a^2+b^2)/(ab)$. The data identifying the location of the small, solid-black markers along this sequence have been drawn from §48, Table VI of [EFE].

Riemann S-type ellipsoids all lie between or on the two (self-adjoint) curves marked "X = -1" and "X = +1" in the EFE Diagram. The yellow circular markers in the diagram shown here, on the right, identify four Riemann S-type ellipsoids that were examined by Ou (2006) and that we have also chosen to use as examples.

Four example models of equilibrium, Riemann S-Type ellipsoids:

• Riemann Meets COLLADA & Oculus Rift S: (b/a, c/a) = (0.41, 0.385)
• (b/a, c/a) = (0.90, 0.333)
• (b/a, c/a) = (0.74, 0.692)
• (b/a, c/a) = (0.28, 0.256)

See Also

• A library of ellipsoidal harmonics may be available from George Dassios,