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. Note that a good summary of the research efforts that preceded (and inspired) the work of 📚 Lebovitz & Lifschitz (1996) can be found in the introductory section of S. Ou, J. E. Tohline, & P. M. Motl (2007, ApJ, Vol. 665, pp. 1074 - 1083).

We were prompted to tackle this review in response to 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 left, identify four Riemann S-type ellipsoids that were examined by 📚 S. Ou (2006, ApJ, Vol. 639, pp. 549 - 558) and that we have also chosen to use as examples.

Four example models of equilibrium Riemann S-Type ellipsoids (click each parameter-pair to go to a related chapter discussion):

• (b/a, c/a) = (0.41, 0.385); the chapter name is "Riemann Meets COLLADA & Oculus Rift S"
• (b/a, c/a) = (0.90, 0.333)
• (b/a, c/a) = (0.74, 0.692)
• (b/a, c/a) = (0.28, 0.256)

 
 
 

Self-Adjoint Sequences

What are the expressions that define the upper $(x=-1)$ and lower $(x=+1)$ boundaries of the horned shaped region of equilibrium S-Type Riemann Ellipsoids? Well, as we have discussed in an associated chapter, the value of the parameter, $x$, that is associated with each point $(b/a,c/a)$ within the horned shaped region is given by the expression,

$1+2Cx+x^2$	$=$	$0,$
📚 Lebovitz & Lifschitz (1996), §2, Eq. (5)

where,

$C$	$=$	$\left[\frac{ab{B}_{12}}{c^2{A}_3-a^2{b}^2{A}_{12}}\right],$
📚 Lebovitz & Lifschitz (1996), §2, Eq. (6)

$A_{12}$	$\equiv$	$-\frac{A_1-A_2}{(a^2-b^2)},$
[ EFE, §21, Eq. (107) ]
$B_{12}$	$\equiv$	$A_2-a^2{A}_{12}.$
[ EFE, §21, Eq. (105) ] See also the note immediately following §21, Eq. (127)

The upper boundary of the horn-shaped region is obtained by setting $x=-1$. That is, it is associated with coordinate pairs $(b/a,c/a)$ for which,

$1-2C+1$	$=$	$0$
$\Rightarrow C$	$=$	$+1$
$\Rightarrow \left[\frac{ab{B}_{12}}{c^2{A}_3-a^2{b}^2{A}_{12}}\right]$	$=$	$+1$
$\Rightarrow ab{B}_{12}$	$=$	$c^2{A}_3-a^2{b}^2{A}_{12}$
$\Rightarrow c^2{A}_3$	$=$	$ab[A_2-a^2{A}_{12}]+a^2{b}^2{A}_{12}$
	$=$	$ab{A}_2+b{a}^2{A}_{12}(b-a)$
	$=$	$ab{A}_2+b{a}^2(a-b)\left[\frac{A_1-A_2}{a^2-b^2}\right]$
$\Rightarrow \left[\frac{c^2}{ab}\right]A_3$	$=$	$A_2+a\left[\frac{A_1-A_2}{a+b}\right]$
$\Rightarrow \left[\frac{c^2(a+b)}{ab}\right]A_3$	$=$	$a{A}_1+b{A}_2.$

Now, from the expressions for A₁, A₂, and A₃, we can furthermore write,

$c^2(a+b)A_3$	$=$	$a^{2}b{A}_1+a{b}^2[2-(A_1+A_3)]$
	$=$	$a^{2}b{A}_1+2a{b}^2-a{b}^2{A}_1-a{b}^2{A}_3$
$\Rightarrow c^2(a+b)A_3+a{b}^2{A}_3$	$=$	$2a{b}^2+a^{2}b{A}_1-a{b}^2{A}_1$
$\Rightarrow \frac{1}{ab}[c^2(a+b)+a{b}^2]A_3$	$=$	$2b+(a-b)A_1$

Similarly, the lower boundary is obtained by setting $x=+1$, that is, it is associated with coordinate pairs $(b/a,c/a)$ for which,

$\Rightarrow C$

$=$

$-1.$

See Also

• A library of ellipsoidal harmonics may be available from George Dassios
• 📚 Ou (2006)
• S. Ou, J. E. Tohline, & P. M. Motl (2007, ApJ, Vol. 665, pp. 1074 - 1083) titled, Further Evidence for an Elliptical Instability in Rotating Fluid Bars and Ellipsoidal Stars.
• N. R. Lebovitz (1974, ApJ, 190, 121 - 130) titled, The Fission Theory of Binary Stars. II. Stability to Third-Harmonics Disturbances
• N. R. Lebovitz (1989a, Geophysical and Astrophysical Fluid Dynamics, Vol. 46, Issue 4, pp. 221 - 243) titled, The Stability Equations for Rotating, Inviscid Fluids:   Galerkin Methods and Orthogonal Bases
• N. R. Lebovitz (1989b, Geophysical and Astrophysical Fluid Dynamics, Vol. 47, Issue 1, pp. 225 - 236) titled, Lagrangian Perturbations of Riemann Ellipsoids