Appendix/Ramblings/ForCohlHoward

Discussions With Howard Cohl

These discussions began in late 2021, when Howard asked if I would be interested in working with him on establishing a better understanding of the stability of Riemann S-Type Ellipsoids. This discussion relates directly to our study of the work by Lebovitz & Lifschitz (1996).

Understanding the Dimensionality of EFE Index Symbols

Howard put together a Mathematica script intended to provide — for any specification of the semi-axis length triplet $(a,b,c)$ — very high-precision, numerical evaluations of any of the index symbols, $A_{ijk\dots }$ and $B_{ijk\dots }$ as defined by Eqs. (103 - 104) in §21 of [EFE]. Originally I suggested that, without loss of generality, he should only need to specify the pair of length ratios, $(1,b/a,c/a)$. In response, Howard pointed out that evaluation of all but a few of the lowest-numbered index symbols — as defined by [EFE] — does explicitly depend on specification of (various powers of) the semi-axis length, $a$.

Joel's response:  Howard is correct! He should leave the explicit dependence of $a$ — to various powers — in his Mathematica notebook's determination of all the EFE index symbols.

Instead, what we should expect is that the evaluation of various physically relevant parameters will produce results that are independent of the semi-axis length, $a$; these evaluations should involve combining various index symbols in such a way that the dependence on $a$ disappears. Consider, for example, our accompanying discussion (click to see relevant expressions) of the virial-equilibrium-based determination of the frequency ratio, $f\equiv \zeta /{\mathrm{\Omega }}_f$, in equilibrium S-Type Riemann Ellipsoids. Although most of the required index symbols, $A_1,A_2,A_3$ and $B_{12}$, are dimensionless parameters, the index symbol $A_{12}$ has the unit of inverse-length-squared. Notice, however, that when $A_{12}$ appears along with any of these other dimensionless parameters in the definition of $f$, it is accompanied by an extra "length-squared" factor, such as $a^2$. Hence, although I strongly agree that Howard should continue to include various powers of $a$ (etc.) in his Mathematica notebook expressions, I suspect that, without loss of generality, in the end we will always be able to set $a=1$ and only need to specify the pair of length ratios, $(1,b/a,c/a)$.

Evaluation of Index Symbols

Three Lowest-Order Expressions

In our accompanying derivation of expressions for the three lowest-order index symbols $A_i$, we have used subscripts $(\ell ,m,s)$ instead of $(1,2,3)$ in order to identify which associated semi-axis length is (largest, medium-length, smallest). We have derived the following expressions:

$\frac{A_{\ell }}{a_{\ell }{a}_m{a}_s}$ $=$ $\frac{2}{a_{\ell }^3{p}^2{\sin }^3\alpha }[F(\alpha ,p)-E(\alpha ,p)];$ $\frac{A_m}{a_{\ell }{a}_m{a}_s}$ $=$ $\frac{2}{a_{\ell }^3}\left[\frac{E(\alpha ,p)-(1-p^2)F(\alpha ,p)-(a_s/a_m)p^2\sin \alpha }{p^2(1-p^2){\sin }^3\alpha }\right];$ $\frac{A_s}{a_{\ell }{a}_m{a}_s}$ $=$ $\frac{2}{a_{\ell }^3}\left[\frac{(a_m/a_s)\sin \alpha -E(\alpha ,p)}{(1-p^2){\sin }^3\alpha }\right].$

The corresponding expressions that appear in Howard's Mathematica notebook are:

$A_1$ $=$ $\frac{2a_2{a}_3}{a_1^{2}m{\sin }^3(\varphi )}[\mathrm{E}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{F}[\varphi ,m]-\mathrm{E}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{E}[\varphi ,m]];$ $A_2$ $=$ $\frac{2a_2{a}_3}{a_1^{2}m(1-m){\sin }^3(\varphi )}[\mathrm{E}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{E}[\varphi ,m]-{\cos }^2\theta \cdot \mathrm{E}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{F}[\varphi ,m]-\frac{a_3}{a_2}\cdot {\sin }^2\theta \sin \varphi ];$ $A_3$ $=$ $\frac{2a_2{a}_3}{a_1^2(1-m){\sin }^3(\varphi )}[\frac{a_2}{a_3}\cdot \sin (\varphi )-\mathrm{E}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{E}[\varphi ,m]].$

With a little study it should be clear that our derived expressions for $A_i$ precisely match Howard's Mathematica-notebook expressions when $\ell =1$, $m=2$, and $s=3$, that is, in all cases for which $a_1>a_2>a_3$. But there will be models to consider (for example, in the uppermost region of the so-called "horn-shaped" region for S-Type Riemann Ellipsoids) for which $a_1>a_3>a_2$, in which case care must be taken in assigning the proper expressions to $A_2$ and $A_3$. Similarly note that most of the Riemann models of Type I, II, and III — see, for example, Figure 16 (p. 161) in Chapter 7 of [EFE] — have either $a_2>a_1$ or $a_3>a_1$.

Determination of Higher-Order Expressions

Howard's Mathematica notebook performs brute-force integrations to evaluate various higher-order index-symbol expressions. Why doesn't he instead use recurrence relations, which point back to the elliptic-integral-based expressions for $A_1,A_2,A_3$? Specifically …

Index-Symbol Recurrence Relations
$B_{ijk\ell \dots }$	$=$	$A_{jkl\dots }-a_i^2{A}_{ijkl\dots }$
[EFE], §21, p. 54, Eq. (105)
$a_i^2{A}_{ikl\dots }-a_j^2{A}_{jkl\dots }$	$=$	$+(a_i^2-a_j^2)B_{ijk\ell \dots }$
[EFE], §21, p. 54, Eq. (106)
$A_{ikl\dots }-A_{jkl\dots }$	$=$	$-(a_i^2-a_j^2)A_{ijk\ell \dots }$
[EFE], §21, p. 54, Eq. (107)

For example, setting $i=1$ and $j=2$ in the third of these expressions gives,

$A_1-A_2$	$=$	$-(a_1^2-a_2^2)A_{12}$
$\Rightarrow A_{12}$	$=$	$\frac{A_2-A_1}{(a_1^2-a_2^2)}$
$\Rightarrow a_1^2{A}_{12}$	$=$	$\frac{A_2-A_1}{(1-a_2^2/a_1^2)};$

and, from the first of the relations,

$B_{12}$	$=$	$A_2-a_1^2{A}_{12}$
	$=$	$A_2-a_1^2\left[\frac{A_2-A_1}{(a_1^2-a_2^2)}\right]$
	$=$	$\frac{1}{(a_1^2-a_2^2)}[(a_1^2-a_2^2)A_2-a_1^2(A_2-A_1)]$
	$=$	$\frac{1}{(a_1^2-a_2^2)}[a_1^2{A}_1-a_2^2{A}_2]$
	$=$	$\frac{1}{(1-a_2^2/a_1^2)}[A_1-\frac{a_2^2}{a_1^2}\cdot A_2].$

Also, consider using the set of relations labeled "LEMMA 7" on p. 54 of [EFE].

Example Test Evaluations

Some of Howard's 20-digit-precision evaluations of various index symbols have been recorded, for comparison with our separate lower-precision evaluations, as follows:

• Values of $(A_1,A_2,A_3)$ are recorded for a model with $(a_1,a_2,a_3)=(1,0.9,0.641)$ in the table titled, TEST (part 1), near the top of our chapter on Riemann S-Type ellipsoids.
• Values of $(A_{12},B_{12})$ are recorded for a model with $(a_1,a_2,a_3)=(1,0.9,0.641)$ in the table titled, TEST (part 2) in our chapter on Riemann S-Type ellipsoids.

Figures from January 2022