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.

Understanding EFE's 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 the simplest, 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 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 a $f$, it is accompanied by an extra "length-squared" factor, such as $a^2$.