Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids

In an accompanying chapter titled, Properties of Homogeneous Ellipsoids (1), we have shown how analytic expressions may be derived for the gravitational potential inside of uniform-density ellipsoids. In that discussion, we largely followed the derivations of EFE. In the latter part of the nineteenth-century, N. M. Ferrers, (1877, Quarterly Journal of Pure and Applied Mathematics, 14, 1 - 22) showed that very similar analytic expressions can be derived for ellipsoids that have certain, specific inhomogeneous mass distributions. Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,

SUMMARY — copied from accompanying, Trial #2 Discussion

After studying the relevant sections of both EFE and BT87 — this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription. As is discussed in a separate chapter, the potential that it generates is sometimes referred to as a Ferrers potential, for the exponent, n = 1.

In our accompanying discussion we find that,

<math>~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)} </math>

<math>~ \frac{1}{2} I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) ~+ \biggl( A_{12} x^2y^2 + A_{13} x^2z^2 + A_{23} y^2z^2\biggr) ~+ \frac{1}{6} \biggl(3A_{11}x^4 + 3A_{22}y^4 + 3A_{33}z^4 \biggr) \, , </math>

where,

for <math>~i \ne j</math>
<math>~A_{ij}</math>	<math>~\equiv</math>	<math>~-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
[ EFE, §21, Eq. (107) ]

for <math>~i = j</math>
<math>~2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>	<math>~=</math>	<math>~\frac{2}{a_i} </math>
[ EFE, §21, Eq. (109) ]

More specifically, in the three cases where the indices, <math>~i=j</math>,

<math>~3A_{11}</math>	<math>~=</math>	<math>~ \frac{2}{a_1^2} - (A_{12} + A_{13}) \, , </math>
<math>~3A_{22}</math>	<math>~=</math>	<math>~ \frac{2}{a_2^2} - (A_{21} + A_{23}) \, , </math>
<math>~3A_{33}</math>	<math>~=</math>	<math>~ \frac{2}{a_3^2} - (A_{31} + A_{32}) \, . </math>

Background

Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and T5 (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids.

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Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids

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