Difference between revisions of "User:Tohline/ThreeDimensionalConfigurations/FerrersPotential"
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In an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids|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 [[User:Tohline/Appendix/References#EFE|EFE]]. In the latter part of the nineteenth-century, N. M. Ferrers (1877) 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, | |||
==Background== | ==Background== | ||
Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosines|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[User:Tohline/Appendix/Ramblings/T3Integrals|T3]] (concentric oblate-spheroidal) and [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]]. | Building on our [[User:Tohline/Appendix/Ramblings/DirectionCosines|general introduction to ''Direction Cosines'']] in the context of orthogonal curvilinear coordinate systems, and on our previous development of [[User:Tohline/Appendix/Ramblings/T3Integrals|T3]] (concentric oblate-spheroidal) and [[User:Tohline/Appendix/Ramblings/EllipticCylinderCoordinates#T5_Coordinates|T5]] (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232|desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids]]. | ||
=See Also= | =See Also= | ||
Revision as of 02:03, 11 December 2020
Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids
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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) 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,
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.
See Also
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