User:Tohline/ThreeDimensionalConfigurations/FerrersPotential - Revision history

Tohline: /* See Also */

2020-12-11T03:53:16Z

Tohline: /* Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids */

2020-12-11T03:42:25Z

Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids

← Older revision		Revision as of 03:42, 11 December 2020
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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, [https://babel.hathitrust.org/cgi/pt?id=uc1.$b417536&view=1up&seq=15 N. M. Ferrers, (1877, Quarterly Journal of Pure and Applied Mathematics, 14, 1 - 22)] showed that very similar analytic expressions can be derived for ellipsoids that have certain, specific inhomogeneous mass distributions. Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,		In an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids\|accompanying chapter]] titled, ''Properties of Homogeneous Ellipsoids (1),'' we have shown how analytic expressions may be derived for the gravitational potential inside of uniform-density ellipsoids. In that discussion, we largely followed the derivations of [[User:Tohline/Appendix/References#EFE\|EFE]]. In the latter part of the nineteenth-century, [https://babel.hathitrust.org/cgi/pt?id=uc1.$b417536&view=1up&seq=15 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,

			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\rho</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ \rho_c \biggl[ 1 - \biggl( \frac{x^2}{a_1^2} + \frac{y^2}{a_2^2} + \frac{z^2}{a_3^2}\biggr) \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

	<table border="1" align="center" width="90%" cellpadding="10">		<table border="1" align="center" width="90%" cellpadding="10">
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	Q.E.D.		Q.E.D.


	=See Also=		=See Also=

Tohline at 03:39, 11 December 2020

2020-12-11T03:39:38Z

Show changes

Tohline: /* See Also */

2020-12-11T03:30:51Z

Tohline at 03:28, 11 December 2020

2020-12-11T03:28:26Z

← Older revision		Revision as of 03:28, 11 December 2020
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	=See Also=		=See Also=
			<ul>
			<li>Our [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Speculation6\|''Speculation6'' ]] effort to develop a "Concentric Ellipsoidal (T6) Coordinate System."</li>
			</ul>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids */

2020-12-11T03:09:25Z

Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids

← Older revision		Revision as of 03:09, 11 December 2020
Line 7:		Line 7:

	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, [https://babel.hathitrust.org/cgi/pt?id=uc1.$b417536&view=1up&seq=15 N. M. Ferrers, (1877, Quarterly Journal of Pure and Applied Mathematics, 14, 1 - 22)] showed that very similar analytic expressions can be derived for ellipsoids that have certain, specific inhomogeneous mass distributions. Here we specifically discuss the case of configurations that exhibit concentric ellipsoidal iso-density surfaces of the form,		In an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids\|accompanying chapter]] titled, ''Properties of Homogeneous Ellipsoids (1),'' we have shown how analytic expressions may be derived for the gravitational potential inside of uniform-density ellipsoids. In that discussion, we largely followed the derivations of [[User:Tohline/Appendix/References#EFE\|EFE]]. In the latter part of the nineteenth-century, [https://babel.hathitrust.org/cgi/pt?id=uc1.$b417536&view=1up&seq=15 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,


			<table border="1" align="center" width="90%" cellpadding="10">
			<tr><td align="center">SUMMARY — copied from [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Trial_.232\|accompanying, ''Trial #2'' Discussion]]</td></tr>

			<tr><td align="left">
			After studying the relevant sections of both [[User:Tohline/Appendix/References#EFE\|EFE]] and [[User:Tohline/Appendix/References#BT87\|BT87]] — this is an example of a heterogeneous density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials\| separate chapter]], the potential that it generates is sometimes referred to as a ''Ferrers'' potential, for the exponent, n = 1.

			In our [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#GravFor1\|accompanying discussion]] we find that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{ \Phi_\mathrm{grav}(\bold{x})}{(-\pi G\rho_c)} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<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>
			</td>
			</tr>
			</table>

			where,
			<table border="0" align="center" cellpadding="10" width="80%">
			<tr>
			<td align="center" width="50%">

			<table border="0" cellpadding="5" align="center">
			<tr><td align="center" colspan="3">for <math>~i \ne j</math></td></tr>
			<tr>
			<td align="right">
			<math>~A_{ij}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~-\frac{A_i-A_j}{(a_i^2 - a_j^2)} </math>
			</td>
			</tr>
			<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§21, Eq. (107)</font> ]</td></tr>
			</table>

			</td>
			<td align="center" width="50%">

			<table border="0" cellpadding="5" align="center">
			<tr><td align="center" colspan="3">for <math>~i = j</math></td></tr>
			<tr>
			<td align="right">
			<math>~2A_{ii} + \sum_{\ell = 1}^3 A_{i\ell}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{2}{a_i} </math>
			</td>
			</tr>
			<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE\|EFE]], <font color="#00CC00">§21, Eq. (109)</font> ]</td></tr>
			</table>

			</td>
			</tr>
			</table>
			More specifically, in the three cases where the indices, <math>~i=j</math>,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~3A_{11}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2}{a_1^2} - (A_{12} + A_{13}) \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~3A_{22}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2}{a_2^2} - (A_{21} + A_{23}) \, ,
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~3A_{33}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2}{a_3^2} - (A_{31} + A_{32}) \, .
			</math>
			</td>
			</tr>
			</table>

			</td></tr></table>

Tohline: /* Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids */

2020-12-11T02:07:44Z

Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids

← Older revision		Revision as of 02:07, 11 December 2020
Line 6:		Line 6:


	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,		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, [https://babel.hathitrust.org/cgi/pt?id=uc1.$b417536&view=1up&seq=15 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,

Tohline: /* Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids */

2020-12-11T02:03:20Z

Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids

← Older revision		Revision as of 02:03, 11 December 2020
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	{{LSU_HBook_header}}		{{LSU_HBook_header}}


			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=

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=Ferrers (1877) Gravitational Potential for Inhomogeneous Ellipsoids=

{{LSU_HBook_header}}

==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]].

=See Also=

{{LSU_HBook_footer}}

← Older revision		Revision as of 03:53, 11 December 2020
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	<li>Our [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Speculation6\|''Speculation6'' ]] effort to develop a "Concentric Ellipsoidal (T6) Coordinate System."</li>		<li>Our [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Speculation6\|''Speculation6'' ]] effort to develop a "Concentric Ellipsoidal (T6) Coordinate System."</li>
	<li>[[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Challenges Constructing Ellipsoidal-Like Configurations]]</li>		<li>[[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Challenges Constructing Ellipsoidal-Like Configurations]]</li>
			<li>[[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Properties_of_Homogeneous_Ellipsoids_.281.29\|Properties of Homogeneous Ellipsoids (1)]] — The Gravitational Potential (A<sub>i</sub> Coefficients)</li>
	</ul>		</ul>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

← Older revision		Revision as of 03:30, 11 December 2020
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	<ul>		<ul>
	<li>Our [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Speculation6\|''Speculation6'' ]] effort to develop a "Concentric Ellipsoidal (T6) Coordinate System."</li>		<li>Our [[User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates#Speculation6\|''Speculation6'' ]] effort to develop a "Concentric Ellipsoidal (T6) Coordinate System."</li>
			<li>[[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations\|Challenges Constructing Ellipsoidal-Like Configurations]]</li>
	</ul>		</ul>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}