ThreeDimensionalConfigurations/HomogeneousEllipsoids - Revision history

Jet53man: /* Evaluating Aℓℓ */

2022-03-25T19:29:57Z

Evaluating Aℓℓ

← Older revision		Revision as of 15:29, 25 March 2022
Line 1,730:		Line 1,730:

	==Evaluating A<sub>ℓℓ</sub>==		==Evaluating A<sub>ℓℓ</sub>==
	~~First, let's focus on~~ the ~~coefficient associated with the longest axis~~ <math>~~~(i =~~ \ell)</math>:
			In the case of <math>A_{\ell\ell}</math>, we have,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{A_\ell}{a_\ell a_m a_s}</math>		<math>~\frac{A_{\ell\ell}}{a_\ell a_m a_s}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,742:		Line 1,743:
	<td align="left">		<td align="left">
	<math>		<math>
	~\int_0^\infty \biggl[ (a_\ell^2 + u)^3(a_m^2 + u)(a_s^2 + u) \biggr]^{-1 / 2} du		~\int_0^\infty \biggl[ (a_\ell^2 + u)^5(a_m^2 + u)(a_s^2 + u) \biggr]^{-1 / 2} du \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	Changing the integration variable to <math>~x \equiv -u</math>, we obtain a definite integral expression that appears as equation (3.~~133~~.1) in [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ I. W. Gradshteyn & I. M. Ryzhik (2007; 7<sup>th</sup> Edition)], ''Table of Integrals, Series, and Products'' — hereafter, GR7<sup>th</sup> — namely,		Changing the integration variable to <math>~x \equiv -u</math>, we obtain a definite integral expression that appears as equation (3.134.1) in [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ I. W. Gradshteyn & I. M. Ryzhik (2007; 7<sup>th</sup> Edition)], ''Table of Integrals, Series, and Products'' — hereafter, GR7<sup>th</sup> — namely,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{A_\ell}{a_\ell a_m a_s}</math>		<math>~\frac{A_{\ell\ell}}{a_\ell a_m a_s}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 1,759:		Line 1,760:
	<td align="left">		<td align="left">
	<math>		<math>
	~\int_{-\infty}^0 \biggl[ (a_\ell^2 - x)^3(a_m^2 - x)(a_s^2 - x) \biggr]^{-1 / 2} dx		~\int_{-\infty}^0 \biggl[ (a_\ell^2 - x)^5(a_m^2 - x)(a_s^2 - x) \biggr]^{-1 / 2} dx
	</math>		</math>
	</td>		</td>
Line 1,770:		Line 1,771:
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>		<math>
	~\frac{2}{(a_\ell^2 - a_m^2~~) \sqrt{~~a_\ell^2 - a_s^2}}		\frac{2}{3(a_\ell - a_m)^2 (a_\ell - a_s)^{3/2}} \biggl[
	\biggl[ F(\alpha, p) - E(\alpha, p) \biggr]		(3a_\ell - a_m - 2a_s) F(\alpha, p) - 2(2a_\ell - a_m - a_s) E(\alpha, p)
			\biggr]
			+ \frac{2}{3(a_\ell - a_s)(a_\ell -a_m)}\biggl[ \frac{a_s a_m}{a_\ell^3} \biggr]^{1 / 2}
			\, ,
	</math>		</math>
	</td>		</td>
Line 1,781:		Line 1,785:
	</tr>		</tr>
	<tr>		<tr>
	<td align="center" colspan="4">[https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], p. ~~255~~, Eq. (3.~~133~~.1)</td>		<td align="center" colspan="4">[https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], p. 257, Eq. (3.134.1)</td>
	</tr>		</tr>
	</table>		</table>
Line 1,809:		Line 1,813:
	</tr>		</tr>
	</table>		</table>
	and where, <math>E(\alpha, p)</math> and <math>F(\alpha, p)</math> are [https://dlmf.nist.gov/19.2#ii Legendre incomplete elliptic integrals of the first and second kind], respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,		and where, <math>E(\alpha, p)</math> and <math>F(\alpha, p)</math> are [https://dlmf.nist.gov/19.2#ii Legendre incomplete elliptic integrals of the first and second kind], respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.)

			<!--
			Recognizing that,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,847:		Line 1,854:
	</tr>		</tr>
	</table>		</table>
			-->

	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

Jet53man at 19:15, 25 March 2022

2022-03-25T19:15:29Z

Show changes

Jet53man: /* Evaluating Aℓℓ */

2022-03-25T19:11:18Z

Evaluating Aℓℓ

← Older revision		Revision as of 15:11, 25 March 2022
Line 867:		Line 867:
	This matches the expression that we have provided for <math>~A_1</math>, [[#Triaxial_Configurations\|above in the context of triaxial configurations]].		This matches the expression that we have provided for <math>~A_1</math>, [[#Triaxial_Configurations\|above in the context of triaxial configurations]].

	=Evaluating A<sub>ℓℓ</sub>=		==Evaluating A<sub>ℓℓ</sub>==
	First, let's focus on the coefficient associated with the longest axis <math>~(i = \ell)</math>:		First, let's focus on the coefficient associated with the longest axis <math>~(i = \ell)</math>:
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,849:		Line 1,849:

	==Derivation of Selected 2<sup>nd</sup>-Order Index Symbols==		==Derivation of Selected 2<sup>nd</sup>-Order Index Symbols==


	=Work In Progress=		=Work In Progress=

Jet53man: /* Work In Progress */

2022-03-25T19:09:33Z

Work In Progress

← Older revision		Revision as of 15:09, 25 March 2022
Line 1,847:		Line 1,847:

	{{ SGFworkInProgress }}		{{ SGFworkInProgress }}

			==Derivation of Selected 2<sup>nd</sup>-Order Index Symbols==


	=Work In Progress=		=Work In Progress=

Jet53man: /* Evaluating Aℓ */

2022-03-25T19:05:59Z

Evaluating Aℓ

@@ Line 866: / Line 866: @@
 </table>
 This matches the expression that we have provided for <math>~A_1</math>, [[#Triaxial_Configurations|above in the context of triaxial configurations]].
 ==Evaluating A<sub>m</sub>==

Jet53man: /* Evaluating Aℓ */

2022-03-15T17:26:07Z

Evaluating Aℓ

← Older revision		Revision as of 13:26, 15 March 2022
Line 827:		Line 827:
	</tr>		</tr>
	</table>		</table>
	and where, <math>E(\alpha, p)</math> and <math>F(\alpha, p)</math> are [https://~~functions~~.~~wolfram~~.~~com/EllipticIntegrals~~/ Legendre incomplete elliptic integrals of the first and second kind], respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,		and where, <math>E(\alpha, p)</math> and <math>F(\alpha, p)</math> are [https://dlmf.nist.gov/19.2#ii Legendre incomplete elliptic integrals of the first and second kind], respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Evaluating Aℓ */

2022-03-15T17:21:30Z

Evaluating Aℓ

← Older revision		Revision as of 13:21, 15 March 2022
Line 827:		Line 827:
	</tr>		</tr>
	</table>		</table>
	and where, <math>E(\alpha, p)</math> and <math>F(\alpha, p)</math> are Legendre incomplete elliptic integrals of the first and second kind, respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,		and where, <math>E(\alpha, p)</math> and <math>F(\alpha, p)</math> are [https://functions.wolfram.com/EllipticIntegrals/ Legendre incomplete elliptic integrals of the first and second kind], respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man: /* Evaluating Aℓ */

2022-03-15T17:17:33Z

Evaluating Aℓ

← Older revision		Revision as of 13:17, 15 March 2022
Line 827:		Line 827:
	</tr>		</tr>
	</table>		</table>
	and where, <math>~E(\alpha, p)</math> and <math>~F(\alpha, p)</math> are elliptic integrals of the first and second kind, respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,		and where, <math>E(\alpha, p)</math> and <math>F(\alpha, p)</math> are Legendre incomplete elliptic integrals of the first and second kind, respectively. (Note that in the notation convention adopted by [https://www.academia.edu/36550954/I_S_Gradshteyn_and_I_M_Ryzhik_Table_of_integrals_series_and_products_Academic_Press_2007_ GR7<sup>th</sup>], the order of the argument list, <math>~(\alpha, p)</math>, is flipped relative to the convention that we have adopted [[#Evaluation_of_Coefficients\|above]] and elsewhere throughout our online, MediaWiki-based chapters.) Recognizing that,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 866:		Line 866:
	</table>		</table>
	This matches the expression that we have provided for <math>~A_1</math>, [[#Triaxial_Configurations\|above in the context of triaxial configurations]].		This matches the expression that we have provided for <math>~A_1</math>, [[#Triaxial_Configurations\|above in the context of triaxial configurations]].


	==Evaluating A<sub>m</sub>==		==Evaluating A<sub>m</sub>==

Jet53man: /* Prolate Spheroids ~(a_1 > a_2 = a_3) */

2021-11-05T22:07:47Z

Prolate Spheroids ~(a_1 > a_2 = a_3)

← Older revision		Revision as of 18:07, 5 November 2021
Line 483:		Line 483:
	</math>		</math>
	</div>		</div>


			<font color="red">NOTE:</font>  If, instead, the longest (and, in this case, symmetry) axis of the prolate mass distribution is aligned with the <math>z</math>-axis — in which case, <math>a_1 = a_2 < a_3</math> — then, <math>e = (1 - a_1^2/a_3^2)^{1 / 2}</math> and the mathematical expressions for the <math>A_i</math> coefficients must be altered; they are essentially "swapped." This modified set of coefficient expressions can be found in a [[Aps/MaclaurinSpheroidFreeFall#Prolate_Spheroids\|parallel discussion]] of the potential inside and on the surface of prolate-spheroidal mass distributions, as well as in the second column of Table 2-1 (p. 57) of [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>].

	==Example Evaluations==		==Example Evaluations==

Jet53man: /* Gravitational Potential */

2021-11-05T02:33:16Z

Gravitational Potential

← Older revision		Revision as of 22:33, 4 November 2021
Line 20:		Line 20:
	</div>		</div>

	As has been explicitly demonstrated in Chapter 3 of [[Appendix/References#EFE\|EFE]] and summarized in Table 2-2 (p. 57) of [[Appendix/References#BT87\|BT87]], for an homogeneous ellipsoid this volume integral can be evaluated analytically in closed form. Specifically, at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>~(x,y,z) = (a_1,a_2,a_3)</math>,		As has been explicitly demonstrated in Chapter 3 of [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] and summarized in Table 2-2 (p. 57) of [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], for an homogeneous ellipsoid this volume integral can be evaluated analytically in closed form. Specifically, at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>~(x,y,z) = (a_1,a_2,a_3)</math>,

	<div align="center">		<div align="center">