AxisymmetricConfigurations/PoissonEq - Revision history

Jet53man: /* Setup */

2023-05-09T22:35:53Z

Setup

← Older revision		Revision as of 18:35, 9 May 2023
Line 3,438:		Line 3,438:

	<table border="1" align="center" width="90%" cellpadding="8"><tr><td align="left">		<table border="1" align="center" width="90%" cellpadding="8"><tr><td align="left">
	'''~~Prolate~~ Spheroidal Coordinates:'''		'''Oblate Spheroidal Coordinates:'''

	According to § 245 of [https://books.google.com/books?hl=en&lr=&id=b2s3AAAAIAAJ&oi=fnd&pg=PA10&dq=the+theory+of+spherical+and+ellipsoidal+harmonics+pdf&ots=oi0kxX8IL9&sig=R65gUh5oWZf2924f3HqL5GQDoUY#v=onepage&q=the%20theory%20of%20spherical%20and%20ellipsoidal%20harmonics%20pdf&f=false E. W. Hobson (1931)],

			According to the ''Erratum'' of …
			<div align="center">{{ CTRS00figure }}</div>
			the "expansion formula" that is relevant to oblate spheroidal coordinates is given correctly on p. 218 (Eq. 41) of MacRobert (1947). It is,

	<table border="0" align="center">		<table border="0" align="center">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~P_n~~^~~m(i\xi)~~</math>		<math>\frac{1}{\|\vec{x} - \vec{x}^{~'}\|} </math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 3,452:		Line 3,453:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>(-i)^{-(n+m)} ~~p_n~~^m(\xi) \, ,</math>		<math>
			\frac{i}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
			~ \sum_{m=0}^{n}~ \epsilon_m (-1)^{m}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
			\cos[m(\phi - \phi')]P_n^m(\cos\theta') \cdot
			P_n^m(\cos\theta)\biggl\{ \begin{array}{ll}
			P_n^m(i\sinh\sigma') Q_n^m(i\sinh\sigma)\, ; ~~~ \sinh\sigma > \sinh\sigma' \\
			P_n^m(i\sinh\sigma) Q_n^m(i\sinh\sigma')\, ; ~~~ \sinh\sigma' > \sinh\sigma \\
			\end{array}
			</math>
	</td>		</td>
	</tr>		</tr>

	</table>		</table>

Jet53man: /* Setup */

2023-05-07T22:21:28Z

Setup

@@ Line 3,435: / Line 3,435: @@
 </tr>
 </table>
 ===Expression for the Potential due to General 3D Mass Distribution===

Jet53man: /* Axisymmetric Mass Distribution */

2023-05-04T23:25:22Z

Axisymmetric Mass Distribution

← Older revision		Revision as of 19:25, 4 May 2023
Line 3,577:		Line 3,577:
	===Axisymmetric Mass Distribution===		===Axisymmetric Mass Distribution===

	For an axisymmetric mass distribution, we need only consider the <math>m = 0</math> contribution. ~~That is~~,		For an axisymmetric mass distribution, we need only consider the <math>m = 0</math> contribution. Also switching the integer index notation from <math>n</math> to <math>\ell</math> gives,

	<table border="0" align="center">		<table border="0" align="center">
Line 3,591:		Line 3,591:
	<math>		<math>
	-2a_0^2 G		-2a_0^2 G
	\sum_{n=0}^{\infty}~ (2n+1)		\sum_{\ell=0}^{\infty}~ (2\ell+1)
	~~q_n~~(\xi)~~P_n~~(\eta) \cdot		q_\ell(\xi)P_\ell(\eta) \cdot
	\int_0^\xi d\xi' \int_{-1}^1 d\eta'		\int_0^\xi d\xi' \int_{-1}^1 d\eta'
	\biggl\{		\biggl\{
	~~p_n~~(\xi')~~P_n~~(\eta')		p_\ell(\xi')P_\ell(\eta')
	\biggr\}		\biggr\}
	[(\xi')^2 + (\eta')^2] \rho(\xi', \eta')		[(\xi')^2 + (\eta')^2] \rho(\xi', \eta')
Line 3,613:		Line 3,613:
	<math>		<math>
	-2a_0^2 G		-2a_0^2 G
	~ \sum_{n=0}^{\infty}~ (2n+1)		~ \sum_{\ell=0}^{\infty}~ (2\ell+1)
	~~p_n~~(\xi)~~P_n~~(\eta) \cdot \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'		p_\ell(\xi)P_\ell(\eta) \cdot \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'
	\biggl\{		\biggl\{
	~~q_n~~(\xi')~~P_n~~(\eta')		q_\ell(\xi')P_\ell(\eta')
	\biggr\}		\biggr\}
	[(\xi')^2 + (\eta')^2] \rho(\xi', \eta')		[(\xi')^2 + (\eta')^2] \rho(\xi', \eta')
Line 3,634:		Line 3,634:
	<math>		<math>
	-		-
	\sum_{n=0}^{\infty}~ (2n+1)a_0^2		\sum_{\ell=0}^{\infty}~ (2\ell+1)a_0^2
	~~q_n~~(\xi)~~P_n~~(\eta) \cdot		q_\ell(\xi)P_\ell(\eta) \cdot
	\int_0^\xi d\xi' \int_{-1}^1 d\eta'		\int_0^\xi d\xi' \int_{-1}^1 d\eta'
	\biggl\{		\biggl\{
	~~p_n~~(\xi')~~P_n~~(\eta')		p_\ell(\xi')P_\ell(\eta')
	\biggr\}		\biggr\}
	[(\xi')^2 + (\eta')^2] \biggl[4\pi G \rho(\xi', \eta') \biggr]		[(\xi')^2 + (\eta')^2] \biggl[4\pi G \rho(\xi', \eta') \biggr]
Line 3,655:		Line 3,655:
	<math>		<math>
	-		-
	~ \sum_{n=0}^{\infty}~ (2n+1)a_0^2		~ \sum_{\ell=0}^{\infty}~ (2\ell+1)a_0^2
	~~p_n~~(\xi)~~P_n~~(\eta) \cdot \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'		p_\ell(\xi)P_\ell(\eta) \cdot \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'
	\biggl\{		\biggl\{
	~~q_n~~(\xi')~~P_n~~(\eta')		q_\ell(\xi')P_\ell(\eta')
	\biggr\}		\biggr\}
	[(\xi')^2 + (\eta')^2]		[(\xi')^2 + (\eta')^2]
	\biggl[4\pi G \rho(\xi', \eta') \biggr]		\biggl[4\pi G \rho(\xi', \eta') \biggr] \, ,
	</math>		</math>
			</td>
			</tr>

			<tr>
			<td align="center" colspan="3">
			{{ Bardeen71 }}, p. 429, Eq. (15)
			</td>
			</tr>
			</table>
			which matches Eq. (15) of {{ Bardeen71 }} except, again, our expression gives a value for the potential that is a factor of two larger. Finally, if we only consider even values of the index, <math>\ell</math> — in which case we make the replacement, <math>\ell \rightarrow 2n</math> — the expression for the potential becomes,
			<table border="0" align="center">

			<tr>
			<td align="right">
			<math>\Phi(\xi, \eta)</math>
			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			-
			\sum_{n=0}^{\infty}~ (4n+1)a_0^2
			q_{2n}(\xi)P_{2n}(\eta) \cdot
			\int_0^\xi d\xi' \int_{-1}^1 d\eta'
			\biggl\{
			p_{2n}(\xi')P_{2n}(\eta')
			\biggr\}
			[(\xi')^2 + (\eta')^2] \biggl[4\pi G \rho(\xi', \eta') \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">

			</td>
			<td align="left">
			<math>
			-
			~ \sum_{\ell=0}^{\infty}~ (4n+1)a_0^2
			p_{2n}(\xi)P_{2n}(\eta) \cdot \int_\xi^\infty d\xi' \int_{-1}^1 d\eta'
			\biggl\{
			q_{2n}(\xi')P_{2n}(\eta')
			\biggr\}
			[(\xi')^2 + (\eta')^2]
			\biggl[4\pi G \rho(\xi', \eta') \biggr] \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="center" colspan="3">
			{{ HE83 }}, §A.1, p. 587, Eq. (2)
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Axisymmetric Mass Distribution */

2023-05-04T22:36:59Z

Axisymmetric Mass Distribution

@@ Line 3,580: / Line 3,580: @@
 <table border="0" align="center">
 <tr>
    <td align="right">
@@ Line 3,619: / Line 3,620: @@
 [(\xi')^2 + (\eta')^2] \rho(\xi', \eta')
 \int_0^{2\pi} d\phi'
 </math>
    </td>

Jet53man: /* Oblate Spheroidal Coordinates */

2023-05-04T22:23:47Z

Oblate Spheroidal Coordinates

@@ Line 3,177: / Line 3,177: @@
 ==Oblate Spheroidal Coordinates==
 Here we adopt the cartesian to spheroidal coordinate transformation,
 <table border="0" align="center" cellpadding="5">
@@ Line 3,435: / Line 3,436: @@
 </table>
 Hence, in spheroidal coordinates, the integral representation of the Poisson equation can be written as the sum of two terms &#8212; one in which the "radial" component of the volume integral covers the region, <math>0 < \xi' \le \xi</math>, while the other covers the region, <math>\xi < \xi' \le \infty</math> &#8212; namely,
@@ Line 3,483: / Line 3,485: @@
 <tr>
    <td align="right">
-<math>\Phi(\vec{x})</math>
+<math>\Phi(\xi, \eta, \phi)</math>
    </td>
    <td align="center">
@@ Line 3,561: / Line 3,563: @@
 \biggr\}
 [(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') \, .
 </math>
    </td>

Jet53man: /* Oblate Spheroidal Coordinates */

2023-05-04T22:05:17Z

Oblate Spheroidal Coordinates

@@ Line 3,325: / Line 3,325: @@
 <math>
 \frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
-\epsilon_m i^{m+1}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
+~ \sum_{m=0}^{n}~ \epsilon_m i^{m+1}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
 \cos[m(\phi - \phi')]P_n^m(\eta') \cdot
 P_n^m(\eta)\biggl\{ \begin{array}{ll}
@@ Line 3,351: / Line 3,351: @@
 <math>
 \frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
-\epsilon_m i^{m+1}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
+~ \sum_{m=0}^{n}~ \epsilon_m i^{m+1}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
 \cos[m(\phi - \phi')]P_n^m(\eta') \cdot
 P_n^m(\eta)\biggl\{ \begin{array}{ll}
@@ Line 3,371: / Line 3,371: @@
 <math>
 \frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
-\epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
+~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
 \cos[m(\phi - \phi')]P_n^m(\eta') \cdot
 P_n^m(\eta)\biggl\{ \begin{array}{ll}
@@ Line 3,493: / Line 3,493: @@
 \biggl\{
 \frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
-\epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
+~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
 \cos[m(\phi - \phi')]P_n^m(\eta') \cdot
 P_n^m(\eta) \cdot p_n^m(\xi') q_n^m(\xi)
@@ Line 3,514: / Line 3,514: @@
 \biggl\{
 \frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
-\epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
+~ \sum_{m=0}^{n}~ \epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
 \cos[m(\phi - \phi')]P_n^m(\eta') \cdot
 P_n^m(\eta) \cdot p_n^m(\xi) q_n^m(\xi')
 \biggr\}
 [(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') \, .

Jet53man: /* Oblate Spheroidal Coordinates */

2023-05-04T21:47:23Z

Oblate Spheroidal Coordinates

@@ Line 3,435: / Line 3,435: @@
 </table>
-Hence, in spheroidal coordinates, the integral representation of the Poisson equation becomes,
+Hence, in spheroidal coordinates, the integral representation of the Poisson equation can be written as the sum of two terms &#8212; one in which the "radial" component of the volume integral covers the region, <math>0 < \xi' \le \xi</math>, while the other covers the region, <math>\xi < \xi' \le \infty</math> &#8212; namely,
 <table border="0" align="center">
@@ Line 3,473: / Line 3,473: @@
    <td align="left">
 <math>
--a_0^3 G \int_\xi^\infty d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' \biggl\{ \frac{1}{R} \biggr\}_{\xi' > \xi} [(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi')
+-a_0^3 G \int_\xi^\infty d\xi' \int_{-1}^1 d\eta' \int_0^{2\pi} d\phi' \biggl\{ \frac{1}{R} \biggr\}_{\xi' > \xi} [(\xi')^2 + (\eta')^2] \rho(\xi', \eta', \phi') \, .
 </math>
    </td>

Jet53man: /* Oblate Spheroidal Coordinates */

2023-05-04T19:08:30Z

Oblate Spheroidal Coordinates

@@ Line 3,305: / Line 3,305: @@
    <td align="center"><math>=</math></td>
    <td align="left">
-<math>a_0^3 (\xi^2 + \eta^2)
+<math>
-d\xi~ d\eta~ d\phi
+a_0^3 (\xi^2 + \eta^2) d\xi~ d\eta~ d\phi
 </math>
    </td>
@@ Line 3,446: / Line 3,446: @@
    </td>
    <td align="left">
-<math>-G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, ,</math>
+<math>-G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^'</math>
    </td>
 </tr>

Jet53man: /* Oblate Spheroidal Coordinates */

2023-05-04T18:52:45Z

Oblate Spheroidal Coordinates

← Older revision		Revision as of 14:52, 4 May 2023
Line 3,338:		Line 3,338:
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], Ch. 10, p. 1296, Eq. (10.3.63)		[<b>[[Appendix/References#MF53\|<font color="red">MF53</font>]]</b>], Ch. 10, p. 1296, Eq. (10.3.63)
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
			\epsilon_m i^{m+1}\biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
			\cos[m(\phi - \phi')]P_n^m(\eta') \cdot
			P_n^m(\eta)\biggl\{ \begin{array}{ll}
			(-i)^{-(n+m)} p_n^m(\xi') (-i)^{n - 2m + 1} q_n^m(\xi)\, ; ~~~ \xi > \xi' \\
			(-i)^{-(n+m)} p_n^m(\xi) (-i)^{n - 2m + 1} q_n^m(\xi')\, ; ~~~ \xi' > \xi \\
			\end{array}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>=</math>
			</td>
			<td align="left">
			<math>
			\frac{2}{a_0} ~ \sum_{n=0}^{\infty}~ (2n+1)
			\epsilon_m \biggl[ \frac{(n-m)!}{(n+m)!} \biggr]^2
			\cos[m(\phi - \phi')]P_n^m(\eta') \cdot
			P_n^m(\eta)\biggl\{ \begin{array}{ll}
			p_n^m(\xi') q_n^m(\xi)\, ; ~~~ \xi > \xi' \\
			p_n^m(\xi) q_n^m(\xi')\, ; ~~~ \xi' > \xi \\
			\end{array}
			</math>
	</td>		</td>
	</tr>		</tr>
Line 3,355:		Line 3,395:
	\biggl\{ \begin{array}{ll}		\biggl\{ \begin{array}{ll}
	1~~~(m=0)\\		1~~~(m=0)\\
	2~~~(m > 0) ~~\, .~~\\		2~~~(m > 0)\\
	\end{array}		\end{array}
	</math>		</math>
Line 3,367:		Line 3,407:
	</tr>		</tr>
	</table>		</table>
			and, following {{ HE84 }}, we have made the substitutions,
	~~Hence~~, ~~in spheroidal coordinates~~, the ~~integral representation of the Poisson equation becomes~~,

	<table border="0" align="center">		<table border="0" align="center">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~\Phi~~(\~~vec{x}~~)</math>		<math>P_n^m(i\xi)</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 3,379:		Line 3,418:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>-~~G \int \frac~~{~~\rho~~(~~\vec{x}^{~'}~~)}~~{\|\vec{x}~~^~~{~'} -~~ \~~vec{x}\|} d^3x^'~~ \, ,</math>		<math>(-i)^{-(n+m)} p_n^m(\xi) \, ,</math>
	</td>		</td>
	</tr>		</tr>
	~~</table>~~
	~~where, following {{ HE84 }}, we have made the substitutions,~~

	~~<table border="0" align="center">~~
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~P_n~~^m(i\xi)</math>		<math>Q_n^m(i\xi)</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 3,394:		Line 3,430:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>(-i)^{-(n+m)} ~~p_n~~^m(\xi) \, ,</math>		<math>(-i)^{n - 2m + 1} q_n^m(\xi) \, .</math>
	</td>		</td>
	</tr>		</tr>
			</table>

			Hence, in spheroidal coordinates, the integral representation of the Poisson equation becomes,

			<table border="0" align="center">
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~Q_n^m~~(i\xi)</math>		<math>\Phi(\vec{x})</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 3,406:		Line 3,446:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>(-i)^{n - ~~2m + 1~~} ~~q_n~~^~~m(\xi)~~ \, .</math>		<math>-G \int \frac{\rho(\vec{x}^{~'})}{\|\vec{x}^{~'} - \vec{x}\|} d^3x^' \, ,</math>
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Oblate Spheroidal Coordinates */

2023-05-04T18:31:59Z

Oblate Spheroidal Coordinates

@@ Line 3,341: / Line 3,341: @@
 </tr>
 </table>
-in which case the integral representation of the Poisson equation becomes,
+where,
 <table border="0" align="center">
@@ Line 3,352: / Line 3,379: @@
    </td>
    <td align="left">
-<math>-G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
+<math>-G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, ,</math>
    </td>
 </tr>