User:Tohline/Apps/Ostriker64 - Revision history

Tohline: /* Relationship to Toroidal Coordinate */

2018-08-20T16:46:48Z

Relationship to Toroidal Coordinate

← Older revision		Revision as of 16:46, 20 August 2018
Line 658:		Line 658:
	4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 \, .		4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 \, .
	</math>		</math>
			</td>
			</tr>
			</table>

			===Summary===

			<table border="1" cellpadding="10" align="center" width="85%">
			<tr>
			<td align="left" width="50%">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{r^2}{R^2} </math>
			</td>
			<td align="center">
			<math>~=</math>
			<td align="left">
			<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] </math>
			</td>
			</tr>
			<tr>
			<td align="right">
			<math>~\cot\phi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\cot\theta + \frac{\sinh\eta - \cosh\eta }{\sin\theta} </math>
			</td>
			</tr>
			</table>

			</td>
			<td align="left">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~e^{2\eta}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\cos\theta</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}}
			</math>
			</td>
			</tr>
			</table>

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

Tohline: /* The Dimensionless Radial Coordinate, ξ, and Smallness Parameter, β */

2018-08-18T21:22:50Z

The Dimensionless Radial Coordinate, ξ, and Smallness Parameter, β

← Older revision		Revision as of 21:22, 18 August 2018
Line 2,010:		Line 2,010:
	But, of course, in the context of Ostriker's presentation, <math>~r</math> is not a spherical radial coordinate but is, rather, as [[#Coordinate_System\|defined above]]; and <math>~\alpha</math> is of the same order as the minor, cross-sectional radius of the torus.		But, of course, in the context of Ostriker's presentation, <math>~r</math> is not a spherical radial coordinate but is, rather, as [[#Coordinate_System\|defined above]]; and <math>~\alpha</math> is of the same order as the minor, cross-sectional radius of the torus.

	[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 17 August 2018: There appears to be a typographical error in the definition of β that is provided by equation (6) in §IIa of Ostriker's ''Paper II''. The published equation defines β as the ratio of α to ''r'' rather than, as we have indicated here, as the ratio of α to ''R''.]]In eq. (6) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker also defines the dimensionless parameter,		[[File:CommentButton02.png\|right\|100px\|Comment by J. E. Tohline on 17 August 2018: There appears to be a typographical error in the definition of β that is provided by equation (6) in §IIa of Ostriker's ''Paper II''. The published equation defines β as the ratio of α to ''r'' rather than, as we have indicated here, as the ratio of α to ''R''. Equation (77) on p. 1078 of Paper II confirms this suspicion.]]In eq. (6) of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker also defines the dimensionless parameter,
	<div align="center">		<div align="center">
	<math>~\beta \equiv \frac{\alpha}{R} \, ,</math>		<math>~\beta \equiv \frac{\alpha}{R} \, ,</math>

Tohline: /* Back to Basics */

2018-08-18T20:33:48Z

Back to Basics

← Older revision		Revision as of 20:33, 18 August 2018
Line 534:		Line 534:
	</table>		</table>
	</div>		</div>
	and <math>~\theta</math> has the same sign as <math>~z</math>.		and <math>~\theta</math> has the same sign as <math>~z</math>. Now, given that Ostriker's <math>~(r,\phi)</math> coordinates are related to cylindrical coordinates via the expressions,

			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\varpi </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~R+r\cos\phi \, ,</math>
			</td>
			<td align="center">    and
			<td align="right">
			<math>~z</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~r\sin\phi \, ,</math>
			</td>
			</tr>
			</table>
			we can write,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~r_1^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~[2R + r\cos\phi]^2 + r^2\sin^2\phi </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~4R^2 + 4Rr\cos\phi + r^2 \, ;</math>
			</td>
			</tr>
			</table>
			and,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~r_2^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~r^2 \, .</math>
			</td>
			</tr>
			</table>
			Hence,

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

			<tr>
			<td align="right">
			<math>~\cos\theta</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{1}{2r[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \biggl[ 4Rr\cos\phi + 2r^2 \biggr]
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{2R\cos\phi + r }{[4R^2 + 4Rr\cos\phi + r^2]^{1 / 2}} \, ;
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~e^{2\eta}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{ 4R^2 + 4Rr\cos\phi + r^2 }{r^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			4\biggl( \frac{R}{r}\biggr)^2 + 4\biggl(\frac{R}{r}\biggr)\cos\phi + 1 \, .
			</math>
			</td>
			</tr>
			</table>

	===Second Attempt===		===Second Attempt===

Tohline: /* Back to Basics */

2018-08-18T20:10:54Z

Back to Basics

← Older revision		Revision as of 20:10, 18 August 2018
Line 502:		Line 502:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2~~} \, ,</math>~~		<math>~\frac{(r_1^2 + r_2^2 - 4R^2)}{2r_1 r_2} \, ,</math>
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\tan\psi</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\frac{y}{x~~} \, ,</math>
	</td>		</td>
	</tr>		</tr>
Line 531:		Line 519:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~[~~(x^2 + y^2)^{1 / 2}~~ + a]^2 + (z~~-Z_0)~~^2 \, ,</math>		<math>~[\varpi + R]^2 + z^2 \, ,</math>
	</td>		</td>
	<~~/tr>~~		<td align="center">      and

	~~<tr~~>
	<td align="right">		<td align="right">
	<math>~r_2^2 </math>		<math>~r_2^2 </math>
Line 543:		Line 529:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~[~~(x^2 + y^2)^{1 / 2}~~ - a]^2 + (z~~-Z_0)~~^2 \, ,</math>		<math>~[\varpi - R]^2 + z^2 \, ,</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>
	and <math>~\theta</math> has the same sign as <math>~(z~~-Z_0)~~</math>.		and <math>~\theta</math> has the same sign as <math>~z</math>.


	===Second Attempt===		===Second Attempt===

Tohline: /* Coordinate System */

2018-08-18T20:08:03Z

Coordinate System

← Older revision		Revision as of 20:08, 18 August 2018
Line 165:		Line 165:
	</table>		</table>

	assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (R,0)</math>. Let's determine how to transform between these two sets of coordinate pairs. First, eliminating reference to Ostriker's "polar angle" <math>~\phi</math>, we see that,		assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (R,0)</math>. Let's determine how to transform between these two sets of coordinate pairs.

			====Independent Exploration====
			First, eliminating reference to Ostriker's "polar angle" <math>~\phi</math>, we see that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Line 471:		Line 474:
	</tr>		</tr>
	</table>		</table>
			Now that I think about it, this is all a bit silly because from the [[User:Tohline/2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System\|basic elements of a toroidal coordinate system]] we already know how to shift from cylindrical to toroidal coordinates.

			====Back to Basics====

			Mapping the other direction [see equations 2.13 - 2.15 of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] ], we have,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\eta</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\cos\theta</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\tan\psi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{y}{x} \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~r_1^2 </math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + (z-Z_0)^2 \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~r_2^2 </math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + (z-Z_0)^2 \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.


	===Second Attempt===		===Second Attempt===

Tohline: /* Relationship to Toroidal Coordinate */

2018-08-18T18:24:08Z

Relationship to Toroidal Coordinate

Show changes

Tohline: /* Relationship to Toroidal Coordinate */

2018-08-18T17:37:20Z

Relationship to Toroidal Coordinate

← Older revision		Revision as of 17:37, 18 August 2018
Line 144:		Line 144:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{\varpi}{a}</math>		<math>~\frac{\varpi}{R}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 154:		Line 154:
	<td align="center">      and,		<td align="center">      and,
	<td align="right">		<td align="right">
	<math>~\frac{z}{a}</math>		<math>~\frac{z}{R}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 165:		Line 165:
	</table>		</table>

	assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (a,0)</math>.		assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (R,0)</math>. Let's determine how to transform between these two sets of coordinate pairs. First, eliminating reference to Ostriker's "polar angle" <math>~\phi</math>, we see that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{r^2}{R^2} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl(\frac{\varpi}{R} - 1 \biggr)^2 + \biggl(\frac{z}{R}\biggr)^2</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{\sinh\eta}{\cosh\eta - \cos\theta} - 1 \biggr]^2 + \biggl[ \frac{\sin\theta}{\cosh\eta - \cos\theta} \biggr]^2</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{(\sinh\eta - \cosh\eta + \cos\theta)^2 + \sin^2\theta}{(\cosh\eta - \cos\theta)^2} \biggr] \, .</math>
			</td>
			</tr>
			</table>
			Then, eliminating reference to Ostriker's radial coordinate <math>~r</math>, we find,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\cot\phi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\varpi/R - 1}{z/R}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\sinh\eta - \cosh\eta + \cos\theta}{\sin\theta}</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\cot\theta + \frac{\sinh\eta - \cosh\eta }{\sin\theta} \, .</math>
			</td>
			</tr>
			</table>

	===Second Attempt===		===Second Attempt===

Tohline: /* Coordinate System */

2018-08-18T17:20:18Z

Coordinate System

← Older revision		Revision as of 17:20, 18 August 2018
Line 16:		Line 16:
	==Coordinate System==		==Coordinate System==

			===Basics===
	In §IIa of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker defines a set of orthogonal coordinates, <math>~(r,\phi,\theta)</math>, that is related to the traditional Cartesian coordinate system, <math>~(x,y,z)</math>, via the relations,		In §IIa of [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], Ostriker defines a set of orthogonal coordinates, <math>~(r,\phi,\theta)</math>, that is related to the traditional Cartesian coordinate system, <math>~(x,y,z)</math>, via the relations,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>

			===Relationship to Toroidal Coordinate===
			Referring back to our separate discussion of the [[User:Tohline/2DStructure/ToroidalGreenFunction#Basic_Elements_of_a_Toroidal_Coordinate_System\|basic elements of a toroidal coordinate system]], we know that, the meridional-plane toroidal coordinates <math>~(\eta,\theta)</math> are related to traditional meridional-plane cylindrical coordinate pair <math>~(\varpi,z)</math> via the expressions,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\frac{\varpi}{a}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\sinh\eta}{\cosh\eta - \cos\theta} \, ,</math>
			</td>
			<td align="center">      and,
			<td align="right">
			<math>~\frac{z}{a}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\frac{\sin\theta}{\cosh\eta - \cos\theta} \, ,</math>
			</td>
			</tr>
			</table>

			assuming that the cylindrical-coordinate location of the ''anchor ring'' is <math>~(\varpi,z) = (a,0)</math>.

	===Second Attempt===		===Second Attempt===

Tohline: /* The Dimensionless Radial Coordinate, ξ, and Smallness Parameter, β */

2018-08-18T16:39:04Z

The Dimensionless Radial Coordinate, ξ, and Smallness Parameter, β

← Older revision		Revision as of 16:39, 18 August 2018
Line 1,496:		Line 1,496:
	<math>~\frac{R}{r} = \frac{1}{\beta\xi} ~~~\Rightarrow ~~~ \ln\frac{8R}{r} = \biggl[ \ln\frac{8}{\beta} - \ln\xi \biggr] </math>		<math>~\frac{R}{r} = \frac{1}{\beta\xi} ~~~\Rightarrow ~~~ \ln\frac{8R}{r} = \biggl[ \ln\frac{8}{\beta} - \ln\xi \biggr] </math>
	</div>		</div>
	— Ostriker ~~rewrites~~ his derived expression for <math>~\Phi_\mathrm{TR}</math> ~~to obtain~~,		— we can follow Ostriker's lead and rewrite his derived expression for <math>~\Phi_\mathrm{TR}</math> in the form,

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 1,508:		Line 1,508:
	<td align="left">		<td align="left">
	<math>~		<math>~
	-\frac{GM}{\pi R} \biggl\{\ln \frac{8R}{r}		-\frac{GM}{\pi R} \biggl\{ \ln\frac{8}{\beta} - \ln\xi
	- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi		+ \frac{\beta\xi}{2}\biggl[ - \biggl( \ln\frac{8}{\beta} -1 \biggr) + \ln\xi \biggr]\cos\phi
	~+~ \frac{r^2}{2^~~4R^2~~} \biggl[ \biggl(2\ln\frac{8R}{r} - 3 \biggr) + \biggl( 3\ln\frac{8R}{r} - 4 \biggr)\cos 2\phi		</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			</td>
			<td align="left">
			<math>~
			+~ \frac{\beta^2\xi^2}{2^4} \biggl[ \biggl(2 \ln\frac{8}{\beta} - 3 - 2\ln\xi \biggr) + \biggl( 3 \ln\frac{8}{\beta} - 4 - 3\ln\xi \biggr)\cos 2\phi
	\biggr] ~+ ~\cdots \biggr\} \, .		\biggr] ~+ ~\cdots \biggr\} \, .
	</math>		</math>

Tohline: /* Gravitational Potential */

2018-08-18T04:05:15Z

Gravitational Potential

← Older revision		Revision as of 04:05, 18 August 2018
Line 1,491:		Line 1,491:
	</div>		</div>
	where <math>~R</math> is associated with the major radius of the ring. Then he states that, <font color="darkgreen">"… since <math>~\alpha \ll R</math> (by hypothesis), we may be sure that <math>~\beta \ll 1</math> …"</font>		where <math>~R</math> is associated with the major radius of the ring. Then he states that, <font color="darkgreen">"… since <math>~\alpha \ll R</math> (by hypothesis), we may be sure that <math>~\beta \ll 1</math> …"</font>

			With the definitions of these two dimensionless parameters in hand — and, more specifically, after appreciating that,
			<div align="center">
			<math>~\frac{R}{r} = \frac{1}{\beta\xi} ~~~\Rightarrow ~~~ \ln\frac{8R}{r} = \biggl[ \ln\frac{8}{\beta} - \ln\xi \biggr] </math>
			</div>
			— Ostriker rewrites his derived expression for <math>~\Phi_\mathrm{TR}</math> to obtain,

			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\Phi_\mathrm{TR}(r,\phi)\biggr\|_\mathrm{JPO}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			-\frac{GM}{\pi R} \biggl\{\ln \frac{8R}{r}
			- \frac{r}{2R}\biggl[ \ln \frac{8R}{r} - 1\biggr]\cos\phi
			~+~ \frac{r^2}{2^4R^2} \biggl[ \biggl(2\ln\frac{8R}{r} - 3 \biggr) + \biggl( 3\ln\frac{8R}{r} - 4 \biggr)\cos 2\phi
			\biggr] ~+ ~\cdots \biggr\} \, .
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			Ostriker's (1964) [http://adsabs.harvard.edu/abs/1964ApJ...140.1067O Paper II], p. 1071, Eq. (26)
			</td>
			</tr>
			</table>

	=See Also=		=See Also=