2DStructure/ToroidalCoordinates - Revision history

Jet53man: /* See Also */

2021-09-13T21:56:25Z

Jet53man: /* See Also */

2021-09-11T21:31:03Z

Jet53man: /* Total Mass */

2021-09-11T21:30:03Z

Total Mass

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	</table>		</table>
	</div>		</div>
	which means that <math>~V_\mathrm{green}</math> is indeed half of the total torus volume, [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Total_Volume\|as derived earlier]].		which means that <math>~V_\mathrm{green}</math> is indeed half of the total torus volume, [[2DStructure/ToroidalCoordinates#Total_Volume\|as derived earlier]].

	===Using Toroidal Coordinates with Special Alignment===		===Using Toroidal Coordinates with Special Alignment===
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	===Move General Case===		===Move General Case===
	<font color="red"><b>NOTE:</b></font> A complete prescription of the toroidal-coordinate integration limits that are appropriate for a determination of the volume or the gravitational potential of a circular torus can be found in [[~~User:Tohline/~~2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits\|an accompanying discussion]].		<font color="red"><b>NOTE:</b></font> A complete prescription of the toroidal-coordinate integration limits that are appropriate for a determination of the volume or the gravitational potential of a circular torus can be found in [[2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits\|an accompanying discussion]].

	In the more general case, the expression for the volume integral should be the same; all we should have to do is incorporate the more general integration ''limits'' as specified in our above evaluation of the gravitational potential. Hence, in the more general case we should have,		In the more general case, the expression for the volume integral should be the same; all we should have to do is incorporate the more general integration ''limits'' as specified in our above evaluation of the gravitational potential. Hence, in the more general case we should have,

Jet53man: /* Special Case */

2021-09-11T21:23:12Z

Special Case

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	As we have detailed in an [[~~User:Tohline/~~Apps/DysonWongTori#The_Coulomb_Potential\|accompanying discussion]], using toroidal coordinates [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] derived an expression for the potential at any point inside or outside of a uniform-density torus that has a circular cross-section — as illustrated by the pink torus, above. At some point, we should compare the expression that Wong derived to our independent derivation, as presented in this chapter; in particular, our "special case" should be compared with Wong's equation (2.57), which gives an expression for the potential after integration over both of the angular toroidal coordinates, <math>~(\theta, \psi)</math>, but before the radial integration has been completed. [[~~User:Tohline/~~Apps/DysonWongTori#CompareWithCohl\|His expression is]],		As we have detailed in an [[Apps/DysonWongTori#The_Coulomb_Potential\|accompanying discussion]], using toroidal coordinates [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)] derived an expression for the potential at any point inside or outside of a uniform-density torus that has a circular cross-section — as illustrated by the pink torus, above. At some point, we should compare the expression that Wong derived to our independent derivation, as presented in this chapter; in particular, our "special case" should be compared with Wong's equation (2.57), which gives an expression for the potential after integration over both of the angular toroidal coordinates, <math>~(\theta, \psi)</math>, but before the radial integration has been completed. [[Apps/DysonWongTori#CompareWithCohl\|His expression is]],

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

Jet53man: /* Perform Angular Integration Over Test Mass Distribution */

2021-09-11T21:20:13Z

Perform Angular Integration Over Test Mass Distribution

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	and     <math>~\theta_\mathrm{max} = + \pi \, ,</math>		and     <math>~\theta_\mathrm{max} = + \pi \, ,</math>
	</div>		</div>
	in practice, the limits should be set to reflect the volume boundaries of the mass distribution that is responsible for generating the gravitational potential. Here we present analytic expressions for these limits in the case of the (pink) [[#Chosen_Test_Mass_Distribution\|toroidal test mass distribution specified above]]; details regarding the derivation of these expressions can be found in [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Yet_Again\|our accompanying set of notes]].		in practice, the limits should be set to reflect the volume boundaries of the mass distribution that is responsible for generating the gravitational potential. Here we present analytic expressions for these limits in the case of the (pink) [[#Chosen_Test_Mass_Distribution\|toroidal test mass distribution specified above]]; details regarding the derivation of these expressions can be found in [[Appendix/Ramblings/ToroidalCoordinates#Yet_Again\|our accompanying set of notes]].

	===Identifying Limits of Integration===		===Identifying Limits of Integration===
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	The animation shown here in Figure 2 builds upon the configuration displayed in [[#THH12Figure4\|our Figure 1, above]]. It shows a meridional cross-section through the selected (pink) uniform-density, toroidal mass distribution, whose geometric properties are fully determined by specifying values for <math>~\varpi_t</math> and <math>~r_t</math>. (For the example illustrated in Figure 2, we have specified the same values used by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)] to construct their [[#THH12Figure4\|Figure 4, as reprinted above]]; namely, <math>~\varpi_t = 3/4</math> and <math>~r_t = 1/4</math>.) Throughout the Figure 2 animation sequence, these two parameters have been fixed — thereby fixing the properties of the (pink) torus — and, in addition, we have fixed the location of the origin of a toroidal coordinate system — identified by the red-filled circular dot. We explicitly associate this coordinate-system origin with the (cylindrical) coordinates of the point in space at which we choose to evaluate the gravitational potential, namely, <math>~(R_, Z_) = (a, Z_0)</math>. [For illustration purposes, in Figure 2 we have set <math>~(a, Z_0) = (1/3, 3/4)</math>.]		The animation shown here in Figure 2 builds upon the configuration displayed in [[#THH12Figure4\|our Figure 1, above]]. It shows a meridional cross-section through the selected (pink) uniform-density, toroidal mass distribution, whose geometric properties are fully determined by specifying values for <math>~\varpi_t</math> and <math>~r_t</math>. (For the example illustrated in Figure 2, we have specified the same values used by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)] to construct their [[#THH12Figure4\|Figure 4, as reprinted above]]; namely, <math>~\varpi_t = 3/4</math> and <math>~r_t = 1/4</math>.) Throughout the Figure 2 animation sequence, these two parameters have been fixed — thereby fixing the properties of the (pink) torus — and, in addition, we have fixed the location of the origin of a toroidal coordinate system — identified by the red-filled circular dot. We explicitly associate this coordinate-system origin with the (cylindrical) coordinates of the point in space at which we choose to evaluate the gravitational potential, namely, <math>~(R_, Z_) = (a, Z_0)</math>. [For illustration purposes, in Figure 2 we have set <math>~(a, Z_0) = (1/3, 3/4)</math>.]

	While the values of the four primary model parameters <math>~(a, Z_0, \varpi_t, r_t)</math> are held fixed, Figure 2 depicts in a quantitatively precise manner how the size of a <math>~\xi_1</math>= constant toroidal surface (the off-center circle traced by the sequence of black dots) varies as the value of the radial coordinate, <math>~\xi_1</math>, is varied. In each frame of the animation sequence, the value of <math>~\xi_1</math> that was used to define the (black) <math>\xi_1</math>-circle is printed in the lower-right corner of the image; additional quantitative details associated with each animation frame can be obtained from the [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Example2\|table titled, "Example 2" in our accompanying notes]].		While the values of the four primary model parameters <math>~(a, Z_0, \varpi_t, r_t)</math> are held fixed, Figure 2 depicts in a quantitatively precise manner how the size of a <math>~\xi_1</math>= constant toroidal surface (the off-center circle traced by the sequence of black dots) varies as the value of the radial coordinate, <math>~\xi_1</math>, is varied. In each frame of the animation sequence, the value of <math>~\xi_1</math> that was used to define the (black) <math>\xi_1</math>-circle is printed in the lower-right corner of the image; additional quantitative details associated with each animation frame can be obtained from the [[Appendix/Ramblings/ToroidalCoordinates#Example2\|table titled, "Example 2" in our accompanying notes]].


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	<math>~\xi_1\|_\mathrm{max} > \xi_1 > ~\xi_1\|_\mathrm{min} \, ,</math>		<math>~\xi_1\|_\mathrm{max} > \xi_1 > ~\xi_1\|_\mathrm{min} \, ,</math>
	</div>		</div>
	the <math>\xi_1</math>-circle intersects the surface of the torus in two locations, defined by two different values of the associated angular coordinate, <math>~\xi_2</math>. In each frame of the animation, points of intersection are marked with small yellow diamonds; the coordinates of these points of intersection are listed in the table associated with [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Example2\|example 2, in our accompanying notes]]. For each relevant value of <math>~\xi_1</math>, these are the limiting values of the toroidal angular coordinate to be used in the integration that produces <math>~q_0</math>. It should be realized that, ''at'' the first and final points of contact, the two values of <math>~\xi_2</math> are degenerate.		the <math>\xi_1</math>-circle intersects the surface of the torus in two locations, defined by two different values of the associated angular coordinate, <math>~\xi_2</math>. In each frame of the animation, points of intersection are marked with small yellow diamonds; the coordinates of these points of intersection are listed in the table associated with [[Appendix/Ramblings/ToroidalCoordinates#Example2\|example 2, in our accompanying notes]]. For each relevant value of <math>~\xi_1</math>, these are the limiting values of the toroidal angular coordinate to be used in the integration that produces <math>~q_0</math>. It should be realized that, ''at'' the first and final points of contact, the two values of <math>~\xi_2</math> are degenerate.
	<!-- COMMENT OUT		<!-- COMMENT OUT
	Next we derive the mathematical relations that give the values of <math>~\xi_1\|_\mathrm{max}</math> and <math>~\xi_1\|_\mathrm{min}</math> for all .		Next we derive the mathematical relations that give the values of <math>~\xi_1\|_\mathrm{max}</math> and <math>~\xi_1\|_\mathrm{min}</math> for all .
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	for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>. For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> — as recorded in the top row of numbers in the Table, below — the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1\|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1\|_\mathrm{min} = 1.0449467</math>. The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame. These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values. For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.		for a toroidal coordinate system whose origin (filled, red dot) remains fixed at the (cylindrical) coordinate location, <math>~(\varpi, z) = (a, Z_0) = (\tfrac{1}{3}, \tfrac{3}{4})</math>. For a toroidal coordinate system with this specified origin and an equatorial-plane torus having <math>~\varpi_t = \tfrac{3}{4}</math> and <math>~r_t = \tfrac{1}{4}</math> — as recorded in the top row of numbers in the Table, below — the <math>\xi_1</math>-circle makes ''first contact'' with the torus when <math>~\xi_1 = \xi_1\|_\mathrm{max} = 1.1927843</math> and it makes ''final contact'' when <math>~\xi_1 = \xi_1\|_\mathrm{min} = 1.0449467</math>. The animation sequence contains ten unique frames: The value of <math>~\xi_1</math> that is associated with the <math>\xi_1</math>-circle in each case appears near the bottom-right corner of the animation frame. These parameter values have also been recorded in the first column of ten separate rows in the following table, along with other relevant parameter values. For example, in each frame of the animation, the points of intersection between the surface of the torus and the <math>\xi_1</math>-circle are identified by filled, green diamonds; the (cylindrical) coordinates associated with these points of intersection, <math>~(\varpi_i, z_i)</math>, are listed in each table row, along with the corresponding value of the toroidal coordinate system's angular, <math>~\xi_2</math> coordinate.

	Specific values for these parameters are tabulated in a [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Example2\|Table titled, ''Example 2'']] in our accompanying notes.		Specific values for these parameters are tabulated in a [[Appendix/Ramblings/ToroidalCoordinates#Example2\|Table titled, ''Example 2'']] in our accompanying notes.
	END DELETED COMMENT -->		END DELETED COMMENT -->

Jet53man: /* Toroidal Coordinates */

2021-09-11T21:12:29Z

Toroidal Coordinates

Jet53man: /* Statement of the Problem */

2021-09-11T21:10:09Z

Statement of the Problem

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	As has been explained in [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Confirmation_Provided_by_Trova.2C_Hur.C3.A9_and_Hersant\|an accompanying set of notes]], this is precisely the same expression for the gravitational potential that [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T A. Trova, J.-M. Huré and F. Hersant (2012; MNRAS, 424, 2635)] used in their study of the potential of self-gravitating, axisymmetric discs.		As has been explained in [[Appendix/Ramblings/ToroidalCoordinates#Confirmation_Provided_by_Trova.2C_Hur.C3.A9_and_Hersant\|an accompanying set of notes]], this is precisely the same expression for the gravitational potential that [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T A. Trova, J.-M. Huré and F. Hersant (2012; MNRAS, 424, 2635)] used in their study of the potential of self-gravitating, axisymmetric discs.

	Our objective, here, is to examine whether or not it might be advantageous to transform this expression to one in which the double integral is performed on a toroidal, rather than a cylindrical, coordinate system.		Our objective, here, is to examine whether or not it might be advantageous to transform this expression to one in which the double integral is performed on a toroidal, rather than a cylindrical, coordinate system.
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	<tr><td align="center">		<tr><td align="center">
	Figure 4 extracted without modification from p. 2640 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)]<p></p>		Figure 4 extracted without modification from p. 2640 of [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, Huré & Hersant (2012)]
	"''The Potential of Discs from a 'Mean Green Function~~' '~~'"<p></p>		<br />
			<i>"The Potential of Discs from a 'Mean Green Function'"</i><br />
	Monthly Notices of the Royal Astronomical Society, vol. 424, pp. 2635-2645 © RAS		Monthly Notices of the Royal Astronomical Society, vol. 424, pp. 2635-2645 © RAS
	</td>		</td>
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	<td align="center">		<td align="center">
	[[File:Figure4THH2012.png\|350px\|~~Figure~~ 4 from Trova, Huré & Hersant (2012)]]		[[File:Figure4THH2012.png\|350px\|To be inserted: Fig. 4 from Trova, Huré & Hersant (2012)]]
	</td>		</td>
	<td align="center">		<td align="center">

Jet53man: /* Preface */

2021-09-11T21:04:28Z

Preface

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	<table border="0" cellpadding="8" align="center" width="60%">		<table border="0" cellpadding="8" align="center" width="60%">
	<tr><td align="center">		<tr><td align="center">
	<font size="+2"><b>Abstract</b></font><~~p><~~/p>Joel E. Tohline (September, 2017)		<font size="+1"><b>Abstract</b></font><br />Joel E. Tohline (September, 2017)
	</td></tr>		</td></tr>
	</table>		</table>
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	</td></tr></table>		</td></tr></table>

	It appears as though we have found an answer to this question posed by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)]. The detailed derivations and associated scratch-work that support the ''summary'' discussion of this chapter can be found under the [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates\|Appendix/Ramblings category]] of this H_Book and in our accompanying discussion of [[~~User:Tohline/~~Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori\|Dyson-Wong toroids]].		It appears as though we have found an answer to this question posed by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)]. The detailed derivations and associated scratch-work that support the ''summary'' discussion of this chapter can be found under the [[Appendix/Ramblings/ToroidalCoordinates\|Appendix/Ramblings category]] of this H_Book and in our accompanying discussion of [[Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori\|Dyson-Wong toroids]].

	<table border="0" align="right"><tr><td align="center">		<table border="0" align="right"><tr><td align="center">
	[[File:~~ThinRing72cropped~~.png\|225px\|center\|Gravitational Potential surface for infinitesimally thin hoop]]		[[File:TThinRing72cropped.png\|225px\|center\|Gravitational Potential surface for infinitesimally thin hoop]]
	</td></tr>		</td></tr>
	<tr><td align="center">Infinitesimally Thin Hoop</td></tr></table>		<tr><td align="center">Infinitesimally Thin Hoop</td></tr></table>
	As has been known for approximately a century, at any meridional-plane coordinate location, <math>~(\varpi,z)</math>, the gravitational potential due to an axisymmetric, infinitesimally thin ring (TR) of radius, <math>~a</math>, and total mass, <math>~M</math>, is given by the "key expression,"		As has been known for approximately a century, at any meridional-plane coordinate location, <math>~(\varpi,z)</math>, the gravitational potential due to an axisymmetric, infinitesimally thin ring (TR) of radius, <math>~a</math>, and total mass, <math>~M</math>, is given by the "key expression,"
	<table border="0" align="center"><tr><td align="center">{{ ~~User:Tohline/~~Math/EQ TRApproximation }}</td></tr></table>		<table border="0" align="center"><tr><td align="center">{{ Math/EQ TRApproximation }}</td></tr></table>
	and <math>~K(k)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind]. Making the coordinate-name substitutions, <math>~(\varpi,z) \rightarrow (R_,z_)</math>, to match much of this chapter's variable notation, we have alternatively,		and <math>~K(k)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind]. Making the coordinate-name substitutions, <math>~(\varpi,z) \rightarrow (R_,z_)</math>, to match much of this chapter's variable notation, we have alternatively,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	If the configuration's density is constant, then, as is [[#Integrate\|shown below]], the integral over the angular coordinate variable, <math>~\theta</math>, can be completed analytically. Hence, the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced a problem of carrying out a single, line integration. We specifically illustrate how this approach can be used to evaluate the gravitational potential of a uniform-density torus with a circular cross-section and aspect ratio, <math>~R/d = 3</math>.		If the configuration's density is constant, then, as is [[#Integrate\|shown below]], the integral over the angular coordinate variable, <math>~\theta</math>, can be completed analytically. Hence, the task of evaluating the gravitational potential (both inside and outside) of a uniform-density, axisymmetric configuration having ''any'' surface shape has been reduced a problem of carrying out a single, line integration. We specifically illustrate how this approach can be used to evaluate the gravitational potential of a uniform-density torus with a circular cross-section and aspect ratio, <math>~R/d = 3</math>.

	Note:  As we have [[~~User:Tohline/~~Apps/DysonWongTori#Wong_.281973.2C_1974.29\|reviewed separately]], it appears as though over forty years ago [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973)] successfully employed toroidal coordinates to derive a closed-form expression for the potential (inside as well as outside) of a uniform-density torus with a circular cross-section; see, for example, his Figure 7. That is to say, he was even able to carry out the final line integral in closed form. We have not (yet) been able to demonstrate that our expression for the potential of a circular-cross-section torus is identical to his.		Note:  As we have [[Apps/DysonWongTori#Wong_.281973.2C_1974.29\|reviewed separately]], it appears as though over forty years ago [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973)] successfully employed toroidal coordinates to derive a closed-form expression for the potential (inside as well as outside) of a uniform-density torus with a circular cross-section; see, for example, his Figure 7. That is to say, he was even able to carry out the final line integral in closed form. We have not (yet) been able to demonstrate that our expression for the potential of a circular-cross-section torus is identical to his.
	</td></tr></table>		</td></tr></table>

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

	It appears as though we have found an answer to this question posed by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)]. The detailed derivations and associated scratch-work that support the ''summary'' discussion of this chapter can be found under the [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates\|Appendix/Ramblings category]] of this H_Book and in our accompanying discussion of [[~~User:Tohline/~~Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori\|Dyson-Wong toroids]].		It appears as though we have found an answer to this question posed by [http://adsabs.harvard.edu/abs/2012MNRAS.424.2635T Trova, et al. (2012)]. The detailed derivations and associated scratch-work that support the ''summary'' discussion of this chapter can be found under the [[Appendix/Ramblings/ToroidalCoordinates\|Appendix/Ramblings category]] of this H_Book and in our accompanying discussion of [[Apps/DysonWongTori#Self-Gravitating.2C_Incompressible_.28Dyson-Wong.29_Tori\|Dyson-Wong toroids]].

	[[File:MacMillanFigure61.png\|250px\|right\|MacMillan (1958, ''The Theory of the Potential'', New York: McGraw-Hill)]]As has been known for approximately a century, at any meridional-plane coordinate location, <math>~(R_, Z_)</math>, the gravitational potential due to an infinitesimally thin, axisymmetric "hoop" of radius, <math>~a</math>, and total mass, <math>~M</math>, is given by the expression,		[[File:MacMillanFigure61.png\|250px\|right\|MacMillan (1958, ''The Theory of the Potential'', New York: McGraw-Hill)]]As has been known for approximately a century, at any meridional-plane coordinate location, <math>~(R_, Z_)</math>, the gravitational potential due to an infinitesimally thin, axisymmetric "hoop" of radius, <math>~a</math>, and total mass, <math>~M</math>, is given by the expression,
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	-->		-->

	{\| class="~~wikitable~~" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="Chap25A" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 150px; width: 150px;" \|[[~~User:Tohline/~~H_BookTiledMenu#Axisymmetric_Equilibrium_Structures\|Exploring the Use of<br />Toroidal Coordinates<br />to Determine the<br />Gravitational<br /> Potential]]		! style="height: 150px; width: 150px;" \|[[H_BookTiledMenu#Axisymmetric_Equilibrium_Structures\|Exploring the Use of<br />Toroidal Coordinates<br />to Determine the<br />Gravitational<br /> Potential]]
	\|}		\|}
	As I have studied the structure and analyzed the stability of (both self-gravitating and non-self-gravitating) toroidal configurations over the years, I have often wondered whether it might be useful to examine such systems mathematically using a toroidal — or at least a toroidal-like — coordinate system. Is it possible, for example, to build an equilibrium torus for which the density distribution is one-dimensional as viewed from a well-chosen toroidal-like system of coordinates?		As I have studied the structure and analyzed the stability of (both self-gravitating and non-self-gravitating) toroidal configurations over the years, I have often wondered whether it might be useful to examine such systems mathematically using a toroidal — or at least a toroidal-like — coordinate system. Is it possible, for example, to build an equilibrium torus for which the density distribution is one-dimensional as viewed from a well-chosen toroidal-like system of coordinates?

	I should begin by clarifying my terminology. In volume II (p. 666) of their treatise on [[~~User:Tohline/~~Appendix/References#MF53\|''Methods of Theoretical Physics'', Morse & Feshbach (1953; hereafter MF53)]] define an orthogonal toroidal coordinate system in which the Laplacian is separable.<sup>1</sup> (See details, below.) It is only this system that I will refer to as ''the'' toroidal coordinate system; all other functions that trace out toroidal surfaces but that don't conform precisely to Morse & Feshbach's coordinate system will be referred to as ''toroidal-like.''		I should begin by clarifying my terminology. In volume II (p. 666) of their treatise on [[Appendix/References#MF53\|''Methods of Theoretical Physics'', Morse & Feshbach (1953; hereafter MF53)]] define an orthogonal toroidal coordinate system in which the Laplacian is separable.<sup>1</sup> (See details, below.) It is only this system that I will refer to as ''the'' toroidal coordinate system; all other functions that trace out toroidal surfaces but that don't conform precisely to Morse & Feshbach's coordinate system will be referred to as ''toroidal-like.''

	I became particularly interested in this idea while working with Howard Cohl (when he was an LSU graduate student). Howie's dissertation research uncovered a ''Compact Cylindrical Greens Function'' technique for evaluating Newtonian potentials of rotationally flattened (especially axisymmetric) configurations.<sup>2,3</sup> The technique involves a multipole expansion in terms of half-integer-degree Legendre functions of the <math>2^\mathrm{nd}</math> kind — see [http://dlmf.nist.gov/14.19 NIST digital library discussion] — where, we have discovered, the argument of this special function is a function only of the ''radial'' coordinate of Morse & Feshbach's orthogonal toroidal coordinate system — see more on this, [[#Connection_With_the_Physical_Problem\|below]].		I became particularly interested in this idea while working with Howard Cohl (when he was an LSU graduate student). Howie's dissertation research uncovered a ''Compact Cylindrical Greens Function'' technique for evaluating Newtonian potentials of rotationally flattened (especially axisymmetric) configurations.<sup>2,3</sup> The technique involves a multipole expansion in terms of half-integer-degree Legendre functions of the <math>2^\mathrm{nd}</math> kind — see [http://dlmf.nist.gov/14.19 NIST digital library discussion] — where, we have discovered, the argument of this special function is a function only of the ''radial'' coordinate of Morse & Feshbach's orthogonal toroidal coordinate system — see more on this, [[#Connection_With_the_Physical_Problem\|below]].

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References

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	=References=		=References=
	# [[~~User:Tohline/~~Appendix/References#MF53\|Morse, P.M. & Feshmach, H. 1953]], ''Methods of Theoretical Physics'' — Volumes I and II		# [[Appendix/References#MF53\|Morse, P.M. & Feshmach, H. 1953]], ''Methods of Theoretical Physics'' — Volumes I and II
	# Cohl, H.S. & Tohline, J.E. [http://adsabs.harvard.edu/abs/1999ApJ...527...86C 1999, ApJ, 527, 86-101]		# Cohl, H.S. & Tohline, J.E. [http://adsabs.harvard.edu/abs/1999ApJ...527...86C 1999, ApJ, 527, 86-101]
	# Cohl, H.S., Rau, A.R.P., Tohline, J.E., Browne, D.A., Cazes, J.E. & Barnes, E.I. [http://adsabs.harvard.edu/abs/2001PhRvA..64e2509C 2001, Phys. Rev. A, 64, 052509]		# Cohl, H.S., Rau, A.R.P., Tohline, J.E., Browne, D.A., Cazes, J.E. & Barnes, E.I. [http://adsabs.harvard.edu/abs/2001PhRvA..64e2509C 2001, Phys. Rev. A, 64, 052509]

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 =See Also=
-* [[2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits|Detailed Illustration of Toroidal Coordinates and Integration Limits]]
+<ul>
-* Ramblings Appendix Chapter: [[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Configurations_and_Related_Coordinate_Systems|Determining the Relationships between Toroidal Coordinates and other Coordinate Systems]]
+  <li>Common Theme:
 * Examine the expressions for the gravitational potential derived in &sect;2.1 of [http://adsabs.harvard.edu/abs/1992ApJ...394..248W Woodward, Sankaran &amp; Tohline (1992, ApJ, 394, 248)].
 * Documentation for the gnu/GSL [http://www.gnu.org/software/gsl/manual/html_node/Special-Functions.html#Special-Functions Special Function Library]

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	=See Also=		=See Also=

	* [[~~User:Tohline/~~2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits\|Detailed Illustration of Toroidal Coordinates and Integration Limits]]		* [[2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits\|Detailed Illustration of Toroidal Coordinates and Integration Limits]]
	* Ramblings Appendix Chapter: [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Toroidal_Configurations_and_Related_Coordinate_Systems\|Determining the Relationships between Toroidal Coordinates and other Coordinate Systems]]		* Ramblings Appendix Chapter: [[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Configurations_and_Related_Coordinate_Systems\|Determining the Relationships between Toroidal Coordinates and other Coordinate Systems]]
	* Examine the expressions for the gravitational potential derived in §2.1 of [http://adsabs.harvard.edu/abs/1992ApJ...394..248W Woodward, Sankaran & Tohline (1992, ApJ, 394, 248)].		* Examine the expressions for the gravitational potential derived in §2.1 of [http://adsabs.harvard.edu/abs/1992ApJ...394..248W Woodward, Sankaran & Tohline (1992, ApJ, 394, 248)].
	* Documentation for the gnu/GSL [http://www.gnu.org/software/gsl/manual/html_node/Special-Functions.html#Special-Functions Special Function Library]		* Documentation for the gnu/GSL [http://www.gnu.org/software/gsl/manual/html_node/Special-Functions.html#Special-Functions Special Function Library]

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	==Toroidal Coordinates==		==Toroidal Coordinates==

	[[~~User:Tohline/~~2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits\|Click here]] to find a supporting, detailed illustration of Toroidal Coordinates and relevant integration limits.		[[2DStructure/ToroidalCoordinateIntegrationLimits#Toroidal-Coordinate_Integration_Limits\|Click here]] to find a supporting, detailed illustration of Toroidal Coordinates and relevant integration limits.

	===Properties===		===Properties===
	Here we highlight certain properties and features of the [[~~User:Tohline/~~Appendix/References#MF53\|MF53]] toroidal coordinate system; more details can be found in a [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Toroidal_Coordinates\|related set of our online notes]]. Most importantly in the context of our discussion, if (at all azimuthal angles) the origin of the toroidal coordinate system is placed at the ''cylindrical-coordinate'' location, <math>~(a, Z_0), </math> the pair of orthogonal coordinates, <math>~(\xi_1, \xi_2)</math>, is related to the cylindrical coordinate pair, <math>~(\varpi, Z)</math>, via the expressions,		Here we highlight certain properties and features of the [[Appendix/References#MF53\|MF53]] toroidal coordinate system; more details can be found in a [[Appendix/Ramblings/ToroidalCoordinates#Toroidal_Coordinates\|related set of our online notes]]. Most importantly in the context of our discussion, if (at all azimuthal angles) the origin of the toroidal coordinate system is placed at the ''cylindrical-coordinate'' location, <math>~(a, Z_0), </math> the pair of orthogonal coordinates, <math>~(\xi_1, \xi_2)</math>, is related to the cylindrical coordinate pair, <math>~(\varpi, Z)</math>, via the expressions,
	<table align="center" border="0" cellpadding="5">		<table align="center" border="0" cellpadding="5">
	<tr><th align="center" colspan="1"> </th>		<tr><th align="center" colspan="1"> </th>
	<th align="center" colspan="2">[<font color="red"><b>[[~~User:Tohline/~~Appendix/References#MF5\|MF53]]</b></font>]</th>		<th align="center" colspan="2">[<font color="red"><b>[[Appendix/References#MF5\|MF53]]</b></font>]</th>
	<th align="center" colspan="2">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>		<th align="center" colspan="2">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
	</tr>		</tr>
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	<tr>		<tr>
	<th align="center" colspan="1">[<font color="red"><b>[[~~User:Tohline/~~Appendix/References#MF5\|MF53]]</b></font>]</th>		<th align="center" colspan="1">[<font color="red"><b>[[Appendix/References#MF5\|MF53]]</b></font>]</th>
	<td align="center">       </td>		<td align="center">       </td>
	<th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>		<th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
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	<tr>		<tr>
	<th align="center" colspan="1">[<font color="red"><b>[[~~User:Tohline/~~Appendix/References#MF5\|MF53]]</b></font>]</th>		<th align="center" colspan="1">[<font color="red"><b>[[Appendix/References#MF5\|MF53]]</b></font>]</th>
	<td align="center">       </td>		<td align="center">       </td>
	<th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>		<th align="center" colspan="1">[https://en.wikipedia.org/wiki/Toroidal_coordinates Wikipedia]</th>
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	This, indeed, equals the right-hand side of the relation, which is, <math>~r_0^2</math>. It all nicely checks out!		This, indeed, equals the right-hand side of the relation, which is, <math>~r_0^2</math>. It all nicely checks out!

	Next, taking a hint from the EUREKA! moment recorded in [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#Relating_CCGF_Expansion_to_Toroidal_Coordinates\|our accompanying notes]], let's rewrite the function <math>~\Chi</math> in terms of toroidal rather than cylindrical coordinates, where <math>~\Chi</math> is the argument of the special function, <math>~Q_{-1/2}</math>, that appears in the [[~~User:Tohline/~~2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|above definition of <math>~q_0</math>]]. More specifically, let's assume that the coordinate location at which the gravitational potential is to be evaluated, <math>~(R_, Z_)</math>, is taken to be the cylindrical-coordinate location of the origin of the toroidal coordinate system, <math>~(a, Z_0)</math>. Given this association, we can write,		Next, taking a hint from the EUREKA! moment recorded in [[Appendix/Ramblings/ToroidalCoordinates#Relating_CCGF_Expansion_to_Toroidal_Coordinates\|our accompanying notes]], let's rewrite the function <math>~\Chi</math> in terms of toroidal rather than cylindrical coordinates, where <math>~\Chi</math> is the argument of the special function, <math>~Q_{-1/2}</math>, that appears in the [[2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|above definition of <math>~q_0</math>]]. More specifically, let's assume that the coordinate location at which the gravitational potential is to be evaluated, <math>~(R_, Z_)</math>, is taken to be the cylindrical-coordinate location of the origin of the toroidal coordinate system, <math>~(a, Z_0)</math>. Given this association, we can write,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	This is the key result motivating the use of a toroidal coordinate system to evaluate the gravitational potential: When expressed in an appropriately defined toroidal coordinate system, the modulus of the special function is a function of one, rather than two, spatial coordinates. This gives some hope that the integral over the second (angular) coordinate, <math>~\xi_2</math>, can be completed analytically, giving rise to an expression for the gravitational potential whose evaluation only requires numerical integration over a single (radial) coordinate, <math>~\xi_1</math>.		This is the key result motivating the use of a toroidal coordinate system to evaluate the gravitational potential: When expressed in an appropriately defined toroidal coordinate system, the modulus of the special function is a function of one, rather than two, spatial coordinates. This gives some hope that the integral over the second (angular) coordinate, <math>~\xi_2</math>, can be completed analytically, giving rise to an expression for the gravitational potential whose evaluation only requires numerical integration over a single (radial) coordinate, <math>~\xi_1</math>.

	Finally, drawing on discussion in [[~~User:Tohline/~~Appendix/Ramblings/ToroidalCoordinates#ToroidalScaleFactors\|our accompanying set of notes]], we recognize that, expressed in terms of toroidal coordinates, the differential area element in the meridional plane is,		Finally, drawing on discussion in [[Appendix/Ramblings/ToroidalCoordinates#ToroidalScaleFactors\|our accompanying set of notes]], we recognize that, expressed in terms of toroidal coordinates, the differential area element in the meridional plane is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">