User:Tohline/AxisymmetricConfigurations/PoissonEq - Revision history

Tohline: /* Stahler (1983) */

2018-05-28T04:14:58Z

Stahler (1983)

← Older revision		Revision as of 04:14, 28 May 2018
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	</td></tr></table>		</td></tr></table>

	[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] used a [[User:Tohline/AxisymmetricConfigurations/HSCF#Introduction\|self-consistent-field]] technique to construct equilibrium sequences of rotationally flattened, isothermal gas clouds. At each iteration step, the method that he adopted to evaluate the gravitational potential along the outer boundary of his computational mesh was essentially the same as the method ~~that~~ [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] ~~used~~ — see our [[#Deupree_.281974.29\|above description]] — to model the time-dependent behavior of pulsating stars. It appears as though Stahler was unaware of Deupree's earlier development of this method to evaluate the boundary potential, as Deupree's (1974) paper is not among Stahler's list of references.		[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] used a [[User:Tohline/AxisymmetricConfigurations/HSCF#Introduction\|self-consistent-field]] technique to construct equilibrium sequences of rotationally flattened, isothermal gas clouds. At each iteration step, the method that he adopted to evaluate the gravitational potential along the outer boundary of his computational mesh was essentially the same as the method used by [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] — see our [[#Deupree_.281974.29\|above description]] — to model the time-dependent behavior of pulsating stars. It appears as though Stahler was unaware of Deupree's earlier development of this method to evaluate the boundary potential, as Deupree's (1974) paper is not among Stahler's list of references.

	<!--		<!--

Tohline: /* Stahler (1983) */

2018-05-28T04:09:53Z

Stahler (1983)

← Older revision		Revision as of 04:09, 28 May 2018
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	</td></tr></table>		</td></tr></table>

	[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] used a self-consistent-field technique to construct equilibrium sequences of rotationally flattened, isothermal gas clouds. At each iteration step, the method that he adopted to evaluate the gravitational potential along the outer boundary of his computational mesh was essentially the same as the method that Deupree (~~1975~~) used — ~~and that we have described,~~ above — to model the time-dependent behavior of pulsating stars. It appears as though Stahler was unaware of Deupree's earlier development of this method to evaluate the boundary potential, as Deupree's (~~1975~~) paper is not among Stahler's list of references. ~~We therefore assume that Stahler's numerical method was developed independently.~~		[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] used a [[User:Tohline/AxisymmetricConfigurations/HSCF#Introduction\|self-consistent-field]] technique to construct equilibrium sequences of rotationally flattened, isothermal gas clouds. At each iteration step, the method that he adopted to evaluate the gravitational potential along the outer boundary of his computational mesh was essentially the same as the method that [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] used — see our [[#Deupree_.281974.29\|above description]] — to model the time-dependent behavior of pulsating stars. It appears as though Stahler was unaware of Deupree's earlier development of this method to evaluate the boundary potential, as Deupree's (1974) paper is not among Stahler's list of references.


			<!--
	In the chapter of this H_Book that focuses on a discussion of [[User:Tohline/Apps/DysonWongTori\|Dyson-Wong tori]], we have included the expression for the [[User:Tohline/Apps/DysonWongTori#RingPotential\|gravitational potential of a thin ring]] of mass, <math>~M</math>, that passes through the meridional plane at coordinate location, <math>~(\varpi^', z^') = (a, 0)</math>, as derived, for example, by [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)] and by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential W. D. MacMillan (1958; originally, 1930)], namely,		In the chapter of this H_Book that focuses on a discussion of [[User:Tohline/Apps/DysonWongTori\|Dyson-Wong tori]], we have included the expression for the [[User:Tohline/Apps/DysonWongTori#RingPotential\|gravitational potential of a thin ring]] of mass, <math>~M</math>, that passes through the meridional plane at coordinate location, <math>~(\varpi^', z^') = (a, 0)</math>, as derived, for example, by [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)] and by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential W. D. MacMillan (1958; originally, 1930)], namely,
	<div align="center">		<div align="center">
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	</table>		</table>
	</div>		</div>
			-->
			In describing [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)] method, we will first draw upon our "[[User:Tohline/Appendix/Equation_templates#Other_Equations_with_Assigned_Templates\|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
	~~Notice that the~~ first ~~of these two rewritten expressions aligns perfectly with~~ our "[[User:Tohline/Appendix/Equation_templates#Other_Equations_with_Assigned_Templates\|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
	<table border="0" align="center" cellpadding="10"><tr><td align="center">		<table border="0" align="center" cellpadding="10"><tr><td align="center">
	{{ User:Tohline/Math/EQ TRApproximation }}		{{ User:Tohline/Math/EQ TRApproximation }}
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	<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png\|225px\|link=User:Tohline/Apps/DysonWongTori#ThinRingContours]]</td>		<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png\|225px\|link=User:Tohline/Apps/DysonWongTori#ThinRingContours]]</td>
	</tr></table>		</tr></table>
	(See our [[User:Tohline/Apps/DysonWongTori#ThinRingContours\|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.)		(See our [[User:Tohline/Apps/DysonWongTori#ThinRingContours\|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.) Stahler has argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring element — positioned at various meridional coordinate locations throughout the mass distribution. According to his independent derivation, the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is (see his equation 11 and the explanatory text that follows it):


	~~[http://adsabs.harvard.edu/abs/1983ApJ...268..155S~~ Stahler ~~(1983a)]~~ has argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring element — positioned at various meridional coordinate locations throughout the mass distribution. According to his independent derivation, the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is (see his equation 11 and the explanatory text that follows it):
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	</table>		</table>
	</div>		</div>
	Stahler's expression for each ''thin ring'' contribution is clearly the same as ~~the expressions presented by [https:/~~/~~archive.org/details/foundationsofpot033485mbp Kellogg (1929)] and [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=~~the~~+theory+of+the+potential MacMillan (1958)] for a~~ ring ~~that~~ cuts through the meridional plane at <math>~(\varpi^', z^') = (a, 0)</math>.		Stahler's expression for each ''thin ring'' contribution is clearly the same as our "key equation" expression for <math>~\Phi_\mathrm{TR}</math> if the individual ring being considered cuts through the meridional plane at <math>~(\varpi^', z^') = (a, 0)</math>.

	In the context of our [[#For_Axisymmetric_Systems\|above discussion of a Green's function expression written in terms of toroidal functions]], we have shown that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,		In the context of our [[#For_Axisymmetric_Systems\|above discussion of a Green's function expression written in terms of toroidal functions]], we have shown that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,

Tohline: /* Stahler (1983) */

2018-05-28T00:13:42Z

Stahler (1983)

← Older revision		Revision as of 00:13, 28 May 2018
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	</tr></table>		</tr></table>
	</td></tr></table>		</td></tr></table>

			[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] used a self-consistent-field technique to construct equilibrium sequences of rotationally flattened, isothermal gas clouds. At each iteration step, the method that he adopted to evaluate the gravitational potential along the outer boundary of his computational mesh was essentially the same as the method that Deupree (1975) used — and that we have described, above — to model the time-dependent behavior of pulsating stars. It appears as though Stahler was unaware of Deupree's earlier development of this method to evaluate the boundary potential, as Deupree's (1975) paper is not among Stahler's list of references. We therefore assume that Stahler's numerical method was developed independently.


	In the chapter of this H_Book that focuses on a discussion of [[User:Tohline/Apps/DysonWongTori\|Dyson-Wong tori]], we have included the expression for the [[User:Tohline/Apps/DysonWongTori#RingPotential\|gravitational potential of a thin ring]] of mass, <math>~M</math>, that passes through the meridional plane at coordinate location, <math>~(\varpi^', z^') = (a, 0)</math>, as derived, for example, by [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)] and by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential W. D. MacMillan (1958; originally, 1930)], namely,		In the chapter of this H_Book that focuses on a discussion of [[User:Tohline/Apps/DysonWongTori\|Dyson-Wong tori]], we have included the expression for the [[User:Tohline/Apps/DysonWongTori#RingPotential\|gravitational potential of a thin ring]] of mass, <math>~M</math>, that passes through the meridional plane at coordinate location, <math>~(\varpi^', z^') = (a, 0)</math>, as derived, for example, by [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)] and by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential W. D. MacMillan (1958; originally, 1930)], namely,
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	</table>		</table>
	</div>		</div>


			Notice that the first of these two rewritten expressions aligns perfectly with our "[[User:Tohline/Appendix/Equation_templates#Other_Equations_with_Assigned_Templates\|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
			<table border="0" align="center" cellpadding="10"><tr><td align="center">
			{{ User:Tohline/Math/EQ TRApproximation }}
			</td>
			<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png\|225px\|link=User:Tohline/Apps/DysonWongTori#ThinRingContours]]</td>
			</tr></table>
			(See our [[User:Tohline/Apps/DysonWongTori#ThinRingContours\|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.)


	[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] has argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring element — positioned at various meridional coordinate locations throughout the mass distribution. According to his independent derivation, the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is (see his equation 11 and the explanatory text that follows it):		[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] has argued that a reasonably good approximation to the gravitational potential due to any extended axisymmetric mass distribution can be obtained by adding up the contributions due to many ''thin rings'' — <math>~\delta M(\varpi^', z^')</math> being the appropriate differential mass contributed by each ring element — positioned at various meridional coordinate locations throughout the mass distribution. According to his independent derivation, the differential contribution to the potential, <math>~\delta\Phi_g(\varpi, z)</math>, due to each differential mass element is (see his equation 11 and the explanatory text that follows it):

Tohline: /* Comparison With Other Related Derivations */

2018-05-27T03:37:42Z

Comparison With Other Related Derivations

← Older revision		Revision as of 03:37, 27 May 2018
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	</tr>		</tr>
	</table>		</table>
	we see that the second of these rewritten expressions for Deupree's <math>~\delta\Phi_B</math> aligns perfectly with our derived expression for the differential contribution to the potential of any axisymmetric mass distribution. It is therefore fair to say that Deupree~~'s expression for~~ the gravitational potential is derivable from a 3D Green's function that is written in terms of ''toroidal functions''.		we see that the second of these rewritten expressions for Deupree's <math>~\delta\Phi_B</math> aligns perfectly with our derived expression for the differential contribution to the potential of any axisymmetric mass distribution. It is therefore fair to say that the expression that Deupree used to determine the gravitational potential along the boundary of his modeled configurations is derivable from a 3D Green's function that is written in terms of ''toroidal functions''.

	===Cook (1977)===		===Cook (1977)===

Tohline: /* Deupree (1974) */

2018-05-27T03:34:08Z

Deupree (1974)

← Older revision		Revision as of 03:34, 27 May 2018
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	Apparently, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D R. G. Deupree (1974, ApJ, 194, 393 - 402)] was the first astronomer to employ self-gravitating, numerical hydrodynamic techniques to model ''nonlinear, nonradial stellar pulsations.'' He chose to carry out his simulations on a spherical coordinate mesh with his earliest simulations being restricted to the examination of 2D (axisymmetric) configurations. The outer boundary of his computational grid was identified as (see his §IIb) <font color="darkgreen">a spherical surface completely exterior to the star.</font>		Apparently, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D R. G. Deupree (1974, ApJ, 194, 393 - 402)] was the first astronomer to employ self-gravitating, numerical hydrodynamic techniques to model ''nonlinear, nonradial stellar pulsations.'' He chose to carry out his simulations on a spherical coordinate mesh with his earliest simulations being restricted to the examination of 2D (axisymmetric) configurations. The outer boundary of his computational grid was identified as (see his §IIb) <font color="darkgreen">a spherical surface completely exterior to the star.</font>

			====His Derived Expression for the Boundary Potential====
	In designing an algorithm to solve the Poisson equation, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] considered <font color="darkgreen">the major difficulty [to be] the evaluation of the potential boundary conditions.</font> He decided to determine the boundary potential via an evaluation of the ''integral representation'' of the Poisson equation; <font color="darkgreen">once the boundary conditions [were] specified, the potential inside the star [was] evaluated by [employing] the [http://adsabs.harvard.edu/abs/1964ApJ...139..306H Henyey method]</font> to solve the ''differential representation'' of the Poisson equation. The assumption of azimuthal symmetry meant that, for example, the density was specified at various meridional-plane locations, <math>~\rho(r,\theta)</math>, with <font color="darkgreen">each zone [being] considered as a uniform density circular ring.</font> He argued that <font color="darkgreen">the potential of [each such] ring at any point on the spherical boundary [could] easily be evaluated by</font> treating it as an infinitesimally thin ring of mass, <math>~\delta M = 2\pi a \rho ~\delta A</math> — where, <math>~\delta A</math> <font color="darkgreen">is the cross-sectional area of the [grid] zone, and <math>~a</math> is the radius of [that] ring</font> — then drawing upon the analytic analysis described by [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)] or, equivalently, by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential MacMillan (1958; originally 1930)]. Then, <font color="darkgreen">the total potential at the boundary is the sum of the potentials from all of the [meridional-plane] zones.</font>		In designing an algorithm to solve the Poisson equation, [http://adsabs.harvard.edu/abs/1974ApJ...194..393D Deupree (1974)] considered <font color="darkgreen">the major difficulty [to be] the evaluation of the potential boundary conditions.</font> He decided to determine the boundary potential via an evaluation of the ''integral representation'' of the Poisson equation; <font color="darkgreen">once the boundary conditions [were] specified, the potential inside the star [was] evaluated by [employing] the [http://adsabs.harvard.edu/abs/1964ApJ...139..306H Henyey method]</font> to solve the ''differential representation'' of the Poisson equation. The assumption of azimuthal symmetry meant that, for example, the density was specified at various meridional-plane locations, <math>~\rho(r,\theta)</math>, with <font color="darkgreen">each zone [being] considered as a uniform density circular ring.</font> He argued that <font color="darkgreen">the potential of [each such] ring at any point on the spherical boundary [could] easily be evaluated by</font> treating it as an infinitesimally thin ring of mass, <math>~\delta M = 2\pi a \rho ~\delta A</math> — where, <math>~\delta A</math> <font color="darkgreen">is the cross-sectional area of the [grid] zone, and <math>~a</math> is the radius of [that] ring</font> — then drawing upon the analytic analysis described by [https://archive.org/details/foundationsofpot033485mbp Kellogg (1929)] or, equivalently, by [https://www.amazon.com/Theory-Potential-W-D-Macmillan/dp/0486604861/ref=sr_1_2?s=books&ie=UTF8&qid=1503444466&sr=1-2&keywords=the+theory+of+the+potential MacMillan (1958; originally 1930)]. Then, <font color="darkgreen">the total potential at the boundary is the sum of the potentials from all of the [meridional-plane] zones.</font>

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	where, <math>~K(k)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind] for the modulus, <math>~k = \sqrt{1-c^2}</math>.		where, <math>~K(k)</math> is the [http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html complete elliptic integral of the first kind] for the modulus, <math>~k = \sqrt{1-c^2}</math>.

			====Comparison With Other Related Derivations====
	Now, given that Deupree chose to construct and evolve his models using a spherical coordinate system, he would have specified the relevant lengths, <math>~p</math> and <math>~q</math>, and each ring's differential cross-section, <math>~\delta A</math>, in terms of spherical coordinates. In an effort to more clearly illustrate the connection between Deupree's expression for the (differential contribution to the) boundary potential and the expression for the boundary potential that we have [[#For_Axisymmetric_Systems\|derived above for axisymmetric systems]], we will insert expressions for these terms that apply, instead, to a cylindrical-coordinate mesh.		Now, given that Deupree chose to construct and evolve his models using a spherical coordinate system, he would have specified the relevant lengths, <math>~p</math> and <math>~q</math>, and each ring's differential cross-section, <math>~\delta A</math>, in terms of spherical coordinates. In an effort to more clearly illustrate the connection between Deupree's expression for the (differential contribution to the) boundary potential and the expression for the boundary potential that we have [[#For_Axisymmetric_Systems\|derived above for axisymmetric systems]], we will insert expressions for these terms that apply, instead, to a cylindrical-coordinate mesh.

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	<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr] K(k) </math>		<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr] K(k) </math>
	</td>		</td>
	~~<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png\|225px\|link=User:Tohline/Apps/DysonWongTori#ThinRingContours]]</td>~~
	</tr>		</tr>

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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~- \frac{2G (~~2\pi a \rho ~~~\delta A) }{\pi } \cdot \frac{K(k) }{ \sqrt{(\varpi + ~~\varpi^'~~)^2 + (z - z^')^2 }} \, , </math>		<math>~- \frac{2G (\delta M) }{\pi } \cdot \frac{K(k) }{ \sqrt{(\varpi + a)^2 + (z - z^')^2 }} \, , </math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	~~(see our [[User:Tohline/Apps/DysonWongTori#ThinRingContours\|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation)~~ or it may be rewritten as,		or it may be rewritten as,

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

	~~Referring~~ back to the expression that was [[#For_Axisymmetric_Systems\|derived above for axisymmetric systems from a toroidal-function-based Green's function]], namely,		Notice that the first of these two rewritten expressions aligns perfectly with our "[[User:Tohline/Appendix/Equation_templates#Other_Equations_with_Assigned_Templates\|key equation]]" that gives the gravitational potential of an axisymmetric torus in the thin ring (TR) approximation, namely,
			<table border="0" align="center" cellpadding="10"><tr><td align="center">
			{{ User:Tohline/Math/EQ TRApproximation }}
			</td>
			<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png\|225px\|link=User:Tohline/Apps/DysonWongTori#ThinRingContours]]</td>
			</tr></table>
			(See our [[User:Tohline/Apps/DysonWongTori#ThinRingContours\|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.) Next, referring back to the expression that was [[#For_Axisymmetric_Systems\|derived above for axisymmetric systems from a toroidal-function-based Green's function]], namely,

	<table border="0" align="center">		<table border="0" align="center">
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	</tr>		</tr>
	</table>		</table>
			we see that the second of these rewritten expressions for Deupree's <math>~\delta\Phi_B</math> aligns perfectly with our derived expression for the differential contribution to the potential of any axisymmetric mass distribution. It is therefore fair to say that Deupree's expression for the gravitational potential is derivable from a 3D Green's function that is written in terms of ''toroidal functions''.
	we see that our expression for the differential contribution to the potential ~~exactly matches the expression that has been obtained, here, from Deupree's work~~. It is therefore fair to say that Deupree's expression for the gravitational potential is derivable from a 3D Green's function that is written in terms of ''toroidal functions''.

	===Cook (1977)===		===Cook (1977)===

Tohline: /* Deupree (1974) */

2018-05-27T02:52:56Z

Deupree (1974)

← Older revision		Revision as of 02:52, 27 May 2018
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	</td>		</td>
	<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png\|225px\|link=User:Tohline/Apps/DysonWongTori#ThinRingContours]]</td>		<td align="center" rowspan="2">[[File:FlatColorContoursCropped.png\|225px\|link=User:Tohline/Apps/DysonWongTori#ThinRingContours]]</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{2G (2\pi a \rho ~\delta A) }{\pi } \cdot \frac{K(k) }{ \sqrt{(\varpi + \varpi^')^2 + (z - z^')^2 }} \, , </math>
			</td>
			</tr>
			</table>
			(see our [[User:Tohline/Apps/DysonWongTori#ThinRingContours\|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation) or it may be rewritten as,

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

			<tr>
			<td align="right">
			<math>~\delta\Phi_B(\varpi,z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{G (2\pi a \rho ~\delta A) }{\pi } \biggl[ \frac{k }{\sqrt{\varpi \varpi^'}} \biggr] K(k) </math>
			</td>
	</tr>		</tr>

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	</table>		</table>

	~~(See our [[User:Tohline/Apps/DysonWongTori#ThinRingContours\|accompanying discussion]] for more information on the meridional-plane contour plot that is displayed to the right of this equation.)~~ Referring back to the expression that was [[#For_Axisymmetric_Systems\|derived above for axisymmetric systems from a toroidal-function-based Green's function]], namely,		Referring back to the expression that was [[#For_Axisymmetric_Systems\|derived above for axisymmetric systems from a toroidal-function-based Green's function]], namely,

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

Tohline: /* Toroidal Functions */

2018-05-08T03:45:24Z

Toroidal Functions

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	Note that, in this Guatschi (~~2965~~) article, a description of the "<b>procedure</b> <i>toroidal</i>" begins on p. 490, near the bottom of the right-hand column.		Note that, in this Guatschi (1965) article, a description of the "<b>procedure</b> <i>toroidal</i>" begins on p. 490, near the bottom of the right-hand column.
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Tohline: /* Toroidal Functions */

2018-05-08T03:37:48Z

Toroidal Functions

← Older revision		Revision as of 03:37, 8 May 2018
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	* [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G A. Gil, J. Segura & N. M. Temme (2000, Journal of Computational Physics, 161, 204 - 217)] — ''Computing Toroidal Functions for Wide Ranges of the Parameters''		* [http://adsabs.harvard.edu/abs/2000JCoPh.161..204G A. Gil, J. Segura & N. M. Temme (2000, Journal of Computational Physics, 161, 204 - 217)] — ''Computing Toroidal Functions for Wide Ranges of the Parameters''
	* [http://adsabs.harvard.edu/abs/2000CoPhC.124..104S J. Segura & A. Gil (2000, Computer Physics Communications, 124, 104 - 122)] — '' Evaluation of Toroidal Harmonics''		* [http://adsabs.harvard.edu/abs/2000CoPhC.124..104S J. Segura & A. Gil (2000, Computer Physics Communications, 124, 104 - 122)] — '' Evaluation of Toroidal Harmonics''
			* [http://adsabs.harvard.edu/abs/1997JMP....38.3679B J. W. Bates (1997, Journal of Mathematical Physics, vol. 38, issue 7, 3679-3691)] — ''On Toroidal Green's Functions''
	* [https://dl-acm-org.libezp.lib.lsu.edu/citation.cfm?id=365474&picked=prox W. Guatschi (1965, Communications of the ACM, 8, 488 - 492)] — ''Algorithms (Algorithm 259): Legendre Functions for Arguments Larger than One''		* [https://dl-acm-org.libezp.lib.lsu.edu/citation.cfm?id=365474&picked=prox W. Guatschi (1965, Communications of the ACM, 8, 488 - 492)] — ''Algorithms (Algorithm 259): Legendre Functions for Arguments Larger than One''
	<table border="0" align="center" width="92%">		<table border="0" align="center" width="92%">

Tohline: /* Toroidal Functions */

2018-05-08T03:20:55Z

Toroidal Functions

← Older revision		Revision as of 03:20, 8 May 2018
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	* [https://www.jstor.org/stable/2369515?seq=1#page_scan_tab_contents A. B. Basset (1893, American Journal of Mathematics, vol. 15, No. 4, pp. 287 - 302)] — ''On Toroidal Functions''		* [https://www.jstor.org/stable/2369515?seq=1#page_scan_tab_contents A. B. Basset (1893, American Journal of Mathematics, vol. 15, No. 4, pp. 287 - 302)] — ''On Toroidal Functions''
			* [http://www.jstor.org/stable/109363?seq=1#page_scan_tab_contents W. M. Hicks (1881, Philosophical Transactions of the Royal Society of London, vol. 172, pp. 609 - 652)] — ''On Toroidal Functions''

	==Reviews==		==Reviews==

Tohline: /* Toroidal Functions */

2018-05-07T21:26:26Z

Toroidal Functions

← Older revision		Revision as of 21:26, 7 May 2018
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			* [https://www.jstor.org/stable/2369515?seq=1#page_scan_tab_contents A. B. Basset (1893, American Journal of Mathematics, vol. 15, No. 4, pp. 287 - 302)] — ''On Toroidal Functions''

	==Reviews==		==Reviews==