User:Tohline/Apps/ReviewStahler83 - Revision history

Tohline: /* Determining the Gravitational Potential */

2018-04-05T22:07:13Z

Determining the Gravitational Potential

← Older revision		Revision as of 22:07, 5 April 2018
Line 173:		Line 173:
	===Determining the Gravitational Potential===		===Determining the Gravitational Potential===

	In the chapter of this H_Book ~~whose focus is the~~ 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">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 186:		Line 186:
	<td align="left">		<td align="left">
	<math>~		<math>~
	- \biggl[ \frac{2GMc}{\pi\rho_1}\biggr] K(k^2)		- \biggl[ \frac{2GMc}{\pi\rho_1}\biggr] K(k)
	</math>		</math>
	</td>		</td>
Line 200:		Line 200:
	<td align="left">		<td align="left">
	<math>~		<math>~
	- \biggl[ \frac{2GM}{\pi } \biggr]\frac{1}{\sqrt{(\varpi+a)^2 + z^2}} \times K(k^2) \, ,		- \biggl[ \frac{2GM}{\pi } \biggr]\frac{1}{\sqrt{(\varpi+a)^2 + z^2}} \times K(k) \, ,
	</math>		</math>
	</td>		</td>
Line 213:		Line 213:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~k^2</math>		<math>~k</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 220:		Line 220:
	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{4a\varpi }{(\varpi+a)^2 + z^2} \, .		\biggl[ \frac{4a\varpi }{(\varpi+a)^2 + z^2} \biggr]^{1 / 2} \, .
	</math>		</math>
	</td>		</td>
Line 277:		Line 277:
	</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)].		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>.

	From our broad analysis of the integral Poisson equation expressed in cylindrical coordinates, we can independently state — see the discussion of [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|our original derivation]], or [[User:Tohline/Apps/DysonWongTori#Our_Integral_Expressions\|separate summary]] — that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,		From our broad analysis of the integral Poisson equation expressed in cylindrical coordinates, we can independently state — see the discussion of [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|our original derivation]], or [[User:Tohline/Apps/DysonWongTori#Our_Integral_Expressions\|separate summary]] — that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,

Tohline: /* Determining the Gravitational Potential */

2018-04-05T16:24:44Z

Determining the Gravitational Potential

← Older revision		Revision as of 16:24, 5 April 2018
Line 279:		Line 279:
	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)].		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)].

	~~We now know~~ — see the discussion of [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|our original derivation]], or [[User:Tohline/Apps/DysonWongTori#Our_Integral_Expressions\|separate summary]] — that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,		From our broad analysis of the integral Poisson equation expressed in cylindrical coordinates, we can independently state — see the discussion of [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|our original derivation]], or [[User:Tohline/Apps/DysonWongTori#Our_Integral_Expressions\|separate summary]] — that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 315:		Line 315:
	</table>		</table>
	</div>		</div>
	Recognizing that, for axisymmetric structures, the differential mass element is, <math>~dM^' = 2\pi \rho(\varpi^', z^') \varpi^' d\varpi^' dz^'</math>, this integral ~~expressions~~ may be rewritten as,		Recognizing that, for axisymmetric structures, the differential mass element is, <math>~dM^' = 2\pi \rho(\varpi^', z^') \varpi^' d\varpi^' dz^'</math>, this integral expression may be rewritten as,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Solution Technique */

2018-04-05T16:16:39Z

Solution Technique

← Older revision		Revision as of 16:16, 5 April 2018
Line 277:		Line 277:
	</table>		</table>
	</div>		</div>
	Stahler's expression 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)].		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)].

	We now know — see the discussion of [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|our original derivation]], or [[User:Tohline/Apps/DysonWongTori#Our_Integral_Expressions\|separate summary]] — that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,		We now know — see the discussion of [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|our original derivation]], or [[User:Tohline/Apps/DysonWongTori#Our_Integral_Expressions\|separate summary]] — that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,
Line 315:		Line 315:
	</table>		</table>
	</div>		</div>
			Recognizing that, for axisymmetric structures, the differential mass element is, <math>~dM^' = 2\pi \rho(\varpi^', z^') \varpi^' d\varpi^' dz^'</math>, this integral expressions may be rewritten as,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Phi(\varpi, z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{2G}{\varpi^{1 / 2}} \int\int (\varpi^')^{1 / 2} \mu K(\mu) \biggl[ \frac{dM^'}{2\pi \varpi^'} \biggr] </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{G}{\pi} \int\int \biggl[ \frac{1}{\varpi^'\varpi}\biggr]^{1 / 2} \biggl[\frac{4\varpi^' \varpi}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} K(\mu) dM^' </math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{2G}{\pi} \int\int \biggl[\frac{1}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} K(\mu) dM^' \, .</math>
			</td>
			</tr>
			</table>
			</div>
			We see, therefore, that our differential contribution to the potential exactly matches [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler's (1983a)].


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Determining the Gravitational Potential */

2018-04-05T04:28:27Z

Determining the Gravitational Potential

← Older revision		Revision as of 04:28, 5 April 2018
Line 278:		Line 278:
	</div>		</div>
	Stahler's expression 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)].		Stahler's expression 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)].

			We now know — see the discussion of [[User:Tohline/2DStructure/ToroidalCoordinates#Expression_for_the_Axisymmetric_Potential\|our original derivation]], or [[User:Tohline/Apps/DysonWongTori#Our_Integral_Expressions\|separate summary]] — that the exact integral expression for the gravitational potential due to any axisymmetric mass-density distribution, <math>~\rho(\varpi^', z^')</math>, is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Phi(\varpi, z)</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \frac{2G}{\varpi^{1 / 2}} \int\int (\varpi^')^{1 / 2} \mu K(\mu) \rho(\varpi^', z^') d\varpi^' dz^' \, ,</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\mu^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[\frac{4\varpi^' \varpi}{(\varpi + \varpi^')^2 + (z^' - z)^2} \biggr] \, .
			</math>
			</td>
			</tr>
			</table>
			</div>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Determining the Gravitational Potential */

2018-04-05T04:05:13Z

Determining the Gravitational Potential

← Older revision		Revision as of 04:05, 5 April 2018
Line 227:		Line 227:
	</div>		</div>

	[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] has ~~reasoned~~ that 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">

Tohline: /* Determining the Gravitational Potential */

2018-04-05T03:56:32Z

Determining the Gravitational Potential

← Older revision		Revision as of 03:56, 5 April 2018
Line 173:		Line 173:
	===Determining the Gravitational Potential===		===Determining the Gravitational Potential===

	In the chapter of this H_Book whose focus is the 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 MacMillan (1958)], namely,		In the chapter of this H_Book whose focus is the 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">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 271:		Line 271:
	<math>~		<math>~
	- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}		- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}
	\times K\biggl\{ \biggl[ \frac{4\varpi \varpi^'}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .		\times K\biggl\{ \biggl[ \frac{4\varpi^' \varpi}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .
	</math>		</math>
	</td>		</td>
Line 277:		Line 277:
	</table>		</table>
	</div>		</div>
	Stahler's expression is clearly the same as the expressions presented by [https://archive.org/details/foundationsofpot033485mbp 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 MacMillan (1958)].		Stahler's expression 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)].

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Determining the Gravitational Potential */

2018-04-05T03:48:09Z

Determining the Gravitational Potential

← Older revision		Revision as of 03:48, 5 April 2018
Line 173:		Line 173:
	===Determining the Gravitational Potential===		===Determining the Gravitational Potential===

	In ~~our extensive~~ 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 MacMillan (1958)], namely,		In the chapter of this H_Book whose focus is the 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 MacMillan (1958)], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 227:		Line 227:
	</div>		</div>

	~~According to~~ [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)], the ~~contribution~~ to the ~~gravitational potential at coordinate location~~ <math>~(\varpi, z)</math> ~~due to a~~ differential mass element, <math>~\delta ~~M</math>, located at coordinate <math>~~~(\varpi^', z^')</math> is (see his equation 11 and the explanatory text that follows it):		[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)] has reasoned that 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">
Line 240:		Line 240:
	<td align="left">		<td align="left">
	<math>~		<math>~
	- \biggl[\frac{2}{\pi \varpi^'}\biggr] \frac{\delta M}{[(\alpha + 1)^2 + \beta^2]^{1 / 2}}		- \biggl[\frac{2G}{\pi \varpi^'}\biggr] \frac{\delta M}{[(\alpha + 1)^2 + \beta^2]^{1 / 2}}
	\times K\biggl\{ \biggl[ \frac{4\alpha}{(\alpha+1)^2 + \beta^2} \biggr]^{1 / 2} \biggr\}		\times K\biggl\{ \biggl[ \frac{4\alpha}{(\alpha+1)^2 + \beta^2} \biggr]^{1 / 2} \biggr\}
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			<!--
	<tr>		<tr>
	<td align="right">		<td align="right">
Line 255:		Line 255:
	<td align="left">		<td align="left">
	<math>~		<math>~
	- \biggl[\frac{2}{\pi }\biggr] \frac{\delta M}{[(\varpi^' \alpha + \varpi^')^2 + (\varpi^' \beta)^2]^{1 / 2}}		- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi^' \alpha + \varpi^')^2 + (\varpi^' \beta)^2]^{1 / 2}}
	\times K\biggl\{ \biggl[ \frac{4\alpha (\varpi^')^2}{(\varpi^' \alpha+\varpi^')^2 + (\varpi^' \beta)^2} \biggr]^{1 / 2} \biggr\}		\times K\biggl\{ \biggl[ \frac{4\alpha (\varpi^')^2}{(\varpi^' \alpha+\varpi^')^2 + (\varpi^' \beta)^2} \biggr]^{1 / 2} \biggr\}
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			-->
	<tr>		<tr>
	<td align="right">		<td align="right">
Line 270:		Line 270:
	<td align="left">		<td align="left">
	<math>~		<math>~
	- \biggl[\frac{2}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}		- \biggl[\frac{2G}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}
	\times K\biggl\{ \biggl[ \frac{4\varpi \varpi^'}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .		\times K\biggl\{ \biggl[ \frac{4\varpi \varpi^'}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .
	</math>		</math>
Line 277:		Line 277:
	</table>		</table>
	</div>		</div>
			Stahler's expression is clearly the same as the expressions presented by [https://archive.org/details/foundationsofpot033485mbp 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 MacMillan (1958)].

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Determining the Gravitational Potential */

2018-04-05T01:26:01Z

Determining the Gravitational Potential

Show changes

Tohline: /* Determining the Gravitational Potential */

2018-04-04T23:17:58Z

Determining the Gravitational Potential

← Older revision		Revision as of 23:17, 4 April 2018
Line 224:		Line 224:
	</div>		</div>

			In our extensive discussion of [[User:Tohline/Apps/DysonWongTori#RingPotential\|Dyson-Wong tori]], we have included the expression for the gravitational potential of a thin ring as derived, for example, 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)], §102, Eq. (5) and by [https://archive.org/details/foundationsofpot033485mbp O. D. Kellogg (1929)], §III.4, Exercise (4)

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

			<tr>
			<td align="right">
			<math>~ \biggl[ \frac{2GMc}{\pi \rho_1} \biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{2GM}{\pi \rho_2} \biggr]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{2GM}{\pi } \biggr]\frac{1}{\sqrt{(\varpi+a)^2 + z^2}} \, ,</math>
			</td>
			</tr>
			</table>
			</div>

			and,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~k^2 = 1-c^2</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			1 - \frac{\rho_1^2}{\rho_2^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

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

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{[(\varpi+a)^2 + z^2] - [(\varpi-a)^2 + z^2]}{(\varpi+a)^2 + z^2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{4a\varpi }{(\varpi+a)^2 + z^2} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>


	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: /* Determining the Gravitational Potential */

2018-04-04T22:58:02Z

Determining the Gravitational Potential

← Older revision		Revision as of 22:58, 4 April 2018
Line 173:		Line 173:
	===Determining the Gravitational Potential===		===Determining the Gravitational Potential===

	~~[http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)]~~		According to [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)], the contribution to the gravitational potential at coordinate location <math>~(\varpi, z)</math> due to a differential mass element, <math>~\delta M</math>, located at coordinate <math>~(\varpi^', z^')</math> is (see his equation 11 and the explanatory text that follows it):
	According to [http://adsabs.harvard.edu/abs/1983ApJ...268..155S Stahler (1983a)], the contribution to the gravitational potential at coordinate location <math>~(\varpi, z)</math> due to a differential mass element, <math>~\delta M</math>, is (see his equation 11):
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 189:		Line 188:
	- \biggl[\frac{2}{\pi \varpi^'}\biggr] \frac{\delta M}{[(\alpha + 1)^2 + \beta^2]^{1 / 2}}		- \biggl[\frac{2}{\pi \varpi^'}\biggr] \frac{\delta M}{[(\alpha + 1)^2 + \beta^2]^{1 / 2}}
	\times K\biggl\{ \biggl[ \frac{4\alpha}{(\alpha+1)^2 + \beta^2} \biggr]^{1 / 2} \biggr\}		\times K\biggl\{ \biggl[ \frac{4\alpha}{(\alpha+1)^2 + \beta^2} \biggr]^{1 / 2} \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[\frac{2}{\pi }\biggr] \frac{\delta M}{[(\varpi^' \alpha + \varpi^')^2 + (\varpi^' \beta)^2]^{1 / 2}}
			\times K\biggl\{ \biggl[ \frac{4\alpha (\varpi^')^2}{(\varpi^' \alpha+\varpi^')^2 + (\varpi^' \beta)^2} \biggr]^{1 / 2} \biggr\}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			- \biggl[\frac{2}{\pi }\biggr] \frac{\delta M}{[(\varpi + \varpi^')^2 + (z^' - z)^2]^{1 / 2}}
			\times K\biggl\{ \biggl[ \frac{4\varpi \varpi^'}{(\varpi +\varpi^')^2 + (z^' - z)^2} \biggr]^{1 / 2} \biggr\} \, .
	</math>		</math>
	</td>		</td>