ParabolicDensity/GravPot - Revision history

Joel2: /* Parabolic Density Distribution */

2024-08-03T17:42:32Z

Parabolic Density Distribution

← Older revision		Revision as of 13:42, 3 August 2024
Line 406:		Line 406:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>\frac{2}{a_i} </math>		<math>\frac{2}{a_i^2} </math>
	</td>		</td>
	</tr>		</tr>

Joel2: /* Parabolic Density Distribution */

2024-07-30T22:58:44Z

Parabolic Density Distribution

← Older revision		Revision as of 18:58, 30 July 2024
Line 352:		Line 352:
	After studying {{ Ferrers1877full }} and the relevant sections of both [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] and [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], we present here an example of a parabolic density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials\| separate chapter]], the potential that it generates is sometimes referred to as a [[ThreeDimensionalConfigurations/FerrersPotential\|''Ferrers'' potential]], for the exponent, n = 1.		After studying {{ Ferrers1877full }} and the relevant sections of both [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] and [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], we present here an example of a parabolic density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials\| separate chapter]], the potential that it generates is sometimes referred to as a [[ThreeDimensionalConfigurations/FerrersPotential\|''Ferrers'' potential]], for the exponent, n = 1.

	In our [[~~SSC~~/~~Structure/OtherAnalyticModels#ParabolicPotential~~\|accompanying discussion]] we find that,		In our [[ThreeDimensionalConfigurations/FerrersPotential\|accompanying discussion]] we find that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: /* Parabolic Density Distribution */

2024-07-30T22:55:50Z

Parabolic Density Distribution

← Older revision		Revision as of 18:55, 30 July 2024
Line 352:		Line 352:
	After studying {{ Ferrers1877full }} and the relevant sections of both [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] and [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], we present here an example of a parabolic density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials\| separate chapter]], the potential that it generates is sometimes referred to as a [[ThreeDimensionalConfigurations/FerrersPotential\|''Ferrers'' potential]], for the exponent, n = 1.		After studying {{ Ferrers1877full }} and the relevant sections of both [<b>[[Appendix/References#EFE\|<font color="red">EFE</font>]]</b>] and [<b>[[Appendix/References#BT87\|<font color="red">BT87</font>]]</b>], we present here an example of a parabolic density distribution whose gravitational potential has an analytic prescription. As is discussed in a [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#Inhomogeneous_Ellipsoids_Leading_to_Ferrers_Potentials\| separate chapter]], the potential that it generates is sometimes referred to as a [[ThreeDimensionalConfigurations/FerrersPotential\|''Ferrers'' potential]], for the exponent, n = 1.

	In our [[~~ThreeDimensionalConfigurations~~/~~HomogeneousEllipsoids~~#~~GravFor1~~\|accompanying discussion]] we find that,		In our [[SSC/Structure/OtherAnalyticModels#ParabolicPotential\|accompanying discussion]] we find that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Joel2: /* Parabolic Density Distribution */

2024-07-30T22:49:49Z

Parabolic Density Distribution

← Older revision		Revision as of 18:49, 30 July 2024
Line 633:		Line 633:
	</table>		</table>

	This matches the gravitational potential [[SSC/Structure/OtherAnalyticModels#ParabolicPotential\|derived for a parabolic density distribution using spherical coordinates]].		<font color="red">This matches the gravitational potential</font> [[SSC/Structure/OtherAnalyticModels#ParabolicPotential\|derived for a parabolic density distribution using spherical coordinates]].

	=See Also=		=See Also=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2 at 22:40, 30 July 2024

2024-07-30T22:40:52Z

← Older revision		Revision as of 18:40, 30 July 2024
Line 632:		Line 632:
	</tr>		</tr>
	</table>		</table>

			This matches the gravitational potential [[SSC/Structure/OtherAnalyticModels#ParabolicPotential\|derived for a parabolic density distribution using spherical coordinates]].

	=See Also=		=See Also=

	{{ SGFfooter }}		{{ SGFfooter }}

Joel2 at 22:32, 30 July 2024

2024-07-30T22:32:46Z

← Older revision		Revision as of 18:32, 30 July 2024
Line 613:		Line 613:
	</tr>		</tr>
	</table>		</table>
	Hence, for a sphere with a parabolic density distribution, we find,		<span id="ParabolicPotential">Hence, for a sphere with a parabolic density distribution, we find,</span>

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

Joel2: /* Parabolic Density Distribution */

2024-07-30T00:32:12Z

Parabolic Density Distribution

@@ Line 578: / Line 578: @@
 </tr>
 </table>
-These results match our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#For_Spheres_(aℓ_=_am_=_as)|separate derivations in the case of a sphere.  Specifically,
+These results match our [[ThreeDimensionalConfigurations/HomogeneousEllipsoids#For_Spheres_(aℓ_=_am_=_as)|separate derivations in the case of a sphere]].  Specifically,
 <table border="0" cellpadding="5" align="center">
@@ Line 609: / Line 609: @@
 <math>
 \frac{2}{5a_1^2} \, .
 </math>
    </td>

Joel2: /* Parabolic Density Distribution */

2024-07-30T00:19:02Z

Parabolic Density Distribution

@@ Line 570: / Line 570: @@
    <td align="center">
 <math>\approx</math>
    </td>
    <td align="left">

Joel2: /* Uniform-Density Reminders */

2024-07-29T18:34:19Z

Uniform-Density Reminders

← Older revision		Revision as of 14:34, 29 July 2024
Line 344:		Line 344:
	</table>		</table>
	This matches the expression for the gravitational potential inside (and on the surface) of a uniform-density sphere, as we have derived in an [[SSC/Structure/UniformDensity#UniformSpherePotential\|accompanying chapter]].		This matches the expression for the gravitational potential inside (and on the surface) of a uniform-density sphere, as we have derived in an [[SSC/Structure/UniformDensity#UniformSpherePotential\|accompanying chapter]].


	===Parabolic Density Distribution===		===Parabolic Density Distribution===

Joel2: /* Gravitational Potential */

2024-07-29T18:33:16Z

Gravitational Potential

@@ Line 492: / Line 492: @@
 In the case of an axisymmetric <math>(a_m = a_\ell)</math>, but nearly spherical configuration,
 =See Also=
 {{ SGFfooter }}