User:Tohline/Apps/RotatingPolytropes - Revision history

Tohline: /* Principle Governing Equations */

2019-08-04T00:29:04Z

Principle Governing Equations

← Older revision		Revision as of 00:29, 4 August 2019
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	===Principle Governing Equations===		===Principle Governing Equations===
			<!--
	We begin with an [[User:Tohline/AxisymmetricConfigurations/PGE#Governing_Equations_.28SPH..29\|Eulerian formulation of the principle governing equations written in spherical coordinates for an axisymmetric configuration]], namely,		We begin with an [[User:Tohline/AxisymmetricConfigurations/PGE#Governing_Equations_.28SPH..29\|Eulerian formulation of the principle governing equations written in spherical coordinates for an axisymmetric configuration]], namely,

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	Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:		Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:
			-->

	<table align="center" border="1" cellpadding="10"><tr><td align="center">		<table align="center" border="1" cellpadding="10">
			<tr>
			<th align="center">Spherical Coordinate Base</th>
			</tr>
			<tr><td align="center">

	<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />		<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

Tohline: /* Principle Governing Equations */

2019-08-03T23:46:57Z

Principle Governing Equations

← Older revision		Revision as of 23:46, 3 August 2019
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	Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:		Given that our aim is to construct steady-state configurations, we should set the partial time-derivative of all scalar quantities to zero; in addition, we will assume that both meridional-plane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the left-hand-sides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:

	<~~div~~ align="center">		<table align="center" border="1" cellpadding="10"><tr><td align="center">
	<span id="~~Continuity~~"><font color="#770000">'''Equation ~~of Continuity~~'''</font></span><br />
			<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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	<td align="right">		<td align="right">
	<math>~		<math>~
	\frac{\partial ~~\rho~~}{\partial t}		\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr]
	+ \biggl[ ~~\frac{1}{~~r^2} \frac{\partial ~~(\rho r^2~~ \~~dot{r})~~}{\partial r}		+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr)
	+ \frac{1}{r\sin\theta} \frac{\partial }{\partial\theta} \biggl( \~~rho~~ \~~dot~~\~~theta r~~ \~~sin~~\theta \biggr)		</math>
	~~\biggr]~~</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~0</math>		<math>~4\pi G\rho</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>


	<span id="PGE:Euler">The Two Relevant Components of the<br />		<span id="PGE:Euler">The Two Relevant Components of the<br />
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	<td align="right">		<td align="right">
	<math>		<math>
	~~\biggl\{ \frac{\partial \dot{r}}{\partial t} + \biggl[ \dot{r} \frac{\partial \dot{r}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\}~~		~0
	~~- r {\dot\theta}^2~~
	</math>		</math>
	</td>		</td>
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	<td align="right">		<td align="right">
	<math>		<math>
	r \biggl\{ \frac{\partial \dot{\theta}}{\partial t} + \biggl[ \dot{\theta} \frac{\partial \dot{\theta}}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \dot{r}}{\partial\theta} \biggr] \biggr\} + 2\dot{r} \dot\theta		~0
	</math>		</math>
	</td>		</td>
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	</table>		</table>

	~~<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />~~		</td></tr></table>
	~~<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />~~

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

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~~~
	~~\biggl\{ \frac{\partial \epsilon}{\partial t} + \biggl[ \dot{r} \frac{\partial \epsilon}{\partial r} \biggr] + \biggl[ \dot\theta \frac{\partial \epsilon}{\partial\theta} \biggr] \biggr\}~~
	~~+ P\biggl\{ \frac{\partial }{\partial t} \biggl( \frac{1}{\rho}\biggr)~~
	~~+ \biggl[ \dot{r} \frac{\partial }{\partial r} \biggl( \frac{1}{\rho}\biggr) \biggr]~~
	~~+ \biggl[ \dot\theta \frac{\partial }{\partial\theta} \biggl( \frac{1}{\rho}\biggr) \biggr] \biggr\}~~
	~~</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~0</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~


	~~<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />~~

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

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~~~
	~~\frac{1}{r^2} \frac{\partial }{\partial r} \biggl[ r^2 \frac{\partial \Phi }{\partial r} \biggr]~~
	~~+ \frac{1}{r^2 \sin\theta} \frac{\partial }{\partial \theta}\biggl(\sin\theta ~ \frac{\partial \Phi}{\partial\theta}\biggr)~~
	~~</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	</td>
	~~<td align="left">~~
	~~<math>~4\pi G\rho</math>~~
	~~</td>~~
	</tr>
	</table>

	~~</div~~>

	===Choices and Adaptations===		===Choices and Adaptations===

Tohline: /* Principle Governing Equations */

2019-08-03T23:40:40Z

Principle Governing Equations

Show changes

Tohline: /* Principle Governing Equations */

2019-08-03T23:32:56Z

Principle Governing Equations

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Tohline: /* Chandrasekhar's Approach */

2019-08-02T21:42:54Z

Chandrasekhar's Approach

← Older revision		Revision as of 21:42, 2 August 2019
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	</div>		</div>

	this "fundamental equation" can be rewritten strictly in terms of the density, <math>~\rho</math>. (~~We will return~~ to this, ~~below~~.)		this "fundamental equation" can be rewritten strictly in terms of the configuration's axisymmetric density distribution, <math>~\rho(r,\theta)</math>. Chandrasekhar first adopts a dimensionless function, <math>~\Theta(\xi, \mu)</math>, that is related to the normalized density via the expression,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Theta</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl( \frac{\rho}{\rho_c} \biggr)^{1/n}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ P = K \rho^{1+1/n}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\rho_c^{1+1/n} K \Theta^{n+1} \, .
			</math>
			</td>
			</tr>
			</table>
			(In doing this, Chandrasekhar is obviously adopting an a path toward solution that parallel's the familiar method used to determine the [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres\|structure of isolated, nonrotating polytropes]].) Adopting the additional pair of dimensionless variables,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\xi</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~r \biggl[ \frac{(n+1)K}{4\pi G} ~\rho_c^{1/n - 1} \biggr]^{-1 / 2} </math>
			</td>
			<td align="center">      and
			<td align="right">
			<math>~v</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\frac{\omega_0^2}{2\pi G \rho_c} </math>
			</td>
			</tr>
			</table>

			Chandrasekhar's "fundamental equation" becomes,
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~
			\frac{1}{\xi^2} \frac{\partial }{\partial \xi} \biggl[\xi^2 ~\frac{\partial \Theta}{\partial \xi} \biggr]
			+ \frac{1}{\xi^2} \frac{\partial }{\partial \mu} \biggl[ (1-\mu^2) ~\frac{\partial \Theta}{\partial\mu} \biggr]
			</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~v -\Theta^n \, .</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 392, Eq. (11)
			</td>
			</tr>
			</table>

	===Milne's Additional Specificity===		===Milne's Additional Specificity===

Tohline: /* Distortions Introduced by Uniform Rotation */

2019-08-02T21:20:09Z

Distortions Introduced by Uniform Rotation

← Older revision		Revision as of 21:20, 2 August 2019
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	* In place of <math>~P</math>, Milne uses <math>~W</math>;		* In place of <math>~P</math>, Milne uses <math>~W</math>;
	* For the gravitational potential, Milne and Chandrasekhar both adopt the convention, <math>~V \equiv -\Phi </math>, while James adopts the notation, <math>~\Psi \equiv - \Phi</math>;		* For the gravitational potential, Milne and Chandrasekhar both adopt the convention, <math>~V \equiv -\Phi </math>, while James adopts the notation, <math>~\Psi \equiv - \Phi</math>;
	* James includes terms with azimuthal derivatives in his equation set; these terms are set to zero (as reflected here) when ~~imposing~~ axisymmetric ~~conditions~~.		* James includes terms with azimuthal derivatives in his equation set; these terms are set to zero (as reflected here) when seeking axisymmetric structures.
	</td>		</td>
	</tr>		</tr>
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	</td></tr></table>		</td></tr></table>


			===Chandrasekhar's Approach===

	Using the two components of the Euler equation to express spatial derivatives of the gravitational potential in terms of spatial derivatives of the gas pressure, that is, rewriting them in the form,		Using the two components of the Euler equation to express spatial derivatives of the gravitational potential in terms of spatial derivatives of the gas pressure, that is, rewriting them in the form,

Tohline: /* Choices and Adaptations */

2019-08-02T21:16:32Z

Choices and Adaptations

← Older revision		Revision as of 21:16, 2 August 2019
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	===Choices and Adaptations===		===Choices and Adaptations===

	Among the set of [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|simple rotation profiles]] that have been adopted by various research groups over the years, ~~both~~ [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)] ~~and~~ [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)] ~~chose~~ what would generally be considered the simplest, which is the assumption of uniform rotation <math>~(\dot\varphi = \omega_0)</math>. This means that,		Among the set of [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential\|simple rotation profiles]] that have been adopted by various research groups over the years, [https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], [https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)] and [https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)] all choose what would generally be considered the simplest, which is the assumption of uniform rotation <math>~(\dot\varphi = \omega_0)</math>. This means that,

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

	And in place of the ~~polar coordinate~~, <math>~\theta</math>, they ~~both adopted~~ the coordinate,		And in place of the co-latitude, <math>~\theta</math>, they all adopt the coordinate,
	<div align="center">		<div align="center">
	<math>~\mu \equiv \cos\theta ~~~\Rightarrow ~~~ \frac{\partial}{\partial\mu} = - \frac{\partial}{\sin\theta \partial\theta} \, .</math>		<math>~\mu \equiv \cos\theta ~~~\Rightarrow ~~~ \frac{\partial}{\partial\mu} = - \frac{\partial}{\sin\theta \partial\theta} \, .</math>
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	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (4')		[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (4')<br />
			[https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)], p. 553, Eq. (2.1)
	</td>		</td>
	</tr>		</tr>
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	<tr>		<tr>
	<td align="center" colspan="4">		<td align="center" colspan="4">
	[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)]], top of p. 126<br />		[https://ui.adsabs.harvard.edu/abs/1923MNRAS..83..118M/abstract Milne (1923)], top of p. 126<br />
	[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (3')		[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 391, Eq. (3')<br />
			[https://ui.adsabs.harvard.edu/abs/1964ApJ...140..552J/abstract James (1964)], p. 553, Eqs. (2.2) & (2.3)
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="left" colspan="4">		<td align="left" colspan="4">
	~~NOTE~~: In place of <math>~P</math>, Milne uses <math>~W</math>; ~~and for~~ the gravitational potential, Milne and Chandrasekhar both adopt the convention, <math>~V \equiv -\Phi \~~, .~~</math>		NOTES:
			* In place of <math>~P</math>, Milne uses <math>~W</math>;
			* For the gravitational potential, Milne and Chandrasekhar both adopt the convention, <math>~V \equiv -\Phi </math>, while James adopts the notation, <math>~\Psi \equiv - \Phi</math>;
			* James includes terms with azimuthal derivatives in his equation set; these terms are set to zero (as reflected here) when imposing axisymmetric conditions.
	</td>		</td>
	</tr>		</tr>

Tohline: /* Solutions */

2019-07-31T17:09:50Z

Solutions

← Older revision		Revision as of 17:09, 31 July 2019
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	<td align="center" colspan="3">		<td align="center" colspan="3">
	[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eqs. (37<sub>1</sub> & 37<sub>2</sub>)		[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eqs. (37<sub>1</sub> & 37<sub>2</sub>)
			</td>
			</tr>
			</table>

			Realizing that "<font color="darkgreen">the boundary <math>~\xi_0</math> is given by <math>~\Theta = 0</math></font>," Chandrasekhar deduced as well that,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\xi_0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\xi_1 + \frac{v}{\|\theta_1^'\|} \biggl[
			\psi_0(\xi_1) - \frac{5}{6} ~ \frac{\xi_1^2 \psi_2(\xi_1) P_2(\mu)}{3\psi_2(\xi_1) + \xi_1 \psi_2^'(\xi_1)} \biggr] \, ,
			</math>
			</td>
			</tr>
			<tr>
			<td align="center" colspan="3">
			[https://ui.adsabs.harvard.edu/abs/1933MNRAS..93..390C/abstract Chandrasekhar (1933)], p. 395, Eq. (38)
	</td>		</td>
	</tr>		</tr>

Tohline: /* Solutions */

2019-07-31T17:00:15Z

Solutions

← Older revision		Revision as of 17:00, 31 July 2019
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	</table>		</table>

	He deduces that, to lowest order in a Legendre series ~~expansion~~,		He deduces that, to lowest order in a [https://en.wikipedia.org/wiki/Legendre_polynomials#Rodrigues'_formula_and_other_explicit_formulas Legendre polynomial series],
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>
	where, <math>~\xi_1</math> is "<font color="darkgreen">the first zero of the Emden's function with index <math>~n</math></font>," and <math>~\psi_0</math> and <math>~\psi_2</math> satisfy the 2<sup>nd</sup>-order ODEs,		where: <math>~P_2(\mu) = \tfrac{1}{2}(3\mu^2 - 1)</math>; <math>~\xi_1</math> is "<font color="darkgreen">the first zero of the Emden's function with index <math>~n</math></font>"; and <math>~\psi_0</math> and <math>~\psi_2</math> satisfy the 2<sup>nd</sup>-order ODEs,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Tohline: /* Solutions */

2019-07-29T17:16:47Z

Solutions

← Older revision		Revision as of 17:16, 29 July 2019
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	</tr>		</tr>
	</table>		</table>
			(In doing this, Chandrasekhar is obviously adopting an a path toward solution that parallel's the familiar method used to determine the [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres\|structure of isolated, nonrotating polytropes]].) He then argues that, for slowly rotating configurations, the function, <math>~\Theta(\xi,\mu)</math> can be effectively expressed as a small perturbation (in the two-dimensional meridional plane), <math>~v\Psi(\xi,\mu) \ll 1</math>, added to the radially dependent "Emden's function," <math>~\theta(\xi)</math>, that defines the structure of non-rotating polytropic configurations; that is,
	He then argues that, for slowly rotating configurations, the function, <math>~\Theta(\xi,\mu)</math> can be effectively expressed as a small perturbation (in the two-dimensional meridional plane), <math>~v\Psi(\xi,\mu) \ll 1</math>, added to the radially dependent function, <math>~\theta(\xi)</math>, that defines the structure of non-rotating polytropic configurations; that is,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</table>		</table>

	He deduces that, to lowest order in ~~the~~ Legendre series,		He deduces that, to lowest order in a Legendre series expansion,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>
	where <math>~\psi_0</math> and <math>~\psi_2</math> satisfy the 2<sup>nd</sup>-order ODEs,		where, <math>~\xi_1</math> is "<font color="darkgreen">the first zero of the Emden's function with index <math>~n</math></font>," and <math>~\psi_0</math> and <math>~\psi_2</math> satisfy the 2<sup>nd</sup>-order ODEs,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">