User:Tohline/Apps/OstrikerBodenheimerLyndenBell66 - Revision history

Tohline: /* Further Implementation by OB68 */

2019-08-10T21:44:18Z

Further Implementation by OB68

← Older revision		Revision as of 21:44, 10 August 2019
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~H(\vec{x}) - H_s</math>		<math>~H(\vec{x}) - H_s + \Phi_\mathrm{eff}(\vec{x}) - \Phi_\mathrm{eff,s}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</td>		</td>
	<td align="left">		<td align="left">
	<math>~~~- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}~~</math>		<math>~0</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Further Implementation by OB68 */

2019-08-10T21:39:41Z

Further Implementation by OB68

← Older revision		Revision as of 21:39, 10 August 2019
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	</tr>		</tr>
	</table>		</table>
			(Note that, as with OBL66, a different sign convention was adopted by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68] for the effective potential than we have used; that is, <math>~\mathfrak{B} = - \Phi_\mathrm{eff}</math>.)

	=See Also=		=See Also=

Tohline: /* Further Implementation by OB68 */

2019-08-10T21:34:56Z

Further Implementation by OB68

← Older revision		Revision as of 21:34, 10 August 2019
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\Rightarrow ~~~ \frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl\{\biggl[ (1 + \~~chi~~^2 )^{1 / 2} \biggr] - 1\~~biggr\}~~</math>		<math>~\Rightarrow ~~~ \frac{8A_\mathrm{F}}{B_\mathrm{F}} \Biggl[ \biggl\{ \biggl[1 + \biggl(\frac{\rho(\vec{x})}{B_\mathrm{F}}\biggr)^{2/3} \biggr]^{1 / 2} \biggr\} - 1 \Biggr]</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<math>~- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}</math>		<math>~- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}</math>
	</td>		</td>
			</tr>
			</table>
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \frac{\rho(\vec{x})}{B_\mathrm{F}} </math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\Biggl[ \biggl\{ \frac{B_\mathrm{F}}{8A_\mathrm{F}} \biggl[- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}\biggr] + 1 \biggr\}^2 - 1 \Biggr]^{3/2} \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="center" colspan="3">[https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68], p. 1090, Eq. (5)</td>
	</tr>		</tr>
	</table>		</table>

Tohline: /* Further Implementation by OB68 */

2019-08-10T21:09:15Z

Further Implementation by OB68

← Older revision		Revision as of 21:09, 10 August 2019
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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[ (1 + \chi^2 )^{1 / 2} \biggr] ~~\, .~~		\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[ (1 + \chi^2 )^{1 / 2} \biggr]
	</math>		</math>
	</td>		</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl\{ \biggl[1 + \biggl(\frac{\rho}{B_\mathrm{F}}\biggr)^{2/3} \biggr]^{1 / 2} \biggr\} \, .
			</math>
			</td>
			</tr>

			<tr>
			<td align="center" colspan="6">[https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68], p. 1090, Eq. (4)</td>
	</tr>		</tr>
			</table>
			Note that, using this expression, the enthalpy at the surface <math>~(\rho = 0)</math> is <math>~H_s = 8A_\mathrm{F}/B_\mathrm{F}</math>. ([[User:Tohline/SR#Barotropic_Structure\|Our tabulated expression for the enthalpy]] has been shifted by this constant value so that the enthalpy naturally goes to zero at the surface.) If we use <math>~\Phi_\mathrm{eff,s}</math> to denote the surface value of the effective potential, the constant in the algebraic hydrostatic-balance expression must be,
			<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="center" ~~colspan~~="3">~~[https:~~//~~ui.adsabs.harvard.edu~~/~~abs~~/~~1968ApJ...151.1089O~~/~~abstract OB68]~~, ~~p. 1090~~, ~~Eq.~~ (4)</td>		<td align="right">
			<math>~C_B</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~H_s + \Phi_\mathrm{eff,s}</math>
			</td>
			</tr>
			</table>
			Then, at every other spatial location, <math>~\vec{x}</math>, we must have,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~H(\vec{x}) - H_s</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~ \frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl\{\biggl[ (1 + \chi^2 )^{1 / 2} \biggr] - 1\biggr\}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~- \Phi_\mathrm{eff}(\vec{x}) + \Phi_\mathrm{eff,s}</math>
			</td>
	</tr>		</tr>
	</table>		</table>
	(Note that, using this expression, the enthalpy at the surface <math>~(\chi = 0)</math> is <math>~H_s = 8A_\mathrm{F}/B_\mathrm{F}</math>. [[User:Tohline/SR#Barotropic_Structure\|Our tabulated expression for the enthalpy]] has been shifted by this constant value so that the enthalpy naturally goes to zero at the surface.)

	=See Also=		=See Also=

Tohline: /* Further Implementation by OB68 */

2019-08-10T20:35:50Z

Further Implementation by OB68

← Older revision		Revision as of 20:35, 10 August 2019
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	</tr>		</tr>
	</table>		</table>
	(Note that ~~in order to retrieve [[User:Tohline/SR#Barotropic_Structure\|our tabulated~~ expression ~~for~~ the enthalpy~~]],~~ the ~~constant,~~ <math>~~~H_0~~ = 8A_\mathrm{F}/B_\mathrm{F}</math>~~, must be subtracted from this~~ [~~https~~:/~~/ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68~~] enthalpy ~~expression~~.)		(Note that, using this expression, the enthalpy at the surface <math>~(\chi = 0)</math> is <math>~H_s = 8A_\mathrm{F}/B_\mathrm{F}</math>. [[User:Tohline/SR#Barotropic_Structure\|Our tabulated expression for the enthalpy]] has been shifted by this constant value so that the enthalpy naturally goes to zero at the surface.)

	=See Also=		=See Also=

Tohline: /* Further Implementation by OB68 */

2019-08-09T21:41:42Z

Further Implementation by OB68

← Older revision		Revision as of 21:41, 9 August 2019
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	</math>		</math>
	</td>		</td>
			</tr>

			<tr>
			<td align="center" colspan="3">[https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68], p. 1090, Eq. (4)</td>
	</tr>		</tr>
	</table>		</table>
	(Note that in order to retrieve [[User:Tohline/SR#Barotropic_Structure\|our tabulated expression for the enthalpy]], the constant, <math>~H_0 = 8A_\mathrm{F}/B_\mathrm{F}</math>, must be subtracted from this OB68 enthalpy expression.)		(Note that in order to retrieve [[User:Tohline/SR#Barotropic_Structure\|our tabulated expression for the enthalpy]], the constant, <math>~H_0 = 8A_\mathrm{F}/B_\mathrm{F}</math>, must be subtracted from this [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract OB68] enthalpy expression.)

	=See Also=		=See Also=

Tohline: /* Further Implementation by OB68 */

2019-08-09T21:37:35Z

Further Implementation by OB68

← Older revision		Revision as of 21:37, 9 August 2019
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	</tr>		</tr>
	</table>		</table>
	and recall that, for a [[User:Tohline/SR#Barotropic_Structure\|barotropic equation of state]], we can make the substitution, <math>~\nabla P \rightarrow \rho\nabla H</math>, where <math>~H</math> is the fluid ~~entropy~~, then OBL66's hydrostatic balance equation — their equation (3) — can be rewritten as,		and recall that, for a [[User:Tohline/SR#Barotropic_Structure\|barotropic equation of state]], we can make the substitution, <math>~\nabla P \rightarrow \rho\nabla H</math>, where <math>~H</math> is the fluid enthalpy, then OBL66's hydrostatic balance equation — their equation (3) — can be rewritten as,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	</tr>		</tr>
	</table>		</table>
	where, <math>~C_B</math> is a constant (i.e., independent of position). Rotationally flattened, steady-state (equilibrium) configurations can be constructed by finding spatial density distributions that simultaneously satisfy the Poisson equation and this deceptively simple algebraic relation. [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract Ostriker & Bodenheimer (1968; hereinafter, OB68)] used this "self-consistent field" technique to obtain models of rotationally flattened white dwarfs; it is a technique of choice that we [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Technique\|broadly ~~recommend~~]] as well.		where, <math>~C_B</math> is a constant (i.e., independent of position). Rotationally flattened, steady-state (equilibrium) configurations can be constructed by finding spatial density distributions that simultaneously satisfy the Poisson equation and this deceptively simple algebraic relation. [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract Ostriker & Bodenheimer (1968; hereinafter, OB68)] used this "self-consistent field" technique to obtain models of rotationally flattened white dwarfs; it is a technique of choice that we [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Technique\|broadly promote]] as well.


	In the specific case of a zero-temperature Fermi (degenerate electron) gas — see our [[User:Tohline/SR#Barotropic_Structure\|related discussion of barotropic structures]] — the enthalpy associated with <math>~P_\mathrm{deg}</math> is ~~given by the expression~~,		In the specific case of a zero-temperature Fermi (degenerate electron) gas — see our [[User:Tohline/SR#Barotropic_Structure\|related discussion of barotropic structures]] — to within an additive constant, the enthalpy associated with <math>~P_\mathrm{deg}</math> is,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	<td align="left">		<td align="left">
	<math>~		<math>~
	\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[ (\chi^2 ~~+ 1~~)^{1 / 2} ~~- 1~~ \biggr] \, .		\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[ (1 + \chi^2 )^{1 / 2} \biggr] \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
			(Note that in order to retrieve [[User:Tohline/SR#Barotropic_Structure\|our tabulated expression for the enthalpy]], the constant, <math>~H_0 = 8A_\mathrm{F}/B_\mathrm{F}</math>, must be subtracted from this OB68 enthalpy expression.)

	=See Also=		=See Also=

Tohline: /* Approach Outlined by Ostriker, Bodenheimer & Lynden-Bell (1966) */

2019-08-09T20:42:49Z

Approach Outlined by Ostriker, Bodenheimer & Lynden-Bell (1966)

← Older revision		Revision as of 20:42, 9 August 2019
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	This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>~P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>~j(\varpi)</math>. How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer & Lynden-Bell (1966)] and further detailed and executed by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract J. P. Ostriker & P. Bodenheimer (1968)]?		This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>~P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>~j(\varpi)</math>. How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer & Lynden-Bell (1966)] and further detailed and executed by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract J. P. Ostriker & P. Bodenheimer (1968)]?

	===Approach Outlined by ~~Ostriker, Bodenheimer & Lynden-Bell (1966)~~===		===Approach Outlined by OBL66===

	* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966; hereinafter OBL66)], Phys. Rev. Letters, 17, 816: ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''		* [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract J. P. Ostriker, P. Bodenheimer & D. Lynden-Bell (1966; hereinafter OBL66)], Phys. Rev. Letters, 17, 816: ''Equilibrium Models of Differentially Rotating Zero-Temperature Stars''
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	[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 816, Eq. (1)		[https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract OBL66], p. 816, Eq. (1)
	</div>		</div>

			===Further Implementation by OB68===

			If we define an effective potential,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\Phi_\mathrm{eff}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\Phi + \Psi \, ,</math>
			</td>
			</tr>
			</table>
			and recall that, for a [[User:Tohline/SR#Barotropic_Structure\|barotropic equation of state]], we can make the substitution, <math>~\nabla P \rightarrow \rho\nabla H</math>, where <math>~H</math> is the fluid entropy, then OBL66's hydrostatic balance equation — their equation (3) — can be rewritten as,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\nabla [H + \Phi_\mathrm{eff} ]</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~0 </math>
			</td>
			<td align="center">    <math>~\Rightarrow</math>    </td>
			<td align="right">
			<math>~H + \Phi_\mathrm{eff}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~C_B \, , </math>
			</td>
			</tr>
			</table>
			where, <math>~C_B</math> is a constant (i.e., independent of position). Rotationally flattened, steady-state (equilibrium) configurations can be constructed by finding spatial density distributions that simultaneously satisfy the Poisson equation and this deceptively simple algebraic relation. [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract Ostriker & Bodenheimer (1968; hereinafter, OB68)] used this "self-consistent field" technique to obtain models of rotationally flattened white dwarfs; it is a technique of choice that we [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Technique\|broadly recommend]] as well.


			In the specific case of a zero-temperature Fermi (degenerate electron) gas — see our [[User:Tohline/SR#Barotropic_Structure\|related discussion of barotropic structures]] — the enthalpy associated with <math>~P_\mathrm{deg}</math> is given by the expression,
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~H</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[ (\chi^2 + 1)^{1 / 2} - 1 \biggr] \, .
			</math>
			</td>
			</tr>
			</table>

	=See Also=		=See Also=

Tohline: /* Our Approach */

2019-08-09T19:12:29Z

Our Approach

← Older revision		Revision as of 19:12, 9 August 2019
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	This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>~P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>~j(\varpi)</math>. How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer & Lynden-Bell (1966]?		This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>~P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>~j(\varpi)</math>. How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer & Lynden-Bell (1966)] and further detailed and executed by [https://ui.adsabs.harvard.edu/abs/1968ApJ...151.1089O/abstract J. P. Ostriker & P. Bodenheimer (1968)]?

	===Approach Outlined by Ostriker, Bodenheimer & Lynden-Bell (1966)===		===Approach Outlined by Ostriker, Bodenheimer & Lynden-Bell (1966)===

Tohline: /* Our Approach */

2019-08-09T19:10:16Z

Our Approach

← Older revision		Revision as of 19:10, 9 August 2019
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	<table align="center" border="1" cellpadding="10">		<table align="center" border="1" cellpadding="10">
			<tr><th align="center" colspan="2"><font size="+0">Table 1:   Simplified Set of Three Coupled PDEs</font></th></tr>
	<tr>		<tr>
	<th align="center" width="50%">Cylindrical Coordinate Base</th>		<th align="center" width="50%">Cylindrical Coordinate Base</th>
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	This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>~P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>~j(\varpi)</math>. How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer & Lynden-Bell (1966]?		This set of simplified governing relations must then be supplemented by a specification of: (a) a barotropic equation of state, <math>~P(\rho)</math>; and (b) the equilibrium configurations's radial specific angular momentum profile <math>~j(\varpi)</math>. How does this recommended modeling approach compare to the approach outlined by [https://ui.adsabs.harvard.edu/abs/1966PhRvL..17..816O/abstract Ostriker, Bodenheimer & Lynden-Bell (1966]?


	===Approach Outlined by Ostriker, Bodenheimer & Lynden-Bell (1966)===		===Approach Outlined by Ostriker, Bodenheimer & Lynden-Bell (1966)===