User:Tohline/SSC/VirialStability - Revision history

Tohline: /* Discussion with Kundan Kadam */

2020-11-17T21:50:10Z

Discussion with Kundan Kadam

← Older revision		Revision as of 21:50, 17 November 2020
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	==Discussion with Kundan Kadam==		==Discussion with Kundan Kadam==
			See particularly … [https://ui.adsabs.harvard.edu/abs/2016MNRAS.462.2237K/abstract K. Kadam ''et al.'' (2016, MNRAS, 462, pp. 2237-2245)]

	The comments provided inside the following box are for Kundan Kadam, providing feedback on a derivation he sent to me ( J. E. Tohline) on Monday, 27 January 2014.		The comments provided inside the following box are for Kundan Kadam, providing feedback on a derivation he sent to me ( J. E. Tohline) on Monday, 27 January 2014.

Tohline: Insert "work in progress" sign

2014-03-05T01:24:04Z

Insert "work in progress" sign

← Older revision		Revision as of 01:24, 5 March 2014
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	</td></tr>		</td></tr>
	</table>		</table>


			{{LSU_WorkInProgress}}

	=BiPolytrope Structural Relations=		=BiPolytrope Structural Relations=

Tohline: /* Discussion with Kundan Kadam */ Add comment dated 25 February 2014

2014-02-25T20:39:31Z

Discussion with Kundan Kadam: Add comment dated 25 February 2014

← Older revision		Revision as of 20:39, 25 February 2014
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	[<font color="red">17 February 2014; by J. E. Tohline</font>] Using derivatives of the free energy function, <math>~\mathfrak{G}</math>, I have finished deriving the equilibrium radius for bipolytropic configurations where the density of the core <math>~(\rho_0)</math> and the density of the envelope <math>~(\rho_e)</math> are both assumed to be uniform. I have also finished deriving the condition for the dynamical stability of such configurations assuming that, as the equilibrium structure undergoes a radial perturbation, the core compresses along a <math>~\gamma_c</math> adiabat while the envelope compresses along a <math>~\gamma_e</math> adiabat. The [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Equilibrium_Condition\|equilibrium radius is derived here]] and the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_Condition\|stability condition is described here]].		[<font color="red">17 February 2014; by J. E. Tohline</font>] Using derivatives of the free energy function, <math>~\mathfrak{G}</math>, I have finished deriving the equilibrium radius for bipolytropic configurations where the density of the core <math>~(\rho_0)</math> and the density of the envelope <math>~(\rho_e)</math> are both assumed to be uniform. I have also finished deriving the condition for the dynamical stability of such configurations assuming that, as the equilibrium structure undergoes a radial perturbation, the core compresses along a <math>~\gamma_c</math> adiabat while the envelope compresses along a <math>~\gamma_e</math> adiabat. The [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Equilibrium_Condition\|equilibrium radius is derived here]] and the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_Condition\|stability condition is described here]].

			<br />

			<div align="center">
			<hr width="85%" />
			</div>

			<br />
			[<font color="red">25 February 2014; by J. E. Tohline</font>] I have extended the free-energy-based stability analysis to models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes. As is [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy\|detailed here]], I have been able to derive analytic, closed form expressions for the gravitational potential energy and internal thermal energies of these configurations. As is [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Equilibrium_Condition\|shown here]], an examination of the first derivative of the free energy identifies equilibrium configurations that exactly match those derived from the earlier detailed force-balance derivation; and, finally, as [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Stability_Condition\|shown here (see especially Figure 3)]], the second derivative of the free energy expression allows us to determine which equilibrium models are stable and which are dynamically unstable. Hooray!
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Discussion with Kundan Kadam */ Enter note to Kundan dated 2/17/2014

2014-02-17T23:27:29Z

Discussion with Kundan Kadam: Enter note to Kundan dated 2/17/2014

← Older revision		Revision as of 23:27, 17 February 2014
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	</ol>		</ol>
	Items #2 and #3 are the next things I will be working on.		Items #2 and #3 are the next things I will be working on.

			<br />

			<div align="center">
			<hr width="85%" />
			</div>

			<br />
			[<font color="red">17 February 2014; by J. E. Tohline</font>] Using derivatives of the free energy function, <math>~\mathfrak{G}</math>, I have finished deriving the equilibrium radius for bipolytropic configurations where the density of the core <math>~(\rho_0)</math> and the density of the envelope <math>~(\rho_e)</math> are both assumed to be uniform. I have also finished deriving the condition for the dynamical stability of such configurations assuming that, as the equilibrium structure undergoes a radial perturbation, the core compresses along a <math>~\gamma_c</math> adiabat while the envelope compresses along a <math>~\gamma_e</math> adiabat. The [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Equilibrium_Condition\|equilibrium radius is derived here]] and the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Stability_Condition\|stability condition is described here]].

	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Discussion with Kundan Kadam */ Update notes based on derivations made over the past couple of days

2014-02-12T20:12:41Z

Discussion with Kundan Kadam: Update notes based on derivations made over the past couple of days

← Older revision		Revision as of 20:12, 12 February 2014
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	My initial derivation gives,		My initial derivation gives,
	<~~div~~ align="center">		<table border="1" align="center" width="85%">
			<tr><td align="left">
			<font color="darkblue">
			Do not use these expressions as they are flawed. The correct expressions follow in the comments dated 12 February 2014.
			</font>
			</td></tr>
			<tr><td>
	<math>		<math>
	S_\mathrm{core} = 2\pi P_0 R^3 \biggl[q^3 - q^5 \biggl( \frac{2\pi G\rho_0^2 R^2}{5P_0} \biggr) \biggr] \, ;		S_\mathrm{core} = 2\pi P_0 R^3 \biggl[q^3 - q^5 \biggl( \frac{2\pi G\rho_0^2 R^2}{5P_0} \biggr) \biggr] \, ;
	</math>		</math>
			</td></tr>
			<tr><td>
	<math>		<math>
	S_\mathrm{env} = \frac{4\pi}{3} R^3 \biggl\{ \frac{3P_i}{2} (1-q^3) + \pi G \rho_0^2 R^2 \biggl(\frac{\rho_e}{\rho_c} \biggr)		S_\mathrm{env} = \frac{4\pi}{3} R^3 \biggl\{ \frac{3P_i}{2} (1-q^3) + \pi G \rho_0^2 R^2 \biggl(\frac{\rho_e}{\rho_c} \biggr)
	\biggl[ \frac{1}{6}(6q^3 - q^2 - 5q^5) - \frac{1}{10}\biggl(\frac{\rho_e}{\rho_c} \biggr) (2 + 10q^3 - 5q^2-7q^5) \biggr] \biggr\} \, .		\biggl[ \frac{1}{6}(6q^3 - q^2 - 5q^5) - \frac{1}{10}\biggl(\frac{\rho_e}{\rho_c} \biggr) (2 + 10q^3 - 5q^2-7q^5) \biggr] \biggr\} \, .
	</math>		</math>
			</td></tr>
			</table>

			<br />

			<div align="center">
			<hr width="85%" />
			</div>

			<br />
			[<font color="red">12 February 2014</font>] In an [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Thermal_Energy_Content\|accompanying discussion of the thermal energy content of an <math>(n_c, n_e) = (0, 0)</math> bipolytrope]], I've re-derived and cross-checked the expressions for <math>S_\mathrm{core}</math> and <math>S_\mathrm{env}</math>. I am quite confident that the correct expressions are:

			<div align="center">
			<table border="0">
			<tr>
			<td align="right">
			<math>~S_\mathrm{core}</math>
			</td>
			<td align="center">
			<math>~=~</math>
			</td>
			<td align="left">
			<math>\biggl( \frac{4\pi}{5} \biggr) R^3 q^5 \biggl (\frac{5P_i}{2q^2} + \Pi \biggr) \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~S_\mathrm{env}</math>
			</td>
			<td align="center">
			<math>~=~</math>
			</td>
			<td align="left">
			<math>
			\frac{\pi R^3}{5} \biggl\{ 10 P_i (1-q^3)
			+ 10 \Pi \biggl( \frac{\rho_e}{\rho_0} \biggr) \biggl[-2q^2 + 3q^3 - q^5 \biggr]
			+ 6 \Pi \biggl( \frac{\rho_e}{\rho_0} \biggr)^2 \biggl[ -1 + 5q^2 -5q^3 + q^5 \biggr] \biggr\} \, ,
			</math>
			</td>
			</tr>
			</table>
	</div>		</div>
	~~These results need to be checked~~.
			where, <math>P_i</math> is the pressure at the interface,
			<div align="center">
			<math>
			\Pi \equiv \frac{3}{2^3 \pi}
			\biggl( \frac{GM_\mathrm{tot}^2}{R^4} \biggr) \biggl( \frac{\nu^2}{q^6} \biggr) \, ,
			</math>
			</div>
			and the relationship between the central pressure and the pressure at the interface is,
			<div align="center">
			<math>~P_0 = P_i + \Pi q^2 \, .</math>
			</div>

	<br />		<br />

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	</math>		</math>
	</div>		</div>
	Since we have analytic expressions for both <math>S</math> and <math>W</math>, we should check to see if they sum appropriately to zero as expected from this simple statement of virial equilibrium.		Since we have analytic expressions for both <math>S</math> and <math>W</math>, we should check to see if they sum appropriately to zero as expected from this simple statement of virial equilibrium. [Note inserted <font color="red">12 February 2014</font>: An [[User:Tohline/SSC/Structure/BiPolytropes/Analytic0_0#Virial_Equilibrium\|accompanying discussion of the virial equilibrium of this particular bipolytrope]] shows that the relation does hold precisely.]
	</ol>		</ol>
	<ol start="3">		<ol start="3">

Tohline: /* Discussion with Kundan Kadam */ Add in central pressure definitions in terms of R and M

2014-02-09T02:33:50Z

Discussion with Kundan Kadam: Add in central pressure definitions in terms of R and M

← Older revision		Revision as of 02:33, 9 February 2014
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	\biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl(1- q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\}^{1/2}\, .		\biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl(1- q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\}^{1/2}\, .
	</math>		</math>
			</div>
			We will also find it useful to combine these two expressions to eliminate direct reference to the central density, <math>\rho_0</math>, obtaining,
			<div align="center">
			<table border="0">
			<tr>
			<td align="right">
			<math>
			\biggl[ \frac{R^4}{GM_\mathrm{tot}^2} \biggr] P_0</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>\biggl( \frac{1}{2^3\pi} \biggr) \frac{3\nu^2 g^2}{q^4} \, .</math>
			</td>
			</tr>
			</table>
	</div>		</div>

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	<td align="left">		<td align="left">
	<math>\biggl( \frac{2}{\pi} \biggr)^{1/2} \frac{A\eta_s}{\theta_i} \, ,</math>		<math>\biggl( \frac{2}{\pi} \biggr)^{1/2} \frac{A\eta_s}{\theta_i} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>
			\biggl[ \frac{R^4}{GM^2_\mathrm{tot}} \biggr]^{1/2} K_c \rho_0^{6/5}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>\biggl( \frac{1}{2^3\pi} \biggr) \frac{\eta_s^2}{A^2\theta_i^6} \, ,</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Discussion with Kundan Kadam */ Correct sign error in definition of function g

2014-02-07T22:25:57Z

Discussion with Kundan Kadam: Correct sign error in definition of function g

← Older revision		Revision as of 22:25, 7 February 2014
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	</table>		</table>
	</div>		</div>
	where the function,		where the function (<font color="red">note that, for a few days, there was a sign error in this definition of <math>g</math></font>),
	<div align="center">		<div align="center">
	<math>		<math>
	g(q, \rho_e/\rho_0) \equiv \biggl\{ 1 +		g(q, \rho_e/\rho_0) \equiv \biggl\{ 1 +
	\biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl( q -		\biggl(\frac{\rho_e}{\rho_0}\biggr) \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl(1- q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\}^{1/2}\, .
	1\biggr) - \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\}^{1/2}\, .
	</math>		</math>
	</div>		</div>

Tohline: /* Discussion with Kundan Kadam */ Finished Feb 7 comments, for now at least

2014-02-07T19:53:52Z

Discussion with Kundan Kadam: Finished Feb 7 comments, for now at least

← Older revision		Revision as of 19:53, 7 February 2014
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	<br />		<br />
	[<font color="red">7 February 2014</font>] More succinctly, this <math>(n_c, n_e) = (0,0)</math> bipolytrope is very interesting because it provides three independent ways of deriving the proper equilibrium structure. If we work through the detailed derivation in all three cases, we ~~will~~ have a very firm foundation from which to perform the stability analysis. The three derivations are …		[<font color="red">7 February 2014</font>] More succinctly, this <math>(n_c, n_e) = (0,0)</math> bipolytrope is very interesting because it provides three independent ways of deriving the proper equilibrium structure. If we work through the detailed derivation in all three cases, we should have a very firm foundation from which to perform the stability analysis of a broad range of bipolytropic configurations. The three derivations are …
	# Detailed force balance (Kundan has already done this): Use the hydrostatic balance equation to determine how the pressure varies with radius throughout the two-component (core-envelope) structure. For a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math>, the radius and, hence, total mass of the configuration is determined by the surface boundary condition, that is, <math>P = 0</math>. These results are summarized above.		<ol>
	# Virial equilibrium: The structures derived from demanding detailed force balance should also be in virial equilibrium. (The virial equilibrium condition is less demanding than detailed force balance because it is a global statement of energy balance and is not concerned with internal structural details. Virial equilibrium is a necessary but not sufficient condition; the force-balance solution gives the more detailed information but must, itself, also define a structure that has virial energy balance.)		<li>Detailed force balance (Kundan has already done this): Use the hydrostatic balance equation to determine how the pressure varies with radius throughout the two-component (core-envelope) structure. For a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math>, the radius and, hence, total mass of the configuration is determined by the surface boundary condition, that is, <math>P = 0</math> at <math>r = R</math>. These results are summarized above.
	# Extrema in the Free Energy:		<li>Virial equilibrium: The structures derived from demanding detailed force balance should also be in virial equilibrium. (The virial equilibrium condition is less demanding than detailed force balance because it is a global statement of energy balance and is not concerned with internal structural details. Virial equilibrium is a necessary but not sufficient condition; the force-balance solution gives the more detailed information but must, itself, also define a structure that has virial energy balance.) It is well known that, for the simplest spherically symmetric stars, the statement of virial balance is,
			<div align="center">
			<math>
			2S + W = 0 \, .
			</math>
			</div>
			Since we have analytic expressions for both <math>S</math> and <math>W</math>, we should check to see if they sum appropriately to zero as expected from this simple statement of virial equilibrium.
			</ol>
			<ol start="3">
			<li>Extrema in the Free Energy: We should also expect to be able to define the free energy of this bipolytrope by the expression,
			<div align="center">
			<math>
			\mathfrak{G} = U + W \, .
			</math>
			</div>
			Then, for a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math> (perhaps we should choose <math>M_\mathrm{tot}</math> instead of <math>P_0</math>), the equilibrium radius should be defined by the radius at which,
			<div align="center">
			<math>
			\frac{\partial \mathfrak{G}}{\partial R} = 0 \, .
			</math>
			</div>
			We should check to make sure that this is also true.
			</ol>
			Items #2 and #3 are the next things I will be working on.
	</td></tr>		</td></tr>
	</table>		</table>

Tohline: /* Discussion with Kundan Kadam */ More content on 7 February

2014-02-07T16:30:47Z

Discussion with Kundan Kadam: More content on 7 February

← Older revision		Revision as of 16:30, 7 February 2014
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	<br />		<br />
	[<font color="red">7 February 2014</font>] More succinctly, this <math>(n_c, n_e) = (0,0)</math> bipolytrope is very interesting because it provides three independent ways of deriving the proper equilibrium structure. If we work through the detailed derivation in all three cases, we will have a very firm foundation from which to perform the stability analysis. The three derivations are …		[<font color="red">7 February 2014</font>] More succinctly, this <math>(n_c, n_e) = (0,0)</math> bipolytrope is very interesting because it provides three independent ways of deriving the proper equilibrium structure. If we work through the detailed derivation in all three cases, we will have a very firm foundation from which to perform the stability analysis. The three derivations are …
	# Detailed force balance (Kundan has already done this): Use the hydrostatic balance equation to determine how the pressure varies with radius throughout the two-component (core-envelope) structure. For a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math>, the radius and, hence, total mass of the configuration is determined by the surface boundary condition, that is, <math>P = 0</math>.		# Detailed force balance (Kundan has already done this): Use the hydrostatic balance equation to determine how the pressure varies with radius throughout the two-component (core-envelope) structure. For a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math>, the radius and, hence, total mass of the configuration is determined by the surface boundary condition, that is, <math>P = 0</math>. These results are summarized above.
	# Virial equilibrium:		# Virial equilibrium: The structures derived from demanding detailed force balance should also be in virial equilibrium. (The virial equilibrium condition is less demanding than detailed force balance because it is a global statement of energy balance and is not concerned with internal structural details. Virial equilibrium is a necessary but not sufficient condition; the force-balance solution gives the more detailed information but must, itself, also define a structure that has virial energy balance.)
	# Extrema in the Free Energy:		# Extrema in the Free Energy:

Tohline: /* Discussion with Kundan Kadam */ Begin February 7th discussion

2014-02-07T16:12:24Z

Discussion with Kundan Kadam: Begin February 7th discussion

← Older revision		Revision as of 16:12, 7 February 2014
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	</div>		</div>
	These results need to be checked.		These results need to be checked.
			<br />

			<div align="center">
			<hr width="85%" />
			</div>

			<br />
			[<font color="red">7 February 2014</font>] More succinctly, this <math>(n_c, n_e) = (0,0)</math> bipolytrope is very interesting because it provides three independent ways of deriving the proper equilibrium structure. If we work through the detailed derivation in all three cases, we will have a very firm foundation from which to perform the stability analysis. The three derivations are …
			# Detailed force balance (Kundan has already done this): Use the hydrostatic balance equation to determine how the pressure varies with radius throughout the two-component (core-envelope) structure. For a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math>, the radius and, hence, total mass of the configuration is determined by the surface boundary condition, that is, <math>P = 0</math>.
			# Virial equilibrium:
			# Extrema in the Free Energy:

	</td></tr>		</td></tr>
	</table>		</table>