User:Tohline/SSC/Stability/MurphyFiedler85 - Revision history

Tohline: /* Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b) */

2019-01-07T23:45:38Z

Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b)

← Older revision		Revision as of 23:45, 7 January 2019
Line 175:		Line 175:

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

			==Aside Regarding Convectively Unstable Core==
			It is worth highlighting that, in their effort to determine the eigenvectors associated with radial pulsations in <math>~(n_c, n_e) = (1, 5)</math> bipolytropes, [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] assumed that fluid elements ''throughout the entire spherical configuration'' expand and contract along <math>~\gamma_g = 5/3</math> adiabats. Referencing separately the ''structural'' polytropic index of the core and of the envelope of the equilibrium bipolytropic models, we see that,
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\gamma_g</math>
			</td>
			<td align="center">
			<math>~< </math>
			</td>
			<td align="left">
			<math>~\frac{n_c+1}{n_c} = 2 \, ,</math>
			</td>
			</tr>
			</table>
			while,
			<table border="0" cellpadding="5" align="center">
			<tr>
			<td align="right">
			<math>~\gamma_g</math>
			</td>
			<td align="center">
			<math>~></math>
			</td>
			<td align="left">
			<math>~\frac{n_e+1}{n_e} = \frac{6}{5} \, .</math>
			</td>
			</tr>
			</table>

			According to the so-called ''Schwarzschild criterion'' — see, for example, our [[User:Tohline/2DStructure/AxisymmetricInstabilities#Modeling_Implications_and_Advice\|accompanying discussion titled, ''Axisymmetric Instabilities to Avoid'']] — it therefore seems that the core of each of their equilibrium models should have been convectively unstable. [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] did not comment on the impact that the presence of a convective core should have had on their radial pulsation analysis.

	==More Detailed Setup==		==More Detailed Setup==

Tohline: /* Profile */

2018-12-06T20:53:22Z

Profile

Show changes

Tohline: /* Foundation */

2018-12-06T20:45:07Z

Foundation

← Older revision		Revision as of 20:45, 6 December 2018
Line 859:		Line 859:
	</tr>		</tr>
	</table>		</table>

			This is the form of the LAWE that we will integrate from the center of the configuration to its surface <math>~(r^* = R^)</math> in order to identify various eigenvectors that are associated with radial oscillations in <math>~(n_c, n_e) = (1, 5)</math> bipolytropes. Before performing the numerical integrations, we need only specify the underlying dimensionless structural functions, <math>~\rho^(r^)</math>, <math>~P^(r^)</math>, and <math>~M_r^(r^*)</math>, throughout the underlying equilibrium configuration.

	==Profile==		==Profile==

Tohline: /* An Alternate Approach */

2018-12-06T20:35:11Z

An Alternate Approach

← Older revision		Revision as of 20:35, 6 December 2018
Line 662:		Line 662:
	</div>		</div>

	whose solution identifies eigenvectors that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. In shifting ''from'' this more general LAWE expression to the so-called ''polytropic'' LAWE — as [[#Overview\|presented above]] — the functions that quantify the structure of the underlying equilibrium configuration, <math>~\rho_0(r_0)</math>, <math>~P_0(r_0)</math>, and <math>~g_0(r_0)</math>, are re-expressed in terms of the polytropic function, <math>~\theta(\xi)</math> [or, instead, Φ(η)] and its derivative, and the dimensional Lagrangian radial coordinate, <math>~r_0</math>, is abandoned in favor the dimensionless Lagrangian radial coordinate, <math>~\xi</math> (or, instead, η), that is familiarly associated with a chosen polytropic ~~configuration~~. In order to avoid confusion that might be associated with switching from one polytropic function to another at the core-envelope interface, here we have chosen to stick with the single Lagrangian radial coordinate, <math>~r_0</math>, throughout the configuration.		whose solution identifies eigenvectors that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. In shifting ''from'' this more general LAWE expression to the so-called ''polytropic'' LAWE — as [[#Overview\|presented above]] — the functions that quantify the structure of the underlying equilibrium configuration, <math>~\rho_0(r_0)</math>, <math>~P_0(r_0)</math>, and <math>~g_0(r_0)</math>, are re-expressed in terms of the polytropic function, <math>~\theta(\xi)</math> [or, instead, Φ(η)] and its derivative, and the dimensional Lagrangian radial coordinate, <math>~r_0</math>, is abandoned in favor the dimensionless Lagrangian radial coordinate, <math>~\xi</math> (or, instead, η), that is familiarly associated with a chosen polytropic index. In order to avoid confusion that might be associated with switching from one polytropic function to another at the core-envelope interface, here we have chosen to stick with the single Lagrangian radial coordinate, <math>~r_0</math>, throughout the configuration.

	==Foundation==		==Foundation==

Tohline: /* An Alternate Approach */

2018-12-06T20:34:20Z

An Alternate Approach

← Older revision		Revision as of 20:34, 6 December 2018
Line 654:		Line 654:

	=An Alternate Approach=		=An Alternate Approach=

	~~==Foundation==~~
	In an [[User:Tohline/SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,		In an [[User:Tohline/SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,

	<div align="center" id="2ndOrderODE">		<div align="center" id="2ndOrderODE">
	<font color="#770000">'''Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />		<font color="#770000">'''Linear Adiabatic Wave''' (or ''Radial Pulsation'') '''Equation'''</font><br />

	{{User:Tohline/Math/EQ_RadialPulsation01}}		{{User:Tohline/Math/EQ_RadialPulsation01}}
	</div>		</div>

	<!--		whose solution identifies eigenvectors that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations. In shifting ''from'' this more general LAWE expression to the so-called ''polytropic'' LAWE — as [[#Overview\|presented above]] — the functions that quantify the structure of the underlying equilibrium configuration, <math>~\rho_0(r_0)</math>, <math>~P_0(r_0)</math>, and <math>~g_0(r_0)</math>, are re-expressed in terms of the polytropic function, <math>~\theta(\xi)</math> [or, instead, Φ(η)] and its derivative, and the dimensional Lagrangian radial coordinate, <math>~r_0</math>, is abandoned in favor the dimensionless Lagrangian radial coordinate, <math>~\xi</math> (or, instead, η), that is familiarly associated with a chosen polytropic configuration. In order to avoid confusion that might be associated with switching from one polytropic function to another at the core-envelope interface, here we have chosen to stick with the single Lagrangian radial coordinate, <math>~r_0</math>, throughout the configuration.
	~~<div align="center" id="2ndOrderODE">~~
	~~<font color="#770000">~~'''~~Adiabatic Wave Equation~~'''</~~font~~><br />

	<math>
	~~\frac{d^2x}{dr_0^2} + \biggl[\frac{4}{~~r_0} - \~~biggl(\frac{g_0 \rho_0}{P_0}\biggr) \biggr] \frac{dx}{dr_0} + \biggl~~(\~~frac{\rho_0}{\gamma_\mathrm{g} P_0} \biggr~~)~~\biggl~~[~~\omega^2 +~~ (~~4 - 3\gamma_\mathrm{g}~~)~~\frac{g_0}{r_0} \biggr~~] ~~x = 0 \~~, ,
	</math>
	</~~div~~>
	-->

	~~whose solution gives eigenfunctions that describe various radial modes of oscillation in spherically symmetric, self-gravitating fluid configurations.~~ Assuming that the underlying equilibrium structure is that of a [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|bipolytrope having <math>~(n_c, n_e) = (1, 5)</math>]], it makes sense to adopt the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Normalization\|normalizations used when defining the equilibrium structure]], namely,		==Foundation==
			Assuming that the underlying equilibrium structure is that of a [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|bipolytrope having <math>~(n_c, n_e) = (1, 5)</math>]], it makes sense to adopt the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Normalization\|normalizations used when defining the equilibrium structure]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">

Tohline: /* Our Confession */

2018-12-06T19:45:59Z

Our Confession

← Older revision		Revision as of 19:45, 6 December 2018
Line 651:		Line 651:
	==Our Confession==		==Our Confession==

	When we tried to integrate the governing LAWEs in the piecemeal fashion described by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] — as we have just detailed — we initially failed to match their published eigenvector solutions. In retrospect, it appears as though we did not correctly implement the interface-matching conditions. In an effort to diagnose this problem, we backed up to a more generalized prescription of the LAWE that allowed us to smoothly integrate a ''single'' equation from the center to the surface of the configuration without having to mess with interface-matching conditions. In what follows, we describe this alternate approach. ~~We show how this~~ approach has allowed us to derive radial-oscillation eigenvectors that match in detail the results published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)].		When we tried to integrate the governing LAWEs in the piecemeal fashion described by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] — as we have just detailed — we initially failed to match their published eigenvector solutions. In retrospect, it appears as though we did not correctly implement the interface-matching conditions. In an effort to diagnose this problem, we backed up to a more generalized prescription of the LAWE that allowed us to smoothly integrate a ''single'' equation from the center to the surface of the configuration without having to mess with interface-matching conditions. In what follows, we describe this alternate approach. This approach has allowed us to derive radial-oscillation eigenvectors that match in detail the results published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)].

	=An Alternate Approach=		=An Alternate Approach=

Tohline: /* Overview */

2018-12-06T19:40:46Z

Overview

← Older revision		Revision as of 19:40, 6 December 2018
Line 51:		Line 51:


	[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] ~~realized~~ that they could not simply integrate the above-presented ''polytropic'' LAWE from the center of the configuration to its surface because the underlying bipolytropic equilibrium structure of the envelope and the core are defined by two different polytropic indexes. Instead, they separated the problem into two pieces — integrating the relevant ''core'' LAWE from the center to the core-envelope interface, then integrating the relevant ''envelope'' LAWE from that interface to the surface — being careful to properly ''match'' the two solutions at the interface.		[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] apparently decided that they could not simply integrate the above-presented ''polytropic'' LAWE from the center of the configuration to its surface because the underlying bipolytropic equilibrium structure of the envelope and the core are defined by two different polytropic indexes. Instead, they separated the problem into two pieces — integrating the relevant ''core'' LAWE from the center to the core-envelope interface, then integrating the relevant ''envelope'' LAWE from that interface to the surface — being careful to properly ''match'' the two solutions at the interface.

	They also realized that the above-specified surface boundary condition is not applicable to bipolytropes. Instead, they ~~use~~ what we ~~can describ~~ as the [[User:Tohline/SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations\|original, more general expression of the surface boundary condition]]:		They also realized that the above-specified surface boundary condition is not applicable to bipolytropes. Instead, they used what we will refer to as the [[User:Tohline/SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations\|original, more general expression of the surface boundary condition]]:

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 175:		Line 175:

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


	==More Detailed Setup==		==More Detailed Setup==

Tohline: /* Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b) */

2018-12-06T03:32:28Z

Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b)

← Older revision		Revision as of 03:32, 6 December 2018
Line 175:		Line 175:

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


	==More Detailed Setup==		==More Detailed Setup==
			Here we describe in more detail the steps that [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] employed in order to numerically determine the radial-oscillation eigenvectors of <math>~(n_c, n_e) = (1, 5)</math> bipolytropic spheres.

	===Core Layers With n = 1===		===Core Layers With n = 1===
Line 651:		Line 653:

	When we tried to integrate the governing LAWEs in the piecemeal fashion described by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] — as we have just detailed — we initially failed to match their published eigenvector solutions. In retrospect, it appears as though we did not correctly implement the interface-matching conditions. In an effort to diagnose this problem, we backed up to a more generalized prescription of the LAWE that allowed us to smoothly integrate a ''single'' equation from the center to the surface of the configuration without having to mess with interface-matching conditions. In what follows, we describe this alternate approach. We show how this approach has allowed us to derive radial-oscillation eigenvectors that match in detail the results published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)].		When we tried to integrate the governing LAWEs in the piecemeal fashion described by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] — as we have just detailed — we initially failed to match their published eigenvector solutions. In retrospect, it appears as though we did not correctly implement the interface-matching conditions. In an effort to diagnose this problem, we backed up to a more generalized prescription of the LAWE that allowed us to smoothly integrate a ''single'' equation from the center to the surface of the configuration without having to mess with interface-matching conditions. In what follows, we describe this alternate approach. We show how this approach has allowed us to derive radial-oscillation eigenvectors that match in detail the results published by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)].


	=An Alternate Approach=		=An Alternate Approach=

Tohline: /* Overview */

2018-12-06T03:17:08Z

Overview

← Older revision		Revision as of 03:17, 6 December 2018
Line 49:		Line 49:
	</tr>		</tr>
	</table>		</table>
	~~But this~~ surface condition is not applicable to bipolytropes. Instead, ~~let's return to~~ the [[User:Tohline/SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations\|original, more general expression of the surface boundary condition]]:

			[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] realized that they could not simply integrate the above-presented ''polytropic'' LAWE from the center of the configuration to its surface because the underlying bipolytropic equilibrium structure of the envelope and the core are defined by two different polytropic indexes. Instead, they separated the problem into two pieces — integrating the relevant ''core'' LAWE from the center to the core-envelope interface, then integrating the relevant ''envelope'' LAWE from that interface to the surface — being careful to properly ''match'' the two solutions at the interface.

			They also realized that the above-specified surface boundary condition is not applicable to bipolytropes. Instead, they use what we can describ as the [[User:Tohline/SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations\|original, more general expression of the surface boundary condition]]:

	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 171:		Line 175:

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


	[http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] realized that they could not simply integrate the above-presented ''polytropic'' LAWE from the center of the configuration to its surface because the underlying bipolytropic equilibrium structure of the envelope and the core are defined by two different polytropic indexes. Instead, they separated the problem into two pieces — integrating the relevant ''core'' LAWE from the center to the interface, then integrating the relevant ''envelope'' LAWE from the interface to the surface — being careful to properly ''match'' the two solutions at the interface.

	==More Detailed Setup==		==More Detailed Setup==

Tohline at 22:22, 5 December 2018

2018-12-05T22:22:16Z

Show changes