SSC/Stability/MurphyFiedler85 - Revision history

Jet53man at 22:12, 1 August 2021

2021-08-01T22:12:28Z

← Older revision		Revision as of 18:12, 1 August 2021
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	=An Alternate Approach=		=An Alternate Approach=
	In an [[~~User:Tohline/~~SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,		In an [[SSC/Perturbations#2ndOrderODE\|accompanying discussion]], we derived the so-called,

	<div align="center" id="2ndOrderODE">		<div align="center" id="2ndOrderODE">
	<font color="#770000">'''Linear 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}}		{{Math/EQ_RadialPulsation01}}
	</div>		</div>

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	==Foundation==		==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,		Assuming that the underlying equilibrium structure is that of a [[SSC/Structure/BiPolytropes/Analytic15#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 [[SSC/Structure/BiPolytropes/Analytic15#Normalization\|normalizations used when defining the equilibrium structure]], namely,
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	<table border="0" cellpadding="3">		<table border="0" cellpadding="3">
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	</div>		</div>

	We [[~~User:Tohline/~~SSC/Stability/Polytropes#Groundwork\|note as well]] that,		We [[SSC/Stability/Polytropes#Groundwork\|note as well]] that,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	==Profile==		==Profile==

	Referencing the relevant [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Profile\|derived bipolytropic model profile]], we should incorporate the following relations:		Referencing the relevant [[SSC/Structure/BiPolytropes/Analytic15#Profile\|derived bipolytropic model profile]], we should incorporate the following relations:

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	<tr>		<tr>
	<td align="left" colspan="6">		<td align="left" colspan="6">
	<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Step_6:__Envelope_Solution\|presented in Step 6 of our accompanying discussion]].		<sup>&dagger;</sup>In order to obtain the various envelope profiles, it is necessary to evaluate <math>~\phi(\eta)</math> and its first derivative using the information [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution\|presented in Step 6 of our accompanying discussion]].
	</td>		</td>
	</tr>		</tr>
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	==Model 10==		==Model 10==
	As we have [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Murphy_and_Fiedler_.281985.29\|reviewed in an accompanying discussion]], equilibrium Model 10 from [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)] is defined by setting <math>~(\xi_i, m) = (2.5646, 1)</math>. Drawing directly from [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Murphy_and_Fiedler_.281985.29\|our reproduction of their Table 1]], we see that a few relevant structural parameters of Model 10 are,		As we have [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29\|reviewed in an accompanying discussion]], equilibrium Model 10 from [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)] is defined by setting <math>~(\xi_i, m) = (2.5646, 1)</math>. Drawing directly from [[SSC/Structure/BiPolytropes/Analytic15#Murphy_and_Fiedler_.281985.29\|our reproduction of their Table 1]], we see that a few relevant structural parameters of Model 10 are,
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	</tr>		</tr>
	</table>		</table>
	Adopting the same approach [[~~User:Tohline/~~SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|as before when we integrated the LAWE for pressure-truncated polytropes]], we will enlist the finite-difference approximations,		Adopting the same approach [[SSC/Stability/Polytropes#Numerical_Integration_from_the_Center.2C_Outward\|as before when we integrated the LAWE for pressure-truncated polytropes]], we will enlist the finite-difference approximations,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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	===Special Handling at the Center===		===Special Handling at the Center===
	In order to kick-start the integration, we set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then draw on the [[~~User:Tohline/~~Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \delta\xi</math>, away from the center. Specifically, we set,		In order to kick-start the integration, we set the displacement function value to <math>~x_1 = 1</math> at the center of the configuration <math>~(\xi_1 = 0)</math>, then draw on the [[Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|derived power-series expression]] to determine the value of the displacement function at the first radial grid line, <math>~\xi_2 = \delta\xi</math>, away from the center. Specifically, we set,
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Jet53man: /* Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b) */

2021-08-01T22:08:24Z

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

← Older revision		Revision as of 18:08, 1 August 2021
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	==Overview==		==Overview==
	In the stability analysis presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)], the relevant polytropic indexes are, <math>~(n_c, n_e) = (1,5)</math>. Structural properties of the underlying equilibrium models have been reviewed in [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|our accompanying discussion]].		In the stability analysis presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)], the relevant polytropic indexes are, <math>~(n_c, n_e) = (1,5)</math>. Structural properties of the underlying equilibrium models have been reviewed in [[SSC/Structure/BiPolytropes/Analytic15#BiPolytrope_with_nc_.3D_1_and_ne_.3D_5\|our accompanying discussion]].

	The ''Linear Adiabatic Wave Equation'' (LAWE) that is relevant to polytropic spheres may be written as,		The ''Linear Adiabatic Wave Equation'' (LAWE) that is relevant to polytropic spheres may be written as,
	<div align="center">		<div align="center">
	{{ ~~User:Tohline/~~Math/EQ_RadialPulsation02 }}		{{ Math/EQ_RadialPulsation02 }}
	</div>		</div>

	<table border="1" align="center" width="85%" cellpadding="10"><tr><td align="left">		<table border="1" align="center" width="85%" cellpadding="10"><tr><td align="left">
	See also …		See also …
	* Accompanying chapter showing [[~~User:Tohline/~~SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|derivation]] and overlap with [[~~User:Tohline/~~SSC/Perturbations#Classic_Papers_that_Derive_.26_Use_this_Relation\|multiple classic papers]]:		* Accompanying chapter showing [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|derivation]] and overlap with [[SSC/Perturbations#Classic_Papers_that_Derive_.26_Use_this_Relation\|multiple classic papers]]:
	** [https://archive.org/details/TheInternalConstitutionOfTheStars A. S. Eddington (1926)], especially equation (127.6) on p. 188 — ''The Internal Constitution of Stars''		** [https://archive.org/details/TheInternalConstitutionOfTheStars A. S. Eddington (1926)], especially equation (127.6) on p. 188 — ''The Internal Constitution of Stars''
	** [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941, ApJ, 94, 124)] — ''Radial Pulsations of Stars''		** [http://adsabs.harvard.edu/abs/1941ApJ....94..124L P. Ledoux & C. L. Pekeris (1941, ApJ, 94, 124)] — ''Radial Pulsations of Stars''
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	** [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1966, Annual Reviews of Astronomy & Astrophysics, 4, 353)] — ''Pulsation Theory''		** [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C R. F. Christy (1966, Annual Reviews of Astronomy & Astrophysics, 4, 353)] — ''Pulsation Theory''
	** [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974, Reports on Progress in Physics, 37, 563)] — ''Pulsating Stars''		** [http://adsabs.harvard.edu/abs/1974RPPh...37..563C J. P. Cox (1974, Reports on Progress in Physics, 37, 563)] — ''Pulsating Stars''
	* Accompanying chapter detailing [[~~User:Tohline/~~SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|specific application to polytropes]] along with a couple of additional key references:		* Accompanying chapter detailing [[SSC/Stability/Polytropes#Radial_Oscillations_of_Polytropic_Spheres\|specific application to polytropes]] along with a couple of additional key references:
	** [http://adsabs.harvard.edu/abs/1966ApJ...143..535H M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, 143, 535)] — ''The Oscillations of Gas Spheres''		** [http://adsabs.harvard.edu/abs/1966ApJ...143..535H M. Hurley, P. H. Roberts, & K. Wright (1966, ApJ, 143, 535)] — ''The Oscillations of Gas Spheres''
	** [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] — ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models''		** [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] — ''Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models''
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	As we have [[~~User:Tohline/~~SSC/Stability/Polytropes#Boundary_Conditions\|detailed separately]], the boundary condition at the center of a polytropic configuration is,		As we have [[SSC/Stability/Polytropes#Boundary_Conditions\|detailed separately]], the boundary condition at the center of a polytropic configuration is,
	<div align="center">		<div align="center">
	<math>~\frac{dx}{d\xi} \biggr\|_{\xi=0} = 0 \, ;</math>		<math>~\frac{dx}{d\xi} \biggr\|_{\xi=0} = 0 \, ;</math>
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	[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.		[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 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]]:		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 [[SSC/Perturbations#Ensure_Finite-Amplitude_Fluctuations\|original, more general expression of the surface boundary condition]]:

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	<table border="1" align="center" width="85%" cellpadding="10"><tr><td align="left">		<table border="1" align="center" width="85%" cellpadding="10"><tr><td align="left">
	Utilizing an [[~~User:Tohline/~~SSC/Stability/Polytropes#Groundwork\|accompanying discussion]], let's examine the frequency normalization used by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] (see the top of the left-hand column on p. 223):		Utilizing an [[SSC/Stability/Polytropes#Groundwork\|accompanying discussion]], let's examine the frequency normalization used by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] (see the top of the left-hand column on p. 223):
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	</table>		</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.		According to the so-called ''Schwarzschild criterion'' — see, for example, our [[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==
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	</table>		</table>
	which matches the expression presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] (see middle of the left column on p. 223 of their article) if we set <math>~\theta_c = 1</math> and <math>~\gamma_g/\gamma_\mathrm{core} = 1</math>. This LAWE also appears in our [[~~User:Tohline/~~SSC/Stability/n1PolytropeLAWE#MurphyFiedler1985b\|separate discussion of radial oscillations in n = 1 polytropic spheres]].		which matches the expression presented by [http://adsabs.harvard.edu/abs/1985PASAu...6..222M Murphy & Fiedler (1985b)] (see middle of the left column on p. 223 of their article) if we set <math>~\theta_c = 1</math> and <math>~\gamma_g/\gamma_\mathrm{core} = 1</math>. This LAWE also appears in our [[SSC/Stability/n1PolytropeLAWE#MurphyFiedler1985b\|separate discussion of radial oscillations in n = 1 polytropic spheres]].

	===Envelope Layers With n = 5===		===Envelope Layers With n = 5===
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	</table>		</table>

	From our [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Step_6:__Envelope_Solution\|accompanying discussion of the underlying equilibrium structure of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes]], we know that,		From our [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution\|accompanying discussion of the underlying equilibrium structure of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes]], we know that,

	<div align="center">		<div align="center">
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	===Surface Boundary Condition===		===Surface Boundary Condition===
	Next, pulling from our [[~~User:Tohline/~~SSC/Stability/Polytropes#Boundary_Conditions\|accompanying discussion of the stability of polytropes]] and an [[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic1_5~~#Parameter_Values\|accompanying table that details the properties of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes]], the surface boundary condition is,		Next, pulling from our [[SSC/Stability/Polytropes#Boundary_Conditions\|accompanying discussion of the stability of polytropes]] and an [[SSC/Structure/BiPolytropes/Analytic15#Parameter_Values\|accompanying table that details the properties of <math>~(n_c, n_e) = (1, 5)</math> bipolytropes]], the surface boundary condition is,

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	===Interface Conditions===		===Interface Conditions===
	Here, we will simply copy the discussion already provided in the context of our attempt to analyze the stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes; specifically, we will draw from [[~~User:Tohline/~~SSC/Stability/~~BiPolytrope0_0~~#Piecing_Together\|<font color="red">'''STEP 4:'''</font> in the ''Piecing Together'' subsection]]. Following the discussion in §§57 & 58 of [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)], the proper treatment is to ensure that fractional perturbation in the gas pressure (see their equation 57.31),		Here, we will simply copy the discussion already provided in the context of our attempt to analyze the stability of <math>~(n_c, n_e) = (0, 0)</math> bipolytropes; specifically, we will draw from [[SSC/Stability/BiPolytrope00#Piecing_Together\|<font color="red">'''STEP 4:'''</font> in the ''Piecing Together'' subsection]]. Following the discussion in §§57 & 58 of [http://adsabs.harvard.edu/abs/1958HDP....51..353L P. Ledoux & Th. Walraven (1958)], the proper treatment is to ensure that fractional perturbation in the gas pressure (see their equation 57.31),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

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2021-08-01T22:02:57Z

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SSC/Stability/MurphyFiedler85 - Revision history

Jet53man at 22:12, 1 August 2021

Jet53man: /* Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b) */

Jet53man: Created page with "__FORCETOC__ =Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b)= ==Overview== In the stability analysis pres..."

Jet53man: Created page with "FORCETOC =Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b)= ==Overview== In the stability analysis pres..."