User:Tohline/SSC/Perturbations - Revision history

Tohline: /* Ledoux and Pekeris (1941) */

2017-07-06T20:16:31Z

Ledoux and Pekeris (1941)

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	This is clearly the [[User:Tohline/SSC/Perturbations#2ndOrderODE\|same <math>2^\mathrm{nd}</math>-order, ordinary differential equation as the one we have derived]], but with a more general definition of the adiabatic exponent that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.		This is clearly the [[User:Tohline/SSC/Perturbations#2ndOrderODE\|same <math>2^\mathrm{nd}</math>-order, ordinary differential equation as the one we have derived]], but with a [[User:Tohline/SR#Adiabatic_Exponent\|more general definition of the adiabatic exponent]] that allows consideration of a situation where the total pressure is a sum of both gas and radiation pressure.

	===Schwarzschild (1941)===		===Schwarzschild (1941)===

Tohline: /* Schwarzschild (1941) */

2017-03-09T02:26:43Z

Schwarzschild (1941)

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	Appreciating that, in Schwarzschild's expression, <math>~a \equiv - (4-3\gamma_g)/\gamma_g</math>, we see that our expression matches his if and only if <math>~n=3</math>. This is, indeed, precisely the "standard model" that Schwarzschild was considering.		Appreciating that, in Schwarzschild's expression, <math>~a \equiv - (4-3\gamma_g)/\gamma_g</math>, we see that our expression matches his if and only if <math>~n=3</math>. This is, indeed, precisely the "standard model" that Schwarzschild was considering.

			In a separate chapter that discusses the [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Radial_Oscillations_of_n_.3D_3_Polytropic_Spheres\|radial oscillations of n =3 polytropes]], we provide a much more in-depth review of this published investigation by Schwarzshild.

	===Review Article by Christy (1966)===		===Review Article by Christy (1966)===

Tohline: /* What Makes This an Eigenvalue Problem? */

2017-03-09T02:08:08Z

What Makes This an Eigenvalue Problem?

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	===What Makes This an Eigenvalue Problem?===		===What Makes This an Eigenvalue Problem?===
	Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has fostered the following appreciation. Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center — see, for example, the expansion that is appropriate for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|polytropic]] or for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement\|isothermal]] spheres — the [[#2ndOrderODE\|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (or at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>. As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition. This is illustrated, for example, by Figure 2 in our [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Figure2\|discussion of ~~oscillations of~~ n = 3 polytropes]]. Typically, at low frequencies, the displacement function exhibits no, or only a few, radial nodes; but the number of radial nodes that reside within the volume of the configuration steadily grows as the specified frequency increases.		Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has fostered the following appreciation. Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center — see, for example, the expansion that is appropriate for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|polytropic]] or for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement\|isothermal]] spheres — the [[#2ndOrderODE\|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (or at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>. As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition. This is illustrated, for example, by Figure 2 in our [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Figure2\|discussion of oscillating n = 3 polytropes]]. Typically, at low frequencies, the displacement function exhibits no, or only a few, radial nodes; but the number of radial nodes that reside within the volume of the configuration steadily grows as the specified frequency increases.

	A discrete spectrum of oscillation ''modes'' ~~emerges~~ when the outward integration is forced to generate a displacement function whose behavior ''at the configuration's surface'' ~~that~~ matches a specific surface boundary condition. For example, as [[#~~Ensure_Finite-Amplitude_Fluctuations~~\|described above]], ~~a boundary condition that ensures~~ that fluctuations are ~~finite~~ at the surface.		The continuous spectrum is transformed into a discrete spectrum of oscillation ''modes'' when the outward integration is forced to generate a displacement function whose behavior ''at the configuration's surface'' matches a specific surface boundary condition. For example, as [[#Implications\|described above]], if the aim is to ensure that there are no pressure fluctuations at the surface throughout an oscillation, then the only physically relevant displacement functions are the ones whose logarithmic radial derivative at the surface is negative three. And each of these relevant displacement functions — now, an ''eigenfunction'' — will be associated with a ''discrete'' value the oscillation frequency — the associated ''eigenfrequency.'' Understanding this interplay between a derived solution to the LAWE and the imposed boundary condition helps clarify why this is an eigenvalue problem.

	==Classic Papers that Derive & Use this Relation==		==Classic Papers that Derive & Use this Relation==

Tohline: /* What Makes This an Eigenvalue Problem? */

2017-03-09T01:52:46Z

What Makes This an Eigenvalue Problem?

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	===What Makes This an Eigenvalue Problem?===		===What Makes This an Eigenvalue Problem?===
	Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has ~~led to~~ the following appreciation. Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center — see, for example, the expansion that is appropriate for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|polytropic]] or for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement\|isothermal]] spheres — the [[#2ndOrderODE\|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>. As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition. This is illustrated, for example, by Figure 2 in our [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Figure2\|discussion of oscillations of n = 3 polytropes]]. Typically, at low frequencies, the displacement function exhibits no, or only a few, radial nodes; but the number of radial nodes that reside within the volume of the configuration steadily grows as the specified frequency increases.		Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has fostered the following appreciation. Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center — see, for example, the expansion that is appropriate for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|polytropic]] or for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement\|isothermal]] spheres — the [[#2ndOrderODE\|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (or at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>. As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition. This is illustrated, for example, by Figure 2 in our [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Figure2\|discussion of oscillations of n = 3 polytropes]]. Typically, at low frequencies, the displacement function exhibits no, or only a few, radial nodes; but the number of radial nodes that reside within the volume of the configuration steadily grows as the specified frequency increases.

	A discrete spectrum of oscillation ''modes'' emerges when the outward integration is forced to generate a displacement function whose behavior ''at the configuration's surface'' that matches a specific surface boundary condition. For example, as [[#Ensure_Finite-Amplitude_Fluctuations\|described above]], a boundary condition that ensures that fluctuations are finite at the surface.		A discrete spectrum of oscillation ''modes'' emerges when the outward integration is forced to generate a displacement function whose behavior ''at the configuration's surface'' that matches a specific surface boundary condition. For example, as [[#Ensure_Finite-Amplitude_Fluctuations\|described above]], a boundary condition that ensures that fluctuations are finite at the surface.

Tohline: /* What Makes This an Eigenvalue Problem? */

2017-03-08T23:34:06Z

What Makes This an Eigenvalue Problem?

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	===What Makes This an Eigenvalue Problem?===		===What Makes This an Eigenvalue Problem?===
	Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has led to the following appreciation. Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center — see, for example, the expansion that is appropriate for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|polytropic]] or for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement\|isothermal]] spheres — the [[#2ndOrderODE\|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>. As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition. This is illustrated, for example, by Figure 2 in our [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Figure2\|discussion of oscillations of n = 3 polytropes]].		Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has led to the following appreciation. Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center — see, for example, the expansion that is appropriate for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|polytropic]] or for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement\|isothermal]] spheres — the [[#2ndOrderODE\|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>. As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition. This is illustrated, for example, by Figure 2 in our [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Figure2\|discussion of oscillations of n = 3 polytropes]]. Typically, at low frequencies, the displacement function exhibits no, or only a few, radial nodes; but the number of radial nodes that reside within the volume of the configuration steadily grows as the specified frequency increases.

			A discrete spectrum of oscillation ''modes'' emerges when the outward integration is forced to generate a displacement function whose behavior ''at the configuration's surface'' that matches a specific surface boundary condition. For example, as [[#Ensure_Finite-Amplitude_Fluctuations\|described above]], a boundary condition that ensures that fluctuations are finite at the surface.

	==Classic Papers that Derive & Use this Relation==		==Classic Papers that Derive & Use this Relation==

Tohline: /* The Eigenvalue Problem */

2017-03-08T23:17:03Z

The Eigenvalue Problem

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	<math>~x = \biggl( \frac{R}{r_0}\biggr)^{3-4/\gamma_g} \, .</math>		<math>~x = \biggl( \frac{R}{r_0}\biggr)^{3-4/\gamma_g} \, .</math>
	</div>		</div>

			===What Makes This an Eigenvalue Problem?===
			Our own study of radial oscillations in spherically symmetric, self-gravitating fluids has led to the following appreciation. Generally, with knowledge only of the boundary condition at the center of the configuration and an associated power-series expansion that provides a description of the displacement function, <math>x(r_0)</math>, near the center — see, for example, the expansion that is appropriate for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#PolytropicDisplacement\|polytropic]] or for [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#IsothermalDisplacement\|isothermal]] spheres — the [[#2ndOrderODE\|LAWE]] can be straightforwardly integrated (numerically) from the center, outward to (at least very ''near'' to) the surface for practically ''any'' specified value of the (square of the) oscillation frequency, <math>~\omega^2</math>. As a result, integration from the center, outward, can very naturally generate a ''continuous spectrum'' of displacement functions, if the integration is unconstrained by specification of an outer boundary condition. This is illustrated, for example, by Figure 2 in our [[User:Tohline/SSC/Stability/n3PolytropeLAWE#Figure2\|discussion of oscillations of n = 3 polytropes]].

	==Classic Papers that Derive & Use this Relation==		==Classic Papers that Derive & Use this Relation==

Tohline: /* Implications */

2017-03-08T22:31:52Z

Implications

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	But, rearranging terms in the full ''adiabatic wave equation'', we see that, throughout the entire structure (including the surface),		But, rearranging terms in the full ''[[#2ndOrderODE\|linear adiabatic wave equation]]'', we see that, throughout the entire structure (including the surface),
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	at the surface of the oscillating configuration in order to satisfy this "second case" boundary condition.		at the surface of the oscillating configuration in order to satisfy this "second case" boundary condition.

	'''Third Case:''' If we follow the lead of [<~~font color="red"~~>~~'''~~[[User:Tohline/Appendix/References#KW94\|KW94]]~~'''~~</~~font~~>] and choose to establish a surface boundary condition that effectively ignores the inertia of the configuration's atmosphere, then, as we have just determined,		'''Third Case:''' If we follow the lead of [<b>[[User:Tohline/Appendix/References#KW94\|<font color="red">KW94</font>]]</b>] and choose to establish a surface boundary condition that effectively ignores the inertia of the configuration's atmosphere, then, as we have just determined,
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Tohline: /* Ignore Atmospheric Inertia */

2017-03-08T22:25:26Z

Ignore Atmospheric Inertia

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	This is identical to the boundary condition presented as equation (38.12) in [<b>[[User:Tohline/Appendix/References#KW94\|<font color="red">KW94</font>]]</b>]. Combining this condition with the [[User:Tohline/SSC/Perturbations#~~Summary_2~~\|above "linearized adiabatic form of the first law of thermodynamics"]] allows us to write, as well,		This is identical to the boundary condition presented as equation (38.12) in [<b>[[User:Tohline/Appendix/References#KW94\|<font color="red">KW94</font>]]</b>]. Combining this condition with the [[User:Tohline/SSC/Perturbations#Summary_Set_of_Linearized_Equations\|above "linearized adiabatic form of the first law of thermodynamics"]] allows us to write, as well,
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	and, in combination with the [[User:Tohline/SSC/Perturbations#~~Summary_2~~\|above "linearized continuity equation"]], to also conclude that,		and, in combination with the [[User:Tohline/SSC/Perturbations#Summary_Set_of_Linearized_Equations\|above "linearized continuity equation"]], to also conclude that,
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Tohline: /* Ignore Atmospheric Inertia */

2017-03-08T22:23:16Z

Ignore Atmospheric Inertia

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	This is identical to the boundary condition presented as equation (38.12) in ~~'''~~[[User:Tohline/Appendix/References#KW94\|KW94]]~~'''~~. Combining this condition with the [[User:Tohline/SSC/Perturbations#Summary_2\|above "linearized adiabatic form of the first law of thermodynamics"]] allows us to write, as well,		This is identical to the boundary condition presented as equation (38.12) in [<b>[[User:Tohline/Appendix/References#KW94\|<font color="red">KW94</font>]]</b>]. Combining this condition with the [[User:Tohline/SSC/Perturbations#Summary_2\|above "linearized adiabatic form of the first law of thermodynamics"]] allows us to write, as well,
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	This matches the boundary condition, as presented in equation (38.13) of ~~'''~~[[User:Tohline/Appendix/References#KW94\|KW94]]~~'''~~.		This matches the boundary condition, as presented in equation (38.13) of [<b>[[User:Tohline/Appendix/References#KW94\|<font color="red">KW94</font>]]</b>].

	=====Implications=====		=====Implications=====

Tohline: /* Ignore Atmospheric Inertia */

2017-03-08T22:20:44Z

Ignore Atmospheric Inertia

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	=====Ignore Atmospheric Inertia=====		=====Ignore Atmospheric Inertia=====
	Here we echo the discussion presented in §38.1 of [<~~font color="red"~~>~~'''~~[[User:Tohline/Appendix/References#KW94\|KW94]]~~'''~~</~~font~~>], where an alternative boundary condition at the surface of our spherically symmetric, oscillating configuration is recommended; text drawn verbatim from this reference is shown here in green. <font color="green">We simplify the atmosphere by assuming its mass <math>~m_a</math> to be comprised in a thin layer at <math>~r_0 = R(t)</math>, which follows the changing <math>~R</math> during the oscillations and provides the outer boundary condition at each moment by its weight</font>. This amounts to ignoring the inertia of this very thin, nearly massless layer and is accomplished, in practice, by setting to zero the second time-derivative on the left-hand side of the [[User:Tohline/SSC/Perturbations#PGE:Euler\|above "Euler + Poisson Equation"]]. Hence, the pressure, <math>~P_b</math>, at the base of the outermost (atmospheric) layer of the oscillating configuration is described, at all times, by,		Here we echo the discussion presented in §38.1 of [<b>[[User:Tohline/Appendix/References#KW94\|<font color="red">KW94</font>]]</b>], where an alternative boundary condition at the surface of our spherically symmetric, oscillating configuration is recommended; text drawn verbatim from this reference is shown here in green. <font color="green">We simplify the atmosphere by assuming its mass <math>~m_a</math> to be comprised in a thin layer at <math>~r_0 = R(t)</math>, which follows the changing <math>~R</math> during the oscillations and provides the outer boundary condition at each moment by its weight</font>. This amounts to ignoring the inertia of this very thin, nearly massless layer and is accomplished, in practice, by setting to zero the second time-derivative on the left-hand side of the [[User:Tohline/SSC/Perturbations#PGE:Euler\|above "Euler + Poisson Equation"]]. Hence, the pressure, <math>~P_b</math>, at the base of the outermost (atmospheric) layer of the oscillating configuration is described, at all times, by,
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