SSC/Structure/OtherAnalyticModels - Revision history

Jet53man: /* Overview */

2022-12-07T00:23:58Z

Overview

← Older revision		Revision as of 20:23, 6 December 2022
Line 1,427:		Line 1,427:

	====Overview====		====Overview====
	~~[http://adsabs.harvard.edu/abs/1949MNRAS.109..103P C. Prasad (1949, MNRAS, 109, 103)]~~ performed a semi-analytic analysis of the radial oscillations and stability of structures having a parabolic density distribution. Let's examine his tabulated results to see if they help us understand more fully whether or not our analysis is on the right track. For example, from his Table I, we see that <math>~\mathfrak{F} = 0</math> when <math>~\alpha = 0</math>, where, according to his equation (3),		{{ Prasad49 }} performed a semi-analytic analysis of the radial oscillations and stability of structures having a parabolic density distribution. Let's examine his tabulated results to see if they help us understand more fully whether or not our analysis is on the right track. For example, from his Table I, we see that <math>~\mathfrak{F} = 0</math> when <math>~\alpha = 0</math>, where, according to his equation (3),

	<div align="center">		<div align="center">
Line 1,547:		Line 1,547:

	Interesting!		Interesting!


	====Third Constraint====		====Third Constraint====

Jet53man: /* Parabolic Density Distribution */

2022-12-07T00:21:47Z

Parabolic Density Distribution

← Older revision		Revision as of 20:21, 6 December 2022
Line 563:		Line 563:

	===Equilibrium Structure===		===Equilibrium Structure===
	In an article titled, "Radial Oscillations of a Stellar Model," ~~[http://adsabs.harvard.edu/abs/1949MNRAS.109..103P C. Prasad (1949, MNRAS, 109, 103)]~~ investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,		In an article titled, "Radial Oscillations of a Stellar Model," {{ Prasad49full }} investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,
	<div align="center">		<div align="center">
	<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>		<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>

Jet53man: /* Eulerian Approach */

2022-12-07T00:18:32Z

Eulerian Approach

← Older revision		Revision as of 20:18, 6 December 2022
Line 555:		Line 555:
	</div>		</div>

	where, following {{ SF67 }},		where, following {{ SF67 }} (see their equation 3 on p. 305),
	<div align="center">		<div align="center">
	<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>		<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>

Jet53man: /* Eulerian Approach */

2022-12-07T00:12:48Z

Eulerian Approach

← Older revision		Revision as of 20:12, 6 December 2022
Line 555:		Line 555:
	</div>		</div>

	where, following ~~[http://adsabs.harvard.edu/abs/1967ApJ...148..305S R. Stothers & J. A. Frogel (1967, ApJ, 148, 305)]~~,		where, following {{ SF67 }},
	<div align="center">		<div align="center">
	<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>		<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>

Jet53man: /* Lagrangian Approach */

2022-12-07T00:11:21Z

Lagrangian Approach

← Older revision		Revision as of 20:11, 6 December 2022
Line 268:		Line 268:
	</table>		</table>
	</div>		</div>
	where, following ~~[http://adsabs.harvard.edu/abs/1967ApJ...148..305S R. Stothers & J. A. Frogel (1967, ApJ, 148, 305)]~~,		where, following {{ SF67full }},
	<div align="center">		<div align="center">
	<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>		<math>~\Omega^2 \equiv \frac{12}{\gamma_\mathrm{g}} \biggl(\frac{\omega^2 R^3}{GM_\mathrm{tot}}\biggr) \, .</math>

Jet53man: /* Linear Density Distribution */

2022-12-06T23:51:15Z

Linear Density Distribution

← Older revision		Revision as of 19:51, 6 December 2022
Line 4:		Line 4:

	==Linear Density Distribution==		==Linear Density Distribution==
	In an article titled, "~~Stellar Evolution:~~ A Survey with Analytic Models," [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19660024135.pdf R. F. Stein] (1966, in ''Stellar Evolution, Proceedings of an International Conference held at the Goddard Space Flight Center, Greenbelt'', MD, U.S.A., edited by R. F. Stein & A. G. W. Cameron, pp. 1-105) defines the "Linear Stellar Model" as a star whose density "varies linearly from the center to the surface," that is (see his equation 3.1),		In an article titled, "A Survey with Analytic Models," {{ Stein66full }} defines the "Linear Stellar Model" as a star whose density "varies linearly from the center to the surface," that is (see his equation 3.1),
	<div align="center">		<div align="center">
	<math>\rho(r) = \rho_c\biggl( 1 - \frac{r}{R} \biggr) \, ,</math>		<math>\rho(r) = \rho_c\biggl( 1 - \frac{r}{R} \biggr) \, ,</math>
Line 58:		Line 58:
	</table>		</table>
	</div>		</div>
	Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, ~~[http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19660024135.pdf Stein (1966)]~~ determines that (see his equation 3.5),		Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, {{ Stein66 }} determines that (see his equation 3.5),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

Jet53man at 17:54, 15 October 2021

2021-10-15T17:54:46Z

← Older revision		Revision as of 13:54, 15 October 2021
Line 8:		Line 8:
	<math>\rho(r) = \rho_c\biggl( 1 - \frac{r}{R} \biggr) \, ,</math>		<math>\rho(r) = \rho_c\biggl( 1 - \frac{r}{R} \biggr) \, ,</math>
	</div>		</div>
	where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Solution_Strategies\|general solution strategy]] for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,		where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our [[SSCpt2/SolutionStrategies#Solution_Strategies\|general solution strategy]] for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 58:		Line 58:
	</table>		</table>
	</div>		</div>
	Hence, proceeding via what we have labeled as [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19660024135.pdf Stein (1966)] determines that (see his equation 3.5),		Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19660024135.pdf Stein (1966)] determines that (see his equation 3.5),
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 96:		Line 96:
	====Lagrangian Approach====		====Lagrangian Approach====

	As has been derived in [[~~User:Tohline/~~SSC/Perturbations#Eigen_Value_Problem\|an accompanying discussion]], the second-order ODE that defines the relevant Eigenvalue problem is,		As has been derived in [[SSC/Perturbations#Eigen_Value_Problem\|an accompanying discussion]], the second-order ODE that defines the relevant Eigenvalue problem is,

	<div align="center">		<div align="center">
Line 567:		Line 567:
	<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>		<math>\rho(r) = \rho_c\biggl[ 1 - \biggl(\frac{r}{R} \biggr)^2 \biggr] \, ,</math>
	</div>		</div>
	where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Solution_Strategies\|general solution strategy]] for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,		where, <math>~\rho_c</math> is the central density and, <math>~R</math> is the radius of the star. Both the mass distribution and the pressure distribution can be obtained analytically from this specified density distribution. Specifically, following our [[SSCpt2/SolutionStrategies#Solution_Strategies\|general solution strategy]] for determining the equilibrium structure of spherically symmetric, self-gravitating configurations,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 614:		Line 614:
	</table>		</table>
	</div>		</div>
	Hence, proceeding via what we have labeled as [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P Prasad (1949)] determines that,		Hence, proceeding via what we have labeled as [[SSCpt2/SolutionStrategies#Technique_1\|"Technique 1"]], and enforcing the surface boundary condition, <math>~P(R) = 0</math>, [http://adsabs.harvard.edu/abs/1949MNRAS.109..103P Prasad (1949)] determines that,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 688:		Line 688:
	===Stabililty===		===Stabililty===

	As has been derived in [[~~User:Tohline/~~SSC/Perturbations#Eigen_Value_Problem\|an accompanying discussion]], the second-order ODE that defines the relevant Eigenvalue problem is,		As has been derived in [[SSC/Perturbations#Eigen_Value_Problem\|an accompanying discussion]], the second-order ODE that defines the relevant Eigenvalue problem is,

	<div align="center">		<div align="center">
Line 774:		Line 774:

	==Ramblings==		==Ramblings==
	The material originally contained in this "Ramblings" subsection has been moved to generate [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Ramblings~~\|a separate chapter that stands on its own.]]		The material originally contained in this "Ramblings" subsection has been moved to generate [[SSC/Structure/OtherAnalyticRamblings\|a separate chapter that stands on its own.]]

	==Promising Avenue of Exploration==		==Promising Avenue of Exploration==
	What follows is a direct extension of what is referred to in our "Ramblings" chapter as [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Ramblings~~#Third_Guess\|the ''third guess'' under "Exploration2"]]. We pursue this line of reasoning, here, because it appears to be a particularly promising avenue of exploration.		What follows is a direct extension of what is referred to in our "Ramblings" chapter as [[SSC/Structure/OtherAnalyticRamblings#Third_Guess\|the ''third guess'' under "Exploration2"]]. We pursue this line of reasoning, here, because it appears to be a particularly promising avenue of exploration.

	In the case of a parabolic density distribution, the LAWE becomes,		In the case of a parabolic density distribution, the LAWE becomes,
Line 816:		Line 816:
	</tr>		</tr>
	</table>		</table>
	where, for a given value of <math>~\alpha</math>, the four parameters, <math>~a_0</math>, <math>~a_2</math>, <math>~n</math> and <math>~m</math> are to be determined in concert with a value of the square of the eigenfrequency, <math>~\sigma^2</math>. From the [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Ramblings~~#Third_Guess\|accompanying discussion]] we have determined that the following five coefficient expressions must independently be zero in order for this trial eigenfunction to satisfy the LAWE:		where, for a given value of <math>~\alpha</math>, the four parameters, <math>~a_0</math>, <math>~a_2</math>, <math>~n</math> and <math>~m</math> are to be determined in concert with a value of the square of the eigenfrequency, <math>~\sigma^2</math>. From the [[SSC/Structure/OtherAnalyticRamblings#Third_Guess\|accompanying discussion]] we have determined that the following five coefficient expressions must independently be zero in order for this trial eigenfunction to satisfy the LAWE:
	<div align="center" id="FirstTable">		<div align="center" id="FirstTable">
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
Line 1,296:		Line 1,296:
	===Remaining Group of Three Constraints===		===Remaining Group of Three Constraints===

	Let's adopt another approach. Let's assume that the parameter, <math>~\alpha</math>, is also initially unspecified and replace it in all three remaining constraint expressions, in favor of <math>~s_{nm}</math>, using the [[~~User:Tohline/SSC/Structure/Other_Analytic_Models~~#First_Constraint\|above-specified, first constraint]], namely,		Let's adopt another approach. Let's assume that the parameter, <math>~\alpha</math>, is also initially unspecified and replace it in all three remaining constraint expressions, in favor of <math>~s_{nm}</math>, using the [[#First_Constraint\|above-specified, first constraint]], namely,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,556:		Line 2,556:

	==More General Approach==		==More General Approach==
	The ''specific'' trial eigenfunction that we have just examined does not appear to simultaneously satisfy all constraints prescribed by the LAWE. So, in [[~~User:Tohline/~~SSC/Stability/MoreGeneralApproach#More_General_Approach_to_the_Parabolic_Eigenvalue_Problem\|a separate chapter]], we will examine an even more general trial eigenfunction. It is the one that also has previously been introduced in our "Ramblings" chapter under the subheading, [[~~User:Tohline/~~SSC/Structure/~~Other_Analytic_Ramblings~~#Consider_Parabolic_Case\|"Consider Parabolic Case"]], having the form,		The ''specific'' trial eigenfunction that we have just examined does not appear to simultaneously satisfy all constraints prescribed by the LAWE. So, in [[SSC/Stability/MoreGeneralApproach#More_General_Approach_to_the_Parabolic_Eigenvalue_Problem\|a separate chapter]], we will examine an even more general trial eigenfunction. It is the one that also has previously been introduced in our "Ramblings" chapter under the subheading, [[SSC/Structure/OtherAnalyticRamblings#Consider_Parabolic_Case\|"Consider Parabolic Case"]], having the form,
	<div>		<div>
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 2,572:		Line 2,572:
	</tr>		</tr>
	</table>		</table>
	In this [[~~User:Tohline/~~SSC/Stability/MoreGeneralApproach#More_General_Approach_to_the_Parabolic_Eigenvalue_Problem\|accompanying chapter]], we will be examining whether or not it satisfies the (same) version of the LAWE that describes stability in structures having a parabolic density profile, namely,		In this [[SSC/Stability/MoreGeneralApproach#More_General_Approach_to_the_Parabolic_Eigenvalue_Problem\|accompanying chapter]], we will be examining whether or not it satisfies the (same) version of the LAWE that describes stability in structures having a parabolic density profile, namely,

	<div align="center">		<div align="center">

Jet53man: Created page with "FORCETOC =Other Analytically Definable, Spherical Equilibrium Models= ==Linear Density Distribution== In an article titled, "Stellar..."

2021-10-15T17:48:36Z

Created page with "__FORCETOC__  =Other Analytically Definable, Spherical Equilibrium Models= ==Linear Density Distribution== In an article titled, "Stellar..."

Show changes

SSC/Structure/OtherAnalyticModels - Revision history

Jet53man: /* Overview */

Jet53man: /* Parabolic Density Distribution */

Jet53man: /* Eulerian Approach */

Jet53man: /* Eulerian Approach */

Jet53man: /* Lagrangian Approach */

Jet53man: /* Linear Density Distribution */

Jet53man at 17:54, 15 October 2021

Jet53man: Created page with "__FORCETOC__ =Other Analytically Definable, Spherical Equilibrium Models= ==Linear Density Distribution== In an article titled, "Stellar..."

Jet53man: Created page with "FORCETOC =Other Analytically Definable, Spherical Equilibrium Models= ==Linear Density Distribution== In an article titled, "Stellar..."