SSC/Structure/Polytropes - Revision history

Jet53man: /* Related Discussions */

2023-12-06T19:52:17Z

Related Discussions

← Older revision		Revision as of 15:52, 6 December 2023
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	* [https://books.google.com/books/about/Mathematical_Tables.html?id=VeJdxwEACAAJ Mathematical Tables: Emden functions being solutions of emden's equation together with certain associated functions. II, Volume 2]		* [https://books.google.com/books/about/Mathematical_Tables.html?id=VeJdxwEACAAJ Mathematical Tables: Emden functions being solutions of emden's equation together with certain associated functions. II, Volume 2]

			* [[SSC/Structure/PowerLawDensity\|Power-Law density distribution]]


	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: /* Lane-Emden Equation */

2023-07-25T20:55:36Z

Lane-Emden Equation

← Older revision		Revision as of 16:55, 25 July 2023
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	<br />		<br />
	</div>		</div>
	Our task is to solve this ODE to determine the behavior of the function <math>~\Theta_H(\xi)</math> — and, from it in turn, determine the radial distribution of various dimensional physical variables — for various values of the polytropic index, {{Math/MP_PolytropicIndex}}. In particular, from time to time we will find it useful to realize that the mass interior to <math>r</math> is given by the expression,		Our task is to solve this ODE to determine the behavior of the function <math>~\Theta_H(\xi)</math> — and, from it in turn, determine the radial distribution of various dimensional physical variables — for various values of the polytropic index, {{Math/MP_PolytropicIndex}}. <span id="IntegralMass">In particular, from time to time we will find it useful to realize that the mass interior to <math>r</math> is given by the expression, </span>

	<table border="0" align="center" cellpadding="5">		<table border="0" align="center" cellpadding="5">

Jet53man: /* Lane-Emden Equation */

2023-07-25T20:52:32Z

Lane-Emden Equation

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 <br />
 {{Math/EQ_SSLaneEmden01}}
 </div>
-Our task is to solve this ODE to determine the behavior of the function <math>~\Theta_H(\xi)</math> &#8212; and, from it in turn, determine the radial distribution of various dimensional physical variables &#8212; for various values of the polytropic index, {{Math/MP_PolytropicIndex}}.
+Our task is to solve this ODE to determine the behavior of the function <math>~\Theta_H(\xi)</math> &#8212; and, from it in turn, determine the radial distribution of various dimensional physical variables &#8212; for various values of the polytropic index, {{Math/MP_PolytropicIndex}}.  In particular, from time to time we will find it useful to realize that the mass interior to <math>r</math> is given by the expression,
 <div align="center" id="MassRadiusRelation">

Jet53man: /* Polytropic Spheres */

2022-10-26T19:13:49Z

Polytropic Spheres

← Older revision		Revision as of 15:13, 26 October 2022
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	It will be useful to note as well that, for any polytropic gas, the three key state variables are always related to one another through the simple expression,		It will be useful to note as well that, for any polytropic gas, the three key state variables are always related to one another through the simple expression,
	<div align="center~~"><font size="+1~~">		<div align="center">
	<math> (n+1) P = H\rho</math> .		<math> (n+1) P = H\rho</math> .
	~~</font>~~</div>		</div>

	In his effort to model the Sun's interior, [https://archive.org/details/mobot31753002152772/page/56 J. Homer Lane (1870)] was the first to couch an examination of stellar structure in the context of, what is now usually referred to as, "polytropic structures." He examined the structural properties of spherically symmetric models having, effectively, indexes of <math>~n = \tfrac{3}{2}</math> and <math>~n = \tfrac{5}{2}</math>. In an [[SSC/Structure/Lane1870#Lane.27s_1870_Work\|accompanying chapter]], we review this early groundbreaking work highlighting the quantitative care with which Lane carried out his analysis. A much more expansive study of polytropic (and isothermal) structures was subsequently published by [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)], who was aware of Lane's work — see, for example, the footnote on p. 462 of his book. It is largely Emden's notation — especially as employed by [[Appendix/References#C67\|[<font color="red">C67</font>] ]] — that is adopted in current discussions of polytropic structures, including our discussion which follows.		In his effort to model the Sun's interior, [https://archive.org/details/mobot31753002152772/page/56 J. Homer Lane (1870)] was the first to couch an examination of stellar structure in the context of, what is now usually referred to as, "polytropic structures." He examined the structural properties of spherically symmetric models having, effectively, indexes of <math>~n = \tfrac{3}{2}</math> and <math>~n = \tfrac{5}{2}</math>. In an [[SSC/Structure/Lane1870#Lane.27s_1870_Work\|accompanying chapter]], we review this early groundbreaking work highlighting the quantitative care with which Lane carried out his analysis. A much more expansive study of polytropic (and isothermal) structures was subsequently published by [https://books.google.com/books?id=MiDQAAAAMAAJ&printsec=frontcover#v=onepage&q&f=true Emden (1907)], who was aware of Lane's work — see, for example, the footnote on p. 462 of his book. It is largely Emden's notation — especially as employed by [[Appendix/References#C67\|[<font color="red">C67</font>] ]] — that is adopted in current discussions of polytropic structures, including our discussion which follows.

Jet53man: /* Lane-Emden Equation */

2022-10-03T16:37:03Z

Lane-Emden Equation

← Older revision		Revision as of 12:37, 3 October 2022
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	</table>		</table>
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	This matches the expression presented in an [[SSC/FreeEnergy/~~PowerPoint~~#Case_M_Equilibrium_Conditions\|accompanying summary supporting a PowerPoint presentation]].		This matches the expression presented in an [[SSC/FreeEnergy/Powerpoint#Case_M_Equilibrium_Conditions\|accompanying summary supporting a PowerPoint presentation]].

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

Jet53man: /* HSCF Technique */

2021-07-29T03:50:58Z

HSCF Technique

← Older revision		Revision as of 23:50, 28 July 2021
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	===Techniques===		===Techniques===
	====HSCF Technique====		====HSCF Technique====
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>via<br />Self-Consistent<br />Field (SCF)<br />Technique</b>]]</font>
			\|}
	On the first spreadsheet within the workbook, we establish the following columns of number:		On the first spreadsheet within the workbook, we establish the following columns of number:
	<ul>		<ul>

Jet53man: /* Straight Numerical Integration */

2021-07-29T03:47:50Z

Straight Numerical Integration

← Older revision		Revision as of 23:47, 28 July 2021
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	====Straight Numerical Integration====		====Straight Numerical Integration====
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>via<br />Direct<br />Numerical<br />Integration</b>]]</font>
			\|}
	The [[#LaneEmdenEquation\|above governing relation]] may be rewritten as,		The [[#LaneEmdenEquation\|above governing relation]] may be rewritten as,
	<div align="center">		<div align="center">

Jet53man: /* Known Analytic Solutions */

2021-07-29T03:43:32Z

Known Analytic Solutions

← Older revision		Revision as of 23:43, 28 July 2021
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	==Known Analytic Solutions==		==Known Analytic Solutions==
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Known<br />Analytic<br />Solutions</b>]]</font>
			\|}
	While the Lane-Emden equation has been studied for over 100 years, to date, analytic solutions to the equation (subject to the above specified boundary conditions) have been found only for three values of the polytropic index, {{Math/MP_PolytropicIndex}}. We will review these three solutions here.		While the Lane-Emden equation has been studied for over 100 years, to date, analytic solutions to the equation (subject to the above specified boundary conditions) have been found only for three values of the polytropic index, {{Math/MP_PolytropicIndex}}. We will review these three solutions here.
			<br />
			<br />
			<br />
			<br />
			<br />

	===n = 0 Polytrope===		===n = 0 Polytrope===

Jet53man: /* Polytropic Spheres */

2021-07-29T03:31:09Z

Polytropic Spheres

← Older revision		Revision as of 23:31, 28 July 2021
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	=Polytropic Spheres=		=Polytropic Spheres=

	[[Image:LSU_Structure_still.gif\|74px\|left]]		{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:#ffff99;" \|
			<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Isolated<br />Polytropes</b>]]</font>
			\|}
			<!-- [[Image:LSU_Structure_still.gif\|74px\|left]]-->
	Here we will supplement the [[SSCpt1/PGE\|simplified set of principal governing equations]] with a polytropic equation of state, as defined in our overview of [[SR#Time-Independent\|supplemental relations for time-independent problems]]. Specifically, we will assume that {{Math/VAR_Density01}} is related to {{Math/VAR_Enthalpy01}} through the relation,		Here we will supplement the [[SSCpt1/PGE\|simplified set of principal governing equations]] with a polytropic equation of state, as defined in our overview of [[SR#Time-Independent\|supplemental relations for time-independent problems]]. Specifically, we will assume that {{Math/VAR_Density01}} is related to {{Math/VAR_Enthalpy01}} through the relation,

Jet53man: /* Plotted Structural Profiles */

2021-07-26T22:56:11Z

Plotted Structural Profiles

← Older revision		Revision as of 18:56, 26 July 2021
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	</div>		</div>

	has been drawn from an [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|accompanying discussion of pressure-truncated polytropic configurations]]. In four of the Figure 4 panels, we have compared the profile of our numerically determined polytropic function (curve defined by <math>~\sim 200</math> small, black circular markers) to results (7 - 9 larger, blue circular markers) taken from Table 2.5.1 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] — see, specifically the segment of his table on pp. 74 - 75 that applies to polytropic spheres — in an effort to demonstrate that our numerically determined solutions are accurate.		has been drawn from an [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation\|accompanying discussion of pressure-truncated polytropic configurations]]. In four of the Figure 4 panels, we have compared the profile of our numerically determined polytropic function (curve defined by <math>~\sim 200</math> small, black circular markers) to results (7 - 9 larger, blue circular markers) taken from Table 2.5.1 of [http://adsabs.harvard.edu/abs/2004ASSL..306.....H Horedt (2004)] — see, specifically the segment of his table on pp. 74 - 75 that applies to polytropic spheres — in an effort to demonstrate that our numerically determined solutions are accurate.

	<div align="center" id="Fig4">		<div align="center" id="Fig4">
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	</table>		</table>
	</div>		</div>


	====Emden's (1907) Tabulated Data====		====Emden's (1907) Tabulated Data====