User:Tohline/SSC/Structure/PolytropesEmbedded - Revision history

Tohline: /* Some Tabulated Values */

2017-04-16T16:40:42Z

Some Tabulated Values

← Older revision		Revision as of 16:40, 16 April 2017
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	===Some Tabulated Values===		===Some Tabulated Values===

	<div align="center" id="~~Table1~~">		<div align="center" id="Table3">
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
	<tr><th align="center" colspan="14">Table 1:   Turning-Point Locations along M-R Sequences of Pressure-Truncated Polytropes</th></tr>		<tr><th align="center" colspan="14">Table 3:   Turning-Point Locations along M-R Sequences of Pressure-Truncated Polytropes</th></tr>
	<tr>		<tr>
	<td align="center" rowspan="2">n</td>		<td align="center" rowspan="2">n</td>

Tohline: /* Tabular Summary (n=5) */

2017-04-16T16:40:04Z

Tabular Summary (n=5)

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	<tr>		<tr>
	<th align="center" colspan="3">		<th align="center" colspan="3">
	Properties of <math>~n=5</math> Polytropes Embedded in an External Medium of Pressure <math>~P_e</math>		Table 2:  Properties of <math>~n=5</math> Polytropes Embedded in an External Medium of Pressure <math>~P_e</math>
	<br>		<br>
	(and, accordingly, truncated at radius <math>~\xi_e</math>)		(and, accordingly, truncated at radius <math>~\xi_e</math>)

Tohline: /* Tabular Summary (n=1) */

2017-04-16T16:39:28Z

Tabular Summary (n=1)

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	<tr>		<tr>
	<th align="center" colspan="3">		<th align="center" colspan="3">
	Properties of <math>~n=1</math> Polytropes Embedded in an External Medium of Pressure <math>~P_e</math>		Table 1:  Properties of <math>~n=1</math> Polytropes Embedded in an External Medium of Pressure <math>~P_e</math>
	<br>		<br>
	(and, accordingly, truncated at radius <math>~\xi_e</math>)		(and, accordingly, truncated at radius <math>~\xi_e</math>)

Tohline: /* Turning Points */

2017-04-16T16:37:52Z

Turning Points

← Older revision		Revision as of 16:37, 16 April 2017
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	==Turning Points==		==Turning Points==

	===Limiting Pressure Along M<sub>1</sub> Sequence===		===Limiting Pressure Along M<sub>1</sub> Sequence===
	As is illustrated in the figures presented above, when an equilibrium sequence is constructed for any bounded (pressure-truncated) configuration having <math>~n > 3</math>, the sequence exhibits multiple "turning points." For example, when moving along the R-P sequence [[User:Tohline/SSC/Structure/PolytropesEmbedded#WhitworthFig1b\|displayed in Figure 1]] for <math>~n=5</math> configurations, the external pressure monotonically climbs to a maximum value, <math>~P_\mathrm{max}</math>, then "turns around" and steadily decreases thereafter. [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)] separately derived an expression that pinpoints the location of the <math>~P_\mathrm{max}</math> turning point along an R-P sequence — Kimura refers to this as an "M<sub>1</sub> sequence" because the configuration's mass is held fixed while the external pressure and the system's corresponding equilibrium radius is varied. The turning point is located along the sequence at the point where,		As is illustrated in the figures presented above, when an equilibrium sequence is constructed for any bounded (pressure-truncated) configuration having <math>~n > 3</math>, the sequence exhibits multiple "turning points." For example, when moving along the R-P sequence [[User:Tohline/SSC/Structure/PolytropesEmbedded#WhitworthFig1b\|displayed in Figure 1]] for <math>~n=5</math> configurations, the external pressure monotonically climbs to a maximum value, <math>~P_\mathrm{max}</math>, then "turns around" and steadily decreases thereafter. [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)] separately derived an expression that pinpoints the location of the <math>~P_\mathrm{max}</math> turning point along an R-P sequence — Kimura refers to this as an "M<sub>1</sub> sequence" because the configuration's mass is held fixed while the external pressure and the system's corresponding equilibrium radius is varied. The turning point is located along the sequence at the point where,
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	<div align="center">		<div align="center">
	<math>~\frac{\xi (-\theta^')}{\theta} = \frac{2}{(n-1)} \, .</math>		<math>~\frac{\xi (-\theta^')}{\theta} = \frac{2}{(n-1)} \, .</math>
			</div>

			===Some Tabulated Values===

			<div align="center" id="Table1">
			<table border="1" cellpadding="8" align="center">
			<tr><th align="center" colspan="14">Table 1:   Turning-Point Locations along M-R Sequences of Pressure-Truncated Polytropes</th></tr>
			<tr>
			<td align="center" rowspan="2">n</td>
			<td align="center" colspan="6"><font color="yellow" size="+2">●</font> Maximum Radius <font color="yellow" size="+2">●</font></td>
			<td align="center" colspan="6"><font color="darkgreen" size="+2">●</font> Maximum Mass <font color="darkgreen" size="+2">●</font></td>
			</tr>
			<tr>
			<td align="center"><math>~\tilde\xi</math></td>
			<td align="center"><math>~\tilde\theta</math></td>
			<td align="center"><math>~\biggl\|\frac{d\theta}{d\xi}\biggr\|_\tilde\xi</math></td>
			<td align="center"><math>~\frac{(n-1)}{2}\biggl[ \frac{\xi}{\theta} \biggl\|\frac{d\theta}{d\xi}\biggr\|~\biggr]_\tilde\xi</math></td>
			<td align="center"><math>~\frac{R}{R_\mathrm{SWS}}</math>
			<td align="center"><math>~\frac{M}{M_\mathrm{SWS}}</math>
			<td align="center"><math>~\tilde\xi</math></td>
			<td align="center"><math>~\tilde\theta</math></td>
			<td align="center"><math>~\biggl\|\frac{d\theta}{d\xi}\biggr\|_\tilde\xi</math></td>
			<td align="center"><math>~\frac{(n-3)}{2}\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2 \biggr]_\tilde\xi</math></td>
			<td align="center"><math>~\frac{R}{R_\mathrm{SWS}}</math>
			<td align="center"><math>~\frac{M}{M_\mathrm{SWS}}</math>
			</tr>
			<tr>
			<td align="center">3</td>
			<td align="center">2.172</td>
			<td align="center">0.5387</td>
			<td align="center">0.2496</td>
			<td align="center">1.006</td>
			<td align="center">0.5717</td>
			<td align="center">1.726</td>
			<td align="center">6.89684862</td>
			<td align="center">0.0</td>
			<td align="center">-0.04242976</td>
			<td align="center">--</td>
			<td align="center">0.0</td>
			<td align="center">2.9583456</td>
			</tr>
			<tr>
			<td align="center">3.05</td>
			<td align="center">2.162</td>
			<td align="center">0.5437</td>
			<td align="center">0.2479</td>
			<td align="center">1.010</td>
			<td align="center">0.5704</td>
			<td align="center">1.715</td>
			<td align="center">5.034</td>
			<td align="center">0.1152</td>
			<td align="center">0.07842</td>
			<td align="center">0.973</td>
			<td align="center">0.2707</td>
			<td align="center">2.829</td>
			</tr>
			<tr>
			<td align="center">3.5</td>
			<td align="center">2.050</td>
			<td align="center">0.5930</td>
			<td align="center">0.2340</td>
			<td align="center">1.011</td>
			<td align="center">0.5630</td>
			<td align="center">1.594</td>
			<td align="center">3.910</td>
			<td align="center">0.2788</td>
			<td align="center">0.1126</td>
			<td align="center">0.994</td>
			<td align="center">0.4180</td>
			<td align="center">2.311</td>
			</tr>
			<tr>
			<td align="center">5</td>
			<td align="center"><math>~\sqrt{3}</math></td>
			<td align="center"><math>~\frac{1}{\sqrt{2}}</math></td>
			<td align="center"><math>~\frac{1}{\sqrt{24}}</math></td>
			<td align="center"><math>~1</math></td>
			<td align="center"><math>~\biggl( \frac{3\cdot 5}{2^4\pi}\biggr)^{1 / 2}</math></td>
			<td align="center"><math>~\biggl( \frac{3\cdot 5^3}{2^6\pi}\biggr)^{1 / 2}</math></td>
			<td align="center"><math>~3</math></td>
			<td align="center"><math>~\frac{1}{2}</math></td>
			<td align="center"><math>~\frac{1}{8}</math></td>
			<td align="center"><math>~1</math></td>
			<td align="center"><math>~\biggl( \frac{3^2\cdot 5}{2^6\pi}\biggr)^{1 / 2}</math></td>
			<td align="center"><math>~\biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1 / 2}</math></td>
			</tr>
			<tr>
			<td align="center">6</td>
			<td align="center">1.6</td>
			<td align="center">0.7510</td>
			<td align="center">0.1884</td>
			<td align="center">1.003</td>
			<td align="center">0.5404</td>
			<td align="center">1.301</td>
			<td align="center">2.7</td>
			<td align="center">0.5811</td>
			<td align="center">0.1221</td>
			<td align="center">0.999</td>
			<td align="center">0.4802</td>
			<td align="center">1.635</td>
			</tr>
			</table>
	</div>		</div>

Tohline: /* Additional, Numerically Constructed Polytropic Configurations */

2017-04-14T20:24:43Z

Additional, Numerically Constructed Polytropic Configurations

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	<div align="center" id="DFBsequences">		<div align="center" id="DFBsequences">
	<table border="1" cellpadding="8" align="center">		<table border="1" cellpadding="8" align="center">
			<tr>
			<th align="center" colspan="2"><br />[[File:DataFileButton02.png\|right\|75px\|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2]]Figure 3:   Mass-Radius Behavior of Various Polytropic Sequences</th>
			</tr>
	<tr>		<tr>
	<td align="center">		<td align="center">

Tohline: /* Additional, Numerically Constructed Polytropic Configurations */ Insert complete set of DFB sequences for comparison with Stahler's schematic

2017-04-14T16:45:00Z

Additional, Numerically Constructed Polytropic Configurations: Insert complete set of DFB sequences for comparison with Stahler's schematic

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	As has been detailed in an [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres\|accompanying chapter]], using numerical techniques we have solved the Lane-Emden equation, and thereby discerned the internal structural profiles, for polytropes having a wide variety of polytropic indexes. The righthand panel of Figure 3 presents a diagram in which the mass-radius "sequences" corresponding to eight different polytropic indexes have been drawn.		As has been detailed in an [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres\|accompanying chapter]], using numerical techniques we have solved the Lane-Emden equation, and thereby discerned the internal structural profiles, for polytropes having a wide variety of polytropic indexes. The righthand panel of Figure 3 presents a diagram in which the mass-radius "sequences" corresponding to eight different polytropic indexes have been drawn.

			<div align="center" id="DFBsequences">
			<table border="1" cellpadding="8" align="center">
			<tr>
			<td align="center">
			[[File:Stahler_MRdiagram1.png\|350px\|center\|Stahler (1983) Figure 17 (edited)]]
			</td>
			<td align="center">
			[[File:DFBsequenceB.png\|350px\|Combined DFB Sequences]]
			</td>
			</tr>
			</table>
			</div>

	==Turning Points==		==Turning Points==

Tohline at 00:18, 14 April 2017

2017-04-14T00:18:31Z

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	</div>		</div>
	</span>		</span>

			==Additional, Numerically Constructed Polytropic Configurations==
			As has been detailed in an [[User:Tohline/SSC/Structure/Polytropes#Polytropic_Spheres\|accompanying chapter]], using numerical techniques we have solved the Lane-Emden equation, and thereby discerned the internal structural profiles, for polytropes having a wide variety of polytropic indexes. The righthand panel of Figure 3 presents a diagram in which the mass-radius "sequences" corresponding to eight different polytropic indexes have been drawn.


	==Turning Points==		==Turning Points==

Tohline: /* Chieze's Presentation */

2016-11-11T02:15:01Z

Chieze's Presentation

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	</div>		</div>

	In this case, the expressions for the physical variable normalizations have been defined in terms of — in addition to <math>~G</math> and/or <math>~K</math> — the equilibrium configuration's central density, <math>~\rho_c</math>, instead of in terms of <math>~M_\mathrm{tot}</math> or <math>~P_e</math>. These expressions for <math>~~~R_\mathrm{eq}~~(\~~tilde\xi)</math> and <math>~M_\mathrm{tot}(\tilde\xi~~)</math> ~~are precisely the expressions for, respectively~~, <math>~R_s(\xi_s)</math> and <math>~M_s(\xi_s)</math> ~~presented~~ in the appendix of [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&A, 171, 225-232)] — see his equations (A5) and (A6). [Note that, for the polytropic systems of interest to us, here — that is, systems having <math>~0 \le n < \infty</math> — Chieze's parameter <math>~\epsilon \equiv \sgn(n+1) = 1</math>.]		In this case, the expressions for the physical variable normalizations have been defined in terms of — in addition to <math>~G</math> and/or <math>~K</math> — the equilibrium configuration's central density, <math>~\rho_c</math>, instead of in terms of <math>~M_\mathrm{tot}</math> or <math>~P_e</math>. These are precisely the expressions for, respectively, <math>~P_s(\xi_s)</math>, <math>~R_s(\xi_s)</math>, and <math>~M_s(\xi_s)</math> that appear in the appendix of [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&A, 171, 225-232)] — see, respectively, his equations (A7), (A5), and (A6). [Note that, for the polytropic systems of interest to us, here — that is, systems having <math>~0 \le n < \infty</math> — Chieze's parameter <math>~\epsilon \equiv \sgn(n+1) = 1</math>.]

	==Polytropic Configurations with n = 1==		==Polytropic Configurations with n = 1==

Tohline: /* Chieze's Presentation */ Show relationship between the Chieze equilibrium expressions and the ones obtained directly from C67

2016-11-11T02:00:38Z

Chieze's Presentation: Show relationship between the Chieze equilibrium expressions and the ones obtained directly from C67

← Older revision		Revision as of 02:00, 11 November 2016
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	===Chieze's Presentation===		===Chieze's Presentation===
			From equations (8), (10), and (68) in Chapter IV of [<b>[[User:Tohline/Appendix/References#C67\|<font color="red">C67</font>]]</b>], we can immediately formulate the following expressions for, respectively, <math>~P_e(\tilde\xi), R_\mathrm{eq}(\tilde\xi)</math>, and <math>~M_\mathrm{tot}(\tilde\xi)</math>:
	~~In the appendix of his paper~~, ~~[http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze~~ (~~1987~~, ~~A&A, 171, 225-232~~)~~] defined the normalizations of physical variables~~ in ~~terms~~ of &#~~8212; in addition to~~ <~~math~~>~G</~~math~~> ~~and <math>~K~~</~~math~~> ~~—~~ the ~~equilibrium configuration's central density~~, <math>~\~~rho_c</math>~~, ~~instead of <math>~M_~~\mathrm{~~tot~~}~~</math> or <math>~P_e</math>. For polytropic systems having <math>~0~~ \~~le n <~~ \~~infty~~</math> ~~— in which case Chieze's parameter~~ <math>~\~~epsilon~~ \~~equiv~~ \~~sgn(n+1~~) ~~= 1~~</math> ~~— his expressions for the equilibrium system's radius and total mass are, respectively (see his equations A5 and A6)~~:

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{R_\mathrm{eq}}{R_\mathrm{~~Chieze~~}}</math>		<math>~\frac{P_e}{P_\mathrm{Ch}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~ {\tilde\theta}^{n+1} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{R_\mathrm{eq}}{R_\mathrm{Ch}}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{M_\mathrm{tot}}{M_\mathrm{~~Chieze~~}}</math>		<math>~\frac{M_\mathrm{tot}}{M_\mathrm{Ch}}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~R_\mathrm{~~Chieze~~}</math>		<math>~P_\mathrm{Ch}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~K\rho_c^{(n+1)/n} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~R_\mathrm{Ch}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~M_\mathrm{~~Chieze~~}</math>		<math>~M_\mathrm{Ch}</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</table>		</table>
	</div>		</div>

			In this case, the expressions for the physical variable normalizations have been defined in terms of — in addition to <math>~G</math> and/or <math>~K</math> — the equilibrium configuration's central density, <math>~\rho_c</math>, instead of in terms of <math>~M_\mathrm{tot}</math> or <math>~P_e</math>. These expressions for <math>~R_\mathrm{eq}(\tilde\xi)</math> and <math>~M_\mathrm{tot}(\tilde\xi)</math> are precisely the expressions for, respectively, <math>~R_s(\xi_s)</math> and <math>~M_s(\xi_s)</math> presented in the appendix of [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&A, 171, 225-232)] — see his equations (A5) and (A6). [Note that, for the polytropic systems of interest to us, here — that is, systems having <math>~0 \le n < \infty</math> — Chieze's parameter <math>~\epsilon \equiv \sgn(n+1) = 1</math>.]

	==Polytropic Configurations with n = 1==		==Polytropic Configurations with n = 1==

Tohline: /* General Properties */ Begin summarizing the work of Chieze (1987)

2016-11-10T23:05:54Z

General Properties: Begin summarizing the work of Chieze (1987)

← Older revision		Revision as of 23:05, 10 November 2016
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	</td></tr>		</td></tr>
			</table>
			</div>

			===Chieze's Presentation===

			In the appendix of his paper, [http://adsabs.harvard.edu/abs/1987A%26A...171..225C J. P. Chieze (1987, A&A, 171, 225-232)] defined the normalizations of physical variables in terms of — in addition to <math>~G</math> and <math>~K</math> — the equilibrium configuration's central density, <math>~\rho_c</math>, instead of <math>~M_\mathrm{tot}</math> or <math>~P_e</math>. For polytropic systems having <math>~0 \le n < \infty</math> — in which case Chieze's parameter <math>~\epsilon \equiv \sgn(n+1) = 1</math> — his expressions for the equilibrium system's radius and total mass are, respectively (see his equations A5 and A6):

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

			<tr>
			<td align="right">
			<math>~\frac{R_\mathrm{eq}}{R_\mathrm{Chieze}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~\biggl[ \frac{n+1}{4\pi} \biggr]^{1 / 2} \tilde\xi</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\frac{M_\mathrm{tot}}{M_\mathrm{Chieze}}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\biggl[ \frac{(n+1)^3}{4\pi} \biggr]^{1 / 2}(- {\tilde\xi}^2 {\tilde\theta}^') \, ,
			</math>
			</td>
			</tr>
			</table>
			</div>
			where,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~R_\mathrm{Chieze}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl[\biggl(\frac{K}{G}\biggr) \rho_c^{1/n-1}\biggr]^{1 / 2} \, ,</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~M_\mathrm{Chieze}</math>
			</td>
			<td align="center">
			<math>~\equiv</math>
			</td>
			<td align="left">
			<math>~\biggl[\biggl(\frac{K}{G}\biggr)^3 \rho_c^{(3-n)/n}\biggr]^{1 / 2} \, .</math>
			</td>
			</tr>
	</table>		</table>
	</div>		</div>