User:Tohline/SSC/Structure/BiPolytropes/Analytic1 5 - Revision history

Tohline: /* Murphy's Example Model Characteristics */

2019-03-11T21:48:15Z

Murphy's Example Model Characteristics

← Older revision		Revision as of 21:48, 11 March 2019
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	<tr>		<tr>
	~~<!--~~<td align="center" colspan="9">[[Image:AAAwaiting01.png\|~~500px~~\|Columns extracted directly from Murphy (1983)]]</td>-->		<td align="center" colspan="9">[[Image:AAAwaiting01.png\|600px\|Columns extracted directly from Murphy (1983)]]</td>
			<!--
	<td align="center">[[File:Table3CompleteCol1.png\|center\|50px\|Column 1 from Table 3 of Murphy (1983)]]</td>		<td align="center">[[File:Table3CompleteCol1.png\|center\|50px\|Column 1 from Table 3 of Murphy (1983)]]</td>
	<td align="right" colspan="2">[[File:Table3CompleteCol2.png\|center\|123px\|Column 3 from Table 3 of Murphy (1983)]]</td>		<td align="right" colspan="2">[[File:Table3CompleteCol2.png\|center\|123px\|Column 3 from Table 3 of Murphy (1983)]]</td>
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	<td align="right">[[File:Table3CompleteCol5.png\|center\|71px\|Column 5 from Table 3 of Murphy (1983)]]</td>		<td align="right">[[File:Table3CompleteCol5.png\|center\|71px\|Column 5 from Table 3 of Murphy (1983)]]</td>
	<td align="right">[[File:Table3CompleteCol6.png\|center\|83px\|Column 6 from Table 3 of Murphy (1983)]]</td>		<td align="right">[[File:Table3CompleteCol6.png\|center\|83px\|Column 6 from Table 3 of Murphy (1983)]]</td>
			-->
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Murphy and Fiedler (1985) */

2019-03-11T21:45:39Z

Murphy and Fiedler (1985)

← Older revision		Revision as of 21:45, 11 March 2019
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:MF85Table1.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]]		<!-- [[File:MF85Table1.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]] -->
	<!-- [[Image:~~AAAwaiting01~~.png\|center\|~~600px~~\|Murphy & Fiedler (1985) Table 1]] ~~-->~~		[[Image:AAAwaiting02.png\|center\|300px\|Murphy & Fiedler (1985) Table 1]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Murphy's Example Model Characteristics */ Unveil images of columns of data from Murphy's Table 3

2016-12-03T20:02:03Z

Murphy's Example Model Characteristics: Unveil images of columns of data from Murphy's Table 3

← Older revision		Revision as of 20:02, 3 December 2016
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	<tr>		<tr>
	<td align="center" colspan="9">[[Image:AAAwaiting01.png\|500px\|Columns extracted directly from Murphy (1983)]]</td>		<!--<td align="center" colspan="9">[[Image:AAAwaiting01.png\|500px\|Columns extracted directly from Murphy (1983)]]</td>-->
	<!--
	<td align="center">[[File:Table3CompleteCol1.png\|center\|50px\|Column 1 from Table 3 of Murphy (1983)]]</td>		<td align="center">[[File:Table3CompleteCol1.png\|center\|50px\|Column 1 from Table 3 of Murphy (1983)]]</td>
	<td align="right" colspan="2">[[File:Table3CompleteCol2.png\|center\|123px\|Column 3 from Table 3 of Murphy (1983)]]</td>		<td align="right" colspan="2">[[File:Table3CompleteCol2.png\|center\|123px\|Column 3 from Table 3 of Murphy (1983)]]</td>
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	<td align="right">[[File:Table3CompleteCol5.png\|center\|71px\|Column 5 from Table 3 of Murphy (1983)]]</td>		<td align="right">[[File:Table3CompleteCol5.png\|center\|71px\|Column 5 from Table 3 of Murphy (1983)]]</td>
	<td align="right">[[File:Table3CompleteCol6.png\|center\|83px\|Column 6 from Table 3 of Murphy (1983)]]</td>		<td align="right">[[File:Table3CompleteCol6.png\|center\|83px\|Column 6 from Table 3 of Murphy (1983)]]</td>
	~~-->~~
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Murphy and Fiedler (1985) */ Unveil image of Murphy & Fiedler's Table 1

2016-12-03T19:58:08Z

Murphy and Fiedler (1985): Unveil image of Murphy & Fiedler's Table 1

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	<tr>		<tr>
	<td align="center">		<td align="center">
	~~<!--~~ [[File:MF85Table1.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]] -->		[[File:MF85Table1.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]]
	[[Image:AAAwaiting01.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]]		<!-- [[Image:AAAwaiting01.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]] -->
	</td>		</td>
	</tr>		</tr>

Tohline: /* Murphy and Fiedler (1985) */ Comment-out Table from Murphy & Fiedler

2015-05-24T20:22:03Z

Murphy and Fiedler (1985): Comment-out Table from Murphy & Fiedler

← Older revision		Revision as of 20:22, 24 May 2015
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:MF85Table1.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]]		<!-- [[File:MF85Table1.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]] -->
			[[Image:AAAwaiting01.png\|center\|600px\|Murphy & Fiedler (1985) Table 1]]
	</td>		</td>
	</tr>		</tr>

Tohline: /* Murphy's Example Model Characteristics */ Comment-out individual columns from Murphy's Table 3

2015-05-24T20:18:16Z

Murphy's Example Model Characteristics: Comment-out individual columns from Murphy's Table 3

← Older revision		Revision as of 20:18, 24 May 2015
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	<tr>		<tr>
			<td align="center" colspan="9">[[Image:AAAwaiting01.png\|500px\|Columns extracted directly from Murphy (1983)]]</td>
			<!--
	<td align="center">[[File:Table3CompleteCol1.png\|center\|50px\|Column 1 from Table 3 of Murphy (1983)]]</td>		<td align="center">[[File:Table3CompleteCol1.png\|center\|50px\|Column 1 from Table 3 of Murphy (1983)]]</td>
	<td align="right" colspan="2">[[File:Table3CompleteCol2.png\|center\|123px\|Column 3 from Table 3 of Murphy (1983)]]</td>		<td align="right" colspan="2">[[File:Table3CompleteCol2.png\|center\|123px\|Column 3 from Table 3 of Murphy (1983)]]</td>
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	<td align="right">[[File:Table3CompleteCol5.png\|center\|71px\|Column 5 from Table 3 of Murphy (1983)]]</td>		<td align="right">[[File:Table3CompleteCol5.png\|center\|71px\|Column 5 from Table 3 of Murphy (1983)]]</td>
	<td align="right">[[File:Table3CompleteCol6.png\|center\|83px\|Column 6 from Table 3 of Murphy (1983)]]</td>		<td align="right">[[File:Table3CompleteCol6.png\|center\|83px\|Column 6 from Table 3 of Murphy (1983)]]</td>
			-->
	</tr>		</tr>
	<tr>		<tr>

Tohline: /* Parameter Values / Corrected expression for R_tot^ by removing dependence on (mu_e/mu_c)

2015-05-17T20:11:18Z

Parameter Values: Corrected expression for R_tot^* by removing dependence on (mu_e/mu_c)

← Older revision		Revision as of 20:11, 17 May 2015
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	<tr>		<tr>
	<td align="center">		<td align="center">
	<math>~ ~~\biggl( \frac{\mu_e}{\mu_c}\biggr)^{-2}~~ R^*_s</math>		<math>~ R^*_s</math>
	</td>		</td>
	<td align="center">		<td align="center">
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	</div>		</div>
	STOP OMISSION -->		STOP OMISSION -->


	==Profile==		==Profile==

Tohline: /* Profile */ Finished detailing "Profile" relations

2015-05-15T19:31:29Z

Profile: Finished detailing "Profile" relations

← Older revision		Revision as of 19:31, 15 May 2015
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	==Profile==		==Profile==

	Once the values of the key set of parameters have been determined as illustrated in the preceding formula table, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in [[User:Tohline/SSC/Structure/BiPolytropes/~~Analytic5_1~~#~~Steps_2_~~.~~26_3~~\|step #4]] and [[User:Tohline/SSC/Structure/BiPolytropes/~~Analytic5_1~~#~~Step_7~~\|step #8]], above. ~~Table 2~~ summarizes the mathematical expressions that define the profile throughout the core (column 2) and throughout the envelope (column 3) of the normalized mass density, <math>\rho^(r^)</math>, the normalized gas pressure, <math>P^(r^)</math>, and the normalized mass interior to <math>r^</math>, <math>M_r^(r^)</math>. For all profiles, the relevant normalized radial coordinate is <math>r^</math>, as defined in the <~~math~~>~~2^\mathrm{~~nd}</~~math~~> row of ~~Table 2~~. Graphical illustrations of these resulting profiles can be viewed by clicking on the thumbnail images posted in the last few columns of ~~Table 2~~.		Once the values of the key set of parameters have been determined as illustrated in the preceding formula table, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Step_4:__Throughout_the_core_.28.29\|step #4]] and [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Step_8:__Envelope.27s_Physical_Profile\|step #8]], above. The following table summarizes the mathematical expressions that define the profile throughout the core (column 2) and throughout the envelope (column 3) of the normalized mass density, <math>~\rho^(r^)</math>, the normalized gas pressure, <math>~P^(r^)</math>, and the normalized mass interior to <math>~r^</math>, <math>~M_r^(r^)</math>. For all profiles, the relevant normalized radial coordinate is <math>~r^</math>, as defined in the 2<sup>nd</sup> row of the table. Graphical illustrations of these resulting profiles can be viewed by clicking on the thumbnail images posted in the last few columns of the table.

	<div align="center">		<div align="center">
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	<tr>		<tr>
	<td align="center">		<td align="center">
	<math>r^*</math>		<math>~r^*</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\biggl( \frac{3}{2\pi} \biggr)^{1/2} \xi</math>		<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \~~theta^~~{-2}~~_i (~~2\pi)^{-1/2}\eta</math>		<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \biggl(\frac{3}{2\pi}\biggr)^{1/2}\eta</math>
	</td>		</td>

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	<tr>		<tr>
	<td align="center">		<td align="center">
	<math>\rho^*</math>		<math>~\rho^*</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~~\biggl( 1 +~~ \frac{1}{3}\xi~~^2 \biggr)^{-5/2~~}</math>		<math>\frac{\sin\xi}{\xi}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \~~theta^{5}_i~~ \phi(\eta)</math>		<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i [\phi(\eta)]^5</math>
	</td>		</td>

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	<tr>		<tr>
	<td align="center">		<td align="center">
	<math>P^*</math>		<math>~P^*</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\biggl( ~~1 +~~ \frac{1}{3}\xi^2 \biggr)^~~{-3}~~</math>		<math>\biggl( \frac{\sin\xi}{\xi} \biggr)^2</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\theta^{6}_i [\phi(\eta)]^{2}</math>		<math>\theta^{2}_i [\phi(\eta)]^{6}</math>
	</td>		</td>

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	<tr>		<tr>
	<td align="center">		<td align="center">
	<math>M_r^*</math>		<math>~M_r^*</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\biggl( \frac{2~~\cdot 3~~}{\pi } \biggr)^{1/2} \~~biggl[~~ \xi^3 \~~biggl( 1 +~~ \~~frac{1}{3}~~\xi~~^2 \biggr~~)~~^{-3/2} \biggr]~~</math>		<math>\biggl( \frac{2}{\pi}\biggr)^{1/2} (\sin\xi - \xi\cos\xi)</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} ~~\theta^{-1}_i~~ \biggl( \frac{2}{\pi} \biggr)^{1/2} \biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr)</math>		<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-2} \biggl( \frac{2\cdot 3^3 }{\pi} \biggr)^{1/2} \theta_i
			\biggl(-\eta^2 \frac{d\phi}{d\eta} \biggr) </math>
	</td>		</td>

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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 presented in Step 6, above.		<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, above]].
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	</div>		</div>


	==Murphy and Fiedler (1985)==		==Murphy and Fiedler (1985)==

Tohline: /* Step 7: Identifying the Surface */ Type in corresponding expression for eta_s

2015-05-15T19:20:09Z

Step 7: Identifying the Surface: Type in corresponding expression for eta_s

← Older revision		Revision as of 19:20, 15 May 2015
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	\xi_i e^{2(\pi - \Delta_i )} \, .		\xi_i e^{2(\pi - \Delta_i )} \, .
	</math>		</math>
			</td>
			</tr>
			</table>
			</div>

			In terms of the natural radial coordinate of the envelope, this is,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\eta_s \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-1}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~3^{-1/2} \xi_i e^{2(\pi - \Delta_i)} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: Redefine various subsection headings and whether they are one, two, or three-level indentations

2015-05-15T18:58:28Z

Redefine various subsection headings and whether they are one, two, or three-level indentations

← Older revision		Revision as of 18:58, 15 May 2015
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	</div>		</div>

	~~Furthermore, because Shrivastava's function — and, along with it, the envelope's density — drops to zero when,~~
	~~<div align="center">~~
	~~<math>\Delta = \Delta_s \equiv \pi \, ,</math>~~
	~~</div>~~
	~~we know that the radius, <math>~\xi_s</math>, of the bipolytropic configuration is given by the expression,~~
	~~<div align="center">~~
	~~<table border="0" cellpadding="5" align="center">~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\pi</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\Delta_i + \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Rightarrow ~~~~ \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\pi - \Delta_i~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Rightarrow ~~~~ \xi_s</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\xi_i e^{2(\pi - \Delta_i )} \, .~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~

	====Obtained from Chandrasekhar's U-constraint====		====Obtained from Chandrasekhar's U-constraint====
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	rather than simply our derived value of <math>~B_0</math>.		rather than simply our derived value of <math>~B_0</math>.

	===Analytic Solution of Key Interface Relation===		==Step 7: Identifying the Surface==

			Because Shrivastava's function — and, along with it, the envelope's density — drops to zero when,
			<div align="center">
			<math>\Delta = \Delta_s \equiv \pi \, ,</math>
			</div>
			we know that the radius, <math>~\xi_s</math>, of the bipolytropic configuration is given by the expression,
			<div align="center">
			<table border="0" cellpadding="5" align="center">

			<tr>
			<td align="right">
			<math>~\pi</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\Delta_i + \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~ \ln\biggl( \frac{\xi_s}{\xi_i} \biggr)^{1/2}</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\pi - \Delta_i
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">
			<math>~\Rightarrow ~~~~ \xi_s</math>
			</td>
			<td align="center">
			<math>~=</math>
			</td>
			<td align="left">
			<math>~
			\xi_i e^{2(\pi - \Delta_i )} \, .
			</math>
			</td>
			</tr>
			</table>
			</div>

			==Analytic Solution of Key Interface Relation==

	Returning to our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#KeyInterfaceRelation\|previously derived]],		Returning to our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#KeyInterfaceRelation\|previously derived]],
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	The points of intersection that are highlighted in this figure are also what [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] refers to as roots of his <math>~F2 = 0</math> function. The values of ten successive coordinate roots of this function are listed in the right-hand column of Table 1 in [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)]. We have re-listed the values of six of these roots in a column immediately to the right of the above figure and have marked with orange circles the locations of these six points in the figure. As they should, Murphy's identified roots lie precisely at the intersection of the red line with the other curves.		The points of intersection that are highlighted in this figure are also what [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)] refers to as roots of his <math>~F2 = 0</math> function. The values of ten successive coordinate roots of this function are listed in the right-hand column of Table 1 in [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy (1983)]. We have re-listed the values of six of these roots in a column immediately to the right of the above figure and have marked with orange circles the locations of these six points in the figure. As they should, Murphy's identified roots lie precisely at the intersection of the red line with the other curves.

	===Physically Viable Parameter Values===		==Physically Viable Parameter Values==
			===Interface Mapping===

	As has already been stated in our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Steps_2_.26_3\|above description of the <math>~n_c = 1</math> core]] of these bipolytropic configurations, the structure of the core will be physically viable as long as the radial location of the interface, <math>~\xi_i</math>, between the core and the envelope is positioned somewhere within the range,		As has already been stated in our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Steps_2_.26_3\|above description of the <math>~n_c = 1</math> core]] of these bipolytropic configurations, the structure of the core will be physically viable as long as the radial location of the interface, <math>~\xi_i</math>, between the core and the envelope is positioned somewhere within the range,
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	We initially suspected that the "Model 4" mismatch resulted from a typographical or transcription error in the value of <math>~\zeta_J</math> — in our terminology, <math>~\xi_i</math> — that appears in Murphy's published Table 3. So we tried a variety of values of <math>~\xi_i</math>, looking for a set of generated characteristics that matched Murphy's published Model 4 characteristics. We were unsuccessful in this effort. The most tantalizing results from this search are displayed as cells with light-green backgrounds in the bottom segment of the above table: (Model 4b) Setting <math>~\xi_i = 0.048681~(m = 0)</math> gives <math>~B=3.5999</math>, which matches Murphy's "Model 4" value of <math>~B</math>; (Model 4c) setting <math>~\xi_i = 1.14988~(m = 0)</math> gives values of <math>~A</math> and <math>~\zeta_J (\mathrm{i.e.,}~\xi_s)</math> that both match Murphy's values; (Model 4d) setting <math>~\xi_i = 1.77027~(m = 0)</math> also gives values of <math>~A</math> and <math>~\zeta_J</math> that both match Murphy's values; and (Model 4e) setting <math>~\xi_i = 2.10961~(m = 1)</math> gives a value of <math>~(A_0\eta)_\mathrm{root}</math> that matches Murphy's value of <math>~\xi_J</math>. As we have been unable to find a value of <math>~\xi_i</math> whose associated parameter values match Murphy's "Model 4" values across the board, we remain at a loss to explain the mismatch between our "Model 4" and Murphy's.		We initially suspected that the "Model 4" mismatch resulted from a typographical or transcription error in the value of <math>~\zeta_J</math> — in our terminology, <math>~\xi_i</math> — that appears in Murphy's published Table 3. So we tried a variety of values of <math>~\xi_i</math>, looking for a set of generated characteristics that matched Murphy's published Model 4 characteristics. We were unsuccessful in this effort. The most tantalizing results from this search are displayed as cells with light-green backgrounds in the bottom segment of the above table: (Model 4b) Setting <math>~\xi_i = 0.048681~(m = 0)</math> gives <math>~B=3.5999</math>, which matches Murphy's "Model 4" value of <math>~B</math>; (Model 4c) setting <math>~\xi_i = 1.14988~(m = 0)</math> gives values of <math>~A</math> and <math>~\zeta_J (\mathrm{i.e.,}~\xi_s)</math> that both match Murphy's values; (Model 4d) setting <math>~\xi_i = 1.77027~(m = 0)</math> also gives values of <math>~A</math> and <math>~\zeta_J</math> that both match Murphy's values; and (Model 4e) setting <math>~\xi_i = 2.10961~(m = 1)</math> gives a value of <math>~(A_0\eta)_\mathrm{root}</math> that matches Murphy's value of <math>~\xi_J</math>. As we have been unable to find a value of <math>~\xi_i</math> whose associated parameter values match Murphy's "Model 4" values across the board, we remain at a loss to explain the mismatch between our "Model 4" and Murphy's.

	===Step 8: Envelope's Physical Profile===		==Step 8: Envelope's Physical Profile==

	Combining the set of expressions provided in the right-most column of the [[User:Tohline/SSC/Structure/BiPolytropes#TableSetup\|accompanying bipolytrope "Table 1: Setup"]] with the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Step_5:_Interface_Conditions\|interface conditions already established, above]], we can now specify how the density, pressure, and integral mass vary throughout the envelope, that is, over the radial extent <math>\xi_i \le \xi \le \xi_s</math>.		Combining the set of expressions provided in the right-most column of the [[User:Tohline/SSC/Structure/BiPolytropes#TableSetup\|accompanying bipolytrope "Table 1: Setup"]] with the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic1_5#Step_5:_Interface_Conditions\|interface conditions already established, above]], we can now specify how the density, pressure, and integral mass vary throughout the envelope, that is, over the radial extent <math>\xi_i \le \xi \le \xi_s</math>.