SSC/Structure/BiPolytropes/Analytic51/Pt2 - Revision history

Joel2: /* Parameter Values */

2025-12-15T17:01:04Z

Parameter Values

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	~~xxx~~		0.538858190
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Joel2: /* Parameter Values */

2025-12-15T16:56:19Z

Parameter Values

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Joel2: /* Parameter Values */

2025-12-11T19:19:59Z

Parameter Values

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Joel2: /* Parameter Values */

2025-12-11T19:10:07Z

Parameter Values

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Joel2: /* Parameter Values */

2025-12-11T19:02:18Z

Parameter Values

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	1.66864602		1.66864602<sup>&dagger;</sup>
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	3.0		3.0
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	<td rowspan="20">		<td rowspan="21">
	[[File:Bipolytrope51Boundaries02.png\|500px\|Examples]]<br />		[[File:Bipolytrope51Boundaries02.png\|500px\|Examples]]<br />
	<table border="0" align="left" cellpadding="10"><tr><td align="left">		<table border="0" align="left" cellpadding="10"><tr><td align="left">
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	0.61751		0.61751
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			</tr>

			<tr>
			<td align="left" colspan="6">
			<sup>&dagger;</sup>This choice of the value of <math>\xi_i = 1.66864602</math> is motivated by our [[SSC/Stability/BiPolytropes/Pt3#Fundamental_Modes\|discussion of the fundamental mode of oscillation]] of the marginally unstable model that has <math>\mu_e/\mu_c = 1.0</math>; see especially row 1 of Table 2 in that associated chapter.
	</tr>		</tr>

Joel2: /* Parameter Values */

2025-12-11T18:45:42Z

Parameter Values

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	1.~~6686460157~~		1.66864602
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Joel2: /* Parameter Values */

2025-12-11T18:42:51Z

Parameter Values

Show changes

Joel2: /* BiPolytrope with nc = 5 and ne = 1 */

2024-01-16T04:59:06Z

BiPolytrope with nc = 5 and ne = 1

← Older revision		Revision as of 00:59, 16 January 2024
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	__FORCETOC__		__FORCETOC__
	=BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1=		=BiPolytrope with n<sub>c</sub> = 5 and n<sub>e</sub> = 1 (Pt 2)=
	<table border="1" align="center" width="100%" colspan="8">		<table border="1" align="center" width="100%" colspan="8">
	<tr>		<tr>
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	</table>		</table>
	</div>		</div>


	=Related Discussions=		=Related Discussions=

Joel2: /* Parameter Values */

2024-01-16T03:49:27Z

Parameter Values

← Older revision		Revision as of 23:49, 15 January 2024
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	<!-- BEGIN TABLE OF PARAMETERS --->		<!-- BEGIN TABLE OF PARAMETERS --->
	<div align="center">		<div align="center">
	<b>Table 1: Properties of <math>(n_c, ~~n_3~~) = (5, 1)</math> BiPolytrope Having Various Interface Locations, <math>\xi_i</math></b><br>		<b>Table 1: Properties of <math>(n_c, n_e) = (5, 1)</math> BiPolytrope Having Various Interface Locations, <math>\xi_i</math></b><br>
	[[File:BiPolytropeParametersV01.xml\|Accompanying spreadsheet with parameter values]]		[[File:BiPolytropeParametersV01.xml\|Accompanying spreadsheet with parameter values]]

Joel2 at 03:38, 16 January 2024

2024-01-16T03:38:52Z

← Older revision		Revision as of 23:38, 15 January 2024
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	</table>		</table>

			==Examples==

	=~~Examples~~=		===Normalization===

	==~~Normalization~~==
	The dimensionless variables used in Tables 1 & 2 are defined as follows:		The dimensionless variables used in Tables 1 & 2 are defined as follows:
	<div align="center">		<div align="center">
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	</div>		</div>

	==Parameter Values==		===Parameter Values===

	The <math>2^\mathrm{nd}</math> column of Table 1 catalogues the analytic expressions that define various parameters and physical properties (as identified, respectively, in column 1) of the <math>(n_c, n_e) = (5, 1)</math> bipolytrope. We have evaluated these expressions for various choices of the dimensionless interface radius, <math>\xi_i</math>, and have tabulated the results in the last few columns of the table. The tabulated values have been derived assuming <math>\mu_e/\mu_c = 1</math>, that is, assuming that the core and the envelope have the same mean molecular weights.		The <math>2^\mathrm{nd}</math> column of Table 1 catalogues the analytic expressions that define various parameters and physical properties (as identified, respectively, in column 1) of the <math>(n_c, n_e) = (5, 1)</math> bipolytrope. We have evaluated these expressions for various choices of the dimensionless interface radius, <math>\xi_i</math>, and have tabulated the results in the last few columns of the table. The tabulated values have been derived assuming <math>\mu_e/\mu_c = 1</math>, that is, assuming that the core and the envelope have the same mean molecular weights.
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	</table>		</table>

	==Profile==		===Profile===

	Once the values of the key set of parameters have been determined as illustrated in Table 1, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in [[#Steps_2_.26_3\|step #4]] and [[#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 Table 1, the radial profile of various physical variables can be determined throughout the bipolytrope as detailed in [[#Steps_2_.26_3\|step #4]] and [[#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.
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	For a given choice of the interface parameter, <math>\xi_i</math>, the (normalized) mass that is contained in the core is independent of the choice of the molecular weight ratio. However, the (normalized) total mass, <math>M_\mathrm{tot}^</math>, varies significantly with the choice of <math>\mu_e/\mu_c</math>; as suggested by the expression provided in row 17 of Table 1, the variation is in rough proportion to <math>(\mu_e/\mu_c)^{-2}</math> but, as with <math>R^</math>, this proportional relation is not exact because the parameters <math>\eta_s</math> and <math>A</math> which also appear in the formula for <math>M_\mathrm{tot}^*</math> harbor an implicit dependence on the molecular weight ratio.		For a given choice of the interface parameter, <math>\xi_i</math>, the (normalized) mass that is contained in the core is independent of the choice of the molecular weight ratio. However, the (normalized) total mass, <math>M_\mathrm{tot}^</math>, varies significantly with the choice of <math>\mu_e/\mu_c</math>; as suggested by the expression provided in row 17 of Table 1, the variation is in rough proportion to <math>(\mu_e/\mu_c)^{-2}</math> but, as with <math>R^</math>, this proportional relation is not exact because the parameters <math>\eta_s</math> and <math>A</math> which also appear in the formula for <math>M_\mathrm{tot}^*</math> harbor an implicit dependence on the molecular weight ratio.

	==Model Sequences==		===Model Sequences===


	For a given choice of <math>\mu_e/\mu_c</math> a physically relevant sequence of models can be constructed by steadily increasing the value of <math>\xi_i</math> from zero to infinity — or at least to some value, <math>\xi_i \gg 1</math>. Figure 1 shows how the fractional core mass, <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, varies with the fractional core radius, <math>q \equiv r_\mathrm{core}/R</math>, along sequences having six different values of <math>\mu_e/\mu_c</math>, as detailed in the figure caption. The natural expectation is that an increase in <math>\xi_i</math> along a given sequence will correspond to an increase in the relative size — both the radius and the mass — of the core. This expectation is realized along the sequences marked by blue diamonds (<math>\mu_e/\mu_c = 1</math>) and by red squares (<math>\mu_e/\mu_c = </math>½). But the behavior is different along the other four illustrated sequences. For sufficiently large <math>\xi_i</math>, the relative radius of the core begins to decrease; then, as <math>\xi_i</math> is pushed to even larger values, eventually the relative core mass begins to decrease. Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence\|an accompanying chapter]].		For a given choice of <math>\mu_e/\mu_c</math> a physically relevant sequence of models can be constructed by steadily increasing the value of <math>\xi_i</math> from zero to infinity — or at least to some value, <math>\xi_i \gg 1</math>. Figure 1 shows how the fractional core mass, <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, varies with the fractional core radius, <math>q \equiv r_\mathrm{core}/R</math>, along sequences having six different values of <math>\mu_e/\mu_c</math>, as detailed in the figure caption. The natural expectation is that an increase in <math>\xi_i</math> along a given sequence will correspond to an increase in the relative size — both the radius and the mass — of the core. This expectation is realized along the sequences marked by blue diamonds (<math>\mu_e/\mu_c = 1</math>) and by red squares (<math>\mu_e/\mu_c = </math>½). But the behavior is different along the other four illustrated sequences. For sufficiently large <math>\xi_i</math>, the relative radius of the core begins to decrease; then, as <math>\xi_i</math> is pushed to even larger values, eventually the relative core mass begins to decrease. Additional properties of these equilibrium sequences are discussed in [[SSC/FreeEnergy/PolytropesEmbedded#Behavior_of_Equilibrium_Sequence\|an accompanying chapter]].