SSC/Structure/BiPolytropes/Analytic15/Pt2 - Revision history

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

2024-01-03T19:09:44Z

BiPolytrope with nc = 1 and ne = 5

← Older revision		Revision as of 15:09, 3 January 2024
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	<table border="1" align="center" width="100%" colspan="8">		<table border="1" align="center" width="100%" colspan="8">
	<tr>		<tr>
	<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic15\|Part I:   Steps 2 thru 7]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15\|Part I:   Steps 2 thru 7]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt2\|Part II:  Analytic Solution of Interface Relation]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt2\|Part II:  Analytic Solution of Interface Relation]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt3\|III:  ~~Examples~~]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="25%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt3\|III:  Modeling]]<br /> </td>
	<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/MurphyUVplane\|IV:  Murphy's UV Plane]]<br /> </td>		<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/MurphyUVplane\|IV:  Murphy's UV Plane]]<br /> </td>
	</tr>		</tr>

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

2024-01-03T19:07:35Z

BiPolytrope with nc = 1 and ne = 5

← Older revision		Revision as of 15:07, 3 January 2024
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	<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt2\|Part II:  Analytic Solution of Interface Relation]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt2\|Part II:  Analytic Solution of Interface Relation]]<br /> </td>
	<td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt3\|III:  Examples]]<br /> </td>		<td align="center" rowspan="1" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/Analytic15/Pt3\|III:  Examples]]<br /> </td>
			<td align="center" bgcolor="lightblue"><br />[[SSC/Structure/BiPolytropes/MurphyUVplane\|IV:  Murphy's UV Plane]]<br /> </td>
	</tr>		</tr>
	</table>		</table>

Joel2: /* Murphy's Example Model Characteristics */

2024-01-03T00:16:30Z

Murphy's Example Model Characteristics

← Older revision		Revision as of 20:16, 2 January 2024
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	As in an [[SSC/Structure/BiPolytropes/Analytic15#MurphyF2Function\|earlier figure]], the following log-log plot shows what value(s) of <math>~(A_0\eta)_\mathrm{root} </math> (vertical axis) result from a wide range of interface coordinate specifications, <math>~\xi_i</math> (horizontal axis), under the simplifying assumption that <math>~(\mu_e/\mu_c) = 1</math>. The solid blue diamonds present exactly the same information as in the previous figure, that is, they identify roots resulting from setting <math>~m=0</math> in the phase-shift specification. But here, focusing on a regime that highlights larger roots of the first constraint equation, solid black squares identify roots for which <math>~m=+1</math>, and solid maroon triangles identify roots for which <math>~m=+2</math>.		As in an [[SSC/Structure/BiPolytropes/Analytic15#MurphyF2Function\|earlier figure]], the following log-log plot shows what value(s) of <math>~(A_0\eta)_\mathrm{root} </math> (vertical axis) result from a wide range of interface coordinate specifications, <math>~\xi_i</math> (horizontal axis), under the simplifying assumption that <math>~(\mu_e/\mu_c) = 1</math>. The solid blue diamonds present exactly the same information as in the previous figure, that is, they identify roots resulting from setting <math>~m=0</math> in the phase-shift specification. But here, focusing on a regime that highlights larger roots of the first constraint equation, solid black squares identify roots for which <math>~m=+1</math>, and solid maroon triangles identify roots for which <math>~m=+2</math>.

			<span id="SecondMurphyF2Function">In this version of the figure</span>, colored dashed-line segments have been drawn to enclose two regions of parameter space in which solutions of the [[SSC/Structure/BiPolytropes/Analytic15#KeyInterfaceRelation\|key nonlinear interface relation]] produce a [[SSC/Structure/BiPolytropes/Analytic15#Physically_Viable_Parameter_Values\|physically viable model]] — that is, <math>~0 \le \xi_i \le \pi</math> and <math>~\tan^{-1}(1+2^{1/3}) \le \Delta_i \le \pi</math>. The portion of the red-outlined rectangular region that lies to the left of the vertical, <math>~\xi_i = \xi_\mathrm{trans}</math> line accommodates some models that arise from specifying an <math>~m=0</math> phase shift and the portion that lies to the right of this demarcation line accommodates some models that arise from specifying an <math>~m=+1</math> phase shift. Similarly, the portion of the green-outlined rectangular region that lies to the left of the vertical, <math>~\xi_i = \xi_\mathrm{trans}</math> line accommodates additional models that arise from specifying an <math>~m=+1</math> phase shift and the portion that lies to the right of this demarcation line accommodates models that arise from specifying an <math>~m=+2</math> phase shift. Given the inherent periodicity of Srivastava's <math>~\phi_{5F}</math> function, it should be clear that we could equally well have identified other rectangular regions at both higher <math>~(m \ge 2)</math> and lower <math>~(m \le 0)</math> values of <math>~(A_0 \eta)_\mathrm{root}</math> in which additional physically viable models exist. It should also be clear that, because the figure presents a log-log plot and the lower limit of viable parameter values for the horizontal axis parameter, <math>~\xi_i</math>, is zero, the identified rectangular regions should be viewed as extending to the left, all the way to minus infinity.
	In this version of the figure, colored dashed-line segments have been drawn to enclose two regions of parameter space in which solutions of the [[SSC/Structure/BiPolytropes/Analytic15#KeyInterfaceRelation\|key nonlinear interface relation]] produce a [[SSC/Structure/BiPolytropes/Analytic15#Physically_Viable_Parameter_Values\|physically viable model]] — that is, <math>~0 \le \xi_i \le \pi</math> and <math>~\tan^{-1}(1+2^{1/3}) \le \Delta_i \le \pi</math>. The portion of the red-outlined rectangular region that lies to the left of the vertical, <math>~\xi_i = \xi_\mathrm{trans}</math> line accommodates some models that arise from specifying an <math>~m=0</math> phase shift and the portion that lies to the right of this demarcation line accommodates some models that arise from specifying an <math>~m=+1</math> phase shift. Similarly, the portion of the green-outlined rectangular region that lies to the left of the vertical, <math>~\xi_i = \xi_\mathrm{trans}</math> line accommodates additional models that arise from specifying an <math>~m=+1</math> phase shift and the portion that lies to the right of this demarcation line accommodates models that arise from specifying an <math>~m=+2</math> phase shift. Given the inherent periodicity of Srivastava's <math>~\phi_{5F}</math> function, it should be clear that we could equally well have identified other rectangular regions at both higher <math>~(m \ge 2)</math> and lower <math>~(m \le 0)</math> values of <math>~(A_0 \eta)_\mathrm{root}</math> in which additional physically viable models exist. It should also be clear that, because the figure presents a log-log plot and the lower limit of viable parameter values for the horizontal axis parameter, <math>~\xi_i</math>, is zero, the identified rectangular regions should be viewed as extending to the left, all the way to minus infinity.

Joel2 at 03:23, 1 January 2024

2024-01-01T03:23:44Z

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Joel2 at 03:22, 1 January 2024

2024-01-01T03:22:26Z

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 ==Analytic Solution of Key Interface Relation==
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Joel2: Created page with " ==Analytic Solution of Key Interface Relation== Returning to our previously derived,
'''Key Nonlinear Interface Relation'''

$~\kappa_i$ $~=$ $~\frac{3 - 2\sin^2\Del..."$
2024-01-01T03:13:57Z

Created page with " ==Analytic Solution of Key Interface Relation== Returning to our previously derived, <div align="center" id="KeyInterfaceRelation"> <font color="#770000">'''Key Nonlinear Interface Relation'''</font> <br /> <table border="0" cellpadding="5" align="center"> <tr> <td align="right"> <math>~\kappa_i</math> </td> <td align="center"> <math>~=</math> </td> <td align="left"> <math>~\frac{3 - 2\sin^2\Del..."
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