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Part I: Steps 2 thru 7

Part II: Analytic Solution of Interface Relation

2024-01-01T03:20:58Z

Created page with "=Examples= <table border="1" align="center" width="100%" colspan="8"> <tr> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part I: Steps 2 thru 7<br /> </td> <td align="center" rowspan="1" bgcolor="lightblue" width="33%"><br />Part II: Analytic Solution of Interface Relation<br /> </td> <td align="center" rowspan="1" bgcolor="lightblue..."

New page
=Examples=
<table border="1" align="center" width="100%" colspan="8">
<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="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>
</tr>
</table>

==Normalization==
The dimensionless variables used in Tables 1 & 2 are defined as follows:
<div align="center">
<table border="0" cellpadding="3">
<tr>
<td align="right">
<math>~\rho^</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{\rho}{\rho_0}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>~r^</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{r}{(K_c/G)^{1/2}}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~P^</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{P}{K_c\rho_0^{2}}</math>
</td>

<td align="center">;    </td>

<td align="right">
<math>~M_r^</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{M_r}{\rho_0 (K_c/G)^{3/2}}</math>
</td>
</tr>

<tr>
<td align="right">
<math>~H^</math>
</td>
<td align="center">
<math>~\equiv</math>
</td>
<td align="left">
<math>~\frac{H}{K_c\rho_0}</math>
</td>

<td align="center">.    </td>

<td align="right" colspan="3">
 
</td>
</tr>

</table>
</div>

==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=1</math>, <math>~n_e=5</math> bipolytrope.


<div align="center">
<table border="1" cellpadding="5">
<tr>
<td align="center" colspan="2">
'''Properties of <math>~n_c=1</math>, <math>~n_e=5</math>, BiPolytrope Having Various Interface Locations, <math>~\xi_i</math>'''
</td>
</tr>

<tr>
<td align="center">
Parameter
</td>
<td align="center">
<math>~\xi_i</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\theta_i</math>
</td>
<td align="center">
<math>~\frac{\sin\xi_i}{\xi_i} </math>
</td>
</tr>

<tr>
<td align="center">
<math>~-\biggl(\frac{d\theta_i}{d\xi}\biggr)_i</math>
</td>
<td align="center">
<math>~\frac{1}{\xi_i^2}(\sin\xi - \xi_i \cos\xi_i) </math>
</td>
</tr>

<tr>
<td align="center">
<math>~r^_\mathrm{core} \equiv r^_i</math>
</td>
<td align="center">
<math>~\biggl( \frac{1}{2\pi} \biggr)^{1/2} \xi_i</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\rho^_i \biggr|_c = \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1} \rho^_i \biggr|_e</math>
</td>
<td align="center">
<math>~\theta_i</math>
</td>
</tr>

<tr>
<td align="center">
<math>P^_i</math>
</td>
<td align="center">
<math>~\theta_i^2</math>
</td>
</tr>

<tr>
<td align="center">
<math>H^_i \biggr|_c = \frac{n_c+1}{n_e+1} \biggl( \frac{\mu_e}{\mu_c} \biggr) H^_i \biggr|_e</math>
</td>
<td align="center">
<math>~2\theta_i</math>
</td>
</tr>

<tr>
<td align="center">
<math>M^_\mathrm{core}</math>
</td>
<td align="center">
<math>~\biggl( \frac{2}{\pi} \biggr)^{1/2} (\sin\xi_i - \xi_i \cos\xi_i)</math>
</td>
</tr>

<tr>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}\eta_i</math>
</td>
<td align="center">
<math>\frac{\xi_i}{\sqrt{3}}</math>
</td>
</tr>

<tr>
<td align="center">
<math>~ -\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
</td>
<td align="center">
<math>~\frac{1}{\sqrt{3} \theta_i}\biggl( - \frac{d\theta}{d\xi} \biggr)_i = \frac{1}{\sqrt{3}}\biggl( \frac{1}{\xi_i}- \cot\xi_i\biggr)</math>
</td>
</tr>

<tr>
<td align="center">
<math>~p</math>
</td>
<td align="center">
<math>~\frac{3}{ 3-2(\mu_e/\mu_c)(1-\xi_i \cot\xi_i)}</math>
</td>
</tr>

<tr>
<td align="center">
<math>~~y_\mathrm{root}</math>
</td>
<td align="center">
<math>~p\biggl( 1 + \{1 + [1 + (p^{-2}-1)^3]^{1/2}\}^{1/3} + \{1 - [1 + (p^{-2}-1)^3]^{1/2}\}^{1/3} \biggr)</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\Delta_i - m\pi</math>
</td>
<td align="center">
<math>~\tan^{-1}(y_\mathrm{root})</math>
</td>
</tr>

<tr>
<td align="center">
<math>~(A_0\eta)_\mathrm{root}</math>
</td>
<td align="center">
<math>~e^{2\Delta_i}</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\xi_s</math>
</td>
<td align="center">
<math>~\xi_ie^{2(\pi-\Delta_i)}</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\biggl[ \frac{A_0}{3^{1/2}} \biggl( \frac{\mu_e}{\mu_c} \biggr) \biggr]</math>
</td>
<td align="center">
<math>~\xi_i^{-1} e^{2\Delta_i}</math>
</td>
</tr>

<tr>
<td align="center">
<math>~\biggl[\frac{B_0}{3^{1/4}} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{1/2} \biggr]</math>
</td>
<td align="center">
<math>~\biggl[ \xi_i \biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr) \biggr]^{-1/2}
= \theta_i \biggl[ \frac{\xi_i^{1/2}}{\sin\xi_i}\biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1/2} \biggr]
</math>
</td>
</tr>

<tr>
<td align="center">
<math>~ -\biggl( \frac{d\phi}{d\eta} \biggr)_s \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
</td>
<td align="center">
<math>~\frac{1}{ 2\xi_i e^{3(\pi-\Delta_i)}} \biggl( \frac{3}{\sin^2\Delta_i} - 2 \biggr)^{1/2}</math>
</td>
</tr>

<tr>
<td align="center">
<math>~ R^_s</math>
</td>
<td align="center">
<math>~(2\pi)^{-1/2} \xi_i e^{2(\pi - \Delta_i)}</math>
</td>
</tr>

<tr>
<td align="center">
<math>~ \biggl( \frac{\mu_e}{\mu_c}\biggr) M^_\mathrm{tot}</math>
</td>
<td align="center">
<math>~\biggl( \frac{3}{2\pi} \biggr)^{1/2} \sin\xi_i \biggl( \frac{3}{\sin^2\Delta_i} - 2 \biggr)^{1/2} e^{(\pi - \Delta_i)}</math>
</td>
</tr>
</table>
</div>



==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 [[SSC/Structure/BiPolytropes/Analytic15#Step_4:__Throughout_the_core_.28.29|step #4]] and [[SSC/Structure/BiPolytropes/Analytic15#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">
<b>Table 2: Radial Profile of Various Physical Variables</b>
<table border="1" cellpadding="6">
<tr>
<td align="center" rowspan="2">
Variable
</td>
<td align="center" rowspan="2">
Throughout the Core<br>
<math>0 \le \xi \le \xi_i</math>
</td>
<td align="center" rowspan="2">
Throughout the Envelope<sup>&dagger;</sup><br>
<math>\eta_i \le \eta \le \eta_s</math>
</td>

<td align="center" colspan="3">
Plotted Profiles
</td>
</tr>

<tr>
<td align="center">
<math>\xi_i = 0.5</math>
</td>
<td align="center">
<math>\xi_i = 1.0</math>
</td>
<td align="center">
<math>\xi_i = 3.0</math>
</td>
</tr>

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

<td align="center" colspan="3">
 
</td>
</tr>

<tr>
<td align="center">
<math>~\rho^</math>
</td>
<td align="center">
<math>\frac{\sin\xi}{\xi}</math>
</td>
<td align="center">
<math>\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i [\phi(\eta)]^5</math>
</td>

<td align="center">

[[Image:DenXi05.jpg|thumb|75px]]
</td>
<td align="center">
[[Image:DenXi10.jpg|thumb|75px]]
</td>
<td align="center">
[[Image:DenXi30.jpg|thumb|75px]]
</td>
</tr>

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

<td align="center">

[[Image:PresXi05.jpg|thumb|75px]]
</td>
<td align="center">
[[Image:PresXi10.jpg|thumb|75px]]
</td>
<td align="center">
[[Image:PresXi30.jpg|thumb|75px]]
</td>
</tr>

<tr>
<td align="center">
<math>~M_r^</math>
</td>
<td align="center">
<math>\biggl( \frac{2}{\pi}\biggr)^{1/2} (\sin\xi - \xi\cos\xi)</math>
</td>
<td align="center">
<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 align="center">

[[Image:MassXi05.jpg|thumb|75px]]
</td>
<td align="center">
[[Image:MassXi10.jpg|thumb|75px]]
</td>
<td align="center">
[[Image:MassXi30.jpg|thumb|75px]]
</td>
</tr>

<tr>
<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 [[SSC/Structure/BiPolytropes/Analytic15#Step_6:__Envelope_Solution|presented in Step 6, above]].
</td>
</tr>
</table>
</div>

==Murphy and Fiedler (1985)==

<div align="center">
<table border="1" cellpadding="8" align="center">
<tr>
<td align="center" bgcolor="lightblue">
Table 1 from [http://adsabs.harvard.edu/abs/1985PASAu...6..219M Murphy & Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)]
</td>
</tr>
<tr>
<td align="center">

[[Image:AAAwaiting02.png|center|300px|Murphy & Fiedler (1985) Table 1]]
</td>
</tr>
<tr>
<td align="center" bgcolor="lightblue">
Reproduction of Table 1 from MF85 Using Excel and Analytic Expressions Derived Here
</td>
</tr>
<tr>
<td align="center">
[[File:MF85Table1byTohline02.png|center|600px|Excel Regeneration of MF85 Table 1]]
</td>
</tr>
</table>
</div>

=Key References=
[http://adsabs.harvard.edu/abs/1962ApJ...136..680S S. Srivastava (1968, ApJ, 136, 680)] ''A New Solution of the Lane-Emden Equation of Index n = 5''
* [http://adsabs.harvard.edu/abs/1978AuJPh..31..115B H. A. Buchdahl (1978, Australian Journal of Physics, 31, 115)]: ''Remark on the Polytrope of Index 5'' — the result of this work by Buchdahl has been [[SSC/Structure/BiPolytropes/Analytic51#Buchdahl1978|highlighted inside our discussion of bipolytropes with <math>~(n_c, n_e) = (5, 1)</math>]].
* [http://adsabs.harvard.edu/abs/1980PASAu...4...37M J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37)]: ''A Finite Radius Solution for the Polytrope Index 5''
* [http://adsabs.harvard.edu/abs/1980PASAu...4...41M J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41)]: ''On the F-Type and M-Type Solutions of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1981PASAu...4..205M J. O. Murphy (1981, Proc. Astr. Soc. of Australia, 4, 205)]: ''Physical Characteristics of a Polytrope Index 5 with Finite Radius''
* [http://adsabs.harvard.edu/abs/1982PASAu...4..376M J. O. Murphy (1982, Proc. Astr. Soc. of Australia, 4, 376)]: ''A Sequence of E-Type Composite Analytical Solutions of the Lane-Emden Equation''
* [http://adsabs.harvard.edu/abs/1983AuJPh..36..453M J. O. Murphy (1983, Australian Journal of Physics, 36, 453)]: ''Structure of a Sequence of Two-Zone Polytropic Stellar Models with Indices 0 and 1''
* [http://adsabs.harvard.edu/abs/1983PASAu...5..175M J. O. Murphy (1983, Proc. Astr. Soc. of Australia, 5, 175)]: ''Composite and Analytical Solutions of the Lane-Emden Equation with Polytropic Indices n = 1 and n = 5''
* [http://adsabs.harvard.edu/abs/1985PASAu...6..219M J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219)]: ''Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models''
* [http://adsabs.harvard.edu/abs/1985PASAu...6..222M J. O. Murphy & R. Fiedler (1985b, Proc. Astr. Soc. of Australia, 6, 222)]: ''Radial Pulsations and Vibrational Stability of a Sequence of Two-Zone Polytropic Stellar Models''

=Related Discussions=
* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|Polytropes emdeded in an external medium]]
* [[SSC/Structure/BiPolytropes#BiPolytropes|Constructing BiPolytropes]]

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SSC/Structure/BiPolytropes/Analytic15/Pt3 - Revision history

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