SSC/Structure/BiPolytropes/Analytic15/Pt3

Examples

Part I:   Steps 2 thru 7

Part II:  Analytic Solution of Interface Relation

III:  Modeling

IV:  Murphy's UV Plane

Normalization

The dimensionless variables used in Tables 1 & 2 are defined as follows:

${\rho }^{*}$	$\equiv$	$\frac{\rho }{{\rho }_0}$	;	$r^{*}$	$\equiv$	$\frac{r}{(K_c/G{)}^{1/2}}$
$P^{*}$	$\equiv$	$\frac{P}{K_c{\rho }_0^2}$	;	$M_r^{*}$	$\equiv$	$\frac{M_r}{{\rho }_0(K_c/G{)}^{3/2}}$
$H^{*}$	$\equiv$	$\frac{H}{K_c{\rho }_0}$	.

Parameter Values

The $2^{\mathrm{n}\mathrm{d}}$ column of Table 1 catalogues the analytic expressions that define various parameters and physical properties (as identified, respectively, in column 1) of the $n_c=1$, $n_e=5$ bipolytrope.

Properties of $n_c=1$, $n_e=5$, BiPolytrope Having Various Interface Locations, ${\xi }_i$
Parameter	${\xi }_i$
${\theta }_i$	$\frac{\sin {\xi }_i}{{\xi }_i}$
$-(\frac{d{\theta }_i}{d\xi }{)}_i$	$\frac{1}{{\xi }_i^2}(\sin \xi -{\xi }_i\cos {\xi }_i)$
$r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}\equiv r_i^{*}$	$(\frac{1}{2\pi }{)}^{1/2}{\xi }_i$
${\rho }_i^{*}{|}_c=(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\rho }_i^{*}{|}_e$	${\theta }_i$
$P_i^{*}$	${\theta }_i^2$
$H_i^{*}{|}_c=\frac{n_c+1}{n_e+1}\left(\frac{{\mu }_e}{{\mu }_c}\right)H_i^{*}{|}_e$	$2{\theta }_i$
$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$	$(\frac{2}{\pi }{)}^{1/2}(\sin {\xi }_i-{\xi }_i\cos {\xi }_i)$
$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\eta }_i$	$\frac{{\xi }_i}{\sqrt{3}}$
$-(\frac{d\varphi }{d\eta }{)}_i$	$\frac{1}{\sqrt{3}{\theta }_i}(-\frac{d\theta }{d\xi }{)}_i=\frac{1}{\sqrt{3}}(\frac{1}{{\xi }_i}-\cot {\xi }_i)$
$p$	$\frac{3}{3-2({\mu }_e/{\mu }_c)(1-{\xi }_i\cot {\xi }_i)}$
$y_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$	$p(1+\{1+[1+(p^{-2}-1{)}^3{]}^{1/2}{\}}^{1/3}+\{1-[1+(p^{-2}-1{)}^3{]}^{1/2}{\}}^{1/3})$
${\mathrm{\Delta }}_i-m\pi$	${\tan }^{-1}(y_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}})$
$(A_0\eta {)}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}$	$e^{2{\mathrm{\Delta }}_i}$
${\xi }_s$	${\xi }_i{e}^{2(\pi -{\mathrm{\Delta }}_i)}$
$\left[\frac{A_0}{3^{1/2}}\right(\frac{{\mu }_e}{{\mu }_c}\left)\right]$	${\xi }_i^{-1}{e}^{2{\mathrm{\Delta }}_i}$
$\left[\frac{B_0}{3^{1/4}}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}]$	$\left[{\xi }_i\right(\frac{3}{{\sin }^2{\mathrm{\Delta }}_i}-2){]}^{-1/2}={\theta }_i[\frac{{\xi }_i^{1/2}}{\sin {\xi }_i}(\frac{3}{{\sin }^2{\mathrm{\Delta }}_i}-2{)}^{-1/2}]$
$-\left(\frac{d\varphi }{d\eta }{)}_s\right(\frac{{\mu }_e}{{\mu }_c})$	$\frac{1}{2{\xi }_i{e}^{3(\pi -{\mathrm{\Delta }}_i)}}(\frac{3}{{\sin }^2{\mathrm{\Delta }}_i}-2{)}^{1/2}$
$R_s^{*}$	$(2\pi {)}^{-1/2}{\xi }_i{e}^{2(\pi -{\mathrm{\Delta }}_i)}$
$\left(\frac{{\mu }_e}{{\mu }_c}\right)M_{\mathrm{t}\mathrm{o}\mathrm{t}}^{*}$	$(\frac{3}{2\pi }{)}^{1/2}\sin {\xi }_i(\frac{3}{{\sin }^2{\mathrm{\Delta }}_i}-2{)}^{1/2}{e}^{(\pi -{\mathrm{\Delta }}_i)}$

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 step #4 and 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, ${\rho }^{*}(r^{*})$, the normalized gas pressure, $P^{*}(r^{*})$, and the normalized mass interior to $r^{*}$, $M_r^{*}(r^{*})$. For all profiles, the relevant normalized radial coordinate is $r^{*}$, as defined in the 2^nd 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.

Table 2: Radial Profile of Various Physical Variables

Variable	Throughout the Core $0\le \xi \le {\xi }_i$	Throughout the Envelope† ${\eta }_i\le \eta \le {\eta }_s$	Plotted Profiles
			${\xi }_i=0.5$	${\xi }_i=1.0$	${\xi }_i=3.0$
$r^{*}$	$(\frac{1}{2\pi }{)}^{1/2}\xi$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right(\frac{3}{2\pi }{)}^{1/2}\eta$
${\rho }^{*}$	$\frac{\sin \xi }{\xi }$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i[\varphi (\eta ){]}^5$
$P^{*}$	$(\frac{\sin \xi }{\xi }{)}^2$	${\theta }_i^2[\varphi (\eta ){]}^6$
$M_r^{*}$	$(\frac{2}{\pi }{)}^{1/2}(\sin \xi -\xi \cos \xi )$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}\right(\frac{2\cdot 3^3}{\pi }{)}^{1/2}{\theta }_i(-{\eta }^2\frac{d\varphi }{d\eta })$
†In order to obtain the various envelope profiles, it is necessary to evaluate $\varphi (\eta )$ and its first derivative using the information presented in Step 6, above.

Murphy and Fiedler (1985)

Table 1 from Murphy & Fiedler (1985, Proc. Astr. Soc. of Australia, 6, 219)

Murphy & Fiedler (1985) Table 1 Murphy & Fiedler (1985) Table 1

Reproduction of Table 1 from MF85 Using Excel and Analytic Expressions Derived Here

Excel Regeneration of MF85 Table 1

Key References

• S. Srivastava (1968, ApJ, 136, 680) A New Solution of the Lane-Emden Equation of Index n = 5
• 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 highlighted inside our discussion of bipolytropes with $(n_c,n_e)=(5,1)$.
• J. O. Murphy (1980a, Proc. Astr. Soc. of Australia, 4, 37): A Finite Radius Solution for the Polytrope Index 5
• J. O. Murphy (1980b, Proc. Astr. Soc. of Australia, 4, 41): On the F-Type and M-Type Solutions of the Lane-Emden Equation
• J. O. Murphy (1981, Proc. Astr. Soc. of Australia, 4, 205): Physical Characteristics of a Polytrope Index 5 with Finite Radius
• J. O. Murphy (1982, Proc. Astr. Soc. of Australia, 4, 376): A Sequence of E-Type Composite Analytical Solutions of the Lane-Emden Equation
• 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
• 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
• J. O. Murphy & R. Fiedler (1985a, Proc. Astr. Soc. of Australia, 6, 219): Physical Structure of a Sequence of Two-Zone Polytropic Stellar Models
• 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

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