SSC/Structure/BiPolytropes/Analytic51Renormalize/Pt3

BiPolytrope with (n_c, n_e) = (5, 1) Renormalized

Part I:   (5, 1) Analytic Renormalize

Part II:  Envelope

III:  Interface Pressure Gradient

Examine Pressure Gradient at the Interface

Determine the interface-pressure-gradient from two different perspectives: (1) Look at the behavior of the pressure as determined when the hydrostatic-balance models have been constructed; and (2) Look at the behavior of the specific entropy at the interface.

From Hydrostatic Balance

Pressure Gradient at Core Interface

Step 4: Throughout the core … we have,

$P^{*}\equiv \frac{P}{[K_c{\rho }_0^{6/5}]}$	$=$	$(1+\frac{1}{3}{\xi }^2{)}^{-3}$
$\Rightarrow \frac{d{P}^{*}}{d\xi }$	$=$	$-2\xi (1+\frac{1}{3}{\xi }^2{)}^{-4};$

and,

$r^{*}\equiv \left[\frac{G{\rho }_0^{4/5}}{K_c}\right]r$	$=$	$(\frac{3}{2\pi }{)}^{1/2}\xi$
$\Rightarrow \frac{\xi }{r^{*}}$	$=$	$(\frac{2\pi }{3}{)}^{1/2}.$

Hence, from the perspective of the core, at the interface $({\xi }_i)$ the radial pressure derivative is,

$\frac{d{P}^{*}}{d{r}^{*}}{|}_i$

$=$

$-2{\xi }_i(1+\frac{1}{3}{\xi }_i^2{)}^{-4}(\frac{2\pi }{3}{)}^{1/2}.$

Note, as well, that,

$[\frac{r^{*}}{P^{*}}\cdot \frac{d{P}^{*}}{d{r}^{*}}{]}_i$	$=$	$-2{\xi }_i(1+\frac{1}{3}{\xi }_i^2{)}^{-4}(\frac{2\pi }{3}{)}^{1/2}\left[\right(\frac{3}{2\pi }{)}^{1/2}{\xi }_i\left]\right(1+\frac{1}{3}{\xi }_i^2{)}^3$
	$=$	$-2{\xi }_i^2(1+\frac{1}{3}{\xi }_i^2{)}^{-1}.$

Pressure Gradient at Envelope Interface

Step 8: Throughout the envelope … we have,

$P^{*}\equiv \frac{P}{[K_c{\rho }_0^{6/5}]}$	$=$	${\theta }_i^6{\varphi }^2=A^2[1+\frac{{\xi }_i^2}{3}{]}^{-3}[\frac{\sin (\eta -B)}{\eta }{]}^2$
$\Rightarrow \frac{d{P}^{*}}{d\eta }$	$=$	$2A^2[1+\frac{{\xi }_i^2}{3}{]}^{-3}[\frac{\sin (\eta -B)}{\eta }\left]\right\{\frac{\cos (\eta -B)}{\eta }-\frac{\sin (\eta -B)}{{\eta }^2}\};$

and,

$r^{*}\equiv \left[\frac{G{\rho }_0^{4/5}}{K_c}\right]r$	$=$	$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta$
$\Rightarrow \frac{\eta }{r^{*}}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2(2\pi {)}^{1/2}.$

Hence, from the perspective of the envelope, at the interface $({\eta }_i)$ the radial pressure derivative is,

$\frac{d{P}^{*}}{d{r}^{*}}{|}_i$

$=$

$\left(\frac{{\mu }_e}{{\mu }_c}\right)(2^3\pi {)}^{1/2}{A}^2[1+\frac{{\xi }_i^2}{3}{]}^{-4}\left[\frac{\sin ({\eta }_i-B)}{{\eta }_i^2}\right]\{{\eta }_i\cos ({\eta }_i-B)-\sin ({\eta }_i-B)\},$

where,

$A$

$=$

$\frac{{\eta }_i}{\sin ({\eta }_i-B)},$

and,

${\eta }_i$

$=$

$3^{1/2}{\xi }_i\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2.$

Simplifying this last expression a bit, we have,

$\frac{d{P}^{*}}{d{r}^{*}}{|}_i$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)(2^3\pi {)}^{1/2}[1+\frac{{\xi }_i^2}{3}{]}^{-4}[{\eta }_i\cot ({\eta }_i-B)-1]$
	$=$	$-2{\xi }_i\left(\frac{2\pi }{3}{)}^{1/2}\right(1+\frac{{\xi }_i^2}{3}{)}^{-4}\{1-{\eta }_i\cot ({\eta }_i-B)\}\left(\frac{{\mu }_e}{{\mu }_c}\right)\frac{\sqrt{3}}{{\xi }_i}.$

Note, as well, that,

$[\frac{r^{*}}{P^{*}}\cdot \frac{d{P}^{*}}{d{r}^{*}}{]}_i$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right)(2^3\pi {)}^{1/2}{A}^2[1+\frac{{\xi }_i^2}{3}{]}^{-4}\left[\frac{\sin ({\eta }_i-B)}{{\eta }_i^2}\right]\{{\eta }_i\cos ({\eta }_i-B)-\sin ({\eta }_i-B)\}$
		$\times (\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}{\eta }_i\left\{A^2\right[1+\frac{{\xi }_i^2}{3}{]}^{-3}[\frac{\sin ({\eta }_i-B)}{{\eta }_i}{]}^2{\}}^{-1}$
	$=$	$2[1+\frac{{\xi }_i^2}{3}{]}^{-1}\{{\eta }_i\cot ({\eta }_i-B)-1\}{\theta }_i^{-2}{\eta }_i$
	$=$	$-2{\xi }_i^2(1+\frac{{\xi }_i^2}{3}{)}^{-1}\{1-{\eta }_i\cot ({\eta }_i-B)\left\}\frac{3^{1/2}}{{\xi }_i}\right(\frac{{\mu }_e}{{\mu }_c}).$

Pressure Gradient Summary

Whether viewed from the perspective of the core or the envelope, we have shown that the pressure at the interface is the same. However, at the interface, the first derivative (or the logarithmic derivative) of the pressure as viewed from the envelope is "larger" than what is viewed from the perspective of the core by the following factor:

factor

$=$

$\{1-{\eta }_i\cot ({\eta }_i-B)\}\frac{\sqrt{3}}{{\xi }_i}\left(\frac{{\mu }_e}{{\mu }_c}\right).$

We note, for later use, that averaging these two pressure-gradients at the interface gives,

$\frac{1}{2}\left\{\right[\frac{r^{*}}{P^{*}}\cdot \frac{d{P}^{*}}{d{r}^{*}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+[\frac{r^{*}}{P^{*}}\cdot \frac{d{P}^{*}}{d{r}^{*}}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}{\}}_i$

$=$

$-{\xi }_i(1+\frac{1}{3}{\xi }_i^2{)}^{-1}\{{\xi }_i+[1-{\eta }_i\cot ({\eta }_i-B)]3^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right)\}.$

From Step-Function Behavior of Specific Entropy

Strategically Incorporate Step Function

As we have discussed separately, a useful expression for the specific entropy of any individual Lagrangian fluid element is,

$\frac{s}{\mathfrak{\Re }}$

$=$

$\frac{1}{\mu (\gamma -1)}\cdot \ln \left(\frac{P}{{\rho }^{\gamma }}\right).$

How does $s/\mathfrak{\Re }$ vary as a function of the Lagrangian mass shell (or Lagrangian radial coordinate)? In the case of a spherical bipolytropic configuration: $s=s_c$ (a constant) throughout the core; $s=s_e$ (another constant) throughout the envelope; and a unit step function,

$H(\zeta )=\{\begin{array}{ll}0;& \zeta <0\\ 1;& \zeta >0\end{array}$ [MF53], Part I, §2.1 (p. 123), Eq. (2.1.6)

can be introduced to accomplish the instantaneous jump from $s_c$ to $s_e$ at the core/envelope interface. Specifically, after defining,

$\zeta$

$\equiv$

$\frac{r}{r_i}-1,$

we obtain the correct physical description of the variation of specific entropy with mass shell, $m$, via the expression,

$s(\zeta )$

$=$

$s_c+H(\zeta )\cdot (s_e-s_c).$

Adopting the half-maximum convention — which states that $H(\zeta =0)=\frac{1}{2}$ — we acknowledge that the functional value of the specific entropy at the interface is, $s_i=\frac{1}{2}(s_c+s_e)$. Also, from our accompanying brief discussion of the behavior of the unit step function, we appreciate that,

$\frac{dH(\zeta )}{d\zeta }$

$=$

$\delta (\zeta ),$

where, $\delta (\zeta )$ is the Dirac delta function. We conclude, therefore, that precisely at the interface,

$\frac{ds(\zeta )}{d\zeta }{|}_i$

$=$

$[(s_e-s_c)\cdot \delta (\zeta ){]}_i=(s_e-s_c).$

Generally speaking, the two parameters, $\mu ,\gamma$, and the mass density, $\rho$, also will exhibit a step-function behavior at the interface of each equilibrium bipolytrope. The following table summarizes how we model the radial variation of these quantities.

Quantity	Functional Behavior	At Interface
		Value	Derivative wrt $\zeta$	Derivative wrt $r$
Specific Entropy	$s(\zeta )=s_c+H(\zeta )\cdot (s_e-s_c)$	$s_i\equiv s(0)=\frac{1}{2}(s_c+s_e)$	$\frac{ds(\zeta )}{d\zeta }{|}_i=(s_e-s_c)$	$\frac{ds}{dr}{|}_i=(s_e-s_c)/r_i$
Mean Molecular Weight	$\mu (\zeta )={\mu }_c+H(\zeta )\cdot ({\mu }_e-{\mu }_c)$	${\mu }_i\equiv \mu (0)=\frac{1}{2}({\mu }_c+{\mu }_e)$	$\frac{d\mu (\zeta )}{d\zeta }{|}_i=({\mu }_e-{\mu }_c)$	$\frac{d\mu }{dr}{|}_i=({\mu }_e-{\mu }_c)/r_i$
Ratio of Specific Heats	$\gamma (\zeta )={\gamma }_c+H(\zeta )\cdot ({\gamma }_e-{\gamma }_c)$	${\gamma }_i\equiv \gamma (0)=\frac{1}{2}({\gamma }_c+{\gamma }_e)$	$\frac{d\gamma (\zeta )}{d\zeta }{|}_i=({\gamma }_e-{\gamma }_c)$	$\frac{d\gamma }{dr}{|}_i=({\gamma }_e-{\gamma }_c)/r_i$

The step-function that arises in a proper description of the density distribution must be handled with a bit more care. Throughout the core,

${\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}$

$=$

$(1+\frac{{\xi }^2}{3}{)}^{-5/2},$

and,

$r^{*}$

$=$

$(\frac{3}{2\pi }{)}^{1/2}\xi ;$

and throughout the envelope,

${\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}$

$=$

$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5\left[\frac{{\eta }_i}{\sin ({\eta }_i-B)}\right]\left[\frac{\sin (\eta -B)}{\eta }\right],$

and,

$r^{*}$

$=$

$(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta .$

The complete functional expression for the normalized mass density can therefore be written as,

${\rho }^{*}(\zeta )$	$=$	${\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )+H(\zeta )\cdot [{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )].$
$\Rightarrow \frac{d}{d{r}^{*}}[{\rho }^{*}(\zeta )]$	$=$	$\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}+[{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )]\frac{1}{r_i}\frac{dH(\zeta )}{d\zeta }+H(\zeta )\cdot [\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}-\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}].$

Sanity check:

• $\zeta <0:$  

  $\frac{d}{d{r}^{*}}[{\rho }^{*}(\zeta )]$

  $=$

  $\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}+[{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )]\frac{1}{r_i}{\xcancel{\frac{dH(\zeta )}{d\zeta }}}^0+{\xcancel{H(\zeta )}}^0\cdot [\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}-\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}]=\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}.$

• $\zeta >0:$  

  $\frac{d}{d{r}^{*}}[{\rho }^{*}(\zeta )]$

  $=$

  $\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}+[{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )]\frac{1}{r_i}{\xcancel{\frac{dH(\zeta )}{d\zeta }}}^0+{\xcancel{H(\zeta )}}^1\cdot [\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}-\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}]=\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}.$

• $\zeta =0:$  

  $\frac{d}{d{r}^{*}}[{\rho }^{*}(\zeta )]$	$=$	$[\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}{]}_i+[{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta ){]}_i\frac{1}{r_i}{\xcancel{\frac{dH(\zeta )}{d\zeta }}}^1+{\xcancel{H(\zeta )}}^{1/2}\cdot [\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}-\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}{]}_i$
  	$=$	$\frac{1}{2}[\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}+\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}{]}_i+\frac{1}{r_i}[{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta ){]}_i.$

Now Take Radial Derivative of Pressure

Solving for $P$ in the expression for specific entropy, we have,

$\ln \left[\frac{P^{*}}{({\rho }^{*}{)}^{\gamma }}\right]$	$=$	$\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}$
$\Rightarrow P^{*}$	$=$	$({\rho }^{*}{)}^{\gamma }\cdot \exp [\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}].$

Hence,

$\frac{d{P}^{*}}{d{r}^{*}}$	$=$	$\gamma ({\rho }^{*}{)}^{\gamma -1}\{\exp \left[\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}\right]\}\frac{d{\rho }^{*}}{d{r}^{*}}+({\rho }^{*}{)}^{\gamma }\{\exp [\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}\left]\right\}\frac{d}{d{r}^{*}}\left[\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}\right]$
$\Rightarrow \frac{1}{P^{*}}\cdot \frac{d{P}^{*}}{d{r}^{*}}$	$=$	$\frac{\gamma }{{\rho }^{*}}\frac{d{\rho }^{*}}{d{r}^{*}}+\frac{d}{d{r}^{*}}\left[\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}\right].$

We will need to recognize that, unless we are sitting exactly at the interface — that is, unless $\zeta =0$ precisely —

$\frac{d}{d{r}^{*}}\left[\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}\right]$

$=$

$0.$

Hence, for two of the separate physical regimes …

• $\zeta <0:$

  $\frac{d{P}^{*}}{d{r}^{*}}$	$=$	$\gamma ({\rho }^{*}{)}^{\gamma -1}\{\exp \left[\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}\right]\}\frac{d{\rho }^{*}}{d{r}^{*}}$
  $\Rightarrow \frac{1}{P_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}\cdot \frac{d{P}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}$	$=$	$\frac{{\gamma }_c}{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}\cdot \frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}$

• $\zeta >0:$

  $\frac{1}{P_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}\cdot \frac{d{P}_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}$

  $=$

  $\frac{{\gamma }_e}{{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}\cdot \frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}$

However, at the interface where $\zeta =0$ precisely, we find,

$\frac{{\mathfrak{\mathbb{Z}}}_0}{r_i}\equiv \left\{\frac{d}{d{r}^{*}}\right[\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}]{\}}_{\zeta =0}$	$=$	$\frac{1}{\mathfrak{\Re }}\{(\gamma -1)s\frac{d\mu }{d{r}^{*}}+\mu s\frac{d\gamma }{d{r}^{*}}+\mu (\gamma -1)\frac{ds}{d{r}^{*}}{\}}_{\zeta =0}$
	$=$	$\frac{1}{\mathfrak{\Re }}\{(\gamma -1)s\frac{d\mu }{d{r}^{*}}+\mu s\frac{d\gamma }{d{r}^{*}}+\mu (\gamma -1)\frac{ds}{d{r}^{*}}{\}}_{\zeta =0}$
	$=$	$\frac{1}{\mathfrak{\Re }}[\frac{1}{2}({\mu }_c+{\mu }_e)][\frac{1}{2}({\gamma }_c+{\gamma }_e)-1][\frac{1}{2}(s_c+s_e)]\left\{\right[\frac{2}{({\mu }_c+{\mu }_e)}]\frac{d\mu }{d{r}^{*}}+[\frac{2}{({\gamma }_c+{\gamma }_e)-1}]\frac{d\gamma }{d{r}^{*}}+[\frac{2}{(s_c+s_e)}]\frac{ds}{d{r}^{*}}{\}}_{\zeta =0}$
	$=$	$\frac{1}{\mathfrak{\Re }}[\frac{1}{2}({\mu }_c+{\mu }_e)][\frac{1}{2}({\gamma }_c+{\gamma }_e)-1][\frac{1}{2}(s_c+s_e)]\{\frac{2({\mu }_e-{\mu }_c)}{({\mu }_c+{\mu }_e)}+\frac{2({\gamma }_e-{\gamma }_c)}{({\gamma }_c+{\gamma }_e)-1}+\frac{2(s_e-s_c)}{(s_c+s_e)}\}\frac{1}{r_i^{*}}$
	$=$	$\frac{1}{2^2\mathfrak{\Re }}\{({\mu }_e-{\mu }_c)[({\gamma }_c+{\gamma }_e)-1](s_c+s_e)+({\gamma }_e-{\gamma }_c)({\mu }_c+{\mu }_e)(s_c+s_e)+(s_e-s_c)({\mu }_c+{\mu }_e)[({\gamma }_c+{\gamma }_e)-1]\}\frac{1}{r_i^{*}}.$

At the interface, then, we have,

$[\frac{1}{P^{*}}\cdot \frac{d{P}^{*}}{d{r}^{*}}{]}_i$	$=$	$\{\frac{\gamma }{{\rho }^{*}}\frac{d{\rho }^{*}}{d{r}^{*}}+\frac{d}{d{r}^{*}}[\frac{\mu (\gamma -1)s}{\mathfrak{\Re }}]{\}}_{\zeta =0}$
	$=$	$\frac{{\mathfrak{\mathbb{Z}}}_0}{r_i}+\{\frac{\gamma }{{\rho }^{*}}{\}}_{\zeta =0}\cdot \{\frac{1}{2}[\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}+\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}{]}_i+\frac{1}{r_i}[{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta ){]}_i\}.$

Finally, we see that,

$\{\frac{\gamma }{{\rho }^{*}}{\}}_{\zeta =0}$	$=$	$\frac{1}{2}({\gamma }_c+{\gamma }_e)\{{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )+H(\zeta )\cdot [{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )]{\}}_{\zeta =0}^{-1}$
	$=$	$\frac{1}{2}({\gamma }_c+{\gamma }_e)\left\{\frac{1}{2}\right[{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )+{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )]{\}}_{\zeta =0}^{-1}$
	$=$	$({\gamma }_c+{\gamma }_e)[{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )+{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta ){]}_i^{-1},$

so, at the interface,

$[\frac{1}{P^{*}}\cdot \frac{d{P}^{*}}{d{r}^{*}}{]}_i$

$=$

$\frac{{\mathfrak{\mathbb{Z}}}_0}{r_i}+\left\{\frac{1}{2}\right[\frac{d{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}}{d{r}^{*}}+\frac{d{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{d{r}^{*}}{]}_i+\frac{1}{r_i}[{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta )-{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta ){]}_i\}({\gamma }_c+{\gamma }_e)[{\rho }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}(\zeta )+{\rho }_{\mathrm{e}\mathrm{n}\mathrm{v}}^{*}(\zeta ){]}_i^{-1}.$

Try Again

Additional studies of radial oscillations in models that lie along the "51 Renormalized" sequences can be found here.

Example BiPolytrope Sequence 0.3100

For the case of $(n_c,n_e)=(5,1)$ and ${\mu }_e/{\mu }_c=0.3100$, we consider here the examination of models with three relatively significant values of the core/envelope interface:

• $({\xi }_i,\overline{\rho }/{\rho }_c,q,\nu )\approx (2.06061,1.1931E+02,0.16296,0.13754)$: Approximate location along the sequence of the model with the maximum fractional core radius.
• $({\xi }_i,\overline{\rho }/{\rho }_c,q,\nu )\approx (2.69697,3.0676E+02,0.15819,0.19161)$: Approximate location along the sequence of the onset of fundamental-mode instability.
• $({\xi }_i,\overline{\rho }/{\rho }_c,q,\nu )\approx (9.0149598,1.1664E+06,0.075502255,0.337217006)$: Exact location along the sequence of the model with the maximum fractional core mass.

See Also

• Prasad, C. (1953), Proc. Natn. Inst. Sci. India, Vol. 19, 739, Radial Oscillations of a Composite Model.
• Singh, Manmohan, (1969), Proc. Natn. Inst. Sci., India, Part A, Vol. 35, pp. 586 - 589, Radial Oscillations of Composite Polytropes — Part I
• Singh, Manmohan, (1969), Proc. Nat. Inst. Sci., India, Part A, Vol. 35, pp. 703 - 708, Radial Oscillations of Composite Polytropes — Part II
• Kumar, S., Saini, S., Singh, K. K., Bhatt, V., & Vashishta, L. (2021), Astronomical & Astrophysical Transactions, Vol. 32, Issue 4, pp. 371-382, Radial Pulsations of distorted Polytropes of Non-Uniform Density.