SSC/Structure/BiPolytropes/51RenormaizePart3

BiPolytrope with n_c = 5 and n_e = 1

After studying 📚 S. Yabushita (1975, MNRAS, Vol. 172, pp. 441 - 453) in depth, here we renormalize our original construction of bipolytropic models with $(n_c,n_e)=(5,1)$ such that both entropy values, $(K_c,K_e)$, are held fixed along each model sequence.

Original Derivation

Throughout the Core

Drawing from our original derivation, throughout the core …

Specify: $K_c$ and ${\rho }_0\Rightarrow$
$\rho$	$=$	${\rho }_0{\theta }^{n_c}$	$=$	${\rho }_0(1+\frac{1}{3}{\xi }^2{)}^{-5/2}$
$P$	$=$	$K_c{\rho }_0^{1+1/n_c}{\theta }^{n_c+1}$	$=$	$K_c{\rho }_0^{6/5}(1+\frac{1}{3}{\xi }^2{)}^{-3}$
$r$	$=$	$[\frac{(n_c+1)K_c}{4\pi G}{]}^{1/2}{\rho }_0^{(1-n_c)/(2n_c)}\xi$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{3}{2\pi }{)}^{1/2}\xi$
$M_r$	$=$	$4\pi \left[\frac{(n_c+1)K_c}{4\pi G}{]}^{3/2}{\rho }_0^{(3-n_c)/(2n_c)}\right(-{\xi }^2\frac{d\theta }{d\xi })$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }^3\right(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]$

Throughout the Envelope

And throughout the envelope,

					Knowing: $K_e/K_c$ and ${\rho }_e/{\rho }_0$ from Step 5   $\Rightarrow$
$\rho$	$=$	${\rho }_e{\varphi }^{n_e}$	$=$	${\rho }_0\left(\frac{{\rho }_e}{{\rho }_0}\right)\varphi$	$=$	${\rho }_0\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5\varphi$
$P$	$=$	$K_e{\rho }_e^{1+1/n_e}{\varphi }^{n_e+1}$	$=$	$K_c{\rho }_0^{6/5}\left(\frac{K_e{\rho }_0^{4/5}}{K_c}\right)(\frac{{\rho }_e}{{\rho }_0}{)}^2{\varphi }^2$	$=$	$K_c{\rho }_0^{6/5}{\theta }_i^6{\varphi }^2$
$r$	$=$	$[\frac{(n_e+1)K_e}{4\pi G}{]}^{1/2}{\rho }_e^{(1-n_e)/(2n_e)}\eta$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{K_e{\rho }_0^{4/5}}{K_c}{)}^{1/2}(2\pi {)}^{-1/2}\eta$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta$
$M_r$	$=$	$4\pi \left[\frac{(n_e+1)K_e}{4\pi G}{]}^{3/2}{\rho }_e^{(3-n_e)/(2n_e)}\right(-{\eta }^2\frac{d\varphi }{d\eta })$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{K_e{\rho }_0^{4/5}}{K_c}{)}^{3/2}\left(\frac{{\rho }_e}{{\rho }_0}\right)\left(\frac{2}{\pi }{)}^{1/2}\right(-{\eta }^2\frac{d\varphi }{d\eta })$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\left(\frac{2}{\pi }{)}^{1/2}\right(-{\eta }^2\frac{d\varphi }{d\eta })$

Interface Conditions

And at the interface …

			Setting $n_c=5$, $n_e=1$, and ${\varphi }_i=1\Rightarrow$
$\frac{{\rho }_e}{{\rho }_0}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{n_c}{\varphi }_i^{-n_e}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5$
$\left(\frac{K_e}{K_c}\right)$	$=$	${\rho }_0^{1/n_c-1/n_e}(\frac{{\mu }_e}{{\mu }_c}{)}^{-(1+1/n_e)}{\theta }_i^{1-n_c/n_e}$	$=$	${\rho }_0^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}$
$\frac{{\eta }_i}{{\xi }_i}$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^{(n_c-1)/2}{\varphi }_i^{(1-n_e)/2}$	$=$	$3^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2$
$(\frac{d\varphi }{d\eta }{)}_i$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}{\theta }_i^{-(n_c+1)/2}{\varphi }_i^{(n_e+1)/2}\right(\frac{d\theta }{d\xi }{)}_i$	$=$	$3^{1/2}{\theta }_i^{-3}(\frac{d\theta }{d\xi }{)}_i$

New Normalization

From one of the interface conditions, we see that,

$\left(\frac{K_e}{K_c}\right)$	$=$	${\rho }_0^{-4/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}$
$\Rightarrow {\rho }_0$	$=$	$\left[\right(\frac{K_e}{K_c}{)}^{-1}(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-4}{]}^{5/4}=[\left(\frac{K_e}{K_c}{)}^{-5/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}]={\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}\right].$

Hence, throughout the core, we have,

$\rho$	$=$	${\rho }_0(1+\frac{1}{3}{\xi }^2{)}^{-5/2}$	$=$	${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}\left]\right(1+\frac{1}{3}{\xi }^2{)}^{-5/2}$
$P$	$=$	$K_c{\rho }_0^{6/5}(1+\frac{1}{3}{\xi }^2{)}^{-3}$	$=$	$K_c\left[\right(\frac{K_e}{K_c}{)}^{-5/4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}{]}^{6/5}\right(1+\frac{1}{3}{\xi }^2{)}^{-3}$
			$=$	$K_c\left[\right(\frac{K_e}{K_c}{)}^{-3/2}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{\theta }_i^{-6}\right](1+\frac{1}{3}{\xi }^2{)}^{-3}=P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}[\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{\theta }_i^{-6}\right](1+\frac{1}{3}{\xi }^2{)}^{-3}$
$r$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{3}{2\pi }{)}^{1/2}\xi$	$=$	$\left[\frac{K_c}{G}{]}^{1/2}\right[\left(\frac{K_e}{K_c}{)}^{-5/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}{]}^{-2/5}(\frac{3}{2\pi }{)}^{1/2}\xi$
			$=$	$\left[\frac{K_c}{G}{]}^{1/2}\right[\left(\frac{K_e}{K_c}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}\left){\theta }_i^2\right](\frac{3}{2\pi }{)}^{1/2}\xi =r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}[\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^2\left]\right(\frac{3}{2\pi }{)}^{1/2}\xi$
$M_r$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }^3\right(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]$	$=$	$\left[\frac{K_c^3}{G^3}{]}^{1/2}\right[\left(\frac{K_e}{K_c}{)}^{-5/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}{]}^{-1/5}\left(\frac{2\cdot 3}{\pi }{)}^{1/2}\right[{\xi }^3(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]$
			$=$	$\left[\frac{K_c^3}{G^3}{]}^{1/2}\right[\left(\frac{K_e}{K_c}{)}^{1/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i\left]\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }^3\right(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]$
			$=$	$M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i\left]\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }^3\right(1+\frac{1}{3}{\xi }^2{)}^{-3/2}]$

And, throughout the envelope …

$\rho$	$=$	${\rho }_0\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5\varphi$	$=$	$\left[\right(\frac{K_e}{K_c}{)}^{-5/4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}\right]\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^5\varphi$
			$=$	${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}]\varphi$
$P$	$=$	$K_c{\rho }_0^{6/5}{\theta }_i^6{\varphi }^2$	$=$	$K_c\left[\right(\frac{K_e}{K_c}{)}^{-5/4}(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}{]}^{6/5}{\theta }_i^6{\varphi }^2$
			$=$	$K_c\left[\right(\frac{K_e}{K_c}{)}^{-3/2}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}\right]{\varphi }^2=P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}]{\varphi }^2$
$r$	$=$	$\left[\frac{K_c}{G{\rho }_0^{4/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta$	$=$	$\left[\frac{K_c}{G}{]}^{1/2}\right[\left(\frac{K_e}{K_c}{)}^{-5/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}{]}^{-2/5}(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}{\theta }_i^{-2}(2\pi {)}^{-1/2}\eta$
			$=$	$\left[\frac{K_c}{G}{]}^{1/2}\right(\frac{K_e}{K_c}{)}^{1/2}(2\pi {)}^{-1/2}\eta =r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}(2\pi {)}^{-1/2}\eta$
$M_r$	$=$	$\left[\frac{K_c^3}{G^3{\rho }_0^{2/5}}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\left(\frac{2}{\pi }{)}^{1/2}\right(-{\eta }^2\frac{d\varphi }{d\eta })$	$=$	$\left[\frac{K_c^3}{G^3}{]}^{1/2}\right[\left(\frac{K_e}{K_c}{)}^{-5/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}{]}^{-1/5}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-2}{\theta }_i^{-1}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })$
			$=$	$\left[\frac{K_c^3}{G^3}{]}^{1/2}\right(\frac{K_e}{K_c}{)}^{1/4}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })$
			$=$	$M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })$

Adopted Normalizations

${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $(\frac{K_c}{K_e}{)}^{5/4};$     $P_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $K_c(\frac{K_e}{K_c}{)}^{-3/2}=[K_c^5{K}_e^{-3}{]}^{1/2};$ $r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $\left[\frac{K_c}{G}{]}^{1/2}\right(\frac{K_e}{K_c}{)}^{1/2}=(\frac{K_e}{G}{)}^{1/2};$     $M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}$ $\equiv$ $\left[\frac{K_c^3}{G^3}{]}^{1/2}\right(\frac{K_e}{K_c}{)}^{1/4}=[K_c^5{K}_e{G}^{-6}{]}^{1/4}.$

Note that the configuration's mean density is,

$\overline{\rho }\equiv \frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi R^3}$			$=$	$\left(\frac{3}{4\pi }\right)M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta }{)}_s[r_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}(2\pi {)}^{-1/2}{\eta }_s{]}^{-3}$
			$=$	$M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}{r}_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}^{-3}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{3}{4\pi })(2\pi {)}^{3/2}\left(\frac{2}{\pi }{)}^{1/2}\right(-\frac{1}{\eta }\cdot \frac{d\varphi }{d\eta }{)}_s$
			$=$	$\left[K_c^5{K}_e{G}^{-6}{]}^{1/4}\right(\frac{K_e}{G}{)}^{-3/2}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right[\left(\frac{3^2}{2^4{\pi }^2}\right)(2\pi {)}^3(\frac{2}{\pi }\left){]}^{1/2}\right(-\frac{1}{\eta }\cdot \frac{d\varphi }{d\eta }{)}_s$
			$=$	$3\left(\frac{K_c}{K_e}{)}^{5/4}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}(-\frac{1}{\eta }\cdot \frac{d\varphi }{d\eta }{)}_s.$

Hence, the central-to-mean density of each equilibrium configuration is,

$\frac{{\rho }_0}{\overline{\rho }}$			$=$	${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}\left]\right\{3{\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(-\frac{1}{\eta }\cdot \frac{d\varphi }{d\eta }{)}_s{\}}^{-1}.$
			$=$	$\left\{3\right(\frac{{\mu }_e}{{\mu }_c}\left){\theta }_i^5\right(-\frac{1}{\eta }\cdot \frac{d\varphi }{d\eta }{)}_s{\}}^{-1}.$

Yabushita75 Plot

Specify Desired Abscissa and Ordinate

Here our desire is to generate a plot that is analogous to the one that appears as Fig. 1 (p. 445) of 📚 Yabushita (1975). We need to plot the core mass versus the central density, and the total mass versus central density where,

$M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i\left]\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }_i^3\right(1+\frac{1}{3}{\xi }_i^2{)}^{-3/2}],$
$M_{\mathrm{t}\mathrm{o}\mathrm{t}}$	$=$	$M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta }{)}_s,$
${\rho }_0$	$=$	${\rho }_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-5/2}{\theta }_i^{-5}].$

As a check against earlier derivations, note as well that,

$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	$M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left[\right(\frac{{\mu }_e}{{\mu }_c}{)}^{1/2}{\theta }_i\left]\right(\frac{2\cdot 3}{\pi }{)}^{1/2}\left[{\xi }_i^3\right(1+\frac{1}{3}{\xi }_i^2{)}^{-3/2}\left]\right\{M_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3/2}\right(\frac{2}{\pi }{)}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta }{)}_s{\}}^{-1}$
	$=$	$3^{1/2}\left(\frac{{\mu }_e}{{\mu }_c}{)}^2{\theta }_i\right[{\xi }_i^3(1+\frac{1}{3}{\xi }_i^2{)}^{-3/2}](-{\eta }^2\frac{d\varphi }{d\eta }{)}_s^{-1}.$

Compare with Earlier Derivation

From our earlier derivation, we know that,

$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$=$	$\sqrt{3}\left(\frac{{\mu }_e}{{\mu }_c}{)}^2\right[\frac{{\xi }_i^3{\theta }_i^4}{A{\eta }_s}]$
	$=$	$\sqrt{3}\left(\frac{{\mu }_e}{{\mu }_c}{)}^2\right[\frac{{\xi }_i^3{\theta }_i^4}{{\eta }_s}\left]\right[-{\eta }_s(\frac{d\varphi }{d\eta }{)}_s{]}^{-1}$
	$=$	$\sqrt{3}\left(\frac{{\mu }_e}{{\mu }_c}{)}^2{\theta }_i\right[{\xi }_i^3(1+\frac{1}{3}{\xi }_i^2{)}^{-3/2}][-{\eta }_s^2(\frac{d\varphi }{d\eta }{)}_s{]}^{-1}.$

Also, our earlier derivation gave,

$\frac{{\rho }_c}{\overline{\rho }}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right[\frac{{\eta }_s^2}{3A{\theta }_i^5}]$
	$=$	$\frac{1}{3}\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-1}\right(1+\frac{1}{3}{\xi }_i^2{)}^{5/2}[-\frac{1}{{\eta }_s}\cdot (\frac{d\varphi }{d\eta }{)}_s{]}^{-1}.$

Hooray! These both match our "new normalization" derivation.

See Also

• M. Gabriel & M. L. Roth (1974, A&A, Vol. 32, p. 309) … On the Secular Stability of Models with an Isothermal Core
• M. Gabriel & P. Ledoux (1967, Annales d'Astrophysique, Vol. 30, p. 975) … Sur la Stabilité Séculaire des Modeéles a Noyaux Isothermes
  

  In § 1 (p. 442) of 📚 Yabushita (1975) we find the following reference:   "A somewhat similar problem has been investigated by Gabriel & Ledoux (1967). Gaseous configurations with an isothermal core and polytropic envelope of index 3 were studied by 📚 Henrich & Chandrasekhar (1941) and by 📚 Schönberg & Chandrasekhar (1942). As is well known there is an upper limit (Schönberg-Chandrasekhar limit) to the mass of the core for the configurations to be in hydrostatic equilibria. Gabriel & Ledoux have investigated the stability of these configurations and have shown that secular stability is lost at the configuration that corresponds to the Schönberg-Chandrasekhar limit."