SSC/Structure/BiPolytropes/Analytic1.53/Pt3

BiPolytrope with (n_c, n_e) = (3/2, 3)

Part I:  Milne's (1930) EOS

Part II:  Point-Source Model

Part III:  Our Derivation

Our Derivation

Steps 2 & 3

Throughout the core, the properties of this bipolytrope can be expressed in terms of the Lane-Emden function, $\theta (\xi )$, which derives from a solution of the 2^nd-order ODE,

$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left[{\xi }^2\frac{d\theta }{d\xi }\right]=-{\theta }^{3/2},$

subject to the boundary conditions,

$\theta =1$       and       $\frac{d\theta }{d\xi }=0$       at       $\xi =0$.

The first zero of the function $\theta (\xi )$ and, hence, the surface of the corresponding isolated $n=\frac{3}{2}$ polytrope is located at ${\xi }_s=3.65375$ (see Table 4 in chapter IV on p. 96 of [C67]). Hence, the interface between the core and the envelope can be positioned anywhere within the range, $0<{\xi }_i<{\xi }_s=3.65375$.

Step 4: Throughout the core (0 ≤ ξ ≤ ξ_i)

Specify: $K_c$ and ${\rho }_0\Rightarrow$
$\rho$	$=$	${\rho }_0{\theta }^{n_c}$	$=$	${\rho }_0{\theta }^{3/2}$
$P$	$=$	$K_c{\rho }_0^{1+1/n_c}{\theta }^{n_c+1}$	$=$	$K_c{\rho }_0^{5/3}{\theta }^{5/2}$
$r$	$=$	$[\frac{(n_c+1)K_c}{4\pi G}{]}^{1/2}{\rho }_0^{(1-n_c)/(2n_c)}\xi$	$=$	$[\frac{5K_c}{8\pi G}{]}^{1/2}{\rho }_0^{-1/6}\xi$
$M_r$	$=$	$4\pi \left[\frac{(n_c+1)K_c}{4\pi G}{]}^{3/2}{\rho }_0^{(3-n_c)/(2n_c)}\right(-{\xi }^2\frac{d\theta }{d\xi })$	$=$	$\frac{1}{2^2(2\pi {)}^{1/2}}\left[\frac{5K_c}{G}{]}^{3/2}{\rho }_0^{1/2}\right(-{\xi }^2\frac{d\theta }{d\xi })$

By comparison, the expressions that 📚 Milne (1930) derived for the run of $\rho$, $r$, and $M_r$ throughout the core are presented in his paper as, respectively, equations (90), (88), and (87). In an effort to facilitate this comparison, Milne's expressions — which also specifically apply to the outer edge of the core, whose identity is associated with primed variable names in Milne's notation — are reprinted as extracted equations in the following boxed-in table.

Equations extracted† from E. A. Milne (1930) The Analysis of Stellar Structure Monthly Notices of the Royal Astronomical Society, Vol. 91, pp. 4 - 55

$\rho$ $=$ ${\lambda }_2{\psi }^{3/2}$ (90) $r^{\text{'}}$ $=$ ${\eta }^{\text{'}}(\frac{5K}{8\pi G\beta }{)}^{1/2}{\lambda }_2^{-1/6}$ (88) $M(r^{\text{'}})$ $=$ $-\frac{1}{4(2\pi {)}^{1/2}}(\frac{5K}{G\beta }{)}^{3/2}{\lambda }_2^{1/2}({\eta }^{\text{'}}{)}^2(\frac{d\psi }{d\eta }{)}_{\eta ={\eta }^{\text{'}}}$ (87)

†Equations displayed here, with presentation order & layout modified from the original publication.

It is clear that the agreement between our derivation and Milne's is exact, once it is realized that Milne has used $\psi (\eta )$ to represent the Lane_Emden function for the $n_c=\frac{3}{2}$ core, whereas we have represented this function by $\theta (\xi )$; and Milne has identified the configuration's central density as ${\lambda }_2$, whereas we have used the notation, ${\rho }_0$.

Step 5: Interface Conditions

			Setting $n_c=\frac{3}{2}$, $n_e=3$, and ${\varphi }_i=1\Rightarrow$
$\frac{{\rho }_e}{{\rho }_0}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{n_c}{\varphi }_i^{-n_e}$	$=$	$\left(\frac{{\mu }_e}{{\mu }_c}\right){\theta }_i^{3/2}$
$\left(\frac{K_e}{K_c}\right)$	$=$	${\rho }_0^{1/n_c-1/n_e}(\frac{{\mu }_e}{{\mu }_c}{)}^{-(1+1/n_e)}{\theta }_i^{1-n_c/n_e}$	$=$	${\rho }_0^{1/3}(\frac{{\mu }_e}{{\mu }_c}{)}^{-4/3}{\theta }_i^{1/2}$
$\frac{{\eta }_i}{{\xi }_i}$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^{(n_c-1)/2}{\varphi }_i^{(1-n_e)/2}$	$=$	$\left(\frac{5}{8}{)}^{1/2}\right(\frac{{\mu }_e}{{\mu }_c}){\theta }_i^{1/4}$
$(\frac{d\varphi }{d\eta }{)}_i$	$=$	$\left[\frac{n_c+1}{n_e+1}{]}^{1/2}{\theta }_i^{-(n_c+1)/2}{\varphi }_i^{(n_e+1)/2}\right(\frac{d\theta }{d\xi }{)}_i$	$=$	$\left(\frac{5}{8}{)}^{1/2}{\theta }_i^{-5/4}\right(\frac{d\theta }{d\xi }{)}_i$

Step 8: Throughout the envelope (η_i ≤ η ≤ ξ_s)

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

Instead of working completely across this table in order to relate the envelope's density, radial coordinate, and mass to properties of the core, it is worth pausing to insert into the leftmost set of relations the expressions for ${\rho }_e$ and $K_e$ that were derived above. In doing this, we obtain,

$\rho$	$=$	${\rho }_e{\varphi }^3$	$=$	$\left[\right(\frac{\mathrm{\Re }}{{\mu }_e}{)}^{-1}\frac{\beta }{(1-\beta )}\cdot \frac{1}{3}{a}_{\mathrm{r}\mathrm{a}\mathrm{d}}]{\lambda }^3{\varphi }^3,$
$r$	$=$	$[\frac{K_e}{\pi G}{]}^{1/2}{\rho }_e^{-1/3}\eta$	$=$	$\frac{1}{(\pi G{)}^{1/2}}\left[\right(\frac{\mathrm{\Re }}{{\mu }_e}{)}^4\left(\frac{1-\beta }{{\beta }^4}\right)\frac{3}{a_{\mathrm{r}\mathrm{a}\mathrm{d}}}{]}^{1/6}\left\{{\lambda }^3\right[(\frac{\mathrm{\Re }}{{\mu }_e}{)}^{-1}\frac{\beta }{(1-\beta )}\cdot \frac{1}{3}{a}_{\mathrm{r}\mathrm{a}\mathrm{d}}]{\}}^{-1/3}\eta$
			$=$	$\frac{1}{(\pi G{)}^{1/2}}\left[\right(\frac{\mathrm{\Re }}{{\mu }_e}{)}^4\left(\frac{1-\beta }{{\beta }^4}\right)\frac{3}{a_{\mathrm{r}\mathrm{a}\mathrm{d}}}{]}^{1/6}\left[\right(\frac{\mathrm{\Re }}{{\mu }_e}{)}^2\frac{(1-\beta {)}^2}{{\beta }^2}(\frac{3}{a_{\mathrm{r}\mathrm{a}\mathrm{d}}}{)}^2{]}^{1/6}\frac{\eta }{\lambda }$
			$=$	$\left(\frac{3}{\pi a_{\mathrm{r}\mathrm{a}\mathrm{d}}G}{)}^{1/2}\right(\frac{\mathrm{\Re }}{{\mu }_e})\frac{(1-\beta {)}^{1/2}}{\beta }\cdot \frac{\eta }{\lambda },$
$M_r$	$=$	$4\pi \left[\frac{K_e}{\pi G}{]}^{3/2}\right(-{\eta }^2\frac{d\varphi }{d\eta })$	$=$	$\frac{4\pi }{(\pi G{)}^{3/2}}\left[\right(\frac{\mathrm{\Re }}{{\mu }_e}{)}^4\left(\frac{1-\beta }{{\beta }^4}\right)\frac{3}{a_{\mathrm{r}\mathrm{a}\mathrm{d}}}{]}^{1/2}(-{\eta }^2\frac{d\varphi }{d\eta })$
			$=$	$4\left(\frac{3}{\pi a_{\mathrm{r}\mathrm{a}\mathrm{d}}{G}^3}{)}^{1/2}\right(\frac{\mathrm{\Re }}{{\mu }_e}{)}^2\frac{(1-\beta {)}^{1/2}}{{\beta }^2}(-{\eta }^2\frac{d\varphi }{d\eta }).$

By comparison, the expressions that 📚 Milne (1930) derived for the run of $\rho$, $r$, and $M_r$ throughout the envelope are presented in his paper as, respectively, equations (89), (86), and (85). In an effort to facilitate this comparison, Milne's expressions — which also specifically apply to the base of the envelope, whose identity is associated with primed variable names in Milne's notation — are reprinted as extracted equations in the following boxed-in table.

Equations extracted† from E. A. Milne (1930) The Analysis of Stellar Structure Monthly Notices of the Royal Astronomical Society, Vol. 91, pp. 4 - 55

$\rho$ $=$ $\frac{\frac{1}{3}a}{R/\mu }\left[\frac{\beta }{1-\beta }\right]{\lambda }_1^3{\theta }^3$ (89) $r^{\text{'}}$ $=$ $\frac{{\xi }^{\text{'}}}{{\lambda }_1}\frac{R}{\mu }\left[\frac{(1-\beta {)}^{1/2}}{\beta }\right]\frac{1}{(\frac{1}{3}\pi aG{)}^{1/2}}$ (86) $M(r^{\text{'}})$ $=$ $-\frac{4}{(\frac{1}{3}\pi a{G}^3{)}^{1/2}}\left(\frac{R}{\mu }{)}^2\right[\frac{(1-\beta {)}^{1/2}}{{\beta }^2}]({\xi }^{\text{'}}{)}^2(\frac{d\theta }{d\xi }{)}_{\xi ={\xi }^{\text{'}}}$ (85)

†Equations displayed here, with presentation order & layout modified from the original publication.

The agreement between our derivation and Milne's is exact, once it is realized that Milne has used $\theta (\xi )$ to represent the Lane_Emden function for the $n_e=3$ envelope, whereas we have represented this function by $\varphi (\eta )$; and in place of Milne's coefficient, ${\lambda }_1$, we have simply written, $\lambda$.

See Also