Appendix/Ramblings/51BiPolytropeStability/RethinkEnvelope

Rethink Handling of n = 1 Envelope

Solution Steps

Drawing from an accompanying discussion …

• Step 1: Choose $n_c$ and $n_e$.
• Step 2: Adopt boundary conditions at the center of the core ($\theta =1$ and $d\theta /d\xi =0$ at $\xi =0$), then solve the Lane-Emden equation to obtain the solution, $\theta (\xi )$, and its first derivative, $d\theta /d\xi$ throughout the core; the radial location, $\xi ={\xi }_s$, at which $\theta (\xi )$ first goes to zero identifies the natural surface of an isolated polytrope that has a polytropic index $n_c$.
• Step 3 Choose the desired location, $0<{\xi }_i<{\xi }_s$, of the outer edge of the core.
• Step 4: Specify $K_c$ and ${\rho }_0$; the structural profile of, for example, $\rho (r)$, $P(r)$, and $M_r(r)$ is then obtained throughout the core — over the radial range, $0\le \xi \le {\xi }_i$ and $0\le r\le r_i$ — via the relations shown in the $2^{\mathrm{n}\mathrm{d}}$ column of Table 1.
• Step 5: Specify the ratio ${\mu }_e/{\mu }_c$ and adopt the boundary condition, ${\varphi }_i=1$; then use the interface conditions as rearranged and presented in Table 3 to determine, respectively:
  • The gas density at the base of the envelope, ${\rho }_e$;
  • The polytropic constant of the envelope, $K_e$, relative to the polytropic constant of the core, $K_c$;
  • The ratio of the two dimensionless radial parameters at the interface, ${\eta }_i/{\xi }_i$;
  • The radial derivative of the envelope solution at the interface, $(d\varphi /d\eta {)}_i$.
• Step 6: The last sub-step of solution step 5 provides the boundary condition that is needed — in addition to our earlier specification that ${\varphi }_i=1$ — to derive the desired particular solution, $\varphi (\eta )$, of the Lane-Emden equation that is relevant throughout the envelope; knowing $\varphi (\eta )$ also provides the relevant structural first derivative, $d\varphi /d\eta$, throughout the envelope.
• Step 7: The surface of the bipolytrope will be located at the radial location, $\eta ={\eta }_s$ and $r=R$, at which $\varphi (\eta )$ first drops to zero.
• Step 8: The structural profile of, for example, $\rho (r)$, $P(r)$, and $M_r(r)$ is then obtained throughout the envelope — over the radial range, ${\eta }_i\le \eta \le {\eta }_s$ and $r_i\le r\le R$ — via the relations provided in the $3^{\mathrm{r}\mathrm{d}}$ column of Table 1.

Setup

Drawing from the accompanying Table 1, we have …

Core		Envelope
Isothermal $(n_c=\mathrm{\infty })$	$n=n_c$	$n=n_e$
$\frac{1}{{\chi }^2}\frac{d}{d\chi }\left({\chi }^2\frac{d\psi }{d\chi }\right)=e^{-\psi }$ sol'n: $\psi (\chi )$	$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d\theta }{d\xi }\right)=-{\theta }^{n_c}$ sol'n: $\theta (\xi )$	$\frac{1}{{\eta }^2}\frac{d}{d\eta }\left({\eta }^2\frac{d\varphi }{d\eta }\right)=-{\varphi }^{n_e}$ sol'n: $\varphi (\eta )$
Specify: $c_s^2$ and ${\rho }_0\Rightarrow$ $\rho$ $=$ ${\rho }_0{e}^{-\psi }$ $P$ $=$ $c_s^2{\rho }_0{e}^{-\psi }$ $r$ $=$ $[\frac{c_s^2}{4\pi G{\rho }_0}{]}^{1/2}\chi$ $M_r$ $=$ $\left[\frac{c_s^6}{4\pi G^3{\rho }_0}{]}^{1/2}\right({\chi }^2\frac{d\psi }{d\chi })$	Specify: $K_c$ and ${\rho }_0\Rightarrow$ $\rho$ $=$ ${\rho }_0{\theta }^{n_c}$ $P$ $=$ $K_c{\rho }_0^{1+1/n_c}{\theta }^{n_c+1}$ $r$ $=$ $[\frac{(n_c+1)K_c}{4\pi G}{]}^{1/2}{\rho }_0^{(1-n_c)/(2n_c)}\xi$ $M_r$ $=$ $4\pi \left[\frac{(n_c+1)K_c}{4\pi G}{]}^{3/2}{\rho }_0^{(3-n_c)/(2n_c)}\right(-{\xi }^2\frac{d\theta }{d\xi })$	Knowing: $K_e$ and ${\rho }_e\Rightarrow$ $\rho$ $=$ ${\rho }_e{\varphi }^{n_e}$ $P$ $=$ $K_e{\rho }_e^{1+1/n_e}{\varphi }^{n_e+1}$ $r$ $=$ $[\frac{(n_e+1)K_e}{4\pi G}{]}^{1/2}{\rho }_e^{(1-n_e)/(2n_e)}\eta$ $M_r$ $=$ $4\pi \left[\frac{(n_e+1)K_e}{4\pi G}{]}^{3/2}{\rho }_e^{(3-n_e)/(2n_e)}\right(-{\eta }^2\frac{d\varphi }{d\eta })$

See Also

• Rappaport, Verbunt, & Joss (1983, ApJ, 275, 713) — A New Technique for Calculations of Binary Stellar Evolution, with Application to Magnetic Braking.