SSC/Stability/BiPolytropes/Pt2

Review of the Analysis by Murphy & Fiedler (1985b)

Part I:   The Search	Part II:  Review of MF85b	III:  (5,1) Radial Oscillations	IV:  Reconciliation
These four chapters, labeled Parts I - IV, are segments of the much longer chapter titled, SSC/Stability/BiPolytropes/PlannedApproach. An accompanying organizational index has helped us write this chapter succinctly.

As we have detailed separately, the boundary condition at the center of a polytropic configuration is,

$\frac{dx}{d\xi }{|}_{\xi =0}=0;$

and the boundary condition at the surface of an isolated polytropic configuration is,

$\frac{d\ln x}{d\ln \xi }$

$=$

$-\alpha +\frac{{\omega }^2}{{\gamma }_g}\left(\frac{1}{4\pi G{\rho }_c}\right)\frac{\xi }{(-{\theta }^{\text{'}})}$         at         $\xi ={\xi }_s.$

But this surface condition is not applicable to bipolytropes. Instead, let's return to the original, more general expression of the surface boundary condition:

$\frac{d\ln x}{d\ln \xi }{|}_s$

$=$

$-\alpha +\frac{{\omega }^2{R}^3}{{\gamma }_{g}G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}.$

Utilizing an accompanying discussion, let's examine the frequency normalization used by 📚 Murphy & Fiedler (1985b) — see the top of the left-hand column on p. 223: ${\mathrm{\Omega }}^2$ $\equiv$ ${\omega }^2\left[\frac{R^3}{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}}\right]$   $=$ ${\omega }^2\left[\frac{3}{4\pi G\overline{\rho }}\right]={\omega }^2\left[\frac{3}{4\pi G{\rho }_c}\right]\frac{{\rho }_c}{\overline{\rho }}=\frac{3{\omega }^2}{(n_c+1)}\left[\frac{(n_c+1)}{4\pi G{\rho }_c}\right]\frac{{\rho }_c}{\overline{\rho }}$   $=$ $\frac{3{\omega }^2}{(n_c+1)}[\frac{a_n^2{\rho }_c}{P_c}\cdot {\theta }_c]\frac{{\rho }_c}{\overline{\rho }}=\frac{3\gamma }{(n_c+1)}\frac{{\rho }_c}{\overline{\rho }}[\frac{a_n^2{\rho }_c}{P_c}\cdot \frac{{\omega }^2{\theta }_c}{\gamma }].$ For a given radial quantum number, $k$, the factor inside the square brackets in this last expression is what 📚 Murphy & Fiedler (1985b) refer to as ${\omega }_k^2{\theta }_c$. Keep in mind, as well, that, in the notation we are using, ${\sigma }_c^2$ $\equiv$ $\frac{3{\omega }^2}{2\pi G{\rho }_c}$ $\Rightarrow {\sigma }_c^2$ $=$ $\left(\frac{2\overline{\rho }}{{\rho }_c}\right){\mathrm{\Omega }}^2=\frac{6\gamma }{(n_c+1)}[\frac{a_n^2{\rho }_c}{P_c}\cdot \frac{{\omega }^2{\theta }_c}{\gamma }]=\frac{6\gamma }{(n_c+1)}\left[{\omega }_k^2{\theta }_c\right].$ This also means that the surface boundary condition may be rewritten as, $\frac{d\ln x}{d\ln \xi }{|}_s$ $=$ $\frac{{\mathrm{\Omega }}^2}{{\gamma }_g}-\alpha .$

Let's apply these relations to the core and envelope, separately.

Interface Conditions

Here, we will simply copy the discussion already provided in the context of our attempt to analyze the stability of $(n_c,n_e)=(0,0)$ bipolytropes; specifically, we will draw from STEP 4: in the Piecing Together subsection. Following the discussion in §§57 & 58 of 📚 Ledoux & Walraven (1958), the proper treatment is to ensure that fractional perturbation in the gas pressure (see their equation 57.31),

$\frac{\delta P}{P}$

$=$

$-\gamma x(3+\frac{d\ln x}{d\ln \xi }),$

is continuous across the interface.

Reaffirmation:   In our introductory discussion of the eigenvalue problem, we adopted the following expression for the time-dependent pressure, $P(m,t)$ $=$ $P_0(m)+P_1(m,t)=P_0(m)[1+p(m)e^{i\omega t}].$ In this expression, $P_0(m)$ is the function that details how the unperturbed pressure varies with Lagrangian mass shell $(m)$, and $P_1=\delta P(m)e^{i\omega t}$ traces the variation of the pressure away from its equilibrium value at each mass shell. The time-dependent and spatially dependent behavior of $P_1$ has been separated, with $\delta P$ carrying information about the function's spatial dependence. Furthermore, we have adopted the shorthand notation, $p(m)\equiv \frac{\delta P(m)}{P_0(m)}.$ We can just as well use $r_0(m)$ to tag the (initial, unperturbed location of the) Lagrangian mass shells, in which case we write, $p(r_0)=\frac{\delta P(r_0)}{P_0(r_0)},$     and, similarly,    $d(r_0)=\frac{\delta \rho (r_0)}{{\rho }_0(r_0)},$     and,     $x(r_0)=\frac{\delta r(r_0)}{r_0}.$ These three spatially dependent quantities — $p,d,$ and $x$ — are related to one another via the set of linearized governing relations, namely, Linearized Equation of Continuity $r_0\frac{dx}{d{r}_0}=-3x-d,$ Linearized Euler + Poisson Equations $\frac{P_0}{{\rho }_0}\frac{dp}{d{r}_0}=(4x+p)g_0+{\omega }^2{r}_{0}x,$ Linearized Adiabatic Form of the First Law of Thermodynamics $p={\gamma }_{\mathrm{g}}d.$ Using the third of these expressions to replace $d$ in favor of $p$ in the first expression, we find that, $r_0\frac{dx}{d{r}_0}$ $=$ $-3x-\frac{p}{{\gamma }_g}$ $\Rightarrow p$ $=$ $-{\gamma }_{g}x[3+\frac{d\ln x}{d\ln r_0}],$ which is identical to the pressure-perturbation expression used by 📚 Ledoux & Walraven (1958) and referenced above. As they state, the function, $p=\delta P/P_0$, should be continuous across the core-envelope interface.

That is to say, at the interface $(\xi ={\xi }_i)$, we need to enforce the relation,

$0$	$=$	$\left[{\gamma }_c{x}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}\right(3+\frac{d\ln x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{d\ln \xi })-{\gamma }_e{x}_{\mathrm{e}\mathrm{n}\mathrm{v}}(3+\frac{d\ln x_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\ln \xi }){]}_{\xi ={\xi }_i}$
	$=$	${\gamma }_e\left[\frac{{\gamma }_c}{{\gamma }_e}\right(3+\frac{d\ln x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{d\ln \xi })-(3+\frac{d\ln x_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\ln \xi }){]}_{\xi ={\xi }_i}$
$\Rightarrow \frac{d\ln x_{\mathrm{e}\mathrm{n}\mathrm{v}}}{d\ln \xi }{|}_{\xi ={\xi }_i}$	$=$	$3(\frac{{\gamma }_c}{{\gamma }_e}-1)+\frac{{\gamma }_c}{{\gamma }_e}(\frac{d\ln x_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{d\ln \xi }{)}_{\xi ={\xi }_i}.$

In the context of this interface-matching constraint (see their equation 62.1), 📚 Ledoux & Walraven (1958) state the following:   In the static (i.e., unperturbed equilibrium) model … discontinuities in $\rho$ or in $\gamma$ might occur at some [radius]. In the first case — that is, a discontinuity only in density, while ${\gamma }_e={\gamma }_c$ — the interface conditions imply the continuity of $\frac{1}{x}\cdot \frac{dx}{d\xi }$ at that [radius]. In the second case — that is, a discontinuity in the adiabatic exponent — the dynamical condition may be written as above. This implies a discontinuity of the first derivative at any discontinuity of $\gamma$.

The algorithm that 📚 Murphy & Fiedler (1985b) used to "… [integrate] through each zone …" was designed "… with continuity in $x$ and $dx/d\xi$ being imposed at the interface …" Given that they set ${\gamma }_c={\gamma }_e=5/3$, their interface matching condition is consistent with the one prescribed by 📚 Ledoux & Walraven (1958).