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Discussing Patrick Motl's 2019 Simulations

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May 1 (from Patrick)

 Hi Joel, I hope things are well for you and yours. I did finally have a couple research students this semester that were able to slog their way into unix, programming, etc. far enough to do some useful things. They ran two spherical bipolytropes with my old cylindrical code. These were n = 5 cores with n = 1 envelopes. No density discontinuity. One model is a little below the stability boundary in xi, the other is a little above the stability boundary. What I can see from the evolutions, especially now that I made plots of the entropy, is that they are convectively unstable and that is just a mess with the core convecting into the envelope. Did you happen to have any thoughts on what might be a better case to run? When I get done grading finals I am going to work through the equations with kappa set explicitly in the envelope so the entropy profile is flat and just live with whatever density discontinuity that gives me. cheers, Patrick

May 4 (from Joel)

 Patrick, My initial response to your 1 May email follows. In an effort to make sure we are on the same page when referencing the structural properties and the stability properties of various configurations, my response includes links to various chapters/subsections of my online wiki-based book. Thanks for pursuing this problem. I am still very interested in Its solution. All the best, Joel

Chosen Initial Models

I presume that when you constructed your pair of initial (spherical) models, you used the analytically prescribed properties found in my chapter titled, "BiPolytrope with nc = 5 and ne = 1" and that when you reference the "stability boundary in $~\xi$" you are drawing from the summary (Table 3 and Figure 3) found near the end of this chapter in the subsection titled, "Stability Condition." In particular, based on the information contained in this chapter, I presume that you are assuming that the marginally unstable model along the $~\mu_e/\mu_c = 1$ sequence is the configuration having the properties …

$~(\xi_i, q, \nu) = (2.416, 0.5952, 0.6830) \, ,$

and that your pair of initial models have values of $~\xi_i$ that are somewhat greater than and somewhat smaller than 2.416. If this does not properly describe your choice of initial models, please clarify.

Instability Toward Convection

In my chapter titled, "Axisymmetric Instabilities to Avoid," I discuss — among other things — the Schwarzschild Criterion. Toward the very end of that chapter, in a subsection titled, Modeling Implications and Advice, I specifically describe how convection might arise when modeling spherical polytropic configurations. Would you please read this short subsection and let me know whether you agree with my explanation or not?

In the context of this discussion of the Schwarzschild criterion (i.e., convective instabilities) in polytropes, my question to you is: What value of the adiabatic index, $~\gamma$, did you use when you evolved the bipolytrope? Did you use the same index for the entire configuration, or did you use a value in the envelope that is different from the value used in the core?

If you used $~\gamma = 6/5$ for the core, then the core should have uniform specific entropy and therefore would be (only) marginally stable against convection. Likewise, if you used $~\gamma = 2$ for the envelope, it should be (only) marginally stable against convection. Incidentally, these are the two separate values of the adiabatic index that I used when I used a free-energy analysis to examine stability.

NOTE: In yet another chapter of my book, I discuss the bipolytropic stability analysis that was performed by Murphy & Fiedler (1985b). They examined the stability of bipolytropes having the core/envelope polytropic indexes swapped, that is, their equilibrium models had, $~(n_c, n_e) = (1, 5)$. Their stability analysis was performed while assuming that the adiabatic index was $~\gamma = 5/3$ throughout the entire configuration. In a short subsection near the beginning of my chapter titled, "Aside Regarding Convectively Unstable Core," I have pointed out that the cores of these models should have all been convectively unstable, according to the Schwarzschild criterion. Please read this short subsection and let me know if you agree with my stated expectation for those models.

Other Approaches to Stability Analysis

After you send me feedback on the comments & questions I have presented, above, I have a good deal more to tell you about my analysis of the stability of our $~(n_c, n_e) = (5, 1)$ bipolytropic configurations. I am still a bit confused, but I'm pretty sure that the transition from stable to unstable configurations along the $~\mu_e/\mu_c = 1$ occurs at a value of $~\xi_i \approx 1.67$, rather than the value of $~\xi_i = 2.416$ that I obtained via the free-energy analysis. But, as I've said, there is a lot to tell you about this.

Other Suggestions

You asked what alternative models might be examined. It might be smarter to first evolve some simpler models, such as pressure-truncated polytropes — not bipolytropes. But we should discuss this after I read your feedback on the above.

May 4 (from Patrick)

 Hi Joel, Replying to your web page The models have xi_interface values of 2.39184 and 2.44016 For the evolutions, I used passive scalars to identify fluids as core or envelope and use effective polytropic indices and exponents based on the proportion of the fluid in a given cell that is core or envelope. There is slightly more to it than that as the “vacuum” is n = 1.5 and a third fluid type. When I do that, the stars are definitely convectively unstable. s.ps is a quick plot of entropy vs radial cell number. The fluid in the envelope is lower in entropy than fluid in the core. The entropy plotted is c_p ln( tau / rho) = R (gamma/(gamma-1)) ln (tau / rho) where R is the gas constant. The movie shows the meridional cross section of density through the star, the core envelope boundary corrugates pretty quickly in the run and then everything goes off the rails. There may be more than one problem but the convective instability is definitely one problem. cheers, Patrick

May 5 (following a phone conversation with Patrick)

Tying Expressions into H_Book Context

In our wiki-based chapter titled, "First Law of Thermodynamics," we have introduced the concept of an entropy tracer, $~\tau$. In the subsubsection of this chapter that is titled, "Substantiation," we show that an expression for the specific entropy of a fluid element is,

$~s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant} \, .$

In addition, from our wiki-based chapter titled, "Ideal Gas Equation of State," we find the relations,

 $~c_P - c_V$ $~=$ $~\frac{\Re}{\bar\mu}$ and, $~\gamma_g$ $~=$ $~\frac{c_P}{c_V}$ $~\Rightarrow$ $~c_P$ $~=$ $~\frac{\gamma_g}{(\gamma_g-1)} \biggl( \frac{\Re}{\bar\mu} \biggr) \, .$

Hence this expression for the entropy may be rewritten as,

$~s = \frac{\gamma_g}{\gamma_g-1} \biggl( \frac{\Re}{\bar\mu} \biggr) \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant} \, .$

Aside from the factor of $~({\bar\mu})^{-1}$ that appears on the RHS — more on this below — these are the expressions that Patrick has used to generate the s.ps plot, where the (unlabeled) ordinate is the normalized specific entropy, $~s/\Re$.

At the end of another subsubsection titled, "Initial Recognition," we also find a relevant expression, namely,

$~\tau \equiv (\rho\epsilon)^{1/\gamma_g} = \biggl[ \frac{P}{(\gamma_g - 1)} \biggr]^{1/\gamma_g} \, .$

Hence, ignoring the additive constant, in general we may write,

 $~\frac{s}{\Re/\bar{\mu}}$ $~=$ $~ \frac{1}{(\gamma_g-1)}\ln \biggl(\frac{\tau}{\rho}\biggr)^{\gamma_g}$ $~=$ $~ \frac{1}{(\gamma_g-1)}\ln \biggl[ \frac{P}{(\gamma_g-1)\rho^{\gamma_g}} \biggr] \, .$

Understanding the Step Function at the Core-Envelope Interface

Now, turning to the accompanying tabular summary of the Radial Profile of Various Physical Variables, we are able to determine how the specific entropy behaves throughout the core and, separately, throughout the envelope in $~(n_c, n_e) = (5, 1)$ bipolytropes.

CORE:   Throughout the core, we see that,

 $~P^*$ $~=$ $~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-3}$ and $~\rho^*$ $~=$ $~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-5/2}$ and $~\gamma_g$ $~=$ $~\frac{6}{5} \, .$

Hence, independent of the radial location, $~\xi$, throughout the core,

 $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}$ $~=$ $~ 5\ln (5) \, .$

ENVELOPE:   Throughout the envelope, we see that,

 $~P^*$ $~=$ $~\theta_i^6 [\phi(\eta)]^2$ and $~\rho^*$ $~=$ $~\biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)]$ and $~\gamma_g$ $~=$ $~2 \, .$

Hence, independent of the radial location, $~\eta$, throughout the envelope,

 $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}$ $~=$ $~ \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \theta_i^{-4} \biggr]$ $~=$ $~ \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr] \, .$

It is therefore clear that the core is a uniform specific-entropy sphere and the envelope is a uniform specific-entropy spherical shell, but in general the specific entropy of material in the envelope is different from the specific entropy of material in the core.

According to our free-energy based evaluation of the stability of bipolytropes having $~(n_c, n_e) = (5, 1)$, the marginally unstable model along the $~\mu_e/\mu_c = 1$ sequence has the following properties: $~(\xi_i, q, \nu, \rho_c/\bar\rho) = (2.41610822, 0.59520261, 0.68306067, 16.3788)$; the red arrow in the following diagram points to this model's position in the $~q-\nu$ parameter plane. In an effort to test whether or not this model does identify the transition from stable to unstable configurations along the $~\mu_e/\mu_c = 1$ sequence, Patrick picked a pair of models (highlighted in green in the following table) that straddle the location of the marginally unstable model, and followed the dynamical evolution of both models using a fully 3-D hydrodynamics code. In particular, for the pair of models that Patrick evolved, we find:

 Figure 2 $~\gamma_c = 6/5$     and     $~\gamma_e = 2$
 Figure 1 Initial Model ParametersforPatrick's Pair of Simulations(green background) $~\frac{\mu_e}{\mu_c}$ $~\xi_i$ $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}$ $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}$ 1 2.39184 8.04719 2.13422 1 2.41610822 8.04719 2.16080 1 2.44016 8.04719 2.18706 For more details, see the accompanying discussion titled, Free-Energy Stability Evaluation

These tabulated values of the normalized specific entropy in the core and, separately, in the envelope — also see the plot shown here on the right — appear to be consistent with Patrick's s.ps plot of specific entropy. In particular, this confirms that a step function should appear at the core-envelope interface and that the specific entropy of the envelope material should be lower than the specific entropy of the core material. Therefore, the Schwarzschild criterion is violated at the interface and we should not have been surprised to see convective motions develop — initially, only at the interface.

Can the Step Function be Flipped or Erased

Assume a Mean-Molecular-Weight Ratio of Unity

If we continue to examine equilibrium models that have $~\mu_e/\mu_c = 1$, is there a value of the interface radius, $~\xi_i$, for which the entropy step-function disappears and above which the step function flips? The answer appears to be, "Yes." It occurs along the equilibrium sequence where the normalized specific entropy has the same value in the core and as in the envelope. That is, it occurs when,

 Figure 3 $~\gamma_c = 6/5$     and     $~\gamma_e = 2$
 $~ \ln \biggl[ \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr]_\mathrm{smooth}$ $~=$ $~5\ln(5)$ $~ \Rightarrow ~~~ \biggl( 1 + \frac{\xi_i^2}{3}\biggr)_\mathrm{smooth}$ $~=$ $~5^{5 / 2}$ $~ \Rightarrow ~~~ [\xi_i ]_\mathrm{smooth}$ $~=$ $~\biggl[3 \biggl(5^{5 / 2} - 1 \biggr)\biggr]^{1 / 2} \approx 12.83375$ …

Prediction:   Any initial model with $~\mu_e/\mu_c = 1$ and $~\xi_i > [\xi_i]_\mathrm{smooth}$ will be stable against convection, but will be globally dynamically unstable.

Pick a Different Molecular-Weight Ratio

Caution:   This subsection contains a derivation that is based on an interpretation of the specific entropy normalization at the interface that may not be physically correct. It needs more thought … After more thought:  In order to determine whether the entropy of the envelope is greater than the entropy of the core it would be wise to always plot the specific entropy relative to $~\mu_c$; therefore, when the ratio $~(\tau/\rho)^{\gamma_g}$ is used to calculate the envelope entropy, $~[s/(\Re/\bar\mu)]_\mathrm{env}$, as above, the calculated answer should then be divided by $~(\mu_e/\mu_c)$ in order to obtain the proper normalization.

In principle, we can determine in a similar fashion the values of $~[\xi_i]_\mathrm{smooth}$ that are relevant to equilibrium model sequences having $~\mu_e/\mu_c < 1$. But in doing this, we must take into account that in most of our above derivations the mean-molecular-weight appears in the denominator of the (LHS) expression for the normalized specific entropy. More generally, the prescription for $~[\xi_i]_\mathrm{smooth}$ should come from the demand that,

 $~0$ $~=$ $~\frac{s_\mathrm{env}}{\Re} - \frac{s_\mathrm{core}}{\Re}$ $~\Rightarrow ~~~ 0$ $~=$ $~\frac{1}{\mu_c} \biggl\{\biggl( \frac{\mu_c}{\mu_e} \biggr) \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr]_\mathrm{smooth} - 5 \ln(5) \biggr\}$ $~\Rightarrow ~~~ \ln \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-2} \biggl( 1 + \frac{\xi_i^2}{3}\biggr)^2 \biggr]_\mathrm{smooth}$ $~=$ $~ \ln\biggl[ 5 \biggr]^{5(\mu_e/\mu_c)}$ $~\Rightarrow ~~~ \biggl( 1 + \frac{\xi_i^2}{3}\biggr)_\mathrm{smooth}$ $~=$ $~ 5^{2.5(\mu_e/\mu_c)} \biggl(\frac{\mu_e}{\mu_c}\biggr)$ $~\Rightarrow ~~~ [\xi_i]_\mathrm{smooth}$ $~=$ $~ 3^{1 / 2}\biggl[ 5^{2.5(\mu_e/\mu_c)} \biggl(\frac{\mu_e}{\mu_c}\biggr) -1\biggr]^{1 / 2} \, .$

Here are a few examples:

 Figure 4 $~\gamma_c = 6/5$     and     $~\gamma_e = 2$
 $~\frac{\mu_e}{\mu_c}$ $~[\xi_i]_\mathrm{smooth}$ 1 12.83375 1/2 2.86620 1/3 0.90754

Use the Same Ratio of Specific Heats Throughout

Let's examine the initial model's entropy profile under the assumption that the system is evolved with $~\gamma_g = 5/3$ throughout the bipolytrope. From the above analysis, in this case the relevant general expression for the specific entropy profile should be,

 $~\frac{s}{\Re/\bar{\mu}}$ $~=$ $~ \frac{3}{2} \ln \biggl[ \frac{3P}{2\rho^{5/3}} \biggr]$ $~=$ $~ \ln \biggl[ \biggl(\frac{3P}{2}\biggr)^{3 / 2} \rho^{-5/2} \biggr] \, .$

CORE:   Given that, throughout the core,

 $~P^*$ $~=$ $~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-3}$ and $~\rho^*$ $~=$ $~\biggl(1+ \frac{\xi^2}{3}\biggr)^{-5/2} \, ,$

we have,

 $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}$ $~=$ $~ \ln \biggl\{ \biggl(\frac{3}{2}\biggr)^{3 / 2} \biggl[ \biggl(1+ \frac{\xi^2}{3}\biggr)^{-9/2} \biggr] \biggl[ \biggl(1+ \frac{\xi^2}{3}\biggr)^{25/4} \biggr] \biggr\} = \frac{1}{4}\cdot\ln \biggl[ \biggl(\frac{3}{2}\biggr)^{6} \biggl(1+ \frac{\xi^2}{3}\biggr)^{7} \biggr] \, .$

ENVELOPE:   Given that, throughout the envelope,

 $~P^*$ $~=$ $~\theta_i^6 [\phi(\eta)]^2$ and $~\rho^*$ $~=$ $~\biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)] \, ,$

we have,

 $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}$ $~=$ $~ \ln \biggl\{ \biggl(\frac{3}{2}\biggr)^{3 / 2} \biggl[ \theta_i^6 [\phi(\eta)]^2 \biggr]^{3/2} \biggl[ \biggl(\frac{\mu_e}{\mu_c}\biggr)\theta_i^5 [\phi(\eta)] \biggr]^{-5/2} \biggr\}$ $~=$ $~ \frac{1}{2} \cdot \ln \biggl\{ \biggl(\frac{3}{2}\biggr)^{3 } \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-5} \theta_i^{-7} [\phi(\eta)] \biggr\}$ $~=$ $~ \frac{1}{4} \cdot \ln \biggl\{ \biggl(\frac{3}{2}\biggr)^{6 } \biggl(1+ \frac{\xi_i^2}{3}\biggr)^{7} \biggl(\frac{\mu_e}{\mu_c}\biggr)^{-10} [\phi(\eta)]^2 \biggr\} \, .$
 Figure 5

Notice that, because $~[\gamma_c = 5/3] > [(n_c + 1)/n_c = 6/5]$, the specific entropy increases with radius throughout the core, so according to the Schwarzschild criterion the core is stable against convection. However, because $~[\gamma_e = 5/3] < [(n_e + 1)/n_e = 2]$, the specific entropy decreases with radius throughout the envelope, so according to the Schwarzschild criterion the envelope must be unstable toward convection.

Note: Murphy & Fiedler (1985b) have examined radial oscillation modes in bipolytropic configurations that have a flipped set of indexes — that is, they studied equilibrium structures having $~(n_c, n_e) = (1, 5)$ — assuming, as we have examined here, that oscillations in both the core and the envelope are governed by $~\gamma_g = 5/3$. The chapter of this H_Book in which we discuss the detailed analysis presented by Murphy & Fiedler, we have inserted a short subsection titled, Aside Regarding Convectively Unstable Core, where we point out that the cores of the Murphy & Fiedler models should be convectively unstable whereas their envelopes should be stable against convection.

Eigenvectors of Marginally Unstable Models

In preparation for our examination of the relative stability of bipolytropic structures having $~(n_c, n_e) = (5, 1)$ — via numerical integration of the Linear-Adiabatic Wave Equation (LAWE) — we have demonstrated that we understand technically how to solve this type of eigenvalue problem by quantitatively reproducing related analyses that have been previously published by other groups.

Good Comparisons With Previously Published Studies

•  Schwarzschild (1941) Taff & Van Horn (1974) Murphy & Fiedler (1985b)

Schwarzschild (1941) numerically integrated the LAWE for isolated $~n=3$ polytropic spheres to find eigenvectors (i.e., the spatially discrete eigenfunction and corresponding eigenfrequency) for five separate oscillation modes (the fundamental mode, plus the 1st, 2nd, 3rd, and 4th overtones) for models having four different adopted adiabatic indexes $~\gamma_g = \tfrac{4}{3}, \tfrac{10}{7}, \tfrac{20}{13}, \tfrac{5}{3})$. In an accompanying chapter of this H_Book, we demonstrate that we have been able to reproduce in detail Schwarzschild's results for the specific case of $~\gamma_g = \tfrac{20}{13}$.

• Taff & Van Horn (1974) examined radial oscillations in pressure-truncated isothermal spheres, assuming that the configurations remain isothermal during the oscillations. For models having nine different truncation radii — chosen to straddle the position along the equilibrium sequence where the marginally unstable model was expected to arise — they determined and published the fundamental-mode eigenvalues. For three of these models they also determined and published eigenvalues for the first harmonic mode of oscillation; the radial eigenfunctions associated with both the fundamental mode and the first harmonic mode of these three models also has been displayed in their Figure 1. In a separate accompanying discussion, we demonstrate that we have been able to reproduce in detail the subset of eigenfunctions and associated eigenvalues that have been previously published by Taff & Van Horn (1974).
• In their published study of bipolytropes having $~(n_c, n_e) = (1,5)$ with $~\mu_e/\mu_c = 1$, Murphy & Fiedler (1985b) integrated a coupled pair of LAWEs — one for the core and another for the envelope — to determine the eigenfunctions and corresponding eigenvalues of various radial modes of oscillation in more than a dozen different equilibrium models, assuming that during the oscillations, $~\gamma_g = 5/3$ throughout both the core and the envelope. In an accompanying chapter of this H_Book titled, Review of the BiPolytrope Stability Analysis by Murphy & Fiedler (1985b), we show that we have been able to duplicate in quantitative detail the eigenvectors associated with their equilibrium Models 10 and 17.

Building upon this set of successful comparisons with stability analyses published by other groups, we have carried out numerical integrations of the relevant LAWE to identify the eigenvectors associated with the fundamental-mode of radial oscillation in pressure-truncated, $~n = 5$ polytropic configurations. Details of this analysis are provided in yet another chapter of this H_Book. The following animation sequence illustrates the results of this analysis. As far as we have been able to determine, an analysis of this type has not previously been conducted for pressure-truncated, $~n = 5$ polytropes.

Figure 6

In a subsection of this separate chapter, we have also shown that, at the maximum-mass turning point along the pressure-truncated $~n=5$ equilibrium sequence — identified by the green circular marker in the left-hand panel of this animation — the fundamental-mode eigenfrequency is precisely zero and the associated eigenfunction is described exactly by the formula for a parabola.

Our Numerical Analysis of Bipolytropes Having (nc, ne) = (5, 1)

In an accompanying discussion we have shown that we can integrate the linear adiabatic wave equation (LAWE) — that is, we effectively have been able to solve the eigenvalue problem — to obtain the eigenvector associated with marginally unstable models along equilibrium sequences for bipolytropes having $~(n_c, n_e) = (5, 1)$. The marginally unstable models that have been identified via this more rigorous approach — see the orange triangles in the following figure — fall at different points along each equilibrium sequence than the points that were identified via our free-energy analysis — see the red-dashed demarcation curve. Assuming that the algorithm that we developed to integrate the LAWE was basically error-free, we trust the model identifications generated via this more rigorous technique.

Our analysis indicates that, along the $~\mu_e/\mu_c = 1$ equilibrium sequence, the core-envelope interface of the marginally unstable model is located at $~\xi_i = 1.6686460157$; in the following figure, the red arrow points to this location along that sequence. In the brief table that accompanies the figure, we have listed values for the dimensionless specific entropy in the core and, separately, in the envelope, along with the interface location, $~\xi_i$.

 Figure 8 $~\gamma_c = 6/5$     and     $~\gamma_e = 2$
 Figure 7 Initial Model ParametersforLAWE-DeterminedMarginally Unstable Model $~\frac{\mu_e}{\mu_c}$ $~\xi_i$ $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{core}$ $~\frac{s}{\Re/\bar{\mu}}\biggr|_\mathrm{env}$ 1 1.6686460157 8.04719 1.31310 For more details, see the accompanying discussion titled, Eigenvectors from Solution of LAWE

Summary

For three separate equilibrium sequences — specifically, for the cases of $~\mu_e/\mu_c = 1, \tfrac{1}{2}, \tfrac{1}{3}$ — the table on the left, below, identifies the location $~(\xi_i)$ along the sequence where the model is marginally unstable toward a global, dynamical instability as estimated by the our free-energy analysis (column 2) and as determined by our numerical integration of the coupled pair of governing LAWEs (column 6). For each identified model, the corresponding values of $~q, \nu$ and $~\rho_c/\bar\rho$ are also provided. Additional details can be found in our accompanying discussion of, respectively, our free-energy analysis and our numerical solution of the relevant eigenvalue problem.

 Marginally Unstable (dynamically)assuming $~\gamma_c = \tfrac{6}{5}$ and $~\gamma_e = 2$ $~\frac{\mu_e}{\mu_c}$ Determined viaFree-Energy Analysis Determined viaEigenvector Analysis $~\xi_i$ $~q$ $~\nu$ $~\frac{\rho_c}{\bar\rho}$ $~\xi_i$ $~q$ $~\nu$ $~\frac{\rho_c}{\bar\rho}$ 1 2.416 0.595 0.683 16.4 1.66865 0.539 0.498 8.42 $~\tfrac{1}{2}$ 4.186 0.328 0.701 354. 2.27926 0.185 0.234 63.3 $~\tfrac{1}{3}$ 8.548 0.099 0.479 63,000. 2.58201 0.176 0.218 230.
 Figure 9 $~\gamma_c = 6/5$     and     $~\gamma_e = 2$

In the figure shown here on the right, the orange-dashed curve shows how $~\xi_i$ (table column 6) varies with $~\mu_e/\mu_c$ (table column 1) for marginally unstable models as determined by our numerical eigenvector analysis. The square of the fundamental-mode eigenfrequency is negative for all models above this orange-dashed curve, indicating that they should be globally dynamically unstable. The solid blue curve shows how $~[\xi_i]_\mathrm{smooth}$ varies with $~\mu_e/\mu_c$, as derived above. Models above and to the left of this blue curve should be stable against convection at the core-envelope interface.

May 11 (from Joel)

 Patrick, Following our phone conversation earlier this week, I have added a number of paragraphs, derivations, and tables/figures to my VisTrails wiki page — see the above subsection dated, May 5. Here is a quick summary, assuming evolutions are conducted with $~(\gamma_c, \gamma_e) = (6/5, 2)$: I completely agree with your expression for the specific entropy, except I believe that the gas constant needs to be divided by the mean-molecular weight $~(\Re/\bar\mu)$. I concur with your s.ps plot. In particular, for the bipolytropic model having $~\mu_e/\mu_c = 1$ and $~\xi_i = 2.4161$ (this is the model that my free-energy-based analysis identifies as the marginally unstable model), I get $~[s/(\Re/\bar\mu)]_\mathrm{core} = 8.04719$ and $~[s/(\Re/\bar\mu)]_\mathrm{env} = 2.16080$. See Figure 2 and the table accompanying my Figure 1. The size of the entropy jump **does** depend on the location of the interface. If we stay on the $~\mu_e/\mu_c = 1$ sequence, the entropy of the envelope will equal the entropy of the core if $~\xi_i = 12.83375$. See Figure 3. (I now label this critical value, $~[\xi_i]_\mathrm{smooth}$.) At values of $~\xi_i > [\xi_i]_\mathrm{smooth}$, the envelope's entropy will be larger than the core's entropy. If you choose a sequence for which $~\mu_e/\mu_c = 0.5$, then my analysis says that $~[\xi_i]_\mathrm{smooth} = 2.86620$. See Figure 4. If you set $~\mu_e/\mu_c = 1$ but assume evolutions are conducted with $~\gamma_c = \gamma_e = 5/3$: The entropy discontinuity disappears at the interface, but the entropy is no longer constant throughout the core or the envelope. See Figure 5. Moving radially outward from the center, the entropy steadily increases through the core (convectively stable) then it steadily drops through the envelope (convectively unstable). Finally, we should consider the following before embarking on more 3D simulations: Along any given equilibrium sequence, a free-energy analysis only provides an **approximate** value of $~\xi_i$ at which a dynamical instability sets in. The **precise** location can be identified by solving the relevant eigenvalue problem and searching for the model at which the fundamental-mode oscillation frequency goes to zero. I have taught myself how to solve this type of eigenvalue problem; as described briefly above — accompanied by a few animated-gifs — in other chapters of my wiki-book, I show that I can quantitatively match related, previously published analyses. I have tentative eigenvalue-analysis results for $~(n_c, n_e) = (5, 1)$ bipolytropes; all marginally unstable models arise **sooner** along each sequence — that is at smaller values of $~\xi_i$ (see Figure 7) — than was predicted by the free-energy analysis. Other things to discuss: Instead of bipolytropic structures, it might make more sense to test (with 3D code) the stability of pressure-truncated, n = 5 polytropes. See Figure 6. Analytic solutions of a limited — but physically quite interesting — sample of eigenvalue problems. I am very interested in discussing all of this with you, if you are interested and when you have the time. Just let me know. Cheers, Joel

May 14 (from Patrick)

 Hi Joel, There are some movies and plots on line at http://patrickmotl.net if you follow the links for bipolytropic stars with n_core = 5 and n_envelope = 1

Patrick and Joel also had a long phone conversation on this date, 14 May 2019.

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