SSC/Stability/BiPolytropes

Organizational Index

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.

In Search of Marginally Unstable (n_c,n_e) = (5,1) Bipolytropes

Overview

file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined2 Figure 1:  Equilibrium Sequences of Pressure-Truncated Polytropes

We expect the content of this chapter — which examines the relative stability of bipolytropes — to parallel in many ways the content of an accompanying chapter in which we have successfully analyzed the relative stability of pressure-truncated polytopes. Figure 1, shown here on the right, has been copied from a closely related discussion. The curves show the mass-radius relationship for pressure-truncated model sequences having a variety of polytropic indexes, as labeled, over the range $1\le n\le 6$. (Another version of this figure includes the isothermal sequence.) On each sequence for which $n\ge 3$, the green filled circle identifies the model with the largest mass. We have shown analytically that the oscillation frequency of the fundamental-mode of radial oscillation is precisely zero^† for each one of these maximum-mass models. As a consequence, we know that each green circular marker identifies the point along its associated sequence that separates dynamically stable (larger radii) from dynamically unstable (smaller radii) models.

^†In each case, the fundamental-mode oscillation frequency is precisely zero if, and only if, the adiabatic index governing expansions/contractions is related to the underlying structural polytropic index via the relation, ${\gamma }_g=(n+1)/n$, and if a constant surface-pressure boundary condition is imposed.

Key Realization: Along sequences of pressure-truncated polytropes, the maximum-mass models identify precisely where the onset of dynamical instability occurs.

---

In another accompanying chapter, we have used purely analytic techniques to construct equilibrium sequences of spherically symmetric bipolytropes that have, $(n_c,n_e)=(5,1)$. For a given choice of ${\mu }_e/{\mu }_c$ — the ratio of the mean-molecular weight of envelope material to the mean-molecular weight of material in the core — a physically relevant sequence of models can be constructed by steadily increasing the value of the dimensionless radius at the core/envelope interface, ${\xi }_i$, from zero to infinity. Figure 2, whose content is essentially the same as Figure 1 of this separate chapter, shows how the fractional core mass, $\nu \equiv M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/M_{\mathrm{t}\mathrm{o}\mathrm{t}}$, varies with the fractional core radius, $q\equiv r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/R$, along sequences having seven different values of ${\mu }_e/{\mu }_c$, as labeled: 1 (black), ½ (dark blue), 0.345 (brown), ⅓ (dark green), 0.316943 (purple), 0.309 (orange), and ¼ (light blue).

When modeling bipolytropes, the default expectation is that an increase in ${\xi }_i$ along a given sequence will correspond to an increase in the relative size — both the radius and the mass — of the core. This expectation is realized along the Figure 2 sequences that have the largest mean-molecular weight ratios: ${\mu }_e/{\mu }_c$ = 1 and ½. But the behavior is different along the other five illustrated sequences. For sufficiently large ${\xi }_i$, the relative radius of the core begins to decrease; along each sequence, a solid purple circular marker identifies the location of this turning point in radius. Furthermore, along sequences for which ${\mu }_e/{\mu }_c<\frac{1}{3}$, eventually the fractional mass of the core reaches a maximum and, thereafter, decreases even as the value of ${\xi }_i$ continues to increase. In Figure 2, a solid green circular marker identifies the location of this maximum mass turning point along each of these sequences; the analytically determined values of ${\xi }_i,q$ and $\nu$ that are associated with each of these turning points are provided in the table adjacent to Figure 2. (Additional properties of these equilibrium sequences are discussed in yet another accompanying chapter.)

The principal question is: Along bipolytropic sequences, are maximum-mass models associated with the onset of dynamical instabilities?

Figure 2: Equilibrium Sequences of Bipolytropes with $(n_c,n_e)=(5,1)$ and Various ${\mu }_e/{\mu }_c$	Analytically Determined Parameters† for Models that have the Maximum Fractional Core Mass (solid green circular markers) Along Various Equilibrium Sequences
	$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R}$	$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$
	$\frac{1}{3}$	$\mathrm{\infty }$	0.0	$\frac{2}{\pi }$
	0.33	24.00496	0.038378833	0.52024552
	0.316943	10.744571	0.068652714	0.382383875
	0.31	9.014959766	0.0755022550	0.3372170064
	0.3090	8.8301772	0.076265588	0.331475715
	$\frac{1}{4}$	4.9379256	0.084824137	0.139370157
	†Additional model parameters can be found here.

Planned Approach(es)

In an effort to answer the principal question posed above, we have pursued each stability-analysis approach described in the introductory section of 📚 G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974, Astron. & Astrophy., Vol. 31, pp. 391 - 404).

"Three different approaches are used in the study of the hydrodynamical stability of stars and other gravitating objects …"   "The first approach is based on the use of the equations of small oscillations. In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations. The solutions consist of a set of eigenfrequencies and eigenfunctions." Second, one can derive "a variational principle from the equations of small oscillations … With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations. In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated …" The third approach is what we usually refer to as a free-energy — and associated virial theorem — analysis. "When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient. The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability." "If one wants to know from a stability analysis the answer to only one question — whether the model is stable or not — then the most straightforward procedure is to use the third, static method … For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation." "Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point."

— Drawn from pp. 391 - 392 of 📚 Bisnovatyi-Kogan & Blinnikov (1974)

Supplemental Chapters

1. Contains Displacement Functions Summary
2. Earlier Planned Approach
3. Headscratching
4. Succinct Discussions
5. 51Models

Equilibrium Models

The original text contained in this section has been commented out.

Third Approach: Free-Energy Analysis

Old Material

Virial Stability01

The following set of menu tiles include links to chapters where this approach has been applied to: (a) uniform-density configurations, (b) pressure-truncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressure-truncated n = 5 configurations, and (e) bipolytropes having $(n_c,n_e)=(1,5)$.

One menu tile, below, links to a chapter in which an analytic (exact) demonstration of the variational principle's utility is provided in the context pressure-truncated n = 5 polytropes.

Ideally we would like to answer the just-stated "principal question" using purely analytic techniques. But, to date, we have been unable to fully address the relevant issues analytically, even in what would be expected to be the simplest case:   bipolytropic models that have $(n_c,n_e)=(0,0)$. Instead, we will streamline the investigation a bit and proceed — at least initially — using a blend of techniques. We will investigate the relative stability of bipolytropic models having $(n_c,n_e)=(5,1)$ whose equilibrium structures are completely defined analytically; then the eigenvectors describing radial modes of oscillation will be determined, one at a time, by solving the relevant LAWE(s) numerically. We are optimistic that this can be successfully accomplished because we have had experience numerically integrating the LAWE that governs the oscillation of:

• Isolated n = 3 polytropes — including a quantitative comparison against the published work of 📚 M. Schwarzschild (1941, ApJ, Vol. 94, pp. 245 - 252);
• Pressure-truncated isothermal spheres — including a quantitative comparison against the published analysis of 📚 L. G. Taff, & H. M. van Horn (1974, MNRAS, Vol. 168, pp. 427 - 432); and
• Pressure-truncated n = 5 polytropes.

A key reference throughout this investigation will be the paper by 📚 J. O. Murphy & R. Fiedler (1985b, Proc. Astron. Soc. Australia, Vol. 6, no. 2, pp. 222 - 226). They studied Radial Pulsations and Vibrational Stability of a Sequence of Two Zone Polytropic Stellar Models. Specifically, their underlying equilibrium models were bipolytropes that have $(n_c,n_e)=(1,5)$. In an accompanying chapter, we describe in detail how 📚 Murphy & Fiedler (1985b) obtained these equilibrium bipolytropic structures and detail some of their equilibrium properties.

Here are the steps we initially plan to take:

• Governing LAWEs:
  • Identify the relevant LAWEs that govern the behavior of radial oscillations in the $n_c=5$ core and, separately, in the $n_e=1$ envelope. Check these LAWE specifications against the published work of 📚 Murphy & Fiedler (1985b).
  • Determine the matching conditions that must be satisfied across the core/envelope interface. Be sure to take into account the critical interface jump conditions spelled out by 📚 P. Ledoux, & Th. Walraven (1958, Handbuch der Physik, Vol. 51, pp. 353 - 604), as we have already discussed in the context of an analysis of radial oscillations in zero-zero bipolytropes.
• Determine what surface boundary condition should be imposed on physically relevant LAWE solutions, i.e., on the physically relevant radial-oscillation eigenvectors.
• Initial Analysis:
  • Choose a maximum-mass model along the bipolytropic sequence that has, for example, ${\mu }_e/{\mu }_c=1/4$. Hopefully, we will be able to identify precisely (analytically) where this maximum-mass model lies along the sequence. Yes! Our earlier analysis does provide an analytic prescription of the model that sits at the maximum-mass location along the chosen sequence.
  • Solve the relevant eigenvalue problem for this specific model, initially for $({\gamma }_c,{\gamma }_e)=(6/5,2)$ and initially for the fundamental mode of oscillation.

New Material

Drawing from our accompanying detailed discussion — see also an accompanying summary — the normalized free-energy associated with each of our spherically symmetric, bipolytropic configurations is given by the expression,

$\mathfrak{𝔤}\equiv \frac{\mathfrak{𝔊}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$

$=$

$-\left(\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{\chi }\right)\mathfrak{𝔴}+\left[\frac{2}{3({\gamma }_c-1)}\right](\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_c}{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+[\frac{2}{3({\gamma }_e-1)}\left]\right(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_e}{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}},$

where: ${\chi }_{\mathrm{e}\mathrm{q}}$ is the dimensionless radius of the configuration when it is in equilibrium; ${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$ and ${\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$ are the appropriately normalized thermal energy content of the $n=5$ core and of the $n=1$ envelope, respectively, in the configuration's equilibrium state; and $\mathfrak{𝔴}$ is the absolute value of the normalized total gravitational potential energy of the equilibrium configuration. For every $(n_c,n_e)=(5,1)$ bipolytropic model, the values of these four terms can be obtained via our derived analytic expressions. And the value of $\mathfrak{𝔤}$ in an equilibrium state is obtained by setting the configuration's dimensionless radius, $\chi$, equal to ${\chi }_{\mathrm{e}\mathrm{q}}$.

This expression for $\mathfrak{𝔤}$ has been written in such a way that we can readily assess how the free energy varies while the configuration undergoes homologous (${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}},{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}},\mathfrak{𝔴}$ all held fixed) expansion/contraction $\chi \gtrless {\chi }_{\mathrm{e}\mathrm{q}}$ about its equilibrium state. Specifically the first variation is,

$\frac{d\mathfrak{𝔤}}{d\chi }$

$=$

$+\left(\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{{\chi }^2}\right)\mathfrak{𝔴}-\frac{2}{{\chi }_{\mathrm{e}\mathrm{q}}}\left[\right(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{2-3{\gamma }_c}{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{2-3{\gamma }_e}{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}\right];$

and the second variation is,

$\frac{d^2\mathfrak{𝔤}}{d{\chi }^2}$

$=$

$-2\left(\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{{\chi }^3}\right)\mathfrak{𝔴}-\frac{2}{{\chi }_{\mathrm{e}\mathrm{q}}^2}[(2-3{\gamma }_c)(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{1-3{\gamma }_c}{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(2-3{\gamma }_e)\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{1-3{\gamma }_e}{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}\right].$

The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability.

Virial Analysis

In an accompanying chapter we have examined the viral stability of $(n_c,n_e)=(5,1)$ bipolytropes — see also an associated discussion of the free-energy of these configurations. We present, here, an overview of this separate discussion in an effort to provide a broader perspective to the focus of this current chapter.

Various Energy Expressions

Drawing from our accompanying detailed discussion, we have,

$\mathfrak{𝔊}$	$=$	$W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+{\mathfrak{𝔖}}_A{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}+{\mathfrak{𝔖}}_A{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}$
	$=$	$W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}{|}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+\left[\frac{2}{3({\gamma }_c-1)}\right]S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}{|}_{\mathrm{e}\mathrm{n}\mathrm{v}}+\left[\frac{2}{3({\gamma }_e-1)}\right]S_{\mathrm{e}\mathrm{n}\mathrm{v}}.$

Recognizing that,

$q$	$\equiv$	$\frac{r_i}{R}=\frac{{\eta }_i}{{\eta }_s},$
$\nu$	$\equiv$	$\frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}},$
$\ell$	$\equiv$	$\frac{\xi }{\sqrt{3}},$
${\chi }_{\mathrm{e}\mathrm{q}}$	$=$	$\left(\frac{\pi }{2^3\cdot 3^6}{)}^{1/2}\right(\frac{\nu }{q^3}{)}^2(1+{\ell }_i^2{)}^3(\frac{q}{{\ell }_i}{)}^5,$

we have,

$-\chi [\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$\left[\right(\frac{3}{2^4}\left)\right(\frac{q}{{\ell }_i}{)}^5\left(\frac{\nu }{q^3}{)}^2\right(1+{\ell }_i^2{)}^3{]}_{\mathrm{e}\mathrm{q}}[{\ell }_i({\ell }_i^4-\frac{8}{3}{\ell }_i^2-1)({\ell }_i^2+1{)}^{-3}+{\tan }^{-1}{\ell }_i]$
$\Rightarrow [\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$-\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{\chi }\left(\frac{3^8}{2^5\pi }{)}^{1/2}\right[{\ell }_i({\ell }_i^4-\frac{8}{3}{\ell }_i^2-1)({\ell }_i^2+1{)}^{-3}+{\tan }^{-1}{\ell }_i].$

Next, given that,

$b_{\eta }={\eta }_s$

and

$b_{\eta }q={\eta }_i=3\left(\frac{{\mu }_e}{{\mu }_c}\right){\ell }_i(1+{\ell }_i^2{)}^{-1},$

we have,

$[\frac{W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{]}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$=$	$-\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{\chi }\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}\right(\frac{1}{2^3\pi }{)}^{1/2}{A}^2[6b_{\eta }x-3\sin [2(b_{\eta }x-B)]-4b_{\eta }x{\sin }^{}(b_{\eta }x-B){]}_q^1$
	$=$	$-\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{\chi }\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}\right(\frac{1}{2^3\pi }{)}^{1/2}{A}^2\left\{\right[6{\eta }_s-3\sin [2({\eta }_s-B)]-4{\eta }_s{\sin }^{}({\eta }_s-B)]-[6{\eta }_i-3\sin [2({\eta }_i-B)]-4{\eta }_i{\sin }^{}({\eta }_i-B)\left]\right\}.$

Also, given that,

$q{a}_{\xi }^{1/2}$

$=$

$\frac{{\xi }_i}{\sqrt{3}},$

we have,

$(\frac{{\mathfrak{𝔖}}_A}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$=$	$\frac{2}{3({\gamma }_c-1)}\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_c}\right\{\left(\frac{3^8}{2^7\pi }{)}^{1/2}\right[{\tan }^{-1}[a_{\xi }^{1/2}q]-a_{\xi }^{1/2}q\frac{(1-a_{\xi }{q}^2)}{(1+a_{\xi }{q}^2{)}^2}\left]\right\}$
	$=$	$\frac{2}{3({\gamma }_c-1)}\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_c}\right\{\left(\frac{3^8}{2^7\pi }{)}^{1/2}\right[{\tan }^{-1}\left(\frac{{\xi }_i}{\sqrt{3}}\right)-\left(\frac{{\xi }_i}{\sqrt{3}}\right)\frac{(1-{\xi }_i^2/3)}{(1+{\xi }_i^2/3{)}^2}\left]\right\}$

Finally, then, we can write,

$(\frac{{\mathfrak{𝔖}}_A}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}{)}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$=$		$\frac{2}{3({\gamma }_e-1)}\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_e}\right\{\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\right(\frac{3^2}{2^5\pi }{)}^{1/2}[2b_{\eta }x-\sin [2(b_{\eta }x-B)]{]}_q^1\}$
	$=$		$\frac{2}{3({\gamma }_e-1)}\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_e}\right(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\left(\frac{3^2}{2^5\pi }{)}^{1/2}\right\{[2{\eta }_s-\sin [2({\eta }_s-B)]]-[2{\eta }_i-\sin [2({\eta }_i-B)]]\}.$

Free Energy and Its Derivatives

We can now rewrite the free-energy expression in the form,

$\mathfrak{𝔤}\equiv \frac{\mathfrak{𝔊}}{E_{\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}}}$

$=$

$-\left(\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{\chi }\right)\mathfrak{𝔴}+\left[\frac{2}{3({\gamma }_c-1)}\right](\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_c}{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+[\frac{2}{3({\gamma }_e-1)}\left]\right(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{3-3{\gamma }_e}{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}},$

where,

${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	$\equiv$	$\left(\frac{3^8}{2^7\pi }{)}^{1/2}\right[{\tan }^{-1}\left(\frac{{\xi }_i}{\sqrt{3}}\right)-\left(\frac{{\xi }_i}{\sqrt{3}}\right)\frac{(1-{\xi }_i^2/3)}{(1+{\xi }_i^2/3{)}^2}],$
${\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$\equiv$	$\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}{A}^2\right(\frac{3^2}{2^5\pi }{)}^{1/2}\left\{\right[2{\eta }_s-\sin [2({\eta }_s-B)]]-[2{\eta }_i-\sin [2({\eta }_i-B)]\left]\right\},$
$\mathfrak{𝔴}$	$\equiv$	$\left(\frac{3^8}{2^5\pi }{)}^{1/2}\right[{\ell }_i({\ell }_i^4-\frac{8}{3}{\ell }_i^2-1)({\ell }_i^2+1{)}^{-3}+{\tan }^{-1}{\ell }_i]$
		$+\left(\frac{{\mu }_e}{{\mu }_c}{)}^{-3}\right(\frac{1}{2^3\pi }{)}^{1/2}{A}^2\left\{\right[6{\eta }_s-3\sin [2({\eta }_s-B)]-4{\eta }_s{\sin }^{}({\eta }_s-B)]-[6{\eta }_i-3\sin [2({\eta }_i-B)]-4{\eta }_i{\sin }^{}({\eta }_i-B)\left]\right\}.$

The first derivative is,

$\frac{d\mathfrak{𝔤}}{d\chi }$

$=$

$+\left(\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{{\chi }^2}\right)\mathfrak{𝔴}-\frac{2}{{\chi }_{\mathrm{e}\mathrm{q}}}\left[\right(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{2-3{\gamma }_c}{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{2-3{\gamma }_e}{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}\right].$

And the second derivative is,

$\frac{d^2\mathfrak{𝔤}}{d{\chi }^2}$

$=$

$-2\left(\frac{{\chi }_{\mathrm{e}\mathrm{q}}}{{\chi }^3}\right)\mathfrak{𝔴}-\frac{2}{{\chi }_{\mathrm{e}\mathrm{q}}^2}[(2-3{\gamma }_c)(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{1-3{\gamma }_c}{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(2-3{\gamma }_e)\left(\frac{\chi }{{\chi }_{\mathrm{e}\mathrm{q}}}{)}^{1-3{\gamma }_e}{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}\right].$

What to Expect for Equilibrium Configurations

In equilibrium we should set $d\mathfrak{𝔤}/d\chi =0$ and $\chi ={\chi }_{\mathrm{e}\mathrm{q}}$. In which case we should expect to find that,

$0$

$=$

$\frac{1}{{\chi }_{\mathrm{e}\mathrm{q}}}[\mathfrak{𝔴}-2({\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}})].$

In addition, an evaluation of the second derivative should give,

$\frac{{\chi }_{\mathrm{e}\mathrm{q}}^2}{2}\cdot \frac{d^2\mathfrak{𝔤}}{d{\chi }^2}$

$=$

$-\mathfrak{𝔴}-[(2-3{\gamma }_c){\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(2-3{\gamma }_e){\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}].$

Then, the transition from stable to unstable configurations occurs when $d^2\mathfrak{𝔤}/d{\chi }^2=0$, that is, when,

$\mathfrak{𝔴}$	$=$	$-[(2-3{\gamma }_c){\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(2-3{\gamma }_e){\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}]$
$\Rightarrow 0$	$=$	$(4-3{\gamma }_c){\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(4-3{\gamma }_e){\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$
$\Rightarrow \frac{{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}}$	$=$	$-\frac{(4-3{\gamma }_e)}{(4-3{\gamma }_c)}.$

For example, if we set ${\gamma }_c=6/5$ and ${\gamma }_e=2$, this implies,

$\frac{{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}}$

$=$

$\frac{2}{(4-18/5)}=5.$

Free-Energy Stability Evaluation

Here we pull together excerpts from several different H_Book Chapters in which we have presented, from several different perspectives, an analysis of the free-energy of bipolytropes.

• In one chapter, using purely analytic techniques, we have derived expressions that detail the structural properties of bipolytropes having $(n_c,n_e)=(5,1)$. Among these are analytic expressions for various terms that make up the free-energy expression: ${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$, ${\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$, ${\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$, ${\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$, and $P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$. Equilibrium model sequences are defined by fixing the ratio, ${\mu }_e/{\mu }_c$, then varying the radial location, $r_i$, of the core-envelope interface; note that the volume of the core is, then, $V_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}\equiv 4\pi r_i^3/3$.

• [Virial Equilibrium] In a subsection of this same chapter titled, Equilibrium Condition: Global, we have shown that a statement of virial equilibrium — obtained by setting the first derivative of the free-energy expression to zero — is,

  $(2S+W{)}_{\mathrm{t}\mathrm{o}\mathrm{t}}=2(S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+S_{\mathrm{e}\mathrm{n}\mathrm{v}})+(W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{e}\mathrm{n}\mathrm{v}})$

  $=$

  $0.$

  In another subsection of this same chapter titled, Equilibrium Condition: In Parts, we showed that for each bipolytropic equilibrium structure, the statements

  $2S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

  and

  $2S_{\mathrm{e}\mathrm{n}\mathrm{v}}+W_{\mathrm{e}\mathrm{n}\mathrm{v}}=-3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}},$

  also hold separately. Therefore, for every equilibrium configuration we should expect the CASE1 expression (see Table XXX) to precisely sum to unity.

• [Marginally Unstable Model] Near the bottom of this same chapter, in a subsection titled, Stability Condition, we point out that the model along each sequence that is marginally (dynamically) unstable — obtained setting the second derivative of the free-energy expression to zero — is identified by the configuration for which,

  $2({\gamma }_e-{\gamma }_c){\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+({\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+{\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}})({\gamma }_e-\frac{4}{3})$

  $=$

  $0.$

  Therefore, along each equilibrium sequence, the marginally unstable model can be identified by the configuration for which the CASE2 expression (see Table XXX) precisely sums to zero. Immediately above, in a subsection titled, What to Expect for Equilibrium Configurations, we have shown that this same marginally unstable model can be identified by the configuration for which the CASE3 energy ratio, ${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$, has a value that is precisely 5. And, thirdly, as highlighted in our accompanying Tabular Overview, this same marginally unstable model can be identified by the configuration for which the CASE4 expression precisely sums to zero.

Table XXX: Properties of Marginally Unstable Bipolytropes Having $(n_c,n_e)=(5,1)$ and $({\gamma }_c,{\gamma }_e)=(\frac{6}{5},2)$ Determined from Free-Energy Arguments
$\frac{{\mu }_e}{{\mu }_c}$	${\xi }_i$	$R_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}^{*}$	$q\equiv \frac{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{R_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}}}$	$\nu \equiv \frac{M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{M_{\mathrm{t}\mathrm{o}\mathrm{t}}}$	$\frac{{\rho }_c}{\overline{\rho }}$	$P_i$	${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	${\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	${\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	${\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}}$	$3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$	CASE1	CASE2	CASE3	CASE4
1	2.41610822	2.8049	0.59520261	0.68306067	16.3788	0.039116848	4.446748782	- 6.606135366	0.889349762	- 4.066061722	2.287362198	1	0	5.0000000	2.1 × 10-8
$\frac{1}{2}$	4.1853093	8.8058	0.328419479	0.70131896	354.089	0.003126324	5.76978580	- 10.58931853	1.153956968	- 3.258165567	0.95025163	1	6.3 × 10-8	5.0000002	0
0.345	7.64325	44.116	0.119714454	0.52700045	2.85 × 104	0.000116533	6.230343527	- 12.24495934	1.1246068658	- 2.707865028	0.215727713	1	6.4 × 10-8	5.0000002	0
$\frac{1}{3}$	8.548103	59.643	0.099032423	0.47901529	6.30 × 104	6.1337 × 10-5	6.261548334	- 12.36425897	1.252309682	- 2.663457063	0.158837699	1	0	4.9999999	6.0 × 10-8
0.316943	10.7441565	108.14	0.068655205	0.38238387	2.93 × 105	1.6252 × 10-5	6.301810768	- 12.52005323	1.260362204	- 2.604292714	0.083568307	1	0	4.9999998	2.0 × 10-7
0.309	12.77156	166.06	0.053145011	0.31696879	8.70 × 105	5.8905 × 10-6	6.318902171	- 12.58692884	1.26378042	- 2.57843634	0.050875500	1	1.9 × 10-8	5.0000001	0
CASE1 $\equiv$ $\frac{1}{6P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}[(2{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+{\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}})-(2{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+{\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}})]$ CASE2 $\equiv$ $2({\gamma }_e-{\gamma }_c){\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+({\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+{\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}})({\gamma }_e-\frac{4}{3})$ CASE3 $\equiv$ $\frac{{\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}}$ CASE4 $\equiv$ $(4-3{\gamma }_c){\mathfrak{𝔴}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(4-3{\gamma }_e){\mathfrak{𝔴}}_{\mathrm{e}\mathrm{n}\mathrm{v}}+3^2({\gamma }_c-{\gamma }_e)P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$

The left-hand panel of Figure 3 is identical to Figure 2, above. It displays in the $q-\nu$ parameter space, the behavior of $(n_c,n_e)=(5,1)$ bipolytropic equilibrium sequences that have, as labeled, seven different values of the ratio of mean-molecular-weights, ${\mu }_e/{\mu }_c$. Using a numerical root-finding technique, we have determined where the virial stability condition, ${\mathfrak{𝔰}}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}/{\mathfrak{𝔰}}_{\mathrm{e}\mathrm{n}\mathrm{v}}=5$, is satisfied along each of these sequences — as well as along a number of additional equilibrium sequences. Key properties of each of these identified models have been recorded in Table 1 of an accompanying discussion; see also an associated discussion of the free-energy of these configurations. Pulling from this tabulated data, the solid-red circular markers that appear in the right-hand panel of Figure 3 identify where this virial stability condition is satisfied along the separate equilibrium sequences while the accompanying dashed red curve identifies more broadly how the $q-\nu$ parameter space is divided into stable (below and to the right) versus unstable (above and to the left) regions.

Figure 3

In what follows we use a complementary — and more quantitatively rigorous — approach to evaluating the stability of equilibrium models, and contrast the results of that analysis with the virial-analysis results presented graphically here in Figure 3.

See Also

• K. De et al. (12 October 2018, Science, Vol. 362, No. 6411, pp. 201 - 206), A Hot and Fast Ultra-stripped Supernova that likely formed a Compact Neutron Star Binary.