SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt3

Virial Equilibrium of Adiabatic Spheres (Summary)

Part I:   Force Balance, Free Energy, & Virial

Part II:  Mass-Radius Relation

III:  Discussion & Other Model Sequences

Discussion

A spherically symmetric, self-gravitating gas cloud whose effective adiabatic exponent is $\gamma <4/3$ — equivalently, $n>3$ — cannot exist in a dynamically stable equilibrium state, in isolation. Such clouds can be stabilized, however, if they are embedded in a hot, tenuous external medium and effectively confined by an external pressure, $P_e$. The pressure-truncated, $n=5(\gamma =6/5)$ polytropic configurations being discussed here provide examples of such embedded clouds. A direct analogy can be drawn between this discussion and discussions of pressure-truncated isothermal $(\gamma =1;n=\mathrm{\infty })$ clouds — see, for example, our review of isothermal cloud structures in the context of Bonnor-Ebert spheres.

Part I: Physical Significance of the Two Curves

The "Stahler" mass-radius relation, plotted as a continuous curve in the above figure and reproduced as a sequence of discrete points in each panel of the subsequent comparison figure, identifies the precise mass $({𝒴})$ and associated radius $({𝒳})$ of physically allowed pressure-truncated, $n=5$ polytropic configurations over the full range of values of the dimensionless truncation radius, $0<\tilde{\xi }<\mathrm{\infty }$. Each model along the curve has an internal structure that ensures detailed force balance throughout the configuration; because this internal structure varies from model to model, the values of the structural form-factors — ${\mathfrak{𝔣}}_M,{\mathfrak{𝔣}}_W$, and ${\mathfrak{𝔣}}_A$ — and the corresponding values of the coefficients associated with the free-energy function — ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$ — will also vary from model to model along the Stahler curve.

If the values of the coefficients, ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$ (as well as the external pressure and, hence, the additional coefficient, ${𝒟}$) are held fixed, the algebraic free-energy function defines how a configuration's free energy will change as its overall size is varied. Extrema in the free energy will identify equilibrium configurations. Based on this understanding, our derived virial theorem expression for $n=5$ polytropic configurations identifies equilibrium radii $({𝒳})$ associated with various configuration masses $({𝒴})$. The "Virial" curve that has been plotted in each panel of the above comparison figure shows how the equilibrium radius varies with configuration mass, as dictated by the virial theorem — and, hence, as identified by extrema in the free-energy function — assuming that the relevant free-energy coefficients are held fixed. In each figure panel, this "Virial" curve qualitatively resembles the quantitatively correct, "Stahler" mass-radius relationship that has been derived from the properties of detailed force-balance models. The two curves overlap, and cross, wherever the coefficients used to define the "Virial" relation are identical to the coefficient values that are associated with a specific model along the "Stahler" relation. The two curves do not trace out identical mass-radius relationships simply because the structural form factors vary from model to model along the "Stahler" sequence.

In the context of star formation, the Stahler sequence can be viewed as an evolutionary sequence for cold protostellar gas clouds that are embedded in a hot, tenuous interstellar medium. An initially low-mass cloud is represented by an equilibrium configuration that has been truncated at a very small Lane-Emden radius, $\tilde{\xi }$; such clouds will appear near the origin of the displayed ${𝒳}-{𝒴}$ plane, at a point along the "lower" segment of the Stahler mass-radius relation. Over time, as the cloud grows in mass (through collisions with and accretion of other low-mass clouds, for example), it will slide up the lower segment of the Stahler curve, moving in a counter-clockwise direction further and further away from the plot origin. The mass-accretion process that drives the cloud's evolution presumably occurs on a time scale that is long compared to the local dynamical-readjustment time of the cloud, allowing the cloud's internal structure time to readjust and establish the properties defined by Stahler's detailed force-balance analysis.

Part II: Curve Intersections

Early Thoughts

Notice that, in each frame of the above comparison figure, the "Virial" curve intersects and crosses the "Stahler" curve at two locations. In each plot these two crossing points are identified by filled black circles and, in each plot, the crossing point that lies farthest along the curve — again, starting from near the origin and moving around the curve in a counter-clockwise direction — is associated with the equilibrium model on the Stahler curve that is defined by the same value of $\tilde{\xi }$ that was used to define the coefficients of the virial theorem mass-radius relation. This is not surprising, as the virial theorem should be precisely satisfied by every one of the equilibrium models along the Stahler sequence, as long as the value of $\tilde{\xi }$ that is used to define the coefficients of the free-energy function and, in turn, the virial theorem mass-radius relation is identical to the value of $\tilde{\xi }$ that defines the truncation radius of the detailed force-balance model. For example, the "Virial" curve that appears in the top-right panel of the comparison figure — a panel whose title includes the notation, $\xi =3$ — intersects the "Stahler" curve at ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, that is, at the location of the detailed force-balance model that, as previously explained, has a truncation radius, $\tilde{\xi }=3$.

It is not (yet) clear to us what physical significance should be ascribed to the model along the Stahler sequence that is identified by the second crossing of the "Virial" curve, given that the value of $\tilde{\xi }$ associated with the truncation radius of this second detailed force-balance model is not the same as the value of $\tilde{\xi }$ that was used to define the coefficients of the "Virial" curve. We note that, at least for the range of values of $\tilde{\xi }$ sampled in the above figure, this second crossing point seems to hover around the same limited segment of the Stahler sequence.

By direct analogy with discussions of Bonnor-Ebert spheres, the "maximum mass" model associated with ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ along the Stahler mass-radius relation has important physical significance in astrophysics. For a given applied external pressure, however, no models exist above some limiting mass — identified, here, by ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$.

Analysis Philosophy

The mass-radius relationship that derives from detailed force-balanced models is a physically meaningful and reliable statement of how a configuration's equilibrium radius will vary if its mass is changed. (It must be accepted that the configuration's structural form factors will change as it settles into each new equilibrium state, so such an "evolution" must occur on a secular time scale.) From the outset, however, the mass-radius relationship derived via the virial theorem — which, itself, derives from an analysis of the free energy function — should not be relied upon for the same physical insight. Consider, for example, that the scalar virial theorem is obtained from an analysis of the free-energy function by varying a system's size while holding constant all coefficients in the free-energy expression; this means that the system's mass as well as its structural form factors is held fixed while searching for an extremum in the free energy. The temptation, then, is to use the virial theorem to predict what the configuration's new equilibrium size will be if the system's mass is changed while holding the coefficients in the virial theorem constant. This means holding the structural form factors constant but not simultaneously holding the mass constant, and this differs from the constraints put on the free-energy function analysis that led to the virial theorem expression in the first place!

But we can combine the two analyses — the detailed force-balance analysis and the free-energy analysis — in the following meaningful way. Use the detailed force-balance analysis to identify the properties of an equilibrium state, specifically, for a given mass, determine the system's equilibrium radius and its accompanying structural form factors. (The virial theorem will be satisfied by this same set of determined parameter values.) Then, holding both the mass and the structural form factors constant, see how the free energy of the system varies as the configuration's size changed. In this manner the system's dynamical stability can be ascertained.

In summary: The mass-radius relationship determined from an analysis of detailed force-balanced models defines the physically correct secular evolutionary track for the system; while, an analysis of the free energy variations about an equilibrium state will answer the question of dynamical stability.

Quantitative Study

The preceding philosophical statements not withstanding, it is still worth understanding the relationship — if any — between the pair of models that are identified by the "second crossing" of the Stahler sequence by the "Virial" curve.

More Information on Secondary Overlap Points
From above table		Determined here
${𝒳}\equiv \frac{R_{\mathrm{e}\mathrm{q}}}{R_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	${𝒴}\equiv \frac{M_{\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{t}}}{M_{\mathrm{S}\mathrm{W}\mathrm{S}}}$	${\tilde{\xi }}_{-}$	${{𝒴}}_{-}$	${{𝒜}}_{M_{\ell }}$	${{ℬ}}_{M_{\ell }}$
0.388938	0.289568	0.72447	0.289568	0.20491	0.75191
0.468918	0.570916	0.98268	0.570916	0.20889	0.75395
0.507387	0.798441	1.17380	0.798441	0.21252	0.75652
0.515168	0.859518	1.22572	0.859520	0.21360	0.75740
0.518588	0.888969	1.25104	0.888968	0.21414	0.75785
0.520269	0.904143	1.26419	0.904142	0.21443	0.75810

Marginal Stability

As mentioned above, it is widely appreciated that the model having the largest mass — that is, the model that sits at ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ — along the Stahler sequence is of considerable astrophysical significance. Viewed in terms of a cloud's secular evolution, counter-clockwise along the sequence, something rather catastrophic must happen once the cloud acquires the mass associated with ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, because no equilibrium structure is available to the cloud if it gains any additional mass. It is tempting to associate this point along the Stahler sequence with a dynamical instability, imagining for example that the cloud will begin to dynamically collapse once it reaches this ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ configuration. But the "detailed force-balance" technique that is used to define the structure of equilibrium models along the Stahler sequence does not give us any insight regarding a configuration's dynamical stability.

Our free-energy analysis does provide this additional insight. The mass-radius relationship derived from the scalar virial theorem — which, itself, was derived via a free-energy analysis — is qualitatively similar to the mass-radius relationship defined (from a detailed force-balance analysis) by the Stahler sequence; in particular, it also exhibits an upper mass limit. And our free-energy analysis reveals that this "maximum mass" point associated with the virial theorem separates dynamically stable from dynamically unstable models along the sequence. This realization fuels the temptation just mentioned; that is, it seems to support the idea that the configuration at ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ along Stahler's sequence is associated with the onset of a dynamical instability along the sequence. But this is not the case! Our free-energy analysis has also shown that, when the structural form-factors — and, most specifically, the coefficients ${{𝒜}}_{M_{\ell }}$ and ${{ℬ}}_{M_{\ell }}$ — are assigned the values appropriate to the configuration at ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ along Stahler's sequence, the point of maximum mass associated with the corresponding expression for the virial theorem does not coincide with the configuration at ${{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$. The configuration at ${𝒴}={{𝒴}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=1.774078$ (also identified as the model having $\tilde{\xi }=3.0$) is found to be dynamically stable. Both of these realizations are illustrated graphically in the above figure.

Our analysis has shown, instead, that the marginally unstable configuration appears farther along the Stahler sequence when moving in a counter-clockwise direction. It corresponds to the model having $\tilde{\xi }=3.850652$ instead of $\tilde{\xi }=3.0$. While this can be illustrated graphically — for example, by carefully analyzing and comparing the bottom-center panel with the top-right panel in the above figure ensemble — an algebraic demonstration is more definitive. Our stability analysis has shown that, for any pressure-truncated polytropic configuration, the equilibrium structure associated with the point of marginal instability has,

$(\frac{{𝒴}}{{{𝒳}}^2}{)}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$

$=$

$[\frac{4\pi n}{{{𝒜}}_{M_{\ell }}(n-3)}{]}^{1/2}.$

For $n=5$ configurations, this means that the critical model along the equilibrium sequence will have,

${{𝒳}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^4$

$=$

$\left[\frac{{{𝒜}}_{M_{\ell }}}{10\pi }\right]{{𝒴}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2.$

But all configurations along Stahler's equilibrium sequence must also obey the mass-radius relationship,

${{𝒴}}^2-5{𝒴}{𝒳}+\frac{20\pi }{3}{{𝒳}}^4$

$=$

$0.$

Combining these two requirements means,

${{𝒴}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2-5({𝒴}{𝒳}{)}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}+(\frac{2{{𝒜}}_{M_{\ell }}}{3}){{𝒴}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2$	$=$	$0$
$\Rightarrow {{𝒴}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2[1+\frac{2}{3}\cdot {{𝒜}}_{M_{\ell }}]$	$=$	$5({𝒴}{𝒳}{)}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$
$\Rightarrow \frac{{{𝒳}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}}{{{𝒴}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}}$	$=$	$\frac{1}{5}[1+\frac{2}{3}\cdot {{𝒜}}_{M_{\ell }}].$

Now, taking into detailed account the internal structure of pressure-truncated, $n=5$ polytropic structures as represented in our summary table of Stahler's equilibrium configurations, we know that, along Stahler's entire sequence,

$\frac{{𝒳}}{{𝒴}}$	$=$	$\left\{\right(\frac{3\cdot 5}{2^2\pi })\frac{{\ell }^2}{(1+{\ell }^2{)}^2}\cdot (\frac{2^2\pi }{3\cdot 5^3})\frac{(1+{\ell }^2{)}^4}{({\ell }^2{)}^3}{\}}^{1/2}$
	$=$	$\frac{1+{\ell }^2}{5{\ell }^2},$

where we have again adopted the shorthand notation,

${\ell }^2\equiv \frac{{\tilde{\xi }}^2}{3}.$

We conclude, therefore, that in the marginally unstable model along the Stahler equilibrium sequence,

$1+\frac{2}{3}\cdot ({{𝒜}}_{M_{\ell }}{)}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$	$=$	$\frac{1+{\ell }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2}{{\ell }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^2}$
$\Rightarrow ({{𝒜}}_{M_{\ell }}{)}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$	$=$	$\frac{3}{2}\cdot {\ell }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}^{-2}.$

Given that the general expression for ${{𝒜}}_{M_{\ell }}$ along the Stahler sequence is,

${{𝒜}}_{M_{\ell }}$

$=$

$\frac{1}{2^4}\left[{\ell }^{-4}\right({\ell }^4-\frac{8}{3}{\ell }^2-1)+{\ell }^{-5}(1+{\ell }^2{)}^3{\tan }^{-1}(\ell )],$

we deduce that,

${\ell }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=2.2231751$       or, equivalently,       ${\tilde{\xi }}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}=3.850652.$

Hence, also,

$({{𝒳}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}},{{𝒴}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}})=(0.408738,1.699778).$

Part III

From our above, detailed analysis of the mass-radius relation for pressure-truncated polytropes, we concluded that configurations along "Stahler's" equilibrium sequence become dynamically unstable at a point that does not coincide with the maximum-mass configuration. Instead, the onset of dynamical instability is associated with the critical point on the mass-radius relation that arises from the free-energy-based virial theorem. In drawing this conclusion, we have implicitly assumed that the proper way to analyze an equilibrium configuration's stability is to vary its radius while, not only holding its mass, specific entropy, and surface pressure $(P_e)$ constant, but also assuming that the configuration's structural form factors are invariable.

This seems like a reasonable assumption, given that we're asking how a configuration's characteristics will vary dynamically when perturbed about an equilibrium state. While oscillating about an equilibrium state, it seems more reasonable to assume that the system will expand and contract in a nearly homologous fashion than that its internal structure will readily readjust to produce a different and desirable set of form factors. In support of this argument, we point to the paper by 📚 P. Goldreich & S. V. Weber (1980, ApJ, Vol. 238, pp. 991 - 997) which explicitly derives a self-similar solution for the homologous collapse of stellar cores that can be modeled as $n=3$ polytropes; an associated chapter of this H_Book details the Goldreich & Weber derivation. Goldreich & Weber use linear perturbation techniques to analyze the stability of their homologously collapsing configurations. In §IV of their paper, they describe the eigenvalues and eigenfunctions that result from this analysis. They discovered, for example, that "the lowest radial mode can be found analytically ... [and it] corresponds to a homologous perturbation of the entire core." Our assumption that the structural form factors remain constant when pressure-truncated polytropic configurations undergo radial size variations therefore appears not to be unreasonable. (Based on the Goldreich & Weber discussion, we should also look at the published work of 📚 M. Schwarzschild (1941, ApJ, Vol. 94, pp. 245 - 252), who has evaluated radial modes, and of 📚 T. G. Cowling (1941, MNRAS, Vol. 101, pp. 367 - 375), who has obtained eigenvalues of some low-order nonradial modes.)

In addition, it would seem that a certain amount of dissipation would be required for the system to readjust to new structural form factors. In order to test this underlying assumption, following Goldreich & Weber (1980), it would be desirable to carry out a full-blown perturbation analysis that involves looking for, for example, the eigenvector associated with the system's fundamental radial mode of pulsation. Ideally, we should be using the structural form factors associated with this pulsation-mode eigenfunction in our free-energy analysis of stability. Better yet, the sign of the eigenfrequency associated with the system's pulsation-mode eigenvector should signal whether the system is dynamically stable or unstable.

---

Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
|   Go Home   |

---

Other Model Sequences

Relating and Reconciling Two Mass-Radius Relationships for n = 4 Polytropes

For pressure-truncated $n=4$ polytropes, 📚 Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models (summarized above), we appreciate that the governing pair of parametric relations is,

${𝒳}$	$=$	$(\frac{1}{\pi }{)}^{1/2}\tilde{\xi }{\tilde{\theta }}^{3/2},$
${𝒴}$	$=$	$(\frac{2^4}{\pi }{)}^{1/2}{\tilde{\theta }}^{1/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}).$

On the other hand, the polynomial that results from plugging $n=4$ into the general mass-radius relation that is obtained via the virial theorem is,

$\frac{4\pi }{3}{{𝒳}}^4-[\frac{{𝒳}{{𝒴}}^5}{4\pi }{]}^{1/4}{\mathfrak{𝔟}}_{n=4}+\frac{5}{4}{{𝒴}}^2=0,$

where,

${\mathfrak{𝔟}}_{n=4}=[5(-{\tilde{\theta }}^{\text{'}}{)}^2+\frac{1}{3}{\tilde{\theta }}^5\left]\right(\frac{\tilde{\xi }}{-{\tilde{\theta }}^{\text{'}}}{)}^{5/4}.$

[For the record we note that, throughout the structure of an $n=4$ polytrope, ${\mathfrak{𝔟}}_{n=4}$ is a number of order unity. Its value is never less than $3^{1/4}$, which pertains to the center of the configuration; its maximum value of $\approx 5.098$ occurs at $\tilde{\xi }\approx 4.0$; and ${\mathfrak{𝔟}}_{n=4}\approx 3.946$ at its (zero pressure) surface, $\tilde{\xi }={\xi }_1\approx 14.97$. A plot showing the variation with $P_e$ of the closely allied parameter, ${ℬ}{|}_{n=4}=(4\pi {)}^{1/4}{\mathfrak{𝔟}}_{n=4}$ is presented in the righthand panel of the above parameter summary figure.]

In both panels of the following figure, the blue curve displays the mass-radius relation for pressure-truncated $n=4$ polytropes, ${𝒴}({𝒳})$, that is generated by Stahler's pair of parametric equations. The coordinates of discrete points along the curve have been determined from the tabular data provided on p. 399 of 📚 G. P. Horedt (1986, Astrophysics and Space Science, Vol. 126, Issue 2, pp. 357 - 408) while Excel has been used to generate the "smooth," continuous blue curve connecting the points; this set of points and accompanying blue curve are identical in both figure panels. In both figure panels, a set of discrete, triangle-shaped points traces the mass-radius relation, ${𝒴}({𝒳})$, that is obtained via the virial theorem, assuming that the coefficient, ${\mathfrak{𝔟}}_{n=4}$, is constant along the sequence. The "green" sequence in the lefthand panel results from setting ${\mathfrak{𝔟}}_{n=4}=3.4205$, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at $\tilde{\xi }=1.4$; the "orange" sequence in the righthand panel results from setting ${\mathfrak{𝔟}}_{n=4}=4.8926$, which is the value of the constant that results from Horedt's tabulated data if the configuration is truncated at $\tilde{\xi }=2.8$.

Comparing Two Separate Mass-Radius Relations for Pressure-Truncated n = 4 Polytropes
Comparison of Two Mass-Radius Relations

According to 📚 Horedt (1986), the surface of an isolated $(P_e=0)$, spherically symmetric, $n=4$ polytrope occurs at the dimensionless (Lane-Emden) radius, ${\xi }_1=14.9715463$. In both panels of the above figure, this isolated configuration is identified by the discrete (blue diamond) point at the origin, that is, at $({𝒳},{𝒴})=(0,0)$. As we begin to examine pressure-truncated models and $\tilde{\xi }$ is steadily decreased from ${\xi }_1$, the mass-radius coordinate of equilibrium configurations "moves" away from the origin, upward along the upper branch of the displayed (blue) mass-radius relation. A maximum mass of ${𝒴}\approx 2.042$ (corresponding to a radius of ${𝒳}\approx 0.4585$) is reached from the left as $\tilde{\xi }$ drops to a value of approximately $3.4$. As $\tilde{\xi }$ continues to decrease, the mass-radius coordinates of equilibrium configurations move along the lower branch of the displayed (blue) curve, reaching a maximum radius at $({𝒳},{𝒴})\approx (0.555,1.554)$ — corresponding to $\tilde{\xi }\approx 2.0$ — then decreasing in radius until, once again, the origin is reached, but this time because $\tilde{\xi }$ drops to zero.

If we set ${\mathfrak{𝔟}}_{n=4}=3.4205$ (corresponding to a choice of $\tilde{\xi }=1.4$), the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane as depicted by the set of green, triangle-shaped points in the lefthand panel of the above figure. While the (green) curve corresponding to this relation does not overlay the blue mass-radius relation, the two curves do intersect. They intersect precisely at the coordinate location along the blue curve (emphasized by the black filled circle) corresponding to a detailed force-balanced model having $\tilde{\xi }=1.4$. In an analogous fashion, in the righthand panel of the figure, the curve delineated by the set of orange triangle-shaped points shows how the virial theorem mass-radius relation maps onto the "Stahler" mass-radius coordinate plane when we set ${\mathfrak{𝔟}}_{n=4}=4.8926$ (corresponding to a choice of $\tilde{\xi }=2.8$); it intersects the blue mass-radius relation precisely at the coordinate location, $({𝒳},{𝒴})\approx (0.5108,1.965)$ — again, emphasized by a black filled circle — that corresponds to a detailed force-balanced model having $\tilde{\xi }=2.8$. Hence, the two relations give the same mass-radius coordinates when the value of ${\mathfrak{𝔟}}_{n=4}$ that is plugged into the virial theorem matches the value of ${\mathfrak{𝔟}}_{n=4}$ that reflects the structural form factor that is properly associated with a detailed force-balanced model.

When we mapped the virial theorem mass-radius relation onto Stahler's mass-radius coordinate plane using a value of ${\mathfrak{𝔟}}_{n=4}=4.8926$ (as traced by the orange triangle-shaped points in the righthand panel of the above figure), we expected it to intersect the blue curve at the point along the blue sequence where $\tilde{\xi }=2.8$, for the reason just discussed. After constructing the plot, it became clear that the two curves also intersect at the coordinate location, $({𝒳},{𝒴})\approx (0.255,1.67)$ — also highlighted by a black filled circle — that corresponds to a detailed force-balanced model having $\tilde{\xi }\approx 6.0$. This makes it clear that it is the equality of the structural form factors, not the equality of the dimensionless (Lane-Emden) radius, $\tilde{\xi }$, that assures precise agreement between the two different mass-radius expressions.

As is detailed in our above discussion of the dynamical stability of pressure-truncated polytropes, an examination of free-energy variations can not only assist us in identifying the properties of equilibrium configurations (via a free-energy derivation of the virial theorem) but also in determining which of these configurations are dynamically stable and which are dynamically unstable. We showed that, for a certain range of polytropic indexes, there is a critical point along the corresponding model sequence where the transition from stability to instability occurs. As has been detailed in our above groundwork derivations, for $n=4$ polytropic structures, the critical point is identified by the dimensionless parameters,

${\eta }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}{|}_{n=4}=\frac{1}{15};$          ${\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}{|}_{n=4}=\frac{15^{15}}{16^{16}};$      and      ${\mathrm{X}}_{\mathrm{m}\mathrm{i}\mathrm{n}}{|}_{n=4}=(\frac{16}{15}{)}^4.$

In the context of the above figure, independent of the chosen value of ${\mathfrak{𝔟}}_{n=4}$, this critical point always corresponds to the maximum mass that occurs along the mass-radius relationship established via the virial theorem. In both panels of the figure, a horizontal red-dotted line has been drawn tangent to this critical point and identifies the corresponding critical value of ${𝒴}$; a vertical red-dashed line drawn through this same point helps identify the corresponding critical value of ${𝒳}$. We have deduced (details of the derivation not shown) that, for pressure-truncated $n=4$ polytropes, the coordinates of this critical point in Stahler's ${𝒳}-{𝒴}$ plane depends on the choice of ${\mathfrak{𝔟}}_{n=4}$ as follows:

${{𝒳}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$	$=$	${\pi }^{-1/2}{2}^{-16/5}(3{\mathfrak{𝔟}}_{n=4}{)}^{4/5},$
${{𝒴}}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$	$=$	${\pi }^{-1/2}{2}^{-22/5}(3{\mathfrak{𝔟}}_{n=4}{)}^{8/5}.$

In practice, for a given plot of the type displayed in the above figure — that is, for a given choice of the structural parameter, ${\mathfrak{𝔟}}_{n=4}$ — it only makes sense to compare the location of this critical point to the location of points that have been highlighted by a filled black circle, that is, points that identify the intersection between the two mass-radius relations. If, in a given figure panel, a filled black circle lies to the right of the vertical dashed line, the equilibrium configuration corresponding to that black circle is dynamically stable. On the other hand, if the filled black circle lies to the left of the vertical dashed line, its corresponding equilibrium configuration is dynamically unstable. We conclude, therefore, that the equilibrium configuration marked by a filled black circle in the lefthand panel of the above figure is stable; however, both configurations identified by filled black circles in the righthand panel are unstable.

It is significant that the critical point identified by our free-energy-based stability analysis does not correspond to the equilibrium configuration having the largest mass along "Stahler's" (blue) equilibrium model sequence. One might naively expect that a configuration of maximum mass along the blue curve is the relevant demarcation point and that, correspondingly, all models along this sequence that fall "to the right" of this maximum-mass point are stable. But the righthand panel of our above figure contradicts this expectation. While both of the black filled circles in the righthand panel of the above figure lie to the left of the vertical dashed line and therefore, as just concluded, are both unstable, one of the two configurations lies to the right of the maximum-mass point along the blue "Stahler" sequence. This finding is related to the curiosity raised earlier in our discussion of the structural properties of pressure-truncated, $n=4$ polytropes.

Relating and Reconciling Two Mass-Radius Relationships for n = 3 Polytropes

For pressure-truncated $n=3$ polytropes, 📚 Stahler (1983) did not identify a polynomial relationship between the mass and radius of equilibrium configurations. However, from his analysis of detailed force-balance models (summarized above), we appreciate that the governing pair of parametric relations is,

${𝒳}$	$=$	$(\frac{3}{4\pi }{)}^{1/2}\tilde{\xi }\tilde{\theta },$
${𝒴}$	$=$	$(\frac{3^3}{4\pi }{)}^{1/2}(-{\tilde{\xi }}^2{\tilde{\theta }}^{\text{'}}).$

On the other hand, the polynomial that results from plugging $n=3$ into the general mass-radius relation that is obtained via the virial theorem is,

$\frac{2^3\pi }{3}{{𝒳}}^4-[\frac{{{𝒴}}^4}{4\pi }{]}^{1/3}{\mathfrak{𝔟}}_{n=3}+\frac{4}{3}{{𝒴}}^2=0,$

where,

${\mathfrak{𝔟}}_{n=3}=[4(-{\tilde{\theta }}^{\text{'}}{)}^2+\frac{2}{3}{\tilde{\theta }}^4\left]\right(\frac{\tilde{\xi }}{-{\tilde{\theta }}^{\text{'}}}{)}^{4/3}.$

See Also