# Pressure-Bounded Isothermal Sphere

Bonnor-Ebert
(Isothermal)
Spheres

(1955-56)

As has been derived and discussed elsewhere, an isolated isothermal sphere has a density profile that extends to infinity and, correspondingly, an unbounded total mass. In an astrophysical context, neither of these properties is desirable. A more realistic isothermal configuration can be constructed by embedding the structure in a low density, but hot external medium whose pressure, ${\displaystyle ~P_{e}}$, confines the isothermal configuration to a finite size. In a mathematical model, this can be accomplished by ripping off an outer layer of the isolated isothermal configuration down to the radius — label it ${\displaystyle ~\xi _{e}}$ — at which the configuration's original (internal) pressure equals ${\displaystyle ~P_{e}}$; the interior of the configuration that remains — containing mass ${\displaystyle ~M_{\xi _{e}}}$ — should be unaltered and in equilibrium. (This will work only for spherically symmetric configurations, as the gravitational acceleration at any location only depends on the mass contained inside that radius.) Ebert (1955) and Bonnor (1956) are credited with constructing the first such models and, most significantly, discovering that, for any specified sound speed and applied external pressure, there is a mass above which no equilibrium configuration exists. We present, here, the salient elements of these (essentially equivalent) derivations.

Prior to studying this discussion of pressure-bounded isothermal spheres, we recommend studying our related discussion of pressure-bounded ${\displaystyle n}$ = 5 polytropes. As with isolated isothermal spheres, isolated ${\displaystyle n}$ = 5 polytropes extend to infinity. But, unlike their isothermal counterparts, the structure of ${\displaystyle n}$ = 5 polytropes is describable analytically. Hence, an analysis of their structure and its extension to pressure-bounded configurations avoids the clutter introduced by a model — such as the isothermal sphere — that can only be described numerically. As it turns out, the pressure-bounded ${\displaystyle n}$ = 5 polytrope exhibits a Bonnor-Ebert type limiting mass that is analytically prescribable. Its derivation is mathematically quite clean and provides a firm foundation for understanding the better known — but only numerically prescribable — Bonnor-Ebert limiting mass.

## Governing Relation

The equilibrium structure of an isolated isothermal sphere, as derived by Emden (1907), has been discussed elsewhere. From this separate discussion we appreciate that the governing ODE is,

${\displaystyle {\frac {1}{r^{2}}}{\frac {d}{dr}}{\biggl (}r^{2}{\frac {d\ln \rho }{dr}}{\biggr )}=-{\frac {4\pi G}{c_{s}^{2}}}\rho \,,}$

where,

${\displaystyle c_{s}^{2}={\frac {\Re T}{\bar {\mu }}}={\frac {kT}{m_{u}{\bar {\mu }}}}\,,}$

is the square of the isothermal sound speed. In their studies of pressure-bounded isothermal spheres, Ebert (1955, ZA, 37, 217) and Bonnor (1956, MNRAS, 116, 351) both started with this governing ODE, but developed its solution in different ways. Here we present both developments while highlighting transformations between the two.

Derivation by Bonnor (1956) (edited) translation Derivation by Ebert (1955) (edited)
 ${\displaystyle ~{\frac {1}{r^{2}}}{\frac {d}{dr}}{\biggl (}{\frac {r^{2}}{\rho }}{\frac {d\rho }{dr}}{\biggr )}=-{\frac {4\pi Gm\rho }{kT}}\,.}$ (2.3)

Let us now transform (2.3) by making the following substitutions:

 ${\displaystyle ~\rho =\lambda e^{-\psi }\,,}$   ${\displaystyle r=\beta ^{1/2}\lambda ^{-1/2}\xi \,,}$ (2.6)

where ${\displaystyle ~\lambda }$ is an arbitrary constant [we choose ${\displaystyle ~\lambda =\rho _{c}}$], and

 ${\displaystyle ~\beta ={\frac {kT}{4\pi Gm}}\,.}$ (2.7)

Then (2.3) becomes

 ${\displaystyle ~\xi ^{-2}{\frac {d}{d\xi }}{\biggl (}\xi ^{2}{\frac {d\psi }{d\xi }}{\biggr )}=e^{-\psi }\,.}$ (2.8)
${\displaystyle G\Leftrightarrow \gamma }$
 ${\displaystyle ~\ell _{0}={\biggl (}{\frac {\Re T_{0}}{4\pi \mu \gamma \rho _{0}}}{\biggr )}^{1/2}\,.}$ (4)

Wir setzen unter Verwendung von (4) ${\displaystyle ~r=\ell _{0}\xi }$ und ${\displaystyle ~\rho =\rho _{0}\eta }$ mit ${\displaystyle ~\rho _{0}=\rho (0)}$. Für ${\displaystyle ~\eta }$ ergibt sich die Differential-gleichung der isothermen Gaskugel:

 ${\displaystyle ~\eta ^{''}-{\frac {(\eta ^{'})^{2}}{\eta }}+{\frac {2\eta ^{'}}{\xi }}+\eta ^{2}=0\,.}$ (17)

Der Strich bezeichnet die Differentiation nach ${\displaystyle ~\xi }$.

${\displaystyle \rho _{c}\Leftrightarrow \rho _{0}}$
${\displaystyle {\frac {kT}{m}}\Leftarrow c_{s}^{2}\Rightarrow {\frac {\Re T_{0}}{\mu }}}$
${\displaystyle \beta ^{1/2}\lambda ^{-1/2}\Leftrightarrow l_{0}}$
${\displaystyle e^{-\psi }\Leftrightarrow \eta }$

Both of these dimensionless governing ODEs — Bonnor's Eq. (2.8) and Ebert's Eq. (17) — are identical to the dimensionless expression derived by Emden (see the presentation elsewhere), namely,

${\displaystyle {\frac {d^{2}v_{1}}{d{\mathfrak {r}}_{1}^{2}}}+{\frac {2}{{\mathfrak {r}}_{1}}}{\frac {dv_{1}}{d{\mathfrak {r}}_{1}}}+e^{v_{1}}=0\,.}$

The translation from Emden-to-Bonnor-to-Ebert is straightforward:

${\displaystyle {\mathfrak {r}}_{1}=\xi |_{\mathrm {Bonner} }=\xi |_{\mathrm {Ebert} }~~~~\mathrm {and} ~~~~e^{v_{1}}=e^{-\psi }=\eta \,.}$

In much of what follows, we will use Bonnor's ${\displaystyle ~(\xi ,\psi )}$ notation rather than Emden's ${\displaystyle ~({\mathfrak {r}}_{1},v_{1})}$ notation. This means that we will be referring to the isothermal Lane-Emden function, ${\displaystyle ~\psi (\xi )}$, which provides a solution to the governing,

Isothermal Lane-Emden Equation

 ${\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}{\biggl (}\xi ^{2}{\frac {d\psi }{d\xi }}{\biggr )}=e^{-\psi }}$

## P-V Diagram

Given a value for the isothermal sound speed, ${\displaystyle ~c_{s}}$, and the central density, ${\displaystyle ~\rho _{c}}$, our accompanying summary of the properties of an isothermal sphere, as derived by Emden (1907), provides the following structural relations:

• Radial Coordinate Position & Associated Volume:

${\displaystyle r={\biggl (}{\frac {c_{s}^{2}}{4\pi G\rho _{c}}}{\biggr )}^{1/2}\xi }$ ;

${\displaystyle V\equiv {\frac {4\pi }{3}}r^{3}={\frac {4\pi }{3}}{\biggl (}{\frac {c_{s}^{2}}{4\pi G\rho _{c}}}{\biggr )}^{3/2}\xi ^{3}}$ ;

• Pressure Profile:

${\displaystyle P=(c_{s}^{2}\rho _{c})e^{-\psi }}$;

• Enclosed Mass:

${\displaystyle M_{r}={\biggl (}{\frac {c_{s}^{6}}{4\pi G^{3}\rho _{c}}}{\biggr )}^{1/2}{\biggl [}\xi ^{2}{\frac {d\psi }{d\xi }}{\biggr ]}\,.}$

Bonnor and Ebert both asked how the surface pressure of a truncated isothermal sphere of a given sound speed — or, equivalently, how the external pressure, ${\displaystyle ~P_{e}}$, required to confine that truncated configuration — will vary with volume if the mass ${\displaystyle ~M_{\xi _{e}}}$ of the configuration, rather than the central density, is held fixed. The above expression for the enclosed mass can be inverted to give ${\displaystyle ~\rho _{c}}$ in terms of ${\displaystyle ~M_{\xi _{e}}}$; specifically,

${\displaystyle \rho _{c}={\biggl (}{\frac {c_{s}^{6}}{4\pi G^{3}M_{\xi _{e}}^{2}}}{\biggr )}{\biggl [}\xi ^{2}{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}^{2}\,,}$

where the subscript ${\displaystyle ~e}$ denotes a truncated configuration of radius ${\displaystyle ~\xi _{e}}$ confined by an external pressure. This allows us to replace ${\displaystyle ~\rho _{c}}$ in favor of ${\displaystyle ~M_{\xi _{e}}}$ in the expressions for both the pressure and volume, obtaining,

• Pressure:

${\displaystyle P_{e}={\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}M_{\xi _{e}}^{2}}}{\biggr )}~\xi _{e}^{4}{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}_{e}^{2}e^{-\psi _{e}}\,,}$

or, adopting Ebert's (1955) normalization pressure, ${\displaystyle P_{0}\equiv c_{s}^{8}/(4\pi G^{3}M_{\xi _{e}}^{2})}$ — see his equation (22),

${\displaystyle {\frac {P_{e}}{P_{0}}}=\xi _{e}^{4}{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}_{e}^{2}e^{-\psi _{e}}\,;}$

• Volume:

${\displaystyle V^{2}={\biggl (}{\frac {4\pi }{3}}{\biggr )}^{2}{\biggl (}{\frac {c_{s}^{2}}{4\pi G}}{\biggr )}^{3}{\biggl (}{\frac {4\pi G^{3}M_{\xi _{e}}^{2}}{c_{s}^{6}}}{\biggr )}^{3}{\biggl [}\xi ^{2}{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}^{-6}\xi _{e}^{6}={\biggl (}{\frac {4\pi }{3}}{\biggr )}^{2}{\biggl (}{\frac {G^{6}M_{\xi _{e}}^{6}}{c_{s}^{12}}}{\biggr )}{\biggl [}\xi ^{2}{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}^{-6}\xi _{e}^{6}\,,}$

${\displaystyle \Rightarrow ~~~~~V={\frac {4\pi }{3}}{\biggl (}{\frac {GM_{\xi _{e}}}{c_{s}^{2}}}{\biggr )}^{3}{\biggl [}\xi {\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}^{-3}\,;}$

or, adopting a normalization volume, ${\displaystyle V_{0}\equiv (4\pi /3)R_{0}^{3}}$, defined in terms of Ebert's (1955) normalization radius, ${\displaystyle R_{0}\equiv GM_{\xi _{e}}/c_{s}^{2}}$ — again, see his equation (22),

${\displaystyle {\frac {V}{V_{0}}}={\biggl [}\xi {\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}^{-3}\,;}$

the corresponding normalized radius of the truncated configuration is,

${\displaystyle {\frac {R}{R_{0}}}={\biggl (}{\frac {V}{V_{0}}}{\biggr )}^{1/3}={\biggl [}\xi {\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}^{-1}\,.}$

The expressions shown here for ${\displaystyle ~P_{e}}$ and ${\displaystyle ~V}$ are identical to those presented by Bonnor (1956) for "${\displaystyle ~p}$" and "${\displaystyle ~V}$," immediately following his equation (2.17). They are also the set of parametric equations that lead to Bonnor's Fig. 1 P-V diagram, which is reproduced here as the left-hand panel of our Figure 1 (with permission, as our caption to this figure documents).

Figure 1: Bonnor's P-V Diagram

Pertaining to Reprinted Figure (immediately above)

 Author (year): W. B. Bonnor (1956) Article Title.: Boyle's Law and Gravitational Instability Journal Title: Monthly Notices of the Royal Astronomical Society Volume & Pages: 116, pp. 351 - 359 Figure No.: 1 (p. 355) Copyright: Royal Astronomical Society — Permission to reuse this figure has been granted implicitly as summarized here and as detailed by the publisher here.
Pertaining to Our Original Figure (immediately above)

Caption: In an effort to demonstrate more explicitly how Bonnor's 1956 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve in the above panel displays a plot of ${\displaystyle ~P_{e}/P_{0}}$ versus ${\displaystyle ~V/V_{0}}$ that we have constructed using data drawn directly from Table 14 of Emden (1907).

Ebert chose to illustrate the behavior of the sequence of pressure-bounded isothermal spheres by plotting ${\displaystyle ~\log(P/P_{0})}$ versus ${\displaystyle ~\log(R/R_{0})}$, rather than ${\displaystyle ~P}$ versus ${\displaystyle ~V}$. The result — Figure 2 in Ebert (1955) — is reproduced here as the left-hand panel of our Figure 2. It should be clear that the information contained in Ebert's plot is identical to the information contained in the curve that is displayed in Bonner's P-V diagram.

Figure 2: Ebert's P-R Diagrams

The reader may readily view this referenced image/text by opening (preferably in a separate browser window) the relevant article and scrolling to the relevant figure/page, as detailed immediately below.

Pertaining to Reprinted Figure (immediately above)

 Author (year): R. Ebert (1955) Article Title.: Über die Verdichtung von HI-Gebieten Journal Title: Zeitschrift für Astrophysik Volume & Pages: 37, pp. 217 - 232 Figure No.: 2 (p. 222) Copyright: Springer-Verlag — Required reuse permissions summarized here and detailed by the publisher here.
Pertaining to Our Original Figure (immediately above)

Caption: In an effort to demonstrate more explicitly how Ebert's 1955 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve in the above panel displays a plot of ${\displaystyle ~\log(P_{e}/P_{0})}$ versus ${\displaystyle ~\log(R/R_{0})}$ that we have constructed using data drawn directly from Table 14 of Emden (1907).

In an effort to demonstrate more explicitly how Bonnor's 1956 work and Ebert's 1955 work tie back to the original work on isolated isothermal spheres that was published by Emden in 1907, the right-hand panel of our Figure 1 shows a plot of ${\displaystyle ~P_{e}/P_{0}}$ versus ${\displaystyle ~V/V_{0}}$, and the right-hand panel of our Figure 2 shows a plot of ${\displaystyle ~\log(P_{e}/P_{0})}$ versus ${\displaystyle ~\log(R/R_{0})}$ that we have constructed using data drawn directly from Table 14 of Emden (1907).

### Whitworth's Normalization

Whitworth (1981, MNRAS, 195, 967) examined the behavior of model sequences that result from embedding polytropes with various effective adiabatic indexes ${\displaystyle ~(\gamma _{g}=1/3,~2/3,~1,~4/3,~5/3)}$ in an external medium of pressure ${\displaystyle ~P_{e}}$. Following Ebert's lead, in his Figure 1b, Whitworth showed how the size of the configuration varies with ${\displaystyle ~P_{e}}$ along each sequence. A reproduction of Whitworth's Figure 1b has been displayed in our separate discussion of pressure-bounded polytropic configurations that have analytic prescriptions, but we have again displayed the figure here, in the right-most panel of our Figure 2, because Whitworth included pressure-bounded isothermal configurations ${\displaystyle ~(\gamma _{g}=1)}$ among his models. The red arrow identifies the isothermal sequence. Whitworth's Figure 1b effectively serves as an extension of Ebert's Figure 2, shown above in the left-most panel of our Figure 2. In order to assist comparison with Ebert's figure, this time we have rotated and flipped Whitworth's figure relative to its original orientation because Whitworth chose to plot ${\displaystyle ~R}$ versus ${\displaystyle ~P_{e}}$ whereas Ebert plotted ${\displaystyle ~P_{e}}$ versus ${\displaystyle ~R}$.

Figure 3: Whitworth's P-R Diagrams

Pertaining to Reprinted Figure
(both on the top-left and on the right)

 Author (year): A. Whitworth (1981) Article Title.: Global Gravitational Stability for One-Dimensional Polytropes Journal Title: Monthly Notices of the Royal Astronomical Society Volume & Pages: 195, pp. 967 - 977 Figure No.: 1b (p. 971) Copyright: Royal Astronomical Society — Permission to reuse this figure has been granted implicitly as summarized here and as detailed by the publisher here.

 Pertaining to Reprinted Figure (on the right) Same as reprinted figure on the top-left, except image has been flipped, both vertically and horizontally, so that the axes correspond with the axes of the diagrams presented above in our Figure 2a; a red arrow (our modification) points to the most relevant (isothermal) equilibrium sequence.

 Pertaining to Our Original Figure (on the bottom-left) Caption: In an effort to demonstrate more explicitly how Whitworth's 1981 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve displays a plot of ${\displaystyle ~P_{e}/P_{\mathrm {rf} }}$ versus ${\displaystyle ~V/V_{\mathrm {rf} }}$ that we have constructed using data drawn directly from Table 14 of Emden (1907).

It should be noted that the scaling adopted along both axes in Whitworth's Figure 1b is different from the scaling used by Ebert. As is detailed in the accompanying ASIDE, Whitworth "referenced" ${\displaystyle P_{e}}$ and ${\displaystyle R}$ to, respectively,

${\displaystyle ~P_{\mathrm {rf} }={\biggl (}{\frac {3^{4}\cdot 5^{3}}{2^{8}}}{\biggr )}P_{0}\,,}$

${\displaystyle ~R_{\mathrm {rf} }={\biggl (}{\frac {2^{2}}{3\cdot 5}}{\biggr )}R_{0}\,.}$

In generating the right-hand panel of our Figure 1, we have adopted Whitworth's scaling. That is to say, we have plotted ${\displaystyle ~P_{e}/P_{\mathrm {rf} }}$ versus ${\displaystyle ~V/V_{\mathrm {rf} }}$, where,

 ${\displaystyle ~{\frac {P_{e}}{P_{\mathrm {rf} }}}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\frac {P_{e}}{P_{0}}}\cdot {\frac {P_{0}}{P_{\mathrm {rf} }}}={\biggl (}{\frac {2^{8}}{3^{4}\cdot 5^{3}}}{\biggr )}\xi _{e}^{4}{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}_{e}^{2}e^{-\psi _{e}}\,,}$

and,

 ${\displaystyle ~{\frac {V}{V_{\mathrm {rf} }}}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\frac {V}{V_{0}}}{\biggl (}{\frac {R_{0}}{R_{\mathrm {rf} }}}{\biggr )}^{3}={\biggl (}{\frac {3\cdot 5}{2^{2}}}{\biggr )}^{3}{\biggl [}\xi {\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}^{-3}\,.}$

## Limiting Pressure and Maximum Mass

No matter how you look at the bounded isothermal sphere sequence — whether plotted as a curve in Bonnor's P-V diagram or as a curve in Whitworth's R-P diagram — it is clear that, for a given choice of the sound speed and the mass, there is a value of the pressure above which no equilibrium configuration exists. (Alternatively, for a given sound speed and external pressure, there is a limiting mass above which no equilibrium configuration exists; see below.) The configuration identifying this limiting pressure resides at the position along Bonnor's P-V diagram sequence where ${\displaystyle ~dP_{e}/dV}$ — or, in Whitworth's discussion, ${\displaystyle ~dP_{e}/dR}$ — first goes to zero.

### Following Bonnor's Presentation

As is shown in his equation (3.1), Bonnor (1956) determined that, at a fixed mass and sound speed, ${\displaystyle ~dP/dV}$ goes to zero when,

${\displaystyle ~1-{\frac {1}{2}}e^{\psi }{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}^{2}=0\,.}$

In the following figure, the function defined by the left-hand-side of this expression is plotted versus ${\displaystyle ~\ln \xi }$ using Emden's (1907) tabulated data. As Bonnor noted, the function first crosses zero when ${\displaystyle ~\xi \approx 6.5}$ ${\displaystyle ~(\ln \xi =1.87)}$.

An isothermal sphere that is truncated at this location will have a radius,

${\displaystyle R={\biggl (}{\frac {c_{s}^{2}}{G\rho _{c}}}{\biggr )}^{1/2}{\frac {6.5}{\sqrt {4\pi }}}=1.83{\biggl (}{\frac {c_{s}^{2}}{G\rho _{c}}}{\biggr )}^{1/2}\,,}$

which matches equation (3.3) of Bonnor (1956); and, drawing on function values provided in Emden's (1907) Table 14, it will have a total mass,

${\displaystyle ~M_{R}={\biggl (}{\frac {c_{s}^{6}}{4\pi G^{3}\rho _{c}}}{\biggr )}^{1/2}{\biggl [}\xi ^{2}{\frac {d\psi }{d\xi }}{\biggr ]}_{e}\approx {\biggl (}{\frac {c_{s}^{6}}{G^{3}\rho _{c}}}{\biggr )}^{1/2}{\biggl [}{\frac {(14.353+17.214)/2}{\sqrt {4\pi }}}{\biggr ]}=4.45{\biggl (}{\frac {c_{s}^{6}}{G^{3}\rho _{c}}}{\biggr )}^{1/2}\,.}$

Dividing this expression for ${\displaystyle ~M_{R}}$ by the expression for ${\displaystyle ~R}$ gives,

${\displaystyle {\frac {M_{R}}{R}}=2.43{\biggl (}{\frac {c_{s}^{2}}{G}}{\biggr )}\,,}$

which matches equation (3.6) of Bonnor (1956). In order to maintain an equilibrium structure while truncating the isothermal model at this radius requires applying an external pressure of the following magnitude:

${\displaystyle P_{e}=c_{s}^{2}\rho _{c}e^{-\psi }\approx c_{s}^{2}\rho _{c}(0.08493+0.05833)/2=0.0716~c_{s}^{2}\rho _{c}\,,}$

where, again, numerical values have been drawn from Emden's (1907) Table 14.

Finally, using this relation to eliminate ${\displaystyle ~\rho _{c}}$ from the expression for ${\displaystyle ~M_{R}}$ gives,

 ${\displaystyle ~M_{R}}$ = ${\displaystyle {\biggl (}{\frac {c_{s}^{8}}{G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}{\frac {1}{4\pi }}{\biggr ]}^{1/2}{\biggl [}\xi ^{2}e^{\psi /2}{\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}_{e}}$ ${\displaystyle \approx }$ ${\displaystyle {\biggl (}{\frac {c_{s}^{8}}{G^{3}P_{e}}}{\biggr )}^{1/2}{\frac {1}{2\pi ^{1/2}}}{\biggl \{}{\frac {1}{2}}{\biggl [}14.353(0.08493)^{1/2}+17.214(0.05833)^{1/2}{\biggr ]}{\biggr \}}}$ ${\displaystyle \approx }$ ${\displaystyle 1.18{\biggl (}{\frac {c_{s}^{8}}{G^{3}P_{e}}}{\biggr )}^{1/2}\,.}$

This is the mass upper limit for a stable, pressure-bounded isothermal sphere — the so-called Bonnor-Ebert mass; see, for example, equation (1) of Shu (1977). In a separate discussion, we compare this result to the determinations of other related mass upper-limits.

### Ebert's Corresponding Presentation

The expression derived by Bonnor for the ratio ${\displaystyle ~M_{R}/R}$ in the limiting configuration can be inverted to give,

${\displaystyle ~R\approx 0.41{\biggl (}{\frac {GM_{R}}{c_{s}^{2}}}{\biggr )}\,.}$

This matches the expression for the critical radius, ${\displaystyle ~R_{\mathrm {kr} }}$, that appears as equation (23) in Ebert's (1955) published derivation. Similarly, the above relation that expresses the Bonnor-Ebert limiting mass in terms of ${\displaystyle ~c_{s}}$, ${\displaystyle ~G}$, and ${\displaystyle ~P_{e}}$ can be inverted to give,

${\displaystyle ~P_{e}\approx 4\pi (1.18)^{2}{\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}M_{R}^{2}}}{\biggr )}=17.5{\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}M_{R}^{2}}}{\biggr )}\,,}$

which agrees with the limiting pressure that was derived by Ebert (1955) and that is also presented in his equation (23).

## Mass versus Radius Equilibrium Sequence

### Old Derivation

For use in a related discussion, let's examine how the mass varies with radius — at fixed ${\displaystyle ~c_{s}^{2}}$ and fixed ${\displaystyle ~P_{e}}$ — in the limit as ${\displaystyle ~\xi _{e}\rightarrow 0}$. From above, we have,

 ${\displaystyle ~{\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}{\biggl [}\xi ^{2}\psi ^{'}{\biggr ]}}$ ${\displaystyle ~=}$ ${\displaystyle ~e^{-\psi }}$ ${\displaystyle ~\Rightarrow ~~~\psi ^{''}}$ ${\displaystyle ~=}$ ${\displaystyle ~e^{-\psi }-{\frac {2}{\xi }}\psi ^{'}\,,}$

and,

 ${\displaystyle ~P_{e}}$ ${\displaystyle ~=}$ ${\displaystyle ~(c_{s}^{2}\rho _{c})e^{-\psi }}$ ${\displaystyle ~\Rightarrow ~~~\rho _{c}}$ ${\displaystyle ~=}$ ${\displaystyle ~(c_{s}^{2}P_{e})e^{-\psi }}$

Hence,

 ${\displaystyle ~M}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{6}}{4\pi G^{3}\rho _{c}}}{\biggr )}^{1/2}{\biggl [}\xi ^{2}{\frac {d\psi }{d\xi }}{\biggr ]}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{4}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}\xi ^{2}\psi ^{'}{\biggr ]}e^{\psi /2}}$ ${\displaystyle ~\Rightarrow ~~~{\frac {dM}{d\xi }}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{4}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}{\frac {2}{\xi }}\psi ^{'}+\psi ^{''}+{\frac {1}{2}}(\psi ^{'})^{2}{\biggr ]}\xi ^{2}e^{\psi /2}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{4}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}e^{-\psi }+{\frac {1}{2}}(\psi ^{'})^{2}{\biggr ]}\xi ^{2}e^{\psi /2}}$

Also,

 ${\displaystyle ~R}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{2}}{4\pi G\rho _{c}}}{\biggr )}^{1/2}\xi }$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {1}{4\pi GP_{e}}}{\biggr )}^{1/2}\xi e^{\psi /2}}$ ${\displaystyle ~\Rightarrow ~~~{\frac {dR}{d\xi }}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {1}{4\pi GP_{e}}}{\biggr )}^{1/2}{\biggl [}1+{\frac {\xi }{2}}\cdot \psi ^{'}{\biggr ]}e^{\psi /2}}$

So the derivative of interest is,

 ${\displaystyle ~{\frac {dM}{dR}}={\frac {dM/d\xi }{dR/d\xi }}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{4}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}e^{-\psi }+{\frac {1}{2}}(\psi ^{'})^{2}{\biggr ]}\xi ^{2}e^{\psi /2}{\biggl \{}{\biggl (}{\frac {1}{4\pi GP_{e}}}{\biggr )}^{1/2}{\biggl [}1+{\frac {\xi }{2}}\cdot \psi ^{'}{\biggr ]}e^{\psi /2}{\biggr \}}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{2}}{G}}{\biggr )}{\biggl [}2e^{-\psi }+(\psi ^{'})^{2}{\biggr ]}\xi ^{2}{\biggl [}2+\xi \psi ^{'}{\biggr ]}^{-1}}$ ${\displaystyle ~\Rightarrow ~~~{\frac {d\ln M}{d\ln R}}}$ ${\displaystyle ~=}$ ${\displaystyle ~\xi e^{\psi /2}{\biggl [}\xi ^{2}\psi ^{'}e^{\psi /2}{\biggr ]}^{-1}{\biggl [}2e^{-\psi }+(\psi ^{'})^{2}{\biggr ]}\xi ^{2}{\biggl [}2+\xi \psi ^{'}{\biggr ]}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~\xi {\biggl [}\psi ^{'}{\biggr ]}^{-1}{\biggl [}2e^{-\psi }+(\psi ^{'})^{2}{\biggr ]}{\biggl [}2+\xi \psi ^{'}{\biggr ]}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\frac {\xi [2e^{-\psi }+(\psi ^{'})^{2}]}{\psi ^{'}[2+\xi \psi ^{'}]}}}$

Now, we can also write,

 ${\displaystyle ~\psi ^{'}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {M}{m_{0}}}{\biggr )}\xi ^{-2}e^{-\psi /2}}$

and,

 ${\displaystyle ~e^{\psi }}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {R}{r_{0}}}{\biggr )}\xi ^{-2}}$

Hence,

 ${\displaystyle {\frac {d\ln M}{d\ln R}}}$ ${\displaystyle ~=}$ ${\displaystyle ~\xi {\biggl [}2e^{-\psi }+{\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}\xi ^{-4}e^{-\psi }{\biggr ]}{\biggl [}2{\biggl (}{\frac {M}{m_{0}}}{\biggr )}\xi ^{-2}e^{-\psi /2}+\xi {\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}\xi ^{-4}e^{-\psi }{\biggr ]}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl [}2\xi ^{4}+{\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}{\biggr ]}{\biggl [}2{\biggl (}{\frac {M}{m_{0}}}{\biggr )}{\biggl (}{\frac {R}{r_{0}}}{\biggr )}^{1/2}+{\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}{\biggr ]}^{-1}}$

### New Derivation

For use in a related discussion, let's examine how the mass, ${\displaystyle ~M_{R}}$, varies with radius, ${\displaystyle ~R}$, at fixed ${\displaystyle ~c_{s}^{2}}$ and ${\displaystyle ~P_{e}}$. From above, we have,

 ${\displaystyle ~{\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}{\biggl [}\xi ^{2}\psi ^{'}{\biggr ]}}$ ${\displaystyle ~=}$ ${\displaystyle ~e^{-\psi }}$ ${\displaystyle ~\Rightarrow ~~~\psi ^{''}}$ ${\displaystyle ~=}$ ${\displaystyle ~e^{-\psi }-{\frac {2}{\xi }}\psi ^{'}\,,}$

and,

${\displaystyle R^{2}={\biggl (}{\frac {c_{s}^{2}}{4\pi G\rho _{c}}}{\biggr )}\xi _{e}^{2}}$ ;

${\displaystyle M_{R}^{2}={\biggl (}{\frac {c_{s}^{6}}{4\pi G^{3}\rho _{c}}}{\biggr )}{\biggl [}\xi ^{2}{\frac {d\psi }{d\xi }}{\biggr ]}_{e}^{2}\,.}$

Given that, ${\displaystyle P_{e}=(c_{s}^{2}\rho _{c})e^{-\psi _{e}}}$, we can also write,

 ${\displaystyle ~R}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{4}}{4\pi GP_{e}}}{\biggr )}^{1/2}\xi _{e}e^{-\psi _{e}/2}\,,}$ ${\displaystyle ~M_{R}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}\xi ^{2}\psi ^{'}{\biggr ]}_{e}e^{-\psi _{e}/2}\,.}$

Hence,

 ${\displaystyle ~{\frac {dM_{R}}{d\xi _{e}}}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}2\xi \psi ^{'}+\xi ^{2}\psi ^{''}-{\frac {1}{2}}\xi ^{2}(\psi ^{'})^{2}{\biggr ]}_{e}e^{-\psi _{e}/2}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}2\xi \psi ^{'}+\xi ^{2}{\biggl (}e^{-\psi _{e}}-{\frac {2}{\xi _{e}}}\psi _{e}^{'}{\biggr )}-{\frac {1}{2}}\xi ^{2}(\psi ^{'})^{2}{\biggr ]}_{e}e^{-\psi _{e}/2}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}e^{-\psi _{e}}-{\frac {1}{2}}(\psi ^{'})^{2}{\biggr ]}_{e}\xi _{e}^{2}e^{-\psi _{e}/2}\,.}$

Also,

 ${\displaystyle ~{\frac {dR}{d\xi _{e}}}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{4}}{4\pi GP_{e}}}{\biggr )}^{1/2}{\biggl [}1-{\frac {1}{2}}\xi _{e}\psi _{e}^{'}{\biggr ]}e^{-\psi _{e}/2}\,.}$

So the derivative of interest is,

 ${\displaystyle ~{\frac {dM}{dR}}={\frac {dM/d\xi _{e}}{dR/d\xi _{e}}}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{8}}{4\pi G^{3}P_{e}}}{\biggr )}^{1/2}{\biggl [}e^{-\psi _{e}}-{\frac {1}{2}}(\psi ^{'})^{2}{\biggr ]}_{e}\xi _{e}^{2}e^{-\psi _{e}/2}{\biggl \{}{\biggl (}{\frac {c_{s}^{4}}{4\pi GP_{e}}}{\biggr )}^{1/2}{\biggl [}1-{\frac {1}{2}}\xi _{e}\psi _{e}^{'}{\biggr ]}e^{-\psi _{e}/2}{\biggr \}}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {c_{s}^{4}}{G^{2}}}{\biggr )}^{1/2}{\biggl [}e^{-\psi _{e}}-{\frac {1}{2}}(\psi ^{'})^{2}{\biggr ]}_{e}\xi _{e}^{2}{\biggl [}1-{\frac {1}{2}}\xi _{e}\psi _{e}^{'}{\biggr ]}^{-1}}$ ${\displaystyle ~\Rightarrow ~~~{\frac {d\ln M}{d\ln R}}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\frac {\xi _{e}}{\psi ^{'}}}{\biggl [}e^{-\psi _{e}}-{\frac {1}{2}}(\psi ^{'})^{2}{\biggr ]}_{e}{\biggl [}1-{\frac {1}{2}}\xi _{e}\psi _{e}^{'}{\biggr ]}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\frac {\xi _{e}}{\psi ^{'}e^{\psi _{e}}}}{\biggl [}2-(\psi ^{'})^{2}e^{\psi _{e}}{\biggr ]}_{e}{\biggl [}2-\xi _{e}\psi _{e}^{'}{\biggr ]}^{-1}\,.}$

Now, we can also write,

 ${\displaystyle ~\psi ^{'}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {M}{m_{0}}}{\biggr )}\xi ^{-2}e^{-\psi /2}}$

and,

 ${\displaystyle ~e^{\psi }}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl (}{\frac {R}{r_{0}}}{\biggr )}\xi ^{-2}}$

Hence,

 ${\displaystyle {\frac {d\ln M}{d\ln R}}}$ ${\displaystyle ~=}$ ${\displaystyle ~\xi {\biggl [}2e^{-\psi }+{\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}\xi ^{-4}e^{-\psi }{\biggr ]}{\biggl [}2{\biggl (}{\frac {M}{m_{0}}}{\biggr )}\xi ^{-2}e^{-\psi /2}+\xi {\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}\xi ^{-4}e^{-\psi }{\biggr ]}^{-1}}$ ${\displaystyle ~=}$ ${\displaystyle ~{\biggl [}2\xi ^{4}+{\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}{\biggr ]}{\biggl [}2{\biggl (}{\frac {M}{m_{0}}}{\biggr )}{\biggl (}{\frac {R}{r_{0}}}{\biggr )}^{1/2}+{\biggl (}{\frac {M}{m_{0}}}{\biggr )}^{2}{\biggr ]}^{-1}}$

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