User:Tohline/SSC/Structure/BonnorEbert

Pressure-Bounded Isothermal Sphere

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, <math>P_e</math>, 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 <math>\xi_e</math> — at which the configuration's original (internal) pressure equals <math>P_e</math>; the interior of the configuration that remains — containing mass <math>M_{\xi_e}</math> — 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 <math>~n</math> = 5 polytropes. As with isolated isothermal spheres, isolated <math>~n</math> = 5 polytropes extend to infinity. But, unlike their isothermal counterparts, the structure of <math>~n</math> = 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 <math>~n</math> = 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,

<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{d\ln\rho}{dr} \biggr) =- \frac{4\pi G}{c_s^2} \rho \, ,</math>

where,

<math>c_s^2 = \frac{\Re T}{\bar{\mu}} = \frac{k T}{m_u \bar{\mu}} \, ,</math>

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 (edited)	translation	Derivation by Ebert (edited)
	<math>G \Leftrightarrow \gamma</math>
	<math>\rho_c \Leftrightarrow \rho_0</math>
	<math>\frac{kT}{m} \Leftarrow c_s^2 \Rightarrow \frac{\Re T_0}{\mu}</math>
	<math>\beta^{1/2}\lambda^{-1/2} \Leftrightarrow l_0</math>
	<math>e^{-\psi} \Leftrightarrow \eta</math>

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,

<math> \frac{d^2v_1}{d\mathfrak{r}_1^2} +\frac{2}{\mathfrak{r}_1} \frac{dv_1}{d\mathfrak{r}_1} + e^{v_1} = 0 \, . </math>

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

<math> \mathfrak{r}_1 = \xi|_\mathrm{Bonner} = \xi|_\mathrm{Ebert}~~~~\mathrm{and}~~~~e^{v_1} = e^{-\psi} = \eta \, . </math>

In much of what follows, we will use Bonnor's <math>(\xi, \psi)</math> notation rather than Emden's <math>(\mathfrak{r}_1, v_1)</math> notation.

P-V Diagram

Given a value for the isothermal sound speed, <math>c_s</math>, and the central density, <math>\rho_c</math>, 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:

<math>r = \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi </math> ;

<math>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 </math> ;

• Pressure Profile:

<math>P = (c_s^2 \rho_c) e^{-\psi}</math>;

• Enclosed Mass:

<math> 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] \, . </math>

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, <math>P_e</math>, required to confine that truncated configuration — will vary with volume if the mass <math>M_{\xi_e}</math> of the configuration, rather than the central density, is held fixed. The above expression for the enclosed mass can be inverted to give <math>\rho_c</math> in terms of <math>M_{\xi_e}</math>; specifically,

<math> \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 \, , </math>

where the subscript <math>e</math> denotes a truncated configuration of radius <math>\xi_e</math> confined by an external pressure. This allows us to replace <math>\rho_c</math> in favor of <math>M_{\xi_e}</math> in the expressions for both the pressure and volume:

• Pressure:

<math>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)^2_e e^{-\psi_e} \, ,</math>

or, normalized to the isothermal reference pressure, <math>P_\mathrm{rf} = (3^4 \cdot 5^3/2^{10}\pi) c_s^8 G^{-3} M_{\xi_e}^{-2}</math>, as defined by Whitworth (1981) — for clarification, read the accompanying ASIDE —

<math>\frac{P_e}{P_\mathrm{rf}} = \biggl( \frac{2^8}{3^4\cdot 5^3} \biggr) ~\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e}</math>;

• Volume:

<math>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 \, , </math>

<math> \Rightarrow ~~~~~ V = \frac{4\pi}{3} \biggl( \frac{G M_{\xi_e}}{c_s^{2}} \biggr)^3 \biggl[ - \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-3} \, ; </math>

or, normalized to the isothermal reference volume, <math>V_\mathrm{rf} \equiv (4\pi /3) R_\mathrm{rf}^3</math>, where, as defined by Whitworth (1981) <math>R_\mathrm{rf} = (2^2 / 3 \cdot 5) G M_{\xi_e} c_s^{-2} </math> — for clarification, read the accompanying ASIDE —

<math> \frac{V}{V_\mathrm{rf}} = \frac{3^3\cdot 5^3}{2^6} \biggl[ - \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-3} \, ; </math>

the corresponding normalized radius of the truncated configuration is,

<math> \frac{R}{R_\mathrm{rf}} = \biggl(\frac{V}{V_\mathrm{rf}}\biggr)^{1/3} = \frac{3\cdot 5}{2^2} \biggl[ - \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-1} \, . </math>

The expressions shown here for <math>P_e</math> and <math>V</math> are identical to those presented by Bonnor for "<math>p</math>" and "<math>V</math>," 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.

Figure 1: Bonnor's P-V Diagram

As another point of reference, the expressions we have presented for <math>P_e/P_\mathrm{rf}</math> and <math>R/R_\mathrm{rf}</math> are identical to the pair of equations labeled (9b) in Whitworth (1981). In the context of his discussion of the global stability of one-dimensional polytropes, Whitworth chose to illustrate the behavior of the sequence of pressure-bounded isothermal spheres by plotting <math>\log(R/R_\mathrm{rf})</math> versus <math>\log(P/P_\mathrm{rf})</math>, rather than "P" versus "V;" the result is the curve labeled "1" in his Fig. 1b, which is reproduced here as the left-hand panel of our Figure 2. It should be clear that the information contained in Whitworth's plot (specifically, his curve labeled "1") is identical to the information contained in the curve that is displayed in Bonner's P-V diagram.

Figure 2: Whitworth's R-P Diagram

In an effort to demonstrate more explicitly how Bonnor's 1956 work and Whitworth's 1981 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 <math>P_e/P_\mathrm{rf}</math> versus <math>V/V_\mathrm{rf}</math>, and the right-hand panel of our Figure 2 shows a plot of <math>\log(R/R_\mathrm{rf})</math> versus <math>\log(P_e/P_\mathrm{rf})</math> that we have constructed using data drawn directly from Table 14 of Emden (1907).

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 <math>dP_e/dV</math> — or, in Whitworth's discussion, <math>dP_e/dR</math> — first goes to zero.

As is shown in his equation (3.1), Bonnor (1956) determined that, at a fixed mass and sound speed, <math>dP/dV</math> goes to zero when,

<math>1-\frac{1}{2} e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2 = 0 \, .</math>

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

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

<math> 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} \, , </math>

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,

<math>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}\, .</math>

Dividing this expression for <math>M_R</math> by the expression for <math>R</math> gives,

<math> \frac{M_R}{R} = 2.4 \biggl( \frac{c_s^2}{G} \biggr) \, , </math>

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:

<math>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 \, ,</math>

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

Finally, using this relation to eliminate <math>\rho_c</math> from the expression for <math>M_R</math> gives,

<math>M_R</math>	=	<math>\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</math>
	<math>\approx</math>	<math> \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\} </math>
	<math>\approx</math>	<math> 1.18 \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \, . </math>

This is the mass upper limit — the so-called Bonnor-Ebert mass — for a stable, pressure-bounded isothermal sphere. In a separate discussion, we compare this result to the determinations of other related mass upper-limits.

Related Wikipedia Discussions

• Bonnor-Ebert Mass

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation