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Virial Equilibrium of Isothermal Spheres

Whitworth's (1981) Isothermal Free-Energy Surface
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Review

In an introductory discussion of the virial equilibrium structure of spherically symmetric configurations — see especially the section titled, Energy Extrema — we deduced that a system's equilibrium radius, ~R_\mathrm{eq}, measured relative to a reference length scale, ~R_0, i.e., the dimensionless equilibrium radius,

~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_0} \, ,

is given by the root(s) of the following equation:


2C \chi^{-2}  + ~ (1-\delta_{1\gamma_g})~3 B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} 3B_I ~-~3A\chi^{-1}  -~ 3D\chi^3 = 0 \, ,

where the definitions of the various coefficients are,

~A

~\equiv

\frac{1}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,

~B

~\equiv


K M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot} }{4\pi R_0^3} \biggr)^{\gamma_g - 1}  
\cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}}  
= \bar{c_s}^2 M_\mathrm{tot} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma_g}} \, ,

~B_I

~\equiv


c_s^2 M_\mathrm{tot}  \, ,

~C

~\equiv


\frac{5J^2}{4M_\mathrm{tot} R_0^2} \cdot \frac{\mathfrak{f}_M}{\mathfrak{f}_T} \, ,

~D

~\equiv


\frac{4}{3} \pi R_0^3 P_e \, .

Once the pressure exerted by the external medium (~P_e), and the configuration's mass (~M_\mathrm{tot}), angular momentum (~J), and specific entropy (via ~K) — or, in the isothermal case, sound speed (~c_s) — have been specified, the values of all of the coefficients are known and ~\chi_\mathrm{eq} can be determined.

Isolated, Nonrotating Configuration

For a nonrotating configuration ~(C=J=0) that is not influenced by the effects of a bounding external medium ~(D=P_e = 0), the statement of virial equilibrium is,

~
(1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1}   = 0 \, .

Isothermal Evolutions

For isothermal configurations ~(\delta_{1\gamma_g} = 1), one and only one equilibrium state arises where,

~
B_I = A\chi^{-1} \, ,

that is,

~
R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \frac{A}{B_I}\cdot R_0 = \frac{GM}{5c_s^2} \, .

Nonrotating Configuration Embedded in an External Medium

For a nonrotating configuration (C = J = 0) that is embedded in, and is influenced by the pressure Pe of, an external medium, the statement of virial equilibrium is,


(1-\delta_{1\gamma_g})~3 B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} 3B_I ~-~3A\chi^{-1}  -~ 3D\chi^3 = 0 \, .

Bounded Isothermal

For isothermal configurations (\delta_{1\gamma_g} = 1), we deduce that equilibrium states exist at radii given by the roots of the equation,


3B_I ~-~3A\chi^{-1}  -~ 3D\chi^3 = 0 \, .

Bonnor's (1956) Equivalent Relation

Inserting the expressions for the coefficients ~B_I, ~A, and ~D gives,

~
3Mc_s^2 ~- \frac{3}{5} \frac{GM^2}{R}  = 3 P_e \biggl( \frac{4\pi}{3} R^3\biggr) \, ,

or, because the volume ~V = (4\pi R^3/3) for a spherical configuration, we can write,


~3P_e V = 3Mc_s^2 ~- \frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/3} \frac{GM^2}{V^{1/3}}  \, .

It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that was presented in 1956 by Bonnor (1956, MNRAS, 116, 351) as equation (1.2) in a paper titled, "Boyle's Law and Gravitational Instability":

Bonnor (1956, MNRAS, 116, 351)

Once we realize that, for an isothermal configuration, twice the thermal energy content, ~2S, can be written as ~(3NkT) just as well as via the product, ~(3Mc_s^2), we see that our expression is identical to Bonnor's if we set the prefactor on Bonnor's last term, ~\alpha = (4\pi/3)^{1/3}/5. (Indeed, later on the first page of his paper, Bonnor points out that this is the appropriate value for ~\alpha when considering a uniform density sphere.)

P-V Diagram

Returning to the dimensionless form of this expression and multiplying through by ~[-\chi/(3D)], we obtain,


\chi^4 - \frac{B_I}{D} \chi + \frac{A}{D} = 0 \, .

Now, taking a cue from the solution presented above for an isolated isothermal configuration, we choose to set the previously unspecified scale factor, ~R_0, to,


R_0 = \frac{GM}{5c_s^2} \, ,

in which case ~B_I = A, and the quartic equation governing the radii of equilibrium states becomes, simply,


\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, ,

where,


\Pi \equiv \frac{D}{B_I} = \frac{4\pi R_0^3 P_e}{3Mc_s^2} = \frac{4\pi P_e G^3 M^2}{3\cdot 5^3 c_s^8} \, .

For a given choice of ~P_e and ~c_s, ~\Pi^{1/2} can represent a dimensionless mass, in which case,


M = \Pi^{1/2} \biggl( \frac{3\cdot 5^3}{2^2\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .

Alternatively, for a given choice of configuration mass and sound speed, this parameter, ~\Pi, can be viewed as a dimensionless external pressure; or, for a given choice of ~M and ~P_e, ~\Pi^{-1/8} can represent a dimensionless sound speed. In most of what follows we will view ~\Pi as a dimensionless external pressure.

The above quartic equation can be rearranged immediately to give the external pressure that is required to obtain a particular configuration radius, namely,


\Pi = \frac{(\chi - 1)}{\chi^4} \, .

The resulting behavior is shown by the black curve in Figure 2.

Figure 2: Equilibrium Isothermal P-V Diagram

The black curve traces out the function,

~
\Pi = (\chi - 1)/\chi^4 \, ,

and shows the dimensionless external pressure, ~\Pi, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius ~\chi. The pressure becomes negative at radii ~\chi < 1, hence the solution in this regime is unphysical.

Figure 1 displays the free energy surface that "lies above" the two-dimensional parameter space ~(1.2 < \chi < 1.51; ~0.103 < \Pi < 0.104) that is identified here by the thin, red rectangle.

Equilibrium P-R Diagram

In the absence of self-gravity (i.e., ~A=0), the product of the external pressure and the volume should be constant. The corresponding relation, ~\Pi = \chi^{-3}, is shown by the blue dashed curve in the figure. As the figure illustrates, when gravity is included the P-V relationship pulls away from the PV = constant curve at sufficiently small volumes. Indeed, the curve turns over at a finite pressure, ~\Pi_\mathrm{max}, and for every value of ~\Pi < \Pi_\mathrm{max} a second, more compact equilibrium configuration appears. The location of ~\Pi_\mathrm{max} along the curve is identified by setting ~\partial\Pi/\partial\chi = 0, that is, it occurs where,


\frac{\partial\Pi}{\partial\chi} = -4 \chi^{-5}(\chi - 1) + \chi^{-4} = 0 \, ,


\Rightarrow ~~~~~ \chi = \frac{2^2}{3} \approx 1.333333 \, .

Hence,

~\Pi_\mathrm{max} = \biggl( \frac{2^2}{3} \biggr)^{-4} \biggl( \frac{2^2}{3}-1 \biggr) = \frac{3^3}{2^8} \approx 0.105469\, ;

therefore, from above,

~
M_\mathrm{max} = \biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2}
\approx 1.77408 \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .

Quartic Solution

In the above ~P-V diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give ~\Pi for any chosen value of ~\chi. Alternatively, the four roots of the quartic equation — ~\chi_1, ~\chi_2, ~\chi_3 and ~\chi_4 in the presentation that follows — will identify the radii at which a spherical configuration will be in equilibrium for any choice of the external pressure, ~\Pi, assuming the roots are real.

Roots of the quartic equation: χ4 − χΠ − 1 + Π − 1 = 0

~\chi_1

~=

~
+\frac{1}{2} y_r^{1/2} + \frac{1}{2} D_q \, ;

~\chi_2

~=

~
+\frac{1}{2} y_r^{1/2} - \frac{1}{2} D_q \, ;

~\chi_3

~=

~
-\frac{1}{2} y_r^{1/2} + \frac{1}{2} E_q \, ;

~\chi_4

~=

~
-\frac{1}{2} y_r^{1/2} - \frac{1}{2} E_q \, ,

where,

~D_q

~\equiv

~
y_r^{1/2} \biggl[ \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2}  \, ,

~E_q

~\equiv

~
y_r^{1/2} \biggl[ - \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2}  \, ,

and,

~
y_r \equiv \biggl( \frac{1}{2\Pi^2} \biggr)^{1/3} \biggl\{ \biggl[ 1 + \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3} + \biggl[ 1 - \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3}  \biggr\} \, ,

is the real root of the cubic equation,

~
y^3 - \frac{4y}{\Pi} - \frac{1}{\Pi^{2}} = 0 \, .

Because ~\Pi must be positive in physically realistic solutions, we conclude that the two roots involving ~E_q — that is, ~\chi_3 and ~\chi_4 — are imaginary and, hence, unphysical. The other two roots — ~\chi_1 and ~\chi_2 — will be real only if the arguments inside the radicals in the expression for ~y_r are positive. That is, ~\chi_1 and ~\chi_2 will be real only for values of the dimensionless external pressure,

~\Pi \leq \Pi_\mathrm{max} \equiv \frac{3^3}{2^8} \, .

This is the same upper limit on the external pressure that was derived above, via a different approach, and translates into a maximum mass for a pressure-bounded isothermal configuration of,

~M_\mathrm{max} = \Pi_\mathrm{max}^{1/2}  \biggl(\frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2}
= \biggl(\frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \, .


When combined, a plot of ~\chi_1 versus ~\Pi and ~\chi_2 versus ~\Pi will reproduce the solid black curve shown in Figure 2, but with the axes flipped. The top-right quadrant of Figure 3 presents such a plot, but in logarithmic units along both axes; also ~\Pi is normalized to ~\Pi_\mathrm{max} and ~\chi is normalized to the equilibrium radius ~(4/3) at that pressure. This is the manner in which Whitworth (1981, MNRAS, 195, 967) chose to present this result for uniform-density, spherical isothermal ~(\gamma_\mathrm{g}=1) configurations. Our solid and dashed curve segments — identifying, respectively, the ~\chi_1(\Pi) and ~\chi_2(\Pi) solutions to the above quadratic equation — precisely match the solid and dashed curve segments labeled "1" in Whitworth's Figure 1a (replicated here in the bottom-right quadrant of Figure 3).

Figure 3: Equilibrium R-P Diagram

Top: The solid curve traces the function ~\chi_1(\Pi) and the dashed curve traces the function ~\chi_2(\Pi), where ~\chi_1 and ~\chi_2 are the two real roots of the quartic equation,

~
\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, .

Logarithmic units are used along both axes; ~\Pi is normalized to ~\Pi_\mathrm{max}; and ~\chi is normalized to the equilibrium radius ~(4/3) at ~\Pi_\mathrm{max}.

Bottom: A reproduction of Figure 1a from Whitworth (1981, MNRAS, 195, 967). The solid and dashed segments of the curve labeled "1" identify the equilibrium radii, ~R_\mathrm{eq}, that result from embedding a uniform-density, isothermal ~(\gamma_\mathrm{g} = 1) gas cloud in an external medium of pressure ~P_\mathrm{ex}.

Comparison: The curve shown above that traces out ~\chi_1(\Pi) and ~\chi_2(\Pi) should be identical to the "Whitworth" curve labeled "1".

To be compared with Whitworth (1981)
Whitworth (1981) Figure 1a

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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