# Virial Equilibrium of Isothermal Spheres

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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":

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.

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".