SSC/Virial/Isothermal

Virial Equilibrium of Isothermal Spheres

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, $R_{\mathrm{e}\mathrm{q}}$, measured relative to a reference length scale, $R_0$, i.e., the dimensionless equilibrium radius,

${\chi }_{\mathrm{e}\mathrm{q}}\equiv \frac{R_{\mathrm{e}\mathrm{q}}}{R_0},$

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

$2C{\chi }^{-2}+(1-{\delta }_{1{\gamma }_g})3B{\chi }^{3-3{\gamma }_g}+{\delta }_{1{\gamma }_g}3B_I-3A{\chi }^{-1}-3D{\chi }^3=0,$

where the definitions of the various coefficients are,

$A$	$\equiv$	$\frac{1}{5}\frac{G{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}^2}{R_0}\cdot \frac{{\mathfrak{𝔣}}_W}{{\mathfrak{𝔣}}_M^2},$
$B$	$\equiv$	$K{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}(\frac{3M_{\mathrm{t}\mathrm{o}\mathrm{t}}}{4\pi R_0^3}{)}^{{\gamma }_g-1}\cdot \frac{{\mathfrak{𝔣}}_A}{{\mathfrak{𝔣}}_M^{{\gamma }_g}}={\overline{c_s}}^2{M}_{\mathrm{t}\mathrm{o}\mathrm{t}}\cdot \frac{{\mathfrak{𝔣}}_A}{{\mathfrak{𝔣}}_M^{{\gamma }_g}},$
$B_I$	$\equiv$	$c_s^2{M}_{\mathrm{t}\mathrm{o}\mathrm{t}},$
$C$	$\equiv$	$\frac{5J^2}{4M_{\mathrm{t}\mathrm{o}\mathrm{t}}{R}_0^2}\cdot \frac{{\mathfrak{𝔣}}_M}{{\mathfrak{𝔣}}_T},$
$D$	$\equiv$	$\frac{4}{3}\pi R_0^3{P}_e.$

Once the pressure exerted by the external medium ($P_e$), and the configuration's mass ($M_{\mathrm{t}\mathrm{o}\mathrm{t}}$), 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 ${\chi }_{\mathrm{e}\mathrm{q}}$ 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,

$R_{\mathrm{e}\mathrm{q}}=R_0{\chi }_{\mathrm{e}\mathrm{q}}=\frac{A}{B_I}\cdot R_0=\frac{GM}{5c_s^2}.$

Nonrotating Configuration Embedded in an External Medium

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

$(1-{\delta }_{1{\gamma }_g})3B{\chi }^{3-3{\gamma }_g}+{\delta }_{1{\gamma }_g}3B_I-3A{\chi }^{-1}-3D{\chi }^3=0.$

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,

$3M{c}_s^2-\frac{3}{5}\frac{G{M}^2}{R}=3P_e\left(\frac{4\pi }{3}{R}^3\right),$

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

$3P_{e}V=3M{c}_s^2-\frac{3}{5}(\frac{4\pi }{3}{)}^{1/3}\frac{G{M}^2}{V^{1/3}}.$

It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that appears as Eq. (1.2) in 📚 W. B. Bonnor (1956, MNRAS, Vol. 116, pp. 351 - 359):

Reprint of the opening (introductory) paragraph from … W. B. Bonnor (1956) Boyle's Law and gravitational instability Monthly Notices of the Royal Astronomical Society Vol. 116, pp. 351 - 359

"It has recently been suggested by Terletsky† that for a large mass $M$ of gas, of volume $V$ and temperature $T$, containing $N$ molecules under boundary pressure $p$, the equation of state should be not $PV$ $=$ $NkT$ (1.1) but $PV$ $=$ $NkT-\alpha G{M}^2{V}^{-1/3},$ (1.2) where $k$ is Boltzmann's constant, $G$ is Newton's constant of gravitation, and $\alpha$ is a constant depending on the shape of the mass. The proposed correction of Boyle's Law arises because, for a large mass, one has to take account of the gravitational interactions between the molecules."

†Y. P. Terletsky (1952, Zh. Eksper. Teor. Fiz., Vol. 22, p. 506) Notes from J. E. Tohline regarding this referenced article: The full title of this (Russian language) journal is, Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki, sometimes abbreviated as, ZhETF. English translations of ZhETF articles dating back to 1967 can be found in the Journal of Experimental and Theoretical Physics (JETP); I have been unable to find an English translation (or even the original Russian-language version) of Terletsky's 1952 article. A more accessible article by I. L. Genkin (1966, Astronomicheskii Zhurnal, Vol. 43, p. 96) heavily references Terletsky's work.

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, $(3M{c}_s^2)$, we see that our expression is identical to the one derived by 📚 Bonnor (1956) if we set the prefactor on his last term, $\alpha =(4\pi /3{)}^{1/3}/5$. (Indeed, later on the first page of his paper, 📚 Bonnor (1956) 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,

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

where,

$\mathrm{\Pi }\equiv \frac{D}{B_I}=\frac{4\pi R_0^3{P}_e}{3M{c}_s^2}=\frac{4\pi P_e{G}^3{M}^2}{3\cdot 5^3{c}_s^8}.$

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

$M={\mathrm{\Pi }}^{1/2}\left(\frac{3\cdot 5^3}{2^2\pi }{)}^{1/2}\right(\frac{c_s^8}{P_e{G}^3}{)}^{1/2}.$

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

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

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, $\mathrm{\Pi }=(\chi -1)/{\chi }^4,$ and shows the dimensionless external pressure, $\mathrm{\Pi }$, 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 $(1.2<\chi <1.51$; $0.103<\mathrm{\Pi }<0.104)$ 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, $\mathrm{\Pi }={\chi }^{-3}$, 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, ${\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and for every value of $\mathrm{\Pi }<{\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ a second, more compact equilibrium configuration appears. The location of ${\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ along the curve is identified by setting $\partial \mathrm{\Pi }/\partial \chi =0$, that is, it occurs where,

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

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

Hence,

${\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}=\left(\frac{2^2}{3}{)}^{-4}\right(\frac{2^2}{3}-1)=\frac{3^3}{2^8}\approx 0.105469;$

therefore, from above,

$M_{\mathrm{m}\mathrm{a}\mathrm{x}}=\left(\frac{3^4\cdot 5^3}{2^{10}\pi }{)}^{1/2}\right(\frac{c_s^8}{P_e{G}^3}{)}^{1/2}\approx 1.77408(\frac{c_s^8}{P_e{G}^3}{)}^{1/2}.$

Quartic Solution

In the above $P-V$ diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give $\mathrm{\Pi }$ 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, $\mathrm{\Pi }$, assuming the roots are real.

Roots of the quartic equation: ${\chi }^4-\chi {\mathrm{\Pi }}^{-1}+{\mathrm{\Pi }}^{-1}=0$

${\chi }_1$ $=$ $+\frac{1}{2}{y}_r^{1/2}+\frac{1}{2}{D}_q;$ ${\chi }_2$ $=$ $+\frac{1}{2}{y}_r^{1/2}-\frac{1}{2}{D}_q;$ ${\chi }_3$ $=$ $-\frac{1}{2}{y}_r^{1/2}+\frac{1}{2}{E}_q;$ ${\chi }_4$ $=$ $-\frac{1}{2}{y}_r^{1/2}-\frac{1}{2}{E}_q,$ where, $D_q$ $\equiv$ $y_r^{1/2}[\frac{2}{\mathrm{\Pi }}{y}_r^{-3/2}-1{]}^{1/2},$ $E_q$ $\equiv$ $y_r^{1/2}[-\frac{2}{\mathrm{\Pi }}{y}_r^{-3/2}-1{]}^{1/2},$ and, $y_r\equiv \left(\frac{1}{2{\mathrm{\Pi }}^2}{)}^{1/3}\right\{[1+\sqrt{1-\frac{2^8}{3^3}\mathrm{\Pi }}{]}^{1/3}+[1-\sqrt{1-\frac{2^8}{3^3}\mathrm{\Pi }}{]}^{1/3}\},$ is the real root of the cubic equation, $y^3-\frac{4y}{\mathrm{\Pi }}-\frac{1}{{\mathrm{\Pi }}^2}=0.$

Because $\mathrm{\Pi }$ 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,

$\mathrm{\Pi }\le {\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\equiv \frac{3^3}{2^8}.$

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,

$M_{\mathrm{m}\mathrm{a}\mathrm{x}}={\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}^{1/2}\left(\frac{3\cdot 5^3}{2^2\pi }{)}^{1/2}\right(\frac{c_s^8}{G^3{P}_e}{)}^{1/2}=\left(\frac{3^4\cdot 5^3}{2^{10}\pi }{)}^{1/2}\right(\frac{c_s^8}{G^3{P}_e}{)}^{1/2}.$

When combined, a plot of ${\chi }_1$ versus $\mathrm{\Pi }$ and ${\chi }_2$ versus $\mathrm{\Pi }$ 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 $\mathrm{\Pi }$ is normalized to ${\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ 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 ${\chi }_1(\mathrm{\Pi })$ and ${\chi }_2(\mathrm{\Pi })$ 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 ${\chi }_1(\mathrm{\Pi })$ and the dashed curve traces the function ${\chi }_2(\mathrm{\Pi })$, where ${\chi }_1$ and ${\chi }_2$ are the two real roots of the quartic equation, ${\chi }^4-\frac{\chi }{\mathrm{\Pi }}+\frac{1}{\mathrm{\Pi }}=0.$ Logarithmic units are used along both axes; $\mathrm{\Pi }$ is normalized to ${\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$; and $\chi$ is normalized to the equilibrium radius $(4/3)$ at ${\mathrm{\Pi }}_{\mathrm{m}\mathrm{a}\mathrm{x}}$. 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, $R_{\mathrm{e}\mathrm{q}}$, that result from embedding a uniform-density, isothermal $({\gamma }_{\mathrm{g}}=1)$ gas cloud in an external medium of pressure $P_{\mathrm{e}\mathrm{x}}$. Comparison: The curve shown above that traces out ${\chi }_1(\mathrm{\Pi })$ and ${\chi }_2(\mathrm{\Pi })$ should be identical to the "Whitworth" curve labeled "1".	To be compared with Whitworth (1981)
	Whitworth (1981) Figure 1a Whitworth (1981) Figure 1a

See Also

• Index to a Variety of Free-Energy and/or Virial Analyses