User:Tohline/VE

Virial Equation

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Free Energy Expression

Associated with any isolated, self-gravitating, gaseous configuration we can identify a total Gibbs-like free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,

<math> \mathfrak{G} = W - \mathfrak{W}_\mathrm{therm} + T_\mathrm{rot} + P_e V + \cdots </math>

Here, we have explicitly included the gravitational potential energy, <math>W</math>, the rotational kinetic energy, <math>T_\mathrm{rot}</math>, a term that accounts for surface effects if the configuration of volume <math>V</math> is embedded in an external medium of pressure <math>P_e</math>, and <math>\mathfrak{W}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts. [See Chandrasekhar & Fermi (1953, ApJ, 118, 116) and Mestel & Spitzer (1956, MNRAS, 116, 503) for early discussions that also take into account the energy associated with a magnetic field that threads through the configuration.]

Expressions for each of the three component energies, <math>~W, \mathfrak{W}_\mathrm{therm},</math> and <math>~T_\mathrm{rot},</math> are obtained by first defining an expression for the relevant energy per unit mass, then integrating that function across the configuration's mass distribution. We begin by discussing <math>~\mathfrak{W}_\mathrm{therm},</math> which is probably the least familiar term in our expression for the free energy, <math>\mathfrak{G}</math>.

Reservoir of Thermodynamic Energy

<math>~\mathfrak{W}_\mathrm{therm}</math> derives from the differential, "PdV" work that is often discussed in the context of thermodynamic systems. It should be made clear that, here, "dV" refers to the differential volume per unit mass, so it should be written as "<math>~d(\rho^{-1})</math>", to be consistent with the notation used throughout this H_Book. Therefore, the differential thermodynamic work is,

<math>dw = Pd(1/\rho) = - \biggl( \frac{P}{\rho^2} \biggr) d\rho \, .</math>

Isothermal Systems

If each element of gas maintains its temperature when the system undergoes compression or expansion — that is, if the compression/expansion is isothermal — then,

<math>~P = c_s^2 \rho \, ,</math>

where the constant, <math>~c_s</math>, is the isothermal sound speed and the expression for the differential thermodynamic work becomes,

<math>dw = - \biggl( \frac{c_s^2}{\rho} \biggr) d\rho = - c_s^2 d\ln\rho \, .</math>

Hence, to within an additive constant, we have,

<math>w = - c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) \, ,</math>

where, <math>~\rho_0</math> is a (as yet unspecified) reference density, and integration throughout the configuration gives (for the isothermal case),

<math>\mathfrak{W}_\mathrm{therm} = - \int c_s^2 \ln \biggl(\frac{\rho}{\rho_0}\biggr) dm \, .</math>

Adiabatic Systems

If, upon compression or expansion, the gaseous configuration behaves adiabatically, the pressure will vary with density as,

<math>P = K \rho^{\gamma_g} \, ,</math>

where, <math>K</math> specifies the specific entropy of the gas and <math>~\gamma_\mathrm{g}</math> is the ratio of specific heats that is relevant to the phase of compression/expansion. In this case, the expression for the differential thermodynamic work becomes,

<math>dw = - K \rho^{{\gamma_g}-2} d\rho = - \frac{K}{({\gamma_g}-1)} d\rho^{{\gamma_g}-1} \, .</math>

Hence, to within an additive constant, we have,

<math>w = - \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) \, ,</math>

and integration throughout the configuration gives (for the adiabatic case),

<math>\mathfrak{W}_\mathrm{therm} = - \int \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) dm = \frac{2}{3({\gamma_g}-1)} S = U \, ,</math>

where, <math>~S</math> is the system's total thermal energy, and <math>~U</math> is the system's corresponding total internal energy.

Uniform-density, Uniformly Rotating Sphere

For a uniform-density, uniformly rotating, spherically symmetric configuration of mass <math>M</math> and radius <math>R</math>,

<math> W </math>	<math>=</math>	<math> - \frac{3}{5} \frac{GM^2}{R} \, , </math>
<math> T_\mathrm{rot} </math>	<math>=</math>	<math> \frac{1}{2} I \omega^2 = \frac{J^2}{2I} = \frac{5}{4} \frac{J^2}{MR^2} \, , </math>
<math> V </math>	<math>=</math>	<math> \frac{4}{3} \pi R^3 \, , </math>

where, <math>~G</math> is the gravitational constant, <math>I=(2/5)MR^2</math> is the moment of inertia, <math>\omega</math> is the angular frequency of rotation, and <math>J=I\omega</math> is the total angular momentum.

Adiabatic

Consider the situation where, upon compression or expansion, the gaseous configuration behaves adiabatically, in which case the pressure will vary with density as,

<math>P = K \rho^{\gamma_g} \, ,</math>

where, <math>K</math> specifies the specific entropy of the gas and <math>~\gamma_\mathrm{g}</math> is the ratio of specific heats. The total thermal energy is, then,

<math> S = \frac{3}{2} NkT = \frac{3}{2} \frac{\mathfrak{R}}{\mu} TM = \frac{3}{2} \frac{P}{\rho} M = \frac{3}{\gamma_g} \biggl[ \frac{1}{2} M a_s^2 \biggr] \, , </math>

and the total internal energy is,

<math> U = \frac{2}{3 (\gamma_g - 1)} S </math>

<math>=</math>

<math> \frac{M a_s^2}{\gamma_g (\gamma_g - 1)} \, , </math>

where the square of the (adiabatic) sound speed,

<math>a_s^2 \equiv \frac{\partial P}{\partial\rho} = \gamma_g \frac{P}{\rho} = \gamma_g K \rho^{\gamma_g-1} \, .</math>

Appreciating that <math>\rho = M/V</math> for the case being considered here (i.e., for a uniform-density sphere), the adiabatic free energy can be written as,

<math> \mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, , </math>

where, <math>R_0</math> is an, as yet unspecified, scale length,

<math>A</math>	<math>\equiv</math>	<math>\frac{3}{5} \frac{GM^2}{R_0} \, ,</math>
<math>B</math>	<math>\equiv</math>	<math> \biggl[ \frac{K}{(\gamma_g-1)} \biggl( \frac{3}{4\pi R_0^3} \biggr)^{\gamma_g - 1} \biggr] M^{\gamma_g} \, , </math>
<math>C</math>	<math>\equiv</math>	<math> \frac{5J^2}{4MR_0^2} \, , </math>
<math>D</math>	<math>\equiv</math>	<math> \frac{4}{3} \pi R_0^3 P_e \, . </math>

Isothermal

If, upon compression or expansion, the configuration remains isothermal — in which case <math>\gamma_g =1</math> — then both the (isothermal) sound speed, <math>c_s</math>, and the total thermal energy, <math>S=(3/2)c_s^2 M</math>, are constant. But as pointed out, for example, in Appendix A of Stahler (1983, ApJ, 268, 16), the total internal energy will vary according to the relation,

<math> U </math>

<math>=</math>

<math> \frac{2}{3} S \ln\rho \, . </math>

Again appreciating that <math>\rho = M/V</math> for the case being considered here (i.e., for a uniform-density sphere), to within an additive constant the isothermal free energy can be written as,

<math> \mathfrak{G} = -A \biggl( \frac{R}{R_0} \biggr)^{-1} - B_I \ln \biggl( \frac{R}{R_0} \biggr) + C \biggl( \frac{R}{R_0} \biggr)^{-2} + D\biggl( \frac{R}{R_0} \biggr)^3 \, , </math>

where, aside from the coefficient definitions provided above in association with the adiabatic case,

<math>B_I</math>

<math>\equiv</math>

<math> 3c_s^2 M =3 KM \, , </math>

Summary

We can combine the two cases — adiabatic and isothermal — into a single expression for <math>\mathfrak{G}</math> through a strategic use of the Kroniker delta function, <math>\delta_{1\gamma_g}</math>, as follows:

<math> \mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +~ (1-\delta_{1\gamma_g})B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} -~ \delta_{1\gamma_g} B_I \ln \biggl( \frac{R}{R_0} \biggr) +~ C \biggl( \frac{R}{R_0} \biggr)^{-2} +~ D\biggl( \frac{R}{R_0} \biggr)^3 \, , </math>

Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M</math>), angular momentum (<math>J</math>), and specific entropy (via <math>K</math>) — or, in the isothermal case, sound speed (<math>c_s</math>) — have been specified, the values of all of the coefficients are known and this algebraic expression for <math>\mathfrak{G}</math> describes how the free energy of the configuration will vary with the configuration's size (<math>R</math>) for a given choice of <math>\gamma_g</math>.

Whitworth (1981)

The above presentation closely parallels Whitworth's (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polyropes." He introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"

<math> \mathfrak{u} </math>	<math> = </math>	<math> \mathfrak{g} + \mathfrak{B}_\mathrm{in} + \mathfrak{B}_\mathrm{ex} </math>
	<math>=</math>	<math> ~~~ - \frac{3}{5} \frac{GM_0^2}{R_0} \biggl(\frac{R}{R_0} \biggr)^{-1} + (1-\delta_{1\eta})\biggl[ \frac{KM_0^\eta}{(\eta - 1)} V_0^{(1-\eta)} \biggr] \biggl(\frac{R}{R_0}\biggr)^{3(1-\eta)} - \delta_{1\eta} \biggl[ 3KM_0 \ln\biggl(\frac{R}{R_0} \biggr) \biggr] </math>
		<math> + P_\mathrm{ex} V_0 \biggl( \frac{R}{R_0} \biggr)^{3} </math>

Clearly Whitworth's global potential function, <math>\mathfrak{u}</math>, is what we have referred to as the configuration's "Gibbs-like" free energy, with <math>\eta</math> being used rather than <math>\gamma_g</math> to represent the ratio of specific heats in the adiabatic case. Our expression for <math>\mathfrak{G}</math> would precisely match his expression for <math>\mathfrak{u}</math> if we chose to examine the free energy of a nonrotating configuration, that is, if we set <math>C=J=0</math>.

© 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