User:Tohline/VE

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 + U + T_\mathrm{rot} + P_e V + \cdots </math>

Here, we have explicitly included the gravitational potential energy, <math>W</math>, the total internal energy, <math>U</math>, the rotational kinetic energy, <math>T_\mathrm{rot}</math>, and 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>.

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 \, , </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] + 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