User:Tohline/SphericallySymmetricConfigurations/Virial

Virial Equilibrium

Free Energy Expression (review)

As has been explained elsewhere, associated with any 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>

where, for the purposes of this discussion, 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>. For spherically symmetric configurations that have a uniform density and are uniformly rotating, each of the terms contributing to this free-energy expression can be written as a product of a scalar coefficient and a function of the configuration's radius, <math>R</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>

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

As written here, the coefficient, <math>B</math>, that appears in the definition of the configuration's total internal energy, comes from assuming that the configuration will expand or contract adiabatically, that is, that internally the pressure scales 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> <math>\ne 1</math> is the ratio of specific heats. (Note that the Kroniker delta function <math>\delta_{1\gamma_g} = 0</math>, since <math>\gamma_g \ne 1</math>.) If compressions/expansions occur isothermally (<math>\gamma_g = 1</math>, hence, <math>\delta_{1\gamma_g} = 1</math>), the relevant <math>P-\rho</math> relationship is,

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

Energy Extrema

The free energy "surface" that is mapped out by the function <math>\mathfrak{G}(R/R_0)</math> generally will exhibit multiple local minima and local maxima, and may also possess one or more points of inflection. The locations along the energy surface where these special points arise identify equilibrium states, and the associated values of <math>(R/R_0)_\mathrm{eq}</math> give the radii of the equilibrium configurations.

For a given choice of the set of physical parameters <math>M</math>, <math>K</math>, <math>J</math>, <math>P_e</math>, and <math>\gamma_g</math>, extrema occur wherever,

<math> \frac{d\mathfrak{G}}{dR} = 0 \, . </math>

For the free energy function identified above,

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