User:Tohline/SphericallySymmetricConfigurations/Virial
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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,
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<math>A</math> |
<math>\equiv</math> |
<math>\frac{3}{5} \frac{GM^2}{R_0} \, ,</math> |
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<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> |
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<math>B_I</math> |
<math>\equiv</math> |
<math> 3c_s^2 M =3 KM \, , </math> |
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<math>C</math> |
<math>\equiv</math> |
<math> \frac{5J^2}{4MR_0^2} \, , </math> |
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<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} = \frac{1}{R_0} \biggl[ A\chi^{-2} +~ (1-\delta_{1\gamma_g})~3(1 - \gamma_g) B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} B_I \chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2 \biggr] \, . </math>
where,
<math>\chi \equiv \frac{R}{R_0} \, .</math>
So <math>\chi_\mathrm{eq} \equiv (R/R_0)_\mathrm{eq}</math> is obtained from the real root(s) of the equation,
<math> A \chi^{-2} +~ (1-\delta_{1\gamma_g})~3(1 - \gamma_g) B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} B_I \chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2 = 0 \, , </math>
or, equivalently, from the roots of the equation,
<math> 2C \chi^{-2} + ~ (1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1} -~ 3D\chi^3 = 0 \, . </math>
As a definition of equilibrium states, this last expression is also the well-known scalar virial equation, derivable from the first moment of the equation of motion. A more recognizable expression can be obtained by replacing each of the terms by the energy contents that they represent:
<math> 2(T_\mathrm{rot} + S) + W - 3P_e V = 0 \, . </math>
In this expression, <math>S</math> is the thermal energy content of the configuration; the relationship between <math>S</math> and the configuration's total internal energy, <math>U</math>, is provided in our associated derivation of both the adiabatic and isothermal free energy functions.
Examples
Isolated, Nonrotating Configuration
For a nonrotating configuration <math>(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>(D=P_e = 0)</math>, the statement of virial equilibrium is,
<math> (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 \, . </math>
Adiabatic
For adiabatic configurations <math>(\delta_{1\gamma_g} = 0)</math>, one equilibrium state exists and it occurs where,
<math> 3(\gamma_g-1) B\chi^{3 -3\gamma_g} = A\chi^{-1} \, , </math>
that is, where,
<math> R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \biggl[ \frac{3(\gamma_g-1) B}{A} \cdot R_0^{(3\gamma_g-4)} \biggr]^{1/(3\gamma_g-4)} = \biggl[ 5\biggl( \frac{3}{4\pi} \biggr)^{\gamma_g-1} \cdot \frac{KM^{(\gamma_g-2)}}{G} \biggr]^{1/(3\gamma_g-4)} \, . </math>
Notice that, for <math>\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass. Conversely, notice that, for <math>\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius. For all other values of <math>\gamma_g</math>, the equilibrium mass-radius relationship for adiabatic configurations is,
<math> M^{(\gamma_g - 2)} \propto R_\mathrm{eq}^{(3\gamma_g -4)} \, . </math>
This means that, for <math>\gamma_g</math> > <math> 2</math> or <math>\gamma_g </math>< <math>4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii. However, for <math>\gamma_g</math> in the range, <math>2</math> > <math>\gamma_g </math> > <math>4/3</math>, configurations with larger mass have smaller equilibrium radii.
Isothermal
For isothermal configurations <math>(\delta_{1\gamma_g} = 1)</math>, one and only one equilibrium state arises where,
<math> B_I = A\chi^{-1} \, , </math>
that is,
<math> R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \frac{A}{B_I}\cdot R_0 = \frac{GM}{5c_s^2} \, . </math>
Note that this result could have been obtained by setting <math>\gamma_g = 1</math> in the adiabatic solution, because <math>K = c_s^2</math> when <math>\gamma_g = 1</math>.
Nonrotating Configuration Embedded in an External Medium
For a nonrotating configuration <math>(C=J=0)</math> that is embedded in, and is influenced by the pressure <math>P_e</math> of, an external medium, the statement of virial equilibrium is,
<math> (1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1} -~ 3D\chi^3 = 0 \, . </math>
Adiabatic
Hence, for adiabatic configurations <math>(\delta_{1\gamma_g} = 0)</math>, equilibrium states exist at radii given by the roots of the following expression:
<math> 3(\gamma_g-1) B\chi^{3 -3\gamma_g} ~-~A\chi^{-1} -~ 3D\chi^3 = 0 \, . </math>
This is precisely the same condition that derives from setting equation (3) to zero in Whitworth's (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polyropes." The overlap with Whitworth's narative is perhaps clearer after introducing the algebraic expressions for the coefficients <math>A</math>, <math>B</math>, and <math>D</math>, dividing the equation through by <math>(3\chi^3 V_0) = (4\pi R^3)</math>, and rewriting <math>R</math> as <math>R_\mathrm{eq}</math> to obtain,
<math> P_e = K \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} - \biggl( \frac{3GM^2}{20\pi R_\mathrm{eq}^4} \biggr) \, . </math>
This exactly matches equation (5) of Whitworth, which reads:
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Ideally we would like to invert this equation to obtain an analytic expression for the configuration's equilibrium radius in terms of the physical parameters, <math>M</math>, <math>K</math>, and <math>P_e</math>. However, this cannot be accomplished for an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>.
Isothermal
Show that derived result is, essentially, the Bonnor-Ebert sphere. Should we also draw analogy with collapse of isothermal core of red giant, or leave this to the later stability discussion?
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© 2014 - 2021 by Joel E. Tohline |