Global Energy Considerations

The astrophysics community relies heavily on the virial equations — most often in the context of the scalar virial theorem — to ascertain the basic properties of equilibrium systems. As is described below, fundamentally the virial equations are obtained by taking moments of the Euler equation. By examining the balance among various relevant energy reservoirs, the mathematical expression that defines virial equilibrium provides a means by which, for example, the radius of a configuration can be estimated, given a total system mass and mean system temperature. It can also be used to estimate a system's maximum allowed rotation frequency and whether or not the properties of the equilibrium configuration will be significantly modified if the system is embedded in a hot tenuous external medium.

As is also discussed, below, it can be even more informative to examine how a system's global, Gibbs-like free energy, <math>\mathfrak{G}</math>, varies under contraction or expansion. Extrema in the free energy identify equilibrium configurations, for example. For spherically symmetric systems, in particular, the scalar virial theorem is "derived" by identifying under what conditions <math>~d\mathfrak{G}/dR = 0</math>. Furthermore, the sign of the second derivative, <math>~d^2\mathfrak{G}/dR^2</math>, tells whether or not the equilibrium state is stable or unstable. Here we define relevant energy reservoirs that contribute to a system's global free energy. In separate chapters we use the free energy function to help identify the properties of equilibrium systems and to examine their relative stability.

Virial Equations

Most of the material presented here has been drawn from Chandrasekhar's Ellipsoidal Figures of Equilibrium [hereafter, EFE], first published in 1969. Relying heavily on EFE's in-depth treatment of the topic, our aim is to highlight key aspects of the tensor-virial equations and to present them in a form that serves as a foundation for our separate discussions of the equilibrium and stability of self-gravitation fluid systems. Strong parallels are drawn between the EFE presentation and our own so that it will be relatively straightforward for the reader to consult the EFE publication to obtain details of the various derivations. Text that appears in a green font has been drawn verbatim from this reference.

Setting the Stage

[EFE, §8, p. 15] A standard technique for treating the integro-differential equations of mathematical physics is to take the moments of the equations concerned and consider suitably truncated sets of the resulting equations. The virial method … is essentially the method of the moments applied to the solution of hydrodynamical problems in which the gravitational field of the prevailing distribution of matter is taken into account. The virial equations of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations. In this context, Chandrasekhar's focus is on two of the four principal governing equations that serve as the foundation of our entire H_Book, namely,

Euler Equation
(Momentum Conservation)

<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math>

Poisson Equation

<math>\nabla^2 \Phi = 4\pi G \rho</math>

In Chandrasekhar's EFE presentation, the Euler equation first appears in §11 (p. 20) as equation (38) and is written as,

<math>~- \frac{\partial p}{\partial x_i} + \rho \frac{\partial \mathfrak{B}}{\partial x_i} \, ,</math>

and the Poisson equation appears in §10 (p. 20) as the left-most component of equation (37) as,

<math>~\nabla^2 \mathfrak{B}</math>

It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of <math>~\vec{v}</math> to represent the inertial velocity field. More importantly, he adopts a different variable name and a different sign convention to represent the gravitational potential, specifically,

<math>~ - \Phi = \mathfrak{B} </math>

<math>~ G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math>

Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from EFE into this H_Book.

As we have explained elsewhere, when examining the equilibrium, stability, and dynamical behavior of configurations that are rotating with angular velocity, <math>~\vec\Omega_f</math>, it is useful to reference the

Lagrangian Representation
of the Euler Equation
as viewed from a Rotating Reference Frame

<math>\biggl[ \frac{d\vec{v}}{dt}\biggr]_{rot} = - \frac{1}{\rho} \nabla P - \nabla \Phi - 2{\vec{\Omega}}_f \times {\vec{v}}_{rot} - {\vec{\Omega}}_f \times ({\vec{\Omega}}_f \times \vec{x}) \, .</math>

Chandrasekhar also adopts this tactic. In his EFE presentation, the equivalent expression first appears in §12 as equation (62) and has the form,

<math>~- \frac{\partial p}{\partial x_i} + \rho \frac{\partial \mathfrak{B}}{\partial x_i} + 2\rho \epsilon_{i \ell m}u_\ell \Omega_m + \frac{1}{2} \rho \frac{\partial}{\partial x_i}|\vec\Omega \times \vec{x}|^2 \, ,</math>

where, as noted in EFE [§12, p. 25], <math>~|\vec\Omega \times \vec{x}|^2/2</math> and <math>~2\vec{u} \times \vec\Omega</math> represent the centrifugal potential and the Coriolis acceleration, respectively — also see our related discussion of the centrifugal and Coriolis accelerations.

First-Order Virial Equations

[EFE, §11(a), p. 21] The [virial] equations of the first order are obtained by simply integrating [the Euler equation] over the instantaneous volume, <math>~V</math>, occupied by the fluid. Specifically, using our variable notation,

<math>~\int\limits_V \rho \frac{dv_i}{dt} d^3x</math>

<math>~- \int\limits_V \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho \frac{\partial \Phi}{\partial x_i} d^3x \, ,</math>

leads to (see EFE for details),

where the moments of inertia about the three separate principal axes <math>(i = 1,2,3)</math> are defined by the expressions,

<math>~\equiv</math>

<math>~\int\limits_V \rho x_i d^3x \, .</math>

Thus, the first-order virial equation(s) expresses the uniform motion of the center of mass of the system.

Second-Order Tensor Virial Equations

Derivation

[EFE, §11(b), p. 22] The second-order (tensor) virial equations are obtained by multiplying [the Euler equation] by <math>~x_j</math> and integrating over the volume, <math>~V</math>. Specifically, again using our variable notation,

<math>~\int\limits_V \rho \frac{dv_i}{dt} x_j d^3x</math>

<math>~- \int\limits_V x_j \frac{\partial P}{\partial x_i} d^3x - \int\limits_V \rho x_j \frac{\partial \Phi}{\partial x_i} d^3x \, .</math>

The antisymmetric part of this tensor expression gives (see EFE for details),

<math>~\int\limits_V \rho (v_ix_j - v_j x_i) d^3x</math>

which expresses simply the conservation of the angular momentum of the system. The symmetric part of the tensor expression gives what is generally referred to as (see EFE for details) the,

Tensor Virial Equation

<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math>

where, by definition,

						EFE Reference
<math>~I_{ij}</math>	<math>~\equiv</math>	<math>~\int\limits_V \rho x_i x_j d^3x </math>	…	is the moment of inertia tensor	…	[Eq. (4), p. 16]
<math>~\mathfrak{T}_{ij}</math>	<math>~\equiv</math>	<math>~\frac{1}{2} \int\limits_V \rho v_i v_j d^3x </math>	…	is the kinetic energy tensor	…	[Eq. (9), p. 17]
<math>~\mathfrak{W}_{ij}</math>	<math>~\equiv</math>	<math>~\frac{1}{2} \int\limits_V \rho \Phi_{ij} d^3x </math>			…	[Eq. (15), p. 17]
	<math>~=</math>	<math>~- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_j} d^3x </math>	…	is the gravitational potential energy tensor	…	[Eq. (18), p. 18]
<math>~\Pi</math>	<math>~\equiv</math>	<math>~\int\limits_V P d^3x </math>	…	is two-thirds of the total thermal energy	…	[Eq. (7), p. 16]

Note that, in the definition of the gravitational potential energy tensor, Chandrasekhar has introduced a tensor generalization of the gravitational potential [see his Eq. (14), p. 17], namely,

<math>~ - \Phi_{ij} = \mathfrak{B}_{ij}</math>

<math>~ G\int\limits_V \rho(\vec{x}^') \frac{ (x_i - x_i^')(x_j - x_j^') }{|\vec{x} - \vec{x}^{~'}|^3} d^3x^' \, .</math>

Steady State (Virial Equilibrium)

[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,

<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>

<math>~- \delta_{ij}\Pi \, .</math>

[This] provides six integral relations which must obtain whenever the conditions are stationary.

Scalar Virial Theorem

The trace of the tensor virial equation (TVE), which is obtained by identifying the trace of each term in the TVE, produces the scalar virial equation, which is widely referenced and used by the astrophysics community. Specifically,

						EFE Reference
<math>~I = \sum\limits_{i=1,3} I_{ii}</math>	<math>~=</math>	<math>~\int\limits_V \rho (\vec{x}) \|\vec{x}\|^2 d^3x </math>	…	is the scalar moment of inertia	…	[Eqs. (3) & (5), p. 16]
<math>~\mathfrak{T} = \sum\limits_{i=1,3} \mathfrak{T}_{ii}</math>	<math>~=</math>	<math>~\frac{1}{2} \int\limits_V \rho \|\vec{v}\|^2 d^3x </math>	…	is the total kinetic energy	…	[Eq. (8), p. 16]
<math>~\mathfrak{W} = \sum\limits_{i=1,3} \mathfrak{W}_{ii}</math>	<math>~\equiv</math>	<math>~- \int\limits_V \rho x_i \frac{\partial \Phi}{\partial x_i} d^3x </math>	…	is the gravitational potential energy	…	[Eq. (18), p. 18]
<math>~2\mathfrak{S} = \sum\limits_{i=1,3} \delta_{ii}\Pi</math>	<math>~=</math>	<math>~3\int\limits_V P d^3x </math>	…	is twice the total thermal energy

So, the scalar virial equation is,

<math>~2 \mathfrak{T} + \mathfrak{W} + 3 \Pi \, ;</math>

and, for a stationary state, we have the, quite broadly used, virial equilibrium condition,

<math>~2 \mathfrak{T} + \mathfrak{W} + 3 \Pi </math>

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

After an evolutionary equation of state has been adopted, this differential relationship can be integrated to give an expression for the energy per unit mass, <math>~\mathfrak{w}</math>, that is potentially available for work. Then we define the thermodynamic energy reservoir as,

<math>\mathfrak{W}_\mathrm{therm} = - \int \mathfrak{w} ~dm \, .</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, the relevant evolutionary equation of state is,

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

<math>d\mathfrak{w} = - \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>\mathfrak{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 evolves 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>d\mathfrak{w} = - 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>\mathfrak{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)} \int \frac{3}{2} \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.

Relationship to the System's Internal Energy

It is instructive to tie this introductory material to the classic discussion of thermodynamic systems, which relates a change in the system's internal energy per unit mass, <math>~\Delta u</math>, to the differential work, <math>~\Delta \mathfrak{w}</math>, via the expression,

<math>~\Delta u = \Delta Q - \Delta \mathfrak{w} \, ,</math>

where, <math>~\Delta Q</math> is the change in heat content of the system.

Isothermal Evolutions: Because the internal energy is only a function of the temperature, we can set <math>~\Delta u = 0</math> for expansions or contractions that occur isothermally. Hence, for isothermal evolutions the change in heat content can immediately be deduced from the expression derived for the differential work; specifically, <math>~\Delta Q = \Delta \mathfrak{w}</math>.

Adiabatic Evolutions: By definition, <math>~\Delta Q = 0</math> for adiabatic evolutions, in which case we find <math>~\Delta u = - \Delta \mathfrak{w}</math>. The definition of the thermodynamic energy reservoir can therefore be rewritten as,

<math>\mathfrak{W}_\mathrm{therm} = - \int \mathfrak{w} ~dm = + \int u ~dm \, .</math>

Quite generally, then — in sync with the above derivation — we can replace <math>~\mathfrak{W}_\mathrm{therm}</math> by <math>~U</math> in the expression for the free energy when analyzing adiabatic evolutions.

Illustration

As is derived in an accompanying discussion, 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_0} \biggl( \frac{R}{R_0} \biggr)^{-1} \, , </math>
<math> ~ T_\mathrm{rot} </math>	<math>~=</math>	<math> ~\frac{5}{4} \frac{J^2}{MR_0^2} \biggl( \frac{R}{R_0} \biggr)^{-2} \, , </math>
<math> ~V </math>	<math>~=</math>	<math> ~\frac{4}{3} \pi R_0^3 \biggl( \frac{R}{R_0} \biggr)^{3} \, , </math>

where, <math>~J</math> is the system's total angular momentum and <math>~R_0</math> is a reference length scale.

Adiabatic Systems: If, upon compression or expansion, the gaseous configuration behaves adiabatically, the reservoir of thermodynamic energy is,

<math> ~\mathfrak{W}_\mathrm{therm} = U = \frac{M K \rho^{\gamma_g-1}}{(\gamma_g - 1)} = \frac{M K }{(\gamma_g - 1)} \biggl( \frac{3M}{4\pi R_0^3} \biggr)^{\gamma_g-1} \biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} \, . </math>

Hence, 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>~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 Systems: If, upon compression or expansion, the configuration remains isothermal, [see, also, Appendix A of Stahler (1983, ApJ, 268, 16)], the reservoir of thermal energy is,

<math> ~\mathfrak{W}_\mathrm{therm} </math>

<math> M c_s^2\ln \biggl( \frac{\rho}{\rho_0} \biggr) = - 3 M c_s^2 \biggl( \frac{R}{R_0} \biggr) \, . </math>

Hence, 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>~\equiv</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 relative size (<math>~R/R_0</math>) for a given choice of <math>~\gamma_g</math>.

Whitworth (1981)

The above formulation of a Gibbs-like free energy has been motivated by Stahler's (1983, ApJ, 268, 16) analysis of the stability of isothermal gas clouds, and it closely parallels Whitworth's (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polytropes." Whitworth 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>.

Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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