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Whitworth's (1981) Isothermal Free-Energy Surface
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Supplemental Relations

Examining the set of principal governing equations that describe adiabatic flows, we see that — apart from the independent variables <math>~t</math> and <math>~\vec{x}</math> — the equations involve the vector velocity <math>~\vec{v}</math>, and the four scalar variables, <math>~\Phi</math>, <math>~P</math>, <math>~\rho</math>, and <math>~\epsilon</math>. Because the variables outnumber the equations by one, a supplemental relationship between the physical variables must be specified in order to close the set of equations. (If nonadiabatic flows are considered, additional supplemental relations must be specified because the scalar variables <math>~T</math> and <math>~s</math> enter the discussion as well.) Also, in order to complete the unique specification of a particular physical problem, either a steady-state flow field or initial conditions must be specified, depending on whether one is studying a time-independent (structure) or time-dependent (stability or dynamics) problem, respectively. Throughout this H_Book, the following strategy will be adopted in order to complete the physical specification of each examined system: (A) For time-independent problems, we will adopt a structural relationship between <math>~P</math> and <math>~\rho</math>, and specify a steady-state flow-field. (B) For time-dependent problems, we will adopt an equation of state, and specify initial conditions.


Time-Independent Problems

Barotropic Structure

For time-independent problems, a structural relationship between <math>~P</math> and <math>~\rho</math> is required to close the system of principal governing equations. [Tassoul (1978) refers to this as a "geometrical" rather than a "structural" relationship; see the discussion associated with his Chapter 4, Eq. 14.] Generally throughout this H_Book, we will assume that all time-independent configurations can be described as barotropic structures; that is, we will assume that <math>~P</math> is only a function of <math>~\rho</math> throughout such structures. More specifically, we generally will adopt one of the two analytically prescribable <math>~P</math>(<math>~\rho</math>) relationships displayed in the following Table.

Barotropic Relations

Polytropic Zero-temperature Fermi (degenerate electron) Gas

<math>~P = K_\mathrm{n} \rho^{1+1/n}</math>

LSU Key.png

<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math>

where:  <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>

and:   

<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>


Reference (original): Chandraskehar, S. (1935)

In the polytropic relation, the "polytropic index" <math>~n</math> and the "polytropic constant" <math>~K_\mathrm{n}</math> are assumed to be independent of both <math>~\vec{x}</math> and <math>~t</math>. In the zero-temperature Fermi gas relation, the two constants <math>~A_\mathrm{F}</math> and <math>~B_\mathrm{F}</math> are expressible in terms of various fundamental physical constants, as detailed in the accompanying variables appendix.

Steady-state Flow-Field Specification

A steady-state velocity flow-field must be specified for time-independent problems. The specification can be as simple as stating that <math>~\vec{v}</math> = 0 everywhere in space. Other examples are uniform rotation; or, for axisymmetric configurations, uniform specific angular momentum. In this H_Book, the flow-field specification will generally vary from chapter to chapter.

Time-Dependent Problems

Equation of State

For time-dependent problems we usually will supplement the set of principal governing equations by adopting an equation of state — i.e., a relationship between the state variables <math>~P</math>, <math>~\rho</math>, and <math>~T</math> — that is given by one of the expressions in the following Table, or by some combination of these expressions. (For example, we could write <math>P_\mathrm{total} = P_\mathrm{gas} + P_\mathrm{deg} + P_\mathrm{rad}</math>.)

Analytic Equations of State

Ideal Gas Degenerate Electron Gas Radiation

LSU Key.png

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

LSU Key.png

<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math>

where:  <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>

and:   

<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>

LSU Key.png

<math>~P_\mathrm{rad} = \frac{1}{3} a_\mathrm{rad} T^4</math>

In the so-called ideal gas equation of state, <math>~\Re</math> is the gas constant and <math>~\bar{\mu}</math> is the mean molecular weight of the gas. In the equation that gives the electron degeneracy pressure, <math>P_\mathrm{deg}</math>, <math>~A_\mathrm{F}</math> is the characteristic Fermi pressure and <math>~B_\mathrm{F}</math> is the characteristic Fermi density. And in the expression for the photon radiation pressure, <math>P_\mathrm{rad}</math>, <math>~a_\mathrm{rad}</math> is the radiation constant. The value of each of these identified physical constants can be found by simply scrolling the computer mouse over the symbol for the constant found in the text of this paragraph, and a definition of each constant can be found in the Variables Appendix of this H_Book.

All three of these equations of state are among the set of key physical equations that provide the foundation for our discussion of the structure, stability, and dynamics of self-gravitating systems. A discussion of the physical principles that underpin each of these relations can be found in any of a number of different published texts — see, for example, the set of parallel references identified in the Equations Appendix of this H_Book — or in the Wiki pages that can be accessed by clicking the linked "origin" buttons in the above Table. See also Tassoul (1978) — specifically the discussion associated with his Chapter 4, Eq. 13 — for a more general statement related to the proper specification of a supplemental, equation of state relationship.

Initial Conditions

For time-dependent problems, the principal governing equations must be supplemented further through the specification of initial conditions. Frequently throughout this H_Book, we will select as initial conditions a specification of <math>~\rho</math>( <math>~\vec{x}</math>, <math>~t</math> = 0), <math>~P</math>( <math>~\vec{x}</math>, <math>~t</math> = 0), and <math>~\vec{v}</math>( <math>~\vec{x}</math>, <math>~t</math> = 0) that, as a group themselves, define a static or steady-state equilibrium structure. Perturbation or computational fluid dynamic (CFD) techniques can be used to test the stability of such structures.

Whitworth's (1981) Isothermal Free-Energy Surface

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