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Supplemental Relations

 "The equilibrium configuration of a stellar system depends crucially on the equation of state. Similarly, the time evolution of the perturbation will depend on the relationship that is assumed between the perturbed values of pressure, temperature, and density. The simplest assumption regarding these perturbations will be to involve an adiabatic relationship between perturbed pressure and density." — Drawn from §7, p. 172 of [P00]
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Examining the set of principal governing equations that describe adiabatic flows, we see that — apart from the independent variables $~t$ and $~\vec{x}$ — the equations involve the vector velocity $~\vec{v}$, and the four scalar variables, $~\Phi$, $~P$, $~\rho$, and $~\epsilon$. 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 $~T$ and $~s$ 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 $~P$ and $~\rho$, 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 $~P$ and $~\rho$ is required to close the system of principal governing equations. [T78] 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 $~P$ is only a function of $~\rho$ throughout such structures. (The Poincaré-Wavre theorem provides additional support for, as well as additional implications of, this assumption.) More specifically, we generally will adopt one of the two analytically prescribable $~P(\rho)$ relationships displayed in the first row of the following Table.

Barotropic Relations

Polytropic Zero-temperature Fermi (degenerate electron) Gas

$~P = K_\mathrm{n} \rho^{1+1/n}$

 $~P_\mathrm{deg} = A_\mathrm{F} F(\chi)$ where:  $F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi$ and: $\chi \equiv (\rho/B_\mathrm{F})^{1/3}$

Reference (original): S. Chandraskehar (1935)

$~H = (n+1)K_n \rho^{1/n} = \frac{(n+1)P}{\rho}$

$H = \frac{8A_\mathrm{F}}{B_\mathrm{F}} \biggl[(\chi^2 + 1)^{1/2} - 1 \biggr]$

$~\rho = \biggl[ \frac{H}{(n+1)K_\mathrm{n}} \biggr]^n$

$\rho = B_\mathrm{F} \biggl[\biggl(\frac{HB_\mathrm{F}}{8A_\mathrm{F}} + 1 \biggr)^{2} - 1 \biggr]^{3/2}$

In the polytropic relation, the "polytropic index" $~n$ and the "polytropic constant" $~K_\mathrm{n}$ are assumed to be independent of both $~\vec{x}$ and $~t$. In the zero-temperature Fermi gas relation, the two constants $~A_\mathrm{F}$ and $~B_\mathrm{F}$ are expressible in terms of various fundamental physical constants, as detailed in the accompanying variables appendix. This table also displays (2nd row) the enthalpy as a function of mass density, $~H(\rho)$, and (3rd row) the inverted relation $~\rho(H)$ for both barotropic relations, where

$H = \int\frac{dP}{\rho}$ .

In both cases, we have chosen an integration constant such that $~H$ is zero when $~\rho$ is zero.

Nonrelativistic ZTF Gas

At sufficiently low densities, specifically, when

$~\chi \ll 1 \, ,$

the zero-temperature Fermi (ZTF) equation of state describes the pressure-density behavior of a nonrelativistic (NR) degenerate electron gas. We can determine the expression for $~P_\mathrm{deg}$ in this limit by writing the function, $~F(\chi)$, in terms of two relevant series expansions — for the inverse hyperbolic sine function, see for example, the Wikipedia presentation — then keeping only the highest order terms.

 $~F(\chi)$ $~=$ $~- 3\chi \biggl(1 - \frac{2}{3}\chi^2 \biggr) \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7} \chi^8+ \cdots \biggr) +3 \displaystyle\sum_{m=0}^{\infty} \biggl[ \frac{(-1)^m (2m)!}{2^{2m}(m!)^2} \cdot \frac{\chi^{2m+1}}{(2m + 1)}\biggr]$ $~=$ $~- 3\chi \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7}\chi^8 + \cdots \biggr) + 2\chi^3 \biggl( 1 + \frac{1}{2}\chi^2 - \frac{1}{2^3}\chi^4 +\frac{1}{2^4}\chi^6 - \frac{5}{2^7}\chi^8 + \cdots \biggr)$ $~ +3\biggl\{\chi - \frac{1}{6}\chi^3 + \frac{3}{2^3 \cdot 5} \chi^5 - \frac{5}{2^4\cdot 7} \chi^7 + \cdots \biggr\}$ $~=$ $~-3\chi + 3\chi + \chi^3\biggl(-\frac{3}{2} + 2 - \frac{1}{2} \biggr) + \chi^5 \biggl(\frac{3}{8} +1 +\frac{3^2}{2^3\cdot 5} \biggr) +\chi^7 \biggl( -\frac{3}{2^4} -\frac{1}{2^2} - \frac{3\cdot 5}{2^4\cdot 7} \biggr) + \cdots$ $~=$ $~\chi^5 \biggl(\frac{15 + 40 + 9}{2^3\cdot 5} \biggr) -\chi^7 \biggl( \frac{21 + 28 + 15}{2^4\cdot 7} \biggr) + \cancelto{0}{\cdots}$ $~\approx$ $~\chi^5 \biggl(\frac{2^3}{5} \biggr) -\chi^7 \biggl( \frac{2^2}{7} \biggr) \, .$

This agrees with, for example, the asymptotic form presented as equation (24) in §X.1 (p. 361) of [C67]. Keeping only the leading term leads to the expression,

 $~P_\mathrm{deg}\biggr|_\mathrm{NR}$ $~=$ $~\frac{2^3}{5} A_F \biggl( \frac{\rho}{B_F}\biggr)^{5/3}$ $~=$ $~\frac{2^3}{5} \biggl( \frac{\pi m_e^4 c^5}{3h^3} \biggr) \biggl[ \frac{8\pi m_p}{3} \biggl( \frac{m_e c}{h} \biggr)^3 \mu_e \biggr]^{-5/3} \rho^{5/3}$ $~=$ $~\mu_e^{-5/3} \biggl[ \frac{2^9 \cdot 3^5 \pi^3}{2^{15}\cdot 3^3\cdot 5^3 \pi^5} \biggr]^{1/3} \biggl( \frac{m_e^4 c^5}{h^3} \biggr) \biggl( \frac{h}{m_e c} \biggr)^5 \biggl( \frac{\rho}{m_p}\biggr)^{5/3}$ $~=$ $~\frac{1}{2^2 \cdot 5}\biggl( \frac{3}{\pi} \biggr)^{2/3} \biggl( \frac{h^2}{m_e} \biggr) \biggl( \frac{\rho}{m_p \mu_e}\biggr)^{5/3} \, .$

This matches equation (27) in §X.1 (p. 362) of [C67]; see also, equation (2-32) of D. D. Clayton (1968), equation (11.42) of [H87], equation (15.23) of [KW94], the upper expression labeled as equation (5.163) in [P00], and equations (5.9)-(5.10) of A. R. Choudhuri (2010).

ZTF Gas in Relativistic Limit

At sufficiently high densities, specifically, when

$~\chi \gg 1 \, ,$

the zero-temperature Fermi (ZTF) equation of state describes the pressure-density behavior of a degenerate electron gas in the (special) relativistic limit (RL). We can determine the expression for $~P_\mathrm{deg}$ in this limit by writing the function, $~F(\chi)$, in terms of two relevant series expansions — for the inverse hyperbolic sine function, see for example, NIST's Digital Library of Mathematical Functions — then keeping only the highest order terms.

 $~F(\chi)$ $~=$ $~\biggl(2\chi^4 - 3\chi^2 \biggr) \biggl( 1 + \frac{1}{2\chi^2} - \frac{1}{2^3\chi^4} +\frac{1}{2^4\chi^6} - \frac{5}{2^7\chi^8} + \cdots \biggr) +3 \biggl[ \ln(2\chi) + \frac{1}{2^2 \chi^2} - \frac{3}{2^5\chi^4} + \frac{5}{2^5\cdot 3\chi^6} - \cdots \biggr]$ $~=$ $~ \biggl( 2\chi^4 + \chi^2 - \frac{1}{2^2} +\frac{1}{2^3\chi^2} - \frac{5}{2^6\chi^4} + \cdots \biggr) - \biggl( 3\chi^2 + \frac{3}{2} - \frac{3}{2^3\chi^2} +\frac{3}{2^4\chi^4} + \cdots \biggr)$ $~ + \biggl[ 3\ln(2\chi) + \frac{3}{2^2 \chi^2} - \frac{3^2}{2^5\chi^4} + \cdots \biggr]$ $~=$ $~ 2\chi^4 - 2 \chi^2 + 3\ln(2\chi) - \frac{7}{4} +\frac{5}{4\chi^2} - \frac{35}{2^6\chi^4} + \cdots$

This agrees with, for example, the asymptotic form presented as equation (25) in §X.1 (p. 361) of [C67]. Keeping only the leading term leads to the expression,

 $~P_\mathrm{deg}\biggr|_\mathrm{RL}$ $~=$ $~2 A_F \biggl( \frac{\rho}{B_F}\biggr)^{4/3}$ $~=$ $~2 \biggl[ \frac{\pi m_e^4 c^5}{3h^3} \biggr] \biggl[ \frac{8\pi m_p \mu_e}{3} \biggl( \frac{m_e c}{h}\biggr)^3\biggr]^{-4/3} \rho^{4/3}$ $~=$ $~\frac{1}{2^3}\biggl(\frac{3}{\pi}\biggr)^{1/3} (hc) \biggl(\frac{\rho}{m_p \mu_e}\biggr)^{4/3} \, .$

This matches equation (28) in §X.1 (p. 362) of [C67]; see also, equation (11.43) of [H87], equation (15.26) of [KW94], the lower expression labeled as equation (5.163) in [P00], and equations (5.11)-(5.12) of A. R. Choudhuri (2010).

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 $~\vec{v}=0$ everywhere in space, or that the system has uniform ("solid body") rotation. Throughout the literature, efforts to generate equilibrium, axisymmetric configurations have adopted a variety of different simple rotation profiles. In this H_Book, the flow-field specification will generally vary from chapter to chapter.

Time-Dependent Problems

Equation of State

Components of the Total Pressure

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

Analytic Equations of State

Radiation Pressure
Ideal Gas Degenerate Electron Gas
 $~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T$

 $~P_\mathrm{deg} = A_\mathrm{F} F(\chi)$ where:  $F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi$ and: $\chi \equiv (\rho/B_\mathrm{F})^{1/3}$
 $~P_\mathrm{rad} = \frac{1}{3} a_\mathrm{rad} T^4$
Normalized Total Pressure:
 $~p_\mathrm{total} = \biggl(\frac{\mu_e m_p}{\bar{\mu} m_u} \biggr) 8 \chi^3 \frac{T}{T_e} + F(\chi) + \frac{8\pi^4}{15} \biggl( \frac{T}{T_e} \biggr)^4$

In the so-called ideal gas equation of state, $~\Re$ is the gas constant and $~\bar{\mu}$ is the mean molecular weight of the gas. In the equation that gives the electron degeneracy pressure, $~A_\mathrm{F}$ is the characteristic Fermi pressure and $~B_\mathrm{F}$ is the characteristic Fermi density. And in the expression for the photon radiation pressure, $~a_\mathrm{rad}$ 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 are among the set of key physical equations that provide the foundation for our discussion of the structure, stability, and dynamics of self-gravitating fluids. 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 "other forms" buttons in the above Table. See also [T78] — 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.

The Parameter, β

It should be pointed out that, in the astrophysics community, the dimensionless quantity $~(1-\beta)$ is sometimes used to denote the relative importance of radiation pressure in a gaseous configuration; specifically,

$~1-\beta = \frac{P_\mathrm{rad}}{P} \, .$

Hence, in the context of our present discussions, the parameter, $~\beta$, itself is,

$~\beta = \frac{P_\mathrm{gas} + P_\mathrm{deg}}{P} \, .$

Examples include our discussion of bipolytropic configurations with $~(n_c, n_e) = (\tfrac{3}{2}, 3)$, as introduced by E. A. Milne (1930); the study of rotating, supermassive stars, especially as introduced by J. R. Bond, W. D. Arnett, & B. J. Carr (1984); and our reference to a derivation of the linear adiabatic wave equation by P. Ledoux & C. L. Pekeris (1941).

Adiabatic Exponent

Consider, first, the evolution of a system that is composed entirely of an ideal gas; that is, $~P_\mathrm{rad} = P_\mathrm{deg} = 0$ and $~\beta = 1$. It is widely appreciated that if the entropy of such a system remains constant during a phase of expansion/contraction, then the variation of pressure with density can be properly described by the expression,

 $~\frac{d\ln P}{d\ln \rho}$ $~=$ $~\gamma_\mathrm{g} \, ,$

where the value of the adiabatic exponent, $~\gamma_\mathrm{g}$, is given by the ratio of specific heats of the (ideal) gas. Now, according to the ideal-gas equation of state, changes in the three state variables must always be related to one another via the differential expression,

 $~dP\biggr|_\mathrm{gas}$ $~=$ $~\frac{\Re}{\bar{\mu}} \biggl[ \rho dT + T d\rho \biggr]_\mathrm{gas}$ $~\Rightarrow ~~~ d\ln P$ $~=$ $~d\ln \rho + d\ln T \, .$

We therefore also deduce that,

 $~\frac{d\ln P}{d\ln \rho}$ $~=$ $~1 + \frac{d\ln T}{d\ln\rho}$ $~\Rightarrow ~~~ \frac{d\ln T}{d\ln\rho}$ $~=$ $~\gamma_\mathrm{g}-1 \, ;$

and,

 $~\frac{d\ln P}{d\ln T}$ $~=$ $~\frac{d\ln \rho}{d\ln T} +1 = \biggl( \frac{1}{\gamma_\mathrm{g} -1}\biggr) + 1$ $~=$ $~\frac{\gamma_\mathrm{g}}{\gamma_\mathrm{g}-1 } \, .$

Following the lead of [C67], the astrophysics community has found that, when radiation pressure is included in the mix — that is, when we consider situations in which,

$~P = P_\mathrm{gas} + P_\mathrm{rad}$

— it can be useful to characterize the adiabatic compression/expansion of fluid elements in terms of three separate adiabatic exponents, $~\Gamma_1, \Gamma_2, \Gamma_3$, that are defined via similar differential expressions, namely,

 $~\frac{d\ln P}{d\ln \rho}$ $~=$ $~\Gamma_1 \, ;$ $~\frac{d\ln P}{d\ln T}$ $~=$ $~\frac{\Gamma_2}{\Gamma_2-1} \, ;$ $~\frac{d\ln T}{d\ln \rho}$ $~=$ $~\Gamma_3-1 \, .$

In this case, though, each of the three adiabatic exponents is a function of $~\beta$ as well as $~\gamma_\mathrm{g}$; specifically,

 $~\Gamma_1$ $~=$ $~ \beta + \frac{(4-3\beta)^2 (\gamma_\mathrm{g}-1)}{\beta + 12(\gamma_\mathrm{g}-1)(1-\beta)} \, ;$ $~\Gamma_2$ $~=$ $~ 1 + \frac{(4-3\beta)(\gamma_\mathrm{g} - 1)}{\beta^2 + 3(\gamma_\mathrm{g} - 1)(1-\beta)(4+\beta)} \, ;$ $~\Gamma_3$ $~=$ $~ 1 + \frac{(4-3\beta)(\gamma_\mathrm{g}-1)}{\beta+12(\gamma_\mathrm{g} - 1)(1-\beta)} \, .$

Entropy Tracer

We begin with the basic equation of state,

$~P = (\gamma_g - 1)\epsilon\rho \, ,$

and the 1st Law of Thermodynamics,

 $T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)$

Adopting the concept of an entropy tracer,

$~\tau \equiv (\epsilon\rho)^{1/\gamma_g} \, ,$

the 1st becomes,

 $~\frac{\tau}{c_p} \frac{ds}{dt}$ $~=$ $~ \frac{\partial\tau}{\partial t} + \nabla\cdot (\tau \vec{v}) \, .$

See equation (33) of Motl et al. (2002)

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 $~\rho(\vec{x}, t=0)$, $~P(\vec{x}, t=0)$, and $~\vec{v}(\vec{x}, t=0)$ 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 or nonlinear dynamical behavior of such structures.

See Also

For derivations and additional discussions of the triplet of adiabatic exponents, $~\Gamma_1, \Gamma_2, \Gamma_3$, see:

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