1.4: Equation of State for an Ideal Gas and Degenerate Matter

Formulation of the Boltzmann transport equation also provides an ideal setting for the formulation of the equation of state for a gas under wide-ranging conditions. The statistical distribution functions developed in Section 1.1 give us the distribution functions for particles which depend largely on how filled phase space happens to be. Those functions relate the particle energy and the kinetic temperature to the distribution of particles in phase space. This is exactly what is meant by \(\mathrm{f}(\vec{\mathrm{v}})\). Thus we can calculate the expected relationship between the pressure as given by the pressure tensor and the state variables of the distribution function. The result is known as the equation of state.

As given in equation \ref{1.2.24}, the pressure tensor is \(\mathbf{p}(\overleftrightarrow{\mathrm{u}}-\overrightarrow{\mathrm{u}} \overrightarrow{\mathrm{u}})\). If \(\mathrm{f}(\overrightarrow{\mathrm{v}})\) is symmetric in \(\mathrm{\vec{v}}\), then \(\vec{\mathrm{u}}\) must be zero (or there must exist an inertial coordinate system in which \(\vec{\mathrm{u}}\) is zero), and the divergence of the pressure tensor can be replaced by the gradient of a scalar, which we call the gas pressure, and will be given by \[P=\rho \frac{\int_{-\infty}^{+\infty} v^2 f(v) d \vec{v}}{\int_{-\infty}^{+\infty} f(v) d \vec{v}}\label{1.3.1}\]

From Maxwell-Boltzmann statistics, the distribution function of particles, in terms of their velocity, was given by equation \ref{1.1.16}. If we regard the number of cells of phase space to be very large, we can replace \(\mathrm{N_i}\) by \(\mathrm{dN}\) and consider equation \ref{1.1.16} to give the distribution function \(\mathrm{f(\vec{v})}\), so that \[f(v)=\text { constant } \times e^{-m v^2 /(2 k T)}\label{1.3.2}\]

Now, in general, \[\int_0^{+\infty} x^{2 n} e^{-a x^2} d x=\frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2^{n+1} a^n}\left(\frac{\pi}{a}\right)^{1 / 2}\label{1.3.3}\]

Substitution of equation \ref{1.3.2} into equation \ref{1.3.1} therefore yields \[P=\frac{\rho k T}{m}=n k T\label{1.3.4}\]

This is known as the ideal-gas law and it is the appropriate equation of state for a gas obeying Maxwell-Boltzmann statistics. That is, we may confidently expect that this simple formula will provide the correct relation among P, T, and ρ as long as the cells of phase space do not become overly filled and quantum effects become important. If the density is increased without a corresponding increase in particle energy, a point will come when the available cells of phase space begin to fill up in accordance with the Pauli Exclusion Principle. As the most "popular" cells in phase space become filled, the particles will have to spill over into adjacent cells, producing a distortion in the distribution function (see Figure 1.5). When this happens, the gas is said to become partially degenerate. Figure 1.5 shows this effect and indicates a way to quantify the effect. We define a momentum \(\boldsymbol{p}_0\) as that momentum above which there are just enough particles to fill the remaining phase space cells below \(\boldsymbol{p}_0\). Thus \[\int_0^{p_0}\left[N_{\max }-N(p)\right] d p=\int_{p_0}^{\infty} N(p) d p\label{1.3.5}\]

If we make the approximation that all the spaces in phase space are filled (i.e., a negligible number of particles exist with momentum above \(\boldsymbol{p_0}\)), then the momentum distribution of the particles can be represented by a section of a sphere in momentum

space so that \[n(p) d p=2\left(4 \pi p^2 \frac{d p}{h^3}\right)\label{1.3.6}\]

The factor of 2 arises because the electron can have two spin states in a cell of size \(\mathrm{h^3}\). The number density of particles can then be given in terms of \(\boldsymbol{p_0}\) as \[n=2 \int_0^{p_0} \frac{4 \pi p^2}{h^3} d p=\frac{8 \pi p_0^3}{3 h^3}\label{1.3.7}\]

Figure 1.5 Shows a momentum cross section of phase space with different particle densities. As the volume saturates, the distribution function departs further and further from Maxwellian. The Fermi momentum p₀, represents that momentum such that the particles having greater momentum would just fill the momentum states below it.

We have already developed a relation for the scalar pressure in terms of the velocity distribution under the assumption of an isotropic velocity field in equation \ref{1.3.1}, and we need only replace the velocity distribution \(\mathrm{f(\vec{v})}\) with a distribution function of momentum. However, we must remember that the integral in equation \ref{1.3.1} is actually three integrals over each velocity coordinate which will all have the same value for an isotropic velocity field. The three integrals corresponding to the three components of velocity in equation \ref{1.3.1} are equal for spherical momentum space. Therefore one-third of the scalar form of equation \ref{1.3.1} will represent the total contribution of the momentum to the pressure. Thus the pressure can be expressed in terms of the maximum momentum p₀, often known as the Fermi momentum, as \[\mathrm{P}=\frac{1}{3} \int_0^{\mathrm{p}_0} \frac{\mathrm{p}^2}{\mathrm{m}} \mathrm{n}(\mathrm{p}) \mathrm{dp}=\frac{1}{3} \int_0^{\mathrm{p}_0} \frac{\mathrm{p}^2}{\mathrm{m}}\left(\frac{8 \pi \mathrm{p}^2}{\mathrm{h}^3}\right) \mathrm{dp}=\frac{8 \pi \mathrm{p}_0^5}{15 \mathrm{mh}^3}=\frac{\mathrm{h}^2}{20 \mathrm{m}}\left(\frac{3}{\pi}\right)^{2 / 3} \mathrm{n}^{5 / 3}\label{1.3.8}\]

Using equation \ref{1.3.7}, we can eliminate p₀ and obtain a relationship between the pressure and the density. This, then, is the equation of state for totally degenerate matter, and since the electrons tend to become degenerate before any other particles, it is common to write the equation of state for electron degeneracy alone.

\[P=\left(\frac{h^2}{20 m_e m_h}\right)\left(\frac{3}{\pi m_h}\right)^{2 / 3}\left(\frac{\rho}{\mu_e}\right)^{5 / 3}=9.9 \times 10^{12}\left(\frac{\rho}{\mu_e}\right)^{5 / 3}\quad
\text {cgs}\label{1.3.9}\]

Here m_e and m_h are the mass of the electron and hydrogen atom, respectively, while m_e is the mean molecular weight of the free electrons.

If we consider a gas under extreme pressure, not only will the cells of phase space be filled, but also the maximum momentum will become very large. Although the mass m and momentum p both approach infinity as the particle energy increases, their ratio p/m does not. It remains finite and approaches the speed of light c. Since these particles also make the largest contribution to the pressure, we can estimate the effect of having a relativistically degenerate gas by replacing p/m by c in equation \ref{1.3.8}, and we get \[P=\frac{1}{3} \int_0^{p_0} \frac{p^2}{m} n(p) d p=\frac{1}{3} \int_0^{p_0} \frac{8 \pi p^3 c}{h^3} d p=\frac{2 \pi p_0^4 c}{3 h^3}\label{1.3.10}\]

which leads to an equation of state that depends on \(\boldsymbol{p}^{4 / 3}\) instead of \(\boldsymbol{p}^{5 / 3}\), as in the case of nonrelativistic degeneracy. Eliminating \(\boldsymbol{p}_0\), we obtain for the electron degeneracy \[\boldsymbol{P}=\left(\mathrm{hc} / 8 \mathrm{~m}_{\mathrm{h}}\right)\left(3 / \pi \mathrm{m}_{\mathrm{h}}\right)^{1 / 3}\left(\mathrm{p} / \mathrm{m}_{\mathrm{e}}\right)^{4 / 3}=1.231 \times 10^{15}\left(\mathrm{p} / \mathrm{m}_{\mathrm{e}}\right)^{4 / 3}\quad\text{cgs}\label{1.3.11}\]

The equations of state for degenerate matter that we have derived represent limiting conditions and are never exactly realized. In real situations we must consider the transition between the ideal-gas law and total degeneracy as well as the transition between nonrelativistic and completely relativistic degeneracy. One way to identify the range of state variables for which we can expect a transition zone is to equate the various equations of state and to solve for the range of state variables involved. Equating the ideal-gas law [equation \ref{1.3.4}] with the equation for a totally degenerate gas [equation \ref{1.3.9}], we can determine the range of density \(\rho_{\mathrm{t}}\) and temperature \(\mathrm{T_t}\) which lie in the transition zone between the two equations of state \[\frac{\rho_t}{\mu_e}=\left(2.40 \times 10^{-8}\right)\left(T_t^{3 / 2}\right) \quad \mathrm{g} / \mathrm{cm}^3\label{1.3.12}\]

For a metal at \(100\mathrm{K}\), \(\rho_{\mathrm{t}} / \mathrm{m}_{\mathrm{e}}=6 \times 10^{-5} \mathrm{gm} / \mathrm{cm}^3\), which implies that the electrons in such a conductor follow the degenerate equation of state and that virtually all the cells in phase space are full. This accounts for the high conductivity of metals, since the saturation of phase space cells implies that free electrons cannot scatter off the other particles in the metal (for in doing so they would have to move to a new cell in phase space, which is more than likely filled). Thus, they travel relatively unhindered through the conductor. In general, a totally degenerate gas proves to be an excellent conductor.

For temperatures on the order of \(10^7 \mathrm{~K}\) which prevail in the center of the sun, the transition densities occur at about \(8 \times 10^2 \mathrm{~g} / \mathrm{cm}^3\) which is significantly higher than we find in the sun. Thus, we may be assured that the ideal-gas law will be appropriate throughout the interior of the sun and most stars. However, white dwarfs do exceed the transition density for the temperatures we may expect in these stars. Therefore, we can expect a transition from the ideal-gas law which will prevail in the surface regions to total degeneracy in the interior. In this transition region the equation of state becomes more complex. A complete discussion of this region can be found in Cox and Giuli⁶ and Chandrasekhar⁷. The basic philosophy is to write the equation of state in parametric form in terms of a degeneracy parameter y, where the equation of state becomes the ideal-gas law when \(\mathrm{y<<0}\) and the equation of state approaches the totally degenerate equation of state if \(\mathrm{y>>0}\). This parametric form can be written as \[\begin{aligned}
\frac{\rho}{\mu_e} & =\left(\frac{4 \pi}{h^3}\right)(2 m k T)^{3 / 2} m_h F_{1 / 2}(\psi) \\
P_e & =\left(\frac{8 \pi}{3 h^3}\right)(2 m k T)^{3 / 2} k T F_{3 / 2}(\psi) \\
F_n(\psi) & =\int_0^{\infty} \frac{x^n d x}{e^{x-\psi}+1}
\end{aligned}\label{1.3.13}\]

In the transition zone between nonrelativistic and relativistic degeneracy, S. Chandrasekhar⁷ also gives a parametric equation of state in terms of the "maximum" momentum \(\boldsymbol{p}_0\) of the Fermi sea: \[\begin{aligned}
\frac{\rho}{\mu_e} & =\left(\frac{8 \pi}{3 h^3}\right)\left(m_e c\right)^3 m_h x^3 \\
P_e & =\frac{\pi}{3 h^3} m_e^4 c^5 f(x) \\
f(x) & =x\left(2 x^2-3\right)\left(x^2+1\right)^{1 / 2}+3 \sinh ^{-1} x
\end{aligned}\label{1.3.14}\]

As x approaches zero, the nonrelativistic equation of state is obtained whereas as x approaches infinity, the fully relativistic limit is obtained. In the rare case where the gas occupies both transition regions at the same time, the equation of state becomes quite complicated. Refer to Cox and Giuli for a detailed description of this situation⁸.

Before leaving this discussion of the equation of state and degenerate matter, we want to explore some consequences of the most notable aspect of the degenerate equation of state. Nowhere in either the nonrelativistic or the relativistic degenerate equation of state does the temperature appear. This complete lack of temperature dependence implies a unique relationship between the pressure and the density.

Hydrostatic equilibrium [equation \ref{1.2.28}] implies a relation among the pressure, mass, and radius. Since the mass, density, and radius are related by definition, these three relationships should allow us to find a unique relation between the mass of the configuration and its radius. Although a detailed investigation of the relation requires the solution of a differential equation coupled with some extremely nonlinear equations, we can get a sense of the mass-radius relation by considering the form of the equations that constrain the solution.

For spherical stars, hydrostatic equilibrium as expressed by equation \ref{1.2.28} implies that \[\frac{d P}{d r} \sim \frac{M(r) \rho}{r^2} \sim \frac{M\left(M / R^3\right)}{R^2}\label{1.3.15}\]

Since we can also expect the pressure gradient to be proportional to P/R, the internal pressure in a star should vary as \[P \sim \frac{M^2}{R^4}\label{1.3.16}\]

For a totally degenerate gas, the equation of state requires that \[P \sim \rho^{5 / 3} \sim \frac{M^{5 / 3}}{R^5}\label{1.3.17}\]

Thus, we eliminate the pressure from these two expressions to get \[R \sim M^{-1 / 3} \quad \text { or } \quad M \sim R^{-3}\label{1.3.18}\]

We arrive at a curious result: As the mass of the configuration increases, the radius decreases. This situation, then, must prevail for white dwarfs. The more massive the white dwarf, the smaller its radius. In a situation where mass is added to a white dwarf, thereby causing its radius to decrease, the internal pressure must increase, which leads to an increase in the Fermi momentum p₀. Sooner or later the equation of state must change over to the fully relativistic equation of state. Here \[P \sim \rho^{4 / 3} \sim \frac{M^{4 / 3}}{R^4}\label{1.3.19}\]

If we again eliminate the pressure by using equation \ref{1.3.16}, then the radius also disappears and \[\mathrm{M} \sim \text { constant. }\label{1.3.20}\]

Thus, for a fully relativistic degenerate gas, there is a unique mass for which the configuration is stable. Should mass be added beyond this point, the star would be forced into a state of unrestrained gravitational collapse. Much later we shall see that a further change in the equation of state, which occurs when the density approaches that of nuclear matter, can halt the collapse, allowing the formation of a neutron star. But for "normal" matter a limit is set by quantum mechanics, and this prevents the formation of white dwarfs with masses greater than about \(1.4 \mathrm{M}_{\odot}\). This is the limit found by S. Chandrasekhar in the late 1930s and for which he received the Nobel Prize in 1983. The configuration described by the fully relativistically degenerate equation of state is a strange one indeed, and we shall explore it in some detail later. For now, let us turn to the most basic assumptions that must be made for the study of stellar structure and to what they imply about the nature of stars.