Equations of state

The equation of state (EOS) defines the relationship between the pressure (\(P\)), density (\(\rho\)), and temperature (\(T\)) of matter. It determines how gas responds to compression or heating and thus governs the structure, stability, and evolution of stars and planets.

Different physical processes dominate in different regimes of temperature and density. Four principal forms of pressure occur in astrophysical interiors: ideal gas, radiation, nonrelativistic degeneracy, and relativistic degeneracy.

At low to moderate densities and high temperatures, matter behaves as a classical gas. The ideal gas law gives

$$ P = \frac{\rho kT}{m_{av}}, $$

where \(k\) is the Boltzmann constant and \(m_{av}\) is the average particle mass. This law describes the interiors of most main-sequence stars, where thermal motion dominates the pressure.

At even higher temperatures, photons contribute significantly to the total pressure. The radiation pressure is given by

$$ P_{rad} = \frac{aT^4}{3}, $$

where \(a\) is the radiation constant. In massive, luminous stars, radiation pressure may equal or even exceed the gas pressure, driving stellar winds and influencing stability.

At very high densities and comparatively low temperatures, quantum effects dominate. Fermions such as electrons or neutrons fill nearly all available low-energy quantum states, creating a degenerate gas. Its pressure arises from the Pauli exclusion principle, not from thermal motion. This degeneracy pressure supports compact objects like white dwarfs, neutron stars, and the dense cores of giant planets.

For a nonrelativistic electron gas, the average energy per particle is

$$ E_{av} = \frac{3}{5}\frac{p_F^2}{2m_e}, $$

where \(p_F\) is the Fermi momentum and \(m_e\) the electron mass. The pressure is obtained by integrating over all occupied momentum states:

$$ P_e = \frac{2}{3}n_e E_{av} = \frac{1}{20}\left(\frac{3}{\pi}\right)^{2/3}\frac{h^2}{m_e} n_e^{5/3}, $$

where \(n_e\) is the electron number density. Substituting \(n_e = \rho / (\mu_e m_p)\), where \(\mu_e\) is the electron molecular weight, gives

$$ P_e = \frac{1}{20}\left(\frac{3}{\pi}\right)^{2/3} \frac{h^2}{m_e}\left(\frac{\rho}{\mu_e m_p}\right)^{5/3}. $$

Thus, for nonrelativistic degeneracy, \(P_e \propto \rho^{5/3}\). This pressure is independent of temperature, providing the main support for white dwarfs of low and intermediate mass.

At extremely high densities, the electrons’ momenta become relativistic (\(p_F \gtrsim m_e c\)). The average energy per particle approaches

$$ E_{av} = \frac{3}{4}c p_F. $$

Substituting this into the pressure relation gives

$$ P_e = \frac{1}{8}\left(\frac{3}{\pi}\right)^{1/3} ch\left(\frac{\rho}{\mu_e m_p}\right)^{4/3}. $$

In this relativistic regime, \(P_e \propto \rho^{4/3}\). Because pressure rises more slowly with density, the gas becomes softer — it cannot indefinitely oppose gravity. This softening leads to the Chandrasekhar limit (\(\approx 1.4\,M_\odot\)), above which electron degeneracy pressure fails and collapse ensues, forming a neutron star or, at still higher densities, a black hole.

The figure above shows the dominant pressure regimes across temperature–density space:

1. Radiation pressure: \(P = aT^4 / 3\)
2. Ideal gas pressure: \(P = \rho kT / m_{av}\)
3. Nonrelativistic degeneracy: \(P \propto \rho^{5/3}\)
4. Relativistic degeneracy: \(P \propto \rho^{4/3}\)

At low density and high temperature, matter behaves as a radiative or ideal gas. At high density and low temperature, it becomes degenerate, with quantum mechanical pressure independent of temperature. In intermediate regions, multiple contributions coexist — for example, in massive white dwarfs, both degeneracy and radiation pressures shape the stellar structure.

As density increases along an isotherm, the effective equation of state transitions smoothly from \(P \propto \rho T\) to \(P \propto \rho^{5/3}\), and finally to \(P \propto \rho^{4/3}\). This sequence determines how stars evolve, collapse, and reach equilibrium at different stages of their life cycles.

1. The equation of state defines how pressure responds to density and temperature, determining stellar structure.
2. Classical gases follow \(P \propto \rho T\); quantum-degenerate matter follows power laws with fixed exponents.
3. Nonrelativistic degeneracy yields \(P \propto \rho^{5/3}\); relativistic degeneracy yields \(P \propto \rho^{4/3}\).
4. Degeneracy pressure arises from the exclusion principle and depends only on density, not temperature.
5. The Chandrasekhar limit results from the softening of the relativistic EOS.
6. Transitions between radiation, gas, and degeneracy regimes govern stellar formation, evolution, and endpoints.

1. Derive \(P_e \propto \rho^{5/3}\) for a nonrelativistic degenerate gas.
2. Explain why relativistic degeneracy produces \(P_e \propto \rho^{4/3}\).
3. How does the Chandrasekhar limit emerge from the relativistic EOS?
4. Under what conditions does radiation pressure dominate over degeneracy pressure?
5. Discuss how changes in the EOS determine the final states of stars.