E10 Thermodynamics and Astrophysical Equations of State

Topic

The central element of this model is the equation of state—the relationship linking the pressure, density, and temperature of the astrophysical material.

The ideal gas, characterized by its mean molecular weight, is the approximation this equation of state takes in most astrophysical environments. If the hydrogen in the gas is fully ionized rather than neutral, the mean molecular weight decreases, because ionization doubles the number of particles per unit mass compared to a neutral gas.

Polytropic models are simplified solutions of this same equation of state applied to hydrostatic equilibrium, wherein pressure depends solely on density raised to a fixed exponent; specific values ​​of this exponent allow the polytropic equation to represent—depending on the case—either a pure radiation gas or the degenerate relativistic limit previously described for white dwarfs and neutron stars.

Radiation pressure, which increases with the fourth power of temperature, becomes a significant term in the equation of state precisely in the most massive stars—those whose luminosity approaches the Eddington limit defined for any massive astrophysical object. As a star's luminosity nears this limit, radiation pressure ceases to be a negligible term within its equation of state.

The speed of sound represents the characteristic velocity scale for acoustic perturbations within the gas itself; the equation of state dictates that this speed increases when pressure is high relative to density. Measuring this speed of sound directly—based on oscillations observed at the solar surface—is precisely the function of helioseismology. Thus, from the equation of state with which this document begins to the helioseismic measurement with which it concludes—encompassing the ideal gas, polytropic models, and radiation pressure—the thermodynamics of astrophysical material ensures that a single relationship between pressure, density, and temperature describes, without changing its form, everything from the typical interior of a star to the extreme case of a white dwarf or a neutron star.

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