Equation of State in the Presence of Gravity

TL;DR

This paper explores how gravity influences the equation of state of an ideal gas, introducing a new relation that accounts for gravitational effects, especially near stellar surfaces, while confirming the conventional law's validity in most astrophysical contexts.

Contribution

It derives a modified equation of state for an ideal gas under gravity, incorporating a new characteristic variable and analyzing its implications for astrophysical systems.

Findings

The ideal gas law holds on average despite density and pressure non-uniformity.

A new relation between averaged pressure and density is established under adiabatic conditions.

Gravity effects are significant near the surface of Newtonian stars.

Abstract

We investigate how an equation of state for matter is affected when a gravity is present. For this purpose, we consider a box of ideal gas in the presence of Newtonian gravity. In addition to the ordinary thermodynamic quantities, a characteristic variable that represents a weight per unit area relative to the average pressure is required in order to describe a macroscopic state of the gas. Although the density and the pressure are not uniform due to the presence of gravity, the ideal gas law itself is satisfied for the thermodynamic quantities when averaged over the system. Assuming that the system follows an adiabatic process further, we obtain a {\it new} relation between the averaged pressure and density, which differs from the conventional equation of state for the ideal gas in the absence of gravity. Applying our results to a small volume in a Newtonian star, however, we find that the conventional one is reliable for most astrophysical situations when the characteristic scale is small. On the other hand, gravity effects become significant near the surface of a Newtonian star.