Equations of State for Photon Gas and Relativistic Electron Gas

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Start date May 18, 2019
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Electron Electron gas Gas Photon Photon gas Relativistic State

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SUMMARY

This discussion focuses on the development of equations of state for photon gas and relativistic electron gas, essential for cosmological calculations and stellar interiors. Specifically, it establishes that for an anisotropic, monochromatic photon gas, the relationship between pressure (p) and mass-energy density (ρ) is defined as p=ρ/3, as derived from Exercise 22 on page 108 of Bernard Schutz’s ‘The First Course in General Relativity (Second Edition)’. The discussion also outlines the calculation of pressure based on the impulse delivered by photons striking a surface, emphasizing the significance of frequency (ν) and photon density (n) in determining the mass-energy density (ρ=n h ν).

PREREQUISITES

Understanding of general relativity concepts, particularly from Bernard Schutz’s textbook.
Familiarity with the properties of photon gas and relativistic gases.
Knowledge of basic thermodynamics and equations of state.
Ability to perform calculations involving impulse and pressure in a physical context.

NEXT STEPS

Study the derivation of the equation of state for relativistic electron gas.
Explore the implications of photon gas equations in cosmological models.
Investigate the role of frequency (ν) in determining the properties of photon gas.
Learn about the applications of equations of state in astrophysics and stellar dynamics.

USEFUL FOR

Astrophysicists, cosmologists, and students of general relativity seeking to deepen their understanding of the thermodynamic properties of photon and relativistic gases in the context of stellar and cosmological phenomena.

May 18, 2019

andrewkirk

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This Insight develops equations of state that are useful in calculations about cosmology and about the insides of stars. The first calculation is for a photon gas and the second is for a ‘relativistic’ gas of particles with mass.
Photon Gas
Exercise 22 on p108 of Bernard Schutz’s ‘The first course in General Relativity (Second Edition) is to prove that, for anisotropic, monochromatic, photon gas, p=ρ/3, where p is pressure and ρ is mass-energy density.
Say all photons have frequency ##\nu## and the number of photons per cubic metre is ##n##. Then ##\rho=n h \nu##.
Now consider one face of a cube of side length 1m. The pressure on that face is the component, parallel to the normal to the face, of the impulse delivered to that face in one second, by photons striking it from outside the cube. We can ignore components of impulse that are parallel to the face because the isotropy will make such components from different particles cancel each other out.
We measure that impulse by...

Last edited by a moderator: Apr 20, 2021