2nd Week: Properties of Astrophysical Plasmas

The remnant of "Tycho's Supernova", a huge ball of expanding plasma.

Plasma is a state of matter in which the atoms and the molecules are so hot, that they have ionized into negatively charged electrons and positively charged ions. The plasma found in the universe, whose physical properties are studied in astrophysics is known as astrophysical plasma. To study the properties of astrophysical plasma the equation of state of matter is very important. Here, we will bring some basic tools from thermodynamics and derive the equation of state for non-relativistic and relativistic plasma.

Basic thermodynamics for quantum systems

The particle density is determined as follows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = \frac{N}{V} =\int_{0}^{\infty }{w(p)f(p)dp} \ , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p)} is the occupation probability and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w(p)} is the state density per unit volume.

The energy density is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u = \frac{U}{V} =\int_{0}^{\infty }{Ew(p)f(p)dp} \ , }

where $E$ in the energy of the particle.

The pressure is defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P = \frac{1}{3}\int_{0}^{\infty }{pvw(p)f(p)dp} \ , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is the value of the momentum of the particle and $v$ is its velocity.

Distribution functions

In statistical physics the density of states of a system describes the number of states at each energy level that are available to be occupied. The general form of the density of state is as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(E) = \frac{d\Phi}{dE} \ , }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} is the number of states.

For the non-relativistic system

$\Phi (E)={\frac {4\pi }{3}}{\frac {g}{h^{3}}}(2mE)^{3/2}\ ,$

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g=2s+1} is the statistical weight.

There are four probability distribution functions in statistical physics that are known with infinite support: Gibbs distribution, Maxwell-Boltzmann distribution, Fermi-Dirac distribution and Bose-Einstein distribution. Herein we provide the expressions for the last three of them.