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

$n={\frac {N}{V}}=\int _{0}^{\infty }{w(p)f(p)dp}\ ,$

where $f(p)$ is the occupation probability and $w(p)$ is the state density per unit volume.

The energy density is given by

$u={\frac {U}{V}}=\int _{0}^{\infty }{Ew(p)f(p)dp}\ ,$

where $E$ in the energy of the particle.

The pressure is defined as

$P={\frac {1}{3}}\int _{0}^{\infty }{pvw(p)f(p)dp}\ ,$

where $p$ 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:

$\omega (E)={\frac {d\Phi }{dE}}\ ,$

where $\Phi$ is the number of states.

For the non-relativistic system

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

where $g=2s+1$ is the statistical weight. Then the density of state is

$\omega (E)=2{\frac {g}{h^{3}}}(2mE)^{1/2}\ .$

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. The occupation probability, i.e. the probability of occupation of a given energy states are as follows.

• Maxwell-Boltzmann distribution

$f(p)=e^{-{\frac {E(p)-\mu }{kT}}}\ ,$

• Bose-Einstein distribution

$f(p)={\frac {1}{e^{\frac {E(p)-\mu }{kT}}-1}}\ ,$

• Fermi-Dirac distribution

$f(p)={\frac {1}{e^{\frac {E(p)-\mu }{kT}}+1}}\ ,$

where $k$ is the Boltzmann constant.

Thermodynamical variables and potentials

Now we provide with the expressions of thermodynamic potentials. The internal energy is given by

$U=TS-PV+\sum _{i}\mu _{i}N_{i}\ ,$

where $T$ is the temperature, $S$ is the entropy, $P$ is the pressure and $V$ is the volume of the system, while $N_{i}$ is the number of particles in the state $i$ of the system and $\mu _{i}$ is is the chemical potential associated to those particles. These quantities are also known as thermodynamical variables.

The so called Helmholtz free energy is given by

$F=U-TS\ .$

The enthalpy of the system can be found using the expression

$H=U+PV\ .$

The Gibbs free energy is

$G=U+PV-TS\ .$

The so called Landau potential is expressed by

$\Omega =U-TS-\sum _{i}\mu _{i}N_{i}\ .$

Equation of state

Comparsion of the Fermi-Dirac, Bose-Einstein, and Boltzmann distributions. All for the same value $\mu$.

In physics equations of state are thermodynamic equations that describe the state of matter under a given set of physical conditions. these equations provide relationships between two or more macroscopically measurable quantities (such as pressure, volume, temperature and Energy) in a given state of matter.

A classic example of an EOS is the Ideal Gas Law by Benoît Paul Émile Clapeyron,

${\ pV=nRT}$

Important Equations of State for Nuclear Astrophysics as given in class:

Particle density for a given system is,

$n=\sum {n_{i}}$,

$n_{i}={\frac {4\pi g_{i}}{h^{3}}}\int _{0}^{\infty }{\frac {p_{i}^{2}dp_{i}}{exp[{\frac {E_{i}-\mu _{i}}{kT}}]\pm 1}}$ where, $E_{i}={\sqrt {p_{i}^{2}c^{2}+(m_{i}c^{2})^{2}}}$

Pressure for a given system is,

$P=\sum {P_{i}}$

$P_{i}={\frac {4\pi c^{2}g_{i}}{3h^{3}}}\int _{0}^{\infty }{\frac {p_{i}^{4}dp_{i}}{E_{i}(exp[{\frac {E_{i}-\mu _{i}}{kT}}]\pm 1)}}$ where again, $E_{i}={\sqrt {p_{i}^{2}c^{2}+(m_{i}c^{2})^{2}}}$