Equations of State I

Notes following Ch. 3 of Pols Lectures here & HKT, Ch. 3, Available Online.

Day 3 - September, 4, 2025

Agenda:

• Updates/Reminders (2m)
• Lecture (25m)
• In-Class Activity 3 - Groups of 2 - Due end of day (30m)
• Report out on ICA in ica Slack Channel (10m)

Recap

• Mass Continuity

• Hydrostatic Equilibrium

Q: Do we have what we need to solve these equations?

The equation of state

A: No, we need some relation between the pressure and the density - a so-called equation of state (EOS).

• describes the microscopic properties of stellar matter, for given density , temperature and composition

Ideal gas EOS:

Equation of state for a gas of free particles

Aim: Derive the equation of state for a perfect gas from the principles of statistical mechanics. This provides a description of the ions, the electrons, as well as the photons in the deep stellar interior.

Assuming that the star is in local thermodynamic equilibrium

• radiation field becomes isotropic
• the photon energy distribution is described by the Planck function
• the average time between particle interactions (the mean free time) is much shorter than the timescale for changes of the macroscopic properties

Equation of state for a gas of free particles

To obtain the pressure, we need to consider the distribution in velocity space of these particles with mass and momentum

The momentum distribution between and is of particles is a Maxwell-Boltzmann distribution:

where the prefactor of the exponential is the normalization constant, and the exponential comes from assuming a Gaussian distribution of kinetic energies for each momentum component.

Equation of state for a gas of free particles

Using the known , we can compute the following:

Total number density (per unit volume):

Internal energy density (per unit volume):

Equation of state for a gas of free particles

Pressure

where is the kinetic energy of the particle with momentum and velocity .

The energy and momenta relate according to special relativity:

and

Equation of state for a gas of free particles

For the non-reletivisitc limit (NR) limit , and

For the extremely relativistic limit (ER) limit , and

Mixture of (non-relativistic, classical) gases

Each gas will contribute to the pressure:

where is the number density of the ions , which have mass
with the atomic mass unit: .

We can relate the number density of the ions of species with the mass density that already appears in the equations we already have with .

Mixture of (non-relativistic, classical) gases

Some definitions and vocabulary:

• - the number of free particles in a gas
• - nuclear charge
• - nuclear mass number in (amu)
• - fraction by mass of a species (aka mass fraction)
• - the ion number density in units of cm of a given species

Mixture of (non-relativistic, classical) gases

The ion number density takes the form of (HKT 1.40)

Summing over all ions gives the and defining as the total mean molecular weight, we have the ion number density for all ions:

where, mole.

Mixture of (non-relativistic, classical) gases

As such, we can rewrite the total mean molecular weight of ions

the ion mean molecular weight is then a sort of mean mass of an “average” ion in the mixture.

Mixture of (non-relativistic, classical) gases

Following a similar derivation, we can also define the mean molecular weight per free electron,

where is the ionization fraction, a value of means the gas is completely ionized (all electrons removed) while implies the gas is completely neutral.

Mixture of (non-relativistic, classical) gases

We finally arrive at expressions for the total mean molecular weight,

The mass fractions for often defined as for H and for He and all else referred to as "metals" denoted by (not ion charge!).

This leads to the composition of a star, .

Mixture of (non-relativistic, classical) gases

Now, lets consider some scenarios where we can approximate values for the molecular weights.

When ,, and are completely ionized and (metals are a small mass fraction)

Q: What is the mean electon molecular weight for a Solar like star on the main-sequence assuming full ionization?

Mixture of (non-relativistic, classical) gases

If we further assume that is small compared the

combining for an approximate form for the total mean molecular weight:

Q: What is the total mean molecular weight for a Solar like star on the main-sequence making the above assumptions?

Can we really use an ideal gas EOS in a plasma?

The dominant interaction between the particles (ions and electrons) is going to be through the Coulomb force,

for particles of charge (the ions) and average distance with number density. We want to compare this with the kinetic energy, which for point-like particles is .

Can we really use an ideal gas EOS in a plasma?

The ratio of these two is often called the Coulomb parameter ()
(neglecting constants of order unity):

We can assume the ideal gas situation if , which is the case
for the average and ρ of the Sun.

Q: Which stars might not follow an ideal gas law and require a different description of the plasma?

In-Class Assignment 3

In class: Work on ICA here with partner, I will ask one or two people to share and describe plots at the end of class.

After Class: End of day today, September, 4, 2025

• Submit as a PDF, preferrably using nbconvert to D2L, the progress you have made.

ICAs are not always designged to be completed but rather worked on in class, submit what you have when you leave the class even if you did not make much progress.