4. Equation of State and Related Descriptions

Mostly from Hansen and Kawaler.

4.1. General considerations

In almost all stellar environments, we may assume a strict thermodynamic equilibrium applies. Exceptions include optically thin regions and rapidly changing regions such as one might find in the initial stages of a supernova.

There are two dominant ways of considering how an assembly of particles behaves. One is called the physical picture, where one keeps track of all elementary particles whether they are bound together in atoms or not. The other is called the chemical picture, where one keeps track of 'chemical' species, eg. different ionization states of elements or in some cases even different excitation states. For all intents and purposes, these chemical species are treated as interacting, continuous fluids. Virtually all stellar (and most other) astronomy uses the chemical picture. Only in non-thermodynamic conditions (plasma waves, etc.) do astronomers sometimes use the physical picture. There is no real reason except historical for this preference.

4.2. Thermodynamic variables of interest

At some point or another we will require knowing the values of the following quantities:
P: pressure (total or partial) in erg/cm3
T: temperature in K
ρ: density in g/cm3 (also Vρ =1/ρ)
S: entropy in ergs/gK
E: internal energy in ergs/g
Q: heat in ergs/g
ni: number density of species i in /cm3
Ni: number density of species i in /g.
In addition, we will be interested in several Γ 's and specific heats to be defined later.

Of particular interest in the chemical picture of things is the chemical potential of species i

μi ≡ (∂E/∂Ni)S,V ;

e.g. how the internal energy per gram changes when the number of species i per gram changes at constant entropy and volume. Thermodynamic and chemical equilibrium demands that ∑i μi dNi = 0.

Consider hydrogen: H+ + e ↔ H0 + γ,
which we may write generically as

i νi Ci = 0

where Ci is the species label and νi is the number of particles of the species involved (1 for all the species in the hydrogen equation). In equilibrium, it must be that any small change dNi of a species in this reaction must be related to changes in the other species via

dNi /νi = dNj /νj .

Exercise: show that the above relation holds true for the reaction 2H↔H2.

Substituting the relation among differential changes into the thermodynamic equilibrium relation gives the chemical equilibrium equation

i νi μi = 0.

Because photons are not conserved, dNγ is arbitrary, and as a result μγ = 0.

In a perfect gas, the interaction process itself happens in a short time and is independent of the state variables. In a real gas, particularly one that is dense enough that internuclear distances approach electron or molecular binding distances, calculations are difficult and generally heuristic.

4.3 Perfect gas statistical equilibrium equations

Elementary statistical mechanics leads to the following expression for the number of particles of species i in phase-space volume d 3r d 3p :

ni(p) d 3r d 3p = (d 3r d 3p / h3) ∑ j gj {e [−μi + εj + ε(p)]/kT ± 1}−1
[+1: Fermi-Dirac; −1: Bose-Einstein; 0: distinguishable particles].

ε(p) = (p2c2 + m2c4)1/2mc2, velocity v = ∂ε/∂p, and

Pi = (1/3) ∫ ni(p) pvp2 dp

for isotropic momenta.

Hansen and Kawaler are rather cavalier with their units and notation. I have specifically included the d 3r d 3p on both sides to show that so far, the units of ni(p) are cm−3 (g cm/s)−3. The distribution ni(p) in the pressure equation is the same. I have used bold-face type to distinguish this phase space distribution.

When the authors start talking about blackbody and other distributions, they suddenly shift units for ni(p) to cm−3 (g cm/s)−1, having carried out the integration over momenta directions internally. I will use normal-face type to distinguish the lower dimensioned distribution.

ni(p)dpni(p)4πp2dp /cm3.

H&K make no notational difference and never say that one n is not the same as another. Since we are assuming isotropy, the internal integration is straightforward. I don't think the 6D distribution will ever come up again.

4.4 Special cases

A. Blackbody radiation
B. Monatomic gas
C. The Saha equation
D. Fermi-Dirac degeneracy

4.5 Adiabatic exponents