§12 Polytropic gases

In subsequent discussions we will consider gases with a particular behaviour in the way they change their thermodynamical quantities under quasi-statical processes. This gas, the so called polytropic gas was first introduced in thermodynamics by G. Zeuner (Chandrasekhar, 1958) and it is used extensively in Astrophysics.

A polytropic change on the thermodynamical quantities of a gas is said to occur if the change is done quasi-statical and is such that its specific heat remains constant during the entire process. From this definition it follows that (Chandrasekhar, 1958):

$$p \propto n^{\kappa},$$ (12.1)

where the polytropic index $\kappa$ is a constant and has a very well known value of $5/3$ for an adiabatic mono atomic gas (Landau & Lifshitz, 1994b) in which relativistic effects are not taken into account. In the case of an ultrarelativistic photon gas it has a value of $4/3$. The first law of thermodynamics, eq.(9.3), can be rewritten as (Stanyuokovich, 1960):

$$\frac{ \mathrm{d} \ln e }{ 1 + p/e } = \mathrm{d} \ln n.$$ (12.2)

The speed of sound $a$ and the enthalpy per unit mass, specific enthalpy, $w$ of a polytropic gas can then be written accordingly (Stanyuokovich, 1960):

$$a^2 = \ensuremath{\mathsf{c}}^2 \frac{ \kappa p }{ e \left( 1 + p/e \right) },$$ (12.3)

$$\frac{ \ensuremath{\mathsf{c}}^2 }{ w } = 1 - \frac{ 1 }{ \kappa - 1 } \frac{ a^2 }{ \ensuremath{\mathsf{c}}^2 }.$$ (12.4)

In the case of an ultrarelativistic gas, that is, when $p \! \sim \! e$ -for example, a photon gas in which $p \! = \! e / 3$, it follows that (Stanyuokovich, 1960):

$$p = \left( \kappa - 1 \right) e$$ (12.5)

$$a = \sqrt{ \kappa - 1 } \ensuremath{\mathsf{c}}$$ (12.6)

For the case in which relativistic effects in the macroscopic motion of the gas are not considered, eqs.(12.3)-(12.4) become

$$c^2 = \kappa \frac{ p }{ \rho },$$ (12.7)

$$w = \frac{ \kappa }{ \kappa - 1 } \frac{ p }{ \rho }.$$ (12.8)

according to eq.(10.1).