Why is a star unstable if it's adiabatic exponents are less than 4/3?

Question

In "Introduction to Stellar Structure" by Walter J. Maciel at page 76 it is said that for a partially ionized non degenerate hydrogen gas, the star is unstable if the adiabatic exponents are less than 4/3.

Why is this the case and where does this limit value of 4/3 come from? Is it related to the fact that for a photon gas all the adiabatic exponents equals 4/3?

For example, the first adiabatic exponent $$\Gamma_1 = \frac{\partial \ln(P)}{\partial \ln(\rho)}|_{Q=0}$$ describes how pressure reponds to compression. The higher this value is, the more the star will emit a pressure to oppose a density increase. I can therefore see why there is a treshold value from which the star is no longer able to resist an increase in density and collapses. But why 4/3?

Answer 1

The fact that the instability appears for $\gamma = 4/3$ is an approximate Newtonian result and refers to some "average" value of the adiabatic index in the body (e.g., a generalization of the classic Chandrasekhar's analysis can be found here, the value is not always $\gamma = 4/3$).

In short (you can find the mathematical details here, see also this very good answer):

1 - For a given polytropic index, the (Newtonian) hydrostatic equilibrium is given by the Lane-Emden equation.

2 - Consider an adiabatic (i.e. you can associate an adiabatic index $\gamma $ for the pressure response that coincides with the "equilibrium" one), homologous, adiabatic perturbation of the hydrostatic solution.

3 - The perturbed configuration no longer satisfies the equation of hydrostatic equilibrium, but the Euler equation of motion for an auto-gravitating fluid (in spherical symmetry).

4 - For small perturbations, the solution to the equation of motion is expressible in terms of exponentials. The complex nature of the exponents defines whether the star oscillates or "implodes", this depends on the value of the adiabatic index. The "threshold" turns out to be $4/3$.

5 - General relativity tends to give "stronger" gravity than Newtonian gravity, so in a relativistic context, the instability may appear for slightly larger values of the adiabatic index (it is "simpler" to collapse). See this for a general relativistic discussion.