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Take a degenerate object, such as a white dwarf, embedded in a gas cloud. The cloud is small enough such that the size of the object is non-negligible in comparison - that is, the white dwarf takes up a decent amount of space in the cloud. The white dwarf is stationary and is not rotating.

If I were the model the accreting gas as a polytrope - i.e. as $$p=K \rho^{1+\frac{1}{n}}$$ where $p$ is pressure, $\rho$ is density, $K$ is a constant and $n$ is the polytropic index - what value of $n$ should I choose?

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The gas outside the white dwarf (and actually, in the interior of the white dwarf but close to the surface) is non-degenerate. So I don't see what the presence of the white dwarf has to do with the question unless it is significantly photo-ionising the cloud?

Anyway for spherically symmetric accretion onto an object, one approach is to make an assumption that no heat is conducted or radiated away, in which case the adiabatic index is 5/3 for a perfect monatomic gas ($n=3/2$).

For other assumptions, or types of gas, things will change.

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