How Is the Constant k0 Calculated in the Stellar Opacity Power Law?

[mccizmt2]

I am trying to write some code to produce a model of a star. I've done loads of research and come to the conclusion that for ease i want to represent the opacity in my model with the power law.

k=k(0)*(rho^alpha)*(T^beta)

this seems very straight forward apart from the fact that in every source I've looked at it states how easy this equation is to use etc but never actually tells you how to calculate the constant k(0). Which i presume is opacity at the centre of the star. Could somebody please help me.

Thanks

 

[QuantumQuest]

The detailed form of opacity involves some tedious calculations. The relation which the OP wants to use for opacity, is k = k₀ ρ^α T^β, which can represent some good approximation to the results of a detailed calculation, provided that we put some constraints on the factors of density (ρ) and temperature (T), basically confining their ranges. We also have to clarify, that α and β are functions with slow variation of their respective components (ρ and T). The constant k₀ for which the OP asks, is a constant for stars of given chemical composition.
Now, in order to calculate k₀ , we have to construct a log - log diagram of a star's opacity (k) vs, temperature (T), for a given star density and chemical composition. Because such a diagram is specific for a certain star, below is given a rough sketch of such a diagram

The numbers -3 and -1 correspond to two different densities 10^-3 and 10^-1 respectively.
We see that the opacity is low at high temperatures and remains roughly constant, as temperature increases. This is explained by the fact that most atoms are fully ionized there, photons have high energy, so their absorption (free - free) is not as easy, as in lower energies. Hence there, opacity works through electron scattering, which is independent of temperature (T). This results in a form k = k₀. (α and β are zero).
At lower temperatures, the opacity is also low, most atoms are not ionized and there are not enough electrons to scatter radiation and photons have not sufficient energy to ionize atoms. The approximate form becomes k = k₀ ρ^1/2 κ⁴.
Finally, we see that opacity reaches a maximum at intermediate temperatures. A rough analytical approximation there is $$k = \frac {k_{0} ρ}{T^{3.5}}$$.
These analytical approximations can be used together with some expressions of stellar structure, in order to model a star.