Calculating Solar Core Radius Using Emden's Equation

[Helios]

I've begun a study of the sun's interior and Emden's equation. Yet nowhere on the web does anyone divulge how to calculate the core radius or the inner core variables. One thing I did was to suppose a homologous polytropic core ( n = 3 ) with:
epsilon = energy generation rate
mean epsilon = L / M
rho = density
mean rho = M / V
so I postulated the homology:
epsilon (center) / epsilon (mean) = rho (center) / rho (mean) = eta
This immediately leads to results that closely match one computer model I found, with
core radius = .2643 R = eta^(-1/3) R,
but my approach seems ad hoc. Any help?

 

[SpaceTiger]

The sun isn't very well approximated by an n=3 polytrope (or any polytrope, for that matter), but if you wish to model it that way, you shouldn't need to guess the central density or temperature. These should fall out naturally when you insist that the total mass be equal to the observed mass of the sun (assuming the usual solar composition).

 

[Helios]

A 3-polytrope is a very good model of the sun. If you disagree, please explain. I did not "guess" at the central density or temperature. I know these derivations. I asked for a way to determine the core radius.

 

[SpaceTiger]

A 3-polytrope is a very good model of the sun. If you disagree, please explain.

The sun has both radiative and convective regions, so has a non-uniform equation of state. Of the simple models, however, n=3 probably is the best.

I did not "guess" at the central density or temperature.

Where did your value of "eta" come from?

I know these derivations. I asked for a way to determine the core radius.

Once given a mass, radius, and polytropic index, I was under the impression you would be able to model the entire star and take from that whatever parameters you want (though I've never done it myself). How are you defining the core radius?

 

[Helios]

Eta is used as a some ratio like epsilon/ mean epsilon or the like.
Chandrasekhar ( in 1939 ) uses it when presenting several star models in his book " Intro to Stellar Structure" at the time when fusion was not well understood.
Lamda is a constant of the polytrope formulation ( = 54.1825 for n=3 ) and ( by proof ) equals rho ( center ) / rho ( mean )
So I set Eta = epsilon (center) / epsilon (mean)
and supposed Eta = Lamda, a homology.
How do I define the core radius? It's the radius where the energy generation equals zero and luminosity becomes constant. The core I defined borrows the polytrope mathematics of the sun at large and replaces density with energy generation rate and mass with luminosity -- a sort of sun within a sun. Then as easy as comparing similar triangles, a core radius can be computed, which looks good ( core = .2643 R ), but is somewhat ad hoc.

I would now rather use lamda instead of eta to avoid any confusion, hence;
epsilon (center) / epsilon (mean) = rho (center) / rho (mean) = lamda = 54.1825