Calculating the central temperature of the Sun using the ideal gas law

[accalternata]

Homework Statement

Assuming an ideal gas law with no radiation pressure, P = ρkT/µmH, find
an expression for T(r).
Given mu = 0.61 for the Sun (you can use the Sun’s mass and radius), what
is the central temperature of the Sun in this model?

Relevant Equations

See below

I derived the equation for P so I substituted that into this equation. I'm struggling with finding rho_c (central density) and rho.
Am I supposed to use the average density for rho (can calculate this since I know the radius of Sun and mass)? That still leaves the problem with the central density though.

 

[Steve4Physics]

Am I supposed to use the average density for rho (can calculate this since I know the radius of Sun and mass)? That still leaves the problem with the central density though.

It's hard to know you are supposed to do. This is not my area but (in the absence of other replies) how about this...

It may be acceptable to assume some simple form for ##\rho(r)##. For example (as used in section 5.1 of this link) ##\rho(r) = \rho_{centre} ( 1 -\frac rR)##.

You can then integrate to get an expression for ##M## in terms of ##R## and ##\rho_{centre}## and hence find a value for ##\rho_{centre}##. This should be in the right 'ball park' which (in the context of the question) is probably acceptable.