Estimation of central temperature of a star

[Darkkk]

Problem:

Relatively low mass stars can be shown to have their radii dependent on their mass as;

R\proptoM^1/2

From this, show that the temperature in kelvin at the centre of such a star can be estimated to be

\approx24\times10⁶(M/M_solar)^1/2


Relevant Equations:

T_c=(2/3ℝ)\timesP_c\times(R³/M)


Attempt at a Solution:

I tried to sub in the relation between R and M of the star, but to be quite honest I didn't see myself getting very far...Any pointers at all would be much appreciated! Not too sure on this one.

 

[zhermes]

What did you get when you plugged in?
Do you know P_c and \mathcal{R}

 

[Darkkk]

I do know R=8.314 joules per Kelvin but Pc is unknown, I am assuming the proportion relation will be used at the end of the problem to get the temperature in terms of the solar mass. I am just not quite sure what the first step is. Perhaps I ought to sub the Tc given in the question into the central pressure equation Pc. What do you think? sounds like circular logic to me...

 

[zhermes]

The first step is to plug in the mass-radius relation. That way you eliminate the radius from the equation. You know R, so all you need to find is the central pressure to solve for the temperature.

 

[Darkkk]

Ah yes I see! I have it now. Thank you for the advice :)