What Is the Density at the Center of the Sun?

[maximus123]

Hello, here is the question I have to answer;

Calculate the total gravitational potential energy U of a gravitating sphere of mass M with a density profile \rho(r) given by

\rho(r)=\rho_{center}\left(1-\frac{r}{R_{star}}\right)​

where R_{star} is the radius of the star and \rho_{center} is the density at r=0. First give an expression for the center density \rho_{center} in terms of R_{star} and M, then compute a value for the sun. Calculate the total gravitational potential energy of the sun.

I am aware that the gravitational energy of one layer of thickness dr is

dU=-\frac{GM(r)dm}{r}​

and that ultimately I will have to integrate this over all radii but I am unclear about the expression for \rho_{center}. The only thing that springs to mind is

\rho=\frac{M}{\frac{4}{3}\pi R^3}​

but this must be for an average density over the whole star. Can anyone point me in the direction of how to establish an expression for \rho_{center}?

Thanks a lot

 

[jedishrfu]

It says in the problem that it is the density at the center ie r=0 its just a constant scalar.

so you must construct a function M(r) using p(r) for the shell.

 

[maximus123]

However the question says "give an expression for \rho_{center}, so presumably I have to calculate the value of \rho_{center} from scratch rather than looking it up. For example I know that the value for the center density quoted from many sources is 1.622\times10^5\textrm{ kg m}^{-3} however it is clear from the question I cannot simply use this value but I need to form an expression and then set r=0, but anything I try always ends up with r in the denominator thus resulting in infinity.