Density at the centre of the sun

[maximus123]

Hello, here is the question I have to answer;

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

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

where [itex]R_{star}[/itex] is the radius of the star and [itex]\rho_{center}[/itex] is the density at [itex]r=0[/itex]. First give an expression for the center density [itex]\rho_{center}[/itex] in terms of [itex]R_{star}[/itex] and [itex]M[/itex], 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

[itex]dU=-\frac{GM(r)dm}{r}[/itex]​

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

[itex]\rho=\frac{M}{\frac{4}{3}\pi R^3}[/itex]​

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 [itex]\rho_{center}[/itex]?

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 [itex]\rho_{center}[/itex], so presumably I have to calculate the value of [itex]\rho_{center}[/itex] from scratch rather than looking it up. For example I know that the value for the center density quoted from many sources is [itex]1.622\times10^5\textrm{ kg m}^{-3}[/itex] 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.