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maximus123

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

[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