Pressure in Hydrostatic Equlibrium

[zachzach]

Homework Statement

Find the pressure as a function of r for the region r<< R where R is the radius of the object. The density goes as $$\rho = \rho_o {(\frac{r_o}{r})}^2$$.

Homework Equations

I know how to get to the answer my problem is dealing with the infinite density at r = 0.

The Attempt at a Solution

My pressure integrates to an equation that is inversely proportional to the square of r. But at r = 0 the central pressure, will be infinite. How can you deal with this infinite. It makes sense since the density is infinite at r = 0, but there must be an answer. If the $$r_o$$ was an R you could do a taylor expansion but it is not.

 

[hikaru1221]

Then just take it as an approximate model A better model may look somewhat like this: $$\rho = \rho _0 (\frac{r_o}{r+\epsilon})^2$$ where $$\epsilon << R$$ (I devise it, so don't take it for real ). If you're interested in the pressure at r=R, r=R/2, etc, then the model given by the problem might be sufficient. If you're interested in the pressure near the center of the object, then there is a need for another model.
In short, don't take the formula given in the problem too seriously

 

[hikaru1221]

Hi,
Sorry, I overlooked the r<<R part, which made my reply above rather stupid But then, my conclusion is still the same: this model is not sufficient for calculating pressure near r=0. It's very non-intuitive to have the density to go to infinity, and thus, this non-intuitive model will probably lead to non-intuitive result.