How Is the Radius of a Star Determined from Its Density Profile?

[Poirot]

Homework Statement

P=Kρ^2 is a solution to the equation of the combination of the Hydrostatic Support equation and the mass continuity equation. Find the radius of the star.

Homework Equations

ρ(r) = (A / r) sin (root( 2πG/K) r)

The Attempt at a Solution

The first part of this was to prove first it was a solution which I have done fairly easily, however the last part about the radius has left me confused.
I figured the density at the surface (r=R) was equal to zero therefore:

0=(A / r) sin (root( 2πG/K) r)

And for the non trivial solution:

sin (root( 2πG/K) r)=0

so root(2πG/K) r)=nπ (for n integer)

However this would give a range of radii for the star which doesn't seem right.
Can you see what I've done wrong, thanks?

 

[mfb]

You get a position of zero density at root(2πG/K) r)=π. Do you expect matter outside this region? What would support it?
Is ρ(r) = (A / r) sin (root( 2πG/K) r) even valid outside that region?

 

[Poirot]

What happens to the other solutions? Surely root(2πG/K) r)=2π etc is still valid? and the density should only hold for the star up to radius R, and no there wouldn't be matter outside the region.

You get a position of zero density at root(2πG/K) r)=π. Do you expect matter outside this region? What would support it?
Is ρ(r) = (A / r) sin (root( 2πG/K) r) even valid outside that region?

 

[mfb]

It is a mathematical solution, but the density profile is not described by a sine in that area any more. The density is zero after the function hits its first zero.