Gravity/density equation? Giant ball of water

[Vree]

Can you help me find the equation for determining density when also taking the object's own center of gravity into account?
For example, say I have an Earth-sized volume of water (or pick any simple spherical object). You have to account for the increase of density as you go closer to the center of mass. Is there a simple equation to determine the change of density from R distance from the center of mass in this case?

Thanks!

 

[Quantum Defect]

Can you help me find the equation for determining density when also taking the object's own center of gravity into account?
For example, say I have an Earth-sized volume of water (or pick any simple spherical object). You have to account for the increase of density as you go closer to the center of mass. Is there a simple equation to determine the change of density from R distance from the center of mass in this case?

Thanks!

There are good analogies with calculating the change in density of the atmosphere with altitude. Google "adiabatic lapse" and "barometric formula". The difference between the atmosphere and a "big ball of water" is obviously two-fold -- the denisty and the compressibility are very different for the two.