Galactic Dynamics (spherical potential)

[gupster]

This is an adaptation of question 2.1. in Galactic Dynamics by Binney and Tremaine in case anyone owns it.

Homework Statement

Astronauts orbiting a planet find that
(i) the surface of the planet is precisely spherical.
(ii) the potential exterior to the planetary surface is $$\phi = \frac{-GM}{r}$$ exactly, that is, perfectly spherical.

Can you conclude from these observations that the mass distribution in the interior of the planet is spherically symmetric?

Homework Equations

The Attempt at a Solution

I think that it doesn't matter whether it is a point mass inside at the centre or whether there is a symmetric mass distribution but I don't really know why I think that!

 

[astrorob]

Interesting question,

I'm wondering if you'd need to know how isolated the planet is since the potential will be the cumulative effect of all bodies in proximity..

That's to say, the mass distribution mightn't be symmetric, but by some random chance the potential is due to the presence of other planets etc..

On another note, are you the guy who posts on punktastic?

 

[gupster]

I'm the girl who posts on punktastic...

 

[astrorob]

Aha, sorry! I don't post on there, I've just read a few threads like

 

[Barny]

Its been a while since I've done gravitation so might be a few meters off the mark, however i would guess that...

If you make the assumption that the planet is in deep space so all other potentials can be neglected then that leaves you with a lump of mass making a perfectly spherically symmetric gravaitaional potential.

The gradient of this potential will give you the gravitational field - just your usual 1/r^2 field.

Could you then not just use Gauss's law to infer the charge distribution?

Hope that's of some help, but has been ages since I've approached these kinda problems.

 

[Ataman]

Well, you could prove by brute force integration that the planet acts as a point source and the gravity field is symmetric.

-Ataman