Gravitational field strength for irregular object

[donaldosaurus]

Hi all
I'm trying to work out what the surface gravitational field strength of an irregularly shaped body would be (for example Mars' moon Phobos). I know that for a sphere, any point outside it can consider all the mass to be at a point inside it, but for something that's potato shaped, how would you work out what g is?
For example, Phobos is a max of 27km long, and a min of 15km. Gravity would be weaker the further away you are from the centre, and stronger the closer you are. I'm presuming it's more complicated than just plugging the relevant radii into Newton's Gravitational Equation, and probably involves some horrible integrals. Is there any simple equation for an ellipsoid which approximates irregularly shaped objects like these?
Any links to further reading material would be appreciated.
(For anyone interested, I'm trying to work out the best place to jump from a moon of Mars to the surface).

Thanks all

Don

 

[mfb]

You can use Newton's formula for point masses and convert them to an integral over the mass density. Integrate over your whole object, and you get the total gravitational force.
Even for ellipsoids, those integrals can become ugly (unless you use a numeric approximation). No idea if there is an analytic expression for the force.

If you want to jump, use the point with the largest distance to the center of mass, this will work for any reasonably shaped object. Deimos is nice, see this xkcd-comic.

 

[donaldosaurus]

Thanks mfb - I freakin love those xkcd infographics.

 

[D H]

The standard approach for modeling the non-spherical nature of the gravitational field of larger bodies such as the Earth, the Moon, Mars, and Venus is to use spherical harmonics. Developing such models requires a lot of observations of relative position and velocity, plus a sufficiently large gravity field so as to make those observations anything but noise, plus some massive numerical infrastructure to grind those observations down to spherical harmonics coefficients.

There are number of problems with using a spherical harmonics model for smaller bodies such as Phobos and asteroids. One is that there just aren't enough relative positive/velocity observations for such bodies to make it possible to develop a set of spherical harmonics coefficients for those bodies. Even if there were a good number of observations, they would be pretty noisy with respect to extracting those coefficients. Another problem is that even if a good set of coefficients could be developed, they wouldn't be of much use inside the Gaussian sphere used to develop the coefficients. Yet another problem is that the coefficients don't go to zero very rapidly. This means a whole lot of coefficients are needed, which in turn means that the computations of the gravitational force will be very expensive.

The reason the spherical coefficients don't go to zero rapidly is because those small bodies are far from spherical gravitationally. This is true even for the Moon. The five large mass concentrations ("mascons") on the near side of the Moon, plus the disparity between the crust on the near side and far side of the Moon, plus the two kilometer offset between the Moon's center of mass and it's geometrical center all make the spherical harmonics model of the Moon not quite a perfect fit. A spherical harmonics approach is even more iffy for bodies smaller than the Moon.

Those lunar mascons suggests an alternative approach for these smaller bodies: Simply model the object as a bunch of rigidly connected point masses. There's no problem with convergence for points on or above the object's surface so long as those mascons are inside the object. However, the same problem of lack of observations that makes it tough to develop a spherical harmonics model is also a problem for developing a mascon model. Another problem is the shear number of mascons that are needed to develop a model of reasonable fidelity.

There is one kind of observation for which there do exist lots of data, and that's photographs. These photographs can be used to create geometrical models of the surfaces of these small bodies. By assuming a density distribution within the body (typically uniform), it is possible to use these surface models as a gravity model. The math is hairy, the computations are expensive, but at least the requisite data do exist.