Keplerian scattering through an array of mass lumps

[FunkyDwarf]

Hey guys,

Hit a bit of a snag in something I'm working on and need some help. I've attached an image so hopefully that will help a bit. Basically I'm trying to model a mass m moving through a potential that is, currently, a grid of point masses and seeing how the overall distance traveled in a given time changes as you vary the masses of the grid. I have a model and I'm comparing to some simulations I'm running (which theoretically should be correct).

The model:
Ok so basically I have some code that treats the incoming mass as coming along the positive x-axis at some velocity and some height, impact parameter b, which is unperturbed until it enters the interaction region, at which point it undergoes motion according to the solution to the Kepler problem, and then leaves. I'm assuming conservation of energy here (ridgid central body) and so the outgoing velocity is the same as the incoming, or close enough anyway. It's basically the impulse approximation. I realize there are problems with this namely with masses of comparible size to the incoming mass and i think that may be where my problem lies but i'll continue anyway. So for a given incoming mass, point mass...mass, impact parameter, radius of interaction region and velocity i can output a scattering angle. From this i assume some distance and travel time between scattering events and get an overall path distance, which i then compare to an unperturbed path difference (ultimately its the difference I am interested in).

Now here's my problem: the parameters i was having trouble fixing or relating to other parameters are the impact parameter and interaction region radius. Now i figure the IR radius s must be related to the mass of the point mass so i did some simulations and found that it follows a power law pretty well ie s proportional to aM^b. Now physically i would assume that this should be sufficient, i mean any change in velocity or the incoming mass should be taken into account by the rest of the mathematics and the 'physical size' of this mass/its IR should be independent of incoming velocity and mass, however when i start fiddling with those i get inconsistent results. I fear this may be because I am considering central masses that are too small to be considered stationary wrt the other mass. I want my method to be as general as possible so should i rewrite my model in terms of centre of mass scattering instead?

I was wondering if anyone had any insights on this matter =(

Cheers
-G

EDIT: Sorry perhaps this should be in the CM forum, its to do with stars and galaxies so i figured i'd put it here =P

 

[bombadil]

Sounds like your having fun. I'm not sure I totally understand your question, but maybe this will help:

Interaction distance, $s$, can be found by setting kinetic energy equal to potential energy:

$$\frac{1}{2} m v^2 = \frac{GMm}{s^2}$$

So:

$$s=\frac{GM}{v^2}$$

This sort of problem was looked at analytically by Chandrasekhr (sp?) and others. When a large body moves through other smaller bodies it actually slows because of gravitational effects. This effect is known as http://en.wikipedia.org/wiki/Dynamical_friction" :

$$F_{dyn} \propto \frac{M^2 \rho}{v^2}$$

You can compare your simulation to the dynamical friction formula found in the above link to check for consistency.

Good luck,
bombadil

 

[FunkyDwarf]

Ah of course. Yes i tried PE = KE but being the muppet that i often am i had the wrong equation and had s = GM/m v^2 which gave a constant scattering angle regardless of other paramters. I think what you've given there is the hard scattering distance which i heard my supervisor mention, i will give it a go, cheers!