Gravity simulation with arbitrary geometries

[debilo]

Hello,

I want to study the trajectory of planets (considered as points) around a body with arbitrary geometry (cube,...). I would like to restrict the simulation to a 2D plane, which would be a plane of symmetry of the 3D object (for example a plane that would cut the cube into two equal parts). I thought I'd solve Poisson's equation in 3D. But my computer is not very efficient (and I only know Finite difference methods).

Are there methods to avoid calculating the field in all the pure space to focus on the field in the plane? I couldn't find any articles on this subject.

Thanks.

Edit : I want to do the simulation with Python.

 

[Ibix]

When I did this for 2d polygonal gravity sources, it turned out to be possible to write an analytical expression for the gravitational field of a parallelogram. Any polygon can be divided into parallelograms, and then calculating the total field was just the sum of the parallelogram fields.

Can you find an algorithm to break your shape into standard simpler shapes - triangular prisms with sloped ends, for example? Can you write down an expression for the gravitational field of that kind of simple shape?

 

[debilo]

Hello,

thank you for your answer.

Can you find an algorithm to break your shape into standard simpler shapes - triangular prisms with sloped ends, for example?

I really don't know how to do these things.

I had a new idea. Instead of solving the Poisson equation I consider that each point of my grid is Dirac-mass and I calculate the potential on the whole space using the well-known formula : $$V(r)=\sum Gm(r')/|r-r'|$$ I don't know if it's a good idea.

 

[Ibix]

It depends what you want. You can, of course, split your object up into a cubical grid and pretend each little cube is a point mass at its center. That will get more accurate the smaller your cubes are - in fact, the limiting case of infinitely small cubes is an integral over the volume, which is exactly what I was proposing.

Algorithms to divide a polygon up into triangles are easy to find online. Imagine drawing a wireframe of your object looking down on it from above - you immediately get a set of polygons that you can triangulate. That gives you the simple polygons. The only question then is if you can write down and/or do the integral to get the gravitational force from such a prism.

However, if you are not going for an analytical solution, then you will have to balance precision (more and smaller cubes) versus the computational cost of doing so.

 

[DrStupid]

I thought I'd solve Poisson's equation in 3D.

How about a 2D integration over the surface of the body using the divergence theorem?