Asteroid Gravity Models

I did this a while back but decided only now to post it. Right now I have 3 gravity models (2 working). The problem with gravity models is that for concave structures, a tetrahedron from a face to the origin may have elements external to the body or overlap other tetrahedra. This "creates" mass/volume outside of the actual object volume which can cause huge errors for simulating an orbiting satellite.

A few ways around this are:
   1). to create two "origins" and set the origin for each face as the one where the inner product (dot product) of the CCW face normal and radius vector are positive (if both are positive take the one with the smaller radius).
   2). Define the mass to exclusively be on the surface.
   3). Do some math and define the gravity as a field affected by the body volume.

Originally, I was only using a point mass tetrahedra model. This model took the faces, connected them to the origin (creating a tetrahedron), found the volume, and put a point mass at the center of the tetrahedron proportional to the volume and density. This model does not work for very irregularly shaped objects, however it works best for the majority of asteroids.

The other working model is a shell model. It takes all of the mass at the surface of the asteroid where each unit of surface area contains the same amount of mass. While this model is better for more irregularly shaped objects, by putting all the mass at the surface it creates a gravity singularity at the surface and can mess up data if trying to analyze an orbit close to the asteroid. This model is best for analyzing the gravity field for very irregularly shaped asteroids at a reasonable distance from the surface.

The following plots are for validating these two models. I will discuss the differences in the plots and what denotes a good "fit" at the end.

It should be noted that the scale at the bottom is the number of points in the analysis direction and does not correlate to the size of the objects.

Eros
    

        

Left: Point Gravity Tetrahedra Model

Right: Surface Area Model

Churyumov-Gerasimenko (67P)

Left: Point Gravity Tetrahedra Model

Right: Surface Area Model

What to take from this:

Point Mass Tetrahedra

• Underestimates radius since it takes the center of the tetrahedra.
• This conserves mass while avoiding the gravity singularity at the surface
• For very irregularly shaped objects (67P), the volume is disconfigured and does not give accurate information about the shape/gravity of the object.

Surface Area

• Preserves shape of irregularly shaped objects (67P)

Conclusions

• The Point Mass Tetrahedra model should be used for regular shaped objects.
• The Surface Area model should be used for irregularly shaped objects. But may create large errors is simulating a close approach.
• Another model needs to be created to better represent this.
• A mixture of the two may work. I am also looking into Spherical Harmonics as well as a triangulation method I read about a while ago. But for now these will do.