Two-body interactions with surface integrals

Two-body interactions between extended, non-spherical bodies can be challenging to compute.
In the most general case, a full three-dimensional treatment requires a sextuple integral
to be solved because a volume integration is carried out over both bodies
(a double volume integral). This is computationally intensive, and approximations and
simplifications are normally done at the cost of accuracy. Here, we are presenting a new model
for computing forces, torques and the mutual gravitational potential
between two bodies with ellipsoidal shapes. The model allows for an exact solution of the
full rigid two-body problem using surface integrals, and is therefore more efficient than
computing volume integrals, as the double volume integral is reduced to a double surface integral.
Compared to more commonly used methods where the mutual potential is expanded in a
series and truncated at e.g. the 2nd or 4th order and hence suffer from truncation errors,
our method is exact and can also be used for bodies in close proximity.

We argue that the model is particularly applicable for closely interacting
asteroids that can be approximated by ellipsoids, such as binary systems, or for studying the dynamical
evolution of asteroids directly after a rotational fission event.

In particular, we show results from some simulations of binary systems formed via rotationial fission of
contact binaries. Asteroid binaries among Near-Earth Objects are believed to have formed by
rotational fission, and we simulate several cases with varying initial conditions
where the components have different shapes and densities. More than 80 per cent
of the simulations end with the two bodies impacting, and collisions between the
bodies are more common when the density of the secondary is lower, or when it becomes more elongated.
When comparing with data on asteroid pairs from Pravec et al. (2019) we
find that variations in density and shape of the secondary can account for some of the spread
seen in the rotation period for observed pairs.

Since our model allows for accurate calculations of the force and torque between two rigid, ellipsoidal
bodies, we make a comparison with models from the literature that uses 2nd and 4th order
approximations of the mutual potential. This allows us to point out which configurations between two asteroids
that will result in significant errors both in the force and torque when using truncated potentials.