Gravitational force outside an irregular shaped shell

Question

Shell theorem states that -

A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre.

Proving this involves integrating gravitational force due to each element of the spherical shell.

My question is, can this theorem be extended to any arbitrary shape (whether solid or hollow) with respect to their centre of mass?

In other words, does the following statement hold true -

Any body (symmetric/non-symmetric/hollow/solid) affects external objects gravitationally as though all of its mass were concentrated at its centre of mass.

Answer 1

This turns out not to be true. Here's a paper that derives the gravitational field of a cubic planet. Key quote:

If we look at the changing direction of the field as we move across a face, then we observe that the field vector only points towards the center of the cube at the center of each face, at the corners, and at the center of each edge, which could also be deduced by symmetry arguments, refer Fig. 2.

FIG. 2: The field through a cube sliced in half through the faces. We can observe the slight distortion of the field lines between the edges and the center of each face.

If you are standing on a face cubic planet midway between the center and an edge, then the local gravity field will not point towards the center of the cube's volume, but in a direction closer to perpendicular to the face.

Orbits around such a planet are also discussed. The ellipse of the orbit of a satellite precesses much more than around a spherical planet because of the enhanced gravity near the edges of the cube. The following shows the orbit of a satellite around a rotating cube. The energy gain of the satellite comes from coupling to the stronger field at the corners.

FIG. 6: Following from the previous figure except that the cube is now rotating faster with a 10 hour day, in order to increase the resonance with the 4.8 hour satellite orbit. We can see that with this resonance the satellite rapidly acquires energy, colliding with the cube near the end of the 8th orbit as shown. From the graph of energy gain, we can see that by the 8th orbit the satellite gains nearly 17% more energy at perigee.

This is very different from orbits around spherical bodies, which are simple, non-precessing ellipses (for an isolated, two-body system).

Answer 2

Given point masses $m_1, \ldots, m_N$ at positions ${\bf r}_1, \ldots, {\bf r}_N$, then OP is asking if the gravitational field ${\bf g}$ at a point (which we w.l.o.g. can assume is the origin)

$${\bf g}~=~G\sum_{i=1}^N\frac{m_i{\bf r}_i}{|{\bf r}_i|^3}~\stackrel{?}{=}~G \frac{M{\bf R}}{|{\bf R}|^3},$$

where $$M~:=~\sum_{i=1}^Nm_i,\qquad {\bf R}~:=~\frac{1}{M}\sum_{i=1}^Nm_i{\bf r}_i.$$

It is not hard to find counterexamples already for only $N=2$ point particles.