What is the most “non-point” gravitating or electrically charged object?

Abstract

In this paper we search the shape of an aspherical body and the direction in space, for which the greatest deviations from the point mass
field (the difference from the inverse-square law) take place for large distances from the field source. It turns out to be a system of two equal
point-like masses at the poles of a fixed sphere (giving the greatest positive deviations from the point mass field) and uniform distribution
of point-like masses (discrete or continuous) around the sphere equator (giving the greatest negative deviations from the point mass field).
In these cases the extremal direction of the field measurement respectively passes through point-like particles and coincides with the axis of
symmetry of a ring, which is perpendicular to its plane. Our numerical estimations show that any body can be considered with reasonable
accuracy (the relative error in the determination of the field strength is less than 5%) as point-like mass if the distance to the observation
point is more than an order of magnitude larger than its characteristic sizes. The problem considered in this paper can help readers to probe
the limits of applicability of the field point source model.