On Asteroid Engineering

TL;DR

This paper investigates the optimal shape of a malleable, homogeneous body to maximize surface gravity at a point, deriving a specific shape that exceeds the gravity of a sphere by 2.6%, with broader shape analyses.

Contribution

It formulates and solves a variational problem to find the shape that maximizes surface gravity on a homogeneous body of fixed mass and volume.

Findings

The optimal shape is given by a specific boundary curve involving a cubic relation.

The maximum surface gravity exceeds that of a sphere by 2.6%.

Additional shape families and their maxima are also analyzed.

Abstract

I pose the question of maximal Newtonian surface gravity on a homogeneous body of a given mass and volume but with variable shape. In other words, given an amount of malleable material of uniform density, how should one shape it in order for a microscopic creature on its surface to experience the largest possible weight? After evaluating the weight on an arbitrary cylinder, at the axis and at the equator and comparing it to that on a spherical ball, I solve the variational problem to obtain the shape which optimizes the surface gravity in some location. The boundary curve of the corresponding solid of revolution is given by $\ (x^2+z^2)^3-(4\,z)^2=0\ $ or $\ r(\theta)=2\sqrt{\cos\theta}$, and the maximal weight (at $x=z=0$) exceeds that on a solid sphere by a factor of $\frac35\root3\of5$, which is an increment of $2.6\%$. Finally, the values and the achievable maxima are computed for three other families of shapes.