theory of figures

The phrase theory of figures (TOF or ToF) indicates a model that calculates the shape of astronomical bodies (planets, moons, stars) specifically aiming to include both the effects of gravity and rotation, in particular, the expected oblation (and J₂). The word figure refers to the shape of a planet (figure of a planet, an early example of interest being the figure of the Earth). The phrase theory of figures is sometimes used generally for such models, but often the author is referring to a particular existing method or set of equations: a method developed centuries ago and refined over the years, which might be termed the classic TOF. Such theory may relate known physics and known constituents of a planet to its figure, or may be oriented toward determining the figure from observables, such as measured gravity at different places on the surface, and ideally a theory would tie the latter to the former.

The figure of a non-rotating body is comparatively straightforward: spherical in the simpler cases, and if the rotation is small enough, this is likely taken as a practical simplification. The effects of rotation make the mathematics far more complicated, and methods of calculation are still a topic of research. One set of simplifications is to assume it consists solely of a non-compressible fluid with virtually no viscosity, which leads to a Maclaurin spheroid, with a known equation. Dealing with multiple constituents and their equations of state complicates the picture, as does the need to use observation data, such as in the case of solar system interplanetary exploration. Modeling a planet as nested Maclaurin spheroids is one approach.

The classic TOF offers a series expansion allowing an approximation using the first few terms, i.e., first-order TOF, second-order TOF, etc. and the higher the order, the more spherical harmonics of the gravity are accurately reflected. The simplification such an approximation offers allows the equations to be solved for the figure, based upon spherical harmonic measurements, and the higher the order, the more spherical harmonics measured contribute to the accuracy of the figure determination. The third-order TOF approximation is reasonable to relate to J₂ measurements.