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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 (J_{2}).
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 straight-forward:
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 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 J_{2} measurements.

http://en.wikipedia.org/wiki/Maclaurin_spheroid

https://ui.adsabs.harvard.edu/abs/1978SvA....22..733E/abstract

https://ui.adsabs.harvard.edu/abs/2014Icar..242..138H/abstract

https://ui.adsabs.harvard.edu/abs/2013ApJ...768...43H/abstract