gravitational potential model

I use the term gravitational potential model for a common type of model of the gravitational potential of an astronomical body (e.g., a planet or moon), essentially defining a map showing the acceleration that the body's gravity would produce in different directions and altitudes, that describes this map using spherical harmonics. The term geopotential model is common specifically for Earth, and the terms gravitational model and gravity model are also used (but be warned gravity model is more often seen regarding economics models with a similarity to the effects of gravitational potential). Such gravitational potential models use spherical harmonic coefficients to describe the gravitational pull in each direction in terms of overall patterns in the variation in different directions, in a similar manner to that in which a Fourier transform describes a periodic function (or equivalently, a function over an interval). The function representing the body's gravitational pull is the delta between a selected equipotential surface (altitudes around the body at which the pull of gravity is constant, in the Earth's case, using sea level) and a hypothetical spherical surface which has that equipotential surface's mean. The coefficients are generally derived from many measurements, e.g., with gravimeters, or by tracking objects orbiting the body, such as space probes, or are modeled and calculated based on analysis of factors such as rotation period and constituents.

The coefficients used generally are based on harmonics aligned with the body's axis of rotation. The coefficients most likely to be significant and of interest for objects massive enough to be nearly spherical are zonal coefficients (C_{n 0}, for n=1,2,3,..., often written C₀₀, C₁₀, etc., for single-digit integers), corresponding to patterns that are the same along any meridian, the other coefficient of most interest being one particular sectorial coefficient, C_2 2 (often written C₂₂). C_0 0 is merely the average gravity, and C_{1 0} is generally minor. C_{2 0} indicates the oblateness (greater or lesser gravity around the equator), and C_2 2 is similar, but "sideways", indicating some greater or lesser amount at two opposite points on the equator, such as from extra mass toward the end points of a line between them. The zonal coefficients are also commonly cited as J_n (the J_n value is defined as -√5 times the C_{n 0} value), J₂ being the most significant.