Geopotential spherical harmonic model
The Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate.
However, a spherical harmonics series expansion captures the actual field with increasing fidelity.
If Earth's shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), it could be integrated numerically (when combined with a reciprocal distance kernel) to find an accurate model for Earth's gravitational field.
However, the situation is in fact the opposite: by observing the orbits of spacecraft and the Moon, Earth's gravitational field can be determined quite accurately.
The best estimate of Earth's mass is obtained by dividing the product GM as determined from the analysis of spacecraft orbit with a value for the gravitational constant G, determined to a lower relative accuracy using other physical methods.
From the defining equations (1) and (2) it is clear (taking the partial derivatives of the integrand) that outside the body in empty space the following differential equations are valid for the field caused by the body: Functions of the form
[Note that the sign convention differs from the one in the page about the associated Legendre polynomials, here
and the coordinates (8) are relative to the standard geodetic reference system extended into space with origin in the center of the reference ellipsoid and with z-axis in the direction of the polar axis.
Gravity does not physically exhibit any dipole character and so the integral characterizing n = 1 must be zero.
The different coefficients Jn, Cnm, Snm, are then given the values for which the best possible agreement between the computed and the observed spacecraft orbits is obtained.
As P0n(x) = −P0n(−x) non-zero coefficients Jn for odd n correspond to a lack of symmetry "north–south" relative the equatorial plane for the mass distribution of Earth.
Non-zero coefficients Cnm, Snm correspond to a lack of rotational symmetry around the polar axis for the mass distribution of Earth, i.e. to a "tri-axiality" of Earth.
In the literature it is common to introduce some arbitrary "reference radius" R close to Earth's radius and to work with the dimensionless coefficients and to write the potential as The spherical harmonics are derived from the approach of looking for harmonic functions of the form where (r, θ, φ) are the spherical coordinates defined by the equations (8).
By straightforward calculations one gets that for any function f Introducing the expression (16) in (17) one gets that As the term only depends on the variable
with one must have that If Pn(x) is a solution to the differential equation one therefore has that the potential corresponding to m = 0 which is rotationally symmetric around the z-axis is a harmonic function If
is selected to make Pn(−1) = −1 and Pn(1) = 1 for odd n and Pn(−1) = Pn(1) = 1 for even n. The first six Legendre polynomials are: The solutions to differential equation (26) are the associated Legendre functions One therefore has that The dominating term (after the term −μ/r) in (9) is the J2 coefficient, the second dynamic form factor representing the oblateness of Earth: Relative the coordinate system illustrated in figure 1 the components of the force caused by the "J2 term" are In the rectangular coordinate system (x, y, z) with unit vectors (x̂ ŷ ẑ) the force components are: The components of the force corresponding to the "J3 term" are and The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly.
The negative value of J3 implies that for a point mass in Earth's equatorial plane the gravitational force is tilted slightly towards the south due to the lack of symmetry for the mass distribution of Earth's "north–south".
Spacecraft orbits are computed by the numerical integration of the equation of motion.
Efficient recursive algorithms have been designed to compute the gravitational force for any
(the max degree of zonal and tesseral terms) and such algorithms are used in standard orbit propagation software.
The earliest Earth models in general use by NASA and ESRO/ESA were the "Goddard Earth Models" developed by Goddard Space Flight Center (GSFC) denoted "GEM-1", "GEM-2", "GEM-3", and so on.
Later the "Joint Earth Gravity Models" denoted "JGM-1", "JGM-2", "JGM-3" developed by GSFC in cooperation with universities and private companies became available.
The newer models generally provided higher order terms than their precursors.
For a normal Earth satellite requiring an orbit determination/prediction accuracy of a few meters the "JGM-3" truncated to Nz = Nt = 36 (1365 coefficients) is usually sufficient.
