where M = ∫Vρ(s)dxdydz is the total mass of the sphere.
The deviations of Earth's gravitational field from that of a homogeneous sphere
In reality, Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. If this shape would have been perfectly known together with the exact mass density ρ = ρ(x, y, z) the integrals (1) and (2) could have been evaluated with numerical methods to find a more accurate model for Earth's gravitational field, but 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 and 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 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:
The first spherical harmonics with n = 0,1,2,3 are presented in the table below.
The model for Earth's gravitational potential is a sum
where and the coordinates (8) are relative 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.
The zonal terms refer to terms of the form:
and the tesseral terms terms refer to terms of the form:
The zonal and tesseral terms for n = 1 are left out in (9).
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.
For large values of n the coefficients above (that are divided by r(n + 1) in (9)) take very large values when for example kilometers and seconds are used as units. 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 dominating term (after the term −μ/r) in (9) is the "J2 term":
Relative the coordinate system
Figure 1: The unit vectors
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"
The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly.
For JGM-3 the values are:
μ = 398600.440 km3⋅s−2
J2 = 1.7555 × 1010 km5⋅s−2
J3 = −2.619 × 1011 km6⋅s−2
With a "reference radius" R of 6378.1363 km corresponding dimensionless parameters are
For example, at a radius of 6600 km (about 200 km above Earth's surface) J3/(J2r) is about 0.002, i.e the correction to the "J2 force" from the "J3 term" is in the order of 2 permille. 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".
Recursive algorithms used for the numerical propagation of spacecraft orbits
Spacecraft orbits are computed by the numerical integration of the equation of motion. For this the gravitational force, i.e. the gradient of the potential, must be computed. Efficient recursive algorithms have been designed to compute the gravitational force for any and 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 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 Goddard Space Flight Center in cooperation with universities and private companies became available. The newer models generally provided higher order terms than their precursors. The EGM96 uses Nz = Nt = 360 resulting in 130317 coefficients. An EGM2008 model is available as well.
For a normal Earth satellite for which an orbit determination/prediction accuracy of a few meters the "JGM-3" truncated to Nz = Nt = 36 (1365 coefficients) is usually sufficient. Inaccuracies from the modeling of the air-drag and to a lesser extent the solar radiation pressure will exceed the inaccuracies caused by the gravitation modeling errors.
The following is a compact account of the spherical harmonics used to model Earth's gravitational field. 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 and the sum
only depends on the variables θ and φ. One gets that φ is harmonic if and only if
El'Yasberg "Theory of flight of artificial earth satellites", Israel program for Scientific Translations (1967)
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Lerch, F.J., Wagner, C.A., Putney, M.L., Sandson, M.L., Brownd, J.E., Richardson, J.A., Taylor, W.A., "Gravitational Field Models GEM3 and 4" ,Report X59272476, Goddard Space Flight Center, Greenbelt/Maryland, 1972
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Lerch, F.J., Nerem, R.S., Putney, B.H., Felsentreger, T.L., Sanchez, B.V., Klosko, S.M., Patel, G.B., Williamson, R.G., Chinn, D.S., Chan, J.C., Rachlin, K.E., Chandler, N.L., McCarthy, J.J., Marshall, J.A., Luthcke, S.B., Pavlis, D.W., Robbins, J.W., Kapoor, S., Pavlis, E.C., " Geopotential Models of the Earth from Satellite Tracking, Altimeter and Surface Gravity Observations: GEMT3 and GEMT3S", NASA Technical Memorandum 104555, Goddard Space Flight Center , Greenbelt/Maryland, 1992