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title	tags	summary
Higher order gravity	gravity, geodesy, harmonics, potential, tesseral	What the oblateness term leaves out, how the Earth's gravity field is expanded, and why the expansion does not converge where you would like it to.

Higher order gravity

Treating the Earth as a point mass is the first approximation of orbital mechanics and the source of its largest single error. The second is to model the Earth as an axisymmetric ellipsoid, which captures the oblateness and nothing else. The third is to expand the full potential, which is what a high-accuracy model does and where its difficulties begin.

Definition

The geopotential of a body is the potential energy per unit mass due to its gravitational field. Near the Earth it is conventionally written


U(r, \theta, \lambda) = \frac{\mu}{r} \left[ 1
  + \sum_{n} \left( \frac{R}{r} \right)^n \sum_{m} P_{nm}(\sin\theta)
      \left( C_{nm} \cos m\lambda + S_{nm} \sin m\lambda \right) \right]

r is the distance from the centre, theta the geocentric colatitude and lambda the longitude.
P_nm are the Legendre functions, real polynomials on sin(theta).
C_nm and S_nm are the geopotential coefficients; the normalised ones are what a model publishes.
R is a reference radius.

For n = 0 there is one coefficient, the gravitational parameter. For each higher degree there are 2n + 1 zonal, tesseral and sectorial terms.

J_2 is the second zonal coefficient, about 1.0826e-3, and it dominates everything above it. The next largest are J_3, J_4, each smaller than the last.

Mathematics

What each term does

Zonal coefficients (m = 0) are axisymmetric and cause secular drift in the node and the argument of perigee, and no change in inclination. This is the classical J2 problem.
Tesseral coefficients (0 < m < n) are longitude dependent and cause short-period variation in every element including the inclination. This is why real orbits wobble rather than precessing smoothly.
Sectorial coefficients (m = n) couple the radius and the longitude directly and are the smallest contribution.

The truncation problem

The series is a power series in (R/r), which converges only for r > R, that is, outside the Earth. For a satellite at r above the surface the ratio is R/r < 1 and the series converges, but slowly, because R/r is close to one at low altitude. The number of terms needed for a given accuracy grows as the altitude falls, which is why a low Earth orbit model truncates hard and a geostationary one can carry many terms.

This is the opposite of intuition: the model is worse where the field is strongest, not where it is weakest.

Normalisation

The coefficients published by a model are normalised so that the unnormalised form does not grow with degree:


\begin{bmatrix} C_{nm} \\ S_{nm} \end{bmatrix}
  = \frac{1}{\sqrt{(1 - \delta)\,(n - m)!\,(n + m)!}}
  \begin{bmatrix} \bar{C}_{nm} \\ \bar{S}_{nm} \end{bmatrix}

A model is defined by its degree N and by the coefficients below it. Degree 180 with the first few hundred coefficients is typical.

Unit conventions

Potential and coefficients dimensionless; mu is the gravitational parameter.
R is the equatorial radius, about 6378.137 km.
Angle in radians; the Legendre functions are polynomials on the sine of the colatitude.

Applicability

A truncated gravity field is a compromise. Cutting at a low degree saves computation and costs accuracy, and the cost is largest at low altitude where (R/r) approaches one. The numerical consequences are worse than the positional ones: a truncated field is smooth, but its derivatives, which drive the short-period terms, are not.

A model fitted to a force function rather than to a potential is a alternative used when the field is known from measurements along a specific orbit; it does not generalise to other altitudes or inclinations.

The distinction between a normalised and unnormalised coefficient is a frequent source of silent error, because the two differ by a factor that depends on the degree.

Standards

EGM2008 and its successors: the standard Earth gravity models, published by the National Geospatial-Intelligence Agency
The degree and normalisation convention: IERS Conventions, section 5

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