113 lines
4.5 KiB
Markdown
113 lines
4.5 KiB
Markdown
---
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title: Higher order gravity
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tags: gravity, geodesy, harmonics, potential, tesseral
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summary: 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.
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---
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# Higher order gravity
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Treating the Earth as a point mass is the first approximation of orbital
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mechanics and the source of its largest single error. The second is to model
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the Earth as an axisymmetric ellipsoid, which captures the oblateness and
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nothing else. The third is to expand the full potential, which is what a
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high-accuracy model does and where its difficulties begin.
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## Definition
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The **geopotential** of a body is the potential energy per unit mass due to its
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gravitational field. Near the Earth it is conventionally written
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$$
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U(r, \theta, \lambda) = \frac{\mu}{r} \left[ 1
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+ \sum_{n} \left( \frac{R}{r} \right)^n \sum_{m} P_{nm}(\sin\theta)
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\left( C_{nm} \cos m\lambda + S_{nm} \sin m\lambda \right) \right]
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$$
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- $r$ is the distance from the centre, $\theta$ the geocentric colatitude and
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$\lambda$ the longitude.
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- $P_{nm}$ are the Legendre functions, real polynomials on $\sin\theta$.
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- $C_{nm}$ and $S_{nm}$ are the **geopotential coefficients**; the normalised ones
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are what a model publishes.
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- $R$ is a reference radius.
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For $n = 0$ there is one coefficient, the gravitational parameter. For each
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higher degree there are `2n + 1` zonal, tesseral and sectorial terms.
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$J_2$ is the second zonal coefficient, about `1.0826e-3`, and it dominates
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everything above it. The next largest are $J_3$, $J_4$, each smaller than
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the last.
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## Mathematics
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### What each term does
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- **Zonal** coefficients ($m = 0$) are axisymmetric and cause secular drift
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in the node and the argument of perigee, and no change in inclination.
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This is the classical J2 problem.
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- **Tesseral** coefficients ($0 < m < n$) are longitude dependent and cause
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short-period variation in every element including the inclination. This is
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why real orbits wobble rather than precessing smoothly.
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- **Sectorial** coefficients ($m = n$) couple the radius and the longitude
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directly and are the smallest contribution.
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### The truncation problem
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The series is a power series in $(R/r)$, which converges only for $r > R$,
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that is, outside the Earth. For a satellite at $r$ above the surface the
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ratio is $R/r < 1$ and the series converges, but **slowly**, because $R/r$ is
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close to one at low altitude. The number of terms needed for a given accuracy
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grows as the altitude falls, which is why a low Earth orbit model truncates
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hard and a geostationary one can carry many terms.
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This is the opposite of intuition: the model is *worse* where the field is
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strongest, not where it is weakest.
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### Normalisation
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The coefficients published by a model are normalised so that the unnormalised
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form does not grow with degree:
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$$
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\begin{bmatrix} C_{nm} \\ S_{nm} \end{bmatrix}
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= \frac{1}{\sqrt{(1 - \delta)\,(n - m)!\,(n + m)!}}
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\begin{bmatrix} \bar{C}_{nm} \\ \bar{S}_{nm} \end{bmatrix}
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$$
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A model is defined by its degree $N$ and by the coefficients below it. Degree
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180 with the first few hundred coefficients is typical.
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## Unit conventions
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- Potential and coefficients dimensionless; $\mu$ is the gravitational
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parameter.
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- $R$ is the equatorial radius, about 6378.137 km.
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- Angle in radians; the Legendre functions are polynomials on the sine of the
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colatitude.
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## Applicability
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A truncated gravity field is a compromise. Cutting at a low degree saves
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computation and costs accuracy, and the cost is largest at low altitude where
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$(R/r)$ approaches one. The numerical consequences are worse than the
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positional ones: a truncated field is smooth, but its derivatives, which
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drive the short-period terms, are not.
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A model fitted to a force function rather than to a potential is a
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alternative used when the field is known from measurements along a specific
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orbit; it does not generalise to other altitudes or inclinations.
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The distinction between a normalised and unnormalised coefficient is a
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frequent source of silent error, because the two differ by a factor that
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depends on the degree.
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## Standards
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- EGM2008 and its successors: the standard Earth gravity models, published by
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the National Geospatial-Intelligence Agency
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- The degree and normalisation convention: *IERS Conventions*, section 5
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## See also
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- [[perturbations]] for where the J2 term sits among the other perturbations
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- [[geodesy-wgs84]] for the ellipsoid that the second-order model assumes
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- [[numerical-methods]] for the series as a method with a truncation error |