Английская Википедия:Geopotential model
Шаблон:Short description Шаблон:No footnotes In geophysics and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field (the geopotential).
Newton's law
Newton's law of universal gravitation states that the gravitational force F acting between two point masses m1 and m2 with centre of mass separation r is given by
- <math>\mathbf{F} = - G \frac{m_1 m_2}{r^2}\mathbf{\hat{r}}</math>
where G is the gravitational constant and r̂ is the radial unit vector. For a non-pointlike object of continuous mass distribution, each mass element dm can be treated as mass distributed over a small volume, so the volume integral over the extent of object 2 gives:
Шаблон:NumBlk = - Gm_1 \int\limits_V \frac{\rho_2 }{r^2}\mathbf{\hat{r}}\,dx\,dy\,dz </math>|Шаблон:EquationRef}}
with corresponding gravitational potential
where ρШаблон:Sub = ρ(x, y, z) is the mass density at the volume element and of the direction from the volume element to point mass 1. <math>u</math> is the gravitational potential energy per unit mass.
The case of a homogeneous sphere
In the special case of a sphere with a spherically symmetric mass density then ρ = ρ(s); i.e., density depends only on the radial distance
- <math>s = \sqrt{x^2 + y^2 + z^2} \,.</math>
These integrals can be evaluated analytically. This is the shell theorem saying that in this case:
with corresponding potential
where M = ∫Vρ(s)dxdydz is the total mass of the sphere.
Spherical harmonics representation
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 were perfectly known together with the exact mass density ρ = ρ(x, y, z), the integrals (Шаблон:EquationNote) and (Шаблон:EquationNote) could be evaluated with numerical methods to find a more 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 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.
Background
Шаблон:Main From the defining equations (Шаблон:EquationNote) and (Шаблон:EquationNote) 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: Шаблон:NumBlk Шаблон:NumBlk
Functions of the form <math>\phi = R(r)\, \Theta(\theta)\, \Phi(\varphi)</math> where (r, θ, φ) are the spherical coordinates which satisfy the partial differential equation (Шаблон:EquationNote) (the Laplace equation) are called spherical harmonic functions.
They take the forms: Шаблон:NumBlk P^m_n(\sin \theta) \cos m\varphi \,,&
0 &\le m \le n \,,&
n &= 0, 1, 2, \dots \\
h(x, y, z) &= \frac{1}{r^{n+1}} P^m_n(\sin \theta) \sin m\varphi \,,&
1 &\le m \le n \,,&
n &= 1, 2, \dots
\end{align}</math>|Шаблон:EquationRef}}
where spherical coordinates (r, θ, φ) are used, given here in terms of cartesian (x, y, z) for reference: Шаблон:NumBlk
also P0n are the Legendre polynomials and Pmn for Шаблон:Nowrap are the associated Legendre functions.
The first spherical harmonics with n = 0, 1, 2, 3 are presented in the table below.
n Spherical harmonics 0 <math>\frac{1}{r}</math> 1 <math>\frac{1}{r^2} P^0_1(\sin\theta) = \frac{1}{r^2} \sin\theta</math> <math>\frac{1}{r^2} P^1_1(\sin\theta) \cos\varphi= \frac{1}{r^2} \cos\theta \cos\varphi</math> <math>\frac{1}{r^2} P^1_1(\sin\theta) \sin\varphi= \frac{1}{r^2} \cos\theta \sin\varphi</math> 2 <math>\frac{1}{r^3} P^0_2(\sin\theta) = \frac{1}{r^3} \frac{1}{2} (3\sin^2\theta - 1)</math> <math>\frac{1}{r^3} P^1_2(\sin\theta) \cos\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta\ \cos\varphi</math> <math>\frac{1}{r^3} P^1_2(\sin\theta) \sin\varphi = \frac{1}{r^3} 3 \sin\theta \cos\theta \sin\varphi</math> <math>\frac{1}{r^3} P^2_2(\sin\theta) \cos2\varphi = \frac{1}{r^3} 3 \cos^2 \theta\ \cos2\varphi</math> <math>\frac{1}{r^3} P^2_2(\sin\theta) \sin2\varphi = \frac{1}{r^3} 3 \cos^2 \theta \sin 2\varphi</math> 3 <math>\frac{1}{r^4} P^0_3(\sin\theta) = \frac{1}{r^4} \frac{1}{2} \sin\theta\ (5\sin^2\theta -3)</math> <math>\frac{1}{r^4} P^1_3(\sin\theta) \cos\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \cos\varphi</math> <math>\frac{1}{r^4} P^1_3(\sin\theta) \sin\varphi = \frac{1}{r^4} \frac{3}{2}\ (5\ \sin^2\theta - 1) \cos\theta \sin\varphi</math> <math>\frac{1}{r^4} P^2_3(\sin\theta) \cos 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \cos 2\varphi</math> <math>\frac{1}{r^4} P^2_3(\sin\theta) \sin 2\varphi = \frac{1}{r^4} 15 \sin\theta \cos^2 \theta \sin 2\varphi</math> <math>\frac{1}{r^4} P^3_3(\sin\theta) \cos 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \cos 3\varphi</math> <math>\frac{1}{r^4} P^3_3(\sin\theta) \sin 3\varphi = \frac{1}{r^4} 15 \cos^3 \theta \sin 3\varphi</math>
Application
The model for Earth's gravitational potential is a sum
Шаблон:NumBlk + \sum_{n=2}^{N_t} \sum_{m=1}^n \frac{ P^m_n(\sin\theta) \left(C_n^m \cos m\varphi + S_n^m \sin m\varphi\right)}{r^{n+1}} </math>|Шаблон:EquationRef}}
where <math>\mu = GM</math> and the coordinates (Шаблон:EquationNote) 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.
The zonal terms refer to terms of the form:
- <math>\frac{P^0_n(\sin\theta)}{r^{n+1}} \quad n=0,1,2,\dots</math>
and the tesseral terms terms refer to terms of the form:
- <math>\frac{P^m_n(\sin\theta) \cos m\varphi}{r^{n+1}}\,, \quad 1 \le m \le n \quad n=1,2,\dots</math>
- <math>\frac{P^m_n(\sin\theta) \sin m\varphi}{r^{n+1}}</math>
The zonal and tesseral terms for n = 1 are left out in (Шаблон:EquationNote). The coefficients for the n=1 with both m=0 and m=1 term correspond to an arbitrarily oriented dipole term in the multi-pole expansion. 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.
For large values of n the coefficients above (that are divided by r(n + 1) in (Шаблон:EquationNote)) 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
- <math>\begin{align}
\tilde{J_n} &= -\frac{J_n}{\mu\ R^n}, &
\tilde{C_{n}^m} &= -\frac{C_{n}^m}{\mu\ R^n}, &
\tilde{S_{n}^m} &= -\frac{S_{n}^m}{\mu\ R^n}
\end{align}</math>
and to write the potential as
Шаблон:NumBlk\ P_n(\sin\theta)</math>
which is rotationally symmetric around the z-axis is a harmonic function
If <math>P_{n}^{m}(x)</math> is a solution to the differential equation Шаблон:NumBlk{dx}\right)\ +\ \left(n(n + 1) - \frac{m^2}{1 - x^2} \right)\ P_{n}^{m}\ =\ 0 </math>|Шаблон:EquationRef}}
with m ≥ 1 one has the potential
Шаблон:NumBlk\ P_{n}^{m}(\sin\theta)\ (a\ \cos m\varphi\ +\ b\ \sin m\varphi) </math>|Шаблон:EquationRef}}
where a and b are arbitrary constants is a harmonic function that depends on φ and therefore is not rotationally symmetric around the z-axis
The differential equation (Шаблон:EquationNote) is the Legendre differential equation for which the Legendre polynomials defined Шаблон:NumBlk
are the solutions.
The arbitrary factor 1/(2nn!) is selected to make Шаблон:Nowrap and Шаблон:Nowrap for odd n and Шаблон:Nowrap for even n.
The first six Legendre polynomials are:
The solutions to differential equation (Шаблон:EquationNote) are the associated Legendre functions Шаблон:NumBlk
One therefore has that
- <math>
P_n^m (\sin\theta) = \cos^m \theta\ \frac{d^n P_n}{dx^n} (\sin\theta)
</math> Шаблон:Collapse bottom
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 <math>N_z</math> and <math>N_t</math> (the max degree of zonal and tesseral terms) and such algorithms are used in standard orbit propagation software.
Available models
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 requiring 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 dimensionless coefficients <math>\tilde{J_n} = -\frac{J_n}{\mu\ R^n}</math>, <math>\tilde{C_{n}^m} = -\frac{C_{n}^m}{\mu\ R^n}</math>, <math>\tilde{S_{n}^m} = -\frac{S_{n}^m}{\mu\ R^n}</math> for the first zonal and tesseral terms (using <math>R</math> = Шаблон:Val and <math>\mu</math> = Шаблон:Val) of the JGM-3 model are
| n | |
|---|---|
| 2 | Шаблон:Val |
| 3 | Шаблон:Val |
| 4 | Шаблон:Val |
| 5 | Шаблон:Val |
| 6 | Шаблон:Val |
| 7 | Шаблон:Val |
| 8 | Шаблон:Val |
| n | m | C | S |
|---|---|---|---|
| 2 | 1 | Шаблон:Val | Шаблон:Val |
| 2 | Шаблон:Val | Шаблон:Val | |
| 3 | 1 | Шаблон:Val | Шаблон:Val |
| 2 | Шаблон:Val | Шаблон:Val | |
| 3 | Шаблон:Val | Шаблон:Val | |
| 4 | 1 | Шаблон:Val | Шаблон:Val |
| 2 | Шаблон:Val | Шаблон:Val | |
| 3 | Шаблон:Val | Шаблон:Val | |
| 4 | Шаблон:Val | Шаблон:Val |
According to JGM-3 one therefore has that JШаблон:Sub = Шаблон:Val × 6378.1363Шаблон:Sup × Шаблон:Val = Шаблон:Val and JШаблон:Sup = Шаблон:Val × 6378.1363Шаблон:Sup × Шаблон:Val = Шаблон:Val.
Further reading
- El'Yasberg Theory of flight of artificial earth satellites, Israel program for Scientific Translations (1967)
- Lerch, F.J., Wagner, C.A., Smith, D.E., Sandson, M.L., Brownd, J.E., Richardson, J.A.,"Gravitational Field Models for the Earth (GEM1&2)", Report X55372146, Goddard Space Flight Center, Greenbelt/Maryland, 1972
- 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
- Lerch, F.J., Wagner, C.A., Richardson, J.A., Brownd, J.E., "Goddard Earth Models (5 and 6)", Report X92174145, Goddard Space Flight Center, Greenbelt/Maryland, 1974
- Lerch, F.J., Wagner, C.A., Klosko, S.M., Belott, R.P., Laubscher, R.E., Raylor, W.A., "Gravity Model Improvement Using Geos3 Altimetry (GEM10A and 10B)", 1978 Spring Annual Meeting of the American Geophysical Union, Miami, 1978
- Lerch, F.J., Klosko, S.M., Laubscher, R.E., Wagner, C.A., "Gravity Model Improvement Using Geos3 (GEM9 and 10)", Journal of Geophysical Research, Vol. 84, B8, p. 3897-3916, 1979
- Lerch, F.J., Putney, B.H., Wagner, C.A., Klosko, S.M. ,"Goddard earth models for oceanographic applications (GEM 10B and 10C)", Marine-Geodesy, 5(2), p. 145-187, 1981
- Lerch, F.J., Klosko, S.M., Patel, G.B., "A Refined Gravity Model from Lageos (GEML2)", 'NASA Technical Memorandum 84986, Goddard Space Flight Center, Greenbelt/Maryland, 1983
- 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
- Lerch, F.J., Nerem, R.S., Putney, B.H., Felsentreger, T.L., Sanchez, B.V., Marshall, J.A., Klosko, S.M., Patel, G.B., Williamson, R.G., Chinn, D.S., Chan, J.C., Rachlin, K.E., Chandler, N.L., McCarthy, J.J., Luthcke, S.B., Pavlis, N.K., Pavlis, D.E., Robbins, J.W., Kapoor, S., Pavlis, E.C., "A Geopotential Model from Satellite Tracking, Altimeter and Surface Gravity Data: GEMT3", Journal of Geophysical Research, Vol. 99, No. B2, p. 2815-2839, 1994
- Nerem, R.S., Lerch, F.J., Marshall, J.A., Pavlis, E.C., Putney, B.H., Tapley, B.D., Eanses, R.J., Ries, J.C., Schutz, B.E., Shum, C.K., Watkins, M.M., Klosko, S.M., Chan, J.C., Luthcke, S.B., Patel, G.B., Pavlis, N.K., Williamson, R.G., Rapp, R.H., Biancale, R., Nouel, F., "Gravity Model Developments for Topex/Poseidon: Joint Gravity Models 1 and 2", Journal of Geophysical Research, Vol. 99, No. C12, p. 24421-24447, 1994a
- Tapley, B. D. M. M. Watkins, J. C. Ries, G. W. Davis, R. J. Eanes, S. R. Poole, H. J. Rim, B. E. Schutz, C. K. Shum, R. S. Nerem, F. J. Lerch, J. A. Marshall, S. M. Klosko, N. K. Pavlis, and R. G. Williamson, “The Joint Gravity Model 3,” J. Geophys. Res., vol. 101, No. B12, ,December 1996
External links
- http://cddis.nasa.gov/lw13/docs/papers/sci_lemoine_1m.pdf
- http://geodesy.geology.ohio-state.edu/course/refpapers/Tapley_JGR_JGM3_96.pdf
