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| Coordinate systems | ||
The need for the use of more than one coordinate system arises from the fact that many different physical phenomena are easier calculated or understood in a system that is appropriate for the phenomenon. Frequently, it is necessary to transform from one coordinate system to another. The transformations needed are described in Smart (1944), Mead (1970), Goldstein (1950), Olson (1970) and by the Magnetic and Electric Fields Branch, GSFC(1970). The definitions used for the various coordinate systems described here come from Russell (1971). The transformations were taken from Hapgood (1991).
The coordinate systems and transformations described on this page are all geocentric coordinates. This means that the centre of the Earth is taken as origin, and the transformations do not include any translations. Additional information on these and other coordinate systems (such as heliocentric and boundary normal systems) is available at the Space Plasma Group website at Ral, maintained by Mike Hapgood.
For the definition of a coordinate system in threedimensional space, one has only to specify the direction of one of the axes, and the orientation of one of the other axes in the plane perpendicular to this direction. The third axis follows automatically in order to complete a right-handed orthogonal set.
Transformations may conveniently be performed using matrix arithmetic. For each transformation, there is a transformation matrix T such that Qb = TQa, where Qa is a vector in the first coordinate system, and Qb is the same vector in the second coordinate system. The transformation is thus totally described by the nine components of the matrix T.
One fortunate feature of transformation matrices is that the inverse is equal to the transpose of the matrix, i.e., if :
In some coordinate systems, especially GEO and MAG, position is often specified in terms of colatitude theta, longitude phi and radial distance r. These are related to Cartesian components using :
The Geocentric Equatorial Inertial Systam (GEI) has its X-axis pointing from the Earth towards the first point of Aries (i.e. the position of the sun at the vernal equinox). This direction is the intersection of the Earth's equatorial plane and the ecliptic plane. The Z-axis is parallel to the rotation axis of the Earth and Y completes the right-handed orthogonal set (Y = Z x X).
The normal GEI coordinate system changes slowly in time owing to the effects of astronomical precession and the nutation of the Earth's rotation axis. The transformations described here are strictly correct if the epoch-of-date inertial system is used. Hapgood (1995) describes how the transformations should be adjusted to take account of this time depence of the GEI system.
Calculation of transformation matrices to and from other coordinate systems.
The Geographic Coordinate system (GEO) is defined so that its X-axis is in the Earth's equatorial plane but is fixed with the rotation of the Earth so that it passes through the Greenwich meridian (0° longitude). Its Z-axis is parallel to the rotation axis of the Earth, and its Y-axis completes a right-handed orthogonal set (Y = Z x X).
Calculation of transformation matrices to and from other coordinate systems.
The geodetic coordinate system defines a position in terms of latitude, longitude and altitude above the ellipsoidal surface of the Earth (see Figure 1).
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| Figure 1. Cross section of ellipsoid (taken from Kelso) |
The ellipsoidal surface is a surface of resolution obtained by rotating an ellips around the minor axis. Thus, the geodetic longitude is the same as the geographic longitude, and only a meridional section must be considered.
The local horizon is defined as the plane that is tangent to the Earth's surface at a given position. The surface considered is the reference ellipsoid. The local zenith is the direction away from the point on the Earth's surface perpendicular to the local horizon. On a sphere, this direction is always directly away from the Earth's centre, but on an ellipsoid, this is not the case (except on the equator and at the poles).
The geodetic latitude, phi is the angle between the local zenith and the equatorial plane. Except at the poles and the equator, phi differs from the geocentric latitude phi'.
The point on the Earth surface directly below a given point above the surface is not on a line joining the given point and the centre of the Earth. It is the point where the local zenith points to the given point (see Figure 2). The geodetic altitude h is the distance from the point to the surface along the local zenith direction.
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| Figure 2. Sub-point and altitude (taken from Kelso) |
The reference ellipsoid is defined by two parameters, a, the
semi-major axis, and f, the flattening, defined as:
Global ellipsoidal parameters are derived from satellite data. Historically, local, regional and global best fitting ellipsoids have been considered. Table 1 lists some of these reference ellipsoids.
| Name | semi-major axis [m] |
1/flattening | Application |
|---|---|---|---|
| WGS 84 | 6378137 | 298.257 | DoD (GPS) |
| GRS 80 | 6378137 | 298.257 | IAG (Geo Ref Sys) |
| WGS 72 | 6378135 | 298.26 | DoD (Doppler) |
| GRS 67 | 6378160 | 298.25 | Australia 1966, South America 1969 |
| IAU (1964) | 6378160 | 298.25 | |
| Krassovsky (1940) | 6378245 | 298.3 | Russia |
| International (1924) | 6378388 | 297 | Europe (ReTrig) |
| Clarke (1880) | 6378249 | 293 | France, Africa |
| Clarke (1866) | 6378206 | 294.98 | North America |
| Bessel (1841) | 6377397 | 299.15 | German DHDN |
| Airy (1830) | 6376542 | 299 | Great Britain |
| Everest (1830) | 6377276 | 300 | India |
The implementations in SPENVIS and UNILIB use the IAU (1964) reference ellipsoid.
The conversion from ellipsoidal coordinates to cartesian coordinates is given by:
| X | = | (N + h) cos(phi) cos(lambda) |
| Y | = | (N + h) cos(phi) sin(lambda) |
| Z | = | [N(1 - e2) + h] sin(phi) |
with:
The inverse conversion can be iteratively computed from:
| h | = | (X2 + Y2)1/2 / cos(phi) - N |
| tan(phi) | = | Z (X2 + Y2)-1/2 [1 - e2 N / (N + h)]-1 |
| tan(lambda) | = | Y / X |
The Geomagnetic Coordinate system (MAG) is defined so that its Z-axis is parallel to the magnetic dipole axis. The Y-axis of this system is perpendicular to the geographic poles such that if D is the dipole position and S is the south pole Y = D x S. Finally, the X-axis completes a right-handed orthogonal set.
The Geographic coordinates of the dipole axis derived from the International Geomagnetic Reference Field 1995 (IGRF-1995) are 79.30°N and 288.59°E for 1995. The values for the other IGRF epochs are listed in Table 2. It should be noted that the magnetic pole is moving with a speed of 2.6 km per year in the direction 15.6°N, 150.9°E. More information on the magnetic dipole and its variations can be found in Fraser-Smith (1987).
| Year | Latitude | Longitude |
|---|---|---|
| 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 |
78.47 78.47 78.46 78.51 78.53 78.59 78.69 78.81 78.97 79.13 79.30 |
291.47 291.15 290.84 290.53 290.15 289.82 289.53 289.24 289.10 288.89 288.59 |
Calculation of transformation matrices to and from other coordinate systems.
The Geocentric Solar Ecliptic System (GSE) has its X-axis pointing from the Earth towards the sun and its Y-axis is chosen to be in the ecliptic plane pointing towards dusk (thus opposing planetary motion). Its Z-axis is parallel to the ecliptic pole. Relative to an inertial system this system has a yearly rotation.
Calculation of transformation matrices to and from other coordinate systems.
The Geocentric Solar Equatorial System (GSEQ) as with the GSE system has its X-axis pointing towards the Sun from the Earth. However, instead of having its Y-axis in the ecliptic plane, the GSEQ Y-axis is parallel to the Sun's equatorial plane which is inclined to the ecliptic. We note that since the X-axis is in the ecliptic plane and therefore is not necessarily in the Sun's equatorial plane, the Z-axis of this system will not necessarily be parallel to the Sun's axis of rotation. However, the Sun's axis of rotation must lie in the X-Z plane. The Z-axis is chosen to be in the same sense as the ecliptic pole, i.e. northwards.
The Geocentric Solar Magnetospheric System (GSM), as with both the GSE and GSEQ systems, has its X-axis from the Earth to the Sun. The Y-axis is defined to be perpendicular to the Earth's magnetic dipole so that the X-Z plane contains the dipole axis. The positive Z-axis is chosen to be in the same sense as the northern magnetic pole. The difference between the GSM system and the GSE and GSEQ is simply a rotation about the X-axis.
Calculation of transformation matrices to and from other coordinate systems.
In Solar Magnetic Coordinates (SM) the Z-axis is chosen parallel to the north magnetic pole and the Y-axis perpendicular to the Earth-Sun line towards dusk. The difference between this system and the GSM system is a rotation about the Y-axis. The amount of rotation is simply the dipole tilt angle as defined in the previous section. We note that in this system the X-axis does not point directly at the Sun. As with the GSM system, the SM system rotates with both a yearly and daily period with respect to inertial coordinates.
Calculation of transformation matrices to and from other coordinate systems.
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| Figure 3. Coordinate transformations for Brussels. The red ring is the geocentric equator, the black one the magnetic equator and the magenta ring is the ecliptic plane (Click on the gif to see an animation [213 Kbytes]). |
| To | From | |||||
| GEI | GEO | GSE | GSM | SM | MAG | |
| GEI | 1 | T1-1 | T2-1 | T2-1T3-1 | T2-1T3-1T4-1 | T1-1T5-1 |
| GEO | T1 | 1 | T1T2-1 | T1T2-1T3-1 | T1T2-1T3-1T4-1 | T5-1 |
| GSE | T2 | T2T1-1 | 1 | T3-1 | T3-1T4-1 | T2T1-1T5-1 |
| GSM | T3T2 | T3T2T1-1 | T3 | 1 | T4-1 | T3T2T1-1T5-1 |
| SM | T4T3T2 | T4T3T2T1-1 | T4T3 | T4 | 1 | T4T3T2T1-1T5-1 |
| MAG | T5T1 | T5 | T5T1T2-1 | T5T1T2-1T3-1 | T5T1T2-1T3-1T4-1 | 1 |
This matrix corresponds to a rotation in the plane of the Earth's geographic equator from the First Point of Aries to Greenwich meridian. The rotation angle theta is the Greenwich mean sidereal time. this can be calculated using the following formula (U.S. Naval Observatory, 1989):
These two matrices correspond to :
These two angles are calculated as follows (U.S. Naval Observatory, 1989). First epsilon, the obliquity of the ecliptic:
To obtain Qe we simply apply matrix arithmetic thus:
The angles phi and lambda are defined in section T3.
Fraser-Smith, A. C., Centered and Eccentric Geomagnetic Dipoles and Their Poles, 1600-1985, Rev. Geophys., 25, pp. 1-16, 1987.
Goldstein, H., Classical Mechanics, Addison-Wesley Publ. Co., Inc., Reading Massachusetts, 1950.
Hapgood, M. A., Space physics coordinate transformations: A user guide, Planet. Space Sci., 40 (5), pp. 711-717, 1992.
Hapgood, M. A., Space physics coordinate transformations: the role of precession, Ann. Geophysicae, 13, pp. 713-716, 1995.
Hapgood, M. A., Corrigendum, Planet. Space Sci., 45 (8), pp. 1047, 1997.
Kelso, T. S., Orbital Coordinate Systems, Part III, Satellite Times, January/February 1996.
Magnetic and Electric Fields Branch, Coordinate Transformations Used in OGO Satellite Data Analysis, Goddard Space Flight Center Report, X-645-70-29, 1970.
Mead, G. D., J. Geophys. Res., 72 (11), 2737, 1970.
Olson, W. P., Coordinate Transformations Used in Magnetopheric Physics, McDonnell-Douglas Astronautics Company Paper WD1145, 1970.
Peddie, N. W., International Geomagnetic Reference Field : The Third Generation, J. Geomag. Geoelectr., 34, pp. 309-326, 1985.
Russell, C. T., Geophysical Coordinate Transformations, Cosmic Electrodynamics, 2 , pp. 184-196, 1971.
Smart, W, M., Text-Book On Spherical Astronomy, Fourth Edition, Cambridge Univ. Press, Cambridge, 1944.
U.S. Naval Observatory, Almanac for Computers 1990. Nautical Almanac Office, U.S. Naval Observatory, Washington, D.C., 1989.