GEOPHYSICAL COORDINATE TRANSFORMATIONS
CHRISTOPHER T. RUSSELL
NInstitute of Geophysics and Planetary Physics,
University of Calififornia, Los Angeles, CA 90095, U.S.A.
(Received l2 January, 1971; in revised form 26
March, 1971)
Absract. The use of a vector-matrix formalism to describe
the transformation from one cartesiancoordinate system to
anothcr results in simple-to-use and easy-to-understand
relationships.Furthermore, the required matrix
transformations may be derived directly. The
commoncoordinate systems in use in solar-terrestrial
relationships are described and then thetransformation
matrices required to convert vectors in one system to
another are derived.
Introduction
Many differcnt coordinate systems are used in experimental
and theoretical work onsolar-terrestrial relationships.
These coordinate systems are used to display satellite
trajectories,boundary locations, and vector field
measurements. The need for more than one coordinatesystem
arises from the fact that often various physical processes
are more understood,experimental data more ordered, or
calculations more easily preformed in one or another of
thevarious systems. Frequently, it is necessary to
transform from one to another of these systems.It is
possible to derive 1he transformation from one coordinate
system to another in terms oftrigonometric relations
between angles measured in each system by means of the
formulas ofspherical trigonometry (Smart, 1944). However,
the use of this technique can be very tricky and can result
in rather complex relationships. However, this method is at
times used. A recentexample of the use of this technique to
transform from geographic to geomagnetic coordinatescan be
found in Mead (1970).
Another technique is to find the required Euler rotation
angles and construct the associatedrotation matrices. Then
these rotation matrices can be multiplied to give a single
transformationmatrix (Goldstein, 1950). The vector-matrix
formalism is attractive not only because it permitsa
shorthand representation of the transformation, but also
because it permits multipletransformations to be preformed
readily by matrix multiplication and the inverse
trsnsformationto be derived readily.
The matrices required for coordinate transformations need
not be derived from Euler rotationangles, however, It is
the purpose of this note to explain how to derive these
coordinate transformations without deriving the rcquired
Euler rotation angles as well as to describe the mostcommon
coordinate systems in use in the field of solar terrestrial
relationships.
Discussions of the transformations for some of the
coordinate systems to be treated in this reportmay aIso be
found in papers by Olson (1970), and by the Magnetic and
Electric Fields FieldsBranch (1970) of the Goddard Space
Flight Center. The former paper differs from the
present(0*0*0*work primarily in notation and the number of
systems treated. Anothcr difference is that the Earth's
orbit is considlered to be
TABLE I
Names and abbreviations given to the coordinate systems
common to this paper, to the reportof Olson (1970), and to
the report of the Magnetic Electric Field Branch (1970).
_____________________________________________________________________________
Present Work Olson (1970) MEFB
(1970)
______________________________________________________________________________
Coordinate System Abbr. Coordinate System-Abbr.
Coordinate SystemxxAAbbr.
_______________________________________________________________________________
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