Coordinate transformations between heliocentric systems
Overview
The transformations between the three
heliocentric coordinate
systems:
can be broken down into two fundamental transformations as shown in the
figure below. The symbol beside each arrow refers to the matrix for the
transformation associated with the arrow (and the direction of the arrow
indicates the sense of the transformation). These symbols and the matrices are
specified in more detail below.
The fundamental transformations
The Sn matrices are specified here using the
notation of
Hapgood (1992):
- The HAE to HEE transformation is given by the matrix S1
= <lambdaO + 180o,Z>, where the rotation
angle lambdaO is the
Sun's ecliptic longitude.
This transformation is a rotation in the plane of the ecliptic from the First
Point of Aries to the Sun-Earth direction.
- The HAE to HEEQ transformation is given by the matrix S2
= <theta0,Z>*<i,X>*<Omega,Z>,
where the rotation angle theta0 is the
the longitude of the Sun's
central meridian, i is the
the inclination of the
Sun's equator and Omega is the
the ecliptic longitude of
the ascending node of the Sun's equator. This transformation comprises a
rotation in the plane of the eclipticfrom the First Point of Aries to the
ascending node of the solar equator, then a rotation from the plane of the
ecliptic to the plane of the equator and finally a rotation in the plane of the
solar equator from the ascending node to the central meridian.
Index of all tranformations
The full set of tranformation matrices between the various heliocentric
coordinate systems can be obtained by multiplication of the matrices for these
fundamental transformations, Sn, as shown in the table
below.
| From |
| To |
HAE | HEE | HEEQ |
| HAE |
1 | S1-1 |
S2-1 |
| HEE | S1 | 1 |
S1S2-1 |
| HEEQ | S2 |
S2S1-1 | 1 |
Last updated 29 July 1997 by Mike
Hapgood (Email:
M.Hapgood@rl.ac.uk)
