Originally published in *Cosmic Electrodynamics, 2*, 184-196,
1971. All rights reserved. Copyright 1971 by D. Reidel,
Publishing Company Dordrecht-Holland.
(Received 12 January, 1971; in revised form 26 March, 1971)

- Abstract
- Introduction
- Section 2: General Remarks
- Section 3: Coordinate Systems
- Section 3.1: The Geocentric Equatorial Inertial System
- Section 3.2: Geographic Coordinates
- Section 3.3: Geomagnetic Coordinates
- Section 3.4: Geocentric Solar Ecliptic System
- Section 3.5: Geocentric Solar Equatorial System
- Section 3.6: Geocentric Solar Magnetospheric System
- Section 3.7: Solar Magnetic Coordinates
- Section 3.8: Dipole Meridian System
- Section 3.9: ATS-1 Coordinate Systems
- Section 3.10: Other Coordinate Systems

- Appendix 1. The Cartesian Representation of a Dipole Magnetic Field
- Appendix 2. The Calculation of the Position of the Sun
- Acknowledgements
- References

Many different coordinate systems are used in
experimental and theoretical work on solar- terrestrial
relationships. These coordinate systems are used to
display satellite trajectories, boundary locations, and
vector field measurements. The need for more than one
coordinate system arises from the fact that often various
physical processes are more understood, experimental data
more ordered, or calculations more easily performed in one
or another of the various systems. Frequently, it is
necessary to transform from one to another of these
systems. It is possible to derive the transformation from
one coordinate system to another in terms of trigonometric
relations between angles measured in each system by means
of the formulas of spherical 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 recent example of the use of
this technique to transform from geographic to geomagnetic
coordinates can be found in *Mead* (1970).

Another technique is to find the required Euler rotation
angles and construct the associated rotation matrices.
Then these rotation matrices can be multiplied to give a
single transformation matrix (*Goldstein*, 1950). The
vector-matrix formalism is attractive not only because it
permits a shorthand representation of the transformation,
but also because it permits multiple transformations to be
performed readily by matrix multiplication and the inverse
transformation to be derived readily.

The matrices required for coordinate transformations need not be derived from Euler rotation angles, however. It is the purpose of this note to explain how to derive these coordinate transformations without deriving the required Euler rotation angles as well as to describe the most common coordinate systems in use in the field of solar terrestrial relationships.

Discussions of the coordinate transformations for some of
the coordinate systems to be treated in this report may
also be found in papers by *Olson* (1970), and by the
*Magnetic and Electric Fields Branch* (1970) of the Goddard
Space Flight Center. The former paper differs from the
present work primarily in notation and the number of
systems treated. Another difference is that the Earth's
orbit is considered to be circular in Olson's treatment.
The latter paper describes coordinate systems and presents
the required transformation matrices but without
derivation. Since the same coordinate system may receive a
different name in each treatment, Table I lists the names
and abbreviations used in these two papers and the present
work for those systems common to two or more of them.

______________________________________________________________________________ Present Work Olson (1970) MEFB(1970) ______________________________________________________________________________ Coordinate System Abbr. Coordinate System Abbr. Coordinate System Abbr. ______________________________________________________________________________ Geocentric Equatorial GEI - - Geocentric Celestial GCI Inertial Inertial Geographic GEO Geographic G Geographic GD Geomagnetic MAG - - Geomagnetic GM Geocentric Solar Ecliptic GSE Ecliptic EC Solar Ecliptic SE Geocentric Solar GSM Solar SM Solar SM Magnetospheric Magnetospheric Magnetospheric Solar Magnetic SM Solar Magnetic MG Solar Geomagnetic SGM ______________________________________________________________________________

In defining a coordinate system, in general, you choose two
quantities: the direction of one of the axes and the
orientation of the other two axes in the plane
perpendicular to this direction. This latter orientation
is often specified by requiring one of the two remaining
axes to be perpendicular to some direction. A fortunate
feature of rotation matrices (the matrix that transforms a
vector from one system to another) is that the inverse is
simply its transpose. Thus, if the matrix *A* transforms the
vector V
measured in system *a* to V
measured in system *b*,
then the matrix that transforms V
into V is A.
Thus we
may write

A · V =
V

A· V =
V

The simplest way to obtain the transformation matrix *A* is
to find the directions of the three new coordinate axes for
system b in the old system (system *a*). If the direction
cosines of the new X-direction expressed in the old system
are (X
, X,
X),
of the new Y-direction are (Y,
Y
Y)
and the new Z-direction are (Z
, Z,
Z), then the rotation
matrix is formed by these three vectors as rows, i.e.

(X
X
X )
(V)
= (V)

(YY
Y )
(V)
= (V)

(Z Z
Z ) (V ) =
(V)

Similarly the transformation from system b to a is

(X Y
Z )
(V)
= (V)

(X Y
Z )
(V) =
(V)

(X
Y Z )
(V)
= (V)

The following properties of rotation matrices are useful for error checking. (1) Each row and column is a unit vector. (2) The dot products of any two rows or any two columns is zero. (3) The cross product of any two rows or columns equals the third row or column or its negative. (Row 1 cross row 2 equals row 3; row 2 cross row 1 equals minus row 3.)

3.1. THE GEOCENTRIC EQUATORIAL INERTIAL SYSTEM

3.1.1.
The Geocentric Equatorial Inertial System (GEI) has its
*X*-axis pointing from the Earth towards the first point of
Aries (the position of the Sun at the vernal equinox).
This direction is the intersection of the Earth's
equatorial plane and the ecliptic plane and thus the X-axis
lies in both planes. The *Z*-axis is parallel to the
rotation axis of the Earth and *Y* completes the right-handed
orthogonal set (Y = Z X).

3.1.2. *Uses*

This is the system commonly used in astronomy and satellite
orbit calculations. The angles right ascension and
declination are measured in this system. If
(*V*,
*V*
,*V*) is a
vector in GEI with magnitude *V*, then its right ascension,
, is tan (V
/ V),
0^{o}
180^{o} if V
0
,180^{o} 360^{o}
if V 0.
Its declination, , is sin
V / V,
-90^{o} 90^{o}.

*
3.2.1. Definition
*

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^{o} 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).

3.2.2. *Uses*

This system is used for defining the positions of ground observatories and transmitting and receiving stations. Longitude and latitude in this system are defined in the same way as right ascension and declination in GEI.

3.2.3. *Transformations*

Since the GEO and GEI coordinate systems have their *Z*-axes
in common, we need only to know the position of the first
point in Aries (the *X*-axis of GEI) relative to the
Greenwich meridian to determine the required
transformation. If we let the angle between the Greenwich
meridian and the first point of Aries measured eastwards
from the first point of Aries in the Earth's equator be *0*,
then the first point of Aries is at (cos
, -sin , 0) in the
geographic system and the transformation from geographic to
GEI is

(cos -sin
0)
(V) =
(V)

(sin
cos 0)
(V) =
(V)

( 0 0
1)
(V)=
(V)

and the inverse transformation is

(cos -sin 0)
(V) =
(V)

(sin
cos 0)
(V) =
(V)

( 0 0 1)
(V)
= (V)

The angle is, of course, a function of the time of day and the time of year, since the Earth spins 366.25 times per year around its axis in inertial space, rather than 365.25 times. Thus, the duration of a day, relative to inertial space, (a sidereal day) is less than 24 h. The angle is called Greenwich Mean Sidereal Time, and can be calculated by means of the formulas given in Appendix 2.

The geomagnetic coordinate system (MAG) is defined so that
its *Z*-axis is parallel to the magnetic dipole axis. The
geographic coordinates of the dipole axis from the
International Geomagnetic Reference Field 1965.0 (IGRF) are
11.435^{o} colatitude and 69.761^{o} east longitude (Mead,
1970). Thus the *Z*-axis is (0.06859, -0.18602, 0.98015) in
geographic coordinates. 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 S. Finally,
the *X*-axis completes a right-handed orthogonal set.

3.3.2. *Uses*

This system is often used for defining the position of
magnetic observatories. Also it is a convenient system in
which to do field line tracing when current systems, in
addition to the Earth's internal field, are being
considered (*Mead*, 1970). The magnetic longitude is
measured eastwards from the *X*-axis and magnetic latitude is
measured from the equator in magnetic meridians, positive
northward and negative southwards. Thus, if (V,
V, V) is
a vector in the MAG system with magnitude *V* then its
magnetic longitude, , is

tan (V
/V), 0^{o}
180^{o} if V
0, 180^{o}
360^{o} if V 0^{o}.
Its magnetic latitude, , is
sin V
/ V, -90^{o}
90^{o} .

Except near the poles, magnetic longitude will generally be
about 70^{o} greater than geographic longitude. We note that
a simple cartesian representation of the dipole magnetic
field exists in this system (see Appendix 1).

*
3.3.3. Transformations
*

This system is fixed in the rotating Earth and thus the transformation from the geographic coordinate system to the geomagnetic system is constant. From the definitions above we obtain

(0.33907, -0.91964, -0.19826) (V)
= (V)

(0.93826, 0.34594,
0 ) (V)
= (V)

(0.06859, 0.18602, 0.98015 )
(V)
= (V)

*
3.4.1. Definition
*

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.

*
3.4.2. Uses
*

This system has been used to display satellite
trajectories, interplanetary magnetic field observations,
and solar wind velocity data. The system is useful for the
latter display since the aberration of the solar wind can
easily be removed in this system because the velocity of
the Earth is approximately 30 km/s in the minus *Y*
direction. However, since the only important effect of the
Earth's orbital motion in solar terrestrial relationships
is to cause the aberration, other choices of the
orientation of the *Y* and *Z*-axes about the *X*-axis have been
used. These will be discussed later.

Longitude, as with the geographic system, is measured in
the *X-Y* plane from the *X*-axis toward the *Y*-axis and
latitude is the angle out of the *X-Y* plane, positive for
positive *Z* components.

*
3.4.3. Transformations
*

The most common required transformation into the GSE system
of those discussed so far is from the GEI system. The
direction of the ecliptic pole (0, -0.398, 0.917) is
constant in the GEI system. The *X*-axis, the direction of
the Sun, may be obtained in GEI from the equations in
Appendix 2. If this direction is
(S,
S,
S), then the
*Y*-axis in GEI (Y,
Y, Y) is

(0, -0.398, 0.917) (S, S, S)

and the transformation is

(S
S
S )
(V )
= (V)

(Y
Y
Y )
(V)
= (V)

(0 -0.398 0.917)
(V)
= (V)

*
3.5.1. Definition
*

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.

*
3.5.2. Uses
*

This system has been used extensively to display
interplanetary magnetic field data by the Ames magnetometer
group (*Colburn*, 1969). We note that this system is useful
for ordering data controlled by the Sun and therefore is an
improvement over the use of the GSE system for studying the
interplanetary magnetic field and the solar wind. However,
for studying the interaction of the interplanetary medium
with the Earth yet a third system is more relevant.

*
3.5.3. Transformations
*

The rotation axis of the Sun, R, has a right ascension of
-74.0^{o} and a declination of 63.8^{o}. Thus R is (0.122,
-0.424, 0.899) in GEI. To transform from GEI to GSEQ, we
must know the position of the Sun (S
, S,
S) in GEI (see
Appendix 2). Then the *Y*-axis in GEI
(Y, Y,
Y) is
parallel to R S. Note that since the cross product of two
unit vectors is not a unit vector unless they are
perpendicular to each other, this cross product must be
normalized. Finally the *Z*-axis in GEI
(Z, Z,
Z) = S
Y. Then

(S
S
S)
(V)
= (V)

(Y
Y
Y)
(V)
= (V)

(Z
Z
Z)
(V)
= (V)

Since both GSE and GSEQ coordinate systems have their
*X*-axes directed towards the Sun, they differ only by a
rotation about the *X*-axis. Thus the transformation matrix
from GSE to GSEQ must be of the form

(1 0 0 )
(V)
= (V)

(0 cos
-sin )
(V)
= (V)

(0 sin
cos )
(V)
= (V)

If the transformations from GEI to GSE and GEI to GSEQ are
both known, then the angle may be determined by examining
the angle between the *Y*-axes in the two systems or the *Z*-
axes (i.e. the angle between the vectors formed by the
second row of each matrix or the third row). If these
transformation matrices are not available, may be
calculated from the following formula

Sin = S.(0.031, -0.112, -0.049)/ |(0.122, -0.424, 0.899)| S

where S is the position of the Sun in GEI and can be
calculated from the formulas in Appendix 2. Since the
Sun's spin axis is inclined 7.25^{o} to the ecliptic, ranges
from -7.25^{o} (on approximately Dec. 5) to 7.25^{o} (on June 5)
each year. The Sun's spin axis is directed most towards
the Earth on approximately Sept. 5 at which time the Earth
reaches its most northerly heliographic latitude. At this
time equals 0.

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.

*
3.6.2. Uses
*

This system is useful for displaying magnetopause and shock
boundary positions, magnetosheath and magnetotail magnetic
fields and magnetosheath solar wind velocities because the
orientation of the magnetic dipole axis alters the
otherwise cylindrical symmetry of the solar wind flow. It
also is used in models of magnetopause currents (*Olson*,
1969). It reduces the three dimensional motion of the
Earth's dipole in GEI, GSE, etc., to motion in a plane (the
*X-Z* plane). The angle of the north magnetic pole to the
GSM *Z*-axis is called the dipole tilt angle and is positive
when the north magnetic pole is tilted towards the Sun. In
addition to a yearly period due to the motion of the Earth
about the Sun, this coordinate system rocks about the solar
direction with a 24 h period. We note that since the
*Y*-axis is perpendicular to the dipole axis, the *Y*-axis is
always in the magnetic equator and since it is
perpendicular to the Earth-Sun-line, it is in the dawn-dusk
meridian (pointing towards dusk). GSM longitude is
measured in the *X-Y* plane from *X* towards *Y* and latitude is
the angle northward from the *X-Y* plane. However, another
set of spherical polar angles is sometimes used. Here the
angle, between the vector and the *X*-axis, called the
Sun-Earth probe angle (SEP) or the Sun-Earth-satellite
angle (SES) is the polar angle and the angle of the
projected vector in the *Y-Z* plane is the azimuthal angle.
It is measured from the positive *Y*-axis towards the
positive *Z*-axis.

*
3.6.3. Transformations
*

To transform from GEI to GSM we need to know both the
position of the Sun in GEI and the position of the Earth's
dipole axis. The position of the Sun S (S,
S, S) can be
obtained from Appendix 2. The position of the dipole D
must be obtained by transforming from geographic
coordinates (see Section 2). In geographic coordinates,
the dipole is at 11.435^{o} colatitude and 69 .761^{o} east
longitude (IGRF epoch 1965.0). Thus, D in geographic
coordinates is (0.06859 -0.18602,0.98015). If D' is D
transformed into GEI, the *Y*-axis is

D' S/ |D' S|

We note that the normalizing factor occurs because D' and S are not necessarily perpendicular. Finally, Z is S Y and the transformation becomes

(S
S
S )
(V)
= (V)

(Y
Y
Y)
(V)
= (V)

(Z
Z
Z )
(V)
= (V)

The transformation matrix between GSM and GSE or GSEQ is of the form

(1 0 0 )

(0 cos
-sin )

(0 sin
cos )

However, since changes both with time of day and time of
year, it is not derivable from a simple equation. However,
if the transformation matrix from GEI to GSE, A and from
GEI to GSM, A are both known, then the transformation
from GSM to GSE is simple A, A
where A is the
transpose of A. An analogous formula holds for the
transformation from GSM to GSEQ. We note that the
amplitude of the diurnal variation of is 11.4^{o} which is
added to an annual variation of 23.5^{o}.

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.

*
3.7.2. Uses
*

The solar magnetic system is useful for ordering data
controlled more strongly by the Earth's dipole field than
by the solar wind. It has been used for magnetopause cross
sections and magnetospheric magnetic fields. We note that
since the dipole axis and the *Z*-axis of this system are
parallel the cartesian components of the dipole magnetic
field are particularly simple in this system (see Appendix
1).

*
3.7.3. Transformations
*

As for GSM, the transformation from GEI to SM requires a knowledge of the Earth Sun direction S, and the dipole direction D in GEI. Having obtained these as in Section 3.6, we find Y=(D S)/ ( D S ) and X=Y D. Then the transformation becomes

(X
X
X )
(V)
= (V)

(Y
Y
Y)
(V)
= (V)

(D
D
D)
(V)
= (V)

The transformation from GSM to SM is simply a rotation
about the *Y*-axis by the dipole tilt angle . Thus

(cosµ 0 -sinµ )
(V)
= (V)

( 0 1 0 )
(V)
= (V)

(sinµ 0 cosµ )
(V )
= (V)

As with the solar magnetic system, the *Z*-axis of the dipole
meridian system (DM) is chosen along the north magnetic
dipole axis. However, the *Y*-axis is chosen to be
perpendicular to a radius vector to the point of
observation rather than the Sun. The positive *Y* direction
is chosen to be eastwards, so that the *X*-axis is directed
outwards from the dipole. This is a local coordinate
system, in that it varies with position, however, since the
*X-Z* plane contains the dipole magnetic field it is quite
useful.

*
3.8.2. Uses
*

It is used to order data controlled by the dipole magnetic
field where the influence of the solar wind interaction
with the magnetosphere is weak. It has been used
extensively to describe the distortions of the
magnetospheric field in terms of the two angles declination
and inclination which can be easily derived from
measurements in this system (*Mead and Cahill*, 1967). The
inclination,** I**, is simply the angle that the field makes
with the radius vector minus 90. Thus, if

*
3.8.3. Transformations
*

To transform from any system to the dipole meridian system
we must know the dipole axis, D, in this system, and the
unit position vector of the point of observation relative
to the center of the Earth. Since **Y** is perpendicular to **R**
and **D** then **Y** = (D R)/( D R )
and X=D Y. Thus

(X
X
X )
(V)
= (V)

(Y
Y
Y)
(V)
= (V)

(D
D
D)
(V )
= (V)

We note that this transformation usually is particularly
straight forward from geographic coordinates because the
geographic latitude and longitude of a point of observation
is often known and the dipole is fixed in geographic
coordinates. From geomagnetic coordinates it is simple
rotation about the *Z*-axis by the magnetic longitude. From
solar magnetic co-ordinates, it is a rotation about the
*Z*-axis by the angle between the projections of the Sun and
the local radius vector in the magnetic equator.

Two coordinate systems have been used extensively in the
analysis of the magnetometer data from the ATS-1 satellite
which differ slightly from previously described coordinate
systems. The ATS *XYZ* system is the coordinate system in
which the ATS magnetometer data are originally obtained.
The *Z*-axis is parallel to the Earth's rotation axis. Thus,
it is parallel to the *Z*-axis of the geographic, and GEI
systems. However, the *Y*-axis is chosen perpendicular to
the Earth-Sun line towards dusk. The *X*-axis completes a
right handed orthogonal set. Thus the *X-Y* plane is the
Earth's rotational equator with *X* in the noon meridian.

The other ATS coordinate system is ATS *VDH*. In this
system *H* is chosen parallel to the Earth's spin axis. *V*
is the local vertical. Since ATS-1 is in the Earth's
equatorial plane, *V* is perpendicular to *H*. Finally, *D*,
completes the right-handed set (D = H
V) and is
azimuthal, eastwards in the equatorial plane. The
transformation between the ATS *XYZ* and ATS *VDH* systems is

( cos
sin 0)
(B)
= (B)

(-sin
cos 0)
(B)
= (B)

( 0 0 1)
(B)
= (B)

where (deg) = 15 (UT + 2 h).

All the coordinate systems described so far have been
geocentric and, with the exception of the dipole meridian
system and the ATS VDH system, have been independent of the
position of the point of observation. When considering
measurements far from the Earth, it is often useful to
choose coordinate systems which are dependent on the
position of the observation point rather than the position
of the Earth. For example, *Coleman et al*. (1969) use a
system analogous to the GSEQ system but with the Mariner 4
- Sun line as the *X*-axis. We note, however, they have
chosen their three axes anti-parallel to the axes of the
analogous GSEQ system and thus their right-handed triad of
coordinates is a noncyclic permutation of these three
antiparallel vectors. For studying solar-planetary
interactions, the required modifications to alter the
transformations given in the previous sections to those
relevant to the problem being considered should be
obvious.

However, there is another class of cartesian coordinate
systems that can be used: those based on a local
measurement. For example, one may wish to define a
coordinate system in which the solar wind flow is parallel
to one of the coordinate axes. This could be done in
coordinate systems such as GSE, GSEQ and GSM by replacing
the position of the Sun by the vector antiparallel to the
observed solar wind flow. The second condition for
choosing the coordinate system would be that the *Y*-axis is
perpendicular to the solar wind and the ecliptic pole (for
GSE) and the Sun's rotation axis (for GSEQ) and the Earth's
dipole (for GSM). However, we note that in GSE, the *Z*-axis
will no longer necessarily be parallel to the ecliptic pole
since the solar wind flow need not be in the ecliptic
plane.

Another way of choosing the system is to choose one axis along the measured magnetic field. As before, we are now left with the choice of the orientation of the other two axes about this one. In the solar wind it is often useful to choose one of these two axes perpendicular to the plane defined by the magnetic field and the solar wind flow velocity. In the magnetosphere, it is convenient to choose one of these two axes to be perpendicular to a dipole magnetic meridian.

Finally, since it is much easier to visualize data and
spacecraft trajectories in two dimensions rather than
three, mention should be made of a two dimensional
coordinate system in common use. Since the solar wind,
neglecting the magnetic field is approximately
cylindrically symmetric about the radial direction from the
Sun, if it interacts with a figure of revolution about the
Earth-Sun line such as a planet, the interaction should be
the same in every plane containing the planet-Sun line. In
other words, while the interaction may be a function of
radial distance and the angle away from the planet-Sun line
(SES or SEP angle in the case of the Earth), it is not a
function of the azimuthal angle around the planet Sun
line. The Earth's magnetosphere is not cylindrically
symmetric about the solar wind flow. However, in the
dawn-dusk plane the calculated magnetopause position should
deviate less than about 20% from cylindrical (*Olson*,
1969). Thus, it is not unreasonable at times to assume
cylindrical symmetry for the interaction.

This coordinate system may be thought of in several ways.
(1) It is a cylindrical coordinate system with the
variables *r*,
, *X* where *r* is the distance from the axis of
the cylinder, *X* is the distance along the axis, and
is
the angle around the axis. In plotting a spacecraft
trajectory in this system, we would plot *r* vs *X*. (2) It is
a polar coordinate system where we plot the magnitude of
the vector versus the angle between the vector and the
planet-Sun line. (3) It is a two dimensional cartesian
coordinate system where we plot the component along the
planet-Sun line versus the square root of the sum of the
squares of the other two components. This system has been
used to describe the trajectory of spacecraft near
encounters with other planets and to plot the positions of
magnetopause and bow shock crossings by Earth orbiting
spacecraft.

The usual representation of a dipole magnetic field is one
which separates the field into a radial and tangential
component. This gives the magnetic field in a local two
dimensional coordinate system. However, a very simple
representation of the field exists in a cartesian
coordinate system also (*Alfven and Falthammar*, 1963). If
(*X, Y, Z*) is the location of the point of observation in
solar magnetic coordinates, the field due to the Earth's
dipole is

B = 3XZ (B/R)

B = 3YZ (B/R)

B = (3Z - R) (B/R)

where R = X + Y +Z and B is the magnetic moment of the Earth. B is numerically equal to the field at the equator on the surface of the Earth if distances are measured in Earth radii.

We note that the same formula is valid for any coordinate system which is a rotation about the dipole from the solar magnetic coordinate system. In particular, it is valid for the dipole meridian system in which case B =0. With the knowledge of the dipole tilt angle the above representation also allows a simple derivation of the dipole field in GSM coordinates (cf. Section 7).

G.D. Mead (private communication) has written a simple subroutine to calculate the position of the Sun in GEI coordinates. It is accurate for years 1901 through 2099, to within 0.006 deg. The input is the year, day of year and seconds of the day in UT. The output is Greenwich Mean Sideral Time in degrees, the ecliptic longitude, apparent right ascension and declination of the Sun in degrees. The listing of this program follows. We note that the cartesian coordinates of the vector from the Earth to the Sun are:

X = cos(SRASN) cos(SDEC) Y = sin(SRASN) cos(SDEC) Z = sin(SDEC)

SUBROUTINE SUN(IYR, IDAY, SECS, GST, SLONG, SRASN, SDEC) C PROGRAM TO CALCULATE SIDEREAL TIME AND POSITION OF THE SUN. C GOOD FOR YEARS 1901 THROUGH 2099. ACCURACY 0.006 DEGREE. C INPUT IS IYR, IDAY (INTEGERS), AND SECS, DEFINING UN. TIME. C OUTPUT IS GREENWICH MEAN SIDEREAL TIME (GST) IN DEGREES, C LONGITUDE ALONG ECLIPTIC (SLONG), AND APPARENT RIGHT ASCENSION C AND DECLINATION (SRASN, SDEC) OF THE SUN, ALL IN DEGREES DATA RAD /57.29578/ DOUBLE PRECISION DJ, FDAY IF(IYR. LT. 1901. OR. IYR. GT. 2099) RETURN FDAY = SECS/86400 DJ = 365* (IYR-1900) + (IYR-1901)/4 + IDAY + FDAY -0.5D0 T = DJ / 36525 VL = DMOD (279.696678 + 0.9856473354*DJ, 360.D0) GST = DMOD (279.690983 + 0.9856473354*DJ + 360.*FDAY + 180., 360.D0) G = DMOD (358.475845 + 0.985600267*DJ, 360.D0) / RAD SLONG = VL + (1.91946 -0.004789*T)*SIN(G) + 0.020094*SIN (2.*G) OBLIQ = (23.45229 -0.0130125*T) / RAD SLP = (SLONG -0.005686) / RAD SIND = SIN (OBLIQ)*SIN (SLP) COSD = SQRT(1.-SIND**2) SDEC = RAD * ATAN (SIND/COSD) SRASN = 180. -RAD*ATAN2 (COTAN (OBLIQ)*SIND/COSD, -COS (SLP)/COSD) RETURN END

I am indebted to G. D. Mead for allowing the inclusion of his subroutine for the determination of the position of the Sun. I also wish to acknowledge many useful discussions of coordinate transformations with P. J. Coleman, Jr., D. S. Colburn, M. G. McLeod, G. D. Mead, W. P. Olson and R. L. Rosenberg. This work was carried out in support of the data reduction program of the UCLA OGO5 flux gate magnetometer and was supported by the National Aeronautics and Space Administration under NASA contract NAS 59098.

Colburn, D. S.: 1969, 'Description of Ames Magnetometer Data from Explorer 33 and 35 Deposited in the Data Bank', NASA/Ames Research Center Report.

Coleman, P. J., Jr., Smith, E. J., Davis, L.,
Jr., and Jones, D. E.: 1969, *J. Geophys. Res.*
74 (11), 26.

Goldstein, H.: 1950, *Classical Mechanics*,
Addison Wesley Publ. Co., Inc., Reading
Massachusetts.

Magnetic and Electric Fields Branch: 1970, `Coordinate Transformations Used in OGO Satellite Data Analysis', Goddard Space Flight Center Report, X-645-70-29.

Mead, G. D.: 1970,* J. Geophys. Res.*
75, 4372.

Mead, G. D. and Cahill, L. J.: 1967, *J. Geophys.
Res.*, 72 (11), 2737.

Olson, W. P.: 1969, *J. Geophys. Res*.
74 (24), 5642.

Olson, W. P.: 1970, `Coordinate Transformations Used in Magnetospheric Physics', McDonnell Douglas Astronautics Company Paper WD1145.

Smart, W. M.: 1944, * Text-Book on Spherical Astronomy*,
Fourth Edition, Cambridge Univ.
Press, Cambridge.

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