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Heliospheric coordinate systems M . Fr"anz , D. Harper ∗
Planetary and Space Science 50 (2002) 217 – 233
www.elsevier.com/locate/planspasci
Heliospheric coordinate systems
M. Fr"anza;∗ , D. Harperb
b The
a Astronomy Unit, Queen Mary, University of London, London, E1 4NS, UK
Sanger Centre, Wellcome Trust Genome Campus, Hinxton, Cambs CB10 1SA, UK
Received 18 January 2001; received in revised form 18 July 2001; accepted 18 September 2001
Abstract
This article gives an overview and reference to the most common coordinate systems currently used in space science. While coordinate
systems used in near-Earth space physics have been described in previous work we extend that description to systems used for physical
observations of the Sun and the planets and to systems based on spacecraft location. For all systems, we de4ne the corresponding
transformation in terms of Eulerian rotation matrices. We also give 4rst order Keplerian elements for planetary orbits and determine their
precision for the period 1950 –2050 and describe methods to improve that precision. We also determine the Keplerian orbital elements
for most major interplanetary missions and discuss their precision. We also give reference to a large set of web-sources relevant to the
subject. ? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Coordinate systems; Orbital elements; Coordinate transformations
1. Introduction
Coordinate systems used in near-Earth space physics have
been well covered by the works of Russell (1971) and
Hapgood (1992). But there has been a lack of publicly available documentation on coordinate systems used in heliospheric space missions and in many cases the information
does not seem comprehensive enough for reference purposes. 1 Speci4cally descriptions of systems based on the
physical ephemeris of the Sun and planets and systems based
on spacecraft position are currently not available in a form
that makes the relation between both systems easy to understand. Experience shows that this de4ciency leads to misunderstandings and errors in the production of spacecraft
data sets. Another problem is the lack of information on the
precision of transformations. This document tries to collect
all information necessary for the calculation of coordinate
transformations in space science and determines the precision of these transformations whenever possible.
∗
Corresponding
author.
MPI
f"ur
Aeronomie,
D-37191
Katlenburg-Lindau,
Germany.
Tel.:
+49-555-697944;
fax:
+49-555-697924.
E-mail address: [email protected] (M. Fr"anz).
1 The American National Space Science Data Center (NSSDC) maintains a webpage at http://nssdc.gsfc.nasa.gov/space/helios/coor des.html.
We base all calculations on the current edition of the
Astronomical Almanac (2000), hereafter cited as A 2 and
the Expl. Suppl. (1992), hereafter cited as S. This means
that the base system of astronomical constants used is the
IAU 1976 system described in Astr. Alm. Suppl. (1984) implemented in the numerically integrated ephemeris DE200
(Standish, 1990). In general, this paper does not describe
methods applicable for spatial resolutions below the level
of 1 arcsecond but the reader will be able to 4nd the information necessary to achieve higher precision in the cited
sources.
To achieve the highest precision in planetary positions
one can either (1) implement the numerically integrated
ephemeris DE200 or its more precise sequel DE405 (Standish, 1998a), 3 (2) implement a polynomial expansion of
the ephemeris, for example the VSOP87 model (Bretagnon
and Francou, 1988), 4 which is an expansion of DE200, or
(3) extend the formulae given in this paper to higher order
in time using the values given by Simon et al. (1994) which
are also based on VSOP87. Since the extraction code for
2
See the Nautical Almanac OHces webpage for details:
http://www.usno.navy.mil/ and http://www.nao.rl.ac.uk/.
3 JPL Horizons System and DE200 at http://ssd.jpl.nasa.gov/
horizons.html.
4 Data are available at the Institut de Mecanique Celeste at
http://www.bdl.fr/.
0032-0633/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 3 2 - 0 6 3 3 ( 0 1 ) 0 0 1 1 9 - 2
218
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
DE200 is available in diJerent computer languages, its implementation is easy (see e.g. Heafner, 1999) but the size
of the corresponding data 4les may prevent its inclusion in
distributed software. For the implementation of VSOP87,
we recommend the book by Meeus (2000). In this paper,
we include 4rst order mean orbital elements from Simon et
al. (1994) and give the resulting precision with respect to
DE200. The deviations are on the order of arcseconds while
diJerences between DE200 and DE405 are only a few milliarcseconds.
We should point out that for purposes of spacecraft navigation or problems of planetary encounters it is recommended to install a tested software system whenever this
is provided by the respective spacecraft navigation team.
For most NASA missions such a system is available in the
form of the JPL SPICE system. 5 The SPICE system is a
software library which implements DE200 and other reference systems in the form of position and attitude data 4les
(‘SPICE kernel 4les’) for solar system bodies and spacecraft. Unfortunately, SPICE kernels do not cover all NASA
missions and the precision of reconstructed trajectory data
is usually not provided. Detailed documentation on SPICE
is only available via software 4le headers, this paper may
provide a useful introduction to the principles implemented
in SPICE and similar software packages. Before considering implementing formulae given in this paper, in your own
software package, you might consider implementing the systems cited above, though these will not contain all the coordinate systems de4ned in our paper. Most data in this paper
have been cross-checked by recopying them from the text
into our software and comparing the results with tested data.
To ease the software implementation of formulae given in
this paper, we are providing all data contained in the paper
on our website, 6 and will provide corrections and updates
on that site as long as possible. The website also contains
orbital plots used to determine the precision of data given
in this paper.
We also cite the formulae and methods given by Hapgood (1992) for geocentric systems, which are based on the
Astronomical Almanac for Computers (1988) which is no
longer updated by the Nautical Almanac OHces. The formulae used by Hapgood (1992) are 4rst order approximations
of the third order formulae given in Expl. Suppl. (1961). We
show later that they achieve a precision of about 34 for the
timespan 1950 –2050 if precession and nutation are included.
For many practical purposes, the 4rst order approximation
is suHcient, but a geocentric error of 34 corresponds to a
distance of 230 km at the L1 Lagrangian point which might
be of importance for relative timings between spacecraft for
geocentric systems. To keep the paper as compact as possible, we will give formulae for planetary orbits to 4rst order
only but will point the reader to the sources for improving
the precision. The formulae for nutation and precession are
5
6
JPL SPICE at http://naif.jpl.nasa.gov/naif.html.
http://http://www.space-plasma.qmul.ac.uk/heliocoords/.
given to a precision of at least 2 for the period 1950 –2050,
which allows a higher accuracy transformation between inertial systems. Numerical values are either given in deci◦
mal degree ( ) or arcseconds ( ). Throughout this paper,
we use Eulerian matrix rotations to describe transformations
denoted E(; ; ) (see Appendix A). A concise explanation of many terms and systems used in this paper may be
found in Section L of the Astronomical Almanac (2000).
2. Time
The time-scales relevant for coordinate transformations
are de4ned in Table 1. Formulae in the J2000.0 reference
system are in ephemeris time Teph [S: 2:26; but see also Standish, 1998b], but for most purposes of space data analysis
one may neglect the diJerence of less than 2 ms between
Teph , Barycentric Dynamical Time (TDB) and Terrestrial
Time (TT) [A: B5] and less than 0:1 s between the two Universal Times (UTC, UT1). A diJerence between Atomic
Time (TAI) and Coordinated Universal Time (UTC) is introduced by leap-seconds tabulated in Table 2 [A. K9 for
current table]. 7
Thus, TDB or TT can be approximated from UTC by
TDB = UTC + 32s :184 + PA where PA is the number of elapsed leap seconds to date. For earlier dates,
Meeus (2000) gives diJerent approximation formulae for
UTC-TDB. Spacecraft data are usually given in UTC. Relative velocities of solar system objects are small enough
(¡ 100 km=s) to neglect the diJerence in time systems.
Care must only be taken for problems of relative timing.
If high precision timing (¡ 0:1 s) is requested the reader
should refer to McCarthy (1996) and to the documentation
of the SPICE system (see above). The reference points
in time (epochs) for the ephemeris are given in Table 3.
Before 1984, the ephemeris referred to B1950.0 and many
spacecraft trajectory data are still given in the older system
(see the appendices). The actual position of solar system
objects and spacecraft is usually given in an epoch of date
system which means that coordinates refer to the orientation
of the Earth equator or ecliptic at the time of measurement.
We give formulae to convert from the reference epoch to
the epoch of date in the following Section 2.1.
The Julian Day Number (JD) starts at Greenwich mean
noon 4713 Jan. 1, B.C. [S. 2.26]. The epoch day number
is de4ned in this paper as the fractional number of days of
86 400 s from the epoch:
d0 = (JD − 2451545:0):
(1)
Formulae from S. and A. use Julian centuries (T0 ) from
J2000.0. One Julian century has 36525 days, one Julian year
has 365.25 days, s.t. [S. T3.222.2]
T0 = d0 =36525:0
7
and
y0 = d0 =365:25:
(2)
See also the webpage of the International Earth Rotation Service
(IERS) at http://www.iers.org/.
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
219
Table 1
Time-scales relevant in space science [see A. B4]
UT1
TAI
UTC
TT
TDB
universal time, de4ned by the mean solar day
international atomic time, de4ned by SI seconds
coordinated universal time, TAI — leap seconds, broadcast
standard
terrestrial time, TT = TAI + 32s :184, basis for geocentric
ephemeris
barycentric dynamical time, de4ned by the mean solar day
at the
solar system barycentre, basis for solar system ephemeris
Table 2
Leap seconds PA = TAI − UTC [see A. K9]
1972=1=1 + 10 s
1974=1=1 + 13 s
1977=1=1 + 16 s
1980=1=1 + 19 s
1983=7=1 + 22 s
1990=1=1 + 25 s
1993=7=1 + 28 s
1997=7=1 + 31 s
1972=7=1 + 11 s
1975=1=1 + 14 s
1978=1=1 + 17 s
1981=7=1 + 20 s
1985=7=1 + 23 s
1991=1=1 + 26 s
1994=7=1 + 29 s
1999=1=1 + 32 s
1973=1=1 + 12 s
1976=1=1 + 15 s
1979=1=1 + 18 s
1982=7=1 + 21 s
1988=1=1 + 24 s
1992=7=1 + 27 s
1996=1=1 + 30 s
The mean obliquity of the ecliptic of date with respect to
the mean equator of date is given by [S: 3:222 − 1; A: B18]
0D = 0J 2000 − 46 :8150T0 − 0 :00059T02 + 0 :001813T03
Table 3
Epoch de4nitions [S Table 15:3; A. B4]
◦
◦
2.1. Precession and nutation
The two fundamental celestial reference systems used in
heliospheric science are the ecliptic system de4ned by the
mean orbit of the Earth at J2000.0 and the equatorial system de4ned by the mean orientation of the Earth equator at
J2000.0 (see Fig. 1).
The intersection of the Earth equatorial plane and the
Earth orbital plane (ecliptic) de4nes the line of the equinoxes
(Fig. 1). The ascending node of the geocentric ecliptic de4nes the vernal equinox (:rst point of Aries). The obliquity
of the ecliptic at epoch J2000.0 with respect to the mean
equator at epoch J2000.0 is given by [A. K6]
◦
◦
−0 :000000164T02 + 0 :000000504T03 :
We use this notation throughout the paper. When the astronomical reference systems eventually switch to the next
epoch (presumably J2050.0) formulae given in this paper
have to be adapted.
◦
◦
≈ 23 :439291111 − 0 :013004167T0
J1900:0 = 1900 January 1, 12:00TDB = JD 2415020:0
J1950:0 = 1950 January 1, 00:00TDB = JD 2433282:5
J2000:0 = 2000 January 1, 12:00TDB = JD 2451545:0
B1950:0=
JD 2433282:42345905
0J 2000 = 23 26 21 :448 ≈ 23 :439291111:
Fig. 1. Ecliptic and Equatorial Systems: the ecliptic plane is inclined by
the obliquity towards the Earth equatorial plane. The vernal equinox R
de4nes the common +X -axis, the +Z-axes are de4ned by the Northern
poles P and K. The position of an object S is de4ned by Right Ascension
and Declination in the equatorial system, by ecliptic longitude and
latitude in the ecliptic system.
(3)
The orientation of both planes changes over time by solar,
lunar and planetary gravitational forces on the Earth axis and
orbit. The continuous change is called ‘general precession’,
the periodic change ‘nutation’. Mean quantities include precessional corrections, true quantities both precessional and
nutational corrections.
(4)
The true obliquity of date D = 0D + P includes the eJects
of nutation which are given to a precision of 2 for the
period 1950 –2050 by [S. 3.225-4]
◦
◦
◦
P = 0 :0026 cos(125 :0 − 0 :05295d0 )
◦
◦
◦
+0 :0002 cos(200 :9 + 1 :97129d0 ):
(5)
For the calculation of true equatorial positions one also needs
the longitudinal nutation which is given to 4rst order by [S.
3.225-4]
◦
◦
◦
◦
◦
◦
P = −0 :0048 sin(125 :0 − 0 :05295d0 )
−0 :0004 sin(200 :9 + 1 :97129d0 ):
(6)
The corresponding rotation matrix from the mean equator of
date to the true equator of date is then given by [S: 3:222:3]
◦
◦
N (GEID ; GEIT ) = E(0 ; −D ; 0 )
◦
◦
◦
◦
∗E(−P ; 0 ; 0 ) ∗ E(0 ; 0D ; 0 ):
(7)
To achieve higher precision one has to add further terms for
the series expansion for nutation from [S. Tables 3:222:1–
3:224:2]. 8
The orientation of the ecliptic plane of date (D ) with respect to the ecliptic plane at another date (F ) is de4ned by
the inclination A , the ascending node longitude A of the
8 Note that there is a typographic error in the mean lunar ascending
longitude in [S. Table 3:222:2], the 4rst argument should read =
◦
125 02 40 :280.
220
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
plane of date D relative to the plane of date F, and the difference in the angular distances pA of the vernal equinoxes
from the ascending node. Values for J2000.0 are given in
[S. Table 3:211:1]
A = (47 :0029 − 0 :06603T0 + 0 :000598T02 )t
+(−0 :03302 + 0 :000598T0 )t 2 + 0 :000060t 3 ;
A
◦
= 174 52 34 :982 + 3289 :4789T0 + 0 :60622T02
sian axes in euclidean space and the position of its origin,
relative to some other system. The +Z-axis always de4nes
the polar axis of the respective spherical coordinates: lati◦
tudes are counted from the XY -plane (polar axis 90 ), colatitudes from the polar axis, longitudes are counted from the
+X -axis (prime meridian) clockwise (left handed, +Y -axis
◦
◦
−90 ) or counter-clockwise (right handed, +Y -axis 90 ) as
speci4ed.
+(−869 :8089 − 0 :50491T0 )t + 0 :03536t 2 ;
3.1. Celestial systems
pA = (5029 :0966 + 2 :22226T0 − 0 :000042T02 )t
+(1 :11113 − 0 :000042T0 )t 2 − 0 :000006t 3 ;
(8)
where T0 = F − J 2000 and t = D − F are the distances in
Julian centuries between the 4xed epoch F and J2000.0 and
between D and F , respectively. The corresponding Eulerian
rotation matrix is
P(HAEJ 2000 ; HAED ) = E(
A ; A ; −pA
−
A ):
(9)
Coordinates de4ned on the equator of epoch are transformed
to the equator of date by the Eulerian precession matrix
◦
◦
P(F ; D ) = E(90 − $A ; A ; −zA − 90 ):
(10)
The Eulerian angles are de4ned in [S. Table 3:211:1]
A = (2004 :3109 − 0 :85330T0 − 0 :000217T02 )t
+(−0 :42665 − 0 :000217T0 )t 2 − 0 :041833t 3 ;
$A = (2306 :2181 + 1 :39656T0 − 0 :000139T02 )t
+(0 :30188 − 0 :000344T0 )t 2 + 0 :017998t 3 ;
zA = (2306 :2181 + 1 :39656T0 − 0 :000139T02 )t
+(1 :09468 + 0 :000066T0 )t 2 + 0 :018203t 3 ;
(11)
where t and T0 are de4ned as above. These formulae de4ne
the precession to the precision used for the Astronomical
Almanac but may be easily reduced to lower order.
Hapgood (1997) gives only the 4rst order transformation
between epoch of J2000.0 and epoch of date which is a
reduction of the above formulae and also given to higher
precision in [A. B18]
◦
◦
◦
◦
◦
◦
A = 0 :55675T0 − 0 :00012T02 ;
$A = 0 :64062T0 + 0 :0008T02 ;
zA = 0 :64062T0 + 0 :00030T02 :
(12)
For the heliocentric position of the Earth a complete neglect
◦
of precession results in an error of 1 :0 for the period 1950
–2050, a neglect of nutation results in an error of 20 . Using
4rst order nutation and precession reduces the error to 2 :0.
3. Description of coordinate systems
Each coordinate system, we describe in the following is
de4ned by the orientation of its three right handed carte-
• Geocentric Earth Equatorial GEIJ 2000 (Hapgood 1995).
This system is realized through the International Celestial
Reference Frame (ICRF), which is the base system for
star catalogues and reference values of planetary positions
(see the IERS webpage cited above).
XY -plane: Earth mean equator at J2000.0.
+X -axis: 4rst point of Aries, i.e. vector (Earth–Sun) of
vernal equinox at epoch J2000.0.
Angles: declination and right ascension right handed.
• Mean Geocentric Earth Equatorial GEID (Hapgood,
1995)
XY -plane: Earth mean equator of date.
+X -axis: 4rst point of Aries, i.e. vector (Earth–Sun) of
vernal equinox of date.
Transform: T (GEIJ 2000 ; GEID ) = P(D ; 0 ) as de4ned in
Eq. (10).
• True Geocentric Earth Equatorial GEIT (Hapgood, 1995)
Base system for actual position of objects.
XY -plane: Earth true equator of date.
+X -axis: 4rst point of Aries, i.e. vector (Earth–Sun) of
vernal equinox of date.
Transform: T (GEID ; GEIT ) as de4ned in Eq. (7).
• Heliocentric Aries Ecliptic HAEJ 2000 (Fig. 1)
XY -plane: Earth mean ecliptic at J2000.0.
+X -axis: 4rst point of Aries, i.e. vector (Earth–Sun) of
vernal equinox at epoch J2000.0.
Angles: celestial latitude and longitude right handed.
Transform: T (GEIJ 2000 ; HAEJ 2000 ) = 0 ; X = E(0; 0 ; 0)
and subtraction of solar position vector if necessary.
• Heliocentric Aries Ecliptic HAED
XY -plane: Earth mean ecliptic of date
+X -axis: 4rst point of Aries, i.e. vector (Earth–Sun) of
vernal equinox of date.
Transform: T (HAEJ 2000 ; HAED )=E( A ; A ; −pA − A )
as de4ned in Eq. (9) and T (GEID ; HAED ) = E(0; D ; 0)
where D is de4ned by Eq. (5).
3.2. Heliographic systems
3.2.1. Solar Pole and prime meridian
Heliographic coordinate systems use the position of the
solar rotation axis which is de4ned by its declination M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
and the right ascension with respect to the celestial pole
(GEIJ 2000 + Z). Values for J2000.0 are [S. Table 15:7]:
◦
◦
= 63 :87;
= 286 :13:
(13)
The traditional de4nition refers to the ecliptic of date with
the values for the inclination i of the solar equator and
longitude of the ascending node [S. 7.2, note the typo]:
◦
i = 7 :25;
◦
◦
= 75 :76 + 1 :397T0 :
(14)
The ecliptic values for the polar axis have been in use since
their 4rst determination by Carrington. Newer measurements
show that the axis direction is less well de4ned (Balthasar et
al., 1987) but for the purpose of coordinate transformations
one sticks with the original values. The same is true for the
Solar rotation period for which the adopted values are [A.
C3]:
rsid = 25:38 days
and
rsyn = 27:2753 days;
(15)
where the sidereal period rsid is relative to the celestial
sphere, and the synodic relative to the rotating Earth (see
also Rosa et al., 1995). The time dependence in takes
approximate account of the ecliptic precession such that no
further precessional transformation should be applied but
there is of course a small diJerence between the ecliptic
and the equatorial de4nition. In transformation of datasets
always the equatorial values should be used.
Physical observations of the Sun refer to the apparent
center of the visible disk from Earth (subterrestrial point)
whose heliocentric ecliptic longitude is the apparent longitude of the Earth =geo — a de4ned in Eq. (36) corrected
for light aberration (a ≈ 20 , see Appendix A:3).
3.2.2. Systems
As pointed out in Section 4.3 heliographic systems should
refer to a solar reference ellipsoid, but since the oblateness of
the Sun is diHcult to measure (Stix, 1989), for the following
de4nitions the Sun is assumed to be spherical.
• Heliographic Coordinates HGC (Expl. Suppl., 1961; Stix,
1989)
Physical features on the surface of the Sun are located
in Heliographic coordinates (Expl. Suppl., 1961, 11.B).
Heliographic latitude is measured from the solar equator positive towards North, Heliographic longitude is de4ned westward (i.e. in the direction of planetary motion)
from the solar prime meridian which passed through the
ascending node on the ecliptic of date on 1854 Jan 1,
noon (JD 239 8220.0). Heliographic longitude is sometimes identi4ed with Carrington longitude, but this usage
should be avoided since there have been diJerent de4nitions of the later term over time.
XY -plane: solar equator of date.
+X -axis: ascending node on 1854 Jan 1, noon (JD 239
8220.0).
Angles: heliographic latitude + and longitude , right
handed.
221
◦
◦
Transform: T (GEIJ 2000 ; HGCJ 2000 ) = E( + 90 ; 90 −
; W0 )
◦
with the values from Eq. (13) and W0 = 84 :10 +
◦
14 :1844d0 [S. Table 15:7].
Alternatively (but less exact) one may use the transformation from ecliptic coordinates:
Transform: T (HAEJ 2000 ; HGCD ) = E( ; i ; w0 ) where
and i are de4ned in Eq. (14) and the prime meridian
angle is given by
◦
w0 = (d0 + 2415020:0 − 2398220:0)=25:38 × 360 :
(16)
• Solar Rotations (Expl. Suppl., 1961)
Rotations of the Sun are counted in Carrington rotations
R; a rotation starts when the heliographic prime meridian
crosses the subterrestrial point of the solar disc. The angular oJset between this point and the ascending node
can be calculated from (Hapgood, 1992):
= arctan(cos i tan( − ))
(17)
such that the quadrant of is opposite that of − .
Note that is called L0 − M in Expl. Suppl. (1961).
The 4rst Carrington rotation started on 1853 Nov 9 (JD
2398167.329), later start points can be calculated using
the synodic period rsyn =27:2753 days. The term Carrington Time has been used for the pair of numbers (R; L0 ),
where L0 is the heliographic longitude of the subterrestrial point. For geophysical eJects, Bartels rotations have
been used which start at 1832 Feb 8.00 (JD 239 0190.50)
with a period of 27.0 days (Bartels, 1952).
• Heliocentric Earth Ecliptic HEE (Hapgood, 1992)
XY -plane: Earth mean ecliptic of date.
+X -axis: vector (Sun–Earth).
◦
◦
Transform: T (HAED ; HEED ) = E(0 ; 0 ; geo ),
where geo is the geometric ecliptic longitude of the Earth
which can be determined by one of the methods described
in Section 4.2.1 or directly from Eq. (36) to a precision
of 34 .
• Heliocentric Earth Equatorial HEEQ (Hapgood, 1992)
XY -plane: Solar equator of date.
+X -axis: intersection between solar equator and solar
central meridian of date.
Angles: heliocentric latitude + and central longitude (increasing eastward) right handed.
Transform: T (HAED ; HEEQ) = E( ; i ; );
where is de4ned in Eq. (17).
• Heliocentric Inertial HCI (Burlaga, 1984)
Burlaga (1984) originally de4ned a system, called heliographic inertial (HGI ), with reference to the orientation
of the Solar equator in J1900.0. We propose to call the
system heliocentric and base it on J2000.0 instead:
XY -plane: solar equator of J2000.0.
+X -axis: solar ascending node on ecliptic of J2000.0.
◦
Transform: T (HAEJ 2000 ; HCI ) = E( (T0 = 0); i ; 0 ).
222
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
• Heliocentric of Date HCD
XY -plane: solar equator of date.
+X -axis: solar ascending node on ecliptic of date.
◦
Transform: T (HAED ; HCD) = E( ; i ; 0 ).
(Hapgood, 1992, 1997; Kertz, 1969):
D = arctan(h11 =g11 );
,D = 90 − arctan
3.3. Geocentric systems
Geocentric systems have been described by Russell
(1971) and Hapgood (1992) with corrections given in Hapgood (1995) and Hapgood (1997). 9 You will also 4nd a
comprehensive introduction in Appendix 3 of Kivelson and
Russell (1995). The ESA SPENVIS system contains an
extensive description of geocentric systems. 10 A software
package by J.-C. Kosik is also maintained and documented
at the Centre de DonnTees de la Physique des Plasmas. 11
We do not describe systems relevant for observations from
the Earth surface, see [S., Chapter 4] for a description of
these systems.
3.3.1. Greenwich mean sidereal time
The Greenwich mean sidereal time is de4ned by the hour
angle between the meridian of Greenwich and mean equinox
of date at 0h UT1: [A. B6]
4GMST = 24110s :54841 + 8640184s :812866TU
+0s :093104TU2 − 6s :2 × 10−6 TU3 ;
(18)
in seconds of a day of 86 400 s UT1, where TU is the time
diJerence in Julian centuries of Universal Time (UT1) from
J2000.0.
From this, the hour angle in degree GMST at any instant
of time d0 (Julian days from J2000.0) can be calculated by
◦
GMST = 4GMST (TU (0h )) × 360 =86400s
◦
◦
+180 + 360 ∗ d0 :
(19)
For the precision needed in this paper, we may neglect the
diJerence between TU and T0 , such that (Meeus, 2000):
◦
◦
GMST ≈ 280 :46061837 + 360 :98564736629d0
◦
◦
+0 :0003875T02 − 2 :6 × 10−8 T03 :
(20)
3.3.2. Earth magnetic pole
The geographic position of the Earth magnetic pole and
the dipole moment ME can be calculated from the 4rst
three coeHcients of the International Geomagnetic Reference Field (IGRF) published 5-yearly by IAGA Working Group 8. 12 For full precision interpolate the values
g10 ; g11 ; and h11 for the date requested and determine the
geographic longitude D , latitude ,D and moment ME by
9 See also their webpage at http://sspg1.bnsc.rl.ac.uk/Share/Coordinates/
ct home.htm.
10 ESA SPENVIS webpage at http://www.spenvis.oma.be/spenvis/.
11 See under MAGLIB at http://cdpp.cesr.fr.
12 See their webpage at http://www.ngdc.noaa.gov/IAGA/wg8/.
ME =
g11 cos D + h11 sin D
;
g10
2 + g 2 + h2 ∗ R3 ;
g10
E
11
11
(21)
where RE = 6378:14 km is the Earth equatorial radius and
D lies in the fourth quadrant.
For the period 1975 –2000, we derive following linear
◦
approximations with a precision of 0 :05:
◦
◦
D = 288 :44 − 0 :04236y0 ;
◦
◦
,D = 79 :53 + 0 :03556y0 ;
ME = 3:01117 − 0:00226y0 [10−6 T R3E ];
(22)
where y0 are Julian years from J2000.0.
3.3.3. Systems
The following systems are referred to the true Earth equator or ecliptic of date, that is corrections for nutation and
precession should be applied in transformations. We also
give the bracket notation ; used by Hapgood (1992) (see
the appendices).
• Geographic Coordinates GEO (Hapgood, 1992)
XY -plane: true Earth equator of date.
+X -axis: intersection of Greenwich meridian and Earth
equator.
Angles: geographic latitude and longitude (increasing
westward) right handed, in the sense of a planetographic
system (see Section 4.3).
◦
◦
Transform: T (GEIT ; GEO)=GMST ; Z=E(0 ; 0 ; GMST );
where the GMST is given by Eq. (20).
• Geocentric Solar Ecliptic GSE (Hapgood, 1992)
XY -plane: Earth mean ecliptic of date.
+X -axis: vector Earth–Sun of date.
◦
Transform: T (HAED ; GSE) = geo + 180 ; Z =
◦
◦
◦
E(0 ; 0 ; geo + 180 )
with geo from Eq. (36) and subtraction of solar position
vector if necessary. Also
T (GEID ; GSED ) = T (GEID ; HAED )−1 ∗ T (HAED ; GSE)−1 :
• Geocentric Solar Magnetospheric GSM (Hapgood, 1992)
+Z-axis: projection of northern dipole axis on GSED YZ
plane.
+X -axis: vector Earth–Sun of date.
◦
◦
Transform: T (GSED ; GSM ) = − ; X = E(0 ; − ; 0 ),
where = arctan(ye =ze ) and Qe = (xe ; ye ; ze ) is the Earth
dipole vector in GSE-coordinates. This can be calculated
from the geographic position Qg given in Eq. (22) by
Qe = T (GEID ; GSED ) ∗ T (GEID ; GEO)−1 Qg :
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
• Boundary Normal Coordinates LMN (Russell and Elphic,
1978) 13
+Z-axis: normal vector to Earth Magnetopause.
+Y -axis: cross-product of +Z-axis and GSM-Z-axis.
The normal vector may be determined by a model or by
minimum-variance analysis of data.
• Solar Magnetic SM (Chapman and Bartels, 1962)
+Z-axis: Northern Earth dipole axis of date.
+Y -axis: cross-product of +Z-axis and Earth–Sun vector
of date.
◦
◦
Transform: T (GSM; SM ) = −9; Y = E(90 ; −9; −90 ),
2
2
where 9 = arctan xe = ye + ze with Qe given above. The
longitude of this system is also called magnetic local time
(MLT ) increasing eastwards from the anti-solar (0h ) to
the solar (12h ) direction.
• Geomagnetic MAG (Chapman and Bartels, 1962)
+Z-axis: Northern Earth dipole axis of date.
+Y -axis: cross-product of Geographic North Pole of date
and +Z-axis.
◦
Transform: T (GEO; MAG) = ,D − 90 ; Y ∗ D ; Z =
◦
◦
◦
E(D + 90 ; 90 − ,D ; −90 ),
where ,D and D are given in Eq. (22). Geomagnetic
latitude m and longitude m (increasing eastward) refer
to this system.
• Invariant magnetic shells (Bd ; Ld ) (McIlwain, 1966)
These coordinates are used for functions of the magnetic
4eld which are constant along the lines of force. For a
position of radial distance R from the dipole center and
magnetic latitude m in a dipolar 4eld the magnetic 4eld
strength Bd and equatorial distance Ld of the line of forth
are given by
M
R
;
(23)
Bd = 3 1 + 3 sin2 m ; Ld =
R
cos2 m
where M is the magnetic moment of the dipole (see Eq.
(22) for Earth value). The oJset between dipole center
and gravity center (≈ 500 km for Earth) has been neglected (Kertz, 1969).
• Other magnetospheric coordinates 14–16
Many coordinate systems depend on a speci4c magnetic
4eld model. For example, Corrected Magnetic Coordinates (CGM ) 17 are constructed by 4eld line tracing.
Magnetospheric Equatorial Coordinates (GME) use speci4c magnetotail models (Dunlop and Cargill, 1999). For
a 4eld model again (B; L) coordinates may be derived
for which particle drift shells can be de4ned (McIlwain,
1966). See the references for details.
13
See also C.T. Russell’s page at http://www-ssc.igpp.ucla.edu/
ssc/tutorial/magnetopause.html.
14 See also the APL Superdarn webpage http://superdarn.jhuapl.edu/
aacgm/.
15 See also the University of Oulu spaceweb at http://spaceweb.oulu.4/.
16 See also S. Haaland’s page at http://gluon.4.uib.no/ haaland/.
17 See also the NSSDC Modelweb at http://nssdc.gsfc.nasa.gov/
space/cgm/.
223
3.4. Position dependent systems
For the study of the local plasma environment of a spacecraft, it is common to choose an axis system which depends
on the position of the spacecraft. Widely used are Radial–
Tangential–Normal systems de4ned by the radial vector
from a central body to the spacecraft and the magnetic or
rotational normal axis of that body. For highest precision
one should use reference systems at the epoch of date.
• Heliocentric RTN System HGRTN (Burlaga, 1984)
This system was, for example, used by the Ulysses mission.
+X -axis: vector (Sun-S=C).
+Y -axis: cross-product of (heliographic polar axis) and
+X -axis.
◦
◦
Transform: T (HCD; HGRTN ) = E(S=C − 90 ; S=C ; 90 ),
where S=C and S=C are the longitude and latitude of
the spacecraft in the HCD system. Cartesian coordinates
of this system are commonly called Radial, Tangential,
Normal (RTN) coordinates.
• Dipole Meridian System DM (Kivelson and Russell,
1995)
This system can be used in any dipolar 4eld to separate
radial and angular motions.
+X -axis: vector (dipole Center-S=C).
+Y -axis: cross-product of (dipole polar axis) and
+X -axis.
◦
◦
Transform: T (MAG; DM ) = E(S=C − 90 ; S=C ; 90 ),
where S=C and S=C are the longitude and latitude of the
spacecraft in the MAG system.
• Spacecraft solar ecliptic SSE [F. Neubauer (pers. comm.)]
This system was for example used by the Helios mission.
XY -plane: Earth mean ecliptic of date.
+X -axis: projection of vector S=C-Sun on XY -plane.
+Z-axis: ecliptic South pole.
◦
◦
◦
Transform: T (HAED ; SSE) = E(S=C − 90 ; 180 ; 90 ).
• Spin axis ecliptic SAE [NSSDC, Pioneer data pages]
This spacecraft centered system was for example used by
the Pioneer missions (under the acronym PE).
+Z-axis: spacecraft spin axis vA (towards Earth).
+X -axis: cross-product of ecliptic polar axis of date and
vA .
◦
Transform: T (HAED ; SAE) = E( + 90 ; ; 0:0),
where and are the ecliptic longitude and colatitude
of the spacecraft spin axis.
• Spin axis Sun pulse SAS
This system is a fundamental reference system for most
spinning spacecraft since the S=C-Sun meridian can easily
be determined on board using a narrow-slit sun sensor.
Thus, a spacecraft-4xed instrumental system has only a
longitudinal oJset with respect to SAS linear in time.
+Z-axis: spacecraft spin axis vA , right-handed orientation.
+Y -axis: cross-product between +Z-axis and S=C-Sun
vector vS .
224
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
◦
Transform: T (HAED ; SAS) = E( + 90 ; ; ,(vS )),
where and are the ecliptic longitude and colatitude
of the spacecraft spin axis and ,(vS ) is the longitude of
◦
◦
the vector vS0 = E( + 90 ; ; 0 ) ∗ vS and vS is given in
the ecliptic system.
4. Planetary systems
4.1. Planetary orbits
As pointed out in the introduction transformations based
on classical Keplerian elements can only achieve a limited
precision. But for many applications it is useful to have
approximate positions available. For this reason, we describe
in the following the calculation of position and velocity of
objects in Keplerian orbits. There are many textbooks on
this subject — we recommend Murray and Dermott (2000)
but e.g. Bate et al. (1971), Danby (1988) or Heafner (1999)
are also very useful. There are also some good web sites
devoted to the subject. 18
The gravitational motion of two bodies of mass M and m
and position vectors rM and rm can be described in terms of
the three invariants: gravitational parameter 9=<0 (M +m),
speci:c mechanical energy E = (v2 =2) − 9=r, and speci:c
angular momentum h = |r × v|, where r = rM − rm , r = |r|
and v = ṙ. <0 is the constant of gravitation whose IAU1976
value is determined by [A. K6]
<0 = k 2
with k = 0:01720209895
(24)
when masses are given in solar masses, distances in AU
[1 AU = 149 597 870 km], and times in days.
The elements of the conical orbit (shown in Fig. 2) are
then given √
as semi-major axis a = −9=E and semi-minor
axis b = h= −E, or alternatively as semi-latus
rectum p =
2
2
2
b
=a
=
h
=9
=
a(1
−
e
)
and
eccentricity
e
=
1 − b2 =a2 =
2
2
1 + Eh =9 . Let the origin be at the focus rM , the vector r
then describes the motion of the body rm . The true anomaly
v is the angle between r and the direction to the closest point
of the orbit (periapsis) and can be determined from
p
:
(25)
r=
1 + e cos @
If there are two focal points (ellipse, hyperbola) their distance is given by c = ea, the distances of the periapsis and
apoapsis are given by rp = a(1 − e) and
ra = a(1 + e). An
elliptical orbit has the period P = 2a a=9.
Mean elements of a body in an elliptical orbit (e ¡ 1)
are de4ned by the motion of a point rqon a concentric
circle with constant angular velocity n = 9=a3 and radius
√
ab, such that the orbital period P = 2a a=9 is the same
for rq and rm . The mean anomaly M = 9=a3 (t − T ) is
de4ned as the angle between periapsis and rq . Unfortunately,
there is no simple relation between M and the true anomaly
18
For example, K. Burnett’s site at http://www.btinternet.com/
∼ kburnett/kepler/.
Fig. 2. Keplerian orbital elements for the elliptical orbit of the point rm
around the focus F with true anomaly @. Parameters of the ellipse are the
axes a and b, the focal distance c = ae, the semi-latus rectum p and the
point of periapsis Pa at distance ra from F. Also shown are the concentric
circles for the eccentric motion of the point rq with eccentric anomaly
E, and mean motion of the point rq with mean anomaly M (dashed).
@. To construct a relation one introduces another auxiliary
concentric circle with radius a and de4nes rq as the point on
that circle which has the same perifocal x-coordinate as rm .
The eccentric anomaly E is the angular distance between rq
and the periapsis measured from the center and is related to
the mean and true anomalies by the set of equations:
M = E − e sin E
(Kepler equation);
cos @ =
e − cos E
;
e cos E − 1
cos E =
r
e + cos @
= (e + cos @);
1 + e cos @ p
r = a(1 − e cos E):
(26)
(27)
Thus, if the orbital position is given as an expansion in t0
of the mean longitude = + ! + M , the true longitude
0 [email protected] can be found by an integration of the transcendental Kepler equation. In most cases a Newton–Raphson
integration converges quickly (see Danby, 1988; or Herrick,
1971 for methods). For hyperbolic orbits
(e ¿ 1), one can
as well de4ne a mean anomaly Mh = 9=|a|3 (t − T ) but
this quantity has no direct angular interpretation. The hyperbolic eccentric anomaly Eh is related to Mh and the true
anomaly @ by
Mh = e sinh Eh − Eh ;
cos @ =
e + cos @
e − cosh Eh
; cosh Eh =
;
e cosh Eh − 1
1 + e cos @
r = a(1 − e cosh Eh ):
(28)
The orientation of an orbit with respect to a reference plane
(e.g. ecliptic) with origin at the orbital focus is de4ned by
the inclination i of the orbital plane, the longitude of the
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
Fig. 3. Orientation of a Keplerian orbit of the point rm around the focus
O with respect to the ecliptic plane. Symbols are given for the equinox
V, the ascending node V and its longitude , the periapsis rp and its
argument !, the inclination i, and the true anomaly @. The perifocal
system is denoted by (X ; Y ; Z ).
ascending node , and the argument of periapsis ! which
is the angle between ascending node and periapsis rp (see
Fig. 3). The position of the body on the orbit can then be
de4ned by its time of periapsis passage T , its true anomaly
@0 at epoch t0 , or its true longitude 0 = + ! + @0 at
epoch t0 . The perifocal coordinate system has its X -axis
from the focus to the periapsis, and its Z-axis right-handed
perpendicular to the orbital plane in the sense of orbital
motion. In this system, the position and velocity vector are
given by
r = r(cos @; sin @; 0)
v = 9=p(−sin @; e + cos @; 0):
(29)
These might directly be expressed by the eccentric anomaly
E:
r = a(cos E − e; 1 − e2 sin E; 0);
9=a
v=
(30)
(−sin E; 1 − e2 cos E; 0):
r
225
–2050. The resulting precisions in relation to the DE200
ecliptic position are given in Table 5. The positions are
calculated from the mean elements by determining the
true anomaly from Eq. (26) and applying Eqs. (30). The
last three columns of Table 5 are not corrected for disturbances by Jupiter and Saturn, while these disturbances
are included in the 4rst four columns using Table 6 of
Simon et al. (1994). As one can see from Table 5 it is
— at least for the outer planets — recommendable to apply these corrections. To save space we do not give the
numerical values in this paper but refer the reader to Simon et al. (1994) or to our web-page (cited above). The
last row of Table 5 gives the loss in precision when using
mean elements (not solving Eq. (26)) instead of true elements for the EMB: mean and true position diJer by up to
◦
2 :7.
4.2.1. Position of Earth and Moon
Table 5 gives also ecliptic positions of Earth and Moon.
DE200 and VSOP87 give only the heliocentric position
rEMB of the EMB. DE200 gives in addition the geocentric position rgM of the Moon. If both values are given the
position of the Earth rE can be calculated exactly (within
the IAU1976 system) using the mass ratio Moon=Earth of
9M = 0:01230002[A. K6] (or its respective value used for
the ephemeris) by
9M
:
(31)
rE = rEMB − rgM
1 + 9M
The velocity vector has the same transformation. For the
VSOP87 system, we apply following formula by J.L. Simon
(pers. comm.) describing the rotation of the Earth around
the EMB:
E = EMB + 6 :468 sin D;
rE = rEMB + 4613 cos D (km);
(32)
where D is the Delauney argument from Eq. (3:5) in Simon
et al. (1994):
In the hyperbolic case replace cos by cosh and sin by sinh.
The transformation from the reference system to the perifocal system is given by the Eulerian rotation E(; i; !) as
de4ned in the appendices. The ecliptic position of a planet
is then given by re = E(; i; !)r.
D = 297 :8502 + P · T0 ;
4.2. Planetary positions
where rD =rE −rEMB and the ecliptic angular velocity vector
is given by E = 2=P · (0; 0; 1).
If only rEMB and rE are given, the ecliptic position and
velocity of the Moon can then be calculated by
Table 4 gives the six orbital elements a; e; ; $; i; and
their time development for the seven major planets and
the Earth–Moon barycentre (EMB), where $ = + !
is the longitude of the periapsis. Values are reduced to
a relative precision of 10−7 from Table 5:8 in Simon et
al. (1994). This precision is suHcient for the calculation
of planetary positions to the highest precision possible
with a single set of mean elements for the period 1950
◦
(33)
◦
where the rotation period is P = 445267 :11=century. The
Earth velocity vector vE can be calculated by
vE = vEMB + E × rD ;
rM = ((1 + 9M )rEMB − rE )=9M :
(34)
(35)
But the precision of the resulting lunar velocity is rather
low (190 m=s). Neglecting the diJerence between rEMB and
rE increases the total error in the Earth position to 14
(Earth-EMB in Table 5).
226
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
Table 4
Heliocentric mean orbital elements of the planets in HAEJ 2000 to a precision of 10−7 or 1 for 1950 –2050 after Simon et al. (1994)a
Mercury
Venus
EMB
Mars
Jupiter
Saturn
Uranus
Neptune
a [AU ]
e (10−7 )
( )
6023600
408523.5
328900.5
3098710
1047.355
3498.5
22869
19314
0.38709831
0.72332982
1.0000010
1.5236793
5.2026032
9.5549092
19.2184461
30.1103869
2056318 + 204T0
67719 − 478T0
167086 − 420T0
934006 + 905T0
484979 + 1632T0
555481 − 3466T0
463812 − 273T0
94557 + 60T0
252:2509055 + 149472:6746358T0
181:9798009 + 58517:8156760T0
100:4664568 + 35999:3728565T0
355:4329996 + 19140:2993039T0
34:3515187 + 3034:9056606T0
50:0774443 + 1222:1138488T0
314:0550051 + 428:4669983T0
304:3486655 + 218:4862002T0
◦
Mercury
Venus
EMB
Mars
Jupiter
Saturn
Uranus
Neptune
◦
9
◦
◦
$( )
i( )
( )
77:4561190 + 0:1588643T0
131:5637030 + 0:0048746T0
102:9373481 + 0:3225654T0
336:0602340 + 0:4439016T0
14:3312069 + 0:2155209T0
93:0572375 + 0:5665415T0
173:0052911 + 0:0893212T0
48:1202755 + 0:0291866T0
7:0049863 − 0:0059516T0
3:3946619 − 0:0008568T0
0:0
+ 0:0130548T0
1:8497265 − 0:0081477T0
1:3032670 − 0:0019877T0
2:4888788 + 0:0025514T0
0:7731969 − 0:0016869T0
1:7699526 + 0:0002256T0
48:3308930 − 0:1254227T0
76:6799202 − 0:2780134T0
174:8731758 − 0:2410908T0
49:5580932 − 0:2950250T0
100:4644070 + 0:1767232T0
113:6655025 − 0:2566722T0
74:0059570 + 0:0741431T0
131:7840570 − 0:0061651T0
a The
elements are semi-major axis a [1 AU = 149 597 870 km], eccentricity e, mean longitude , longitude of periapsis $, inclination i, and
ascending node . The time parameter T0 is scaled in Julian centuries of 36 525 days from J2000.0. EMB denotes the Earth–Moon barycentre.
The 4rst column 9 gives the IAU1976 mass ratio Sun=planet.
Table 5
Precision of planetary positions derived from orbital elements (Table 4) for the period 1950 –2060 compared to DE200 positions on the ecliptic of datea
With disturbances
Mercury
Venus
EMB
Mars
Jupiter
Saturn
Uranus
Neptune
Earth
Moon
Earthapprox
Earth-EMB
EMBmean
( )
0.8
0.9
0.6
1.0
5.5
14
4.9
1.7
1.0
51
1.1
1.1
1.1
Without disturbances
( )
6.0
5.5
7.6
26
46
81
86
10
7.9
64
16
14
6900
r (1000 km)
v (m=s)
( )
( )
r (1000 km)
0.51
1.0
1.2
8.1
71
170
510
170
1.8
62
8.6
5.8
2500
1.2
1.8
1.9
4.5
15
37
27
22
3.6
180
260
14
3.2
1.6
0.6
4.3
20
62
44
69
1.1
51
1.1
1.1
26
28
29
160
830
2100
3600
2400
29
84
34
29
1.6
5.0
7.0
39
990
6700
8800
11000
7.2
67
8.6
7.2
a Maximal diJerences are given for heliocentric ecliptic latitude , longitude , and distance r and orbital velocity v. Values ‘with disturbances’
use the corrections given in Table 6 of Simon et al. (1994).
For slightly lower precision without solving the Kepler
equation (26) the geometric ecliptic longitude of the Earth
can be calculated by the approximation given for the Solar
longitude in [A. C24]:
◦
also been used by Hapgood (1992). The respective precision
is 34 (Earthapprox in Table 5).
4.3. Planetocentric systems
◦
geo = mean + 1 :915 sin g + 0 :020 sin 2g;
rgeo = 1:00014 − 0:01671 cos g − 0:00014 cos 2g[AU];
(36)
where mean and the mean anomaly g = mean − $ for the
EMB can be taken from Table 4. This approximation has
For solar system bodies, the IAU diJerentiates between
planetocentric and planetographic body-4xed coordinates:
planetocentric latitude refers to the equatorial plane and the
polar axis, planetographic latitude is de4ned as the angle
between equatorial plane and a vector through the point
of interest that is normal to the biaxial ellipsoid reference
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
227
Table 6
˙ its change per Julian
Physical ephemeris of the planets GEIJ 2000 [S. Table 15:7, A. E87]: (0 ; 0 ) is the position of the North pole in GEIJ 2000 , (;
˙ )
century T , W0 is the position of the prime meridian at GEIJ 2000 , Ẇ its change per day
◦
◦
Name
0 ( )
Sun
Mercury
Venus
Earth
Mars
Jupiter III
Saturn III
Uranus III
Neptune
286.13
281.01
272.72
0.00
317.681
268.05
40.58
257.43
299.36
(where N =
Pluto
359.28
313.02
◦
˙ ( =T )
0 ( )
63.87
61.45
67.15
90.00
52.886
64.49
83.54
−15:10
43.46
−0:003
−0:641
−0:108
−0:009
−0:036
+0:70 sin N
+54:308)
◦
T (GEIJ 2000 ; PLAJ 2000 ) = E(0 + 90 ; 90 − 0 ; W0 );
◦
◦
T (GEIJ 2000 ; PLAD ) = E(0 + T
˙ 0 + 90 ; 90
˙ 0 ; W0 + Ẇ d0 );
−0 − T
(37)
where d0 and T0 are de4ned in Eqs. (1) and (2).
4.3.1. Jovian systems
Since, we have used diJerent Jovian coordinate systems
in previous work (Krupp et al., 1993) we include a description of these systems. Most of these systems are discussed
in Dessler (1983). The Jovian pole of rotation is de4ned
by the values (; ) given for ‘Jupiter III’ in Table 6. The
transformation from GEIJ 2000 can be calculated from
◦
◦
T (GEIJ 2000 ; JUPX ) = E( + 90 ; 90 − ; w0 );
◦
−0:005
−0:557
−0:061
+0:003
−0:004
−0:51 cos N
9.09
surface of the body. Both latitudes are identical for a spherical body. Planetocentric longitude is measured eastwards
(i.e. positive in the sense of rotation) from the prime meridian. Planetographic longitude of the sub-observation point
increases with time, i.e. to the west for prograde rotators
and to the east for retrograde rotators. All systems de4ned
in the following are planetocentric.
Table 6 gives the orientation of the planetary rotation systems for all major planets at epoch GEIJ 2000 and their change
with time. These are de4ned by the equatorial attitude (; )
of the rotation axis and the prime meridian angle w0 . Data
are taken from [S. Table 15:7] which is identical to the table given by Davies et al. (1996). The ascending node right
◦
ascensions are given by = + 90 . The respective transformation matrices are
◦
◦
˙ ( =T )
(38)
where the prime meridian angle w0 is given in the following list. Note that longitudes are counted left-handed (clockwise) from the prime meridian in the following Jovian systems:
• System I JUPI , mean atmospheric equatorial rotation
[S. Table 15:7]
◦
+Z-axis: pole of rotation. p-angle: w0I = 67 :1 +
◦
877 :900d0 .
◦
W0 ( )
Ẇ ( =d)
84.10
329.71
160.26
190.16
176.868
284.95
38.90
203.81
253.18
+14:1844000
+6:1385025
−1:4813596
+360:9856235
+350:8919830
+870:5360000
+810:7939024
−501:1600928
536.3128492
−0:48\sin N
236.77
−56:3623195
• System II JUPII , mean atmospheric polar rotation [S.
Table 15:7]
◦
+Z-axis: pole of rotation. p-angle: w0II = 43 :3 +
◦
870 :270d0 .
• System III JUPIII , magnetospheric rotation [S. Table
15:7]
◦
+Z-axis: pole of rotation. p-angle: w0III = 284 :95 +
◦
870 :536d0 .
◦
◦
Transform: T (GEIJ 2000 ; JUPIII ) = E( + 90 ; 90 −
; w0III ).
This is the 1965 de4nition of System III, the Pioneer
missions used the 1957 de4nition:
◦
w01957 = w0III + 106 :31209 + 0:0083169d0
which can be calculated from Eq. (7c) in Seidelmann and
Divine (1977) and was originally de4ned by the magnetospheric rotation period.
• System III 4x Sun Line
+Z-axis: pole of rotation. +Y -axis: cross-product of
+Z-axis and vector (Jupiter–Sun).
• Magnetic Dipole System JUPD (Dessler, 1983)
+Z-axis: dipole axis de4ned by its System III latitude
and longitude:
◦
◦
latD = (90 − 9 :8);
◦
D = 200 :
+X -axis: intersection of System III prime meridian and
magnetic equator.
◦
◦
Transform: T (JUPIII ; JUPD ) = E(D + 90 ; 9 :8; −D −
◦
90 ) (approximately).
• Centrifugal System JUPC (Dessler, 1983)
+Z-axis: centrifugal axis de4ned by its System III latitude
and longitude:
◦
◦
latC = (90 − 7 :0);
◦
C = 200 :
+X -axis: intersection of System III prime meridian and
centrifugal equator.
◦
◦
Transform: T (JUPIII ; JUPC ) = E(C + 90 ; 7 :0; −C −
◦
90 ) (approximately).
• Magnetic Dipole System 4x Sun line
+Z-axis: dipole axis. +Y -axis: cross-product of +Z-axis
and vector (Jupiter–Sun).
228
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
• Magnetic Dipole r System
+X -axis: vector (Jupiter-S=C) +Z-axis: (dipole axis)×+
X -axis. This system depends on the S=C-position.
5. Spacecraft elements
To determine approximate positions of spacecraft relative
to each other or to planets without using positional data 4les
it is useful to have orbital elements of spacecraft in Keplerian orbits. This excludes most near Earth missions since
their orbits are not Keplerian. In Table 7, we list orbital elements for most major interplanetary missions. We have 4tted these elements to trajectory data provided by NSSDC. 19
Not much accuracy is claimed by NSSDC for the propagated trajectories of any heliospheric spacecraft. But random
cross-comparison with published papers had revealed mis◦
matches of ¡ 0 :1 in angles or ¡ 1% in radial distance HEE
(R. Parthasarathy, pers. comm.). We used the vector method
given in Chapter 2 of Bate et al. (1971) to calculate initial
values for the elements which we then 4tted to achieve the
smallest maximal deviation from the position data. The deviations are listed in the last three columns of Table 7. The
◦
spatial resolution of the NSSDC position data is only 0:1
and the temporal resolution 1 day. This results in a poor precision of the orbital elements at perihelion speci4cally for
◦
the Helios mission where the spacecraft moves 8 =day. For
this reason, we re-calculated the Helios orbits by integration
from cartesian state vectors provided by JPL and then 4tted
elements to the re-calculated orbits. See also the JPL Voyager home page 20 for more Voyager orbital elements, and
the ESA Ulysses home page 21 for a discussion of Ulysses
orbital elements.
6. Summary
We have collected formulae relevant for the transformation between planetocentric and heliocentric coordinate systems and determined the precision of these transformations
for the period 1950 –2060, most relevant for space science.
We give a very short but complete description of orbit determination from Keplerian orbital elements. With the simple set of formulae given in this paper (and adapted from
Simon et al., 1994) the positions of the inner planets can
be determined to 160 though for the Earth this precision
can be increased to 29 . Adding disturbance terms from Simon et al. (1994) increases the precision to 8 . This sets
the limits for the precision to be achieved by a single set of
Keplerian elements. For higher precision, the installation of
an integrated ephemeris is recommended.
We also determined Keplerian orbital elements for major
interplanetary spacecraft within the precision limits given
by the NSSDC data source. These allow quick approximate
calculations of spacecraft positions and also allow to cross
check existing position data sets. Formulae given in this
Table 7
Heliocentric mean orbital elements of major interplanetary spacecraft in HAEJ 2000 4tted to data provided by NSSDCa
Mission
Period
a (AU )
Galileo
Galileo
Galileo
Helios1
Helios2
Pioneer10
Pioneer10
Pioneer11
Pioneer11
Pioneer11
Ulysses
Ulysses
Voyager1
Voyager1
Voyager1
Voyager2
Voyager2
Voyager2
Voyager2
Voyager2
1990:4 − 1990:9
0.982
1.572
1991:2 − 1992:8
3.113
1993:8 − 1996:0
0.6472
1977:0 − 1986:0
0.6374
1977:0 − 1981:0
3.438
1972:4 − 1973:9
1974:3 − 2005:0 −6:942
3.508
1973:5 − 1974:8
16.729
1975:0 − 1979:6
1979:7 − 2000:0 −8:059
9.035
1991:1 − 1992:1
3.375
1992:2 − 2005:0
5.020
1978:0 − 1979:1
1979:2 − 1980:8 −4:109
1980:9 − 2005:0 −3:203
3.624
1977:9 − 1979:4
1979:6 − 1981:6 −17:345
1981:7 − 1986:0 −3:913
1986:1 − 1989:3 −2:902
1990:7 − 2000:0 −4:021
◦
◦
e
( )
$( )
0.298
0.439
0.700
0.5216
0.5436
0.715
1.727
0.7166
0.7767
2.161
0.8905
0.6032
0.8009
2.258
3.742
0.7244
1.2905
3.4537
6.0618
6.2853
195:36 + 366:670y0
304:32 + 181:146y0
180:16 + 64:938y0
126:77 + 691:475y0
147:76 + 707:453y0
291:99 + 56:479y0
111:81 + 19:700y0
220:69 + 54:797y0
180:91 + 5:264y0
127:99 + 15:668y0
143:48 + 13:272y0
256:31 + 58:073y0
332:66 + 31:820y0
302:05 + 43:088y0
332:47 + 62:642y0
65:98 + 52:225y0
216:12 + 5:000y0
324:52 + 46:379y0
7:18 + 72:400y0
256:56 + 44:661y0
182.17
−240:47
−277:61
−101:84
294.58
160.02
−42:02
195.46
55.05
173.21
21.13
−22:93
−17:71
112.12
157.35
−20:65
110.80
189.87
−144:23
231.66
◦
i( )
◦
( )
3.39
76.51
4.57 −103:37
1.68 −105:39
0.004
70.18
0.024
121.85
2.08
−17:06
3.14
−28:57
3.05
16.64
15.29
−5:24
16.63
160.40
1.99
13.57
79.15
−21:85
0.93
−11:4
2.46
113.23
35.71
178.95
0.84
−33:03
2.58
120.05
2.66
77.65
2.81
−98:07
78.92
101.65
◦
◦
Pr (AU ) P ( ) P ( )
0.014
0.036
0.014
0.001
0.002
0.019
0.019
0.013
0.012
0.06
0.014
0.007
0.010
0.010
0.034
0.013
0.017
0.017
0.034
0.045
a The
1.2
0.7
0.8
0.39
0.89
0.18
0.02
0.10
0.25
0.23
0.07
0.92
0.44
0.23
0.10
0.13
0.10
0.12
0.12
0.13
0.10
0.09
0.09
0.016
0.004
0.07
0.007
0.06
0.20
0.12
0.06
0.50
0.12
0.06
0.11
0.06
0.06
0.05
0.12
0.10
elements are semi-major axis a, eccentricity e, mean longitude , longitude of periapsis $, inclination i, and ascending node . The
time parameter y0 is scaled in Julian years of 365.25 days from J2000.0, periods are given in decimal Julian years from J2000:0 + 2000:0: The
last three columns contain the precision of positions determined from the elements relative to NSSDC position data: maximal diJerence in HAE
distance, longitude and latitude over the period given.
20
19
NSSDC at http://nssdc.gsfc.nasa.gov/space/helios/heli.html.
21
Voyager at http://vraptor.jpl.nasa.gov.
Ulysses at http://helio.estec.esa.nl/ulysses/.
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
229
paper can easily be implemented in software. Programs used
in preparation of this paper have been written in the IDL
language and are available from our website. 22
Acknowledgements
We would like to thank D. Burgess of QMUL for helpful discussions and the encouragement to start this work,
J.-P. Simon of the Bureau de Longitude for providing data
and information on the VSOP87 system, R.S. Parthasarathy
of NSSDC for providing spacecraft position data and a discussion on Helios coordinate systems, K. Ackerson of the
University of Iowa, and T. Mukai of ISAS for a discussion
of Geotail coordinates, R. Skoug of LANL for a discussion
on ACE coordinates. We also would like to thank E. Wright
and C. Acton of JPL for information on the SPICE system
and the referees for very helpful comments and corrections.
This work was partially supported by PPARC grants GR=L
29903 and PPA=Z=R=1999=00606.
Appendix A.
Fig. 4. Eulerian rotation E(; 4; ,) (after Madelung, 1964): the transformation between system S(X; Y; Z) and system S (X ; Y ; Z ) can be
expressed by the three right-handed principal rotations: 1. ; Z around
the Z-axis towards the ascending node V, 2. #; X around the ascending
node axis, 3. ; Z around the Z -axis towards the +X -axis.
Note that all rotation matrices are orthogonal, s.t. E −1 = E T
and transformations between all systems de4ned in this paper can easily be calculated by a series of matrix multiplications.
A.1. Eulerian rotation
In this paper, we describe transformations between cartesian coordinate systems in Euclidean space. Let system S
be de4ned by the orthonormal right-handed basis vectors
X; Y; Z and system S by the orthonormal right-handed basis
vectors X ; Y ; Z with a common origin O. The position of
system S in system S is then de4ned by the angular coor◦
dinates of its pole (Z = (; + = − 90 )) and the prime
meridian angle (see Fig. 4) which is the angular distance
between prime meridian X and ascending node V. The Eulerian transformation matrix from S to S is then de4ned by
(Madelung, 1964)

cos cos − sin sin cos A.2. Velocity transformations
While position and magnetic 4eld vectors are independent
of the relative motion of the coordinate system this is not
true for other vectors for example for the solar wind velocity
vector. Usually, this vector is originally given in a spacecraft reference frame. For solar wind studies, it is advisable
to subtract the eJect of the spacecraft motion relative to a
heliocentric inertial system. If the spacecraft velocity vector
is not provided together with the positional data the velocity
can be calculated from the temporal derivative of the
cos sin + sin cos cos sin sin 


E(; ; ) =  −sin cos − cos sin cos −sin sin + cos cos cos cos sin  :
sin sin −cos sin (A:1)
cos Such that a vector v given in S has coordinates v = E ∗ v in
S . This corresponds to three principal rotations:
E = R3 () ∗ R1 () ∗ R3 () = ; Z ∗ ; X ∗ ; Z
(A.2)
in the notation of Hapgood (1992) where ‘∗’ denotes matrix
multiplication. The three principal rotations are on the other
hand given by
R1 ($) = $; X = E(0; $; 0);
◦
◦
R2 ($) = $; Y = E(90 ; $; −90 );
R3 ($) = $; Z = E(0; 0; $):
22
http://www.space-plasma.qmul.ac.uk/heliocoords/.
(A.3)
position time series. The velocity vector in the transformed
system is generally given by
v = Ėr + Ev − vc ;
(A.4)
where vc is the relative speed of the system origins and Ė is
the temporal derivative of the rotation matrix:
˙
Ė(; ; ) = A ∗ E ˙ + B ∗ E ˙ + E ∗ A;


0 1 0


A = Ṙ3 RT3 =  −1 0 0  ; B = R3 ()Ṙ1 RT1 RT3 ()
0 0 1
230

M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
cos2 cos sin sin 


=  −cos sin −sin2 cos  :
sin −cos (A.5)
0
System
For the transformation into planetocentric systems is
the only angle changing rapidly such that Ė(; ; ) ≈
˙
E ∗ A.
One of the most common transformations is the transformation from a heliocentric inertial system like HAED to a
◦
geocentric rotating system like GSED . Since ˙E ≈ 1 =day ≈
2 × 10−7 rad=s the rotational part of the velocity transformation can be neglected for geocentric distances of less
than 5 × 106 km to keep an accuracy of ≈ 1 km=s. In that
case, the transformation reduces to the subtraction of the
orbital velocity of the Earth which in the ecliptic system is
given by
◦
Table 8
Numerical example for a geocentric S=C position vector in diJerent
coordinate systemsa
◦
◦
vEHAE = v0 ∗ (cos(geo + 90 ); sin(geo + 90 ); 0 );
(A.6)
where v0 = 29:7859 km=s is the mean orbital velocity of the
Earth and geo the Earth longitude de4ned in Eq. (36).
A.3. Light aberration
For physical eJects which depend not on the geometric
relative position of two objects B1 ; B2 but on the apparent
position of B1 relative to B2 one has to take light travel into
account. The relativistic deZection of light by the Sun is only
larger than 1 for angular distances from the Sun of less then
◦
0 5 (see [S. Table 3:26:1]) and may be neglected for our
purposes. The change in position during the light travel time
(for example 20 between Sun and Earth) can be calculated
by iteration by determining the geometric position at time
t1 = t0 − R(t0 )=c where R(t0 ) is the distance between B1 and
B2 at t0 and c is the speed of light [S. 3:314–315]. The light
aberration is caused by the relative speeds of the observer B1
to the light coming from object B2 and the aberrated position
of B2 moving with relative speed v can be calculated by
r2 = r20 + Rv=c [S. 3.317].
Appendix B. Numerical example
In the following, we give a numerical example for the
application of some formulas given in the paper for comparison with software implementations. As pointed out in
the introduction all numerical values in this paper will be
available through our website.
B.1. Position transforms
We assume that a spacecraft position is given in true geographic coordinates (GEOT ) on the date Aug 28, 1996
16:46:00 UT (JD 2450324.19861111). Numerical results
GEOT
GEIT
GEID
HAED
HAEJ 2000
GEIJ 2000
HGCJ 2000
HEED
HEEQD
HCD
GSED
GSMD
SMD
MAGD
HGRTNE
X (RE )
6.9027400
−5:7864335
−5:7864918
−5:7864918
−5:7840451
−5:7840451
−5:4328785
−4:0378470
−4:4132668
−4:3379628
Y (Re )
Z (RE )
−1:6362400
−4:1039357
−4:1039136
−3:0028771
−3:0076174
−4:1082375
1.9166900
1.9166900
1.9165612
3.3908764
3.3908496
1.9146822
2.7493786
3.3908764
2.7496187
2.7496187
3.3908764
1.2681645
2.5733108
2.5732497
−3:2771992
4.1138243
−5:1182566
−5:1924440
5.2555187
5.1182566
6.0071917
6.0071917
6.0215108
5.1931904
4.0378470
4.0378470
3.3601371
3.3344557
4.0360303
a Positions are in geocentric cartesian coordinates in units of Earth
equatorial radii (RE =6378:14 km) for the date Aug 28, 1996 16:46:00
UT.
are given in Table 8 (We have chosen this date and position
because software by M. Hapgood (pers. comm.) uses these
values as a reference set.) The Julian century for this date
is T0 = −0:0334237204350195 (Eq. (2)). In the following
we apply the formulas of Section 3.3.3. To convert from
◦
GEOT to GEIT we calculate GMST = 228 :68095 by Eq.
(20). To convert from the true equator of date to the mean
equator of date, we have to apply the nutation matrix (Eq.
◦
◦
(7)) with 0D = 23 :439726; P = 0 :0011126098; P =
◦
−0 :0024222837. Then, we apply E(0; 0D ; 0) to transform to the mean ecliptic of date (HAED ), the precession
matrix (Eq. (9)) to transform to the ecliptic of J2000
(HAEJ 2000 ) and E(0; 0J 2000 ; 0) to transform to the equator of J2000 (GEIJ 2000 ). (The vector is still geocentric
since we did not apply a translation. So it might better be
called GAED , etc. but we stick with the H to avoid more
acronyms.)
We use T (GEIJ 2000 ; HGCJ 2000 ) and T (HAED ; HCD) of
Section 3.2.2 to transform to the heliographic systems.
To transform to geocentric Earth Ecliptic (HEED ) coordinates we use T (HAED ; HEED ) from Section 3.2.2, for
◦
HEEQD we use = 259 :89919 (Eq. (17)). To transform to GSED with low precision we use the ecliptic
◦
longitude of the Earth geo = −24 :302838 (Eq. (36)).
To transform to GSMD , we use the Earth dipole position
◦
◦
D = 288 :58158; ,D = 79 :411145 (Eq. (22)) and angles
◦
◦
D = −21 :604166; 9D = 20 :010247.
To proceed to position-dependent systems we, now determine the Earth position to a higher precision using the
orbital elements of the EMB from Table 4 corrected by
Table 6 of Simon et al. (1994) (values available on our
website):
a = 1:0000025;
e = 0:016710039;
◦
= −22 :769425;
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
231
Table 9
Heliocentric position and velocity vectors of the Earth and the Ulysses spacecraft on Jul 31, 1994 23:59 UT
Vector
Source
System
Units
X
Y
Z
rU N
vUN
rUP
vUP
rUJ
vUJ
rUB
vUB
rUS
vUS
rEMB
vEMB
rE
vE
rEJ
rEAA
NSSDC
NSSDC
Table 7
Table 7
Table 7
Table 7
Table 7
Table 7
SPICE
SPICE
Table 4
Table 4
Table 4
Table 4
Table 4
AstrAlm
GEIB1950
GEIB1950
HAEJ 2000
HAEJ 2000
GEIJ 2000
GEIJ 2000
GEIB1950
GEIB1950
GEIB1950
GEIB1950
HAEJ 2000
HAEJ 2000
HAEJ 2000
HAEJ 2000
GEIJ 2000
GEIJ 2000
km
km=s
km
km=s
km
km=s
km
km=s
km
km=s
km
km=s
km
km=s
AU
AU
−135 927 895:1
126 880 660.0
−8:287477214
−208 469 160
−6:2275842
125 262 820
−8:0959738
126 773 410
−8:3037482
126 877 772
−8:287645
−11865004
18.477
−11865391
18.471
−0:727 6995
−0:727 6944
−340 567 928:0
◦
$ = 102 :92657;
18.54622396
−134 999 360
118.624156
−134 999 360
18.624156
−135 247 550
18.546930
−135 922 227
18.546466
94751599
22.792
94748833
22.802
0.633 3568
0.633 3616
◦
i = −0 :00043635047;
= 174 :88123:
(B.1)
Using Eqs. (26) and (30) with 9E = 1=332946 (Table 4),
we get the EMB position in HAEJ 2000 :
◦
EMB = −24 :305587;
◦
EMB = −0 :00014340633;
rEMB = 1:0099340[AU]:
(B.2)
◦
Using the Delauney argument D = −184 :63320, we get
the Earth position in HAEJ 2000 (Eq. (32)):
◦
5.9889740
−341 330 080
3.0175855
−340 673 490
2.9272750
−340 564 861
2.894987
−1355
0.00025
−1355
0.00025
−0:315 5062
−0:315 5035
B.2. State vectors
◦
E = −24 :305442;
2.89468231
−362 990 910
rE = 1:0099033[AU]:
The position and velocity vectors (state vector, rUN ; vUN
in Table 9) of the Ulysses spacecraft which we used to
determine the orbital elements in Table 7 was provided by
NSSDC for the Julian date JD = 2449565:49930556 (Jul 31,
1994 23:59 UT) in heliocentric earth-equatorial coordinates
for epoch B1950 .
In the following, we describe how to derive the state vectors for Ulysses and Earth from the orbital elements for
this date and compare the values with the respective data of
the JPL SPICE system. The Julian century for this date is
T0 =−0:0541957753441315 (Eq. (2)). From Table 7 we take
the values for the orbital elements for Ulysses in HAEJ 2000 :
◦
◦
(B.3)
a = 3:375d;
= 256 :31 + 58 :073T0 100;
We apply the precession matrix (Eq. (9)) to get the Earth
position vector in HAED (1 AU = 149 597 870 km; 1RE =
6378:14 km):
e = 0:6032;
$ = −22 :93;
◦
= 79 :15;
◦
◦
i = −21 :84:
(B.6)
Using Eqs. (26) and (30) and 1 AU = 149 597 870 km we
calculate the HAEJ 2000 state vector (rUP ; vUP ).
ZE = 0:000016[RE ]:
(B.4)
This position is in agreement with the ecliptic position
available from the spacecraft Situation Center for day
Adding this vector to the geocentric position (GAED ) and
◦
◦
213,
1994: (HAEJ 2000 = 188 :8; HAEJ 2000 = −69 :4; r =
transforming to HCD we get the HCD longitude and latitude
2:59 AU). To compare this vector with the NSSDC
of the spacecraft:
value (rUN ; vUN ) we have 4rst to transform from the
◦
◦
,S=C = −100 :11050; 4S=C = 7 :1466473;
(B.5)
ecliptic HAEJ 2000 system to the equatorial GEIJ 2000 system using T (HAEJ 2000 ; GEIJ 2000 ) = E(0; −0 ; 0). Since,
from which we calculate the S=C-centered position vector
GEIB1950 refers to the orientation of the Earth equator at
of the Earth HGRTNE .
B1950 (T0B1950 = −0:50000210) we have to calculate the
precession matrix using Eq. (10):


0:99992571
0:011178938
0:0048590038


0:99993751
−2:7157926 · 10−5  :
(B:7)
P(0:0; B1950 ) =  −0:011178938
XE = 21579:585[RE ];
YE = −9767:205[RE ];
−0:0048590038 −2:7162595 · 10−5
0:99998819
232
M. Fr.anz, D. Harper / Planetary and Space Science 50 (2002) 217–233
Finally, we derive the Ulysses state vector in GEIB1950
(rUB ; vUB ).
The distance to the original NSSDC position (rUN ; vUN ) is
69 6790 km (0:0046 AU), the diJerence in velocity 36 m=s
in agreement with the precision cited in Table 7 for the
orbital elements. The respective position provided by the
JPL SPICE system is (rUS ; vUS ), which deviates by 7062 km
and 0:42 m=s from the NSSDC state vector.
Now, we calculate the HEIJ 2000 state vector of the Earth at
the same time. From Table 4 we get the undisturbed orbital
elements of the EMB:
a = 1:0000010;
◦
= −50 :547467;
e = 0:016710876;
◦
$ = 102 :91987;
◦
= 174 :88624;
◦
i = −0 :00070751501:
(B.8)
To increase precision we apply the disturbance corrections
by Table 6 of Simon et al. (1994) (values available on our
website) and get:
a = 0:99998900;
◦
= −50 :550224;
e = 0:016710912;
◦
$ = 102 :91987;
◦
◦
= 174 :88624;
i = −0 :00070754248:
(B.9)
Using Eqs. (26) and (30) with 9E = 1=332946 (Table 4), we
get the EMB state vector in HAEJ 2000 (rEMB ; vEMB ). Given
the low precision of the Ulysses position this would already
be good enough to get the geocentric Ulysses state vector
but to compare with SPICE data or the Astronomical Almanac, we now apply Eq. (32) to get the Earth state vector
in HAEJ 2000 (rE ; vE ), where we used the Delauney argument
◦
D = −73 :746062. Finally, we transform from HAEJ 2000 to
GEIJ 2000 using E(0; −0 ; 0) as above to get rEJ . which can
be compared with the value given in Section C22 of the Astronomical Almanac for 1994 (rEAA ,which agrees with the
value given by the SPICE system).
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