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 ). 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