ASTRONOMY & ASTROPHYSICS APRIL II 1998, PAGE 205
SUPPLEMENT SERIES
Astron. Astrophys. Suppl. Ser. 129, 205-217 (1998)
Galilean satellite ephemerides E5J.H. Lieske
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., MS 301-150 Pasadena, 91109 California,U.S.A.e-mail: [email protected]
Received May 26; accepted September 30, 1997
Abstract. New ephemerides of Jupiterβs Galilean satel-lites are produced from an analysis of CCD astrometricdata, Voyager-mission optical navigation images, mutualevent observations, photographic plates, and eclipse tim-ing observations. The resulting parameters, for use in thegalsat computer software, are in the B1950 frame for useby the Galileo space mission. Results in the J2000 systemare also available.
Key words: astrometry β celestial mechanics βephemerides β Planets and satellites: Jupiter
1. Introduction
This paper documents the Galilean satellite ephemeridesdesignated as E5, which were delivered in support of theGalileo space mission to Jupiter. The E5 ephemerides su-persede the E4 ephemerides, which were developed (Lieske1994a) without using CCD astrometric data in order toassess the new data type. It is believed that the E5ephemerides are better than the E3 and E4 ephemeridesand they are recommended for general usage. The param-eters of E5 are given in the B1950 system so that thegalsat software (Lieske 1977) can be employed directly tocompute coordinates in the B1950 frame, which has beenadopted for the Galileo mission.
The ephemerides E2 (Lieske 1980) were developedprior to the Voyager mission and were based solely on ananalysis of earth-based observations. The E2 ephemeridesutilized mutual event data from 1973 (Aksnes & Franklin1976), photographic astrometric observations from 1967-1978 (Pascu 1977 1979), and Jovian satellite eclipse tim-ings from 1878-1974 (Pickering 1907; Pierce 1974; Lieske1980).
Post-Voyager mission ephemeris improvements yieldedephemerides E3, which included Voyager optical naviga-tion astrometric data and Voyager-derived physical con-stants (Campbell & Synnott 1985). The E3 ephemerides
employed mutual event data from 1973 and 1979 (Aksneset al. 1984), Voyager optical navigation astrometric mea-surements from 1979 (Synnott et al. 1982), additional pho-tographic observations by D. Pascu from 1973-1979, andeclipse timings from 1652 to 1983 (Lieske 1986, 1987).
The initial pre-Galileo mission ephemerides were des-ignated E4 (Lieske 1994a) and included extended mutualevent data and photographic data, but no CCD observa-tions, since they were still in the process of being evalu-ated. The E4 ephemerides employed the previously men-tioned Voyager data, mutual event data from 1973 and1979 corrected for phase effects by adding Ξ΄t to the ob-servation time (Aksnes et al. 1986), photographic dataand Jovian eclipse timings, as well as additional mu-tual event astrometric measurements from 1985 and 1991(Aksnes et al. 1986; Franklin et al. 1991; Kaas et al.1997; Descamps 1994; Goguen et al. 1988; Goguen 1994;Mallama 1992), and additional photographic observationsfrom Pascu (1993) covering the interval 1980-1991. Three-yearsβ of CCD data from Flagstaff (Monet et al. 1994;Owen 1995) were evaluated, but not employed in develop-ing the E4 ephemerides.
The E5 ephemerides represent the most current evolu-tion of the Galilean satellite ephemerides and incorporateall of the above data types, including an evaluation theDoppler data of Ostro et al. (1992).
The 50 parameters which define the theory of mo-tion of the Galilean satellites (Lieske 1977) could also betransformed in a manner such that the same galsat com-puter program can be employed to compute rectangularcoordinates with their values being in the J2000 system.Documentation and an algorithm for such transformationof all galsat-related ephemerides (e.g., Lieske 1977, 1980;Arlot 1982; Vasundhara 1994) will be issued later. In themeantime the equatorial coordinates can be transformedin the following manner.
For the Galileo mission, all input quantities arein the B1950 frame and Earth equatorial coordinates
206 J.H. Lieske: Galilean satellite ephemerides E5
transformation from B1950 to J2000 when necessary isdone by the matrix multiplication
rJ2000 = ArB1950 , (1)
where the matrix A could be taken from that recom-mended by IAU Commission 20 (West 1992),
A = PIAUR3(β0β²β². 525) (2)
with PIAU being the standard IAU precession matrix fromB1950 to J2000 (Lieske 1979),
PIAU = R3(βzA)R2(ΞΈA)R3(βΞΆA) (3)
or A could be taken from the earlier discussion of Standish(1982), which was developed for transforming from DE118to DE200,
A = R3(+0β²β². 00073)PIAUR3(β0β²β². 53160) . (4)
It essentially consists of a rotation βE in the B1950 equa-torial plane from the FK4 origin to the dynamical equinoxand then precessing from B1950 to J2000 using the IAU1976 equatorial precession parameters PIAU (Lieske et al.1977).
The matrix A could also be derived from Lieskeβs dis-cussion (1994b) on the precession of orbital elements,
A = R1(βΞ΅J2000)R3(Lβ²)R1(βJA)R3(βL)R1(Ξ΅B1950) . (5)
For the Galileo mission, the method of Standish given inEq. (4) is employed to precess from B1950 to J2000.
The rotation matrices Ri are the standard matrices forrotations about the x, y, or z axes for i = 1, 2, 3:
R1 =
1 0 00 cos ΞΈ sin ΞΈ0 β sin ΞΈ cos ΞΈ
R2 =
cos ΞΈ 0 β sin ΞΈ0 1 0
sin ΞΈ 0 cos ΞΈ
(6)
R3 =
cos ΞΈ sin ΞΈ 0β sin ΞΈ cos ΞΈ 0
0 0 1
.
The various matrices mentioned in Eqs. (2), (4) and(5) are presented in Table 1. The maximum differencein satellite coordinates, due to the different precessionaltransformations, is about 1.5 km, so any of the previouslymentioned matrices could be used in a practical situation.
2. The basic parameters
In the galsat-type ephemerides, the Jovicentric Earth-equatorial coordinates of the Galilean satellites are com-puted as a function of 50 βgalsatβ parameters (Lieske
1977). The definitions of the basic parameters upon whichthe theory depends are given in Tables 2 and 3. It is seenthat they are a combination of physical parameters andorbital elements.
In the E5 ephemerides, we employed the satellitemasses (Ξ΅1 β Ξ΅4) and Jupiter pole which were determinedby Campbell & Synnott (1985) from their analysis of theVoyager data. The Jupiter pole is a function of the lon-gitude of the origin of the coordinates Ο [theory param-eter Ξ²15], and the inclination IJ of Jupiterβs equator toJupiterβs orbit [theory parameter Ξ΅25], with some depen-dence upon the Jupiter orbital inclination to the ecliptic[theory parameter Ξ΅26], Jupiterβs node Ξ©J [theory parame-ter Ξ²22], and the obliquity Ξ΅ of the ecliptic [theory param-eter Ξ΅27]. The mass of the Jupiter system was that of JPLephemeris DE140 (Standish & Folkner 1995) Sun/Jupiter-system = 1047.3486. Ephemerides E3 and E4 employedJupiter system masses which are consistent with JPLephemeris DE125 (Standish 1985), Sun/Jupiter-system= 1047.349. The Jupiter pole employed was Ξ±J = 268.001and Ξ΄J = 64.504 at the theory epoch JED 2443000.5 andin the B1950 frame. The rate of Ο [theory parameter Ξ²15]models the secular motion of Jupiterβs pole from the the-ory epoch. Jupiterβs oblateness parameters J2 and J4 werealso taken from the Campbell & Synnott analysis. Theycorrespond to theory parameters Ξ΅11 and Ξ΅12 in Table 2.
Over the years different tables of βT have been usedfor the calculation of Ephemeris Time (barycentric dy-namical time TDB) minus Universal Time. The appropri-ate table of βT values depends upon what model of theMoonβs tidal acceleration one adopts. The Earthβs Moonwas most often used to determine values of βT prior to1955 because of its rapid motion. The derived values of βTeffectively depend upon a partitioning into portions due tolunar tidal effects versus real changes in βT . It essentiallydepends upon the parameter employed to describe the lu-nar tidal acceleration nMoon. The classical determinationof nMoon = β22.44 arcsec/cy2 by Spencer Jones (1939)was employed for the E1 and E2 (Lieske 1980) ephemeridesby means of the Brouwer (1952) and Martin (1969) valuesof βT , which were on the Spencer Jones system.
The Morrison and Ward (1975) value of nMoon =β26.0 arcsec/cy2 was used for E3, E4 and E5. Tables ofβT given by Stephenson & Morrison (1984) can be ad-justed for any nMoon by the technique noted in Lieske(1987) for times prior to 1955.5 by computing
βT (nMoon) = βTMorrison β 0.911(nMoon + 26)T 20 sec (7)
where T0 is measured in centuries from the 1955.5 epochof Morrison (1980). The theory parameters of E1 andE2 are consistent with the Spencer-Jones value of nMoon,while those for E3 through E5 are consistent with that ofMorrison and Ward.
J.H. Lieske: Galilean satellite ephemerides E5 207
Table 1. Matrices for precession from B1950 to J2000
Eq. (2): Commission 20 matrix from PIAUR3(β0β²β². 525)
0.9999256794956877 β0.0111814832204662 β0.00048590038153590.0111814832391717 0.9999374848933135 β0.00002716259471420.0048590037723143 β0.0000271702937440 0.9999881946023742
Eq. (4): Standish matrix from R3(+0β²β². 00073)PIAUR3(-0β²β². 53160)
0.9999256791774783 β0.0111815116768724 β0.00485900381545530.0111815116959975 0.9999374845751042 β0.00002716257751750.0048590037714450 β0.0000271704492210 0.9999881946023742
Eq. (5): Lieske matrix from R1(βΞ΅J2000)R3(Lβ²)R1(βJA)R3(βL)R1(Ξ΅B1950)
0.9999256795268940 β0.0111810778339439 β0.00048599301590150.0111810775053504 0.9999374894281627 β0.00002723825033870.0048599309149990 β0.0000271030297995 0.9999881900987267
Table 2. Definition of theory parameters Ξ΅
Epsilon Parameter Generating value Description
1 m1 449.7 10β7(1 + Ξ΅1) Mass of Satellite I relative to Jupiter2 m2 252.9 10β7(1 + Ξ΅2) Mass of Satellite II relative to Jupiter3 m3 798.8 10β7(1 + Ξ΅3) Mass of Satellite III relative to Jupiter4 m4 450.4 10β7(1 + Ξ΅4) Mass of Satellite IV relative to Jupiter5 S/J 1047.355(1 + Ξ΅5) Mass of Sun relative to Jupiter
6 n1 203.48895 4208(1 + Ξ΅6) Mean motion of Satellite I, deg/day7 n2 101.37472 3445(1 + Ξ΅7) Mean motion of Satellite II, deg/day8 n4 21.57107 1403(1 + Ξ΅8) Mean motion of Satellite IV, deg/day9 Ξ»A 180Ξ΅9/Ο Amplitude of free libration, Ξ»A in deg, Ξ΅9 in rad10 nJ 8.30912 15712 10β2(1 + Ξ΅10) Mean motion of Jupiter, deg/day
11 J2 0.01484 85(1 + Ξ΅11) Jupiter J2
12 J4 β8.107 10β4(1 + Ξ΅12) Jupiter J4
13 RJ 71420(1 + Ξ΅13) Radius of Jupiter, km14 PJ 9.92482 5(1 + Ξ΅14) Period of Jupiter rotation, hr15 3(C βA)/2C 0.111(1 + Ξ΅15) Ratio of Jupiter moments of inertia
16 e11 465 10β7(1 + Ξ΅16) Primary eccentricity of Satellite I, rad17 e22 825 10β7(1 + Ξ΅17) Primary eccentricity of Satellite II, rad18 e33 15164 10β7(1 + Ξ΅18) Primary eccentricity of Satellite III, rad19 e44 73725 10β7(1 + Ξ΅19) Primary eccentricity of Satellite IV, rad20 eJ 0.04846 02472(1 + Ξ΅20) Eccentricity of Jupiter
21 c11 4756 10β7(1 + Ξ΅21) Primary sine inclination of Satellite I22 c22 81490 10β7(1 + Ξ΅22) Primary sine inclination of Satellite II23 c33 31108 10β7(1 + Ξ΅23) Primary sine inclination of Satellite III24 c44 47460 10β7(1 + Ξ΅24) Primary sine inclination of Satellite IV25 IJ 3.10401(1 + Ξ΅25) Inclination of Jupiter orbit to Jupiter equator, deg
26 J 1.30691(1 + Ξ΅26) Inclination of Jupiter orbit to ecliptic, deg27 Ξ΅ 2326β²44β²β². 84(1 + Ξ΅27) Inclination (Obliquity) of ecliptic to Earth equator deg28 nS 3.34597 33896 10β2(1 + Ξ΅28) Mean motion of Saturn, deg/day
208 J.H. Lieske: Galilean satellite ephemerides E5
Table 3. Definition of theory parameters Ξ²
Beta Parameter Epoch value (deg) Description
1 `1 106.03042 + Ξ²1 Mean longitude of Satellite I2 `2 175.74748 + Ξ²2 Mean longitude of Satellite II3 `3 [120.60601β 1
2Ξ²1 + 32Ξ²2] Mean longitude of Satellite III
4 `4 84.51861 + Ξ²4 Mean longitude of Satellite IV5 ΟΞ» Ξ²5 Free Libration Ο1 β 3Ο2 + 2Ο3 = Ο + Ξ΅9 sinΟΞ»
= 180 + Ξ»A sinΟΞ»
6 Ο1 4.51172 + Ξ²6 Proper periapse of Satellite I7 Ο2 74.53051 + Ξ²7 Proper periapse of Satellite II8 Ο3 174.85831 + Ξ²8 Proper periapse of Satellite III9 Ο4 336.02667 + Ξ²9 Proper periapse of Satellite IV10 Ξ J 13.30364 + Ξ²10 Longitude of perihelion of Jupiter
11 Ο1 242.73706 + Ξ²11 Proper node of Satellite I12 Ο2 95.28556 + Ξ²12 Proper node of Satellite II13 Ο3 125.14673 + Ξ²13 Proper node of Satellite III14 Ο4 317.89250 + Ξ²14 Proper node of Satellite IV15 Ο 316.73369 + Ξ²15 Longitude of origin of coordinates (Jupiterβs pole)
16 Gβ² 31.97852 80244 + Ξ²16 Mean anomaly of Saturn17 G 30.37841 20168 + Ξ²17 + Ξ΄G Mean anomaly of Jupiter18 Ο1 172.84(1β 0.014Ξ΅20) + Ξ²18 Phase angle in solar (A/R)
3with angle 2Gβ² βG
19 Ο2 47.03(1β 0.156Ξ΅20) + Ξ²19 Phase angle in solar (A/R)3
with angle 5Gβ² β 2G
20 Ο3 259.18 + Ξ²20 Phase angle in solar (A/R)3
with angle Gβ² βG
21 Ο4 157.12(1 + 0.0014Ξ΅20) + Ξ²21 Phase angle in solar (A/R)3 with angle 2Gβ² β 2G22 Ξ©J 99.95326 + Ξ²22 Longitude ascending node of Jupiterβs orbit on ecliptic
Beta Symbol Rate (deg/day) Description
1 Λ1 203.48895 4208(1 + Ξ΅6) Mean motion of Satellite I
2 Λ2 101.37472 3445(1 + Ξ΅7) Mean motion of Satellite II
3 Λ3 [50.31760 806351β 2Ξ΅6 + 3Ξ΅7 Mean motion of Satellite III
β0.02204 51849 7(Ξ΅6 β Ξ΅7)]4 Λ
4 21.57107 1403(1 + Ξ΅8) Mean motion of Satellite IV5 ΟΞ»
βL (= 0.1737 9190 + Β· Β· Β·) Rate of free libration (Fiche Table A.30)
6 Ο1 (0.1613 8586 + Β· Β· Β·) Proper periapse rate of Satellite I7 Ο2 (0.0472 6307 + Β· Β· Β·) Proper periapse rate of Satellite II8 Ο3 (0.0071 2734 + Β· Β· Β·) Proper periapse rate of Satellite III9 Ο4 (0.0018 4000 + Β· Β· Β·) Proper periapse rate of Satellite IV10 Ξ J 0
11 Ο1 (β0.1327 9386 + Β· Β· Β·) Proper node rate of Satellite I12 Ο2 (β0.0326 3064 + Β· Β· Β·) Proper node rate of Satellite II13 Ο3 (β0.0071 7703 + Β· Β· Β·) Proper node rate of Satellite III14 Ο4 (β0.0017 5934 + Β· Β· Β·) Proper node rate of Satellite IV15 Ο (β0.0000 0208 + Β· Β· Β·) Longitude of origin rate
16 Gβ² 3.34597 33896 Β· 10β2(1 + Ξ΅28) Mean motion of Saturn17 G 8.30912 15712 Β· 10β2(1 + Ξ΅10) Mean motion of Jupiter18. . .22 0
J.H. Lieske: Galilean satellite ephemerides E5 209
3. The observations
A variety of different observational data types were em-ployed in developing ephemerides E5. A new and verypowerful data type of CCD observations from the U.S.Naval Observatory Flagstaff Station was used for the firsttime, together with very accurate Voyager optical nav-igation data from 1979 and the mutual event observa-tions 1973-1991, photographic observations of D. Pascufrom 1967-1993 and Jovian eclipse timings from 1652-1983. Doppler observations from 1987-1991 were employedto assess the value of the Doppler data and evaluate theephemerides.
Table 4. Observational data employed for ephemeris E5
Data span observable type observ. % chg
1992-1994 CCD data, Flagstaff ra & dec 870 β52.61979 Voyager opnav ra & dec 366 β19.01973-1991 mutual events ra & dec 860 β55.51967-1993 photographic ra & dec 8462 β3.21652-1983 eclipse timings 15711 +2.71994 CCD data, Table Mountain 72 +68.31987-1991 Doppler 50 β55.6
By intercomparing various data types one learns of thestrengths and weaknesses of each individual type of dataand discovers inconsistencies among the data types. Thedata are described in Table 4, which also gives the per-centage change in weighted sum-of-squares for ephemerisE5 relative to ephemeris E3. A plus sign indicates anincrease and a minus sign indicates a decrease in theweighted residuals. The various data types were com-bined by weighting each observation by the reciprocalof its squared a priori standard deviation. A commondata set (including weights) was employed to evaluate allephemerides so that one can compare the relative merits ofa given ephemeris to a common data set. Thus, althoughno CCD observations were employed in the developmentof ephemeris E3, the residuals of the CCD data employedin this paper are also given for ephemeris E3 so that thereader can make meaningful comparisons.
In order to more closely compare the variousephemerides with the different data types, we present inTable 5 the residuals of unit weight for each data typefor the different ephemerides E2 through E5 by Lieske,as well as for the Bureau des Longitudesβ ephemeris G5(Arlot 1982). In comparing Table 4 with Table 5 it shouldbe remembered that Table 4 is related to the square ofthe residuals while Table 5 employs the square root of thesum-of-squares. The comparison for Flagstaff CCD data,for example, for Table 5 would indicate that the Table 4entry should be about (29/43)2 β 1 = β54.5% for the E5vs E3 comparison.
Table 5. Observational rms residuals for various ephemerides
Observable type E2 G5 E3 E4 E5
CCD Flagstaff, mas 43 40 43 32 29Voyager opnav, mas 1309 1334 929 904 820mutual events, mas 62 53 62 47 46photographic, mas 107 106 106 104 104eclipse timings, sec 55.5 74.5 53.2 53.9 53.9CCD Table Mtn, mas 43 73 41 52 53Doppler, Hz 15.3 18.4 13.7 11.7 11.9
3.1. CCD observations
The new CCD observations were made at the U.S. NavalObservatory Flagstaff Station (A. Monet et al. 1994) dur-ing the years 1993-1995, employing techniques developedby D. Monet and described in Monet et al. (1992) and inMonet & Monet (1992). The Flagstaff data were processedat JPL by W. Owen who produced normal-point residuals,typically from 30β50 CCD βexposuresβ, for the author us-ing ephemeris E3. Those residuals were then employed bythe author to generate pseudo-observable βnormal-pointobservationsβ by adding the residual to an artificially-constructed computed position at the mean time of theCCD exposures using the same ephemeris which was em-ployed in computing the CCD residuals. Such a βnormalpoint observationβ could be employed with other astro-metric data in an analysis of the observations, and shouldrepresent a valid description of the actual CCD observa-tions. Additionally, the pseudo-observations will serve thepurpose of archiving the CCD observations in convenientform. In processing the CCD data Owen would estimatethe pointing and orientation parameters and employ a sin-gle telescope scale factor (modified for refraction and at-mospheric effects) for all the Flagstaff data and he woulduse a single ephemeris (viz. E3) which was not adjustedin the reduction process. If that procedure is valid, thenthe pseudo-observables generated should behave like validobservational data, viz. the residuals should decrease ifone employs a better ephemeris with the original pseudo-observables. It was for this reason that ephemeris E3 wasintentionally employed β it was known to need some cor-rection and we desired to explore the validity of the pro-cess of constructing normal point pseudo-observables. Ifthe normal points were constructed instead on a differentephemeris, then the pseudo-observables differed by lessthan 15 km (0β²β².005) from those generated via ephemerisE3, even though the residuals might actually be signifi-cantly different using the two ephemerides. That 15-kmreproducibility of the normal points is a good indicationof the intrinsic accuracy of the CCD data.
Some less-accurate CCD data from the JPL TableMountain Facility (Owen 1995) were also employed, al-though with hindsight they probably should not have been
210 J.H. Lieske: Galilean satellite ephemerides E5
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Fig. 1. Residuals in right ascension (left) and declination (right) for Flagstaff CCD observations relative to Satellite 1 usingephemeris E5. The observations of Europa relative to Io are indicated by a , those of Ganymede by a 2, and those of Callistoby a
included in developing E5. They did not exhibit the reduc-tion of residuals with a better ephemeris, and that is be-lieved to be due to the fact that there were too few TableMountain data to adequately separate the orbital effectsfrom the telescope effects.
The CCD data were processed using Lambert scat-tering to compute the offset between the center of lightand center of figure (Lindegren 1977) and it is believedthat the dominant remaining unmodeled error source inthese data is due to albedo variations across the disk ofthe satellites. Recent estimates of the albedo variationsby several scientists (Goguen 1994; Mallama 1993; Riedel1994; Gaskell 1995) are not entirely consistent and for theGalileo-mission ephemerides it was decided to limit theprocessing to computation of the difference between cen-ter of light and center of figure due to Lambert scatter-ing only, since it represents a reasonable first approxima-tion to the scattering properties of the satellites if one ex-cludes albedo variations (viz., effects which depend uponfeatures on the satellites and which vary with planetocen-tric longitude of the central disc). The extrapolation ofVoyager-derived scattering properties (which occurred athigh phase angle) to the scattering properties of the satel-lites at low phase angle as observed from the Earth is notentirely satisfactory and the several efforts done to dateare not entirely consistent with one another. It is hopedthat some series of observations made from the HubbleSpace Telescope will resolve the problems. Employmentof Lambert scattering is a useful first-approximation. Thedifferences between Lambert, Minnaert or Hapke scatter-ing laws is minor compared to the albedo variations in-troduced by physical features on the satellites, which mayintroduce center-of-light relative to center-of-figure varia-tions on the order of 75β 100 km.
The Flagstaff CCD data were weighted using a stan-dard deviation of 0β²β².03, which corresponds to about 90 kmfor these earth-based observations. The Table Mountain
data were weighted using a standard deviation of 0β²β². 05,corresponding to about 150 km.
3.2. Voyager optical navigation data
During the Voyager mission in 1979, some optical nav-igation images of the Jovian satellites were taken fromthe spacecraft for use in navigating the spacecraft tothe Jovian encounter. We have 183 observations of theJovian satellites in right ascension and in declination,made during the Voyager I and Voyager II encounters(Synnott et al. 1982). The optical navigation images areanalogous to earth-based astrometric observations of thesatellites except that the βopnavβ images are taken byan βobserverβ much closer to the Jovian system (typ-ically 13 β 95 light seconds from the satellites). At 5106 km from Jupiter, one arcsec corresponds approxi-mately to 25 km. Additionally, the spacecraft-based ob-servations are the result of analyzing extended satelliteimages. By inferring the center of the satellite from ob-servations of the limb, the Voyager data do not have thecenter-of-light vs center-of-figure problems which are com-mon to disk-integrated images such as those contained inCCD observations and photographic plates and mutualevents. The Voyager data were weighted using a stan-dard deviation of 1β²β². 0 (as seen at the spacecraftβs dis-tance from Jupiter). For spacecraft-to-satellite distances of13β 95 light seconds, the 1β²β². 0 corresponds to 19 and 140km respectively for these spacecraft-based observations.The Voyager optical navigation residuals on ephemeris E5are depicted for right ascension and declination in Fig. 2.
3.3. Mutual event astrometric data
Since 1973 there have been successful campaigns to ob-serve the mutual event seasons every six years, when theJovian satellites eclipse and occult one another as the Sunand the Earth pass through the plane of the Jovian equa-tor, in which the satellite orbits lie. Aksnes and colleagues
J.H. Lieske: Galilean satellite ephemerides E5 211
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Fig. 2. Residuals in right ascension (left) and declination (right) for the Voyager optical navigation observations using ephemerisE5. The ordinate is in arcsec with an approximate corresponding linear distance scale on the right. Jupiter-relative observationsof Io are indicated by , Europa by 2, Ganymede by 4, and Callisto by
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Fig. 3. Residuals in right ascension (left) and declination (right) for astrometric mutual event observations using ephemeris E5.The ordinate is in arcsec with an approximate corresponding linear distance scale on the right
(Aksnes 1974, 1984; Aksnes & Franklin 1978, 1990), alongwith Arlot and colleagues (Arlot 1978, 1984, 1990, 1996),have made predictions of such mutual events available toscientists throughout the world and have organized scien-tific programs to observe the mutual events. Aksnesβ teamhas produced astrometric separations of the satellites, attimes near the mid-event times, which are very useful forephemeris development purposes.
The early Galilean satellite ephemerides E1 and E2(Lieske 1980) employed the Aksnes data from 1973(Aksnes & Franklin 1976) and 1979 (Aksnes et al. 1984)and were affected by the phase offsets between eclipsesand occultations which led Aksnes et al. (1986) to recom-mend that Ξ΄t be added to the published observation timesfor the 1973 and 1979 data. The ephemerides E3 weregenerated using the recommended additions of Ξ΄t to theobservation times in processing the 1973 and 1979 mutualevents astrometric data.
In the processing of mutual event observations by theAksnes team in 1985 (Franklin et al. 1991) and 1991 (Kaaset al. 1997), it was intended that no value of Ξ΄t would berequired but that instead the authors would incorporatethe phase effects into their published times and separa-
tions. However, the effects were added in the incorrectdirection for the published data and hence it is recom-mended (Aksnes 1993; Franklin 1993; Lieske 1995) thatthe 1985 and 1991 Aksnes data be employed by addingtwice the published values of the Ξ΄t phase corrections tothe observation times. Essentially the first addition of Ξ΄tremoves the erroneous application of the phase effects withthe incorrect sign and the second application of Ξ΄t actu-ally corrects for the phase problem. Additionally, someinfra-red astrometric mutual event separations were ob-tained from Goguen et al. (1988) in 1985 as well as in1991 (Goguen 1994). Astrometric separations from the1991 mutual event season which were employed in the de-velopment of E5 were also published by Mallama (1992a),Spencer (1993) and by Descamps (1994).
The mutual event data were weighted using standarddeviations of 0β²β². 020 to 0β²β². 045, which corresponds to 60 kmand 140 respectively for these earth-based observations.The typical weight corresponds to a standard deviation of0β²β².030 or 90 km.
The obvious offset in right ascension residuals for the1991 mutual event season depicted in Fig. 3 is believed notto be due to ephemeris errors, but rather is due to albedo
212 J.H. Lieske: Galilean satellite ephemerides E5
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Fig. 4. Residuals in right ascension (left) and declination (right) for photographic observations relative to Io using ephemerisE5. The residuals for exposures of a given satellite on each plate have been combined to produce a normal point for each plate.Observations of Europa relative to Io are indicated by a , those of Ganymede by a 2 and those of Callisto by a
effects since almost all of the 1991 mutual event observa-tions involved Io and were made at comparable longitudeson the satellite disk. The CCD and photographic data, forexample, show no such offset and those data were sampledat various longitudes.
3.4. Photographic observations
The long and valuable series of photographic observa-tions made by D. Pascu of the U.S. Naval Observatoryhave been an essential ingredient of the Galilean satel-lite ephemerides since the first development of the Galsatsoftware. In an extended series of observations 1967β1993,Pascu (1977, 1979, 1993, 1994) provided astrometric ob-servations of the satellites. He pioneered the developmentof neutral density filters to enable the accurate obser-vation of the Galilean satellites on a regular basis. ThePascu data were reduced using a single scale factor (mod-ified by adjustments for refraction for each observation)for the ensemble of observations, as determined by Pascu.Additionally, a correction to the Pascu scale was appliedfor a refraction-related effect, amounting to a relativechange in scale of β58β²β². 2/206265, which probably resultedfrom the manner in which the plate scale was originallydetermined.
The photographic data from 1967 through 1975 wereweighted using a standard deviation of 0β²β². 13 per expo-sure, while those from 1976 onwards were weighted usinga standard deviation of 0β²β². 09 per exposure, correspondingto position uncertainties of 400 km and 275 km, respec-tively, for each exposure. A photographic plate typicallyconsisted of 4 exposures of each satellite.
The residuals on E5 for photographic observations areplotted in Fig. 4. In the figure, normal-point residuals arepresented for each photographic plate, in order to makethe comparison with the normal-point CCD observationsmore feasible. In the plots, the residuals for all exposuresof a given satellite on a single plate are averaged into asingle normal-point residual.
3.5. Jupiter eclipse timings
The Jovian eclipse timings, representing the classical ob-servations of the Galilean satellites back to the 17th cen-tury, were discussed in Lieske (1986a,b). The early dataare from the Pingre 17th century collection later pub-lished by Bigourdan (1901), and from the Delisle collec-tion (Bigourdan 1897). The book on 17th century as-tronomy by Pingre published by Bigourdan was origi-nally scheduled for publication 100 years earlier by Pingre.But Pingreβs death and the French revolution intervened,and the printerβs proof copies were destroyed as scrappaper. It was only 100 years later that a copy of theproofs was found and ultimately published by the ParisAcademy. The manuscript collection of J.-N. Delisle con-tains a wealth of historically and scientifically interestingobservations of Galilean satellite eclipses. These two col-lections effectively re-construct the βlostβ Delambre col-lection.
We employed satellite radii of 1815, 1569, 2631 and2400 km for Io through Callisto, respectively (Davies et al.1985), in reducing the eclipse timings.
Additionally, the series of eclipse observations byPickering from 1878-1903 (Pickering 1907) and those ac-cumulated by Pierce (1974), together with those of manyamateur astronomers, especially those coordinated byB. Loader and J. Westfall, were employed. Finally, a feweclipse timings by Mallama (1992b) taken in 1990β91 wereanalyzed.
The eclipse timing data were employed with averagestandard deviations between 44 s for Io and 150 s forCallisto with a mean of 63 s, which correspond to posi-tion uncertainties of 775 km for Io, 1225 km for Callisto,and 800 km on the average for all satellites. The residualsappear visually similar to those depicted in Lieske (1986a)and therefore they are not presented here again.
J.H. Lieske: Galilean satellite ephemerides E5 213
Table 6. Values of theory parameters Ξ΅ and Ξ² for E5 in B1950 frame [see also Tables 2 and 3]
Parameter Related to Value Parameter Related to Value
Ξ΅1 m1 0.046323 (Β±0.000813) Ξ΅26 J β0.000137 (Β±0.000117)Ξ΅2 m2 β0.000906 (Β±0.001394) Ξ΅27 Ξ΅ 0.000000 (Β±0.000004)Ξ΅3 m3 β0.022997 (Β±0.000276) Ξ΅28 nS 0.000000 (Β±0.000001)Ξ΅4 m4 0.258508 (Β±0.000537) Ξ²1 `1 0.046767 (Β±0.00218)Ξ΅5 S/J 2009.3457E β 07 (Β±8.12E β 07) Ξ²2 `2 β0.015865 (Β±0.000835)
Ξ΅6 n1 7.7760E β 09 (Β±0.549E β 09) Ξ²3 `3 [= β 12Ξ²1 + 3
2Ξ²2]Ξ΅7 n2 12.7230E β 09 (Β±1.04E β 09) Ξ²4 `4 β0.074023 (Β±0.001950)Ξ΅8 n4 β10.4916E β 09 (Β±4.90E β 09) Ξ²5 ΟΞ» 199.676608 (Β±1.57)Ξ΅9 Ξ»A 11.2104E β 04 (Β±0.391E β 04) rad Ξ²6 Ο1 92.576366 (Β±19.9)Ξ΅10 nJ 1.63E β 05 (Β±0.13E β 05) Ξ²7 Ο2 80.335825 (Β±1.35)
Ξ΅11 J2 β0.007576 (Β±0.000066) Ξ²8 Ο3 13.325727 (Β±0.150)Ξ΅12 J4 β0.275934 (Β±0.00631) Ξ²9 Ο4 β0.739863 (Β±0.0152)Ξ΅13 RJ β0.000308 (Β±0.000057) Ξ²10 Ξ J 0.166302 (Β±0.00344)Ξ΅14 PJ 9.5E β 06 (Β±102.E β 06) Ξ²11 Ο1 69.597506 (Β±0.788000)Ξ΅15 3(C βA)/2C β0.170000 (Β±0.0676) Ξ²12 Ο2 5.155556 (Β±0.0495)
Ξ΅16 e11 β0.995346 (Β±0.0291) Ξ²13 Ο3 β5.952489 (Β±0.101)Ξ΅17 e22 0.748031 (Β±0.0221) Ξ²14 Ο4 4.726133 (Β±0.0772)Ξ΅18 e33 β0.051182 (Β±0.00167) Ξ²15 Ο β0.215487 (Β±0.00545)Ξ΅19 e44 β0.002434 (Β±0.000324) Ξ²16 Gβ² 0.000000 (Β±0.407)Ξ΅20 eJ 0.002750 (Β±0.000081) Ξ²17 G β0.140855 (Β±0.00279)
Ξ΅21 c11 0.344275 (Β±0.0196) Ξ²18 Ο1 15.541000 (Β±0.411)Ξ΅22 c22 β0.005970 (Β±0.000872) Ξ²19 Ο2 5.215000 (Β±0.469)Ξ΅23 c33 0.041611 (Β±0.00199) Ξ²20 Ο3 β1.996000 (Β±0.757)Ξ΅24 c44 β0.070074 (Β±0.000810) Ξ²21 Ο4 β7.968000 (Β±0.293)Ξ΅25 IJ 0.005110 (Β±0.000079) Ξ²22 Ξ©J 0.045266 (Β±0.00664)
3.6. Doppler data
The Doppler observations discussed by Ostro et al. (1992)were employed to evaluate the ephemerides and explorethe potential of Doppler data, but they were not includedin analysis and the development of E5. The data are con-sistent with the observations which were analyzed, butthey were not included in the analysis because of possi-ble uncertainty in the radar scattering properties of thesatellites similar to albedo effects which depend upon theplanetocentric longitude. The 50 Doppler observations ofthe outer three Galilean satellites were made between 1987and 1991.
The Doppler data were weighted using standard de-viations of 19 Hz for Europa, 12 Hz for Ganymede and10 Hz for Callisto for the Arecibo 13-cm S-band systemdata.
4. Discussion
The theory parameters which result from the analysis ofthese data are listed in Table 6, which will produce co-ordinates in the B1950 frame when used with the galsatsoftware. A future paper will document how they, and any
other set of galsat parameters, can be transformed to theJ2000 system in a manner such that the galsat softwarewill directly produce J2000 coordinates. In Table 6, theuncertainties listed for the Ξ΅ and Ξ² parameters are theformal errors obtained in the estimation process. By com-paring the coordinates of ephemerides E3 with those ofE5 and interpreting those differences to represent a 1-Οerror, we obtain a scale factor which should be appliedfor the formal uncertainties listed in the table. That scalefactor ranges between 2.5 and 3, so we recommend thatthe formal errors be multiplied by 3. The derived valuesof the angular variables for E5 are given in Table 7. Theseries coefficients for satellite coordinates ΞΎ, Ο and ΞΆ aresummarized in Table 8 for the E5 ephemerides.
Representing the Jupiter-equatorial projection of theorbital radius by Ο, and the true and mean longitudesby Ξ½ and `, respectively, then the equatorial radial com-ponent ΞΎ = (Ο β a)/a consists of cosine terms ΞΎ(t) =Ξ£K1 cos Ξ1(t), while the longitude component Ο = Ξ½ β `consists of sine terms Ο (t) = Ξ£K2 sin Ξ2(t), and the lat-itude component ΞΆ = z/a consists of sine terms ΞΆ(Ο) =
214 J.H. Lieske: Galilean satellite ephemerides E5
Table 7. Derived variables for ephemeris E5
Index Variable Value (deg) Rate (deg/day)
1 `1 106.077187 203.488955790332 `2 175.731615 101.374724734793 `3 120.558829 50.317609207024 `4 84.444587 21.571071176685 ΟΞ» 199.676608 0.17379190461
6 Ο1 97.088086 0.161385861447 Ο2 154.866335 0.047263066098 Ο3 188.184037 0.007127339499 Ο4 335.286807 0.0018399963710 Ξ J 13.469942 0.
11 Ο1 312.334566 β0.1327938594012 Ο2 100.441116 β0.0326306373113 Ο3 119.194241 β0.0071770315514 Ο4 322.618633 β0.0017593388015 Ο 316.518203 β2.08362 Β· 10β6
16 Gβ² 31.978528 0.0334597339017 G 30.237557 0.0830925701018 Ο1 188.374346 0.19 Ο2 52.224824 0.20 Ο3 257.184000 0.
21 Ο4 149.152605 0.22 Ξ©J 99.998526 0.
a1 2.819353 Β· 10β3 a.u.a2 4.485883 Β· 10β3 a.u.a3 7.155366 Β· 10β3 a.u.a4 12.585464 Β· 10β3 a.u.
Ξ£K3 sin Ξ3(Ο). As developed by Sampson (1921, pp. 229β230), the βtime-completedβ Ο may be defined as
Ο = t+ Ο /n, (8)
where t is βephemeris timeβ (TDB). One can employ thetime-completed to compute the latitude quantity s(t) =z/Ο from the shorter series for ΞΆ(t) = z/a via the relation-ship s(t) = ΞΆ(t+Ο /n). It effectively amounts to calculatingthe latitude perturbations as a function of true longituderather than as a function of mean longitude.
The Jupiter equatorial coordinates r = (x, y, z)T arecomputed from the orbital components ΞΎ, Ο , ΞΆ using theequations
x = a(1 + ΞΎ) cos(`β Ο + Ο )y = a(1 + ΞΎ) sin(`β Ο + Ο )z = a(1 + ΞΎ)s. (9)
The Earth-equatorial coordinates r = (x, y, z)T are thencomputed from the Jupiter-equatorial coordinates via therotation matrices
r = R1(βΞ΅)R3(βΞ©)R1(βJ)R3(βΟ + Ξ©)R1(βI)r. (10)
It is these Earth-equatorial coordinates r that are pro-vided by the galsat software.
As described in Theory, the Earth-equatorial coordi-nates are constructed from the series for ΞΎ, Ο and ΞΆ by therelationship
ΞΎ(t) = Ξ£K1 cos Ξ1(t)Ο (t) = Ξ£K2 sin Ξ2(t)
s(t) = ΞΆ(Ο) = Ξ£K3 sin Ξ3(Ο) (11)
where the right-hand sides are the result of computing theseries given in Table 8. The third equation for s(t) employsthe time-completed Ο = t+ Ο /n to evaluate the series forΞΆ(Ο) and thus to obtain s(t).
The adjustable parameters Ξ΅ and Ξ² for ephemeridesE5 in the B1950 frame are given in Table 6. The derivedvalues of the angular variables for E5 are given in Table 7.
Acknowledgements. This paper represents the results of onephase of research conducted at the Jet Propulsion Laboratory,California Institute of Technology, under contract with theNational Aeronautics and Space Administration. The CCDobservations were made by D. and A. Monet of the USNOFlagstaff Station and were processed into right-ascension anddeclination normal-point residuals on a fixed ephemeris byW.M. Owen Jr at JPL.
References
Aksnes K., 1974, Icarus 21, 100Aksnes K., Franklin F., 1976, AJ 81, 464Aksnes K., Franklin F., 1978, Icarus 34, 188Aksnes K., 1984, Icarus 60, 180Aksnes K., Franklin F., Millis R., et al., 1984, AJ 89, 28; AJ
89, 1081Aksnes K., Franklin F., Magnusson P., 1986, AJ 92, 1436Aksnes K., Franklin F., 1990, Icarus 84, 542Aksnes K., 1993 (personal communication)Arlot J.-E., 1978, A&AS 34, 195Arlot J.-E., 1982, A&A 107, 305Arlot J.-E., 1984, A&A 138, 113Arlot J.-E., 1990, A&A 237, 259Arlot J.-E., 1996, A&A 314, 312Bigourdan G., 1897, βInventaire general et sommaire des
manuscrits de la bibliotheque de lβobservatoire Parisβ Ann.Obs. Paris 21, F1-F60. [Delisle manuscripts are filed underthe heading Manuscripts A-5-1 through A-5-8]
Bigourdan G., 1901, A.-G. Pingre: Annales Celestes du dix-septieme siecle. Paris: Gauthier-Villars
Brouwer D., 1952, AJ 57, 126Campbell J.K., Synnott S.P., 1985, AJ 90, 364Davies M.E., Abalakin V.K., Bursa M., et al., 1986, Celest.
Mech. 39, 103
J.H. Lieske: Galilean satellite ephemerides E5 215
Descamps P., 1994, A&A 291, 664Franklin F., et al., βGalilean Satellite Observersβ, 1991, AJ
102, 806Franklin F., 1993 (personal communication)Gaskell R.W., 1995 (personal communication)Goguen J.D., Sinton W.M., Matson D.L., et al., 1988, Icarus
76, 465Goguen J.D., 1994 (personal communication)Kaas A.A., Franklin F., Aksnes K., Lieske J.H., 1997, βMutual
phenomena of the Galilean satellites 1990-1991β, AJ (inpress)
Lieske J.H., 1977, A&A 56, 333 (referred to as Theory)Lieske J.H., Lederle T., Fricke W., Morando B., 1977, A&A
58, 1Lieske J.H., 1979, A&A 73, 28Lieske J.H., 1980, A&A 82, 340 [referred to as E2 ephemerides]Lieske J.H., 1986a, A&A 154, 61Lieske J.H., 1986b, A&AS 63, 143Lieske J.H., 1987, A&A 176, 146Lieske J.H., 1994a, βGalilean Satellite Ephemerides E4β JPL
Engineering Memorandum 314-545 (19 June 1994) (JPL in-ternal publication)
Lieske J.H., 1994b, A&A 281, 281Lieske J.H., 1995, Bull. AAS 27, 1197Lindegren L., 1977, A&A 57, 55Mallama A., 1992a, Icarus 95, 309Mallama A., 1992b, Icarus 97, 298Mallama A., 1993, J. Geophys. Res. 98, p. 18.873-18.876Martin C.F., 1969, Ph.D. Diss., Yale Univ.Monet A.K.B., Stone R.C., Monet D.G., et al., 1994, AJ 107,
2290Monet D.G., Monet A.K.B., 1992, βGalilean satellite astrom-
etryβ, U.S. Naval Observatory Flagstaff Station memoran-dum
Monet D.G., Dahn C.C., Vrba F.J., et al., 1992, AJ 103, 638
Morrison L.V., Ward C.G., 1975, MNRAS 173, 183Morrison L.V., 1980 (personal communication)Ostro S.J., Campbell D.B., Simpson R.A., et al., 1992, J.
Geophys. Res. 97, p. 18.277Owen W.M., 1993 (personal communication)Owen W.M., 1995 (personal communication)Pascu D., 1977, in: Planetary Satellites, Burns J.A. (ed.).
University of Arizona Press, Tucson, p. 63Pascu D., 1979, in: Natural and Artificial Satellite Motion
Nacozy P.E., Ferraz-Mello S. (eds.). University of Texas,Austin, p. 17
Pascu D., 1993 (personal communication)Pascu D., 1994, in: Galactic and Solar System Optical
Astrometry, Morrison L.V., Gilmore G.F. (eds.). CambridgeUniversity Press, p. 304
Pierce D.A., 1980, βObservations of Jupiterβs Satellitesβ, JPLEngineering Memorandum 900-672
Pickering E.C., 1907, Harvard Ann. 52, Part I, 1Riedel J.E., 1994 (personal communication)Sampson R.A., 1921, MRAS 63 (Sampson Theory)Spencer J., 1993 (personal communication)Spencer Jones H., 1939, MNRAS 99, 541Standish E.M., 1982, A&A 114, 297Standish E.M., 1985, βJPL Planetary Ephemerides DE125β,
JPL IOM 314.6-591Standish E.M., Folkner W.M., 1995, βJPL Ephemerides DE400
and DE140β, JPL IOM 314.10-109Stephenson F.R., Morrison L.V., 1984, Phil. Trans. R. Soc.
London, Ser. A, 313, 47Synnott S.P., Donegan A.J., Morabito L.A., 1982, βPosition
Observations of the Galilean Satellites from Voyager Dataβ,Jet Propulsion Laboratory internal document
Vasundhara R., 1994, A&A 281, 565West R.M, 1992, in: Trans IAU XXIB, Bergeron J. (ed.).
Kluwer, Dordrecht, p. 211
216 J.H. Lieske: Galilean satellite ephemerides E5
Table 8. Series coefficients for E5
Index E5 Argument Ratio n/nsat
ββββ XI-1: Series coefficients for ΞΎ1 = (Ο1 β a1)/a1 (cosine) ββββ
1 170 `1 β `2 .501817072 106 `1 β `3 .752725603 β2 `1 β Ο1 .999206914 β2 `1 β Ο2 .999767745 β387 `1 β Ο3 .999964976 β214 `1 β Ο4 .999990967 β66 `1 + Ο3 β 2Ξ J β 2G .999218358 β41339 2`1 β 2`2 1.003634139 3 2`1 β 2`3 1.50545120
10 β131 4`1 β 4`2 2.00726827
ββββ V-1: Series coefficients for Ο 1 = Ξ½1 β `1 (sine) ββββ
1 β26 β2Ξ J + 2Ο β 2G β.000816702 β553 β2Ξ J + 2Ο β.000000023 β240 β2Ξ J + Ο3 + Ο β 2G β.000851964 92 βΟ2 + Ο .000160355 β72 βΟ3 + Ο .000035266 β49 βΟ4 + Ο .000008647 β325 G .000408348 65 2G .000816689 β33 5Gβ² β 2G+ Ο2 .00000547
10 β27 Ο3 β Ο4 β.0000266211 145 Ο2 β Ο3 β.0001250912 30 Ο2 β Ο4 β.0001517113 β38 Ο4 βΞ J .0000090414 β6071 Ο3 β Ο4 .0000259815 282 Ο2 β Ο3 .0001972416 156 Ο2 β Ο4 .0002232217 β38 Ο1 β Ο3 .0007580718 β25 Ο1 β Ο4 .0007840519 β27 Ο1 + Ο4 β 2Ξ J β 2G β.0000145420 β1176 Ο1 + Ο3 β 2Ξ J β 2G .0000114421 1288 ΟΞ» .0008540622 39 3`3 β 7`4 + 4Ο4 β.0001833523 β32 3`3 β 7`4 + Ο3 + 3Ο4 β.0001573724 β1162 `1 β 2`2 + Ο4 .0036431825 β1887 `1 β 2`2 + Ο3 .0036691626 β1244 `1 β 2`2 + Ο2 .0038664027 38 `1 β 2`2 + Ο1 .0044272328 β617 `1 β `2 .5018170729 β270 `1 β `3 .7527256030 β26 `1 β `4 .8939939031 4 `1 β Ο1 .9992069132 5 `1 β Ο2 .9997677433 776 `1 β Ο3 .9999649734 462 `1 β Ο4 .9999909635 149 `1 + Ο3 β 2Ξ J β 2G .9992183536 21 2`1 β 4`2 + Ο2 + Ο3 .0070726437 β200 2`1 β 4`2 + 2Ο2 .0069475638 82483 2`1 β 2`2 1.0036341339 β35 2`1 β 2`3 1.5054512040 β3 3`1 β 4`2 + Ο3 1.0073033041 276 4`1 β 4`2 2.00726827
Index E5 Argument Ratio n/nsat
ββββ LAT-1: Series coefficients for ΞΆ1 = z1/a1 (sine) ββββ
1 46 `1 β 2Ξ J + Ο β 2G .999183312 6393 `1 β Ο1 1.000652593 1825 `1 β Ο2 1.000160364 329 `1 β Ο3 1.000035275 93 `1 β Ο4 1.000008656 β311 `1 β Ο 1.000000017 75 3`1 β 4`2 + Ο2 1.00710791
ββββ XI-2: Series coefficients for ΞΎ2 = (Ο2 β a2)/a2 (cosine) ββββ
1 β18 Ο2 β Ο3 β.000251082 β27 2`3 β 2Ξ J β 2G .991065903 553 `2 β `3 .503647394 45 `2 β `4 .787214505 β102 `2 β Ο1 .998408036 β1442 `2 β Ο2 .999533787 β3116 `2 β Ο3 .999929698 β1744 `2 β Ο4 .999981859 β15 `2 βΞ J βG .99918034
10 β64 2`2 β 2`4 1.5744290111 164 2`2 β 2Ο2 2.0006437612 18 2`2 β Ο2 β Ο3 2.0003926813 β54 5`2 β 5`3 2.5182369514 β30 `1 β 2`2 + Ο4 .0073129315 β67 `1 β 2`2 + Ο3 .0073650916 93848 `1 β `2 1.0072947817 48 `1 β 2`3 + Ο4 1.0146077118 107 `1 β 2`3 + Ο3 1.0146598719 β19 `1 β 2`3 + Ο2 1.0150557820 523 `1 β `3 1.5109421721 30 `1 β Ο3 2.0072244722 β290 2`1 β 2`2 2.0145895623 β91 2`1 β 2`3 3.0218843424 22 4`1 β 4`2 4.02917912
ββββ V-2: Series coefficients for Ο 2 = Ξ½2 β `2 (sine) ββββ
1 98 β2Ξ J + 2Ο β 2G β.001639362 β1353 β2Ξ J + 2Ο β.000000043 551 β2Ξ J + Ο3 + Ο β 2G β.001710134 26 β2Ξ J + Ο2 + Ο β 2G β.001961225 31 βΟ2 + Ο .000321866 255 βΟ3 + Ο .000070787 218 βΟ4 + Ο .000017338 β1845 G .000819669 β253 2G .00163932
10 18 2Gβ² β 2G+ Ο4 β.0009792011 19 2Gβ² βG+ Ο1 β.0001595412 β15 5Gβ² β 3G+ Ο1 β.0008086713 β150 5Gβ² β 2G+ Ο2 .0000109814 102 Ο3 β Ο4 β.0000534415 56 Ο2 β Ο3 β.0002510816 72 Ο4 βΞ J .0000181517 2259 Ο3 β Ο4 .0000521618 β24 Ο3 β Ο4 + Ο3 β Ο4 β.0000012919 β23 Ο2 β Ο3 .0003959120 β36 Ο2 β Ο4 .0004480721 β31 Ο1 β Ο2 .00112575
Index E5 Argument Ratio n/nsat
22 4 Ο1 β Ο3 .0015216723 111 Ο1 β Ο4 .0015738224 β354 Ο1 + Ο3 β 2Ξ J β 2G .0000229625 β3103 ΟΞ» .0017143526 55 2`3 β 2Ξ J β 2G .9910659027 β111 3`3 β 7`4 + 4Ο4 β.0003680528 91 3`3 β 7`4 + Ο3 + 3Ο4 β.0003158929 β25 3`3 β 7`4 + 2Ο3 + 2Ο4 β.0002637330 β1994 `2 β `3 .5036473931 β137 `2 β `4 .7872145032 1 `2 β Ο1 .9984080333 2886 `2 β Ο2 .9995337834 6250 `2 β Ο3 .9999296935 3463 `2 β Ο4 .9999818536 30 `2 βΞ J βG .9991803437 β18 2`2 β 3`3 + Ο4 .5109603238 β39 2`2 β 3`3 + Ο3 .5110124839 98 2`2 β 2`4 1.5744290140 β164 2`2 β 2Ο2 2.0006437641 β18 2`2 β Ο2 β Ο3 2.0003926842 72 5`2 β 5`3 2.5182369543 30 `1 β 2`2 β Ο3 + 2Ξ J + 2G .0088637944 4180 `1 β 2`2 + Ο4 .0073129345 7428 `1 β 2`2 + Ο3 .0073650946 β2329 `1 β 2`2 + Ο2 .0077610047 β19 `1 β 2`2 + Ο1 .0088867548 β185835 `1 β `2 1.0072947849 β110 `1 β 2`3 + Ο4 1.0146077150 β200 `1 β 2`3 + Ο3 1.0146598751 39 `1 β 2`3 + Ο2 1.0150557852 β16 `1 β 2`3 + Ο1 1.0161815353 β803 `1 β `3 1.5109421754 β19 `1 β Ο2 2.0068285655 β75 `1 β Ο3 2.0072244756 β31 `1 β Ο4 2.0072766357 β9 2`1 β 4`2 + Ο3 + Ο .0145187458 4 2`1 β 4`2 + 2Ο3 .0144479759 β14 2`1 β 4`2 + Ο2 + Ο3 .0141968860 150 2`1 β 4`2 + 2Ο2 .0139458061 β11 2`1 β 4`2 + 2Ξ J + 2G .0162288862 β9 2`1 β 4`2 + Ο3 + Ο4 .0146780263 β8 2`1 β 4`2 + 2Ο3 .0147301764 915 2`1 β 2`2 2.0145895665 96 2`1 β 2`3 3.0218843466 β18 4`1 β 4`2 4.02917912
ββββ LAT-2: Series coefficients for ΞΆ2 = z2/a2 (sine) ββββ
1 17 `2 β 2Ξ J + Ο β 3G .997541012 143 `2 β 2Ξ J + Ο β 2G .998360663 β144 `2 β Ο1 1.001309934 81004 `2 β Ο2 1.000321885 4512 `2 β Ο3 1.000070806 1160 `2 β Ο4 1.000017357 β19 `2 β Ο βG .999180368 β3284 `2 β Ο 1.000000029 35 `2 β Ο +G 1.00081968
10 β28 `1 β 2`3 + Ο3 1.0145187611 272 `1 β 2`3 + Ο2 1.01426768
Index E5 Argument Ratio n/nsat
ββββ XI-3: Series coefficients for ΞΎ3 = (Ο3 β a3)/a3 (cosine) ββββ
1 24 βΟ3 + Ο .000142592 β9 Ο3 β Ο4 β.000107673 10 Ο3 β Ο4 .000105084 294 `3 β `4 .571301755 18 `3 β Ο2 .999060716 β14388 `3 β Ο3 .999858357 β7919 `3 β Ο4 .999963438 β23 `3 βΞ J βG .998348649 β20 `3 + Ο4 β 2Ξ J β 2G .99673384
10 β51 `3 + Ο3 β 2Ξ J β 2G .9968389211 39 2`3 β 3`4 + Ο4 .7139418112 β1761 2`3 β 2`4 1.1426034913 β11 2`3 β 2Ο3 1.9997167114 β10 2`3 β Ο3 β Ο4 1.9998217915 β27 2`3 β 2Ξ J β 2G 1.9966972816 24 2`3 β 2Ο3 2.0002852717 9 2`3 β Ο3 β Ο4 2.0001776018 β24 2`3 β Ο3 β Ο 2.0001426819 β16 3`3 β 4`4 + Ο4 1.2852435520 β156 3`3 β 3`4 1.7139052421 β42 4`3 β 4`4 2.2852069922 β11 5`3 β 5`4 2.8565087323 6342 `2 β `3 1.0146967724 9 `2 β Ο3 2.0145551225 39 2`2 β 3`3 + Ο4 1.0294301126 70 2`2 β 3`3 + Ο3 1.0295351927 10 `1 β 2`2 + Ο4 .0147333428 20 `1 β 2`2 + Ο3 .0148384229 β153 `1 β `2 2.0293935430 156 `1 β `3 3.0440903131 11 2`1 β 2`2 4.05878708
ββββ V-3: Series coefficients for Ο 3 = Ξ½3 β `3 (sine) ββββ
1 10 βΟ3 + Ο4 β Ο3 + Ο .000037512 28 β2Ξ J + 2Ο β 2G β.003302813 β1770 β2Ξ J + 2Ο β.000000084 β48 β2Ξ J + Ο3 + Ο β 2G β.003445405 14 βΟ2 + Ο .000648456 411 βΟ3 + Ο .000142597 345 βΟ4 + Ο .000034928 β2338 G .001651369 β66 2G .00330272
10 10 Gβ² βG+ Ο3 β.0009863911 22 2Gβ² β 2G+ Ο4 β.0019727812 26 2Gβ² βG+ Ο1 β.0003214213 11 3Gβ² β 2G+ Ο2 + Ο3 β.0013078114 9 3Gβ² βG+ Ο1 β Ο2 .0003435515 β19 5Gβ² β 3G+ Ο1 β.0016292316 β208 5Gβ² β 2G+ Ο2 .0000221317 159 Ο3 β Ο4 β.0001076718 21 Ο2 β Ο3 β.0005058619 121 Ο4 βΞ J .0000365720 6604 Ο3 β Ο4 .0001050821 β65 Ο3 β Ο4 + Ο3 β Ο4 β.0000025922 β88 Ο2 β Ο3 .0007976523 β72 Ο2 β Ο4 .0009027324 β26 Ο1 β Ο3 .00306570
J.H. Lieske: Galilean satellite ephemerides E5 217
Table 8. continued
Index E5 Argument Ratio n/nsat
25 β9 Ο1 β Ο4 .0031707826 16 Ο1 + Ο4 β 2Ξ J β 2G β.0000588127 125 Ο1 + Ο3 β 2Ξ J β 2G .0000462728 307 ΟΞ» .0034539029 β10 `4 β Ο4 .4286616930 β100 `3 β 2`4 + Ο4 .1426400631 83 `3 β 2`4 + Ο3 .1427451432 β944 `3 β `4 .5713017533 β37 `3 β Ο2 .9990607134 28780 `3 β Ο3 .9998583535 15849 `3 β Ο4 .9999634336 7 `3 β Ο4 + Ο3 β Ο4 .9998557637 46 `3 βΞ J βG .9983486438 51 `3 + Ο4 β 2Ξ J β 2G .9967338439 11 `3 + Ο3 β 2Ξ J β 3G .9951875640 97 `3 + Ο3 β 2Ξ J β 2G .9968389241 1 `3 + Ο1 β 2Ξ J β 2G .9999046242 β101 2`3 β 3`4 + Ο4 .7139418143 13 2`3 β 3`4 + Ο3 .7140468944 3222 2`3 β 2`4 1.1426034945 29 2`3 β 2Ο3 1.9997167146 25 2`3 β Ο3 β Ο4 1.9998217947 37 2`3 β 2Ξ J β 2G 1.9966972848 β24 2`3 β 2Ο3 2.0002852749 β9 2`3 β Ο3 β Ο4 2.0001776050 24 2`3 β Ο3 β Ο 2.0001426851 β174 3`3 β 7`4 + 4Ο4 β.0007415052 140 3`3 β 7`4 + Ο3 + 3Ο4 β.0006364253 β55 3`3 β 7`4 + 2Ο3 + 2Ο4 β.0005313454 27 3`3 β 4`4 + Ο4 1.2852435555 227 3`3 β 3`4 1.7139052456 53 4`3 β 4`4 2.2852069957 13 5`3 β 5`4 2.8565087358 42 `2 β 3`3 + 2`4 β.1279067259 β12055 `2 β `3 1.0146967760 β24 `2 β Ο3 2.0145551261 β10 `2 β Ο4 2.0146602062 β79 2`2 β 3`3 + Ο4 1.0294301163 β131 2`2 β 3`3 + Ο3 1.0295351964 β665 `1 β 2`2 + Ο4 .0147333465 β1228 `1 β 2`2 + Ο3 .0148384266 1082 `1 β 2`2 + Ο2 .0156360667 90 `1 β 2`2 + Ο1 .0179041168 190 `1 β `2 2.0293935469 218 `1 β `3 3.0440903170 2 2`1 β 4`2 + Ο3 + Ο .0292508671 β4 2`1 β 4`2 + 2Ο3 .0291082772 3 2`1 β 4`2 + 2Ο2 .0280965573 2 2`1 β 4`2 + Ο3 + Ο4 .0295717574 2 2`1 β 4`2 + 2Ο3 .0296768375 β13 2`1 β 2`2 4.05878708
ββββ LAT-3: Series coefficients for ΞΆ3 = z3/a3 (sine) ββββ
1 37 `3 β 2Ξ J + Ο β 3G .995045872 321 `3 β 2Ξ J + Ο β 2G .996697243 β15 `3 β 2Ξ J + Ο βG .998348604 β45 `3 β 2Ξ J + Ο .999999965 β2797 `3 β Ο2 1.00064849
Index E5 Argument Ratio n/nsat
6 32402 `3 β Ο3 1.000142637 6847 `3 β Ο4 1.000034968 β45 `3 β Ο βG .998348689 β16911 `3 β Ο 1.00000004
10 51 `3 β Ο +G 1.0016514011 10 2`2 β 3`3 + Ο 1.0293935012 β21 2`2 β 3`3 + Ο3 1.0292509113 30 2`2 β 3`3 + Ο2 1.02874505
ββββ XI-4: Series coefficients for ΞΎ4 = (Ο4 β a4)/a4 (cosine) ββββ
1 -19 βΟ3 + Ο .000332622 167 βΟ4 + Ο .000081463 11 G .003852044 12 Ο3 β Ο4 β.000251165 β13 Ο3 β Ο4 .000245116 1621 `4 β Ο3 .999669597 β24 `4 β Ο4 β 2Ξ J + 2Ο .999914518 β17 `4 β Ο4 βG .996062669 β73546 `4 β Ο4 .99991470
10 15 `4 β Ο4 +G 1.0037667411 30 `4 β Ο4 + 2Ξ J β 2Ο .9999148912 β5 `4 βΞ J β 2G .9922959313 β89 `4 βΞ J βG .9961479614 182 `4 βΞ J 1.0000000015 β6 `4 + Ο4 β 2Ξ J β 4G .9846771516 β62 `4 + Ο4 β 2Ξ J β 3G .9885291917 β543 `4 + Ο4 β 2Ξ J β 2G .9923812218 27 `4 + Ο4 β 2Ξ J βG .9962332619 6 `4 + Ο4 β 2Ξ J 1.0000853020 6 `4 + Ο4 β Ο4 β Ο 1.0001669621 β9 `4 + Ο3 β 2Ο4 1.0001598122 14 `4 + Ο3 β 2Ξ J β 2G .9926263423 13 2`4 β Ο3 β Ο4 1.9995842924 β271 2`4 β 2Ο4 1.9998294025 β25 2`4 β 2Ξ J β 3G 1.9884438926 β155 2`4 β 2Ξ J β 2G 1.9922959327 β12 2`4 β Ο3 β Ο4 2.0004142828 19 2`4 β Ο3 β Ο 2.0003328129 48 2`4 β 2Ο4 2.0001631230 β167 2`4 β Ο4 β Ο 2.0000816631 142 2`4 β 2Ο 2.0000001932 β22 `3 β 2`4 + Ο4 .3327283533 20 `3 β 2`4 + Ο3 .3329734634 974 `3 β `4 1.3326430535 24 2`3 β 3`4 + Ο4 1.6653713936 177 2`3 β 2`4 2.6652860937 4 3`3 β 4`4 + Ο4 2.9980144438 42 3`3 β 3`4 3.9979291439 14 4`3 β 4`4 5.3305721940 5 5`3 β 5`4 6.6632152441 β8 `2 β 3`3 + 2`4 β.2983607342 92 `2 β `4 3.6995684143 105 `1 β `4 8.43341914
Index E5 Argument Ratio n/nsat
ββββ V-4: Series coefficients for Ο 4 = Ξ½4 β `4 (sine) ββββ
1 8 βΟ3 β Ο4 + 2Ο β.000415902 β9 βΟ3 β Ο4 + Ο4 + Ο β.000497373 27 βΟ3 + Ο4 β Ο4 + Ο β.000163654 β409 β2Ο4 + 2Ο β.000170795 310 β2Ο4 + Ο4 + Ο β.000252266 β19 β2Ο4 + Ο3 + Ο β.000503417 8 βΟ4 βΞ J + 2Ο β.000085498 β5 βΟ4 βΞ J + Ο4 + Ο β.000166969 63 βΟ4 + Ξ J β Ο4 + Ο β.00000384
10 8 β2Ξ J + 2Ο β 3G β.0115563011 73 β2Ξ J + 2Ο β 2G β.0077042712 β5768 β2Ξ J + 2Ο β.0000001913 16 β2Ξ J + Ο4 + Ο β 2G β.0077857314 β97 βΟ3 + Ο .0003326215 152 β2Ο4 + 2Ο .0001629316 2070 βΟ4 + Ο .0000814617 β5604 G .0038520418 β204 2G .0077040719 β10 3G .0115561120 24 Gβ² βG+ Ο3 β.0023009021 11 Gβ² + Ο1 β 2Ο2 .0015511422 52 2Gβ² β 2G+ Ο4 β.0046018023 61 2Gβ² βG+ Ο1 β.0007497624 25 3Gβ² β 2G+ Ο2 + Ο3 β.0030506625 21 3Gβ² βG+ Ο1 β Ο2 .0008013826 β45 5Gβ² β 3G+ Ο1 β.0038004227 β495 5Gβ² β 2G+ Ο2 .0000516228 β44 Ο3 β Ο4 β.0002511629 5 Ο4 βΞ J βG β.0037667430 234 Ο4 βΞ J .0000853031 11 2Ο4 β 2Ξ J β 2G β.0075334832 β10 2Ο4 β Ο3 β Ο4 .0005848733 68 2Ο4 β 2Ο4 .0003337234 β13 Ο3 β Ο4 β Ο4 + Ο .0003265835 β5988 Ο3 β Ο4 .0002451136 β47 Ο3 β Ο4 + Ο3 β Ο4 β.0000060437 β3249 `4 β Ο3 .9996695938 48 `4 β Ο4 β 2Ξ J + 2Ο .9999145139 10 `4 β Ο4 β Ο4 + Ο .9999961640 33 `4 β Ο4 βG .9960626641 147108 `4 β Ο4 .9999147042 β31 `4 β Ο4 +G 1.0037667443 β6 `4 β Ο4 + Ο4 β Ο .9998332444 β61 `4 β Ο4 + 2Ξ J β 2Ο .9999148945 10 `4 βΞ J β 2G .9922959346 178 `4 βΞ J βG .9961479647 β363 `4 βΞ J 1.0000000048 5 `4 + Ο4 β 2Ξ J β 5Gβ² + 2Gβ Ο1 1.0000336849 12 `4 + Ο4 β 2Ξ J β 4G .9846771550 124 `4 + Ο4 β 2Ξ J β 3G .9885291951 1088 `4 + Ο4 β 2Ξ J β 2G .9923812252 β55 `4 + Ο4 β 2Ξ J βG .9962332653 β12 `4 + Ο4 β 2Ξ J 1.0000853054 β13 `4 + Ο4 β Ο4 β Ο 1.0001669655 6 `4 + Ο4 β 2Ο 1.0000854956 17 `4 + Ο3 β 2Ο4 1.0001598157 β28 `4 + Ο3 β 2Ξ J β 2G .99262634
Index E5 Argument Ratio n/nsat
58 β33 2`4 β Ο3 β Ο4 1.99958429
59 676 2`4 β 2Ο4 1.99982940
60 36 2`4 β 2Ξ J β 3G 1.98844389
61 218 2`4 β 2Ξ J β 2G 1.99229593
62 β5 2`4 β 2Ξ J βG 1.99614796
63 12 2`4 β Ο3 β Ο4 2.00041428
64 β19 2`4 β Ο3 β Ο 2.00033281
65 β48 2`4 β 2Ο4 2.00016312
66 167 2`4 β Ο4 β Ο 2.00008166
67 β142 2`4 β 2Ο 2.00000019
68 148 `3 β 2`4 + Ο4 .33272835
69 β94 `3 β 2`4 + Ο3 .33297346
70 β390 `3 β `4 1.33264305
71 9 2`3 β 4`4 + 2Ο4 .66545669
72 β37 2`3 β 3`4 + Ο4 1.66537139
73 6 2`3 β 3`4 + Ο3 1.66561651
74 β195 2`3 β 2`4 2.66528609
75 6 3`3 β 7`4 + 2Ο4 + Ο4 + Ο β.0019819276 187 3`3 β 7`4 + 4Ο4 β.0017296677 β149 3`3 β 7`4 + Ο3 + 3Ο4 β.0014845578 51 3`3 β 7`4 + 2Ο3 + 2Ο4 β.0012394379 β10 3`3 β 7`4 + 3Ο3 + Ο4 β.0009943280 6 3`3 β 6`4 + 3Ο4 .99818504
81 β8 3`3 β 4`4 + Ο4 2.99801444
82 β41 3`3 β 3`4 3.99792914
83 β13 4`3 β 4`4 5.33057219
84 β44 `2 β 3`3 + 2`4 β.2983607385 89 `2 β `4 3.69956841
86 106 `1 β `4 8.43341914
ββββ LAT-4: Series coefficients for ΞΆ4 = z4/a4 (sine) ββββ
1 8 `4 β 2Ξ J β Ο4 + 2Ο 1.00008137
2 8 `4 β 2Ξ J + Ο β 4G .98459175
3 88 `4 β 2Ξ J + Ο β 3G .98844379
4 773 `4 β 2Ξ J + Ο β 2G .99229583
5 β38 `4 β 2Ξ J + Ο βG .99614787
6 5 `4 β 2Ξ J + Ο .99999990
7 9 `4 β Ο1 1.00615611
8 β17 `4 β Ο2 1.00151270
9 β5112 `4 β Ο3 1.00033272
10 β7 `4 β Ο4 βG .99622952
11 44134 `4 β Ο4 1.00008156
12 7 `4 β Ο4 +G 1.00393360
13 β102 `4 β Ο βG .99614806
14 β76579 `4 β Ο 1.00000010
15 104 `4 β Ο +G 1.00385213
16 β10 `4 β Ο + 5Gβ² β 2G+ Ο2 1.00005172
17 β11 `3 β 2`4 + Ο .33264295
18 7 `3 β 2`4 + Ο4 .33256149