1The citation for this article is Mallama, A., Stockdale, C., Kroubsek, B., and Nelson, P. (2010)Icarus 210 346–357. DOI 10.1016/j.icarus.2010.06.007Assessment of the Resonant Perturbations Errors in Galilean SatelliteEphemerides Using Precisely Measured Eclipse TimesAbstractAstrometric satellite positions are derived from timings of their eclipses in the shadow of Jupiter.The 548 data points span 20 years and are accurate to about 0.006 arc seconds for Io and Europaand about 0.015 arc seconds or better for Ganymede and Callisto. The precision of the data setand its nearly continuous distribution in time allows measurement of regular oscillations with anaccuracy of 0.001 arc second with a telescope. This level of sensitivity permits detailed evaluation of modernephemerides and reveals anomalies at the 1.3 year period of the resonant perturbations betweenIo, Europa and Ganymede. The E5 ephemeris shows large errors at that period for all threesatellites as well as other significant anomalies. The L1 ephemeris fits the observations muchmore closely than E5 but discrepancies for the resonant satellites are still apparent and themeasured positions of Io are drifting away from the predictions. The JUP230 ephemeris fits theobservations more accurately than L1 although there is still a measurable discordance betweenthe predictions and observations for Europa at the resonance period.Key words: Orbit determination, Io, Europa, Ganymede, Callisto
Satellite orbit theories have numerous important applications including geophysical studies of themoons and their planets. A universally recognized example is the tidal interaction between theEarth and the Moon. Eclipses of satellites by their planets provide important data for orbit theoriesand some geophysical information can be derived immediately from eclipse observations. themselves For example Neugebauer et al., 2005 measured the thermal inertia of Iapetus basedon observations obtained at 20 and 2.2 μm when that satellite was in the shadow of the saturnianring system. Additionally, the state of a planetary atmosphere can be assessed from photometry ofsatellite eclipses. Mallama et al., 1995 deduced the altitude of atmospheric fall-out from the impactof comet Shoemaker-Levy 9 with Jupiter, Nicholson et al., 1997 found that methane absorptionrather than refraction was the principal cause of satellite dimming during 2.3 μm observations ofthe saturnian satellites during eclipses, and Mallama et al., 2000b determined the altitude of thehaze layer at the north and south poles of Jupiter from partial eclipses of the satellite Callisto. Inaddition to geophysics, accurate theories are essential in the production of ephemerides fornavigating interplanetary spacecraft. Satellite positions must be especially precise for close rangeremote sensing and during fly-bys used in gravity assists.The astrometric results in this paper were derived by fitting photometric ingress or egress eclipsedata to a synthetic light curve which is generated using a physical model. Section 2 describes themodel, which accounts for the effects of the Sun-Jupiter-Earth-satellite geometry, refraction ofsunlight in the atmosphere of Jupiter, and the bi-directional reflectance (BDR) functions of thesatellites. Some refinements to the model first developed by Mallama, 1991 and 1992 arereported. The procedures for fitting the synthetic light curve to observed photometry and that ofderiving event mid-times are also discussed. The method of observation is described in Section 3and a listing of mid-event timings that span 20 years is given. The procedure for deriving residualsbetween these observations and ephemeris calculations is explained and the overall methodology is validated. The differences between the along-track satellite observed positions and thosecalculated (that is, O – Cs) with the JUP230 ephemeris of Jacobson et al., 1999 (and later work),the L1 ephemeris of Lainey et al., 2004a, 2004b and 2006, and the E5 ephemeris of Lieske, 1998are evaluated in Section 4. In Section 5 regular O – C oscillations are characterized and identifiedwith the resonant perturbations between Io, Europa and Ganymede. Section 6 concludes withestimation of the uncertainties in the astrometric data.
Astrometric information is derived from eclipse photometry by means of a model based on thegeometry between the Sun, satellite, planet and observer, as well as two physical factors. The firstof these is refraction of rays from the Sun, occurring about 150 km above the clouds at theplanetary limb, which causes the dimming of sunlight that is observed during an eclipse.Atmospheric refraction must be integrated with the geometrical configuration in order to calculatethe intensity of light that impacts a grid of points on the satellite surface. The other factor is thesatellite BDR function which determines how much light from each of the grid of points is directedto the observer. The BDR depends on the distribution of satellite albedo features and on limbdarkening. The center-of-figure of a Galilean satellite is usually offset from its photocenter by 100km or more, so the effect has about the same magnitude as does refraction.The model is used to generate a synthetic light curve tailored to the geometry of the observedingress or egress. Atmospheric refraction, which is fixed in the model, acts on sunlight bound forthe satellite according to the geometry of the Sun, planet and satellite. The BDR, which is alsofixed, allows for computation of the intensity of light reflected toward the observer; so it dependson the geometry of the Sun, satellite and observer.2.1 Atmospheric refractionThe dimming of sunlight is caused by differential refraction in the milli-bar (mbar) pressure domainof the atmosphere where rays are bent inward toward the planet center. The intensity of lighttraversing the atmosphere at lower altitudes becomes fainter because refraction increasesprogressively with atmospheric density, which is exponential. The diminished strength of anyrefracted ray at the satellite location can be computed from Eq. 1 (Baum and Code, 1953)φ0 / φr – 2 + ln (φ0 / φr – 1) = d0 – d(Eq. 1) where φr / φ0 is the ratio of that intensity to its full intensity before refraction and the differentialdistance d0 – d is measured in atmospheric scale heights. The special case where d – d0 = 0 andφr / φ0 = 0.5 is illustrated by the path of this half-intensity ray from the Sun to the satellite in Fig. 1and relates to the radius of curvature, a, in Eq. 2 below. The half-intensity ray strikes the satelliteat the location in space that defines the half-phase of an eclipse ingress or egress.Fig. 1. Differential refraction in the planetary atmosphere reduces the intensity of light from the Sun, and theray diminished to half-intensity is shown impacting the middle of a satellite. The symbols are used in the textto define the bending angle and the radius of this ray.The minimum altitude of the half-intensity ray for Io is taken from a study by Spencer et al., 1990where it is reported to be at the 2.2 mbar pressure level or 133 km above the one-bar level. Thealtitudes for the other satellites were derived relative to Io from Eq. 2, which is also from Baum andCode and is illustrated by Fig. 1.θ = H / D = ν (2πa/H) 0.5(Eq. 2) where angle of deflection, θ, is determined geometrically by the atmospheric scale height, H,divided by the distance from the planetary limb to the satellite, D. Physically, the ray is bent byatmospheric gas for which the refractivity is ν, at the radius a, mentioned above. The minimumaltitudes of the half-intensity rays for Europa, Ganymede and Callisto are thus found to be 145,160 and 172 km, respectively.2.2 Integrated luminosityThe total luminosity at any location within the planetary shadow is a summation over the intensityof all rays from the solar disk that impact on that point. In the geometrical case of a planet with noatmosphere there will be a completely dark umbral region corresponding to locations where astraight line to the Sun intersects the planet. The umbra is surrounded by a penumbral annulus,wherein a portion of the solar disk is visible, and which corresponds in width to the apparentdiameter of the Sun projected from the planet limb to the satellite distance. The computation ofluminosity in the geometric penumbra is straightforward although there is a minor effect from solarlimb-darkening. The dashed line in Fig. 2 shows luminosity increasing from zero at the boundarybetween the umbra and penumbra to unity at the outer edge of the penumbra during an egress.For a planet with an atmosphere, however, luminosity in the shadow is a summation over productsof limb-darkened solar intensity times the effect of refraction as computed from Eq. 1. In this casethe full uneclipsed luminosity for the satellite is attained a short distance outside of the geometricpenumbral boundary, and significant brightness penetrates a long distance inside of the umbralboundary due to refraction as shown by the solid line in Fig. 2. The graph also shows that theluminosity at phase 0.5 is substantially greater than half even though that location is defined by theprojection of a half-intensity ray from the center of the solar disk.
The normalized luminosity of the Sun sensed at the distance of Io from the limb of Jupiter as afunction of position, which is defined geometrically as the umbra (X < 0) and the penumbra (0 < X < 1). Zeroand unity luminosity for the case of no refraction correspond with these geometric boundaries, but whenrefraction is taken into account the curve extends inside and outside of the penumbra. A half-intensity rayfrom the center of the solar disk falls at 0.5 on the horizontal axis.2.3. Satellite characteristicsIn addition to the solar eclipse luminosity described above, the physical size and the BDRcharacteristics of a satellite are taken into account when computing its integrated brightness in theplanetary shadow. The details of BDR functions adopted for the Galilean satellites are listed byMallama, 1991. The albedo differences between the leading and trailing hemispheres of thesesatellites result in significant offsets between their centers-of-luminosity and centers-of-figure.When combined with limb-darkening and phase effects the total offsets are hundreds of km and 8their exact size and direction depends on the geometry of the eclipse event and the viewing anglefrom Earth.The BDR model for Callisto was modified after the results in Mallama et al., 2000a were published.The leading/trailing hemisphere albedo component of the BDR function for that satellite used in theoriginal eclipse model was based on an incorrect interpretation of the photometric results for itssurface reported by Buratti, 1991 and Squyers and Veverka, 1981. After discussion with Buratti(private communication, 2002) the eclipse model BDR was corrected by changing the distributionof brightness as a function of the angle, ρ, which is measured along the satellite surface from theapex of its orbital motion. Specifically, the coefficient of cos(ρ) given in Mallama, 1991 waschanged to +0.057. The original model resulted in observed timings that were consistently tooearly relative to ephemeris predictions, while the corrected model resolves these systematicdifferences as reported in Section 4.4 of this paper.2.4. The synthetic light curve and the method of fitting observationsThe brightness of a satellite is evaluated as a function of its changing position in the solar eclipseregion during ingress or egress. This results in the synthetic light curve for a particular event. Thatbrightness at any one location is computed by integrating the product of solar eclipse luminosityand the BDR function over the hemisphere of the satellite facing the sensor. Satellite eclipsephase space is a useful construct defined such that phase zero of an egress corresponds to theleading edge of the satellite crossing from the geometric umbra into the penumbra and phase oneoccurs when its trailing edge exits the geometric penumbra, and vice-versa for an ingress. Whilehalf-phase in eclipse space corresponds with that shown in Fig. 2 for geometric penumbra space,the limits at phases zero and unity are extended because of the finite radii of the satellite. In order to correctly represent the geometry of an event the solar phase angle and the angle atwhich the shadow crosses the globe of the satellite are input to the model. Figures 3 and 4demonstrate synthetic egress light curves for variations of phase angle and satellite.Fig. 3. The brightness of a satellite during an eclipse depends on phase angle, which is taken to benegative before the opposition of Jupiter from the Sun and positive afterward. In these synthetic egress lightcurves Ganymede attains half-luminosity at phase 0.499 when the event occurs at phase angle -11 degreesor morning quadrature with the Sun. However, the same luminosity is attained at phase 0.446, which is 36seconds of time earlier for an event at evening quadrature. That difference is for eclipses that occur at theequator of Jupiter, but it can exceed one minute when the eclipse latitude is farther from the equator.
Differences between synthetic light curves for the Galilean satellites are evident even when they areevaluated at the same phase angle, +11 degrees in this case. BDR characteristics, Sun-planet-moongeometry, and refraction contribute to the variation. Notice, too, that the luminosity at phase zero is nearzero and that at phase one is near unity, in contrast to the solar eclipse luminosity with refraction shown inFig. 2. This is due to the large size of the satellites relative to the width of the penumbral shadow.Model luminosities are fitted to photometric data in order to establish distances between the centerof the satellite and planet in the plane of the sky as seen from the Sun as a function of time. Thefitting process for light curves is a non-linear least squares problem of the type described byBevington, 1969. The model is parameterized and successive iterations adjust the parametersuntil the differences between modeled and observed data are reduced to a minimum. Forastrometric purposes the critical parameter is the time of half-phase, which can be compared withthat predicted by an ephemeris. The luminosity scale and the time scale are also adjusted.
Observation and ValidationCCD recordings of Galilean satellite eclipses began in 1990 when solid state sensors had becomewidely available. Older visual timings spanned more than three centuries, however they were veryimprecise. Photoelectric photometry did not greatly improve the precision because the steep non-linear sky background near the limb of Jupiter introduced large biases into the satellite luminositymeasurements. However, the linear two dimensional photometric data from a CCD chip proved tobe ideal because the sky background could be accurately isolated and removed. This sectionbegins by describing how the raw CCD images were acquired and how photometric data wereextracted from them. Next, a listing of all the half-phase timings is presented, and finally theeclipse model and data analysis techniques are validated.3.1 Observational methods, photometric analysis and timing resultsA standard procedure has been followed during the 20 years of observation reported here. Theeclipses were recorded through V-band filters with telescopes ranging from 20 to 40 cm inaperture and from 2 to 5 m in focal length. Typically about 100 images were taken during an eventand each image was time tagged based on a clock synchronized to WWV radio broadcasts, GPSsatellites signals, NTP Internet sources or the NIST telephone service.The data were analyzed with customized aperture photometry software that produces accuratesatellite luminosities even in the vicinity of the bright nearby planet. Precise sky backgroundsubtraction was accomplished by reading data from a tailored annulus centered around themeasurement aperture for the satellite. The brightest 25% and the faintest 25% of the luminositiesin the annulus were rejected, thus masking out the sky in the directions toward and away fromJupiter where the gradient is steepest. The remaining 50% of the luminosity data represents thesky background brightness at the same distance from the planetary limb as the satellite. The observed normalized brightness of a satellite when totally eclipsed is generally within 1% of zero,which demonstrates the accuracy of the sky subtraction.Additionally, each CCD image of the eclipsing satellite contains at least one other moon forluminosity and positional reference. The analysis program uses the locations of the two satellitesand their relative motion over the course of the event to track the eclipsing satellite even when it isvery faint or totally eclipsed.Fig. 5. An egress of Europa observed on 2007 July 30 demonstrating the close fit between photometric dataand the synthetic light curve.
The resulting set of luminosities for each ingress or egress is fitted to a synthetic light curve(customized for that event) by the non-linear technique described in Section 2.4 and illustrated inFig. 5. The observed times of half-phase in seconds measured from the epoch of J2000 are listedin Table 1 where the satellites are identified by numbers 1 through 4 for Io, Europa, Ganymedeand Callisto, respectively, and eclipse Ingress or Egress is indicated by I or E.The data in Table 1 are the basis of this paper and its most important result, so the followingexample is given in order to clearly specify the contents. The first timing with a positive number ofseconds, 470546.0, corresponds to half-phase of an event that occurred on 2000 January 6 at22:41:21.8 UTC, which was 470481.8 s later than 2000 January 1.5 UTC. The difference of 64.2 sbetween the numbers quoted above corresponds to the offset between UTC and ephemeris time(ET) at that epoch. The quantity ‘2E’ in this same record of the Table indicates that it was anegress ‘E’ of the satellite Europa ‘2’. Please contact the first author for information about using theeclipse timings as astrometric data.Table 1 is at the end of the manuscript3.2 Validation by comparison with the dimensions of JupiterThe long time span and the large number of observations in this study allow the consistency of theobserving, modeling and fitting techniques to be verified. In particular, the absolute accuracy canbe assessed by comparing satellite positions computed at the times of half-phase with the locationof Jupiter’s limb. While discrepancies for any one event are to be expected because of ephemerisand observational uncertainties, the satellite and limb positions should correspond when the wholedata set is averaged.Three ephemerides are used to perform this validation and are referenced throughout theremainder of this paper. JUP230 is the final ephemeris for the Galileo satellite mission. Jacobson et al., 1999 described the early orbit modeling work that centered on Galileo spacecraft databeginning in 1995 and eventually extending through 2003. JUP230-Long gives the same results asJUP230 but it allows computations for years before 2000.0 and the two are considered equivalenthere. The L1 theory of Lainey et al., 2004a and 2004b fits a long time span of observations,beginning with astrographic observations in 1891. The E5 ephemeris (Lieske, 1998) is ananalytical model that had been the state of the art before JUP230 and L1 were developed usingnumerical integration.These ephemerides were used to calculate satellite positions at the observed ET corrected for thelight-time between the satellite and the Earth. The position of Jupiter relative to the satellite wasprojected to the heliocentric plane of the sky as depicted in Fig. 6. Then the radial distancebetween the centers of Jupiter and the satellite in that plane was determined, and this quantity isreferred to as the satellite’s ephemeris radial distance. Next the radius at the minimum altitude ofthe half-intensity ray, corrected for refraction, was subtracted from the ephemeris radial distance,and this is referred to as the delta radius. Since Jupiter is an ellipsoid, this minimum altitude is theone projected by the pole-tilted ellipsoid on the sky (also depicted in the figure).The very small average delta radius values of -6.0 km (+/- 9.7) for JUP230, -23.8 (+/- 20.6) for L1and -2.6 (+/- 6.8) for E5 validate the methodology. In the next section the individual delta radiicorresponding to each of the Galilean satellites are used in the evaluation of the ephemerides.
The X-axis points out of the plane of this figure toward the Sun. The spin axis of Jupiter is in the XZplane. The latitude at Jupiter’s limb (indicated by the dashed line) that projects onto the satellite is derivedfrom the arc tangent of Z / Y at the center of the planet. The radius of the ellipsoid at that point on the limb iscomputed from the latitude.
Characterization of the Eclipse ResidualsSatellite ephemeris errors are usually dominated by their along-track component because orbitalperturbations act most strongly in that direction. Since the Galilean satellites tend to encounterJupiter’s shadow at a small angle of incidence the eclipses are very useful for evaluating along-track residuals. In particular, the eclipses of Io occur within 18 degrees of Jupiter’s equator andthose of Europa happen within 35 degrees. While the eclipses of Ganymede and Callisto canoccur at higher latitudes they are still very effective for evaluating along-track error because theout-of-plane ephemeris uncertainties are much smaller.Eclipse time O – Cs and distances are related by the common equation for velocity. If the eventwas an ingress and the O – C in time is positive, then the satellite was observed to be at half-phase when its ephemeris calculates a position closer to Jupiter, thus the delta radial distance(defined in the previous section) is negative. Conversely, for an egress, a positive O – C in timeindicates a positive delta radius because the ephemeris indicates a satellite farther from Jupiterthan the location of half-phase.The average delta radius parameters for individual satellites are often significantly different fromzero (though usually less than one percent of the satellite radius) as shown in Table 2. The non-zero averages are most likely due to a combination of uncertainties in the BDR model, theatmosphere model and the ephemerides. Therefore, in the following analysis, the delta radialdistance for each event was corrected for the average delta radius of the appropriate satellite andephemeris before the O – C was derived.4.1. Io - oscillation and driftThe O – C residuals of Io plotted in Fig. 7 show that most timings of half-phase agree withpredictions from all 3 ephemerides within about 5 s. The root-mean-square (rms) residuals rangefrom a best of 2.0 s for JUP230 to 2.7 s for L1 to a worst of 3.7 s for E5 as indicated in Table 2.
The most striking signature in the residuals is a cyclic variation with a period near 1.3 yearsrelative to the E5 ephemeris. Fig. 8 shows the correlation between O – Cs and sinusoids fitted tothem as a function of period. The peak correlation is that at 1.28 years for E5 and there is asmaller one nearby at 1.31 for L1. The periods, amplitudes and phases for these oscillations andthose of the other satellites are listed in Table 3. The other notable feature in the O – Cs is the driftof Io relative to the L1 ephemeris beginning in 2002. That model appears to be predicting eclipsetimes that are increasingly too late and by the end of 2009 the discrepancy was about 65 km.Fig. 7. The residuals between observed half-phase times for Io and calculations from the JUP230, L1 andE5 ephemerides. Those for E5 are fitted with a sine function having a period of 1.28 years. The trend line forL1 beginning in 2002 indicates that observed eclipse times are increasingly early relative to that ephemeris.
The correlation between sine fits and the residuals of Io to JUP230, L1 and E5 for periods around 1.3years. The dashed horizontal line indicates 99% confidence…Table 2. Ephemeris evaluation summaryJUP230L1E5Io, 211 observationsAverage delta radius (km)+9.6 (3.6)-14.2 (5.3) +20.2 (4.3)O – C mean (s)+1.1 (0.1)-1.3 (0.1)-0.7 (0.3)O - Cs rms (s)2.02.53.7Europa, 149 observations
Average delta radius (km)+11.9 (3.2) +20.8 (7.4) +4.7 (20.4)O – C mean (s)-0.3 (0.2)-0.7 (0.3)-1.5 (1.6)O - Cs rms (s)2.03.919.0Ganymede, 149 observationsAverage delta radius (km)-25.1 (5.4)-78.5 (8.9) -41.2 (12.4)O – C mean (s)-3.6 (0.4)+1.7 (0.7)+2.7 (1.3)O - Cs rms (s)220.127.116.11Callisto, 39 observationsAverage delta radius (km)-20.4 (13.2) -23.0 (13.7) -7.0 (10.2)O – C mean (s)-3.0 (1.3)+2.5 (1.5)-2.2 (1.6)O - Cs rms (s)18.104.22.168The numbers in parentheses are uncertainties.4.2 Europa - large O – CsThe residuals of Europa to E5 are more than 5 times as large as those of Io, with an overall rms of19.0 s or 260 km. They are significantly negative in 1990, increase to a large positive value aroundthe year 2000, and then turn negative again as shown in Fig. 9. This large error in the E5ephemeris has been noted before by Lainey et al., 2004b and by Mallama et al., 2000a. Theimprovements of JUP230 and L1 are very evident from the graph. Since this long term variationadds complexity to the analysis of short period oscillations it was fitted with the empirical functionshown by the dashed line in the graph. After this correction the remaining residuals fit a sine curvewith period 1.27 years which is just 0.01 year less than that for Io. The correlation coefficientbetween the observations and the sinusoid for E5 is extremely high as shown in Fig. 10. There is also a strong correlation near the same period for L1 and one that is weaker but still significant forJUP230.Fig. 9. When the long term O – C variations of Europa to E5 are removed with a heuristic function (dashedline) the 1.27 year sine term is very apparent. An oscillation of about the same period but smaller amplitudeis evident for L1.
The correlation between sine fits and the residuals of Europa to JUP230, L1 and E5. There aresignificant correlations for all 3 ephemerides at periods near 1.3 years…4.3 Ganymede - drift and ephemeris biasesThe residuals to E5 for Ganymede shown in Fig. 11 are scattered around zero until 2007 when atrend becomes evident. This behavior mirrors the plot in Fig. 6 of Lainey et al., 2004b whichcompares predictions of L1 with those of E5. The fit of the residuals to E5 shown by the sinusoidcorresponds to a period of 1.30 years, so it is similar to those of Io and Europa. The E5 correlationis strong as shown in Fig. 12 while a peak for L1 just less than 1.3 years is weaker though stillsignificant. The mean O – C values for Ganymede range from -3.6 for JUP230 to +1.7 for L1 and+2.7 for E5. This spread, which corresponds to 67 km, is the largest among the satellites.
The observed position of Ganymede has fallen behind that predicted by the E5 ephemeris in recentyears as shown by the trend line. The L1 and JUP230 ephemerides predict eclipses more accuratelyalthough there is a considerable bias between them. The average eclipse timing is early relative to JUP230and late relative to L1.
The correlation between sine fits and the residuals for Ganymede to the three ephemerides.4.4 Callisto - improved resultsThe leading/trailing hemisphere albedo parameter of the BDR function for Callisto given in themodel paper by Mallama, 1991, and used for analysis of the results reported by Mallama et al.,2000a, has been corrected as described in section 2.3. Consequently, the observed times of half-phase that had appeared to be systematically much too early relative to E5 (open circles in Fig.13) are now far more consistent with it (filled circles). While the previously negative bias revealedthe error in the old BDR, the parameterization was not adjusted empirically to make the data fit theephemeris; rather the new albedo parameter was derived from the corrected understanding of theBDR. So, the better overall fit between ephemeris theory and observations is an independentverification of the corrected satellite model. The O – Cs for E5 in the most recent years seem to be trending negatively while those for JUP230 and L1 remain scattered around zero. Fig. 14 does notindicate high correlations between O – Cs and sine fits for any of the ephemerides.Fig. 13. The O – C residuals for Callisto relative to E5 grow increasingly negative in 2008 and 2009. Theopen circles represent results from an old BDR model of Callisto reported by Mallama et al., 2000a asdiscussed in the text.
Characteristics and interpretation of the oscillationsMismodeling of the 1.3 year along-track oscillations of Io, Europa and Ganymede is reported inthis paper for the first time. Table 3 lists periods, amplitudes and phases for the statisticallysignificant oscillations relative to the three modern ephemerides. The amplitudes are much largerfor E5 (4.6 to 15.0 s) than for L1 (1.2 to 3.7) . The oscillations referenced to JUP230 are onlysignificant for Europa and that amplitude is only 1.1 s. The phase for the oscillation of any givensatellite relative to L1 and JUP230 is roughly opposite to that for E5. This difference is nearest to πradians in the case of Europa where the phases are determined most accurately. The L1 andJUP230 phases for Europa are almost equal. The oscillations of Ganymede are approximately outof phase to those of Io and Europa for both E5 and L1.Table 3. O – C oscillations with periods near 1.3 years Ephemeris Period(years) Amplitude(seconds) Phase(radians) IoE51.28(0.03)4.6(0.2)0.87(0.04) L11.31(0.01)1.2(0.2)4.96(0.19) EuropaE51.27(0.02)15.0(0.5)0.90(0.03)L11.30(0.01)3.6(0.4)4.35(0.09)JUP2301.32(0.01)1.1(0.2)4.26(0.16) GanymedeE51.30(0.05)13.8(1.0)4.28(0.06)L11.25(0.01)3.7(0.8)1.99(0.26)
The periods of the oscillations described above are close to those of the resonant perturbationsbetween Io, Europa and Ganymede. Their existence seems to imply that the perturbations are notrepresented accurately enough in the ephemerides. The π radian phase differences between E5and the other two ephemerides suggest that the amplitudes of the modeled perturbations areeither too large for E5 and too small for L1 and JUP230, or vice-versa. Callisto does not participatein the resonances of the other 3 satellites, so strong sinusoidal O – C variations are not expected.The dynamical importance of the three bodies in resonance is the associated librations that forceorbital eccentricities which, in turn, lead to the dissipation of tidal energy inside the satellites. Theeffect is most pronounced for innermost Io and accounts for its volcanic activity. Yoder and Peale,1981 explained this phenomenon and it is summarized by Murray and Dermott, 1999.
Estimated uncertainty of the timingsThe timings contained in Table 1 are the fundamental observational result of this paper. Thesedata have dynamical and geophysical implications so it is important to know their accuracy. Theoverall uncertainty of a datum is equal to the square root of the sum of the squares (rss) of theuncertainties of the observationally measured timing and of the modeling of the eclipse.The observational uncertainty can be estimated from timing differences where one eclipse wasrecorded by multiple observers. Table 4 shows that the average standard deviation of suchdifferences increases by more than a factor of 3, from 0.80 s for Io to 2.45 s for Callisto. Thesevalues are nearly inversely proportional to the satellites velocities, which decrease from 17.3 km/sfor Io to 8.2 for Callisto. In the last column of the table the standard deviations of the timings aremultiplied by the satellite orbital velocities to convert from times to distances. The much narrowerrange of distances indicates that orbital velocity alone accounts for most of the variation inobservational uncertainty. The average of the standard deviations is 15 km or 4 milli-arc second(mas).Table 4. Statistics for eclipses timed by multiple observers.Eclipses withMultiple TimingsAverage of StandardDeviationsSatelliteEclipsesTimingsSecondsKmIo36740.8014Europa23480.9914Ganymede33721.2113Callisto6122.4520
he uncertainty of the model can be approximated by taking the square root of the average of thesums of squares of the delta radius values. That value for the data in Table 2 corresponding withthe best fitting ephemeris (JUP230) is 18 km or 5 mas. When the observational and modeluncertainties are combined in the rss sense the total uncertainty is 6 mas.This estimated uncertainty should be less than or equal to the rms values of the O - Cs becausethe residuals include a component of ephemeris error. When the JUP230 values in Table 2 areconverted from units of time to distance they are 28 km or 7 mas for Io, 26 km or 7 mas forEuropa, 56 km or 15 mas for Ganymede and 65 or 17 mas for Callisto. Thus the estimated overalluncertainty of 6 mas is validated for Io and Europa. The larger rms residuals for Ganymede andCallisto may indicate that 6 mas is an underestimate of the overall uncertainty for those satellitesunless there is significant ephemeris error. The underestimation may be due to their largerphysical diameters which increases the difficulty of modeling their BDRs to the same level ofaccuracy as for Io and Europa. In any case, the overall uncertainty for Ganymede and Callistomust not be greater than about 15 mas.The high precision of the data and its nearly continuous distribution in time permits very tinyoscillations to be detected. The smallest amplitude in Table 3 is 1.1 +/- 0.2 s for Europa whoseorbital velocity is 13.7 km/s. The amplitude in km is 15 +/- 3 which, at the mean distance fromJupiter, corresponds to 3.9 +/- 0.7 mas.AcknowledgmentsThis research was supported in part by NASA through the American Astronomical Society’s SmallResearch Grant Program. Robert J. Modic, Donald F. Collins and Maurizio Martinengo suppliedobservational data in addition to that obtained by the authors. The reviews by Valéry Lainey andan anonymous referee were very helpful, as were comments by Phillip Nicholson, Stanton Pealeand Robert Jacobson.
ReferencesBaum, W.A. and Code, A.D. 1953. A photometric observation of the occultation of σ Arietis byJupiter. Astron. J., 58, 108-112.Bevington, P.R. 1969. Data reduction and error analysis for the physical sciences. McGraw-Hill,NY.Buratti, B. 1991. Ganymede and Callisto: surface textural dichotomies and photometric analysis.Icarus, 92, 312-323.Jacobson, R., Haw, R., McElrath, T., and Antreasian, P. 1999. A comprehensive orbitreconstruction for the Galileo prime mission in the J2000 system. Presented at AAS/AIAAastrodynamics meeting, Girdwood, Alaska, USA. August 16,1999. Publication number AAS 99-330. http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/17924/1/99-1374.pdf.Lainey, V., Arlot, J.E., Vienne, A. 2004a. New accurate ephemerides for the Galilean satellites ofJupiter, II. Fitting the observations. Astron. Astrophys. 427, 371-376. doi: 10.1051/0004-6361:20041271.Lainey, V., Arlot, J.E., Karatekin, Ö, and Van Hoolst, T. 2009. Strong tidal dissipation in Io andJupiter from astrometric observations. Nature, 459, 957-959. doi:10.1038/nature08108.Lainey, V., Duriez, L., Vienne, A. 2004b. New accurate ephemerides for the Galilean satellites ofJupiter, I. Numerical integration of elaborated equations of motion. Astron. Astrophys. 420,1171-1183. doi: 10.1051/0004-6361:20034565.Lainey, V., Duriez, L., Vienne, A. 2006. Synthetic representation of Galilean satellites’ orbitalmotions from L1 ephemerides. Astron. Astrophys. 456, 783-788. doi: 10.1051/0004-6361:20064941.Lieske, J.H. 1998. Galilean satellite ephemerides E5. Astron. Astrophys. Suppl. 129, 205-217.Mallama, A., 1991. Light curve model for the Galilean satellites during jovian eclipse. Icarus, 92,324-331.
Mallama, A., 1992. CCD photometry for jovian eclipses of the Galilean satellites. Icarus, 97, 298-302.Mallama, A., Nelson, P., and Park, J., 1995. Detection of very high altitude fallout from the cometShoemaker-Levy 9 explosions in Jupiter’s atmosphere. J. Geophys. Res., 100, 16879-16884.Mallama, A. and Krobusek, B., 1996. Eclipses of Saturn’s moons. J. Geophys. Res., 101, 16,901-16,904.Mallama, A., Collins. D.F., Nelson, P., Park, J. and Krobusek, B., 2000a. Precise timings ofGalilean satellite eclipses and assessment of the E5 ephemeris. Icarus, 147, 348-352,doi:10.1006/icar.2000.6455.Mallama, A., Krobusek, B., Collins. D.F., Nelson, P., and Park, J., 2000b. The radius of Jupiter andits polar haze. Icarus, 144, 99-103, doi:10.1006/icar.1999.6276.Mallama, A., Sôma, M., Sada, P.V., Modic, R.J., and Ellington, C.,K. Astrometry of Iapetus, Ariel,Umbriel, and Titania from eclipses and occultations. Icarus, 200, 265-275,doi:10.1016/j.icarus.2008.11.022.Murray, C.D. and Dermott, S.F. , Solar System Dynamics. Cambridge University Press,Cambridge, UK. pp. 396-399.Neugebauer, G., Matthews, K., Nicholson, P.D., Soifer, B.T., Gatley, I. and Beckwith, S.V.W.2005. Thermal response of Iapetus to an eclipse by Saturn’s rings. Icarus, 177, 63–68.doi:10.1016/j.icarus.2005.03.002.Nicholson, P.D., French, R.G., and Matthews, K. 1997. Eclipses and occultations of Saturn’ssatellites observed at Palomar in 1995. Paper presented at the 1997 Workshop on MutualEvents and Astrometry of Planetary Satellites, Catania, Italy.Spencer, J.R., Shure, M. A., Ressler, M.E., Goguen, J.D., Sinton, W.M., Toomey, D.W., Renault,A. and Westfall, J. 1990. Discovery of hotspots on Io using disk-resolved infrared imaging.Nature 348, 618-62I.
Squyers, S.W. and Veverka, J. 1981. Voyager photometry of surface features on Ganymede andCallisto. Icarus, 46, 137-155.Vasundhara, R., Arlot, J.-E., Lainey, V., and Thuillot, W. 2003. Astrometry from mutual events ofthe jovian satellites in 1997. Astron. Astrophys. 410, 337-341. doi: 10.1051/0004-6361:20030.Yoder, C.F. and Peale, S.J. 1981. The tides of Io. Icarus 47, 1-35.
Table 1Satellite number (1-4), ingress or egress (I/E), and seconds after the epoch of J20002I -289105670.61I -288239185.54I -286773407.03E -278930708.04E -278072095.22E -277121540.11E -276762150.51E -275997505.54E -272282254.24I -252037849.11I -251693173.43E -245498726.42E -245493991.62E -244879956.13E -244879625.01E -243427504.24E -243340500.31E -242815797.62E -242730782.02E -242730779.23E -241783903.33E -241783901.23I -241177178.31E -240827738.62I -223087969.41I -218511302.81I -218511299.92I -218482024.93I -217651226.52I -216946805.63E -209592901.21E -208411138.81E -208258216.81E -208258216.43I -207745330.43E -207735501.63I -206507032.42E -206498707.13E -206497321.91E -206270151.51E -203823300.71I -189304643.12I -187470101.41I -187469931.71I -185329455.11I -185329457.11I -184870778.82I -183785292.02I -183171161.01E -175076929.62E -174564911.22E -174564912.01E -174006451.51E -173241819.73I -173075383.93E -173067908.51E -172630103.71E -171253739.33I -168741228.13E -150779918.61I -150006986.22I -149395912.32I -149395909.01I -149395410.51I -149395409.2
1E -140518839.12E -140174018.71E -139142513.11E -139142515.13I -138405858.92E -138331502.11E -138072015.33I -136548404.13E -133443418.13I -119212965.81I -119118128.42I -117769250.22I -116541003.13E -116106840.71I -115448573.22I -115005618.01I -114836980.22I -114391457.01I -114378280.14I -113107430.54I -113107424.33I -113022168.02E -106703478.82E -106396330.32E -105782085.81E -104583707.24E -104411162.84E -104411168.11E -103971994.92E -103632267.91E -103513208.51E -103207350.74E -102963262.11E -101219246.51E -101219247.63E -100007368.33E -100007364.71I -85170774.23E -83907724.93E -83907723.51I -83641713.24E -82695037.03I -82063306.23E -82050257.62I -81227573.82I -81227571.52I -79999258.62I -79077971.91I -78442939.84I -76921953.23E -73381451.51E -72012214.42E -72003546.71E -71400513.74E -71113404.01E -70788806.41E -70330022.13E -70285069.23E -70285070.51E -70024170.01E -69718316.81E -68953663.71E -68647800.84E -68217716.52E -67396362.22E -64325162.32E -64325166.02I -50210731.03I -49862192.64E -49397593.84I -47962493.4
1I -45717768.91I -45717769.91I -44341570.31I -43729915.32I -43148144.22I -43148140.53I -43050600.73I -42431311.81E -38216747.02E -37609830.01E -37605049.01E -37605049.41E -37452124.03E -36846096.71E -36840417.53I -36238200.73E -36226830.72E -36074035.82E -35459715.41E -33475973.91E -33475976.12E -33309530.03E -29414644.12I -13665690.63I -13323250.93E -13314401.21I -13145086.31I -11768867.11I -10545557.72I -10288151.03E -10218353.83I -9607609.73I -9607611.53E -9599169.93E -9599167.91I -9016409.91I -9016408.11I -9016411.02I -8138616.02E -3829752.91E -3656353.71E -3503430.23E -3406758.01E -2891724.13I -2795253.43I -2795245.33E -2787460.43E -2787463.22E -2293956.21E -2280014.02E -1679626.22E 470546.01E 472715.71E 472715.12E 1084895.42E 3849466.33I 4017448.53E 4024745.93E 4024745.21I 19427370.32I 20115663.43I 20118532.82I 22878834.61I 23403089.82I 23492884.13I 23833772.53E 23841028.63E 23841028.11I 24167650.21I 24779307.12I 25028041.8
1I 25390964.32I 25642128.62I 26256225.73E 30652819.41E 31515674.61E 31668604.52E 31793336.72E 32100455.72E 32714740.81E 33656674.91E 33656672.43I 33741330.43I 33741325.93I 33741329.53E 33749324.93E 33749323.71E 34268392.51E 34268390.92E 34864837.33I 37456855.13I 37456853.53I 37456853.73E 37465206.93E 37465210.23E 37465212.12E 37629299.62E 37629298.61E 37632836.41E 37632836.82E 38243630.52E 38243632.61E 38244548.41E 38244550.22I 51746539.32I 51746536.63I 53555774.12I 53895683.91I 53986999.32I 54509709.62I 54509708.31I 55974808.32I 56658738.72I 56658739.23E 56661797.43E 56661794.23E 56661793.11I 58421345.11I 58727163.72I 58807763.51I 59338805.21I 59338805.52I 59421777.04I 60631889.44I 60631885.73E 64711564.91E 66228322.34I 66420804.92E 66493961.52E 66493965.83E 66569329.53E 66569325.53E 66569321.51E 66840030.63E 67188557.63I 67796308.13I 67796307.54I 67868208.84I 67868204.34E 67880251.74E 67880246.31E 68216381.5
371E 68216382.11E 68216383.23E 68427029.32E 68643711.11E 70204444.71E 72192494.42E 74786443.72E 74786443.82E 74786444.44E 89595031.83I 91323689.01I 92521340.91I 93285866.33I 94419130.72I 94428158.14I 95366668.31E 100022499.22E 100271893.22E 100271892.13E 100623831.23E 100623829.82E 100885995.91E 101092982.54E 101173634.91E 101857609.61E 102010537.44E 102621051.63E 104338917.41E 104763225.23I 104944887.11I 121728064.52I 122678959.52I 122678957.13E 124150095.14E 125775078.02E 134662727.12E 134662729.13E 134674328.23E 134674329.42E 135276777.61E 136415587.51E 136415587.72E 136504894.11E 136568508.61E 136568508.71E 137180219.64I 137339438.14E 137350242.52E 137426009.02E 137426008.41E 137638999.43E 137769928.23E 137769931.41E 138097780.51E 138403636.21E 138403632.91E 138556565.93E 139008250.01E 139015343.82E 139268282.63I 139615469.53I 139615473.21E 139779979.11E 139932901.51E 140085831.52E 140189449.01E 140391680.32E 140803566.32E 141724752.51I 159343946.41I 160872899.6
3I 161902935.32I 161981521.42I 163823714.92E 168745771.41E 168832179.21E 168832175.82E 169052791.91E 169138021.81E 169596792.21E 169596793.81E 170514350.11E 170514347.91E 171584834.01E 171584833.62E 171816018.23E 171817971.51E 172961191.52E 173044140.42E 173044141.23I 173047427.83I 173047425.93E 173056272.23E 173056267.12E 174579307.81E 174949249.43I 176762445.33I 176762446.61I 191455557.82I 191458173.53E 191628102.92I 193607642.51I 194666292.91I 194819184.52I 196064177.33I 196573160.13I 196573160.71I 196653934.91I 196653935.82I 196985387.93I 197192206.32E 202521634.52E 202828683.42E 203135733.13E 205247437.02E 205592102.03I 205859880.52E 205899143.61E 206142477.53E 206485774.41E 206754191.51E 206754191.01E 207060050.51E 207518839.91E 207518835.21E 207671772.81E 208130556.81E 208130553.82E 208355468.61E 208742276.83E 208962443.51E 209048137.11E 209506927.21E 210118644.82E 210197684.73E 210200750.63I 210813227.71E 211494997.43I 229385613.23I 230004613.83E 230012399.42I 230761079.2
2I 231989419.01I 232435185.22I 232603592.82E 236298019.01E 236724593.63E 236822895.32E 237219299.22E 237219297.81E 237336276.51E 237336278.11E 237489200.62E 237526376.82E 237833472.01E 238253817.71E 238559668.71E 238559668.21E 238712593.71E 238865518.72E 239061809.33I 239290996.03I 239290994.13I 239290997.02E 239675976.01E 240241865.01E 240700648.53I 241148506.51E 241312369.11E 241924090.92E 242439625.72E 242439629.81E 243300476.71E 243300479.11E 243606341.53I 244244302.42E 244589046.92E 244589045.81E 244676863.81I 258278488.61I 260266185.14E 261806705.12I 262080974.32I 262080973.81I 262253860.92I 262388029.13I 262817867.94I 263243583.53E 264067243.81I 264394441.33I 264675083.92I 264844688.41I 265617644.73E 270258941.63E 272735911.31E 272812718.12E 272839661.82E 272839659.11E 273118570.21E 273118571.31E 273271495.71E 273271497.62E 273761039.11E 273883204.23I 273962288.23I 273962285.53E 273974362.23E 273974362.11E 274189061.33I 274581507.93I 274581507.91E 274800775.64E 274836568.9
1E 275259569.42E 275603723.92E 276217937.94I 276269342.23I 276439218.33I 276439221.91E 276483017.91E 276483019.32E 276525034.13I 277677641.53E 277689981.03E 277689986.24E 277732333.61E 277859408.81E 279235808.31E 279235810.01I 292225927.91I 292837563.51I 292837561.62I 293095763.12I 296166289.22I 296166290.92I 296166288.13I 296253575.23I 296253576.11I 296813117.51I 297118924.03I 297491953.73I 298111110.31I 298953782.11I 298953781.11I 300177027.61I 300329936.91I 300482840.52I 300772550.92I 301079682.82I 301079682.64I 302326081.33E 306174021.11E 306301871.14E 306686709.74E 306686714.62E 307540067.81E 307678195.61E 309054549.93E 309270335.94I 309565798.63I 309876535.02E 309997237.12E 310918670.51E 311042657.03E 311128216.43E 311128216.23I 311734451.41E 312113184.61E 312419052.33E 312986035.8