Astrometry of mutual approximations between natural satellites. Application to the Galilean moons
B. Morgado, M. Assafin, R. Vieira-Martins, J.I.B. Camargo, A. Dias-Oliveira, A. R. Gomes-Júnior
MMon. Not. R. Astron. Soc. , 1–13 (2015) Printed 15 August 2018 (MN L A TEX style file v2.2)
Astrometry of mutual approximations between naturalsatellites. Application to the Galilean moons. (cid:63)
B. Morgado , † , M. Assafin , R. Vieira-Martins , , J.I.B. Camargo ,A. Dias-Oliveira , A. R. Gomes-J´unior Observat´orio Nacional/MCTI, R. General Jos´e Cristino 77, Rio de Janeiro RJ 20.921-400, Brazil Observat´orio do Valongo/UFRJ, Ladeira Pedro Antonio 43, Rio de Janeiro RJ 20080-090, Brazil
Accepted 2016 May 19. Received 2016 May 18; in original form 2016 April 08
ABSTRACT
Typically we can deliver astrometric positions of natural satellites with errors inthe 50-150 mas range. Apparent distances from mutual phenomena, have much smallererrors, less than 10 mas. However, this method can only be applied during the equinoxof the planets. We developed a method that can provide accurate astrometric data fornatural satellites – the mutual approximations. The method can be applied when anytwo satellites pass close by each other in the apparent sky plane. The fundamentalparameter is the central instant t of the passage when the distances reach a minimum.We applied the method for the Galilean moons. All observations were made witha 0.6 m telescope with a narrow-band filter centred at 889 nm with width of 15 nmwhich attenuated Jupiter’s scattered light. We obtained central instants for 14 mutualapproximations observed in 2014-2015. We determined t with an average precision of3.42 mas (10.43 km). For comparison, we also applied the method for 5 occultationsin the 2009 mutual phenomena campaign and for 22 occultations in the 2014-2015campaign. The comparisons of t determined by our method with the results frommutual phenomena show an agreement by less than 1-sigma error in t , typicallyless than 10 mas. This new method is particularly suitable for observations by smalltelescopes. Key words:
Methods: data analysis – Astrometry – Planets and satellites: individual:Io, Europa, Ganymede, Callisto.
The formation of planets occurs in a disk of gas and dustaround the protostar. The formation of regular satellitesaround a giant planet probably occurs in a similar way ina circum-planetary disk around the planet. However, thereare different models for this formation in the literature withpros and cons each (Crida & Charnoz 2012). The orbitalevolution of these regular satellites around a giant planetcan give us hints about their formation.The orbital studies of these celestial bodies demand theobservation of positions, relative distances or other forms ofobservables, like central instants and impact parameters inmutual phenomena, over extended period of time to fit themwith dynamic models (De Sitter 1928; Lieske 1987; Lainey etal. 2009). However, to detect very weak disturbance forces, (cid:63)
Based on observations made at the Laborat´orio Nacional deAstrof´ısica (LNA), Itajub´a-MG, Brazil. † E-mail: Morgado.fi[email protected] such as tidal forces , these observables have to be very precise.This high precision usually cannot be attained with classi-cal methods. For example, Jupiter and Saturn’s brightnessmakes it very difficult to obtain enough number of astro-metric calibration stars in a CCD frame in order to obtaina good individual position for its main satellites. The clas-sical CCD astrometry of a single satellite furnishes posi-tions with precisions (usually the standard deviation from aset of a few hundreds of images) in the range of 50 to 150mas (Stone 2001; Kiseleva et al. 2008). This issue was par-tially solved using relative positions between two (or more)satellites (Veillet & Ratier 1980; Veiga et al. 1987; Veiga &Vieira-Martins 1994; Harper et al. 1997; Vienne et al. 2001;Peng et al. 2008). More recently Peng et al. (2012) achievedrelative positions with precision of 30 mas.Some of these works use the ephemeris of the satellitesas a ruler in order to obtain the reference system orientationand scale in the CCD image. Often, one uses the positionsof bodies with more precise ephemeris to determine the po- c (cid:13) a r X i v : . [ a s t r o - ph . E P ] M a y B. Morgado et al. sition of another body with a worst ephemeris. An exampleis the case of Miranda of Uranus, when the ephemeris ofthe others satellites (mainly Oberon) were used as referenceframe in the position reductions (Veillet & Ratier 1980).Another possibility is the use of the so called precision pre-mium (Peng et al. 2008), first pointed out by Pascu (1994).The idea is that the accuracy of the determination of ap-parent distances is remarkably improved for short apparentdistances (less than 85”), as instrumental and astronomicalissues tend to affect the images of both satellites in the sameway. In principle, we can use the ephemeris distance betweentwo small-separated satellites and the actual observed dis-tance to determine the scale and orientation of the image.This method was mainly used for the Saturnian main satel-lites (Harper et al. 1997; Vienne et al. 2001) and for theGalilean satellites (Peng et al. 2012).Another way to obtain very precise relative positions oftypically a few mas is with the observation of mutual phe-nomena. However this can only be done during the equinoxof these planets – see for example Assafin et al. (2009);Emelyanov (2009); Emelyanov et al. (2011); Dias-Oliveiraet al. (2013); Arlot et al. (2014). Mutual phenomena consistof occultations and eclipses between natural satellites, asseen along the line of sight of an observer. One satellite (orits projected shadow) hides the other causing a drop in thelight flux which can be measured with high precision usingdifferential photometry in a small FOV, when astronomicaland instrumental systematic errors in the measurements ofthe targets and calibration sources tend to cancel out. Thisflux variation is connected to the relative apparent motionof the satellites in the sky, and thus, ultimately, to theirorbits. The apparent relative motion between the satellitesunder mutual phenomena can be described in terms of theinstant of minimum apparent distance (central instant ofthe event), the minimum apparent distance at this instant(impact parameter), and the relative apparent velocity be-tween one satellite (or shadow) and the other (Emelyanov(2003); Dias-Oliveira et al. (2013) and references therein).The particular nature of orbital geometry in mutual phe-nomena offers valuable constrains in the orbital solutions ofthe satellites (Lainey et al. 2004). However, these phenom-ena can be observed only every 6 years for Jupiter, every 15years for Saturn and every 42 years for Uranus (Arlot et al.2012, 2013, 2014).Inspired in the mutual phenomena and based in thesame orbital geometry, we propose the observation of whatwe call mutual approximations between natural satellites inorder to obtain the central instant when the apparent dis-tance between two satellites reaches a minimum.In mutual approximations the bodies only approacheach other in the sky plane at adequate, not too short, dis-tances – we actually avoid occultations, we wish to be able tomeasure the satellites separately. Similarly to the geometricparameters of mutual phenomena, we can also determine theimpact parameter and the relative velocity, but, in this case,only in pixel units. For orbit fitting, these supplementary pa-rameters can only be useful if we can accurately convert thedistances from pixels to arc seconds using a standard refer-ence frame, like an astrometric star catalogue representativeof the ICRS. However, this is frequently not the case, as usu-ally an insufficient number of reference stars is available inthe FOV, for two reasons: short FOV and/or short exposure times to avoid saturation of the satellite images and lightscatter from the central planet. One way to overcome thisproblem is to observe a crowded star field nearby and deter-mine the pixel scale to be used. Another alternative is to usean ephemeris of the pair of satellites as the reference frame.But, by doing that we scale the observed impact parameterand the relative velocity to the used ephemeris, so no trullyindependent new results are really obtained, although theymay serve to check parameters for internal consistency (seemore details and computations in this respect in Section 3).In fact, Arlot et al. (1982) were the first to suggest suchastrometric approach. Also, Mason et al. (1999) reported ob-servations which they called ”close pairings” of the Galileanmoons with a speckle interferometer mounted on the 26-inchrefractor at the Naval Observatory in Washington, DC. Butno further references, results or studies in this subject werepublished in the literature to our knowledge.We give in this paper a complete description of themethod of mutual approximations and present the resultsof its application to an observational campaign of mutualapproximations for the Galilean moons carried out in 2014-2015. Our observations were made with the 0.6 m Zeiss tele-scope of the Observat´orio do Pico dos Dias (OPD, Brazil)with a narrow-band methane filter centred at 889 nm withwidth of 15 nm that attenuates Jupiter’s scattered light.The paper is arranged as follows. In Section 2, we de-scribe the theoretical model of relative motion for the mu-tual approximations and how to obtain the central instantof the event. In Section 3 we show also how to computethe impact parameter and relative velocity. In Section 4 wediscuss what astronomical effects like solar phase-angle, at-mospheric refraction and aberration could cause an offsetin the raw measured central instant. In Section 5 we detailthe observational campaign of the mutual approximationsfor the Galilean satellites, the measurement of the imagesand our results. In Section 6 we compare the results frommutual approximations and mutual phenomena simultane-ously observed during the mutual phenomena campaign of2009. For future comparisons, we also give the results of theapproximations observed during the 2014-2015 equinox ofJupiter. Discussion and conclusions are presented in Section7.
We set here a geometric model that describes the variationwith time of the apparent distance d in the plane of the skybetween two satellites for a short arc of their orbits, typi-cal of mutual approximations. We assume that there is nostrong deformation from the telescope optics on the images.A simple description by a polynomial in t of arbitrary degree n (typically n = 4) is obtained using d . In mutual approx-imations, d always reaches a minimum and the polynomialcurve will present a positive concavity.The same geometric parameters that describe mutualphenomena also describe mutual approximations, i.e. cen-tral instant, impact parameter and relative velocity. Noticehowever that only the central instant can be determined in- c (cid:13) , 1–13 utual approximations: application to the Galilean moons. dependently of any reference frame, even without the knowl-edge of the pixel scale of the images. But astronomical cor-rections such as refraction, aberration and solar phase anglemust be taken into account (see Sect. 4). However, if onedoes want to get all the parameters, one needs to convertdistances from pixels to arcseconds. In this case, the pixelscale must be computed. (see Sect. 3).The vector distance between the two satellites in thesky plane ∆ (cid:126)r ( t ) in a specific instant t is:∆ (cid:126)r ( t ) = (cid:126)r ( t ) − (cid:126)r ( t ) (1)Assuming that the square of the distance between twosatellites in the short arc of their orbits, nearby the centralinstant of the approximation, can be described as a polinonmof arbitrary power n in time, then, considering (1) we obtain: d ( t ) = ∆ (cid:126)r ( t ) . ∆ (cid:126)r ( t ) d ( t ) = a + a t + a t + ... + a n t n (2)where the parameters a , a , a n are related with the kine-matics of the satellites’ motion.Using least squares we fit this model with observed dis-tances. Alternatively, we can fit ephemeris’ apparent dis-tances to study the event and even for testing the most ad-equate polynomial power to use in the fitting.The central instant t of the mutual approximation isobtained when the first time derivative of equation (2) is setequal to zero. a + 2 a t + 3 a t + .. + na n t n − = 0 (3)The central instant t comes from the determination ofthe root of Eq. (3). Using an interactive process we deter-mine the central instant, subtract it in t and then redo theadjustment in order to set t = 0. The central instant errorcomes from around-zero derivative of Eq. (3) as a functionof the error of the other coefficients, computed in the leastsquares fit. This error is given by Eq. (4): δt = − δa a (4)Note that the numerator of Eq. (4) relates to the ob-served distance errors, thus it reflects as expected the noisein the observed apparent distance curve, related with theatmospheric conditions of the night. The error of the cen-tral instant is also dependent on the curve concavity relatedto a , which in turn is dominated by the relative velocity.A smaller central instant error is expected from a more pro-nounced minimum which comes from larger a values, orhigher relative velocities.In practice the observational procedure in mutual ap-proximations consists in observing a number of images, anhour before and after a central instant foreseen from anephemeris. We then measure the apparent distances and,with the help of any ephemeris, apply corrections due to so-lar phase angle, refraction and aberration that could shiftthe distances in a way that could offset the central instant(see Sect. 4). The instant errors can be translated from sec-onds of time to mas or km by using the relative velocity anddistance to the observer given by an ephemeris. In Section Figure 1.
Apparent sky distances between Io and Ganymede inthe mutual approximation of February, 19th 2014. We used theNOE-5-2010-GAL ephemeris obtained in its topocentric form forOPD. Ephemeris distances d are in black and model fitted onesin green. We took ephemeris positions every second. Notice thatthe fit is actually done in d , using Eq. (2). fromLainey et al. (2009). In the Figure 1 we computed the ap-parent distances from the ephemeris and fitted them witha fourth degree polynomial. The comparison between thefitted (green line) and ephemeris (black dots) apparent dis-tances. The residual dispersion is also illustrated in the bot-tom (red crosses), in the sense model minus ephemeris . Thetime resolution of the ephemeris was one second.As seen in Figure 1, our model fits quite well the satelliteephemeris. The residual of the adjustment stays in 1 mas,corresponding to the numerical limit of extracted ephemerispositions, here truncated for computational purposes (noticethat the actual ephemeris precision is worse than this). InFigure 2 we illustrate the discrepancy between the value forthe central instant in mas obtained using different polyno-mial models in time for d . Models with power less than the Institut de M´ecanique C´eleste et de Calcul des ´Eph´em´erides;Website: c (cid:13) , 1–13 B. Morgado et al.
Figure 2.
Central instant offset for each polynomial fitting. Afterthe fourth degree there is no significant improvement in accuracy. fourth degree give incomplete descriptions of the satellite’srelative motion. We show that after the fourth degree thereis no significant improvement in the precision or accuracyof the model, for this example. Others tests showed similarresults.
Although the determination of the central instant comesfrom fits to the raw measured apparent distances given inpixel units (see Sect. 2), the UTC time is recorded in theobservations with a GPS system or precise internet timing.Thus, the central instant derived in UTC is the only trulyindependent parameter obtained in mutual approximationswithout regard to any reference frame, be it a star catalogueor an ephemeris. For this reason, the central instant is themain parameter derived from the mutual approximations.The impact parameter and relative velocity are also de-rived from fits to the observed apparent distances, but inthis case they share the same metric. Thus, these parame-ters can only be obtained in standard units if the observeddistances are also given in standard metrics, such as arcsec-onds. A conversion from pixels to arcseconds must be made.This can only be accomplished with the use of a referenceframe.Notice that if we use a star catalogue as reference frame,the obtained impact parameter and relative velocity are in-dependent from any ephemeris. In this case, like in mutualphenomena, they may be as useful as the central instant inorbit fit work. This is possible if we have enough cataloguereference stars and can make the right ascension and decli-nation reduction of the FOV. If this is not the case, we canalternatively reduce a nearby field with a sufficient numberof reference stars and use the derived pixel scale in the FOVof the satellites to compute the observed distances. Unfortu-nately, this is rarely the case of Jupiter and Saturn. They arecurrently crossing a sky path with few stars. Besides, due to the brightness of their main moons and their proximity tothe planet, the very short exposure times used prevent theimaging of reference stars. But Uranus may be a good andpromising example for the use of this procedure.We can also use an ephemeris to obtain the pixel scale,see Peng et al. (2008). This is very useful when we haveno reference stars, or even when only the two satellites areavailable in the FOV. This is frequently the case of the obser-vations of Jupiter and Saturn moons. The drawback is thatthe observed distances are scaled according to the ephemerisframe, so that the impact parameter and the relative veloc-ity will be dependent of the used ephemeris to some extent,and may be not useful in orbital works. Even so, it is alwaysa good practical procedure to compute the impact param-eter and relative velocity using the ephemeris as reference,for it serves as an extra checking procedure of the results.For this reason, we describe this procedure in Sect. 3.2.
Since the observed satellites images are affected by solarphase angle, aberration and refraction, these effects aretaken into account prior to fitting the observed distanceswith the topocentric ephemeris ones (see Sect. 4).From Eq. (2), when t = 0, we see that the minimumdistance between the satellites, the impact parameters d ofthe approximation, will be related with a in the form of d = √ a (5)and the incertitude of this parameter is δd = δa d (6)The relative velocity is determined as follows: Using twoconsecutives images we determine the instantaneous varia-tion of the relative position in x and y in the CCD frame.With the acurate time of each image we can determine therelative instantaneous velocity between both frames. Thenwe fit a linear function in the velocity curve in time obtainedwith all the images. The relative velocity of interest is theone for the central instant t and its error is obtained fromthe linear fit.Notice that, from our fit, these parameters are in pixelunits and pixels per second. The conversion to arcsecondunits is explained in the following Section. Knowing from an ephemeris the theoretical apparent dis-tances between the satellites (∆ α cos δ, ∆ δ ) in arcseconds,affected by solar phase angle, aberration and atmosphericrefraction, and the instrumental distances (∆ x, ∆ y ) in pix-els, makes it possible to obtain the pixel scale. Notice thatwe projected the ephemeris on the tangent plane using thegnomonic projection, so that the standard coordinates (X,Y)are used in the computation of the pixel scale.We compute the pixel scale P s as the slope of a linear c (cid:13) , 1–13 utual approximations: application to the Galilean moons. Figure 3.
Pixel-scale determined for the approximation betweenIo and Ganymede of February, 19th 2014 in arcseconds perpixel as a function of the time. We used the NOE-5-2010-GALephemeris obtained in its topocentric form for OPD. function fitted to the ratio between the ephemeris distances d e and instrumental distances d o (Eq. 7). P s = d e d o (7)Figure 3 illustrates the variation of the pixel scaleovernight for the mutual approximation between Io andGanymede which occurred at February, 19th, 2014. Forillustration, in this example we computed an error of0 . mas.pixel − for the pixel scale.For comparison, we determined all the parameters ofa mutual approximation, central instant, impact parame-ter, relative velocity and the pixel-scale using two differentephemeris. We chose the NOE-5-2010-GAl provided by IM-CCE from Lainey et al. (2009) and the JPL ephemeris, jup Astronomical effects such as the solar phase angle, the at-mospheric refraction and aberration affect the apparent dis-tance between the satellites. They must be taken into ac-count when we want to obtain all the parameters, includingthe impact parameter and relative velocity, as described inSect 3. But which of these effects can actually offset the cen-tral instant in a mutual approximation? And to what typicalamounts? We address these question in this Section. Jet Propulsion Laboratory;Website: http://ssd.jpl.nasa.gov/
Table 1.
Comparison between the results for the central in-stant, t , impact parameter, d and relative velocity, v r for Ioand Ganymede mutual approximation of February, 19th 2014 andthe calibration parameter, pixel-scale P s , using two ephemeris –NOE-5-2010-GAL and jup t ( hh : mm : ss ) 23:46:35.79 (0.76) 23:46:31.68 (0.00) d ( mas ) 12982.89 (6.08) 12995.12 (0.12) v r ( mas/s ) 7.53 (0.08) 7.54 (0.00) P s ( (cid:48)(cid:48) /px ) 0.3729 (0.0009)Parameters JPL – jup t ( hh : mm : ss ) 23:46:35.79 (0.76) 23:46:35.22 (0.00) d ( mas ) 12974.24 (6.07) 12996.66 (0.12) v r ( mas/s ) 7.52 (0.08) 7.58 (0.00) P s ( (cid:48)(cid:48) /px ) 0.3728 (0.0009) In the case of the solar phase angle, the photocentreof a satellite in an image is shifted relative to its geometriccenter, the center of mass, due the solar phase angle. Ac-cording to Lindegren (1977) a spherical object with a lightscattering in its surface causes an offset in its positions ac-cording to Equations (8) where i is the solar phase angle, r is the apparent radius of the satellite, Q is the positionangle of the sub-solar point in the tangential plane and C ( i )is a parameter related to the reflectance model adopted. (cid:18) ∆ α cos δ ∆ δ (cid:19) = C ( i ) r sin( i/ (cid:18) sin Q cos Q (cid:19) (8)Here we adopted the Lambertian sphere modelling (Lin-degren 1977). But since the satellites’ radii may be different,and considering that the direction of the Sun and of the rel-ative motion may not coincide, the relative distances of thephoto and geometric centres may vary differently, so thatthe associated central instants may be different too, causingan offsset in the observed central instant which is associatedto the photo and not to the geometric center. For example,in the case of the Galilean moons, for an approximation witha phase angle of 10 degrees the central instant can be shiftedby up to 6 seconds, a significant value.The atmospheric refraction causes a shift in the tar-get position towards the zenith direction that increases athigher zenital distances. However, once both satellites havealmost the same zenital distance and very close positions inan approximation it has a small, but non-negligible effect inthe distances and in the central instant (less than 2 secondsfor z ≈ o ). More details about its implementation can befound in Stone (1996).The aberration causes a position shift toward the di-rection of the instantaneous velocity vector of the observer(Green 1985). Due to the small apparent distance betweensatellites, the effect is very small in the central instant, lessthan 0.0005 seconds for diurnal aberration and 0.0145 sec-onds for annual aberration. c (cid:13) , 1–13 B. Morgado et al.
Figure 4.
Inter-satellite apparent sky distances of the Galileansatellites over a week. I stands for Io, E for Europe, G forGanymede and C for Callisto. Every minima in this graphic is amutual approximation between two satellites. We used the NOE-5-2010-GAL ephemeris obtained in its topocentric form for OPD.
The accurate prediction of mutual phenomena depends onhow precise the ephemeris is and upon our knowledge of thesize of the satellites. In the absence of one of these condi-tions, we may miss an event foreseen to be visible at a certainlocation. None of these conditions matter in our case. Onecan easily predict mutual approximations, even using poorephemeris with precision in the arcsecond level only. By set-ting a threshold value of at least some arcseconds, we avoidselecting possible mutual occultations or situations wherethe apparent distances between the satellites are too smallfor centroid measurement purposes.The observational campaign for the mutual approxima-tions between the Galilean satellites was carried out in Brazilbetween 2014 and 2015. The predictions for these eventswere made with the topocentric ephemeris for the OPDobservatory using the NAIF SPICE platform, ephemerisNOE-5-2010-GAL, derived from Lainey et al. (2009), andDE430. Figure 4 plots the inter-satellite apparent distances,in arcseconds, over a week for the six possibles combina-tions between these satellites. Every minima is a mutual ap-proximation. However, in order to not pick up a prohibitivenumber of events, we only selected the approximations forwhich the impact parameter was smaller than 30 arcseconds,improving even further the precision premium.We selected all the visible events for the OPD obser-vatory with elevation above 20 degrees and with a distanceto the Jupiter’s limb greater than 10 arcseconds. The pre-dictions were spread in 58 nights, with 65 approximations,selected in fifteen months between 2014 and 2015. We at- Website: http://naif.jpl.nasa.gov/naif/
Figure 5.
Image of Jupiter, Io and Europa obtained with the0.6 m diameter Zeiss telescope, equipped with a methane filter inFebruary 3 th of 2014. The planet and the satellites show about thesame brightness due to the use of the narrow-band filter, centredat λ = 889 nm with 15 nm width. tempted observations for all these events. From these, only36 events could be observed with success, the others werelost due to bad weather conditions. Our observations were made at Observat´orio do Pico dosDias (OPD, IAU code 874) located at geographical longi-tude − o (cid:48) (cid:48)(cid:48) , latitude − o (cid:48) (cid:48)(cid:48) and an altitude of1864 m . The telescope used was the 0 . m diameter Zeisstelescope. It is a manual pointing Cassegrain telescope withfocal ratio f/12.5.For all the observations in this work the time, in UTC,was calibrated by a GPS and recorded in the FITS image’sheader. For observations which the GPS is not an option, atime’s calibrator software can be used such as Dimension 4 . Tests comparing this software with GPS indicate that thetime precision is on the order of 20 ms .The CCD camera utilized in all observations was theAndor Ikon-L with 2048 x 2048 square pixels of 13.5 µm .This camera added to the Zeiss telescope has a field of viewof 12.63’ x 12.63’. The filter chosen was the narrow-band fil-ter centered at 889 nm (region of methane absorption), witha width of 15 nm . In this specific wavelength, the methane inJupiter’s upper atmosphere strongly absorb the light caus-ing the planet’s albedo to drop to 0.1 in this spectral regionas pointed out by Karkoschka (1994, 1998).Although the observations in this wavelength are veryefficient to eliminate the scattered light from Jupiter, thealbedo of the Galilean satellites did not change much(Karkoschka 1994). Because of this, the brightness of Jupiteris nearly the same of that of the satellites in this wavelengthas can be seen in the Figure 5.We summarized the specifications of the telescope, cam-era and filter utilized in these observations in Table 2.Table 3 contains the observational characteristics forthe 36 nights analysed in this work, where 14 are mutual Website: Website: c (cid:13) , 1–13 utual approximations: application to the Galilean moons. Table 2.
Specifications of the telescope, CCD camera and filter.Diameter of primary mirror 0 . m Focal ratio f/ . . (cid:48) x 12 . (cid:48) Pixel size 13 . . µm Size of CCD array 2048 x 2048Pixel-scale 0 . (cid:48)(cid:48)
34 or 0 . (cid:48)(cid:48) pixel − Filter 889 nm (∆ = 15 nm ) approximations (See Section 5.4) and 22 are mutual approx-imation observed from mutual occultations data of the 2014-2015 mutual phenomena campaign (See Section 6.2). We listthe mutual approximation targeted, the seeing of the night,the zenithal distance ( z ) for the central instant, the solarphase angle ( i ) and the sub-solar point in the tangentialplane ( Q ). The solar phase angles and the sub-solar point inthe tangential plane are the same for both satellites and donot change during the approximation, which lasts typicallyless than two hours. The observational information aboutthe 5 nights in the 2009 mutual phenomena campaign uti-lized in this work can be found in Dias-Oliveira et al. (2013)(See Section 6.1). Firstly, all the images were corrected by bias and flat-fieldby means of standard procedures using the IRAF package(Butcher & Stevens 1981). The centroid of the satellites weredetermined using the PRAIA package described in Assafin etal. (2011). The PRAIA package measures the satellite coor-dinates (x, y) in the image with a two-dimensional circularsymmetric Gaussian fit within a radius of one Full WidthHalf Maximum (FWHM = seeing). The typical error in thecentroid measurement for our images was 12 mas. Figure 6is the normalized histogram for the centroid determinationerror, in mas, for x , y and r = (cid:112) x + y , from all measuredimages. We used the model described in Section 2 to determine thecentral instant t of the observed mutual approximations.The current sky path of Jupiter is not crowded of stars. Also,Jupiters brightness made us use a short exposure time anda narrow-band filter. Because of that there was not enoughnumber of reference stars in the images for an usual CCDastrometry. Even so, for evaluation purposes, we also deter-mined the impact parameter d and the relative velocity v r following the procedures given in Sect. 3 with the help of anephemeris.We separate our results in two different groups. Group 1contains our best results. It consists of observations made ingood sky conditions with no gaps along the event. Group 2has gaps in the distance curve, which may present more noisethan in Group 1 due to poor atmosphere conditions. Someobservations in the later group were virtually made during Website: http://iraf.noao.edu/
Table 3.
Approximations and observation conditionsDate Event Seeing z i Q (d-m-y) (arcsec) ( o ) ( o ) ( o )03-02-2014 IaE 1.9 56.74 5.80 275.4405-02-2014 EaG 2.1 49.58 6.27 275.3607-02-2014 IaE 1.8 51.87 6.61 275.2819-02-2014 IaG 2.8 45.76 8.39 274.8427-02-2014 IaE 1.9 47.23 9.32 274.7118-03-2014 IaE 2.4 46.68 10.73 274.6307-04-2014 IaE 2.4 51.73 10.99 275.1520-04-2014 EaG 2.3 52.12 10.58 275.7521-04-2014 GaC 2.2 51.38 10.54 275.7721-04-2014 IaG 2.2 64.52 10.54 275.7715-10-2014 GaC 1.3 66.32 9.70 108.2615-10-2014 IaE 1.3 66.32 9.70 108.2629-10-2014 IaG 1.5 54.46 10.45 108.6702-11-2014 IaC 2.0 68.93 10.60 108.6719-11-2014 EaC 1.3 41.34 10.69 108.8602-02-2015 GaC 1.7 42.82 1.01 98.0322-02-2015 IaE 1.7 39.58 3.17 290.4724-02-2015 IaG 1.8 40.01 3.59 289.9926-02-2015 IaE 1.6 64.54 4.14 289.4727-02-2015 GaC 1.6 40.35 4.16 289.4627-02-2015 IaG 1.6 40.60 4.17 289.4503-03-2015 IaG 1.6 56.01 4.95 288.9024-03-2015 GaC 2.2 41.53 8.19 287.3825-03-2015 IaE 1.6 40.65 8.46 287.2702-04-2015 IaE 1.5 52.27 9.14 287.0711-04-2015 EaG 2.1 41.50 10.07 286.8213-04-2015 IaE 1.5 43.80 10.21 286.7917-04-2015 IaC 1.4 45.35 10.42 286.7218-04-2015 GaC 1.4 61.45 10.42 286.7218-04-2015 IaG 1.5 45.31 10.45 286.7319-04-2015 EaG 1.4 59.89 10.46 286.7321-04-2015 IaE 1.4 68.34 10.57 286.7025-04-2015 IaG 2.7 48.86 10.41 286.7026-04-2015 IaE 2.0 40.67 10.74 286.6929-04-2015 IaG 1.9 57.72 10.81 286.7003-05-2015 IaE 2.0 52.39 10.84 286.71 Note.
For each event, we have the day, month and year, thesatellites designated by their initials (capital letters), where ’a’stands for approximation. We also give the seeing, the zenithaldistance ( z ), the solar phase angle ( i ) and the position angle ofthe sub-solar point in the tangential plane ( Q ). the 2014-2015 Jupiter equinox. Although they were not mu-tual occultations, the satellites approached each other byvery small distances, about 2 arcseconds. This campaign started in the beginning of 2014. Since themutual phenomena started in the middle of the same year,only a few mutual approximations were observed withoutgaps in the center of the curves due to the apparent prox-imity of the satellites. The group 1 is composed of 8 events.These approximations are similar to the ones that will beobserved after the mutual phenomena campaign, this makesthem the focus of this paper.As an example we display the approximation between Ioand Ganymede that occurred in February, 19th of 2014. Thecomparison between the fitted (green) and observed (black) c (cid:13)000
For each event, we have the day, month and year, thesatellites designated by their initials (capital letters), where ’a’stands for approximation. We also give the seeing, the zenithaldistance ( z ), the solar phase angle ( i ) and the position angle ofthe sub-solar point in the tangential plane ( Q ). the 2014-2015 Jupiter equinox. Although they were not mu-tual occultations, the satellites approached each other byvery small distances, about 2 arcseconds. This campaign started in the beginning of 2014. Since themutual phenomena started in the middle of the same year,only a few mutual approximations were observed withoutgaps in the center of the curves due to the apparent prox-imity of the satellites. The group 1 is composed of 8 events.These approximations are similar to the ones that will beobserved after the mutual phenomena campaign, this makesthem the focus of this paper.As an example we display the approximation between Ioand Ganymede that occurred in February, 19th of 2014. Thecomparison between the fitted (green) and observed (black) c (cid:13)000 , 1–13 B. Morgado et al.
Figure 6.
Normalized histogram for the centroid determinationerror, in mas, for x , y and r = (cid:112) x + y , from all measuredimages. Figure 7.
Apparent sky distances between Io and Ganymede inthe mutual approximation of February, 19th 2014. Inter-satellitedistances, d , are in black and model fitted ones in green. Theexposure time utilized was 3 seconds. Notice that the fit is actuallydone in d , using Eq. (2). apparent distances is shown in Figure 7. The residual dis-persion is also illustrated in the bottom (red crosses), in thesense ”fitted minus observed” .We show in Table 4 the central instant, in hours, min-utes and seconds in UTC and the error in seconds. We alsolist the error in milliarcseconds by the use of the ephemerisrelative velocity in milliarcseconds per second. These valuesare listed in column E t . We compared the results with theephemeris published by the IMCCE, currently consideredthe most accurate representative for the Jovian system, inthe sense ”observations minus ephemeris” .The average precision obtained for the central instant is0.56 seconds or 3.42 mas (using the relative velocity in eachevent obtained with the ephemeris). Table 4.
The central instant for the Group 1 mutual approx-imations and comparison with the ephemeris (see text in Sect.5.4.1).Date Event t E t ∆ t (d-m-y) (hh:mm:ss) (mas) (s) (mas)03-02-14 IaE 03:18:47.42 (0.20) 1.55 +4.37 +33.9205-02-14 EaG 23:27:50.65 (0.66) 4.04 +2.86 +17.5319-02-14 IaG 23:46:35.40 (0.76) 5.78 +2.31 +17.5927-02-14 IaE 22:34:27.89 (0.10) 0.79 –1.16 –9.1607-04-14 IaE 22:35:27.89 (0.19) 1.46 –0.35 –2.6820-04-14 EaG 21:47:40.57 (0.52) 1.85 –9.70 –34.4721-04-14 GaC 21:41:53.56 (1.01) 4.74 +1.71 +8.0421-04-14 IaG 23:13:56.69 (1.01) 5.45 –2.73 –14.74 Note. t is the central instant of the mutual approximations. Col-umn E t lists the central instant error in mas by the use of theephemeris relative velocities in mas per second. For each event,we have the day, month and year, the satellites designated bytheir initials (capital letters), where ’a’ stands for approximation.The column ∆ t is the comparison between the observations andthe ephemeris, here the NOE − − − GAL from IMCCEplus DE430, derived from Lainey et al. (2009), in the sense ”ob-servations minus ephemeris” in seconds and in mas by the useof the relative velocities in each event.
A mutual approximation can last a couple of hours. Thismakes it possible to acquire many observations and fit themodel even without the full coverage of the approximation.Logically, these results will not be as good as if we couldobserve all the event. We have 6 mutual approximations inthis group. A natural cause for the random observationalgaps in the curves were bad weather or very bad seeing.Two examples can be displayed. One is when theweather prevented the observation of part of the event suchas in the approximation between Io and Europa that oc-curred in February, 07th of 2014. In Figure 8 we show thecomparison between the fitted and observed apparent dis-tances. Notice the absence of positions in the left side of theevent.The second example is when the approximation oc-curred during the mutual phenomena campaign at theplanet’s equinox. Near the central instant of the approxi-mation, the satellites are so close together that it is impos-sible to obtain a centroid for each satellite in the images.An example is the approximation between Ganymede andCallisto that occurred in February, 27th of 2015. In Figure9 we show the comparison between the fitted and observedapparent distances. Notice the absence of positions near thecentral instant.We show in Table 5 the central instant for these sixapproximations, in the same manner as in Table 4 for theGroup 1. We also compared the parameters with the IMCCEephemeris.The average precision obtained for the central instantis 2.02 seconds or 14.15 mas (using the relative velocity ineach event obtained with the ephemeris). Notice that evenhere the precisions are still much better than those obtainedwith usual CCD astrometry.How does the gaps in these distance curves affect the de- c (cid:13) , 1–13 utual approximations: application to the Galilean moons. Figure 8.
Apparent sky distances between Io and Europa inthe mutual approximation of February, 07th 2014. Inter-satellitedistances, d , are in black and model fitted ones in green. Theexposure time utilized was 3 seconds. Notice that the fit is actuallydone in d , using Eq. (2). Figure 9.
Apparent sky distances between Ganymede and Cal-listo in the mutual approximation of February, 27th 2015. Inter-satellite distances, d , are in black and model fitted ones in green.The exposure time utilized was 3 seconds. Notice that the fit isactually done in d , using Eq. (2). termination of the central instant? We addressed this ques-tion by simulating gaps in the Group 1 approximations re-ported in the previous section. As example, we illustratethe simulation using the event between Io and Ganymede inFebruary 19th 2014. Similar conclusions are drawn for theother simulations. The event took 1 hour, thus a 6-minutesgap represents 10% of the entire curve.We explore two scenarios. (1) Removing points fromthe beginning, or end, of the curve. This affects the centralinstant error, even though the central instant value is close(within the errors) to the one obtained with the completecurve (Figure 10); (2) Removing points in the central part of Table 5.
Central instant for the Group 2 mutual approximationsand comparison with the ephemeris (see text in Sect. 5.4.2)Date Event t E t ∆ t (d-m-y) (hh:mm:ss) (mas) (s) (mas)07-02-14 IaE 23:00:17.29 (4.94) 40.97 –1.38 –11.4618-03-14 IaE 22:43:23.26 (2.52) 16.06 –7.93 –50.5427-02-15 GaC 01:24:10.29 (1.42) 8.14 +0.71 +4.0711-04-15 EaG 22:07:13.35 (1.13) 5.76 –2.60 –13.2513-04-15 IaE 23:44:40.98 (1.12) 8.60 –1.78 –13.6719-04-15 EaG 01:17:48.89 (1.00) 5.15 –4.10 –21.12 Note.
Same as in Table 4.
Figure 10.
Simulation of gaps in the beginning of the Group1 mutual approximation between Io and Ganymede of February,19th 2014. Offsets are in the sense ”with gap minus without gap” . the curve. The central instant precision and value are nearlyunaffected (Figure 11). The offsets are consistent within theerrors.These simulations confirm the slight deterioration ob-served in the errors of the parameters of the incomplete mu-tual approximations of Group 2, as compared to those fromthe Group 1. But the simulations also indicate that the ob-tained central instant are practically unaffected with regardto the ones that would be derived with complete curves,within the errors. Therefore, in principle even incompletecurves of mutual approximations should not be discarded. In Table 6 we list the impact parameters and the relativevelocity of our 14 mutual approximations. They are listedin milliarcseconds, and milliarcseconds per second, respec-tively. We also list the comparison of these parameters withthe ephemeris in the sense ”observations minus ephemeris” .Both observed and ephemeris results were obtained by fit-ting the model of Section 3.1 to the respective observed andephemeris apparent distances. The pixel scale determined c (cid:13)000
Simulation of gaps in the beginning of the Group1 mutual approximation between Io and Ganymede of February,19th 2014. Offsets are in the sense ”with gap minus without gap” . the curve. The central instant precision and value are nearlyunaffected (Figure 11). The offsets are consistent within theerrors.These simulations confirm the slight deterioration ob-served in the errors of the parameters of the incomplete mu-tual approximations of Group 2, as compared to those fromthe Group 1. But the simulations also indicate that the ob-tained central instant are practically unaffected with regardto the ones that would be derived with complete curves,within the errors. Therefore, in principle even incompletecurves of mutual approximations should not be discarded. In Table 6 we list the impact parameters and the relativevelocity of our 14 mutual approximations. They are listedin milliarcseconds, and milliarcseconds per second, respec-tively. We also list the comparison of these parameters withthe ephemeris in the sense ”observations minus ephemeris” .Both observed and ephemeris results were obtained by fit-ting the model of Section 3.1 to the respective observed andephemeris apparent distances. The pixel scale determined c (cid:13)000 , 1–13 B. Morgado et al.
Figure 11.
Simulation of gaps in the central part of the Group1 mutual approximation between Io and Ganymede of February,19th 2014. Offsets are in the sense ”with gap minus without gap” . is also listed. The nominal pixel scale of the instrumentalset was 0.34 or 0.37 mas per pixel depending of the instru-mental configuration of the night. The last column is anidentification flag where 1 stands for ”Group 1 mutual ap-proximations” and 2 stands for ”Group 2 mutual approxi-mations”. Notice that, in the near future, a new reductioncan be made using more precise ephemeris. This will allowthe confirmation of ours results. Mutual approximations and mutual phenomena - occulta-tions in particular - share the same concepts of orbital ge-ometry, though based in very distinct measuring techniques,with the last being a consolidated and most precise methodfor measuring distances between natural satellites. It wouldbe very interesting if we could compare the performance ofboth methods in an equal basis. Indeed this is possible be-cause, in a broader sense, any mutual occultation is alwayscontained in a mutual approximation. The only drawback isthat the same set of useful observations to be fitted in mu-tual occultations, when the satellites are too close together,is exactly the set that must be discarded in the mutual ap-proximations, and vice-versa. Even so, this still makes aninteresting comparison, because the instrumental and astro-nomical observational conditions are quite the same, and theindependence of the observational sets has a relevance of itsown.
For this comparison, we used the data of the mutual phe-nomena campaign of 2009 of Dias-Oliveira et al. (2013). Weutilized images acquired before and after five occultationsoriginally designed for albedo determination. Arlot et al.(2014) also determined the geometric parameters of these five occultations using the same light curves as Dias-Oliveiraet al. (2013), however obtaining slightly different results.Here we compare the results derived from the mutual ap-proximations with the results of Arlot et al. (2014) and theNOE-5-2010-GAL ephemeris. The comparison is displayedin Table 7. As shown in Table 7, the results for the mu-tual approximation method agrees with the results of Arlotet al. (2014) and the NOE-5-2010-GAL, within the errors.This highlights the strength of mutual approximations.
The results for the mutual phenomena campaign of 2014-2015 are still being processed. However we list here the cen-tral instant for the mutual approximations derived from theobservations before and after the mutual phenomena them-selves. These results can be used, in the near future, forcomparison with the results from the mutual occultations ofthis campaign. It is important to stress that, for this sce-nario, the precision of the mutual approximation results isbelow its capacity, once we will always have an absence ofpoints around the central instant.These results are shown in Table 8, where we list thecentral instant and its error obtained from our analysis for22 mutual approximations and the comparison with theephemeris, similar as in Table 4
In this paper we presented a method to measure the cen-tral instant in an approximation between natural satellitepairs. Instead of being restricted to the particular configu-ration of mutual occultations - which only occur during theequinox of the central planet - in mutual approximations (aswe call the method) we can make observations every timewhen the satellites don’t cross each other in the sky, butrather approach each other up to a minimum apparent dis-tance, which is at least larger than the sum of their radii, orin practice larger than the seeing - a special geometry thatrecurrently occurs for natural satellites. In this method therelative motion of the satellites is essentially described bythe same geometric parameters as in a mutual occultation –central instant, impact parameter and relative velocity. Butonly the central instant can be truly determined indepen-dently of any reference system, and so we consider it themain result of the method. Here astrometry is the techniqueused to directly measure typically very short distances (lessthan 85”) with very small errors due to the precision pre-mium, while differential photometry is the technique usedin mutual phenomena.We successfully applied the method to the Galileanmoons using CCD observations made in 2014 and 2015. Wecompared ours results with the ephemeris NOE-5-2010-GALfrom IMCCE. Using old observations from the 2009 equinoxof Jupiter, we also compared the performance of mutualapproximations with published mutual phenomena resultsfrom that campaign.The frequency of these approximations depends only ofthe orbital period of the satellites. In the case of the Galileanmoons, a couple of days or so. Because there is no need of c (cid:13) , 1–13 utual approximations: application to the Galilean moons. Table 6.
Fitted impact parameter and relative velocity and comparison between observations and the ephemerisDate Event d v r ∆ d ∆ v r P s Id.(d-m-y) (mas) (mas/s) (mas) (mas/s) (”/px)03-02-2014 IaE 9908.66 (07.93) 7.81 (0.31) –17.18 +0.05 0.37214 (0.00011) 105-02-2014 EaG 15014.25 (09.61) 6.24 (0.35) +51.00 +0.04 0.37478 (0.00005) 107-02-2014 IaE 10080.02 (12.18) 8.36 (0.71) –73.26 +0.02 0.37154 (0.00028) 219-02-2014 IaG 12995.06 (06.08) 7.55 (0.22) +0.43 +0.07 0.37288 (0.00087) 127-02-2014 IaE 9809.83 (13.64) 7.96 (0.38) –7.61 –0.03 0.34943 (0.00007) 118-03-2014 IaE 7984.82 (29.31) 6.52 (0.71) +7.69 +0.09 0.34971 (0.00014) 207-04-2014 IaE 8866.39 (11.83) 7.74 (0.34) –24.28 +0.04 0.36115 (0.00012) 120-04-2014 EaG 8425.61 (03.46) 3.56 (0.08) +4.27 +0.02 0.36161 (0.00033) 121-04-2014 GaC 18052.70 (02.97) 4.72 (0.16) –13.71 +0.03 0.36233 (0.00057) 121-04-2014 IaG 9250.29 (07.25) 5.48 (0.52) –24.25 +0.09 0.36235 (0.00064) 127-02-2015 GaC 1763.02 (25.99) 5.76 (0.19) –25.97 +0.03 0.34873 (0.00005) 211-04-2015 EaG 2173.79 (19.42) 5.16 (0.36) –22.31 +0.09 0.34837 (0.00017) 213-04-2015 IaE 1358.27 (42.35) 7.73 (0.25) –61.96 +0.05 0.34860 (0.00011) 219-04-2015 EaG 2126.17 (31.88) 5.19 (0.37) –75.47 +0.07 0.34818 (0.00027) 2
Note.
The impact parameter, d and the relative velocity v r of the mutual approximations. The columns ∆ are the comparisonsbetween the observations and the ephemeris parameters, here the NOE − − − GAL from IMCCE plus DE430, derived fromLainey et al. (2009), in the sense ”observations minus ephemeris” . For each event, we have the day, month and year, the satellitesdesignated by their initials (capital letters), where ’a’ stands for approximation. The Pixel-scale, P s , determined with the ephemeris is,also, listed. In the Id. column 1 stands for ”Group 1 mutual approximations” and 2 stand for ” Group 2 mutual approximations”. Table 7.
Comparison between the central instant for five mutual approximations and occultations observed in the 2009 equinox ofJupiter Date Event [1] error Central instant difference[1] - [2] [1] - [3] [2] - [3](d-m-y) (s) (mas) (s) (mas) (s) (mas) (s) (mas)09-05-09 IaE 3.23 23.84 –0.50 –3.65 +0.40 +2.98 +0.90 +6.6328-05-09 IaE 3.62 22.55 +2.28 +14.16 –2.31 –14.34 –4.59 –20.4922-06-09 IaE 4.72 26.57 –2.25 –12.77 –4.51 –25.59 –2.26 –12.8206-07-09 IaE 3.02 16.01 –2.98 –15.61 –4.66 –24.35 –1.67 –8.7407-08-09 IaE 3.59 13.46 +1.96 +7.33 –7.07 –26.40 –9.03 -33.72
Note. [1] Mutual approximation; [2] Arlot et al. (2014); [3] Ephemeris, NOE-5-2010-GAL from IMCCE plus DE430. The average errorof the approximations are 3.63 seconds (20.47 mas) for the central instant. Notice that the difference between the approximations andthe mutual phenomena of Arlot et al. (2014) is smaller than the errors of the central instant of the mutual approximation, and has thesame order that the difference between Arlot et al. (2014) and the ephemeris. reference stars for the astrometry even small telescopes canbe used. Because the events may last for hours there is noneed of a high cadence in time between the images, eventens of seconds would be ok.The mutual approximations extend the possibility ofobtaining relative distances with precision of a few mas, toperiods where there are no occultations or eclipses. Thismeans getting a relative position with an error about 10mas for every observed event along the visibility period. Inthe case of the Galilean moons we obtained a precision of0.56 seconds for the central instant when the whole approx-imation curve of distances was observed and a precision of1.52 seconds for the central instant when there were gapsalong the curve or around the central instant.The high precision results obtained in this work for theGalilean moons benefited from: (i) the precision premiumfrom very small field astrometry; (ii) from the use of a nar-row band filter centred in a methane absorption region, elim-inating the scattered light of Jupiter (this filter was alsoused in the reported mutual phenomena observations); (iii) the use of an adequate telescope/detector/exposure configu-ration set, allowing for imaging the satellites with high S/N(signal/noise) ratios, but avoiding saturation.The instrumental distortions due to the non flatness andnon parallelness of the filter and the CCD cover glass, andtheir distance from the CCD chip affect the global reductionof the entire FoV of the CCD to some extent. However, dueto the very small distance between both satellites (smallerthan 30 arc seconds) these distortions can be neglected here.The error in the measurement of the centroids, and thusof the distances, due to the effects of low/high albedo re-gions in the surface of the satellites, is presently unknown.However, taking the surface illumination resulting from thesolar phase angle geometry as an extreme example, and us-ing the relations in Section 4, we find that we need nearlya 5 degrees phase angle to change the photocentre by 10mas. This corresponds to a zero albedo circular region of100 km radius. Indeed, craters or volcanoes in many of theGalilean moons are features of this size (Faure & Mensing2007). However, they represent a variation of only 0.1 in the c (cid:13)000
Note. [1] Mutual approximation; [2] Arlot et al. (2014); [3] Ephemeris, NOE-5-2010-GAL from IMCCE plus DE430. The average errorof the approximations are 3.63 seconds (20.47 mas) for the central instant. Notice that the difference between the approximations andthe mutual phenomena of Arlot et al. (2014) is smaller than the errors of the central instant of the mutual approximation, and has thesame order that the difference between Arlot et al. (2014) and the ephemeris. reference stars for the astrometry even small telescopes canbe used. Because the events may last for hours there is noneed of a high cadence in time between the images, eventens of seconds would be ok.The mutual approximations extend the possibility ofobtaining relative distances with precision of a few mas, toperiods where there are no occultations or eclipses. Thismeans getting a relative position with an error about 10mas for every observed event along the visibility period. Inthe case of the Galilean moons we obtained a precision of0.56 seconds for the central instant when the whole approx-imation curve of distances was observed and a precision of1.52 seconds for the central instant when there were gapsalong the curve or around the central instant.The high precision results obtained in this work for theGalilean moons benefited from: (i) the precision premiumfrom very small field astrometry; (ii) from the use of a nar-row band filter centred in a methane absorption region, elim-inating the scattered light of Jupiter (this filter was alsoused in the reported mutual phenomena observations); (iii) the use of an adequate telescope/detector/exposure configu-ration set, allowing for imaging the satellites with high S/N(signal/noise) ratios, but avoiding saturation.The instrumental distortions due to the non flatness andnon parallelness of the filter and the CCD cover glass, andtheir distance from the CCD chip affect the global reductionof the entire FoV of the CCD to some extent. However, dueto the very small distance between both satellites (smallerthan 30 arc seconds) these distortions can be neglected here.The error in the measurement of the centroids, and thusof the distances, due to the effects of low/high albedo re-gions in the surface of the satellites, is presently unknown.However, taking the surface illumination resulting from thesolar phase angle geometry as an extreme example, and us-ing the relations in Section 4, we find that we need nearlya 5 degrees phase angle to change the photocentre by 10mas. This corresponds to a zero albedo circular region of100 km radius. Indeed, craters or volcanoes in many of theGalilean moons are features of this size (Faure & Mensing2007). However, they represent a variation of only 0.1 in the c (cid:13)000 , 1–13 B. Morgado et al.
Table 8.
Results for the mutual approximations for the 22 occultations observed in 2014-2015 and comparison with the ephemeris.Date Event t E t ∆ t (d-m-y) (hh:mm:ss) (mas) (s) (mas)15-10-14 GaC 07:07:07.26 (3.55) +4.80 14.60 +19.7515-10-14 IaE 07:07:54.19 (2.48) +0.02 17.54 +0.1629-10-14 IaG 07:07:24.26 (4.34) –2.05 22.17 –10.4502-11-14 IaC 06:02:34.11 (9.19) –2.82 16.75 –5.1419-11-14 EaC 07:37:52.48 (3.61) +11.79 18.32 +59.8602-02-15 GaC 02:24:31.55 (2.57) –3.56 14.99 –20.7422-02-15 IaE 02:07:53.57 (1.32) +6.24 7.21 +34.0724-02-15 IaG 01:44:46.97 (6.43) +2.72 53.15 +22.4826-02-15 IaE 22:21:26.26 (2.53) +1.32 23.72 +12.4027-02-15 IaG 02:20:23.81 (1.30) –1.63 8.54 –10.6803-03-15 IaG 04:08:13.38 (3.85) –1.46 32.28 –12.2424-03-15 GaC 00:14:15.44 (5.08) –0.02 27.19 –0.0925-03-15 IaE 23:34:49.47 (10.46) –9.13 65.38 –57.0802-04-15 IaE 01:43:57.43 (1.86) +4.26 11.83 +27.0917-04-15 IaC 23:47:03.29 (1.26) +3.00 6.28 +14.9218-04-15 GaC 01:32:21.21 (1.16) –0.58 5.80 –2.8918-04-15 IaG 20:54:40.80 (2.06) +0.07 11.39 +0.3921-04-15 IaE 01:55:02.68 (1.16) –1.61 8.56 –11.8625-04-15 IaG 23:45:24.78 (1.51) +1.23 8.99 +7.3026-04-15 IaE 21:24:57.44 (4.12) +0.12 27.29 +0.7929-04-15 IaG 00:28:55.49 (4.36) –14.96 29.87 –102.5003-05-15 IaG 23:39:20.87 (1.53) +2.35 10.20 +15.70 Note.
Same as in Table 4. surrounding surface albedo. So, in principle, at least for theGalilean moons, this effect can be neglected. In the specialcase of Io, the volcanoes can affect the centroid for infraredobservations in wavelengths such as 3800 nm, as can be seenfrom Descamps et al. (1992). However in the same paper,the authors obtained a lightcurve observed in Pic du Midiat 800 nm for comparison purposes, but in this wavelengththe effect of the volcanoes could not be observed. Since ourobservations were made in the same wavelength (889 nm),we conclude that these effects can be neglected in our im-ages.Mutual approximations is a simple, efficient and suit-able method for small telescopes. It can be used to con-tinually furnish high precision central instants that can beused to strongly constrain orbit fitting. Ultimately, mutualapproximations will significantly contribute to the improve-ment of the orbits of natural satellites, including the consid-eration of weak interactions like tidal forces.
ACKNOWLEDGMENTS
The authors thank the referee Dr. D. Pascu for hisconstructive comments. BM thanks the financial supportby the CAPES/Brazil. MA thanks the CNPq (Grants473002/2013-2 and 308721/2011-0) and FAPERJ (GrantE-26/111.488/2013). RVM acknowledges the followinggrants: CNPq-306885/2013, CAPES/Cofecub-2506/2015,FAPERJ/PAPDRJ-45/2013 and FAPERJ/CNE/05-2015.JIBC acknowledges CNPq for a PQ2 fellowship (processnumber 308489/2013-6). ADO is thankful for the supportof the CAPES (BEX 9110/12-7) FAPERJ/PAPDRJ (E-26/200.464/2015) grants. ARGJ thanks CAPES/Brazil.
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