Cassini ISS Mutual Event Astrometry of the Mid-sized Saturnian Satellites 2005-2012
N.J. Cooper, C.D. Murray, V. Lainey, R. Tajeddine, M.W. Evans, G.A. Williams
aa r X i v : . [ a s t r o - ph . E P ] S e p Astronomy & Astrophysicsmanuscript no. Cooper_25206_2_14f_astroph c (cid:13)
ESO 2018August 16, 2018
Cassini ISS Mutual Event Astrometry of the Mid-sized SaturnianSatellites 2005-2012 ⋆ N.J. Cooper , , C.D. Murray , V. Lainey , R. Tajeddine , , M.W. Evans , and G.A. Williams Astronomy Unit, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK IMCCE, Observatoire de Paris, UMR 8028 du CNRS, UPMC, Université de Lille 1, 77 av. Denfert-Rochereau, 75014 Paris, France Department of Astronomy, Cornell University, Ithaca, NY 14853, USAReceived 8 July 2014 / Accepted 1 September 2014
Abstract
Aims.
We present astrometric observations of the Saturnian satellites Mimas, Enceladus, Tethys, Dione and Rhea from
Cassini
Imaging Science Subsystem (ISS) narrow-angle camera (NAC) images. Image sequences were designed to observe mutual occulta-tions between these satellites.
Methods.
The positions of satellite centres were estimated by fitting ellipsoidal shape models to the measured limbs of the imagedsatellites. Spacecraft pointing corrections were computed using the UCAC2 star catalogue. We compare observed-minus-computed(O − C) residuals based on inter-satellite separations with those based on individual satellite positions, relative to the SAT360 andNOE-6-2012-MAIN ephemerides.
Results.
We provide a total of 2303 astrometric observations, resulting in 976 pairs, the remainder consisting of observations of asingle satellite. We obtain mean residuals for the individual satellite positions relative to the SAT360 ephemeris of 4.3 km in the linedirection and -2.4 km in the sample direction, with standard deviations of 5.6 and 7.0 km respectively, an order of magnitude improve-ment in precision compared to published HST observations. We show that, by considering inter-satellite separations, uncertainties incamera pointing and spacecraft positioning along with possible biases in the individual positions of the satellites can be largely elim-inated, resulting in an order-of-magnitude increase in accuracy compared to that achievable using the individual satellite positionsthemselves. We demonstrate how factors relating to the viewing geometry cause small biases in the individual positions of order 0.28pixel to become systematic across the dataset as a whole and discuss options for reducing their e ff ects. The reduced astrometric dataare provided in the form of individual positions for each satellite, together with the measured positions of reference stars, in order toallow more flexibility in the processing of the observations, taking into account possible future advances in limb-fitting techniques aswell as the future availability of more accurate star catalogues, such as those from the GAIA mission. Key words. astrometry, occultations, planets and satellites:general, methods:observational
1. Introduction
A planned campaign of astrometric observation of the innersatellites of Saturn using the Imaging Science Subsystem (ISS)of the
Cassini orbiter has been ongoing since Saturn OrbitInsertion (SOI) in July 2004. This work has been driven bothby scientific objectives and as a contribution to the operationalnavigation e ff ort of the Cassini project, with regular deliveriesof observations provided to JPL for the updating of satellite or-bit models throughout the mission. Recent scientific results, seefor example Lainey et al. (2012), have highlighted the key roleof high-resolution imaging and astrometry techniques in the so-lution of fundamental problems relating to the structure and dy-namical evolution of planetary satellite systems. The data pre-sented here represent a further contribution towards that widergoal.In terms of previously published Cassini ISS astrometry, ob-servations of the Jovian satellites Amalthea and Thebe using im-ages from Cassini’s Jupiter fly-by were published by Cooper etal. (2006), while Tajeddine et al. (2013) published astrometry ofthe Saturnian satellites Mimas and Enceladus, using a variety ofISS images, including some from the planned programme de- ⋆ Full Table 4 and Table 5 are only available at the CDS via anony-mous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http: // cdsweb.u-strasbg.fr / cgi-bin / qcat?J / A + A / . scribed above. Cooper et al. (2014) present astrometry of thesmall inner satellites of Saturn, Atlas, Prometheus, Pandora,Janus and Epimetheus using images both from the planned pro-gramme and image sequences designed to study Saturn’s F ring.The planned campaign of astrometric data collection hasbeen divided into two parallel programmes: image sequencestargeting specific individual satellites, normally identified by thelabel SATELLORB in the Cassini image sequence name, andsequences designed to capture chance occurrences of more thanone satellite in the NAC field-of-view, typically with sequencename containing MUTUALEVE. The observations presented inthis paper form part of the second programme.Although we describe these observations as mutual events,we use the term somewhat more loosely to describe any partial-occultation of one satellite with another, or the occurrence ofmore than one satellite (without actual or partial occultation)within an image. Since all the target satellites in these imagesare fully resolved, we have reduced the observations using anastrometric approach based on limb-fitting, rather than the tra-ditional photometric approach adopted for ground-based mu-tual phenomena for unresolved satellites, based on light curves(Thuillot et al., 2001). We described our approach in the nextsection.During the course of this work, we have also investigatedpotential sources of bias in the astrometric observations and dis- Figure 1: A portion of selected images from sequenceISS_144RH_MUTUALEVE002_PRIME, showing Dionebeing occulted by Rhea. Images are (a) N1675397022_1.IMG(b) N1675397057_1.IMG (c) N1675397090_1.IMG (d)N1675397192_1.IMG (e) N1675397226_1.IMG and (f)N1675397261_1.IMG. The images are consecutive andtaken approximately 34 seconds apart, except that imagesN1675397124_1.IMG and N1675397159_1.IMG, between (c)and (d) are not shown, since Dione is fully occulted for theseimages, as in image (c). Image exposure lengths are (a) 560 ms(b) 2000 ms (c) 460 ms (d) 460 ms (e) 560 ms (f) 2000 ms.cuss both their origin and possible approaches to reducing theire ff ects.Throughout, we use the Cassini ISS convention of referringto the image pixel coordinate along the x axis as ‘sample’ andthe y coordinate as ‘line’.
2. Observations
The images in each sequence used in this work were de-signed to target a ‘primary’ satellite while taking a se-ries of images as the secondary satellite moved across thefield-of-view. A selection of images from a typical sequence(ISS_144RH_MUTUALEVE002_PRIME) is provided in Fig.1,showing the secondary satellite, in this case Dione, entering thefield-of-view from the bottom of the first image (a) before be-ing fully occulted by the primary satellite, Rhea, which remainsfixed in the centre of the images. Dione then emerges from be-hind Rhea as the sequence progresses. The images are consecu-tive and approximately 34 seconds apart, except that two addi-tional images in the sequence, in which Dione is still fully oc-culted, are not shown.Table 1 summarises some relevant characteristics of all theimage sequences used in this work. Exposure lengths variedfrom 60 to 2000 ms. Solar phase angles (observer-object-Sun)varied from 32.5 to 162.9 degrees, with a mean of 103.1 ± / pixel, with a mean of 13.6 ± / pixel. Imagesat the start and end of each sequence generally only contain theprimary, hence there are typically more observations of each pri-mary satellite than its corresponding secondary (Table 1). Thus,while the total number of individual observations is 2303, thereare 976 pairs of observations (85% of the total) with the remain-ing 15% consisting of observations from images with only onesatellite present. Image size is, in all cases, 1024 by 1024 pixels. Astrometric reduction was performed using the IDL-based Ca v iar software package, developed at Queen Mary Universityof London, and incorporating the NAIF SPICE library (Acton,1996) together with the UCAC2 star catalogue (Zacharias et al.,2004). Reduction consisted of a correction to the camera point-ing direction for each image followed by an independent mea-surement of the centre-of-figure for each satellite using a limb-fitting approach. For this work, we used the Owen Model for theCassini ISS NAC (Owen, 2003; Cooper et al., 2006) to relatethe right ascension and declination of solar system objects andcatalogue reference stars to their equivalent line and sample po-sitions in each image. Tajeddine et al. (2013) developed an alter-native model, which may also be used. Unlike the Owen Model,the latter is more easily invertible, allowing line and sample tobe more readily converted back to inertial positions, should thatbe required. For each image, the nominal camera pointing direction ob-tained from the SPICE ‘C-kernels’ was corrected using an it-erative minimisation of the observed-minus-computed positionsof background reference stars, based on the UCAC2 catalogue.This involved firstly a manual translation of a template of pre-dicted catalogue star positions, graphically, until approximatelyaligned with the imaged stars, followed by an automatic itera-tive search to fine-tune the alignment, typically with an accuracyof 0.1 pixel or less. The imaged star positions were computedusing a centroiding technique based on the DAOPHOT methodof Stetson (1987). A mean of 7.8 stars per image were detected,with a mean magnitude of 11.20 ± / pixel.Based on the size of the largest satellite observed, Rhea, with amaximum radius of 764.30 ± The pixel coordinates of the centre-of-figure of each satellitewere estimated by comparing a shape model, projected ontothe image, with the position of the imaged limb itself. Shapemodels, in the form of ellipsoids, were extracted from the latestCassini SPICE kernels and projected on to the image using thechosen reference ephemeris (SAT360) and the corrected camerapointing information. Shape models were based on Thomas etal. (2007).Each imaged limb position was estimated by computing themaximum of the numerical derivative within a three-by-three ar-ray of pixels centred at a given pixel location. Detected limb
Table 1: Image Sequences
Sequence Time (UTC) a Exposure Mean phase b Mean resolution b Primary No. c Secondary No. c (ms) (deg) (km / pixel) satellite satelliteISS_003DI_MUTUALEVE004_PRIME 2005 FEB 20 12:26:06.294 UTC 180 92.3 9.0 DIONE 32 RHEA 23ISS_014TE_MUTUALEVE004_PRIME 2005 SEP 11 21:01:14.951 UTC 100,1500 87.9 14.4 TETHYS 33 DIONE 33ISS_015DI_MUTUALEVE005_PRIME 2005 SEP 16 08:15:40.484 UTC 100,320,380.1800 65.7 12.6 DIONE 56 TETHYS 23ISS_016EN_MUTUALEVE002_PRIME 2005 OCT 14 02:06:15.145 UTC 150,260 108.5 6.8 ENCELADUS 83 TETHYS 23ISS_016MI_MUTUALEVE007_PRIME 2005 OCT 14 19:36:15.742 UTC 150 90.8 10.0 MIMAS 15 TETHYS 0ISS_018TE_MUTUALEVE001_PRIME 2005 NOV 29 04:00:09.709 UTC 150 122.2 6.6 TETHYS 4 ENCELADUS 4ISS_018RH_MUTUALEVE011_PRIME 2005 DEC 05 03:46:40.402 UTC 180 108.4 16.1 RHEA 38 DIONE 34ISS_019EN_MUTUALEVE008_PRIME 2006 JAN 02 15:30:08.579 UTC 150 97.0 15.8 ENCELADUS 17 DIONE 14ISS_020RH_MUTUALEVE001_PRIME 2006 JAN 27 21:22:09.716 UTC 180 123.2 18.1 RHEA 57 MIMAS 83ISS_021RH_MUTUALEVE002_PRIME 2006 FEB 06 04:11:37.600 UTC 180 110.0 25.8 RHEA 30 TETHYS 30ISS_021MI_MUTUALEVE004_PRIME 2006 FEB 11 23:18:10.437 UTC 150 101.3 22.5 MIMAS 40 TETHYS 40ISS_021EN_MUTUALEVE009_PRIME 2006 MAR 02 12:09:40.192 UTC 180 136.0 12.2 ENCELADUS 39 RHEA 23ISS_021EN_MUTUALEVE005_PRIME 2006 MAR 03 03:20:09.844 UTC 180,220 139.0 15.4 ENCELADUS 35 DIONE 35ISS_022EN_MUTUALEVE020_PRIME 2006 MAR 16 08:42:10.643 UTC 120 99.5 11.3 ENCELADUS 30 TETHYS 12ISS_022RH_MUTUALEVE009_PRIME 2006 MAR 27 07:46:10.329 UTC 680 158.7 13.9 RHEA 10 DIONE 8ISS_023RH_MUTUALEVE001_PRIME 2006 APR 14 14:55:10.507 UTC 180 130.8 20.5 RHEA 10 ENCELADUS 10ISS_023DI_MUTUALEVE006_PRIME 2006 APR 17 05:29:10.052 UTC 220 120.1 20.6 DIONE 10 RHEA 10ISS_023RH_MUTUALEVE006_PRIME 2006 MAY 07 02:13:09.818 UTC 820 159.8 16.1 RHEA 10 DIONE 10ISS_024DI_MUTUALEVE002_PRIME 2006 MAY 14 06:43:40.157 UTC 260,820,1000 133.8 15.9 DIONE 31 RHEA 31ISS_024EN_MUTUALEVE001_PRIME 2006 JUN 06 16:01:40.167 UTC 460 162.9 25.0 ENCELADUS 0 TETHYS 9ISS_024EN_MUTUALEVE009_PRIME 2006 JUN 09 06:00:19.635 UTC 680 161.2 23.3 ENCELADUS 23 RHEA 0ISS_025MI_MUTUALEVE001_PRIME 2006 JUN 11 06:59:09.622 UTC 460 158.5 23.7 MIMAS 8 ENCELADUS 9ISS_025RH_MUTUALEVE005_PRIME 2006 JUN 11 08:30:39.537 UTC 560 156.5 21.9 RHEA 10 TETHYS 8ISS_025EN_MUTUALEVE003_PRIME 2006 JUN 13 15:00:10.227 UTC 680 159.1 24.3 ENCELADUS 8 RHEA 10ISS_025DI_MUTUALEVE003_PRIME 2006 JUN 14 03:38:09.998 UTC 560 159.1 22.7 DIONE 10 TETHYS 10ISS_025RH_MUTUALEVE016_PRIME 2006 JUN 16 10:46:09.733 UTC 560 155.2 20.3 RHEA 9 TETHYS 0ISS_025MI_MUTUALEVE006_PRIME 2006 JUN 21 22:58:09.851 UTC 260 148.7 20.5 MIMAS 2 TETHYS 6ISS_025MI_MUTUALEVE007_PRIME 2006 JUL 03 21:47:35.132 UTC 460 145.6 9.8 MIMAS 7 DIONE 10ISS_025RH_MUTUALEVE004_PRIME 2006 JUL 04 00:50:40.201 UTC 180 141.8 8.1 RHEA 10 ENCELADUS 6ISS_025RH_MUTUALEVE006_PRIME 2006 JUL 08 17:39:05.360 UTC 680 158.4 13.4 RHEA 20 TETHYS 20ISS_047TE_MUTUALEVE004_PRIME 2007 JUN 21 11:52:09.736 UTC 120 64.2 12.3 TETHYS 10 ENCELADUS 10ISS_084OT_MUTGRNGAR001_PRIME 2008 SEP 13 09:17:10.389 UTC 60 32.7 5.3 DIONE 27 ENCELADUS 19ISS_119MI_MUTUALEVE001_PRIME 2009 OCT 19 09:06:10.486 UTC 150 94.9 12.1 MIMAS 20 RHEA 17ISS_119RH_MUTUALEVE001_PRIME 2009 OCT 22 07:31:39.662 UTC 180 98.0 12.6 RHEA 20 DIONE 18ISS_120RH_MUTUALEVE001_PRIME 2009 OCT 26 20:22:09.847 UTC 220 113.1 11.3 RHEA 20 TETHYS 17ISS_120EN_MUTUALEVE001_PRIME 2009 OCT 27 01:47:09.742 UTC 150 117.3 12.7 ENCELADUS 20 TETHYS 12ISS_121DI_MUTUALEVE001_PRIME 2009 NOV 11 22:28:39.988 UTC 260 118.8 14.8 DIONE 20 RHEA 18ISS_121TE_MUTUALEVE001_PRIME 2009 NOV 11 23:25:09.968 UTC 220 115.4 14.1 TETHYS 20 ENCELADUS 17ISS_121EN_MUTUALEVE001_PRIME 2009 NOV 15 12:15:09.846 UTC 120 115.2 13.7 ENCELADUS 20 RHEA 20ISS_121RH_MUTUALEVE002_PRIME 2009 NOV 26 15:18:09.980 UTC 180 103.6 10.5 RHEA 20 TETHYS 18ISS_121DI_MUTUALEVE002_PRIME 2009 NOV 28 13:52:39.787 UTC 180 109.5 13.1 DIONE 20 TETHYS 17ISS_122RH_MUTUALEVE003_PRIME 2009 DEC 01 21:38:09.761 UTC 150 107.3 12.3 RHEA 20 ENCELADUS 16ISS_127DI_MUTUALEVE002_PRIME 2010 FEB 24 06:53:09.886 UTC 150 116.6 12.2 DIONE 22 ENCELADUS 21ISS_128DI_MUTUALEVE002_PRIME 2010 MAR 23 12:11:09.963 UTC 460,680,2000 87.4 7.2 DIONE 27 MIMAS 16ISS_128DI_MUTUALEVE003_PRIME 2010 MAR 26 19:03:09.955 UTC 460,680,2000 89.9 11.7 DIONE 27 TETHYS 26ISS_135DI_MUTUALEVE001_PRIME 2010 JUL 27 00:15:39.574 UTC 150,460,560,2000 78.5 6.6 DIONE 23 RHEA 21ISS_141DI_MUTUALEVE002_PRIME 2010 DEC 06 06:33:09.863 UTC 320,460,1800 75.2 13.1 DIONE 75 TETHYS 75ISS_143RH_MUTUALEVE003_PRIME 2011 JAN 20 18:18:09.919 UTC 560,1000,2000 99.6 17.0 RHEA 32 DIONE 28ISS_144RH_MUTUALEVE002_PRIME 2011 FEB 03 03:12:09.672 UTC 460,560,2000 75.2 7.1 RHEA 27 DIONE 17ISS_147RH_MUTUALEVE006_PRIME 2011 APR 25 19:25:40.172 UTC 320,380,1500 66.9 13.3 RHEA 30 ENCELADUS 33ISS_150DI_MUTUALEVE006_PRIME 2011 JUL 18 01:19:09.625 UTC 560,680,2000 37.3 13.1 DIONE 24 RHEA 13ISS_158RH_MUTUALEVE001_PRIME 2011 DEC 07 04:50:10.103 UTC 460,680,2000 72.4 11.7 RHEA 30 DIONE 30ISS_162TE_MUTUALEVE002_PRIME 2012 MAR 14 01:22:39.857 UTC 150,180,820 51.7 10.1 TETHYS 29 DIONE 20 ( a ) Mid-time for first image in sequence. ( b ) For primary satellite. ( c ) Zero indicates limb-fitting failed points greater than two pixels away from the predicted limb po-sition, based on the reference ellipsoid, were rejected. Di ff erentvalues were tested. Given that the camera pointing correctionwas applied before the limb-finding / fitting the measured limbpositions were considered unlikely to be more than two pixelsfrom their predicted locations based on the best available shapemodels. Using a progressively smaller value than 2.0 pixel re-duced the scatter in the observed residuals (next section), but ar-tificially drove the estimated limb points towards their predictedlocations. It was considered more desirable to avoid this, at theexpense of more random scatter in the residuals.An iterative fitting procedure was used to find the optimumalignment of the position of the limb (based on the shape model)with the imaged limb positions. The observed position of thecentre-of-figure was then computed by correcting the predictedposition by the mean shift required to align, optimally, the mea-sured and predicted limbs. Five ellipse parameters could poten-tially be fitted. However, the ellipsoid projection fixes the sizeof the ellipsoid, while its orientation is fixed by the satellite’sknown orientation to the order of 0.1 degree (Archinal et al.,2011) and the camera’s twist angle (known to order of 60 µ rad (Porco et el., 2004) and 90 µ rad (Tajeddine et al., 2013), leavingthe ellipse’s centre to be fitted only.The measured pixel coordinates (line versus sample) for eachsatellite are shown in Fig. 2. To allow more flexibility in the use of the reduced data, weprovide separate positions for each satellite, rather than inter-satellite coordinates, particularly given that some images ( ∼ α , δ ) in the International CelestialReference Frame (ICRF), we provide their measured pixel co-ordinates and camera pointing directions, so that either of thesesets of measurement can easily be re-estimated independentlyat some future date, if required. This would, for example, allowthe camera pointing corrections to be updated using improvedstar catalogues, such as those soon to become available from Table 2: Sample of Cassini ISS Observations
Image ID Mid-time (UTC) α c δ c TWIST Line a Sample a α δ Body(deg) (deg) (deg) (px) (px) (deg) (deg)N1487595425 2005 FEB 20 12:30:38.277 27.296799 -6.438903 178.585994 595.83 591.92 27.268304 -6.467161 DIONEN1487595459 2005 FEB 20 12:31:12.277 27.289024 -6.438771 178.630998 595.78 591.47 27.260708 -6.467036 DIONEN1487595459 2005 FEB 20 12:31:12.277 27.289024 -6.438771 178.630998 621.95 994.47 27.121289 -6.472685 RHEAN1487595493 2005 FEB 20 12:31:46.276 27.281160 -6.438414 178.595788 596.02 590.78 27.253062 -6.466749 DIONEN1487595493 2005 FEB 20 12:31:46.276 27.281160 -6.438414 178.595788 622.42 947.46 27.129638 -6.472789 RHEA
Notes.
Columns α c , δ c and TWIST refer to the right ascension, declination and twist angle of the camera’s pointing vector in the InternationalCelestial Reference Frame (ICRF), while α and δ are the right ascension and declination in the ICRF for the body listed in the far right-handcolumn. Relative to the SAT360 ephemeris, considering all the individual positions together, we obtain a precision of 0.34 pixels in line and 0.41in sample. The full table is available from the CDS. ( a ) The origin of the line, sample coordinate system is at the top left of the image with line, y,increasing downwards and sample, x, to the right. Image size is 1024 by 1024 pixels.
Figure 2: Observed image positions of satellite centres, line ver-sus sample. Image sizes are 1024 by 1024 pixel.the GAIA mission, or for the astrometric positions themselvesto be re-estimated independently of the pointing corrections, ifadvances in limb measurement or centre-finding became avail-able (see also the section on Sources of Error). Alternatively, adi ff erent NAC camera distortion model, such as that developedby Tajeddine et al. (2013), could also be used.The complete set of reduced data is available at the CDS.A small section of the table showing the satellite positions isreproduced in Table 2. Computed star positions (Table 5) areonly available electronically at the CDS.
3. Analysis of Residuals
In this work, we compared observed minus computed (O − C)residuals relative to two di ff erent ephemerides: JPL’s SAT360ephemeris, based on a fit to Earth-based, Pioneer, Voyager, HSTand Cassini data to the end of 2013, and NOE-6-2012-MAIN,created by IMCCE Paris based on a fit to Earth-based and space-craft data from 1886-2012. In both cases, these are post-fit resid-uals, since the orbit models on which both these ephemeridesare based included the observations presented here. However,SAT360 was generated based on corrected astrometry, which in-cluded a constant sample and line bias due to camera pointingplus additional corrections for the satellite-dependent phase bi-ases (R.A. Jacobson, private communication).In Fig. 3, we show O − C residuals relative to the SAT360ephemeris for the 976 pairs of observations. Fig. 3(a) shows theresiduals, line versus sample, for each individual observation,computed using the absolute satellite positions, while Fig. 3(b)shows the equivalent residuals based on the separation betweenthe primary and secondary satellite in each image. The meanline and sample residuals for the absolute positions in Fig. 3(a)are 0.27 and − σ values of 0.34 and0.37 pixel, while for the inter-satellite separations (Fig. 3(b)),we obtain an order-of-magnitude improvement in accuracy, withmeans of -0.03 and − σ values of0.29 and 0.26 respectively. The equivalent mean residual val-ues across all absolute positions (2303 values) in km are 4.3 km(line) and − σ values of 5.6 and 7.0 km, re-spectively. By comparison, French et al. (2006) obtained typicaluncertainties at Saturn of 80km and 120km respectively, usingthe Planetary Camera and the Wide Field Planetary Camera 2 ofthe Hubble Space Telescope.We discuss these results further in the following section.Line and sample residuals for each individual satellite (ab-solute positions) are plotted as a function of time relative to theSAT360 ephemeris in Fig. 4 and as line versus sample in Fig. 5.
4. Sources of Error
Tajeddine et al. (2013) give a detailed description of the sourcesof uncertainty in the astrometric reduction of Cassini ISS data.Here we focus on possible systematic errors in the absolute posi-tions, that might give rise to the non-zero means of several tenthsof a pixel, mentioned in the previous section. Although these bi-ases are small in comparison to the typical accuracy achievedwith earth-based astrometry (French et al., 2006), and we haveseen in the previous section that they can be largely eliminatedby measuring inter-satellite separations, it is clearly desirable tounderstand their origin.
Figure 3: Comparison between O − C residuals based on absolutepositions and those based on inter-satellite di ff erences, plotted asline residual versus sample residual relative to the JPL SAT360ephemeris. All satellites are shown. Units are NAC pixels.Figure 4: O − C residuals for each satellite, using absolute posi-tions, plotted versus time relative to the JPL SAT360 ephemeris.For each satellite, line residuals are plotted on the left, and sam-ple on the right. Units are NAC pixels.As noted previously, Figs. 3-5 indicate a non-zero mean of0.28 pixels in the line residuals across all five satellites. Thisis also clear quantitatively in Tables 3 and 4 where we showmean values relative to two di ff erent ephemerides. The meanline residuals are consistently positive, although there is somevariation in the magnitude of the mean, both between satellites,and also depending on the reference ephemeris used. The largestpositive mean values occur in the line residuals for Mimasand Enceladus, based on either of the reference ephemerides Figure 5: O − C residuals for each satellite, using absolute po-sitions, relative to the JPL SAT360 ephemeris, plotted as lineresidual versus sample residual. Units are NAC pixels.Table 3: Mean values of residuals in pixels relative to the JPLSAT360 ephemeris, including standard deviations. line σ line sample σ sample Mimas 0.44 0.52 -0.04 0.56Enceladus 0.35 0.32 -0.24 0.47Tethys 0.16 0.35 -0.13 0.36Dione 0.30 0.32 -0.07 0.36Rhea 0.24 0.28 -0.11 0.37All 0.28 0.34 -0.12 0.41All (inter) a -0.03 0.29 -0.01 0.26 Notes. ( a ) Using inter-satellite separations between pairs of satelliteswithin a given image. (SAT360 and NOE-6-2012-MAIN). Given that these are post-fit residuals, the implication is either that there is a missing orpoorly-determined dynamical component in the models used togenerate both ephemerides, or that there is a systematic bias inthe measurements, or a combination of both. Since all five satel-lites show a positive mean at some level, an inadequacy in themodelling would have to a ff ect all five satellites in the same di-rection, which seems less likely than a systematic bias in themeasurements. Also, any inadequately modelled component inthe dynamics would give rise to a bias in both absolute and rela-tive positions, which is not observed. Table 4: Mean values of residuals in pixels relative to the IMCCENOE–6-2012-MAIN ephemeris, including standard deviations. line σ line sample σ sample Mimas 0.55 0.57 -0.25 1.19Enceladus 0.32 0.42 -0.30 0.40Tethys 0.18 0.47 0.01 0.52Dione 0.19 0.38 -0.04 0.35Rhea 0.26 0.31 -0.08 0.39All 0.25 0.42 -0.10 0.51All (inter) a Notes. ( a ) Using inter-satellite separations between pairs of satelliteswithin a given image.
Some measurement bias is inevitable because the phase an-gle is never precisely zero in practice, so that imaged limbsare always one-sided, with the terminator forming what remainsof the boundary of the satellite image. This is true for all re-solved observations based on limb measurement. Tajeddine etal. (2013) also found a positive bias in the residuals for Mimasand Enceladus in the direction towards the Sun, with a signifi-cantly larger bias for Mimas than Enceladus. They put forwardtwo possible explanations: (1) that the greater level of crater-ing on Mimas may distort the limb and (2) that the dimensionsof Mimas could be larger than those based on the shape modelof Thomas et al. (2007). However, in the data presented here,this e ff ect is systematic across all the observations for five dif-ferent satellites, implying that some common geometric charac-teristic of the dataset as a whole is the key contributory factor:if the limbs had randomly distributed phases and illuminationdirections, this e ff ect would not be systematic. A clue that sucha common geometric characteristic exists is evident from Fig.2, showing how the measurements are distributed dominantlyalong the line axis.We investigated this further using synthetically-generatedimages, whose centre-of-figure is known in advance. Sequencesof images were generated, using the Mathematica softwarepackage (Wolfram Research Inc., 2012) for di ff erent illumina-tion directions and phase angles, in order to assess possible mea-surement bias as a function of these parameters. In Figs. 6 and7, we show using these synthetic images how the limb-fittingalgorithm does indeed generate a bias in the computed centre-of-figure that varies as a function of phase angle and illumina-tion direction. The maximum size of the observed bias is morethan 1.5 pixel, which is almost an order of magnitude larger thanthe mean values we obtain for the real images. This discrepancyarises because the illumination model used to generate the syn-thetic images is not a realistic photometric match for the real im-ages. Thus this comparison is illustrative only and serves purelyto demonstrate how a bias can arise and how it may change ac-cording to the imaging geometry.Returning to the question of why this appears as a systematice ff ect in the real data, in Fig. 8(a), we plot the variation of phaseangle across the entire set of real images. This shows clearlythat phase angles across all the observations are clustered around90–100 degrees (see also the mean phase values listed in Table1). Furthermore, Fig. 8(b) shows that the sun directions are alsopreferentially distributed along the positive line direction. Weconclude therefore that the combination of these two geometri-cal characteristics of the images accounts for the systematicallypositive mean values in the absolute positions: from the synthetictests, the maximum possible bias occurs at ∼
90 degree phase an-gle, and the common alignment of the limbs in the real images Figure 6: Synthetic images for di ff erent illumination directions,showing O − C residuals in pixel (right) for centre-of figure posi-tions measured by limb-fitting the images shown left. The biasis always in the illumination direction. All points in the right-hand displays are the same, with the particular O − C residualvalues corresponding to each of the four images represented bythe small circles (directions are, from top to bottom, 0 deg, 90deg, 180 deg, 270 deg).in the positive line direction then accounts for the appearance ofa systematic positive bias in that direction across all the absolutepositions.
5. Discussion and Conclusions
A particular advantage of all mutual event observations, us-ing either ground-based or spacecraft imaging, is that image-dependent errors cancel when computing the separation between
Figure 7: Synthetic images for di ff erent phase angles, showingO − C residuals in pixel (right) for centre-of figure positions mea-sured by limb-fitting the images shown left. The maximum biasoccurs for 90 degree phase and is zero for zero phase. All pointsin the right-hand displays are the same, with the particular O − Cresidual values corresponding to each of the four images repre-sented by the small circles (phase angles are, from top to bottom, +
90 deg, 0 deg, -90 deg, 180 deg).two satellites in the same image, leading to a potentially signifi-cant increase in accuracy. This type of observation is also usefulwhen no suitable reference stars are detectable within a given im-age. Although we did not foresee the tendency of the limb-fittingmethod to introduce a bias in the absolute positions of satellitecentres-of-figure in the direction of the limb (i.e. in the sun di-rection), we have shown that this also cancels when computinginter-satellite separations because the magnitude and directionof the the bias turns out to be dependent on phase angle and sun Figure 8: Distribution of (a) satellite phase angles and (b) sun di-rections for the Cassini images used in this work. Sun directionsare resolved into line and sample directions within each image.direction, and these two quantities, like the camera pointing di-rection, are also e ff ectively constant for a given image.On the other hand, a single measurement of the separationbetween two satellites necessarily contains less information thanthe individual absolute positions combined. Thus there is an in-evitable trade-o ff in information content versus accuracy, de-pending on which of the two approaches to data reduction isused. This is one reason why we have chosen to provide the rawdata in terms of the absolute positions, in order to give the readerthe option to use either approach.As described previously, investigation showed that Sun ori-entations (and phase angles) are preferentially distributed in onedirection across the dataset as a whole and thus the bias, ratherthan being randomly distributed, has a preferred orientation inthat direction (the positive line direction). This is an unintendedconsequence of the particular observing geometry of the imagesequences, Cassini Project standard operational practice and thehabits of the imaging sequence designers.Given that there may be circumstances where it is more de-sirable to make use of the greater information content of the ab-solute positions, the question arises as to what can be done eitherto correct such biases, or to minimise their occurrence in the firstinstance. We have considered four approaches: (a) correcting anexisting bias using information derived from synthetic images,(b) correcting an existing bias by subtracting mean values de-rived for each satellite, (c) changing the observation strategy toprevent the bias from occurring systematically in one directionand (d) improving the performance of the limb-measuring tech-nique to try to reduce the magnitude of any bias.Firstly, considering (a), the synthetic images have provideda useful way of evaluating how the measured centre-of-figurebased on limb-fitting behaves as a function of phase angle andillumination direction. Thus, in principle, a table of correctionscould be generated from these synthetic images, correctionswhich could then be applied to the real observations in order toremove any potential bias generated in the limb-fitting. However,as noted previously, the magnitude of the bias we currently ob-tain from the synthetic images is much larger than any e ff ect wesee in the real observations, so this approach would only be vi-able if a way could be found to model the limb profiles to givea satisfactory photometric representation of the profiles obtainedfrom the real images. Although a photometric approach is rou-tinely used for ground-based observations of mutual phenomenafor unresolved bodies (see for example Noyelles et al. (2003);Arlot & Thuillot (2008)), to our knowledge this has not beenattempted for observations of resolved satellites, where centre-of-figure measurements are based on limb-fitting. Clearly, with an observed mean bias of less than 0.2 pixel for some satellites,any such correction process would have to be precise enough soas not to risk introducing a bias in a di ff erent direction. This willbe the subject of further work.In principle, biases in the absolute positions could also becorrected by applying a global correction for each satellite,prior to orbit modelling, based on the computed mean valuesof the O − C residuals relative to a suitable reference ephemeris.However, clearly this can only be justified if there is su ffi cientconfidence that the mean values do not represent genuine dy-namical e ff ects.A di ff erent approach to observing the mutual events withthe aim of randomising the sun directions would potentially re-duce the tendency of the bias to be systematic in one direction.However this would only randomise the direction of the biasacross the dataset as a whole and would not reduce its magni-tude in a given image. Also any such modification of the observ-ing strategy would still need to satisfy spacecraft flight rules.A possible fourth approach would be to improve the perfor-mance of the limb-measurement algorithms in order to minimisethe magnitude of any possible bias in a given image. Approacheswhich we have investigated so far have included the applicationof a gaussian to the derivative in order to estimate the limb po-sition to sub-pixel precision, the use of other edge detection al-gorithms, such as the Canny algorithm (Canny, 1986), and theinterpolation of the input images to produce finer spatial sam-pling, before limb fitting. Preliminary investigations have shownthat all these approaches, to a greater or lesser degree, can pro-vide some improvement in the overall precision of the data, butdo not significantly increase the accuracy.Although these are small e ff ects, which we already have seencan be largely eliminated by measuring the separations betweenpairs of satellites in each mutual event image, further investi-gation would still be advantageous, particularly given the needfor ever greater astrometric accuracy based on current scientificgoals. Acknowledgements.
The authors thank the reviewer, Dr Marina Brozovic, forhelping us to improve this paper. NJC is grateful to the Paris Observatoryfor funding while he was an invited researcher at the IMCCE. The work wasalso supported by the Science and Technology Facilities Council (Grant No.ST / F007566 /
1) and NJC and CDM are grateful to them for financial assis-tance. CDM is also grateful to The Leverhulme Trust for the award of aResearch Fellowship. VL and RT thank UPMC for funding under grant EME0911. NJC, CDM, VL and RT also thank the CNRS and Royal Society forfunding under the ‘GAME’ proposal and the ESPACE consortium for fund-ing under agreement 263466. The authors thank the members and associatesof the
Cassini
ISS team, particularly Kevin Beurle for assistance in design-ing some of the image sequences used in this work, and also thank their col-leagues in the Encelade working group at the IMCCE of the Paris Observatory(http: // / ∼ lainey / Encelade.htm) and Dr Bob Jacobson at JPL formany discussions.
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