- Email: [email protected]
New Ganymede control point network and global shape model
Planetary and Space Science 117 (2015) 246–249
Contents lists available at ScienceDirect
Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
New Ganymede control point network and global shape model A. Zubarev a,n, I. Nadezhdina a, J. Oberst a,b,c, H. Hussmann b, A. Stark b a b c
Moscow State University of Geodesy and Cartography (MIIGAiK),MIIGAiK Extraterrestrial Laboratory (MExLab), Gorokhovsky per. 4, 105064 Moscow, Russia German Aerospace Center (DLR), Institute of Planetary Research, Berlin, Germany Technical University Berlin, Institute for Geodesy and Geoinformation Sciences, Berlin, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 24 February 2015 Received in revised form 10 June 2015 Accepted 28 June 2015 Available online 10 July 2015
We computed a 3D control point network for Ganymede using combinations of 126 Voyager-1 and -2 and 87 Galileo images, benefiting from reconstructed trajectory data for the three spacecraft and a more complete Galileo image data base than was available for earlier studies. Using more than 3000 control point coordinates, we determine global shape parameters, including mean radius, spheroid- and ellipsoidal axes, and make tests for various equilibrium shape models, constrained by the most recent estimates for gravity field parameters. We confirm that Ganymede has a pronounced ellipsoidal shape, approximately aligned with the Jupiter-direction, in agreement with Ganymede being in tidal equilibrium. The point heights, suffering from large individual errors, do not reveal any large-scale topography below our typical error levels (97% o 5 km). By analysis of data residuals we search for, but cannot detect Ganymede longitudinal forced librations. We conclude that libration amplitudes cannot be larger than 0.1° (corresponding to a lateral displacement of 4.6 km at the equator). & 2015 Elsevier Ltd. All rights reserved.
Keywords: Ganymede Control points network Libration Ellipsoid
1. Introduction While being the largest planetary satellite in the Solar System, information on physical dimensions and reference parameters for Ganymede (e.g., Archinal et al., 2011), are still rather limited. As Ganymede is exposed to strong centrifugal and tidal forces, global shape parameters and physical librations represent useful constraints on the dynamic state and interior structure of Ganymede (Schubert et al., 2004). In this paper, we establish a new geodetic control point network for Ganymede, based on images obtained by the Voyager and Galileo missions, and study global shape and rotational parameters of the satellite. Our study includes several improvements over a previous analysis (Davies et al., 1998), as we benefit from new orbit reconstructions for the Voyager and Galileo spacecraft and a larger image data base than was previously available. Also, the control point measurements in the images were redone. Using a new developed software interface and semi-automatic methods, we worked more effectively and greatly increased the number of measurements and measurement accuracy. The only previous Ganymede control point network (Davies et al., 1998) had used a 2-D block adjustment, i.e., solutions were determined for latitudes and longitudes of the control point coordinates and one common radius only. The ellipsoidal axes were computed analytically from the mean radius, assuming Ganymede's n
Corresponding author. E-mail address: [email protected] (A. Zubarev).
http://dx.doi.org/10.1016/j.pss.2015.06.022 0032-0633/& 2015 Elsevier Ltd. All rights reserved.
shape to be in tidal equilibrium, with the a- and b axes in the equatorial plane and the a-axis aligned with the Jupiter direction. This is in contrast to the approach of this paper, where we determine 3-D Cartesian coordinates of our control points, measure Ganymede's global shape from the control point radii and carry out tests for various equilibrium shape models. In this paper, we also constraint a possible forced longitudinal libration amplitude.
2. Data Our analysis is based on images taken by the Galileo (http://pds-imaging.jpl.nasa.gov/volumes/galileo.html) and Voyager (http://pds-imaging.jpl.nasa.gov/volumes/galileo.html) spacecraft. The two Voyagers made their Ganymede flybys in 1979–03 and 1979 07, at ranges of 186,000 km and 208,000 km, respectively. Both spacecraft were equipped with Vidicon sensors (2 cameras per spacecraft (WAC and NAC): VGR1 NAC f¼ 1500.190 mm, pixel size ¼11.789 mm, WAC f ¼200.465 mm, pixel size¼11.800 mm; VGR2 NAC f¼ 1503.490 mm, pixel size¼11.789 mm, WAC f¼ 200.770 mm, pixel size ¼11.666 mm). Unfortunately, these Vidicon images suffer from significant geometric distortions and require sophisticated geometric calibration schemes, e.g., implemented in the planetary image processing package VICAR (see http://www-mipl.jpl.nasa.gov/external/vicar. html for details). In contrast, the Galileo spacecraft was equipped with the Solid State Imaging Instrument (SSI), which was equipped with a 800 by 800 pixels CCD chip providing geometrically stable
A. Zubarev et al. / Planetary and Space Science 117 (2015) 246–249
images with f ¼1500.47 mm, pixel size ¼ 15.240 mm. During its orbital mission, Galileo obtained 47 Ganymede image sequences from 15 flybys. Spacecraft position and attitude (“image navigation”) data have been supplied by the JPL Navigation Ancillary Facility (NAIF) in the format of SPICE kernels (Acton, 1996). While previous investigators had to cope with large uncertainties of exterior orientations for Voyager (http://naif.jpl.nasa.gov/pub/naif/VOYAGER/kernels/spk) and Galileo (http://naif.jpl.nasa.gov/pub/naif/GLL/kernels/spk), we now used reconstructed spacecraft ephemerides provided by NAIF (Jacobson, 2002). While improvements of the Galileo trajectory data are small ( 100 m), comparison between reconstructed and previous ephemerides for Voyager 1 and 2 revealed shifts as large as 873 km and 221 km, respectively. All position and navigation data were converted to the Ganymede-fixed reference frame defined by the satellite's nominal rotation parameters (Archinal et al., 2011) (with Ganymede assumed in a perfect tidal lock and its rotation axis perpendicular to the orbit plane).
3. Method
247
3.3. Block adjustment All tie-point measurements and associated image navigation data are subjected to a bundle block adjustment (also implemented within the 〈〈PHOTOMOD〉〉 software package), which produces 3D Cartesian coordinates of the control points and improved navigation for each involved image by inversion of the collinearity equations (see Oberst et al. (2014a, 2014b) for details). The bundle block adjustment was carried out repeatedly to identify, possibly re-measure, or remove gross outliers after each step. As a result we obtained coordinates of 3377 control points observed 15655 times with a minimum of 2 and a maximum of 14 observations per point (average: 4.6). Coordinate errors were determined after solving the normal equations from the covariance matrix. These differ substantially, owing to different image resolutions, viewing conditions (notably the different convergence angles), and tiepoint measurement errors. For 97% (1 sigma) of the control points, the height accuracy is better than 5.0 km, with remaining point outliers having errors as large as 33.0 km. We removed all outlier points with height errors larger than 15.0 km (3 sigma). Thus, we eliminated 8 points (0.2%).
3.1. Image block
4. Results
We first carried out a catalog search to assemble an appropriate block of images for the analysis. The goal is to identify images with suitable global coverage and multiple overlaps. We also aim at images possibly obtained with similar resolutions and under similar lighting and viewing conditions, in particular, avoiding images taken under small emission angles. Unfortunately, only rather oblique images are available for the north and south polar area. Furthermore, we have no image coverage at all for small distinct areas (representing gaps of approximately 4000 km2, each) near the north and south pole. The image search was repeated after initial data analysis and correcting the image navigation data. Finally, we selected 126 Voyager and 87 Galileo images, which varied in their resolutions between 0.5 km and 25 km. Resolutions for a few Galileo image were as small as 19 m/pixel.
We first solved for the center point of the control points, which was found to be shifted from the origin of the Cartesian coordinate system by approximately þ 32.5, 6.0, and 3.2 km, probably due to errors in spacecraft flyby trajectories, Ganymede ephemeris, or both. After correction for this offset, cartesian coordinates were converted to latitude, longitude and radius for further analysis. Ranges of the control points from the coordinate center were found to vary from 2621 km to 2644 km. In Fig. 1, we show locations of control points, their associated errors and height residuals with respect to the mean. There are notable data gaps near the South pole, where we have no coverage by any of the three spacecraft, or where there is no image overlap. Also, control points on the trailing hemisphere suffer from larger errors on average. The data reveals benign regional elevation trends (note the yellow-color points). Next, we resampled the control points to form a contiguous grid of 3 3 degrees (7200 points), which we used for our shape studies. In this step we eliminated 77 points with local height peaks more then 10 km and lower than 10 km. Based on the gridded model, we determine a best-fit sphere radius of 2632.6 km (Table 1). Then, we computed best-fit parameters for spheroids and ellipsoids. The spheroid is a notable improvement in the approximation of Ganymede's shape, as the total sum of height squares is reduced by 13.9%. With the equatorial and polar axes differing by 2.4 km, this confirms a polar flattening for Ganymede. The nominal accuracies of the obtained radii are 0.1 km. Even though the errors of individual control point coordinates are large, errors of our best-fit shape model axes are 0.1 km only owing to the large number of data points. In contrast, the ellipsoid description (ellipsoid I) constitutes only a minor improvement by 4.4% over the spheroid model. Finally, we solved for an ellipsoidal model allowing for a shift of the major axis within the equatorial plane (ellipsoid II), which yields an added improvement by 10.5% over the ellipsoid model with fixed orientation (ellipsoid I). With an orientation of 37° West 71°, the ellipsoidal long axis of Ganymede is approximately pointed towards Jupiter. However, this ellipsoid is characterized by an unreasonably large a-axis of 2637 km. We believe, this solution is driven by artifacts in the data. An alternative method to determine planetary shape is from limb point measurements. Unfortunately, owing to the limited
3.2. Measurements The point measurements were carried out by means of the 〈〈PHOTOMOD〉〉 software package (http://www.racurs.ru). The package, originally developed for planar image data from aerial surveys, was upgraded to accommodate true 3-dimensional global control point networks. Also, we added a new data analysis tool to the package, which allowed us to measure large numbers of tiepoint coordinates effectively. Overlapping parts of images were first orthorectified to the same cylindrical projection using the above-mentioned image navigation data and Ganymede reference frame. While operators typically have to cope with different image orientations, resolutions, brightness, and illumination geometry, this process greatly facilitated the visual inspection of image pairs, the tie-point identification and the point measurement. For appropriate image pairs, tie-point measurements were made in stereo mode on a stereo screen. Image brightness correction was used, allowing for more accurate tie-point targeting during measurements. Several images can be uploaded simultaneously for image cross-comparisons and simultaneous measurements of tie-points. Thus, we effectively exclude gross errors and reduce uncertainties of the measurements. After measuring the pixel coordinates of tie-points from orthorectified images were converted back to image location coordinates.
248
A. Zubarev et al. / Planetary and Space Science 117 (2015) 246–249
number of Galileo limb images, only a sphere could be fitted (Thomas et al., 1997). The derived radius is comparably small and associated with large errors of 2631.57 2 km. We also solved for equilibrium shape models, which require as a condition (Zharkov et al., 1985)
(a − c )/(b − c ) = 4
(1)
hence, we determine the ellipsoidal axes a, b, and c, using (1) as a solution constraint (Table 1, ellipsoid III). Finally, following the suggested analysis by Davies et al. (1998) we adopt the mean radius r from our spherical shape fit and form
a = r (1 + 7/6qr h2 ) b = r (1 − 1/3qr h2 ) c = r (1 − 5/6qr h2 )
(2)
where qr is the ratio of the centrifugal to gravitational acceleration at the equator and h2 the tidal Love number (Table 1, ellipsoid IV). The two critical dynamic parameters were updated from what was
previously proposed (Anderson et al., 1996), as h2 ¼ 1.08027 and qr ¼1.906 10 4, using new estimated parameters for mass (1.48186 1023 kg) and C22 (38.26 10 6) (Schubert et al., 2004) as well as using our new reference radius (r ¼2632.63 km, Table 1) for having most recent and internally consistent data. Both equilibrium ellipsoid solutions are similar and show good agreement with ellipsoid I, as well as with the ellipsoid solution by Davies et al. (1998). Indeed, by analysis of the total sum of squares, we find that all ellipsoid solutions constitute significant improvements over the spherical model. After the ellipsoid fit, we take a closer look at residuals in the heights of our control points to possibly identify topographic depressions or elevated areas beyond the ellipsoidal shape models. While these “heights” vary significantly around their mean, high or low data values often correlate with the control point errors. Inspection of the data suggests that we cannot confidently identify any topography at global scale. In contrast, we note that some areas e.g. near 135° longitude (where errors for several hundreds of adjacent control point are acceptably small) are remarkably flat. Hence, our data suggest that the surface of Ganymede is
Fig. 1. Control point distribution and total coordinate uncertainties. The sizes of the circles are proportional to the coordinate errors. Color codes represent heights in kilometers above the best-fit sphere. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Table 1 Ganymede shape parameters, all errors 0.1 km; errors for volume are 0.9 106 km3. Model
Parameters (km)
Volume (106 km3)
Density (kg/m3)
Total sum of squares and rms (km2/km)
Sphere Spheroid
r¼ 2632.63 a¼ b¼2633.77 c¼ 2631.38 a¼ 2634.57 b¼ 2632.97 c¼ 2631.38 a¼ 2636.72 b¼ 2630.81 c¼ 2631.38 a¼ 2634.77 b¼ 2632.38 c¼ 2631.59 a¼ 2633.69 b¼ 2632.33 c¼ 2631.87
76428.76
1939
47760.7/3.82
76458.72
1938
41105.7/3.55
76458.72
1938
39315.0/3.47
76458.61
1938
35207.0/3.28
76453.85
1938
39467.7/3.47
76428.75
Ellipsoid I
Ellipsoid IIa
Equilibrium Ellipsoid IIIb
Equilibrium Ellipsoid IVc
1939
42583.6/3.61
Davies et al. (1998) Ellipsoidd
r¼ 2634.17 0.3 a¼ 2635.16 b¼ 2633.80 c¼ 2633.34
76557.08
1936
48650.3/3.86
76649.41
1933
45306.7/3.72
Thomas et al. (1997)e
r¼ 2631.5 7 2 km
76330.61
1941
56731.8/4.17
a
Ellipsoid orientation: 37° West 7 1° in the equatorial plane. Ellipsoid axes determined constrained by Eq. (1). c Ellipsoid determined from mean radius, see Eq. (2). d Only one digit behind the comma as given in Davies et al. (1998). More digits computed from Eq. (2). e from analysis of Galileo limb data. b
A. Zubarev et al. / Planetary and Space Science 117 (2015) 246–249
249
perturbations by other satellites predicted to cause pronounced librations (Rambaux et al., 2011), we determine upper limits for the amplitude of a simple sinusoidal libration model. A full Ganymede rotation model will require dedicated observational data from future missions, e.g., ESA's JUICE (Jupiter Icy Moon Explorer). As a byproduct of our block adjustment, we have derived improved pointing data for a global block of more than 200 Voyager and Galileo Ganymede images, which will form a basis for geometrically accurate maps, useful for planning and operation of ESA's JUICE mission.
Acknowledgments Fig. 2. Total RMS errors of spacecraft positions plotted vs. libration amplitude parameters of the Ganymede rotation model, used in the bundle block adjustment. The correct libration amplitude is expected to be where the total errors are at a minimum.
smooth on global scale, with no large areas rising above or falling more than 5 km (i.e., the average control point error) below the ellipsoid models. We also made attempts to determine offsets from the nominal rotational model, in particular, we searched for a sinusoidal libration in longitude superimposed to Ganymede's nominal (uniform) rotation. We investigated total errors of our bundle block adjustments, while applying different numerical values for the libration amplitudes to the rotation model (Fig. 2). We demonstrate that the control point method is sensitive to such a librational motion in principle, but subject to limitations in the quality of our image data set. A rather broad minimum of total errors is found, which leads us to give an upper limit for the libration amplitude of 0.1° (corresponding to lateral displacement of 4.6 km at the equator, which is roughly the magnitude of our point errors). Our estimates may be useful to constraint interior structure models for Ganymede. Our finding is in agreement with data by Rambaux et al. (2011) who carried out a spectral analysis of Ganymede's libration function and found an amplitude of 12.685′′ (162 m at the equator) for the dominant orbital period. More recent models of the librational motion of icy satellites suggest that the libration amplitude of Ganymede is even smaller, only 6.4–9.1 m at the equator (Van Hoolst et al., 2013). While Rambaux et al. (2011) used a rigid 3-layer model for Ganymede, Van Hoolst et al. (2013) include elastic tidal deformation.
5. Conclusions/discussions We have derived a control point network for Ganymede and studied the satellite's global shape and rotation. In a previous analysis, only 2D coordinates for approximately 2100 control points and a mean radius were determined; Ganymede was assumed to be in tidal equilibrium shape and orientation. Benefitting from improved spacecraft trajectory data and a much larger set of new measured 3000 control points, we determine 3D-control point coordinates and can measure global shape and orientation parameters directly, or test available shape models for agreement with the data. From our measurements, we confirm that Ganymede's shape is approximated by an ellipsoid, with its long axis near the Jupiter line. Our data agree with Ganymede being in tidal equilibrium shape. Owing to the large errors of our control point heights, we cannot confidently identify any topography at scales smaller than the ellipsoidal model. Our analysis is assuming a simplified uniform rotation model, with the rotational axis perpendicular to the orbit plane. With
A. Zubarev, I. Nadezhdina and J. Oberst have been supported by the Russian Science Foundation, Project #14-22-00197. H. Hussmann has been supported by the European Community's Seventh Framework Program (FP7) under Grant agreement n_ 263466, ESPaCE. The new 3-D Control Points Network can be obtained by the web link: http://cartsrv.mexlab.ru/test3d/#body ¼ganymede &proj¼ sc&loc¼%28-3.33984375%2C3.515625% 29&zoom ¼2&layer ¼Global_3D_control_network&lang ¼ en.
References Acton, C.H., 1996. Planet. Space Sci. 44 (1), 65–70. Archinal, B.A., A’Hearn, M.F., Bowell, E., Conrad, A., Consolmagno, G.J., Courtin, R., Fukushima, T., Hestroffer, D., Hilton, J.L., Krasinsky, G.A., Neumann, G., Oberst, J., Seidelmann, P.K., Stooke, P., Tholen, D.J., Thomas, P.C., Williams, I.P., 2011. Celest. Mech. Dyn. Astron. 109, 101. http://dx.doi.org/10.1007/s10569-010-9320-4. Anderson, J.D., Lau, E.L., Sjogren, W.L., Schubert, G., Moore, W.B., 1996. Gravitational constraints on the internal structure of Ganymede. Nature 384, 541–543. Davies, M.E., Colvin, T.R., Oberst, J., Zeitler, W., Schuster, P., Neukum, G., McEwen, A.S., Phillips, C.B., Thomas, P.C., Veverka, J., Belton, M.J.S., Schubert, G., 1998. The control networks of the galilean satellites and implications for global shape. Icarus 135, 372–376, Article no. IS985982. Jacobson, R.A., 2002. Pioneer and Voyager Jupiter Encounter Orbit Reconstruction in the ICRF System. Paper AAS-02-157, AAS/AIAA Space Flight Mechanics Meeting. San Antonio, Texas. Oberst, J., Zubarev, A., Nadezhdina, I., Shishkina, L., Rambaux, N., 2014a. The Phobos geodetic control point network and rotation model. Planet. Space Sci. 102, 45–50. http://dx.doi.org/10.1016/j.pss.2014.03.006i. Oberst, J., Gwinner, K., Preusker, F., 2014b. Exploration and analysis of planetary shape and topography using stereophotogrammetry. In: Spohn, T., Breuer, D., Johnson, T.V. (Eds.), Encyclopedia of the Solar System. Elsevier, Netherlands, pp. 1223–1233, ISBN: 9780124158450. Rambaux, N., Van Hoolst, T., Karatekin., O., 2011. Libration response of Europa, Ganymede, and Callisto with an ocean for a non-Keplerian orbit. Astron. Astrophys. 527, A118. http://dx.doi.org/10.1051/0004-6361/201015304. Schubert, G., Anderson, J.D., Spohn, T., McKinnon, W.B., 2004. Interior composition, structure, and dynamics of the Galilean satellites. In: Bagenal, F., et al. (Eds.), Jupiter: The Planet, Satellites, and Magnetosphere. Cambridge University Press, United Kingdom, pp. 281–306. Thomas, P.C., Simonelli, D., Burns, J., Davies, M.E., McEwen, A.S., Belton, M., Galileo SSI Team, 1997. Shapes and topography of Galilean satellites from Galileo SSI limb coordinates. In: Proceedings of the 28th Annual Lunar and Planetary Science Conference, Conference Paper, p. 429. Van Hoolst, T., Baland, R.-M., Trinh, A., 2013. On the librations and tides of large icy satellites. Icarus 226, 299–315. http://dx.doi.org/10.1016/j.icarus.2013.05.036. Zharkov, V.N., Leontjev, V.V., Kozenko, A.V., 1985. Models, figures, and gravitational moments of the Galilean satellites of Jupiter and icy satellites of Saturn. Icarus 61, 92–100.
Web references 〈http://pds-imaging.jpl.nasa.gov/volumes/galileo.html〉. 〈http://pds-imaging.jpl.nasa.gov/volumes/voyager.html〉. 〈http://naif.jpl.nasa.gov/pub/naif/GLL/kernels/spk〉. 〈http://naif.jpl.nasa.gov/pub/naif/VOYAGER/kernels/spk〉.
Contents lists available at ScienceDirect
Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
New Ganymede control point network and global shape model A. Zubarev a,n, I. Nadezhdina a, J. Oberst a,b,c, H. Hussmann b, A. Stark b a b c
Moscow State University of Geodesy and Cartography (MIIGAiK),MIIGAiK Extraterrestrial Laboratory (MExLab), Gorokhovsky per. 4, 105064 Moscow, Russia German Aerospace Center (DLR), Institute of Planetary Research, Berlin, Germany Technical University Berlin, Institute for Geodesy and Geoinformation Sciences, Berlin, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 24 February 2015 Received in revised form 10 June 2015 Accepted 28 June 2015 Available online 10 July 2015
We computed a 3D control point network for Ganymede using combinations of 126 Voyager-1 and -2 and 87 Galileo images, benefiting from reconstructed trajectory data for the three spacecraft and a more complete Galileo image data base than was available for earlier studies. Using more than 3000 control point coordinates, we determine global shape parameters, including mean radius, spheroid- and ellipsoidal axes, and make tests for various equilibrium shape models, constrained by the most recent estimates for gravity field parameters. We confirm that Ganymede has a pronounced ellipsoidal shape, approximately aligned with the Jupiter-direction, in agreement with Ganymede being in tidal equilibrium. The point heights, suffering from large individual errors, do not reveal any large-scale topography below our typical error levels (97% o 5 km). By analysis of data residuals we search for, but cannot detect Ganymede longitudinal forced librations. We conclude that libration amplitudes cannot be larger than 0.1° (corresponding to a lateral displacement of 4.6 km at the equator). & 2015 Elsevier Ltd. All rights reserved.
Keywords: Ganymede Control points network Libration Ellipsoid
1. Introduction While being the largest planetary satellite in the Solar System, information on physical dimensions and reference parameters for Ganymede (e.g., Archinal et al., 2011), are still rather limited. As Ganymede is exposed to strong centrifugal and tidal forces, global shape parameters and physical librations represent useful constraints on the dynamic state and interior structure of Ganymede (Schubert et al., 2004). In this paper, we establish a new geodetic control point network for Ganymede, based on images obtained by the Voyager and Galileo missions, and study global shape and rotational parameters of the satellite. Our study includes several improvements over a previous analysis (Davies et al., 1998), as we benefit from new orbit reconstructions for the Voyager and Galileo spacecraft and a larger image data base than was previously available. Also, the control point measurements in the images were redone. Using a new developed software interface and semi-automatic methods, we worked more effectively and greatly increased the number of measurements and measurement accuracy. The only previous Ganymede control point network (Davies et al., 1998) had used a 2-D block adjustment, i.e., solutions were determined for latitudes and longitudes of the control point coordinates and one common radius only. The ellipsoidal axes were computed analytically from the mean radius, assuming Ganymede's n
Corresponding author. E-mail address: [email protected] (A. Zubarev).
http://dx.doi.org/10.1016/j.pss.2015.06.022 0032-0633/& 2015 Elsevier Ltd. All rights reserved.
shape to be in tidal equilibrium, with the a- and b axes in the equatorial plane and the a-axis aligned with the Jupiter direction. This is in contrast to the approach of this paper, where we determine 3-D Cartesian coordinates of our control points, measure Ganymede's global shape from the control point radii and carry out tests for various equilibrium shape models. In this paper, we also constraint a possible forced longitudinal libration amplitude.
2. Data Our analysis is based on images taken by the Galileo (http://pds-imaging.jpl.nasa.gov/volumes/galileo.html) and Voyager (http://pds-imaging.jpl.nasa.gov/volumes/galileo.html) spacecraft. The two Voyagers made their Ganymede flybys in 1979–03 and 1979 07, at ranges of 186,000 km and 208,000 km, respectively. Both spacecraft were equipped with Vidicon sensors (2 cameras per spacecraft (WAC and NAC): VGR1 NAC f¼ 1500.190 mm, pixel size ¼11.789 mm, WAC f ¼200.465 mm, pixel size¼11.800 mm; VGR2 NAC f¼ 1503.490 mm, pixel size¼11.789 mm, WAC f¼ 200.770 mm, pixel size ¼11.666 mm). Unfortunately, these Vidicon images suffer from significant geometric distortions and require sophisticated geometric calibration schemes, e.g., implemented in the planetary image processing package VICAR (see http://www-mipl.jpl.nasa.gov/external/vicar. html for details). In contrast, the Galileo spacecraft was equipped with the Solid State Imaging Instrument (SSI), which was equipped with a 800 by 800 pixels CCD chip providing geometrically stable
A. Zubarev et al. / Planetary and Space Science 117 (2015) 246–249
images with f ¼1500.47 mm, pixel size ¼ 15.240 mm. During its orbital mission, Galileo obtained 47 Ganymede image sequences from 15 flybys. Spacecraft position and attitude (“image navigation”) data have been supplied by the JPL Navigation Ancillary Facility (NAIF) in the format of SPICE kernels (Acton, 1996). While previous investigators had to cope with large uncertainties of exterior orientations for Voyager (http://naif.jpl.nasa.gov/pub/naif/VOYAGER/kernels/spk) and Galileo (http://naif.jpl.nasa.gov/pub/naif/GLL/kernels/spk), we now used reconstructed spacecraft ephemerides provided by NAIF (Jacobson, 2002). While improvements of the Galileo trajectory data are small ( 100 m), comparison between reconstructed and previous ephemerides for Voyager 1 and 2 revealed shifts as large as 873 km and 221 km, respectively. All position and navigation data were converted to the Ganymede-fixed reference frame defined by the satellite's nominal rotation parameters (Archinal et al., 2011) (with Ganymede assumed in a perfect tidal lock and its rotation axis perpendicular to the orbit plane).
3. Method
247
3.3. Block adjustment All tie-point measurements and associated image navigation data are subjected to a bundle block adjustment (also implemented within the 〈〈PHOTOMOD〉〉 software package), which produces 3D Cartesian coordinates of the control points and improved navigation for each involved image by inversion of the collinearity equations (see Oberst et al. (2014a, 2014b) for details). The bundle block adjustment was carried out repeatedly to identify, possibly re-measure, or remove gross outliers after each step. As a result we obtained coordinates of 3377 control points observed 15655 times with a minimum of 2 and a maximum of 14 observations per point (average: 4.6). Coordinate errors were determined after solving the normal equations from the covariance matrix. These differ substantially, owing to different image resolutions, viewing conditions (notably the different convergence angles), and tiepoint measurement errors. For 97% (1 sigma) of the control points, the height accuracy is better than 5.0 km, with remaining point outliers having errors as large as 33.0 km. We removed all outlier points with height errors larger than 15.0 km (3 sigma). Thus, we eliminated 8 points (0.2%).
3.1. Image block
4. Results
We first carried out a catalog search to assemble an appropriate block of images for the analysis. The goal is to identify images with suitable global coverage and multiple overlaps. We also aim at images possibly obtained with similar resolutions and under similar lighting and viewing conditions, in particular, avoiding images taken under small emission angles. Unfortunately, only rather oblique images are available for the north and south polar area. Furthermore, we have no image coverage at all for small distinct areas (representing gaps of approximately 4000 km2, each) near the north and south pole. The image search was repeated after initial data analysis and correcting the image navigation data. Finally, we selected 126 Voyager and 87 Galileo images, which varied in their resolutions between 0.5 km and 25 km. Resolutions for a few Galileo image were as small as 19 m/pixel.
We first solved for the center point of the control points, which was found to be shifted from the origin of the Cartesian coordinate system by approximately þ 32.5, 6.0, and 3.2 km, probably due to errors in spacecraft flyby trajectories, Ganymede ephemeris, or both. After correction for this offset, cartesian coordinates were converted to latitude, longitude and radius for further analysis. Ranges of the control points from the coordinate center were found to vary from 2621 km to 2644 km. In Fig. 1, we show locations of control points, their associated errors and height residuals with respect to the mean. There are notable data gaps near the South pole, where we have no coverage by any of the three spacecraft, or where there is no image overlap. Also, control points on the trailing hemisphere suffer from larger errors on average. The data reveals benign regional elevation trends (note the yellow-color points). Next, we resampled the control points to form a contiguous grid of 3 3 degrees (7200 points), which we used for our shape studies. In this step we eliminated 77 points with local height peaks more then 10 km and lower than 10 km. Based on the gridded model, we determine a best-fit sphere radius of 2632.6 km (Table 1). Then, we computed best-fit parameters for spheroids and ellipsoids. The spheroid is a notable improvement in the approximation of Ganymede's shape, as the total sum of height squares is reduced by 13.9%. With the equatorial and polar axes differing by 2.4 km, this confirms a polar flattening for Ganymede. The nominal accuracies of the obtained radii are 0.1 km. Even though the errors of individual control point coordinates are large, errors of our best-fit shape model axes are 0.1 km only owing to the large number of data points. In contrast, the ellipsoid description (ellipsoid I) constitutes only a minor improvement by 4.4% over the spheroid model. Finally, we solved for an ellipsoidal model allowing for a shift of the major axis within the equatorial plane (ellipsoid II), which yields an added improvement by 10.5% over the ellipsoid model with fixed orientation (ellipsoid I). With an orientation of 37° West 71°, the ellipsoidal long axis of Ganymede is approximately pointed towards Jupiter. However, this ellipsoid is characterized by an unreasonably large a-axis of 2637 km. We believe, this solution is driven by artifacts in the data. An alternative method to determine planetary shape is from limb point measurements. Unfortunately, owing to the limited
3.2. Measurements The point measurements were carried out by means of the 〈〈PHOTOMOD〉〉 software package (http://www.racurs.ru). The package, originally developed for planar image data from aerial surveys, was upgraded to accommodate true 3-dimensional global control point networks. Also, we added a new data analysis tool to the package, which allowed us to measure large numbers of tiepoint coordinates effectively. Overlapping parts of images were first orthorectified to the same cylindrical projection using the above-mentioned image navigation data and Ganymede reference frame. While operators typically have to cope with different image orientations, resolutions, brightness, and illumination geometry, this process greatly facilitated the visual inspection of image pairs, the tie-point identification and the point measurement. For appropriate image pairs, tie-point measurements were made in stereo mode on a stereo screen. Image brightness correction was used, allowing for more accurate tie-point targeting during measurements. Several images can be uploaded simultaneously for image cross-comparisons and simultaneous measurements of tie-points. Thus, we effectively exclude gross errors and reduce uncertainties of the measurements. After measuring the pixel coordinates of tie-points from orthorectified images were converted back to image location coordinates.
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number of Galileo limb images, only a sphere could be fitted (Thomas et al., 1997). The derived radius is comparably small and associated with large errors of 2631.57 2 km. We also solved for equilibrium shape models, which require as a condition (Zharkov et al., 1985)
(a − c )/(b − c ) = 4
(1)
hence, we determine the ellipsoidal axes a, b, and c, using (1) as a solution constraint (Table 1, ellipsoid III). Finally, following the suggested analysis by Davies et al. (1998) we adopt the mean radius r from our spherical shape fit and form
a = r (1 + 7/6qr h2 ) b = r (1 − 1/3qr h2 ) c = r (1 − 5/6qr h2 )
(2)
where qr is the ratio of the centrifugal to gravitational acceleration at the equator and h2 the tidal Love number (Table 1, ellipsoid IV). The two critical dynamic parameters were updated from what was
previously proposed (Anderson et al., 1996), as h2 ¼ 1.08027 and qr ¼1.906 10 4, using new estimated parameters for mass (1.48186 1023 kg) and C22 (38.26 10 6) (Schubert et al., 2004) as well as using our new reference radius (r ¼2632.63 km, Table 1) for having most recent and internally consistent data. Both equilibrium ellipsoid solutions are similar and show good agreement with ellipsoid I, as well as with the ellipsoid solution by Davies et al. (1998). Indeed, by analysis of the total sum of squares, we find that all ellipsoid solutions constitute significant improvements over the spherical model. After the ellipsoid fit, we take a closer look at residuals in the heights of our control points to possibly identify topographic depressions or elevated areas beyond the ellipsoidal shape models. While these “heights” vary significantly around their mean, high or low data values often correlate with the control point errors. Inspection of the data suggests that we cannot confidently identify any topography at global scale. In contrast, we note that some areas e.g. near 135° longitude (where errors for several hundreds of adjacent control point are acceptably small) are remarkably flat. Hence, our data suggest that the surface of Ganymede is
Fig. 1. Control point distribution and total coordinate uncertainties. The sizes of the circles are proportional to the coordinate errors. Color codes represent heights in kilometers above the best-fit sphere. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
Table 1 Ganymede shape parameters, all errors 0.1 km; errors for volume are 0.9 106 km3. Model
Parameters (km)
Volume (106 km3)
Density (kg/m3)
Total sum of squares and rms (km2/km)
Sphere Spheroid
r¼ 2632.63 a¼ b¼2633.77 c¼ 2631.38 a¼ 2634.57 b¼ 2632.97 c¼ 2631.38 a¼ 2636.72 b¼ 2630.81 c¼ 2631.38 a¼ 2634.77 b¼ 2632.38 c¼ 2631.59 a¼ 2633.69 b¼ 2632.33 c¼ 2631.87
76428.76
1939
47760.7/3.82
76458.72
1938
41105.7/3.55
76458.72
1938
39315.0/3.47
76458.61
1938
35207.0/3.28
76453.85
1938
39467.7/3.47
76428.75
Ellipsoid I
Ellipsoid IIa
Equilibrium Ellipsoid IIIb
Equilibrium Ellipsoid IVc
1939
42583.6/3.61
Davies et al. (1998) Ellipsoidd
r¼ 2634.17 0.3 a¼ 2635.16 b¼ 2633.80 c¼ 2633.34
76557.08
1936
48650.3/3.86
76649.41
1933
45306.7/3.72
Thomas et al. (1997)e
r¼ 2631.5 7 2 km
76330.61
1941
56731.8/4.17
a
Ellipsoid orientation: 37° West 7 1° in the equatorial plane. Ellipsoid axes determined constrained by Eq. (1). c Ellipsoid determined from mean radius, see Eq. (2). d Only one digit behind the comma as given in Davies et al. (1998). More digits computed from Eq. (2). e from analysis of Galileo limb data. b
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perturbations by other satellites predicted to cause pronounced librations (Rambaux et al., 2011), we determine upper limits for the amplitude of a simple sinusoidal libration model. A full Ganymede rotation model will require dedicated observational data from future missions, e.g., ESA's JUICE (Jupiter Icy Moon Explorer). As a byproduct of our block adjustment, we have derived improved pointing data for a global block of more than 200 Voyager and Galileo Ganymede images, which will form a basis for geometrically accurate maps, useful for planning and operation of ESA's JUICE mission.
Acknowledgments Fig. 2. Total RMS errors of spacecraft positions plotted vs. libration amplitude parameters of the Ganymede rotation model, used in the bundle block adjustment. The correct libration amplitude is expected to be where the total errors are at a minimum.
smooth on global scale, with no large areas rising above or falling more than 5 km (i.e., the average control point error) below the ellipsoid models. We also made attempts to determine offsets from the nominal rotational model, in particular, we searched for a sinusoidal libration in longitude superimposed to Ganymede's nominal (uniform) rotation. We investigated total errors of our bundle block adjustments, while applying different numerical values for the libration amplitudes to the rotation model (Fig. 2). We demonstrate that the control point method is sensitive to such a librational motion in principle, but subject to limitations in the quality of our image data set. A rather broad minimum of total errors is found, which leads us to give an upper limit for the libration amplitude of 0.1° (corresponding to lateral displacement of 4.6 km at the equator, which is roughly the magnitude of our point errors). Our estimates may be useful to constraint interior structure models for Ganymede. Our finding is in agreement with data by Rambaux et al. (2011) who carried out a spectral analysis of Ganymede's libration function and found an amplitude of 12.685′′ (162 m at the equator) for the dominant orbital period. More recent models of the librational motion of icy satellites suggest that the libration amplitude of Ganymede is even smaller, only 6.4–9.1 m at the equator (Van Hoolst et al., 2013). While Rambaux et al. (2011) used a rigid 3-layer model for Ganymede, Van Hoolst et al. (2013) include elastic tidal deformation.
5. Conclusions/discussions We have derived a control point network for Ganymede and studied the satellite's global shape and rotation. In a previous analysis, only 2D coordinates for approximately 2100 control points and a mean radius were determined; Ganymede was assumed to be in tidal equilibrium shape and orientation. Benefitting from improved spacecraft trajectory data and a much larger set of new measured 3000 control points, we determine 3D-control point coordinates and can measure global shape and orientation parameters directly, or test available shape models for agreement with the data. From our measurements, we confirm that Ganymede's shape is approximated by an ellipsoid, with its long axis near the Jupiter line. Our data agree with Ganymede being in tidal equilibrium shape. Owing to the large errors of our control point heights, we cannot confidently identify any topography at scales smaller than the ellipsoidal model. Our analysis is assuming a simplified uniform rotation model, with the rotational axis perpendicular to the orbit plane. With
A. Zubarev, I. Nadezhdina and J. Oberst have been supported by the Russian Science Foundation, Project #14-22-00197. H. Hussmann has been supported by the European Community's Seventh Framework Program (FP7) under Grant agreement n_ 263466, ESPaCE. The new 3-D Control Points Network can be obtained by the web link: http://cartsrv.mexlab.ru/test3d/#body ¼ganymede &proj¼ sc&loc¼%28-3.33984375%2C3.515625% 29&zoom ¼2&layer ¼Global_3D_control_network&lang ¼ en.
References Acton, C.H., 1996. Planet. Space Sci. 44 (1), 65–70. Archinal, B.A., A’Hearn, M.F., Bowell, E., Conrad, A., Consolmagno, G.J., Courtin, R., Fukushima, T., Hestroffer, D., Hilton, J.L., Krasinsky, G.A., Neumann, G., Oberst, J., Seidelmann, P.K., Stooke, P., Tholen, D.J., Thomas, P.C., Williams, I.P., 2011. Celest. Mech. Dyn. Astron. 109, 101. http://dx.doi.org/10.1007/s10569-010-9320-4. Anderson, J.D., Lau, E.L., Sjogren, W.L., Schubert, G., Moore, W.B., 1996. Gravitational constraints on the internal structure of Ganymede. Nature 384, 541–543. Davies, M.E., Colvin, T.R., Oberst, J., Zeitler, W., Schuster, P., Neukum, G., McEwen, A.S., Phillips, C.B., Thomas, P.C., Veverka, J., Belton, M.J.S., Schubert, G., 1998. The control networks of the galilean satellites and implications for global shape. Icarus 135, 372–376, Article no. IS985982. Jacobson, R.A., 2002. Pioneer and Voyager Jupiter Encounter Orbit Reconstruction in the ICRF System. Paper AAS-02-157, AAS/AIAA Space Flight Mechanics Meeting. San Antonio, Texas. Oberst, J., Zubarev, A., Nadezhdina, I., Shishkina, L., Rambaux, N., 2014a. The Phobos geodetic control point network and rotation model. Planet. Space Sci. 102, 45–50. http://dx.doi.org/10.1016/j.pss.2014.03.006i. Oberst, J., Gwinner, K., Preusker, F., 2014b. Exploration and analysis of planetary shape and topography using stereophotogrammetry. In: Spohn, T., Breuer, D., Johnson, T.V. (Eds.), Encyclopedia of the Solar System. Elsevier, Netherlands, pp. 1223–1233, ISBN: 9780124158450. Rambaux, N., Van Hoolst, T., Karatekin., O., 2011. Libration response of Europa, Ganymede, and Callisto with an ocean for a non-Keplerian orbit. Astron. Astrophys. 527, A118. http://dx.doi.org/10.1051/0004-6361/201015304. Schubert, G., Anderson, J.D., Spohn, T., McKinnon, W.B., 2004. Interior composition, structure, and dynamics of the Galilean satellites. In: Bagenal, F., et al. (Eds.), Jupiter: The Planet, Satellites, and Magnetosphere. Cambridge University Press, United Kingdom, pp. 281–306. Thomas, P.C., Simonelli, D., Burns, J., Davies, M.E., McEwen, A.S., Belton, M., Galileo SSI Team, 1997. Shapes and topography of Galilean satellites from Galileo SSI limb coordinates. In: Proceedings of the 28th Annual Lunar and Planetary Science Conference, Conference Paper, p. 429. Van Hoolst, T., Baland, R.-M., Trinh, A., 2013. On the librations and tides of large icy satellites. Icarus 226, 299–315. http://dx.doi.org/10.1016/j.icarus.2013.05.036. Zharkov, V.N., Leontjev, V.V., Kozenko, A.V., 1985. Models, figures, and gravitational moments of the Galilean satellites of Jupiter and icy satellites of Saturn. Icarus 61, 92–100.
Web references 〈http://pds-imaging.jpl.nasa.gov/volumes/galileo.html〉. 〈http://pds-imaging.jpl.nasa.gov/volumes/voyager.html〉. 〈http://naif.jpl.nasa.gov/pub/naif/GLL/kernels/spk〉. 〈http://naif.jpl.nasa.gov/pub/naif/VOYAGER/kernels/spk〉.