The Astronomical Journal, 132:2520 Y2526, 2006 December # 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A.
THE GRAVITY FIELD OF THE SATURNIAN SYSTEM FROM SATELLITE OBSERVATIONS AND SPACECRAFT TRACKING DATA R. A. Jacobson, P. G. Antreasian, J. J. Bordi, K. E. Criddle, R. Ionasescu, J. B. Jones, R. A. Mackenzie, M. C. Meek, D. Parcher, F. J. Pelletier, W. M. Owen, Jr., D. C. Roth, I. M. Roundhill, and J. R. Stauch Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109-8099;
[email protected] Received 2006 July 12; accepted 2006 August 30
ABSTRACT We present values for the masses of Saturn and its major satellites, the zonal harmonics in the spherical harmonic expansion of Saturn’s gravitational potential, and the orientation of the pole of Saturn. We determined these values using an extensive data set: satellite astrometry from Earth-based observatories and the Hubble Space Telescope; Earth-based, Voyager 1, and Voyager 2 ring occultation measurements; Doppler tracking data from Pioneer 11; and Doppler tracking, radiometric range, and imaging data from Voyager 1, Voyager 2, and Cassini. Key words: gravitation — planets and satellites: general — planets and satellites: individual (Saturn) 1. INTRODUCTION
tracking, radiometric range, imaging, and very long baseline interferometric ( VLBI ) data. We have augmented our data set with additional Earth-based and HST satellite astrometry and with the Saturn ring occultation measurements used by French et al. (1993) to determine the orientation of the Saturn pole. In addition, we added the newly discovered satellite Methone ( Porco et al. 2004, 2005a) to our satellite system. Because Methone’s orbit is strongly perturbed by Mimas, observations of Methone aid in the Mimas mass determination. Our work is a part of the overall Cassini spacecraft navigation effort, which is focused on the determination of precise orbits for the Cassini spacecraft and the Saturnian satellites. Computing those orbits requires a high-quality model for the gravity field of the Saturnian system. By-products of our work are reconstructed orbits for the Pioneer 11, Voyager, and Cassini spacecraft, orbits consistent with the gravity field.
Shortly after the Pioneer 11 encounter with Saturn in September of 1979, Null et al. (1981) performed an analysis of the gravitational field of the Saturnian system. Using the Doppler tracking of the spacecraft together with Saturnian satellite apse and node rate information, they determined the masses of Saturn and three of its satellites, Rhea, Titan, and Iapetus, and the second and fourth zonal harmonics (J2 and J4 ) in the spherical harmonic expansion of Saturn’s gravitational potential. Campbell & Anderson (1989) followed up the work of Null et al., adding the Doppler tracking, radiometric range, and starsatellite imaging acquired from the Voyager 1 and Voyager 2 spacecraft during their respective 1980 November and 1981 August visits to the Saturnian system. Combining Pioneer 11 and Voyager data sets, Campbell & Anderson improved the mass estimates of Saturn and the three satellites, obtained a weak estimate of Tethys’s mass, and revised estimates for J2 and J4 . They also found a value for J6 by using a priori information on the zonal harmonics from the ringlet constraint devised by Nicholson & Porco (1988). In preparation for the Cassini tour of the Saturnian system, Jacobson (2004) repeated Campbell & Anderson’s (1989) work but with a more sophisticated data processing technique and a more extensive spacecraft data set. Moreover, he incorporated a full dynamical model for the Saturnian satellite orbits into the analysis for the first time; the model employed numerical integration for the propagation of the orbits. Consequently, observations of the gravitational interaction among the satellites contributed directly to the determination of the gravity parameters. The satellite system contained not only the eight major satellites but also Phoebe and the Lagrangian satellites, Helene, Telesto, and Calypso; the dynamics of the Lagrangians provide valuable information on the masses of Tethys and Dione ( Dermott & Murray 1981). To help constrain the satellite orbits and gravity parameters, an extensive set of satellite astrometry from Earth-based observatories and the Hubble Space Telescope (HST ) was processed together with the Pioneer 11 and Voyager data. The Cassini project used the satellite ephemerides and gravity parameters obtained from the analysis in tour design and spacecraft navigation on approach to Saturn. Since the arrival of Cassini at Saturn, we have been extending Jacobson’s (2004) work by incorporating the Cassini Doppler
2. TRAJECTORIES AND ORBITS The spacecraft trajectories and satellite orbits are produced by numerical integration of their equations of motion. The equations are formulated in Cartesian coordinates centered at the Saturnian system barycenter and referred to the International Celestial Reference Frame (ICRF ). For the spacecraft we include the pointmass Newtonian accelerations due to the Sun, the planets, and the Saturnian satellites; the relativistic perturbations due to the Sun, Jupiter, and Saturn; and the perturbation due to the oblateness of Saturn. Details of the equations may be found in Moyer (2000). The spacecraft are also subject to a variety of nongravitational forces, as follows: 1. trajectory correction maneuvers (Voyager and Cassini ), 2. precession maneuvers (Pioneer 11, a spin-stabilized spacecraft), 3. small forces from the attitude control system, 4. solar radiation pressure, and 5. nonisotropic thermal radiation from the RTGs (Pu238 radioactive thermal generators provide electrical power for all four spacecraft). Jacobson (2003) and Antreasian et al. (2005) discuss these forces. The satellite dynamics, described in Peters (1981), include the mutual interaction of the satellites and perturbations due to the 2520
GRAVITY FIELD OF SATURNIAN SYSTEM TABLE 1 Cassini Close Satellite Encounters
Date 2004 2004 2004 2004 2004 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2005 2006 2006 2006 2006 2006
Jun 11 ............... Oct 17 ............... Oct 26 ............... Dec 13 .............. Dec 31 .............. Jan 14................ Feb 15............... Feb 17............... Mar 9 ................ Mar 31 .............. Apr 16............... May 2................ Jul 14 ................ Aug 2 ................ Aug 22 .............. Sep 7................. Sep 24............... Sep 26............... Oct 11 ............... Oct 12 ............... Oct 28 ............... Nov 26 .............. Dec 26 .............. Jan 15................ Feb 27............... Mar 19 .............. Apr 30............... May 20..............
Satellite
Distance ( km)
Phoebe Iapetus Titan Titan Iapetus Titan Titan Enceladus Enceladus Titan Titan Tethys Enceladus Mimas Titan Titan Tethys Hyperion Dione Enceladus Titan Rhea Titan Titan Titan Titan Titan Titan
2182 1115086 3749 3767 124121 62578 4154 1511 750 4979 3602 52402 421 63000 6235 3650 2024 620 1059 49332 3928 1266 12986 4617 4387 4524 4431 4454
Sun and Jupiter and to Saturn’s oblateness. The ring perturbations on the spacecraft and satellites are ignored because they are small and the ring masses are unknown. JPL planetary ephemeris DE414 (Standish 2006) provides the positions of the Sun and planets (see Standish et al. [1992] for a discussion of the planetary dynamics). Cassini has flown closer to all of the satellites than any of the other spacecraft. The flyby times and distances appear in Table 1; times and distances for the other spacecraft can be found in Campbell & Anderson (1989). The closest Cassini Saturn periapsis passage, which occurred during the Saturn orbit insertion, was at a distance of 80,245 km, slightly less than the 81,000 km of the Pioneer 11 Saturn periapsis. Most of the Cassini orbits have their Saturn periapsis at distances in the range of 200,000 Y 300,000 km. 3. DATA Our Earth-based satellite astrometry data arc begins with the visual micrometer observations made during the years 1938 Y 1947 at the US Naval Observatory ( USNO), Washington, and extends through the CCD observations made at the USNO Flagstaff Station in 2006. There is a gap in the set of available observations between 1947 and 1966. The HST observations cover the period from the Saturn ring plane crossing in 1995 to January of 2005. We have attempted to include all astrometry acquired within the data arc. The changes to the Jacobson (2004) data set are as follows: 1. the replacement of the original Nikolaev Observatory data (Gorel 1977; Voronenko & Gorel 1982, 1986) by their modern reduction (G. Krasinsky 2000, private communication to M. Standish),
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2. the replacement of the original Pulkovo Observatory data ( Tolbin 1985, 1987) by their modern reduction ( Tolbin 1991a, 1991b), 3. the replacement of much of the HST data by the recalibrated and expanded set ( French et al. 2006), 4. the addition of the USNO visual micrometer data (Hall et al. 1954), 5. the addition of newly available Pulkovo Observatory data ( Kisseleva et al. 1996; Kisseleva & Kalinitchenko 1998, 2000; Kisseleva & Izmailov 2000),1 6. the addition of CCD observations from Sheshan Observatory (Qiao et al. 2004), 7. the addition of CCD observations from Table Mountain Observatory, and 8. the addition of Flagstaff Astrometric Scanning Transit Telescope observations from the USNO Flagstaff Station (R. C. Stone 2004, 2005, private communications; A. K. B. Monet 2006, private communication). The ring occultation measurements, in the form of the occultation times and radii of the occulting rings, are from: 1. the occultation of Voyager 1 observed by the radio science team (RSS) on 1980 November 13, 2. the occultation of the star Sco observed with the Voyager 2 photopolarimeter (PPS) on 1981 August 25/26, and 3. the occultation of the star 28 Sgr observed by major observatories in Hawaii and North and South America on 1989 July 3. We replaced the occulted star positions used by French et al. (1993) with positions taken from the Hipparcos star catalog, corrected for proper motion and parallax. The spacecraft tracking data comprise Doppler and range acquired via NASA’s Deep Space Network (DSN ). Only Doppler tracking is available for Pioneer 11, beginning on 1979 August 20 and extending for 16 days to September 5, 4 days after Saturn periapsis. For Voyager we have both Doppler and range. The Voyager 1 tracking data arc begins on 1980 August 7 and ends 105 days later on November 20. The 106 day arc for Voyager 2 spans 1981 June 8 to September 22. The Voyager imaging observations are the sample and line locations of images of the satellites and background reference stars as seen with the Voyager Vidicon system. The imaging data arcs begin at the same time as the respective tracking data arcs and end on 1980 November 12 for Voyager 1 ( just prior to encounter) and on 1981 August 31 for Voyager 2 (5 days after encounter). Extensive discussions of the Pioneer 11 and Voyager data appear in the references (Null et al. 1981; Campbell & Anderson 1989; Jacobson 2004). The Cassini tracking data arc, from 2004 February 6 to 2006 June 20, includes Doppler and range. For the most part we used coherent two-way Doppler compressed to a 5 minute sample interval and ranging acquired on 5 minute cycle times. During some periods of high spacecraft activity we found it necessary to decrease the sample interval to better monitor that activity. For example, we used a 1 minute sample interval during the Rhea encounter and a 10 s sample interval during the Titan encounter of 2006 February 27. When two-way Doppler was unavailable, i.e., when the roundtrip light time prevented the tracking station that was transmitting the radio signal from also receiving it, we occasionally acquired coherent three-way Doppler or noncoherent one-way Doppler. These data typically proved useful in periods around spacecraft 1 See also T. P. Kisseleva (2002) at http://www.imcce.fr/page.php?nav=en/ ephemerides/donnees/nsdc/nsdf/fsapomai.html.
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JACOBSON ET AL.
maneuvers, and the three-way Doppler collected at Rhea periapsis passage provided critical gravity information. We corrected the tracking data for all four spacecraft for Earth media effects with a seasonal troposphere model and an ionosphere model; for Cassini we also applied daily weather adjustments to the troposphere model. The Cassini three-way data were calibrated for DSN interstation clock offsets. Although the Voyager tracking was primarily at S band, the spacecraft also has X-band downlink capability. Differences between the S-band and X-band downlinks provided interplanetary media calibrations for the S-band Doppler (Green et al. 1980). Analogous calibrations were considered for Cassini, which is tracked at X band but has Ka-band downlink capability (Tortora et al. 2003). However, X-band Doppler is much less sensitive to transmission media effects than S-band. Moreover, the Ka-band downlink is not normally transmitted due to operational considerations (e.g., only a few DSN stations are equipped to receive it, and its use complicates spacecraft power management). Consequently, the Cassini project elected to forgo routine calibration for interplanetary media. The Cassini imaging observations, like those from Voyager, are the sample and line locations of images of the satellites and background reference stars as seen by either the narrow-angle or wide-angle camera. However, the Cassini cameras have CCD detectors rather than the Vidicon system of Voyager. The CCDs are not subject to the electromagnetic distortions that affect the Vidicons, and they have superior noise characteristics, less dark current, and a wider dynamic range than the Vidicons. Moreover, Cassini’s narrow-angle camera has a finer resolution than Voyager’s. Consequently, Cassini’s imaging data are more accurate than those from Voyager. The data set contains 2984 images of the Saturnian satellites. We also introduced VLBI data into the data set used to determine the Cassini orbit. Our VLBI technique involves simultaneous tracking of the spacecraft and a nearby extragalactic radio source with the 10 antennas of the National Radio Astronomy Observatory’s Very Long Baseline Array (VLBA) ( Martin-Mur et al. 2006). The VLBA data were reduced to differential time-ofarrival measurements between the spacecraft and radio source signals at the different antennas. These measurements are related to the angular separation between the spacecraft and radio source and indirectly provide angular positions of Saturn. As noted by Campbell & Anderson (1989), gravity parameter determination can be sensitive to errors in the Saturn ephemeris. Adding the VLBI helps reduce those errors. We acquired the data on 2004 September 8 and October 14 Y20. Cassini tracking data were acquired during encounters with all of the major satellites. However, because the science instruments are fixed to the Cassini bus, tracking data are not normally available during a 1Y2 day period centered on the satellite close encounter time; the science instrument pointing geometries preclude pointing the high-gain antenna toward Earth for telecommunications. There have been two exceptions where there was continuous tracking, the Rhea encounter on 2005 November 26 and the Titan encounter on 2006 February 27. The Cassini tracking data also cover a total of 26 Saturn periapsis passages. More detailed discussions of the Cassini data can be found in Roth et al. (2005) and Antreasian et al. (2005). 4. METHOD OF SOLUTION We determine the gravity field from observational data via a weighted least-squares fit in which we adjust the parameters in our model of the orbits of the four spacecraft (Pioneer 11, Voy-
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ager 1, Voyager 2, and Cassini), the planet, and the satellites. The set of estimated dynamical parameters contains the following: 1. Epoch position and velocity of each spacecraft and satellite. 2. Elements of Saturn’s orbit. 3. GM of the Saturnian system and the satellites (GM is the product of the Newtonian constant of gravitation G and the body’s mass M ). In our model the system GM is a fundamental parameter, and the planet GM is inferred by subtracting the satellite GM values. 4. Gravitational harmonics J2 , J4 , and J6 of Saturn. 5. Right ascension and declination of Saturn’s pole. 6. Spacecraft maneuvers and nongravitational accelerations. We also estimate the following parameters in the observation model: 1. Biases and drift rates in the on-board oscillator frequency for the one-way noncoherent Doppler. 2. Tracking station- dependent range biases. These are included in two forms, a global bias and a bias unique to each pass. 3. Solar corona parameters for the Voyager 1 1980 and Cassini 2004 and 2005 solar conjunctions. These are parameters in a model that corrects range data for the delay caused when the ranging signal passes through the solar corona. 4. Doppler biases during the solar conjunction periods. There is currently no model for the Doppler delay caused by the solar corona, so we use a bias parameter for each pass to correct for the delay. 5. Station-dependent ionosphere and troposphere media corrections for the VLBI data. Unlike Doppler and range, media calibrations are unavailable for the VLBI data. 6. Spacecraft camera pointing angles. The pointing for each picture is adjusted based on the background stars appearing in the picture. 7. Satellite-dependent phase angle biases in the optical data. These biases account for the error in determining the center of the image of a partially illuminated object. 8. Station-dependent timing biases for the ring occultations. 9. Radii of the occulting rings. 10. Positions of occulted stars. 11. Scale and orientation of the CCD detector in some of the Earth-based astrometry. A number of observers calibrated the scale and orientation of their detectors with a procedure that relies on positions of the satellites predicted from preexisting ephemerides, and we correct for errors in those ephemerides. 12. Telescope pointing direction for those CCD observations provided in their raw form, e.g., those from Qiao et al. (2004). We grouped the astrometric data (Earth-based and HST ) according to type, observatory, and the observing period in which they were acquired. Data weights for each group were assigned through an iterative process to be consistent with the rms of the residuals (the differences between the actual observations and their values predicted by our model ) of that group. Similarly, we set separate Doppler data weights for each DSN pass to correspond to an accuracy consistent with the rms of the residuals for that pass. In addition, we applied a scale factor to the rms to account for the fact that the Doppler noise is not a white= noise process (Folkner 1994). The scale factor is 0:468ð86400/ Þ1 3 , where is the sample interval in seconds (for our normal 5 minute sample interval the factor is 3.09). The range data were weighted on a pass-by-pass basis with weights derived from the rms of the data residuals scaled by the square root of the number of points in the pass. The scaling
No. 6, 2006
GRAVITY FIELD OF SATURNIAN SYSTEM TABLE 2 Cassini Imaging Scale Factors Satellite
Scale
Mimas ........................ Enceladus ................... Tethys ......................... Dione.......................... Rhea ........................... Titan ........................... Hyperion..................... Iapetus ........................ Phoebe........................
0.02 0.015 0.01 0.01 0.01 0.02 0.06 0.02 0.04
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postencounter Voyager 2 images, taken with the wide-angle camera, were also set at 1.0Y2.0 pixels because of center finding problems introduced by their high phase angles. For the Cassini imaging data we assigned an accuracy of 0.1 pixel to the stars and computed an accuracy for the major sat2 þ ðCda Þ2 , where is the asellites from the expression 2 ¼ base signed accuracy, base is a base accuracy (0.25 pixel for satellites, except for 0.5 pixel for Titan and Hyperion), da is the satellite’s apparent diameter, and C is an empirical scale factor given in Table 2. An accuracy of 0.5 pixel is set for the Lagrangian satellites and Methone. The occultation observables are the radial distances of various ring features at the recorded occultation times. Based on the residual rms we set the measured distance accuracies at 1.15 km for the Voyager 1 RSS data, 1.18 km for the Voyager 2 PPS data, and 1.23 km for the 28 Sgr data. We performed the parameter estimation with a weighted leastsquares square root information filter (Lawson & Hanson 1974; Bierman 1977). This filter uses Householder transformations to pack the matrix of weighted observation partial derivatives and the weighted residuals into an upper triangular square root information matrix and associated residual vector. The solution for the parameters was generated and analyzed by means of singular value decomposition techniques applied to the square root information array. In keeping with the previous gravity analyses, our estimation process included a priori information on Saturn’s zonal harmonics from the ringlet constraint of Nicholson & Porco (1988) and a priori information on Saturn’s orbital elements provided by E. M. Standish (2006, private communication). We constrained the corrections to the occultation star positions to be within the uncertainties quoted in the Hipparcos catalog. A priori information
suppresses range rate information inferred from the change in the range during the pass; the Doppler data then become the primary source of range rate information. The VLBI data weights represent accuracies, in units of pico= seconds (ps), computed from the formula VLBI ¼ 30ð5N Þ1 2 . Here N is the number of points per baseline, and the factor 5 accounts for the fact that the 10 VLBA stations provide 45 possible but only 9 independent baselines. An accuracy of 20 ps corresponds to a position error of roughly 1 km at Saturn. The accuracy assumed for most of the Voyager imaging data was 0.5 pixel for the stars, major satellites, and Phoebe and 1.0 pixel for the Lagrangian satellites (their locations were not measured as carefully as were the locations of the major satellites). Near the Voyager encounters the accuracy for the major satellites in selected frames was decreased to 1.0 Y2.0 pixels to account for the increased difficulty in finding the image centers as image size grew with decreasing spacecraft distance. The accuracy of the
TABLE 3 Saturnian System Gravity Field Parameter
Null et al. (1981)
Campbell & Anderson (1989)
Jacobson (2004)
This Paper
GMSun /GMsys ............................ GM (km3 s2) ........................... System................................... Saturn .................................... Mimas ................................... Enceladus .............................. Tethys .................................... Dione..................................... Rhea ...................................... Titan ...................................... Hyperion................................ Iapetus ................................... Phoebe................................... J2 ( ; 106 ).................................... J4 ( ; 106 ).................................... J6 ( ; 106 ).................................... J8 ( ; 106 ).................................... p (deg) .................................... p (deg) ..................................... ˙ p (deg century1) ................... ˙p (deg century1).....................
3498.09 0.22
3497.898 0.018
3497.893 0.005
3497.9018 0.0001
37938544.0 2400.0 37929085.0 2400.0 2.50 0.06a 4.9 2.4b 41.5 0.8a 70.2 2.2b 151.0 34.0 9059.0 114.0 1.1 129.0 49.0 ... 16299.0 18.0 916.0 38.0 81.0 ... 40.6076 0.0230a 83.5219 0.0036a ... ...
37940630.0 200.0 37931272.0 200.0 2.50 0.06a 4.9 2.4b 45.0 10.0 70.2 2.2b 154.0 4.0 8978.2 1.0 0.0 106.0 10.0 ... 16298.0 10.0 915.0 40.0 103.0 50.0 10.0c 40.580 0.016d 83.540 0.002d ... ...
37940672.0 59.0 37931284.0 57.0 2.55 0.05 6.9 1.5 41.21 0.08 73.12 0.02 155.0 5.0 8978.1 0.8 0.72 0.35 130.0 17.0 0.5 0.2 16292.0 7.0 931.0 31.0 91.0 31.0 10.0c 40.5955 0.0036e 83.5381 0.0002e 0.04229g 0.00444g
37940585.2 1.1 37931207.7 1.1 2.5023 0.0020 7.2096 0.0067 41.2097 0.0063 73.1127 0.0025 153.9416 0.0049 8978.1356 0.0039 0.3727 0.0045 120.5117 0.0173 0.5534 0.0006 16290.71 0.27 935.83 2.77 86.14 9.64 10.0c 40.58279 0.00201f 83.53763 0.00021f 0.04229g 0.00444g
Note.— Reference radius for the gravitational harmonics: 60,330 km. a Adopted from Kozai (1957). b Adopted from Kozai (1976). c Theoretical estimate used by Nicholson & Porco (1988). d Adopted from Simpson et al. (1983). e At epoch 1980 November 12, 23:46:32, from French et al. (1993). f At epoch 2000 January 1, 12:00:00. g Derived from the precession rate of Nicholson et al. (1999).
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JACOBSON ET AL. TABLE 4 Physical Properties
Body
Radius (km)
Mass (1022 g)
Saturn .............. 58232.0 6.0a 56832592.0 8515.0 Mimas ............. 198.30 0.30b 3.7493 0.0031 10.8022 0.0101 Enceladus ........ 252.10 0.10b Tethys .............. 533.00 0.70b 61.7449 0.0132 Dione............... 561.70 0.45b 109.5452 0.0168 Rhea ................ 764.30 1.10b 230.6518 0.0353 Titan ................ 2575.50 2.00c 13452.0029 2.0155 Hyperion.......... 133.00 8.00d 0.5584 0.0068 Iapetus ............. 735.60 1.50b 180.5635 0.0375 Phoebe............. 106.60 1.00e 0.8292 0.0010 a b c d e
Density (g cm3) 0.6873 1.1479 1.6096 0.9735 1.4757 1.2333 1.8798 0.5667 1.0830 1.6342
0.0002 0.0053 0.0024 0.0038 0.0036 0.0053 0.0044 0.1025 0.0066 0.0460
Lindal et al. (1985). Thomas et al. (2006). Lindal et al. (1983). Thomas et al. (1995). Porco et al. (2005b).
was also included for the spacecraft maneuvers, nongravitational accelerations, and the parameters associated with the spacecraft observation models; see Jacobson (2003), Roth et al. (2005), and Antreasian et al. (2005) for details. 5. RESULTS AND DISCUSSION Table 3 gives our gravity field results along with those from the previous investigations involving spacecraft data. We have determined all GM values to a fraction of 1%. The uncertainties quoted in the table represent our assessment of the actual uncertainties; they span the range of solutions obtained with different data arcs and variations in our modeling. Table 4 contains the masses and densities derived from the GM values; to compute the masses we used G ¼ (6:6742 0:0010) ; 1023 km3 g1 s2 ( Mohr & Taylor 2005). Except for Mimas, Tethys, and Dione, the satellite GM values are found directly from spacecraft tracking during the satellite flybys. The Pioneer 11 and Voyager data aid in determining the Rhea, Titan, and Iapetus GM values, but the dominant source of information is the Cassini data. The Cassini tracking acquired during the Saturn periapsis passages is the primary source of the information on Saturn’s GM and its second and fourth zonal gravity harmonics. The sixth harmonic derives from a combination of the Cassini tracking and the ringlet constraint with a small
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contribution from the Pioneer 11 tracking. The ring occultation measurements supplemented with the Cassini tracking define the Saturn pole orientation. We did not estimate the pole angular rates (˙ p ; ˙p ) but instead continued to use the rates previously derived from a precession rate of 0B51 yr1 ( Nicholson et al. 1999). The mean motions of Mimas and Tethys are in a near 2:1 commensurability, resulting in a mean longitude libration with about a 72 yr period and amplitudes of 43 for Mimas and 2 for Tethys. There is a 2 : 1 mean motion commensurability between Enceladus and Dione as well, which causes an 11 yr period libration and an 3.8 yr period circulation with an amplitude of 1400 for Enceladus and 1B5 for Dione. A number of investigators found GM values of the four satellites from astrometric observations of these resonances. Jacobson (1995) employed a different approach to determining the Tethys and Dione GM values, namely, using the astrometric and Voyager imaging of the Lagrangian satellites, Helene, Telesto, and Calypso. With the discovery of the small satellite Methone, a different approach to the Mimas GM determination also became available. Jacobson et al. (2006) discovered that Mimas is a strong perturber of the orbit of Methone and that observations of the perturbation lead to an improved Mimas GM determination. In our current work the Lagrangian satellites and Methone are members of our satellite system. Consequently, the major satellite mean motion resonances and the perturbations on Telesto, Calypso, Helene, and Methone all contribute to our GM determinations. We find that the contribution from Cassini tracking data during the Mimas, Tethys, and Dione flybys is negligible. On the other hand, the tracking from the three flybys of Enceladus lead to a 2 orders of magnitude more accurate GM determination than that made solely from the resonance with Dione. Table 5 contains a summary of mass determinations of the four inner satellites. Our Mimas and Tethys values agree very well with those of Kozai (1957, 1976). We have effectively confirmed his values and significantly reduced their uncertainty. As is evident from the table entries, there has been considerable disagreement on the mass of Enceladus; Cassini has clearly settled those disagreements. Our estimate of Dione’s mass has not changed since its initial determination from the Dione perturbation of Helene’s orbit. The pre- and post-Cassini Rhea and Titan GM values are in good agreement, an indication of the strength of the Voyager data. The addition of the Cassini data primarily served to reduce the uncertainties.
TABLE 5 Inner Satellite Masses in Units of Saturn’s Mass ; 107 Source
Mimas
Enceladus
Struve (1930) ..................................... Jefferys (1953) ................................... Kozai (1957, 1976)............................ Tyler et al. (1982) .............................. Dourneau (1987) ................................ Campbell & Anderson (1989) ........... Harper & Taylor (1993)..................... Dermott & Thomas (1994)................ Jacobson (1995) ................................. Vienne & Duriez (1995).................... Dourneau & Baratchart (1999).......... Jacobson (2004) ................................. This paper ..........................................
0.040 0.020 0.015 0.095 0.021 ... 0.646 0.011 ... ... 0.640 0.025 0.651 0.021 0.672 0.012 0.6597 0.0006
1.51 0.30 1.27 0.53 1.34 0.61 ... 2.06 0.55 ... 2.13 0.46 1.14 0.03 ... 1.07 0.88 2.02 0.54 1.83 0.39 1.901 0.002
0.669 0.669 0.659 0.800 0.648
Tethys 11.41 11.41 10.95 13.3 10.88 11.9 10.75 ... 10.97 10.68 10.94 10.86 10.864
0.54 0.30 0.22 1.6 0.31 2.6 0.18 0.14 0.42 0.31 0.03 0.003
Dione 18.47 18.25 18.5 ... 19.54 ... 19.16 ... 19.28 19.50 19.23 19.28 19.275
0.92 0.61 0.6 0.58 0.36 0.07 1.21 0.57 0.04 0.004
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GRAVITY FIELD OF SATURNIAN SYSTEM
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TABLE 6 Ringlet Constraint Effect Case
J2 ; 106
J4 ; 106
J6 ; 106
With ringlet constraint ....................... Without ringlet constraint ..................
16290.71 0.27 16290.64 0.31
935.8 2.8 936.8 4.0
86.0 10.0 82.0 16.0
Prior to Cassini, Hyperion’s GM was based on its volume and an assumed density. Hyperion is in a 3 : 4 mean longitude resonance with Titan, but we thought that the predicted GM was so small that the effect of the resonance on Titan’s orbit would be insignificant. However, we found that Hyperion was indeed perturbing Titan’s orbit to the extent that it was having a detrimental effect on the spacecraft navigation to the Titan encounters. By estimating Hyperion’s GM as part of the routine navigation operations, we obtained a smaller GM, a slightly altered Titan orbit, and more consistent and stable predictions of the spacecraft trajectory. The close flyby in September of 2005 provided the definitive GM value (Criddle et al. 2006). Because of its size and distance from Saturn, Iapetus has a significant effect on the location of Saturn relative to the Saturnian system barycenter. Early in the Cassini tour we found that the spacecraft tracking was weakly sensitive to the Iapetus GM through this barycenter location effect. Prior to the first Cassini Iapetus flyby, we obtained a GM value of 120 11 km3 s2 based on the combined Pioneer 11, Voyager, and Cassini data. When we added the data from the first flyby, the value became 121 3, and after the second flyby it was 120:52 0:04. As we have improved our knowledge of Saturn’s barycentric position through the acquisition of more data, we have further reduced the GM uncertainty. Prior to Cassini, as in the case of Hyperion, Phoebe’s GM was based on its volume and an assumed density. However, unlike Hyperion, the predicted GM proved to be close to the value actually obtained from the flyby data. Our Saturnian system GM is nearly 1.5 less than the preCassini value. As with all of the previous analyses, we initially found that we could not easily distinguish between the effects on the spacecraft due to GM errors and those due to errors in Saturn’s orbit. As we continued to accumulate Cassini tracking data, the two error sources slowly became separable, yielding a stronger determination of both the GM and Saturn’s orbital elements. The most significant improvement in the GM occurred when we processed the tracking acquired during the four consecutive Saturn periapsis passages in May and June of 2005. These periapses were at about 3.6 Saturn radii and had little spacecraft activity between them to corrupt the data arc. The GM uncertainty dropped from 21 km3 s2 before the passages to 1.1 km3 s2 after them. The data from an additional 14 periapses since 2005 June have done little to change the GM value or its uncertainty. Our knowledge of Saturn’s gravitational harmonics likewise steadily improved with each succeeding Saturn periapsis passage. During the May-June periapses mentioned above, the uncertainties were reduced almost to their current levels; as more data have been added, the values have varied slightly within the uncertainties. The ringlet constraint continues to be important; Table 6 compares the harmonics obtained with and without the constraint. In the absence of the constraint there is an increase of about 40% in the uncertainty of J4 and a 60% increase in that of J6 . To determine the Saturn pole orientation we essentially reproduce the ‘‘adopted fit’’ of French et al. (1993) within our work but add to it the pole information available from the satellite and spacecraft data. We process the occultation observations with the
‘‘vector calculation’’ procedure described in their appendix. Unlike French et al. we work in the ICRF system and use Hipparcos star positions. As part of their fit they determined Voyager along-track trajectory offsets and Saturn-star offsets; we incorporate the occultation data into our overall correction of the Saturn orbit and Voyager trajectories. For purposes of comparison Table 7 contains the J2000.0 pole from French et al. and our pole propagated to their epoch of 1980 November 12, 23:46:32 UTC. The disagreement in right ascension is about 1.3 and in declination about 1.7 . 6. CONCLUDING REMARKS This paper reports on the gravity field of the Saturnian system based on data acquired through mid-2006. The data include those from all of the spacecraft that have visited the system, as well as an extensive collection of astrometric measurements from Earth-based observatories and HST. We have determined the GM of Saturn to an accuracy of 3 parts in 108 and the GM values of all of the major satellites to less than 1%, an accuracy sufficient to ensure that estimates of the satellites’ bulk densities are now limited by the ability to determine the satellites’ volumes. The Cassini mission calls for only two additional close satellite flybys, Iapetus at 1500 km in September of 2007 and Enceladus at less than 100 km in March of 2008. Data from those flybys are expected to further improve our knowledge of the GM values. We have also significantly improved the estimates of the zonal harmonics of Saturn’s gravitational potential and redetermined the orientation of the pole of Saturn as an integral part of the gravity field analysis. We acknowledge the Cassini Imaging Science Team, in particular the CICLOPS group at the Space Science Institute in Boulder, Colorado, and the Queen Mary group at the University of London, in the planning and execution of imaging observations of Methone, Helene, Calypso, and Telesto. The authors also thank S. D. Gillam, V. Alwar, J. Costello, A. Vaughan, and T-C. M. Wang (JPL) for their efforts in reducing the Cassini optical navigation data and for finding serendipitous Methone and Lagrangian satellite observations among the optical navigation images. Finally, thanks to Jon Romney of the National Radio Astronomy Observatory, Socorro, New Mexico, for his efforts in support of the VLBA observations. The research described in this publication was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. TABLE 7 Pole Orientation Comparison
Case
p (deg)
p (deg)
French et al. (1993) ...... This paper .....................
40.5955 0.0036 40.5909 0.0020
83.53812 0.00018 83.53848 0.00021
Note.—These orientation angles are at epoch 1980 November 12, 23:46:32.
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JACOBSON ET AL.
REFERENCES Antreasian, P. G., et al. 2005, Cassini Orbit Determination Performance During Mohr, P. J., & Taylor, B. N. 2005, Rev. Mod. Phys., 77, 1 the First Eight Orbits of the Saturn Satellite Tour (AAS Paper 05-312; Moyer, T. D. 2000, Formulation for Observed and Computed Values of Deep Springfield: Am. Astronaut. Soc.) Space Network Data Types for Navigation ( Pasadena: JPL) Bierman, G. J. 1977, Factorization Methods for Discrete Sequential Estimation Nicholson, P. D., French, R. G., & Bosh, A. S. 1999, BAAS, 31, 1140 ( New York: Academic Press) Nicholson, P. D., & Porco, C. C. 1988, J. Geophys. Res., 93, 10209 Campbell, J. K., & Anderson, J. D. 1989, AJ, 97, 1485 Null, G. W., Lau, E. L., Biller, E. D., & Anderson, J. D. 1981, AJ, 86, 456 Criddle, K. E., et al. 2006, Cassini Orbit Determination: The Only Targeted Peters, C. F. 1981, A&A, 104, 37 Hyperion Flyby of the Prime Mission (AAS Paper 06-082; Springfield: Am. Porco, C. C., et al. 2004, IAU Circ. 8389 Astronaut. Soc.) ———. 2005a, Science, 307, 1226 Dermott, S. F., & Murray, C. D. 1981, Icarus, 48, 12 ———. 2005b, Science, 307, 1237 Dermott, S., & Thomas, P. 1994, Icarus, 109, 241 Qiao, R. C., Shen, K. X., Harper, D., & Liu, J. R. 2004, A&A, 422, 377 Dourneau, G. 1987, Ph.D. thesis, Univ. Bordeaux I Roth, D. C., et al. 2005, Cassini Orbit Reconstruction from Jupiter to Saturn Dourneau, G., & Baratchart, S. 1999, A&A, 350, 680 (AAS Paper 05-311; Springfield: Am. Astronaut. Soc.) Folkner, W. M. 1994, Effect of Uncalibrated Particles on Doppler Tracking Simpson, R. A., Tyler, G. L., & Holberg, J. B. 1983, AJ, 88, 1531 ( Interoffice Memo. 335.1-94-005; Pasadena: JPL) Standish, E. M. 2006, JPL Planetary Ephemeris DE414 ( Interoffice Memo. French, R. G., McGhee, C. A., Frey, M., Hock, R., Rounds, S., Jacobson, R., & 343R-06-002; Pasadena: JPL) Verbiscer, A. 2006, PASP, 118, 246 Standish, E. M., Newhall, X. X., Williams, J. G., & Yeomans, D. K. 1992, in French, R. G., et al. 1993, Icarus, 103, 163 Explanatory Supplement to the Astronomical Almanac, ed. P. K. Seidelmann Gorel, G. K. 1977, Results of Astrometrical Observations of Saturn Made in ( Mill Valley: University Science Books), 279 Nikolaev with Zone Astrograph in 1973 Y1976 ( Leningrad: Gl. Astron. Obs. Struve, G. 1930, Vero¨ff. Univ.-Sternw. Berlin-Babelsberg, 4, No. 4, 1 Pulkovo) Thomas, P. C., Black, G. J., & Nicholson, P. D. 1995, Icarus, 117, 128 Green, D. W., Lam, V. W., & Royden, H. N. 1980, Effects of the Charged Thomas, P. C., et al. 2006, in 37th Annual Lunar and Planetary Science ConParticle Environment of Voyager Navigation at Jupiter and Saturn (AIAA ference, ed. S. Mackwell & E. Stansbery ( Houston: LPI ), 1639 Paper 80-1650; Reston: AIAA) Tolbin, S. B. 1991a, Results of Positional Photographic Observations of the Hall, Jr., A., Burton, H. E., Lyons, U. S., Raynsford, G. M., Browne, W. H., & Saturn Satellite System Made in Pulkovo with 26-Inch Refractor in Smith, J. L. 1954, Publ. USNO, 17, 93 1975 Y1984 ( Leningrad: Gl. Astron. Obs. Akad. Nauk) Harper, D., & Taylor, D. B. 1993, A&A, 268, 326 ———. 1991b, Results of Positional Photographic Observations of the Saturn Jacobson, R. A. 1995, BAAS, 27, 1202 Satellite System Made in Pulkovo with Normal Astrograph in 1975 Y1984 ———. 2003, Reconstruction of the Voyager Saturn Encounter Orbits in the ( Leningrad: Gl. Astron. Obs. Akad. Nauk) ICRF System (AAS Paper 03-198; Springfield: Am. Astronaut. Soc.) Tolbin, S. V. 1985, Results of Positional Photographic Observations of Saturn’s ———. 2004, AJ, 128, 492 System Made in Pulkovo on 26 Inch Refractor in 1975 ( Leningrad: Gl. Jacobson, R. A., Spitale, J., Porco, C. C., & Owen, W. M., Jr. 2006, AJ, 132, 711 Astron. Obs. Akad. Nauk) Jefferys, F. R. S. 1953, MNRAS, 113, 81 ———. 1987, Izv. Gl. Astron. Obs. Pulkovo, 204, 65 Kisseleva, T. P., & Izmailov, I. S. 2000, Izv. Gl. Astron. Obs., 214, 333 Tortora, P., Less, L., Bordi, J. J., Ekelund, J. E., & Roth, D. C. 2003, Cassini Kisseleva, T. P., & Kalinitchenko, O. A. 1998, Izv. Gl. Astron. Obs., 213, 122 Navigation during Solar Conjunctions via Removal of Solar Plasma Noise ———. 2000, Izv. Gl. Astron. Obs., 214, 344 (AAS Paper 03-200; Springfield: Am. Astronaut. Soc.) Kisseleva, T. P., Kisselev, A. A., Khrutskaya, E. V., & Kalinitchenko, O. A. Tyler, G., Eshleman, V., Anderson, J., Levy, G., Lindal, G., Wood, G., & Croft, 1996, Izv. Gl. Astron. Obs., 210, 76 T. 1982, Science, 215, 553 Kozai, Y. 1957, Ann. Tokyo Astron. Obs., 5, 73 Vienne, A., & Duriez, L. 1995, A&A, 297, 588 ———. 1976, PASJ, 28, 675 Voronenko, V. I., & Gorel, G. K. 1982, Photographic Observations of Satellites Lawson, C. L., & Hanson, R. J. 1974, Solving Least Squares Problems of Jupiter and Saturn Made in Nikolaev 1976 Y1979 ( Leningrad: Gl. Astron. ( Englewood Cliffs: Prentice-Hall ) Obs. Akad. Nauk) Lindal, G. F., Sweetnam, D. N., & Eshleman, V. R. 1985, AJ, 90, 1136 ———. 1986, Photographic Observations of Jupiter’s and Saturn’s Satellites Lindal, G. F., Wood, G. E., Hotz, H. B., Sweetnam, D. N., Eshleman, V. R., & Made in Nikolaev in 1980 Y1982 ( Leningrad: Gl. Astron. Obs. Akad. Nauk) Tyler, G. L. 1983, Icarus, 53, 348 Martin-Mur, T., et al. 2006, Use of Very Long Baseline Array Interferometric Data for Spacecraft Navigation ( Paper ISTS 2006-d-50; Tokyo: Japan Soc. Aeronaut. Space Sci.)
