HUBBLE SPACE TELESCOPE ASTROMETRIC OBSERVATIONS AND ORBITAL MEAN MOTION CORRECTIONS FOR THE INNER SATELLITES OF NEPTUNE

Transcription

1 The Astronomical Journal, 127: , 2004 May # The American Astronomical Society. All rights reserved. Printed in U.S.A. HUBBLE SPACE TELESCOPE ASTROMETRIC OBSERVATIONS AND ORBITAL MEAN MOTION CORRECTIONS FOR THE INNER SATELLITES OF NEPTUNE Dan Pascu and James R. Rohde 1 US Naval Observatory, 3450 Massachusetts Avenue, NW, Washington, DC ; pascu.dan@usno.navy.mil, jrohde@ngs.noaa.gov P. Kenneth Seidelmann Department of Astronomy, University of Virginia, P.O. Box 3818, Charlottesville, VA 22903; pks6n@virginia.edu Eddie N. Wells and John L. Hershey 2 Computer Sciences Corporation, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218; wells@stsci.edu Alex D. Storrs Department of Physics, Astronomy, and Geosciences, Towson University, 8000 York Road, Baltimore, MD 21252; astorrs@towson.edu Ben H. Zellner Department of Physics, Georgia Southern University, Landrum Box 8031, Statesboro, GA 30460; zellner@gasou.edu Amanda S. Bosh Institute for Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA 02215; and Lowell Observatory, 1400 West Mars Hill Road, Flagstaff, AZ 86001; amanda@lowell.edu and Douglas G. Currie Department of Physics, University of Maryland, College Park, MD 20742; currie@hubble.physics.umd.edu Received 2003 December 22; accepted 2002 February 4 ABSTRACT Six small inner satellites of Neptune were imaged in 1989 with Voyager 2. In 1997, we recovered the four outermost with the Hubble Space Telescope (HST ) Wide Field Planetary Camera 2 for astrometric, dynamical, and photometric studies. The ring arcs were not detected in our images. Thirteen exposures were taken in each of three HST orbits: two orbits on July 3 and one on July 6. Exposures were taken in the BVI filters. Measurable images of Neptune and Triton were also obtained on the same PC1 frames with those of the faint satellites. We present here the astrometric observations of these four satellites relative to Neptune, as well as corrected orbital mean motions for them. Field distortions in the PC1 chip were corrected with both the Trauger et al. and the Anderson & King distortion models. Calibration of the scale and orientation was accomplished by comparing the measured positions of Neptune and Triton with an accurate JPL J2000 ephemeris. Separate calibrations were made for both distortion models. Small differences were detected in the calibrations, dependent on wavelength, saturation, and filter, and a small difference was found between the calibrations resulting from both distortion correction models. The resulting separation and position angle observations for the inner satellites were compared with the orbits of Owen et al. and corrections derived to their mean daily motions. A small but significant discrepancy was found for Proteus between the correction derived from the observations of separation and that from the position angles. This was shown not to be due to calibrational errors but, apparently, to the need for improvement of other orbital elements at least for Proteus. Despite this anomaly, the mean motion accuracies were improved by almost 2 orders of magnitude as a result of the longer baseline since the Voyager observations. More HST observations of these satellites are recommended in order to improve their orbits further and for the investigation of satellite-ring interactions. Key words: astrometry planets and satellites: individual (Neptune) 1. INTRODUCTION Five new inner satellites of Neptune were discovered by Voyager 2 in 1989 (Smith et al. 1989), while a sixth inner satellite, Larissa, was an earlier ground-based discovery by Reitsema and others (Reitsema et al. 1982). This inner system is important because its origin and evolution are unlike those of either Triton or Nereid, and so, it represents another 1 Current address: National Geodetic Survey, National Ocean Service, NOAA, N/ NGS6, Station 8138, 1315 East-West Highway, Silver Spring, MD Retired dimension, and perhaps a clue, to the history of the most enigmatic of the satellite systems. Of particular interest at present is the gravitational interaction of Galatea with the five ring arcs. Several years ago, Goldreich, Tremaine, & Borderies (1986) proposed a ring containment model dependent on resonance with an inner satellite. Porco (1991) proposed a resonance model of confinement for Neptune s five ring arcs that depended on the vertical as well as the mean motion of the satellite Galatea. This model was recently abandoned when ground-based observations (Sicardy et al. 1999) and Hubble Space Telescope (HST ) NICMOS observations (Dumas et al. 1999, 2002) indicated a discrepancy between the observed location of the ring arcs and that predicted by the resonance

2 MOTION OF NEPTUNE S SATELLITES 2989 Filter TABLE 1 Exposure Sequence Exposure Duration (s) F439W... 80, 100, 160 F555W... 10, 20, 40, 40, 40 F791W... 16, 35, 160, 160, 160 model. A new resonance model was proposed (Namouni & Porco 2002; Porco & Namouni 2002), which is dependent on Galatea s orbital eccentricity. Such a resonance, if confirmed, would permit the determination of the mass of the ring arcs from the measurement of Galatea s orbital eccentricity. The confirmation of this new model depends strongly on precise astrometric observations. Precise astrometric observations will also result in accurate new ephemerides useful for physical studies of the satellites for years to come. Below, we describe our astrometric observations of the inner satellites with the HST Wide Field Planetary Camera 2 (WFPC2), the corrections of the satellite mean motions, and an assessment of the observational accuracy. Our photometric results will be published elsewhere. 2. OBSERVATIONS Thirteen images of the inner system were obtained with the Planetary Camera (PC1) in each of two HST orbits on 1997 July 3, and an additional 13 in one orbit on July 6 (Pascu et al. 1999). This sequence provided good orbital distribution for the faint satellites, as well as imaging of Triton near both elongations, which is important for calibration. The exposures were taken in three filters to yield some color information, and to accommodate the large dynamic range, a range of exposures were taken in each filter. The distribution of exposures in each orbital set is given in Table 1. The upper exposure limits were set by the 2 pixel motions of the inner satellites or by the blooming threshold of the planet. The middle exposure lengths were determined as the saturation limit of Triton, while the shortest exposures were expected to be fail-safe calibrational frames. Four of the six inner satellites were recovered by electronically blinking together the two longest-exposure F791W frames in each set and identifying the moving objects. The four satellites, Proteus, Larissa, Galatea, and Despina, were found close to their predicted positions. Naiad and Thalassa were much too faint and close to the planet for detection at these wavelengths. Proteus was detected on all 39 frames, Larissa on 13, Galatea on 17, and Despina on six. Images of Neptune and Triton were also on each frame. Their images were saturated, but not blooming, on all but the shortest exposures. There were no stellar configurations on the frames suitable for calibration, nor were the ring arcs detected. 3. ASTROMETRY Our astrometric reduction of the HST observations of the inner Uranian satellites (Pascu et al. 1997, 1998, hereafter Papers I and II, respectively) used JPL ephemerides of Ariel and Miranda to calibrate the scale and orientation of individual PC1 frames. This strategy will not work for the inner Neptunian satellites, because there is only one brighter moon with a high-precision orbit relative to Neptune. This leaves us with the option of using the standard scale (Paper I) and the HST log orientation or measuring the position of Neptune s image, as well as that of Triton, and comparing them with the JPL ephemeris. Both methods have their drawbacks; in the former case, individual frame calibration is systematically more accurate than the application of standard calibrational values, while in the latter case, this involves centroiding of large and saturated images. The images of Neptune, Triton, and the four inner satellites were measured by centroiding with a two-dimensional Gaussian. No images on the frames could be used for pointspread function modeling, but Gaussian fits proved to be robust, especially on the faint inner satellites embedded in the planetary halo. The planetary halo was also modeled using a sloped planar or quadratic function. The saturated images of Triton were centroided, as well as the unsaturated images; however, only unsaturated pixels were included in the leastsquares Gaussian fits. Neptune s image was also fitted with a Gaussian using only the unsaturated pixels. Formal errors better than 0.01 pixels resulted. However, when centroids were performed on the same planetary image with differing pixel data-number limits, the fitting solution returned centers that differed by more than the formal error. To obtain a more reliable center for the planet, we measured it at about 10 different data-number limits and averaged the three to six most significant measurements. The average value had an accuracy of about 0.02 pixels, or less than 1 mas. Since Triton s image could also be measured with an accuracy of better than 1 mas, the scale could reasonably be expected to be derived with an accuracy of a part in 15,000 from the JPL orbit of Triton. Likewise, the orientation can be determined with an accuracy better than It is not clear, however, whether this Neptune centroid measures the center of figure of the planet or something closer to the center of brightness. If the former, the correction for phase defect is less than 0.1 mas, the centroid corresponds to the planet s center, and the scale and orientation calibrations are not affected. If the latter, the centroid will be shifted as much as 1.2 mas toward the eastern limb. While the effect on the scale and orientation calibrations will be minimal, the shift of the coordinate origin by 1.2 mas will affect the inner satellite observations to some extent. For Proteus, for example, the measured separations will have a maximum systematic error of 1.2 mas and a maximum systematic position angle error of Before the calibration can proceed further, the measurements must be corrected for geometric distortion. The measured (x, y) were corrected for geometric distortion in PC1 using both the model of Trauger et al. (1995, hereafter TVEM95) and that of Anderson & King (2003, hereafter AK03). While the AK03 model is an improvement over the TVEM95 model, it applies only to the F555W filter, whereas the TVEM95 model accommodates all filters. A comparison of the two models at 555 nm is shown in Figure 1. The largest difference between the two models is 0.25 pixels (11 mas) in the corners, but the primary feature of the comparison is a 0.03 rotation of one model relative to the other. The distortion-corrected coordinates of Triton relative to Neptune were compared with a JPL J2000 ephemeris of Triton (R. A. Jacobson 1998, private communication) and a scale (arcseconds per pixel) and orientation correction derived for each frame. Figure 2 shows the scale determinations for each of the 39 frames for both the AK03 (Fig. 2a) and the TVEM95 (Fig. 2b) distortion corrections. The mean scale is accurate to 1 part in 23,000; however, the ordinate axis was considerably expanded to show small systematic effects. A comparison of Figures 2a and 2b shows that while the AK03 corrections result in a more

3 2990 PASCU ET AL. Fig. 1. Vector diagram comparing the distortion model of Anderson & King (2003) with that of Trauger et al. (1995) at 555 nm. The sense is Anderson & King minus Trauger et al. (times 100). The striking feature here is the 0.03 rotation of the Anderson & King distortion model with respect to the Trauger et al. model. The largest disagreement between the two models is about0.25pixelsinthecorners. central distribution of values, the TVEM95 corrections give a smaller total range. There is also an increase in scale value with wavelength for the unsaturated images in both distortion models, and a decrease in scale value with wavelength for the saturated images. The pattern within each filter is identical in both models, and the most conspicuous feature of this pattern is the increased scale values for the unsaturated images (the first two exposures in each filter) in the F791W filter and in the F555W filter, but only in the last visit, when Triton was in the opposite corner of the field. This suggests that the saturated F791W images of Triton are considerably distorted, because the centroids of the unsaturated images measure the image cores, while those of the saturated images measure more of the wings. The third visit shows smaller color effects for unsaturated images in both distortion models, probably because Triton s image is more than 100 pixels farther from the edge of the chip than in the first two visits. In fact, for the TVEM95 model, the unsaturated images show no scale or color effect in the third visit. Figure 3 shows the calibration of the rotation offset relative to the equator of J2000 for both the AK03 (Fig. 3a) and TVEM95 (Fig. 3b) distortion corrections. While the orientation correction had an internal precision of 0 o :002, the ordinate axis was expanded, as in the previous figure, to show small systematic effects. Most noticeable is that there are no significant saturation effects in the orientation calibration. This indicates that the saturation effect is radial and supports our suggestion that the saturation scale anomaly was due to the elongation of the image of Triton by the severe field distortion near the edge of the CCD. Secondly, there is a substantial filter effect for both models, noticeable mainly in the second and third visits. This effect is reversed between the second and third visits as Triton moves from one corner of the CCD chip to the opposite corner. We surmise that this is related to the large color difference between Neptune and Triton [(B V ) = 0.3 mag]. The F791W data, however, do not show this effect and have a precision of a millidegree. Finally, there is a 0.03 systematic difference between the two distortion models for all three filters. Table 2 summarizes the scale and rotation/orientation calibrations. Given are the mean (average) of the 39 individual scale and WFPC2 rotation angle determinations for both the AK03 and TVEM95 distortion corrections. The numbers in parentheses are the mean error in the last figure of the average value. The standard deviations are larger by the square root of 39. Astrometric positions for the four inner satellites were derived relative to the position of Neptune s center, rather than Triton s position. Since the scale and orientation calibrations varied (slightly) with filter, exposure, etc., each frame was reduced with its own derived scale and orientation calibrations. Table 3 gives the J2000 positions of the four detected inner satellites relative to Neptune, reduced with the AK03 distortion only. The difference in mean scale value between the AK03 and TVEM95 reductions will make at most 1 mas difference in the separation of Proteus from Neptune. However, the 0.03 rotational difference between the AK03 and TVEM95 distortion models will have the same effect on all satellites regardless of distance from Neptune. Since AK03 did not fit their model to the extreme edges of the chip, we concluded that the position angles relative to the AK03 distortion model were systematically more accurate. We list the HST file number, the date of the observation, the UTC of mid-exposure, X (= cos ), Y (=), and the separation and position angle of the satellite s image relative to Neptune s image all for the equator and equinox of J2000. For the separation observations of Larissa, a colon follows the observations not used in the final orbital correction because they were outliers, or for reasons explained below. 4. CORRECTION OF MEAN MOTIONS While several orbital parameters may be in need of improvement, only the orbital mean motions could be reliably corrected with this small observational set, taken over such a short interval. A more definitive improvement of the orbits should include additional observations, especially the Voyager discovery observations, and those of Dumas et al. (2002), which are unpublished. Such an undertaking is planned by R. A. Jacobson (2003, private communication). Because the calibrations were made in scale and orientation, the mean motion corrections were made in separation and in position angle so that the effects of the two calibrations on the orbital corrections would be independent. While both Neptune and Triton were saturated on most of the frames, Neptune was taken as coordinate origin both for astrometric reasons and for considerations of the orbital analysis. First, it was surmised that the radial distortion of Triton s image was the primary source of astrometric error, not the saturation. Second, the difference in scale between the AK03 and TVEM95 reductions would have a significant effect on the satellite separations if Triton were taken as origin, but negligible effect if Neptune were taken as origin. Furthermore, the position angle solutions would have low weight for position angles relative to Triton, but high weight relative to Neptune. Positions of the faint inner satellites relative to Neptune were computed using the published orbits of Owen, Vaughan, & Synnott (1991), and observed minus computed, O C, residuals were obtained for each observation. Conditional

4 Fig. 2a Fig. 2b Fig. 2. Scale calibration, following (a) Anderson & King (2003) distortion corrections and (b) Trauger et al. (1995) wavelength-dependent distortion corrections. The abscissa scale lists the 39 exposures sequentially (see Table 1 for exposure times). The solid line is the average value. Although the scale is precise to 1 part in 23,000, the ordinate axis has been expanded to more clearly show systematic effects. Common features include an increase in scale with wavelength for unsaturated images (the first two exposures in each filter) and a substantial difference in scale between the saturated and unsaturated images for the F791W filter. The Anderson-King distortion correction increases the scale value over that of Trauger et al. by 1 part in See the text for more details.

5 Fig. 3a Fig. 3b Fig. 3. Orientation calibration, following (a) Anderson & King distortion corrections and (b) Trauger et al. The abscissa scale is identical to that in Fig. 2, while the ordinate axis lists the angle that the PC1 columns are oriented with respect to north. The addition of this offset to a provisional position angle results in the true position angle. Important common features include the absence of saturated vs. unsaturated effects, a systematic difference of 0.03 deg for all filters between the two distortion models, very small scatter in the F791W results, and last, that the F439W and F555W values reverse their displacement from the F791W values as Triton moves from western elongation to eastern (Triton in diagonally opposite corner of PC1).

6 MOTION OF NEPTUNE S SATELLITES 2993 TABLE 2 Scale and Orientation Calibrations Distortion Model Mean Scale Value (m.e.) (arcsec pixel 1 ) Position Angle a Rotation (m.e.) (deg) Anderson & King (2) (2) Trauger et al (2) (2) Paper I results (2)... HST obs. log a Offset plus 180. equations for the least-squares correction to the mean daily motion, n, for each of the four inner satellites were of n ¼ (O C) s in separation, S, and similarly for position angle. Solutions were made separately in separation and position angle, since incompatible solution results might indicate calibrational problems. No more than three iterations were necessary. Final results are listed in Table 4. Listed are the satellite name and V magnitude at mean opposition (Thomas & Veverka 1991), the mean daily motion corrections with mean errors, the number of observations used in the solution, the post-solution rms, and the corrected mean daily motions (with errors) derived from the combined separation and position angle solutions. While the solutions tabulated in Table 4 are for the AK03-reduced observations of Table 3, solutions using the TVEM95-reduced observations gave nearly identical results. The first six separation observations for Larissa were not used, because they yielded a noisy solution. Unfortunately, Larissa was observed at superior conjunction in the first set and inferior conjunction in the second set. Consequently, for those observations the partials went through zero while the observational errors were large. A good solution was obtained after those observations of separation were dropped. 5. DISCUSSION As a result of the longer baseline, the accuracies of the mean motions are improved by more than 2 orders of magnitude over those of Owen et al. (1991). However, an unexpected result is the statistical incompatibility of the separation and position angle solutions for Proteus. This is especially significant, since Proteus is most accurately measured; it appears in all 39 frames and has the smallest residuals after solution. Two common causes for such a discrepancy are systematic calibrational errors and other orbital elements in need of correction. The JPL J2000 ephemeris of Triton was confirmed by comparison with an independent ephemeris. After several solutions for orbital mean motion corrections, using different scale values, it was concluded that the solution was insensitive to any reasonable scale variations. This is mostly the result of using Neptune s image as coordinate origin rather than Triton s image. To verify the position angle calibration, the average rotation of the PC1 columns with respect to north were computed for both the AK03 and TVEM95 distortion corrections. These are listed in Table 2, along with the HST observation log value for comparison. The small differences between these values were insufficient to reduce the discrepancy in the mean motion solutions for Proteus. Working from the conditional equations, it was determined that the discrepancy could be resolved if the observed position angles of Proteus were increased by about R. A. Jacobson & W. M. Owen (2003, private communication) maintain that connecting the Voyager and HST observations will take a more sophisticated model than that used to fit the Voyager data. They suggest that the inclinations of the Laplacian planes of these inner satellites are not well defined and probably the cause of the anomalous results for Proteus. The issue will be resolved only when all observations (Voyager, HST, groundbased) are used in a global fit to a complex orbital model to derive definitive orbits for these satellites. Such an analysis is planned by them. Whether the new analysis is sensitive enough to confirm or reject the confinement theory of Porco & Namouni (2002) and Namouni & Porco (2002) remains to be seen. If not, an effort should be made to obtain new intensive astrometric observations of these satellites with HST especially for Galatea. For now, however, it should be noted that the separations solution for Proteus, though of slightly lower weight than the position angles solution, is not affected by the position angle anomaly. The combined mean motions for the three fainter satellites, and those derived from the separations for Proteus, will give ephemerides accurate to 1 in longitude for at least 45 years for Despina, and for over 600 years for Proteus. The residuals after solution for these satellites are almost half those resulting from our analysis of HST observations of the inner Uranian satellites (Paper I), although the measurement environment and methods were very similar. The principal difference is that Miranda was taken as the origin for the Uranian analysis, while Neptune served as origin in this study. That is, satellite-satellite coordinates were used in the orbital correction solutions for the inner Uranians, while planetsatellite coordinates were used in this investigation. Apparently, the calibrational imprecisions, distortion correction deficiencies, and Miranda s orbital inaccuracies contributed more to the total error in that study than did the measurement of Neptune s overexposed image in this analysis. 6. SUMMARY In summary, we have detected the four brightest of the inner Neptunian satellites with the HST PC1 during three orbits of HST. Gaussian-plus-background models were used to centroid the faint satellites. Both Triton s and Neptune s images were centroided using only unsaturated pixels. Distortion corrections were applied with both the Anderson & King (2003) and the Trauger et al. (1995) models, and the scale and rotation of PC1 were calibrated by comparing the separation and position angle measurements of Triton relative to Neptune with an accurate JPL ephemeris. Small differences were detected in the calibrations, dependent on wavelength, saturation, and filter, and a small difference was found between the AK03 and TVEM95 calibrations. The reductions yielded astrometric separations and position angles relative to Neptune for four

7 TABLE 3 Satellite Positions, J2000 HST Frame (U40N0xxxM) Date UTC Mid- Exposure X Y Sep. P.A. (deg) Despina-Neptune positions: 10B Jul 3 9:43: C Jul 3 9:47: D Jul 3 9:51: B Jul 6 11:56: C Jul 6 12:00: D Jul 6 12:04: Galatea-Neptune positions: Jul 3 9:14: Jul 3 9:18: Jul 3 9:29: Jul 3 9:31: Jul 3 9:33: A Jul 3 9:40: B Jul 3 9:43: C Jul 3 9:47: D Jul 3 9:51: Jul 3 15:58: A Jul 3 16:08: B Jul 3 16:11: C Jul 3 16:15: D Jul 3 16:19: B Jul 6 11:56: C Jul 6 12:00: D Jul 6 12:04: Larissa-Neptune positions: 10B Jul 3 9:43: : C Jul 3 9:47: : D Jul 3 9:51: : A Jul 3 16:08: : B Jul 3 16:11: : C Jul 3 16:15: : D Jul 3 16:19: : Jul 6 11:44: : Jul 6 11:46: A Jul 6 11:53: B Jul 6 11:56: C Jul 6 12:00: D Jul 6 12:04: Proteus-Neptune positions: Jul 3 9:14: Jul 3 9:18: Jul 3 9:21: Jul 3 9:25: Jul 3 9:27: Jul 3 9:29: Jul 3 9:31: Jul 3 9:33: Jul 3 9:38: A Jul 3 9:40: B Jul 3 9:43: C Jul 3 9:47: D Jul 3 9:51: Jul 3 15:41: Jul 3 15:45: Jul 3 15:48: Jul 3 15:52: Jul 3 15:54: Jul 3 15:56: Jul 3 15:58: Jul 3 16:00:

8 TABLE 3 Continued HST Frame (U40N0xxxM) Date UTC Mid- Exposure X Y Sep. P.A. (deg) Jul 3 16:06: A Jul 3 16:08: B Jul 3 16:11: C Jul 3 16:15: D Jul 3 16:19: Jul 6 11:27: Jul 6 11:31: Jul 6 11:34: Jul 6 11:38: Jul 6 11:40: Jul 6 11:42: Jul 6 11:44: Jul 6 11:46: Jul 6 11:51: A Jul 6 11:53: B Jul 6 11:56: C Jul 6 12:00: D Jul 6 12:04: TABLE 4 Mean Motion Corrections Satellite a Correction to Mean Daily Motion (deg day 1 ) N rms (mas) Corrected Mean Daily Motion (deg day 1 ) Despina (V = 22.5): Separation ( ) 6 13 Position angle ( ) 6 14 Combined ( ) ( ) Galatea (V = 22.4): Separation ( ) Position angle ( ) Combined ( ) ( ) Larissa (V = 22.0): Separation ( ) 5 9 Position angle ( ) Combined ( ) ( ) Proteus (V = 20.3): Separation ( ) ( ) Position angle ( ) 39 6 Combined ( ) ( ) a Listed V magnitudes are values at mean opposition. 2995

9 2996 PASCU ET AL. inner satellites, which were used to determine corrections to their mean motions. A small, but significant, discrepancy between the separation and the position angle solutions was detected for Proteus. This disparity was found not to be due to calibration or measurement but probably to an inability to account for the position and motion of the satellite fixed planes. These problems notwithstanding, the mean motion accuracies were improved by almost 2 orders of magnitude because of the longer baseline since the Voyager observations. More observations of these satellites are needed in order to improve their orbits. We thank Robert Jacobson of the Jet Propulsion Laboratory for high-precision ephemerides of Triton and for exploratory computations, and both Jacobson and William Owen for fruitful discussions. We are also grateful to Jay Anderson for his distortion-correcting software. Support for this work was provided by NASA through proposal GO submitted to the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS Anderson, J., & King, I. R. 2003, PASP, 115, 113 (AK03) Dumas, C., Terrile, R. J., Smith, B. A., & Schneider, G. 2002, AJ, 123, 1776 Dumas, C., Terrile, R. J., Smith, B. A., Schneider, G., & Becklin, E. E. 1999, Nature, 400, 733 Goldreich, P., Tremaine, S., & Borderies, N. 1986, AJ, 92, 490 Namouni, F., & Porco, C. 2002, Nature, 417, 45 Owen, W. M., Jr., Vaughan, R. M., & Synnott, S. P. 1991, AJ, 101, 1511 Pascu, D., et al. 1999, BAAS, 31, , in IAU Colloq. 165, Dynamics and Astrometry of Natural and Artificial Celestial Bodies, ed. I. Wytrzyszczak, J. H. Lieske, & F. Mignard (Dordrecht: Kluwer), 517 (Paper I). 1998, AJ, 115, 1190 (Paper II) REFERENCES Porco, C., & Namouni, F. 2002, BAAS, 34, 940 Porco, C. C. 1991, Science, 253, 995 Reitsema, H. J., Hubbard, W. B., Lebofsky, L. A., & Tholen, D. J. 1982, Science, 215, 289 Sicardy, B., Roddier, F., Roddier, C., Perozzi, E., Graves, J. E., Guyon, O., & Northcott, M. J. 1999, Nature, 400, 731 Smith, B. A., et al. 1989, Science, 246, 1422 Thomas, P., & Veverka, J. 1991, J. Geophys. Res., 96, Trauger, J. T., Vaughan, A. H., Evans, R. W., & Moody, D. C. 1995, in Calibrating Hubble Space Telescope: Post Servicing Mission, ed. A. Koratkar & C. Leitherer (Baltimore: STScI), 379 (TVEM95)