Precise mass determination and the nature of Phobos
Abstract
[1] We report independent results from two subgroups of the Mars Express Radio Science (MaRS) team who independently analyzed Mars Express (MEX) radio tracking data for the purpose of determining consistently the gravitational attraction of the moon Phobos on the MEX spacecraft, and hence the mass of Phobos. New values for the gravitational parameter (GM = 0.7127 ± 0.0021 × 10−3 km3/s2) and density of Phobos (1876 ± 20 kg/m3) provide meaningful new constraints on the corresponding range of the body's porosity (30% ± 5%), provide a basis for improved interpretation of the internal structure. We conclude that the interior of Phobos likely contains large voids. When applied to various hypotheses bearing on the origin of Phobos, these results are inconsistent with the proposition that Phobos is a captured asteroid.
1. Introduction
[2] The trajectory of an orbiting planetary spacecraft is sensitive to the gravitational attraction of a co‐orbiting moon. The strength of the moon’s effect can be measured by: i) long‐term secular tracking of the spacecraft orbit at large distances from the moon, and ii) short‐term observation of the changes in the spacecraft velocity during close flybys of the moon. The results can be used to obtain the mass of the perturbing moon.
[3] The eccentric orbit of Mars Express (MEX) provides the first opportunity in twenty years for both long‐ and short‐term observations bearing on the mass MPh of Phobos. Early values for the gravitational parameter GMPh (gravitational constant G times MPh) obtained from radio tracking are based on a few close flybys of the Viking spacecraft in 1977 [Christensen et al., 1977; Tolson et al., 1977, 1978; Williams et al., 1988], and from the Phobos 2 spacecraft in 1989 [Kolyuka et al., 1990], with relative errors of about 10% and 2%, respectively (Figure 1). The values of these early solutions for GMPh vary over a range of ±0.1 × 10−3 km3s−2. Much of this uncertainty results from the use of gravity field models of Mars and ephemerides of Phobos that were relatively less accurate than those currently available.
[4] Mass estimates are also available from secular tracking of Mariner 9, Viking Orbiter, Mars Global Surveyor, and Mars Odyssey, and Mars Express [Smith et al., 1995; Yuan et al., 2001; Rosenblatt et al., 2008]. These secular mass solutions often have small formal errors as a consequence of the large number of data arcs included in the observations, while the effects of systematic errors are problematic.
2. Observations
[5] Apoapsis of the elliptical MEX orbit established in December 2003, intercepts the orbit of Phobos and occasionally results in close encounters. At sufficiently close flybys, the attracting forces of Phobos sensibly perturb the velocity of the MEX spacecraft, which is observed as a Doppler shift of the received radio tracking frequency. The magnitude of the frequency shift depends on the flyby distance, the direction and magnitude of the change in flyby velocity and the flyby geometry [Anderson, 1971; Pätzold et al., 2001]. The MEX Radio Science (MaRS) team [Pätzold et al., 2004] estimated that flyby distances well within 500 km to Phobos were needed to achieve meaningfully improved mass determinations compared to past missions. MaRS obtained good data from a close flyby in 2006 at 460 km and again in 2008 at 275 km distance.
[6] Here we report results of investigators in Munich who worked with these observations of the two close flybys of Phobos, while a separate group in Brussels worked with observations sensitive to both the close flyby and longer term secular effects. The necessary data reduction and analysis software were developed independently by each group, and the application to the MEX observations were also carried out independently. Results independently obtained by the two groups are mutually consistent.
[7] Analysis methods for the close‐flyby using only short‐term observations were developed and applied at the Institute of Space Technology at the University of the German Armed Forces in Munich, Germany. The approach resulted in a least‐mean‐squares solution for the mass of Phobos based on the use of two‐way round trip radio tracking observations from the time interval surrounding closest approach of MEX to Phobos. For this a predicted received frequency was computed based on the spacecraft orbit before the flyby, taking into account the important gravitational and non‐gravitational perturbing forces acting on the spacecraft but excluding the mass of Phobos. This predicted baseline frequency was corrected for contributions from the radio signal propagation through Earth's atmosphere and then subtracted from the observed frequency. The resulting residuals represent the change in the spacecraft velocity along the spacecraft‐to‐Earth line‐of‐sight during the period of closest approach, which is attributed to the attraction of Phobos. The value and the time history of the change in relative spacecraft velocity depends essentially on the mass of Phobos, the relative flyby trajectory and distance, and the projection of the MEX trajectory on the line of sight to Earth [Anderson, 1971; Pätzold et al., 2001].
[8] Solutions for Phobos GMPh from Brussels were obtained using the software package “Géodésie par Intégration Numérique Simultanée (GINS)” developed at the Centre National d’Etudes Spatiales (CNES) and adapted for use in planetary geodesy applications at the Royal Observatory of Belgium. The secular solution is based on precise determination of the MEX orbit over successive 7 day data arcs [Rosenblatt et al., 2008].
[9] Figures 2 and 3 show the 8.4 GHz radio carrier frequency tracking residuals used by the Munich group in obtaining a direct mass determination from the change in the line‐of‐sight spacecraft velocity.
[10] The closest approach distance on 23 March was 460 km, with a relative MEX/Phobos flyby velocity of 2.8 km/s and a subtended angle between the line‐of‐sight (LOS) and the velocity of MEX relative to Phobos that varied from 96° to 105°. For the flyby on 17 July 2008, the corresponding velocity and angles were 3.0 km/s, 38°, and 88°, respectively. Although the distance of the first flyby from Phobos was sub‐optimum, the flyby velocity and the geometry were very favorable for the observation [Pätzold et al., 2001]. As above, it is assumed that any differences between the calculated ‘no‐Phobos’ and actual orbits are due solely to the gravitational attraction of Phobos acting on the spacecraft.
[11] A least‐mean‐squares solution for the mass based on the tracking frequency residuals yields a value of GMPh = (0.7120 ± 0.0120) × 10−3 km3/sec2 (±1.7%) from the first flyby (Figure 2) and GMPh = (0.7127 ± 0.0021) × 10−3 km3/sec2 (±0.3%) from the second flyby (Figure 3); the corresponding value for the mass derived from GMPh of the second flyby is MPh = (1.0668 ± 0.003) × 1016 kg (±0.3%).
[12] The constant offsets of −12 and −21 mHz in the first and second flybys, respectively, were also estimated as part of the solution of the mass. These offsets are the result of small Doppler contributions not included in the predicted received frequency. Using the GINS software package, the group in Brussels obtained a GMPh value from the second close flyby of (0.711 ± 0.003) × 10−3 km3/sec2 (±0.4%) which is very consistent with the secular solution [Rosenblatt et al., 2008] and with the two close flyby solutions from the Munich group.
[13] The above solutions from the close flybys provide the most accurate value currently available for the mass of Phobos. The accuracy of 0.3% is attributed to the use of 8.4 GHz carrier frequency, the most recent gravity field model of Mars [Konopliv et al., 2006], the most recent ephemerides of the planets [Folkner et al., 2008] and of Phobos [Jacobson, 2008], and especially the opportunity to observe close approaches of the MEX spacecraft to Phobos.
[14] While the three independently derived GMPh values from the two MEX close flybys and the GMPh value from the MEX secular changes [Rosenblatt et al., 2008] agree very well, the recently derived value by Jacobson [2009] from the Viking flybys is more precise (0.7104 ± 0.0006 × 10−3 km3/s2) but its error bar overlaps only slightly with that of MEX. The source of this difference is unknown.
3. Discussion
[15] The bulk density of Phobos sheds some light on its internal composition, structure, and probable origin. Willner et al. [2009] derived a value of 5689 ± 60 km3 from MEX High Resolution Stereo Camera (HRSC) images. With this, the bulk density is rPh = 1876 ± 20 kg/m3 (±1.1%) which is an improvement by a factor of three compared to past values. The error in the bulk density is driven by the error in the volume.
[16] The porosity, the ratio of the bulk density and the grain density of an object, represents the percentage of the volume occupied by voids. The porosity of Phobos is computed from its mass, its bulk density and known grain densities of the hydrous chondrites of the CM group and the Tagish‐Lake meteorite samples. The result of 30 ± 5 percent suggests that the interior of Phobos contains large voids. Similar large porosities and low bulk densities have been found in C‐type asteroid such as the asteroid Mathilde [Yeomans et al., 1997] and the Jupiter's small inner moon Amalthea [Anderson et al., 2005]. A similar formation process of these porous bodies, however, is totally unclear.
[17] The interior structure of Phobos could well be the result of its complete shattering and subsequent reassembly, as is thought to have occurred in the history of many asteroids subjected to violent collisions [Richardson et al., 2002]. The existence of the Stickney crater on Phobos would support the conclusion that Phobos contains large voids throughout its interior.
[18] The origin of Phobos can be discussed in terms of its orbital history. Several scenarios ranging from possible to speculative have been proposed. The surface of Phobos shows some spectral similarities to those of various asteroid types. Based on these similarities it was suggested that Phobos is a former asteroid, formed in the outer asteroid belt and later captured by Mars [Burns, 1992]. This scenario, however, does not explain how the energy loss required to change the incoming hyperbolic orbit into an elliptical orbit bound to Mars is accounted for [Burns, 1992; Peale, 2007]. Models of orbit evolution based on tidal interactions between Mars and Phobos cannot account for the current near‐circular and near‐equatorial orbit [Mignard, 1981]. Scenarios of evolution to the current circular orbit require an additional drag by, e.g., the primitive planetary nebulae or the Martian atmosphere [Sasaki, 1990]. An alternative is a Phobos formed in an orbit around Mars. Phobos and Deimos could be remnants of an early, larger body that was broken into parts by gravitational gradient forces during Mars capture [Singer, 2007]. Or Phobos could have formed by the re‐accretion of impact debris lifted into Mars’ orbit [Craddock, 1994]. If Phobos were a remnant of a larger moon, it is not expected to be as porous as is reported here. If Phobos were formed from the re‐accretion of impact debris lifted into Mars’ orbit, the disc would be composed of a mixture of Martian crust and impactor material. The spectral properties, however, of the Phobos surface and the Martian crust do not match very well.
[19] This inconsistency is resolved by a collision between a body already orbiting Mars but formed from the debris disc remaining after formation of Mars and a second body originating from the asteroid belt [Peale, 2007]. This scenario is consistent with the high porosity of Phobos and its spectral properties.
4. Conclusions
[20] Figure 1 compares the Mars Express flyby solutions with previous determinations of GMPh. The MEX results reported here have significantly reduced systematic uncertainties as compared with previous secular solutions. This is largely due to improvements in our knowledge of Mars gravity field, the ephemeris of Phobos, the ephemerides of the other planets important to the mass solution, and to improvements in the spacecraft radio system used for close flybys.
[21] The Phobos’ mass reported here tightens the constraints on our knowledge of the physical structure of Phobos. The derived bulk density is consistent with a loosely consolidated body of 30% porosity, independent of its origin. It appears though highly unlikely that Phobos is a captured asteroid.
[22] The three mass determinations from this work are derived from two close flybys using two different methods and two different software packages. The results are mutually consistent with each other as well as with the most recent secular solution [Rosenblatt et al., 2008]. We note that the latter is based on a different data set from the ‘close flyby’ solution employed here but makes use of the same models for the gravity field of Mars and similar ephemerides of Phobos.
[23] A more precise volume estimate would not change our conclusions as the absolute volume estimate and therefore the bulk density is well constrained. The Phobos porosity value is safely within the range for loosely consolidated bodies.
[24] The gravity coefficient J2 derived from flybys at Phobos closer than 100 km which are planned for MEX in 2010 will give further insight into the interior. The comparison of theoretical J2 assuming constant and homogenous bulk density with an observed J2 also will give evidence as to mass enhancements or voids.
Acknowledgments
[25] The Mars Express Radio Science Experiment MaRS is funded by DLR Bonn under grants 50QP9909 and 50QM0701. P. Rosenblatt is supported by the Belgian PRODEX program managed by the European Space Agency in collaboration with the Belgian Federal Science Policy Office. Support for Mars Express Radio Science at Stanford University is provided by NASA through JPL contract 1217744. J.C. Marty was supported by the CNES Programme Directorate and the Toulouse Space Center. We thank all persons involved in the Mars Express project at ESTEC, ESOC, ESAC, JPL and the ESTRACK and DSN ground station antennas for their continuous support.
