A ballistics analysis of the Deep Impact ejecta plume: Determining Comet Tempel 1's gravity, mass, and density

In July of 2005, the Deep Impact mission collided a 366 kg impactor with the nucleus of Comet 9P/Tempel 1, at a closing speed of 10.2 km s⁻¹. In this work, we develop a first-order, three-dimensional, forward model of the ejecta plume behavior resulting from this cratering event, and then adjust the model parameters to match the flyby-spacecraft observations of the actual ejecta plume, image by image. This modeling exercise indicates Deep Impact to have been a reasonably “well-behaved” oblique impact, in which the impactor-spacecraft apparently struck a small, westward-facing slope of roughly 1/3-1/2 the size of the final crater produced (determined from initial ejecta plume geometry), and possessing an effective strength of not more than . The resulting ejecta plume followed well-established scaling relationships for cratering in a medium-to-high porosity target, consistent with a transient crater of not more than 85-140 m diameter, formed in not more than 250-550 s, for the case of (gravity-dominated cratering); and not less than 22-26 m diameter, formed in not less than 1-3 s, for the case of (strength-dominated cratering). At , an upper limit to the total ejected mass of 1.8×¹⁰⁷ kg (1.5-2.2×¹⁰⁷ kg) is consistent with measurements made via long-range remote sensing, after taking into account that 90% of this mass would have stayed close to the surface and then landed within 45 min of the impact. However, at , a lower limit to the total ejected mass of 2.3×¹⁰⁵ kg (1.5-2.9×¹⁰⁵ kg) is also consistent with these measurements. The expansion rate of the ejecta plume imaged during the look-back phase of observations leads to an estimate of the comet's mean surface gravity of (0.17-0.90 mm s⁻²), which corresponds to a comet mass of mt=4.5×¹⁰¹³ kg (2.3-12.0×¹⁰¹³ kg) and a bulk density of (200-1000 kg m⁻³), where the large high-end error is due to uncertainties in the magnitude of coma gas pressure effects on the ejecta particles in flight.     

Oxygen ion escape at Mars in a hybrid model: High energy and low energy ions

We have studied the escape and energization of several O⁺ populations and an population at Mars by using a hybrid model. The quasi-neutral hybrid model, HYB-Mars model, included five oxygen ion populations making it possible to distinguish photoions from oxygen ions originating from charge exchange processes and from the ionosphere.We have identified two high-energy ion components and one low-energy ion component of oxygen. They have different spatial and energy distributions near Mars. The two high-energy oxygen ion components, consisting of a high-energy “beam” and a high-energy “halo”, have different origins. (1) The high-energy (>∼100 eV) “beam” of O⁺ and ions are originating from the ionosphere. These ions form a highly asymmetric spatial distribution of escaping oxygen ions with respect to the direction of the convective electric field in the solar wind. (2) The high-energy (>∼100 eV) “halo” component contains O⁺ ions which are formed from the oxygen neutral exosphere by extreme ultraviolet radiation (EUV) and by charge exchange processes. These energetic halo ions can be found all around Mars. (3) The low energy O⁺ and ions (<∼100 eV) form a relatively symmetric spatial distribution around the Mars-Sun line. They originate from the ionosphere and from charge exchange processes between protons and exospheric oxygen atoms.The existence of the low- and the high-energy oxygen components is in agreement with recent in situ plasma measurements made by the ASPERA-3 instrument on the Mars Express mission. The analysis of the escaping oxygen ions suggests that the global energization of escaping planetary ions in the martian tail is controlled by the convective electric field.     

Thermal disturbances caused by lander shadowing and the measurability of the martian planetary heat flow

A measurement of the martian planetary heat flow requires the determination of the subsurface temperature gradient, which is affected by surface insolation. I investigate the propagation of thermal disturbances caused by lander shadowing and derive measurement requirements for in situ heat flow experiments. I find that for short term measurements spanning 180 sol, a measurement depth of at least 2 m is needed to guarantee a stable thermal environment directly underneath the lander for Moon-like thermal conductivities of . For extremely large conductivities of , this depth needs to be increased to 4 m, but if the probe can be deployed outside the lander structure, the respective depths can be decreased by 1 m. For long term measurements spanning at least a full martian year heat flow perturbations are smaller than 5% below a depth of 3 m directly underneath the lander. Outside the lander structure, essentially unperturbed measurements may be conducted at depths of 0.5 and 1.5 m for thermal conductivities of 0.02 and , respectively.     

Metallicity of the massive protoplanets around HR 8799 If formed by gravitational instability

The final composition of giant planets formed as a result of gravitational instability in the disk gas depends on their ability to capture solid material (planetesimals) during their ‘pre-collapse’ stage, when they are extended and cold, and contracting quasi-statically. The duration of the pre-collapse stage is inversely proportional roughly to the square of the planetary mass, so massive protoplanets have shorter pre-collapse timescales and therefore limited opportunity for planetesimal capture. The available accretion time for protoplanets with masses of 3, 5, 7, and 10 Jupiter masses is found to be and 5.67×10³ years, respectively. The total mass that can be captured by the protoplanets depends on the planetary mass, planetesimal size, the radial distance of the protoplanet from the parent star, and the local solid surface density. We consider three radial distances, 24, 38, and 68 AU, similar to the radial distances of the planets in the system HR 8799, and estimate the mass of heavy elements that can be accreted. We find that for the planetary masses usually adopted for the HR 8799 system, the amount of heavy elements accreted by the planets is small, leaving them with nearly stellar compositions.     

Fragment properties at the catastrophic disruption threshold: The effect of the parent body’s internal structure

Numerical simulations of asteroid breakups, including both the fragmentation of the parent body and the gravitational interactions between the fragments, have allowed us to reproduce successfully the main properties of asteroid families formed in different regimes of impact energy, starting from a non-porous parent body. In this paper, using the same approach, we concentrate on a single regime of impact energy, the so-called catastrophic threshold usually designated by , which results in the escape of half of the target’s mass. Thanks to our recent implementation of a model of fragmentation of porous materials, we can characterize for both porous and non-porous targets with a wide range of diameters. We can then analyze the potential influence of porosity on the value of , and by computing the gravitational phase of the collision in the gravity regime, we can characterize the collisional outcome in terms of the fragment size and ejection speed distributions, which are the main outcome properties used by collisional models to study the evolutions of the different populations of small bodies. We also check the dependency of on the impact speed of the projectile.In the strength regime, which corresponds to target sizes below a few hundreds of meters, we find that porous targets are more difficult to disrupt than non-porous ones. In the gravity regime, the outcome is controlled purely by gravity and porosity in the case of porous targets. In the case of non-porous targets, the outcome also depends on strength. Indeed, decreasing the strength of non-porous targets make them easier to disrupt in this regime, while increasing the strength of porous targets has much less influence on the value of . Therefore, one cannot say that non-porous targets are systematically easier or more difficult to disrupt than porous ones, as the outcome highly depends on the assumed strength values. In the gravity regime, we also confirm that the process of gravitational reaccumulation is at the origin of the largest remnant’s mass in both cases. We then propose some power-law relationships between and both target’s size and impact speed that can be used in collisional evolution models. The resulting fragment size distributions can also be reasonably fitted by a power-law whose exponent ranges between −2.2 and −2.7 for all target diameters in both cases and independently on the impact velocity (at least in the small range investigated between 3 and 5 km/s). Then, although ejection velocities in the gravity regime tend to be higher from porous targets, they remain on the same order as the ones from non-porous targets.     

Low-speed impacts between rubble piles modeled as collections of polyhedra, 2

We present the results of additional calculations involving the collisions of km-scale rubble piles. In new work, we used the Open Dynamics Engine (ODE), an open-source library for the simulation of rigid-body dynamics that incorporates a sophisticated collision-detection and resolution routine. We found that using ODE resulted in a speed-up of approximately a factor of 30 compared with previous code. In this paper we report on the results of almost 1200 separate runs, the bulk of which were carried out with 1000-2000 elements. We carried out calculations with three different combinations of the coefficients of friction η and (normal) restitution ?: low (η=0,?=0.8), medium (η=0,?=0.5), and high (η=0.5,?=0.5) dissipation.For target objects of ∼1 km in radius, we found reduced critical disruption energy values in head-on collisions from 2 to 100 J kg⁻¹ depending on dissipation and impactor/target mass ratio. Monodisperse objects disrupted somewhat more easily than power-law objects in general. For oblique collisions of equal-mass objects, mildly off-center collisions (b/b₀=0.5) seemed to be as efficient or possibly more efficient at collisional disruption as head-on collisions. More oblique collisions were less efficient and the most oblique collisions we tried (b/b₀=0.866) required up to ∼200 J kg⁻¹ for high-dissipation power-law objects. For calculations with smaller numbers of elements (total impactor or 200 elements) we found that collisions were more efficient for smaller numbers of more massive elements, with values as low as for low-dissipation cases. We also analyzed our results in terms of the relations proposed by Stewart and Leinhardt [Stewart, S.T., Leinhardt, Z.M., 2009. Astrophys. J. 691, L133-L137] where where Q_R is the impact kinetic energy per unit total mass m_i+m_T. Although there is a significant amount of scatter, our results generally bear out the suggested relation.     

On the mean oxygen isotope composition of the Solar System

Since the first discovery of extraordinary oxygen isotope compositions in carbonaceous meteorites by Clayton et al. [Clayton, R.N., Grossman, L., Mayeda, T.K., 1973. Science 182, 485-488], numerous studies have been done to explain the unusual mass-independent isotope fractionation, but the problem is still unresolved to this day. Clayton's latest interpretation [Clayton, R.N., 2002. Nature 415, 860-861] sheds new light on the problem, and possible hypotheses now seem to be fairly well defined. A key issue is to resolve whether the oxygen isotopes in the Solar System represented by the Sun (solar oxygen) are the same as oxygen isotopes in planetary objects such as bulk meteorites, Mars, Earth, and Moon, or whether the solar oxygen is more similar to the lightest oxygen isotopes observed in CAIs (Calcium Aluminum-rich Inclusions) in primitive meteorites. Here, we examined the problem using oxygen isotope analytical data of about 400 bulk meteorite samples of various classes or types (data compiled by K. Lodders). We used in our discussion exclusively the parameter , a direct measure of the degree of mass-independent isotope fractionation of oxygen isotopes. When is arranged according to a characteristic size of their host planetary object, it shows a systematic trend: (1) values scatter around zero; (2) the scatter from the mean () decreases with increasing representative size of the respective host planetary object. This systematic trend is easily understood on the basis of a hierarchical scenario of planetary formation, that is, larger planetary objects have formed by progressive accretion of planetesimals by random sampling over a wide spectrum of proto-solar materials. If this progressive random sampling of planetesimals were the essential process of planetary formation, the isotopic composition of planetary oxygen should approach that of the solar oxygen. To test this random sampling hypothesis, we applied a multiscale, multistep bootstrap statistical method [Shimodaira, H., 2004. Ann. Statist. 32, 2616-2641] to the meteorite oxygen isotope data, and deduced a σ-N relation, where σ is the standard deviation of , and N is the representative size of a host planetary object. If we assign 200 and 500 km as a representative sizes of the chondrite and achondrite parent bodies, the observed σ of agree well with the values predicted by the σ-N relation. A common mean value of for all planetary objects also agrees with the progressive random sampling process. Therefore, we conclude that the solar oxygen is the same as planetary oxygen, but differs from CAI oxygen. The conclusion implies that a massive enrichment in ¹⁷O and ¹⁸O resulting from CO self-shielding, a current influential interpretation of CAI-O, did not occur.     

Mars high resolution gravity fields from MRO, Mars seasonal gravity, and other dynamical parameters

With 2 years of tracking data collection from the MRO spacecraft, there is noticeable improvement in the high frequency portion of the spherical harmonic Mars gravity field. The new JPL Mars gravity fields, MRO110B and MRO110B2, show resolution near degree 90. Additional years of MGS and Mars Odyssey tracking data result in improvement for the seasonal gravity changes which compares well to global circulation models and Odyssey neutron data and Mars rotation and precession (). Once atmospheric dust is accounted for in the spacecraft solar pressure model, solutions for Mars solar tide are consistent between data sets and show slightly larger values (k₂ = 0.164 ± 0.009, after correction for atmospheric tide) compared to previous results, further constraining core models. An additional 4 years of Mars range data improves the Mars ephemeris, determines 21 asteroid masses and bounds solar mass loss (dGM_Sun/dt < 1.6 × 10⁻¹³ GM_Sun year⁻¹).     

Colors and collision rates within the Kuiper belt: problems with the collisional resurfacing scenario

We present a numerical check of the collisional resurfacing (CR) hypothesis proposed to explain the observed color diversity within the Kuiper Belt (where surface reddening due to space weathering is counteracted by regular resurfacing of neutral material after mutual collisions). Deterministic simulations are performed in order to estimate the relative spatial distribution of kinetic energy received by collisions, , for a population of target Kuiper Belt objects (KBOs) embedded in a swarm of impactors distributed within the belt. Four different impactor disks have been considered, depending on the excitation and the external limit of the belt and the density of the scattered KBOs (SKBOs) population. The obtained results are compared to the relative color index distribution within the observed Kuiper Belt, in order to derive possible similarities between the high vs low objects spatial distribution in our simulations and the bluer vs redder KBOs distribution in the “real” Kuiper Belt. Such similarities are found for several important features, in particular the general correlations between highly impacted objects and high rms excitation and low perihelion q values that are in good agreement with equivalent correlations found for the bluest objects of the observed belt. Nevertheless, simulations disagree with observations on two crucial points. (1) The plutinos are significantly more collisionally affected than the rest of our test KBO population, whereas there is no observed tendency toward bluer plutinos. (2) There is always a much stronger correlation between and eccentricities than inclinations, whereas observations show just the opposite feature. The presence of numerous SKBO impactors could significantly damp these problematic features, but cannot erase them. Whether these contradictions invalidate the whole CR scenario or not remains yet uncertain, since the physical processes at play are still far from being fully understood and the sample of available observational data is still relatively limited. But it seems nevertheless that the scenario might not hold in its simple present form.     

Explosion of Comet 17P/Holmes as revealed by the Spitzer Space Telescope

An explosion on Comet 17P/Holmes occurred on 2007 October 23, projecting particulate debris of a wide range of sizes into the interplanetary medium. We observed the comet using the mid-Infrared Spectrograph (5-40 μm), on 2007 November 10 and 2008 February 27, and the imaging photometer (24 and 70 μm), on 2008 March 13, on board the Spitzer Space Telescope. The 2007 November 10 spectral mapping revealed spatially diffuse emission with detailed mineralogical features, primarily from small crystalline olivine grains. The 2008 February 27 spectra, and the central core of the 2007 November 10 spectral map, reveal nearly featureless spectra, due to much larger grains that were ejected from the nucleus more slowly. Optical images were obtained on multiple dates spanning 2007 October 27-2008 March 10 at the Holloway Comet Observatory and 1.5-m telescope at Palomar Observatory. The images and spectra can be segmented into three components: (1) a hemispherical shell fully 28′ on the sky in 2008 March, due to the fastest (262 m s⁻¹), smallest (2 μm) debris, with a mass ; (2) a ‘blob’ or ‘pseudonucleus’ offset from the true nucleus and subtending some 10′ on the sky, due to intermediate speed (93 m s⁻¹) and size (8 μm) particles, with a total mass ; and (3) a ‘core’ centered on the nucleus due to slower (9 m s⁻¹), larger (200 μm) ejecta, with a total mass . This decomposition of the mid-infrared observations can also explain the temporal evolution of the millimeter-wave flux. The orientation of the leading edge of the ejecta shell and the ejecta ‘blob,’ relative to the nucleus, do not change as the orientation of the Sun changes; instead, the configuration was imprinted by the orientation of the initial explosion. The distribution and speed of ejecta implies an explosion in a conical pattern directed approximately in the solar direction on the date of explosion. The kinetic energy of the ejecta >10²¹ erg is greater than the gravitational binding energy of the nucleus. We model the explosion as being due to crystallization and release of volatiles from interior amorphous ice within a subsurface cavity; once the pressure in the cavity exceeded the surface strength, the material above the cavity was propelled from the comet. The size of the cavity and the tensile strength of the upper layer of the nucleus are constrained by the observed properties of the ejecta; tensile strengths on >10 m scale must be greater than 10 kPa (or else the ejecta energy exceeds the binding energy of the nucleus) and they are plausibly 200 kPa. The appearance of the 2007 outburst is similar to that witnessed in 1892, but the 1892 explosion was less energetic by a factor of about 20.     

Estimating the masses of Saturn’s A and B rings from high-optical depth N-body simulations and stellar occultations

We have completed a series of local N-body simulations of Saturn’s B and A rings in order to identify systematic differences in the degree of particle clumping into self-gravity wakes as a function of orbital distance from Saturn and dynamical optical depth (a function of surface density). These simulations revealed that the normal optical depth of the final configuration can be substantially lower than one would infer from a uniform distribution of particles. Adding more particles to the simulation simply piles more particles onto the self-gravity wakes while leaving relatively clear gaps between the wakes. Estimating the mass from the observed optical depth is therefore a non-linear problem. These simulations may explain why the Cassini UVIS instrument has detected starlight at low incidence angles through regions of the B ring that have average normal optical depths substantially greater than unity at some observation geometries [Colwell, J.E., Esposito, L.W., Srem?evi?, M., Stewart, G.R., McClintock, W.E., 2007. Icarus 190, 127-144]. We provide a plausible internal density of the particles in the A and B rings based upon fitting the results of our simulations with Cassini UVIS stellar occultation data. We simulated Cassini-like occultations through our simulation cells, calculated optical depths, and attempted to extrapolate to the values that Cassini observes. We needed to extrapolate because even initial optical depths of >4 (σ > 240 g cm⁻²) only yielded final optical depths no greater than 2.8, smaller than the largest measured B ring optical depths. This extrapolation introduces a significant amount of uncertainty, and we chose to be conservative in our overall mass estimates. From our simulations, we infer the surface density of the A ring to be , which corresponds to a mass of . We infer a minimum surface density of for Saturn’s B ring, which corresponds to a minimum mass estimate of . The A ring mass estimate agrees well with previous analyses, while the B ring is at least 40% larger. In sum, our lower limit estimate is that the total mass of Saturn’s ring system is 120-200% the mass of the moon Mimas, but significantly larger values would be plausible given the limitations of our simulations. A significantly larger mass for Saturn’s rings favors a primordial origin for the rings because the disruption of a former satellite of the required mass would be unlikely after the decay of the late heavy bombardment of planetary surfaces.     

Dynamical erosion of the asteroid belt and implications for large impacts in the inner Solar System

The cumulative effects of weak resonant and secular perturbations by the major planets produce chaotic behavior of asteroids on long timescales. Dynamical chaos is the dominant loss mechanism for asteroids with diameters in the current asteroid belt. In a numerical analysis of the long-term evolution of test particles in the main asteroid belt region, we find that the dynamical loss history of test particles from this region is well described with a logarithmic decay law. In our simulations the loss rate function that is established at persists with little deviation to at least . Our study indicates that the asteroid belt region has experienced a significant amount of depletion due to this dynamical erosion—having lost as much as ∼50% of the large asteroids—since 1 Myr after the establishment of the current dynamical structure of the asteroid belt. Because the dynamical depletion of asteroids from the main belt is approximately logarithmic, an equal amount of depletion occurred in the time interval 10-200 Myr as in 0.2-4 Gyr, roughly ∼30% of the current number of large asteroids in the main belt over each interval. We find that asteroids escaping from the main belt due to dynamical chaos have an Earth-impact probability of ∼0.3%. Our model suggests that the rate of impacts from large asteroids has declined by a factor of 3 over the last 3 Gyr, and that the present-day impact flux of objects on the terrestrial planets is roughly an order of magnitude less than estimates currently in use in crater chronologies and impact hazard risk assessments.     

Low-speed impacts between rubble piles modeled as collections of polyhedra

We present results of modeling rubble piles as collections of polyhedra. The use of polyhedra allows more realistic (irregular) shapes and interactions (e.g. collisions), particularly for objects of different sizes. Rotational degrees of freedom are included in the modeling, which may be important components of the motion. We solved the equations of rigid-body dynamics, including frictional/inelastic collisions, for collections of up to several hundred elements. As a demonstration of the methods and to compare with previous work by other researchers, we simulated low-speed collisions between km-scale bodies with the same general parameters as those simulated by Leinhardt et al. [Leinhardt, Z.M., Richardson, D.C., Quinn, T., 2000. Icarus 146, 133-151]. High-speed collisions appropriate to present-day asteroid encounters require additional treatment of shock effects and fragmentation and are the subject of future work; here we study regimes appropriate to planetesimal accretion and re-accretion in the aftermath of catastrophic events. Collisions between equal-mass objects at low speeds () were simulated for both head-on and off-center collisions between rubble piles made of a power-law mass spectrum of sub-elements. Very low-speed head-on collisions produce single objects from the coalescence of the impactors. For slightly higher speeds, extensive disruption occurs, but re-accretion produces a single object with most of the total mass. For increasingly higher speeds, the re-accreted object has smaller mass, finally resulting in complete catastrophic disruption with all sub-elements on escape trajectories and only small amounts of mass in re-accreted bodies. Off-center collisions at moderately low speeds produce two re-accreted objects of approximately equal mass, separating at greater than escape speed. At high speed, complete disruption occurs as with the high-speed head-on collisions. Head-on collisions at low to moderate speeds result in objects of mostly oblate shape, while higher speed collisions produce mostly prolate objects, as do off-center collisions at moderate and high speeds. Collisions carried out with the same dissipative coefficients (coefficient of restitution ?n=0.8, zero friction) as used by Leinhardt et al. [Leinhardt, Z.M., Richardson, D.C., Quinn, T., 2000. Icarus 146, 133-151] result in a value for specific energy for disruption , somewhat lower than the value of 2 J/kg found by them, while collisions with a lower coefficient of restitution and friction [?n=0.5, ?t=0, μ=0.5, similar to those used by Michel, et al. [Michel, P., Benz, W., Richardson, D.C., 2004. Planet. Space Sci. 52, 1109-1117] for SPH + N-body calculations] yield .     

An investigation of Pluto’s troposphere using stellar occultation light curves and an atmospheric radiative-conductive-convective model

We use a radiative-conductive-convective model to assess the height of Pluto’s troposphere, as well as surface pressure and surface radius, from stellar occultation data from the years 1988, 2002, and 2006. The height of the troposphere, if it exists, is less than 1 km for all years analyzed. Pluto has at most a planetary boundary layer and not a troposphere. As in previous analyses of Pluto occultation light curves, we find that the surface pressure is increasing with time, assuming that latitude and longitude variations in Pluto’s atmosphere are negligible. The surface pressure is found to be slightly higher ( μbar in 1988,  μbar in 2002, and 18.5 ± 4.7 μbar in 2006) than in our previous analyses with the troposphere excluded. The surface radius is determined to be . Comparison of the minimum reduced chi-squared values between the best-fit radiative-conductive-convective (i.e., troposphere-included) model and best-fit radiative-conductive (i.e., troposphere-excluded) shows that the troposphere-included model is only a slightly better fit to the data for all 3 years. Uncertainties in the small-scale physical processes of Pluto’s lower atmosphere and consequently the functional form of the model troposphere lend more confidence to the troposphere-excluded results.     

On the population, physical decay and orbital distribution of Jupiter family comets: Numerical simulations

We study the Jupiter family comet (JFC) population assumed to come from the Scattered Disk and transferred to the Jupiter’s zone through gravitational interactions with the Jovian planets. We shall define as JFCs those with orbital periods and Tisserand parameters in the range 2<T?3.1, while those comets coming from the same source, but that do not fulfill the previous criteria (mainly because they have periods ) will be called ‘non-JFCs’. We performed a series of numerical simulations of fictitious comets with a purely dynamical model and also with a more complete dynamical-physical model that includes besides nongravitational forces, sublimation and splitting mechanisms. With the dynamical model, we obtain a poor match between the computed distributions of orbital elements and the observed ones. However with the inclusion of physical effects in the complete model we are able to obtain good fits to observations. The best fits are attained with four splitting models with a relative weak dependence on q, and a mass loss in every splitting event that is less when the frequency is high and vice versa. The mean lifetime of JFCs with radii and is found to be of about 150-200 revolutions (∼. The total population of JFCs with radii within Jupiter’s zone is found to be of 450±50. Yet, the population of non-JFCs with radii in Jupiter-crossing orbits may be ∼4 times greater, thus leading to a whole population of JFCs + non-JFCs of ∼2250±250. Most of these comets have perihelia close to Jupiter’s orbit. On the other hand, very few non-JFCs reach the Earth’s vicinity (perihelion distances ) which gives additional support to the idea that JFCs and Halley-type comets have different dynamical origins. Our model allows us to define the zones of the orbital element space in which we would expect to find a large number of JFCs. This is the first time, to our knowledge, that a physico-dynamical model is presented that includes sublimation and different splitting laws. Our work helps to understand the role played by these erosion effects in the distribution of the orbital elements and lifetimes of JFCs.     

The fossilized size distribution of the main asteroid belt

Planet formation models suggest the primordial main belt experienced a short but intense period of collisional evolution shortly after the formation of planetary embryos. This period is believed to have lasted until Jupiter reached its full size, when dynamical processes (e.g., sweeping resonances, excitation via planetary embryos) ejected most planetesimals from the main belt zone. The few planetesimals left behind continued to undergo comminution at a reduced rate until the present day. We investigated how this scenario affects the main belt size distribution over Solar System history using a collisional evolution model (CoEM) that accounts for these events. CoEM does not explicitly include results from dynamical models, but instead treats the unknown size of the primordial main belt and the nature/timing of its dynamical depletion using innovative but approximate methods. Model constraints were provided by the observed size frequency distribution of the asteroid belt, the observed population of asteroid families, the cratered surface of differentiated Asteroid (4) Vesta, and the relatively constant crater production rate of the Earth and Moon over the last 3 Gyr. Using CoEM, we solved for both the shape of the initial main belt size distribution after accretion and the asteroid disruption scaling law . In contrast to previous efforts, we find our derived function is very similar to results produced by numerical hydrocode simulations of asteroid impacts. Our best fit results suggest the asteroid belt experienced as much comminution over its early history as it has since it reached its low-mass state approximately 3.9-4.5 Ga. These results suggest the main belt's wavy-shaped size-frequency distribution is a “fossil” from this violent early epoch. We find that most diameter D?120 km asteroids are primordial, with their physical properties likely determined during the accretion epoch. Conversely, most smaller asteroids are byproducts of fragmentation events. The observed changes in the asteroid spin rate and lightcurve distributions near D∼100-120 km are likely to be a byproduct of this difference. Estimates based on our results imply the primordial main belt population (in the form of D<1000 km bodies) was 150-250 times larger than it is today, in agreement with recent dynamical simulations.     

A global solution for the Mars static and seasonal gravity, Mars orientation, Phobos and Deimos masses, and Mars ephemeris

With the collection of six years of MGS tracking data and three years of Mars Odyssey tracking data, there has been a continual improvement in the JPL Mars gravity field determination. This includes the measurement of the seasonal changes in the gravity coefficients (e.g., , , , , , ) caused by the mass exchange between the polar ice caps and atmosphere. This paper describes the latest gravity field MGS95J to degree and order 95. The improvement comes from additional tracking data and the adoption of a more complete Mars orientation model with nutation, instead of the IAU 2000 model. Free wobble of the Mars' spin axis, i.e. polar motion, has been constrained to be less than 10 mas by looking at the temporal history of and . A strong annual signature is observed in , and this is a mixture of polar motion and ice mass redistribution. The Love number solution with a subset of Odyssey tracking data is consistent with the previous liquid outer core determination from MGS tracking data [Yoder et al., 2003. Science 300, 299-303], giving a combined solution of k₂=0.152±0.009 using MGS and Odyssey tracking data. The solutions for the masses of the Mars' moons show consistency between MGS, Odyssey, and Viking data sets; Phobos GM=(7.16±0.005)×¹⁰⁻⁴ km³/s² and Deimos GM=(0.98±0.07)×¹⁰⁻⁴ km³/s². Average MGS orbit errors, determined from differences in the overlaps of orbit solutions, have been reduced to 10-cm in the radial direction and 1.5 m along the spacecraft velocity and normal to the orbit plane. Hence, the ranging to the MGS and Odyssey spacecraft has resulted in position measurements of the Mars system center-of-mass relative to the Earth to an accuracy of one meter, greatly reducing the Mars ephemeris errors by several orders of magnitude, and providing mass estimates for Asteroids 1 Ceres, 2 Pallas, 3 Juno, 4 Vesta, and 324 Bamberga.