The orbit and size distribution of small Solar System objects orbiting the Sun interior to the Earth's orbit

We present the first observational measurement of the orbit and size distribution of small Solar System objects whose orbits are wholly interior to the Earth's (Inner Earth Objects, IEOs, with aphelion <0.983 AU). We show that we are able to model the detections of near-Earth objects (NEO) by the Catalina Sky Survey (CSS) using a detailed parameterization of the CSS survey cadence and detection efficiencies as implemented within the Jedicke et al. [Jedicke, R., Morbidelli, A., Spahr, T., Petit, J.M., Bottke, W.F., 2003. Icarus 161, 17-33] survey simulator and utilizing the Bottke et al. [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H.F., Michel, P., Metcalfe, T.S., 2002. Icarus 156, 399-433] model of the NEO population's size and orbit distribution. We then show that the CSS detections of 4 IEOs are consistent with the Bottke et al. [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H.F., Michel, P., Metcalfe, T.S., 2002. Icarus 156, 399-433] IEO model. Observational selection effects for the IEOs discovered by the CSS were then determined using the survey simulator in order to calculate the corrected number and H distribution of the IEOs. The actual number of IEOs with H<18 (21) is 36±26 (530±240) and the slope of the H magnitude distribution (∝¹⁰αH) for the IEOs is . The slope is consistent with previous measurements for the NEO population of αNEO=0.35±0.02 [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H.F., Michel, P., Metcalfe, T.S., 2002. Icarus 156, 399-433] and αNEO=0.39±0.013 [Stuart, J.S., Binzel, R.P., 2004. Icarus 170, 295-311]. Based on the agreement between the predicted and observed IEO orbit and absolute magnitude distributions there is no indication of any non-gravitational effects (e.g. Yarkovsky, tidal disruption) affecting the known IEO population.     

Realistic survey simulations for kilometer class near Earth objects

We present a new Near Earth Object (NEO) survey simulator which incorporates the four-dimensional population model of 4668 NEOs [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H.F., Michel, P., Metcalfe, T.S., 2002. Icarus 156, 399-433] and the observing strategies of most asteroid search programs. With the recent expansion of survey capabilities, previous simulators focused on a specific survey facility are no longer useful in predicting the future detection rates. Our simulation is a superposition of simplified search patterns adopted by all major wide-field surveys in operation in both hemispheres. We defined five different simulation periods to follow the evolution of survey efficiencies reflecting changes in either search volume as a result of upgrades of telescopes and instruments or in observing schedules. The simulator makes remarkably good reproductions of actual survey results as of December 2005, not only the total number of detections but also (a,e,i,H) (‘H’ means absolute magnitude of an asteroid) distributions. An extended experiment provides excellent predictions for discovery statistics of NEOs (H<18) reported to the Minor Planet Center in 2006. These support that our simulator is a plausible approximation of real surveys. We further confirm that, with the Bottke et al. [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J.-M., Levison, H.F., Michel, P., Metcalfe, T.S., 2002. Icarus 156, 399-433] population model and present survey capability, the 90% completeness level of kilometer-sized NEOs will be achieved by 2010 or 2011. However, about 8% of the kilometer-sized or larger NEOs would remain undetected even after 10-year operation (2007-2016) of all current NEO survey facilities. They are apparently faint, with orbits characterized by large semimajor axis and higher eccentricity; these “hardest-to-find” objects tend to elude the search volume of existing NEO survey facilities. Our simulation suggests that 15% of undetectable objects are Atens and Inner Earth Objects. Because of their orbital characteristics, they will remain within ±45° from the Sun, thus cannot be discovered in the forthcoming decade if our effort is limited to current ground-based telescopes.     

Observed spectral properties of near-Earth objects: results for population distribution, source regions, and space weathering processes

We present new visible and near-infrared spectroscopic measurements for 252 near-Earth (NEO) and Mars-crossing (MC) objects observed from 1994 through 2002 as a complement to the Small Main-Belt Asteroid Spectroscopic Survey (SMASS, http://smass.mit.edu/). Combined with previously published SMASS results, we have an internally consistent data set of more than 400 of these objects for investigating trends related to size, orbits, and dynamical history. These data also provide the basis for producing a bias-corrected estimate for the total NEO population (Stuart and Binzel, 2004, Icarus 170, 295-311). We find 25 of the 26 Bus (1999, PhD thesis) taxonomic types are represented, with nearly 90% of the objects falling within the broad S-, Q-, X-, and C-complexes. Rare A- and E-types are more common in the MC than NEO population (about 5% compared to <1%) and may be direct evidence of slow diffusion into MC orbits from the Flora and Hungaria regions, respectively. A possible family of MC objects (C-types) may reside at the edge of the 5:2 jovian resonance. Distinct signatures are revealed for the relative contributions of different taxonomic types to the NEO population through different source regions. E-types show an origin signature from the inner belt, C-types from the mid to outer belt, and P-types from the outer belt. S- and Q-types have effectively identical main-belt source region profiles, as would be expected if they have related origins. A lack of V-types among Mars-crossers suggests entry into NEO space via rapid transport through the ν6 and 3:1 resonances from low eccentricity main-belt orbits, consistent with a Vesta origin. D-types show the strongest signature from Jupiter family comets (JFC), with a strong JFC component also seen among the X-types. A distinct taxonomic difference is found with respect to the jovian Tisserand parameter T, where C-, D-, and X-type (most likely low albedo P-class) objects predominate for T?3. These objects, which may be extinct comets, comprise 4% of our observed sample, but their low albedos makes this magnitude limited fraction under-representative of the true value. With our taxonomy statistics providing a strong component to the diameter limited bias correction analysis of Stuart (2003, PhD thesis), we estimate 10-18% of the NEO population above any given diameter may be extinct comets, taking into account asteroids scattered into T<3 orbits and comets scattered into T>3 orbits. In terms of possible space weathering effects, we see a size-dependent transition from ordinary chondrite-like (Q-type) objects to S-type asteroids over the size range of 0.1 to 5 km, where the transition is effectively complete at 5 km. A match between the average surface age of 5 km asteroids and the rate of space weathering could constrain models for both processes. However, space weathering may proceed at a very rapid rate compared with collisional timescales. In this case, the presence or absence of a regolith may be the determining factor for whether or not an object appears “space weathered.” Thus 0.1 to 5 km appears to be a critical size range for understanding the processes, timescales, and conditions under which a regolith conducive to space weathering is generated, retained, and refreshed.     

Earth and space-based NEO survey simulations: prospects for achieving the spaceguard goal

We have used an improved model of the orbit and absolute magnitude distribution of Near Earth Objects (NEOs) to simulate the performance of asteroid surveys. Our results support general conclusions of previous studies using preliminary Near Earth Asteroid (NEA) orbit and magnitude distributions and suggest that meeting the Spaceguard Goal of 90% completion for Near Earth Objects (NEOs) greater than 1 km diameter by 2008 is impossible given contemporary surveying capabilities.The NEO model was derived from NEO detections by the Spacewatch Project. For this paper we developed a simulator for the Catalina Sky Survey (CSS) for which we had a complete pointing history and NEO detection efficiency. The good match between the output of the simulator and the actual CSS performance gives confidence that both the NEO model and simulator are correct. Then, in order to determine if existing surveys can meet the Spaceguard Goal, we developed a simulator to mimic the LINEAR survey, for which detailed performance characteristics were unavailable. This simulator serendipitously provided an estimate for the currently undiscovered population of NEOs upon which we base all our estimates of time to 90% completion. We also developed a set of idealized NEO surveys in order to constrain the best possible survey performance in contrast to more realistic systems.A 100% efficient, all-sky, every night survey, subject only to the constraints of detection above a specified air mass and when the Sun is 18° below the horizon provides a benchmark from which to examine the effect of imposing more restrictions and the efficacy of some simple survey strategies. Such a survey must have a limiting V-magnitude of 20.1 ± 0.2 to meet the Spaceguard Goal.More realistic surveys, limited by latitude, the galaxy, minimum rates of NEO motion, etc., require fainter limiting magnitudes to reach the same completion. Our most realistic simulations, which have been normalized to the performance of the LINEAR detector system’s operation in the period 1999-2000, indicate that it would take them another 33 ± 5 years to reach 90% completeness for the larger asteroids (?1 km diameter). They would need to immediately increase the limiting magnitude to about 24 in order to meet the Spaceguard Goal.The simulations suggest that there may be little need for distributing survey telescopes in longitude and latitude as long as there is sufficient sky coverage from a telescope or network of telescopes which may be geographically close. An idealized space-based survey, especially from a satellite orbit much interior to Earth, would offer an advantage over their terrestrial counterparts. We do not consider a cost-benefit analysis for any of the simulations but suspect that a local-area network of telescopes capable of covering much of the sky in a month to V ∼ 21.5 may be administratively, financially, and scientifically the best compromise for reaching 90% completion of NEOs larger than 1 km diameter.     

Treatment of star catalog biases in asteroid astrometric observations

In this paper, we discuss the detection of systematic biases in star positions of the USNO A1.0, A2.0, and B1.0 catalogs, as deduced from the residuals of numbered asteroid observations. We present a technique for the removal of these biases, and validate this technique by illustrating the resulting improvements in numbered asteroid residuals, and by establishing that debiased orbits predict omitted observations more accurately than do orbits derived from non-debiased observations. We also illustrate the benefits of debiasing to high-precision astrometric applications such as asteroid mass determination and collision analysis, including a refined prediction of the impact probability of 99942 Apophis. Specifically, we find the IP of Apophis to be lowered by nearly an order of magnitude to 4.5 × 10⁻⁶ for the 2036 close approach.     

The size-frequency distribution of dormant Jupiter family comets

We estimate the total number and the slope of the size-frequency distribution (SFD) of dormant Jupiter family comets (JFCs) by fitting a one-parameter model to the known population. We first select 61 near-Earth objects (NEOs) that are likely to be dormant JFCs because their orbits are dynamically coupled to Jupiter [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J., Levison, H.F., Michel, P., Metcalfe, T.S., 2002a. Icarus 156, 399-433]. Then, from the numerical simulations of Levison and Duncan [1997. Icarus 127, 13-32], we construct an orbit distribution model for JFCs in the NEO orbital element space. We assume an orbit-independent SFD for all JFCs, the slope of which is our unique free parameter. Finally, we compute observational biases for dormant JFCs using a calibrated NEO survey simulator [Jedicke, R., Morbidelli, A., Spahr, T., Petit, J., Bottke, W.F., 2003. Icarus 161, 17-33]. By fitting the biased model to the data, we estimate that there are ∼75 dormant JFCs with H<18 in the NEO region and that the slope of their cumulative SFD is −1.5±0.3. Our slope for the SFD of dormant JFCs is very close to that of active JFCs as determined by Weissman and Lowry [2003. Lunar Planet. Sci. 34. Abstract 2003]. Thus, we argue that when JFCs fade they are likely to become dormant rather than to disrupt and that the fate of faded comets is size-independent. Our results imply that the size distribution of the JFC progenitors—the scattered disk trans-neptunian population—either (i) has a similar and shallow SFD or (i^′) is slightly steeper and physical processes acting on the comets in a size-dependent manner creates the shallower active comet SFD. Our measured slope, typical of collisionally evolved populations with a size-dependent impact strength [Benz, W., Asphaug, E., 1999. Icarus 142, 5-20], suggests that scattered disk bodies reached collisional equilibrium inside the protoplanetary disk prior to their removal from the planetary region.     

Albedo and size determination of potentially hazardous asteroids: (99942) Apophis

The near-Earth object (99942) Apophis will make an extremely close approach to the Earth in 2029, and currently has approximately a one-in-45,000 chance of impacting our planet in 2036 (JPL Sentry, November 2006). Computation of the orbital evolution of this object is limited by insufficient knowledge of physical properties required to determine the role played by non-gravitational effects. Using polarimetric observations, we have obtained the first reliable determination of the albedo of Apophis, obtaining 0.33±0.08. We also derive an updated estimate of the asteroid's absolute magnitude: H=19.7±0.4. Using this albedo and H, we find that Apophis has a diameter of 270±60 m, slightly smaller than preliminary estimates based upon an assumed albedo. Our observations demonstrate the feasibility of polarimetric observations aimed at obtaining albedos and sizes of small, potentially hazardous asteroids.     

Identifying near-Earth object families

The study of asteroid families has provided tremendous insight into the forces that sculpted the main belt and continue to drive the collisional and dynamical evolution of asteroids. The identification of asteroid families within the NEO population could provide a similar boon to studies of their formation and interiors. In this study we examine the purported identification of NEO families by Drummond [Drummond, J.D., 2000. Icarus 146, 453-475] and conclude that it is unlikely that they are anything more than random fluctuations in the distribution of NEO osculating orbital elements. We arrive at this conclusion after examining the expected formation rate of NEO families, the identification of NEO groups in synthetic populations that contain no genetically related NEOs, the orbital evolution of the largest association identified by Drummond [Drummond, J.D., 2000. Icarus 146, 453-475], and the decoherence of synthetic NEO families intended to reproduce the observed members of the same association. These studies allowed us to identify a new criterion that can be used to select real NEO families for further study in future analyses, based on the ratio of the number of pairs and the size of strings to the number of objects in an identified association.     

Orbital YORP and asteroid orbit evolution, with application to Apophis

Photon thrust from shape alone can produce quasi-secular changes in an asteroid's orbital elements. An asteroid in an elliptical orbit with a north–south shape asymmetry can steadily alter its elements over timescales longer than one orbital trip about the Sun. This thrust, called here orbital YORP (YORP = Yarkovsky–O'Keefe–Radzievskii–Paddack), operates even in the absence of thermal inertia, which the Yarkovsky effects require. However, unlike the Yarkovsky effects, which produce secular orbital changes over millions or billions of years, the change in an asteroid's orbital elements from orbital YORP operates only over the precession timescale of the orbit or of the asteroid's spin axis; this is generally only thousands or tens of thousands of years. Thus while the orbital YORP timescale is too short for an asteroid to secularly journey very far, it is long enough to warrant investigation with respect to 99942 Apophis, which might conceivably impact the Earth in 2036. A near-maximal orbital YORP effect is found by assuming Apophis is without thermal inertia and is shaped like a hemisphere, with its spin axis lying in the orbital plane. With these assumptions orbital YORP can change its along-track position by up to ±245 km, which is comparable to Yarkovsky effects. Though Apophis' shape, thermal properties, and spin axis orientation are currently unknown, the practical upper and lower limits are liable to be much less than the ±245 km extremes. Even so, the uncertainty in position is still likely to be much larger than the 0.5 km “keyhole” Apophis must pass through during its close approach in 2029 in order to strike the Earth in 2036.     

Linking the collisional history of the main asteroid belt to its dynamical excitation and depletion

The main belt is believed to have originally contained an Earth mass or more of material, enough to allow the asteroids to accrete on relatively short timescales. The present-day main belt, however, only contains ∼5×¹⁰⁻⁴ Earth masses. Numerical simulations suggest that this mass loss can be explained by the dynamical depletion of main belt material via gravitational perturbations from planetary embryos and a newly-formed Jupiter. To explore this scenario, we combined dynamical results from Petit et al. [Petit, J. Morbidelli, A., Chambers, J., 2001. The primordial excitation and clearing of the asteroid belt. Icarus 153, 338-347] with a collisional evolution code capable of tracking how the main belt undergoes comminution and dynamical depletion over 4.6 Gyr [Bottke, W.F., Durda, D., Nesvorny, D., Jedicke, R., Morbidelli, A., Vokrouhlický, D., Levison, H., 2005. The fossilized size distribution of the main asteroid belt. Icarus 175, 111-140]. Our results were constrained by the main belt's size-frequency distribution, the number of asteroid families produced by disruption events from diameter D>100 km parent bodies over the last 3-4 Gyr, the presence of a single large impact crater on Vesta's intact basaltic crust, and the relatively constant lunar and terrestrial impactor flux over the last 3 Gyr. We used our model to set limits on the initial size of the main belt as well as Jupiter's formation time. We find the most likely formation time for Jupiter was 3.3±2.6 Myr after the onset of fragmentation in the main belt. These results are consistent with the estimated mean disk lifetime of 3 Myr predicted by Haisch et al. [Haisch, K.E., Lada, E.A., Lada, C.J., 2001. Disk frequencies and lifetimes in young clusters. Astrophys. J. 553, L153-L156]. The post-accretion main belt population, in the form of diameter D?1000 km planetesimals, was likely to have been 160±40 times the current main belt's mass. This corresponds to 0.06-0.1 Earth masses, only a small fraction of the total mass thought to have existed in the main belt zone during planet formation. The remaining mass was most likely taken up by planetary embryos formed in the same region. Our results suggest that numerous D>200 km planetesimals disrupted early in Solar System history, but only a small fraction of their fragments survived the dynamical depletion event described above. We believe this may explain the limited presence of iron-rich M-type, olivine-rich A-type, and non-Vesta V-type asteroids in the main belt today. The collisional lifetimes determined for main belt asteroids agree with the cosmic ray exposure ages of stony meteorites and are consistent with the limited collisional evolution detected among large Koronis family members. Using the same model, we investigated the near-Earth object (NEO) population. We show the shape of the NEO size distribution is a reflection of the main belt population, with main belt asteroids driven to resonances by Yarkovsky thermal forces. We used our model of the NEO population over the last 3 Gyr, which is consistent with the current population determined by telescopic and satellite data, to explore whether the majority of small craters (D<0.1-1 km) formed on Mercury, the Moon, and Mars were produced by primary impacts or by secondary impacts generated by ejecta from large craters. Our results suggest that most small craters formed on these worlds were a by-product of secondary rather than primary impacts.     

Orbital and mission planning constraints for the deflection of NEOs impacting on Earth

This paper is the third in a series. Paper 1 presented the results of numerical modeling of deflections of NEOs in route of collision with the Earth. The model was applied to a variety of dynamical cases including both asteroidal and cometary NEOs. Paper 2 introduced the concept of “distributed deflection,” i.e., the possibility to provide the ΔV necessary to deflect an object with a succession of maneuvers each of which would have been insufficient per se to obtain the desired result. In both papers no assumptions were made on the physical composition and structure of the NEO, nor on the details of the possible deflection maneuvers from the point of view of mission analysis. Moreover, ΔV-plots were computed assuming only along-track impulses (both in the positive and negative directions), because it is easy to demonstrate that in general this is energetically the most favorable configuration. Also in the present paper no assumptions were made on the physical composition and structure of the NEO, even if order of magnitude considerations are made on the physical feasibility of a deflection, in terms of the internal strength of the NEO. We present here the results of an investigation on the mission requirements necessary to deflect an object (or contribute to a succession of deflecting maneuvers) in terms of accessibility of the spacecraft terminal orbit from Earth with the current launchers.     

Solar radiation pressure on (99942) Apophis

Near-Earth asteroid (99942) Apophis currently resides among the top positions on the list of objects with small, yet non-zero impact probability with the Earth. For that reason an unusual observational and theoretical effort has been dedicated to precisely characterize its future orbit. Here we discuss orbital perturbation of Apophis due to incident and reflected solar radiation pressure (SRP). We both revisit recent analytical estimate of the SRP effects for this body and also formulate a numerical approach allowing us to compute the SRP orbital perturbation under general assumptions. Contrary to some previous results, we show that SRP has a much smaller effect on the Apophis trajectory than does the thermal re-radiation force which produces the Yarkovsky effect. When the Yarkovsky effect becomes constrained enough in the future, our approach may be used to improve the orbit determination for this asteroid.     

Comets in the near-Earth object population

Because the lifespan of near-Earth objects (NEOs) is shorter than the age of the Solar System, these objects originate elsewhere. Their most likely sources are the main asteroid belt and comets. Through physical observations we seek to identify potential dormant or extinct comets among “asteroids” catalogued as NEOs and thereby determine the fraction of “comet candidates” within the total NEO population. Both discovery statistics and dynamical models indicate that candidate cometary objects in near-Earth space are predominantly found among those having a jovian Tisserand parameter Tj<3. Therefore, we seek to identify comet candidates among asteroid-like NEOs using three criteria: Tj<3, spectral parameters (C, D, T, or P taxonomic types), and/or low (<0.075) albedos. We present new observations for 20 NEOs having Tj<3, consisting of visible spectra, near-infrared spectra, and/or albedo measurements obtained using the NASA Infrared Telescope Facility, the Kitt Peak National Observatory 4 m, and the Magellan Observatory 6.5-m. Four of our “asteroid” targets have been subsequently confirmed as low activity comets. Thus our sample includes spectra of the nuclei of Comets 2002 EX12 = 169P (NEAT), 2001 WF2 = 182P (LONEOS), 2003 WY25 = D/1891 W1 (Blanplain), and Halley Family Comet 2006 HR30 = P/2006 HR30 (Siding Spring). From the available literature, we tabulate physical properties for 55 NEOs having Tj<3, and after accounting for possible bias effects, we estimate that 54±10% of NEOs in Tj<3 orbits have “comet-like” spectra or albedos. Bias corrected discovery statistics [Stuart, J.S., Binzel, R.P., 2004. Icarus 170, 295-311] estimate 30±5% of the entire NEO population resides in orbits having Tj<3. Combining these two factors suggests that 16±5% of the total discovered “asteroid-like” NEO population has “comet-like” dynamical and physical properties. Outer main-belt asteroids typically have similar taxonomic and albedo properties as our “comet candidates.” Using the model of Bottke et al. [Bottke, W.F., Morbidelli, A., Jedicke, R., Petit, J.M., Levison, H., Michel, P., Metcalfe, T.S., 2002. Icarus 156, 399-433] to evaluate source region probabilities, we conclude that 8±5% of the total asteroid-like NEO population have the requisite orbital properties, physical properties, and dynamical likelihood to have originated as comets from the outer Solar System.     

Evidence-based robust design of deflection actions for near Earth objects

This paper presents a novel approach to the robust design of deflection actions for near Earth objects (NEO). In particular, the case of deflection by means of solar-pumped laser ablation is studied here in detail. The basic idea behind laser ablation is that of inducing a sublimation of the NEO surface, which produces a low thrust thereby slowly deviating the asteroid from its initial Earth threatening trajectory. This work investigates the integrated design of the space-based laser system and the deflection action generated by laser ablation under uncertainty. The integrated design is formulated as a multi-objective optimisation problem in which the deviation is maximised and the total system mass is minimised. Both the model for the estimation of the thrust produced by surface laser ablation and the spacecraft system model are assumed to be affected by epistemic uncertainties (partial or complete lack of knowledge). Evidence Theory is used to quantify these uncertainties and introduce them in the optimisation process. The propagation of the trajectory of the NEO under the laser-ablation action is performed with a novel approach based on an approximated analytical solution of Gauss’ variational equations. An example of design of the deflection of asteroid Apophis with a swarm of spacecraft is presented.     

Feasibility study for near-earth-object tracking by a piggybacked micro-satellite with penetrators

As of August 2007, over 5000 near-earth-objects (NEO) have been discovered. Some already represent a potential danger to the Earth while others might become hazards in the future. The Planetary Society organised in 2007 the “Apophis Mission Design Competition” in response to this potential threat with the objective to identify promising concepts to track NEOs; the asteroid 99942 Apophis was taken as the study case. This paper describes the “Houyi” proposal which was evaluated by the competition jury as an innovative approach to this problem. Instead of launching a large satellite for NEO tracking, this novel concept proposes a miniaturized satellite that is piggybacked onto a larger (scientific) mission. Such mission design would drastically reduce the costs for NEO surveillance. The presented scenario uses the ESA’s SOLO mission as a design baseline for the piggyback option. This paper summarizes the architecture of this CubeSat towards Apophis and extends the previous study by focusing on the feasibility of a piggybacked mission in terms of propulsion requirements.     

Spectral properties and composition of potentially hazardous Asteroid (99942) Apophis

The known close approach of Asteroid (99942) Apophis in April 2029 provides the opportunity for the case study of a potentially hazardous asteroid in advance of its encounter. The visible to near-infrared (0.55 to 2.45 μm) reflectance spectrum of Apophis is compared and modeled with respect to the spectral and mineralogical characteristics of likely meteorite analogs. Apophis is found to be an Sq-class asteroid that most closely resembles LL ordinary chondrite meteorites in terms of spectral characteristics and interpreted olivine and pyroxene abundances, although we cannot rule out some degree of partial melting. A meteorite analog allows some estimates and conjectures of Apophis' possible range of physical properties such as the grain density and micro-porosity of its constituent material. Composition and size similarities of Apophis with (25143) Itokawa suggest a total porosity of 40% as a “current best guess” for Apophis. Applying these parameters to Apophis yields a mass estimate of 2×¹⁰¹⁰ kg with a corresponding energy estimate of 375 Mt for its potential hazard. Substantial unknowns, most notably the total porosity, allow uncertainties in these mass and energy estimates to be as large as factors of two or three.     

Spin axis evolution of two interacting bodies

We consider the solid-solid interactions in the two body problem. The relative equilibria have been previously studied analytically and general motions were numerically analyzed using some expansion of the gravitational potential up to the second order, but only when there are no direct interactions between the orientation of the bodies. Here we expand the potential up to the fourth order and we show that the secular problem obtained after averaging over fast angles, as for the precession model of Boué and Laskar [Boué, G., Laskar, J., 2006. Icarus 185, 312-330], is integrable, but not trivially. We describe the general features of the motions and we provide explicit analytical approximations for the solutions. We demonstrate that the general solution of the secular system can be decomposed as a uniform precession around the total angular momentum and a periodic symmetric orbit in the precessing frame. More generally, we show that for a general n-body system of rigid bodies in gravitational interaction, the regular quasiperiodic solutions can be decomposed into a uniform precession around the total angular momentum, and a quasiperiodic motion with one frequency less in the precessing frame.     

Computing the effects of YORP on the spin rate distribution of the NEO population

The overall change of NEO spin rate due to planetary encounters and YORP is evaluated by using a Monte Carlo model. A large sample of test objects mimicking a source population is evolved over a timescale comparable with the Solar System age until they reach a steady state spin distribution that should reproduce the current NEO distribution. The spin change due to YORP is computed for each body according to a simplified model based on Scheeres [Scheeres, D.J., 2007a. Icarus 188, 430-450].The steady state cumulative distribution of NEO spin rates obtained from our simulation nicely reproduces the observed one, once our results are biased to match the diameter distribution of the sample of objects included in the observational database. The excellent agreement strongly suggests that YORP is responsible for the concentration of spin at low rotation rates. In fact, in the absence of YORP the steady state population significantly deviates from the observed one. The spin evolution due to YORP is also so rapid for NEOs that the initial rotation rate distribution of any source population is quickly relaxed to that of the observed population. This has profound consequences for the study of NEO origin since we cannot trace the sources of NEOs from their rotation rate only.     

Physical investigation of the potentially hazardous Asteroid (144898) 2004 VD17

In this paper we present the observational campaign carried out at ESO NTT and VLT in April and May 2006 to investigate the nature and the structure of the near-Earth object (144898) 2004 VD17. In spite of a great quantity of dynamical information, according to which it will have a close approach with the Earth in the next century, the physical properties of this asteroid are largely unknown. We performed visible and near-infrared photometry and spectroscopy, as well as polarimetric observations. Polarimetric and spectroscopic data allowed us to classify 2004 VD17 as an E-type asteroid. A good agreement was also found with the spectrum of the aubrite meteorite Mayo Belwa. On the basis of the polarimetric albedo (pv=0.45) and of photometric data, we estimated a diameter of about 320 m and a rotational period of about 2 h. The analysis of the results obtained by our complete survey have shown that (144898) 2004 VD17 is a peculiar NEO, since it is close to the breakup limits for fast rotator asteroids, as defined by Pravec and Harris [Pravec, P., Harris, A.W., 2000. Icarus 148, 12-20]. These results suggest that a more robust structure must be expected, as a fractured monolith or a rubble pile in a “strength regime” [Holsapple, K.A., 2002. Speed limits of rubble pile asteroids: Even fast rotators can be rubble piles. In: Workshop on Scientific Requirements for Mitigation of Hazardous Comets and Asteroids, Washington, September, 2002].     

Optical colors of 56 near-Earth objects: trends with size and orbit

We present the results of BVRIZ photometry of 56 near-Earth objects (NEOs) obtained with the 1-m Jacobus Kapteyn telescope on La Palma during 2000 and 2001. Our sample includes many NEOs with particularly deep 1-μm pyroxene/olivine absorption bands, similar to Q-type asteroids. We also classify three NEOs with particularly blue colors. No D-type asteroids were found, placing an upper limit of ∼2% on the fraction of the NEO population originating in the outer main belt or the Trojan clouds. The ratio of dark to bright objects in our sample was found to be 0.40, significantly higher than current theoretical predictions. As well as classifying the NEOs, we have investigated color trends with size and orbit. We see a general trend for larger silicate objects to have shallower absorption bands but find no significant difference in the distribution of taxonomic classes at small and large sizes. Our data clearly show that different taxonomic classes tend to occupy different regions of (a, e) space. By comparing our data with current model predictions for NEO dynamical evolution we see that Q-, R-, and V-type NEOs tend to have orbits associated with “fast track” delivery from the main belt, whereas S-type NEOs tend to have orbits associated with “slow track” delivery. This outcome would be expected if space weathering occurs on time scales of >10⁶ years.