Construction of confidence regions in the problem of asteroid orbit determination

The factors required for estimation of the accuracy of the confidence region construction in the problem of asteroid orbit determination are considered. Blunders and large systematic errors occurring in asteroid observations increase the sizes of confidence regions and cause their noticeable shift in the space of determined parameters. We present the factors that, in addition to analysis of discrepancies (O–C), provide an opportunity to estimate the efficiency of screening observations containing gross systematic errors. The developed factors have been tested for efficiency using simulated observations. The observations have been simulated by parameters set by us and assumed true. It is shown how the sizes of systematic errors and the number of observations with these errors influence the results of screening. All calculations have been performed within the Keplerian model of asteroid motion.     

A Research on the Imaging Strategy and Imaging Simulation of Toutatis in the Chang’e-2 Flyby Mission

For the Chang’e-2 extended mission of asteroid exploration, the illumination conditions for imaging the asteroid Toutatis are calculated in this paper according to the orbital parameters of both the Chang’e-2 detector and the asteroid, as well as the incident angles of sunlight. On this basis, it is suggested to take photographs after flyby, and the orientation of the camera's optical axis in the coordinate system deflned by Earth's mean equator and equinox at J2000.0 is proposed to be (118.02°, 22.03°). Based on the shape model of Toutatis determined by the foreign radar data, the orientation of the asteroid in the inertial space is calculated at the rendezvous time. Using the Oren-Nayar diffuse-reflection model and the relative positions among the sun, the asteroid, and the detector, together with the cameras orientation, the imaging simulations are performed on the starry sky background respectively at the distances of 300 km, 500 km, and 1000 km from the asteroid after flyby. The results of simulations are verified further by the optical images of Toutatis obtained in the mission.     

Yarkovsky/YORP chronology of asteroid families

Asteroid families are the byproducts of catastrophic collisions whose fragments form clusters in proper semimajor axis, eccentricity, and inclination space. Although many families have been observed in the main asteroid belt, only two very young families, Karin and Veritas, have well-determined ages. The ages of other families are needed, however, if we hope to infer information about their ejection velocity fields, space weathering processes, etc. In this paper, we developed a method that allows us to estimate the ages of moderately young asteroid families (approximately in between 0.1 and 1 Gyr). We apply it to four suitable cases—Erigone, Massalia, Merxia, and Astrid—and derive their likely ages and approximate ejection velocity fields. We find that Erigone and Merxia were produced by large catastrophic disruption events (i.e., parent body ?100 km) that occurred approximately 280 and 330 Myr ago, respectively. The Massalia family was likely produced by a cratering event on Asteroid (20) Massalia less than 200 Myr ago. Finally, the Astrid family, which was produced by the disruption of a 60-70 km asteroid, is 100-200 Myr old, though there is considerable uncertainty in this result. We estimate that the initial ejection velocities for these families were only a few tens of meters per second, consistent with numerical hydrocode models of asteroid impacts. Our results help to verify that asteroid families are constantly undergoing dynamical orbital evolution from thermal (Yarkovsky) forces and spin vector evolution from thermal (YORP) torques.     

Radar observations at 3.5 and 12.6 cm wavelength of asteroid 433 Eros

A study of the asteroid 433 Eros using 3.5 and 12.6 cm radar waves indicates that the surface is very much rougher than any planetary or lunar surface observed by this method. A surface completely covered with sharp edges, pits, subsurface holes, or embedded chunks with scale sizes on the order of our wavelengths seems to be indicated. A model based on a rough rotating triaxial ellipsoid having radii in the rotation equator of 18.6 and 7.9 km agrees well with our data, although a strong wobble in the apparent center frequency of the spectra as rotation progresses indicates that one side may be more reflective than the other, or more likely, that the projected axis of rotation does not equally divide the projected area.     

Comparison of feature sizes on the surface of the asteroid 4 Vesta determined using ground-based and space observations

Vesta’s crater sizes that are based on images returned from the Dawn probe orbiting the asteroid 4 Vesta are compared with sizes of spots on its surface derived from hydrosilicate equivalent widths and asteroid color indices B-V and V-R observed spectrophotometrically on earth using a spectral-frequency method. The sizes of craters and spots at the asteroid poles prove to correspond to the sizes of spots that are assumed to be located at latitudes of 40°...45° N and S. Comparative results for the crater sizes ranging from 10 to 100 km are tabulated. We conclude that crater sizes on asteroid surfaces can be determined using the spectral-frequency method.     

The sizes of color patches on the surface of the asteroid 4 Vesta

The sizes of color patches on the surface of the asteroid 4 Vesta were estimated with the spectral-frequency method. We used the digital records of the asteroid spectra obtained on February 2, 3, 4, and 7, 2002, with the TV system and the slitless afocal spectrograph of the MTM-500 telescope at the Research Institute of the Crimean Astrophysical Observatory. The spectral resolution caused by the image quantity was about 40 Å and the exposure duration was 30 s. The energy calibration was performed both with the artificial light-emitting diode standard and three standard stars. The synthetic color indices B-V and V-R were calculated from the asteroid spectra taken out of the atmosphere. From these data, the changes with an asteroid rotation phase were eliminated. After that, the periods that were significantly smaller than the asteroid rotation period were searched for, and each time the data were whitened for the obtained period. Assuming that the patch sizes are determined by the half-obtained period and that the patches are located in the equatorial region of the asteroid, we estimated the sizes of 20 and 19 patches in the long-and short-wavelength ranges, respectively. The smallest found patch was about 9 km across. The statistical estimates of the reddish patches and the comparison with the statistics of old craters allowed us to suggest that the reddish patches on the asteroid Vesta surface are old formations.     

A preliminary analysis of the orbit of the Mars Trojan asteroid (5261) Eureka

Observations and results of orbit determination of the first known Mars Trojan asteroid (5261) Eureka are presented. We have numerically calculated the evolution of the orbital elements, and have analyzed the behavior of the motion during the next 2 Myr. Strong perturbations by planets other than Mars seem to stabilize the eccentricity of the asteroid by stirring the high order resonances present in the elliptic restricted problem. As a result, the orbit appears stable at least on megayear timescales. The difference of the mean longitudes of Mars and Eureka and the semimajor axis of the asteroid form a pair of variables that essentially behave in an adiabatic manner, while the evolution of the other orbital elements is largely determined by the perturbations due to other planets.     

Initial sizes of planetesimals and accretion of the asteroids

The present size frequency distribution (SFD) of bodies in the asteroid belt appears to have preserved some record of the primordial population, with an excess of bodies of diameter D ∼ 100 km relative to a simple power law. The survival of Vesta’s basaltic crust also implies that the early SFD had a shallow slope in the range ∼10-100 km. (Morbidelli, A., Bottke, W.F., Nesvorny, D., Levison, H.F. [2009]. Icarus 204, 558-573) were unable to produce these features by accretion from an initial population of km-sized planetesimals. They concluded that bodies with sizes in the range ∼100-1000 km and a SFD similar to the current population were produced directly from solid particles of sub-meter scale, without experiencing accretion through intermediate sizes. We present results of new accretion simulations in the primordial asteroid region. The requisite SFD can be produced from an initial population of planetesimals of sizes ?0.1 km, smaller than the usual assumption of km-sized bodies. The bump at D ∼ 100 km is produced by a transition from dispersion-dominated runaway growth to a regime dominated by Keplerian shear, before the formation of large protoplanetary embryos. Thus, accretion of the asteroids from an initial population of small (sub-km) planetesimals cannot be ruled out.     

Thermal inertia of near-Earth asteroids and implications for the magnitude of the Yarkovsky effect

Thermal inertia determines the temperature distribution over the surface of an asteroid and therefore governs the magnitude the Yarkovsky effect. The latter causes gradual drifting of the orbits of km-sized asteroids and plays an important role in the delivery of near-Earth asteroids (NEAs) from the main belt and in the dynamical spreading of asteroid families. At present, very little is known about the thermal inertia of asteroids in the km size range. Here we show that the average thermal inertia of a sample of NEAs in the km-size range is . Furthermore, we identify a trend of increasing thermal inertia with decreasing asteroid diameter, D. This indicates that the dependence of the drift rate of the orbital semimajor axis on the size of asteroids due to the Yarkovsky effect is a more complex function than the generally adopted D⁻¹ dependence, and that the size distribution of objects injected by Yarkovsky-driven orbital mobility into the NEA source regions is less skewed to smaller sizes than generally assumed. We discuss how this fact may help to explain the small difference in the slope of the size distribution of km-sized NEAs and main-belt asteroids.     

The impact origin of Eunomia and Themis families

Abstract— Criteria for finding asteroid families (Zappala et al. 1995) are applied to a large (205,770 member) data set of asteroid orbital elements. The cases of the Eunomia and Themis families are considered as examples. This is combined with the cratering criteria for catastrophic disruption of small bodies in the solar system (Leliwa‐Kopystyński et al. 2008). We find that the Eunomia parent body itself was not catastrophically disrupted in the family‐generating impact event; after impact, the current body contains as much as 70% of its primordial mass. However, by contrast with Eunomia, the present mass of 24 Themis is only about 21% of that of its primordial body. Limits are placed on the sizes of the impactors in both examples, and for the case of Eunomia, the radius of the just sub‐critical crater (which may be present on 15 Eunomia) is predicted as <58 km.     

Reliability of estimates for the sizes of spots on asteroid surfaces made by the spectral-frequency method

A new spectral-frequency method (SFM) for the study of solid body surfaces is briefly described. This method allows estimation of the sizes of various spots. Estimates for the sizes of spots on asteroid surfaces made by the SFM and other methods are compared and discussed. The sizes of spots on the surface of asteroid 1620 Geographos determined by the SFM are well consistent with those of the craters obtained from radar data. The sizes of hydrosilicate spots on the surface of asteroid 21 Lutetia found by the SFM agree with those of the craters determined by the Rosetta spacecraft. The size of a blue spot on the surface of asteroid 4 Vesta found by the SFM is consistent with the size of the well-known crater on the south pole of the asteroid. It is inferred that the SFM is a promising method for the estimation of the sizes of spots on asteroid surfaces.     

Yarkovsky depletion and asteroid collisional evolution

The orbital parameters of small asteroids change with time, as a consequence of the so-called Yarkovsky effect. This leads to a steady removal of objects from the Main Belt, which takes place when the objects reach one of the major resonant regions in the orbital elements space. The process may influence the evolution of the inventory and size distribution of Main Belt asteroids, but it has not been taken into account by classical models of the collisional evolution of the asteroid population. In this paper we discuss the role of the Yarkovsky effect in producing the current observed size distribution. We show that adding Yarkovsky effect to purely collisional mechanisms may increase the removal of objects at sizes around 1 km by a factor of about 2 with respect to a purely collisional scenario. Moreover, waves in the size distribution may also be produced. However, taking also into account current uncertainties in the efficiency of purely collisional mechanisms, the role of the Yarkovsky effect seems not dominant, and cannot be unambiguously determined.     

Stability of Surface Motion on a Rotating Ellipsoid

The dynamical environment on the surface of a rotating, massive ellipsoid is studied, with applications to surface motion on an asteroid. The analysis is performed using a combination of classical dynamics and geometrical analysis. Due to the small sizes of most asteroids, their shapes tend to differ from the classical spheroids found for the planets. The tri-axial ellipsoid model provides a non-trivial approximation of the gravitational potential of an asteroid and is amenable to analytical computation. Using this model, we study some properties of motion on the surface of an asteroid. We find all the equilibrium points on the surface of a rotating ellipsoid and we show that the stability of these points is intimately tied to the conditions for a Jacobi or MacLaurin ellipsoid of equilibria. Using geometrical analysis we can define global constraints on motion as a function of shape, rotation rate, and density, we find that some asteroids should have accumulation of material at their ends, while others should have accumulation of surface material at their poles. This study has implications for motion of a rover on an asteroid, and for the distribution of natural material on asteroids, and for a spacecraft hovering over an asteroid.     

Radar observations of the asteroid 2011 UW158

In July 2015 intercontinental bistatic radar observations of the potentially dangerous asteroid 2011 UW158 during its close approach to the Earth were carried out. The asteroid was illuminated at a frequency of 8.4 GHz with the 70-m DSS-14 antenna of the Goldstone Deep Space Communications Complex, while the signal reflected from the asteroid was received with the 32-m radio telescopes of the Quasar VLBI network at the Zelenchukskaya and Badary Observatories. The spectra of the reflected radio signals were obtained. The sizes and rotation period of the asteroid consistent with photometric observations and the ratio of the powers of the reflected signals with left- and right-hand circular polarizations were determined. The derived values suggest that the asteroid has an inhomogeneous surface and a prolate shape. The observations of the Doppler shift of the reflected signal frequency were obtained, which allowed the orbital parameters of the asteroid to be improved.     

Triaxial ellipsoid dimensions and poles of asteroids from AO observations at the Keck-II telescope

Five main belt asteroids, 2 Pallas, 129 Antigone, 409 Aspasia, 532 Herculina, and 704 Interamnia were imaged with the adaptive optics system on the 10 m Keck-II telescope in the near infrared on one night, August 16, 2006. The three axis dimensions and rotational poles were determined for Pallas, Antigone, Aspasia, and Interamnia, from their changing apparent sizes and shapes as measured with parametric blind deconvolution. The rotational pole found for Interamnia is much different from all previous work, including our own at Lick Observatory the previous month. Although images of Herculina were obtained at only two rotational phases, its rotation appears to be opposite to that predicted from the lightcurve inversion model of M. Kaasalainen, J. Torppa, and J. Piironen [2002. Icarus 159, 369-395]. A search for satellites was made in all of the asteroid images, with negative results, but three trailing stars around Herculina (200 km diameter), down to 8.9 magnitudes fainter and between 1 and 115 asteroid radii (100 to 11,500 km) from the asteroid, establishes an upper limit of 3.3 km for any object with the same albedo near Herculina.     

Flattening,pole, and albedo features of 4 vesta from photometric data

A number of large asteroids show irregular lightcurves of relatively small amplitude and/or ambiguous rotational periods. These observations and the fact that their strong gravitational binding probably results in quasi-equilibrium shapes lead to model these bodies as axisymmetric, biaxial ellipsoids covered by albedo markings. We developed a general numerical algorithm for obtaining simulated lightcurves of “spotted” asteroids and varied the most critical geometrical and physical parameters (albedo contrast, size, and position of the spots; polar coordinates, and shape of the asteroid). We then analyzed the case of 4 Vesta by assuming an axisymmetric ellipsoidal shape with a large brighter region on one hemisphere, in agreement with the results of photometric and polarimetric observations. Fitting the numerical simulations to the available data, we obtained the flattening of the ellipsoid (0.79 ± 0.03), the albedo contrast and geometry of the brighter region, and the orientation of the polar axis. If the derived flattenning corresponds to the equilibrium shape of a nearly homogeneous body, a density of 2.4 ± 0.3 g cm⁻³ can be inferred. These results show satisfactory agreement with values by different techniques. We plan to apply the same method both to other large asteroids and to smaller, irregularly shaped ones; in the latter case, this will allow us to test the uncertainties in current pole determination methods.     

Towards initial mass functions for asteroids and Kuiper Belt Objects

Our goal is to understand primary accretion of the first planetesimals. Some examples are seen today in the asteroid belt, providing the parent bodies for the primitive meteorites. The primitive meteorite record suggests that sizeable planetesimals formed over a period longer than a million years, each of which being composed entirely of an unusual, but homogeneous, mixture of millimeter-size particles. We sketch a scenario that might help explain how this occurred, in which primary accretion of 10-100 km size planetesimals proceeds directly, if sporadically, from aerodynamically-sorted millimeter-size particles (generically “chondrules”). These planetesimal sizes are in general agreement with the currently observed asteroid mass peak near 100 km diameter, which has been identified as a “fossil” property of the pre-erosion, pre-depletion population. We extend our primary accretion theory to make predictions for outer Solar System planetesimals, which may also have a preferred size in the 100 km diameter range. We estimate formation rates of planetesimals and explore parameter space to assess the conditions needed to match estimates of both asteroid and Kuiper Belt Object (KBO) formation rates. For parameters that satisfy observed mass accretion rates of Myr-old protoplanetary nebulae, the scenario is roughly consistent with not only the “fossil” sizes of the asteroids, and their estimated production rates, but also with the observed spread in formation ages of chondrules in a given chondrite, and with a tolerably small radial diffusive mixing during this time between formation and accretion. As previously noted, the model naturally helps explain the peculiar size distribution of chondrules within such objects. The optimum range of parameters, however, represents a higher gas density and fractional abundance of solids, and a smaller difference between Keplerian and pressure-supported orbital velocities, than “canonical” models of the solar nebula. We discuss several potential explanations for these differences. The scenario also produces 10-100 km diameter primary KBOs, and also requires an enhanced abundance of solids to match the mass production rate estimates for KBOs (and presumably the planetesimal precursors of the ice giants themselves). We discuss the advantages and plausibility of the scenario, outstanding issues, and future directions of research.     

任意倾角的共轨运动研究

共轨运动天体与摄动天体的半长径相同,处于1:1平运动共振中.太阳系内多个行星的特洛伊天体即为处于蝌蚪形轨道的共轨运动天体,其中一些高轨道倾角特洛伊天体的轨道运动与来源仍未被完全理解.利用一个新发展的适用于处理1:1平运动共振的摄动函数展开方式,对三维空间中的共轨运动进行考察,计算不同初始轨道根数情况下共轨轨道的共振中心、共振宽度,分析轨道类型与初始轨道根数的关系.并将分析方法所得结果与数值方法的结果相互比较验证,得到了广阔初始轨道根数空间内共轨运动的全局图景.     

The peculiar case of the Agnia asteroid family

The Agnia asteroid family, a cluster of asteroids located near semimajor axis a=2.79 AU, has experienced significant dynamical evolution over its lifetime. The family, which was likely created by the breakup of a diameter D∼50 km parent body, is almost entirely contained within the high-order secular resonance z₁. This means that unlike other families, Agnia's full extent in proper eccentricity and inclination is a byproduct of the large-amplitude resonant oscillations produced by this resonance. Using numerical integration methods, we found that the spread in orbital angles observed among Agnia family members would have taken at least 40 Myr to create; this sets a lower limit on the family's age. To determine the upper bound on Agnia's age, we used a Monte Carlo model to track how the small members in the family evolve in semimajor axis by Yarkovsky thermal forces. Our results indicate the family is no more than 140 Myr old, with a best-fit age of 100⁺³⁰₋₂₀ Myr. Using two independent methods, we also determined that the D∼5 km fragments were ejected from the family-forming event at a velocity near 15 m/s. This velocity is consistent with results from numerical hydrocode simulations of asteroid impacts and observations of other similarly sized asteroid families. Finally, we found that 57% of known Agnia fragments were initially prograde rotators. The reason for this limited asymmetry is unknown, though we suspect it is a fluke produced by the stochastic nature of asteroid disruption events.     

Water combinations on the surface of the asteroid 4 Vesta

The spectral-frequency method (Busarev et al., 2007b) allowed us to obtain data on 16 sizes of hydrosilicate spots on the surface of the asteroid 4 Vesta. Large sizes (800 and 750 km) show clusters of small hydrosilicate spots near a well-known crater. Small spots of 50–13 km cover more than 50% of the surface. The predominant number of small-sized spots suggests their recent origin. Our data confirm the presence of water combinations on the surface of 4 Vesta and allow us to draw a conclusion on their recent appearance in collisions of this asteroid with primitive bodies arriving from the zone of Jupiter.