A concept for providing warning of earth impacts by small asteroids

The atmospheric detonation of a 17 m-asteroid above Chelyabinsk, Russia on 2013 February 15 shows that even small asteroids can cause extensive damage. Earth-based telescopes have found smaller harmless objects, such as 2008 TC3, a 4 m-asteroid that was discovered 20h before it exploded over northeastern Sudan (Jenniskens, 2009). 2008 TC3 remains the only asteroid discovered before it hit Earth because it approached Earth from the night side, where it was observed by large telescopes searching for near-Earth objects (NEO’s). The larger object that exploded over Chelyabinsk approached Earth from the day side, from too close to the Sun to be detected from Earth. A sizeable telescope in an orbit about the Sun-Earth L1 (SE-L1) libration point could find objects like the “Chelyabinsk” asteroid approaching approximately from the line of sight to the Sun about a day before Earth closest approach. Such a system would have the astrometric accuracy needed to determine the time and impact zone for a NEO on a collision course. This would give at least several hours, and usually 2–4 days, to take protective measures, rather than the approximately two-minute interval between the flash and shock wave arrival that occurred in Chelyabinsk. A perhaps even more important reason for providing warning of these events, even smaller harmless ones that explode high in the atmosphere with the force of an atomic bomb, is to prevent mistaking such an event for a nuclear attack that could trigger a devastating nuclear war. A concept using a space telescope similar to that needed for an SE-L1 monitoring satellite, is already conceived by the B612 Foundation, whose planned Sentinel Space Telescope could find nearly all 140 m and larger NEO’s, including those in orbits mostly inside the Earth’s orbit that are hard to find with Earth-based telescopes, from a Venus-like orbit (Lu, 2013). Few modifications would be needed to the Sentinel Space Telescope to operate in a SE-L1 orbit, 0.01 AU from Earth towards the Sun, to find most asteroids larger than about 5 meters that approach the Earth from the solar direction. The spacecraft would scan 165 square degrees of the sky around the Earth every hour, finding asteroids when they are brightest (small phase angle) as they approach Earth. We will undertake Monte Carlo studies to see what fraction of asteroids 5 m and larger approaching from the Sun might be found by such a mission, and how much warning time might typically be expected. Also, we will check the overall coverage for all Earth-approaching NEO’s, including ground-based observations and observations by the recently-launched NEOSSat, which may best fill any gaps in coverage between that provided by an SE-L1 telescope and ground-based surveys. Many of the objects as large as 50 m, like the one that created Meteor Crater in Arizona, will not be found by current NEO surveys, while they would usually be seen by this possible mission even if they approached from the direction of the Sun. We should give better warning for future “Bolts out of the blue.”     

The results of positional observations of near earth asteroids using the combined observation method

The high velocity of the apparent motion of near Earth asteroids (NEAs) is the main problem in their observation. This problem is solved at the Research Institute Nikolaev Astronomical Observatory (RI NAO) with a combined observation method using the time delay and integration mode of a CCD array and a camera rotator. A total of 1317 positions of 74 NEAs were obtained at RI NAO in 2008–2012. All the observations were made using the combined-observation method. The error in observations made at RI NAO is compared with the results that were obtained at other observatories in this work.     

Optimal low-thrust trajectories to asteroids through an algorithm based on differential dynamic programming

In this paper an optimisation algorithm based on Differential dynamic programming is applied to the design of rendezvous and fly-by trajectories to near Earth objects. Differential dynamic programming is a successive approximation technique that computes a feedback control law in correspondence of a fixed number of decision times. In this way the high dimensional problem characteristic of low-thrust optimisation is reduced into a series of small dimensional problems. The proposed method exploits the stage-wise approach to incorporate an adaptive refinement of the discretisation mesh within the optimisation process. A particular interpolation technique was used to preserve the feedback nature of the control law, thus improving robustness against some approximation errors introduced during the adaptation process. The algorithm implements global variations of the control law, which ensure a further increase in robustness. The results presented show how the proposed approach is capable of fully exploiting the multi-body dynamics of the problem; in fact, in one of the study cases, a fly-by of the Earth is scheduled, which was not included in the first guess solution.     

On the deflection of asteroids with mirrors

This paper presents an analysis of an asteroid deflection method based on multiple solar concentrators. A model of the deflection through the sublimation of the surface material of an asteroid is presented, with simulation results showing the achievable impact parameter with, and without, accounting for the effects of mirror contamination due to the ejected debris plume. A second model with simulation results is presented analyzing an enhancement of the Yarkovsky effect, which provides a significant deflection even when the surface temperature is not high enough to sublimate. Finally the dynamical model of solar concentrators in the proximity of an irregular celestial body are discussed, together with a Lyapunov-based controller to maintain the spacecraft concentrators at a required distance from the asteroid.     

Observations of asteroids with ALMA

Thermal observations of large asteroids at millimeter wavelengths have revealed high amplitude rotational lightcurves. Such lightcurves are important constraints on thermophysical models of asteroids, and provide unique insight into the nature of their surface and subsurface composition. A better understanding of asteroid surfaces provides insight into the composition, physical structures, and processing history of these surviving remnants from the formation of our solar system. In addition, detailed observations of the larger asteroids, accompanied by thermophysical models with appropriate temporal and spatial resolution, promise to decrease uncertainties in their flux predictions. Of particular interest are the near-Earth objects, which can be observed at large phase angles, permitting better assessment of the thermal response of their unilluminated surfaces. The high sensitivity of ALMA will enable us to detect many small bodies in all the major groups, to obtain lightcurves for a large sample of main-belt and near-Earth objects, to resolve the surfaces of some large objects, and to separate the emission from primary and secondary objects in binary pairs. In addition to the science goals of asteroid studies, these bodies may also prove useful operationally because those with known shapes and well-characterized lightcurves could be employed for flux calibration by ALMA and other high frequency instruments.     

Theory of the Trojan asteroids

In the previously published Parts I and II of the paper, the author has constructed a formal long-periodic solution for the case of 11 resonance in the restricted problem of three bodies to 0(m ^3/2), wherem is the small mass parameter of the system. The time-dependencet(, ,m), where is the mean synodic longitude and is related to the Jacobi constant, has been expressed by ahyperelliptic integral. It is shown here that with the approximationm=0 in the integrand, the functiont(, , 0) can be expanded in a series involving standardelliptic functions. Then the problem of inversion can be formally solved, yielding the function (t, , 0).Similarly, the normalized period (,m) of the motion can be approximated by theHagihara hyperelliptic integral (, 0), corresponding tom=0. This integral is also expanded into elliptic functions. Asymptotic forms for (, 0) are derived for 0 and for 1, corresponding to the extreme members of thetadpole branch of the family of orbits.     

Dynamics of the Hungaria asteroids

To try to understand the dynamical and collisional evolution of the Hungaria asteroids we have built a large catalog of accurate synthetic proper elements. Using the distribution of the Hungaria, in the spaces of proper elements and of proper frequencies, we can study the dynamical boundaries and the internal structure of the Hungaria region, both within a purely gravitational model and also showing the signature of the non-gravitational effects. We find a complex interaction between secular resonances, mean motion resonances, chaotic behavior and Yarkovsky-driven drift in semimajor axis. We also find a rare occurrence of large scale instabilities, leading to escape from the region. This allows to explain the complex shape of a grouping which we suggest is a collisional family, including most Hungaria but by no means all; we provide an explicit list of non-members of the family. There are finer structures, of which the most significant is a set of very close asteroid couples, with extremely similar proper elements. Some of these could have had, in a comparatively recent past, very close approaches with low relative velocity. We argue that the Hungaria, because of the favorable observing conditions, may soon become the best known sub-group of the asteroid population.     

On the origin of the asteroids

Initial temperature distribution of the nebulous disk around the sun led to the bordering region between the asteriods and Jupiter becoming a transition region, in which the “icy” matter changed from uncondensed to condensed state. This continuity in chemical constitution makes the projected surface density of solid condensate in the inner portion of the Jupiter region greater than the mean value in the asteriod region, and the time taken for dust particles to precipitate to become comparable to that in the asteriod region. Owing to gravitational instability, the dust layer then broke up into clusters of particles, each cluster transforming into a planetesimal. The mass of such a planetesimal and the rate of its growth in the Jupiter region are greater than those in the asteriod region. Perturbations and mutual encounters between the larger planetesimals in the inner portion of the Jupiter region changed their orbits and produced rather large relative velocities. Some of these entered the asteriod region, drew out most of the matter there, and increased the random velocity of the asteriods that remained, so that they could not combine to form a planet. We prove the above view by a quantitative analysis in this paper.     

Photometry of asteroids

Photometry is one of the most efficient investigation techniques. It provided a large body of data on albedos, sizes, shapes, rotation, optical properties, and structural characteristics of asteroids and other minor bodies of the solar system. The contribution of photometry to the determination of asteroid parameters was most crucial. This review summarizes main results of asteroid studies in three most important areas: i) determination of the shape and rotation parameters of asteroids, ii) investigation of optical properties of asteroid surfaces, and iii) detection and investigation of binary asteroid systems.     

Taxonomy of asteroids

A taxonomic system was introduced by C. R. Chapman, D. Morrison, and B. Zellner [Icarus25, 104–130 (1975)], in which minor planets are classified according to a few readily observable optical properties, independent of specific mineralogical interpretations. That taxonomy is here augmented to five classes, now precisely defined in terms of seven parameters obtained from polarimetry, spectrophotometry, radiometry, and UBV photometry of 523 objects. We classify 190 asteroids as type C, 141 as type S, 13 as type M, 3 as type E, and 3 as type R; 55 objects are shown to fall outside these five classes and are designated U (unclassifiable). For the remaining 118, the data exclude two or more types but are insufficient for unambiguous classification. Reliable diameters, from radiometry or polarimetry or else from albedos adopted as typical of the types, are listed for 396 objects. We also compare our taxonomy with other ones and discuss how classification efforts are related to the interpretation of asteroid mineralogies.     

Conditions for magnetic interaction of asteroids with the solar wind

Conditions are presented for maintenance of asteroid magnetospheres by dipole moments and for propagation of whistler mode noise in the solar wind at asteroid distances. Surface field intensities less than one thousandth that of the Earth are found adequate for supporting magnetospheres in the quiet solar wind surrounding the larger asteroids. Magnetospheric diameters are likely to be small, however, and difficult to identify without targeted, close-approach flybys. Under most ordinary conditions, whistler noise generated in an asteroidal shock or by other interaction with the solar wind will not propagate back upstream toward the sun, but may form a detectable wake downstream. Pure standing whistler wavefronts could be a unique asteroidal phenomenon.     

Detection of Earth-impacting asteroids with the next generation all-sky surveys

We have performed a simulation of a next generation sky survey’s (Pan-STARRS 1) efficiency for detecting Earth-impacting asteroids. The steady-state sky-plane distribution of the impactors long before impact is concentrated towards small solar elongations (Chesley, S.R., Spahr T.B., 2004. In: Belton, M.J.S., Morgan, T.H., Samarashinha, N.H., Yeomans, D.K. (Eds.), Mitigation of Hazardous Comets and Asteroids. Cambridge University Press, Cambridge, pp. 22-37) but we find that there is interesting and potentially exploitable behavior in the sky-plane distribution in the months leading up to impact. The next generation surveys will find most of the dangerous impactors (>140 m diameter) during their decade-long survey missions though there is the potential to miss difficult objects with long synodic periods appearing in the direction of the Sun, as well as objects with long orbital periods that spend much of their time far from the Sun and Earth. A space-based platform that can observe close to the Sun may be needed to identify many of the potential impactors that spend much of their time interior to the Earth’s orbit. The next generation surveys have a good chance of imaging a bolide like 2008 TC₃ before it enters the atmosphere but the difficulty will lie in obtaining enough images in advance of impact to allow an accurate pre-impact orbit to be computed.     

A theory of the Trojan asteroids

The paper constructs a long-periodic solution for the case of 11 resonance in the restricted problem of three bodies. The solution is smoothed by the exclusion of the internal resonant terms arising from the near-commensurability between the long and the short periods of the asteroid.Although the Brown (1911) conjecture regarding the termination of the family of the tadpoles at the Lagrangian pointL ₃ is not supported by our analysis, the conjecture seems to hold in the limit asm0.     

The angular momentum of the asteroids

Consideration of the basic physics involved in the structure of the object are used to obtain relationships for the radius, period, angular momentum, etc. of a typical asteroid. The mass-angular momentum relation for asteroids would tend to favour the fragmentation hypothesis.     

Dynamical evolution of the Hungaria asteroids

The Hungarias are a stable asteroid group orbiting between Mars and the main asteroid belt, with high inclinations (16–30°), low eccentricities (e < 0.18), and a narrow range of semi-major axes (1.78–2.06 AU). In order to explore the significance of thermally-induced Yarkovsky drift on the population, we conducted three orbital simulations of a 1000-particle grid in Hungaria a–e–i space. The three simulations included asteroid radii of 0.2, 1.0, and 5.0 km, respectively, with run times of 200 Myr. The results show that mean motion resonances—martian ones in particular—play a significant role in the destabilization of asteroids in the region. We conclude that either the initial Hungaria population was enormous, or, more likely, Hungarias must be replenished through collisional or dynamical means. To test the latter possibility, we conducted three more simulations of the same radii, this time in nearby Mars-crossing space. We find that certain Mars crossers can be trapped in martian resonances, and by a combination of chaotic diffusion and the Yarkovsky effect, can be stabilized by them. Therefore, some Hungarias (around 5% of non-family members with absolute magnitudes H < 15.5 and 10% for H < 17) may represent previously transient Mars crossers that have been adopted in this manner.     

A lightcurve survey of asteroids with Schmidt telescopes: Observations of nine asteroids during oppositions in 1977

The results of photographic photometry of nine asteroids with the Schmidt telescope at the Uppsala Southern Station are presented. The results tend to confirm earlier results [Lagerkvist, (1978) Astron. Astrophys. Suppl.31, 361] that the smaller asteroids are collisional products. Rotation periods were determined for the asteroids 700, 1267, 1523, and 1562. Absolute magnitudes were determined for the asteroids 525, 700, 1207, 1267, and 1717.     

Dynamical study of the Hilda asteroids

Schubart's model of a planar, elliptic restricted three-body problem is used to study the orbital motion of the Hilda asteroids from thePalomar-Leiden Survey. The 3:2 resonant coupling to Jupiter of some of these small asteroids are found to be stable. However, some of the small asteroids with absolute magnitudeg>15 have large amplitude of variation in their orbital elements in one libration period. Since the lifetime scales against catastrophic collision of the Hilda asteroids are estimated to be several times larger than those of the main belt objects, a significant portion of these resonant asteroids could be the original members of the Hilda group. From this point of view, it is suggested that such size-dependence of resonant orbital motions might be the result of the cosmogonic effects ofjet stream accretion.     

Orbital evolution of asteroids with the effects of mutual gravitational attraction

The effects of the mutual gravitational attraction between asteroids were analyzed by two N-body calculations, in which N=4,516 (the Sun, the nine planets, and 4,506 asteroids). In one calculation the gravity of the asteroids was taken into account, and in the other it was ignored. These calculations were carried out for a time period of about 100 years. The largest difference in the positions of the asteroids between these two calculations is about 10^–3 AU. For the orbital elements of the semimajor axis, the eccentricity, and the inclination, the largest differences were 9 × 10^–6 AU, 4 × 10^–6, and 5 × 10^–4 degrees, respectively. It was found that the distribution of the differences of the semimajor axis between the two calculations is quite similar to the Cauchy distribution.     

The association of Earth-crossing asteroids with meteoroid streams

Various points are discussed concerning the association of Earth-crossing asteroids (ECAs) with meteoroid streams, including the drawbacks of the techniques used in some previous work. In comparing the theoretical radiants of any ECA (or, indeed, comet) with observed meteor radiants it is necessary that the orbit used be that appropriate for epochs when the ECA has a node at 1 AU; in each precession cycle of the argument of perihelion () there will be four values rendering a node at the Earth's orbit, so that four showers are expected. Precession of the node will result in sets of showers at different times of year from different-precession cycles, whilst for some objects the orbital evolution is more convoluted. For diffuse, low-flux showers a problem is differentiating the meteors associated with any ECA from the sporadic background; a new graphical technique is introduced for illuminating whether such associations exist. A re-evaluation is required of whether ECAs should be thought of as being parent bodies of specific showers. Although this might be the case for some very large ECAs (such as (3200) Phaethon, associated with the Geminid stream), the bodies observed now being extinct or dormant cometary cores, it is suggested that in general the ECAs are better thought of as being large fragments produced in hierarchical cometary disintegrations. That is, some ECAs are just the largest meteoroids in meteoroid streams.     

Tidal encounters of ellipsoidal granular asteroids with planets

We investigate planetary fly-bys of asteroids using an approximate volume-averaged method that offers a relatively simple, but very flexible, approach to study the rotational dynamics of ellipsoids. The asteroid is considered to be a deformable, prolate ellipsoid, with its interior being modeled as a rigid-granular material. Effects due to the asteroid's rotation, its self-gravity and gravitational interaction with the planet are included. Using a simplified approach allows us to explore in detail the mechanics of asteroid's deformations and disruptions during planetary encounters. We also compare our results with those obtained by Richardson et al. [Richardson, D.C., Bottke Jr., W.F., Love, S.G., 1998. Icarus 134, 47-76] who used a large numerical code. We find that many of the features reported by them can indeed be captured by our rather simple methodology, and we discuss the reasons why some of our results differ from theirs.