Accessibility of Main-belt Asteroids and Trajectory Design for Multi-target Exploration

The mission designed to explore asteroids has nowadays become a hot spot of deep space exploration, and the accessibility of the explored objects is the most important problem to make clear. The number of asteroids is large, and it needs an enormous quantity of calculations to evaluate the accessibility for all asteroids. In this paper, based on the direct transfer strategy, we have calculated the accessibility for the different regions of the solar system and compared it with the distribution of asteroids. It is found that most main-belt asteroids are accessible by the direct transfer orbit with the launch energy of C₃ = 50 km²/s², and that with an additional small velocity correction, the designed trajectory is able to realize the multi-target flyby mission. Such a kind of multi-target flyby can reach the same effect of the orbit manoeuvre in the ΔV-EGA trajectory scheme^[1,2]. Being assisted by the earth's gravity, the accompanying flight with asteroids or the exploration of more distant asteroids can be realized with a lower energy. In the end, as an example, a trajectory scheme is given, in which the probe flies by multiple main-belt asteroids at first, then with the assistance of the earth's gravity, it makes the accompanying flight to a more distant asteroid.     

Excitation of Orbital Inclinations of Asteroids during Depletion of a Protoplanetary Disk: Dependence on the Disk Configuration

The pumping up of orbital inclinations of asteroids caused by sweeping secular resonances associated with depletion of a protoplanetary disk is discussed, focusing on the dependence on the disk inclinations and surface density distribution. The asteroids have large mean inclinations that cannot be explained by present planetary perturbations alone. It has been suggested that the sweeping secular resonances caused by disk depletion are responsible for these high inclinations. Nagasawa et al. (2000, Astron. J.119, 1480-1497) showed that the inclinations of asteroids are pumped up if the disk is depleted in an inside-out manner on a time scale longer than 3×10⁵ years. Their assumed disk midplane is not on the invariant plane. However, it should be affected by the inclination of the disk plane. Here we investigate the dependence on the disk inclinations. We assume a disk depletion model in which the disk inside the jovian orbit has been removed and the residual outer disk is uniformly depleted. We calculate the locations of the secular resonances and the excitation magnitude of the inclinations with analytical methods. We found that the inclinations are pumped up to the observational level for a depletion time scale longer than 10⁶ years in the case of the disk plane that coincides with the invariant plane. The required time scale is longest (3×10⁶ years) if the disk plane coincides with the jovian orbital plane. However, it is still within the observationally inferred depletion time scale. We also studied dependence on a disk surface density gradient and found that the results do not change significantly as long as the inner disk depletion is faster than the outer disk one.     

Attitude stability of a spacecraft with two flexible solar arrays on a stationary orbit around an asteroid subjected to gravity gradient torque

In the gravity field of an asteroid with the second order and degree harmonics C ₂₀ and C ₂₂, the attitude stability of a spacecraft with two flexible solar arrays on a stationary orbit subjected to the fourth-order gravity gradient torque is investigated in this paper. The sufficient conditions of attitude stability of the spacecraft are obtained, the effect of the direction of the flexible solar arrays and some special cases are discussed. Taking the asteroids 4769 Castalia, 25143 Itokawa and the imaginary asteroids as examples, the attitude stability domains, determined by the sufficient conditions, of the spacecrafts moving on stationary orbits around them are presented. It is found that the attitude stability domains of the spacecraft with two flexible solar arrays are evidently different when the solar arrays are installed in different directions; the effect of the harmonics C ₂₀ and C ₂₂ of the asteroids has the significant influence on the attitude stability domains of the spacecrafts with flexible appendages moving on stationary orbits; in the certain case, the effect of the harmonics C ₂₀ and C ₂₂ of the asteroids has no influence on the attitude stability domains of the rigid spacecrafts moving on stationary orbits, but in the other cases, the effect of the harmonics C ₂₀ and C ₂₂ of the asteroids has also the significant influence on the attitude stability domains of the rigid spacecrafts moving on stationary orbits; whether the harmonics C ₂₀ and C ₂₂ of the asteroids are considered or not, the effect of flexible appendages decreases the attitude stability domains.     

Model orbits of asteroid 3040 Kozai

The orbital evolutions of the asteroid 3040 Kozai and model asteroids with similar orbits have been investigated. Their osculating orbits for an epoch 1991 December 10 were numerically integrated forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all inner planets, Jupiter, and Saturn.The orbit of the asteroid Kozai is stable. Its motion is affected only by long-period perturbations of planets. With change of the argument of perihelion of the asteroid Kozai, the evolution of the model asteroid orbits changes essentially, too. The model orbits with the argument of perihelion changed by the order of 10% show that asteroids with such orbital parameters may approach the Earth orbit, while asteroids with larger changes may even cross it, at least after 10,000 years. Long-term orbital evolution of asteroids with these orbital parameters is very sensitive on their angular elements.     

Peripheral Structures of Planetary Systems

We analyze the conditions for the formation and time evolution of peripheral comet structures of solar-type planetary systems. In the Solar system, these include the Kuiper belt, the Oort cloud, the comet spear, and the Galactic comet ring that marks the Galactic orbit of the Sun. We consider the role of the viscosity of a protoplanetary gas–dust disk, major planets, field stars, globular clusters, giant molecular clouds, and the Galactic gravitational field in the formation of these peripheral structures marked by comets and asteroids. We give a list of the closest past and future passages of neighboring stars through the solar Oort cloud that perturb the motion of its comets and, thus, contribute to the enhancement of its cometary activity, on the one hand, and to the replenishment of the solar comet spear with new members, on the other hand.     

From comets to asteroids: When hairy stars go bald

We discuss the essential differences between comets and asteroids. Ironically, with the exception of the rocky asteroids in the inner solar system, most of the objects classified as asteroids at and beyond Jupiter's orbit are likely to conceal buried volatiles, and thus are more usefully considered as comets.     

Slow and Fast Diffusion in Asteroid-Belt Resonances: A Review

This paper reviews recent advances in several topics of resonant asteroidal dynamics as the role of resonances in the transportation of asteroids and asteroidal debris to the inner and outer solar system; the explanation of the contrast of a depleted 2/1 resonance (Hecuba gap) and a high-populated 3/2 resonance (Hilda group); the overall stochasticity created in the asteroid belt by the short-period perturbations of Jupiter's orbit, with emphasis in the formation of significant three-period resonances, the chaotic behaviour of the outer asteroid belt, and the depletion of the Hecuba gap. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Laplacian orbit determination of asteroids

A modified Laplacian technique is described for initial orbit determination of asteroids from CCD observations and its applications for orbit determination of the main belt asteroids and near Earth asteroids. The proposed modification is based on a simultaneous improvement of both the orbital elements and the derivatives of spherical coordinates in frames of Laplace's method. It provides an orbit which represents the used observations with the residuals comparable with errors of these observations. The improved values of the derivatives might be used as ephemeris parameters for identification of newly discovered objects.     

From comets to asteroids: When hairy stars go bald

We discuss the essential differences between comets and asteroids. Ironically, with the exception of the rocky asteroids in the inner solar system, most of the objects classified as asteroids at and beyond Jupiter's orbit are likely to conceal buried volatiles, and thus are more usefully considered as comets.     

Axial rotation of near-earth asteroids: The influence of the YORP effect

The distribution of axial rotation velocities of near-Earth asteroids (NEAs) substantially differs from that of the Main-Belt asteroids by an excess of both quickly and slowly rotating objects. Among the possible causes of this difference is the influence of the solar radiation—the so-called YORP effect—that arises from the absorption of solar energy and its reemission in the thermal range by a rotating body of irregular shape. It is known that the magnitude of this effect depends on the asteroid size and the quantity of received solar energy (the insolation). Analysis of the observational data showed that the mean diameter of NEAs decreases from the middle of the distribution to the edges, i.e., the excess of both slowly (ω ≤ 2 rev/day) and quickly (ω = 8–11 rev/day) rotating objects is formed due to the asteroids with sizes smaller than those in the middle of the distribution, which agrees well with the influence of the YORP effect. Moreover, the dependence of the axial rotation velocity of NEAs on the relative insolation shows that, for the NEAs referred to, both excesses are found in orbits where, on average, they receive 8–10% more solar energy than the NEAs in the middle of the distribution. This result also agrees with the character of the influence of the YORP effect and can be considered as an additional argument in its support. Thus, the study showed that one can infer that the currently available observational data suggest the possible influence of the YORP effect on the axial rotation of the near-Earth asteroids having sizes of D ～ 2 km and less. This is the first attempt to find the influence of the YORP effect on the axial rotation of the NEA family as a whole.     

Capture of Phobos and Deimos by photoatmospheric drag

It is suggested that Phobos and Deimos are carbonaceous asteroids captured by drag in an extended protoatmosphere of solar composition. The time scales for regularization of the orbital parameters are estimated, and found to be of the order of a few years. The atmosphere is modeled as a slowly-rotating condensation in the solar nebula; the surface pressure should be a few tenths of a bar. Capture and evolution by such an atmosphere are found to be improbable. The odds are greatly improved if the atmosphere is rapidly rotating or if the pressure is 1 to 2 orders of magnitude greater. Escape of the atmosphere, after removal of the nebular pressure, takes a few years, depending on the solar heat input. But it relaxes much more quickly to a state with negligible density at satellite altitudes. This relaxation is taken as the event that leaves the satellites in stable orbits. Previous candidates presumably were added to the solid body of Mars, and later ones were not captured.     

Sweeping of the Jovian resonances and the evolution of the asteroids

During the formation of the solar system the variation of the gravitational field produced by removal of a nebula with its moderately massive accretion disk led to sweeping of the Jovian commensurability resonances through the asteroid zone. This process produced increased eccentricities and random velocities of the early planetesimals which resulted in collisional comminution rather than accretion. The existence of the asteroids, their low mass density, and their high relative velocities are interpreted as due to disruption of the accretion processes of the terrestrial planets by the influence of Jupiter.     

Current important topics in solar system dynamics

This is a discussion and personal views on current important topics and methods in solar system dynamics. The topics include dynamical models, orbit resonance, planetary rings, chaos and secular evolution, motion of near-earth asteroids, the Kuiper belt, gravitational theory of the solar system and other related problems.     

Populating the asteroid belt from two parent source regions due to the migration of giant planets—“The Grand Tack”

Abstract– The asteroid belt is found today in a dramatically different state than that immediately following its formation. It is estimated that it has been depleted in total mass by a factor of at least 1000 since its formation, and that the asteroids’ orbits evolved from having near‐zero eccentricity and inclination to the complex distributions we find today. The asteroid belt also hosts a wide range of compositions, with the inner regions dominated by S‐type and other water‐poor asteroids and the outer regions dominated by C‐type and other primitive asteroids. We discuss a model of early inner solar system evolution whereby the gas‐driven migration of Jupiter and Saturn brings them inwards to 1.5 AU, truncating the disk of planetesimals in the terrestrial planet region, before migrating outwards toward their current locations. This model, informally titled “The Grand Tack,” examines the planetary dynamics of the solar system bodies during the final million years of the gaseous solar nebula lifetime—a few million years (Myr) after the formation of the first solids, but 20–80 Myr before the final accretion of Earth, and approximately 400–600 Myr before the Late Heavy Bombardment of the inner solar system. The Grand Tack attempts to solve some outstanding problems for terrestrial planet formation, by reproducing the size of Mars, but also has important implications for the asteroid population. The migration of Jupiter causes a very early depletion of the asteroid belt region, and this region is then repopulated from two distinct source regions, one inside the formation region of Jupiter and one between and beyond the giant planets. The scattered material reforms the asteroid belt, producing a population the appropriate mass, orbits, and with overlapping distributions of material from each parent source region.     

地月DRO星载光学测量近地小行星轨道确定

针对地基光学监测系统对近地小行星在近太阳方向的监测存在盲区的问题,提出了远距离逆行轨道(Distant Retrograde Orbit,DRO)天基光学平台对近地小行星进行跟踪定轨的方法.通过可视性分析,筛选仿真观测数据,利用美国宇航局喷气推进实验室(Jet Propulsion Laboratory,JPL)公布的小行星初始轨道信息对不同轨道类型的目标天体进行轨道确定,将计算结果与参考轨道对比分析.仿真结果表明:在测量精度2角秒,定轨弧长3年的情况下,DRO平台对仿真算例中所选择的近地小行星的定轨精度可以达到几十公里量级,其中Atira型轨道精度可达10公里以内.由此可见,DRO天基平台对近地小行星具有较好的监测能力,定轨精度能实现对目标小行星的精确跟踪,并对其进行轨道预报.     

Problems of origin of asteroids and comets

Abstract— Various hypotheses of the origin of asteroids and comets are briefly discussed. Interaction of planetesimals in the asteroid zone (AZ) with the gas, their perturbations by proto-Jupiter, and sweeping them out by more massive Jupiter zone bodies when they penetrated the AZ are considered. If the gas was turbulent, it could prevent a settling of dust particles to the equatorial plane of the disk and formation of dust condensations due to gravitational instability. Then particles grew by sticking upon collision. Gas moved radially due to turbulent viscosity and its dissipation. Small particles moved more-or-less together with the gas. As a result of gas drag, larger particles and bodies moved relative to the gas in the direction of increasing gas pressure. Gas would remove much of the solid material from the AZ if most bodies larger than a few km disintegrated by collisions into fragments smaller than a few tens of meters. Most of these fragments would then move into the Martian zone, and the small mass of Mars would have no explanation. Resonant perturbations of asteroids by Jupiter are discussed. In the model of a small mass disk they could scan through the asteroid belt due to changes in Jupiter's distance from the Sun that occurred when this planet accreted the gas and ejected the bodies from the solar system. Such a scanning considerably accelerated the removal of asteroids from the AZ. Massive Jupiter zone bodies with large orbital eccentricities that crossed the AZ were probably efficient at sweeping out bodies. Larger bodies increased the random velocities of the remaining asteroids at close encounters to the present values ～ 5 km/s. Restrictions on the runaway growth of giant planets, on the relative velocities of bodies and the disk surface density that follow from the consideration of the origin of the asteroid belt and the cometary cloud are considered.     

The missing planet

Ovenden's hypothesis suggesting former existence of a planet of 90 Earth masses which supposedly filled the Titius-Bode gap in the asteroid belt and then suddenly disappeared 16 million years ago, is critically examined by the morphological method. It is shown that an explosive removal, however improbable, could have led to the formation of the asteroids from a non-explosive core (the nuclear charge being placed outside of it), but that life on Earth would have been completely destroyed by three successive blasts-one from the direct impact of the ejecta of the planet, another from the increased radiation suddenly emitted by the Sun when hit by the ejecta, and a third one (arriving, however, first) from the radiation emitted by the nuclear explosion. The geological record of the continuity of Life on Earth for the past 10⁹ years definitely excludes the possibility of such an explosion in the late Tertiary.The other mode of removal of the planet-in a gravitational encounter with an intruder either from interstellar space or from the unexplored outskirts of the solar system, under the condition of not having disturbed the existing regularity of planetary orbits-is not only extremely improbable, to be expected once during 100 million times the age of the solar system; but it would leave no asteroids behind, all of the previously existing primaeval asteroids having been rapidly eliminated in encounters with the hypothetical planet.Whatever the merits of Ovenden's long-range calculations of the secular perturbations of coplanar circularized planetary orbits, the hypothesis of a massive planet to have existed in the asteroidal region and then recently to have suddenly disappeared, belongs to the realm of the impossible. After such a hypothetical event, either we would not be here on Earth, or there would be no asteroids in their present place between Jupiter and Mars.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May 1978.     

The hierarchical stability of the seven known large size ratio triple asteroids using the empirical stability parameters

In this study, the hierarchical stability of the seven known large size ratio triple asteroids is investigated. The effect of the solar gravity and primary’s J ₂ are considered. The force function is expanded in terms of mass ratios based on the Hill’s approximation and the large size ratio property. The empirical stability parameters are used to examine the hierarchical stability of the triple asteroids. It is found that the all the known large size ratio triple asteroid systems are hierarchically stable. This study provides useful information for future evolutions of the triple asteroids.     

Orbital resonance in a dissipative medium

Orbital resonances tend to force bodies into noncircular orbits. If a body is also under the influence of an eccentricity-reducing medium, it will experience a secular change in semimajor axis which may be positive or negative depending on whether its orbit is exterior or interior to that of the perturbing body. Thus a dissipative medium can promote either a loss or a gain in orbital energy. This process may explain the resonant structure of the asteroid belt and of Saturn's rings. For reasonable early solar system parameters, it would clear a gap near the 2:1 resonance with Jupiter on a time scale of a few thousand years; the gap width would be comparable to the Kirkwood gap presently at the location in the asteroid belt. Similarly, a gap comparable in width to Cassini's division would be cleared in Saturn's rings at the 2:1 resonance with Mimas in ～10⁶ yr. Most of the material from the gap would be deposited at the outer edge of ring B. The process would also affect the radial distribution of preplanetary material. Moreover, it provides an explanation for the large amplitude of the Titan-Hyperion libration. Consideration of the effects of dissipation on orbits near the stable L₄ and L₅ points of the restricted three-body problem indicates that energy loss causes particles to move away from these points. This results explains the large amplitude of Trojan asteroids about these points and the possible capture of Trojan into orbit about Jupiter.     

ADVANTAGES OF SEARCHING FOR ASTEROIDS FROM LOW EARTH ORBIT: THE NEOSSat MISSION

Space-based observatories have several advantages over ground-based observatories in searching for asteroids and comets. In particular, the Aten and Interior to Earth’s Orbit (IEO) asteroid classes may be efficiently sought at low solar elongations along the ecliptic plane. A telescope in low Earth orbit has a sufficiently long orbital baseline to determine the parallax for all Aten and IEO class asteroids discovered with this observing strategy. The Near Earth Object Space Surveillance Satellite (NEOSSat) mission will launch a microsatellite to exploit this observing strategy complementing ground-based search programmes.