Orbital Element Space Research Articles

The Asteroid Identification Problem: II. Target Plane Confidence Boundaries

The nominal orbit solution for an asteroid/comet resulting from a least squares fit to astrometric observations is surrounded by a region containing solutions equally compatible with the data, the confidence region. If the observed arc is not too short, and for an epoch close to the observations, the confidence region in the six-dimensional space of orbital elements is well approximated by an ellipsoid. This uncertainty of the orbital elements maps to a position uncertainty at close approach, which can be represented on a Modified Target Plane (MTP), a modification of the one used by Öpik. The MTP is orthogonal to the geocentric velocity at the closest approach point along the nominal orbit. In the linear approximation, the confidence ellipsoids are mapped on the MTP into concentric ellipses, computed by solving the variational equation. For an object observed at only one opposition, however, if the close approach is expected after many revolutions, the ellipses on the MTP become extremely elongated, therefore the linear approximation may fail, and the confidence boundaries on the MTP, by definition the nonlinear images of the confidence ellipsoids, may not be well approximated by the ellipses. In theory the Monte Carlo method by Muinonen and Bowell (1993, Icarus104, 255–279) can be used to compute the nonlinear confidence boundaries, but in practice the computational load is very heavy. We propose a new method to compute semilinear confidence boundaries on the MTP, based on the theory developed by Milani (1999, Icarus137, 269–292) to efficiently compute confidence boundaries for predicted observations. This method is a reasonable compromise between reliability and computational load, and can be used for real time risk assessment.These arguments can be applied to any small body approaching any planet, but in the case of a potentially hazardous object (PHO), either an asteroid or a comet whose orbit comes very close to that of the Earth, the application is most important. We apply this technique to discuss the recent case of asteroid 1997 XF11, which, on the basis of the observations available up to March 11, 1998, appeared to be on an orbit with a near miss of the Earth in 2028. Although the least squares solution had a close approach at 1/8 of the lunar distance, the linear confidence regions corresponding to acceptable size of the residuals are very elongated ellipses which do not include collision; this computation was reported by Chodas and Yeomans. In this paper, we compute the semilinear confidence boundaries and find that they agree with the results of the Monte Carlo method, but differ in a significant way from the linear ellipses, although the differences occur only far from the Earth. The use of the 1990 prediscovery observations has confirmed the impossibility of an impact in 2028 and reduces the semilinear confidence regions to subsets of the regions computed with less data, as expected. The confidence regions computed using the linear approximation, on the other hand, do not reduce to subsets of the regions computed with less data. We also discuss a simulated example (Bowell and Muinonen 1992, Bull. Am. Astron. Soc.24, 965) of an Earth-impacting asteroid. In this hypothetical case the semilinear confidence boundary has a completely different shape from the linear ellipse, and indeed for orbits determined with only few weeks of observational data the semilinear confidence boundary correctly includes possible collisions, while the linear one does not. Free software is available now, allowing everyone to compute target plane confidence boundaries as in this paper; in case a new asteroid with worrisome close approaches is discovered, our method allows to quickly perform an accurate risk assessment.

The Asteroid Identification Problem: I. Recovery of Lost Asteroids

When an asteroid is lost, the observers need to know the portion of the celestial sphere where it could be recovered at a given time. This region is an image of the region, in the space of orbital elements, where the orbit is compatible with the previous observations. The map between these two regions is nonlinear; therefore the classical linear approximation can fail. Indeed it fails by a large amount when both these regions are large, which is precisely when an asteroid has been observed only over a short arc and/or it has been lost for a long time. The recovery, and identification, of asteroids long lost is very difficult if the only available prediction is a single point corresponding to the least squares solution, which could be very far from the real state; thus the availability of an efficient algorithm to bound the recovery region is essential, also to decide if the recovery is worth the effort. This paper proposes three new algorithms to better approximate the recovery region based upon approximations going beyond linearization. It gives the results of tests based upon asteroids which have been recovered by chance and could have been found in the recovery region computed by the new algorithms. Free software is available now, by means of which the new algorithms can be tested, and eventually adopted, by the observers and by the ephemerides computation centers.

Target Plane Confidence Boundaries: Mathematics of The 1997 XF11 Scare

The uncertainty of the close approach distance of a Potentially Hazardous Object (PHO), either an asteroid or a comet, can be represented on the Modified Target Plane (MTP), a modification of the one used by Öpik. The MTP is orthogonal to the geocentric velocity at the closest approach along the nominal orbit, solution of the least square fit to the observations. The confidence regions of this solution in the 6-D space of orbital elements (for an epoch close to the observations) are well approximated by a family of concentric ellipsoids, if the observed arc is not too short. In the linear approximation these ellipsoids are mapped on the MTP into concentric ellipses, which can be computed by solving for the state transition matrix.For a PHO observed at only one opposition, with a close approach expected after many revolutions, the ellipses on the MTP become extremely elongated and the linear approximation may fail. In this case the confidence boundaries on the MTP, i.e. the nonlinear images of the confidence ellipsoids, may not be well approximated by the ellipses. The Monte Carlo method (Muinonen and Bowell, 1993) can be used to find nonlinear confidence regions, but the computational load is very heavy: to estimate a low probability event the number of test cases must be larger than the inverse of the probability. We propose a new method to compute semilinear confidence boundaries on the MTP (Milani and Valsecchi, 1998), based on the theory developed to compute confidence boundaries for predicted observations (Milani, 1999). This method is a good compromise between reliability and computational load, and can be used for real time risk assessment.

Sources of Planetary Rotation: Mapping Planetesimals' Contributions to Angular Momentum

A systematic study of the motion backward and forward in time from a uniform distribution of bodies entering a planet's Hill sphere yields a mapping of the contribution to the planet's rotation from planetesimals as a function of their source in heliocentric orbital element space. The mapping allows determination of the contribution to rotation from any assumed distribution of heliocentric planetesimal orbits. For example, it can systematically reproduce earlier results from Monte-Carlo studies of contributions to rotational angular momentum. With our method of calculation, contributions to planetary rotation can be interpreted in terms of the arrival geometries at the planet's Hill sphere and the subsequent two-body motion inside the Hill sphere leading to impact. Prograde rotation is strongly favored if a planet grows in a relatively quiescent population of planetesimals with accretion nibbling at the edges of its feeding zone. However, if the impacting population were dominated by large bodies with high relative velocity, the direction and magnitude of rotation would be random.

Surface Properties of (6) Hebe: A Possible Parent Body of Ordinary Chondrites

We report for the first time rotationally resolved spectroscopic observations, as well as new photometric and polarimetric measurements, of the large S-type asteroid (6) Hebe, always at near-equatorial aspects. We have found evidence for only minor variations in Hebe's visible reflectance spectrum over a rotational cycle, comparable to our measurement accuracy (a few percent) and consistent with an undifferentiated silicate assemblage on the surface. We have also confirmed previous results of the existence of small polarization changes, but they are not correlated with the (complex and asymmetric) photometric lightcurve of the asteroid. A plausible interpretation of these data is that Hebe's surface is composed of undifferentiated materials, but presents texture, albedo, and spectral changes possibly related to the occurrence of energetic cratering events. Given the comparatively large size of Hebe, its proximity to resonance-related chaotic zones of the orbital element space (Farinellaet al. 1993,Celest. Mech.56,287–305, andIcarus101,174–187), and its assignment by Gaffeyet al. (1993,Icarus106,573–602) to the S(IV) taxonomic subclass, our results support the idea that this asteroid may be the source of a significant fraction of the ordinary chondrite meteorites.

Meteoritic, Asteroidal, and Theoretical Constraints on the 500 Ma Disruption of theLChondrite Parent Body

The high abundance of heavily shocked and degassedLchondrites with Ar–Ar ages around 500 Myr shows that theLchondrite parent body suffered a major impact 500 Myr ago. We infer from constraints on the thermal evolution of impact heated rocks after the 500-Myr event and the high abundance of shockedLchondrites that the parent body was catastrophically disrupted. The slow cooling rates of some shocked and degassedLchondrites (0.01–1°C year−1) show that they were derived from kilometer-sized impact-heated fragments or rubble piles that were ejected from near the impact point.We suggest that the catastrophic dispersion of the parent body provided some fragments with sufficiently high velocities to put them into resonances and that this initiated the orbital evolution which resulted in the high flux ofLchondrite meteoroids impacting the Earth at present. It appears likely that this was a “slow-track” type of dynamical evolution, with most objects avoiding drastic resonant changes of orbital eccentricity, and undergoing a slow random walk in orbital element space, driven by a sequence of random encounters with Mars and, later on, with the Earth. The ν6secular resonance provides a plausible mechanism to start this evolution, since fragments inserted into it in the main belt frequently have their eccentrici_ties raised to values of about 0.4, sufficient for Mars-crossing but not for Earth-crossing orbits; on the other hand, recent numerical work has shown that the 3:1 mean motion resonance leads almost always to a fast-track evolution, ending up with a solar collision or a hyperbolic ejection within a few Myr.

An Asteroid on the Brink

Asteroid 2953 Vysheslavia, a Koronis family member about 15 km in diameter, very close to the outer edge of the 5/2 Kirkwood gap, has been shown by numerical integrations to fall into the resonance; then it approaches Jupiter and ends up into a hyperbolic orbit. The typical dynamical lifetime is of the order of 10 Myr. This "marginally unstable" region of the orbital element space bordering the 5/2 resonance is only ≈ 10-3 AU wide in semimajor axis. Thus Vysheslavia has stayed in its current "dangerous" location in orbital element space for a time much shorter than the age of the Solar System. We discuss several hypotheses on the origin of this object, including a comparatively recent collision in the Koronis family and the possible implantation into a quasi-stable main-belt orbit of a stray body on a comet-type orbit. We conclude that physical observations are needed to discriminate among these possibilities.

The evolution of the zodiacal dust cloud under plasma drag and lorentz forces in the latitudinally asymmetric solar wind

Orbital evolutions of zodiacal dust particles are mainly influenced by electromagnetic and by plasma Poynting-Robertson effects. Whereas the first effect acts radially symmetric, the second one depends on heliographic latitude caused by the dynamical friction of the dust particles in the ambient asymmetric solar wind flow. The solar wind mass and momentum flows vary with heliographic latitude by about 50% from the ecliptic towards the poles. Here we take into account related asymmetries in the plasma Poynting-Robertson drag. In addition, we derive formulae for a competing effect due to systematic Lorentz forces and to Lorentz scatterings of the charged dust particles at the global interplanetary magnetic field. We describe the orbital element changes occurring under these conditions. Whereas Lorentz forces by means of induced inclination drifts only cause negligible evolutionary effects, we can show that the radial migration rates due to asymmetric plasma friction forces are greater by about 5–10% for particles close to the ecliptic compared to those at high inclinations. This causes a typical variation of the inclination distribution with decreasing semimajor axis. In addition, it reveals that the dust distribution function cannot be factorized into parts solely dependent on radial distance or on inclination. We also discuss the kinetic equation for the distribution function of dust particles, and find the distribution in orbital element space from which the spatial density of the particles can then be found as an integral over this distribution function after application of a Jacobian transformation of element space differentials into configuration space differentials. Some observational consequences are drawn from this comparison and are discussed in some more detail.

Evolution of Dust Particle Orbits under the Influence of Solar Wind Outflow Asymmetries and the Formation of the Zodiacal Dust Cloud

The secular orbital evolution of zodiacal dust particles is mainly influenced by the electromagnetic and the plasma Poynting-Robertson effects. Whereas the first effect is radially symmetric, the second one, caused by the dynamical friction of the dust particles in the ambient solar wind flow, depends on heliographic latitude. This is because the solar wind appears to have pronounced mass and momentum flow variations with respect to heliographic latitudes, with mass flows decreasing by about 50% from the ecliptic toward the poles. We take into account related asymmetries in the plasma Poynting-Robertson effect and study orbital evolution of dust particles orbiting with different inclinations. We derive equations describing the secular changes of the orbital elements under these conditions. We show that the radial migration rates are greater by about 5 to 10% for particles close to the ecliptic as compared to those at higher inclinations. This leads to typical changes in the distribution of inclinations with decreasing semimajor axes of the particles and also shows that the dust distribution function cannot be factored into parts solely dependent on radial distance or inclination alone. Furthermore, the kinetic equation for the distribution function of dust particles is solved numerically, yielding the distribution in orbital element space. The spatial density of the particles can then be found as an integral over this distribution function after application of a Jacobian transformation of element space differentials into configuration space differentials and can be compared with earlier results. Some observational consequences are drawn from this comparison and are discussed in some more detail. It is also pointed out that the plasma Poynting-Robertson effect in a two-temperature solar wind produces normal components of perturbation forces and thus tends to change the inclination of the dust particle orbits. This is of special importance in the solar wind regime inside the critical point where low-Mach-number flows prevail.

Dynamics, interrelations and evolution of the systems of asteroids and comets

The dynamics of larger interplanetary bodies is reviewed, with emphasis on evolutionary problems, interrelations, and open questions. Observational biases distinguishing the sample of known objects from the whole population are briefly discussed. A schematic division of the solar system into zones with different regimes of motion, and a rough taxonomy of orbit types are attempted. The role of individual major planets in controlling the dynamical evolution of interplanetary objects, in particular by stabilizing resonances and destabilizing close encounters, is compared. There are significant evolution asymmetries due to boundary conditions and preferential evolutionary paths; individual major displacements in the phase space of orbital elements conserve the Tisserand invariant with respect to the planet responsible, thus favouring certain evolutionary sequences against others. Very limited lifetimes of some orbit types imply a long-term balance between source and sink, and require a continuous supply of objects from other types of orbits. In this respect, the ultimate fate of extinct comets is of particular interest. Under very specific conditions, nongravitational effects of mass loss can result in stabilization of a formerly unstable orbit. Since the dividing line between the two basic interplanetary populations distinguished by origin and composition — the asteroids and the comets — is essentially that between stable and unstable motion, orbital data can be used to specify which of the known asteroid-like objects may be devolatilized cometary nuclei.

Optimal low-thrust, limited power transfers between arbitrary elliptical orbits

This study is a generalization of Edelbaum's work on coplanar and on coaxial transfers. As before, the solution is valid only if the fixed transfer duration is sufficiently long to allow a large number of revolutions around the planet. The determination of the optimal trajectories in the orbital elements space is facilitated by the use of the notion of “mean Hamiltonian” and can be reduced to the solution of a differential system of order two, followed by two quadratures. This integration, as well as the determination of the conjugate point, generally necessitates a numerical treatment. The global optimum can be determined for the transfers starting from a circle and possibly for the transfers between symmetrical orbits.