Light pressure effect on the orbital evolution of objects moving in the neighborhood of low-order resonances

The hardware-software complex of the SBG telescope of the Kourovka astronomical observatory of the Ural Federal University is described. The area-to-mass ratio of high-orbit objects was estimated from the results of positional observations at the SBG telescope. Positions and sizes of resonance zones are determined from the numerical simulation results depending on the area-to-mass ratio in the case of low-order resonances (1 : 1, 1 : 2, and 1 : 3). Secular perturbations of semimajor axes of orbits, caused by the Poynting-Robertson effect, are estimated in the neighborhood of low-order resonance zones at different area-to-mass ratios.     

On Stable Chaos in the Asteroid Belt

The twenty most chaotic objects found among first hundred of numbered asteroids are studied. Lyapunov time calculated with and without inner planets indicates that for eleven of those asteroids the strongest chaotic effect results from the resonances with Mars. The filtered semimajor axis displays an abrupt variation only when a close approach to Mars takes place. The study of the behaviour of the critical argument for candidate resonances can reveal which is responsible for the semimajor axis variation. We have determined these resonances for the asteroids in question. For the asteroids chaotic even without the inner planets we have determined the most important resonances with Jupiter, or three-body resonances. This revised version was published online in July 2006 with corrections to the Cover Date.     

Investigation of Light Pressure Influence on Dynamics of Near-Earth Objects in Resonant Orbits

The results of an analysis of changes in resonant orbital dynamics of near-Earth objects, which are caused by light pressure, are presented. We describe features of the orbital evolution of the objects subjected to the combined effect of secular resonances and light pressure. It was shown that the transfer of objects to highly elliptical and hyperbolic orbits is possible when the object’s area-to-mass ratio is large.     

Identification of Asteroids in Two-Body Resonances

A catalog of asteroids in two-body orbital resonances with the planets of the Solar System has been created. The AstDyS database was a source of the input data, and all the numbered objects (467303 objects at the time of the analysis) were considered. The orbits were integrated in the framework of a pure gravitational problem considering all the planets of the Solar System and Pluto. From the analysis of the behavior of the resonant argument and the semimajor axis on the 100-kyr interval, the resonance membership and the libration type (pure or transient) were verified for each of the asteroids. A more accurate method to identify the resonant argument librations was developed on the basis of the analysis of mutual periodograms. We found 23251 resonant asteroids, ~36% of which (8397 objects) are in pure resonances.     

Reviewing the Yarkovsky effect: New light on the delivery of stone and iron meteorites from the asteroid belt

Abstract— We give a nonmathematical review of recent work regarding the Yarkovsky effect on asteroidal fragments. This effect may play a critical, but underappreciated, role in delivering meteorites to Earth. Two variants of the effect cause drifts in orbital elements, notably semimajor axes. The “classic” or “diurnal” Yarkovsky effect is associated with diurnal rotation at low obliquity. More recently, a “seasonal” effect has also been described, associated with high obliquity. Studies of these Yarkovsky effects are combined with studies of resonance effects to clarify meteorite delivery. If there were no Yarkovsky drift, asteroid fragments could reach a resonance only if produced very near that resonance. However, objects in resonances typically reach Earth-crossing orbits within a few million years, which is inconsistent with stone meteorites' cosmic-ray exposure (CRE) ages (5–50 Ma) and iron meteorites' CRE ages (100–1000 Ma). In the new view, on the other hand, large objects in the asteroid belt are “fixed” in semimajor axis, but bodies up to 100 m in diameter are in a constant state of mixing and flow, especially if the thermal conductivity of their surface layers is low. Thus, small asteroid fragments may reach the resonances after long periods of drift in the main belt. Yarkovsky drift effects, combined with resonance effects, appear to explain many meteorite properties, including: (1) the long CRE ages of iron meteorites (due to extensive drift lifetimes in the belt); (2) iron meteorites' sampling of numerous parent bodies; (3) the shorter CRE ages of most stone meteorites (due to faster drift, coupled with weaker strength and more rapid collisional erosion); and (4) the abundance of falls from discrete impact events near resonances, such as the 8 Ma CRE age of H chondrites. Other consequences include: the delivery of meteorite parent bodies to resonances is enhanced; proportions of stone and iron meteorites delivered to Earth may be different from the proportions at the same sizes left in the belt, which in turn may differ from the ratio produced in asteroidal collisions; Rabinowitz's 10–100 m objects may be preferentially delivered to near-Earth space; and the delivery of C-class fragments from the outer belt may be inhibited, compared to classes in other parts of the belt. Thus, Yarkovsky effects may have important consequences in meteoritics and asteroid science.     

A secular theory of coplanar, non-resonant planetary system

We present the secular theory of coplanar N -planet system, in the absence of mean motion resonances between the planets. This theory relies on the averaging of a perturbation to the two-body problem over the mean longitudes. We expand the perturbing Hamiltonian in Taylor series with respect to the ratios of semimajor axes which are considered as small parameters, without direct restrictions on the eccentricities. Next, we average out the resulting series term by term. This is possible thanks to a particular but in fact quite elementary choice of the integration variables. It makes it possible to avoid Fourier expansions of the perturbing Hamiltonian. We derive high-order expansions of the averaged secular Hamiltonian (here, up to the order of 24) with respect to the semimajor axes ratio. The resulting secular theory is a generalization of the octupole theory. The analytical results are compared with the results of numerical (i.e. practically exact) averaging. We estimate the convergence radius of the derived expansions, and we propose a further improvement of the algorithm. As a particular application of the method, we consider the secular dynamics of three-planet coplanar system. We focus on stationary solutions in the HD 37124 planetary system.     

Theory of satellite orbit-orbit resonance

On the basis of the strong mathematical and physical parallels between orbit-orbit and spin-orbit resonances, the dynamics of mutual orbit perturbations between two satellites about a massive planet are examined, exploiting an approach previously adopted in the study of spin-orbit coupling. The satellites are assumed to have arbitrary mass ratio and to move in non-intersecting orbits of arbitrary size and eccentricity. Resonances are found to exist when the mean orbital periods are commensurable with respect to some rotating axis, which condition also involves the apsidal and nodal motions of both satellites. In any resonant state the satellites are effectively trapped in separate potential wells, and a single variable is found to describe the simultaneous librations of both satellites. The librations in longitude are 180° out-of-phase, with fixed amplitude ratio that depends only on their relative masses and semimajor axes. At the same time the stroboscopic longitude of conjunction also librates about the commensurate axis with the same period. The theory is applicable to Saturn's resonant pairs Titan-Hyperion and Mimas-Tethys, and in these cases our calculated libration periods are in reasonably good agreement with the observed periods.This research supported under a grant from the California Institute of Technology President's Fund and NASA Contract NAS 7-100.     

Natural highways for end-of-life solutions in the LEO region

We present the main findings of a dynamical mapping performed in the Low Earth Orbit region. The results were obtained by propagating an extended grid of initial conditions, considering two different epochs and area-to-mass ratios, by means of a singly averaged numerical propagator. It turns out that dynamical resonances associated with high-degree geopotential harmonics, lunisolar perturbations and Solar radiation pressure can open natural deorbiting highways. For area-to-mass ratios typical of the orbiting intact objects, these corridors can be exploited only in combination with the action exerted by the atmospheric drag. For satellites equipped with an area augmentation device, we show the boundary of application of the drag, and where the Solar radiation pressure can be exploited.     

An analysis of the region of the Phocaea dynamical family

In this work, I conduct a preliminary analysis of the Phocaea family region. I obtain families and clumps in the space of proper elements and proper frequencies, study the taxonomy of the asteroids for which this information is available, analyse the albedo and absolute magnitude distribution of objects in the area, obtain a preliminary estimate of the possible family age, study the cumulative size distribution and collision probabilities of asteroids in the region, the rotation rate distribution and obtain dynamical map of averaged elements and Lyapunov times for grids of objects in the area.
Among my results, I identified the first clump visible only in the frequency domain, the (6246) Komurotoru clump, obtained a higher limit for the possible age of the Phocaea family of 2.2 Byr, identified a class of Phocaea members on Mars-crossing orbits characterized by high Lyapunov times and showed that an apparently stable region on time-scales of 20 Myr near the  ν₆  secular resonance is chaotic, possibly because of the overlapping of secular resonances in the region. The Phocaea dynamical group seems to be a real S-type collisional family, formed up to 2.2 Byr ago, whose members with a large semimajor axis have been dynamically eroded by the interaction with the local web of mean-motion and secular resonances. Studying the long-term stability of orbits in the chaotic regions and the stability of family and clumps identified in this work remain challenges for future works.     

Additional criteria for identifying the asteroid families and confirmation of the effect of spatial separation of family members according to their albedos

The D(a) distribution of asteroid sizes by their semimajor axes and the N(p) distribution of the number of asteroids by their albedo values for individual families are used to isolate the asteroid families more clearly. The families identified by Masiero et al. (2013) are analyzed with the use of these distributions, and correctly and incorrectly isolated families are found. A reduction in the mean albedo with increasing semimajor axis is observed for almost all correctly identified families that are not truncated by resonances. This reduction is statistically significant for the majority of these families. Not a single family exhibits a statistically significant increase in albedo. This confirms our previous conclusions that a nongravitational effect acting in the asteroid belt results in the spatial separation of asteroids with different albedos.     

Tidal evolution of close binary asteroid systems

We provide a generalized discussion of tidal evolution to arbitrary order in the expansion of the gravitational potential between two spherical bodies of any mass ratio. To accurately reproduce the tidal evolution of a system at separations less than 5 times the radius of the larger primary component, the tidal potential due to the presence of a smaller secondary component is expanded in terms of Legendre polynomials to arbitrary order rather than truncated at leading order as is typically done in studies of well-separated system like the Earth and Moon. The equations of tidal evolution including tidal torques, the changes in spin rates of the components, and the change in semimajor axis (orbital separation) are then derived for binary asteroid systems with circular and equatorial mutual orbits. Accounting for higher-order terms in the tidal potential serves to speed up the tidal evolution of the system leading to underestimates in the time rates of change of the spin rates, semimajor axis, and mean motion in the mutual orbit if such corrections are ignored. Special attention is given to the effect of close orbits on the calculation of material properties of the components, in terms of the rigidity and tidal dissipation function, based on the tidal evolution of the system. It is found that accurate determinations of the physical parameters of the system, e.g., densities, sizes, and current separation, are typically more important than accounting for higher-order terms in the potential when calculating material properties. In the scope of the long-term tidal evolution of the semimajor axis and the component spin rates, correcting for close orbits is a small effect, but for an instantaneous rate of change in spin rate, semimajor axis, or mean motion, the close-orbit correction can be on the order of tens of percent. This work has possible implications for the determination of the Roche limit and for spin-state alteration during close flybys.     

Dynamical evolution of NEAs: Close encounters, secular perturbations and resonances

We discuss the main mechanisms affecting the dynamical evolution of Near-Earth Asteroids (NEAs) by analyzing the results of three numerical integrations over 1 Myr of the NEA (4179) Toutatis. In the first integration the only perturbing planet is the Earth. So the evolution is dominated by close encounters and looks like a random walk in semimajor axis and a correlated random walk in eccentricity, keeping almost constant the perihelion distance and the Tisserand invariant. In the second integration Jupiter and Saturn are present instead of the Earth, and the 3/1 (mean motion) and v ₆ (secular) resonances substantially change the eccentricity but not the semimajor axis. The third, most realistic, integration including all the three planets together shows a complex interplay of effects, with close encounters switching the orbit between different resonant states and no approximate conservation of the Tisserand invariant. This shows that simplified 3-body or 4-body models cannot be used to predict the typical evolution patterns and time scales of NEAs, and in particular that resonances provide some “fast-track” dynamical routes from low-eccentricity to very eccentric, planet-crossing orbits.     

The positions of secular resonance surfaces

The surfaces for the three strongest secular resonances have been located as a function of proper semimajor axis, eccentricity, and inclination for semimajor axes between 1.25 and 3.5 AU. The results are presented graphically. The ν₅ resonance only occurs at high inclinations (?23°). The ν₆ resonance passes through both the main belt and Mars-crossing space. The ν₁₆ resonance starts near the inner edge of the belt and, at low inclinations at least, folds around a portion of the Mars-crossing space until it runs nearly parallel with the Earth-crossing boundary.     

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.     

Determination of the area and mass distribution of orbital debris fragments

An important factor in modeling the orbital debris environment is the loss rate of debris due to atmospheric drag and luni/solar perturbations. An accurate knowledge of the area-to-mass ratio of debris fragments is required for the calculation of the effect of atmospheric drag. In general, this factor is unknown and assumed values are used. However, this ratio can be calculated for fragments for which changes in the orbital elements due to atmospheric drag as a function of time are known. This is the inverse of the technique used to determine the atmospheric density from the decay of satellites with accurately known area-to-mass ratios. These kinds of propagation programs are routinely used in predicting the decay of an orbiting vehicle. In this work the area-to-mass ratio of about 2600 fragments arising from the breakup of 24 artificial satellites have been determined. An analysis of the data on about 200 objects (rocket bodies, scientific satellites, etc.) with known mass, size, and shape has also been made. The value of the radar cross-section (RCS), as measured by the Eglin radar operating at 70 cm wavelength, has been correlated to the effective area of these objects. The measurements of the area-to-mass ratio of these objects then provide a calibration of the actual to the calculated mass. It has been shown that the debris mean mass, m, is related to the mean effective area, A, by a power law relation, m = k A ^1.86. However, for a given effective area the mass distribution is very broad. Moreover, the cumulative mass distribution, N(>m), can be expressed as N(>m) = D(m + b), where D, b, and c are constants. The asymptotic slope, c, of low intensity explosions is on the average lower than the slope for high intensity explosions, but there is considerable spread of this slope in each class. Part of the flattening, as indicated by the finite value of the parameter, b, can be understood as arising out of the spread in the RCS values due to the tumbling motion of the fragments and effects related to the detectability of the fragment by the Eglin radar. It has been established that the mass in a given breakup calculated using this technique is in good agreement with the expected mass value. These results can be used in modeling the breakups of other artificial earth satellites and safety analysis.     

The origin of TNO 2004 XR190 as a primordial scattered object

Numerical integrations of the equations of motion of the giant planets and scattering particles show that there is a possible orbital itinerary that a particle may follow from a scattering mode up to a stable position near the orbit of 2004 XR₁₉₀. This orbital evolution requires that the particle gets trapped in a mean motion resonance with Neptune coupled with the Kozai resonance. Imposing migration on Neptune while a particle is experiencing both resonances can entail an escape from resonance at a low particle’s eccentricity. This eccentricity and the associated inclination are always similar to those of 2004 XR₁₉₀. I conclude that 2004 XR₁₉₀ was most likely a scattered object that went through those resonance processes and was eventually deposited at its current position. By the same argument, it is expected that there must exist several other objects with similar semimajor axis, eccentricity and inclination as those of 2004 XR₁₉₀.     

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.     

East-west stationkeeping requirements of nearly synchronous satellites due to Earth's triaxiality and luni-solar effects

A closed form solution, for longitude and semimajor axis deviations in the neighborhood of a prespecified station, is obtained for nearly synchronous satellites. The model use includes the important terms in Earth's zonal and tesseral harmonics as well as the luni-solar perturbations. The initial semimajor axis for two-maneuver east-west stationkeeping is then deduced. Due to the luni-solar effects, it is found that the initial semimajor axis deviation from synchronous orbit value is highly dependent on the initial position of the satellite relative to the Moon and the Sun. Verifications of the results by means of numerical integrations are also included.     

Chaotic zone boundary for low free eccentricity particles near an eccentric planet

We consider particles with low free or proper eccentricity that are orbiting near planets on eccentric orbits. Through collisionless particle integration, we numerically find the location of the boundary of the chaotic zone in the planet's corotation region. We find that the distance in semimajor axis between the planet and boundary depends on the planet mass to the 2/7 power and is independent of the planet eccentricity, at least for planet eccentricities below 0.3. Our integrations reveal a similarity between the dynamics of particles at zero eccentricity near a planet in a circular orbit and with zero free eccentricity particles near an eccentric planet. The 2/7th law has been previously explained by estimating the semimajor at which the first-order mean motion resonances are large enough to overlap. Orbital dynamics near an eccentric planet could differ due to first-order corotation resonances that have strength proportional to the planet's eccentricity. However, we find that the corotation resonance width at low free eccentricity is small; also the first-order resonance width at zero free eccentricity is the same as that for a zero-eccentricity particle near a planet in a circular orbit. This accounts for insensitivity of the chaotic zone width to planet eccentricity. Particles at zero free eccentricity near an eccentric planet have similar dynamics to those at zero eccentricity near a planet in a circular orbit.     

Detailed survey of the phase space around Nix and Hydra

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto–Charon system. The spatial elliptic restricted three-body problem was used as model and stability maps were created by chaos indicators. The orbital elements of the moons are in the stable domain on the semimajor axis, eccentricity and inclination spaces. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps. They do not contain the positions of the moons, confirming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericentre or longitude of node falls in a certain range. The results strongly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of pericentre or longitude of node.