Capture Probability in the 3:1 Mean Motion Resonance with Jupiter

We study the capture and crossing probabilities into the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100%, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (1995) and Quillen (2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations

On the dynamics of Extrasolar Planetary Systems under dissipation. Migration of planets

We study the dynamics of planetary systems with two planets moving in the same plane, when frictional forces act on the two planets, in addition to the gravitational forces. The model of the general three-body problem is used. Different laws of friction are considered. The topology of the phase space is essential in understanding the evolution of the system. The topology is determined by the families of stable and unstable periodic orbits, both symmetric and non symmetric. It is along the stable families, or close to them, that the planets migrate when dissipative forces act. At the critical points where the stability along the family changes, there is a bifurcation of a new family of stable periodic orbits and the migration process changes route and follows the new stable family up to large eccentricities or to a chaotic region. We consider both resonant and non resonant planetary systems. The 2/1, 3/1 and 3/2 resonances are studied. The migration to larger or smaller eccentricities depends on the particular law of friction. Also, in some cases the semimajor axes increase and in other cases they are stabilized. For particular laws of friction and for special values of the parameters of the frictional forces, it is possible to have partially stationary solutions, where the eccentricities and the semimajor axes are fixed.Comment: Accepted in Celestial Mechanics and Dynamical Astronom

Stable manifolds and homoclinic points near resonances in the restricted three-body problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem

The mass-period distribution of close-in exoplanets

The lower limit to the distribution of orbital periods P for the current population of close-in exoplanets shows a distinctive discontinuity located at approximately one Jovian mass. Most smaller planets have orbital periods longer than P~2.5 days, while higher masses are found down to P~1 day. We analyze whether this observed mass-period distribution could be explained in terms of the combined effects of stellar tides and the interactions of planets with an inner cavity in the gaseous disk. We performed a series of hydrodynamical simulations of the evolution of single-planet systems in a gaseous disk with an inner cavity mimicking the inner boundary of the disk. The subsequent tidal evolution is analyzed assuming that orbital eccentricities are small and stellar tides are dominant. We find that most of the close-in exoplanet population is consistent with an inner edge of the protoplanetary disk being located at approximately P>2 days for solar-type stars, in addition to orbital decay having been caused by stellar tides with a specific tidal parameter on the order of Q'*=10^7. The data is broadly consistent with planets more massive than one Jupiter mass undergoing type II migration, crossing the gap, and finally halting at the interior 2/1 mean-motion resonance with the disk edge. Smaller planets do not open a gap in the disk and remain trapped in the cavity edge. CoRoT-7b appears detached from the remaining exoplanet population, apparently requiring additional evolutionary effects to explain its current mass and semimajor axis.Comment: 8 Pages, 8 figures, accepted for publication in A&

Taxonomy of asteroid families among the Jupiter Trojans: Comparison between spectroscopic data and the Sloan Digital Sky Survey colors

We present a comparative analysis of the spectral slope and color distributions of Jupiter Trojans, with particular attention to asteroid families. We use a sample of data from the Moving Object Catalogue of the Sloan Digital Sky Survey, together with spectra obtained from several surveys. A first sample of 349 observations, corresponding to 250 Trojan asteroids, were extracted from the Sloan Digital Sky Survey, and we also extracted from the literature a second sample of 91 spectra, corresponding to 71 Trojans. The spectral slopes were computed by means of a least-squares fit to a straight line of the fluxes obtained from the Sloan observations in the first sample, and of the rebinned spectra in the second sample. In both cases the reflectance fluxes/spectra were renormalized to 1 at 6230 $\textrm{\AA}$. We found that the distribution of spectral slopes among Trojan asteroids shows a bimodality. About 2/3 of the objects have reddish slopes compatible with D-type asteroids, while the remaining bodies show less reddish colors compatible with the P-type and C-type classifications. The members of asteroid families also show a bimodal distribution with a very slight predominance of D-type asteroids, but the background is clearly dominated by the D-types. The L4 and L5 swarms show different distributions of spectral slopes, and bimodality is only observed in L4. These differences can be attributed to the asteroid families since the backgraound asteroids show the same slope distribtuions in both swarms. The analysis of individual families indicates that the families in L5 are taxonomically homogeneous, but in L4 they show a mixture of taxonomic types. We discuss a few scenarios that might help to interpret these results.Comment: 20 pages, 15 figures, 2 table

Resonances of low orders in the planetary system of HD37124

The full set of published radial velocity data (52 measurements from Keck + 58 ones from ELODIE + 17 ones from CORALIE) for the star HD37124 is analysed. Two families of dynamically stable high-eccentricity orbital solutions for the planetary system are found. In the first one, the outer planets c and d are trapped in the 2/1 mean-motion resonance. The second family of solutions corresponds to the 5/2 mean-motion resonance between these planets. In both families, the planets are locked in (or close to) an apsidal corotation resonance. In the case of the 2/1 MMR, it is an asymmetric apsidal corotation (with the difference between the longitudes of periastra $\Delta\omega\sim 60^\circ$), whereas in the case of the 5/2 MMR it is a symmetric antialigned one ($\Delta\omega = 180^\circ$). It remains also possible that the two outer planets are not trapped in an orbital resonance. Then their orbital eccentricities should be relatively small (less than, say, 0.15) and the ratio of their orbital periods is unlikely to exceed $2.3-2.5$.Comment: 28 pages, 10 figures, 3 tables; Accepted to Celestial Mechanics and Dynamical Astronom

A Peculiar Family of Jupiter Trojans: the Eurybates

The Eurybates family is a compact core inside the Menelaus clan, located in the L4 swarm of Jupiter Trojans. Fornasier et al. (2007) found that this family exhibits a peculiar abundance of spectrally flat objects, similar to Chiron-like Centaurs and C-type main belt asteroids. On the basis of the visible spectra available in literature, Eurybates family's members seemed to be good candidates for having on their surfaces water/water ice or aqueous altered materials. To improve our knowledge of the surface composition of this peculiar family, we carried out an observational campaign at the Telescopio Nazionale Galileo (TNG), obtaining near-infrared spectra of 7 members. Our data show a surprisingly absence of any spectral feature referable to the presence of water, ices or aqueous altered materials on the surface of the observed objects. Models of the surface composition are attempted, evidencing that amorphous carbon seems to dominate the surface composition of the observed bodies and some amount of silicates (olivine) could be present.Comment: 23 pages, 2 figures, paper accepted for publication in Icaru

Orbital structure of the GJ876 extrasolar planetary system, based on the latest Keck and HARPS radial velocity data

We use full available array of radial velocity data, including recently published HARPS and Keck observatory sets, to characterize the orbital configuration of the planetary system orbiting GJ876. First, we propose and describe in detail a fast method to fit perturbed orbital configuration, based on the integration of the sensitivity equations inferred by the equations of the original $N$-body problem. Further, we find that it is unsatisfactory to treat the available radial velocity data for GJ876 in the traditional white noise model, because the actual noise appears autocorrelated (and demonstrates non-white frequency spectrum). The time scale of this correlation is about a few days, and the contribution of the correlated noise is about 2 m/s (i.e., similar to the level of internal errors in the Keck data). We propose a variation of the maximum-likelihood algorithm to estimate the orbital configuration of the system, taking into account the red noise effects. We show, in particular, that the non-zero orbital eccentricity of the innermost planet \emph{d}, obtained in previous studies, is likely a result of misinterpreted red noise in the data. In addition to offsets in some orbital parameters, the red noise also makes the fit uncertainties systematically underestimated (while they are treated in the traditional white noise model). Also, we show that the orbital eccentricity of the outermost planet is actually ill-determined, although bounded by $\sim 0.2$. Finally, we investigate possible orbital non-coplanarity of the system, and limit the mutual inclination between the planets \emph{b} and \emph{c} orbits by $5^\circ-15^\circ$, depending on the angular position of the mutual orbital nodes.Comment: 36 pages, 11 figures, 3 tables; Accepted to Celestial Mechanics and Dynamical Astronom

A new analysis of the GJ581 extrasolar planetary system

We have done a new analysis of the available observations for the GJ581 exoplanetary system. Today this system is controversial due to choices that can be done in the orbital determination. The main ones are the ocurrence of aliases and the additional bodies - the planets f and g - announced in Vogt et al. 2010. Any dynamical study of exoplanets requires the good knowledge of the orbital elements and the investigations involving the planet g are particularly interesting, since this body would lie in the Habitable Zone (HZ) of the star GJ581. This region,for this system, is very attractive of the dynamical point of view due to several resonances of two and three bodies present there. In this work, we investigate the conditions under which the planet g may exist. We stress the fact that the planet g is intimately related with the orbital elements of the planet d; more precisely, we conclude that it is not possible to disconnect its existence from the determination of the eccentricity of the planet d. Concerning the planet f, we have found one solution with period $\approx 450$ days, but we are judicious about any affirmation concernig this body because its signal is in the threshold of detection and the high period is in a spectral region where the ocorruence of aliases is very common. Besides, we outline some dynamical features of the habitable zone with the dynamical map and point out the role played by some resonances laying there.Comment: 12 pages, 9 figure

The HARPS search for southern extra-solar planets XIX. Characterization and dynamics of the GJ876 planetary system

Precise radial-velocity measurements for data acquired with the HARPS spectrograph infer that three planets orbit the M4 dwarf star GJ876. In particular, we confirm the existence of planet "d", which orbits every 1.93785 days. We find that its orbit may have significant eccentricity (e=0.14), and deduce a more accurate estimate of its minimum mass of 6.3 Earth masses. Dynamical modeling of the HARPS measurements combined with literature velocities from the Keck Observatory strongly constrain the orbital inclinations of the "b" and "c" planets. We find that i_b = 48.9 degrees and i_c = 48.1 degrees, which infers the true planet masses of M_b = 2.64 Jupiter masses and M_c = 0.83 Jupiter masses, respectively. Radial velocities alone, in this favorable case, can therefore fully determine the orbital architecture of a multi-planet system, without the input from astrometry or transits. The orbits of the two giant planets are nearly coplanar, and their 2:1 mean motion resonance ensures stability over at least 5 Gyr. The libration amplitude is smaller than 2 degrees, suggesting that it was damped by some dissipative process during planet formation. The system has space for a stable fourth planet in a 4:1 mean motion resonance with planet "b", with a period around 15 days. The radial velocity measurements constrain the mass of this possible additional planet to be at most that of the Earth.Comment: 10 pages, 10 figures, accepted for publication in Astronomy & Astrophysic