A dynamical mechanism for establishing apsidal resonance

We show that in a system of two planets initially in nearly circular orbits, an impulse perturbation that imparts a finite eccentricity to one planet's orbit causes the other planet's orbit to become eccentric as well, and also naturally results in a libration of their relative apsidal longitudes for a wide range of initial conditions. We suggest that such a mechanism may explain orbital eccentricities and apsidal resonance in some exo-planetary systems. The eccentricity impulse could be caused by the ejection of a planet from these systems, or by torques from a primordial gas disk. The amplitude of secular variations provides an observational constraint on the dynamical history of such systems.Comment: to appear in ApJ-Letter

The Mass Distribution Function of Planets

The distribution of orbital period ratios of adjacent planets in extra-solar planetary systems discovered by the {\it Kepler} space telescope exhibits a peak near $\sim1.5$--$2$, a long tail of larger period ratios, and a steep drop-off in the number of systems with period ratios below $\sim1.5$. We find from this data that the dimensionless orbital separations have an approximately log-normal distribution. Using Hill's criterion for the dynamical stability of two planets, we find an upper bound on planet masses such that the most common planet mass does not exceed $10^{-3.2}m_*$, or about two-thirds Jupiter mass for solar mass stars. Assuming that the mass ratio and the dynamical separation (orbital spacings in units of mutual Hill radius) of adjacent planets are independent random variates, and adopting empirical distributions for these, we use Hill's criterion in a statistical way to estimate the planet mass distribution function from the observed distribution of orbital separations. We find that the planet mass function is peaked in logarithm of mass, with a peak value and standard deviation of $\log m/M_\oplus$ of $\sim(0.6-1.0)$ and $\sim(1.1-1.2)$, respectively.Comment: Updated analysis with debiased period ratio data and updated discussion; accepted to Ap

Prospects for the habitability of OGLE-2006-BLG-109L

The extrasolar system OGLE-2006-BLG-109L is the first multiple-planet system to be discovered by gravitational microlensing (Gaudi et al., 2008); the two large planets that have been detected have mass ratios, semimajor axis ratios, and equilibrium temperatures that are similar to those of Jupiter and Saturn; the mass of the host star is only 0.5 M_sun, and the system is more compact than our own Solar system. We find that in the habitable zone of the host star, the two detected planets resonantly excite large orbital eccentricities on a putative earth-mass planet, driving such a planet out of the habitable zone. We show that an additional inner planet of ~>0.3M_earth at <~0.1 AU would suppress the eccentricity perturbation and greatly improve the prospects for habitability of the system. Thus, the planetary architecture of a potentially habitable OGLE-2006-BLG-109L planetary system -- with two ``terrestrial'' planets and two jovian planets -- could bear very close resemblance to our own Solar system.Comment: 11 pages including 4 figures; accepted for publication in ApJ-Letter

Asymmetric impacts of near-Earth asteroids on the Moon

Recent lunar crater studies have revealed an asymmetric distribution of rayed craters on the lunar surface. The asymmetry is related to the synchronous rotation of the Moon: there is a higher density of rayed craters on the leading hemisphere compared with the trailing hemisphere. Rayed craters represent generally the youngest impacts. The purpose of this paper is to test the hypotheses that (i) the population of Near-Earth asteroids (NEAs) is the source of the impactors that have made the rayed craters, and (ii) that impacts by this projectile population account quantitatively for the observed asymmetry. We carried out numerical simulations of the orbital evolution of a large number of test particles representing NEAs in order to determine directly their impact flux on the Moon. The simulations were done in two stages. In the first stage we obtained encounter statistics of NEAs on the Earth's activity sphere. In the second stage we calculated the direct impact flux of the encountering particles on the surface of the Moon; the latter calculations were confined within the activity sphere of the Earth. A steady-state synthetic population of NEAs was generated from a debiased orbital distribution of the known NEAs. We find that the near-Earth asteroids do have an asymmetry in their impact flux on the Moon: apex-to-antapex ratio of 1.32 +/- 0.01. However, the observed rayed crater distribution's asymmetry is significantly more pronounced: apex-to-antapex ratio of 1.65 +/- 0.16. Our results suggest the existence of an undetected population of slower (low impact velocity) projectiles, such as a population of objects nearly coorbiting with Earth; more observational study of young lunar craters is needed to secure this conclusion.Comment: 16 pages, 7 figures, accepted for publication in Astronomy & Astrophysic

Neptune's resonances in the Scattered Disk

The Scattered Disk Objects (SDOs) are thought to be a small fraction of the ancient population of leftover planetesimals in the outer solar system that were gravitationally scattered by the giant planets and have managed to survive primarily by capture and sticking in Neptune's exterior mean motion resonances (MMRs). In order to advance understanding of the role of MMRs in the dynamics of the SDOs, we investigate the phase space structure of a large number of Neptune's MMRs in the semi-major axis range 33--140~au by use of Poincar\'e sections of the circular planar restricted three body model for the full range of particle eccentricity pertinent to SDOs. We find that, for eccentricities corresponding to perihelion distances near Neptune's orbit, distant MMRs have stable regions with widths that are surprisingly large and of similar size to those of the closer-in MMRs. We identify a phase-shifted second resonance zone that exists in the phase space at planet-crossing eccentricities but not at lower eccentricities; this second resonance zone plays an important role in the dynamics of SDOs in lengthening their dynamical lifetimes. Our non-perturbative measurements of the sizes of the stable resonance zones confirm previous results and provide an additional explanation for the prominence of the $N$:1 sequence of MMRs over the $N$:2, $N$:3 sequences and other MMRs in the population statistics of SDOs; our results also provide a tool to more easily identify resonant objects.Comment: 20 pages, 15 figures, 1 table. Some re-organization and minor revisions; to be published in CMD