An Overview of the 13:8 Mean Motion Resonance between Venus and Earth

It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore we did numerical experiments in different dynamical models with different initial conditions -- on one hand the couple Venus-Earth was set close to different mean motion resonances (MMR), and on the other hand Venus' orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth's semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in eccentricity and/or inclination of Venus in these resonances. Even escapes for the innermost planet are possible which may happen quite rapidly.Comment: 14 pages, 11 figures, submitted to CMD

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

Dynamics of two planets in co-orbital motion

We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities, also we assume that both planets share the same orbital plane. Initially we perform numerical simulations over a grid of osculating initial conditions to map the regions of stable/chaotic motion and identify equilibrium solutions. These results are later analyzed in more detail using a semi-analytical model. Apart from the well known quasi-satellite (QS) orbits and the classical equilibrium Lagrangian points L4 and L5, we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at $(\sigma,\Delta\omega) = (\pm 60\deg, \mp 120\deg)$, where \sigma is the difference in mean longitudes and \Delta\omega is the difference in longitudes of pericenter. The position of these Anti-Lagrangian solutions changes with the mass ratio and the orbital eccentricities, and are found for eccentricities as high as ~ 0.7. Finally, we also applied a slow mass variation to one of the planets, and analyzed its effect on an initially asymmetric periodic orbit. We found that the resonant solution is preserved as long as the mass variation is adiabatic, with practically no change in the equilibrium values of the angles.Comment: 9 pages, 11 figure

Dust in the wind: the role of recent mass loss in long gamma-ray bursts

We study the late-time (t>0.5 days) X-ray afterglows of nearby (z<0.5) long Gamma-Ray Bursts (GRB) with Swift and identify a population of explosions with slowly decaying, super-soft (photon index Gamma_x>3) X-ray emission that is inconsistent with forward shock synchrotron radiation associated with the afterglow. These explosions also show larger-than-average intrinsic absorption (NH_x,i >6d21 cm-2) and prompt gamma-ray emission with extremely long duration (T_90>1000 s). Chance association of these three rare properties (i.e. large NH_x,i, super-soft Gamma_x and extreme duration) in the same class of explosions is statistically unlikely. We associate these properties with the turbulent mass-loss history of the progenitor star that enriched and shaped the circum-burst medium. We identify a natural connection between NH_x,i Gamma_x and T_90 in these sources by suggesting that the late-time super-soft X-rays originate from radiation reprocessed by material lost to the environment by the stellar progenitor before exploding, (either in the form of a dust echo or as reprocessed radiation from a long-lived GRB remnant), and that the interaction of the explosion's shock/jet with the complex medium is the source of the extremely long prompt emission. However, current observations do not allow us to exclude the possibility that super-soft X-ray emitters originate from peculiar stellar progenitors with large radii that only form in very dusty environments.Comment: 6 pages, Submitted to Ap

Is the outer Solar System chaotic?

The existence of chaos in the system of Jovian planets has been in question for the past 15 years. Various investigators have found Lyapunov times ranging from about 5 millions years upwards to infinity, with no clear reason for the discrepancy. In this paper, we resolve the issue. The position of the outer planets is known to only a few parts in 10 million. We show that, within that observational uncertainty, there exist Lyapunov timescales in the full range listed above. Thus, the ``true'' Lyapunov timescale of the outer Solar System cannot be resolved using current observations.Comment: 8 pages, 2 figure

Variational Integrators for Almost-Integrable Systems

We construct several variational integrators--integrators based on a discrete variational principle--for systems with Lagrangians of the form L = L_A + epsilon L_B, with epsilon << 1, where L_A describes an integrable system. These integrators exploit that epsilon << 1 to increase their accuracy by constructing discrete Lagrangians based on the assumption that the integrator trajectory is close to that of the integrable system. Several of the integrators we present are equivalent to well-known symplectic integrators for the equivalent perturbed Hamiltonian systems, but their construction and error analysis is significantly simpler in the variational framework. One novel method we present, involving a weighted time-averaging of the perturbing terms, removes all errors from the integration at O(epsilon). This last method is implicit, and involves evaluating a potentially expensive time-integral, but for some systems and some error tolerances it can significantly outperform traditional simulation methods.Comment: 14 pages, 4 figures. Version 2: added informative example; as accepted by Celestial Mechanics and Dynamical Astronom