On the relationship between instability and Lyapunov times for the 3-body problem

In this study we consider the relationship between the survival time and the Lyapunov time for 3-body systems. It is shown that the Sitnikov problem exhibits a two-part power law relationship as demonstrated previously for the general 3-body problem. Using an approximate Poincare map on an appropriate surface of section, we delineate escape regions in a domain of initial conditions and use these regions to analytically obtain a new functional relationship between the Lyapunov time and the survival time for the 3-body problem. The marginal probability distributions of the Lyapunov and survival times are discussed and we show that the probability density function of Lyapunov times for the Sitnikov problem is similar to that for the general 3-body problem.Comment: 9 pages, 19 figures, accepted for publication in MNRA

Noise-assisted spike propagation in myelinated neurons

We consider noise-assisted spike propagation in myelinated axons within a multi-compartment stochastic Hodgkin-Huxley model. The noise originates from a finite number of ion channels in each node of Ranvier. For the subthreshold internodal electric coupling, we show that (i) intrinsic noise removes the sharply defined threshold for spike propagation from node to node, and (ii) there exists an optimum number of ion channels which allows for the most efficient signal propagation and it corresponds to the actual physiological values.Comment: 8 pages, 12 figures, accepted for publication in Phys. Rev.

Halting Planet Migration in the Evacuated Centers of Protoplanetary Disks

Precise Doppler searches for extrasolar planets find a surfeit of planets with orbital periods of 3-4 days, and no planets with orbital periods less than 3 days. The circumstellar distance, R_0, where small grains in a protoplanetary disk reach sublimation temperature (~1500 K) corresponds to a period of ~6 days. Interior to R_0, turbulent accretion due to magneto-rotational instability may evacuate the disk center. We suggest that planets with orbital periods of 3-4 days are so common because migrating planets halt once this evacuated region contains the sites of their exterior 2:1 Lindblad resonances.Comment: 9 pages, 1 figure, to appear in ApJ letter

Instability of the Gravitational N-Body Problem in the Large-N Limit

We use a systolic N-body algorithm to evaluate the linear stability of the gravitational N-body problem for N up to 1.3 x 10^5, two orders of magnitude greater than in previous experiments. For the first time, a clear ~ln N-dependence of the perturbation growth rate is seen. The e-folding time for N = 10^5 is roughly 1/20 of a crossing time.Comment: Accepted for publication in The Astrophysical Journa

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. Via 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 semi-major 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 2/7 law has been previously explained by estimating the semi-major 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 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.Comment: accepted for publication in MNRA

On the Snow Line in Dusty Protoplanetary Disks

The snow line, in Hayashi's (1981) model, is where the temperature of a black body that absorbed direct sunlight and re-radiated as much as it absorbed, would be 170~K. It is usually assumed that the cores of the giant planets, e.g., Jupiter, form beyond the snow line. Since Hayashi, there have been a series of more detailed models of the absorption by dust of the stellar radiation, and of accretional heating, which alter the location of the snow line. We have attempted a "self-consistent" model of a T Tauri disk in the sense that we used dust properties and calculated surface temperatures that matched observed disks. We then calculated the midplane temperature for those disks, with no accretional heating or with small (<10^-8) accretion rates. Our models bring the snow line in to the neighbourhood of 1 AU; not far enough to explain the close planetary companions to other stars, but much closer than in recent starting lines for orbit migration scenarios.Comment: 9 pages, 1 figure, to appear in ApJ,528,200

Do we expect to find the Super-Earths close to the gas giants?

We have investigated the evolution of a pair of interacting planets embedded in a gaseous disc, considering the possibility of the resonant capture of a Super-Earth by a Jupiter mass gas giant. First, we have examined the situation where the Super-Earth is on the internal orbit and the gas giant on the external one. It has been found that the terrestrial planet is scattered from the disc or the gas giant captures the Super-Earth into an interior 3:2 or 4:3 mean-motion resonance. The stability of the latter configurations depends on the initial planet positions and on eccentricity evolution. The behaviour of the system is different if the Super-Earth is the external planet. We have found that instead of being captured in the mean-motion resonance, the terrestrial planet is trapped at the outer edge of the gap opened by the gas giant. This effect prevents the occurrence of the first order mean-motion commensurability. These results are particularly interesting in light of recent exoplanet discoveries and provide predictions of what will become observationally testable in the near future.Comment: 7 pages, to appear in the proceedings of the conference "Extra-solar Planets in Multi-body Systems: Theory and Observations"; eds. K. Gozdziewski, A. Niedzielski and J. Schneider, EAS Publication Serie

Stochastic resonance as a collective property of ion channel assemblies

By use of a stochastic generalization of the Hodgkin-Huxley model we investigate both the phenomena of stochastic resonance (SR) and coherence resonance (CR) in variable size patches of an excitable cell membrane. Our focus is on the challenge how internal noise stemming from individual ion channels does affect collective properties of the whole ensemble. We investigate both an unperturbed situation with no applied stimuli and one in which the membrane is stimulated externally by a periodic signal and additional external noise. For the nondriven case, we demonstrate the existence of an optimal size of the membrane patch for which the internal noise causes a most regular spike activity. This phenomenon shall be termed intrinsic CR. In presence of an applied periodic stimulus we demonstrate that the signal-to-noise ratio (SNR) exhibits SR vs. decreasing patch size, or vs. increasing internal noise strength, respectively. Moreover, we demonstrate that conventional SR vs. the external noise intensity occurs only for sufficiently large membrane patches, when the intensity of internal noise is below its optimal level. Thus, biological SR seemingly is rooted in the collective properties of large ion channel ensembles rather than in the individual stochastic dynamics of single ion channels.Comment: 9 pages, 2 figure