Protoplanet Dynamics in a Shear-Dominated Disk

The velocity dispersion, or eccentricity distribution, of protoplanets interacting with planetesimals is set by a balance between dynamical friction and viscous stirring. We calculate analytically the eccentricity distribution function of protoplanets embedded in a cold, shear-dominated planetesimal swarm. We find a distinctly non-Rayleigh distribution with a simple analytical form. The peak of the distribution lies much lower than the root-mean-squared value, indicating that while most of the bodies have similarly small eccentricities, a small subset of the population contains most of the thermal energy. We also measure the shear-dominated eccentricity distribution using numerical simulations. The numerical code treats each protoplanet explicitly and adds an additional force term to each body to represent the dynamical friction of the planetesimals. Without fitting any parameters, the eccentricity distribution of protoplanets in the N-body simulation agrees with the analytical results. This distribution function provides a useful tool for testing hybrid numerical simulations of late-stage planet formation.Comment: 8 pages, 2 figure

A Unified Theory for the Effects of Stellar Perturbations and Galactic Tides on Oort Cloud Comets

We examine the effects of passing field stars on the angular momentum of a nearly radial orbit of an Oort cloud comet bound to the Sun. We derive the probability density function (PDF) of the change in angular momentum from one stellar encounter, assuming a uniform and isotropic field of perturbers. We show that the total angular momentum follows a Levy flight, and determine its distribution function. If there is an asymmetry in the directional distribution of perturber velocities, the marginal probability distribution of each component of the angular momentum vector can be different. The constant torque attributed to Galactic tides arises from a non-cancellation of perturbations with an impact parameter of order the semimajor axis of the comet. When the close encounters are rare, the angular momentum is best modeled by the stochastic growth of stellar encounters. If trajectories passing between the comet and sun occur frequently, the angular momentum exhibits the coherent growth attributed to the Galactic tides.Comment: 8 pages, 2 figures; accepted to A

The Self-Similarity of Shear-Dominated Viscous Stirring

We examine the growth of eccentricities of a population of particles with initially circular orbits around a central massive body. Successive encounters between pairs of particles increase the eccentricities in the disk on average. As long as the epicyclic motions of the particles are small compared to the shearing motion between Keplerian orbits, there is no preferred scale for the eccentricities. The simplification due to this self-similarity allows us to find an analytic form for the distribution function; full numerical integrations of a disk with 200 planetesimals verify our analytical self-similar distribution. The shape of this non-equilibrium profile is identical to the equilibrium profile of a shear-dominated population whose mutual excitations are balanced by dynamical friction or Epstein gas drag.Comment: 8 pages, 2 figure

Co-orbital Oligarchy

We present a systematic examination of the changes in semi-major axis caused by the mutual interactions of a group of massive bodies orbiting a central star in the presence of eccentricity dissipation. For parameters relevant to the oligarchic stage of planet formation, dynamical friction keeps the typical eccentricities small and prevents orbit crossing. Interactions at impact parameters greater than several Hill radii cause the protoplanets to repel each other; if the impact parameter is instead much less than the Hill radius, the protoplanets shift slightly in semi-major axis but remain otherwise unperturbed. If the orbits of two or more protoplanets are separated by less than a Hill radius, they are each pushed towards an equilibrium spacing between their neighbors and can exist as a stable co-orbital system. In the shear-dominated oligarchic phase of planet formation we show that the feeding zones contain several oligarchs instead of only one. Growth of the protoplanets in the oligarchic phase drives the disk to an equilibrium configuration that depends on the mass ratio of protoplanets to planetesimals, $\Sigma/\sigma$. Early in the oligarchic phase, when $\Sigma/\sigma$ is low, the spacing between rows of co-orbital oligarchs are about 5 Hill radii wide, rather than the 10 Hill radii cited in the literature. It is likely that at the end of oligarchy the average number of co-orbital oligarchs is greater than unity. In the outer solar system this raises the disk mass required to form the ice giants. In the inner solar system this lowers the mass of the final oligarchs and requires more giant impacts than previously estimated. This result provides additional evidence that Mars is not an untouched leftover from the oligarchic phase, but must be composed of several oligarchs assembled through giant impacts.Comment: 10 pages, 8 figures. v2 includes major revisions including additional results motivated by the referee's comment

Levy Flights of Binary Orbits due to Impulsive Encounters

We examine the evolution of an almost circular Keplerian orbit interacting with unbound perturbers. We calculate the change in eccentricity and angular momentum that results from a single encounter, assuming the timescale for the interaction is shorter than the orbital period. The orbital perturbations are incorporated into a Boltzmann equation that allows for eccentricity dissipation. We present an analytic solution to the Boltzmann equation that describes the distribution of orbital eccentricity and relative inclination as a function of time. The eccentricity and inclination of the binary do not evolve according to a normal random walk but perform a Levy flight. The slope of the mass spectrum of perturbers dictates whether close gravitational scatterings are more important than distant tidal ones. When close scatterings are important, the mass spectrum sets the slope of the eccentricity and inclination distribution functions. We use this general framework to understand the eccentricities of several Kuiper belt systems: Pluto, 2003 EL 61, and Eris. We use the model of Tholen et al (2007) to separate the non-Keplerian components of the orbits of Pluto's outer moons Nix and Hydra from the motion excited by interactions with other Kuiper belt objects. Our distribution is consistent with the observations of Nix, Hydra, and the satellites of 2003 EL 61 and Eris. We address applications of this work to objects outside of the solar system, such as extrasolar planets around their stars and millisecond pulsars.Comment: 14 pages, 2 figure

Reproducibility of left ventricular mass measurements by two-dimensional and M-mode echocardiography

AbstractBoth two-dimensional and M-mode echocardiography provide accurate estimates of left ventricular mass. However, their reproducibility in serial studies has not been compared, although this issue is critical to evaluation of regression of hypertrophy. To determine which technique provides more reproducible estimates of left ventricular mass, three serial studies were performed prospectively in each of eight normal adults over 5 months. Both two-dimensional and M-mode echocardiograms were obtained at each of these 24 studies. Measurements were performed by two independent observers who did not know patient identity. For the two-dimensional method, left ventricular mass was determined with use of a computer light-pen system and the truncated ellipsoid formula. For the M-mode method, mass was calculated from Penn convention measurements with use of the cube formula.At study 1 the group mean left ventricular mass by two-dimensional echocardiography (115 ± 20 g) did not differ from that by M-mode study (127± 37 g, p = NS). However, serial estimates of left ventricular mass were more reproducible by two-dimensional echocardiography. The mean difference among the three serial two-dimensional studies in each individual was 4.8 ± 4 g (4.2 ± 3%) by the two-dimensional method, but was 18.5 ± 13 g (14.9 ± 10%) by the M-mode method (p = 0.01). Interobserver results for left ventricular mass by two-dimensional echocardiography correlated closely (r = 0.95, n = 24, p < 0.001).The superior reproducibility of two-dimensional echocardiographic estimates of left ventricular mass in normal adults supports the use of two-dimensional echocardiography when serial studies are to be performed

Ultra-compact optical auto-correlator based on slow-light enhanced third harmonic generation in a silicon photonic crystal waveguide

The ability to use coherent light for material science and applications is directly linked to our ability to measure short optical pulses. While free-space optical methods are well-established, achieving this on a chip would offer the greatest benefit in footprint, performance, flexibility and cost, and allow the integration with complementary signal processing devices. A key goal is to achieve operation at sub-Watt peak power levels and on sub-picosecond timescales. Previous integrated demonstrations require either a temporally synchronized reference pulse, an off-chip spectrometer, or long tunable delay lines. We report the first device capable of achieving single-shot time-domain measurements of near-infrared picosecond pulses based on an ultra-compact integrated CMOS compatible device, with the potential to be fully integrated without any external instrumentation. It relies on optical third-harmonic generation in a slow-light silicon waveguide. Our method can also serve as a powerful in-situ diagnostic tool to directly map, at visible wavelengths, the propagation dynamics of near-infrared pulses in photonic crystals.Comment: 20 pages, 6 figures, 38 reference

Integrated spatial multiplexing of heralded single photon sources

The non-deterministic nature of photon sources is a key limitation for single photon quantum processors. Spatial multiplexing overcomes this by enhancing the heralded single photon yield without enhancing the output noise. Here the intrinsic statistical limit of an individual source is surpassed by spatially multiplexing two monolithic silicon correlated photon pair sources, demonstrating a 62.4% increase in the heralded single photon output without an increase in unwanted multi-pair generation. We further demonstrate the scalability of this scheme by multiplexing photons generated in two waveguides pumped via an integrated coupler with a 63.1% increase in the heralded photon rate. This demonstration paves the way for a scalable architecture for multiplexing many photon sources in a compact integrated platform and achieving efficient two photon interference, required at the core of optical quantum computing and quantum communication protocols.Comment: 10 pages, 3 figures, comments welcom

Stationary models for the extra-planar gas in disc galaxies

The kinematics of the extra-planar neutral and ionised gas in disc galaxies shows a systematic decline of the rotational velocity with height from the plane (vertical gradient). This feature is not expected for a barotropic gas, whilst it is well reproduced by baroclinic fluid homogeneous models. The problem with the latter is that they require gas temperatures (above $10^5$ K) much higher than the temperatures of the cold and warm components of the extra-planar gas layer. In this paper, we attempt to overcome this problem by describing the extra-planar gas as a system of gas clouds obeying the Jeans equations. In particular, we consider models having the observed extra-planar gas distribution and gravitational potential of the disc galaxy NGC 891: for each model we construct pseudo-data cubes and we compare them with the HI data cube of NGC 891. In all cases the rotational velocity gradients are in qualitative agreement with the observations, but the synthetic and the observed data cubes of NGC 891 show systematic differences that cannot be accommodated by any of the explored models. We conclude that the extra-planar gas in disc galaxies cannot be satisfactorily described by a stationary Jeans-like system of gas clouds.Comment: 14 pages, 7 figures, accepted for pubblication in MNRA