The long-term dynamical behavior of short-period comets

The orbits of the known short-period comets under the influence of the Sun and all the planets except Mercury and Pluto are numerically integrated. The calculation was undertaken in order to determine the dynamical lifetimes for these objects as well as explaining the current orbital element distribution. It is found that a comet can move between Jupiter-family and Halley-family comets several times in its dynamical lifetime. The median lifetime of the known short-period comets from the time they are first injected into a short-period comet orbit to ultimate ejection is approximately 50,000 years. The very flat inclination distribution of Jupiter-family comets is observed to become more distended as it ages. The only possible explanation for the observed flat distribution is that the comets become extinct before their inclination distribution can change significantly. It is shown that the anomalous concentration of the argument of perihelion of Jupiter-family comets near 0 and 180 deg is a direct result of their aphelion distance being close to 5.2AU and the comet being recently perturbed onto a Jupiter-family orbit. Also the concentration of their aphelion near Jupiter's orbit is a result of the conservation of the Tisserand invariant during the capture process

A Lagrangian Integrator for Planetary Accretion and Dynamics (LIPAD)

We presented the first particle based, Lagrangian code that can follow the collisional/accretional/dynamical evolution of a large number of km-sized planetesimals through the entire growth process to become planets. We refer to it as the 'Lagrangian Integrator for Planetary Accretion and Dynamics' or LIPAD. LIPAD is built on top of SyMBA, which is a symplectic $N$-body integrator. In order to handle the very large number of planetesimals required by planet formation simulations, we introduce the concept of a `tracer' particle. Each tracer is intended to represent a large number of disk particles on roughly the same orbit and size as one another, and is characterized by three numbers: the physical radius, the bulk density, and the total mass of the disk particles represented by the tracer. We developed statistical algorithms that follow the dynamical and collisional evolution of the tracers due to the presence of one another. The tracers mainly dynamically interact with the larger objects (`planetary embryos') in the normal N-body way. LIPAD's greatest strength is that it can accurately model the wholesale redistribution of planetesimals due to gravitational interaction with the embryos, which has recently been shown to significantly affect the growth rate of planetary embryos . We verify the code via a comprehensive set of tests which compare our results with those of Eulerian and/or direct N-body codes.Comment: Accepted to the Astronomical Journal. See http://www.boulder.swri.edu/~hal/LIPAD.html for more detail including animation

Modeling the Formation of Giant Planet Cores I: Evaluating Key Processes

One of the most challenging problems we face in our understanding of planet formation is how Jupiter and Saturn could have formed before the the solar nebula dispersed. The most popular model of giant planet formation is the so-called 'core accretion' model. In this model a large planetary embryo formed first, mainly by two-body accretion. This is then followed by a period of inflow of nebular gas directly onto the growing planet. The core accretion model has an Achilles heel, namely the very first step. We have undertaken the most comprehensive study of this process to date. In this study we numerically integrate the orbits of a number of planetary embryos embedded in a swarm of planetesimals. In these experiments we have included: 1) aerodynamic gas drag, 2) collisional damping between planetesimals, 3) enhanced embryo cross-sections due to their atmospheres, 4) planetesimal fragmentation, and 5) planetesimal driven migration. We find that the gravitational interaction between the embryos and the planetesimals lead to the wholesale redistribution of material - regions are cleared of material and gaps open near the embryos. Indeed, in 90% of our simulations without fragmentation, the region near that embryos is cleared of planetesimals before much growth can occur. The remaining 10%, however, the embryos undergo a burst of outward migration that significantly increases growth. On timescales of ~100,000 years, the outer embryo can migrate ~6 AU and grow to roughly 30 Earth-masses. We also find that the inclusion of planetesimal fragmentation tends to inhibit growth.Comment: Accepted to AJ, 62 pages 11 figure

The Calibration of the HST Kuiper Belt Object Search: Setting the Record Straight

The limiting magnitude of the HST data set used by Cochran et al. (1995) to detect small objects in the Kuiper belt is reevaluated, and the methods used are described in detail. It is shown, by implanting artificial objects in the original HST images, and re-reducing the images using our original algorithm, that the limiting magnitude of our images (as defined by the 50% detectability limit) is $V=28.4$. This value is statistically the same as the value found in the original analysis. We find that $\sim50$ of the moving Kuiper belt objects with $V=27.9$ are detected when trailing losses are included. In the same data in which these faint objects are detected, we find that the number of false detections brighter than $V=28.8$ is less than one per WFPC2 image. We show that, primarily due to a zero-point calibration error, but partly due to inadequacies in modeling the HST'S data noise characteristics and Cochran et al.'s reduction techniques, Brown et al. 1997 underestimate the SNR of objects in the HST dataset by over a factor of 2, and their conclusions are therefore invalid.Comment: Accepted to ApJ Letters; 10 pages plus 3 figures, LaTe

Effects of Type I Migration on Terrestrial Planet Formation

Planetary embryos embedded in a gas disc suffer a decay in semimajor axis -- type I migration -- due to the asymmetric torques produced by the interior and exterior wakes raised by the body (Goldreich & Tremaine 1980; Ward 1986). This presents a challenge for standard oligarchic approaches to forming the terrestrial planets (Kokubo & Ida 1998) as the timescale to grow the progenitor objects near 1 AU is longer than that for them to decay into the Sun. In this paper we investigate the middle and late stages of oligarchic growth using both semi-analytic methods (based upon Thommes et al. 2003) and N-body integrations, and vary gas properties such as dissipation timescale in different models of the protoplanetary disc. We conclude that even for near-nominal migration efficiencies and gas dissipation timescales of ~1 Myr it is possible to maintain sufficient mass in the terrestrial region to form Earth and Venus if the disc mass is enhanced by factors of ~2-4 over the minimum mass model. The resulting configurations differ in several ways from the initial conditions used in previous simulations of the final stages of terrestrial accretion (e.g. Chambers 2001), chiefly in (1) larger inter-embryo spacings, (2) larger embryo masses, and (3) up to ~0.4 Earth masses of material left in the form of planetesimals when the gas vanishes. The systems we produce are reasonably stable for ~100 Myr and therefore require an external source to stir up the embryos sufficiently to produce final systems resembling the terrestrial planets.Comment: 49 pages, 22 figures; accepted in AJ, expected Dec '0

Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node v equal to 2{\pi}/d(v). We show: 1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution. 3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure

Planetesimal-driven planet migration in the presence of a gas disk

We report here on an extension of a previous study by Kirsh et al. (2009) of planetesimal-driven migration using our N-body code SyMBA (Duncan et al., 1998). The previous work focused on the case of a single planet of mass Mem, immersed in a planetesimal disk with a power-law surface density distribution and Rayleigh distributed eccentricities and inclinations. Typically 10^4-10^5 equal-mass planetesimals were used, where the gravitational force (and the back-reaction) on each planetesimal by the Sun and planetwere included, while planetesimal-planetesimal interactions were neglected. The runs reported on here incorporate the dynamical effects of a gas disk, where the Adachi et al. (1976) prescription of aerodynamic gas drag is implemented for all bodies. In some cases the Papaloizou and Larwood (2000) prescription of Type-I migration for the planet are implemented, as well as a mass distribution. In the gas-free cases, rapid planet migration was observed - at a rate independent of the planet's mass - provided the planet's mass was not large compared to the mass in planetesimals capable of entering its Hill sphere. In such cases, both inward and outward migrations can be self-sustaining, but there is a strong propensity for inward migration. When a gas disk is present, aerodynamic drag can substantially modify the dynamics of scattered planetesimals. For sufficiently large or small mono-dispersed planetesimals, the planet typically migrates inward. However, for a range of plausible planetesimal sizes (i.e. 0.5-5.0 km at 5.0 AU in a minimum mass Hayashi disk) outward migration is usually triggered, often accompanied by substantial planetary mass accretion. The origins of this behaviour are explained in terms of a toy model. The effects of including a size distribution and torques associated with Type-I migration are also discussed.Comment: 37 pages, 17 figures, Accepted for publication in Icaru

Coherent Versus Incoherent Ladar Detection at 2.09 μm

A 2.09-μm ladar system is built to compare coherent to incoherent detection. The 2.09-μm wavelength is of interest because of its high atmospheric transmission and because it is eyesafe. The 2.09-μm system presented is capable of either a coherent or incoherent operational mode, is tunable in a small region around 2.09 μm, and is being used to look at the statistical nature of the ladar return pulses for typical glint and speckle targets. To compare coherent to incoherent detection the probability of detection is investigated as the primary performance criterion of interest. The probability of detection is dependent on both the probability of false alarm and the probability density function, representing the signal current output from the detector. These probability distributions are different for each detection technique and for each type of target. Furthermore, the probability of detection and the probability of false alarm are both functions of the dominating noise source(s) in the system. A description of the theoretical expectations of this system along with the setup of the ladar system and how it is being used to collect data for both coherent and incoherent detection is presented