Resonant capture by inward migrating planets

We investigate resonant capture of small bodies by planets that migrate inwards, using analytic arguments and three-body integrations. If the orbits of the planet and the small body are initially circular and coplanar, the small body is captured when it crosses the 2:1 resonance with the planet. As the orbit shrinks it becomes more eccentric, until by the time its semimajor axis has shrunk by a factor of four, its eccentricity reaches nearly unity (1-e<<10^{-4}). In typical planetary systems, bodies in this high-eccentricity phase are likely to be consumed by the central star. If they can avoid this fate, as migration continues the inclination flips from 0 to i=180 degrees; thereafter the eccentricity declines until the semimajor axis is a factor of nine smaller than at capture, at which point the small body is released from the 2:1 resonance on a nearly circular retrograde orbit. Small bodies captured into resonance from initially inclined or eccentric orbits can also be ejected from the system, or released from the resonance on highly eccentric polar orbits (i\simeq 90 degrees) that are stabilized by a secular resonance. We conclude that migration can drive much of the inner planetesimal disk into the star, and that post-migration multi-planet systems may not be coplanar.Comment: 12 pages, 5 figures, submitted to Astronomical Journa

Gravitational Collapse in One Dimension

We simulate the evolution of one-dimensional gravitating collisionless systems from non- equilibrium initial conditions, similar to the conditions that lead to the formation of dark- matter halos in three dimensions. As in the case of 3D halo formation we find that initially cold, nearly homogeneous particle distributions collapse to approach a final equilibrium state with a universal density profile. At small radii, this attractor exhibits a power-law behavior in density, {\rho}(x) \propto |x|^(-{\gamma}_crit), {\gamma}_crit \simeq 0.47, slightly but significantly shallower than the value {\gamma} = 1/2 suggested previously. This state develops from the initial conditions through a process of phase mixing and violent relaxation. This process preserves the energy ranks of particles. By warming the initial conditions, we illustrate a cross-over from this power-law final state to a final state containing a homogeneous core. We further show that inhomogeneous but cold power-law initial conditions, with initial exponent {\gamma}_i > {\gamma}_crit, do not evolve toward the attractor but reach a final state that retains their original power-law behavior in the interior of the profile, indicating a bifurcation in the final state as a function of the initial exponent. Our results rely on a high-fidelity event-driven simulation technique.Comment: 14 Pages, 13 Figures. Submitted to MNRA

Comparison of simple mass estimators for slowly rotating elliptical galaxies

We compare the performance of mass estimators for elliptical galaxies that rely on the directly observable surface brightness and velocity dispersion profiles, without invoking computationally expensive detailed modeling. These methods recover the mass at a specific radius where the mass estimate is expected to be least sensitive to the anisotropy of stellar orbits. One method (Wolf et al. 2010) uses the total luminosity-weighted velocity dispersion and evaluates the mass at a 3D half-light radius $r_{1/2}$, i.e., it depends on the GLOBAL galaxy properties. Another approach (Churazov et al. 2010) estimates the mass from the velocity dispersion at a radius $R_2$ where the surface brightness declines as $R^{-2}$, i.e., it depends on the LOCAL properties. We evaluate the accuracy of the two methods for analytical models, simulated galaxies and real elliptical galaxies that have already been modeled by the Schwarzschild's orbit-superposition technique. Both estimators recover an almost unbiased circular speed estimate with a modest RMS scatter ($\lesssim 10 \%$). Tests on analytical models and simulated galaxies indicate that the local estimator has a smaller RMS scatter than the global one. We show by examination of simulated galaxies that the projected velocity dispersion at $R_2$ could serve as a good proxy for the virial galaxy mass. For simulated galaxies the total halo mass scales with $\sigma_p(R_2)$ as $M_{vir} \left[M_{\odot}h^{-1}\right] \approx 6\cdot 10^{12} \left( \frac{\sigma_p(R_2)}{200\, \rm km\, s^{-1}} \right)^{4}$ with RMS scatter $\approx 40 \%$.Comment: 19 pages, 14 figures, 4 tables, accepted for publication in MNRA

Lattice Stellar Dynamics

We describe a technique for solving the combined collisionless Boltzmann and Poisson equations in a discretised, or lattice, phase space. The time and the positions and velocities of `particles' take on integer values, and the forces are rounded to the nearest integer. The equations of motion are symplectic. In the limit of high resolution, the lattice equations become the usual integro-differential equations of stellar dynamics. The technique complements other tools for solving those equations approximately, such as $N$-body simulation, or techniques based on phase-space grids. Equilibria are found in a variety of shapes and sizes. They are true equilibria in the sense that they do not evolve with time, even slowly, unlike existing $N$-body approximations to stellar systems, which are subject to two-body relaxation. They can also be `tailor-made' in the sense that the mass distribution is constrained to be close to some pre-specified function. Their principal limitation is the amount of memory required to store the lattice, which in practice restricts the technique to modeling systems with a high degree of symmetry. We also develop a method for analysing the linear stability of collisionless systems, based on lattice equilibria as an unperturbed model.Comment: Accepted for publication in Monthly Notices. 18 pages, compressed PostScript, also available from http://www.cita.utoronto.ca/~syer/papers

Made-to-measure N-body systems

We describe an algorithm for constructing N-body realisations of equilibrium stellar systems. The algorithm complements existing orbit-based modelling techniques using linear programming or other optimization algorithms. The equilibria are constructed by integrating an N-body system while slowly adjusting the masses of the particles until the time-averaged density field and other observables converge to a prescribed value. The procedure can be arranged to maximise a linear combination of the entropy of the system and the $\chi^2$ statistic for the observables. The equilibria so produced may be useful as initial conditions for N-body simulations or for modelling observations of individual galaxies.Comment: 8 pages, tex, figures included, accepted by MNRAS, also available at http://www.mpa-garching.mpg.de/~syer/papers