Relativistic Dynamical Friction in the Weak Scattering Limit

A test mass, $M$, moving through an ambient medium of light particles with lower average kinetic energy than itself suffers a deceleration caused by its scattering of the light particles. The phenomenon is usually referred to as dynamical friction. The velocity, \v, of the test mass decays on a timescale independent of \v in the non-relativistic case. We derive expressions for dynamical friction in the case that the test mass and the light particles are relativistic, and that the scattering is weak (with impact parameter, $b\gg M$). In the case that the light particles are ultra-relativistic, and isotropic in the frame in which $M$ moves with velocity $v$, we find an explicit expression for the dynamical friction. The well known factor of 2 correcting the Newtonian scattering of photons to give the Einstein angle, $4M/b$, has the largest effect on the resulting friction, which is modified by a factor of roughly $16 / 3\gamma_v$ over the simple non-relativistic case. In the non-relativistic case, the largest contribution to the friction comes from light particles moving slower than $v$. We find that this is not the case for ultra-relativistic scattering, essentially because the scattering angle is independent of \v. Some astrophysical implications are discussed. (Accepted for publication in Monthly Notices.)Comment: 10 pages (no figures), self-unpacking uuencoded PostScript (uufiles), RDF#

Implications of Neutrino Balls as the Source of Gamma-Ray Bursts

(To appear in the Astrophysical Journal) Holdom and Malaney (1994) have suggested a mechanism for gamma-ray bursts which requires that stars be captured by a neutrino ball. Neutrino balls would be, for the most part, denser than main sequence stars, but their density would decrease as their mass increased. We show that small neutrino balls would subject stars to tidal forces sufficient to disrupt them. We thus argue that if neutrino balls existed at the centres of galaxies, only the largest would be able to act as a source of gamma-ray bursts. Such neutrino balls would have a mass of order 10^7\Msun. Tidal capture of stars by a neutrino ball would not be important, but dynamical friction against the neutrinos or star-disc interactions could both be important capture mechanisms. We find that a gamma-ray burst would occur in a galaxy containing such a neutrino ball roughly every 10^2\y, and the fraction of all galaxies contributing to the gamma-ray burst flux would be $\sim 10^{-4}$, assuming that this was the mechanism of all gamma-ray bursts. These numbers have implications for neutrino ball models of active galaxies, assuming that all gamma-ray bursts and all AGN come from neutrino balls. Either a small fraction $\sim 10^{-2}$ of the lifetime of such an object could be spent as an AGN, or that the probability of a neutrino ball becoming an AGN would be $10^{-2}$. It is not possible to rule out the possibility that neutrino balls might exist at the centres of galaxies through direct ground-based observation of stellar kinematics.Comment: 10 pages uuencoded PostScript (no figures), NB-0

Tidal disruption rates of stars in observed galaxies

We derive the rates of capture, Ndot, of main sequence turn off stars by the central massive black hole in a sample of galaxies from Magorrian et al. 1998. The disruption rates are smaller than previously believed with Ndot ~ 10^-4 - 10^-7 per galaxy. A correlation between Ndot and black hole mass, M, is exploited to estimate the rate of tidal disruptions in the local universe. Assuming that all or most galaxies have massive black holes in their nuclei, this rate should be dominated by sub-Lstar galaxies. The rate of tidal disruptions could be high enough to be detected in supernova (or similar) monitoring campaigns---we estimate the rate of tidal disruptions to be 0.01 - 0.1 times the supernova rate. We have also estimated the rates of disruption of red giants, which may be significant (Ndot ~> 10^-4 y^-1 per galaxy) for M ~> 10^8 Msun, but are likely to be harder to observe---only of order 10^-4 times the supernova rate in the local universe. In calculating capture rates, we advise caution when applying scaling formulae by other authors, which are not applicable in the physical regime spanned by the galaxies considered here.Comment: MNRAS, Accepted; 9 pages, Late

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

Density cusps: restrictions on non-axisymmetric models

Galactic nuclei are now generally thought to have density cusps in their centres, and the evidence is mounting that as a consequence they are unlikely to be triaxial. Self-consistent stellar dynamical models of non-axisymmetric cusps would be an interesting counter-argument to this conclusion. We consider 2-d analogues of triaxial cusps: a sequence of non-axisymmetric, cuspy discs first described by Sridhar & Touma (1997). Scale-free models with potential {}$\Phi\propto r^\alpha$ are examined in detail. It is shown analytically for 0<\alpha \la 0.43 that self-consistent models with positive phase-space density do not exist. Numerical solutions of the combined Vlasov and Poisson equations suggest that the whole sequence of models with $0<\alpha<1$ are also unphysical. Along with existing work on cusps, we conclude on purely theoretical grounds that galactic nuclei are not expected to be triaxial.Comment: Corrected some minor typos in equations. 8 pages, tex, with figures, submitted to MNRAS, also at http://www.mpa-garching.mpg.de/~syer/paper

Massive planet migration: Theoretical predictions and comparison with observations

We quantify the utility of large radial velocity surveys for constraining theoretical models of Type II migration and protoplanetary disk physics. We describe a theoretical model for the expected radial distribution of extrasolar planets that combines an analytic description of migration with an empirically calibrated disk model. The disk model includes viscous evolution and mass loss via photoevaporation. Comparing the predicted distribution to a uniformly selected subsample of planets from the Lick / Keck / AAT planet search programs, we find that a simple model in which planets form in the outer disk at a uniform rate, migrate inward according to a standard Type II prescription, and become stranded when the gas disk is dispersed, is consistent with the radial distribution of planets for orbital radii 0.1 AU < a < 2.5 AU and planet masses greater than 1.65 Jupiter masses. Some variant models are disfavored by existing data, but the significance is limited (~95%) due to the small sample of planets suitable for statistical analysis. We show that the favored model predicts that the planetary mass function should be almost independent of orbital radius at distances where migration dominates the massive planet population. We also study how the radial distribution of planets depends upon the adopted disk model. We find that the distribution can constrain not only changes in the power-law index of the disk viscosity, but also sharp jumps in the efficiency of angular momentum transport that might occur at small radii.Comment: ApJ, in press. References updated to match published versio