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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 MNRAS

We analyze six X-ray bright elliptical galaxies, observed with Chandra and XMM-Newton, and approximate their gravitational potentials by isothermal spheres phi(r)=v_c^2 ln(r) over a range of radii from ~0.5 to ~25 kpc. We then compare the circular speed v_c derived from X-ray data with the estimators available from optical data. In particular we discuss two simple and robust procedures for evaluating the circular speed of the galaxy using the observed optical surface brightness and the line-of-sight velocity dispersion profiles. The best fitting relation between the circular speeds derived from optical observations of stars and X-ray observations of hot gas is v_{c,opt}~ \eta * v_{c,X}, where \eta=1.10-1.15 (depending on the method), suggesting, albeit with large statistical and systematic uncertainties, that non-thermal pressure on average contributes ~20-30% of the gas thermal pressure., Comment: 24 pages, 15 figures; Accepted for publication in MNRAS

Near-future surveys promise a dramatic improvement in the number and precision of astrometric, photometric and spectroscopic measurements of stars in the Milky Way's disk. We examine the impact of such surveys on our understanding of the Galaxy by "observing" particle realizations of non-axisymmetric disk distributions orbiting in an axisymmetric halo with appropriate errors and then attempting to recover the underlying potential using a Markov Chain Monte Carlo (MCMC) approach. We demonstrate that the azimuthally averaged gravitational force field in the Galactic plane--and hence, to a lesser extent, the Galactic mass distribution--can be tightly constrained over a large range of radii using a variety of types of surveys so long as the error distribution of the measurements of the parallax, proper motion and radial velocity are well-understood and the disk is surveyed globally. One advantage of our method is that the target stars can be selected non-randomly in real or apparent-magnitude space to ensure just such a global sample without biasing the results. Assuming we can always measure the line-of-sight velocity of a star with at least 1 km/s precision, we demonstrate that the force field can be determined to better than ~1% for Galactocentric radii in the range R=4-20 kpc We conclude that near-future surveys, like SIM Lite, Gaia, and VERA, will provide the first precise mapping of the gravitational force field in the region of the Galactic disk., Comment: 41 pages and 10 figures, accepted for publication in ApJ

We show that the algorithm proposed by Gauss to compute the secular evolution of gravitationally interacting Keplerian rings extends naturally to softened gravitational interactions. The resulting tool is ideal for the study of the secular dynamical evolution of nearly Keplerian systems such as stellar clusters surrounding black holes in galactic nuclei, cometary clouds, or planetesimal discs. We illustrate its accuracy, efficiency and versatility on a variety of configurations. In particular, we examine a secularly unstable unstable system of counter-rotating disks, and follow the unfolding and saturation of the instability into a global, uniformly precessing, lopsided (m=1) mode., Comment: 28 pages, 11 figures; minor changes to text and figures. To be published in MNRAS

We compare the gravitational potential profiles of the elliptical galaxies NGC 4486 (M87) and NGC 1399 (the central galaxy in the Fornax cluster) derived from X-ray and optical data. This comparison suggests that the combined contribution of cosmic rays, magnetic fields and micro-turbulence to the pressure is ~10% of the gas thermal pressure in the cores of NGC 1399 and M87, although the uncertainties in our model assumptions (e.g., spherical symmetry) are sufficiently large that the contribution could be consistent with zero. In the absence of any other form of non-thermal pressure support, these upper bounds translate into upper limits on the magnetic field of ~10-20 muG at a distance of 1'-2' from the centers of NGC1399 and M87. We show that these results are consistent with the current paradigm of cool cluster cores, based on the assumption that AGN activity regulates the thermal state of the gas by injecting energy into the intra-cluster medium. The limit of ~10-20% on the energy density in the form of relativistic protons applies not only to the current state of the gas, but essentially to the entire history of the intra-cluster medium, provided that cosmic ray protons evolve adiabatically and that their spatial diffusion is suppressed., Comment: Accepted for MNRAS. 19 pages; 14 figures; expanded version in response to comments from the referee

We explore the possibility that the observed eccentricity distribution of extrasolar planets arose through planet-planet interactions, after the initial stage of planet formation was complete. Our results are based on ~3250 numerical integrations of ensembles of randomly constructed planetary systems, each lasting 100 Myr. We find that for a remarkably wide range of initial conditions the eccentricity distributions of dynamically active planetary systems relax towards a common final equilibrium distribution, well described by the fitting formula dn ~ e exp[-1/2 (e/0.3)^2] de. This distribution agrees well with the observed eccentricity distribution for e > 0.2, but predicts too few planets at lower eccentricities, even when we exclude planets subject to tidal circularization. These findings suggest that a period of large-scale dynamical instability has occurred in a significant fraction of newly formed planetary systems, lasting 1--2 orders of magnitude longer than the ~1 Myr interval in which gas-giant planets are assembled. This mechanism predicts no (or weak) correlations between semimajor axis, eccentricity, inclination, and mass in dynamically relaxed planetary systems. An additional observational consequence of dynamical relaxation is a significant population of planets (>10%) that are highly inclined (>25deg) with respect to the initial symmetry plane of the protoplanetary disk; this population may be detectable in transiting planets through the Rossiter-McLaughlin effect., Comment: Accepted to ApJ, conclusions updated to reflect the current observational constraints

One of the parameters fitted by Doppler radial velocity measurements of extrasolar planetary systems is omega, the argument of pericenter of a given planet's orbit referenced to the plane of the sky. Curiously, the omega's of the outer two planets orbiting Upsilon Andromedae are presently nearly identical: Delta-omega = omega_D - omega_C = 4.8 deg +/- 4.8 deg (1 sigma). This observation is least surprising if planets C and D occupy orbits that are seen close to edge-on (sin i_C, sin i_D > 0.5) and whose mutual inclination Theta does not exceed 20 deg. In this case, planets C and D inhabit a secular resonance in which Delta-omega librates about 0 deg with an amplitude of 30 deg and a period of 4000 yr. The resonant configuration spends about one-third of its time with |Delta-omega| < 10 deg. If Theta > 40 deg, either Delta-omega circulates or the system is unstable. This instability is driven by the Kozai mechanism which couples the eccentricity of planet C to Theta to drive the former quantity to values approaching unity. Our expectation that Theta < 20 deg suggests that planets C and D formed in a flattened, circumstellar disk, and may be tested by upcoming astrometric measurements with the FAME satellite., Comment: Refereed version, accepted by AJ, to appear in September 2001 issue

Partially phase-mixed structures in galaxies occupy a complex surface of dimension D in six-dimensional phase space. The appearance of such structures to observers is determined by their projection into a subspace whose dimensionality K is determined by the number of observables (e.g. sky position, distance, radial velocity, etc.). We discuss the expected dimensionality of phase-space structures and suggest that the most prominent features in surveys with K=D or K>D will be stable singularities (catastrophes). The simplest of these are the shells seen in the outer parts of elliptical galaxies., Comment: 14 pages, Latex, 4 figures. Submitted to MNRAS

Why do galactic bars rotate with high pattern speeds, when dynamical friction should rapidly couple the bar to the massive, slowly rotating dark halo? This longstanding paradox may be resolved by considering the dynamical interactions between the galactic disk and structures in the dark halo. Dynamical friction between small-scale halo structure and the disk spins up and flattens the inner halo, thereby quenching the dynamical friction exerted by the halo on the bar; at the same time the halo heats and thickens the disk, perhaps forming a rapidly rotating bulge. Two possible candidates for the required halo structures are massive black holes and tidal streamers from disrupted precursor halos. More generally, gravitational scattering from phase-wrapped inhomogeneities represents a novel relaxation process in stellar systems, intermediate between violent relaxation and two-body relaxation, which can isotropize the distribution function at radii where two-body relaxation is not effective., Comment: 7 pages, Latex, no figures. Submitted to MNRAS

Black holes, an extreme consequence of the mathematics of General Relativity, have long been suspected of being the prime movers of quasars, which emit more energy than any other objects in the Universe. Recent evidence indicates that supermassive black holes, which are probably quasar remnants, reside at the centers of most galaxies. As our knowledge of the demographics of these relics of a violent earlier Universe improve, we see tantalizing clues that they participated intimately in the formation of galaxies and have strongly influenced their present-day structure., Comment: 20 pages, - This is a near-duplicate of the paper in Nature 395, A14, 1998 (Oct 1)

This report provides a summary and bibliography for the research activities at the Princeton University Observatory and Department of Astrophysical Sciences during the period July 1, 1997 to June 30, 1998., Comment: 39 pages, no figures, LaTeX

We construct a simple symplectic map to study the dynamics of eccentric orbits in non-spherical potentials. The map offers a dramatic improvement in speed over traditional integration methods, while accurately representing the qualitative details of the dynamics. We focus attention on planar, non-axisymmetric power-law potentials, in particular the logarithmic potential. We confirm the presence of resonant orbit families (``boxlets'') in this potential and uncover new dynamics such as the emergence of a stochastic web in nearly axisymmetric logarithmic potentials. The map can also be applied to triaxial, lopsided, non-power-law and rotating potentials., Comment: 29 pages, 9 figures, Latex, submitted to MNRAS

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/

We present a set of structural parameters for the central parts of 57 early-type galaxies observed with the Planetary Camera of the Hubble Space Telescope. These parameters are based on a new empirical law that successfully characterizes the centers of early type galaxies. This empirical law assumes that the surface brightness profile is a combination of two power laws with different slopes gamma and beta for the inner and outer regions. Conventional structural parameters such as core radius and central surface brightness are replaced by break radius r_b, where the transition between power-law slopes takes place, and surface brightness mu_b at that radius. An additional parameter alpha describes the sharpness of the break. The structural parameters are derived using a chi-squared minimization process applied to the mean surface brightness profiles. The resulting model profiles generally give very good agreement to the observed profiles out to the radius of 10 arcseconds imaged by the Planetary Camera. Exceptions include galaxies which depart from pure power-laws at large radius, those with strong nuclear components, and galaxies partly obscured by dust. The uncertainties in the derived parameters are estimated using Monte-Carlo simulations which test the stability of solutions in the face of photon noise and the effects of the deconvolution process. The covariance of the structural parameters is examined by computing contours of constant chi squared in multi-dimensional parameter space., Comment: 14 pages, 1 table, 8 postscript figures, aaspp4.sty included, accepted for publication in the Astronomical Journal

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/

This paper contains all known errata in the book "Galactic Dynamics"., Comment: 3 pages, TeX

It is proposed that the incomplete rings of Neptune consist of a number of short arcs centered on the corotation resonances of a single satellite. The satellite must have a radius of the order of 100 km or more and move on an inclined orbit. Corotation resonances are located at potential maxima. Thus, mechanical energy dissipated by interparticle collisions must be continually replenished to prevent the arcs from spreading. It is shown that each corotation resonance is associated with a nearby Lindblad resonance, which excites the ring particles' orbital eccentricity, thus supplying the energy required to maintain the arc. The ultimate energy reservoir is the satellite's orbital energy. Therefore, interaction with the arcs damps the satellite's orbital inclination. The self-gravity of the arcs limits their contraction and enforces a relation between arc length and mass. The estimated arc masses are so small, of the order of 10 to the 16th g, that the satellite's orbital inclination suffers negligible decay over the age of the solar system. The inferred surface mass densities are comparable to those found in the major rings of Saturn and Uranus.

The proposal of Harwit and Salpeter (1973) that gamma-ray bursts are due to impacts of comets onto neutron stars is examined further. It is assumed that most stars are formed with comet clouds similar to the Oort comet cloud which surrounds the sun, and it is suggested that there are at least four mechanisms by wich neutron stars may be formed while retaining their comet clouds: a spherically symmetric supernova explosion in an isolated star, accretion-induced collapse of a white dwarf in a cataclysmic variable with a very low mass secondary, accretion-induced collapse of a white dwarf in a wide binary with a low-mass giant companion, and coalescence of a close binary composed of two white dwarfs. Estimates are given of the cometary impact rates for such systems. It is suggested that if the wide binary scenario is correct, optical bursts may arise from the impact of comets onto the white dwarf remnant of the giant companion.

The effects of the gravitational interactions between Saturn's rings and the coorbital satellites, Janus and Epimetheus, on the 1:1 horseshoe resonance between these moons is examined. It is shown that the 7:6 resonance of these moons, which presumably maintains the sharp outer edge of the A ring, leads to a rapid tightening of the coorbital lock. The results lead to the prediction that the orbital configuration might evolve from the current horseshoe-type lock to one of tadpole orbits around a single Lagrangian point in about 20 myr.

In the present study of the viscosity of a differentially rotating particle disk, in the limiting case where the particles are densely packed and their collective behavior resembles that of a liquid, the pressure tensor is derived from both the equations of hydrodynamics and a simple kinetic model of collisions due to Haff (1983). Density waves and narrow circular rings are unstable if the liquid approximation applies, and the consequent nonlinear perturbations may generate 'splashing' of the ring material in the vertical direction. These results are pertinent to the origin of the ellipticities of ringlets, the nonaxisymmetric features near the outer edge of the Saturn B ring, and unexplained residuals in kinematic models of the Saturn and Uranus rings.

It is suggested that the mass of the individual unseen 'disk things' which furnish unseen mass satisfies an inequality that is based on the demonstrated existence of a large number of wide binary stars in the Galaxy whose projected separations are as large as 0.1 pc. Since these binaries are weakly bound, with orbital velocities of 0.3 km/sec and periods of the order of 1 million years, they can be disrupted by passing stars or other pointlike objects. This implies that the mass of the unseen disk objects, which provide about half the local mass density, must be less than 2 solar masses.

Resonant gravitational interactions between a ring and a satellite produce secular variations of their orbital inclinations. Interactions at vertical resonances, analogous to Lindblad resonances but involving inclinations instead of eccentricities, excite inclinations. There is no inclination analog of the corotation resonance. An equatorial ring changes the inclination of a nearby satellite in qualitatively the same way that a satellite in an equatorial orbit changes the inclination of a nearby ring. Viscous dissipation in a ring leads to an equilibrium value of its inclination. These results provide a basis for discussing the origins of the inclinations of planetary rings.

Voyager photographic data are used for microscopic and macroscopic analyses of the dynamics of the Saturn rings. Attention is given to particle collisions, local velocity dispersion, the particle size distribution, mechanisms governing the ring structure and interactive effects with Saturn satellite orbits. The mechanical properties and collision dynamics of the particles are discussed, along with models for perturbed and unperturbed disks. The forces involved and true scale of particle erosion are considered, as are instabilities in the rings, satellite torques, resonances, and the time scale for the evolution of the satellite orbits. Finally, the possibility that angular momentum exchanges are occurring between individual rings and satellite orbits is examined.

The paper investigates the evolution of eccentric rings under the influence of (1) differential precession due to the planetary quadrupole moment; (2) self-gravity; (3) viscous forces due to interparticle collisions; and (4) eccentricity excitation by shepherd satellites. The principal conclusions are that: (1) uniform precession can be enforced by self-gravity; the resulting configuration is both dynamically and secularly stable. (2) due to viscous forces the line of apsides at the inner ring edge is not exactly aligned with the line of apsides at the outer edge; the apse shift may be detectable in the alpha and beta rings of Uranus; (3) the mean eccentricity is determined by a balance between viscous damping and excitation by shepherds; (4) and the dimensionless eccentricity gradient is expected to be positive and of order unity in most eccentric rings, as observed.

The two-dimensional analogs to Riemann S-type ellipsoids are considered as simple fluid models for galactic bars. Their equilibrium is discussed and their evolution under the effects of viscosity, gravitational radiation, and external torques from other components of the galaxy is computed. As a bar loses angular momentum due to external torques, it generally becomes less elongated, and its figure speed may either increase or decrease.

The velocity ellipsoid in a particle disk near an isolated satellite resonance is determined by solving the Boltzmann moment equations, and solutions are obtained that are stationary functions of the azimuthal angle in a coordinate frame which rotates with the pattern speed of the perturbation potential. The magnitude of the deformation rate tensor in a perturbed particle disk is bounded from above by an expression which includes the orbital angular velocity, the optical depth, and a dimensionless constant of order unity. It is also found that, in sufficiently perturbed regions, there are ranges of azimuthal angle over which the radial component of the angular momentum flux is negative. It is also possible for the angular momentum luminosity to be negative. These results are pertinent to the understanding of sharp edges and density wave decay in planetary rings.

Published data on the orbits of the rings and satellites of Uranus are analyzed in terms of their dynamic interaction, in an attempt to explain discrepancies between current kinematic models of the ring system and recent, very accurate occultation observations. Apse precession, short-term perturbations, satellite resonance, and the movement of the planet and rings with respect to the barycenter of the planet-satellite system are investigated. It is found that the five known satellites could cause the observed ring residuals by shifting the barycenter, but not by direct or resonance effects. Hypothetical interring shepherd satellites of limited mass (such as about 10 to the 19th g for a shepherd between rings 4 and 5), possibly affected by known-satellite resonances, could influence the rings directly.

It is proposed that node alignment across an inclined ring is maintained by the self-gravity of the ring, which cancels the tendency for differential node precession due to the planet's multiple moments. The analysis is confined to circular rings, which is a self-consistent approximation because the interactions among circular and inclined ringlets do not generate secular perturbations of eccentricity.

The first-order perturbations of orbital eccentricity and apse precession rate for the case of a narrow ring, due to a close satellite whose orbit is also eccentric, are described by means of a Hamiltonian. The present treatment covers cases in which the satellite crosses the ring, and the level curves of the Hamiltonian are displayed for several parameter values. The results obtained are applied to the interaction of Saturn's F ring with its inner shepherd satellite.

A multiple three-body technique is used to study the orbital evolution of satellite galaxies which is similar to the N-body method but neglects two-body forces between stars in the halo of the parent galaxy. It is found that, for satellites orbiting within the halo, Chandrasekhar's (1960) dynamical friction formula accurately describes the orbital decay rate, including its variation with satellite mass and size and with the number density and mass of halo stars. Significant frictional forces are present even outside the halo, and the orbital decay rate, instead of depending on the procedure used to place the satellite in its orbit, is determined only by the current orbital parameters. This semirestricted N-body method is sufficiently fast to have permitted the running of 200 simulations to date, many more than would have been possible by means of the conventional N-body technique.

The ring systems of Saturn and Uranus exhibit several sharp edges, across which the optical depth drops from order unity to essentially zero. At least two and perhaps all of these features are associated with the location of orbital resonances between a satellite and the ring particles. It is remarkable that the optical depth varies on a distance scale which is much finer than that over which angular momentum can be transferred between a satellite and the ring material. The important features of this phenomenon are: (1) a perturbed band of width Delta a/a of about (satellite mass/planetary mass) to the 1/2 power adjacent to the edge within which the angular momentum transfer occurs, (2) streamlines perturbed such that the angular momentum luminosity decreases smoothly across the band to zero at the edge even though the optical depth remains constant, and (3) dynamical equilibrium requires a relation between the random velocity, the rate of deformation and the optical depth.

Self-consistent dynamic models of the M31 nucleus were constructed which satisfy Stratoscope II and ground-based photometry, as well as ground-based velocity dispersion measurements. Two models designed to yield the highest and lowest mass-to-light ratios (M/L) consistent with observations are concentrated on. It is found that self-consistent static models of the M31 nucleus can be constructed with a nuclear mass-to-blue magnitude ratio between 0 and 50, and that it is very difficult to construct high M/L models with plausible evolutionary histories. The low values of M/L that were found suggest that the nucleus must be composed of stars on nearly radial orbits whose apocenters lie well outside the observable part of the nucleus. Such nuclei may be formed either during core collapse of a preexisting dense nucleus or by orbital decay of globular clusters in a triaxial bulge.

The physical processes that occur in planetary rings are discussed. The theoretical arguments leading to the conclusion that Saturn's rings are solid particles in nonuniform rotation are summarized, and the optical depth, thickness, and particle size of the rings are discussed. The influence of nearby satellites on the rings is analyzed, and asymmetries in the rings are briefly discussed. What is known of the rings of Uranus and of Jupiter's ring is summarized. Some of the dynamical processes and influences that are expected to be incorporated in more advanced theories of planetary rings are reviewed in detail, including radiation drag, plasma drag, interparticle collisions and viscosity, resonances with external satellites, shepherd satellites and moonlets. Finally, the orbital evolution of the shepherd satellites caused by the rings is estimated.

The effect of gravitational perturbations from a nearby satellite on the eccentricity e of a narrow particulate ring is considered. The perturbations near a resonance in an eccentric ring may be divided into corotation and Lindblad terms. For small e, the corotation terms damp e, whereas the Lindblad terms excite e. In the absence of saturation the corotation terms win by a small margin, and e damps. However, if the perturbations open gaps at the strongest resonances, then the Lindblad terms win, and e grows. This result offers an explanation for the existence of both circular and eccentric rings around Uranus. It is also shown that eccentricity changes induced by circular rings on eccentric satellite orbits are similar to those induced by satellites with circular orbits on eccentric rings.

The rate at which angular momentum and energy are transferred between a disk and a satellite which orbit a central mass is calculated. It is shown that the angular momentum and energy transfer at Lindblad resonances tends to increase the satellite's orbit to lowest order in eccentricity, whereas the transfer at corotation resonances tends to decrease it. The results are applied to the interaction between Jupiter and the protoplanetary disk. The angular momentum transfer is shown to be so rapid that substantial changes in both the structure of the disk and the orbit of Jupiter must have taken place on a time scale of a few thousand years.

The linear response of a differentially rotating two-dimensional gas disk, both with and without self-gravity, to a rigidly rotating external potential is calculated on the assumptions that the speed of sound is much smaller that the orbital velocity and that the external potential varies on the scale of the disk radius. The results show that: (1) the external potential exerts torques on the disk only at the Lindblad and corotation resonances; (2) the torque is positive at the outer Lindblad resonance and negative at the inner Lindblad resonance; (3) the torque at corotation has the sign of the radial vorticity gradient; and (4) the torques are of the same order of magnitude at both types of resonance and independent of the speed of sound in the disk. It is found that the external potential also excites density waves in the vicinity of the Lindblad and corotation resonances, that the long trailing wave is excited at a Lindblad resonance, and that short trailing waves are excited at the corotation resonance. The behavior of particle disks is briefly discussed, and the external torques on particle disks are proven to be identical to those on gas disks