Dynamical evolution of thin dispersion-dominated planetesimal disks

We study the dynamics of a vertically thin, dispersion-dominated disk of planetesimals with eccentricities $e$ and inclinations $i$ (normalized in Hill units) satisfying $e >> 1$, $i << e^{-2} << 1$. This situation may be typical for e.g. a population of protoplanetary cores in the end of the oligarchic phase of planet formation. In this regime of orbital parameters planetesimal scattering has an anisotropic character and strongly differs from scattering in thick ($i ~ e$) disks. We derive analytical expressions for the planetesimal scattering coefficients and compare them with numerical calculations. We find significant discrepancies in the inclination scattering coefficients obtained by the two approaches and ascribe this difference to the effects not accounted for in the analytical calculation: multiple scattering events (temporary captures, which may be relevant for the production of distant planetary satellites outside the Hill sphere) and distant interaction of planetesimals prior to their close encounter. Our calculations show that the inclination of a thin, dispersion-dominated planetesimal disk grows exponentially on a very short time scale implying that (1) such disks must be very short-lived and (2) planetesimal accretion in this dynamical phase is insignificant. Our results are also applicable to the dynamics of shear-dominated disks switching to the dispersion-dominated regime.Comment: 16 pages, 12 figures, submitted to A

Fast accretion of small planetesimals by protoplanetary cores

We explore the dynamics of small planetesimals coexisting with massive protoplanetary cores in a gaseous nebula. Gas drag strongly affects the motion of small bodies leading to the decay of their eccentricities and inclinations, which are excited by the gravity of protoplanetary cores. Drag acting on larger ($\gtrsim 1$ km), high velocity planetesimals causes a mere reduction of their average random velocity. By contrast, drag qualitatively changes the dynamics of smaller ($\lesssim 0.1-1$ km), low velocity objects: (1) small planetesimals sediment towards the midplane of the nebula forming vertically thin subdisk; (2) their random velocities rapidly decay between successive passages of the cores and, as a result, encounters with cores typically occur at the minimum relative velocity allowed by the shear in the disk. This leads to a drastic increase in the accretion rate of small planetesimals by the protoplanetary cores, allowing cores to grow faster than expected in the simple oligarchic picture, provided that the population of small planetesimals contains more than roughly 1% of the solid mass in the nebula. Fragmentation of larger planetesimals ($\gtrsim 1$ km) in energetic collisions triggered by the gravitational scattering by cores can easily channel this amount of material into small bodies on reasonable timescales ($< 1$ Myr in the outer Solar System), providing a means for the rapid growth (within several Myr at 30 AU) of rather massive protoplanetary cores. Effects of inelastic collisions between planetesimals and presence of multiple protoplanetary cores are discussed.Comment: 17 pages, 8 figures, additional clarifications, 1 more figure and table adde

Dynamical evolution of planetesimals in protoplanetary disks

The current picture of terrestrial planet formation relies heavily on our understanding of the dynamical evolution of planetesimals -- asteroid-like bodies thought to be planetary building blocks. In this study we investigate the growth of eccentricities and inclinations of planetesimals in spatially homogeneous protoplanetary disks using methods of kinetic theory. We explore disks with a realistic mass spectrum of planetesimals evolving in time, similar to that obtained in self-consistent simulations of planetesimal coagulation. We calculate the behavior of planetesimal random velocities as a function of the planetesimal mass spectrum both analytically and numerically; results obtained by the two approaches agree quite well. Scaling of random velocity with mass can always be represented as a combination of power laws corresponding to different velocity regimes (shear- or dispersion-dominated) of planetesimal gravitational interactions. For different mass spectra we calculate analytically the exponents and time dependent normalizations of these power laws, as well as the positions of the transition regions between different regimes. It is shown that random energy equipartition between different planetesimals can only be achieved in disks with very steep mass distributions (differential surface number density of planetesimals falling off steeper than m^{-4}), or in the runaway tails. In systems with shallow mass spectra (shallower than m^{-3}) random velocities of small planetesimals turn out to be independent of their masses. We also discuss the damping effects of inelastic collisions between planetesimals and of gas drag, and their importance in modifying planetesimal random velocities.Comment: 20 pages, 17 figures, 1 table, submitted to A

The growth of planetary embryos: orderly, runaway, or oligarchic?

We consider the growth of a protoplanetary embryo embedded in a planetesimal disk. We take into account the dynamical evolution of the disk caused by (1) planetesimal-planetesimal interactions, which increase random motions and smooth gradients in the disk, and (2) gravitational scattering of planetesimals by the embryo, which tends to heat up the disk locally and repels planetesimals away. The embryo's growth is self-consistently coupled to the planetesimal disk dynamics. We demonstrate that details of the evolution depend on only two dimensionless parameters incorporating all the physical characteristics of the problem: the ratio of the physical radius to the Hill radius of any solid body in the disk and the number of planetesimals inside the annulus of the disk with width equal to the planetesimal Hill radius. The results of exploration in the framework of our model of several situations typical for protosolar nebula can be summarized as follows: initially, the planetesimal disk dynamics is not affected by the presence of the embryo and the growth of the embryo's mass proceeds very rapidly in the runaway regime. Later on, when the embryo starts being dynamically important, its accretion slows down similar to the ``oligarchic'' growth picture. The scenario of orderly growth suggested by Safronov (1972) is never realized in our calculations; scenario of runaway growth suggested by Wetherill & Stewart (1989) is only realized for a limited range in mass. Slow character of the planetesimal accretion on the oligarchic stage of the embryo's accumulation leads to a considerable increase of the protoplanetary formation timescale compared to that following from a simple runaway accretion picture valid in the homogeneous planetesimal disks.Comment: 42 pages, 13 figures, submitted to A

Planetesimal disk evolution driven by embryo-planetesimal gravitational scattering

The process of gravitational scattering of planetesimals by a massive protoplanetary embryo is explored theoretically. We propose a method to describe the evolution of the disk surface density, eccentricity, and inclination caused by the embryo-planetesimal interaction. It relies on the analytical treatment of the scattering in two extreme regimes of the planetesimal epicyclic velocities: shear-dominated (dynamically ``cold'') and dispersion-dominated (dynamically ``hot''). In the former, planetesimal scattering can be treated as a deterministic process. In the latter, scattering is mostly weak because of the large relative velocities of interacting bodies. This allows one to use the Fokker-Planck approximation and the two-body approximation to explore the disk evolution. We compare the results obtained by this method with the outcomes of the direct numerical integrations of planetesimal orbits and they agree quite well. In the intermediate velocity regime an approximate treatment of the disk evolution is proposed based on interpolation between the two extreme regimes. We also calculate the rate of embryo's mass growth in an inhomogeneous planetesimal disk and demonstrate that it is in agreement with both the simulations and earlier calculations. Finally we discuss the question of the direction of the embryo-planetesimal interaction in the dispersion-dominated regime and demonstrate that it is repulsive. This means that the embryo always forms a gap in the disk around it, which is in contrast with the results of other authors. The machinery developed here will be applied to realistic protoplanetary systems in future papers.Comment: 40 pages, 9 figures, submitted to A

Boundary Layers of Accretion Disks: Wave-Driven Transport and Disk Evolution

Astrophysical objects possessing a material surface (white dwarfs, young stars, etc.) may accrete gas from the disc through the so-called surface boundary layer (BL), in which the angular velocity of the accreting gas experiences a sharp drop. Acoustic waves excited by the supersonic shear in the BL play an important role in mediating the angular momentum and mass transport through that region. Here we examine the characteristics of the angular momentum transport produced by the different types of wave modes emerging in the inner disc, using the results of a large suite of hydrodynamic simulations of the BLs. We provide a comparative analysis of the transport properties of different modes across the range of relevant disc parameters. In particular, we identify the types of modes which are responsible for the mass accretion onto the central object. We find the correlated perturbations of surface density and radial velocity to provide an important contribution to the mass accretion rate. Although the wave-driven transport is intrinsically non-local, we do observe a clear correlation between the angular momentum flux injected into the disc by the waves and the mass accretion rate through the BL. We find the efficiency of angular momentum transport (normalized by thermal pressure) to be a weak function of the flow Mach number. We also quantify the wave-driven evolution of the inner disc, in particular the modification of the angular frequency profile in the disc. Our results pave the way for understanding wave-mediated transport in future three-dimensional, magnetohydrodynamic studies of the BLs.Comment: 16 pages, 9 figures, submitted to MNRA

Atmospheres of protoplanetary cores: critical mass for nucleated instability

We study quasi-static atmospheres of accreting protoplanetary cores for different opacity behaviors and realistic planetesimal accretion rates in various parts of protoplanetary nebula. Atmospheres segregate into those having outer convective zone which smoothly merges with the nebular gas, and those having almost isothermal outer radiative region decoupling atmospheric interior from the nebula. Specific type of atmosphere depends only on the relations between the Bondi radius of the core, photon mean free path in the nebular gas, and the luminosity radius (roughly the size of the sphere which can radiate luminosity of the core at effective temperature equal to the nebular temperature). Cores in the inner parts of protoplanetary disk (within roughly 0.3 AU from the Sun) have large luminosity radii resulting in the atmospheres of the first type, while cores in the giant planet region (beyond several AU) have small luminosity radii and always accumulate massive atmospheres of the second type. Critical core mass for nucleated instability is found to vary as a function of distance from the Sun. It is 5-20 M_Earth at 0.1-1 AU which is too large to permit the formation of ``hot Jupiters'' by nucleated instability near the cores that have grown in situ. In the region of giant planets critical mass is 20-60 M_Earth (for opacity 0.1 cm^2/g) if planetesimal accretion was fast enough for protoplanetary cores to form prior to the nebular gas dissipation. This might indicate that giant planets in the Solar System have gained their atmospheres by nucleated instability only after their cores have accumulated most of the mass in solids during the epoch of oligarchic growth, subsequent to which planetesimal accretion slowed down and cores became supercritical.Comment: 19 pages, 7 figures, submitted to Ap

The Double Pulsar Eclipses I: Phenomenology and Multi-frequency Analysis

The double pulsar PSR J0737-3039A/B displays short, 30 s eclipses that arise around conjunction when the radio waves emitted by pulsar A are absorbed as they propagate through the magnetosphere of its companion pulsar B. These eclipses offer a unique opportunity to probe directly the magnetospheric structure and the plasma properties of pulsar B. We have performed a comprehensive analysis of the eclipse phenomenology using multi-frequency radio observations obtained with the Green Bank Telescope. We have characterized the periodic flux modulations previously discovered at 820 MHz by McLaughlin et al., and investigated the radio frequency dependence of the duration and depth of the eclipses. Based on their weak radio frequency evolution, we conclude that the plasma in pulsar B's magnetosphere requires a large multiplicity factor (~ 10^5). We also found that, as expected, flux modulations are present at all radio frequencies in which eclipses can be detected. Their complex behavior is consistent with the confinement of the absorbing plasma in the dipolar magnetic field of pulsar B as suggested by Lyutikov & Thompson and such a geometric connection explains that the observed periodicity is harmonically related to pulsar B's spin frequency. We observe that the eclipses require a sharp transition region beyond which the plasma density drops off abruptly. Such a region defines a plasmasphere which would be well inside the magnetospheric boundary of an undisturbed pulsar. It is also two times smaller than the expected standoff radius calculated using the balance of the wind pressure from pulsar A and the nominally estimated magnetic pressure of pulsar B.Comment: 9 pages, 7 figures, 3 tables, ApJ in pres