The Formation of Ice Giants in a Packed Oligarchy: Instability and Aftermath

As many as 5 ice giants--Neptune-mass planets composed of 90% ice and rock and 10% hydrogen--are thought to form at heliocentric distances of 10-25 AU on closely packed orbits spaced ~5 Hill radii apart. Such oligarchies are ultimately unstable. Once the parent disk of planetesimals is sufficiently depleted, oligarchs perturb one another onto crossing orbits. We explore both the onset and the outcome of the instability through numerical integrations, including dynamical friction cooling of planets by a planetesimal disk whose properties are held fixed. To trigger instability and the ejection of the first ice giant in systems having an original surface density in oligarchs of Sigma ~ 1 g/cm^2, the disk surface density s must fall below 0.1 g/cm^2. Ejections are predominantly by Jupiter and occur within 10 Myr. To eject more than 1 oligarch requires s < 0.03 g/cm^2. Systems starting with up to 4 oligarchs in addition to Jupiter and Saturn can readily yield solar-system-like outcomes in which 2 surviving ice giants lie inside 30 AU and have their orbits circularized by dynamical friction. Our numerical simulations support the idea that planetary systems begin in more crowded and compact configurations, like those of shear-dominated oligarchies. In contrast to previous studies, we identify s < 0.1 Sigma as the regime relevant for understanding the evolution of the outer solar system, and we encourage future studies to concentrate on this regime while relaxing our assumption of a fixed planetesimal disk.Comment: Accepted to ApJ Jan 27. Incorporates comments from the referee and community at large. 15 pages, 14 figures, including 7 colo

Angular Momentum Transport in Particle and Fluid Disks

We examine the angular momentum transport properties of disks composed of macroscopic particles whose velocity dispersions are externally enhanced (``stirred''). Our simple Boltzmann equation model serves as an analogy for unmagnetized fluid disks in which turbulence may be driven by thermal convection. We show that interparticle collisions in particle disks play the same role as fluctuating pressure forces and viscous dissipation in turbulent disks: both transfer energy in random motions associated with one direction to those associated with another, and convert kinetic energy into heat. The direction of angular momentum transport in stirred particle and fluid disks is determined by the direction of external stirring and by the properties of the collision term in the Boltzmann equation (or its analogue in the fluid problem). In particular, our model problem yields inward transport for vertically or radially stirred disks, provided collisions are suitably inelastic; the transport is outwards in the elastic limit. Numerical simulations of hydrodynamic turbulence driven by thermal convection find inward transport; this requires that fluctuating pressure forces do little to no work, and is analogous to an externally stirred particle disk in which collisions are highly inelastic.Comment: 15 pages; final version accepted by ApJ; minor changes, some clarificatio

Particle Pile-ups and Planetesimal Formation

Solid particles in protoplanetary disks that are sufficiently super-solar in metallicity overcome turbulence generated by vertical shear to gravitationally condense into planetesimals. Super-solar metallicities result if solid particles pile up as they migrate starward as a result of aerodynamic drag. Previous analyses of aerodynamic drift rates that account for mean flow differences between gas and particles yield particle pile-ups. We improve on these studies not only by accounting for the collective inertia of solids relative to that of gas, but also by including the transport of angular momentum by turbulent stresses within the particle layer. These turbulent stresses are derived in a physically self-consistent manner from the structure of marginally Kelvin-Helmholtz turbulent flows. They are not calculated using the usual plate drag formulae, whose use we explain is inappropriate. Accounting for the relative inertia of solids to gas retards, but does not prevent, particle pile-ups, and generates more spatially extended regions of metal enrichment. Turbulent transport hastens pile-ups. We conclude that particle pile-up is a robust outcome in sufficiently passive protoplanetary disks. Connections to observations of circumstellar disks, including the Kuiper Belt, and the architectures of planetary systems are made.Comment: Final revised version, accepted to ApJ. Error corrected in density dependence of Epstein drift rate; correction caused quantitative changes, particularly in high particle density limit. Qualitative conclusion that particle pile-ups can trigger planetesimal formation within protostellar disk lifetime is unaffecte

Three-Dimensional Dynamics of Narrow Planetary Rings

Narrow planetary rings are eccentric and inclined. Particles within a given ring must therefore share the same pericenter and node. We solve for the three-dimensional geometries and mass distributions that enable the Uranian Alpha and Beta rings, and the Saturnian Maxwell and Colombo (Titan) rings, to maintain simultaneous apsidal and nodal lock. Ring self-gravity, interparticle collisions, and the quadrupole field of the host planet balance each other to achieve this equilibrium. We prove that such an equilibrium is linearly stable. Predictions for the Saturnian ringlets to be tested by the Cassini spacecraft include: (1) ringlet masses are of order 1e19 g, (2) surface mass densities should increase from ring midline to ring edges, and (3) rings are vertically warped such that the fractional variation of inclination across the ring is of order 10%. Analogous predictions are made for the Uranian rings. Simultaneous apsidal and nodal locking forces the narrowest portion of the ring--its ``pinch,'' where self-gravitational and collisional forces are strongest--to circulate relative to the node, and introduces previously unrecognized time-varying forces perpendicular to the planet's equator plane. We speculate that such periodic stressing might drive kilometer-scale bending waves at a frequency twice that of apsidal precession. Such flexing might be observed over a few weeks by Cassini.Comment: Final revised version, ApJ, in pres

The Circumbinary Ring of KH 15D

The light curves of the pre-main-sequence star KH 15D from the years 1913--2003 can be understood if the star is a member of an eccentric binary that is encircled by a vertically thin, inclined ring of dusty gas. Eclipses occur whenever the reflex motion of a star carries it behind the circumbinary ring; the eclipses occur with period equal to the binary orbital period of 48.4 days. Features of the light curve--including the amplitude of central reversals during mid-eclipse, the phase of eclipse with respect to the binary orbit phase, the level of brightness out-of-eclipse, the depth of eclipse, and the eclipse duty cycle--are all modulated on the timescale of nodal regression of the obscuring ring, in accord with the historical data. The ring has a mean radius near 3 AU and a radial width that is likely less than this value. While the inner boundary could be shepherded by the central binary, the outer boundary may require an exterior planet to confine it against viscous spreading. The ring must be vertically warped to maintain a non-zero inclination. Thermal pressure gradients and/or ring self-gravity can readily enforce rigid precession. In coming years, as the node of the ring regresses out of our line-of-sight toward the binary, the light curve from the system should cycle approximately back through its previous behavior. Near-term observations should seek to detect a mid-infrared excess from this system; we estimate the flux densities from the ring to be 3 mJy at wavelengths of 10--100 microns.Comment: Final version, ApJ, v607, 913 (June 1); includes prediction for full spectral energy distribution (new Figure 5

Brownian Motion in Planetary Migration

A residual planetesimal disk of mass 10-100 Earth masses remained in the outer solar system following the birth of the giant planets, as implied by the existence of the Oort cloud, coagulation requirements for Pluto, and inefficiencies in planet formation. Upon gravitationally scattering planetesimal debris, planets migrate. Orbital migration can lead to resonance capture, as evidenced here in the Kuiper and asteroid belts, and abroad in extra-solar systems. Finite sizes of planetesimals render migration stochastic ("noisy"). At fixed disk mass, larger (fewer) planetesimals generate more noise. Extreme noise defeats resonance capture. We employ order-of-magnitude physics to construct an analytic theory for how a planet's orbital semi-major axis fluctuates in response to random planetesimal scatterings. To retain a body in resonance, the planet's semi-major axis must not random walk a distance greater than the resonant libration width. We translate this criterion into an analytic formula for the retention efficiency of the resonance as a function of system parameters, including planetesimal size. We verify our results with tailored numerical simulations. Application of our theory reveals that capture of Resonant Kuiper belt objects by a migrating Neptune remains effective if the bulk of the primordial disk was locked in bodies having sizes 1000 km was less than a few percent. Coagulation simulations produce a size distribution of primordial planetesimals that easily satisfies these constraints. We conclude that stochasticity did not interfere with, nor modify in any substantive way, Neptune's ability to capture and retain Resonant Kuiper belt objects during its migration

Binaries in the Kuiper Belt

Binaries have played a crucial role many times in the history of modern astronomy and are doing so again in the rapidly evolving exploration of the Kuiper Belt. The large fraction of transneptunian objects that are binary or multiple, 48 such systems are now known, has been an unanticipated windfall. Separations and relative magnitudes measured in discovery images give important information on the statistical properties of the binary population that can be related to competing models of binary formation. Orbits, derived for 13 systems, provide a determination of the system mass. Masses can be used to derive densities and albedos when an independent size measurement is available. Angular momenta and relative sizes of the majority of binaries are consistent with formation by dynamical capture. The small satellites of the largest transneptunian objects, in contrast, are more likely formed from collisions. Correlations of the fraction of binaries with different dynamical populations or with other physical variables have the potential to constrain models of the origin and evolution of the transneptunian population as a whole. Other means of studying binaries have only begun to be exploited, including lightcurve, color, and spectral data. Because of the several channels for obtaining unique physical information, it is already clear that binaries will emerge as one of the most useful tools for unraveling the many complexities of transneptunian space.Comment: Accepted for inclusion in "The Kuiper Belt", University of Arizona Press, Space Science Series Corrected references in Table

Dust Dynamics, Surface Brightness Profiles, and Thermal Spectra of Debris Disks: The Case of AU Mic

AU Microscopii is a 12 Myr old M dwarf that harbors an optically thin, edge-on disk of dust. The scattered light surface brightness falls with projected distance b from the star as b^-a; within b = 43 AU, a = 1-2, while outside 43 AU, a = 4-5. We devise a theory to explain this profile. At a stellocentric distance r = r_BR = 43 AU, we posit a ring of parent bodies on circular orbits: the "birth ring," wherein micron-sized grains are born from the collisional attrition of parent bodies. The "inner disk" at r < r_BR contains grains that migrate inward by corpuscular and Poynting-Robertson (CPR) drag. The "outer disk" at r > r_BR comprises grains just large enough to remain bound to the star, on orbits rendered highly eccentric by stellar wind and radiation pressure. How the vertical optical depth tau scales with r depends on the fraction of grains that migrate inward by CPR drag without suffering a collision. If this fraction is large, the inner disk and birth ring share the same optical depth, and tau scales as r^-5/2 in the outer disk. By contrast, under collision-dominated conditions, the inner disk is empty, and tau scales as r^-3/2 outside. These scaling relations, which we derive analytically and confirm numerically, are robust against uncertainties in the grain size distribution. By simultaneously modeling the surface brightness and thermal spectrum, we break model degeneracies to establish that the AU Mic system is collision-dominated, and that its narrow birth ring contains a lunar mass of decimeter-sized bodies. The inner disk is devoid of micron-sized grains; the surface brightness at b < 43 AU arises from light forward scattered by the birth ring. Inside b = 43 AU, the disk's V-H color should not vary with b; outside, the disk must become bluer as ever smaller grains are probed.Comment: Final proofed version to be published in ApJ; no significant changes from version