The Distortion of the Cosmic Microwave Background by the Milky Way

The Milky Way can act as a large-scale weak gravitational lens of the cosmic microwave background (CMB). We study this effect using a photon ray-tracing code and a Galactic mass distribution with disk, bulge and halo components. For an observer at the Sun's coordinates in the Galaxy, the bending of CMB photon paths is limited to less than one arcsecond, and only for rays that pass within a few degrees of the Galactic Center. However, the entire sky is affected, resulting in global distortions of the CMB on large angular scales. These distortions can cause the low-order multipoles of a spherical harmonic expansion of the CMB sky temperature to leak into higher-order modes. Thus the component of the CMB dipole that results from the Local Group's motion relative to the local cosmic frame of rest contributes to higher-order moments for an observer in the solar system. With our ray-tracing code we show that the phenomenon is not sensitive to the specific choice of Galactic potential. We also quantitatively rule it out as a contributor to CMB anomalies such as power asymmetry or correlated alignment of low-order multipole moments.Comment: 4 pages, 3 Figures, Brief Report in Physical Review D, accepted for publicatio

Evolution of a ring around the Pluto-Charon binary

We consider the formation of satellites around the Pluto-Charon binary. An early collision between the two partners likely produced the binary and a narrow ring of debris, out of which arose the moons Styx, Nix, Kerberos and Hydra. How the satellites emerged from the compact ring is uncertain. Here we show that a particle ring spreads from physical collisions and collective gravitational scattering, similar to migration. Around a binary, these processes take place in the reference frames of "most circular" orbits, akin to circular ones in a Keplerian potential. Ring particles damp to these orbits and avoid destructive collisions. Damping and diffusion also help particles survive dynamical instabilities driven by resonances with the binary. In some situations, particles become trapped near resonances that sweep outward with the tidal evolution of the Pluto-Charon binary. With simple models and numerical experiments, we show how the Pluto-Charon impact ring may have expanded into a broad disk, out of which grew the circumbinary moons. In some scenarios, the ring can spread well beyond the orbit of Hydra, the most distant moon, to form a handful of smaller satellites. If these small moons exist, New Horizons will find them.Comment: 44 pages of text, 10 figures, ApJ, accepted. (In this version, Section 2 and Figure 1 are new, outlining our approach to the problem of satellite formation around the Pluto-Charon binary.

Making Planet Nine: Pebble Accretion at 250--750 AU in a Gravitationally Unstable Ring

We investigate the formation of icy super-Earth mass planets within a gravitationally unstable ring of solids orbiting at 250-750 AU around a 1 solar mass star. Coagulation calculations demonstrate that a system of a few large oligarchs and a swarm of pebbles generates a super-Earth within 100-200 Myr at 250 AU and within 1-2 Gyr at 750 AU. Systems with more than ten oligarchs fail to yield super-Earths over the age of the solar system. As these systems evolve, destructive collisions produce detectable debris disks with luminosities of $10^{-5}$ to $10^{-3}$ relative to the central star.Comment: 24 pages of text, 1 table, 8 figures, ApJ submitted, comments welcom

Numerical Simulations of Collisional Cascades at the Roche Limits of White Dwarf Stars

We consider the long-term collisional and dynamical evolution of solid material orbiting in a narrow annulus near the Roche limit of a white dwarf. With orbital velocities of 300 km/sec, systems of solids with initial eccentricity $e \gtrsim 10^{-3}$ generate a collisional cascade where objects with radii $r \lesssim$ 100--300 km are ground to dust. This process converts 1-100 km asteroids into 1 $\mu$m particles in $10^2 - 10^6$ yr. Throughout this evolution, the swarm maintains an initially large vertical scale height $H$. Adding solids at a rate $\dot{M}$ enables the system to find an equilibrium where the mass in solids is roughly constant. This equilibrium depends on $\dot{M}$ and $r_0$, the radius of the largest solid added to the swarm. When $r_0 \lesssim$ 10 km, this equilibrium is stable. For larger $r_0$, the mass oscillates between high and low states; the fraction of time spent in high states ranges from 100% for large $\dot{M}$ to much less than 1% for small $\dot{M}$. During high states, the stellar luminosity reprocessed by the solids is comparable to the excess infrared emission observed in many metallic line white dwarfs.Comment: 37 pages of text, 12 figures, ApJ, accepte

Rapid Formation of Icy Super-Earths and the Cores of Gas Giant Planets

We describe a coagulation model that leads to the rapid formation of super-Earths and the cores of gas giant planets. Interaction of collision fragments with the gaseous disk is the crucial element of this model. The gas entrains small collision fragments, which rapidly settle to the disk midplane. Protoplanets accrete the fragments and grow to masses of at least 1 Earth mass in roughly 1 Myr. Our model explains the mass distribution of planets in the Solar System and predicts that super-Earths form more frequently than gas giants in low mass disks.Comment: ApJLetters, accepted; 10 pages of text and 2 figure

Making Planet Nine: A Scattered Giant in the Outer Solar System

Correlations in the orbits of several minor planets in the outer solar system suggest the presence of a remote, massive Planet Nine. With at least ten times the mass of the Earth and a perihelion well beyond 100 AU, Planet Nine poses a challenge to planet formation theory. Here we expand on a scenario in which the planet formed closer to the Sun and was gravitationally scattered by Jupiter or Saturn onto a very eccentric orbit in an extended gaseous disk. Dynamical friction with the gas then allowed the planet to settle in the outer solar system. We explore this possibility with a set of numerical simulations. Depending on how the gas disk evolves, scattered super-Earths or small gas giants settle on a range of orbits, with perihelion distances as large as 300 AU. Massive disks that clear from the inside out on million-year time scales yield orbits that allow a super-Earth or gas giant to shepherd the minor planets as observed. A massive planet can achieve a similar orbit in a persistent, low-mass disk over the lifetime of the solar system.Comment: 14 pages of text, 2 tables, 5 figures, ApJ, submitte

Formation of Super-Earth Mass Planets at 125-250 AU from a Solar-type Star

We investigate pathways for the formation of icy super-Earth mass planets orbiting at 125-250 AU around a 1 solar mass star. An extensive suite of coagulation calculations demonstrates that swarms of 1 cm to 10 m planetesimals can form super-Earth mass planets on time scales of 1-3 Gyr. Collisional damping of 0.01-100 cm particles during oligarchic growth is a highlight of these simulations. In some situations, damping initiates a second runaway growth phase where 100-3000 km protoplanets grow to super-Earth sizes. Our results establish the initial conditions and physical processes required for in situ formation of super-Earth planets at large distances from the host star. For nearby dusty disks in HD 107146, HD 202628, and HD 207129, ongoing super-Earth formation at 80-150 AU could produce gaps and other structures in the debris. In the solar system, forming a putative planet X at a 1000 AU) requires a modest (very massive) protosolar nebula.Comment: 48 pages of text, 23 figures, ApJ in press, revised version contains new text on aspects of the calculations and a more comprehensive description of the origin of the second phase of runaway growt

Migration of small moons in Saturn's rings

The motions of small moons through Saturn's rings provide excellent tests of radial migration models. In theory, torque exchange between these moons and ring particles leads to radial drift. We predict that moons with Hill radii r_H ~ 2-24 km should migrate through the A ring in 1000 yr. In this size range, moons orbiting in an empty gap or in a full ring eventually migrate at the same rate. Smaller moons or moonlets -- such as the propellers (e.g., Tiscareno et al. 2006) -- are trapped by diffusion of disk material into corotating orbits, creating inertial drag. Larger moons -- such as Pan or Atlas -- do not migrate because of their own inertia. Fast migration of 2-24 km moons should eliminate intermediate-size bodies from the A ring and may be responsible for the observed large-radius cutoff of r_H ~ 1-2 km in the size distribution of the A ring's propeller moonlets. Although the presence of Daphnis (r_H ~ 5 km) inside the Keeler gap challenges this scenario, numerical simulations demonstrate that orbital resonances and stirring by distant, larger moons (e.g., Mimas) may be important factors. For Daphnis, stirring by distant moons seems the most promising mechanism to halt fast migration. Alternatively, Daphnis may be a recent addition to the ring that is settling into a low inclination orbit in ~10^3 yr prior to a phase of rapid migration. We provide predictions of observational constraints required to discriminate among possible scenarios for Daphnis.Comment: ApJ, accepted; 47 pages, 14 figure

A Pluto-Charon Sonata: The Dynamical Architecture of the Circumbinary Satellite System

Using a large suite of n-body simulations, we explore the discovery space for new satellites in the Pluto-Charon system. For the adopted masses and orbits of the known satellites, there are few stable prograde or polar orbits with semimajor axes $a \lesssim 1.1~a_H$, where $a_H$ is the semimajor axis of the outermost moon Hydra. Small moons with radii $r \lesssim$ 2 km and $a \lesssim 1.1~a_H$ are ejected on time scales ranging from several yr to more than 10 Myr. Orbits with $a \gtrsim 1.1~a_H$ are stable on time scales exceeding 100 Myr. Near-IR and mid-IR imaging with JWST and ground-based occultation campaigns with 2-3-m class telescopes can detect 1-2 km satellites outside the orbit of Hydra. Searches for these moons enable new constraints on the masses of the known satellites and on theories for circumbinary satellite formation.Comment: 34 pages of text, 2 tables, 12 figures, submitted to AAS journals, comments welcome. Animations associated with the paper are available at https://www.cfa.harvard.edu/~kenyon/Media/PCSonata.htm

Variations on Debris Disks IV. An Improved Analytical Model for Collisional Cascades

We derive a new analytical model for the evolution of a collisional cascade in a thin annulus around a single central star. In this model, $r_{max}~$ the size of the largest object declines with time (t); $r_{max} \propto t^{-\gamma}$, with $\gamma$ = 0.1-0.2. Compared to standard models where $r_{max}~$ is constant in time, this evolution results in a more rapid decline of $M_d$, the total mass of solids in the annulus and $L_d$, the luminosity of small particles in the annulus: $M_d \propto t^{-(\gamma + 1)}~$ and $L_d \propto t^{-(\gamma/2 + 1)}~~$. We demonstrate that the analytical model provides an excellent match to a comprehensive suite of numerical coagulation simulations for annuli at 1 AU and at 25 AU. If the evolution of real debris disks follows the predictions of the analytical or numerical models, the observed luminosities for evolved stars require up to a factor of two more mass than predicted by previous analytical models.Comment: ApJ, in press, 22 pages of text and 14 figure