Toward an initial mass function for giant planets

The distribution of exoplanet masses is not primordial. After the initial stage of planet formation is complete, the gravitational interactions between planets can lead to the physical collision of two planets, or the ejection of one or more planets from the system. When this occurs, the remaining planets are typically left in more eccentric orbits. Here we use present-day eccentricities of the observed exoplanet population to reconstruct the initial mass function of exoplanets before the onset of dynamical instability. We developed a Bayesian framework that combines data from N-body simulations with present-day observations to compute a probability distribution for the planets that were ejected or collided in the past. Integrating across the exoplanet population, we obtained an estimate of the initial mass function of exoplanets. We find that the ejected planets are primarily sub-Saturn type planets. While the present-day distribution appears to be bimodal, with peaks around $\sim 1 M_{\rm J}$ and $\sim 20 M_\oplus$, this bimodality does not seem to be primordial. Instead, planets around $\sim 60 M_\oplus$ appear to be preferentially removed by dynamical instabilities. Attempts to reproduce exoplanet populations using population synthesis codes should be mindful of the fact that the present population has been depleted of intermediate-mass planets. Future work should explore how the system architecture and multiplicity might alter our results.Comment: 10 pages, 9 figures; submitted to MNRA

Survival of habitable planets in unstable planetary systems

Many observed giant planets lie on eccentric orbits. Such orbits could be the result of strong scatterings with other giant planets. The same dynamical instability that produces these scatterings may also cause habitable planets in interior orbits to become ejected, destroyed, or be transported out of the habitable zone. We say that a habitable planet has resilient habitability if it is able to avoid ejections and collisions and its orbit remains inside the habitable zone. Here we model the orbital evolution of rocky planets in planetary systems where giant planets become dynamically unstable. We measure the resilience of habitable planets as a function of the observed, present-day masses and orbits of the giant planets. We find that the survival rate of habitable planets depends strongly on the giant planet architecture. Equal-mass planetary systems are far more destructive than systems with giant planets of unequal masses. We also establish a link with observation; we find that giant planets with present-day eccentricities higher than 0.4 almost never have a habitable interior planet. For a giant planet with an present-day eccentricity of 0.2 and semimajor axis of 5 AU orbiting a Sun-like star, 50% of the orbits in the habitable zone are resilient to the instability. As semimajor axis increases and eccentricity decreases, a higher fraction of habitable planets survive and remain habitable. However, if the habitable planet has rocky siblings, there is a significant risk of rocky planet collisions that would sterilize the planet.Comment: Accepted to MNRA

How to form planetesimals from mm-sized chondrules and chondrule aggregates

The size distribution of asteroids and Kuiper belt objects in the solar system is difficult to reconcile with a bottom-up formation scenario due to the observed scarcity of objects smaller than $\sim$100 km in size. Instead, planetesimals appear to form top-down, with large $100-1000$ km bodies forming from the rapid gravitational collapse of dense clumps of small solid particles. In this paper we investigate the conditions under which solid particles can form dense clumps in a protoplanetary disk. We use a hydrodynamic code to model the interaction between solid particles and the gas inside a shearing box inside the disk, considering particle sizes from sub-millimeter-sized chondrules to meter-sized rocks. We find that particles down to millimeter sizes can form dense particle clouds through the run-away convergence of radial drift known as the streaming instability. We make a map of the range of conditions (strength of turbulence, particle mass-loading, disk mass, and distance to the star) which are prone to producing dense particle clumps. Finally, we estimate the distribution of collision speeds between mm-sized particles. We calculate the rate of sticking collisions and obtain a robust upper limit on the particle growth timescale of $\sim$$10^5$ years. This means that mm-sized chondrule aggregates can grow on a timescale much smaller than the disk accretion timescale ($\sim$$10^6 - 10^7$ years). Our results suggest a pathway from the mm-sized grains found in primitive meteorites to fully formed asteroids. We speculate that asteroids may form from a positive feedback loop in which coagualation leads to particle clumping driven by the streaming instability. This clumping, in turn reduces collision speeds and enhances coagulation.} Future simulations should model coagulation and the streaming instability together to explore this feedback loop further.Comment: 20 pages. Accepted for publication in A&

The SU(2)SU(2) Global Anomaly Through Level Circling

We discuss a novel manifestation of the $SU(2)$ global anomaly in an $SU(2)$ gauge theory with an odd number of chiral quark doublets and arbitrary Yukawa couplings. We argue that the massive 4-dim.($D=4$) Euclidean Dirac operator is nonhermitean with its spectrum of eigenvalues $(\lambda,-\lambda)$ lying in pairs in the complex plane. Consequently the existence of an odd number of normalizable zero modes of the 5-dim.($D=5$) massive Dirac operator is equivalent to a fermionic level exchange phenomenon, level ``circling'', under continuous topologically nontrivial deformations of the external gauge field. More generally global anomalies are a manifestation of fermionic level ``circling'' in any $SP(2n)$ gauge theory with an odd number of massive fermions in the spinor representation and arbitrary Yukawa couplings.Comment: 14 pages, NBI-HE-93-5

On sphaleron deformations induced by Yukawa interactions

Due to the presence of the chiral anomaly sphalerons with Chern-Simons number a half (CS=1/2) are the only static configurations that allow for a fermion level crossing in the two-dimensional Abelian-Higgs model with massless fermions, i.e. in the absence of Yukawa interactions. In the presence of fermion-Higgs interactions we demonstrate the existence of zero energy solutions to the one-dimensional Dirac equation at deformed sphalerons with CS$\neq 1/2 .$ Induced level crossing due to Yukawa interactions illustrates a non-trivial generalization of the Atiyah-Patodi-Singer index theorem and of the equivalence between parity anomaly in odd and the chiral anomaly in even dimensions. We discuss a subtle manifestation of this effect in the standard electroweak theory at finite temperatures.Comment: 14 pages, Latex, NBI-HE-93-7

The destruction of inner planetary systems during high-eccentricity migration of gas giants

Hot Jupiters are giant planets on orbits a few hundredths of an AU. They do not share their system with low-mass close-in planets, despite these latter being exceedingly common. Two migration channels for hot Jupiters have been proposed: through a protoplanetary gas disc or by tidal circularisation of highly-eccentric planets. We show that highly-eccentric giant planets that will become hot Jupiters clear out any low-mass inner planets in the system, explaining the observed lack of such companions to hot Jupiters. A less common outcome of the interaction is that the giant planet is ejected by the inner planets. Furthermore, the interaction can implant giant planets on moderately-high eccentricities at semimajor axes $<1$ AU, a region otherwise hard to populate. Our work supports the hypothesis that most hot Jupiters reached their current orbits following a phase of high eccentricity, possibly excited by other planetary or stellar companions.Comment: Replaced with accepted versio

The effects of external planets on inner systems: multiplicities, inclinations, and pathways to eccentric warm Jupiters

We study how close-in systems such as those detected by Kepler are affected by the dynamics of bodies in the outer system. We consider two scenarios: outer systems of giant planets potentially unstable to planet--planet scattering, and wide binaries that may be capable of driving Kozai or other secular variations of outer planets' eccentricities. Dynamical excitation of planets in the outer system reduces the multiplicity of Kepler-detectable planets in the inner system in $\sim20-25\%$ of our systems. Accounting for the occurrence rates of wide-orbit planets and binary stars, $\approx18\%$ of close-in systems could be destabilised by their outer companions in this way. This provides some contribution to the apparent excess of systems with a single transiting planet compared to multiple, however, it only contributes at most $25\%$ of the excess. The effects of the outer dynamics can generate systems similar to Kepler-56 (two coplanar planets significantly misaligned with the host star) and Kepler-108 (two significantly non-coplanar planets in a binary). We also identify three pathways to the formation of eccentric warm Jupiters resulting from the interaction between outer and inner systems: direct inelastic collision between an eccentric outer and an inner planet, secular eccentricity oscillations that may "freeze out" when scattering resolves in the outer system; and scattering in the inner system followed by "uplift", where inner planets are removed by interaction with the outer planets. In these scenarios, the formation of eccentric warm Jupiters is a signature of a past history of violent dynamics among massive planets beyond $\sim1$ au.Comment: 24 pages, 19 figures. Accepted to MNRA

Planetesimal formation by the streaming instability in a photoevaporating disk

Recent years have seen growing interest in the streaming instability as a candidate mechanism to produce planetesimals. However, these investigations have been limited to small-scale simulations. We now present the results of a global protoplanetary disk evolution model that incorporates planetesimal formation by the streaming instability, along with viscous accretion, photoevaporation by EUV, FUV, and X-ray photons, dust evolution, the water ice line, and stratified turbulence. Our simulations produce massive (60-130 $M_\oplus$) planetesimal belts beyond 100 au and up to $\sim 20 M_\oplus$ of planetesimals in the middle regions (3-100 au). Our most comprehensive model forms 8 $M_\oplus$ of planetesimals inside 3 au, where they can give rise to terrestrial planets. The planetesimal mass formed in the inner disk depends critically on the timing of the formation of an inner cavity in the disk by high-energy photons. Our results show that the combination of photoevaporation and the streaming instability are efficient at converting the solid component of protoplanetary disks into planetesimals. Our model, however, does not form enough early planetesimals in the inner and middle regions of the disk to give rise to giant planets and super-Earths with gaseous envelopes. Additional processes such as particle pileups and mass loss driven by MHD winds may be needed to drive the formation of early planetesimal generations in the planet forming regions of protoplanetary disks.Comment: 20 pages, 12 figures; accepted to Ap