Hilke E. Schlichting

OVERSTABLE LIBRATIONS CAN ACCOUNT FOR THE PAUCITY OF MEAN MOTION RESONANCES AMONG EXOPLANET PAIRS

We assess the multi-planet systems discovered by the Kepler satellite in terms of current ideas about orbital migration and eccentricity damping due to planet-disk interactions. Our primary focus is on first order mean motion resonances, which we investigate analytically to lowest order in eccentricity. Only a few percent of planet pairs are in close proximity to a resonance. However, predicted migration rates (parameterized by ) imply that during convergent migration most planets would have been captured into first order resonances. Eccentricity damping (parameterized by ) offers a plausible resolution. Estimates suggest ? e /? n ~ (h/a)2 ~ 10?2, where h/a is the ratio of disk thickness to radius. Together, eccentricity damping and orbital migration give rise to an equilibrium eccentricity, e eq ~ (? e /? n )1/2. Capture is permanent provided e eq ?1/3, where ? denotes the planet to star mass ratio. But for e eq ?1/3, capture is only temporary because librations around equilibrium are overstable and lead to passage through resonance on timescale ? e . Most Kepler planet pairs have e eq > ?1/3. Since ? n ? e is the timescale for migration between neighboring resonances, only a modest percentage of pairs end up trapped in resonances after the disk disappears. Thus the paucity of resonances among Kepler pairs should not be taken as evidence for in situ planet formation or the disruptive effects of disk turbulence. Planet pairs close to a mean motion resonance typically exhibit period ratios 1%-2% larger than those for exact resonance. The direction of this shift undoubtedly reflects the same asymmetry that requires convergent migration for resonance capture. Permanent resonance capture at these separations from exact resonance would demand ?(? n /? e )1/2 0.01, a value that estimates of ? from transit data and (? e /? n )1/2 from theory are insufficient to match. Plausible alternatives involve eccentricity damping during or after disk dispersal. The overstability referred to above has applications beyond those considered in this investigation. It was discovered numerically by Meyer & Wisdom in their study of the tidal evolution of Saturns satellites.

A single sub-kilometre Kuiper belt object from a stellar occultation in archival data

The Kuiper belt is a remnant of the primordial Solar System. Measurements of its size distribution constrain its accretion and collisional history, and the importance of material strength of Kuiper belt objects. Small, sub-kilometre-sized, Kuiper belt objects elude direct detection, but the signature of their occultations of background stars should be detectable. Observations at both optical and X-ray wavelengths claim to have detected such occultations, but their implied abundances are inconsistent with each other and far exceed theoretical expectations. Here we report an analysis of archival data that reveals an occultation by a body with an approximately 500-metre radius at a distance of 45 astronomical units. The probability of this event arising from random statistical fluctuations within our data set is about two per cent. Our survey yields a surface density of Kuiper belt objects with radii exceeding 250 metres of , ruling out inferred surface densities from previous claimed detections by more than 5σ. The detection of only one event reveals a deficit of sub-kilometre-sized Kuiper belt objects compared to a population extrapolated from objects with radii exceeding 50 kilometres. This implies that sub-kilometre-sized objects are undergoing collisional erosion, just like debris disks observed around other stars.

The Formation of Super-Earths and Mini-Neptunes with Giant Impacts

The majority of discovered exoplanetary systems harbour a new class of planets, bodies that are typically several times more massive than the Earth but that orbit their host stars well inside the orbit of Mercury. The origin of these close-in super-Earths and mini-Neptunes is one of the major unanswered questions in planet formation. Unlike the Earth, whose atmosphere contains less than 10 6 of its total mass, a large fraction of close-in planets have significant gaseous envelopes, containing 1 10 per cent or more of their total mass. It has been proposed that close-in super-Earths and mini-Neptunes formed in situ either by delivery of 50 100M of rocky material to the inner regions of the protoplanetary disc, or in a disc enhanced relative to the minimum mass solar nebula. In both cases, the final assembly of the planets occurs via giant impacts. Here we test the viability of these scenarios. We show that atmospheres that can be accreted by isolation masses are small (typically 10 3 10 2 of the core mass) and that the atmospheric mass-loss during giant impacts is significant, resulting in typical postgiant impact atmospheres that are 8 10 4 of the core mass. Such values are consistent with terrestrial planet atmospheres but more than an order of magnitude below atmospheric masses of 1 10 per cent inferred for many close-in exoplanets. In the most optimistic scenario in which there is no core luminosity from giant impacts and/or planetesimal accretion, we find that post-giant impact envelope accretion from a depleted gas disc can yield atmospheric masses that are several per cent the core mass. If the gravitational potential energy resulting from the last mass doubling of the planet by giant impacts is released over the disc dissipation time-scale as core luminosity, then the accreted envelope masses are reduced by about an order of magnitude. Finally we show that, even in the absence of type I migration, radial drift time-scales due to gas drag for many isolation masses are shorter than typical disc lifetimes for standard gas-to-dust ratios. Given these challenges, we conclude that most of the observed close-in planets with envelopes larger than several per cent of their total mass likely formed at larger separations from their host stars.

THE LAST STAGES OF TERRESTRIAL PLANET FORMATION: DYNAMICAL FRICTION AND THE LATE VENEER

Thefinal stage of terrestrial planet formation consists of the clean-up of residual planetesimals after the giant impact phase. Dynamically, a residual planetesimal population is needed to damp the high eccentricities and inclinations of the terrestrial planets to circular and coplanar orbits after the giant impact stage. Geochemically, highly siderophile element (HSE) abundance patterns inferred for the terrestrial planets and the Moon suggest that a total of about 0.01 M⊕ of chondritic material was delivered as “late veneer” by planetesimals to the terrestrial planets after the end of giant impacts. Here, we combine these two independent lines of evidence for a leftover population of planetesimals and show that: (1) a residual population of small planetesimals containing 0.01 M⊕ is able to damp the high eccentricities and inclinations of the terrestrial planets after giant impacts to their observed values. (2) At the same time, this planetesimal population can account for the observed relative amounts of late veneer added to the Earth, Moon, and Mars provided that the majority of the accreted late veneer was delivered by small planetesimals with radii 10m. These small planetesimal sizes are required to ensure efficient damping of the planetesimal’s velocity dispersion by mutual collisions, which in turn ensures sufficiently low relative velocities between the terrestrial planets and the planetesimals such that the planets’ accretion cross sections are significantly enhanced by gravitational focusing above their geometric values. Specifically, we find that, in the limit that the relative velocity between the terrestrial planets and the planetesimals is significantly less than the terrestrial planets’ escapevelocities,gravitationalfocusingyieldsamassaccretionratioofEarth/Mars ∼ (ρ⊕/ρmars)(R⊕/Rmars) 4 ∼17, which agrees well with the mass accretion ratio inferred from HSEs of 12‐23. For the Earth‐Moon system, we find a mass accretion ratio of ∼200, which, as we show, is consistent with estimates of 150‐700 derived from HSE abundances that include the lunar crust as well as the mantle component. We conclude that small residual planetesimals containing about ∼1% of the mass of the Earth could provide the dynamical friction needed to relax the terrestrial planet’s eccentricities and inclinations after giant impacts, and also may have been the dominant source for the late veneer added to Earth, Moon, and Mars.

Atmospheric mass loss during planet formation: The importance of planetesimal impacts

Quantifying the atmospheric mass loss during planet formation is crucial for understanding the origin and evolution of planetary atmospheres. We examine the contributions to atmospheric loss from both giant impacts and planetesimal accretion. Giant impacts cause global motion of the ground. Using analytic self-similar solutions and full numerical integrations we find (for isothermal atmospheres with adiabatic index γ=5/3) that the local atmospheric mass loss fraction for ground velocities v_g ≲ 0.25v_esc is given by χ_loss = (1.71v_g/v_esc)^4.9, where v_esc is the escape velocity from the target. Yet, the global atmospheric mass loss is a weaker function of the impactor velocity v_Imp and mass m_Imp and given by X_loss≃0.4x + 1.4x^2 - 0.8x^3 (isothermal atmosphere) and X_loss≃0.4x + 1.8x^2 - 1.2x^3 (adiabatic atmosphere), where x=(v_Impm/v_escM). Atmospheric mass loss due to planetesimal impacts proceeds in two different regimes: (1) large enough impactors m≳√(2)ρ_0(πhR)^(3/2) (25 km for the current Earth), are able to eject all the atmosphere above the tangent plane of the impact site, which is h/2R of the whole atmosphere, where View the MathML sourceh,R and ρ0ρ0 are the atmospheric scale height, radius of the target, and its atmospheric density at the ground. (2) Smaller impactors, but above m>4πρ_(0)h^3(1 km for the current Earth) are only able to eject a fraction of the atmospheric mass above the tangent plane. We find that the most efficient impactors (per unit impactor mass) for atmospheric loss are planetesimals just above that lower limit (2 km for the current Earth). For impactor flux size distributions parametrized by a single power law, N(>r)∝r^(-q+1), with differential power law index q, we find that for 1 3 the mass loss is dominated by regime (2). Impactors with m≲4πρ_(0)h^3 are not able to eject any atmosphere. Despite being bombarded by the same planetesimal population, we find that the current differences in Earth’s and Venus’ atmospheric masses can be explained by modest differences in their initial atmospheric masses and that the current atmosphere of the Earth could have resulted from an equilibrium between atmospheric erosion and volatile delivery to the atmosphere from planetesimal impacts. We conclude that planetesimal impacts are likely to have played a major role in atmospheric mass loss over the formation history of the terrestrial planets.

FORMATION OF CLOSE IN SUPER-EARTHS AND MINI-NEPTUNES: REQUIRED DISK MASSES AND THEIR IMPLICATIONS

Recent observations by the {\it Kepler} space telescope have led to the discovery of more than 4000 exoplanet candidates consisting of many systems with Earth- to Neptune-sized objects that reside well inside the orbit of Mercury, around their respective host stars. How and where these close-in planets formed is one of the major unanswered questions in planet formation. Here we calculate the required disk masses for {\it in situ} formation of the {\it Kepler} planets. We find that, if close-in planets formed as {\it isolation masses}, then standard gas-to-dust ratios yield corresponding gas disks that are gravitationally unstable for a significant fraction of systems, ruling out such a scenario. We show that the maximum width of a planets accretion region in the absence of any migration is

Measuring the Abundance of Sub-kilometer-sized Kuiper Belt Objects Using Stellar Occultations

2 v_{esc}/\Omega

Runaway Growth During Planet Formation: Explaining the Size Distribution of Large Kuiper Belt Objects

, where

INITIAL PLANETESIMAL SIZES AND THE SIZE DISTRIBUTION OF SMALL KUIPER BELT OBJECTS

v_{esc}

The Creation of Haumea's Collisional Family

is the escape velocity of the planet and