Self-consistent size and velocity distributions of collisional cascades

The standard theoretical treatment of collisional cascades derives a steady-state size distribution assuming a single constant velocity dispersion for all bodies regardless of size. Here we relax this assumption and solve self-consistently for the bodies' steady-state size and size-dependent velocity distributions. Specifically, we account for viscous stirring, dynamical friction, and collisional damping of the bodies' random velocities in addition to the mass conservation requirement typically applied to find the size distribution in a steady-state cascade. The resulting size distributions are significantly steeper than those derived without velocity evolution. For example, accounting self-consistently for the velocities can change the standard q=3.5 power-law index of the Dohnanyi (1969) differential size spectrum to an index as large as q=4. Similarly, for bodies held together by their own gravity, the corresponding power-law index range 2.88<q<3.14 of Pan & Sari (2005) can steepen to values as large as q=3.26. Our velocity results allow quantitative predictions of the bodies' scale heights as a function of size. Together with our predictions, observations of the scale heights for different sized bodies for the Kuiper belt, the asteroid belt, and extrasolar debris disks may constrain the mass and number of large bodies stirring the cascade as well as the colliding bodies' internal strengths.Comment: 23 pages, 3 figures, 1 table; submitted to Ap

Atmospheric Mass Loss During Planet Formation: The Importance of Planetesimal Impacts

We quantify the atmospheric mass loss during planet formation by examining 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 ($\gamma=5/3$) that the local atmospheric mass loss fraction for ground velocities $v_g < 0.25 v_{esc}$ is given by $\chi_{loss}=(1.71 v_g/v_{esc})^{4.9}$, 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 $x=(v_{Imp}m/v_{esc}M)$. Atmospheric mass loss due to planetesimal impacts proceeds in two different regimes: 1) Large enough impactors $m > \sqrt{2} \rho_0 (\pi h R)^{3/2}$ (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 $h$, $R$ and $\rho_0$ are the atmospheric scale height, radius of the target, and its atmospheric density at the ground. 2) Smaller impactors, but above $m>4 \pi \rho_0 h^3$ (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 and that the current atmosphere of the Earth could have resulted from an equilibrium between atmospheric erosion and volatile delivery to the atmosphere from planetesimals. 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. (Abridged)Comment: Submitted to Icarus, 39 pages, 16 figure

Atmospheric mass loss due to giant impacts: the importance of the thermal component for hydrogen-helium envelopes

Systems of close-in super-Earths display striking diversity in planetary bulk density and composition. Giant impacts are expected to play a role in the formation of many of these worlds. Previous works, focused on the mechanical shock caused by a giant impact, have shown that these impacts can eject large fractions of the planetary envelope, offering a partial explanation for the observed spread in exoplanet compositions. Here, we examine the thermal consequences of giant impacts, and show that the atmospheric loss caused by these effects can significantly exceed that caused by mechanical shocks for hydrogen-helium (H/He) envelopes. When a giant impact occurs, part of the impact energy is converted into thermal energy, heating the rocky core and the envelope. We find that the ensuing thermal expansion of the envelope can lead to a period of sustained, rapid mass loss through a Parker wind, resulting in the partial or complete erosion of the H/He envelope. The fraction of the envelope lost depends on the planet's orbital distance from its host star and its initial thermal state, and hence age. Planets closer to their host stars are more susceptible to thermal atmospheric loss triggered by impacts than ones on wider orbits. Similarly, younger planets, with rocky cores which are still hot and molten from formation, suffer greater atmospheric loss. This is especially interesting because giant impacts are expected to occur $10{-}100~\mathrm{Myr}$ after formation. For planets where the thermal energy of the core is much greater than the envelope energy, the impactor mass required for significant atmospheric removal is $M_\mathrm{imp} / M_p \sim \mu / \mu_c \sim 0.1$, approximately the ratio of the heat capacities of the envelope and core. When the envelope energy dominates the total energy budget, complete loss can occur when the impactor mass is comparable to the envelope mass.Comment: 10 pages, 9 figure

Super-Earth Atmospheres: Self-Consistent Gas Accretion and Retention

Some recently discovered short-period Earth to Neptune sized exoplanets (super Earths) have low observed mean densities which can only be explained by voluminous gaseous atmospheres. Here, we study the conditions allowing the accretion and retention of such atmospheres. We self-consistently couple the nebular gas accretion onto rocky cores and the subsequent evolution of gas envelopes following the dispersal of the protoplanetary disk. Specifically, we address mass-loss due to both photo-evaporation and cooling of the planet. We find that planets shed their outer layers (dozens of percents in mass) following the disk's dispersal (even without photo-evaporation), and their atmospheres shrink in a few Myr to a thickness comparable to the radius of the underlying rocky core. At this stage, atmospheres containing less particles than the core (equivalently, lighter than a few % of the planet's mass) can be blown away by heat coming from the cooling core, while heavier atmospheres cool and contract on a timescale of Gyr at most. By relating the mass-loss timescale to the accretion time, we analytically identify a Goldilocks region in the mass-temperature plane in which low-density super Earths can be found: planets have to be massive and cold enough to accrete and retain their atmospheres, while not too massive or cold, such that they do not enter runaway accretion and become gas giants (Jupiters). We compare our results to the observed super-Earth population and find that low-density planets are indeed concentrated in the theoretically allowed region. Our analytical and intuitive model can be used to investigate possible super-Earth formation scenarios.Comment: Updated (refereed) versio

Understanding the Origin of Planetary Systems: Studying the Kuiper Belt and the Dynamics of Planet Formation

This thesis presents theoretical and observational studies pertaining to the early solar system, planet formation and extrasolar planets. First, we explore the dynamics of protoplanet formation. We find that the growth of protoplanets may be dominated by the accretion of a planetesimal disk that forms from planetesimal-planetesimal collisions, rather than direct planetesimal impacts onto the protoplanet. This has far reaching implications for the formation of planets, their growth rate and dynamics. We focus on the implications for planetary spins: it can explain the prevalence of prograde spins of planets and asteroids in the solar system, which is commonly believed to be an accident. Second, we present a series of investigations of the formation of multiple systems in the Kuiper Belt. Two of our studies are concerned with the formation of comparable mass binaries. We find that in a dynamically cold Kuiper Belt, binaries become bound predominantly by dynamical friction. This leads to a binary population with mostly retrograde mutual binary orbits. In a dynamically hot Kuiper Belt three-body gravitational interactions dominate the binary formation producing a roughly equal number of prograde and retrograde binaries. We propose a new formation scenario for Haumea’s collisional family. In our scenario, the family members are ejected while in orbit around Haumea rather than directly from Haumea’s surface as previously proposed. Our formation scenario offers an explanation for the observed velocity dispersion among the family members which is much smaller than Haumea’s escape velocity. It is consistent with detecting just one collisional family in the Kuiper Belt and aids with explaining Haumea’s initial giant impact. We conclude with observational work that aims to detect sub-km sized Kuiper Belt objects and to measure their size-distribution. Our results provide the best constraint on the surface density of small Kuiper Belt objects to date. Our findings support the idea that small Kuiper Belt objects underwent collisional evolution that modified their size distribution. We present our first candidate occultation event and show that it is unlikely to be due to instrumental artifacts or statistical fluctuations in the data.</p

The Self-Similarity of Shear-Dominated Viscous Stirring

We examine the growth of eccentricities of a population of particles with initially circular orbits around a central massive body. Successive encounters between pairs of particles increase the eccentricities in the disk on average. As long as the epicyclic motions of the particles are small compared to the shearing motion between Keplerian orbits, there is no preferred scale for the eccentricities. The simplification due to this self-similarity allows us to find an analytic form for the distribution function; full numerical integrations of a disk with 200 planetesimals verify our analytical self-similar distribution. The shape of this non-equilibrium profile is identical to the equilibrium profile of a shear-dominated population whose mutual excitations are balanced by dynamical friction or Epstein gas drag.Comment: 8 pages, 2 figure