Direct N-body Simulations of Rubble Pile Collisions

There is increasing evidence that many km-sized bodies in the Solar System are piles of rubble bound together by gravity. We present results from a project to map the parameter space of collisions between km-sized spherical rubble piles. The results will assist in parameterization of collision outcomes for Solar System formation models and give insight into fragmentation scaling laws. We use a direct numerical method to evolve the positions and velocities of the rubble pile particles under the constraints of gravity and physical collisions. We test the dependence of the collision outcomes on impact parameter and speed, impactor spin, mass ratio, and coefficient of restitution. Speeds are kept low (< 10 m/s, appropriate for dynamically cool systems such as the primordial disk during early planet formation) so that the maximum strain on the component material does not exceed the crushing strength. We compare our results with analytic estimates and hydrocode simulations. Off-axis collisions can result in fast-spinning elongated remnants or contact binaries while fast collisions result in smaller fragments overall. Clumping of debris escaping from the remnant can occur, leading to the formation of smaller rubble piles. In the cases we tested, less than 2% of the system mass ends up orbiting the remnant. Initial spin can reduce or enhance collision outcomes, depending on the relative orientation of the spin and orbital angular momenta. We derive a relationship between impact speed and angle for critical dispersal of mass in the system. We find that our rubble piles are relatively easy to disperse, even at low impact speed, suggesting that greater dissipation is required if rubble piles are the true progenitors of protoplanets.Comment: 30 pages including 4 tables, 8 figures. Revised version to be published in Icarus

Are exoplanetesimals differentiated?

Metals observed in the atmospheres of white dwarfs suggest that many have recently accreted planetary bodies. In some cases, the compositions observed suggest the accretion of material dominantly from the core (or the mantle) of a differentiated planetary body. Collisions between differentiated exoplanetesimals produce such fragments. In this work, we take advantage of the large numbers of white dwarfs where at least one siderophile (core-loving) and one lithophile (rock-loving) species have been detected to assess how commonly exoplanetesimals differentiate. We utilise N-body simulations that track the fate of core and mantle material during the collisional evolution of planetary systems to show that most remnants of differentiated planetesimals retain core fractions similar to their parents, whilst some are extremely core-rich or mantle-rich. Comparison with the white dwarf data for calcium and iron indicates that the data are consistent with a model in which $66^{+4}_{-6}\%$ have accreted the remnants of differentiated planetesimals, whilst $31^{+5}_{-5}\%$ have Ca/Fe abundances altered by the effects of heating (although the former can be as high as $100\%$, if heating is ignored). These conclusions assume pollution by a single body and that collisional evolution retains similar features across diverse planetary systems. These results imply that both collisions and differentiation are key processes in exoplanetary systems. We highlight the need for a larger sample of polluted white dwarfs with precisely determined metal abundances to better understand the process of differentiation in exoplanetary systems

Planetesimal collisions in binary systems

We study the collisional evolution of km-sized planetesimals in tight binary star systems to investigate whether accretion towards protoplanets can proceed despite the strong gravitational perturbations from the secondary star. The orbits of planetesimals are numerically integrated in two dimensions under the influence of the two stars and gas drag. The masses and orbits of the planetesimals are allowed to evolve due to collisions with other planetesimals and accretion of collisional debris. In addition, the mass in debris can evolve due to planetesimal-planetesimal collisions and the creation of new planetesimals. We show that it is possible in principle for km-sized planetesimals to grow by two orders of magnitude in size if the efficiency of planetesimal formation is relatively low. We discuss the limitations of our two-dimensional approach.Comment: 5 pages, 5 figures, accepted for publication in MNRA

Modelling circumbinary protoplanetary disks II. Gas disk feedback on planetesimal dynamical and collisional evolution in the circumbinary systems Kepler-16 and 34

Aims. We investigate the feasibility of planetesimal growth in circumbinary protoplanetary disks around the observed systems Kepler- 16 and Kepler-34 under the gravitational influence of a precessing eccentric gas disk. Methods. We embed the results of our previous hydrodynamical simulations of protoplanetary disks around binaries into an N-body code to perform 3D, high-resolution, inter-particle gravity-enabled simulations of planetesimal growth and dynamics that include the gravitational force imparted by the gas. Results. Including the full, precessing asymmetric gas disk generates high eccentricity orbits for planetesimals orbiting at the edge of the circumbinary cavity, where the gas surface density and eccentricity have their largest values. The gas disk is able to efficiently align planetesimal pericenters in some regions leading to phased, non-interacting orbits. Outside of these areas eccentric planetesimal orbits become misaligned and overlap leading to crossing orbits and high relative velocities during planetesimal collisions. This can lead to an increase in the number of erosive collisions that far outweighs the number of collisions that result in growth. Gravitational focusing from the static axisymmetric gas disk is weak and does not significantly alter collision outcomes from the gas free case. Conclusions. Due to asymmetries in the gas disk, planetesimals are strongly perturbed onto highly eccentric orbits. Where planetesimals orbits are not well aligned, orbit crossings lead to an increase in the number of erosive collisions. This makes it difficult for sustained planetesimal accretion to occur at the location of Kepler-16b and Kepler-34b and we therefore rule out in-situ growth. This adds further support to our initial suggestions that most circumbinary planets should form further out in the disk and migrate inwards.Comment: 12 pages and 12 figure

The TAOS Project: Upper Bounds on the Population of Small KBOs and Tests of Models of Formation and Evolution of the Outer Solar System

We have analyzed the first 3.75 years of data from TAOS, the Taiwanese American Occultation Survey. TAOS monitors bright stars to search for occultations by Kuiper Belt Objects (KBOs). This dataset comprises 5e5 star-hours of multi-telescope photometric data taken at 4 or 5 Hz. No events consistent with KBO occultations were found in this dataset. We compute the number of events expected for the Kuiper Belt formation and evolution models of Pan & Sari (2005), Kenyon & Bromley (2004), Benavidez & Campo Bagatin (2009), and Fraser (2009). A comparison with the upper limits we derive from our data constrains the parameter space of these models. This is the first detailed comparison of models of the KBO size distribution with data from an occultation survey. Our results suggest that the KBO population is comprised of objects with low internal strength and that planetary migration played a role in the shaping of the size distribution.Comment: 18 pages, 16 figures, Aj submitte

Rings in the Solar System: a short review

Rings are ubiquitous around giant planets in our Solar System. They evolve jointly with the nearby satellite system. They could form either during the giant planet formation process or much later, as a result of large scale dynamical instabilities either in the local satellite system, or at the planetary scale. We review here the main characteristics of rings in our solar system, and discuss their main evolution processes and possible origin. We also discuss the recent discovery of rings around small bodies.Comment: Accepted for the Handbook of Exoplanet

Modelling circumbinary protoplanetary disks I. Fluid simulations of the Kepler-16 and 34 systems

S.L and Z.M.L are supported by the STFC. P.J.C is grateful to NERC Grant NE/K004778/1. S.J.P. is supported by a Royal Society University Research Fellowship

Investigation of eighth-grade students' understanding of the slope of the linear function

This study aimed to investigate eighth-grade students' difficulties and misconceptions and their performance of translation between the different representation modes related to the slope of linear functions. The participants were 115 Turkish eighth-grade students in a city in the eastern part of the Black Sea region of Turkey. Data was collected with an instrument consisting of seven written questions and a semi-structured interview protocol conducted with six students. Students' responses to questions were categorized and scored. Quantitative data was analyzed using the SPSS 17.0 statistical packet program with cross tables and one-way ANOVA. Qualitative data obtained from interviews was analyzed using descriptive analytical techniques. It was found that students' performance in articulating the slope of the linear function using its algebraic representation form was higher than their performance in using transformation between graphical and algebraic representation forms. It was also determined that some of them had difficulties and misunderstood linear function equations, graphs, and slopes and could not comprehend the connection between slope and the x- and y-intercepts

A super-massive Neptune-sized planet

Neptune-sized planets exhibit a wide range of compositions and densities, depending onf cators related to their formation and evolution history, such as the distance from their host stars and atmospheric escape processes. They can vary from relatively low-density planets with thick hydrogen-helium atmospheres to higher-density planets with a substantial amount of water or a rocky interior with a thinner atmosphere, such as HD 95338 b, TOI-849 b and TOI-2196 b. The discovery of exoplanets in the hot-Neptune desert, a region close to the host stars with a deficit of Neptune-sized planets, provides insights into the formation and evolution of planetary systems, including the existence of this region itself. Here we show observations of the transiting planet TOI-1853 b, which has a radius of 3.46 +- 0.08 Earth radii and orbits a dwarf star every 1.24 days. This planet has a mass of 73.2 +- 2.7 Earth masses, almost twice that of any other Neptune-sized planet known so far, and a density of 9.7 +- 0.8 grams per cubic centimetre. These values place TOI-1853 b in the middle of the Neptunian desert and imply that heavy elements dominate its mass. The properties of TOI-1853 b present a puzzle for conventional theories of planetary formation and evolution, and could be the result of several proto-planet collisions or the final state of an initially high-eccentricity planet that migrated closer to its parent star.Comment: Preprint submitted to Nature. Please refer to the published version for the final parameters estimation