The GENGA Code: Gravitational Encounters in N-body simulations with GPU Acceleration

We describe an open source GPU implementation of a hybrid symplectic N-body integrator, GENGA (Gravitational ENcounters with Gpu Acceleration), designed to integrate planet and planetesimal dynamics in the late stage of planet formation and stability analyses of planetary systems. GENGA uses a hybrid symplectic integrator to handle close encounters with very good energy conservation, which is essential in long-term planetary system integration. We extended the second order hybrid integration scheme to higher orders. The GENGA code supports three simulation modes: Integration of up to 2048 massive bodies, integration with up to a million test particles, or parallel integration of a large number of individual planetary systems. We compare the results of GENGA to Mercury and pkdgrav2 in respect of energy conservation and performance, and find that the energy conservation of GENGA is comparable to Mercury and around two orders of magnitude better than pkdgrav2. GENGA runs up to 30 times faster than Mercury and up to eight times faster than pkdgrav2. GENGA is written in CUDA C and runs on all NVIDIA GPUs with compute capability of at least 2.0.Comment: Accepted by ApJ. 18 pages, 17 figures, 4 table

A universal velocity distribution of relaxed collisionless structures

Several general trends have been identified for equilibrated, self-gravitating collisionless systems, such as density or anisotropy profiles. These are integrated quantities which naturally depend on the underlying velocity distribution function (VDF) of the system. We study this VDF through a set of numerical simulations, which allow us to extract both the radial and the tangential VDF. We find that the shape of the VDF is universal, in the sense that it depends only on two things namely the dispersion (radial or tangential) and the local slope of the density. Both the radial and the tangential VDF's are universal for a collection of simulations, including controlled collisions with very different initial conditions, radial infall simulation, and structures formed in cosmological simulations.Comment: 13 pages, 6 figures; oversimplified analysis corrected; changed abstract and conclusions; significantly extended discussio

Forming iron-rich planets with giant impacts

We investigate mantle stripping giant impacts (GI) between super-Earths with masses between 1 and 20M⊕⁠. We infer new scaling laws for the mass of the largest fragment and its iron mass fraction, as well as updated fitting coefficients for the critical specific impact energy for catastrophic disruption, Q∗RD⁠. With these scaling laws, we derive equations that relate the impact conditions, i.e. target mass, impact velocity, and impactor-to-target mass ratio, to the mass and iron mass fraction of the largest fragment. This allows one to predict collision outcomes without performing a large suite of simulations. Using these equations we present the maximum and minimum planetary iron mass fraction as a result of collisional stripping of its mantle for a given range of impact conditions. We also infer the radius for a given mass and composition using interior structure models and compare our results to observations of metal-rich exoplanets. We find good agreement between the data and the simulated planets suggesting that GI could have played a key role in their formation. Furthermore, using our scaling laws we can further constrain the impact conditions that favour their masses and compositions. Finally, we present a flexible and easy-to-use tool that allows one to predict mass and composition of a planet after a GI for an arbitrary range of impact conditions, which, in turn, allows to assess the role of GI in observed planetary systems

GENGA. II. GPU Planetary N-body Simulations with Non-Newtonian Forces and High Number of Particles

We present recent updates and improvements of the graphical processing unit (GPU) N-body code GENGA. Modern state-of-the-art simulations of planet formation require the use of a very high number of particles to accurately resolve planetary growth and to quantify the effect of dynamical friction. At present the practical upper limit is in the range of 30,000–60,000 fully interactive particles; possibly a little more on the latest GPU devices. While the original hybrid symplectic integration method has difficulties to scale up to these numbers, we have improved the integration method by (i) introducing higher level changeover functions and (ii) code improvements to better use the most recent GPU hardware efficiently for such large simulations. We added treatments of non-Newtonian forces such as general relativity, tidal interaction, rotational deformation, the Yarkovsky effect, and Poynting–Robertson drag, as well as a new model to treat virtual collisions of small bodies in the solar system. We added new tools to GENGA, such as semi-active test particles that feel more massive bodies but not each other, a more accurate collision handling and a real-time openGL visualization. We present example simulations, including a 1.5 billion year terrestrial planet formation simulation that initially started with 65,536 particles, a 3.5 billion year simulation without gas giants starting with 32,768 particles, the evolution of asteroid fragments in the solar system, and the planetesimal accretion of a growing Jupiter simulation. GENGA runs on modern NVIDIA and AMD GPUs

Density Profiles of Cold Dark Matter Substructure: Implications for the Missing Satellites Problem

The structural evolution of substructure in cold dark matter (CDM) models is investigated combining ``low-resolution'' satellites from cosmological N-body simulations of parent halos with N=10^7 particles with high-resolution individual subhalos orbiting within a static host potential. We show that, as a result of mass loss, convergence in the central density profiles requires the initial satellites to be resolved with N=10^7 particles and parsec-scale force resolution. We find that the density profiles of substructure halos can be well fitted with a power-law central slope that is unmodified by tidal forces even after the tidal stripping of over 99% of the initial mass and an exponential cutoff in the outer parts. The solution to the missing-satellites problem advocated by Stoehr et al. in 2002 relied on the flattening of the dark matter (DM) halo central density cusps by gravitational tides, enabling the observed satellites to be embedded within DM halos with maximum circular velocities as large as 60 km/s. In contrast, our results suggest that tidal interactions do not provide the mechanism for associating the dwarf spheroidal satellites (dSphs) of the Milky Way with the most massive substructure halos expected in a CDM universe. We compare the predicted velocity dispersion profiles of Fornax and Draco to observations, assuming that they are embedded in CDM halos. Models with isotropic and tangentially anisotropic velocity distributions for the stellar component fit the data only if the surrounding DM halos have maximum circular velocities in the range 20-35 km/s. If the dSphs are embedded within halos this large then the overabundance of satellites within the concordance LCDM cosmological model is significantly alleviated, but this still does not provide the entire solution.Comment: Accepted for publication in ApJ, 17 pages, 9 figures, LaTeX (uses emulateapj5.sty

Tidal stirring and the origin of dwarf spheroidals in the Local Group

N-Body/SPH simulations are used to study the evolution of dwarf irregular galaxies (dIrrs) entering the dark matter halo of the Milky Way or M31 on plunging orbits. We propose a new dynamical mechanism driving the evolution of gas rich, rotationally supported dIrrs, mostly found at the outskirts of the Local Group (LG), into gas free, pressure supported dwarf spheroidals (dSphs) or dwarf ellipticals (dEs), observed to cluster around the two giant spirals. The initial model galaxies are exponential disks embedded in massive dark matter halos and reproduce nearby dIrrs. Repeated tidal shocks at the pericenter of their orbit partially strip their halo and disk and trigger dynamical instabilities that dramatically reshape their stellar component. After only 2-3 orbits low surface brightness (LSB) dIrrs are transformed into dSphs, while high surface brightness (HSB) dIrrs evolve into dEs. This evolutionary mechanism naturally leads to the morphology-density relation observed for LG dwarfs. Dwarfs surrounded by very dense dark matter halos, like the archetypical dIrr GR8, are turned into Draco or Ursa Minor, the faintest and most dark matter dominated among LG dSphs. If disks include a gaseous component, this is both tidally stripped and consumed in periodic bursts of star formation. The resulting star formation histories are in good qualitative agreement with those derived using HST color-magnitude diagrams for local dSphs.Comment: 5 pages, 5 figures, to appear on ApJL. Simulation images and movies can be found at the Local Group web page at http://pcblu.uni.mi.astro.it/~lucio/LG/LG.htm

The velocity anisotropy - density slope relation

One can solve the Jeans equation analytically for equilibrated dark matter structures, once given two pieces of input from numerical simulations. These inputs are 1) a connection between phase-space density and radius, and 2) a connection between velocity anisotropy and density slope, the \alpha-\beta relation. The first (phase-space density v.s. radius) has already been analysed through several different simulations, however the second (\alpha-\beta relation) has not been quantified yet. We perform a large set of numerical experiments in order to quantify the slope and zero-point of the \alpha-\beta relation. We find strong indication that the relation is indeed an attractor. When combined with the assumption of phase-space being a power-law in radius, this allows us to conclude that equilibrated dark matter structures indeed have zero central velocity anisotropy \beta_0 = 0, central density slope of \alpha_0 = -0.8, and outer anisotropy of \beta_\infty = 0.5.Comment: 15 pages, 7 figure

The formation of ultra-compact dwarf galaxies and nucleated dwarf galaxies

Ultra compact dwarf galaxies (UCDs) have similar properties as massive globular clusters or the nuclei of nucleated galaxies. Recent observations suggesting a high dark matter content and a steep spatial distribution within groups and clusters provide new clues as to their origins. We perform high-resolution N-body / smoothed particle hydrodynamics simulations designed to elucidate two possible formation mechanisms for these systems: the merging of globular clusters in the centre of a dark matter halo, or the massively stripped remnant of a nucleated galaxy. Both models produce density profiles as well as the half light radii that can fit the observational constraints. However, we show that the first scenario results to UCDs that are underluminous and contain no dark matter. This is because the sinking process ejects most of the dark matter particles from the halo centre. Stripped nuclei give a more promising explanation, especially if the nuclei form via the sinking of gas, funneled down inner galactic bars, since this process enhances the central dark matter content. Even when the entire disk is tidally stripped away, the nucleus stays intact and can remain dark matter dominated even after severe stripping. Total galaxy disruption beyond the nuclei only occurs on certain orbits and depends on the amount of dissipation during nuclei formation. By comparing the total disruption of CDM subhaloes in a cluster potential we demonstrate that this model also leads to the observed spatial distribution of UCDs which can be tested in more detail with larger data sets.Comment: 8 pages, 8 figures, final version accepted for publication in MNRA

The Structural Evolution of Substructure

We investigate the evolution of substructure in cold dark matter halos using N-body simulations of tidal stripping of substructure halos (subhalos) within a static host potential. We find that halos modeled following the Navarro, Frenk & White (NFW) mass profile lose mass continuously due to tides from the massive host, leading to the total disruption of satellite halos with small tidal radii. The structure of stripped NFW halos depends mainly on the fraction of mass lost, and can be expressed in terms of a simple correction to the original NFW profile. We apply these results to substructure in the Milky Way, and conclude that the dark matter halos surrounding its dwarf spheroidal (dSph) satellites have circular velocity curves that peak well beyond the luminous radius at velocities significantly higher than expected from the stellar velocity dispersion. Our modeling suggests that the true tidal radii of dSphs lie well beyond the putative tidal cutoff observed in the surface brightness profile, suggesting that the latter are not really tidal in origin but rather features in the light profile of limited dynamical relevance. For Draco, in particular, our modeling implies that its tidal radius is much larger than derived by Irwin & Hatzidimitriou (1995), lending support to the interpretation of recent Sloan survey data by Odenkirchen et al. (2001). Similarly, our model suggests that Carina's halo has a peak circular velocity of ~55 km/s, which may help explain how this small galaxy has managed to retain enough gas to undergo several bursts of star formation. Our results imply a close correspondence between the most massive subhalos expected in a CDM universe and the known satellites of the Milky Way, and suggest that only subhalos with peak circular velocities below 35 km/s lack readily detectable luminous counterparts.Comment: 28 pages, 14 figures; accepted for publication in the Astrophysical Journa

Dark Matter Substructure in Galactic Halos

We use numerical simulations to examine the substructure within galactic and cluster mass halos that form within a hierarchical universe. Clusters are easily reproduced with a steep mass spectrum of thousands of substructure clumps that closely matches observations. However, the survival of dark matter substructure also occurs on galactic scales, leading to the remarkable result that galaxy halos appear as scaled versions of galaxy clusters. The model predicts that the virialised extent of the Milky Way's halo should contain about 500 satellites with circular velocities larger than Draco and Ursa-Minor i.e. bound masses > 10^8Mo and tidally limited sizes > kpc. The substructure clumps are on orbits that take a large fraction of them through the stellar disk leading to significant resonant and impulsive heating. Their abundance and singular density profiles has important implications for the existence of old thin disks, cold stellar streams, gravitational lensing and indirect/direct detection experiments.Comment: Astrophysical Journal Letters. 4 pages, latex. Simulation images and movies at http://star-www.dur.ac.uk:80/~moore