HELIOS-K: An Ultrafast, Open-source Opacity Calculator for Radiative Transfer

We present an ultrafast opacity calculator that we name HELIOS-K. It takes a line list as an input, computes the shape of each spectral line and provides an option for grouping an enormous number of lines into a manageable number of bins. We implement a combination of Algorithm 916 and Gauss-Hermite quadrature to compute the Voigt profile, write the code in CUDA and optimise the computation for graphics processing units (GPUs). We restate the theory of the k-distribution method and use it to reduce $\sim 10^5$ to $10^8$ lines to $\sim 10$ to $10^4$ wavenumber bins, which may then be used for radiative transfer, atmospheric retrieval and general circulation models. The choice of line-wing cutoff for the Voigt profile is a significant source of error and affects the value of the computed flux by $\sim 10\%$. This is an outstanding physical (rather than computational) problem, due to our incomplete knowledge of pressure broadening of spectral lines in the far line wings. We emphasize that this problem remains regardless of whether one performs line-by-line calculations or uses the k-distribution method and affects all calculations of exoplanetary atmospheres requiring the use of wavelength-dependent opacities. We elucidate the correlated-k approximation and demonstrate that it applies equally to inhomogeneous atmospheres with a single atomic/molecular species or homogeneous atmospheres with multiple species. Using a NVIDIA K20 GPU, HELIOS-K is capable of computing an opacity function with $\sim 10^5$ spectral lines in $\sim 1$ second and is publicly available as part of the Exoclimes Simulation Platform (ESP; www.exoclime.org).Comment: Accepted by ApJ. 8 pages, 5 figure

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

Stochasticity & Predictability in Terrestrial Planet Formation

Terrestrial planets are thought to be the result of a vast number of gravitational interactions and collisions between smaller bodies. We use numerical simulations to show that practically identical initial conditions result in a wide array of final planetary configurations. This is a result of the chaotic evolution of trajectories which are highly sensitive to minuscule displacements. We determine that differences between systems evolved from virtually identical initial conditions can be larger than the differences between systems evolved from very different initial conditions. This implies that individual simulations lack predictive power. For example, there is not a reproducible mapping between the initial and final surface density profiles. However, some key global properties can still be extracted if the statistical spread across many simulations is considered. Based on these spreads, we explore the collisional growth and orbital properties of terrestrial planets which assemble from different initial conditions (we vary the initial planetesimal distribution, planetesimal masses, and giant planet orbits). Confirming past work, we find that the resulting planetary systems are sculpted by sweeping secular resonances. Configurations with giant planets on eccentric orbits produce fewer and more massive terrestrial planets on tighter orbits than those with giants on circular orbits. This is further enhanced if the initial mass distribution is biased to the inner regions. In all cases, the outer edge of the system is set by the final location of the $\nu_6$ resonance and we find that the mass distribution peaks at the $\nu_5$ resonance. Using existing observations, we find that extrasolar systems follow similar trends. Although differences between our numerical modelling and exoplanetary systems remain, we suggest that CoRoT-7, HD 20003, and HD 20781 may host undetected giant planets.Comment: replaced to match published version, 20 pages, 11 figures, published in MNRAS, simulation outputs available at https://cheleb.net/astro/sp15

THOR 2.0: Major Improvements to the Open-Source General Circulation Model

THOR is the first open-source general circulation model (GCM) developed from scratch to study the atmospheres and climates of exoplanets, free from Earth- or Solar System-centric tunings. It solves the general non-hydrostatic Euler equations (instead of the primitive equations) on a sphere using the icosahedral grid. In the current study, we report major upgrades to THOR, building upon the work of Mendon\c{c}a et al. (2016). First, while the Horizontally Explicit Vertically Implicit (HEVI) integration scheme is the same as that described in Mendon\c{c}a et al. (2016), we provide a clearer description of the scheme and improved its implementation in the code. The differences in implementation between the hydrostatic shallow (HSS), quasi-hydrostatic deep (QHD) and non-hydrostatic deep (NHD) treatments are fully detailed. Second, standard physics modules are added: two-stream, double-gray radiative transfer and dry convective adjustment. Third, THOR is tested on additional benchmarks: tidally-locked Earth, deep hot Jupiter, acoustic wave, and gravity wave. Fourth, we report that differences between the hydrostatic and non-hydrostatic simulations are negligible in the Earth case, but pronounced in the hot Jupiter case. Finally, the effects of the so-called "sponge layer", a form of drag implemented in most GCMs to provide numerical stability, are examined. Overall, these upgrades have improved the flexibility, user-friendliness, and stability of THOR.Comment: 57 pages, 31 figures, revised, accepted for publication in ApJ

HELIOS-Retrieval: An Open-source, Nested Sampling Atmospheric Retrieval Code, Application to the HR 8799 Exoplanets and Inferred Constraints for Planet Formation

We present an open-source retrieval code named HELIOS-Retrieval (hereafter HELIOS-R), designed to obtain chemical abundances and temperature-pressure profiles from inverting the measured spectra of exoplanetary atmospheres. In the current implementation, we use an exact solution of the radiative transfer equation, in the pure absorption limit, in our forward model, which allows us to analytically integrate over all of the outgoing rays (instead of performing Gaussian quadrature). Two chemistry models are considered: unconstrained chemistry (where the mixing ratios are treated as free parameters) and equilibrium chemistry (enforced via analytical formulae, where only the elemental abundances are free parameters). The nested sampling algorithm allows us to formally implement Occam's Razor based on a comparison of the Bayesian evidence between models. We perform a retrieval analysis on the measured spectra of the HR 8799b, c, d and e directly imaged exoplanets. Chemical equilibrium is disfavored by the Bayesian evidence for HR 8799b, c and d. We find supersolar C/O, C/H and O/H values for the outer HR 8799b and c exoplanets, while the inner HR 8799d and e exoplanets have substellar C/O, substellar C/H and superstellar O/H values. If these retrieved properties are representative of the bulk compositions of the exoplanets, then they are inconsistent with formation via gravitational instability (without late-time accretion) and consistent with a core accretion scenario in which late-time accretion of ices occurred differently for the inner and outer exoplanets. For HR 8799e, we find that spectroscopy in the K band is crucial for constraining C/O and C/H. HELIOS-R is publicly available as part of the Exoclimes Simulation Platform (ESP; www.exoclime.org).Comment: 27 pages, 21 figures, 3 tables, published in A

Aperiodic Ising Quantum Chains

Some years ago, Luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In this article, we show how Luck's criterion can be derived within an exact renormalisation scheme for Ising quantum chains with coupling constants modulated according to substitution rules. Luck's conjectures for this case are confirmed and refined. Among other outcomes, we give an exact formula for the correlation length critical exponent for arbitrary two-letter substitution sequences with marginal fluctuations of the coupling constants.Comment: 27 pages, LaTeX, 1 Postscript figure included, using epsf.sty and amssymb.sty (one error corrected, some minor changes

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

Interior Characterization in Multiplanetary Systems: TRAPPIST-1

Interior characterization traditionally relies on individual planetary properties, ignoring correlations between different planets of the same system. For multi-planetary systems, planetary data are generally correlated. This is because, the differential masses and radii are better constrained than absolute planetary masses and radii. We explore such correlations and data specific to the multiplanetary-system of TRAPPIST-1 and study their value for our understanding of planet interiors. Furthermore, we demonstrate that the rocky interior of planets in a multi-planetary system can be preferentially probed by studying the most dense planet representing a rocky interior analogue. Our methodology includes a Bayesian inference analysis that uses a Markov chain Monte Carlo scheme. Our interior estimates account for the anticipated variability in the compositions and layer thicknesses of core, mantle, water oceans and ice layers, and a gas envelope. Our results show that (1) interior estimates significantly depend on available abundance proxies and (2) that the importance of inter-dependent planetary data for interior characterization is comparable to changes in data precision by 30 %. For the interiors of TRAPPIST-1 planets, we find that possible water mass fractions generally range from 0-25 %. The lack of a clear trend of water budgets with orbital period or planet mass challenges possible formation scenarios. While our estimates change relatively little with data precision, they critically depend on data accuracy. If planetary masses varied within ±24 %, interiors would be consistent with uniform (~7 %) or an increasing water mass fractions with orbital period (~2-12 %)