We present TES, a new n-body integration code for the accurate and rapid propagation of planetary systems in the presence of close encounters. TES builds upon the classic Encke method and integrates only the perturbations to Keplerian trajectories to reduce both the error and runtime of simulations. Variable step size is used throughout to enable close encounters to be precisely handled. A suite of numerical improvements are presented that together makes TES optimal in terms of energy error. Lower runtimes are found in all test problems considered when compared to direct integration using ias15. TES is freely available.
A planetary instability occurring at time $<100$ My after formation of the giant planets in our solar system can be responsible for some characteristics of the inner solar system. However, the actual influence of the instability on the terrestrial planet formation is not well understood. The simulations of terrestrial planet formation are very CPU-expensive, and this limits the exploration of different instability scenarios. To include the effects of the giant planets instability in the simulations of terrestrial planets formation in a feasible way, we approach the problem in two steps. First,…
Fast and Furious: Long-term orbit integrations with collocation integrator Lobbie
University of Washington（华盛顿大学）；Institute of Applied Astronomy of the Russian Academy of Sciences（俄罗斯科学院应用天体物理研究所）；NSF-DOE Vera C. Rubin Observatory/NSF NOIRLab（美国国家科学基金会-能源部薇拉·C·鲁宾天文台/美国国家科学基金会NOIRLab）；The Grainger College of Engineering Department of Aerospace Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷格工程学院航空航天工程系）
To date, a handful of exoplanets have been photometrically mapped using phase-modulated reflection or emission from their surfaces, but the small amplitudes of such signals have limited previous maps almost exclusively to coarse dipolar features on hot giant planets. In this work, we uncover a signal using recently released data from the Transiting Exoplanet Survey Satellite (TESS), which we show corresponds to time-variable reflection from a terrestrial planet with a rotation period of 0.9972696 days. Using a spherical harmonic-based reflection model developed as an extension of the STARRY pa…
We present a tool for the comparison and validation of the integration packages suitable for Solar System dynamics. iCompare, written in Python, compares the ephemeris prediction accuracy of a suite of commonly-used integration packages (JPL/HORIZONS, OpenOrb, OrbFit at present). It integrates a set of test particles with orbits picked to explore both usual and unusual regions in Solar System phase space and compares the computed to reference ephemerides. The results are visualized in an intuitive dashboard. This allows for the assessment of integrator suitability as a function of population, …
We present a GPU-accelerated numerical integrator specifically optimized for stability calculations of small bodies in planetary systems. Specifically, the integrator is designed for cases when large numbers of test particles (tens or hundreds of thousands) need to be followed for long durations (millions of orbits) to assess the orbital stability of their initially "close-encounter free" orbits. The GLISSE (Gpu Long-term Integrator for Solar System Evolution) code implements several optimizations to achieve a roughly factor of 100 speed increase over running the same code on a CPU. We explain…
We present a simple algorithm to switch between $N$-body time integrators in a reversible way. We apply it to planetary systems undergoing arbitrarily close encounters and highly eccentric orbits, but the potential applications are broader. Upgrading an ordinary non-reversible switching integrator to a reversible one is straightforward and introduces no appreciable computational burden in our tests. Our method checks if the integrator during the time step violates a time-symmetric selection condition and redoes the step if necessary. In our experiments a few percent of steps would have violate…
We present EnckeHH, a new, highly accurate code for orbital dynamics of perturbed Keplerian systems such as planetary systems or galactic centre systems. It solves Encke's equations of motion, which assume perturbed Keplerian orbits. By incorporating numerical techniques, we have made the code follow optimal roundoff error growth. In a $10^ {12}$ day integration of the outer Solar System, EnckeHH was $3.5$ orders of magnitude more accurate than IAS15 in a fixed time step test. Adaptive steps are recommended for IAS15. Through study of efficiency plots, we show that EnckeHH reaches significantl…
For terrestrial exoplanets with thin atmospheres or no atmospheres, the surface contributes light to the reflected light signal of the planet. Measurement of the variety of disk-integrated brightnesses of bodies in the Solar System and the variation with illumination and wavelength is essential for both planning imaging observations of directly imaged exoplanets and interpreting the eventual datasets. Here we measure the change in brightness of the Galilean satellites as a function of planetocentric longitude, illumination phase angle, and wavelength. The data span a range of wavelengths from …
Hamiltonian systems such as the gravitational N-body problem have time-reversal symmetry. However, all numerical N-body integration schemes, including symplectic ones, respect this property only approximately. In this paper, we present the new N-body integrator JANUS, for which we achieve exact time-reversal symmetry by combining integer and floating point arithmetic. JANUS is explicit, formally symplectic and satisfies Liouville's theorem exactly. Its order is even and can be adjusted between two and ten. We discuss the implementation ofJANUS and present tests of its accuracy and speed by per…
Symplectic integrators have made it possible to study the long-term evolution of planetary systems with direct N-body simulations. In this paper we reassess the accuracy of such simulations by running a convergence test on 20Myr integrations of the Solar System using various symplectic integrators. We find that the specific choice of metric for determining a simulation's accuracy is important. Only looking at metrics related to integrals of motions such as the energy error can overestimate the accuracy of a method. As one specific example, we show that symplectic correctors do not improve the …
The goal of this research is to study how the fragmentation of planetary embryos can affect the physical and dynamical properties of terrestrial planets around solar-type stars. Our work focuses on the formation and evolution of planets and water delivery in the Habitable Zone. We distinguish class A and class B HZ planets, which have an accretion seed initially located inside and beyond the snowline, respectively. We develop an Nbody integrator that incorporates fragmentation and hit-and-run collisions, which is called D3 N-body code. From this, we perform 46 numerical simulations of planetar…
Symplectic integrators are a foundation to the study of dynamical $N$-body phenomena, at scales ranging from from planetary to cosmological. These integrators preserve the Poincaré invariants of Hamiltonian dynamics. The $N$-body Hamiltonian has another, perhaps overlooked, symmetry: it is smooth, or, in other words, it has infinite differentiability class order (DCO) for particle separations greater than $0$. Popular symplectic integrators, such as hybrid methods or block adaptive stepping methods do not come from smooth Hamiltonians and it is perhaps unclear whether they should. We investiga…
Terrestrial planets in temperate orbits around very low mass stars are likely to have evolved in a very different way than solar system planets, and in particular Earth. However, because these are the first planets that are and will be accessible for in-depth atmosphere, clouds and surface characterizations with existing and forthcoming telescopes, we need to develop the best possible observational strategies to maximize the scientific return from these characterizations. Here I discuss and expand on the recent works of Bean et al. (2017) and Turbet et al. (2019) to show that terrestrial plane…
We model the assembly of planets from planetary embryos under the conditions suggested by various scenarios for the formation of the planetary system around the millisecond pulsar B1257+12. We find that the most likely models fall at the low angular momentum end of the proposed range. Models that invoke supernova fallback produce such disks, although we find that a solar composition disk produces a more likely evolution than one composed primarily of heavy elements. Furthermore, we find that dust sedimentation must occur rapidly as the disk cools, in order that the solid material be confined t…
We presented the first particle based, Lagrangian code that can follow the collisional/accretional/dynamical evolution of a large number of km-sized planetesimals through the entire growth process to become planets. We refer to it as the 'Lagrangian Integrator for Planetary Accretion and Dynamics' or LIPAD. LIPAD is built on top of SyMBA, which is a symplectic $N$-body integrator. In order to handle the very large number of planetesimals required by planet formation simulations, we introduce the concept of a `tracer' particle. Each tracer is intended to represent a large number of disk particl…
Many asteroids that make close encounters with terrestrial planets are in a binary configuration. Here we calculate the relevant encounter timescales and investigate the effects of encounters on a binary's mutual orbit. We use a combination of analytical and numerical approaches with a wide range of initial conditions. Our test cases include generic binaries with close, moderate, and wide separations, as well as seven well-characterized near-Earth binaries. We find that close approaches (<10 Earth radii) occur for almost all binaries on 1-10 million year timescales. At such distances, our resu…
WHFast: A fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long term gravitational simulations
University of Toronto at Scarborough（多伦多大学士嘉堡校区）；University of Toronto（多伦多大学）；Canadian Institute for Theoretical Astrophysics（加拿大理论天体物理研究所）；Centre for Planetary Sciences（行星科学中心）
Terrestrial exoplanets orbiting M-dwarf stars are promising targets for transmission spectroscopy with existing or near-future instrumentation. The atmospheric composition of such rocky planets remains an open question, especially given the high X-ray and ultraviolet flux from their host M dwarfs that can drive atmospheric escape. The 1.3 $R_\oplus$ exoplanet GJ 486b ($T_{\rm{eq}} \sim$ 700 K), orbiting an M3.5 star, is expected to have one of the strongest transmission spectroscopy signals among known terrestrial exoplanets. We observed three transits of GJ 486b using three different high-res…
Hurricanes are one of the most extreme storm systems that occur on Earth, characterized by strong rainfall and fast winds. The terrestrial exoplanets that will be characterized with future infrared space telescopes orbit M dwarf stars. As a result, the best observable terrestrial exoplanets have vastly different climates than Earth, with a large dayside-to-nightside irradiation contrast and relatively slow rotation. Hurricanes may affect future observations of terrestrial exoplanets because they enhance the vertical transport of water vapor and could influence ocean heat transport. In this wor…
The macroturbulent atmospheric circulation of Earth-like planets mediates their equator-to-pole heat transport. For fast-rotating terrestrial planets, baroclinic instabilities in the mid-latitudes lead to turbulent eddies that act to transport heat poleward. In this work, we derive a scaling theory for the equator-to-pole temperature contrast and bulk lapse rate of terrestrial exoplanet atmospheres. This theory is built on the work of Jansen & Ferrari (2013), and determines how unstable the atmosphere is to baroclinic instability (the baroclinic "criticality") through a balance between the bar…
