arXiv subjects

David M. Hernandez

Publications and source records attributed to David M. Hernandez.

At least 19 recordsLinked to original sources

arXiv2026

Artificial precession and instability in solar system and planetary simulations: analytic and numerical results

Wisdom--Holman (WH) methods are algorithms used as a basis for a wide range of codes used to solve problems in solar system and planetary dynamics. The problems range from the growth and migration of planets to the stability of the solar system. In many cases, these codes work with Democratic Heliocentric Coordinates (DHC) which offer some advantages. However, it has been noted these coordinates affect the dynamics of solar system bodies in simulations, in particular Mercury's, and introduce artificial precession which affects solar system stability. In this work, we analytically derive the two-body artificial precession induced by DHC. We show the effect is small for solar system bodies, but the artificial effect on Jupiter is $242$ times larger than on Mercury. In a two-body Mercury-Sun system with general relativity (GR), artificial precession is negligible compared to GR precession, even with extreme timesteps that amplify the numerical effects. A simple two-planet Mercury--Jupiter system without GR amplifies artificial precession significantly. However, large artificial precession or artificial instability is not a danger unless one uses large timesteps that break the surrogate Hamiltonian approximation.

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arXiv2026

A free-fall-based switching criterion for P^3 T N-body methods in collisional stellar systems

The P$^3$T scheme is a hybrid method for simulating gravitational $N$-body systems. It combines a fast particle-tree (PT) algorithm for long-range forces with a high-accuracy particle-particle (PP, direct $N$-body) solver for short-range interactions. Preserving both PT efficiency and PP accuracy requires a robust PT-PP switching criterion. We introduce a simple free-fall-based switching criterion for general stellar systems, alongside the commonly used velocity-dispersion-based ($σ$-based) criterion. Using the \textsc{petar} code with the P$^3$T scheme and slow-down algorithmic regularization for binaries and higher-order multiples, we perform extensive simulations of star clusters to evaluate how each criterion affects energy conservation and binary evolution. For systems in virial equilibrium, we find that the free-fall-based criterion is generally more accurate for low-$σ$ or loose clusters containing binaries, whereas the $σ$-based criterion is better suited for high-$σ$ systems. Under subvirial or fractal initial conditions, both criteria struggle to maintain high energy conservation; however, the free-fall-based criterion improves as the tree timestep is reduced, whereas the $σ$-based degrades due to its low-accuracy treatment of two-body encounters.

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arXiv2025

No influence of passing stars on paleoclimate reconstructions over the past 56 million years

Passing stars (also called stellar flybys) have notable effects on the solar system's long-term dynamical evolution, injection of Oort cloud comets into the solar system, properties of trans-Neptunian objects, and more. Based on a simplified solar system model, omitting the Moon and the Sun's quadrupole moment J2, it has recently been suggested that passing stars are also an important driver of paleoclimate before ~50 Myr ago, including a climate event called the Paleocene-Eocene Thermal Maximum (~56 Myr ago). In contrast, using a state-of-the-art solar system model, including a lunar contribution and J2, and random stellar parameters (>400 simulations), we find no influence of passing stars on paleoclimate reconstructions over the past 56 Myr. Even in an extreme flyby scenario in which the Sun-like star HD 7977 (m = 1.07 M_Sun) would have passed within ~3,900 au about 2.8 Myr ago (with 5% likelihood), we detect no discernible change in Earth's orbital evolution over the past 70 Myr, compared to our standard model. Our results indicate that a complete physics model is essential to accurately study the effects of stellar flybys on Earth's orbital evolution.

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arXiv2025

Collisional Fragmentation Support in TRACE

We present improved collision support for TRACE, a state-of-the-art hybrid integrator in REBOUND. TRACE now supports collisional fragmentation and can handle both removing and adding particles mid-timestep. We describe the back-end logic implemented for robust collision support, and compare TRACE's performance to other integrators including MERCURIUS on a large-N protoplanetary disk simulation with various collision prescriptions, a system which TRACE previously could not handle. TRACE matches the behavior of these integrators, while offering potentially vast speedups of over 70x. All updates described in this Note are available with the most recent public release of REBOUND.

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arXiv2024

TRACE: a code for Time-Reversible Astrophysical Close Encounters

We present TRACE, an almost time-reversible hybrid integrator for the planetary N-body problem. Like hybrid symplectic integrators, TRACE can resolve close encounters between particles while retaining many of the accuracy and speed advantages of a fixed time-step symplectic method such the Wisdom-Holman map. TRACE switches methods time-reversibly during close encounters following the prescription of Hernandez & Dehnen. In this paper we describe the derivation and implementation of TRACE and study its performance for a variety of astrophysical systems. In all our test cases, TRACE is at least as accurate and fast as the hybrid symplectic integrator MERCURIUS. In many cases, TRACE's performance is vastly superior to that of MERCURIUS. In test cases with planet-planet close encounters, TRACE is as accurate as MECURIUS with a 12x speed-up. If close encounters with the central star are considered, TRACE achieves good error performance while MERCURIUS fails to give qualitatively correct results. In ensemble tests of violent scattering systems, TRACE matches the high-accuracy IAS15 while providing a 15x speed-up. In large N systems simulating lunar accretion, TRACE qualitatively gives the same results as IAS15 but at a 41x speed-up. We also discuss some cases such as von Zeipel-Lidov-Kozai cycles where hybrid integrators perform poorly and provide some guidance on which integrator to use for which system. TRACE is freely available within the REBOUND package.

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arXiv2024

Mercury's chaotic secular evolution as a subdiffusive process

Mercury's orbit can destabilize, generally resulting in a collision with either Venus or the Sun. Chaotic evolution can cause g1 to decrease to the approximately constant value of g5 and create a resonance. Previous work has approximated the variation in g1 as stochastic diffusion, which leads to a phenomological model that can reproduce the Mercury instability statistics of secular and N-body models on timescales longer than 10 Gyr. Here we show that the diffusive model underpredicts the Mercury instability probability by a factor of 3-10,000 on timescales less than 5 Gyr, the remaining lifespan of the Solar System. This is because g1 exhibits larger variations on short timescales than the diffusive model would suggest. To better model the variations on short timescales, we build a new subdiffusive phenomological model for g1. Subdiffusion is similar to diffusion but exhibits larger displacements on short timescales and smaller displacements on long timescales. We choose model parameters based on the behavior of the g1 trajectories in the N-body simulations, leading to a tuned model that can reproduce Mercury instability statistics from 1-40 Gyr. This work motivates fundamental questions in Solar System dynamics: Why does subdiffusion better approximate the variation in g1 than standard diffusion? Why is there an upper bound on g1, but not a lower bound that would prevent it from reaching g5?

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arXiv2024

Multiple timestep reversible $N$-body integrators for close encounters in planetary systems

We present new almost time-reversible integrators for solution of planetary systems consisting of "planets" and a dominant mass ("star"). The algorithms can be considered adaptive generalizations of the Wisdom--Holman method, in which all pairs of planets can be assigned timesteps. These timesteps, along with the global timestep, can be adapted time-reversibly, often at no appreciable additional compute cost, without sacrificing any of the long-term error benefits of the Wisdom--Holman method. The method can also be considered a simpler and more flexible version of the \texttt{SYMBA} symplectic code. We perform tests on several challenging problems with close encounters and find the reversible algorithms are up to $2.6$ times faster than a code based on \texttt{SYMBA}. The codes presented here are available on Github. We also find adapting a global timestep reversibly and discretely must be done in block-synchronized manner or similar.

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arXiv2024

AI can identify Solar System instability billions of years in advance

Rare event schemes require an approximation of the probability of the rare event as a function of system state. Finding an appropriate reaction coordinate is typically the most challenging aspect of applying a rare event scheme. Here we develop an artificial intelligence (AI) based reaction coordinate that effectively predicts which of a limited number of simulations of the Solar System will go unstable using a convolutional neural network classifier. The performance of the algorithm does not degrade significantly even 3.5 billion years before the instability. We overcome the class imbalance intrinsic to rare event problems using a combination of minority class oversampling, increased minority class weighting, and pulling multiple non-overlapping training sequences from simulations. Our success suggests that AI may provide a promising avenue for developing reaction coordinates without detailed theoretical knowledge of the system.

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arXiv2023

ASSIST: An Ephemeris-Quality Test Particle Integrator

We introduce ASSIST, a software package for ephemeris-quality integrations of test particles. ASSIST is an extension of the REBOUND framework and makes use of its IAS15 integrator to integrate test particle trajectories in the field of the Sun, Moon, planets, and 16 massive asteroids, with the positions of the masses coming from the JPL DE441 ephemeris and its associated asteroid perturber file. The package incorporates the most significant gravitational harmonics and general relativistic corrections. ASSIST also accounts for position- and velocity-dependent non-gravitational effects. The first order variational equations are included for all terms to support orbit fitting and covariance mapping. This new framework is meant to provide an open-source package written in a modern language to enable high-precision orbital analysis and science by the small body community. ASSIST is open source, freely distributed under the GNU General Public license, version 3.

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arXiv2023

Switching integrators reversibly in the astrophysical $N$-body problem

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 violated the condition without our corrections. By eliminating them the algorithm avoids long-term error accumulation, of several orders magnitude in some cases.

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arXiv2023

Simple physics and integrators accurately reproduce Mercury instability statistics

The long-term stability of the Solar System is an issue of significant scientific and philosophical interest. The mechanism leading to instability is Mercury's eccentricity being pumped up so high that Mercury either collides with Venus or is scattered into the Sun. Previously, only three five-billion-year $N$-body ensembles of the Solar System with thousands of simulations have been run to assess long-term stability. We generate two additional ensembles, each with 2750 members, and make them publicly available at \texttt{https://archive.org/details/@dorianabbot}. We find that accurate Mercury instability statistics can be obtained by (1) including only the Sun and the 8 planets, (2) using a simple Wisdom-Holman scheme without correctors, (3) using a basic representation of general relativity, and (4) using a time step of 3.16 days. By combining our Solar System ensembles with previous ensembles we form a 9,601-member ensemble of ensembles. In this ensemble of ensembles, the logarithm of the frequency of a Mercury instability event increases linearly with time between 1.3 and 5 Gyr, suggesting that a single mechanism is responsible for Mercury instabilities in this time range and that this mechanism becomes more active as time progresses. Our work provides a robust estimate of Mercury instability statistics over the next five billion years, outlines methodologies that may be useful for exoplanet system investigations, and provides two large ensembles of publicly available Solar System integrations that can serve as testbeds for theoretical ideas as well as training sets for artificial intelligence schemes.

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arXiv2021

Stepsize errors in the $N$-body problem: discerning Mercury's true possible long-term orbits

Numerical integrations of the Solar System have been carried out for decades. Their results have been used, for example, to determine whether the Solar System is chaotic, whether Mercury's orbit is stable, or to help discern Earth's climate history. We argue that all of the past studies we consider in this work are affected by numerical chaos to different degrees, affecting the possible orbits and instability probability of Mercury, sometimes significantly. We show how to eliminate the effects of numerical chaos by resolving Mercury's pericentre passage. We also show that several higher order symplectic maps do not exhibit significant differences in resolving pericentre passage of Mercury (at fixed time step), making their advantages suspect for calculating long-term orbits. Resolving pericentre passage affects a wide array of orbital numerical studies, like exoplanet studies, studies of the galactic centre, and other $N$-body problems.

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arXiv2021

A differentiable N-body code for transit timing and dynamical modeling. I. Algorithm and derivatives

When fitting N-body models to astronomical data - including transit times, radial velocity, and astrometric positions at observed times - the derivatives of the model outputs with respect to the initial conditions can help with model optimization and posterior sampling. Here we describe a general-purpose symplectic integrator for arbitrary orbital architectures, including those with close encounters, which we have recast to maintain numerical stability and precision for small step sizes. We compute the derivatives of the N-body coordinates and velocities as a function of time with respect to the initial conditions and masses by propagating the Jacobian along with the N-body integration. For the first time we obtain the derivatives of the transit times with respect to the initial conditions and masses using the chain rule, which is quicker and more accurate than using finite differences or automatic differentiation. We implement this algorithm in an open source package, NbodyGradient.jl, written in the Julia language, which has been used in the optimization and error analysis of transit-timing variations in the TRAPPIST-1 system. We present tests of the accuracy and precision of the code, and show that it compares favorably in speed to other integrators which are written in C.

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arXiv2021

On the reliability of simulations of collisional stellar systems

It is well known that numerical errors grow exponentially in $N$-body simulations of gravitational bound stellar systems, but it is not well understood how the accuracy parameters of algorithms affect the physical evolution in simulations. By using the hybrid $N$-body code, PeTar, we investigate how escapers and the structure evolution of collisional stellar systems (e.g., star clusters) depend on the accuracy of long-range and short-range interactions. We study a group of simulations of ideal low-mass star clusters in which the accuracy parameters are varied. We find that although the number of escapers is different in individual simulations, its distribution from all simulations can be described by Poisson statistics. The density profile also has a similar feature. By using a self-consistent set-up of the accuracy parameters for long- and short-range interactions, such that orbits are resolved well enough, the physical evolution of the models is identical. But when the short-range accuracy is too low, a nonphysical dynamical evolution can easily occur; this is not the case for long-range interactions. This strengthens the need to include a proper algorithm (e.g. regularization methods) in the realistic modelling of collisional stellar systems. We also demonstrate that energy-conservation is not a good indicator to monitor the quality of the simulations. The energy error of the system is controlled by the hardest binary, and thus, it may not reflect the ensemble error of the global system.

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arXiv2021

Significant improvement in planetary system simulations from statistical averaging

Symplectic integrators are widely used for the study of planetary dynamics and other $N$-body problems. In a study of the outer Solar system, we demonstrate that individual symplectic integrations can yield biased errors in the semi-major axes and possibly other orbital elements. The bias is resolved by studying an ensemble of initial conditions of the outer Solar system. Such statistical sampling could significantly improve measurement of planetary system properties like their secular frequencies. We also compared the distributions of action-like variables between high and low accuracy integrations; traditional statistical metrics are unable to distinguish the distribution functions.

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arXiv2021

Refining the transit timing and photometric analysis of TRAPPIST-1: Masses, radii, densities, dynamics, and ephemerides

We have collected transit times for the TRAPPIST-1 system with the Spitzer Space Telescope over four years. We add to these ground-based, HST and K2 transit time measurements, and revisit an N-body dynamical analysis of the seven-planet system using our complete set of times from which we refine the mass ratios of the planets to the star. We next carry out a photodynamical analysis of the Spitzer light curves to derive the density of the host star and the planet densities. We find that all seven planets' densities may be described with a single rocky mass-radius relation which is depleted in iron relative to Earth, with Fe 21 wt% versus 32 wt% for Earth, and otherwise Earth-like in composition. Alternatively, the planets may have an Earth-like composition, but enhanced in light elements, such as a surface water layer or a core-free structure with oxidized iron in the mantle. We measure planet masses to a precision of 3-5%, equivalent to a radial-velocity (RV) precision of 2.5 cm/sec, or two orders of magnitude more precise than current RV capabilities. We find the eccentricities of the planets are very small; the orbits are extremely coplanar; and the system is stable on 10 Myr timescales. We find evidence of infrequent timing outliers which we cannot explain with an eighth planet; we instead account for the outliers using a robust likelihood function. We forecast JWST timing observations, and speculate on possible implications of the planet densities for the formation, migration and evolution of the planet system.

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arXiv2020

EnckeHH: an integrator for gravitational dynamics with a dominant mass that achieves optimal error behaviour

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 significantly higher accuracy than the Rebound integrators IAS15 and WHCKL for fixed step size.

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arXiv2020

Are long-term $N$-body simulations reliable?

$N$-body integrations are used to model a wide range of astrophysical dynamics, but they suffer from errors which make their orbits diverge exponentially in time from the correct orbits. Over long time-scales, their reliability needs to be established. We address this reliability by running a three-body planetary system over about $200$ e-folding times. Using nearby initial conditions, we can construct statistics of the long-term phase-space structure and compare to rough estimates of resonant widths of the system. We compared statistics for a wide range of numerical methods, including a Runge--Kutta method, Wisdom--Holman method, symplectic corrector methods, and a method by Laskar and Robutel. "Improving" an integrator did not increase the phase space accuracy, but simply increasing the number of initial conditions did. In fact, the statistics of a higher order symplectic corrector method were inconsistent with the other methods in one test.

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