Symplectic integration for the collisional gravitational<i>N</i>-body problem

Hernández, David M.; Bertschinger, Edmund

doi:10.1093/mnras/stv1439

Cited by 33 publications

(58 citation statements)

References 28 publications

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“…As shown in Hernandez & Bertschinger (2015), HB15 conserves angular momentum exactly. WHJ has analogous operators A h , B h , and C h of eq.…”

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 77%

“…For HB15, A † h/2 = A h/2 , and twice as many Kepler solver evaluations will be needed, 8 total. Hernandez & Bertschinger (2015) found that the time-reversible method φ 2 h performed better than φ h , even though they are both second order accurate.…”

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 95%

“…Pseudocode for general HB15 is presented in Hernandez & Bertschinger (2015) Section 4, and we restate it here. A drift of time h updates positions by adding a linear term in the momenta.…”

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 99%

“…We now explore the computation times of the two methods. Hernandez & Bertschinger (2015) found a Kepler solver was ≈ 40% more computationally expensive than a kick, but this rough figure will vary depending on the Kepler solver, the orbit, the time step, the programming language, and so forth. With these caveats, we use it to make timing estimates.…”

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 99%

“…The small dR is reasonable for BS as BS is highly accurate with an accuracy tuned by Tol. We show dR for four other integrators not part of the MERCURY software: symplectic methods HB15 and Drift-Kick-Drift (DKD) leapfrog (see for example (Hernandez & Bertschinger 2015)), and non-symplectic methods SAKURA (Gonçalves Ferrari, Boekholt & Portegies Zwart 2014) and 4th order Runge-Kutta (RK4). The non-symplectic methods SAKURA and RK4 have the largest dR.…”

Section: Symplecticity Of Mercurymentioning

confidence: 99%

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Fast and reliable symplectic integration for planetary systemN-body problems

Hernández

2016

Mon. Not. R. Astron. Soc.

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We apply one of the exactly symplectic integrators, that we call HB15, of Hernandez & Bertschinger (2015), along with the Kepler problem solver of Wisdom & Hernandez (2015), to solve planetary system N -body problems. We compare the method to Wisdom-Holman methods (WH) in the MERCURY software package, the MERCURY switching integrator, and others and find HB15 to be the most efficient method or tied for the most efficient method in many cases. Unlike WH, HB15 solved N -body problems exhibiting close encounters with small, acceptable error, although frequent encounters slowed the code. Switching maps like MERCURY change between two methods and are not exactly symplectic. We carry out careful tests on their properties and suggest they must be used with caution. We then use different integrators to solve a 3-body problem consisting of a binary planet orbiting a star. For all tested tolerances and time steps, MERCURY unbinds the binary after 0 to 25 years. However, in the solutions of HB15, a time-symmetric Hermite code, and a symplectic Yoshida method, the binary remains bound for > 1000 years. The methods' solutions are qualitatively different, despite small errors in the first integrals in most cases. Several checks suggest the qualitative binary behavior of HB15's solution is correct. The Bulirsch-Stoer and Radau methods in the MERCURY package also unbind the binary before a time of 50 years, suggesting this dynamical error is due to a MERCURY bug.

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“…As shown in Hernandez & Bertschinger (2015), HB15 conserves angular momentum exactly. WHJ has analogous operators A h , B h , and C h of eq.…”

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 77%

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 95%

“…Pseudocode for general HB15 is presented in Hernandez & Bertschinger (2015) Section 4, and we restate it here. A drift of time h updates positions by adding a linear term in the momenta.…”

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 99%

Section: Overview Of Hb15 and Comparison With Wisdom-holman Methodsmentioning

confidence: 99%

Section: Symplecticity Of Mercurymentioning

confidence: 99%

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Fast and reliable symplectic integration for planetary systemN-body problems

Hernández

2016

Mon. Not. R. Astron. Soc.

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Magnetic activity in the HARPS M dwarf sample

Astudillo-Defru

Delfosse

Bonfils

et al. 2017

A&A

196

172

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Context. Atmospheric magnetic fields in stars with convective envelopes heat stellar chromospheres, and thus increase the observed flux in the Ca ii H and K doublet. Starting with the historical Mount Wilson monitoring program, these two spectral lines have been widely used to trace stellar magnetic activity, and as a proxy for rotation period (P rot) and consequently for stellar age. Monitoring stellar activity has also become essential in filtering out false-positives due to magnetic activity in extra-solar planet surveys. The Ca ii emission is traditionally quantified through the R HK-index, which compares the chromospheric flux in the doublet to the overall bolometric flux of the star. Much work has been done to characterize this index for FGK-dwarfs, but M dwarfs-the most numerous stars of the Galaxy-were left out of these analyses and no calibration of their Ca ii H and K emission to an R HK exists to date. Aims. We set out to characterize the magnetic activity of the low-and very-low-mass stars by providing a calibration of the R HK-index that extends to the realm of M dwarfs, and by evaluating the relationship between R HK and the rotation period. Methods. We calibrated the bolometric and photospheric factors for M dwarfs to properly transform the S-index (which compares the flux in the Ca ii H and K lines to a close spectral continuum) into the R HK. We monitored magnetic activity through the Ca ii H and K emission lines in the HARPS M dwarf sample. Results. The R HK index, like the fractional X-ray luminosity L X /L bol , shows a saturated correlation with rotation, with saturation setting in around a ten days rotation period. Above that period, slower rotators show weaker Ca ii activity, as expected. Under that period, the R HK index saturates to approximately 10 −4. Stellar mass modulates the Ca ii activity, with R HK showing a constant basal activity above 0.6 M and then decreasing with mass between 0.6 M and the fully-convective limit of 0.35 M. Short-term variability of the activity correlates with its mean level and stars with higher R HK indexes show larger R HK variability, as previously observed for earlier spectral types.

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The nature of the TRAPPIST-1 exoplanets

Grimm

Demory

Gillon

et al. 2018

A&A

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344

412

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Context. The TRAPPIST-1 system hosts seven Earth-sized, temperate exoplanets orbiting an ultra-cool dwarf star. As such, it represents a remarkable setting to study the formation and evolution of terrestrial planets that formed in the same protoplanetary disk. While the sizes of the TRAPPIST-1 planets are all known to better than 5% precision, their densities have significant uncertainties (between 28% and 95%) because of poor constraints on the planet's masses. Aims. The goal of this paper is to improve our knowledge of the TRAPPIST-1 planetary masses and densities using transit-timing variations (TTV). The complexity of the TTV inversion problem is known to be particularly acute in multi-planetary systems (convergence issues, degeneracies and size of the parameter space), especially for resonant chain systems such as TRAPPIST-1. Methods. To overcome these challenges, we have used a novel method that employs a genetic algorithm coupled to a full N-body integrator that we applied to a set of 284 individual transit timings. This approach enables us to efficiently explore the parameter space and to derive reliable masses and densities from TTVs for all seven planets. Results. Our new masses result in a five-to eight-fold improvement on the planetary density uncertainties, with precisions ranging from 5% to 12%. These updated values provide new insights into the bulk structure of the TRAPPIST-1 planets. We find that TRAPPIST-1 c and e likely have largely rocky interiors, while planets b, d, f, g, and h require envelopes of volatiles in the form of thick atmospheres, oceans, or ice, in most cases with water mass fractions less than 5%.

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Symplectic integration for the collisional gravitationalN-body problem

Cited by 33 publications

References 28 publications

Fast and reliable symplectic integration for planetary systemN-body problems

Fast and reliable symplectic integration for planetary systemN-body problems

Magnetic activity in the HARPS M dwarf sample

The nature of the TRAPPIST-1 exoplanets

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