arXiv subjects

Sam Hadden

Publications and source records attributed to Sam Hadden.

34 records · Page 2Linked to original sources

arXiv2021

The Criterion for Chaos in Three-Planet Systems

We establish the criterion for chaos in three-planet systems, for systems similar to those discovered by the Kepler spacecraft. Our main results are as follows: (i) The simplest criterion, which is based on overlapping mean motion resonances MMR's), only agrees with numerical simulations at a very crude level. (ii) Much greater accuracy is attained by considering neighboring MMR's that do not overlap. We work out the width of the chaotic zones around each of the neighbors, and also provide simple approximate expressions for the widths. (iii) Even greater accuracy is provided by the overlap of three-body resonances (3BR's), which accounts for fine-grained structure seen in maps from N-body simulations, and also predicts the Lyapunov times. Previous studies conflict on whether overlap of MMR's or of 3BR's drive interplanetary chaos. We show that both do, and in fact they are merely different ways of looking at the same effect. (iv) We compare both criteria with high-resolution maps of chaos from N-body simulations, and show that they agree at a high level of detail.

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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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arXiv2020

Mildly-Hierarchical triple dynamics and applications to the outer solar system

Three-body interactions are ubiquitous in astrophysics. For instance, Kozai-Lidov oscillations in hierarchical triple systems have been studied extensively and applied to a wide range of astrophysical systems. However, mildly-hierarchical triples also play an important role, but they are less explored. In this work we consider the secular dynamics of a test particle in a mildly-hierarchical configuration. We find the limit within which the secular approximation is reliable, present resonances and chaotic regions using surface of sections, and characterize regions of phase space that allow large eccentricity and inclination variations. Finally, we apply the secular results to the outer solar system. We focus on the distribution of extreme trans-neptunian objects (eTNOs) under the perturbation of a possible outer planet (Planet-9), and find that in addition to a low inclination Planet-9, a polar or a counter-orbiting one could also produce pericenter clustering of eTNOs, while the polar one leads to a wider spread of eTNO inclinations.

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arXiv2020

Modeling Radial Velocity Data of Resonant Planets to Infer Migration Histories

A number of giant planet pairs discovered by the radial velocity method with period ratios $\lesssim 2$ may reside in mean motion resonances. Convergent orbital migration and resonant capture at the time of formation would naturally explain the present-day resonant orbital configurations of these systems. Planets that experience smooth migration and eccentricity damping forces due to a proto-planetary disk should not only capture into mean motion resonances but also end up in a specific dynamical configuration within the resonance, sometimes referred to as apsidal corotation resonance (ACR). Here we develop a method for testing the hypothesis that a planet pair resides in an ACR by directly fitting radial velocity data. The ACR hypothesis strongly restricts the number of free parameters describing the radial velocity signal and we compare fits using this highly restricted model to fits using a more conventional two-planet RV model by using nested sampling simulations. We apply our method to HD 45364 and HD 33844, two systems hosting giant planet pairs in 3:2 and 5:3 resonances, respectively. We demonstrate that the observations of both systems support an ACR configuration and we use the results of our ACR model fits to constrain possible migration histories of these systems.

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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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arXiv2019

An Integrable Model for the Dynamics of Planetary Mean Motion Resonances

I consider the dynamics of mean motion resonances between pairs of co-planar planets and derive a new integrable Hamiltonian model for planets' resonant motion. The new model generalizes previously-derived integrable Hamiltonians for first-order resonances to treat higher-order resonances by exploiting a surprising near-symmetry of the full, non-integrable Hamiltonians of higher-order resonances. Whereas past works have frequently relied on truncated disturbing function expansions to derive integrable approximations to resonant motion, I show that no such expansion is necessary, thus enabling the new model to accurately capture the dynamics of both first- and higher-order resonances for eccentricities up to orbit-crossing. I demonstrate that predictions of the new integrable model agree well with numerical integrations of resonant planet pairs. Finally, I explore the secular evolution of resonant planets' eccentricities. I show that the secular dynamics are governed by conservation of an AMD-like quantity. I also demonstrate that secular frequencies depend on planets' resonant libration amplitude and this generally gives rise to a secular resonance inside the mean motion resonance at large libration amplitudes. Outside of the secular resonance the long-term dynamics are characterized small adiabatic modulations of the resonant motion while inside the secular resonance planets can experience large variations of the resonant trajectory over secular timescales. The integrable model derived in this work can serve as a framework for analyzing the dynamics of planetary MMRs in a wide variety of contexts.

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arXiv2019

Prospects for TTV Detection and Dynamical Constraints with TESS

We consider the potential for the Transiting Exoplanet Survey Satellite (TESS) to detect transit timing variations (TTVs) during both its nominal and extended mission phases. Building on previous estimates of the overall yield of planetary systems from the TESS mission, we predict that during its nominal two-year mission, TESS will observe measurable TTVs in $\sim30$ systems, from which $\mathcal{O}(10)$ planet will get precise mass measurements from TTVs alone, $\sim 5$ planets will have significant constraints placed on their masses from TTVs, and over a dozen systems will be singly-transiting TTV systems. We consider a number of different extended mission scenarios, and predict that in a typical scenario, an extended mission will allow TESS to increase the number of systems with measurable TTVs to a total of $\sim90$, from which $\sim 15$ planets will have precise mass measurements, another $\sim 15$ will have significant constraints placed on their masses, and $\sim60$ will be singly-transiting TTV systems. We also describe how follow-up transit observations of multi-planet systems discovered by the TESS mission can be optimally planned to maximize TTV mass and eccentricity constraints.

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arXiv2019

Near-resonance in a system of sub-Neptunes from TESS

We report the Transiting Exoplanet Survey Satellite ($TESS$) detection of a multi-planet system orbiting the $V=10.9$ K0 dwarf TOI 125. We find evidence for up to five planets, with varying confidence. Three high signal-to-noise transit signals correspond to sub-Neptune-sized planets ($2.76$, $2.79$, and $2.94\ R_{\oplus}$), and we statistically validate the planetary nature of the two inner planets ($P_b = 4.65$ days, $P_c = 9.15$ days). With only two transits observed, we report the outer object ($P_{.03} = 19.98$ days) as a high signal-to-noise ratio planet candidate. We also detect a candidate transiting super-Earth ($1.4\ R_{\oplus}$) with an orbital period of only $12.7$ hours and a candidate Neptune-sized planet ($4.2\ R_{\oplus}$) with a period of $13.28$ days, both at low signal-to-noise. This system is amenable to mass determination via radial velocities and transit timing variations, and provides an opportunity to study planets of similar size while controlling for age and environment. The ratio of orbital periods between TOI 125 b and c ($P_c/P_b = 1.97$) is slightly smaller than an exact 2:1 commensurability and is atypical of multiple planet systems from $Kepler$, which show a preference for period ratios just $wide$ of first-order period ratios. A dynamical analysis refines the allowed parameter space through stability arguments and suggests that, despite the nearly commensurate periods, the system is unlikely to be in resonance.

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arXiv2018

A Compact Multi-Planet System With A Significantly Misaligned Ultra Short Period Planet

We report the discovery of a compact multi-planet system orbiting the relatively nearby (78pc) and bright ($K=8.9$) K-star, K2-266 (EPIC248435473). We identify up to six possible planets orbiting K2-266 with estimated periods of P$_b$ = 0.66, P$_{.02}$ = 6.1, P$_c$ = 7.8, P$_d$ = 14.7, P$_e$ = 19.5, and P$_{.06}$ = 56.7 days and radii of R$_P$ = 3.3 R$_{\oplus}$, 0.646 R$_{\oplus}$, 0.705 R$_{\oplus}$, 2.93 R$_{\oplus}$, 2.73 R$_{\oplus}$, and 0.90 R$_{\oplus}$, respectively. We are able to confirm the planetary nature of two of these planets (d & e) from analyzing their transit timing variations ($m_d= 8.9_{-3.8}^{+5.7} M_\oplus$ and $m_e=14.3_{-5.0}^{+6.4} M_\oplus$), confidently validate the planetary nature of two other planets (b & c), and classify the last two as planetary candidates (K2-266.02 & .06). From a simultaneous fit of all 6 possible planets, we find that K2-266 b's orbit has an inclination of 75.32$^{\circ}$ while the other five planets have inclinations of 87-90$^{\circ}$. This observed mutual misalignment may indicate that K2-266 b formed differently from the other planets in the system. The brightness of the host star and the relatively large size of the sub-Neptune sized planets d and e make them well-suited for atmospheric characterization efforts with facilities like the Hubble Space Telescope and upcoming James Webb Space Telescope. We also identify an 8.5-day transiting planet candidate orbiting EPIC248435395, a co-moving companion to K2-266.

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arXiv2018

Prospects for Refining Kepler TTV Masses using TESS Observations

In this paper we investigate systems previously identified to exhibit transit timing variations (TTVs) in Kepler data, with the goal of predicting the expected improvements to the mass and eccentricity constraints that will arise from combining Kepler data with future data from the TESS mission. We advocate for the use of the Kullback-Leibler (KL) divergence as a means to quantify improvements in the measured constraints. Compared to the original Kepler data, the TESS data will have a lower signal-to-noise ratio, rendering some of the planetary transits undetectable, and lowering the accuracy with which the transit mid-time can be estimated. Despite these difficulties, out of the 55 systems (containing 143 planets) investigated, we predict that the collection of short-cadence data by TESS will be of significant value (i.e. it will improve the mass uncertainty such that the KL divergence is > 0.1) for approximately 6 - 14 planets during the nominal mission, with the range primarily driven by the uncertain precision with which transit mid-times will be recovered from TESS data. In an extended mission this would increase to a total of approximately 12 - 25 planets.

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arXiv2018

A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets

We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom (1980)'s pioneering work. But whereas Wisdom considered only nearly circular planets, and hence examined only first order resonances, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet, and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian (Hadden, in prep). We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.

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arXiv2018

Chaotic Dynamics of Trans-Neptunian Objects Perturbed by Planet Nine

Observations of clustering among the orbits of the most distant trans-Neptunian objects (TNOs) has inspired interest in the possibility of an undiscovered ninth planet lurking in the outskirts of the solar system. Numerical simulations by a number of authors have demonstrated that, with appropriate choices of planet mass and orbit, such a planet can maintain clustering in the orbital elements of the population of distant TNOs, similar to the observed sample. However, many aspects of the rich underlying dynamical processes induced by such a distant eccentric perturber have not been fully explored. We report the results of our investigation of the dynamics of coplanar test-particles that interact with a massive body on an circular orbit (Neptune) and a massive body on a more distant, highly eccentric orbit (the putative Planet Nine). We find that a detailed examination of our idealized simulations affords tremendous insight into the rich test-particle dynamics that are possible. In particular, we find that chaos and resonance overlap plays an important role in particles' dynamical evolution. We develop a simple mapping model that allows us to understand, in detail, the web of overlapped mean-motion resonances explored by chaotically evolving particles. We also demonstrate that gravitational interactions with Neptune can have profound effects on the orbital evolution of particles. Our results serve as a starting point for a better understanding of the dynamical behavior observed in more complicated simulations that can be used to constrain the mass and orbit of Planet 9.

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arXiv2017

The Gold Standard: Accurate Stellar and Planetary Parameters for Eight Kepler M dwarf Systems Enabled by Parallaxes

We report parallaxes and proper motions from the Hawaii Infrared Parallax Program for eight nearby M dwarf stars with transiting exoplanets discovered by Kepler. We combine our directly measured distances with mass-luminosity and radius-luminosity relationships to significantly improve constraints on the host stars' properties. Our astrometry enables the identification of wide stellar companions to the planet hosts. Within our limited sample, all the multi-transiting planet hosts (three of three) appear to be single stars, while nearly all (four of five) of the systems with a single detected planet have wide stellar companions. By applying strict priors on average stellar density from our updated radius and mass in our transit fitting analysis, we measure the eccentricity probability distributions for each transiting planet. Planets in single-star systems tend to have smaller eccentricities than those in binaries, although this difference is not significant in our small sample. In the case of Kepler-42bcd, where the eccentricities are known to be $\simeq$0, we demonstrate that such systems can serve as powerful tests of M dwarf evolutionary models by working in $L_*-ρ_*$ space. The transit-fit density for Kepler-42bcd is inconsistent with model predictions at 2.1$σ$ (22%), but matches more empirical estimates at 0.2$σ$ (2%), consistent with earlier results showing model radii of M dwarfs are under-inflated. Gaia will provide high precision parallaxes for the entire Kepler M dwarf sample, and TESS will identify more planets transiting nearby, late-type stars, enabling significant improvements in our understanding of the eccentricity distribution of small planets and the parameters of late-type dwarfs.

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arXiv2016

Kepler Planet Masses and Eccentricities from TTV Analysis

We conduct a uniform analysis of the transit timing variations (TTVs) of 145 planets from 55 Kepler multiplanet systems to infer planet masses and eccentricities. Eighty of these planets do not have previously reported mass and eccentricity measurements. We employ two complementary methods to fit TTVs: Markov chain Monte Carlo simulations based on N-body integration and an analytic fitting approach. Mass measurements of 49 planets, including 12 without previously reported masses, meet our criterion for classification as robust. Using mass and radius measurements, we infer the masses of planets' gaseous envelopes for both our TTV sample as well as transiting planets with radial velocity observations. Insight from analytic TTV formulae allows us to partially circumvent degeneracies inherent to inferring eccentricities from TTV observations. We find that planet eccentricities are generally small, typically a few percent, but in many instances are non-zero.

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arXiv2015

Numerical and Analytical Modelling of Transit Time Variations

We develop and apply methods to extract planet masses and eccentricities from observed transit time variations (TTVs). First, we derive simple analytic expressions for the TTV that include the effects of both first- and second-order resonances. Second, we use N-body Markov chain Monte Carlo (MCMC) simulations, as well as the analytic formulae, to measure the masses and eccentricities of ten planets discovered by Kepler that have not previously been analyzed. Most of the ten planets have low densities. Using the analytic expressions to partially circumvent degeneracies, we measure small eccentricities of a few percent or less.

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arXiv2014

Densities and Eccentricities of 139 Kepler Planets from Transit Time Variations

We extract densities and eccentricities of 139 sub-Jovian planets by analyzing transit time variations (TTVs) obtained by the Kepler mission through Quarter 12. We partially circumvent the degeneracies that plague TTV inversion with the help of an analytical formula for the TTV. From the observed TTV phases, we find that most of these planets have eccentricities of order a few percent. More precisely, the r.m.s. eccentricity is 0.018^{+0.005}_{-0.004}, and planets smaller than 2.5R_\earth are around twice as eccentric as those bigger than 2.5R_\earth. We also find a best-fit density-radius relationship ρ~3 g/cm^3 \times (R/3R_\earth)^{-2.3} for the 56 planets that likely have small eccentricity and hence small statistical correction to their masses. Many planets larger than 2.5R_\earth are less dense than water, implying that their radii are largely set by a massive hydrogen atmosphere.

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