ABSTRACT We develop and apply methods to extract planet masses and eccentricities from observed transit timing 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 simulations, as well as the analytic formulae, to measure the masses and eccentricities of 10 planets discovered by Kepler that have not previously been analyzed. Most of the 10 planets have low densities. Using the analytic expressions to partially circumvent degeneracies, we measure small eccentricities of a few percent or less.
I consider the dynamics of mean-motion resonances (MMRs) between pairs of coplanar planets and derive a new integrable Hamiltonian model for the planets' resonant motion. The new model generalizes ...integrable Hamiltonians previously derived for first-order resonances to the case of 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 truncated expansion is necessary to derive an integrable model. This enables 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 angular-momentum-deficit-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 MMR at large libration amplitudes. 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.
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 and 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 nonzero.
Abstract Mercury’s orbit can destabilize, generally resulting in a collision with either Venus or the Sun. Chaotic evolution can cause g 1 to decrease to the approximately constant value of g 5 and ...create a resonance. Previous work has approximated the variation in g 1 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 significantly underpredicts the Mercury instability probability on timescales less than 5 Gyr, the remaining lifespan of the solar system. This is because g 1 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 g 1 . 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 g 1 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 g 1 than standard diffusion? Why is there an upper bound on g 1 , but not a lower bound that would prevent it from reaching g 5 ?
Abstract
We introduce self-consistent spin, tidal, and dynamical equations of motion into
REBOUNDx
, a library of additional effects for the popular
N
-body integrator
REBOUND
. The equations of ...motion used are derived from the constant time lag approximation to the equilibrium tide model of tidal friction. These effects will allow the study of a variety of systems of which the full dynamical picture cannot be encapsulated by point particle dynamics. We provide several test cases and benchmark the code’s performance against analytic predictions. The open-source code is available in the most recent release of
REBOUNDx
.
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 the order of a few percent. More precisely, the rms eccentricity is 0.018 super(+0.005) sub(-0.004) , and planets smaller than 2.5 R sub(+ in circle) are around twice as eccentric as those bigger than 2.5 R sub(+ in circle). We also find a best-fit density-radius relationship rho approximately 3 g cm super(-3) x(R/3 R sub(+ in circle)) super(-2.3) for the 56 planets that likely have small eccentricity and hence small statistical correction to their masses. Many planets larger than 2.5 R sub(+ in circle) are less dense than water, implying that their radii are largely set by a massive hydrogen atmosphere.
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's pioneering work. But whereas Wisdom considered the overlap of first-order resonances only, limiting the applicability of his criterion to nearly circular planets, 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. 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.
Abstract
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
https://archive.org/details/@dorianabbot
. We find that accurate Mercury instability statistics can be obtained by (1) including only the Sun and the eight 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 9601-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 test beds for theoretical ideas as well as training sets for artificial intelligence schemes.
We report the Transiting Exoplanet Survey Satellite 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 ...transit signals with high signal-to-noise ratio correspond to sub-Neptune-sized planets (2.76, 2.79, and 2.94 R⊕), and we statistically validate the planetary nature of the two inner planets (Pb = 4.65 days, Pc = 9.15 days). With only two transits observed, we report the outer object (P.03 = 19.98 days) as a planet candidate with high signal-to-noise ratio. We also detect a candidate transiting super-Earth (1.4 R⊕) with an orbital period of only 12.7 hr and a candidate Neptune-sized planet (4.2 R⊕) with a period of 13.28 days, both at low signal-to-noise ratio. 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 (Pc/Pb = 1.97) is slightly lower 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.
ABSTRACT
We derive, and discuss the properties of, a symplectic map for the dynamics of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a circular, coplanar orbit interior ...to the pericentre of the parabolic orbit. The map shows excellent agreement with direct numerical integrations and elucidates how the dynamics depends on perturber mass and pericentre distance. We also use the map to explore the onset of chaos, statistical descriptions of chaotic transport, and sticking in mean-motion resonances. We discuss implications of our mapping model for the dynamical evolution of the Solar system’s scattered disc and other highly eccentric trans-Neptunian objects.
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