Orbital Period Ratios Research Articles

Semi-analytical orbital dynamics around the primary of a binary asteroid system

ABSTRACT The orbital dynamics in the vicinity of a binary asteroid system has been studied extensively, motivated by the special dynamical environment and possible exploration missions. Equilibrium points, periodic orbits, and invariant manifolds have been investigated in many studies based on the model of the Restricted Full Three Body Problem (RF3BP). In this paper, a new semi-analytical orbital dynamical model around the primary of a binary system is developed as a perturbed two-body problem. The solution includes the effect of the primary's oblateness and the secondary's third-body gravity. The semi-analytical dynamical model, also denoted as the averaged model, is obtained by using the averaging process and Lagrange planetary equations in terms of the Milankovitch orbital elements. This semi-analytical model enables much faster orbital propagations than the non-averaged counterpart, and is particularly useful in orbital stability analysis and the design of long-term passively stable orbits and orbits with specific performance, e.g. frozen orbits. The applicability of the semi-analytical model is then discussed. Two parameters describing relative magnitudes of both perturbations w.r.t. the primary's point mass gravity and the third parameter related to the orbital period ratio w.r.t. the secondary are defined to provide indicators for the validity of the averaged model. The validity boundaries in terms of the three parameters are given based on numerical simulations, by comparing with the full orbital model. The application to a real binary system, 2003 YT1, has shown that the averaged solution has a high precision in the long-term orbital propagation.

Forming Diverse Super-Earth Systems In Situ

Abstract Super-Earths and mini-Neptunes exhibit great diversity in their compositional and orbital properties. Their bulk densities span a large range, from those dense enough to be purely rocky to those needing a substantial contribution from volatiles to their volumes. Their orbital configurations range from compact, circular multitransiting systems like Kepler-11 to systems like our solar system’s terrestrial planets with wider spacings and modest but significant eccentricities and mutual inclinations. Here we investigate whether a continuum of formation conditions resulting from variation in the amount of solids available in the inner disk can account for the diversity of orbital and compositional properties observed for super-Earths, including the apparent dichotomy between single and multitransiting systems. We simulate in situ formation of super-Earths via giant impacts and compare to the observed Kepler sample. We find that intrinsic variations among disks in the amount of solids available for in situ formation can account for the orbital and compositional diversity observed among Kepler’s transiting planets. Our simulations can account for the planets’ distributions of orbital period ratios, transit duration ratios, and transit multiplicity; higher eccentricities for planets in single transiting systems than for those in multitransiting systems; smaller eccentricities for larger planets; scatter in the mass–radius relation, including lower densities for planets with masses measured with transit timing variations instead of with radial velocity; and similarities in planets’ sizes and spacings within each system. Our findings support the theory that variation among super-Earth and mini-Neptune properties is primarily locked in by different in situ formation conditions, rather than arising stochastically through subsequent evolution.

Near-resonance in a System of Sub-Neptunes from TESS

Abstract 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 (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 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 (P c /P b = 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.

Doubly eclipsing systems

Context. Only several doubly eclipsing quadruple stellar systems are known to date, and no dedicated effort to characterize population properties of these interesting objects has yet been made. Aims. Our first goal was to increase number of known doubly eclipsing systems such that the resulting dataset would allow us to study this category of objects via statistical means. In order to minimize biases, we used long-lasting, homogeneous, and well-documented photometric surveys. Second, a common problem of basically all known doubly eclipsing systems is the lack of proof that they constitute gravitationally bound quadruple system in the 2+2 architecture (as opposed to two unrelated binaries that are projected onto the same location in the sky by chance). When possible, we thus sought evidence for the relative motion of the two binaries. In that case, we tried to determine the relevant orbital periods and other parameters. Methods. We analysed photometric data for eclipsing binaries provided by the OGLE survey and we focused on the LMC fields. We found a large number of new doubly eclipsing systems (our discoveries are three times more numerous than the previously known cases in this dataset). In order to prove relative motion of the binaries about a common centre of mass, we made use of the fact that OGLE photometry covers several years. With a typical orbital period of days for the observed binaries, we sought eclipse time variations (ETVs) on the timescale comparable to a decade (this is the same method used for an archetype of the doubly eclipsing system, namely V994 Her). In the cases where we were able to detect the ETV period, the difference between the inner and outer periods in the quadruple system is large enough. This allows us to interpret ETVs primarily as the light-time effect, thus providing an interesting constraint on masses of the binaries. Results. In addition to significantly enlarging the database of known doubly eclipsing systems, we performed a thorough analysis of 72 cases. ETVs for 28 of them (39% of the studied cases) showed evidence of relative motion. Among these individual systems, we note OGLE BLG-ECL-145467, by far the most interesting case; it is bright (12.6 mag in I filter), consists of two detached binaries with periods of ≃3.3 d and ≃4.9 d (making it a candidate for a 3:2 resonant system) revolving about each other in only ≃1538 d. Distribution of the orbital period ratio PA/PB of binaries in 2+2 quadruples shows statistically significant excess at ≃1 and ≃1.5. The former is likely a natural statistical preference in weakly interacting systems with periods within the same range. The latter is thought to be evidence of a capture in the 3:2 mean motion resonance of the two binaries. This sets important constraints on evolutionary channels in these systems. Conclusions. The total number of doubly eclipsing systems increased to 146, more than 90% of which are at low declinations on the southern sky. This motivates us to use southern hemisphere facilities to further characterize these systems, and to seek possibilities to complement this dataset with northern sky systems.

Prevalence of Fibonacci numbers in orbital period ratios in solar planetary and satellite systems and in exoplanetary systems

It is shown that orbital period ratios of successive planets in the Solar System, of satellites in giant planet systems and of exoplanets in exoplanetary systems are preferentially closer to irreducible fractions formed with Fibonacci numbers between 1 and 8 than to other fractions, in a ratio of approximately 60% vs 40%. Furthermore, if sets of minor planets are chosen with gradually smaller inclinations and eccentricities, the proximity to Fibonacci fractions of their period ratios with Jupiter or Mars’ period tends to increase. Finally, a simple model explains why the resonance of the form $\frac{P_{1}}{P_{2}} = \frac{p}{p+q}$ , with $P_{1}$ and $P_{2}$ orbital periods of planets or satellites and $p$ and $q$ small integers, are stronger and more commonly observed for $p$ and $( p+q )$ being both small Fibonacci numbers than for other cases.

TOI-216b and TOI-216 c: Two Warm, Large Exoplanets in or Slightly Wide of the 2:1 Orbital Resonance

Abstract Warm, large exoplanets with 10–100 day orbital periods pose a major challenge to our understanding of how planetary systems form and evolve. Although high eccentricity tidal migration has been invoked to explain their proximity to their host stars, a handful reside in or near orbital resonance with nearby planets, suggesting a gentler history of in situ formation or disk migration. Here we confirm and characterize a pair of warm, large exoplanets discovered by the TESS Mission orbiting K-dwarf TOI-216. Our analysis includes additional transits and transit exclusion windows observed via ground-based follow-up. We find two families of solutions, one corresponding to a sub-Saturn-mass planet accompanied by a Neptune-mass planet and the other to a Jupiter in resonance with a sub-Saturn-mass planet. We prefer the second solution based on the orbital period ratio, the planet radii, the lower free eccentricities, and libration of the 2:1 resonant argument, but cannot rule out the first. The free eccentricities and mutual inclination are compatible with stirring by other, undetected planets in the system, particularly for the second solution. We discuss prospects for better constraints on the planets’ properties and orbits through follow-up, including transits observed from the ground.

The California-Kepler Survey. V. Peas in a Pod: Planets in a Kepler Multi-planet System Are Similar in Size and Regularly Spaced*

Abstract We have established precise planet radii, semimajor axes, incident stellar fluxes, and stellar masses for 909 planets in 355 multi-planet systems discovered by Kepler. In this sample, we find that planets within a single multi-planet system have correlated sizes: each planet is more likely to be the size of its neighbor than a size drawn at random from the distribution of observed planet sizes. In systems with three or more planets, the planets tend to have a regular spacing: the orbital period ratios of adjacent pairs of planets are correlated. Furthermore, the orbital period ratios are smaller in systems with smaller planets, suggesting that the patterns in planet sizes and spacing are linked through formation and/or subsequent orbital dynamics. Yet, we find that essentially no planets have orbital period ratios smaller than 1.2, regardless of planet size. Using empirical mass–radius relationships, we estimate the mutual Hill separations of planet pairs. We find that 93% of the planet pairs are at least 10 mutual Hill radii apart, and that a spacing of ∼20 mutual Hill radii is most common. We also find that when comparing planet sizes, the outer planet is larger in 65% ± 0.4% of cases, and the typical ratio of the outer to inner planet size is positively correlated with the temperature difference between the planets. This could be the result of photo-evaporation.

Near Mean-motion Resonances in the System Observed by Kepler: Affected by Mass Accretion and Type I Migration

Abstract The Kepler mission has released over 4496 planetary candidates, among which 3483 planets have been confirmed as of 2017 April. The statistical results of the planets show that there are two peaks around 1.5 and 2.0 in the distribution of orbital period ratios. The observations indicate that plenty of planet pairs could have first been captured into mean-motion resonances (MMRs) in planetary formation. Subsequently, these planets depart from exact resonant locations to be near-MMR configurations. Through type I migration, two low-mass planets have a tendency to be trapped in first-order MMRs (2:1 or 3:2 MMRs); however, two scenarios of mass accretion of planets and potential outward migration play important roles in reshaping their final orbital configurations. Under the scenario of mass accretion, the planet pairs can cross 2:1 MMRs and then enter into 3:2 MMRs, especially for the inner pairs. With such a formation scenario, the possibility that two planets are locked into 3:2 MMRs can increase if they are formed in a flat disk. Moreover, the outward migration can make planets have a high likelihood to be trapped into 3:2 MMRs. We perform additional runs to investigate the mass relationship for those planets in three-planet systems, and we show that two peaks near 1.5 and 2.0 for the period ratios of two planets can be easily reproduced through our formation scenario. We further show that the systems in chain resonances (e.g., 4:2:1, 3:2:1, 6:3:2, and 9:6:4 MMRs), have been observed in our simulations. This mechanism can be applicable to understand the formation of systems of Kepler-48, Kepler-53, Kepler-100, Kepler-192, Kepler-297, Kepler-399, and Kepler-450.

Exoplanet Predictions Based on Harmonic Orbit Resonances

The current exoplanet database includes 5454 confirmed and candidate planets observed with the Kepler mission. We find 932 planet pairs from which we extract distance and orbital period ratios. While earlier studies used a logarithmic spacing, which lacks a physical model, we employ here the theory of harmonic orbit resonances, which contains quantized ratios instead, to explain the observed planet distance ratios and to predict undetected exoplanets. We find that the most prevailing harmonic ratios are (2:1), (3:2), and (5:3) in 73% of the cases, while alternative harmonic ratios of (5:4), (4:3), (5:2), and (3:1) occur in the other 27% of the cases. Our orbital predictions include 171 exoplanets, 2 Jupiter moons, 1 Saturn moon, 3 Uranus moons, and 4 Neptune moons. The accuracy of the predicted planet distances amounts to a few percent, which fits the data significantly better than the logarithmic spacing. This information may be useful for targeted exoplanet searches with Kepler data and to estimate the number of live-carrying planets in habitable zones.

Analytical model of multi-planetary resonant chains and constraints on migration scenarios

Resonant chains are groups of planets for which each pair is in resonance, with an orbital period ratio locked at a rational value (2/1, 3/2, etc.). Such chains naturally form as a result of convergent migration of the planets in the proto-planetary disk. In this article, I present an analytical model of resonant chains of any number of planets. Using this model, I show that a system captured in a resonant chain can librate around several possible equilibrium configurations. The probability of capture around each equilibrium depends on how the chain formed, and especially on the order in which the planets have been captured in the chain. Therefore, for an observed resonant chain, knowing around which equilibrium the chain is librating allows for constraints to be put on the formation and migration scenario of the system. I apply this reasoning to the four planets orbiting Kepler-223 in a 3:4:6:8 resonant chain. I show that the system is observed around one of the six equilibria predicted by the analytical model. Using N-body integrations, I show that the most favorable scenario to reproduce the observed configuration is to first capture the two intermediate planets, then the outermost, and finally the innermost.

Self-organizing systems in planetary physics: Harmonic resonances of planet and moon orbits

The geometric arrangement of planet and moon orbits into a regularly spaced pattern of distances is the result of a self-organizing system. The positive feedback mechanism that operates a self-organizing system is accomplished by harmonic orbit resonances, leading to long-term stable planet and moon orbits in solar or stellar systems. The distance pattern of planets was originally described by the empirical Titius–Bode law, and by a generalized version with a constant geometric progression factor (corresponding to logarithmic spacing). We find that the orbital periods Ti and planet distances Ri from the Sun are not consistent with logarithmic spacing, but rather follow the quantized scaling (Ri+1/Ri)=(Ti+1/Ti)2/3=(Hi+1/Hi)2/3, where the harmonic ratios are given by five dominant resonances, namely (Hi+1:Hi)=(3:2),(5:3),(2:1),(5:2),(3:1). We find that the orbital period ratios tend to follow the quantized harmonic ratios in increasing order. We apply this harmonic orbit resonance model to the planets and moons in our solar system, and to the exo-planets of 55 Cnc and HD 10180 planetary systems. The model allows us a prediction of missing planets in each planetary system, based on the quasi-regular self-organizing pattern of harmonic orbit resonance zones. We predict 7 (and 4) missing exo-planets around the star 55 Cnc (and HD 10180). The accuracy of the predicted planet and moon distances amounts to a few percents. All analyzed systems are found to have ≈ 10 resonant zones that can be occupied with planets (or moons) in long-term stable orbits.

Convergent Migration Renders TRAPPIST-1 Long-lived

Abstract TRAPPIST-1 is a late M-dwarf orbited by seven Earth-sized planets with orbital period ratios near a chain of mean motion resonances. Due to uncertain system parameters, most orbital configurations drawn from the inferred posterior distribution are unstable on short timescales of ∼0.5 Myr, even when including the eccentricity damping effect of tides. By contrast, we show that most physically plausible resonant configurations generated through disk migration are stable even without tidal dissipation on timescales of at least 50 Myr (1010 orbits), an increase of at least two orders of magnitude. This result, together with the remarkable chain of period ratios in the system, provide strong evidence for convergent migration naturally emplacing the system near an equilibrium configuration forced by the resonant chain. We provide an openly available database of physically plausible initial conditions for TRAPPIST-1 generated through parameterized planet–disk interactions, as well as bit-by-bit reproducible N-body integrations over 109–1010 orbits.

Statistical analysis of properties of dwarf novae outbursts

We present a statistical study of all measurable photometric features of a large sample of dwarf novae during their outbursts and superoutbursts. We used all accessible photometric data for all our objects to make the study as complete and up-to-date as possible. Our aim was to check correlations between these photometric features in order to constrain theoretical models which try to explain the nature of dwarf novae outbursts. We managed to confirm a few of the known correlations, that is the Stolz and Schoembs Relation, the Bailey Relation for long outbursts above the period gap, the relations between the cycle and supercycle lengths, amplitudes of normal and superoutbursts, amplitude and duration of superoutbursts, outburst duration and orbital period, outburst duration and mass ratio for short and normal outbursts, as well as the relation between the rise and decline rates of superoutbursts. However, we question the existence of the Kukarkin-Parenago Relation but we found an analogous relation for superoutbursts. We also failed to find one presumed relation between outburst duration and mass ratio for superoutbursts. This study should help to direct theoretical work dedicated to dwarf novae.

Mass transfer and magnetic braking in Sco X-1

Sco X-1 is a low-mass X-ray binary (LMXB) that has one of the most precisely determined set of binary parameters such as the mass accretion rate, companions mass ratio and the orbital period. For this system, as well as for a large fraction of other well-studied LMXBs, the observationally-inferred mass accretion rate is known to strongly exceed the theoretically expected mass transfer rate. We suggest that this discrepancy can be solved by applying a modified magnetic braking prescription, which accounts for increased wind mass loss in evolved stars compared to main sequence stars. Using our mass transfer framework based on {\tt MESA}, we explore a large range of binaries at the onset of the mass transfer. We identify the subset of binaries for which the mass transfer tracks cross the Sco X-1 values for the mass ratio and the orbital period. We confirm that no solution can be found for which the standard magnetic braking can provide the observed accretion rates, while wind-boosted magnetic braking can provide the observed accretion rates for many progenitor binaries that evolve to the observed orbital period and mass ratio.

THE MASS DISTRIBUTION FUNCTION OF PLANETS

The distribution of orbital period ratios of adjacent planets in extra-solar planetary systems discovered by the {\it Kepler} space telescope exhibits a peak near $\sim1.5$--$2$, a long tail of larger period ratios, and a steep drop-off in the number of systems with period ratios below $\sim1.5$. We find from this data that the dimensionless orbital separations have an approximately log-normal distribution. Using Hill's criterion for the dynamical stability of two planets, we find an upper bound on planet masses such that the most common planet mass does not exceed $10^{-3.2}m_*$, or about two-thirds Jupiter mass for solar mass stars. Assuming that the mass ratio and the dynamical separation (orbital spacings in units of mutual Hill radius) of adjacent planets are independent random variates, and adopting empirical distributions for these, we use Hill's criterion in a statistical way to estimate the planet mass distribution function from the observed distribution of orbital separations. We find that the planet mass function is peaked in logarithm of mass, with a peak value and standard deviation of $\log m/M_\oplus$ of $\sim(0.6-1.0)$ and $\sim(1.1-1.2)$, respectively.

Orbital resonances in exoplanetary systems

At present, more than 700 exoplanetary systems are known to have been discovered. They incorporate more than 130 multiplanet systems, i.e. those hosting two or more planets. The orbital resonance and near-resonance phenomena are ubiquitous in them. We present a statistical and dynamical analysis of the resonance structure of the multiplanet systems and planetary systems of binary stars. We have built distributions of the orbital period ratios, separately considering the cases of inner and outer location of the massive perturber. The histograms reveal apparent peaks close to the first order orbital resonances 2/1 and 3/2 in both cases; this confirms previous findings. We have performed analytical modelling of the histograms, and obtained exact positions of the peaks. Moreover, we have built the "period ratio – eccentricity" diagrams, with collision curves superimposed, so that to find anomalous systems.

PHOTO-DYNAMICAL ANALYSIS OF THREE KEPLER OBJECTS OF INTEREST WITH SIGNIFICANT TRANSIT TIMING VARIATIONS

KOI-227, KOI-319 and KOI-884 are identified here as (at least) two planet systems. For KOI-319 and KOI-884, the observed Transit Timing Variations (TTVs) of the inner transiting planet are used to detect an outer non-transiting planet. The outer planet in KOI-884 is 2.6 Jupiter masses and has the orbital period just narrow of the 3:1 resonance with the inner planet (orbital period ratio 2.93). The distribution of parameters inferred from KOI-319.01's TTVs is bimodal with either a 1.6 Neptune-mass (M_N) planet wide of the 5:3 resonance (period 80.1 d) or a 1 Saturn-mass planet wide of the 7:3 resonance (period 109.2 d). The radial velocity measurements can be used in this case to determine which of these parameter modes is correct. KOI-227.01's TTVs with large 10 hour amplitude can be obtained for planetary-mass companions in various major resonances. Based on the Bayesian evidence, the current TTV data favor the outer 2:1 resonance with a companion mass 1.5 M_N, but this solution implies a very large density of KOI-227.01. The inner and outer 3:2 resonance solutions with sub-Neptune-mass companions are physically more plausible, but will need to be verified.

ASYMMETRIC ORBITAL DISTRIBUTION NEAR MEAN MOTION RESONANCE: APPLICATION TO PLANETS OBSERVED BYKEPLERAND RADIAL VELOCITIES

Many multiple-planet systems have been found by the Kepler transit survey and various radial velocity (RV) surveys. Kepler planets show an asymmetric feature, namely, there are small but significant deficits/excesses of planet pairs with orbital period spacing slightly narrow/wide of the exact resonance, particularly near the first order mean motion resonance (MMR), such as 2:1 and 3:2 MMR. Similarly, if not exactly the same, an asymmetric feature (pileup wide of 2:1 MMR) is also seen in RV planets, but only for massive ones. We analytically and numerically study planets' orbital evolutions near and in the MMR. We find that their orbital period ratios could be asymmetrically distributed around the MMR center regardless of dissipation. In the case of no dissipation, Kepler planets' asymmetric orbital distribution could be partly reproduced for 3:2 MMR but not for 2:1 MMR, implying that dissipation might be more important to the latter. The pileup of massive RV planets just wide of 2:1 MMR is found to be consistent with the scenario that planets formed separately then migrated toward the MMR. The location of the pileup infers a K value of 1-100 on the order of magnitude for massive planets, where K is the damping rate ratio between orbital eccentricity and semimajor axis during planet migration.

DISK-PLANETS INTERACTIONS AND THE DIVERSITY OF PERIOD RATIOS INKEPLER'S MULTI-PLANETARY SYSTEMS

The Kepler mission is dramatically increasing the number of planets known in multi-planetary systems. Many adjacent planets have orbital period ratios near resonant values, with a tendency to be larger than required for exact first-order mean-motion resonances. This intriguing feature has been shown to be a natural outcome of orbital circularization of resonant planetary pairs due to star-planet tidal interactions. However, this feature holds in multi-planetary systems with periods longer than ten days, for which tidal circularization is unlikely to provide efficient divergent evolution of the planets orbits. Gravitational interactions between planets and their parent protoplanetary disk may instead provide efficient divergent evolution. For a planet pair embedded in a disk, we show that interactions between a planet and the wake of its companion can reverse convergent migration, and significantly increase the period ratio from a near-resonant value. Divergent evolution due to wake-planet interactions is particularly efficient when at least one of the planets opens a partial gap around its orbit. This mechanism could help account for the diversity of period ratios in Kepler's multiple systems comprising super-Earth to sub-jovian planets with periods greater than about ten days. Diversity is also expected for planet pairs massive enough to merge their gap. The efficiency of wake-planet interactions is then much reduced, but convergent migration may stall with a variety of period ratios depending on the density structure in the common gap. This is illustrated for the Kepler-46 system, for which we reproduce the period ratio of Kepler-46b and c.

KOI-142, THE KING OF TRANSIT VARIATIONS, IS A PAIR OF PLANETS NEAR THE 2:1 RESONANCE

The Transit Timing Variations (TTVs) can be used as a diagnostic of gravitational interactions between planets in a multi-planet system. Many Kepler Objects of Interest (KOIs) exhibit significant TTVs, but KOI-142.01 stands out among them with an unrivaled, 12-hour TTV amplitude. Here we report a thorough analysis of KOI-142.01's transits. We discover periodic Transit Duration Variations (TDVs) of KOI-142.01 that are nearly in phase with the observed TTVs. We show that KOI-142.01's TTVs and TDVs uniquely detect a non-transiting companion with a mass 0.7 that of Jupiter (KOI-142c). KOI-142.01's mass inferred from the transit variations is consistent with the measured transit depth, suggesting a Neptune class planet (KOI-142b). The orbital period ratio P_c/P_b=2.03 indicates that the two planets are just wide of the 2:1 resonance. The present dynamics of this system, characterized here in detail, can be used to test various formation theories that have been proposed to explain the near-resonant pairs of exoplanets.