Mutual Inclination Research Articles

Inner and Outer Orbits in 13 Resolved Hierarchical Stellar Systems

Abstract Orbits of inner and outer subsystems in 13 triple or higher-order stellar systems are computed or updated using position measurements and, in three cases, radial velocities. The goal is to determine mutual orbital inclinations, period ratios, and masses to complement the statistics of hierarchical systems. The effect of the subsystems on the motion in the outer orbits (wobble) is explicitly modeled to determine inner mass ratios. Stars studied here (HD 5408, 8036, 9770, 15089, 29310, 286955, 29316, 140538, 144362, 154621, 156034, 185655, and 213235) are bright and nearby (from 15 pc to 150 pc). Their inner periods range from 1.7 yr to 49 yr, and the outer periods from 83 to 2400 yr. Some long-period outer orbits are poorly constrained. Four astrometric inner orbits and one outer orbit are computed for the first time.

The TESS–Keck Survey. IV. A Retrograde, Polar Orbit for the Ultra-low-density, Hot Super-Neptune WASP-107b

Abstract We measured the Rossiter–McLaughlin effect of WASP-107b during a single transit with Keck/HIRES. We found the sky-projected inclination of WASP-107b’s orbit, relative to its host star’s rotation axis, to be degrees. This confirms the misaligned/polar orbit that was previously suggested from spot-crossing events and adds WASP-107b to the growing population of hot Neptunes in polar orbits around cool stars. WASP-107b is also the fourth such planet to have a known distant planetary companion. We examined several dynamical pathways by which this companion could have induced such an obliquity in WASP-107b. We find that nodal precession and disk dispersal-driven tilting can both explain the current orbital geometry while Kozai–Lidov cycles are suppressed by general relativity. While each hypothesis requires a mutual inclination between the two planets, nodal precession requires a much larger angle, which for WASP-107 is on the threshold of detectability with future Gaia astrometric data. As nodal precession has no stellar type dependence, but disk dispersal-driven tilting does, distinguishing between these two models is best done on the population level. Finding and characterizing more extrasolar systems like WASP-107 will additionally help distinguish whether the distribution of hot-Neptune obliquities is a dichotomy of aligned and polar orbits or if we are uniformly sampling obliquities during nodal precession cycles.

Semianalytical model for planetary resonances

Context. Planetary resonances are a common dynamical mechanism acting on planetary systems. However, no general model for describing their properties exists, particularly for commensurabilities of any order and arbitrary eccentricity and inclination values. Aims. We present a semianalytical model that describes the resonance strength, width, location and stability of fixed points, and periods of small-amplitude librations. The model is valid for any two gravitationally interacting massive bodies, and is thus applicable to planets around single or binary stars. Methods. Using a theoretical framework in the Poincaré and Jacobi reference system, we developed a semianalytical method that employs a numerical evaluation of the averaged resonant disturbing function. Validations of the model are presented that compare its predictions with dynamical maps for real and fictitious systems. Results. The model describes many dynamical features of planetary resonances very well. Notwithstanding the good agreement found in all cases, a small deviation is noted in the location of the resonance centers for circumbinary systems. As a consequence of its application to the HD 31527 system, we found that the updated best-fit solution leads to a high-eccentricity stable libration between the middle and outer planets inside the 16/3 mean-motion resonance (MMR). This is the first planetary system whose long-term dynamics appears dominated by such a high-order commensurability. In the case of circumbinary planets, the overlap of N/1 mean-motion resonances coincides very well with the size of the global chaotic region close to the binary, as well as its dependence on the mutual inclination.

Inclination dynamics of resonant planets under the influence of an inclined external companion

ABSTRACT Recent observations suggest that a large fraction of Kepler super-Earth systems have external giant planet companions (cold Jupiters), which can shape the architecture of the inner planets, in particular their mutual inclinations. The dynamical perturbation from cold Jupiters may account for the population of misaligned planets in the Kepler data. The effectiveness of this mechanism can be hindered by a strong planet–planet coupling in the inner system. In this paper, we study how mean-motion resonances (MMRs) affect this coupling and the expected misalignment. We derive approximate analytical expressions for the mutual inclination excitations in the inner planet system induced by an inclined companion, for various period ratios and perturber properties. In most cases, the mutual inclination is proportional to a dimensionless parameter that characterizes the strength of the perturber relative to the coupling in the inner system. We show that the MMR strengthens the inner coupling, reducing the mutual inclination induced by the perturber by a factor of a few. We find that the resonance is resilient to the perturbation, and derive a criterion for the libration of the resonant angle. Our results have applications for constraining unseen planetary perturbers, and for understanding the architecture of multiplanet systems.

First Detection of Orbital Motion for HD 106906 b: A Wide-separation Exoplanet on a Planet Nine–like Orbit

Abstract HD 106906 is a 15 Myr old short-period (49 days) spectroscopic binary that hosts a wide-separation (737 au) planetary-mass ( ) common proper motion companion, HD 106906 b. Additionally, a circumbinary debris disk is resolved at optical and near-infrared wavelengths that exhibits a significant asymmetry at wide separations that may be driven by gravitational perturbations from the planet. In this study we present the first detection of orbital motion of HD 106906 b using Hubble Space Telescope images spanning a 14 yr period. We achieve high astrometric precision by cross-registering the locations of background stars with the Gaia astrometric catalog, providing the subpixel location of HD 106906 that is either saturated or obscured by coronagraphic optical elements. We measure a statistically significant 31.8 ± 7.0 mas eastward motion of the planet between the two most constraining measurements taken in 2004 and 2017. This motion enables a measurement of the inclination between the orbit of the planet and the inner debris disk of either deg or deg, depending on the true orientation of the orbit of the planet. There is a strong negative correlation between periastron and mutual inclination; orbits with smaller periastra are more misaligned with the disk plane. With a periastron of au, HD 106906 b is likely detached from the planetary region within 100 au radius, showing that a Planet Nine–like architecture can be established very early in the evolution of a planetary system.

Formation of Ultra-short-period Planets by Obliquity-driven Tidal Runaway

Abstract Small, rocky planets have been found orbiting in extreme proximity to their host stars, sometimes down to only ∼2 stellar radii. These ultra-short-period planets (USPs) likely did not form in their present-day orbits, but rather migrated from larger initial separations. While tides are the probable cause of this migration, the tidal source has remained uncertain. Here, we introduce planetary obliquity tides as a natural pathway for the production of USPs within close-in multiplanet systems. The crucial idea is that tidal dissipation generally forces planetary spin vectors to equilibrium configurations called “Cassini states,” in which the planetary obliquities (axial tilts) are nonzero. In these cases, sustained tidal dissipation and inward orbital migration are inevitable. Migration then increases the obliquity and strengthens the tides, creating a positive feedback loop. Thus, if a planet's initial semimajor axis is small enough (a ≲ 0.05 au), it can experience runaway orbital decay, which is stalled at ultra-short orbital periods when the forced obliquity reaches very high values (∼85°) and becomes unstable. We use secular dynamics to outline the parameter space in which the innermost member of a prototypical Kepler multiple-planet system can become a USP. We find that these conditions are consistent with many observed features of USPs, such as period ratios, mutual inclinations, and occurrence rate trends with stellar type. Future detections of stellar obliquities and close-in companions, together with theoretical explorations of the potential for chaotic obliquity dynamics, can help constrain the prevalence of this mechanism.

Architectures of Exoplanetary Systems. III. Eccentricity and Mutual Inclination Distributions of AMD-stable Planetary Systems

AbstractThe angular momentum deficit (AMD) of a planetary system is a measure of its orbital excitation and a predictor of long-term stability. We adopt the AMD-stability criterion to constrain the orbital architectures for exoplanetary systems. Previously, He et al. showed through forward modeling (SysSim) that the observed multiplicity distribution can be well reproduced by two populations consisting of a low and a high mutual inclination component. Here, we show that a broad distribution of mutual inclinations arising from systems at the AMD-stability limit can also match the observed Kepler population. We show that distributing a planetary system’s maximum AMD among its planets results in a multiplicity-dependent distribution of eccentricities and mutual inclinations. Systems with intrinsically more planets have lower median eccentricities and mutual inclinations, and this trend is well described by power-law functions of the intrinsic planet multiplicity (n):and, whereandare the medians of the eccentricity and inclination distributions. We also find that intrinsic single planets have higher eccentricities (σe,1 ∼ 0.25) than multiplanet systems and that the trends with multiplicity appear in the observed distributions of period-normalized transit duration ratios. We show that the observed preferences for planet-size orderings and uniform spacings are more extreme than what can be produced by the detection biases of the Kepler mission alone. Finally, we find that for systems with detected transiting planets between 5 and 10 days, there is another planet with a greater radial velocity signal ≃53% of the time.

Orbits of Five Triple Stars

Abstract Joint analysis of radial velocities and position measurements of five hierarchical stellar systems is undertaken to determine elements of their inner and outer orbits and, whenever possible, their mutual inclinations. The inner and outer periods are 12.9 and 345 yr for HD 12376 (ADS 1613), 1.14 and ∼1500 yr for HD 19971 (ADS 2390), 8.3 and 475 yr for HD 89795 (ADS 7338), 1.11 and 40 yr for HD 152027, 0.69 and 7.4 yr for HD 190412. The latter system with its coplanar and quasi-circular orbits belongs to the family of compact planetary-like hierarchies, while the orbits in HD 12376 have a mutual inclination of 131°.

Mutual inclinations between giant planets and their debris discs in HD 113337 and HD 38529

ABSTRACT HD 113337 and HD 38529 host pairs of giant planets, a debris disc, and wide M-type stellar companions. We measure the disc orientation with resolved images from Herschel and constrain the three-dimensional orbits of the outer planets with Gaia DR2 and Hipparcos astrometry. Resolved disc modelling leaves degeneracy in the disc orientation, so we derive four separate planet–disc mutual inclination (ΔI) solutions. The most aligned solutions give ΔI = 17°–32° for HD 113337 and ΔI = 21°–45○ for HD 38529 (both 1σ). In both systems, there is a small probability (<0.3 per cent) that the planet and disc are nearly aligned (ΔI < 3○). The stellar and planetary companions cause the orbits of disc material to precess about a plane defined by the forced inclination. We determine this as well as the precession time-scale to interpret the mutual inclination results. We find that the debris discs in both systems could be warped via joint influences of the outer planet and stellar companion, potentially explaining the observed misalignments. However, this requires HD 113337 to be old (0.8–1.7 Gyr), whereas if young (14–21 Myr), the observed misalignment in HD 113337 could be inherited from the protoplanetary disc phase. For both systems, the inclination of the stellar spin axis is consistent with the disc and outer planet inclinations, which instead supports system-wide alignment or near alignment. High-resolution observations of the discs and improved constraints on the planetary orbits would provide firmer conclusions about the (mis)alignment status.

CzeV1731: The unique doubly eclipsing quadruple system

We report the discovery of the relatively bright (V = 10.5 mag), doubly eclipsing 2+2 quadruple system CzeV1731. This is the third known system of its kind, in which the masses are determined for all four stars and both the inner and outer orbits are characterized. The inner eclipsing binaries are well-detached systems moving on circular orbits: pair A with period PA = 4.10843 d and pair B with PB = 4.67552 d. The inner binaries contain very similar components (q ≈ 1.0), making the whole system a so-called double twin. The stars in pair B have slightly larger luminosities and masses and pair A shows deeper eclipses. All four components are main-sequence stars of F/G spectral type. The mutual orbit of the two pairs around the system barycenter has a period of about 34 yr and an eccentricity of about 0.38. However, further observations are needed to reveal the overall architecture of the whole system, including the mutual inclinations of all orbits. This is a promising target for interferometry to detect the double at about 59 mas and ΔMbol < 1 mag.

Light-curve Evolution due to Secular Dynamics and the Vanishing Transits of KOI 120.01

Abstract Non-Keplerian dynamics of planetary orbits manifest in the transit light curve as variations of different types. In addition to transit timing variations, the shape of the transits contains additional information on variations in the geometry of the orbit. This study presents an analytic approach to light-curve fitting: dynamical variations in the orbital elements are transformed to a light curve using an analytic function with a restricted set of fitting parameters. Our method requires no N-body integration, resulting in a smaller number of degrees of freedom and a faster calculation. The approach described here is for the case of secular perturbations. By assuming that the orbital motion is dominated by nodal and apsidal precessions, analytic expressions for the light-curve transit parameters are derived as a function of the orbital variations. Detecting and characterizing such dynamical scenarios provides information regarding the possible existence of nontransiting companions, or the nonspherical mass distribution of the host star. The variations may imply forces out of the orbital plane, and thus probe mutual inclinations among components of the system. The derived models successfully reproduce the vanishing transit signals of KOI 120.01, and suggest a possible interesting scenario of a planet orbiting one member of a close-in binary system undergoing unusually rapid nodal regression. The model parameters are degenerate, so we provide relevant information for follow-up observations, which are suggested in order to place further constraints on this unique Kepler object.

A Disk-driven Resonance as the Origin of High Inclinations of Close-in Planets

Abstract The recent characterization of transiting close-in planets has revealed an intriguing population of sub-Neptunes with highly tilted and even polar orbits relative to their host star’s equator. Any viable theory for the origin of these close-in, polar planets must explain (1) the observed stellar obliquities, (2) the substantial eccentricities, and (3) the existence of Jovian companions with large mutual inclinations. In this work, we propose a theoretical model that satisfies these requirements without invoking tidal dissipation or large primordial inclinations. Instead, tilting is facilitated by the protoplanetary disk dispersal during the late stage of planet formation, initiating a process of resonance sweeping and parametric instability. This mechanism consists of two steps. First, a nodal secular resonance excites the inclination to large values; then, once the inclination reaches a critical value, a linear eccentric instability is triggered, which detunes the resonance and ends inclination growth. The critical inclination is pushed to high values by general relativistic precession, making polar orbits an inherently post-Newtonian outcome. Our model predicts that polar, close-in sub-Neptunes coexist with cold Jupiters in low stellar obliquity orbits.

TIC 278825952: a triply eclipsing hierarchical triple system with the most intrinsically circular outer orbit

ABSTRACT We report the discovery of a compact triply eclipsing triple star system in the southern continuous viewing zone of the TESS space telescope. TIC 278825952 is a previously known, but unstudied circular eclipsing binary with a period of 4.781 d with a tertiary component in a wider, circular orbit of 235.55-d period that was found from three sets of third-body eclipses and from light travel-time effect dominated eclipse timing variations. We performed a joint photodynamical analysis of the eclipse timing variation curves, photometric data, and the spectral energy distribution, coupled with the use of PARSEC stellar isochrones. We find that the inner binary consists of slightly evolved, near twin stars of masses of 1.12 and 1.09 M⊙ and radii of 1.40 and 1.31 R⊙. The third, less massive star has a mass of 0.75 M⊙ and radius of 0.70 R⊙. The low mutual inclination and eccentricities of the orbits show that the system is highly coplanar and surprisingly circular.

Orbits and Structure of the Quadruple Systems GJ 225.1 and FIN 332

Only a handful of quadruple systems with two accurate inner visual orbits are known. Architecture of two such systems is studied here to determine period ratios, mutual orbit orientation, and other parameters; updated orbital elements and their errors are derived. Gliese 225.1 (HIP 28442) is composed of three K-type and one M-type dwarfs and has inner orbital periods of 67.2+-0.2 and 23.4+-0.5 yr. Its inner orbits have small mutual inclination and are likely coplanar with the outer orbit of ~2 kyr period. The quadruple system FIN 332 (HIP 92037) consists of four early A type stars with similar masses and magnitudes. Both its inner orbits with periods of 27.6+-0.2 and 39.8+-0.4 yr have large eccentricities (0.82 and 0.84). Their orientation in the sky is remarkably similar. In contrast, the outer orbit with a period of ~5 kyr has a large relative inclination to the inner orbits. Dynamics and formation of these quadruple systems are briefly discussed.

Secular Resonances in Nonrestricted Hierarchical Triple Systems

Abstract In this paper, the averaged Hamiltonian of a nonrestricted hierarchical triple system truncated at the third order is investigated. First, each secular resonant term is studied. For the well-studied secular quadrupole theory, it is analytically reformulated in a different manner in our work. The resonance width is numerically determined and displayed on the plane (also denoted as the plane). In terms of the octupole terms, we show that for a near-planar configuration of the system, considerable variations of both the eccentricities of the inner and outer orbits can be generated by a single resonant term. The resonance width for every secular resonant angle from the octupole terms is also numerically determined and displayed on the plane. The results show that an orbit flip with a near-perpendicular initial mutual inclination is possible for each secular resonance. By displaying the resonance widths of different resonant terms on the same plane, we intuitively show the overlap of different secular resonances. Then, the full averaged Hamiltonian with both quadrupole and octupole terms is investigated using the Poincaré surface of section, with a special focus on the orbit flip. For the cases we exploited, we find that the near-planar flip of the inner orbit can be either regular or chaotic while the outer orbit flip is generally chaotic.

Stellar Oblateness versus Distant Giants in Exciting Kepler Planet Mutual Inclinations

Abstract An overabundance of single-transiting Kepler planets suggests the existence of a subpopulation of intrinsically multiplanet systems possessing large mutual inclinations. However, the origin of these mutual inclinations remains unknown. Recent work has demonstrated that mutual inclinations can be excited soon after protoplanetary disk dispersal owing to the oblateness of the rapidly rotating host star, provided that the star is tilted. Alternatively, distant giant planets, which are common in systems of close-in Kepler planets, could drive up mutual inclinations. The relative importance of each of these mechanisms has not been investigated. Here, we show that the influence of the stellar oblateness typically exceeds that of an exterior giant soon after planet formation. However, the magnitude of the resulting mutual inclinations depends critically on the timescale over which the natal disk disperses. Specifically, we find that if the disk vanishes over a timescale shorter than ∼103–4 yr, comparable to the viscous timescale of the inner ∼0.2 au, the inner planets impulsively acquire misalignments that scale with the stellar obliquity. In contrast, if the disk disperses slowly, the inner planets remain coplanar. They first align with the stellar equator but subsequently realign with the distant giant’s plane as the star spins down. Our findings are consistent with recent observations that giants tend to be aligned with close-in multiplanet systems but misaligned with single-transiting planets. Stellar obliquity measurements offer a promising test of our proposed framework.

A significant mutual inclination between the planets within the π Mensae system

Context. Measuring the geometry of multi-planet extrasolar systems can provide insight into their dynamical history and the processes of planetary formation. These types of measurements are challenging for systems that are detected through indirect techniques such as radial velocity and transit, having only been measured for a handful of systems to date. Aims. We aim to place constraints on the orbital geometry of the outer planet in the π Mensae system, a G0V star at a distance of 18.3 pc that is host to a wide-orbit super-Jovian (M sin i = 10.02 ± 0.15MJup) with a 5.7-yr period and an inner transiting super-Earth (M = 4.82 ± 0.85M⊕) with a 6.3-d period. Methods. The reflex motion induced by the outer planet on the π Mensae star causes a significant motion of the photocenter of the system on the sky plane over the course of the 5.7-year orbital period of the planet. We combined astrometric measurements from the HIPPARCOS and Gaia satellites with a precisely determined spectroscopic orbit in an attempt to measure this reflex motion, and in turn we constrained the inclination of the orbital plane of the outer planet. Results. We measure an inclination of ib = 49.9−4.5+5.3 deg for the orbital plane of π Mensae b, leading to a direct measurement of its mass of 13.01−0.95+1.03 MJup. We find a significant mutual inclination between the orbital planes of the two planets, with a 95% credible interval for imut of between 34.°5 and 140.°6 after accounting for the unknown position angle of the orbit of π Mensae c, strongly excluding a co-planar scenario for the two planets within this system. All orbits are stable in the present-day configuration, and secular oscillations of planet c’s eccentricity are quenched by general relativistic precession. Planet c may have undergone high eccentricity tidal migration triggered by Kozai-Lidov cycles, but dynamical histories involving disk migration or in situ formation are not ruled out. Nonetheless, this system provides the first piece of direct evidence that giant planets with large mutual inclinations have a role to play in the origins and evolution of some super-Earth systems.

Discovery and characterization of the exoplanets WASP-148b and c

We present the discovery and characterization of WASP-148, a new extrasolar system that includes at least two giant planets. The host star is a slowly rotating inactive late-G dwarf with a V = 12 magnitude. The planet WASP-148b is a hot Jupiter of 0.72 RJup and 0.29 MJup that transits its host with an orbital period of 8.80 days. We found the planetary candidate with the SuperWASP photometric survey, then characterized it with the SOPHIE spectrograph. Our radial velocity measurements subsequently revealed a second planet in the system, WASP-148c, with an orbital period of 34.5 days and a minimum mass of 0.40 MJup. No transits of this outer planet were detected. The orbits of both planets are eccentric and fall near the 4:1 mean-motion resonances. This configuration is stable on long timescales, but induces dynamical interactions so that the orbits differ slightly from purely Keplerian orbits. In particular, WASP-148b shows transit-timing variations of typically 15 min, making it the first interacting system with transit-timing variations that is detected on ground-based light curves. We establish that the mutual inclination of the orbital plane of the two planets cannot be higher than 35°, and the true mass of WASP-148c is below 0.60 MJup. We present photometric and spectroscopic observations of this system that cover a time span of ten years. We also provide their Keplerian and Newtonian analyses; these analyses should be significantly improved through future TESS observations.

The origins of nearly coplanar, non-resonant systems of close-in super-Earths

ABSTRACT Some systems of close-in ‘super-Earths’ contain five or more planets on non-resonant but compact and nearly coplanar orbits. The Kepler-11 system is an iconic representative of this class of system. It is challenging to explain their origins given that planet–disc interactions are thought to be essential to maintain such a high degree of coplanarity, yet these same interactions invariably cause planets to migrate into chains of mean motion resonances. Here, we mine a large data set of dynamical simulations of super-Earth formation by migration. These simulations match the observed period ratio distribution as long as the vast majority of planet pairs in resonance become dynamically unstable. When instabilities take place resonances are broken during a late phase of giant impacts, and typical surviving systems have planet pairs with significant mutual orbital inclinations. However, a subset of our unstable simulations matches the Kepler-11 system in terms of coplanarity, compactness, planet-multiplicity, and non-resonant state. This subset has dynamical instability phases typically much shorter than ordinary systems. Unstable systems may keep a high degree of coplanarity post-instability if planets collide at very low orbital inclinations (≲1○) or if collisions promote efficient damping of orbital inclinations. If planetary scattering during the instability takes place at low orbital inclinations (i ≲ 1○), orbital inclinations are barely increased by encounters before planets collide. When planetary scattering pumps orbital inclinations to higher values (≳1○) planets tend to collide at higher mutual orbital inclinations, but depending on the geometry of collisions mergers’ orbital inclinations may be efficiently damped. Each of these formation pathways can produce analogues to the Kepler-11 system.

Evidence for a high mutual inclination between the cold Jupiter and transiting super Earth orbiting π Men

ABSTRACT π Men hosts a transiting super Earth (P ≈ 6.27 d, m ≈ 4.82 M⊕, R ≈ 2.04 R⊕) discovered by TESS and a cold Jupiter (P ≈ 2093 d, msin I ≈ 10.02 MJup, e ≈ 0.64) discovered from radial velocity. We use Gaia DR2 and Hipparcos astrometry to derive the star’s velocity caused by the orbiting planets and constrain the cold Jupiter’s sky-projected inclination (Ib = 41°−65°). From this, we derive the mutual inclination (ΔI) between the two planets, and find that 49° < ΔI < 131° (1σ) and 28° < ΔI < 152° (2σ). We examine the dynamics of the system using N-body simulations, and find that potentially large oscillations in the super Earth’s eccentricity and inclination are suppressed by general relativistic precession. However, nodal precession of the inner orbit around the invariable plane causes the super Earth to only transit between 7 and 22 per cent of the time, and to usually be observed as misaligned with the stellar spin axis. We repeat our analysis for HAT-P-11, finding a large ΔI between its close-in Neptune and cold Jupiter and similar dynamics. π Men and HAT-P-11 are prime examples of systems where dynamically hot outer planets excite their inner planets, with the effects of increasing planet eccentricities, planet–star misalignments, and potentially reducing the transit multiplicity. Formation of such systems likely involves scattering between multiple giant planets or misaligned protoplanetary discs. Future imaging of the faint debris disc in π Men and precise constraints on its stellar spin orientation would provide strong tests for these formation scenarios.