Ultra-short-period planets (USPs) provide important clues to planetary formation and migration. Recently, it is found that the mutual inclinations of the planetary systems are larger if the inner orbits are closer ($\lesssim 5R_*$) and if the planetary period ratios are larger ($P_2/P_1 \gtrsim 5$) (Dai et al. 2018). This suggests that the USPs experienced both inclination excitation and orbital shrinkage. Here we investigate the increase in the mutual inclination due to stellar oblateness. We find that the stellar oblateness (within $\sim 1$Gyr) is sufficient to enhance the mutual inclination to explain the observed signatures. This suggests that the USPs can migrate closer to the host star in a near coplanar configuration with their planetary companions (e.g., disk migration+tides or in-situ+tides), before mutual inclination gets excited due to stellar oblateness.
An overabundance of single-transiting Kepler planets suggests the existence of a sub-population of intrinsically multi-planet 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 due to the oblateness of the rapidly-rotating host star, provided 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 t…
Systems with ultra-short-period planets (USPs) tend to possess larger mutual inclinations compared to those with planets located farther from their host stars. This could be explained due to precession caused by stellar oblateness at early times when the host star was rapidly spinning. However, stellar oblateness reduces over time due to the decrease in the stellar rotation rate, and this may further shape the planetary mutual inclinations. In this work, we investigate in detail how the final mutual inclination varies under the effect of a decreasing $J_2$. We find that different initial param…
The aim of this work is to provide an analytical model to characterize the equilibrium points and the phase space associated with the singly-averaged dynamics caused by the planetary oblateness coupled with the solar radiation pressure perturbations. A two-dimensional differential system is derived by considering the classical theory, supported by the existence of an integral of motion comprising semi-major axis, eccentricity and inclination. Under the single resonance hypothesis, the analytical expressions for the equilibrium points in the eccentricity-resonant angle space are provided, toget…
Previous full-lifetime simulations of single-star multi-planet systems across all phases of stellar evolution have predominately assumed coplanar or nearly-coplanar orbits. Here we assess the consequences of this assumption by removing it and exploring the effect of giant branch mass loss on the stability of two-planet systems with small to moderate non-Kozai (<40 degrees) relative inclinations. We run nearly 10^4 simulations over 14 Gyr for F-star, A-star and B-star planet hosts, incorporating main sequence stellar masses of 1.5, 2.0, 2.5, 3.0 and 5.0 solar masses, and initial planetary semim…
A large proportion of transiting planetary systems appear to possess only a single planet as opposed to multiple transiting planets. This excess of singles is indicative of significant mutual inclinations existing within a large number of planetary systems, but the origin of these misalignments is unclear. Moreover, recent observational characterization reveals that mutual inclinations tend to increase with proximity to the host star. These trends are both consistent with the dynamical influence of a strong quadrupolar potential arising from the host star during its early phase of rapid rotati…
Planet formation is often considered in the context of one circumstellar disk around one star. Yet stellar binary systems are ubiquitous, and thus a substantial fraction of all potential planets must form and evolve in more complex, dynamical environments. We present the results of a five-year astrometric monitoring campaign studying 45 binary star systems that host Kepler planet candidates. The planet-forming environments in these systems would have literally been shaped by the binary orbits that persist to the present day. Crucially, the mutual inclinations of star-planet orbits can only be …
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 1$σ$). In both systems, there is a small probability (<0.3 per cent) that the planet and d…
Early analyses of exoplanet statistics from the Kepler Mission revealed that a model population of multiple-planet systems with low mutual inclinations (${\sim1^{\circ}-2^{\circ}}$) adequately describes the multiple-transiting systems but underpredicts the number of single-transiting systems. This so-called "Kepler dichotomy" signals the existence of a sub-population of multi-planet systems possessing larger mutual inclinations. However, the details of these inclinations remain uncertain. In this work, we derive constraints on the intrinsic mutual inclination distribution by statistically expl…
The {\it Kepler} mission has detected thousands of planetary systems with 1-7 transiting planets packed within 0.7~au from their host stars. There is an apparent excess of single-transit planet systems that cannot be explained by transit geometries alone, when a single planetary mutual inclination dispersion is assumed. This suggests that the observed compact planetary systems have at least two different architectures. We present a scenario where the "Kepler dichotomy" may be explained by the action of an external giant planet or stellar companion misaligned with the inner multi-planet system.…
We study the excitation of mutual inclination between planetary orbits by a novel secular-orbital resonance in multiplanet systems perturbed by binary companions which we call "ivection". The ivection resonance happens when the nodal precession rate of the planet matches a multiple of the orbital frequency of the binary, and its physical nature is similar to the previously-studied evection resonance. Capture into an ivection resonance requires encountering the resonance with slowly increasing nodal precession rate, and it can excite the mutual inclination of the planets without affecting their…
While there have been many studies examining the stability of hierarchical triple systems, the meaning of ``stability'' is somewhat vague and has been interpreted differently in previous literatures. The present paper focuses on ``Lagrange stability'', which roughly refers to the stability against the escape of a body from the system, or ``disruption'' of the triple system, in contrast to ``Lyapunov-like stability'' that is related to the chaotic nature of the system dynamics. We compute the evolution of triple systems using direct $N$-body simulations up to $10^7 P_\mathrm{out}$, which is sig…
In this paper we study numerically the effect of the initial mutual orbital inclination on the stability of hierarchical triple systems with initially circular orbits. Our aim is to investigate the possibility that the stability boundary may be independent of the orbital inclination for certain mass ratios. We integrate numerically the equations of motion of hierarchical triple systems with initially circular orbits and different orbital configurations. The mass ratios cover the range from 0.000001 to 1000000 and the initial mutual inclination angle varies from 0 to 180 degrees. The results fr…
The {\it Kepler} mission revealed a population of compact multiple-planet systems with orbital periods shorter than a year, and occasionally even shorter than a day. By analyzing a sample of 102 {\it Kepler} and {\it K2} multi-planet systems, we measure the minimum difference $ΔI$ between the orbital inclinations, as a function of the orbital distance of the innermost planet. This is accomplished by fitting all the planetary signals simultaneously, constrained by an external estimate of the stellar mean density. We find $ΔI$ to be larger when the inner orbit is smaller, a trend that does not a…
Previous analyses of Doppler and Kepler data have found that Sun-like stars hosting "cold Jupiters" (giant planets with $a\gtrsim 1\,\mathrm{AU}$) almost always host "inner super-Earths" (1-$4\,R_\oplus$, $a\lesssim1\,\mathrm{AU}$). Here, we attempt to determine the degree of alignment between the orbital planes of the cold Jupiters and the inner super-Earths. The key observational input is the fraction of Kepler stars with transiting super-Earths that also have transiting cold Jupiters. This fraction depends on both the probability for cold Jupiters to occur in such systems, and on the mutual…
The 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 criteria to constrain the orbital architectures for exoplanetary systems. Previously, He, Ford, & Ragozzine (2019) (arXiv:1907.07773v2) showed through forward modelling (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 c…
Hierarchical two-planet systems, in which the inner body's semi-major axis is between 0.1 and 0.5 AU, usually present high eccentricity values, at least for one of the orbits. As a result of the formation process, one may expect that planetary systems with high eccentricities also have high mutual inclinations. However, here we show that tidal effects combined with gravitational interactions damp the initial mutual inclination to modest values in timescales that are shorter than the age of the system. This effect is not a direct consequence of tides on the orbits, but it results from a secular…
The long-period giant planet HR 5183 b has one of the most extreme orbits among exoplanets known to date, and represents a test for models of their dynamical evolution. In this work we use Hipparcos-Gaia astrometry to measure the orbital inclination of this planet for the first time and find $i=89.9^{+13.3\circ}_{-13.5}$, fully consistent with edge-on. The long orbital period and high eccentricity of HR 5183 b are supported by our results, with $P=102^{+84}_{-34}$ years and $e=0.87 \pm 0.04$. We confirm that HR 5183 forms a physically bound binary with HIP 67291 at a projected separation of 15…
We study the dynamics of the 3-D three-body problem of a small body moving under the attractions of a star and a giant planet which orbits the star on a much wider and elliptic orbit. In particular, we focus on the influence of an eccentric orbit of the outer perturber on the dynamics of a small highly inclined inner body. Our analytical study of the secular perturbations relies on the classical octupole hamiltonian expansion (third-order theory in the ratio of the semi-major axes), as third-order terms are needed to consider the secular variations of the outer perturber and potential secular …
We study the chaotic and secular evolution of hierarchical quadruple systems in the $3+1$ configuration, focusing on the evolution of mutual inclination of the inner binaries as the system undergoes coupled Lidov-Kozai (LK) oscillations. We include short-range forces (SRF; such as those due to tidal and rotational distortions) that control the eccentricity excitation of the inner binary. The evolution of mutual inclination is described, a priori, by two dimensionless parameters, $\pazocal{R}_0$, the ratio between the inner and outer LK time-scales and $ε_{SRF}$, the ratio between the SRF prece…
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 analytica…
Aims. To date, more than 600 multi-planetary systems have been discovered. Due to the limitations of the detection methods, our knowledge of the systems is usually far from complete. In particular, for planetary systems discovered with the radial velocity (RV) technique, the inclinations of the orbital planes, and thus the mutual inclinations and planetary masses, are unknown. Our work aims to constrain the spatial configuration of several RV-detected extrasolar systems that are not in a mean-motion resonance. Methods. Through an analytical study based on a first-order secular Hamiltonian expa…
We evaluate the orbital evolution and several plausible origins scenarios for the mutually inclined orbits of Upsilon Andromedae c and d. These two planets have orbital elements that oscillate with large amplitudes and lie close to the stability boundary. This configuration, and in particular the observed mutual inclination, demands an explanation. The planetary system may be influenced by a nearby low-mass star, Upsilon And B, which could perturb the planetary orbits, but we find it cannot modify two coplanar orbits into the observed mutual inclination of ~30 deg. However, it could incite eje…
Measuring the geometry of multi-planet extrasolar systems can provide insight into their dynamical history and the processes of planetary formation. Such measurements are challenging for systems detected through indirect techniques such as radial velocity and transit, having only been measured for a handful of systems to-date. We aimed to place constraints on the orbital geometry of the outer planet in the $π$ Mensae system, a G0V star at 18.3 pc host to a wide-orbit super-jovian ($M\sin i = 10.02\pm0.15$ $M_{\rm Jup}$) with a 5.7-year period and an inner transiting super-earth ($M=4.82\pm0.85…
WASP-148 is a recently announced extra-solar system harbouring at least two giant planets. The inner planet transits its host star. The planets travel on eccentric orbits and are near the 4:1 mean-motion resonance, which implies significant mutual gravitational interactions. In particular, this causes transit-timing variations of a few minutes, which were detected based on ground-based photometry. This made WASP-148 one of the few cases where such a phenomenon was detected without space-based photometry. Here, we present a self-consistent model of WASP-148 that takes into account the gravitati…
In this paper we describe an algorithm and deduce the related mathematical formulae that allows the computation of observed fluxes in stellar and planetary systems with arbitrary number of bodies being part of a transit or occultation event. The presented method does not have any limits or constraints for the geometry and can be applied for almost all of the available limb darkening models as well. As a demonstration, we apply this scheme to gather information for the orbital inclinations from multiple transiting planetary systems in cases when mutual transits occur. We also show here that the…
The probability that an exoplanet transits its host star is high for planets in close orbits, but drops off rapidly for increasing semimajor axes. This makes transit surveys for planets with large semimajor axes orbiting bright stars impractical, since one would need to continuously observe hundreds of stars that are spread out over the entire sky. One way to make such a survey tractable is to constrain the inclination of the stellar rotation axes in advance, and thereby enhance the transit probabilities. We derive transit probabilities for stars with stellar inclination constraints, consideri…
We present spatially and spectrally resolved Atacama Large Millimeter/submillimeter Array (ALMA) observations of gas and dust orbiting the pre-main sequence hierarchical triple star system GW Ori. A forward-modeling of the ${}^{13}$CO and C${}^{18}$O $J$=2-1 transitions permits a measurement of the total stellar mass in this system, $5.29 \pm 0.09\,M_\odot$, and the circum-triple disk inclination, $137.6 \pm 2.0^\circ$. Optical spectra spanning a 35 year period were used to derive new radial velocities and, coupled with a spectroscopic disentangling technique, revealed that the A and B compone…
$π$ Men hosts a transiting super Earth ($P\approx6.27$ d, $m\approx4.82$ $M_{\oplus}$, $R\approx2.04$ $R_{\oplus}$) discovered by TESS and a cold Jupiter ($P\approx2093$ d, $m \sin I\approx10.02$ $M_{\rm{Jup}}$, $e\approx0.64$) 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 ($I_b=41-65^{\circ}$). From this we derive the mutual inclination ($ΔI$) between the two planets, and find that $49^{\circ}< ΔI < 131^{\circ}$ (1$σ$), and $28^{\circ} < ΔI < 152^…
Joint analysis of radial velocities and position measurements of five hierarchical stellar systems is undertaken to determine elements of their inner and outer orbits and, whenever possible, their mutual inclinations. The inner and outer periods are 12.9 and 345 yr for HD 12376 (ADS 1613), 1.14 and ~1500 yr for HD 19971 (ADS 2390), 8.3 and 475 yr for HD 89795 (ADS 7338), 1.11 and 40 yr for HD 152027, 0.69 and 7.4 yr for HD 190412. The latter system with its co-planar and quasi-circular orbits belongs to the family of compact planetary-like hierarchies, while the orbits in HD 12376 have mutual …
Breaking the Mass Inclination Degeneracy of Radial Velocity Measurements via Monitoring von Zeipel-Lidov-Kozai Cycles: Implications in the HD 41004 System
This paper considers whether the population of known transiting exoplanets provides evidence for additional outer planets on inclined orbits, due to the perturbing effect of such planets on the orbits of inner planets. As such, we develop a semi-analytical method for calculating the probability that two mutually inclined planets are observed to transit. We subsequently derive a simplified analytical form to describe how the mutual inclination between two planets evolves due to secular interactions with a wide orbit inclined planet and use this to determine the mean probability that the two inn…
Compact multi-planet systems containing super-Earths or sub-Neptunes, commonly found around solar-type stars, may be surrounded by external giant planet or stellar companions, which can shape the architechture and observability of the inner systems. We present a comprehensive study on the evolution of the inner planetary system subject to the gravitational influence of an eccentric, misaligned outer perturber. Analytic results are derived for the inner planet eccentricities ($e_i$) and mutual inclination ($θ_{12}$) of the "2-planet + perturber" system, calibrated with numerical secular and N-b…
We present N-body simulations of resonant planets with inclined orbits that show chaotically evolving eccentricities and inclinations that can persist for at least 10 Gyr. A wide range of behavior is possible, from fast, low amplitude variations to systems in which eccentricities reach 0.9999 and inclinations 179.9 degrees. While the orbital elements evolve chaotically, at least one resonant argument always librates. We show that the HD 73526, HD 45364 and HD 60532 systems may be in chaotically-evolving resonances. Chaotic evolution is apparent in the 2:1, 3:1 and 3:2 resonances, and for plane…
The Kepler Mission has detected dozens of compact planetary systems with more than four transiting planets. This sample provides a collection of close-packed planetary systems with relatively little spread in the inclination angles of the inferred orbits. A large fraction of the observational sample contains limited multiplicity, begging the question whether there is a true diversity of multi transiting systems, or if some systems merely possess high mutual inclinations, allowing them to appear as single-transiting systems in a transit-based survey. This paper begins an exploration of the effe…
All four circumbinary (CB) protoplanetary disks orbiting short-period ($P < 20$ day) double-lined spectroscopic binaries (SB2s)---a group that includes UZ Tau E, for which we present new ALMA data---exhibit sky-plane inclinations $i_{\rm disk}$ which match, to within a few degrees, the sky-plane inclinations $i_\star$ of their stellar hosts. Although for these systems the true mutual inclinations $θ$ between disk and binary cannot be directly measured because relative nodal angles are unknown, the near-coincidence of $i_{\rm disk}$ and $i_\star$ suggests that $θ$ is small for these most compac…
We present a catalog of stellar companions to host stars of Transiting Exoplanet Survey Satellite Objects of Interest (TOIs) identified from a marginalized likelihood ratio test that incorporates astrometric data from the Gaia Early Data Release 3 catalog (EDR3). The likelihood ratio is computed using a probabilistic model that incorporates parallax and proper-motion covariances and marginalizes the distances and 3D velocities of stars in order to identify comoving stellar pairs. We find 172 comoving companions to 170 non-false-positive TOI hosts, consisting of 168 systems with two stars and 2…
The late-stage formation of giant planetary systems is rich in interesting dynamical mechanisms. Previous simulations of three giant planets initially on quasi-circular and quasi-coplanar orbits in the gas disc have shown that highly mutually inclined configurations can be formed, despite the strong eccentricity and inclination damping exerted by the disc. Much attention has been directed to inclination-type resonance, asking for large eccentricities to be acquired during the migration of the planets. Here we show that inclination excitation is also present at small to moderate eccentricities …
We establish the three-dimensional architecture of the Kepler-419 (previously KOI-1474) system to be eccentric yet with a low mutual inclination. Kepler-419b is a warm Jupiter at semi-major axis a = 0.370 +0.007/-0.006 AU with a large eccentricity e=0.85 +0.08/-0.07 measured via the "photoeccentric effect." It exhibits transit timing variations induced by the non-transiting Kepler-419c, which we uniquely constrain to be a moderately eccentric (e=0.184 +/- 0.002), hierarchically-separated (a=1.68 +/- 0.03 AU) giant planet (7.3 +/- 0.4 MJup). We combine sixteen quarters of Kepler photometry, rad…
It is often assumed that the "Kepler dichotomy" -- the apparent excess of planetary systems with a single detected transiting planet in the Kepler catalog -- reflects an intrinsic bimodality in the mutual inclinations of planetary orbits. After conducting 600 simulations of planet formation followed by simulated Kepler observations, we instead propose that the apparent dichotomy reflects a divergence in the amount of migration and the separation of planetary semimajor axes into distinct "clusters". We find that our simulated high-mass systems migrate rapidly, bringing more planets into orbital…
We present a combined analysis of previously published high-precision radial velocities and astrometry for the GJ876 planetary system using a self-consistent model that accounts for the planet-planet interactions. Assuming the three planets so far identified in the system are coplanar, we find that including the astrometry in the analysis does not result in a best-fit inclination significantly different than that found by Rivera and collaborators from analyzing the radial velocities alone. In this unique case, the planet-planet interactions are of such significance that the radial velocity dat…
We consider planetary systems evolving under the effect of a Stokes-type dissipative force mimicking the outcome of a type II migration process. As inward migration proceeds and the planets follow the circular family (they start on circular orbits) and even though they are initially almost coplanar, resonance capture can be realized. Then, at the \textit{vertical critical orbits} (VCOs), that the circular family possesses, the inclination excitation can abruptly take place. The planets are now guided by the spatial elliptic families, which bifurcate from those critical orbits. We herein, perfo…
The Kepler satellite has discovered a number of transiting planets around close binary stars. These circumbinary systems have highly aligned planetary and binary orbits. In this paper, we explore how the mutual inclination between the planetary and binary orbits may reflect the physical conditions of the assembly of protoplanetary disks and the interaction between protostellar binaries and circumbinary disks. Given the turbulent nature of star-forming molecular clouds, it is possible that the gas falling onto the outer region of a circumbinary disk and the central protostellar binary have diff…
We examine the stability of hierarchical triple systems using direct $N$-body simulations without adopting a secular perturbation approximation. We estimate their disruption timescales in addition to the mere stable/unstable criterion, with particular attention to the mutual inclination between the inner and outer orbits. First, we improve the fit to the dynamical stability criterion by \citet{Mardling1999,Mardling2001} widely adopted in the previous literature. Especially, we find that that the stability boundary is very sensitive to the mutual inclination; coplanar retrograde triples and ort…
HD 164604 c: a second giant planet on a 15-yr orbit and the constraint of the planet-planet mutual inclination
State Key Laboratory of Dark Matter Physics, Tsung-Dao Lee Institute & School of Physics and Astronomy, Shanghai Jiao Tong University（暗物质物理国家重点实验室，李政道研究所与物理天文学院，上海交通大学）；Earth and Planets Laboratory, Carnegie Institution for Science（地球与行星实验室，卡内基科学机构）；Observatories of the Carnegie Institution for Science（卡内基科学机构天文台）；Department of Astronomy, Tsinghua University（天文学系，清华大学）
HD 164604 c是一颗围绕K3.5矮星运行的巨行星，其轨道周期为15年，研究发现其与另一颗行星的互倾角约束及动态历史。
Accordling to the theory of Kozai resonance, the initial mutual inclination between a small body and a massive planet in an outer circular orbit is as high as $\sim39.2^{\circ}$ for pumping the eccentricity of the inner small body. Here we show that, with the presence of a residual gas disk outside two planetary orbits, the inclination can be reduced as low as a few degrees. The presence of disk changes the nodal precession rates and directions of the planet orbits. At the place where the two planets achieve the same nodal processing rate, vertical secular resonance would occur so that mutual …
We investigate the inclination-growth mechanisms for two-planet systems during the late protoplanetary disc phase. In previous works, much attention has been directed to the inclination-type resonance, and it has been shown that it asks for high eccentricities to be acquired during the migration of the giant planets. By adopting eccentricity and inclination damping formulae based on hydrodynamical simulations (instead of the K-prescription), we have carried out 20 000 numerical simulations, where we vary the initial planetary eccentricities, the migration rate, and the dispersal time of the ga…
[Abridged] The recent discovery of a transiting planet on a non-circular orbit with a massive highly eccentric companion orbiting HAT-P-13 offers the possibility of probing the structure of the short-period planet. The ability to do this relies on the system being in a quasi-equilibrium state in the sense that the eccentricities are constant on the usual secular timescale, and decay on a timescale which is much longer than the age of the system. Since the equilibrium eccentricity is effectively a function only of observable system parameters and the unknown Love number of the short-period plan…
The hierarchical triple body approximation has useful applications to a variety of systems from planetary and stellar scales to supermassive black holes. In this approximation, the energy of each orbit is separately conserved and therefore the two semi-major axes are constants. On timescales much larger than the orbital periods, the orbits exchange angular momentum which leads to eccentricity and orientation (i.e., inclination) oscillations. The orbits' eccentricity can reach extreme values leading to a nearly radial motion, which can further evolve into short orbit periods and merging binarie…
Kozai-Lidov (KL) oscillations in hierarchical triple systems have found application to many astrophysical contexts, including planet formation, type Ia supernovae, and supermassive black hole dynamics. The period of these oscillations is known at the order-of-magnitude level, but dependences on the initial mutual inclination or inner eccentricity are not typically included. In this work I calculate the period of KL oscillations ($t_{\textrm{KL}}$) exactly in the test particle limit at quadrupole order (TPQ). I explore the parameter space of all hierarchical triples at TPQ and show that except …
