Giant Planets Research Articles

A Novel Method to Constrain Tidal Quality Factor from A Nonsynchronized Exoplanetary System

We propose a novel method to constrain the tidal quality factor, Q′ , from an observed nonsynchronized star–planet system consisting of a slowly rotating low-mass star and a close-in Jovian planet, taking into account the coevolution of stellar spin and planetary orbit due to the tidal interaction and the magnetic braking. On the basis of dynamical system theory, the track of the coevolution of angular momentum from the fast-rotator regime for such a system exhibits the existence of a forbidden region in the Ωorb–Ωspin plane, where Ωspin and Ωorb denote the angular velocity of the stellar spin and planetary orbit, respectively. The forbidden region is determined primarily by the strength of the tidal interaction. By comparing the (Ωorb, Ωspin) of a single star–planet system to the forbidden region, we can constrain the tidal quality factor regardless of the evolutionary history of the system. The application of this method to the star–planet system NGTS-10–NGTS-10 b gives Q′≳108 , leading to a tight upper bound on the tidal torque. Since this cannot be explained by previous theoretical predictions for nonsynchronized star–planet systems, our result requires mechanisms that suppress the tidal interaction in such systems.

Bulk and Atmospheric Metallicities as Direct Probes of Sequentially Varying Accretion Mechanisms of Gas and Solids Onto Planets

Core accretion is the standard scenario of planet formation, wherein planets are formed by sequential accretion of gas and solids, and is widely used to interpret exoplanet observations. However, no direct probes of the scenario have been discussed yet. Here, we introduce an onion-like model as one idealization of sequential accretion and propose that bulk and atmospheric metallicities of exoplanets can be used as direct probes of the process. Our analytical calculations, coupled with observational data, demonstrate that the trend of observed exoplanets supports the sequential accretion hypothesis. In particular, accretion of planetesimals that are ≳100 km in size is most favored to consistently explain the observed trends. The importance of opening gaps in both planetesimal and gas disks following planetary growth is also identified. A new classification is proposed, wherein most observed planets are classified into two interior statuses: globally mixed and locally (well) mixed. Explicit identification of the locally (well) mixed status enables reliable verification of sequential accretion. During the JWST era, the quality and volume of observational data will increase drastically and improve exoplanet characterization. This work provides one key reference of how both the bulk and atmospheric metallicities can be used to constrain gas and solid accretion mechanisms of planets.

Radiative signatures of circumplanetary disks and envelopes during the late stages of giant planet formation

During the late stages of giant planet formation, protoplanets are surrounded by a circumplanetary disk and an infalling envelope of gas and dust. For systems with sufficient cooling, material entering the sphere of influence of the planet falls inward and approaches ballistic conditions. Due to conservation of angular momentum, most of the incoming material falls onto the disk rather than directly onto the planet. This paper determines the spectral energy distributions of forming planets in this stage of evolution. Generalizing previous work, we consider a range of possible geometries for the boundary conditions of the infall and determine the two-dimensional structure of the envelope, as well as the surface density of the disk. After specifying the luminosity sources for the planet and disk, we calculate the corresponding radiative signatures for the system, including the emergent spectral energy distributions and emission maps. These results show how the observational appearance of forming planets depend on the input parameters, including the instantaneous mass, mass accretion rate, semimajor axis of the orbit, and the planetary magnetic field strength (which sets the inner boundary condition for the disk). We also consider different choices for the form of the opacity law and attenuation due to the background circumstellar disk. Although observing forming planets will be challenging, these results show how the observational signatures depend on the underlying properties of the planet/disk/envelope system.

The Star–Planet Composition Connection

The mantra “know thy star, know thy planet” has proven to be very important for many aspects of exoplanet science. Here I review how stellar abundances inform our understanding of planet composition and, thus, formation and evolution. In particular, I discuss how: ▪The strongest star–planet connection is still the giant planet–metallicity correlation, the strength of which may indicate a break point between the formation of planets versus brown dwarfs.▪We do not have very good constraints on the lower metallicity limit for planet formation, although new statistics from TESS are helping, and it appears that, at low [Fe/H], α elements can substitute for iron as seeds for planet formation.▪The depletion of refractory versus volatile elements in stellar photospheres (particularly the Sun) was initially suggested as a sign of small planet formation but is challenging to interpret, and small differences in binary star compositions can be attributed mostly to processes other than planet formation.▪We can and should go beyond comparisons of the carbon-to-oxygen ratio in giant planets and their host stars, incorporating other volatile and refractory species to better constrain planet formation pathways.▪There appears to be a positive correlation between small planet bulk density and host star metallicity, but exactly how closely small planet refractory compositions match those of their host stars—and their true diversity—is still uncertain.

Impact sculpting of the early martian atmosphere.

Intense bombardment of solar system planets in the immediate aftermath of protoplanetary disk dissipation has played a key role in their atmospheric evolution. During this epoch, energetic collisions will have removed substantial masses of gas from rocky planet atmospheres. Noble gases are powerful tracers of this early atmospheric history, xenon in particular, which on Mars and Earth shows significant depletions and isotopic fractionations relative to the lighter noble gasses. To evaluate the effect of impacts on the loss and fractionation of xenon, we measure its ionization and recombination efficiency by laser shock and apply these constraints to model impact-driven atmospheric escape on Mars. We demonstrate that impact bombardment within the first 200 to 300 million years of solar system history generates the observed Xe depletion and isotope fractionation of the modern martian atmosphere. This process may also explain the Xe depletion recorded in Earth's deep mantle and provides a latest date for the timing of giant planet instability.

Global climate modelling of Saturn’s atmosphere, Part V: Large-scale vortices

This paper presents an analysis of large-scale vortices in the atmospheres of gas giants, focusing on a detailed study conducted using the Saturn-DYNAMICO global climate model (GCM). Large-scale vortices, a prominent feature of gas giant atmospheres, play a critical role in their atmospheric dynamics. By employing three distinct methods – manual detection, machine learning via artificial neural networks (ANN), and dynamical detection using the Automated Eddy-Detection Algorithm (AMEDA) – we characterise the spatial, temporal, and dynamical properties of these vortices within the Saturn-DYNAMICO GCM. Our findings reveal a consistent production of vortices due to well-resolved eddy-to-mean flow interactions, exhibiting size and intensity distributions broadly in agreement with observational data. However, notable differences in vortex location, size, and concentration highlight the model’s limitations and suggest areas for further refinement. The analysis underscores the importance of zonal wind conditions in influencing vortex characteristics and suggests that more accurate modelling of giant planet vortices may require improved representation of moist convection and jet structure. This study not only provides insights into the dynamics of Saturn’s atmosphere as simulated by the GCM but also offers a framework for comparing vortex characteristics across observations and models of planetary atmospheres.

The GAPS Programme at TNG

Context. Due to observational biases, a large fraction of known exoplanets are short-period objects. However, the search for planets began more than 20 yr ago, and so it is already possible – with the use of a suitable dataset – to begin exploring a wider range of the parameter space, such as that encompassing long-period planets. Aims. The aim of this paper is to investigate the presence of long-period giant companions in two systems where one or more planets are already known and for which a long-term trend in the radial velocities (RVs) was noted in previous works. Methods. Over the last 11 yr, we have collected 122 spectra of HD 75898 and 72 spectra of HD 11506 with the High Accuracy Radial velocity Planet Searcher for the Northern hemisphere (HARPS-N) in the framework of the Global Architecture of Planetary Systems (GAPS) project, from which we derived precise RV and activity indicator measurements. Additional RV data from the High-Resolution Echelle Spectrometer (HIRES) are also used here to increase the total time span. For our RV analysis, we used PyORBIT, an advanced Python tool for the simultaneous Bayesian analysis of RVs and stellar activity indicators. In addition, we used astrometric (Gaia DR3) and imaging archive data to complete our analysis. In particular, we combined RVs and astrometry to better constrain the mass and period of the new long-period companions. Results. We find evidence for one additional long-period companion (gas giant planet or brown dwarf) in both systems considered. The new candidate for HD 75898 has a period of roughly 18 yr and a true mass of around 8.5 Mj. For HD 11506, we confirm the new object (planet d) recently announced using HIRES data but we find that the period and true mass are both almost double the values based on HIRES results (Pd ~ 72 yr, Md ~ 13 Mj). In addition, for HD 75898, we also find evidence of an activity cycle affecting RVs with a period of one order of magnitude lower than found in the literature.

The MIRI Exoplanets Orbiting White dwarfs (MEOW) Survey: Mid-infrared Excess Reveals a Giant Planet Candidate around a Nearby White Dwarf

The MIRI Exoplanets Orbiting White dwarfs survey is a cycle 2 JWST program to search for exoplanets around dozens of nearby white dwarfs via infrared excess and direct imaging. In this Letter, we present the detection of mid-infrared excess at 18 and 21 μm toward the bright (V = 11.4) metal-polluted white dwarf WD 0310–688. The source of the IR excess is almost certainly within the system; the probability of background contamination is <0.1%. While the IR excess could be due to an unprecedentedly small and cold debris disk, it is best explained by a 3.0−1.9+5.5 M Jup cold (248 −61+84 K) giant planet orbiting the white dwarf within the forbidden zone (the region where planets are expected to be destroyed during the star’s red giant phase). We constrain the source of the IR excess to an orbital separation of 0.1–2 au, marking the first discovery of a white dwarf planet candidate within this range of separations. WD 0310–688 is a young remnant of an A- or late B-type star, and at just 10.4 pc, it is now the closest white dwarf with a known planet candidate. Future JWST observations could distinguish the two scenarios by either detecting or ruling out spectral features indicative of a planet atmosphere.

ALMA high-resolution observations unveil planet formation shaping molecular emission in the PDS 70 disk

With two directly detected protoplanets, the PDS 70 system is a unique source in which to study the complex interplay between forming planets and their natal environment. The large dust cavity carved by the two giant planets can affect the disk chemistry, and therefore the molecular emission morphology. On the other hand, chemical properties of the gas component of the disk are expected to leave an imprint on the planetary atmospheres. In this work, we reconstruct the emission morphology of a rich inventory of molecular tracers in the PDS 70 disk, and we look for possible chemical signatures of the two actively accreting protoplanets, PDS 70b and c. We leverage Atacama Large Millimeter/submillimeter Array (ALMA) band 6 high-angular-resolution and deep-sensitivity line emission observations, together with image and uv-plane techniques, to boost the detection of faint lines. We robustly detect ring-shaped emission from 12CO, 13CO, C18O, H13CN, HC15N, DCN, H2CO, CS, C2H, and H13CO+ lines in unprecedented detail. Most of the molecular tracers show a peak of the emission inside the millimeter dust peak. We interpret this as the direct impact of the effective irradiation of the cavity wall, as a result of the planet formation process. Moreover, we have found evidence of an O-poor gas reservoir in the outer disk, which is supported by the observations of bright C-rich molecules, the non-detection of SO, and a lower limit on the CS/SO ratio of ~1. Eventually, we provide the first detection of the c-C3H2 transitions at 218.73 GHz, and the marginal detection of an azimuthal asymmetry in the higher-energy H2CO (32,1−22,0) line, which could be due to accretion heating near PDS 70b.

Characterisation of the warm-Jupiter TOI-1130 system with CHEOPS and a photo-dynamical approach

Context. Among the thousands of exoplanets discovered to date, approximately a few hundred gas giants on short-period orbits are classified as ‘lonely’ and only a few are in a multi-planet system with a smaller companion on a close orbit. The processes that formed multi-planet systems hosting gas giants on close orbits are poorly understood, and only a few examples of this kind of system have been observed and well characterised. Aims. Within the contest of a multi-planet system hosting a gas giant on short orbits, we characterise the TOI-1130 system by measuring masses and orbital parameters. This is a two-transiting planet system with a Jupiter-like planet (c) on a 8.35 days orbit and a Neptune-like planet (b) on an inner (4.07 days) orbit. Both planets show strong anti-correlated transit timing variations (TTVs). Furthermore, radial velocity (RV) analysis showed an additional linear trend, a possible hint of a non-transiting candidate planet on a far outer orbit. Methods. Since 2019, extensive transit and radial velocity observations of the TOI-1130 have been acquired using TESS and various ground-based facilities. We present a new photo-dynamical analysis of all available transit and RV data, with the addition of new CHEOPS and ASTEP+ data, which achieve the best precision to date on the planetary radii and masses and on the timings of each transit. Results. We were able to model interior structure of planet b constraining the presence of a gaseous envelope of H/He, while it was not possible to assess the possible water content. Furthermore, we analysed the resonant state of the two transiting planets, and we found that they lie just outside the resonant region. This could be the result of the tidal evolution that the system underwent. We obtained both masses of the planets with a precision of less than 1.5%, and radii with a precision of about 1% and 3% for planet b and c, respectively.

ESPRESSO reveals blueshifted neutral iron emission lines on the dayside of WASP-76 b

Context. Ultra hot Jupiters (gas giants with Teq &gt; 2000 K) are intriguing exoplanets due to the extreme physics and chemistry present in their atmospheres. Their torrid daysides can be characterised using ground-based high-resolution emission spectroscopy. Aims. We search for signatures of neutral and singly ionised iron (Fe I and Fe II, respectively) in the dayside of the ultra hot Jupiter WASP-76 b, as these species were detected via transmission spectroscopy in this exoplanet. Furthermore, we aim to confirm the existence of a thermal inversion layer, which has been reported in previous studies, and attempt to constrain its properties. Methods. We observed WASP-76 b on four epochs with ESPRESSO at the VLT, at orbital phases shortly before and after the secondary transit, when the dayside is in view. We present the first analysis of high-resolution optical emission spectra for this exoplanet. We compare the data to synthetic templates created with petitRADTRANS, using cross-correlation function techniques. Results. We detect a blueshifted (−4.7 ± 0.3 km s−1) Fe I emission signature on the dayside of WASP-76 b at 6.0σ. The signal is detected independently both before and after the eclipse, and it is blueshifted in both cases. The presence of iron emission features confirms the existence of a thermal inversion layer. Fe II was not detected, possibly because this species is located in the upper layers of the atmosphere, which are more optically thin. Thus the Fe II signature on the dayside of WASP-76 b is too weak to be detected with emission spectroscopy. Conclusions. We propose that the blueshifted Fe I signature is created by material rising from the hot spot to the upper layers of the atmosphere, and discuss possible scenarios related to the position of the hotspot. This work unveils some of the dynamic processes ongoing on the dayside of the ultra hot Jupiter WASP-76 b through the analysis of the Fe I signature from its atmosphere, and complements previous knowledge obtained from transmission studies. It also highlights the ability of ESPRESSO to probe the dayside of this class of exoplanets.

TESS Investigation—Demographics of Young Exoplanets (TI-DYE). II. A Second Giant Planet in the 17 Myr System HIP 67522

The youngest (<50 Myr) planets are vital to understand planet formation and early evolution. The 17 Myr system HIP 67522 is already known to host a giant (≃10R ⊕) planet on a tight orbit. In their discovery paper, Rizzuto et al. reported a tentative single-transit detection of an additional planet in the system using TESS. Here, we report the discovery of HIP 67522c, a 7.9 R ⊕ planet that matches with that single-transit event. We confirm the signal with ground-based multiwavelength photometry from Sinistro and MuSCAT4. At a period of 14.33 days, planet c is close to a 2:1 mean-motion resonance with b (6.96 days or 2.06:1). The light curve shows distortions during many of the transits, which are consistent with spot-crossing events and/or flares. Fewer stellar activity events are seen in the transits of planet b, suggesting that planet c is crossing a more active latitude. Such distortions, combined with systematics in the TESS light-curve extraction, likely explain why planet c was previously missed.

Suppression of giant planet formation around low-mass stars in clustered environments

Context. Current exoplanet formation studies tend to overlook the birth environment of stars in clustered environments. However, the effects of this environment on the planet formation process are important, especially in the earliest stage. Aims. We investigate the differences in planet populations forming in star-cluster environments through pebble accretion and compare these results with planet formation around isolated stars. We strive to provide potential signatures of the young planetary systems to guide future observations. Methods. We present a new planet population synthesis code designed for clustered environments. This planet formation model is based on pebble accretion and includes migration in the circumstellar disk. The disk’s gas and dust have been evolved via 1D simulations, while considering the effects of photo-evaporation of the nearby stars. Results. Planetary systems in a clustered environment are different than those born in isolation; the environmental effects are important for a wide range of observable parameters and the eventual architecture of the planetary systems. Planetary systems born in a clustered environment lack cold Jupiters, as compared to isolated planetary systems. This effect is more pronounced for low-mass stars (≲0.2 M⊙). On the other hand, planetary systems born in clusters show an excess of cold Neptune around these low-mass stars. Conclusions. In future observations, finding an excess of cold Neptunes and a lack of cold Jupiters could be used to constrain the birth environments of these planetary systems. Exploring the dependence of cold Jupiter’s intrinsic occurrence rate on stellar mass offers insights into the birth environment of their proto-embryos.

Temporal variability in transmission spectra of H2-dominated exoplanets: The influence of thermal evolution and stellar irradiation on atmospheric composition

Planets and their host stars undergo evolutionary changes over time, resulting in variations in internal temperature and incoming radiation, which significantly impact the temperature structure and composition of their atmospheres. These evolving conditions give rise to distinctive features in planetary spectra that are observable only during specific stages of planetary evolution. We aim to understand how the composition of planets with H2-dominated atmospheres changes over longer timescales due to their thermal evolution. We also investigate time-dependent features in the transmission spectra. These features could provide insights in both the formation and evolution of these gaseous planets, as well as the timescales of these changes, enabling us to study the potential variability of exoplanets over time. We evolve a ∼0.04 MJup and ∼0.45 MJup planet around a 1.0 M⊙ and 1.3 M⊙ star respectively for 109.5 years. In both systems, the planets are considered at semi-major axes of 0.1 AU and 1.0 AU. The star-planet systems are evolved by making use of Modules for Experiments in Stellar Astrophysics (MESA). The temperature–pressure profiles are obtained at selected time-steps using an analytical approximation based on the internal and irradiation temperature of the planet at each time step. We then use VULCAN, a photochemical kinetics code, to see how the composition changes with time in the atmosphere due to the thermal evolution of the planets. By making use of the radiative transfer code petitRADTrans, we also simulate the evolution of the transmission spectra of the planets to find potential time-dependent spectral features. Our findings show a prominent change in the CO2 feature at ∼4.3μm. For the 0.45 MJup case, this feature is visible in the pre-main-sequence phase of the host star, regardless of orbital distance from the host star. In the case of the ∼0.04 MJup planet, this CO2 feature is visible until t ≤ 106 years, and then it reappears after t ≥ 108 years when the planet is 0.1 AU away the host star. The CH4 features at ∼3.3μm and ∼7.5μm are only time-dependent when the planet is located at 0.1 AU from the host star and experiences high irradiation since the high temperature at early stages favoured CO2 over CH4. When the planet is 1.0 AU away from its host star, the CH4 features are always visible regardless of the mass and hence internal temperature of the planet.

Ionization by XFEL radiation produces distinct structure in liquid water

In the warm dense matter (WDM) regime, where condensed, gas, and plasma phases coexist, matter frequently exhibits unusual properties that cannot be described by contemporary theory. Experiments reporting phenomena in WDM are therefore of interest to advance our physical understanding of this regime, which is found in dwarf stars, giant planets, and fusion ignition experiments. Using 7.1 keV X-ray free electron laser radiation (nominally 5×105 J/cm2), we produced and probed transient WDM in liquid water. Wide-angle X-ray scattering (WAXS) from the probe reveals a new ~9 Å structure that forms within 75 fs. By 100 fs, the WAXS peak corresponding to this new structure is of comparable magnitude to the ambient water peak, which is attenuated. Simulations suggest that the experiment probes a superposition of two regimes. In the first, fluences expected at the focus severely ionize the water, which becomes effectively transparent to the probe. In the second, out-of-focus pump radiation produces O1+ and O2+ ions, which rearrange due to Coulombic repulsion over 10 s of fs. Our simulations account for a decrease in ambient water signal and an increase in low-angle X-ray scattering but not the experimentally observed 9 Å feature, presenting a new challenge for theory.

An early giant planet instability recorded in asteroidal meteorites

Giant planet migration appears widespread among planetary systems in our Galaxy. However, the timescales of this process, which reflect the underlying dynamical mechanisms, are not well constrained, even within the Solar System. As planetary migration scatters smaller bodies onto intersecting orbits, it would have resulted in an epoch of enhanced bombardment in the Solar System’s asteroid belt. Here, to accurately and precisely quantify the timescales of migration, we interrogate thermochronologic data from asteroidal meteorites, which record the thermal imprint of energetic collisions. We present a database of 40K–40Ar system ages from chondrite meteorites and evaluate it with an asteroid-scale thermal code coupled to a Markov chain Monte Carlo inversion. Simulations require bombardment to reproduce the observed age distribution and identify a bombardment event beginning 11.3−6.6+9.5Myr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$11.{3}_{-6.6}^{+9.5}\\, {\\mathrm{Myr}}$$\\end{document} after the Sun formed (50% credible interval). Our results associate a giant planet instability in our Solar System with the dissipation of the gaseous protoplanetary disk.

Giant planets migrated shortly after the Solar System’s protoplanetary disk dispersed

Giant planets migrated shortly after the Solar System’s protoplanetary disk dispersed

Different Planetary Eccentricity-period (PEP) Distributions of Small and Giant Planets

We used the database of 1040 short-period (1 ≤ P < 200 days) exoplanets radial-velocity orbits to study the planetary eccentricity-period (PEP) distribution. We first divided the sample into low- and high-mass exoplanet subsamples based on the distribution of the (minimum) planetary masses, which displays a clear two-Gaussian distribution, separated at 0.165M J. We then selected 216 orbits, low- and high-mass alike, with eccentricities significantly distinct from circular orbits. The 131 giant-planet eccentric orbits display a clear upper envelope, which we model quantitatively, rises monotonically from zero eccentricity and reaches an eccentricity of 0.8 at P ∼ 100 days. Conversely, the 85 low-mass planetary orbits display a flat eccentricity distribution between 0.1 and 0.5, with almost no dependence on the orbital period. We show that the striking difference between the two PEP distributions is not a result of the detection technique used. The upper envelope of the high-mass planets, also seen in short-period binary stars, is a clear signature of tidal circularization, which probably took place inside the planets, while the small-planet PEP distribution suggests that the circularization was not effective, probably due to dynamical interactions with neighboring planets.

On the potential origin of the circumbinary planet Delorme 1 (AB)b

ABSTRACT Many circumbinary gas giant planets have been recently discovered. The formation mechanism of circumbinary planets on wide orbits is unclear. We investigate the formation of Delorme 1 (AB)b, a $13 \pm 5 \ \mathrm{ M_J}$ planet, orbiting its host binary at 84 au. The planet is accreting while having an estimated age of 40 Myr, which is unexpected, as this process should have ceased due to the dissipation of the protoplanetary disc. Using the smoothed particle hydrodynamics code seren, we model three formation scenarios for this planet. In Scenario I, the planet forms in situ on a wide orbit in a massive disc (by gravitational instability), in Scenario II closer to the binary in a massive disc (by gravitational instability), and in Scenario III much closer to the binary in a less massive disc (by core accretion). Planets in Scenario I stay at the observed separation and have mass accretion rates consistent with observed value, but their final mass is too high. In Scenario II, the planet reaches the observed separation through outward migration or scattering by the binary, and has mass accretion rate comparable to the observed; however, the planet mass is above the observed value. In Scenario III, the planet’s final mass and mass accretion rate are comparable to the observed ones, but the planet’s separation is smaller. We conclude that all models may explain some features of the observations but not all of them, raising questions about how gas is accreted on to the planet from its circumplanetary disc, and the presumed age of the system.

Irregular fixation: I. Fixed points and librating orbits of the Brown Hamiltonian

ABSTRACT In hierarchical triple systems, the inner binary is slowly perturbed by a distant companion, giving rise to large-scale oscillations in eccentricity and inclination, known as von-Zeipel–Lidov–Kozai oscillations. Stable systems with a mild hierarchy, where the period ratio is not too small, require an additional corrective term, known as the Brown Hamiltonian, to adequately account for their long-term evolution. Although the Brown Hamiltonian has been used to accurately describe the highly eccentric systems on circulating orbits where the periapse completes a complete revolution, the analysis near its elliptical fixed points had been overlooked. We derive analytically the modified fixed points including the Brown Hamiltonian and analyse its librating orbits (where the periapse motion is limited in range). We compare our result to the direct three-body integrations of millions of orbits and discuss the regimes of validity. We numerically discover the regions of orbital instability, allowed and forbidden librating zones with a complex, fractal, structure. The retrograde orbits, where the mutual inclination is $\iota \ \gt\ 90\ \rm deg$, are more stable and allowed to librate for larger areas of the parameter space. We find numerical fits for the librating-circulating boundary. Finally, we discuss the astrophysical implications for systems of satellites, stars, and compact objects. In a companion paper (Paper II), we apply our formalism to the orbits of irregular satellites around giant planets.