Secular Spin-orbit Resonance Research Articles

Tidal Dissipation Regimes among the Short-period Exoplanets

The efficiency of tidal dissipation provides a zeroth-order link to a planet’s physical properties. For super-Earth and sub-Neptune planets in the range R ⊕ ≲ R p ≲ 4R ⊕, particularly efficient dissipation (i.e., low tidal quality factors) may signify terrestrial-like planets capable of maintaining rigid crustal features. Here, we explore global constraints on planetary tidal quality factors using a population of planets in multiple-planet systems whose orbital and physical properties indicate susceptibility to capture into secular spin–orbit resonances. Planets participating in secular spin–orbit resonance can maintain large axial tilts and significantly enhanced heating from obliquity tides. When obliquity tides are sufficiently strong, planets in low-order mean-motion resonances can experience resonant repulsion (period ratio increase). The observed distribution of period ratios among transiting planet pairs may thus depend nontrivially on the underlying planetary structures. We model the action of resonant repulsion and demonstrate that the observed distribution of period ratios near the 2:1 and 3:2 commensurabilities implies Q values spanning from Q ≈ 101–107 and peaking at Q ≈ 106. This range includes the expected range in which super-Earth and sub-Neptune planets dissipate (Q ≈ 103–104). This work serves as a proof of concept for a method of assessing the presence of two dissipation regimes, and we estimate the number of additional multitransiting planetary systems needed to place any bimodality in the distribution on a strong statistical footing.

Secular Spin–Orbit Resonances of Black Hole Binaries in AGN Disks

The spin–orbit misalignment of stellar-mass black hole (sBH) binaries provides important constraints on the formation channels of merging sBHs. Here, we study the role of secular spin–orbit resonance in the evolution of an sBH binary component around a supermassive BH (SMBH) in an AGN disk. We consider the sBH’s spin precession due to the J 2 moment introduced by a circum-sBH disk within the warping/breaking radius of the disk. We find that the sBH’s spin–orbit misalignment (obliquity) can be excited via spin–orbit resonance between the sBH binary’s orbital nodal precession and the sBH spin precession driven by a massive circum-sBH disk. Using an α-disk model with Bondi–Hoyle–Lyttleton accretion, the resonances typically occur for sBH binaries with semimajor axis of 1 au and at a distance of ∼1000 au around a 107 M ⊙ SMBH. The spin–orbit resonances can lead to high sBH obliquities and a broad distribution of sBH binary spin–spin misalignments. However, we note that the Bondi–Hoyle–Lyttleton accretion is much higher than that of Eddington accretion, which typically results in spin precession being too low to trigger spin–orbit resonances. Thus, secular spin–orbit resonances can be quite rare for sBHs in AGN disks.

Tilting Uranus via the migration of an ancient satellite

Context. The 98° obliquity of Uranus is commonly attributed to giant impacts that occurred at the end of the planetary formation. This picture, however, is not devoid of weaknesses. Aims. On a billion-year timescale, the tidal migration of the satellites of Jupiter and Saturn has been shown to strongly affect their spin-axis dynamics. We aim to revisit the scenario of tilting Uranus in light of this mechanism. Methods. We analyse the precession spectrum of Uranus and identify the candidate secular spin-orbit resonances that could be responsible for the tilting. We determine the properties of the hypothetical ancient satellite required for a capture and explore the dynamics numerically. Results. If it migrates over 10 Uranus’s radii, a single satellite with minimum mass 4 × 10−4 Uranus’s mass is able to tilt Uranus from a small obliquity and make it converge towards 90°. In order to achieve the tilting in less than the age of the Solar System, the mean drift rate of the satellite must be comparable to the Moon’s current orbital expansion. Under these conditions, simulations show that Uranus is readily tilted over 80°. Beyond this point, the satellite is strongly destabilised and triggers a phase of chaotic motion for the planet’s spin axis. The chaotic phase ends when the satellite collides into the planet, ultimately freezing the planet’s obliquity in either a prograde or a plainly retrograde state (as Uranus today). Spin states resembling that of Uranus can be obtained with probabilities as large as 80%, but a bigger satellite is favoured, with mass 1.7 × 10−3 Uranus’s mass or more. Yet, a smaller ancient satellite is not categorically ruled out, and we discuss several ways to improve this basic scenario in future studies. Interactions among several pre-existing satellites are a promising possibility. Conclusions. The conditions required for the tilting seem broadly realistic, but it remains to be determined whether Uranus could have hosted a big primordial satellite subject to substantial tidal migration. The efficiency of tidal energy dissipation within Uranus is required to be much higher than traditionally assumed, more in line with that measured for the migration of Titan. Hints about these issues would be given by a measure of the expansion rate of Uranus’s main satellites.

Tilting Uranus via Spin–Orbit Resonance with Planet Nine

Uranus’s startlingly large obliquity of 98° has yet to admit a satisfactory explanation. The most widely accepted hypothesis involving a giant impactor that tipped Uranus onto its side encounters several difficulties with regard to Uranus’s spin rate and prograde satellite system. An obliquity increase that was driven by capture of Uranus into a secular spin–orbit resonance remains a possible alternative hypothesis that avoids many of the issues associated with a giant impact. We propose that secular spin–orbit resonance could have excited Uranus’s obliquity to its present-day value if it was driven by the outward migration of an as-yet-undetected outer solar system body commonly known as Planet Nine. We draw support for our hypothesis from an analysis of 123 N-body simulations with varying parameters for Planet Nine and its migration. We find that in multiple instances, a simulated Planet Nine drives Uranus’s obliquity past 98°, with a significant number falling within 10% of this value. We note a significant caveat to our results in that a much faster than present-day spin axis precession rate for Uranus is required in all cases for it to reach high obliquities. We conclude that while it was, in principle, possible for Planet Nine (if it exists) to have been responsible for Uranus’s obliquity, the feasibility of such a result hinges on Uranus’s primordial precession rate.

Low spin-axis variations of circumbinary planets

ABSTRACT Having a massive moon has been considered as a primary mechanism for stabilized planetary obliquity, an example of which being our Earth. This is, however, not always consistent with the exoplanetary cases. This article details the discovery of an alternative mechanism, namely that planets orbiting around binary stars tend to have low spin-axis variations. This is because the large quadrupole potential of the stellar binary could speed up the planetary orbital precession, and detune the system out of secular spin-orbit resonances. Consequently, habitable zone planets around the stellar binaries in low inclination orbits hold higher potential for regular seasonal changes comparing to their single star analogues.

Future destabilisation of Titan as a result of Saturn’s tilting

Context. As a result of Titan’s migration and Saturn’s probable capture in secular spin–orbit resonance, recent works show that Saturn’s obliquity could be steadily increasing today and may reach large values in the next billions of years. Satellites around high-obliquity planets are known to be unstable in the vicinity of their Laplace radius, but the approximations used so far for Saturn’s spin axis are invalidated in this regime. Aims. We aim to investigate the behaviour of a planet and its satellite when the satellite crosses its Laplace radius while the planet is locked in secular spin–orbit resonance. Methods. We expand on previous works and revisit the concept of Laplace surface. We use it to build an averaged analytical model that couples the planetary spin-axis and satellite dynamics. Results. We show that the dynamics is organised around a critical point, S1, at which the phase-space structure is singular, located at 90° obliquity and near the Laplace radius. If the spin-axis precession rate of the planet is maintained fixed by a resonance while the satellite migrates outwards or inwards, then S1 acts as an attractor towards which the system is forced to evolve. When it reaches the vicinity of S1, the entire system breaks down, either because the planet is expelled from the secular spin–orbit resonance or because the satellite is ejected or collides into the planet. Conclusions. Provided that Titan’s migration is not halted in the future, Titan and Saturn may reach instability between a few gigayears and several tens of gigayears from now, depending on Titan’s migration rate. The evolution would destabilise Titan and drive Saturn towards an obliquity of 90°. Our findings may have important consequences for Uranus. They also provide a straightforward mechanism for producing transiting exoplanets with a face-on massive ring, a configuration that is often put forward to explain some super-puff exoplanets.

Exploring Tidal Obliquity Variations with SMERCURY-T

Abstract We introduce our new code, SMERCURY-T, which is based on existing codes SMERCURY and Mercury-T. The result is a mixed-variable symplectic N-body integrator that can compute the orbital and spin evolution of a planet within a multiplanet system under the influence of tidal spin torques from its star. We validate our implementation by comparing our experimental results to that of a secular model. As we demonstrate in a series of experiments, SMERCURY-T allows for the study of secular spin–orbit resonance crossings and captures for planets within complex multiplanet systems. These processes can drive a planet’s spin state to evolve along vastly different pathways on its road toward tidal equilibrium, as tidal spin torques dampen the planet’s spin rate and evolve its obliquity. Additionally, we show the results of a scenario that exemplifies the crossing of a chaotic region that exists as the overlap of two spin–orbit resonances. The test planet experiences violent and chaotic swings in its obliquity until its eventual escape from resonance as it tidally evolves. All of these processes are and have been important over the obliquity evolution of many bodies within the solar system and beyond and have implications for planetary climate and habitability. SMERCURY-T is a powerful and versatile tool that allows for further study of these phenomena.

A Tidal Origin for a Three-body Resonance in Kepler-221

Abstract Over the course of the last two decades, traditional models of planet formation have been repeatedly challenged by the emerging census of extrasolar planets. Key among them is the orbital architecture problem: while standard models of orbital migration predict resonant orbits for short-period objects, most planets do not appear to lie in orbital resonances. Here, we show that the four-planet system Kepler-221, not previously recognized to have active orbital resonances, has a three-body commensurability relation unique within the Kepler sample. Using a suite of numerical experiments as well as a perturbative analysis, we demonstrate that this system likely began as a resonant chain and proceeded to undergo large-scale divergence away from resonance, under the action of tidal dissipation. Our results further indicate that obliquity tides, driven by a secular spin–orbit resonance and mutual inclination, are an excellent candidate for driving this orbital divergence, and that the high tidal luminosity may also explain the anomalous size of planet b, which lies within the Fulton radius gap.

Tilting Uranus: Collisions versus Spin–Orbit Resonance

Abstract In this paper, we investigate whether Uranus’s 98° obliquity was a by-product of a secular spin–orbit resonance assuming that the planet originated closer to the Sun. In this position, Uranus’s spin precession frequency is fast enough to resonate with another planet located beyond Saturn. Using numerical integration, we show that resonance capture is possible in a variety of past solar system configurations but that the timescale required to tilt the planet to 90° is of the order ∼108 yr—a time span that is uncomfortably long. A resonance kick could tilt the planet to a significant 40° in ∼107 yr only if conditions were ideal. We also revisit the collisional hypothesis for the origin of Uranus’s large obliquity. We consider multiple impacts with a new collisional code that builds up a planet by summing the angular momentum imparted from impactors. Because gas accretion imparts an unknown but likely large part of the planet’s spin angular momentum, we compare different collisional models for tilted, untilted, spinning, and nonspinning planets. We find that a 1 M ⊕ strike is sufficient to explain the planet’s current spin state, but that two 0.5 M ⊕ collisions produce better likelihoods. Finally, we investigate hybrid models and show that resonances must produce a tilt of at least ∼40° for any noticeable improvements to the collision model. Because it is difficult for spin–orbit resonances to drive Uranus’s obliquity to 98° even under these ideal conditions, giant impacts seem inescapable.

The future large obliquity of Jupiter

Aims. We aim to determine whether Jupiter’s obliquity is bound to remain exceptionally small in the Solar System, or if it could grow in the future and reach values comparable to those of the other giant planets. Methods. The spin-axis of Jupiter is subject to the gravitational torques from its regular satellites and from the Sun. These torques evolve over time due to the long-term variations of its orbit and to the migration of its satellites. With numerical simulations, we explore the future evolution of Jupiter’s spin axis for different values of its moment of inertia and for different migration rates of its satellites. Analytical formulas show the location and properties of all relevant resonances. Results. Because of the migration of the Galilean satellites, Jupiter’s obliquity is currently increasing, as it adiabatically follows the drift of a secular spin-orbit resonance with the nodal precession mode of Uranus. Using the current estimates of the migration rate of the satellites, the obliquity of Jupiter can reach values ranging from 6° to 37° after 5 Gyr from now, according to the precise value of its polar moment of inertia. A faster migration for the satellites would produce a larger increase in obliquity, as long as the drift remains adiabatic. Conclusions. Despite its peculiarly small current value, the obliquity of Jupiter is no different from other obliquities in the Solar System: It is equally sensitive to secular spin-orbit resonances and it will probably reach comparable values in the future.

Retrograde-rotating Exoplanets Experience Obliquity Excitations in an Eccentricity-enabled Resonance

Abstract Previous studies have shown that planets that rotate retrograde (backward with respect to their orbital motion) generally experience less severe obliquity variations than those that rotate prograde (the same direction as their orbital motion). Here, we examine retrograde-rotating planets on eccentric orbits and find a previously unknown secular spin–orbit resonance that can drive significant obliquity variations. This resonance occurs when the frequency of the planet’s rotation axis precession becomes commensurate with an orbital eigenfrequency of the planetary system. The planet’s eccentricity enables a participating orbital frequency through an interaction in which the apsidal precession of the planet’s orbit causes a cyclic nutation of the planet’s orbital angular momentum vector. The resulting orbital frequency follows the relationship , where and are the rates of the planet’s changing longitude of periapsis and longitude of ascending node, respectively. We test this mechanism by simulating cases of a simple Earth–Jupiter system and confirm the predicted resonance. Over the course of 100 Myr, the test Earths with rotation axis precession rates near the predicted resonant frequency experienced pronounced obliquity variations of the order of 10°–30°. These variations can be significant and suggest that while retrograde rotation is a stabilizing influence most of the time, retrograde rotators can experience large obliquity variations if they are on eccentric orbits and enter this spin–orbit resonance.

Obliquity Constraints on an Extrasolar Planetary-mass Companion

Abstract We place the first constraints on the obliquity of a planetary-mass companion outside of the solar system. Our target is the directly imaged system 2MASS J01225093–2439505 (2M0122), which consists of a 120 Myr 0.4 M ⊙ star hosting a 12–27 M J companion at 50 au. We constrain all three of the system’s angular-momentum vectors: how the companion spin axis, the stellar spin axis, and the orbit normal are inclined relative to our line of sight. To accomplish this, we measure projected rotation rates (v sin i) for both the star and the companion using new near-infrared high-resolution spectra with NIRSPEC at Keck Observatory. We combine these with a new stellar photometric rotation period from TESS and a published companion rotation period from Hubble Space Telescope to obtain spin-axis inclinations for both objects. We also fitted multiple epochs of astrometry, including a new observation with NIRC2/Keck, to measure 2M0122b’s orbital inclination. The three line-of-sight inclinations place limits on the true de-projected companion obliquity and stellar obliquity. We find that while the stellar obliquity marginally prefers alignment, the companion obliquity tentatively favors misalignment. We evaluate possible origin scenarios. While collisions, secular spin–orbit resonances, and Kozai–Lidov oscillations are unlikely, formation by gravitational instability in a gravito-turbulent disk—the scenario favored for brown dwarf companions to stars—appears promising.

Tilting Ice Giants with a Spin–Orbit Resonance

Abstract Giant collisions can account for Uranus’s and Neptune’s large obliquities, yet generating two planets with widely different tilts and strikingly similar spin rates is a low-probability event. Trapping into a secular spin–orbit resonance, a coupling between spin and orbit precession frequencies, is a promising alternative, as it can tilt the planet without altering its spin period. We show with numerical integrations that if Uranus harbored a massive circumplanetary disk at least three times the mass of its satellite system while it was accreting its gaseous atmosphere, then its spin precession rate would increase enough to resonate with its own orbit, potentially driving the planet’s obliquity to 70°. We find that the presence of a massive disk moves the Laplace radius significantly outward from its classical value, resulting in more of the disk contributing to the planet’s pole precession. Although we can generate tilts greater than 70° only rarely and cannot drive tilts beyond 90°, a subsequent collision with an object about 0.5 M ⊕ could tilt Uranus from 70° to 98°. Minimizing the masses and number of giant impactors from two or more to just one increases the likelihood of producing Uranus’s spin states by about an order of magnitude. Neptune, by contrast, needs a less massive disk to explain its 30° tilt, eliminating the need for giant collisions altogether.

Mars in ice ages for ∼25% of post-Noachian geologic history

Mars is currently a hyperarid, hypothermal desert planet, whose surface inventory of water is primarily confined to the north and south polar ice caps. Previous studies have shown that Mars has undergone massive shifts in its spin-axis obliquity (present-day is 25.2°) due to secular spin orbit resonances. During periods of higher obliquity, water-ice from the polar caps is mobilized to the mid-latitudes (∼35° obliquity) and even the equator (≥45° obliquity), where it is deposited as snow and accumulates over time to form thick regional surface ice sheets. Abundant evidence exists today for the remnants of these ice ages in the form of debris-covered glaciers and ice deposits, but due to the chaotic nature of orbital simulations beyond ∼20 Ma, it has remained unclear to what temporal extent Mars has experienced such ice ages. Recent developments have suggested that impact events which formed in martian surface ice deposits exhibit a distinctive double-layered ejecta morphology. In tandem with cratering statistics, this observation offers the potential to better our understanding of the history of ice ages on Mars. This work explores the timing of ice age events by evaluating the size-frequency distribution of craters forming in surface ice. Using Hartmann isochron model ages, this work shows that Mars has experienced mid-latitude ice ages for up to a cumulative ∼680 Myr out of the past 3.6 Ga, and experienced equatorial ice ages for up to a cumulative ∼250 Myr within the same time period. The results of this study indicate that Mars has experienced mid-latitude/equatorial ice age states for up to approximately 25% of its post-Noachian geologic history, emphasizing that much of the geologic history of Mars is dominated by the presence of widespread non-polar surface ice sheets.

Signatures of Obliquity in Thermal Phase Curves of Hot Jupiters

Recent work suggests that many short-period extrasolar planets may have spin obliquities that are significantly tilted with respect to their orbital planes. These large obliquities are a natural outcome of ``secular spin-orbit resonance'', a configuration in which the planetary spin precession frequency matches the frequency of orbit nodal regression, or a Fourier component thereof. While exoplanet spin obliquities have not yet been measured directly, they may be detectable indirectly through their signatures in various observations, such as photometric measurements across the full phase of a planet's orbit. In this work, we employ a thermal radiative model to explore how large polar tilts affect full-phase light curves, and we discuss the range of unique signatures that are expected to result. We show that the well-studied short-period planets HD 149026 b, WASP-12 b, and CoRoT-2 b all exhibit phase curve features that may arise from being in high-obliquity states. We also constrain the parameters and assess the detectability of hypothetical perturbing planets that could maintain the planets in these states. Among the three planets considered, CoRoT-2 b has the tightest constraints on its proposed obliquity ($45.8^{\circ} \pm 1.4^{\circ}$) and axial orientation. For HD 149026 b, we find no significant evidence for a non-zero obliquity, and the phase curve of WASP-12 b is too complicated by strong tidal distortions for a conclusive assessment.

Excitation of Planetary Obliquities through Planet–Disk Interactions

Abstract The tilt of a planet’s spin axis off its orbital axis (“obliquity”) is a basic physical characteristic that plays a central role in determining the planet’s global circulation and energy redistribution. Moreover, recent studies have also highlighted the importance of obliquities in sculpting not only the physical features of exoplanets but also their orbital architectures. It is therefore of key importance to identify and characterize the dominant processes of excitation of nonzero axial tilts. Here we highlight a simple mechanism that operates early on and is likely fundamental for many extrasolar planets and perhaps even solar system planets. While planets are still forming in the protoplanetary disk, the gravitational potential of the disk induces nodal recession of the orbits. The frequency of this recession decreases as the disk dissipates, and when it crosses the frequency of a planet’s spin axis precession, large planetary obliquities may be excited through capture into a secular spin–orbit resonance. We study the conditions for encountering this resonance and calculate the resulting obliquity excitation over a wide range of parameter space. Planets with semimajor axes in the range 0.3 au ≲ a ≲ 2 au are the most readily affected, but large-a planets can also be impacted. We present a case study of Uranus and Neptune, and show that this mechanism likely cannot help explain their high obliquities. While it could have played a role if finely tuned and envisioned to operate in isolation, large-scale obliquity excitation was likely inhibited by gravitational planet–planet perturbations.

Obliquity Tides May Drive WASP-12b’s Rapid Orbital Decay

Abstract Recent analyses have revealed a mystery. The orbital period of the highly inflated hot Jupiter, WASP-12b, is decreasing rapidly. The rate of inspiral, however, is too fast to be explained by either eccentricity tides or equilibrium stellar tides. While dynamical stellar tides are possible, they require a subgiant structure for the star, whereas stellar models point toward a main-sequence host. Here, we show that these hitherto irreconcilable observations might be explained by planetary obliquity tides if planet b’s spin vector is trapped in a high-obliquity state maintained by a secular spin–orbit resonance with an unseen exterior perturbing planet. We derive constraints on the obliquity ( ), reduced tidal quality factor ( ), and perturbing planet parameters ( , ) required to generate the observed orbital decay. Direct N-body simulations that include tidal and spin dynamics reinforce the plausibility of the scenario. Furthermore, we show that the resonance could have been captured when planet b’s obliquity was small, making the proposed sequence of events easy to explain. The hypothetical perturbing planet is within the limits of current radial velocity constraints on the system, yet it is also detectable. If it exists, it could provide evidence in favor of the in situ formation hypothesis for hot Jupiters.

Exo-Milankovitch Cycles. I. Orbits and Rotation States

Abstract The obliquity of the Earth, which controls our seasons, varies by only ∼2.°5 over ∼40,000 years, and its eccentricity varies by only ∼0.05 over 100,000 years. Nonetheless, these small variations influence Earth’s ice ages. For exoplanets, however, variations can be significantly larger. Previous studies of the habitability of moonless Earth-like exoplanets have found that high obliquities, high eccentricities, and dynamical variations can extend the outer edge of the habitable zone by preventing runaway glaciation (snowball states). We expand upon these studies by exploring the orbital dynamics with a semianalytic model that allows us to map broad regions of parameter space. We find that, in general, the largest drivers of obliquity variations are secular spin–orbit resonances. We show how the obliquity varies in several test cases, including Kepler-62 f, across a wide range of orbital and spin parameters. These obliquity variations, alongside orbital variations, will have a dramatic impact on the climates of such planets.

Dynamics of stellar spin driven by planets undergoing Lidov–Kozai migration: paths to spin–orbit misalignment

Many exoplanetary systems containing hot Jupiters (HJs) exhibit significant misalignment between the spin axes of the host stars and the orbital angular momentum axes of the planets ("spin-orbit misalignment"). High-eccentricity migration involving Lidov-Kozai oscillations of the planet's orbit induced by a distant perturber is a possible channel for producing such misaligned HJ systems. Previous works have shown that the dynamical evolution of the stellar spin axis during the high-$e$ migration plays a dominant role in generating the observed spin-orbit misalignment. Numerical studies have also revealed various patterns of the evolution of the stellar spin axis leading to the final misalignment. Here we develop an analytic theory to elucidate the evolution of spin-orbit misalignment during the Lidov-Kozai migration of planets in stellar binaries. Secular spin-orbit resonances play a key role in the misalignment evolution. We include the effects of short-range forces and tidal dissipation, and categorize the different possible paths to spin-orbit misalignment as a function of various physical parameters (e.g. planet mass and stellar rotation period). We identify five distinct spin-orbit evolution paths and outcomes, only two of which are capable of producing retrograde orbits. We show that these paths to misalignment and the outcomes depend only on two dimensionless parameters, which compare the stellar spin precession frequency with the rate of change of the planet's orbital axis, and the Lidov-Kozai oscillation frequency. Our analysis reveals a number of novel phenomena for the stellar spin evolution, ranging from bifurcation, adiabatic advection, to fully chaotic evolution of spin-orbit angles.

TILTING SATURN WITHOUT TILTING JUPITER: CONSTRAINTS ON GIANT PLANET MIGRATION

The migration and encounter histories of the giant planets in our solar system can be constrained by the obliquities of Jupiter and Saturn. We have performed secular simulations with imposed migration and N-body simulations with planetesimals to study the expected obliquity distribution of migrating planets with initial conditions resembling those of the smooth migration model, the resonant Nice model and two models with five giant planets initially in resonance (one compact and one loose configuration). For smooth migration, the secular spin–orbit resonance mechanism can tilt Saturn's spin axis to the current obliquity if the product of the migration timescale and the orbital inclinations is sufficiently large (exceeding 30 Myr deg). For the resonant Nice model with imposed migration, it is difficult to reproduce today's obliquity values, because the compactness of the initial system raises the frequency that tilts Saturn above the spin precession frequency of Jupiter, causing a Jupiter spin–orbit resonance crossing. Migration timescales sufficiently long to tilt Saturn generally suffice to tilt Jupiter more than is observed. The full N-body simulations tell a somewhat different story, with Jupiter generally being tilted as often as Saturn, but on average having a higher obliquity. The main obstacle is the final orbital spacing of the giant planets, coupled with the tail of Neptune's migration. The resonant Nice case is barely able to simultaneously reproduce the orbital and spin properties of the giant planets, with a probability The loose five planet model is unable to match all our constraints (probability <0.08%). The compact five planet model has the highest chance of matching the orbital and obliquity constraints simultaneously (probability ∼0.3%).