A giant impact origin for the Moon is generally accepted, but many aspects of lunar formation remain poorly understood and debated. Ćuk et al. (2016) proposed that an impact that left the Earth-Moon system with high obliquity and angular momentum could explain the Moon's orbital inclination and isotopic similarity to Earth. In this scenario, instability during the Laplace Plane transition, when the Moon's orbit transitions from the gravitational influence of Earth's figure to that of the Sun, would both lower the system's angular momentum to its present-day value and generate the Moon's orbital inclination. Recently, Tian and Wisdom (2020) discovered new dynamical constraints on the Laplace Plane transition and concluded that the Earth-Moon system could not have evolved from an initial state with high obliquity. Here we demonstrate that the Earth-Moon system with an initially high obliquity can evolve into the present state, and we identify a spin-orbit secular resonance as a key dynamical mechanism in the later stages of the Laplace Plane transition. Some of the simulations by Tian and Wisdom (2020) did not encounter this late secular resonance, as their model suppressed obliquity tides and the resulting inclination damping. Our results demonstrate that a giant impact that left Earth with high angular momentum and high obliquity ($θ> 61^{\circ}$) is a promising scenario for explaining many properties of the Earth-Moon system, including its angular momentum and obliquity, the geochemistry of Earth and the Moon, and the lunar inclination.
Accurate estimation of cratering asymmetry on the Moon is crucial for understanding Moon evolution history. Early studies of cratering asymmetry have omitted the contributions of high lunar obliquity and inclination. Here, we include lunar obliquity and inclination as new controlling variables to derive the cratering rate spatial variation as a function of longitude and latitude. With examining the influence of lunar obliquity and inclination on the asteroids population encountered by the Moon, we then have derived general formulas of the cratering rate spatial variation based on the crater sc…
Matija Cuk et.al (2016) have proposed a new model for the birth and tidal evolution of our natural satellite Moon, born from impact generated terrestrial debris in the equatorial plane of high obliquity, high angular momentum Earth. This paper examines their findings critically in the light of advanced kinematic model (AKM) which includes Earth's obliquity (ϕ), Moon's orbital plane inclination (α) , Moon's obliquity (β) and lunar's orbit eccentricity (e). For the real Earth-Moon (E-M) system, the history of evolution of ϕ, α, β, e and (length of month)/(length of day) or LOM/LOD is traced from…
In the giant impact hypothesis for lunar origin, the Moon accreted from an equatorial circum-terrestrial disk; however the current lunar orbital inclination of 5 degrees requires a subsequent dynamical process that is still debated. In addition, the giant impact theory has been challenged by the Moon's unexpectedly Earth-like isotopic composition. Here, we show that tidal dissipation due to lunar obliquity was an important effect during the Moon's tidal evolution, and the past lunar inclination must have been very large, defying theoretical explanations. We present a new tidal evolution model …
We investigate how the variation of the obliquity (the axial tilt) of a hypothetical exo-Earth is effected by the presence of a satellite, an exo-Moon. Namely, we study analytically and numerically how the range of obliquity of the exo-Earth changes if an exo-Moon is added to a system comprised of exo-Sun, exo-Earth and exo-planets. We say that the impact of the exo-Moon is stabilising if upon the addition of the exo-Moon the range of obliquity decreases, while we call the impact destabilising if the range increases as the exo-Moon is added to the system. The problem is considered in a general…
We investigate tidal dissipation of obliquity in hot Jupiters. Assuming an initial random orientation of obliquity and parameters relevant to the observed population, the obliquity of hot Jupiters does not evolve to purely aligned systems. In fact, the obliquity evolves to either prograde, retrograde or 90^{o} orbits where the torque due to tidal perturbations vanishes. This distribution is incompatible with observations which show that hot jupiters around cool stars are generally aligned. This calls into question the viability of tidal dissipation as the mechanism for obliquity alignment of h…
The variation of a planet's obliquity is influenced by the existence of satellites with a high mass ratio. For instance, the Earth's obliquity is stabilized by the Moon, and would undergo chaotic variations in the Moon's absence. In turn, such variations can lead to large-scale changes in the atmospheric circulation, rendering spin-axis dynamics a central issue for understanding climate. The relevant quantity for dynamically-forced climate change is the rate of chaotic diffusion. Accordingly, here we reexamine the spin-axis evolution of a Moonless Earth within the context of a simplified pertu…
The Earth's obliquity is stabilized by the Moon, which facilitates a rapid precession of the Earth's spin-axis, de-tuning the system away from resonance with orbital modulation. It is however, likely that the architecture of the Solar System underwent a dynamical instability-driven transformation, where the primordial configuration was more compact. Hence, the characteristic frequencies associated with orbital perturbations were likely faster in the past, potentially allowing for secular resonant encounters. In this work we examine if at any point in the Earth's evolutionary history, the obliq…
We introduce our new code, SMERCURY-T, which is based on existing codes SMERCURY (Lissauer et al. 2012) and Mercury-T (Bolmont et al. 2015). The result is a mixed-variable symplectic N-body integrator that can compute the orbital and spin evolution of a planet within a multi-planet 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 comp…
Small, rocky planets have been found orbiting in extreme proximity to their host stars, sometimes down to only $\sim 2$ stellar radii. These ultra-short-period planets (USPs) likely did not form in their present-day orbits, but rather migrated from larger initial separations. While tides are the probable cause of this migration, the tidal source has remained uncertain. Here we introduce planetary obliquity tides as a natural pathway for the production of USPs within close-in multi-planet systems. The crucial idea is that tidal dissipation generally forces planetary spin vectors to equilibrium …
As a long-term energy source, tidal heating in subsurface oceans of icy satellites can influence their thermal, rotational, and orbital evolution, and the sustainability of oceans. We present a new theoretical treatment for tidal heating in thin subsurface oceans with overlying incompressible elastic shells of arbitrary thickness. The stabilizing effect of an overlying shell damps ocean tides, reducing tidal heating. This effect is more pronounced on Enceladus than on Europa because the effective rigidity on a small body like Enceladus is larger. For the range of likely shell and ocean thickne…
We present a comprehensive theoretical study on the spin evolution of a planet under the combined effects of tidal dissipation and gravitational perturbation from an external companion. Such a "spin + companion" system (called Colombo's top) appears in many [exo]planetary contexts. The competition between the tidal torque (which drives spin-orbit alignment and synchronization) and the gravitational torque from the companion (which drives orbital precession of the planet) gives rise to two possible spin equilibria ("Tidal Cassini Equilibria", tCE) that are stable and attracting: the "simple" tC…
Tidal force plays an important role in the evolution of the planet-moon system. The tidal force of a moon can excite seismic waves in the planet it is orbiting. A tidal-seismic resonance is expected when a tidal force frequency matches a free-oscillation frequency of the planet. Here we show that when the moon is close to the planet, the tidal-seismic resonance can cause large-amplitude seismic waves, which can change the shape of the planet and in turn exert a negative torque on the moon to cause it to fall rapidly toward the planet. We postulate that the tidal-seismic resonance may be an imp…
New Horizons mission observations show that the small satellites Styx, Nix, Kerberos and Hydra, of the Pluto-Charon system, have not tidally spun-down to near synchronous spin states and have high obliquities with respect to their orbit about the Pluto-Charon binary (Weaver et al. 2016). We use a damped mass-spring model within an N-body simulation to study spin and obliquity evolution for single spinning non-round bodies in circumbinary orbit. Simulations with tidal dissipation alone do not show strong obliquity variations from tidally induced spin-orbit resonance crossing and this we attribu…
Almost all meteorite impacts occur at oblique incidence angles, but the effect of impact angle on crater size is not well understood, especially for large craters. To improve oblique impact crater scaling, we present a suite of simulations of complex crater formation on Earth and the Moon over a range of impact angles, velocities and impactor sizes. We show that crater diameter is larger than predicted by existing scaling relationships for oblique impacts and for impacts steeper than 45$^{\circ}$ shows little dependence on obliquity. Crater depth, volume and diameter depend on impact angle in …
Stellar insolation has been used as the main constraint on a planet's habitability. However, as more Earth-like planets are discovered around low-mass stars (LMSs), a re-examination of the role of tides on the habitability of exoplanets has begun. Those studies have yet to consider the misalignment between a planet's rotational axis and the orbital plane normal, i.e. the planetary obliquity. We apply two equilibrium tide theories to compute the obliquity evolution of terrestrial planets orbiting in the habitable zones around LMSs. The time for the obliquity to decrease from an Earth-like obliq…
The Moon migrated to $r_{\leftmoon}\simeq3.8\times10^{10}$ cm over a characteristic time $r/v=10^{10}$ Gyr by tidal interaction with the Earth's oceans at a present velocity of $v=3.8$ cm yr$^{-1}$. We derive scaling of global dissipation that covers the entire history over the past 4.52 Gyr. Off-resonance tidal interactions at relatively short tidal periods in the past reveal the need for scaling {with amplitude}. The global properties of the complex spatio-temporal dynamics and dissipation in broad spectrum ocean waves is modeled by damping $ε= h F/(2Q_0)$, where $h$ is the tidal wave amplit…
Using viscoelastic mass spring model simulations to track heat distribution inside a tidally perturbed body, we measure the asymmetry of heating in the crust of a spin synchronous Moon in eccentric orbit about the Earth. With the Moon within a few Earth radii of the Earth, we find that tidal heating per unit area in a lunar crustal shell is asymmetric due to the octupole order moment in the Earth's tidal field and is 10 to 20\% higher on its near side than on its far side. Tidal heating reduces the crustal basal heat flux and the rate of magma ocean crystallization. Assuming that the local cru…
Close tidal encounters among large planetesimals and moons should have been more common than grazing or normal impacts. Using a mass spring model within an N-body simulation, we simulate the deformation of the surface of an elastic spherical body caused by a close parabolic tidal encounter with a body that has similar mass as that of the primary body. Such an encounter can induce sufficient stress on the surface to cause brittle failure of an icy crust and simulated fractures can extend a large fraction of the radius of body. Strong tidal encounters may be responsible for the formation of long…
The habitability of exoplanets can be strongly influenced by the presence of an exomoon, and in some cases the exomoon itself could be a possible place for life to develop. For moons outside of the habitable zone, significant tidal heating may raise their surface temperature enough to be considered habitable. Tidal heating of a moon depends on numerous factors such as eccentricity, semimajor axis, size of parent planet, and presence of additional moons. In this work, we explore the degree of tidal heating possible for multi-moon systems in resonance using a combination of semi-analytic and num…
There is compelling evidence for subsurface water oceans among the three outer Galilean satellites, and evidence for an internal magma ocean in the innermost moon, Io. Tidal forces from Jupiter periodically deform these bodies, causing heating and deformation that, if measured, can probe their interior structures. In addition to Jupiter-raised tides, each moon also raises tides on the others. We investigate moon-moon tides for the first time in the Galilean moons, and show that they can cause significant heating through the excitation of high-frequency resonant tidal waves in their subsurface …
Detached circumplanetary disks are unstable to tilting as a result of the stellar tidal potential. We examine how a tilted circumplanetary disk affects the evolution of the spin axis of an oblate planet. The disk is evolved using time-dependent equations for linear wave-like warp evolution, including terms representing the effect of the tidal potential and planetary oblateness. For a disk with a sufficiently large mass, we find that the planet spin quickly aligns to the misaligned disk. The tilt of the planetary spin axis then increases on the same timescale as the disk. This can be an efficie…
We formulated tidal decay lifetimes for hypothetical moons orbiting extrasolar planets with both lunar and stellar tides. Previous work neglected the effect of lunar tides on planet rotation, and are therefore applicable only to systems in which the moon's mass is much less than that of the planet. This work, in contrast, can be applied to the relatively large moons that might be detected around newly-discovered Neptune-mass and super-Earth planets. We conclude that moons are more stable when the planet/moon systems are further from the parent star, the planets are heavier, or the parent stars…
The inner moons of Jupiter and Saturn migrate outwards due to tidal energy dissipation within the planets, the details of which remain poorly understood. We demonstrate that resonance locking between moons and internal oscillation modes of the planet can produce rapid tidal migration. Resonance locking arises due to the internal structural evolution of the planet and typically produces an outward migration rate comparable to the age of the solar system. Resonance locking predicts a similar migration timescale but a different effective tidal quality factor $Q$ governing the migration of each mo…
The Coupled Tidal Evolution of the Moons and Spins of Warm Exoplanets
Princeton University（普林斯顿大学）；Canadian Institute for Theoretical Astrophysics（加拿大理论天体物理研究所）；Observatoire de Paris, Université PSL, Sorbonne Université, Université de Lille, LNE, CNRS（巴黎天文台，巴黎文理研究大学，索邦大学，里尔大学，LNE，法国国家科学研究中心）
It is debated whether the two hot Jupiter populations --- those on orbits misaligned from their host star's spin axis and those well-aligned --- result from two migration channels or from two tidal realignment regimes. Here I demonstrate that equilibrium tides raised by a planet on its star can account for three observed spin-orbit alignment trends: the aligned orbits of hot Jupiters orbiting cool stars, the planetary mass cut-off for retrograde planets, and the stratification by planet mass of cool host stars' rotation frequencies. The first trend can be caused by strong versus weak magnetic …
In recent years, there has been interest in Earth-like exoplanets in the habitable zones of low mass stars ($\sim0.1-0.6\,M_\odot$). Furthermore, it has been argued that a large moon may be important for stabilizing conditions on a planet for life. If these two features are combined, then an exoplanet can feel a similar tidal influence from both its moon and parent star, leading to potentially interesting dynamics. The moon's orbital evolution depends on the exoplanet's initial spin period $P_0$. When $P_0$ is small, transfer of the exoplanet's angular momentum to the moon's orbit can cause th…
Spin-orbit misalignments have been detected in exoplanetary systems and binary star systems. Tidal interactions may have played an important role in the evolution of the spin-orbit angle. In this study, we investigate the tidal interactions in spin-orbit misaligned systems. In particular, we focus on the tidal response of a rotating fluid body to the obliquity tide, which may be important for the evolution of the spin-orbit angle but hardly affects the orbital evolution. The obliquity tide also provides a torque for the mutual precession of the spin and orbital axes around the total angular mo…
Changes in planetary obliquity, or axial tilt, influence the climates on Earth-like planets. In the solar system, the Earth's obliquity is stabilized due to interactions with our moon and the resulting {small amplitude variations ($\sim$2.4\degree)} are beneficial for advanced life. Most Sun-like stars have at least one stellar companion and the habitability of circumstellar exoplanets is shaped by their stellar companion. We show that a stellar companion can dramatically change whether {Earth-like obliquity stability is} possible through planetary orbital precession relative to the binary orb…
Tidal dissipation of kinetic energy, when it is strong enough, tends to synchronize the rotation of planets and moons with the mean orbital motion, or drive it into long-term stable spin-orbit resonances. As the orbital motion undergoes periodic acceleration due to a finite orbital eccentricity, the spin rate oscillates around the equilibrium mean value too, giving rise to the forced, or eccentricity-driven, librations. Both the shape and amplitude of forced librations of synchronous viscoelastic planets and moons are defined by a combination of two different types of perturbative torque, the …
Lainey et al. (2012), by re-analyzing long-baseline astrometry of Saturn's moons, have found that the moons' tidal evolution is much faster than previously thought, implying an order of magnitude stronger tidal dissipation within Saturn. This result is controversial and implies recent formation of at least some of the mid-sized icy moons of Saturn. Here we show that this more intensive tidal dissipation is in full agreement with the evolved state of the Titan-Hyperion resonance. This resonance was previously thought to be non-tidal in origin, as the amount of tidal evolution required for its a…
Using the Andrade-derived Sundberg-Cooper rheology, we apply several improvements to the secular tidal evolution of TRAPPIST-1e and the early history of Pluto-Charon under the simplifying assumption of homogeneous bodies. By including higher-order eccentricity terms (up to and including $e^{20}$), we find divergences from the traditionally used $e^{2}$ truncation starting around $e=0.1$. Order-of-magnitude differences begin to occur for $e>0.6$. Critically, higher-order eccentricity terms activate additional spin-orbit resonances. Worlds experiencing non-synchronous rotation can fall into and …
The (yet-to-be confirmed) discovery of a Neptune-sized moon around the ~3.2 Jupiter-mass planet in Kepler 1625 puts interesting constraints on the formation of the system. In particular, the relatively wide orbit of the moon around the planet, at ~40 planetary radii, is hard to reconcile with planet formation theories. We demonstrate that the observed characteristics of the system can be explained from the tidal capture of a secondary planet in the young system. After a quick phase of tidal circularization, the lunar orbit, initially much tighter than 40 planetary radii, subsequently gradually…
Evolution of a synchronous planet-moon pair due to solar tides. Demise of the synchronous moon that initiated Mars' triaxiality. A possible link to tidalites in Vastitas Borealis and Gale Crater
Motivated by the current search for exomoons, this paper considers the stability of tidal equilibrium for hierarchical three-body systems containing a star, a planet, and a moon. In this treatment, the energy and angular momentum budgets include contributions from the planetary orbit, lunar orbit, stellar spin, planetary spin, and lunar spin. The goal is to determine the optimized energy state of the system subject to the constraint of constant angular momentum. Due to the lack of a closed form solution for the full three-body problem, however, we must use use an approximate description of the…
The Earth-Moon system is suggested to have formed through a single giant collision, in which the Moon accreted from the impact-generated debris disk. However, such giant impacts are rare, and during its evolution the Earth experienced many more smaller impacts, producing smaller satellites that potentially coevolved. In the multiple-impact hypothesis of lunar formation, the current Moon was produced from the mergers of several smaller satellites (moonlets), each formed from debris disks produced by successive large impacts. In the Myrs between impacts, a pre-existing moonlet tidally evolves ou…
Discoveries of extrasolar planets in the habitable zone (HZ) of their parent star lead to questions about the habitability of massive moons orbiting planets in the HZ. Around low-mass stars, the HZ is much closer to the star than for Sun-like stars. For a planet-moon binary in such a HZ, the proximity of the star forces a close orbit for the moon to remain gravitationally bound to the planet. Under these conditions the effects of tidal heating, distortion torques, and stellar perturbations become important considerations for exomoon habitability. Utilizing a model that considers both dynamical…
The Moon is known to have a small liquid core, and it is thought that in the distant past the core may have produced strong magnetic fields recorded in lunar samples. Here we implement a numerical model of lunar orbital and rotational dynamics that includes the effects of a liquid core. In agreement with previous work, we find that the lunar core is dynamically decoupled from the lunar mantle, and that this decoupling happened very early in lunar history. Our model predicts that the lunar core rotates sub-synchronously, and the difference between the core and the mantle rotational rates was si…
Detections of massive extrasolar moons are shown feasible with the Kepler space telescope. Kepler's findings of about 50 exoplanets in the stellar habitable zone naturally make us wonder about the habitability of their hypothetical moons. Illumination from the planet, eclipses, tidal heating, and tidal locking distinguish remote characterization of exomoons from that of exoplanets. We show how evaluation of an exomoon's habitability is possible based on the parameters accessible by current and near-future technology.
The formation and orbital evolution of Saturn's inner mid-sized moons are still debated. The most puzzling aspects are 1) how the Tethys-Dione pair and the Mimas-Enceladus pair passed through their strong 3:2 mean-motion resonances during the tidal orbital evolution, and 2) the current strong heat flow from Enceladus, which is a few orders of magnitude higher than the tidal energy dissipation caused by the present orbital eccentricity of Enceladus. Here we perform N-body simulations of the moons' orbital evolution from various initial conditions -- assuming that the moons were formed from Satu…
We address the thermal history of the Earth after the Moon-forming impact, taking tidal heating and thermal blanketing by the atmosphere into account. The atmosphere sets an upper bound of ~100 W/m^2 on how quickly the Earth can cool. The liquid magma ocean cools over 2-10 Myrs, with longer times corresponding to high angular-momentum events. Tidal heating is focused mostly in mantle materials that are just beginning to freeze. The atmosphere's control over cooling sets up a negative feedback between viscosity-dependent tidal heating and temperature-dependent viscosity of the magma ocean. Whil…
Mercury is expected to deviate from the classical Cassini state since this state is defined for a uniformly precessing rigid planet. We develop an extended Cassini state model that includes the variations (or nutations) in obliquity and deviation induced by the slow precession of the pericenter. The model also describes the constant shift over time in mean obliquity and deviation associated with the short-periodic tidal deformations of Mercury, characterized by the tidal love number k2 and by the ratio k2/Q of the tidal Love number over the tidal quality factor, respectively. This model is the…
It is debated whether close-in giant planets can form in-situ and if not, which mechanisms are responsible for their migration. One of the observable tests for migration theories is the current value of the angle between the stellar equatorial plane and the orbital plane, called the obliquity. After the main migration mechanism has ended, the obliquity and the semi-major axis keep on evolving due to the combined effects of tides and magnetic braking. The observed correlation between effective temperature and measured projected obliquity in known short-period systems has been taken as evidence …
Forming the Moon by a high-angular momentum impact may explain the Earth-Moon isotopic similarities, however, the post-impact angular momentum needs to be reduced by a factor of 2 or more to the current value (1 L_EM) after the Moon forms. Capture into the evection resonance, occurring when the lunar perigee precession period equals one year, could remove the angular momentum excess. However the appropriate angular momentum removal appears sensitive to the tidal model and chosen tidal parameters. In this work, we use a constant-time delay tidal model to explore the Moon's orbital evolution thr…
Each of the giant planets within the Solar System has large moons but none of these moons have their own moons (which we call ${\it submoons}$). By analogy with studies of moons around short-period exoplanets, we investigate the tidal-dynamical stability of submoons. We find that 10 km-scale submoons can only survive around large (1000 km-scale) moons on wide-separation orbits. Tidal dissipation destabilizes the orbits of submoons around moons that are small or too close to their host planet; this is the case for most of the Solar System's moons. A handful of known moons are, however, capable …
An exomoon on a non perfectly circular orbit experiences tidal heating that is capable to significantly contribute to the thermal brightness of the moon. Here we argue that the thermal heat is unevenly distributed on the moon's surface, the emission of the tidal heat is limited to a few hot spots on the surface. A well-known example is the tidally heated Io. Due to their significantly increased temperature, the hot spots enhance the energy emission in thermal wavelengths. We made simulations using Monte-Carlo method to examine this contribution, and to predict about the possible detectability …
The present obliquity of Mercury is very low (less than 0.1 degree), which led previous studies to always adopt a nearly zero obliquity during the planet's past evolution. However, the initial orientation of Mercury's rotation axis is unknown and probably much different than today. As a consequence, we believe that the obliquity could have been significant when the rotation rate of the planet first encountered spin-orbit resonances. In order to compute the capture probabilities in resonance for any evolutionary scenario, we present in full detail the dynamical equations governing the long term…
It has been proposed recently that the first step in the formation of both rocky and gas giant planets is dust sedimentation into a solid core inside a gas clump (giant planet embryo). The clumps are then assumed to migrate closer to the star where their metal poor envelopes are sheared away by the tidal forces or by an irradiation-driven mass loss. We consider the implications of this hypothesis for natal rotation rates of both terrestrial and gas giant planets. It is found that both types of planets may rotate near their break up angular frequencies at birth. The direction of the spin should…
Tidal coplanarization of circumbinary planetary systems through stellar Cassini states
School of Astronomy and Space Science, Nanjing University（南京大学天文与空间科学学院）；Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University（南京大学现代天文与天体物理教育部重点实验室）
Hot stars with hot Jupiters have a wide range of obliquities, while cool stars with hot Jupiters tend to have low obliquities. An enticing explanation for this pattern is tidal realignment of the cool host stars, although this explanation assumes that obliquity damping occurs faster than orbital decay, an assumption that needs further exploration. Here we revisit this tidal realignment problem, building on previous work identifying a low-frequency component of the time-variable tidal potential that affects the obliquity but not the orbital separation. We adopt a recent empirically-based model …
Various theories have been proposed to explain the Moon's current inclined orbit. We test the viability of these theories by reconstructing the thermal-orbital history of the Moon. We build on past thermal-orbital models and incorporate the evolution of the lunar figure including a fossil figure component. Obliquity tidal heating in the lunar magma ocean would have produced rapid inclination damping, making it difficult for an early inclination to survive to the present-day. An early inclination is preserved only if the solid-body of the early Moon were less dissipative than at present. If ins…
