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 complex multi-planet 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.
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…
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 …
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…
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 orbita…
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…
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…
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…
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…
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 …
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…
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-o…
We investigate the obliquity and spin period of Earth-Moon like systems after 4.5 Gyr of tidal evolution with various satellite masses and initial planetary obliquity and discuss their relations to the habitability of the planet. We find three possible outcomes: either i) the system is still evolving, ii) the system is double synchronous or iii) the satellite has collided with the planet. The transition between case i) and ii) is abrupt and occurs at slightly larger satellite mass ($m_s \sim 0.02m_p$) than the lunar mass. We suggest that cases ii) and iii) are less habitable than case i). Usin…
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 …
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 …
We present three-dimensional atmospheric circulation models of a hypothetical "warm Jupiter" planet, for a range of possible obliquities from 0-90 degrees. We model a Jupiter-mass planet on a 10-day orbit around a Sun-like star, since this hypothetical planet sits at the boundary between planets for which we expect that tidal forces should have aligned their rotation axes with their orbital axes (i.e., ones with zero obliquity) and planets whose timescale for tidal alignment is longer than the typical age of an exoplanet system. In line with observational progress, which is pushing atmospheric…
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…
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 …
The obliquity of a star, or the angle between its spin axis and the average orbit normal of its companion planets, provides a unique constraint on that system's evolutionary history. Unlike the Solar System, where the Sun's equator is nearly aligned with its companion planets, many hot Jupiter systems have been discovered with large spin-orbit misalignments, hosting planets on polar or retrograde orbits. We demonstrate that, in contrast to stars harboring hot Jupiters on circular orbits, those with eccentric companions follow no population-wide obliquity trend with stellar temperature. This fi…
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）
A tidally detached super Neptune on a strongly misaligned retrograde orbit
Centro di Ateneo di Studi e Attività Spaziali “G. Colombo” – Università degli Studi di Padova（帕多瓦大学太空研究与活动中心“G. Colombo”）；INAF - Osservatorio Astronomico di Padova（意大利国家天体物理研究所帕多瓦天文台）；Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova（帕多瓦大学伽利略·伽利雷物理与天文学系）；INAF - Osservatorio Astrofisico di Catania（意大利国家天体物理研究所卡塔尼亚天文台）；Dipartimento di Fisica e Astronomia, Università di Padova（帕多瓦大学物理与天文学系）；INAF - Osservatorio Astrofisico di Torino（意大利国家天体物理研究所都灵天文台）；INAF - Osservatorio Astronomico di Roma（意大利国家天体物理研究所罗马天文台）；Fundación Galileo Galilei-INAF（伽利略·伽利雷基金会-INAF）；INAF - Osservatorio Astronomico di Palermo（意大利国家天体物理研究所巴勒莫天文台）
Two formation scenarios have been proposed to explain the tight orbits of hot Jupiters. They could be formed in orbits with a small inclination (with respect to the stellar spin) via disk migration, or in more highly inclined orbits via high-eccentricity migration, where gravitational interactions with a companion and tidal dissipation are at play. Here we target hot Jupiter systems where the misalignment $λ$ has been inferred observationally and we investigate whether their properties are consistent with high-eccentricity migration. Specifically, we study whether stellar tides can be responsi…
Observations of hot Jupiters around solar-type stars with very short orbital periods (~day) suggest that tidal dissipation in such stars is not too efficient so that these planets can survive against rapid orbital decay. This is consistent with recent theoretical works, which indicate that the tidal Q of planet-hosting stars can indeed be much larger than the values inferred from stellar binaries. On the other hand, recent measurements of Rossiter-McLaughlin effects in transiting hot Jupiter systems not only reveal that many such systems have misaligned stellar spin with respect to the orbital…
We report a spin-orbit misalignment for the hot-Jupiter HATS-14b, measuring a projected orbital obliquity of |lambda|= 76 -5/+4 deg. HATS-14b orbits a high metallicity, 5400 K G dwarf in a relatively short period orbit of 2.8 days. This obliquity was measured via the Rossiter-McLaughlin effect, obtained with observations from Keck-HIRES. The velocities were extracted using a novel technique, optimised for low signal-to-noise spectra, achieving a high precision of 4 m/s point-to-point scatter. However, we caution that our uncertainties may be underestimated. Due to the low rotational velocity o…
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 …
The obliquity of a planet is the tilt between its equator and its orbital plane. Giant planets are expected to form with near-zero obliquities. After its formation, some dynamical mechanism must therefore have tilted Saturn up to its current obliquity of 26.7°. This event is traditionally thought to have happened more than 4 Gyrs ago during the late planetary migration because of the crossing of a resonance between the spin-axis precession of Saturn and the nodal orbital precession mode of Neptune. Here, we show that the fast tidal migration of Titan measured by Lainey et al. (2020) is incompa…
Recent work suggests that many short-period super-Earth and sub-Neptune planets may have significant spin axis tilts ("obliquities"). When planets are locked in high-obliquity states, the tidal dissipation rate may increase by several orders of magnitude. This intensified heat deposition within the planets' interiors should generate significant structural consequences, including atmospheric inflation leading to larger transit radii. Using up-to-date radius estimates from Gaia Data Release 2, we show evidence for $\sim50\%$ larger average radii of planets wide of first-order mean-motion resonan…
Many Sun-like stars are observed to host close-in super-Earths (SEs) as part of a multi-planetary system. In such a system, the spin of the SE evolves due to spin-orbit resonances and tidal dissipation. In the absence of tides, the planet's obliquity can evolve chaotically to large values. However, for close-in SEs, tidal dissipation is significant and suppresses the chaos, instead driving the spin into various steady states. We find that the attracting steady states of the SE's spin are more numerous than previously thought, due to the discovery of a new class of "mixed-mode" high-obliquity e…
We present the measured projected obliquity -- the sky-projected angle between the stellar spin axis and orbital angular momentum -- of the inner planet of the HAT-P-17 multi-planet system. We measure the sky-projected obliquity of the star to be λ=19+/-15 degrees by modeling the Rossiter-McLaughlin (RM) effect in Keck/HIRES radial velocities (RVs). The anomalous RV time series shows an asymmetry relative to the midtransit time, ordinarily suggesting a nonzero obliquity -- but in this case at least part of the asymmetry may be due to the convective blueshift, increasing the uncertainty in the …
We review how tides may impact the habitability of terrestrial-like planets. If such planets form around low-mass stars, then planets in the circumstellar habitable zone will be close enough to their host stars to experience strong tidal forces. We discuss 1) decay of semi-major axis, 2) circularization of eccentric orbits, 3) evolution toward zero obliquity, 4) fixed rotation rates (not necessarily synchronous), and 5) internal heating. We briefly describe these effects using the example of a 0.25 solar mass star with a 10 Earth-mass companion. We suggest that the concept of a habitable zone …
Tides may be crucial to the habitability of exoplanets. If such planets form around low-mass stars, then those in the circumstellar habitable zone will be close enough to their host stars to experience strong tidal forces. Tides may result in orbital decay and circularization, evolution toward zero obliquity, a fixed rotation rate (not necessarily synchronous), and substantial internal heating. Due to tidal effects, the range of habitable orbital locations may be quite different from that defined by the traditional concept of a habitable zone (HZ) based on stellar insolation, atmospheric effec…
NASA's Kepler mission revealed that $\sim 30\%$ of Solar-type stars harbor planets with sizes between that of Earth and Neptune on nearly circular and co-planar orbits with periods less than 100 days. Such short-period compact systems are rarely found with planet pairs in mean-motion resonances (MMRs) -- configurations in which the planetary orbital periods exhibit a simple integer ratio -- but there is a significant overabundance of planet pairs lying just wide of the first-order resonances. Previous work suggests that tides raised on the planets by the host star may be responsible for forcin…
The observed amplitude of the rotational photometric modulation of a star with spots should depend on the inclination of its rotational axis relative to our line of sight. Therefore, the distribution of observed rotational amplitudes of a large sample of stars depends on the distribution of their projected axes of rotation. Thus, comparison of the stellar rotational amplitudes of the Kepler KOIs with those of Kepler single stars can provide a measure to indirectly infer the properties of the spin-orbit obliquity of Kepler planets. We apply this technique to the large samples of 993 KOIs and 33…
We find the orbit of the Neptune-sized exoplanet HAT-P-11b to be highly inclined relative to the equatorial plane of its host star. This conclusion is based on spectroscopic observations of two transits, which allowed the Rossiter-McLaughlin effect to be detected with an amplitude of 1.5 m/s. The sky-projected obliquity is 103_{-10}^{+26} degrees. This is the smallest exoplanet for which spin-orbit alignment has been measured. The result favors a migration scenario involving few-body interactions followed by tidal dissipation. This finding also conforms with the pattern that the systems with t…
Recent discoveries of several transiting planets with clearly non-zero eccentricities and some large inclinations started changing the simple picture of close-in planets having circular and well-aligned orbits. Two major scenarios to form such planets are planet migration in a disk, and planet--planet interactions combined with tidal dissipation. The former scenario can naturally produce a circular and low-obliquity orbit, while the latter implicitly assumes an initially highly eccentric and possibly high-obliquity orbit, which are then circularized and aligned via tidal dissipation. We invest…
We provide evidence that the obliquities of stars with close-in giant planets were initially nearly random, and that the low obliquities that are often observed are a consequence of star-planet tidal interactions. The evidence is based on 14 new measurements of the Rossiter-McLaughlin effect (for the systems HAT-P-6, HAT-P-7, HAT-P-16, HAT-P-24, HAT-P-32, HAT-P-34, WASP-12, WASP-16, WASP-18, WASP-19, WASP-26, WASP-31, Gl 436, and Kepler-8), as well as a critical review of previous observations. The low-obliquity (well-aligned) systems are those for which the expected tidal timescale is short, …
Obliquity measurements for stars hosting relatively long-period giant planets with weak star-planet tidal interactions may play a key role in distinguishing between formation theories for shorter-period hot Jupiters. Few such obliquity measurements have been made to date due to the relatively small sample of known wide-orbiting, transiting Jovian-mass planets and the challenging nature of these targets, which tend to have long transit durations and orbit faint stars. We report a measurement of the Rossiter-McLaughlin effect across the transit of K2-140 b, a Jupiter-mass planet with period $P=6…
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 …
We revisit the tidal stability of extrasolar systems harboring a transiting planet and demonstrate that, independently of any tidal model, none but one (HAT-P-2b) of these planets has a tidal equilibrium state, which implies ultimately a collision of these objects with their host star. Consequently, conventional circularization and synchronization timescales cannot be defined because the corresponding states do not represent the endpoint of the tidal evolution. Using numerical simulations of the coupled tidal equations for the spin and orbital parameters of each transiting planetary system, we…
Stellar radiation has conservatively been used as the key constraint to planetary habitability. We review here the effects of tides, exerted by the host star on the planet, on the evolution of the planetary spin. Tides initially drive the rotation period and the orientation of the rotation axis into an equilibrium state but do not necessarily lead to synchronous rotation. As tides also circularize the orbit, eventually the rotation period does equal the orbital period and one hemisphere will be permanently irradiated by the star. Furthermore, the rotational axis will become perpendicular to th…
The Qatar-2 transiting exoplanet system was recently observed by the {\it Kepler} telescope as part of {\it K2} Campaign 6. The photometric time series has one-minute time sampling and a precision of about 690~ppm, after filtering out artifacts and spurious trends. We identify dozens of starspot-crossing events, when the planet eclipsed a relatively dark region of the stellar photosphere. The observed patterns in the sequence of these events demonstrate that the planet always transits over the same range of stellar latitudes, and therefore that the stellar obliquity is less than about 10$^\cir…
The distribution of spin-orbit angles for systems with wide-separation, tidally detached exoplanets offers a unique constraint on the prevalence of dynamically violent planetary evolution histories. Tidally detached planets provide a relatively unbiased view of the primordial stellar obliquity distribution, since they cannot tidally realign within the system lifetime. We present the third result from our Stellar Obliquities in Long-period Exoplanet Systems (SOLES) survey: a measurement of the Rossiter-McLaughlin effect across two transits of the tidally detached warm Jupiter TOI-1478 b with th…
In this paper, we present the consistent evolution of short-period exoplanets coupling the tidal and gravothermal evolution of the planet. Contrarily to previous similar studies, our calculations are based on the complete tidal evolution equations of the Hut model, valid at any order in eccentricity, obliquity and spin. We demonstrate, both analytically and numerically, that, except if the system was formed with a nearly circular orbit (e<0.2), solving consistently the complete tidal equations is mandatory to derive correct tidal evolution histories. We show that calculations based on tidal mo…
An Aligned Very-Low-Mass Star Orbiting an M dwarf and Obliquity Patterns Across Giant Planets, Brown Dwarfs, and Binary Stars
Westlake University（西湖大学）；Université de Montréal（蒙特利尔大学）；Observatoire du Mont-Mégantic, Université de Montréal（蒙特利尔大学马格尼克天文台）；Université de Genève（日内瓦大学）；Observatoire de Genève, Département d’Astronomie, Université de Genève（日内瓦大学天文学系日内瓦天文台）
