Spin-orbit Resonance Research Articles

Thermal and Orbital Evolution of Low-mass Exoplanets

Abstract The thermal, orbital, and rotational dynamics of tidally loaded exoplanets are interconnected by intricate feedback. The rheological structure of the planet determines its susceptibility to tidal deformation and, as a consequence, participates in shaping its orbit. The orbital parameters and the spin state, conversely, control the rate of tidal dissipation and may lead to substantial changes in the interior. We investigate the coupled thermal–orbital evolution of differentiated rocky exoplanets governed by the Andrade viscoelastic rheology. The coupled evolution is treated by a semianalytical model, 1D parameterized heat transfer, and self-consistently calculated tidal dissipation. First, we conduct several parametric studies, exploring the effect of the rheological properties, the planet size, and the orbital eccentricity on tidal locking and dissipation. These tests show that the role of tidal locking into high spin–orbit resonances is most prominent on low eccentric orbits, where it results in substantially higher tidal heating than synchronous rotation. Second, we calculate the long-term evolution of three currently known low-mass exoplanets with nonzero orbital eccentricity and absent or yet-unknown eccentricity forcing (namely GJ 625 b, GJ 411 b, and Proxima Centauri b). The tidal model incorporates the formation of a stable magma ocean and a consistently evolving spin rate. We find that the thermal state is strongly affected by the evolution of eccentricity and spin state and proceeds as a sequence of thermal equilibria. Final despinning into synchronous rotation slows down the orbital evolution and helps to maintain long-term stable orbital eccentricity.

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.

Direct observation of a Feshbach resonance by coincidence detection of ions and electrons in Penning ionization collisions

Observation of molecular dynamics with quantum state resolution is one of the major challenges in chemical physics. Complete characterization of collision dynamics leads to the microscopic understanding and unraveling of different quantum phenomena such as scattering resonances. Here we present an experimental approach for observing molecular dynamics involving neutral particles and ions that is capable of providing state-to-state mapping of the dynamics. We use Penning ionization reaction between argon and metastable helium to generate argon ion and ground state helium atom pairs at separation of several angstroms. The energy of an ejected electron carries the information about the initial electronic state of an ion. The coincidence detection of ionic products provides a state resolved description of the post-ionization ion-neutral dynamics. We demonstrate that correlation between the electron and ion energy spectra enables us to directly observe the spin-orbit excited Feshbach resonance state of HeAr+. We measure the lifetime of the quasi-bound HeAr+A2 state and discuss possible applications of our method.

N-body Simulations of the Ring Formation Process around the Dwarf Planet Haumea

Abstract Haumea is the only known trans-Neptunian object with a ring. The ring is in a position that causes a 3:1 spin-orbit resonance with the rotational period of Haumea, which has a triaxial shape. The non-axisymmetric gravitational field around Haumea is thought to affect the dynamics of the ring; however, the process of ring formation has not been elucidated. In this study, we analyze in some detail a potential ring formation scenario for Haumea. We first calculated the gravitational field around the triaxial ellipsoid and estimated the distance at which an object revolving around Haumea can exist stably using a simulation that incorporated the time-varying gravitational field. The results of this simulation showed that the trajectory of the object became unstable just inside its current ring position. Next, we analytically derived the Roche radius for a rigid body revolving around a triaxial ellipsoid and showed that the Roche radius could be near the current ring position. Furthermore, as a parameter study, we performed N-body simulations using the coefficient of rubble-pile restitution as a variable. Results demonstrated that, according to numerous parameters, the position of the Roche radius was near the current position of Haumea’s ring. Based on these findings, we can assume that there is a high possibility that the ring formed in the region between the boundary of the unstable region of the orbit and the Roche radius. The scenario presented in this study could help explain the process by which Haumea’s ring formed.

Endpoint of the up-down instability in precessing binary black holes

Binary black holes in which both spins are aligned with the binary's orbital angular momentum do not precess. However, the up-down configuration, in which the spin of the heavier (lighter) black hole is aligned (anti-aligned) with the orbital angular momentum, is unstable to spin precession at small orbital separations. We first cast the spin precession problem in terms of a simple harmonic oscillator and provide a cleaner derivation of the instability onset. Surprisingly, we find that following the instability, up-down binaries do not disperse in the available parameter space but evolve toward precise endpoints. We then present an analytic scheme to locate these final configurations and confirm them with numerical integrations. Namely, unstable up-down binaries approach mergers with the two spins coaligned with each other and equally misaligned with the orbital angular momentum. Merging up-down binaries relevant to LIGO/Virgo and LISA may be detected in these endpoint configurations if the instability onset occurs prior to the sensitivity threshold of the detector. As a by-product, we obtain new generic results on binary black hole spin-orbit resonances at 2nd~post-Newtonian order. We finally apply these findings to a simple astrophysical population of binary black holes where a formation mechanism aligns the spins without preference for co- or counteralignment, as might be the case for stellar-mass black holes embedded in the accretion disk of a supermassive black hole.

Dynamical History of the Uranian System

Abstract We numerically simulate the past tidal evolution of the five large moons of Uranus (Miranda, Ariel, Umbriel, Titania, and Oberon). We find that the most recent major mean–motion resonance (MMR) between any two moons, the Ariel–Umbriel 5:3 MMR, had a large effect on the whole system. Our results suggest that this resonance is responsible for the current 4.°3 inclination of Miranda (instead of the previously proposed 3:1 Miranda–Umbriel MMR), and that all five moons had their inclinations excited during this resonance. Miranda experienced significant tidal heating during the Ariel–Umbriel 5:3 MMR, due to its eccentricity being excited by Ariel’s secular perturbations. This tidal heating draws energy from the shrinking of Miranda’s orbit, rather than Ariel’s outward evolution, and can generate heat flows in excess of 100 mW m−2, sufficient to produce young coronae on Miranda. We find that this MMR was followed by a sequence of secular resonances, which reshuffled the moons’ eccentricities and inclinations. We also find that the precession of Oberon’s spin axis is close to a resonance with the precession of Umbriel’s orbital plane, and that this spin–orbit resonance was likely excited during the Ariel–Umbriel 5:3 MMR. After the exit from the MMR, subsequent Ariel–Umbriel secular resonance and Oberon–Umbriel spin–orbit resonance may be able to explain the current low inclinations of Ariel and Umbriel. The age of Miranda’s surface features tentatively suggests Uranian tidal Q = 15,000–20,000, which can be further refined in future work.

Topological resonances, superefficient orbital-angular-momentum control, and spin-orbit-interaction enhancement in fiber-loop resonators

In this paper we study the propagation of optical vortices (OVs) through the loop resonator (LR) on a multimode fiber. Within the framework of a fully vectorial treatment that allows for the spin-orbit interaction (SOI) we demonstrate the existence in such LRs of a special topological resonance, which is in the inversion of the topological charge (TC) of the transmitted OV. We study the angular momentum (AM) of the transmitted field and show that near resonance points the orbital and spin AMs of the output field are sensitive to wavelength-scale variations of the LR's length parameters. The mechanism that underlies such supersensitivity is the multipass interference assisted by topological charge inversion in the coupling area. We show that this property can be used for superefficient control of the output field's OAM. We also demonstrate that such LRs are suitable for creation of combined wavelength and mode comb filters on their basis. The effect of losses is also discussed. We show the existence in LRs of an additional resonance: the spin-orbit resonance. This type of topological resonance arises due to the effect of the SOI enhancement, which nature we also unveil.

Erratum: “Tilting Ice Giants with a Spin–Orbit Resonance” (2020, ApJ, 888, 60)

Erratum: “Tilting Ice Giants with a Spin–Orbit Resonance” (2020, ApJ, 888, 60)

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.

Cassini states of a rigid body with a liquid core

The purpose of this work is to determine the location and stability of the Cassini states of a celestial body with an inviscid fluid core surrounded by a perfectly rigid mantle. Both situations where the rotation speed is either non-resonant or trapped in a p:1 spin-orbit resonance where p is a half integer are addressed. The rotation dynamics is described by the Poincar\'e-Hough model which assumes a simple motion of the core. The problem is written in a non-canonical Hamiltonian formalism. The secular evolution is obtained without any truncation in obliquity, eccentricity nor inclination. The condition for the body to be in a Cassini state is written as a set of two equations whose unknowns are the mantle obliquity and the tilt angle of the core spin-axis. Solving the system with Mercury's physical and orbital parameters leads to a maximum of 16 different equilibrium configurations, half of them being spectrally stable. In most of these solutions the core is highly tilted with respect to the mantle. The model is also applied to Io and the Moon.

On the secondary’s rotation in a synchronous binary asteroid

ABSTRACT This article studies the secondary’s rotation in a synchronous binary asteroid system in which the secondary enters the 1:1 spin-orbit resonance. The model used is the planar full two-body problem, composed of a spherical primary plus a triaxial ellipsoid secondary. Compared with classical spin-orbit work, there are two differences: (1) influence of the secondary’s rotation on the mutual orbit is considered and (2) instead of the Hamiltonian approach, the approach of periodic orbits is adopted. Our studies find the following. (1) The genealogy of the two families of periodic orbits is the same as that of the families around triangular libration points in the restricted three-body problem. That is, the long-period family terminates on to a short-period orbit travelling N times. (2) In the limiting case where the secondary’s mass is negligible, our results can be reduced to classical spin-orbit theory, by equating the long-period orbit with free libration and the short-period orbit with the forced libration caused by orbit eccentricity. However, the two models show obvious differences when the secondary’s mass is non-negligible. (3) By studying the stability of periodic orbits for a specific binary asteroid system, we are able to obtain the maximum libration amplitude of the secondary (which is usually less than 90°) and the maximum mutual orbit eccentricity that does not break the secondary’s synchronous state. We also find an anti-correlation between the secondary’s libration amplitude and the orbit eccentricity. The (65803) Didymos system is taken as an example to show the results.

Excitation of tumbling in Phobos and Deimos

Mass–spring model simulations are used to investigate past spin states of a viscoelastic Phobos and Deimos. From an initially tidally locked state, we find crossing of a spin–orbit resonance with Mars or a mean motion resonance with each other does not excite tumbling in Phobos or Deimos. However, once tumbling our simulations show that these moons can remain so for an extended period and during this time their orbital eccentricity can be substantially reduced. We attribute the tendency for simulations of an initially tumbling viscoelastic body to drop into spin-synchronous state at very low eccentricity to the insensitivity of the tumbling chaotic zone volume to eccentricity. After a tumbling body enters the spin synchronous resonance, it can exhibit long lived non-principal axis rotation and this too can prolong the period of time with enhanced tidally generated energy dissipation. The low orbital eccentricities of Phobos and Deimos could in part be due to spin excitation by nearly catastrophic impacts rather than tidal evolution following orbital resonance excitation.

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.

Rotation and figure evolution in the creep tide theory: a new approach and application to Mercury

This paper deals with the rotation and figure evolution of a planet near the 3/2 spin–orbit resonance and the exploration of a new formulation of the creep tide theory (Folonier et al. in Celest Mech Dyn Astron 130:78, 2018). This new formulation is composed by a system of differential equations for the figure and the rotation of the body simultaneously (which is the same system of equations used in Folonier et al. 2018), different from the original one (Ferraz-Mello in Celest Mech Dyn Astron 116:109–140, 2013; Celest Mech Dyn Astron 122:359–389, 2015a. arXiv: 1505.05384) in which rotation and figure were considered separately. The time evolution of the figure of the body is studied for both the 3/2 and 2/1 spin–orbit resonances. Moreover, we provide a method to determine the relaxation factor $$\gamma $$ of non-rigid homogeneous bodies whose endpoint of rotational evolution from tidal interactions is the 3/2 spin–orbit resonance, provided that (i) an initially faster rotation is assumed and (ii) no permanent components of the flattenings of the body existed at the time of the capture in the 3/2 spin–orbit resonance. The method is applied to Mercury, since it is currently trapped in a 3/2 spin–orbit resonance with its orbital motion and we obtain $$4.8 \times 10^{-8}$$ s$$^{-1} \le \gamma \le 4.8 \times 10^{-9}$$ $$\mathrm{s}^{-1}$$. The equatorial prolateness and polar oblateness coefficients obtained for Mercury’s figure with such range of values of $$\gamma $$ are the same as the ones given by the Darwin–Kaula model (Matsuyama and Nimmo in J Geophys Res 114, E01010, 2009). However, comparing the values of the flattenings obtained for such range of $$\gamma $$ with those obtained from MESSENGER’s measurements (Perry et al. in Geophys. Res. Lett. 42, 6951–6958, 2015), we see that the current values for Mercury’s equatorial prolateness and polar oblateness are 2–3 orders of magnitude larger than the values given by the tidal theories.

Characterization of material around the centaur (2060) Chiron from a visible and near-infrared stellar occultation in 2011

ABSTRACT The centaur (2060) Chiron exhibits outgassing behaviour and possibly hosts a ring system. On 2011 November 29, Chiron occulted a fairly bright star (R ∼ 15 mag) as observed from the 3-m NASA Infrared Telescope Facility (IRTF) on Mauna Kea and the 2-m Faulkes Telescope North (FTN) at Haleakala. Data were taken as visible wavelength images and simultaneous, low-resolution, near-infrared (NIR) spectra. Here, we present a detailed examination of the light-curve features in the optical data and an analysis of the NIR spectra. We place a lower limit on the spherical diameter of Chiron's nucleus of 160.2 ± 1.3 km. Sharp, narrow dips were observed between 280 and 360 km from the centre (depending on event geometry). For a central chord and assumed ring plane, the separated features are 298.5–302 and 308–310.5 km from the nucleus, with normal optical depth ∼0.5–0.9, and a gap of 9.1 ± 1.3 km. These features are similar in equivalent depth to Chariklo's inner ring. The absence of absorbing/scattering material near the nucleus suggests that these sharp dips are more likely to be planar rings than a shell of material. The region of relatively increased transmission is within the 1:2 spin-orbit resonance, consistent with the proposed clearing pattern for a non-axisymmetric nucleus. Characteristics of possible azimuthally incomplete features are presented, which could be transient, as well as a possible shell from ∼900–1500 km: future observations are needed for confirmation. There are no significant features in the NIR light curves, nor any correlation between optical features and NIR spectral slope.

A cartographic study of spin-orbit coupling in binary asteroids

AbstractIn the spin-orbit resonances, we assume that the orbit of the secondary asteroid around the primary is invariant, which is a reasonable assumption at first glance. Owing to the irregularity of asteroids’ geometry and their effect on the mutual orbit, this assumption should be revised. Therefore, we focus on a binary asteroid with a spherical primary and a secondary with an irregular shape. When the shape of a secondary asteroid is not a sphere, the gravitational interaction is important, and we should consider the interaction of orbit and spin. We generate fast Lyapunov indicator (FLI) maps for both spin-orbit resonance and spin-orbit coupling problems and investigate the effect of orbit alternation on the structure of phase space.

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.

Transition from eyeball to snowball driven by sea-ice drift on tidally locked terrestrial planets

Tidally locked terrestrial planets around low-mass stars are the prime targets for future atmospheric characterizations of potentially habitable systems, especially the three nearby ones--Proxima b, TRAPPIST-1e, and LHS 1140b. Previous studies suggest that if these planets have surface ocean they would be in an eyeball-like climate state: ice-free in the vicinity of the substellar point and ice-covered in the rest regions. However, an important component of the climate system--sea ice dynamics has not been well studied in previous studies. A fundamental question is: would the open ocean be stable against a globally ice-covered snowball state? Here we show that sea-ice drift cools the ocean's surface when the ice flows to the warmer substellar region and melts through absorbing heat from the ocean and the overlying air. As a result, the open ocean shrinks and can even disappear when atmospheric greenhouse gases are not much more abundant than on Earth, turning the planet into a snowball state. This occurs for both synchronous rotation and spin-orbit resonances (such as 3 to 2). These results suggest that sea-ice drift strongly reduces the open ocean area and can significantly impact the habitability of tidally locked planets.

Final spin states of eccentric ocean planets

Context. Eccentricity tides generate a torque that can drive an ocean planet towards asynchronous rotation states of equilibrium when enhanced by resonances associated with the oceanic tidal modes. Aims. We investigate the impact of eccentricity tides on the rotation of rocky planets hosting a thin uniform ocean and orbiting cool dwarf stars such as TRAPPIST-1, with orbital periods ~1−10 days. Methods. Combining the linear theory of oceanic tides in the shallow water approximation with the Andrade model for the solid part of the planet, we developed a global model including the coupling effects of ocean loading, self-attraction, and deformation of the solid regions. From this model we derive analytic solutions for the tidal Love numbers and torque exerted on the planet. These solutions are used with realistic values of parameters provided by advanced models of the internal structure and tidal oscillations of solid bodies to explore the parameter space both analytically and numerically. Results. Our model allows us to fully characterise the frequency-resonant tidal response of the planet, and particularly the features of resonances associated with the oceanic tidal modes (eigenfrequencies, resulting maxima of the tidal torque, and Love numbers) as functions of the planet parameters (mass, radius, Andrade parameters, ocean depth, and Rayleigh drag frequency). Resonances associated with the oceanic tide decrease the critical eccentricity beyond which asynchronous rotation states distinct from the usual spin-orbit resonances can exist. We provide an estimation and scaling laws for this critical eccentricity, which is found to be lowered by roughly one order of magnitude, switching from ~0.3 to ~0.06 in typical cases and to ~0.01 in extremal ones.