Spin-orbit Resonance Research Articles

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.

Field-Free Spin-Orbit-Torque Switching in Co / Pt / Co Multilayer with Mixed Magnetic Anisotropies

Spin-orbit-torque (SOT) induced magnetization switching in Co/Pt/Co trilayer, with two Co layers exhibiting magnetization easy axes orthogonal to each other is investigated. Pt layer is used as a source of spin-polarized current as it is characterized by relatively high spin-orbit coupling. The spin Hall angle of Pt, $\theta = 0.08$ is quantitatively determined using spin-orbit torque ferromagnetic resonance technique. In addition, Pt serves as a spacer between two Co layers and depending on it's thickness, different interlayer exchange coupling (IEC) energy between ferromagnets is induced. Intermediate IEC energies, resulting in a top Co magnetization tilted from the perpendicular direction, allows for SOT-induced feld-free switching of the top Co layer. The switching process is discussed in more detail, showing the potential of the system for neuromorphic applications.

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.

Asteroseismic Determination of the Stellar Rotation Period of the Kepler Transiting Planetary Systems and its Implications for the Spin–Orbit Architecture

Abstract We measure the rotation periods of 19 stars in the Kepler transiting planetary systems, P rot,astero from asteroseismology and P rot,phot from the photometric variation of their light curves. Two stars exhibit two clear peaks in the Lomb–Scargle periodogram, neither of which agrees with the seismic rotation period. Other four systems do not show any clear peak, whose stellar rotation period is impossible to estimate reliably from the photometric variation; their stellar equators may be significantly inclined with respect to the planetary orbital plane. For the remaining 13 systems, P rot,astero and agree within 30%. Interestingly, 3 out of the 13 systems are in the spin–orbit resonant state in which with P orb,b being the orbital period of the innermost planet of each system. The corresponding chance probability is (0.2–4.7)% based on the photometric rotation period data for 464 Kepler transiting planetary systems. While further analysis of stars with reliable rotation periods is required to examine the statistical significance, the spin–orbit resonance between the star and planets, if confirmed, has important implications for the star–planet tidal interaction, in addition to the origin of the spin–orbit (mis-)alignment of transiting planetary systems.

Secular spin-axis dynamics of exoplanets

Context. Seasonal variations and climate stability of a planet are very sensitive to the planet obliquity and its evolution. This is of particular interest for the emergence and sustainability of land-based life, but orbital and rotational parameters of exoplanets are still poorly constrained. Numerical explorations usually realised in this situation are therefore in heavy contrast with the uncertain nature of the available data. Aims. We aim to provide an analytical formulation of the long-term spin-axis dynamics of exoplanets, linking it directly to physical and dynamical parameters, but still giving precise quantitative results if the parameters are well known. Together with bounds for the poorly constrained parameters of exoplanets, this analysis is designed to enable a quick and straightforward exploration of the spin-axis dynamics. Methods. The long-term orbital solution is decomposed into quasi-periodic series and the spin-axis Hamiltonian is expanded in powers of eccentricity and inclination. Chaotic zones are measured by the resonance overlap criterion. Bounds for the poorly known parameters of exoplanets are obtained from physical grounds (rotational breakup) and dynamical considerations (equipartition of the angular momentum deficit). Results. This method gives accurate results when the orbital evolution is well known. The detailed structure of the chaotic zones for the solar system planets can be retrieved from simple analytical formulas. For less-constrained planetary systems, the maximal extent of the chaotic regions can be computed, requiring only the mass, the semi-major axis, and the eccentricity of the planets present in the system. Additionally, some estimated bounds of the precession constant allow to classify which observed exoplanets are necessarily out of major spin-orbit secular resonances (unless the precession rate is affected by the presence of massive satellites).

On tidal theories and the rotation of viscous bodies

Celestial bodies are complex systems with the ability to get deformed by tides. Modelling accurately this process is a difficult task even for the planets of our own solar system. This is why tidal theories applied to exoplanets are often reduced to a simple formula accounting for our lack of knowledge. In this work, we identify the three minimal assumptions leading to this level of simplification. We then compare different rheologies compatible with these hypotheses and discuss their qualitative behaviour. Finally, we apply the viscous model to close-in planets and describe their rotation evolution, emphasising the apparition of spin-orbit resonances as the viscous timescale increases.

Charge-spin current conversion in high quality epitaxial Fe/Pt systems: Isotropic spin Hall angle along different in-plane crystalline directions

We report the growth of MgO[001]//Fe(6 nm)/MgO(7 nm) and MgO[001]//Fe(6 nm)/Pt(6 nm) by molecular beam epitaxy and show that the full characterization by spin-orbit ferromagnetic resonance (SO-FMR) allows the determination of magnetic anisotropies by classical FMR-only studies. The spin mixing conductance of the epitaxial Fe/Pt interface was measured to be geffect↑↓=2.6±0.5×1019 m2, and the effective spin Hall angle (SHA) θSHEeffect was estimated at different in-plane crystalline directions. It was found that θSHEeffect is the same in all directions. When taking into account high enough excitation frequencies to achieve uniform precession of magnetization, the effective SHA for epitaxial Pt in Fe/Pt is θSHEeffect=0.051±0.005. We address about the proper conditions to determine those relevant spintronic parameters.

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.

Detection and characterization of spin-orbit resonances in the advanced gravitational wave detectors era

In this paper, we test the performance of templates in detection and characterization of Spin-orbit resonant (SOR) binaries. We use precessing SEOBNRv3 waveforms as well as {\it four} numerical relativity (NR) waveforms to model GWs from SOR binaries and filter them through IMRPhenomD, SEOBNRv4 (non-precessing) and IMRPhenomPv2 (precessing) approximants. We find that IMRPhenomD and SEOBNRv4 recover only $\sim70\%$ of injections with fitting factor (FF) higher than 0.97 (or 90\% of injections with ${\rm FF} >0.9$).However, using the sky-maxed statistic, IMRPhenomPv2 performs magnificently better than their non-precessing counterparts with recovering $99\%$ of the injections with FFs higher than 0.97. Interestingly, injections with $\Delta \phi = 180^{\circ}$ have higher FFs ($\Delta \phi$ is the angle between the components of the black hole spins in the plane orthogonal to the orbital angular momentum) as compared to their $\Delta \phi =0^{\circ}$ and generic counterparts. This implies that we will have a slight observation bias towards $\Delta \phi=180^{\circ}$ SORs while using non-precessing templates for searches. All template approximants are able to recover most of the injected NR waveforms with FFs $>0.95$. For all the injections including NR, the error in estimating chirp mass remains below $<10\%$ with minimum error for $\Delta \phi = 180^{\circ}$ resonant binaries. The symmetric mass ratio can be estimated with errors below $15\%$. The effective spin parameter $\chi_{\rm eff}$ is measured with maximum absolute error of 0.13. The in-plane spin parameter $\chi_p$ is mostly underestimated indicating that a precessing signal will be recovered as a relatively less precessing signal. Based on our findings, we conclude that we not only need improvements in waveform models towards precession and non-quadrupole modes but also better search strategies for precessing GW signals.

Habitable Climate Scenarios for Proxima Centauri b with a Dynamic Ocean.

The nearby exoplanet Proxima Centauri b will be a prime future target for characterization, despite questions about its retention of water. Climate models with static oceans suggest that Proxima b could harbor a small dayside surface ocean despite its weak instellation. We present the first climate simulations of Proxima b with a dynamic ocean. We find that an ocean-covered Proxima b could have a much broader area of surface liquid water but at much colder temperatures than previously suggested, due to ocean heat transport and/or depression of the freezing point by salinity. Elevated greenhouse gas concentrations do not necessarily produce more open ocean because of dynamical regime transitions between a state with an equatorial Rossby-Kelvin wave pattern and a state with a day-night circulation. For an evolutionary path leading to a highly saline ocean, Proxima b could be an inhabited, mostly open ocean planet with halophilic life. A freshwater ocean produces a smaller liquid region than does an Earth salinity ocean. An ocean planet in 3:2 spin-orbit resonance has a permanent tropical waterbelt for moderate eccentricity. A larger versus smaller area of surface liquid water for similar equilibrium temperature may be distinguishable by using the amplitude of the thermal phase curve. Simulations of Proxima Centauri b may be a model for the habitability of weakly irradiated planets orbiting slightly cooler or warmer stars, for example, in the TRAPPIST-1, LHS 1140, GJ 273, and GJ 3293 systems.

Spin-orbit coupling and resonances in the conductance of quantum wires

We investigate a possibility of pair electron-electron ($e\ensuremath{-}e$) collisions in a ballistic wire with spin-orbit coupling and only one populated mode. Unlike in a spin-degenerate system, a combination of spin splitting in momentum space with a momentum-dependent spin precession opens up a finite phase space for pair $e\ensuremath{-}e$ collisions around three distinct positions of the wire's chemical potential. For a short wire, we calculate the corresponding resonant contributions to the conductance, which have different power-law temperature dependencies, and, in some cases, vanish if the wire's transverse confinement potential is symmetric. Our results may explain the recently observed feature at the lower conductance plateau in InAs wires.

Dynamics of Haumea’s dust ring

The particle dynamics of the recently observed ring around dwarf planet Haumea is numerically investigated. The point mass gravitational force, a second degree and order gravity field, and the solar radiation pressure as the main perturbations are considered. The quasi-stationary state of the ring varies for different micron-sized grains and depends also on the spin-orbit resonances between the rotation rate of the main body and the orbital period of the dust particles. The simulations confirm the variable radial width of the ring observed during the transit ingress and egress. Results show that the micron sized grains, initially on circular orbits, become eccentric and form an apse-aligned ring at the observed radial distance near to the 3:1 spin-orbit resonance. It is also demonstrated that this coincidence is only apparent and independent of 3:1 resonance.

Epitaxial Lattice Matching and the Growth Techniques of Compound Semiconductors for their Potential Photovoltaic Applications

This paper presents the recent advances in semiconductor alloys for photovoltaic applications. The two main growth techniques involved in these compounds are metal organic chemical vapor deposition (MOCVD) and molecular beam epitaxy (MBE), that has also been discussed. With these techniques, hetero-structures can be grown with a high efficiency. A combination of more than one semiconductor like GaAs, InGaAs and CuInGaAs increases the range of their electrical and optical properties. A large range of direct band gap, high optical absorption and emission coefficients make these materials optimally suitable for converting the light to electrical energy. Their electronic structures reveal that they are highly suitable for photovoltaic applications also because they exhibit spin orbit resonance and metal/semiconductor transitions. The dissociation energy has also been discussed in reference to the increased stability of these compounds.

Spin-orbital Tidal Dynamics and Tidal Heating in the TRAPPIST-1 Multiplanet System

Abstract We perform numerical simulations of the TRAPPIST-1 system of seven exoplanets orbiting a nearby M dwarf, starting with a previously suggested stable configuration. The long-term stability of this configuration is confirmed, but the motion of planets is found to be chaotic. The eccentricity values are found to vary within finite ranges. The rates of tidal dissipation and tidal evolution of orbits are estimated, assuming an Earth-like rheology for the planets. We find that under this assumption, the planets b, d, and e were captured in the 3:2 or higher spin–orbit resonances during the initial spin-down, but slipped further down into the 1:1 resonance. Depending on its rheology, the innermost planet b may be captured in a stable pseudosynchronous rotation. Nonsynchronous rotation ensures higher levels of tidal dissipation and internal heating. The positive feedback between the viscosity and the dissipation rate—and the ensuing runaway heating—are terminated by a few self-regulation processes. When the temperature is high and the viscosity is low enough, the planet spontaneously leaves the 3:2 resonance. Further heating is stopped either by passing the peak dissipation or by the emergence of partial melt in the mantle. In the post-solidus state, the tidal dissipation is limited to the levels supported by the heat transfer efficiency. The tides on the host star are unlikely to have had a significant dynamical impact. The tides on the synchronized inner planets tend to reduce these planets’ orbital eccentricity, possibly contributing thereby to the system’s stability.

AD Leonis: Radial Velocity Signal of Stellar Rotation or Spin–Orbit Resonance?

Abstract AD Leonis is a nearby magnetically active M dwarf. We find Doppler variability with a period of 2.23 days, as well as photometric signals: (1) a short-period signal, which is similar to the radial velocity signal, albeit with considerable variability; and (2) a long-term activity cycle of 4070 ± 120 days. We examine the short-term photometric signal in the available All-Sky Automated Survey and Microvariability and Oscillations of STars (MOST) photometry and find that the signal is not consistently present and varies considerably as a function of time. This signal undergoes a phase change of roughly 0.8 rad when considering the first and second halves of the MOST data set, which are separated in median time by 3.38 days. In contrast, the Doppler signal is stable in the combined High-Accuracy Radial velocity Planet Searcher and High Resolution Echelle Spectrometer radial velocities for over 4700 days and does not appear to vary in time in amplitude, phase, period, or as a function of extracted wavelength. We consider a variety of starspot scenarios and find it challenging to simultaneously explain the rapidly varying photometric signal and the stable radial velocity signal as being caused by starspots corotating on the stellar surface. This suggests that the origin of the Doppler periodicity might be the gravitational tug of a planet orbiting the star in spin–orbit resonance. For such a scenario and no spin–orbit misalignment, the measured indicates an inclination angle of 15.°5 ± 2.°5 and a planetary companion mass of 0.237 ± 0.047 M Jup.

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.

A Note on the Spin–Orbit, Spin–Spin, and Spin–Orbit–Spin Resonances in the Binary Minor Planet System

Abstract By considering a varying mutual orbit between the two bodies in a binary minor planet system, modified models for the spin–orbit, spin–spin, and spin–orbit–spin resonances are given. For the spin–orbit resonances, our study shows that the resonance center changes with the mass ratio and the mutual distance between the two bodies, and the size of the body in the resonance. The 1:1, 3:2, and 1:2 resonances are taken as examples to show the results. For the spin–spin and spin–orbit–spin resonances, our studies show that the resonance center changes with the rotation states of the two minor planets. The 1:1 spin–spin resonance and the 1:2:1 spin–orbit–spin resonance are discussed in detail. Simple analytical criteria are given to identify the resonance centers, and numerical simulations were ran in order to verify the analytical results.

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

We investigate the stable Cassini state 1 in the p : q spin–orbit resonant problem. Our study includes the effect of the gravitational potential up to degree and order 4 and p : q spin–orbit resonances with \(p,q\le 8\) and \(p\ge q\). We derive new formulae that link the gravitational field coefficients with its secular orbital elements and its rotational parameters. The formulae can be used to predict the orientation of the spin axis and necessary angular momentum at exact resonance. We also develop a simple pendulum model to approximate the dynamics close to resonance and make use of it to predict the libration periods and widths of the oscillatory regime of motions in phase space. Our analytical results are based on averaging theory that we also confirm by means of numerical simulations of the exact dynamical equations. Our results are applied to a possible rotational history of Mercury.

Tidal synchronization of an anelastic multi-layered body: Titan’s synchronous rotation

Tidal torque drives the rotational and orbital evolution of planet–satellite and star–exoplanet systems. This paper presents one analytical tidal theory for a viscoelastic multi-layered body with an arbitrary number of homogeneous layers. Starting with the static equilibrium figure, modified to include tide and differential rotation, and using the Newtonian creep approach, we find the dynamical equilibrium figure of the deformed body, which allows us to calculate the tidal potential and the forces acting on the tide generating body, as well as the rotation and orbital elements variations. In the particular case of the two-layer model, we study the tidal synchronization when the gravitational coupling and the friction in the interface between the layers is added. For high relaxation factors (low viscosity), the stationary solution of each layer is synchronous with the orbital mean motion (n) when the orbit is circular, but the rotational frequencies increase if the orbital eccentricity increases. This behavior is characteristic in the classical Darwinian theories and in the homogeneous case of the creep tide theory. For low relaxation factors (high viscosity), as in planetary satellites, if friction remains low, each layer can be trapped in different spin-orbit resonances with frequencies $$n/2,n,3n/2,2n,\ldots $$ . When the friction increases, attractors with differential rotations are destroyed, surviving only commensurabilities in which core and shell have the same velocity of rotation. We apply the theory to Titan. The main results are: (i) the rotational constraint does not allow us to confirm or reject the existence of a subsurface ocean in Titan; and (ii) the crust-atmosphere exchange of angular momentum can be neglected. Using the rotation estimate based on Cassini’s observation (Meriggiola et al. in Icarus 275:183–192, 2016), we limit the possible value of the shell relaxation factor, when a deep subsurface ocean is assumed, to $$\gamma _s\lesssim 10^{-9}\,\hbox {s}^{-1}$$ , which corresponds to a shell’s viscosity $$\eta _s\gtrsim 10^{18}\,\hbox {Pa}\,\hbox {s}$$ , depending on the ocean’s thickness and viscosity values. In the case in which a subsurface ocean does not exist, the maximum shell relaxation factor is one order of magnitude smaller and the corresponding minimum shell’s viscosity is one order higher.

Tidal effects in differentiated viscoelastic bodies: a numerical approach

The majority of confirmed terrestrial exoplanets orbits close to their host stars and their evolution was likely altered by tidal interaction. Nevertheless, due to their viscoelastic properties on the tidal frequencies, their response cannot be described exactly by standardly employed constant-lag models. We therefore introduce a tidal model based on the numerical evaluation of a continuum mechanics problem describing the deformation of viscoelastic (Maxwell or Andrade) planetary mantles subjected to external force. We apply the method on a model Earth-size planet orbiting a low-mass star and study the effect of the orbital eccentricity, the mantle viscosity and the chosen rheology on the tidal dissipation, the complex Love numbers and the tidal torque. The number of stable spin states (i.e., zero tidal torque) grows with increasing mantle viscosity, similarly to the analytical model of Correia et al. (Astron Astrophys 571:A50, 2014) for homogeneous bodies. This behavior is only slightly influenced by the rheology used. Similarly, the Love numbers do not distinctly depend on the considered rheological model. The increase in viscosity affects the amplitude of their variations. The tidal heating described by the Maxwell rheology attains local minima associated with low spin-orbit resonances, with depth and shape depending on both the eccentricity and the viscosity. For the Andrade rheology, the minima at low resonances are very shallow and the tidal heating for all viscosities resembles a “fluid limit.” The tidal heating is the quantity influenced the most by the rheology, having thus possible impact on the internal thermal evolution.