Tidal Dissipation in WASP-12

Tidal Dissipation in WASP-12
Authors:

Weinberg et al

Abstract:
WASP-12 is a hot Jupiter system with an orbital period of P=1.1 day, making it one of the shortest-period giant planets known. Recent transit timing observations by Maciejewski et al. (2016) and Patra et al. (2017) find a decreasing period with P/|P˙|=3.2 Myr. This has been interpreted as evidence of either orbital decay due to tidal dissipation or a long term oscillation of the apparent period due to apsidal precession. Here we consider the possibility that it is orbital decay. We show that the parameters of the host star are consistent with either a M∗≃1.3M⊙ main sequence star or a M∗≃1.2M⊙ subgiant. We find that if the star is on the main sequence, the tidal dissipation is too inefficient to explain the observed P˙. However, if it is a subgiant, the tidal dissipation is significantly enhanced due to nonlinear wave breaking of the dynamical tide near the star's center. The subgiant models have a tidal quality factor Q′∗≃2×105 and an orbital decay rate that agrees well with the observed P˙. It would also explain why the planet survived for ≃3 Gyr while the star was on the main sequence and yet is now inspiraling on a 3 Myr timescale. Although this suggests that we are witnessing the last ∼0.1% of the planet's life, the probability of such a detection is a few percent given the observed sample of ≃30 hot Jupiters in P less than 3 day orbits around M∗ greater than 1.2M⊙ hosts.

Precise Masses in the WASP-47 System

Precise Masses in the WASP-47 System

Authors:

Vanderburg et al

Abstract:
We present precise radial velocity observations of WASP-47, a star known to host a hot Jupiter, a distant Jovian companion, and, uniquely, two additional transiting planets in short-period orbits: a super-Earth in a ~19 hour orbit, and a Neptune in a ~9 day orbit. We analyze our observations from the HARPS-N spectrograph along with previously published data to measure the most precise planet masses yet for this system. When combined with new stellar parameters and reanalyzed transit photometry, our mass measurements place strong constraints on the compositions of the two small planets. We find unlike most other ultra-short-period planets, the inner planet, WASP-47 e, has a mass (6.83 +/- 0.66 Me) and radius (1.810 +/- 0.027 Re) inconsistent with an Earth-like composition. Instead, WASP-47 e likely has a volatile-rich envelope surrounding an Earth-like core and mantle. We also perform a dynamical analysis to constrain the orbital inclination of WASP-47 c, the outer Jovian planet. This planet likely orbits close to the plane of the inner three planets, suggesting a quiet dynamical history for the system. Our dynamical constraints also imply that WASP-47 c is much more likely to transit than a geometric calculation would suggest. We calculate a transit probability for WASP-47 c of about 10%, more than an order of magnitude larger than the geometric transit probability of 0.6%.

The impact of rotation on turbulent tidal friction in stellar and planetary convective regions

The impact of rotation on turbulent tidal friction in stellar and planetary convective regions

Authors:

Mathis et al

Abstract:

Turbulent friction in convective regions in stars and planets is one of the key physical mechanisms that drive the dissipation of the kinetic energy of tidal flows in their interiors and the evolution of their systems. This friction acts both on the equilibrium/non-wave like tide and on tidal inertial waves in these layers. It is thus necessary to obtain a robust prescription for this friction. In the current state-of-the-art, it is modeled by a turbulent eddy-viscosity coefficient, based on mixing-length theory, applied on velocities of tides. However, none of the current prescriptions take into account the action of rotation that can strongly affects turbulent convection. Therefore, we use theoretical scaling laws for convective velocities and characteristic lengthscales in rotating stars and planets that have been recently confirmed by 3-D high-resolution nonlinear Cartesian numerical simulations to derive a new prescription. A corresponding local model of tidal waves is used to understand the consequences for the linear tidal dissipation. Finally, new grids of rotating stellar models and published values of planetary convective Rossby numbers are used to discuss astrophysical consequences. The action of rotation on convection deeply modifies the turbulent friction applied on tides. In the regime of rapid rotation (with a convective Rossby number below 0.25), the eddy-viscosity may be decreased by several ordres of magnitude. It may lead to a loss of efficiency of the viscous dissipation of the equilibrium tide and to a more efficient complex and resonant dissipation of tidal inertial waves in the bulk of convective regions. Therefore, it is necessary to have a completely coupled treatment of the tidal/rotational evolution of star-planet systems and multiple stars with a coherent treatment of the variations of tidal flows and of their dissipation as a function of rotation.

Resonant capture of multiple planet systems under dissipation and stable orbital configurations

Resonant capture of multiple planet systems under dissipation and stable orbital configurations

Authors:

Voytazis et al

Abstract:

Migration of planetary systems caused by the action of dissipative forces may lead the planets to be trapped in a resonance. In this work we study the conditions and the dynamics of such resonant trapping. Particularly, we are interested in finding out whether resonant capture ends up in a long-term stable planetary configuration. For two planet systems we associate the evolution of migration with the existence of families of periodic orbits in the phase space of the three-body problem. The family of circular periodic orbits exhibits a gap at the 2:1 resonance and an instability and bifurcation at the 3:1 resonance. These properties explain the high probability of 2:1 and 3:1 resonant capture at low eccentricities. Furthermore, we study the resonant capture of three-planet systems. We show that such a resonant capture is possible and can occur under particular conditions. Then, from the migration path of the system, stable three-planet configurations, either symmetric or asymmetric, can be determined.

Tidal Locking Only Predicted for Orbits With Less Than 15 Day Periods?

Nonlinear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability

Author:

Barker

Abstract:

I present results from the first global hydrodynamical simulations of the elliptical instability in a tidally deformed gaseous planet (or star) with a free surface. The elliptical instability is potentially important for tidal evolution of the shortest-period hot Jupiters. I model the planet as a spin-orbit aligned or anti-aligned, and non-synchronously rotating, tidally deformed, homogeneous fluid body. A companion paper presented an analysis of the global modes and instabilities of such a planet. Here I focus on the nonlinear evolution of the elliptical instability. This is observed to produce bursts of turbulence that drive the planet towards synchronism with its orbit in an erratic manner. If the planetary spin is initially anti-aligned, the elliptical instability also drives spin-orbit alignment on a similar timescale as the spin synchronisation. The instability generates differential rotation inside the planet in the form of zonal flows, which play an important role in the saturation of the instability, and in producing the observed burstiness. These results are broadly consistent with the picture obtained using a local Cartesian model (where columnar vortices played the role of zonal flows). I also simulate the instability in a container that is rigid (but stress-free) rather than free, finding broad quantitative agreement. The dissipation resulting from the elliptical instability could explain why the shortest-period hot Jupiters tend to have circular orbits inside about 2-3 days, and predicts spin-synchronisation (and spin-orbit alignment) out to about 10-15 days. However, other mechanisms must be invoked to explain tidal circularisation for longer orbital periods.

Effect of the rotation and tidal dissipation history of stars on the evolution of close-in planets

Effect of the rotation and tidal dissipation history of stars on the evolution of close-in planets

Authors:

Bolmont et al

Abstract:

Since twenty years, a large population of close-in planets orbiting various classes of low-mass stars (from M to A-type stars) has been discovered. In such systems, the dissipation of the kinetic energy of tidal flows in the host star may modify its rotational evolution and shape the orbital architecture of the surrounding planetary system. In this context, recent works demonstrated that the amplitude of this dissipation can vary over several orders of magnitude as a function of stellar mass, age and rotation. In addition, the stellar rotation evolution strongly impacts angular momentum exchanges within star-planet systems. Therefore, it is now necessary to take into account the structural and rotational evolution of stars when studying the orbital evolution of close-in planets. The presence of planets may also modify the rotation of the host stars and as a consequence their evolution, magnetic activity and mixing. In this work, we present the first study of the dynamics of close-in planets of various masses orbiting low-mass stars (from 0.6 to 1.2 M⋆) where we compute the simultaneous evolution of the star's structure, rotation and tidal dissipation in its external convective enveloppe. We find that tidal friction due to the stellar dynamical tide, i.e. tidal inertial waves excited in the convection zone, can be larger by several orders of magnitude than the one of the equilibrium tide, especially during the PMS phase. Consequently, the orbital migration of the planet is now more pronounced and depends more on the stellar mass, rotation and age. This would very weakly affect the planets in the habitable zone because they are located at orbital distances such that stellar tide-induced migration happens on very long timescales. We also demonstrate that the rotational evolution of host stars is only weakly affected by the presence of planets except for massive companions.

Free Inertial Modes in Differentially Rotating Convective Envelopes of Low-mass Stars

Free inertial modes in differentially rotating convective envelopes of low-mass stars : numerical exploration

Authors:

Guenel et al

Abstract:

Tidally-excited inertial waves in stellar convective regions are a key mechanism for tidal dissipation in stars and therefore the evolution of close-in binary or planetary systems. As a first step, we explore here the impact of latitudinal differential rotation on the properties of free inertial modes and identify the different families of modes. We show that they differ from the case of solid-body rotation. Using an analytical approach as well as numerical calculations, we conclude that critical layers (where the Doppler-shifted frequency vanishes) could play a very important role for tidal dissipation.

Forced Libration of Tidally Synchronized ExoPlanets & ExoMoons

Forced libration of tidally synchronized planets and moons

Authors:

Makarov et al

Abstraction:

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 tidal torque and the triaxial torque. Consequently, forced librations can be tidally dominated (e.g., Io and possibly Titan) or deformation-dominated (e.g., the Moon) depending on a set of orbital, rheological, and other physical parameters. With small eccentricities, for the former kind, the largest term in the libration angle can be minus cosine of the mean anomaly, whereas for the latter kind, it is minus sine of the mean anomaly. The shape and the amplitude of tidal forced librations determine the rate of orbital evolution of synchronous planets and moons, i.e., the rate of dissipative damping of semimajor axis and eccentricity. The known super-Earth exoplanets can exhibit both kinds of libration, or a mixture thereof, depending on, for example, the effective Maxwell time of their rigid mantles. Our approach can be extended to estimate the amplitudes of other libration harmonics, as well as the forced libration in non-synchronous spin-orbit resonances.

The Impact on Tidal Dissipation of Melted Gas Giant Cores

Melting the core of giant planets: impact on tidal dissipation

Authors:

Mathis et al

Abstract:

Giant planets are believed to host central dense rocky/icy cores that are key actors in the core-accretion scenario for their formation. In the same time, some of their components are unstable in the temperature and pressure regimes of central regions of giant planets and only ab-initio EOS computations can address the question of the state of matter. In this framework, several works demonstrated that erosion and redistribution of core materials in the envelope must be taken into account. These complex mechanisms thus may deeply modify giant planet interiors for which signatures of strong tidal dissipation have been obtained for Jupiter and Saturn. The best candidates to explain this dissipation are the viscoelastic dissipation in the central dense core and turbulent friction acting on tidal inertial waves in their fluid convective envelope. In this work, we study the consequences of the possible melting of central regions for the efficiency of each of these mechanisms.

Scaling laws to quantify tidal dissipation in star-planet systems

Scaling laws to quantify tidal dissipation in star-planet systems

Authors:

Auclair-Desrotour et al

Abstract:

Planetary systems evolve over secular time scales. One of the key mechanisms that drive this evolution is tidal dissipation. Submitted to tides, stellar and planetary fluid layers do not behave like rocky ones. Indeed, they are the place of resonant gravito-inertial waves. Therefore, tidal dissipation in fluid bodies strongly depends on the excitation frequency while this dependence is smooth in solid ones. Thus, the impact of the internal structure of celestial bodies must be taken into account when studying tidal dynamics. The purpose of this work is to present a local model of tidal gravito-inertial waves allowing us to quantify analytically the internal dissipation due to viscous friction and thermal diffusion, and to study the properties of the resonant frequency spectrum of the dissipated energy. We derive from this model scaling laws characterizing tidal dissipation as a function of fluid parameters (rotation, stratification, diffusivities) and discuss them in the context of star-planet systems.

Orbital Decay of hot Jupiters

Orbital decay of hot Jupiters due to nonlinear tidal dissipation within solar-type hosts

Authors:

Essick et al

Abstract:

We study the orbital evolution of hot Jupiters due to the excitation and damping of tidally driven g-modes within solar-type host stars. Linearly resonant g-modes (the dynamical tide) are driven to such large amplitudes in the stellar core that they excite a sea of other g-modes through weakly nonlinear interactions. By solving the dynamics of large networks of nonlinearly coupled modes, we show that the nonlinear dissipation rate of the dynamical tide is several orders of magnitude larger than the linear dissipation rate. As a result, we find that the orbits of planets with mass Mp greater than 0.5MJ and period P less than 2 days decay on timescales that are small compared to the main-sequence lifetime of their solar-type hosts. This corresponds to stellar tidal quality factors Q′∗≃105−106 for this range of Mp and P. Our results imply that there are ≃10 currently known exoplanetary systems, including WASP-19b and HAT-P-36-b, with orbital decay timescales shorter than a Gyr. Rapid, tide induced orbital decay may explain the observed paucity of planets with Mp greater than MJ and P less than 2 days around solar-type hosts and could generate detectable transit-timing variations in the near future.

Scaling Tidal Dissipation Simulations

Scaling laws to understand tidal dissipation in fluid planetary regions and stars I - Rotation, stratification and thermal diffusivity

Authors:

Auclair-Desrotour et al

Abstract:

Tidal dissipation in planets and stars is one of the key physical mechanisms driving the evolution of star-planet and planet-moon systems. Several signatures of its action are observed in planetary systems thanks to their orbital architecture and the rotational state of their components. Tidal dissipation inside the fluid layers of celestial bodies are intrinsically linked to the dynamics and the physical properties of the latter. This complex dependence must be characterized. We compute the tidal kinetic energy dissipated by viscous friction and thermal diffusion in a rotating local fluid Cartesian section of a star/planet/moon submitted to a periodic tidal forcing. The properties of tidal gravito-inertial waves excited by the perturbation are derived analytically as explicit functions of the tidal frequency and local fluid parameters (i.e. the rotation, the buoyancy frequency characterizing the entropy stratification, viscous and thermal diffusivities) for periodic normal modes. The sensitivity of the resulting possibly highly resonant dissipation frequency-spectra to a control parameter of the system is either important or negligible depending on the position in the regime diagram relevant for planetary and stellar interiors. For corresponding asymptotic behaviors of tidal gravito-inertial waves dissipated by viscous friction and thermal diffusion, scaling laws for the frequencies, number, width, height and contrast with the non-resonant background of resonances are derived to quantify these variations. We characterize the strong impact of the internal physics and dynamics of fluid planetary layers and stars on the dissipation of tidal kinetic energy in their bulk. We point out the key control parameters that really play a role and demonstrate how it is now necessary to develop ab-initio modeling for tidal dissipation in celestial bodies.

Improving the Modeling of Tidal Distortion and Dissipation for a Giant Planet

Analytical Model of Tidal Distortion and Dissipation for a Giant Planet with a Viscoelastic Core

Authors:

Storch et al

Abstract:

We present analytical expressions for the tidal Love numbers of a giant planet with a solid core and a fluid envelope. We model the core as a uniform, incompressible, elastic solid, and the envelope as a non-viscous fluid satisfying the n=1 polytropic equation of state. We discuss how the Love numbers depend on the size, density, and shear modulus of the core. We then model the core as a viscoelastic Maxwell solid and compute the tidal dissipation rate in the planet as characterized by the imaginary part of the Love number k2. Our results improve upon existing calculations based on planetary models with a solid core and a uniform (n=0) envelope. Our analytical expressions for the Love numbers can be applied to study tidal distortion and viscoelastic dissipation of giant planets with solid cores of various rheological properties, and our general method can be extended to study tidal distortion/dissipation of super-earths.

Understanding Tidal Dissipation in Gaseous Giant Planets

Understanding tidal dissipation in gaseous giant planets : the respective contributions of their core and envelope

Authors:

Guenel et al

Abstract:

Tidal dissipation in planetary interiors is one of the key physical mechanisms that drive the evolution of star-planet and planet-moon systems. New constraints are now obtained both in the Solar and exoplanetary systems. Tidal dissipation in planets is intrinsically related to their internal structure. In particular, fluid and solid layers behave differently under tidal forcing. Therefore, their respective dissipation reservoirs have to be compared. In this work, we compute separately the contributions of the potential dense rocky/icy core and of the convective fluid envelope of gaseous giant planets, as a function of core size and mass. We then compare the associated dissipation reservoirs, by evaluating the frequency-average of the imaginary part of the Love numbers k22 in each region. We demonstrate that in general both mechanisms must be taken into account.

Scaling Applied to Tidal Dissipation in Fluid Planetary Layers

Scaling laws to understand tidal dissipation in fluid planetary layers and stars

Authors:

Auclair-Desrotour et al

Abstract:

Tidal dissipation is known as one of the main drivers of the secular evolution of planetary systems. It directly results from dissipative mechanisms that occur in planets and stars' interiors and strongly depends on the structure and dynamics of the bodies. This work focuses on the mechanism of viscous friction in stars and planetary layers. A local model is used to study tidal dissipation. It provides general scaling laws that give a qualitative overview of the different possible behaviors of fluid tidal waves. Furthermore, it highlights the sensitivity of dissipation to the tidal frequency and the roles played by the internal parameters of the fluid such as rotation, stratification, viscosity and thermal diffusivity that will impact the spins/orbital architecture in planetary systems.

Signatures of Tidal Dissipation of the Core in a Two-layer Planet

The surface signature of the tidal dissipation of the core in a two-layer planet

Authors:

Remus et al

Abstract:

Tidal dissipation, which is directly linked to internal structure, is one of the key physical mechanisms that drive systems evolution and govern their architecture. A robust evaluation of its amplitude is thus needed to predict evolution time for spins and orbits and their final states. The purpose of this paper is to refine recent model of the anelastic tidal dissipation in the central dense region of giant planets, commonly assumed to retain a large amount of heavy elements, which constitute an important source of dissipation. The previous paper evaluated the impact of the presence of the static fluid envelope on the tidal deformation of the core and on the associated anelastic tidal dissipation, through the tidal quality factor Qc. We examine here its impact on the corresponding effective anelastic tidal dissipation, through the effective tidal quality factor Qp. We show that the strength of this mechanism mainly depends on mass concentration. In the case of Jupiter- and Saturn-like planets, it can increase their effective tidal dissipation by, around, a factor 2.4 and 2 respectively. In particular, the range of the rheologies compatible with the observations is enlarged compared to the results issued from previous formulations. We derive here an improved expression of the tidal effective factor Qp in terms of the tidal dissipation factor of the core Qc, without assuming the commonly used assumptions. When applied to giant planets, the formulation obtained here allows a better match between the an elastic core's tidal dissipation of a two-layer model and the observations.

Hot Jupiter Spin-Orbit Misalignment due to Tidal Dissipation in Protoplanetary Disk

The Turbulent Origin of Spin-Orbit Misalignment in Planetary Systems

Authors:

Fielding et al

Abstract:

The turbulent environment from which stars form leads to misalignment between the stellar spin and the remnant protoplanetary disk. By using hydrodynamic and magnetohydrodynamic simulations, we demonstrate that a wide range of stellar obliquities are produced as a by-product of forming a star within a turbulent environment. We present a simple semi-analytic model that reveals this connection between the turbulent motions and the orientation of a star and its disk. Our results are consistent with the observed obliquity distribution of hot Jupiters. Migration of misaligned hot Jupiters may, therefore, be due to tidal dissipation in the disk, rather than tidal dissipation of the star-planet interaction.

Explaining the Maddening HD 40307 System Orbital Configuration

How to reach the orbital configuration of the inner three planets in HD 40307 Planet System ?

Authors:

Yuan-Yuan et al

Abstract:

The formation of the present configuration of three hot super-Earths in the planet system HD 40307 is a challenge to dynamical astronomers. With the two successive period ratios both near and slightly larger than 2, the system may have evolved from pairwise 2:1 mean motion resonances (MMRs). In this paper, we investigate the evolutions of the period ratios of the three planets after the primordial gas disk was depleted. Three routines are found to probably result in the current configuration under tidal dissipation with the center star, they are: (i) through apsidal alignment only; (ii) out of pairwise 2:1 MMRs, then through apsidal alignment; (iii) out of the 4:2:1 Laplace Resonance (LR) , then through apsidal alignment. All the three scenarios require the initial eccentricities of planets ∼0.15, which implies a planetary scattering history during and after the gas disk was depleted. All the three routines will go through the apsidal alignment phase, and enter a state with near-zero eccentricities finally. We also find some special characteristics for each routine. If the system went through pairwise 2:1 MMRs at the beginning, the MMR of the outer two planets would be broken first to reach the current state. As for routine (iii), the planets would be out of the Laplace Resonance at the place where some high-order resonances are located. At the high-order resonances 17:8 or 32:15 of the planets c and d, the system will possibly enter the current state as the final equilibrium.

Tidal dissipation in Io, Mercury & Kepler-10b

Tidal dissipation in a homogeneous spherical body. II. Three examples: Io, Mercury, and Kepler-10 b

Authors:

Makarov et al

Abstract:

In Efroimsky & Makarov (2014), we derived from the first principles a formula for the tidal heating rate in a tidally perturbed homogeneous sphere. We compared it with the formulae used in the literature, and pointed out the differences. Using this result, we now present three case studies - Mercury, Kepler-10b, and a triaxial Io. A very sharp frequency-dependence of k2/Q near spin-orbit resonances yields a similarly sharp dependence of k2/Q on the spin rate. This indicates that physical libration may play a major role in tidal heating of synchronously rotating bodies. The magnitude of libration in the spin rate being defined by the planet's triaxiality, the latter should be a factor determining the dissipation rate. Other parameters equal, a synchronously rotating body with a stronger triaxiality should generate more heat than a similar body of a more symmetrical shape. Further in the paper, we discuss scenarios where initially triaxial objects melt and lose their triaxiality. Thereafter, dissipation in them becomes less intensive; so the bodies freeze. The tidal bulge becomes a new permanent figure, with a new triaxiality lower than the original. In the paper, we also derive simplified, approximate expressions for dissipation rate in a rocky planet of the Maxwell rheology, with a not too small Maxwell time. The three expressions derived pertain to the cases of a synchronous spin, a 3:2 resonance, and a nonresonant rotation; so they can be applied to most close-in super-Earth exoplanets detected thus far. In such bodies, the rate of tidal heating outside of synchronous rotation is weakly dependent on the eccentricity and obliquity, provided both these parameters are small or moderate. According to our calculation, Kepler-10b could hardly survive the great amount of tidal heating without being synchronised, circularised and also reshaped through a complete or partial melt-down.

The Methods for Analyzing Tidal Dissipation in a Homogeneous Spherical Body

Tidal dissipation in a homogeneous spherical body. I. Methods

Authors:

Efroimsky et al

Abstract:

A formula for the tidal dissipation rate in a spherical body is derived from first principles, to correct some mathematical inaccuracies found in the literature. The development is combined with the Darwin-Kaula formalism for tides. Our intermediate results are compared with those by Zschau (1978) and Platzman (1984). When restricted to the special case of an incompressible spherical planet spinning synchronously without libration, our final formula can be compared with the commonly used expression from Peale & Cassen (1978, Eqn. 31). The two turn out to differ. In our expression, the contributions from all Fourier modes are positive-definite, this not being the case of the formula from Ibid. (The presence of negative terms in their formula was noticed by Makarov 2013.) Examples of application of our expression for the tidal damping rate are provided in the work by Makarov and Efroimsky (2014).