# Tidal Interactions Between Planets and Host Stars

- Gordon OgilvieGordon OgilvieDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge

### Summary

Hundreds of planets are already known to have orbits only a few times wider than the stars that host them. The tidal interaction between a planet and its host star is one of the main agents shaping the observed distributions of properties of these systems. Tidal dissipation in the planet tends make the orbit circular, as well as synchronizing and aligning the planet’s spin with the orbit, and can significantly heat the planet, potentially affecting its size and structure. Dissipation in the star typically leads to inward orbital migration of the planet, accelerating the star’s rotation, and in some cases destroying the planet.

Some essential features of tidal evolution can be understood from the basic principles that angular momentum and energy are exchanged between spin and orbit by means of a gravitational field and that energy is dissipated. For example, most short-period exoplanetary systems have too little angular momentum to reach a tidal equilibrium state.

Theoretical studies aim to explain tidal dissipation quantitatively by solving the equations of fluid and solid mechanics in stars and planets undergoing periodic tidal forcing. The equilibrium tide is a nearly hydrostatic bulge that is carried around the body by a large-scale flow, which can be damped by convection or hydrodynamic instability, or by viscoelastic dissipation in solid regions of planets. The dynamical tide is an additional component that generally takes the form of internal waves restored by Coriolis and buoyancy forces in a rotating and stratified fluid body. It can lead to significant dissipation if the waves are amplified by resonance, are efficiently damped when they attain a very short wavelength, or break because they exceed a critical amplitude.

Thermal tides are excited in a planetary atmosphere by the variable heating by the star’s radiation. They can oppose gravitational tides and prevent tidal locking, with consequences for the climate and habitability of the planet.

Ongoing observations of transiting exoplanets provide information on the orbital periods and eccentricities as well as the obliquity (spin–orbit misalignment) of the star and the size of the planet. These data reveal several tidal processes at work and provide constraints on the efficiency of tidal dissipation in a variety of stars and planets.

### Introduction

Tides raised in the Earth’s seas and oceans by the gravitational attraction of the Moon and the Sun have been studied for centuries (Cartwright, 1999; Deparis, Legros, & Souchay, 2013). Among the key contributors to the theoretical understanding of tides were Sir Isaac Newton (17th century), Pierre-Simon, Marquis de Laplace (18th century), and Sir George Darwin (19th century).

The most important astronomical consequences of the tidal interaction between the Moon and the Earth are that angular momentum is being transferred from the spin of the Earth to the orbit of the Moon, causing both the day and the month to lengthen as the Earth slows and the Moon retreats, and that the orbit is becoming increasingly elliptical. The current rates of increase of the orbital semimajor axis and eccentricity, determined by lunar laser ranging (Williams & Boggs, 2016), are $\dot{a}=3.8\phantom{\rule{0.2em}{0ex}}\text{cm}{\text{yr}}^{-1}$ and $\dot{e}=1.5\times {10}^{-11}{\text{yr}}^{-1}$, the source of energy for both these processes being the Earth’s rotation. In addition, as a consequence of tides raised by the Earth on the Moon, the spin of the Moon has been synchronized with its orbital motion, so that it presents the same familiar face towards the Earth; this is known as *tidal locking*.

Further afield in the solar system, tidal theory has been applied to explain many properties of the moons of the other planets (e.g., Peale, 1999). In most cases, like the Earth’s Moon, the satellite is tidally locked and its orbit expands as it receives angular momentum from the rotating planet. This process could be the origin of the remarkable resonant configurations seen around Jupiter and Saturn, such as the 1:2:4 ratio of orbital periods between Io, Europa, and Ganymede (Goldreich, 1965). Most moons have nearly circular orbits, because dissipation in the moon outweighs the tendency of the rotating planet to increase the orbital eccentricity. In cases where a significant eccentricity is maintained by an orbital resonance, intense heating from ongoing tidal dissipation occurs; this is thought to explain the dramatic volcanic activity on Jupiter’s moon Io (Peale, Cassen, & Reynolds, 1979) and Saturn’s moon Enceladus (Porco et al., 2006), discovered by the *Voyager* and *Cassini* space missions.

Beyond the solar system, tidal theory has been applied to explain observational properties of close binary stars (Zahn, 1977). A spectroscopic binary star is one in which the orbital period and eccentricity can be measured by observing the periodic Doppler shifting of the spectral lines of at least one of the two stars. Within a cluster of stars, binaries of shorter period tend to have circular orbits, while those of longer period have widely spread eccentricities; this is interpreted as evidence that tidal dissipation has circularized the orbits of the closer, more strongly interacting binaries. The transitional period is longer in older clusters, providing information about the evolution of the circularization process (Meibom & Mathieu, 2005).

The first exoplanet to be discovered around a main-sequence star, 51 Peg b (Mayor & Queloz, 1995), is an example of a *hot Jupiter*. With a mass about half that of Jupiter, it orbits every 4.2 days around a star slightly more massive than the Sun, and of similar age, in a circular orbit whose radius is about nine times that of the star. Soon after its discovery, it was proposed that 51 Peg b was formed in a circumstellar disk at a similar orbital radius to Jupiter, and that its inward migration through disk torques could have been halted by a tidal torque from the star, when the star was younger, larger, and more rapidly rotating (Lin, Bodenheimer, & Richardson, 1996). Since 1995, hundreds of other exoplanets with orbital periods of a few days or less have been discovered by the radial-velocity and transit methods. Evidence for tidal interaction between these planets and their host stars can be seen both in individual systems and in the statistical properties of the population.

### Tidal Theory for Exoplanetary Systems

#### Tidal Forces

A key result of Newtonian dynamics is that two point masses, subject only to their mutual gravitational attraction, move in Keplerian orbits about their center of mass. The orbit is a circle or an ellipse if it is bound; a parabola or a hyperbola if it is unbound. The behavior of extended, deformable bodies such as planets and stars differs from this simple picture and involves an interplay between the external dynamics (orbital motion of the centers of masses of the two bodies) and the internal dynamics (fluid or solid mechanics within each body).

Tidal effects result from the spatial variation, over an extended body, of the gravitational field due to an orbital companion. If the gravitational field at the center of the first body provides just the right acceleration required for its orbital motion, then the variation of the field produces a tidal acceleration that tends to deform the body from its natural shape.

Consider the interaction of a planet of mass ${M}_{\text{p}}$ and radius ${R}_{\text{p}}$ with a star of mass ${M}_{\text{s}}$ and radius ${R}_{\text{s}}$. A simple estimate of the magnitude of the tidal deformation of the star by the planet is provided by the dimensionless *tidal amplitude parameter* ${\u03f5}_{\text{s}}=({M}_{\text{p}}/{M}_{\text{s}}){({R}_{\text{s}}/r)}^{3}$, where $r$ is the orbital separation. This can be understood as the tidal potential due to the planet at the point on the stellar surface closest to it, divided by the star’s own gravitational potential at this point. A similar estimate of the deformation of the planet, neglecting any rigidity, is ${\u03f5}_{\text{p}}=({M}_{\text{s}}/{M}_{\text{p}}){({R}_{\text{p}}/r)}^{3}$. Tidal effects are very sensitive to the orbital separation and are most significant for systems of short orbital period. Values of ${\u03f5}_{\text{s}}$ up to $1.8\times {10}^{-4}$ are found for stars with massive hot Jupiters such as WASP-18. Values of ${\u03f5}_{\text{p}}$ up to $0.06$, implying significant tidal deformation, are found for hot Jupiters such as WASP-12 b and WASP-19 b.

#### Tidal Deformation and Disruption

The main effect of tidal gravity is to deform each body into a spheroidal shape, elongated along an axis that points towards (and away from) its companion. This is in addition to the flattening of each body due to its rotation. If a planet orbits too close to its host star, the tidal deformation will be so great that the planet will be disrupted.

Consider a planet of mass ${M}_{\text{p}}$ and volumetric radius ${R}_{\text{p}}$ (defined as the radius of a sphere with the same volume as the planet) in a circular orbit of radius $a$ around a star of much greater mass ${M}_{\text{s}}$. For a fluid planet in synchronous, aligned rotation, two classical models bracket the expected behavior:

In the case of a *homogeneous* planet of uniform density, the planet can find an equilibrium in the form of a *Roche ellipsoid*, which is elongated by tidal forces and flattened by rotation, provided that $a$ exceeds a critical value known as the *Roche limit*, which can be expressed as $2.46\phantom{\rule{0.2em}{0ex}}{({M}_{\text{s}}/{M}_{\text{p}})}^{1/3}{R}_{\text{p}}$. If $a$ is less than the Roche limit, the planet is disrupted by being drawn out into a needle-like configuration.

In the opposite extreme of a *centrally condensed* planet whose mass is concentrated in a single point, the planet can find a hydrostatic equilibrium in the *Roche potential*, which combines the gravity of the two bodies with the centrifugal potential due to the uniform rotation (Figure 1). The same formula for the Roche limit applies but with the coefficient 2.46 replaced by 2.03. For values of $a$ greater than this critical value, the surface of the planet follows a contour of the Roche potential, whereas for smaller $a$ the planetary envelope overflows the Roche lobe and material is lost. A related concept is the *Hill radius* ${R}_{\text{H}}={({M}_{\text{p}}/3{M}_{\text{s}})}^{1/3}a$, which gives the distances of the Lagrangian points ${\text{L}}_{1}$ and ${\text{L}}_{2}$ from the center of the planet in the limit that the planet is much less massive than the star. The volume of the planet’s Roche lobe (also known as the *Hill sphere*) in this limit is $1.51\phantom{\rule{0.2em}{0ex}}{R}_{\text{H}}^{3}$ (which is considerably less than that of a sphere of radius ${R}_{\text{H}}$).

The coefficient 2.46 in the Roche limit can also be reduced for a solid planet with significant viscosity or material strength (Holsapple & Michel, 2006; Leinhardt, Ogilvie, Latter, & Kokubo, 2012).

The calculation of a hydrostatic tidal (or rotational) bulge of small amplitude in a self-gravitating body is a classical one developed by Clairaut in the 18th century, involving the solution of a linear second-order ordinary differential equation (e.g., Cook, 1980).

#### Tidal Equilibrium

A possible endpoint of tidal evolution is a *tidal equilibrium* in which the orbit is circular and both the star and the planet are tidally locked. Viewed in a frame of reference that rotates with the common angular velocity $\Omega $, the tidal deformation is then static and no dissipation or tidal evolution occurs.

Given the masses ${M}_{\text{s}}$ and ${M}_{\text{p}}$ of the two bodies and their moments of inertia ${I}_{\text{s}}$ and ${I}_{\text{p}}$, the total angular momentum of the system in a tidal equilibrium can be evaluated as the sum $J=L+S$ of the orbital angular momentum $L=\mu \sqrt{GMa}$ and the spin angular momentum $S=I\Omega $, where $M={M}_{\text{s}}+{M}_{\text{p}}$ is the total mass, $\mu ={M}_{\text{s}}{M}_{\text{p}}/M$ is the reduced mass (slightly less than the planet’s mass), $I={I}_{\text{s}}+{I}_{\text{p}}$ is the sum of the moments of inertia, and $a$ is the radius of the orbit, related to $\Omega $ by Kepler’s Third Law, ${\Omega}^{2}=GM/{a}^{3}$. Note that $L\phantom{\rule{0.2em}{0ex}}\propto \phantom{\rule{0.2em}{0ex}}{a}^{1/2}\phantom{\rule{0.2em}{0ex}}\propto \phantom{\rule{0.2em}{0ex}}{\Omega}^{-1/3}$, whereas $S\phantom{\rule{0.2em}{0ex}}\propto \Omega $. Since $J=L+S$ is the sum of a decreasing function of $\Omega $ and an increasing function of $\Omega $, it has a minimum value, ${J}_{\text{c}}=4I{\Omega}_{\text{c}}$, at a critical angular velocity, ${\Omega}_{\text{c}}=\sqrt{GM}{(\mu /3I)}^{3/4}$. It is then possible to show the following (Hut, 1980):

If the angular momentum of the system exceeds the critical value ${J}_{\text{c}}$, then two tidal equilibrium solutions are possible, but only the more slowly rotating one (which has $\Omega <{\Omega}_{\text{c}}$ and $L>3S$) minimizes the energy and is stable.

If $J<{J}_{\text{c}}$, then no tidal equilibrium exists.

Figure 2 shows the stable tidal equilibria and evolutionary tracks of a system consisting of a star and a planet in which the orbit is assumed to be circular and aligned with the stellar spin. The $x$ and $y$ axes represent the orbital angular velocity $n$ and the stellar spin ${\Omega}_{\text{s}}$, both in units of the critical angular velocity ${\Omega}_{\text{c}}$. Each evolutionary track is a curve on which the total angular momentum $J=L+S$ (to which the planetary spin is assumed to make a negligible contribution) is equal to a constant. The direction of evolution, indicated by the arrows, is that in which the total mechanical energy of the system decreases as a result of tidal dissipation. Even though the spin–orbit interaction is frictional in nature, transferring angular momentum from the more rapidly rotating component to the less rapidly rotating one and dissipating energy, this does not always lead to synchronization because of the peculiar property that the orbital angular momentum is a *decreasing* function of the angular velocity.

Most of the short-period exoplanetary systems for which tidal interactions are important have $J<{J}_{\text{c}}$, which implies $n>{\Omega}_{\text{s}}$. Both $n$ and ${\Omega}_{\text{s}}$ are increasing as a result of tides, and the planet will eventually be consumed. Even for those few short-period systems that have $J>{J}_{\text{c}}$ and may be evolving towards a tidal equilibrium, this equilibrium can only be temporary; stars continue to evolve and lose angular momentum through the magnetic torques on outflowing matter as they emit hot, ionized winds along open magnetic field lines.

#### Torque and Dissipation

A deformed body with a spheroidal bulge possesses a gravitational quadrupole moment that causes its external gravitational field to differ from that of a point mass (a gravitational monopole). The quadrupolar component of the field lacks complete rotational symmetry and decays more rapidly with distance than the monopolar component. The tidal interaction between two bodies can be thought of as a coupling of their monopole and quadrupole moments, which allows a torque to be exerted, meaning that angular momentum is exchanged between spin and orbit. If the bulge points towards the companion, which is true of an instantaneous hydrostatic response, as in Figure 1, then this torque vanishes. Misalignment of the bulge can be thought of as resulting from a delay in the response. This requires dissipation and allows a torque to be exerted, leading to an irreversible evolution of spin and orbit; it also causes the tidally deformed body in which the dissipation occurs to be heated.

All the relevant information about the tidal interaction is encoded in the gravitational quadrupole moments of the deformed bodies. The Love number $k$ (actually the potential Love number of second degree, ${k}_{2}$) is a dimensionless measure of how much a body is deformed hydrostatically by a tidal force. For a fluid body, $k$ is determined from the solution of Clairaut’s equation; it is equal to 1.5 for a homogeneous body, but smaller for more centrally condensed bodies such as giant planets and especially for stars. The rigidity of a solid body such as a terrestrial exoplanet also reduces $k$. (Estimates for Jupiter, the Earth, the Sun, and the Moon are 0.59, 0.30, 0.035, and 0.024, respectively.)

The dissipative aspect of the tidal response can be described in a variety of equivalent ways. Most common are the *quality factor*$Q$, the *modified quality factor*${Q}^{\prime}$, and the *time lag*$\tau $. These are related by $\omega \tau =1/Q$ (which is the *phase lag*) and $k/Q=1.5/{Q}^{\prime}$, where $\omega $ is the angular frequency of the tidal forcing in a frame of reference that rotates with the body. There is a long history of describing the damping of waves and oscillations in the Earth in terms of a quality factor $Q$, which is a dimensionless quantity *inversely* proportional to the efficiency of dissipation. An advantage of ${Q}^{\prime}$ is that it combines the parameter $Q$ with the Love number $k$, which itself may be uncertain, in such a way that they need not be considered separately. Both the tidal torque and the dissipation rate are proportional to $k/Q$, or to $1/{Q}^{\prime}$, or to $k\tau $.

It is best not to think of $Q$, ${Q}^{\prime}$, or $\tau $ as constant properties of a star or planet. For small-amplitude tides, these parameters are different ways of quantifying the linear response of the body to tidal forcing at a particular frequency, and can depend significantly on that frequency.

### Tidal Force

In the weak friction approximation (Alexander, 1973; Hut, 1981), the phase lag is assumed to be small and proportional to the tidal frequency; this means that the tidal deformation is identical to an instantaneous hydrostatic one, but delayed by a constant time lag $\tau $. There is some physical justification for this assumption. In a fluid body of mass $M$, radius $R$, and kinematic viscosity $\nu $, the behavior is indeed of this form, with a time lag proportional to $\nu R/GM$, if the tidal frequency is well below the natural frequencies of any relevant normal modes of oscillation.

With this convenient assumption, the tidal acceleration acting on the orbital separation vector $\text{r}=r\phantom{\rule{0.2em}{0ex}}\widehat{\text{r}}$ due to dissipation in the star can be written as $-{\gamma}_{\text{s}}(\dot{\text{r}}-{\text{\Omega}}_{\text{s}}\times \text{r}+2\dot{r}\phantom{\rule{0.2em}{0ex}}\widehat{\text{r}})$ where ${\gamma}_{\text{s}}=3{k}_{\text{s}}{\tau}_{\text{s}}\left(\frac{{M}_{\text{p}}}{{M}_{\text{s}}}\right){\left(\frac{{R}_{\text{s}}}{r}\right)}^{5}\frac{GM}{{r}^{3}}$ is a damping coefficient. Here ${\text{\Omega}}_{\text{s}}$ is the spin angular velocity of the star, and $\dot{\text{r}}-{\text{\Omega}}_{\text{s}}\times \text{r}$ is the velocity of the planet in a frame of reference moving and rotating with the star. The tidal force therefore tends to damp any asynchronism or misalignment between the orbital and rotational motions (as seen in the first two terms in the expression) and also to damp any radial motion (associated with orbital eccentricity). A similar expression, with subscripts “s” and “p” reversed, gives the acceleration due to dissipation in the planet.

#### Tidal Evolution

Away from a tidal equilibrium, each body experiences a tidal torque that exchanges angular momentum between its spin and the orbit; energy is also dissipated, lowering the mechanical energy (spin plus orbit) of the system. This can be conveniently illustrated in the weak friction approximation.

In the simple case of a circular orbit with aligned spins, the following equations determine the evolution of the orbital angular velocity $n=\sqrt{GM/{a}^{3}}$ and the two spins:

where, again, $\mu ={M}_{\text{s}}{M}_{\text{p}}/M$ is the reduced mass (slightly less than the planet’s mass). The smallness of the moment of inertia of the planet (${I}_{\text{p}}\ll {I}_{\text{s}}$ and ${I}_{\text{p}}\ll \mu {a}^{2}$) means that the most rapid tidal process is usually the synchronization (and alignment) of the planetary spin with the orbit. This tidal locking $({\Omega}_{\text{p}}=n)$ can have important implications for the atmospheric dynamics and habitability of terrestrial exoplanets.

Once the planetary spin is synchronous, the stellar asynchronism evolves according to

A feature of hot Jupiter systems is that $\mu {a}^{2}$ and ${I}_{\text{s}}$ can be comparable, leading to an interesting coupled evolution in which both the stellar spin and the orbital radius change significantly. If, for example, the orbital angular momentum of a Jupiter-mass planet in a three-day orbit around a solar-mass star were transferred to the stellar spin, the star would rotate with a period of about three days. For a system consisting of Jupiter and the Sun, the bracket $\left(3-\frac{\mu {a}^{2}}{{I}_{\text{s}}}\right)$ is positive for orbits smaller than about 0.07 AU, or shorter than about seven days, and in such cases the angular velocities of the spin and orbit will diverge from each other (despite the frictional nature of the tidal interaction). This corresponds to the condition $n>{\Omega}_{\text{c}}$, that is, to being on the right-hand side of Figure 2.

If a small orbital eccentricity is allowed for, this evolves according to

The eccentricity is damped, leading to circularization of the orbit, provided that neither body spins significantly faster than the orbit. (The precise critical ratio of $18/11$ suggested by this equation is dependent on the assumptions underlying the weak friction approximation (Darwin, 1880).) The ratio of the rates of circularization due to dissipation in the planet and the star is proportional to

and tends to be dominated by the planet.

If a small misalignment angle ${i}_{\text{s}}$ between the stellar spin and the orbit is also allowed for, this evolves according to

For typical short-period exoplanetary systems, the large bracket is positive and tidal dissipation causes alignment of the spin and orbit.

#### Tidal Encounters

The sensitivity of tides to the orbital separation creates a distinction between situations of small and large orbital eccentricity. In the former case, the tidal interaction involves a small number of Fourier components that vary sinusoidally with time. In the latter case, the interaction is concentrated at the periapsis (where the orbital separation takes its minimum value $r=q$) and takes the form of a succession of impulses or *tidal encounters*. For exoplanets, both situations are relevant; a planet set on a highly elliptical orbit by interactions with companions can have its orbit circularized and shrunk through a sequence of dissipative encounters with the star.

It is relatively easy to calculate the effect of a tidal encounter in the weak friction approximation, for illustrative purposes. Assuming that the planetary spin is aligned with the orbit, the angular momentum ${(\Delta L)}_{\text{p}}$ transferred from spin to orbit due to dissipation in the planet is given by

where, in the limit of a highly eccentric orbit, $L=\mu \sqrt{2GMq}$ is the orbital angular momentum and ${\Omega}_{q}=\sqrt{2GM/{q}^{3}}$ is the orbital angular velocity at the periapsis (where it is maximal, and where the damping coefficient ${\gamma}_{\text{p}}$ is to be evaluated). A similar expression applies for the star, with the subscript “p” replaced by “s.” Since the moment of inertia of the planet is relatively small (${I}_{\text{p}}\ll {I}_{\text{s}}$ and ${I}_{\text{p}}\ll \mu {a}^{2}$), the planetary spin can be assumed to adjust after relatively few encounters to the *pseudosynchronous* value $\frac{33}{40}{\Omega}_{q}$ (Hut, 1981), which represents a suitably weighted average of the orbital angular velocity close to the periapsis, with little change in the orbit. However, since ${I}_{\text{s}}$ and $\mu {a}^{2}$ can be comparable for a hot Jupiter system, adjustment of the stellar spin is strongly coupled to evolution of the orbit. The energy lost from the orbit due to dissipation in the planet causes a change in the orbital eccentricity of

which is negative in the pseudosynchronous state. The corresponding quantity ${(\Delta e)}_{\text{s}}$ due to the star is also negative in the typical situation that ${\Omega}_{\text{s}}<{\Omega}_{q}$. After many such encounters, the orbit will be circularized.

When dynamical tides are considered, each tidal encounter leads to the impulsive excitation of oscillation modes in the star and planet, with a corresponding transfer of energy and angular momentum from the orbit to the modes (Press & Teukolsky, 1977). Damping of these modes, by linear or nonlinear processes, allows their angular momentum to be deposited, affecting the spin of the body, while energy is dissipated. If the modes are damped within a single orbit, then each successive encounter contributes to the tidal evolution of the system in a qualitatively similar way to that found in the weak friction approximation, albeit with differences of detail. If they are not so efficiently damped, chaotic dynamics can ensue (Vick & Lai, 2018; Wu, 2018, and references therein).

If the periapsis $q$ is too small, the planet will be disrupted as a result of the encounter. For a homogeneous fluid planet, disruption occurs if $q<1.69\phantom{\rule{0.2em}{0ex}}{({M}_{\text{s}}/{M}_{\text{p}})}^{1/3}{R}_{\text{p}}$, which is smaller than the Roche limit by a factor of 0.69 (Sridhar & Tremaine, 1992). Numerical simulations of tidal disruptions have been carried out with more realistic models of planets (Guillochon, Ramirez-Ruiz, & Lin, 2011).

#### Equilibrium and Dynamical Tides

The dominant response of a star or planet to tidal forcing is usually a spheroidal bulge in which the pressure and density of the body adjust hydrostatically to the modified gravitational potential. Away from a tidal equilibrium, the bulge is time-dependent and a smooth velocity field is required to move it at the appropriate rate around the body. This *equilibrium tide* (or *non-wavelike tide*) is not an exact solution of the hydrodynamic equations because the inertial terms in the equation of motion are neglected in estimating it. The *dynamical tide* (or *wavelike tide*) is the residual response, which typically takes the form of internal waves.

Internal waves are those restored by buoyancy and Coriolis forces in stratified and rotating fluids. Also known as internal gravity waves, g modes, inertial waves, inertia-gravity waves, and so on, they have been studied extensively in the Earth’s atmosphere and oceans. Their properties are in some ways opposite to those of the more familiar sound waves. First, the frequency of an internal wave is independent of the wavelength and depends only on the direction of propagation; it cannot exceed the buoyancy frequency or twice the spin frequency of the fluid (whichever is greater). The buoyancy frequency $N$ is defined by ${N}^{2}=\frac{g}{\rho}\left({\frac{d\rho}{dr}|}_{\text{ad}}-\frac{d\rho}{dr}\right)$, where $g$ is gravity, $\rho $ is density, $r$ is radius, and “ad” refers to the adiabatic gradient that would occur in the case of uniform composition and entropy per unit mass. Second, the motion in an internal wave is approximately incompressible and therefore transverse to the direction in which the phase of the wave varies, which is in turn perpendicular to the direction of propagation.

In most cases of interest for exoplanetary systems, surface-gravity waves and sound waves (also known as f modes and p modes) in stars and planets have frequencies that are too high (with wave periods of less than an hour) to be excited significantly by tidal forcing (with forcing periods of hours or days). Internal waves occupy the low-frequency end of the spectrum of oscillation modes, and are more naturally excited by tidal forcing.

#### Dissipation of the Equilibrium Tide

Any mechanism that provides a frictional or viscous drag on the equilibrium tide will lead to dissipation and a tidal torque. In stars, where the viscosity of the fluid is usually negligible, the main candidate is an effective “eddy viscosity” resulting from turbulent motion. This could be either the convective motion in regions such as the outer part of the Sun, where the star’s luminosity is being carried to the surface predominantly by convection (e.g., Zahn, 1989), or it could be turbulence arising from an instability of the tidal flow itself, such as the elliptical instability (Kerswell, 2002). In planets such as hot Jupiters, convection is less powerful than in stars because of the weaker sources of heat, but instability of the tidal flow is more likely because of the larger tidal amplitudes (e.g., Barker, 2019).

Solid regions of planets are often modeled as *viscoelastic* materials, which behave like an elastic solid on short timescales but can flow like a viscous fluid on long timescales. The commonly adopted *Maxwell model* is characterized by a viscosity $\eta $ and a relaxation time $\tau $, such that the elastic modulus on short timescales is $\eta /\tau $. In response to periodic strain with angular frequency $\omega $, the effective viscosity of the material is $\frac{\eta}{1+{(\omega \tau )}^{2}}$, which is significantly less than $\eta $ in the “elastic” regime in which the oscillation period is short compared with the relaxation time. Viscoelastic dissipation of the equilibrium tide is thought to be important for rocky bodies in the solar system and is likely to be so for terrestrial exoplanets (e.g., Correia, Boué, Laskar, & Rodríguez, 2014), although there is considerable uncertainty regarding the viscoelastic parameters.

Interestingly, fluid turbulence also has a viscoelastic character, with the relaxation time being related to the turnover time of the turbulent eddies (Ogilvie, 2019; Ogilvie & Lesur, 2012). The effectiveness of convection in dissipating the equilibrium tide in stars is limited because it is often in the “elastic” regime in which the tidal period is short compared with the turnover time (Goodman & Oh, 1997). The same is true of viscoelastic dissipation in solid regions of planets. This ordering of timescales means that the weak friction approximation is not applicable in detail to either situation.

#### Dissipation of the Dynamical Tide

Different mechanisms apply to the dynamical tide, which usually takes the form of internal waves. If these waves develop a sufficiently short wavelength, then linear dissipative mechanisms, in particular thermal diffusion due to radiative transport, can be relevant. For example, internal gravity waves acquire a short radial wavelength if they propagate into strongly stably stratified regions of a star or planet in which the buoyancy frequency is much greater than the wave frequency. In Zahn’s theory of the dynamical tide in stars more massive than the Sun (e.g., Zahn, 1977), which have convective cores and radiative envelopes, internal gravity waves are excited by tidal forcing near the base of the envelope and propagate towards the stellar surface, where radiative diffusion can be effective. Inertial waves in convective zones can also reach short lengthscales through geometrical focusing as a result of multiple reflections (e.g., Ogilvie & Lin, 2004). In these cases the efficiency of tidal dissipation can be strongly dependent on the tidal frequency, as this determines the detailed behavior of the waves.

Alternatively, if the propagation of internal waves causes them to exceed a critical amplitude, they become unstable and break, dissipating through transmission to waves of smaller scale. This mechanism could apply to stars more massive than the Sun, or to exoplanetary atmospheres. In solar-type stars, internal waves forced by a close planetary companion propagate towards the stellar center, where they break if the planet is sufficiently massive and the star sufficiently evolved (Barker & Ogilvie, 2010). This process could absorb the orbital angular momentum of the planet within a few million years, leading to the consumption of massive hot Jupiters.

For a more detailed and quantitative discussion of mechanisms of tidal dissipation in fluid bodies, the reader is referred to the review articles by Ogilvie (2014) and Mathis (2018).

#### Thermal Tides and Planetary Atmospheres

In addition to gravitational tides, planets experience thermal forcing from their host stars, as some of the radiation from the star that is received by the planet is absorbed and heats the atmosphere. Like gravitational tidal forcing, this effect is strongly dependent on the orbital separation, and results in a periodic disturbance if the planet is not tidally locked.

Stellar heating of the atmosphere creates a thermal bulge, on which the star’s gravity exerts a torque. On the Earth, the thermal bulge can be detected through the periodic variation of the atmospheric pressure at ground level (Haurwitz, 1964), which reflects the changing column density of the overlying atmosphere. The relevant Fourier component of this variation (known as the semidiurnal solar atmospheric tide) has a period of 12 hours and an amplitude exceeding one millibar at the equator, where pressure maxima occur a little before 10 o’clock (am and pm). The temperature variation is out of phase, peaking a little before 4 o’clock, in accord with everyday experience. The fact that a pressure maximum leads the position of the Sun by about two hours means that the solar torque on the thermal bulge accelerates the Earth’s rotation (Thomson, 1882).

Similar effects can be expected in other planets, especially those with a solid surface that can rigidly support the pressure fluctuations of the overlying atmosphere (although thermal tides in hot Jupiters have also been discussed (Arras & Socrates, 2010)). A competition between thermal and gravitational tides can result in a stable equilibrium in which the spin of the planet is significantly asynchronous. This process is thought to explain the rotation of Venus, which is closer to the Sun and has a thicker atmosphere than the Earth (Gold & Soter, 1969; Ingersoll & Dobrovolskis, 1978). It has been proposed that many terrestrial exoplanets in the habitable zone may have asynchronous rotation because of this process (Leconte, Wu, Menou, & Murray, 2015), which may be beneficial for their habitability. Tidal locking may be unfavorable for habitability because of the extreme temperature contrasts between the permanent day and night sides of the planet, which can lead to atmospheric collapse (Kasting, Whitmire, & Reynolds, 1993). General circulation models (GCMs) are now used to compute atmospheric dynamics and thermal tides in exoplanets (Pierrehumbert & Hammond, 2019).

### Application to Observed Exoplanetary Systems

At the time of writing (August 2019), more than 400 exoplanetary systems are known with an orbital semimajor axis less than 10 times the stellar radius, and more than 80 of these have $a/{R}_{\text{s}}<5$. The majority of these short-period exoplanets are transiting systems, which allow measurements of a number of stellar and planetary properties. Little can be said so far about the spins of exoplanets, although the question of whether they are tidally locked may be answered by ongoing developments in atmospheric studies. The main observational data relevant to tidal interactions are the orbital size and eccentricity, the stellar spin and obliquity (spin–orbit misalignment), and possible changes in orbital period. The distributions of these properties will be discussed before selected objects are examined in detail.

### Orbital Circularization

There is a clear trend for exoplanets with the shortest periods to have orbits of lower eccentricity. This can be seen in Figure 3, based on a sample of transiting giant planets studied by Bonomo et al. (2017). Orbital circularization could be explained, in principle, by tidal dissipation in either the planet or the star. O’Connor and Hansen (2018) have obtained estimates of the tidal dissipation constant (related to the time lag of the weak friction approximation) in hot Jupiters by modeling this data.

#### The Roche Limit

Figure 4 compares the orbital semimajor axes of observed exoplanets with the Roche limit, in cases where this can be determined to reasonable accuracy. As described in the section “Tidal Deformation and Disruption,” the Roche limit for a fluid planet is expected to lie between $2.03\phantom{\rule{0.2em}{0ex}}{({M}_{\text{s}}/{M}_{\text{p}})}^{1/3}{R}_{\text{p}}$ and $2.46\phantom{\rule{0.2em}{0ex}}{({M}_{\text{s}}/{M}_{\text{p}})}^{1/3}{R}_{\text{p}}$, depending on how centrally condensed it is. It is clear that the observed distribution of $a$ is cut off at, or very close to, the Roche limit as expected theoretically. This suggests that many planets have been destroyed or have lost material as a result of strong tidal forces from their host stars.

There is some evidence for an edge in the distribution close to *twice* the Roche limit, which could be explained if these short-period planets initially had highly elliptical orbits that were circularized while approximately conserving the orbital angular momentum, because the periapsis is doubled during this process (Ford & Rasio, 2006).

#### Orbital Migration and Stellar Spin-Up

If a planet’s orbit decays as a result of tidal dissipation in the host star, its orbital period should gradually decrease. This can be detected through accurate measurements of transit-timing variations if the effect is strong enough and the observational baseline long enough. The best case to date is WASP-12 b, discussed in the section “Selected Case Studies,” but other candidates exist (Bouma et al., 2019; Maciejewski et al., 2018) and the observational constraints are likely to improve in the coming years.

The hosts of several hot Jupiters have been found to be rotating significantly faster than expected for single stars of their mass and age (Brown et al., 2011; Husnoo et al., 2012; Kovács et al., 2014), suggesting that they have gained angular momentum from their planetary companions via tidal torques. This implies in turn that the orbits of these planets have decayed. Penev, Bouma, Winn, and Hartman (2018) have modeled the spin evolution of the hosts of hot Jupiters, finding evidence of a strong dependence of the stellar modified quality factor ${Q}^{\prime}$ on the tidal forcing frequency. Collier-Cameron and Jardine (2018) have modeled the orbital decay of hot Jupiters, finding evidence for enhanced dissipation in situations where the star spins sufficiently fast that the planet can excite inertial waves in it, but pointing also to important selection effects.

#### Stellar Obliquity

The misalignment of the stellar spin and the orbit of a transiting exoplanet, projected onto the plane of the sky, has been measured in a number of systems using the Rossiter–McLaughlin effect (Triaud, 2018). Observations show that significant misalignments are uncommon for cooler, less massive stars but common for hotter, more massive stars, with the transition occurring roughly where the star changes from having a radiative core and convective envelope (like the Sun) to having a convective core and a radiative envelope (Albrecht et al., 2012). The differing efficiencies of tidal dissipation in the two types of star may contribute to this dichotomy. Theories of dynamical tides involving inertial waves can explain how a misalignment can be damped on a timescale shorter than that for the orbit to decay (Lai, 2012; Lin & Ogilvie, 2017).

#### Tidal Heating

A long-standing observational puzzle is that many hot Jupiters are found to have larger radii than expected for their mass and age, even if they are composed purely of hydrogen and helium, which suggests the existence of an internal source of heat. The radius anomaly is correlated with the equilibrium temperature of the planet as a result of stellar irradiation, and is largest for planets of about the mass of Jupiter (e.g., Thorngren & Fortney, 2018). Tidal dissipation could contribute to the heating required for planetary inflation (Bodenheimer, Lin, & Mardling, 2001), but needs to be sustained and connected with the irradiation of the planet (Jermyn, Tout, & Ogilvie, 2017; Socrates, 2013).

#### Selected Case Studies

Here are listed a number of systems of special interest, some of which are illustrated, to scale, in Figure 5. The subscripts $\odot $, $\text{J}$, and $\text{E}$ refer to the Sun, Jupiter, and the Earth, and $\text{AU}$ is the astronomical unit (the mean distance between the Earth and the Sun).

**WASP-19 b**. Currently one of the shortest-period giant planets known, this is a hot Jupiter $({M}_{\text{p}}\approx 1.1\phantom{\rule{0.2em}{0ex}}{M}_{\text{J}},{R}_{\text{p}}\approx 1.4\phantom{\rule{0.2em}{0ex}}{R}_{\text{J}})$ in a very short-period (19 hour, $a\approx 0.016\phantom{\rule{0.2em}{0ex}}\text{AU}\approx 3.4\phantom{\rule{0.2em}{0ex}}{R}_{\text{s}}$) orbit around a G8V star $({M}_{\text{s}}\approx 0.9\phantom{\rule{0.2em}{0ex}}{M}_{\odot},{R}_{\text{s}}\approx 1.0\phantom{\rule{0.2em}{0ex}}{R}_{\odot})$ (Hebb et al., 2010; Mancini et al., 2013; Tregloan-Reed, Southworth, & Tappert, 2013). Observations are consistent with the orbit being circular and in the star’s equatorial plane. The star is thought to have been spun up tidally by the planet (Brown et al., 2011), so this is a promising system in which to look for orbital decay, although this has not been detected (Petrucci et al., 2020).

**WASP-18 b**. This is a very massive hot Jupiter $({M}_{\text{p}}\approx 10\phantom{\rule{0.2em}{0ex}}{M}_{\text{J}},{R}_{\text{p}}\approx 1.2\phantom{\rule{0.2em}{0ex}}{R}_{\text{J}})$ in a very short-period (23 hour, $a\approx 0.02\phantom{\rule{0.2em}{0ex}}\text{AU}\approx 3.6\phantom{\rule{0.2em}{0ex}}{R}_{\text{s}}$), slightly eccentric $(e\approx 0.008)$ orbit around an F6V star $({M}_{\text{s}}\approx 1.3\phantom{\rule{0.2em}{0ex}}{M}_{\odot},{R}_{\text{s}}\approx 1.3\phantom{\rule{0.2em}{0ex}}{R}_{\odot})$ (Hellier et al., 2009; Triaud et al., 2010). Given the relatively large tidal amplitude in the star, dissipation in the star must be relatively inefficient to avoid observable orbital decay.

**WASP-12 b**. This is a hot Jupiter $({M}_{\text{p}}\approx 1.5\phantom{\rule{0.2em}{0ex}}{M}_{\text{J}},{R}_{\text{p}}\approx 1.9\phantom{\rule{0.2em}{0ex}}{R}_{\text{J}})$ in a short-period (26 hour, $a\approx 3.0\phantom{\rule{0.2em}{0ex}}{R}_{\text{s}}$) circular orbit around a late F-type star $({M}_{\text{s}}\approx 1.3\phantom{\rule{0.2em}{0ex}}{M}_{\odot},{R}_{\text{s}}\approx 1.5\phantom{\rule{0.2em}{0ex}}{R}_{\odot})$ (Hebb et al., 2009; Maciejewski et al., 2018; Weinberg, Sun, Arras, & Essick, 2017). Transit-timing variations indicate that the orbital period is decreasing at a rate of $\dot{P}=-(29\pm 2)\phantom{\rule{0.2em}{0ex}}\text{ms}\phantom{\rule{0.2em}{0ex}}{\text{yr}}^{-1}$, corresponding to a timescale of $-P/\dot{P}=3.25\phantom{\rule{0.2em}{0ex}}\text{Myr}$ (Yee et al., 2020). This observation provides the best evidence to date of the orbital decay, or inward migration, of an exoplanet due to tidal dissipation in the host star. The planet is also thought to be overflowing its Roche lobe, predominantly through the inner Lagrangian point ${\text{L}}_{1}$ (Figure 1), into a gas ring around the star (Lai, Helling, & van den Heuvel, 2010; Li, Miller, Lin, & Fortney, 2010); this loss contributes to outward migration of the planet, which is more than compensated for by tidal dissipation in the host star.

**KOI 1843.03**. Currently the shortest-period planet known around a main-sequence star, this is a terrestrial planet in an extremely short-period (4.2 hour) orbit around an M3V star $({M}_{\text{s}}\approx 0.4-0.5\phantom{\rule{0.2em}{0ex}}{M}_{\odot},{R}_{\text{s}}\approx 0.4-0.5\phantom{\rule{0.2em}{0ex}}{R}_{\odot})$ (Rappaport, Sanchis-Ojeda, Rogers, Levine, & Winn, 2013). The star is slowly rotating (34 days) and expected to be fully convective. The tidal period (more than two hours) is still not short enough to resonate with the quadrupolar surface gravity mode (f mode) of the star. The planet’s mass is estimated to lie in the range $0.32-1.06\phantom{\rule{0.2em}{0ex}}{M}_{\text{E}}$, and its mean density must exceed $7\phantom{\rule{0.2em}{0ex}}\text{g}\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{-1}$ (implying a composition of at least 70% iron) to avoid disruption. Extreme proximity to the star ($a/{R}_{\text{s}}$ is estimated to lie between 1.4 and 2.2) implies a molten surface. A very similar object, with an orbital period only four minutes longer, is **K2-137 b** (Smith et al., 2018). These are the most extreme examples currently known of *ultra-short period planets* (USPs) (Winn, Sanchis-Ojeda, & Rappaport, 2018), a population of planets with periods less than one day and radii less than twice that of the Earth. Their tidal deformation has been modeled by Price and Rogers (2020), who find that they should have aspect ratios of between 1.3 and 1.8.

**HD 80606 b**. A prime candidate for *high-eccentricity migration*, this is a massive planet $({M}_{\text{p}}\approx 4\phantom{\rule{0.2em}{0ex}}{M}_{\text{J}},{R}_{\text{p}}\approx 1.0\phantom{\rule{0.2em}{0ex}}{R}_{\text{J}})$ in a highly eccentric $(e\approx 0.93)$ orbit around a G5V star $({M}_{\text{s}}\approx 1.0\phantom{\rule{0.2em}{0ex}}{M}_{\odot},{R}_{\text{s}}\approx 1.0\phantom{\rule{0.2em}{0ex}}{R}_{\odot})$ (Hébrard et al., 2010; Naef et al., 2001). The remarkable 111-day orbit has a minimum separation of $0.030\phantom{\rule{0.2em}{0ex}}\text{AU}$ (about $6.6\phantom{\rule{0.2em}{0ex}}{R}_{\text{s}}$) and a maximum separation of $0.88\phantom{\rule{0.2em}{0ex}}\text{AU}$. The projected stellar spin–orbit misalignment is significant, at $42\pm 8\xb0$. By analyzing the *Spitzer* phase curve, de Wit et al. (2016) deduced a stellar rotation period of 93 hours (with a large error), significantly longer than the standard pseudosynchronous value. The star has a binary companion, HD 80607, with a projected separation of about 1000 AU. This system led Wu and Murray (2003) to propose “Kozai migration,” in which the planet gains a large orbital eccentricity from a binary companion on a highly inclined orbit; the orbit is then circularized progressively through a succession of tidal encounters with the star, eventually producing a hot Jupiter on a compact, circular orbit (see also Fabrycky & Tremaine, 2007; Naoz, Farr, Lithwick, Rasio, & Teyssandier, 2013). High-eccentricity migration can also occur if the planet gains eccentricity through dynamical or secular interactions with other planets (Dawson & Johnson, 2018).

**TRAPPIST-1**. This is a very compact system of at least seven Earth-sized planets around an M8V star (Gillon et al., 2017). The inner six planets are in a chain of orbital resonances. Combining the resonant dynamics with tidal dissipation in each planet, which tends to circularize the orbits, Papaloizou, Szuszkiewicz, and Terquem (2018) have placed constraints on the tidal quality factors of the planets. This is also a system in which planet–planet tidal interactions may be relevant (Hay & Matsuyama, 2019).

### Conclusions

The tidal interaction of two bodies on a close Keplerian orbit is one of the classical problems of planetary science and theoretical astrophysics. Developed originally for the Earth–Moon system and other solar-system bodies, the theory has found a new lease of life in application to exoplanets that interact with their host stars. While the celestial mechanics of tidally interacting systems is fairly well understood, much remains to be learned about the efficiency of tidal dissipation in stars and planets. Significant progress has already been made in identifying a number of relevant mechanisms, many of which involve complicated fluid dynamical processes, often in a nonlinear regime. More work, including advanced linear calculations and nonlinear numerical simulations, is required in order to enable a reliable evaluation of the tidal dissipation rate in realistic applications and so to predict the rates of tidal evolution. Further improvements in the understanding of the interior structure and properties of planets may also be required.

Observations of transiting exoplanets have supplied a wide range of valuable data that provide evidence of a number of tidal processes having occurred, including tidal disruption, orbital circularization, orbital decay, stellar spin-up, spin–orbit alignment, and tidal heating. As this remarkable dataset continues to expand through the work of new and existing facilities, the observational constraints will tighten and more will certainly be learned about tidal interactions in exoplanetary systems.

So far, attempts to model the observational data have, for good reasons, used simple empirical models of tidal dissipation to obtain useful constraints on its efficiency in stars and planets. It is to be hoped that, in the years to come, advances in theory, simulations, and observations will allow a convergence towards a quantitative understanding of tidal evolution with predictive power.

### Acknowledgements

This article was prepared with the use of NASA’s Astrophysics Data System and the Extrasolar Planets Encyclopaedia.

#### Further Reading

- Mathis, S. (2018). Tidal star-planet interactions: A stellar and planetary perspective. In H. Deeg & J. Belmonte (Eds.),
*Handbook of exoplanets*(pp. 1801–1831). Cham, Switzerland: Springer. - Murray, C. D., & Dermott, S. F. (1999).
*Solar system dynamics*. Cambridge, UK: Cambridge University Press. - Ogilvie, G. I. (2014). Tidal dissipation in stars and giant planets.
*Annual Review of Astronomy and Astrophysics**52*, 171–210. - Ogilvie, G. I. (2020). Internal waves and tides in stars and giant planets. In M. Le Bars & D. Lecoanet (Eds.),
*Fluid mechanics of planets and stars*(pp. 1–30). Cham, Switzerland: Springer. - Souchay, J., Mathis, S., & Tokieda, T. (Eds.) (2013).
*Tides in astronomy and astrophysics*. Heidelberg, Germany: Springer.

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