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Register a new userTidal dissipation in a homogeneous spherical body. I. Methods
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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).
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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 planets 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.
We study the orbital evolution of a three planet system with masses in the super-Earth regime resulting from the action of tides on the planets induced by the central star which cause orbital circularization. We consider systems either in or near to a three body commensurability for which adjacent pairs of planets are in a first order commensurability. We develop a simple analytic solution, derived from a time averaged set of equations, that describes the expansion of the system away from strict commensurability as a function of time, once a state where relevant resonant angles undergo small amplitude librations has been attained. We perform numerical simulations that show the attainment of such resonant states focusing on the Kepler 60 system. The results of the simulations confirm many of the scalings predicted by the appropriate analytic solution. We go on to indicate how the results can be applied to put constraints on the amount of tidal dissipation that has occurred in the system. For example, if the system has been in a librating state since its formation, we find that its present period ratios imply an upper limit on the time average of 1/Q, with Q being the tidal dissipation parameter. On the other hand if a librating state has not been attained, a lower upper bound applies.
WASP-12 is a hot Jupiter system with an orbital period of $P= 1.1textrm{ day}$, 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 $P/|dot{P}| = 3.2textrm{ Myr}$. 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 $M_ast simeq 1.3 M_odot$ main sequence star or a $M_ast simeq 1.2 M_odot$ subgiant. We find that if the star is on the main sequence, the tidal dissipation is too inefficient to explain the observed $dot{P}$. However, if it is a subgiant, the tidal dissipation is significantly enhanced due to nonlinear wave breaking of the dynamical tide near the stars center. The subgiant models have a tidal quality factor $Q_astsimeq 2times10^5$ and an orbital decay rate that agrees well with the observed $dot{P}$. It would also explain why the planet survived for $simeq 3textrm{ Gyr}$ 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 $sim 0.1%$ of the planets life, the probability of such a detection is a few percent given the observed sample of $simeq 30$ hot Jupiters in $P<3textrm{ day}$ orbits around $M_ast>1.2 M_odot$ hosts.
Earth-like planets have viscoelastic mantles, whereas giant planets may have viscoelastic cores. The tidal dissipation of such solid regions, gravitationally perturbed by a companion body, highly depends on their rheology and on the tidal frequency. Therefore, modelling tidal interactions presents a high interest to provide constraints on planets properties and to understand their history and their evolution, in our Solar System or in exoplanetary systems. We examine the equilibrium tide in the anelastic parts of a planet whatever the rheology, taking into account the presence of a fluid envelope of constant density. We show how to obtain the different Love numbers that describe its tidal deformation. Thus, we discuss how the tidal dissipation in solid parts depends on the planets internal structure and rheology. Finally, we show how the results may be implemented to describe the dynamical evolution of planetary systems. The first manifestation of the tide is to distort the shape of the planet adiabatically along the line of centers. Then, the response potential of the body to the tidal potential defines the complex Love numbers whose real part corresponds to the purely adiabatic elastic deformation, while its imaginary part accounts for dissipation. This dissipation is responsible for the imaginary part of the disturbing function, which is implemented in the dynamical evolution equations, from which we derive the characteristic evolution times. The rate at which the system evolves depends on the physical properties of tidal dissipation, and specifically on how the shear modulus varies with tidal frequency, on the radius and also the rheological properties of the solid core. The quantification of the tidal dissipation in solid cores of giant planets reveals a possible high dissipation which may compete with dissipation in fluid layers.
Internal dissipation in a tidally perturbed librating body differs from the tidal dissipation in a steadily spinning rotator. First, libration changes the spectral distribution of tidal damping across the tidal modes, as compared to the case of steady spin. This changes both the tidal heating rate and the tidal torque. Second, while a non-librating rotator experiences alternating deformation only due to the potential force exerted on it by the perturber, a librating body is also subject to a toroidal force proportional to the angular acceleration. Third, while the centrifugal force in a steadily spinning body renders only a permanent deformation, in a librating body this force contains two alternating components $-$ one radial, another a degree-2 potential force. Both contribute to heating, as well as to the tidal torque and potential. We build a formalism to describe dissipation in a homogeneous terrestrial body performing small-amplitude libration in longitude. This formalism incorporates a linear rheological law defining the response of the material to forcing. While the formalism can work with an arbitrary linear rheology, we consider a simple example of a Maxwell material. We show that, independent of rheology, the forced libration in longitude can provide a considerable and even leading input in the tidal heating. Based on the observed parameters, this input amounts to 52% in Phobos, 33% in Mimas, 23% in Enceladus, and 96% in Epimetheus. This supports the hypothesis by Makarov & Efroimsky (2014) that the additional damping due to forced libration may have participated in the early heating up of some moons. As one possibility, a moon could have been chipped by collisions $-$ whereby it acquired a higher triaxiality and a higher forced-libration magnitude and, consequently, a higher heating rate. After the moon warms up, its triaxiality reduces, and so does the tidal heating.
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