Figure 1- - uploaded by Rachele Meriggiola
Content may be subject to copyright.
Source publication
Similar publications
We consider a billiard system consisting on an infinite rod (a straight line) and a ball (a massless point) on the plane. The rod uniformly rotates around one of its points. We describe the dynamics of this system.
Citations
... In these theories, the excess of rotation 6ne 2 does not depend on the rheology of the body. However, this prediction is not confirmed for Titan, where the excess provided by the theory is ∼38 • per year, and the Cassini mission, using radar measurement, has not shown discrepancy from synchronous motion larger than ∼0.02 • per year (Meriggiola 2012;Meriggiola et al. 2016). Standard theories circumvent this difficulty by assuming that the satellite has an ad hoc triaxiality, which is permanent and not affected by the tidal forces acting on the body. ...
... in function of γ s , for two dynamical models: (i) tidal forces, gravitational coupling and linear friction (solid black lines); and (ii) tidal forces, gravitational coupling, linear friction and the atmospheric influence (dashed red lines). The horizontal lines show the intervals corresponding to 1σ uncertainties of the observed values: the blue dashed lines, labelled M, correspond to Meriggiola (2012) and Meriggiola et al. (2016) and green dashed lines, labelled S, correspond to the Stiles et al. (2008Stiles et al. ( , 2010. The core relaxation factor γ c increases from γ c = 10 −9 s −1 (top panels) to 10 −6 s −1 (bottom panels) and the ocean thickness h increases from 15 km (left panels) to 250 km (right panels). ...
... We also observe that when γ s < 10 −8 s −1 , independently of the values of γ c and h, the amplitude of oscillation of the shell tends to zero when the relaxation factor γ s decreases. Particularly, if γ s < 10 −9 s −1 , the amplitude of the oscillation of the excess of rotation reproduces the dispersion of the Ω s value of ±0.02 • /year around the synchronous value, observed as reported by Meriggiola (2012) and Meriggiola et al. (2016). The results are not consistent with the previous drift reported by Stiles et al. (2008Stiles et al. ( , 2010. ...
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 spin rates increase if the orbital eccentricity increases. 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,... . We apply the theory to Titan. The main results are: i) the rotational constraint does not allow us 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, we limit the possible value of the shell relaxation factor, when a subsurface ocean is assumed, to 10^-9 Hz, which correspond to a shell's viscosity 10^18 Pa s, depending on the ocean's thickness and viscosity values. In the case in which the 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.
We present a tidal model for treating the rotational evolution in the general three-body problem with arbitrary viscosities, in which all the masses are considered to be extended and all the tidal interactions between pairs are taken into account. Based on the creep tide theory, we present a set of differential equations that describes the rotational evolution of each body, in a formalism that is easily extensible to the N tidally interacting body problem. We apply our model to the case of a circumbinary planet and use a Kepler-38 like binary system as a working example. We find that, in this low planetary eccentricity case, the most likely final stationary rotation state is the 1:1 spin–orbit resonance, considering an arbitrary planetary viscosity inside the estimated range for the Solar System planets. The timescales for reaching the equilibrium state are expected to be approximately millions of years for stiff bodies but can be longer than the age of the system for planets with a large gaseous component. We derive analytical expressions for the mean rotational stationary state, based on high-order power series of the ratio of the semimajor axes a 1 ∕ a 2 and low-order expansions of the eccentricities. These are found to very accurately reproduce the mean behaviour of the low-eccentric numerical integrations for arbitrary planetary relaxation factors, and up to a 1 ∕ a 2 ~ 0.4. Our analytical model is used to predict the stationary rotation of the Kepler circumbinary planets and we find that most of them are probably rotating in a subsynchronous state, although the synchrony shift is much less important than our previous estimations. We present a comparison of our results with those obtained with the Constant Time Lag and find that, as opposed to the assumptions in our previous works, the cross torques have a non-negligible net secular contribution, and must be taken into account when computing the tides over each body in an N -extended-body system from an arbitrary reference frame. These torques are naturally taken into account in the creep theory. In addition to this, the latter formalism considers more realistic rheology that proved to reduce to the Constant Time Lag model in the gaseous limit and also allows several additional relevant physical phenomena to be studied.
Les librations en longitude d'un satellite naturel en rotation synchrone peuvent être définies comme étant des oscillations autour de sa rotation uniforme. Leurs amplitudes dépendent de la structure interne du stallite et des librations peuvent donc être utilisées pour sonder l'intérieur de ces corps célestes. Récemment, l'analyse des observations de la sonde Cassini a permis de suggérer la présence d'un océan interne sur Titan, la lune la plus massive de Saturne. La mesure de ses librations serait utile pour contraindre les caractéristiques de cet océan. Durant cette thèse, nous avons modélisé les librations en longitude d'un satellite de glace à trois couches élastiques évoluant sur une orbite non-Képlerienne et forcé par un couple atmosphérique. L'objectif était de déterminer l'influence de la structure interne et des déformations élastiques de marée sur l'amplitude des librations. Nous avons déterminé que les déformations élastiques des couches de Titan réduisent la libration à la fréquence orbitale au même ordre de grandeur que celle d'un modèle solide. La signature de l'océan est alors difficile à détecter et nécessite une précision de quelques dizaines de mètres sur les observations du mouvement de la surface. Nous avons également identifié deux librations de longues périodes dont l'amplitude est sensible à la présence de l'océan. Le modèle de libration développé ici est général et a été ensuite appliqué avec succès à Mimas. Les récentes observations des librations en longitude de Mimas ne peuvent être expliquées par un modèle solide du satellite. Nous avons montré que les librations d'un modèle de Mimas contenant un océan interne vérifient les amplitudes observées, ce qui permet de suggérer un nouveau scénario pour la structure interne de Mimas et d'ouvrir le débat sur l'histoire de sa formation.
New theory of the dynamical tides of celestial bodies founded on a Newtonian
creep instead of the classical delaying approach of the standard viscoelastic
theories. The results of the theory derive mainly from the solution of a
non-homogeneous ordinary differential equation. Lags appear in the solution,
but as quantities determined from the solution of the equation and are not
arbitrary external quantities plugged on an elastic model. The resulting lag of
each tide component is an increasing function of its frequency (as in Darwin's
theory), and lags are not small quantities. The amplitudes of the tide
components depend on the viscosity of the body and on their frequencies; they
are not constants. The resulting stationary rotations (often called
pseudo-synchronous) have an excess velocity roughly proportional to
6ne^2/(X^2+1/X^2) (X is the mean-motion in units of one relaxation factor
inversely proportional to the viscosity) instead of the exact 6ne^2 of standard
theories. The dissipation in the pseudo-synchronous solution is inversely
proportional to (X+1/X); thus, in the inviscid limit it is roughly proportional
to the frequency (as in standard theories), but that behavior is inverted when
the viscosity is high and the relaxation factor much smaller than the tide
frequency. For free rotating bodies, the dissipation is given by the same law,
but now X is the frequency of the semidiurnal tide in units of the relaxation
factor. This approach fails, however, to reproduce the actual tidal lags on
Earth and on natural satellites. To reconcile theory and observations, in this
case, we had to assume the coexistence of an elastic tide superposed to the
creeping tide. The theory is applied to several Solar System and extrasolar
bodies and values of the relaxation factor \gamma\ are derived for these bodies
on the basis of currently available data.
