Wave Number of Maximal Growth in Viscous Magnetic Fluids of Arbitrary Depth

An analytical method within the frame of linear stability theory is presented for the normal field instability in magnetic fluids. It allows to calculate the maximal growth rate and the corresponding wave number for any combination of thickness and viscosity of the fluid. Applying this method to magnetic fluids of finite depth, these results are quantitatively compared to the wave number of the transient pattern observed experimentally after a jump--like increase of the field. The wave number grows linearly with increasing induction where the theoretical and the experimental data agree well. Thereby a long-standing controversy about the behaviour of the wave number above the critical magnetic field is tackled.Comment: 19 pages, 15 figures, RevTex; revised version with a new figure and references added. submitted to Phys Rev

Is tidal heating sufficient to explain bloated exoplanets? Consistent calculations accounting for finite initial eccentricity

In this paper, we present the consistent evolution of short-period exoplanets coupling the tidal and gravothermal evolution of the planet. Contrarily to previous similar studies, our calculations are based on the complete tidal evolution equations of the Hut model, valid at any order in eccentricity, obliquity and spin. We demonstrate, both analytically and numerically, that, except if the system was formed with a nearly circular orbit (e<0.2), solving consistently the complete tidal equations is mandatory to derive correct tidal evolution histories. We show that calculations based on tidal models truncated at second order in eccentricity, as done in all previous studies, lead to erroneous tidal evolutions. As a consequence, tidal energy dissipation rates are severely underestimated in all these calculations and the characteristic timescales for the various orbital parameters evolutions can be wrong by up to three orders in magnitude. Based on these complete, consistent calculations, we revisit the viability of the tidal heating hypothesis to explain the anomalously large radius of transiting giant planets. We show that, even though tidal dissipation does provide a substantial contribution to the planet's heat budget and can explain some of the moderately bloated hot-Jupiters, this mechanism can not explain alone the properties of the most inflated objects, including HD 209458b. Indeed, solving the complete tidal equations shows that enhanced tidal dissipation and thus orbit circularization occur too early during the planet's evolution to provide enough extra energy at the present epoch. In that case another mechanisms, such as stellar irradiation induced surface winds dissipating in the planet's tidal bulges, or inefficient convection in the planet's interior must be invoked, together with tidal dissipation, to provide all the pieces of the abnormally large exoplanet puzzle.Comment: 14 pages, 10 figures, Accepted for publication in Astronomy and Astrophysics

Parametric Forcing of Waves with Non-Monotonic Dispersion Relation: Domain Structures in Ferrofluids?

Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a non-monotonic dispersion relation. The effect of a parametric driving on such waves is studied within suitable coupled Ginzburg-Landau equations. Due to the non-monotonicity the neutral curve for the excitation of standing waves can have up to three minima. The stability of the waves with respect to long-wave perturbations is determined $via$ a phase-diffusion equation. It shows that the band of stable wave numbers can split up into two or three sub-bands. The resulting competition between the wave numbers corresponding to the respective sub-bands leads quite naturally to patterns consisting of multiple domains of standing waves which differ in their wave number. The coarsening dynamics of such domain structures is addressed.Comment: 23 pages, 6 postscript figures, composed using RevTeX. Submitted to PR

Tidal torques. A critical review of some techniques

We point out that the MacDonald formula for body-tide torques is valid only in the zeroth order of e/Q, while its time-average is valid in the first order. So the formula cannot be used for analysis in higher orders of e/Q. This necessitates corrections in the theory of tidal despinning and libration damping. We prove that when the inclination is low and phase lags are linear in frequency, the Kaula series is equivalent to a corrected version of the MacDonald method. The correction to MacDonald's approach would be to set the phase lag of the integral bulge proportional to the instantaneous frequency. The equivalence of descriptions gets violated by a nonlinear frequency-dependence of the lag. We explain that both the MacDonald- and Darwin-torque-based derivations of the popular formula for the tidal despinning rate are limited to low inclinations and to the phase lags being linear in frequency. The Darwin-torque-based derivation, though, is general enough to accommodate both a finite inclination and the actual rheology. Although rheologies with Q scaling as the frequency to a positive power make the torque diverge at a zero frequency, this reveals not the impossible nature of the rheology, but a flaw in mathematics, i.e., a common misassumption that damping merely provides lags to the terms of the Fourier series for the tidal potential. A hydrodynamical treatment (Darwin 1879) had demonstrated that the magnitudes of the terms, too, get changed. Reinstating of this detail tames the infinities and rehabilitates the "impossible" scaling law (which happens to be the actual law the terrestrial planets obey at low frequencies).Comment: arXiv admin note: sections 4 and 9 of this paper contain substantial text overlap with arXiv:0712.105

Tidal dissipation within hot Jupiters: a new appraisal

Eccentricity or obliquity tides have been proposed as the missing energy source that may explain the anomalously large radius of some transiting ``hot Jupiters''. To maintain a non-zero and large obliquity, it was argued that the planets can be locked in a Cassini state, i.e. a resonance between spin and orbital precessions. We compute the tidal heating within ``inflated'' close-in giant planets with a non-zero eccentricity or obliquity. We further inspect whether the spin of a ``hot Jupiter'' could have been trapped and maintained in a Cassini state during its early despinning and migration. We estimate the capture probability in a spin-orbit resonance between $\sim$ 0.5 AU (a distance where tidal effects become significant) and 0.05 AU for a wide range of secular orbital frequencies and amplitudes of gravitational perturbations. Numerical simulations of the spin evolution are performed to explore the influence of tidal despinning and migration processes on the resonance stability. We find that tidal heating within a non-synchronous giant planet is about twice larger than previous estimates based on the hypothesis of synchronization. Chances of capture in a spin-orbit resonance are very good around 0.5 AU but they decrease dramatically with the semi-major axis. Furthermore, even if captured, both tidal despinning and migration processes cause the tidal torque to become large enough that the obliquity ultimately leaves the resonance and switches to near $0^{\circ}$. Locking a ``hot Jupiter'' in an isolated spin-orbit resonance is unlikely at 0.05 AU but could be possible at larger distances. Another mechanism is then required to maintain a large obliquity and create internal heating through obliquity tidesComment: 4 pages & 2 Figure

The role of chaotic resonances in the solar system

Our understanding of the Solar System has been revolutionized over the past decade by the finding that the orbits of the planets are inherently chaotic. In extreme cases, chaotic motions can change the relative positions of the planets around stars, and even eject a planet from a system. Moreover, the spin axis of a planet-Earth's spin axis regulates our seasons-may evolve chaotically, with adverse effects on the climates of otherwise biologically interesting planets. Some of the recently discovered extrasolar planetary systems contain multiple planets, and it is likely that some of these are chaotic as well.Comment: 28 pages, 9 figure

A seven-planet resonant chain in TRAPPIST-1

The TRAPPIST-1 system is the first transiting planet system found orbiting an ultra-cool dwarf star1. At least seven planets similar to Earth in radius were previously found to transit this host star2. Subsequently, TRAPPIST-1 was observed as part of the K2 mission and, with these new data, we report the measurement of an 18.77 d orbital period for the outermost transiting planet, TRAPPIST-1h, which was unconstrained until now. This value matches our theoretical expectations based on Laplace relations3 and places TRAPPIST-1h as the seventh member of a complex chain, with three-body resonances linking every member. We find that TRAPPIST-1h has a radius of 0.727 R⊕ and an equilibrium temperature of 169 K. We have also measured the rotational period of the star at 3.3 d and detected a number of flares consistent with a low-activity, middle-aged, late M dwarf

The habitability of Proxima Centauri b I. Irradiation, rotation and volatile inventory from formation to the present

International audienceProxima b is a planet with a minimum mass of 1.3 MEarth orbiting within the habitable zone (HZ) of Proxima Centauri, a very low-mass, active star and the Sun's closest neighbor. Here we investigate a number of factors related to the potential habitability of Proxima b and its ability to maintain liquid water on its surface. We set the stage by estimating the current high-energy irradiance of the planet and show that the planet currently receives 30 times more EUV radiation than Earth and 250 times more X-rays. We compute the time evolution of the star's spectrum, which is essential for modeling the flux received over Proxima b's lifetime. We also show that Proxima b's obliquity is likely null and its spin is either synchronous or in a 3:2 spin-orbit resonance, depending on the planet's eccentricity and level of triaxiality. Next we consider the evolution of Proxima b's water inventory. We use our spectral energy distribution to compute the hydrogen loss from the planet with an improved energy-limited escape formalism. Despite the high level of stellar activity we find that Proxima b is likely to have lost less than an Earth ocean's worth of hydrogen before it reached the HZ 100-200 Myr after its formation. The largest uncertainty in our work is the initial water budget, which is not constrained by planet formation models. We conclude that Proxima b is a viable candidate habitable planet

Quantifying the Influence of Jupiter on the Earth's Orbital Cycles

A wealth of Earth-sized exoplanets will be discovered in the coming years, proving a large pool of candidates from which the targets for the search for life beyond the Solar system will be chosen. The target selection process will require the leveraging of all available information in order to maximise the robustness of the target list and make the most productive use of follow-up resources. Here, we present the results of a suite of $n$-body simulations that demonstrate the degree to which the orbital architecture of the Solar system impacts the variability of Earth's orbital elements. By varying the orbit of Jupiter and keeping the initial orbits of the other planets constant, we demonstrate how subtle changes in Solar system architecture could alter the Earth's orbital evolution -- a key factor in the Milankovitch cycles that alter the amount and distribution of solar insolation, thereby driving periodic climate change on our planet. The amplitudes and frequencies of Earth's modern orbital cycles fall in the middle of the range seen in our runs for all parameters considered -- neither unusually fast nor slow, nor large nor small. This finding runs counter to the `Rare Earth' hypothesis, which suggests that conditions on Earth are so unusual that life elsewhere is essentially impossible. Our results highlight how dynamical simulations of newly discovered exoplanetary systems could be used as an additional means to assess the potential targets of biosignature searches, and thereby help focus the search for life to the most promising targets.Comment: 19 pages; 11 figures; accepted for publication in the Astronomical Journal Version 2 - incorporates typo corrections and minor changes noted at the proofing stage, after acceptanc

Linear growth of instabilities on a liquid metal under normal electric field

It is well known that an electric field that is applied normally to the free surface of a conducting fluid has a destabilizing effect. Here we study the linear growth of electro-capillary instabilities in the very general case where the viscosity of the fluid and its thickness are of any value. We derive the asymptotic behaviour in various regimes and give analytical dispersion equations in the case of thin or thick, inviscid or viscous films and compare with previous results. Then we present a diagram of the corresponding simplifications of the dispersion relation and as we derive them directly from the general equation, we are able to derive their conditions of validity explicitly. We also show the similarities and the differences with the case of a liquid falling from a solid flat plane (Rayleigh-Taylor instabilities) without electric field and with ferrofluid instabilities.On sait qu'un champ électrique appliqué normalement à la surface libre d'un fluide conducteur a un effet déstabilisant. En restant dans le cadre d'une théorie linéaire, nous étudions ici le développement des instabilités électrocapillaires, dans le cas très général où la viscosité du fluide et son épaisseur sont quelconques. Nous déduisons différents comportements dans divers régimes et donnons des équations de dispersion analytiques dans le cas des couches épaisses et minces, inertielles et visqueuses. Nous présentons alors un diagramme des différentes simplifications possibles et comme nous déduisons ces simplifications directement de l'équation générale il est plus facile de préciser les limites de validité des hypothèses. Nous montrons aussi les similarités et les différences avec le cas d'un liquide tombant d'un support solide plan (instabilités de Rayleigh-Taylor) en absence de champ électrique ainsi qu'avec les instabilités dans les ferrofluides