Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach

This paper presents a new theory of the dynamical tides of celestial bodies. It is founded on a Newtonian creep instead of the classical delaying approach of the standard viscoelastic theories and 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 in an elastic model. The resulting lags of the tide components are increasing functions of their frequencies (as in Darwin’s theory), but 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 $6ne^2/(\chi ^2+\chi ^{-2})$ ( $\chi $ is the mean-motion in units of one critical frequency—the 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 $(\chi +\chi ^{-1})$ ; 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 tide frequency larger than the critical frequency. For free rotating bodies, the dissipation is given by the same law, but now $\chi $ is the frequency of the semi-diurnal tide in units of the critical frequency. This approach fails, however, to reproduce the actual tidal lags on Earth. In this case, to reconcile theory and observations, we need to assume the existence of an elastic tide superposed to the creeping tide. The theory is applied to several Solar System and extrasolar bodies and currently available data are used to estimate the relaxation factor $\gamma $ (i.e. the critical frequency) of these bodies.     

Tidal synchronization of close-in satellites and exoplanets. III. Tidal dissipation revisited and application to Enceladus

This paper deals with a new formulation of the creep tide theory (Ferraz-Mello in Celest Mech Dyn Astron 116:109, 2013—Paper I) and with the tidal dissipation predicted by the theory in the case of stiff bodies whose rotation is not synchronous but is oscillating around the synchronous state with a period equal to the orbital period. We show that the tidally forced libration influences the amount of energy dissipated in the body and the average perturbation of the orbital elements. This influence depends on the libration amplitude and is generally neglected in the study of planetary satellites. However, they may be responsible for a 27% increase in the dissipation of Enceladus. The relaxation factor necessary to explain the observed dissipation of Enceladus (\(\gamma =1.2{-}3.8\times 10^{-7}\ \mathrm{s}^{-1}\)) has the expected order of magnitude for planetary satellites and corresponds to the viscosity \(0.6{-}1.9 \times 10^{14}\) Pa s, which is in reasonable agreement with the value recently estimated by Efroimsky (Icarus 300:223, 2018) (\(0.24 \times 10^{14}\) Pa s) and with the value adopted by Roberts and Nimmo (Icarus 194:675, 2008) for the viscosity of the ice shell (\(10^{13}{-}10^{14}\) Pa s). For comparison purposes, the results are extended also to the case of Mimas and are consistent with the negligible dissipation and the absence of observed tectonic activity. The corrections of some mistakes and typos of paper II (Ferraz-Mello in Celest Mech Dyn Astron 122:359, 2015) are included at the end of the paper.     

Tidal evolution of close-in exoplanets in co-orbital configurations

In this paper, we study the behavior of a pair of co-orbital planets, both orbiting a central star on the same plane and undergoing tidal interactions. Our goal is to investigate final orbital configurations of the planets, initially involved in the 1/1 mean-motion resonance (MMR), after long-lasting tidal evolution. The study is done in the form of purely numerical simulations of the exact equations of motions accounting for gravitational and tidal forces. The results obtained show that, at least for equal mass planets, the combined effects of the resonant and tidal interactions provoke the orbital instability of the system, often resulting in collision between the planets. We first discuss the case of two hot-super-Earth planets, whose orbital dynamics can be easily understood in the frame of our semi-analytical model of the 1/1 MMR. Systems consisting of two hot-Saturn planets are also briefly discussed.     

Tidal evolution of the Galilean satellites: A linearized theory

The Laplace resonance among the Galilean satellites Io, Europa, and Ganymede is traditionally reduced to a pendulum-like dynamical problem by neglecting short-period variations of several orbital elements. However, some of these variations that can now be neglected may once have had longer periods, comparable to the “pendulum” period, if the system was formerly in deep resonance (pairs of periods even closer to the ratio 2:1 than they are now). In that case, the dynamical system cannot be reduced to fewer than nine dimensions. The nine-dimensional system is linearized here in order to study small variations about equilibrium. When tidal effects are included, the resulting evolution is substantially the same as was indicated by the pendulum approach, except that evolution out of deep resonance is found to be somewhat slower than suggested by extrapolation of the pendulum results. This slower rate helps support my hypothesis that the system may have evolved from deep resonance, although other factors still need to be considered to determine whether that hypothesis is quantitatively viable.     

Tidal decay and stable Roche-lobe overflow of short-period gaseous exoplanets

Many gaseous exoplanets in short-period orbits are on the verge or are in the process of Roche-lobe overflow (RLO). Moreover, orbital stability analysis shows tides can drive many hot Jupiters to spiral inevitably toward their host stars. Thus, the coupled processes of orbital evolution and RLO likely shape the observed distribution of close-in exoplanets and may even be responsible for producing some of the short-period rocky planets. However, the exact outcome for an overflowing planet depends on its internal response to mass loss, and the accompanying orbital evolution can act to enhance or inhibit RLO. In this study, we apply the fully-featured and robust Modules for Experiments in Stellar Astrophysics suite to model RLO of short-period gaseous planets. We show that, although the detailed evolution may depend on several properties of the planetary system, it is largely determined by the core mass of the overflowing gas giant. In particular, we find that the orbital expansion that accompanies RLO often stops and reverses at a specific maximum period that depends on the core mass. We suggest that RLO may often strand the remnant of a gas giant near this orbital period, which provides an observational prediction that can corroborate the hypothesis that short-period gas giants undergo RLO. We conduct a preliminary comparison of this prediction to the observed population of small, short-period planets and find some planets in orbits that may be consistent with this picture. To the extent that we can establish some short-period planets are indeed the remnants of gas giants, that population can elucidate the properties of gas giant cores, the properties of which remain largely unconstrained.     

Tidal friction in triple stars

Tidal friction in close binaries, with periods of a few days, is expected to circularize the orbit on a time-scale long compared with human observation but shorter than, or comparable to, the lifetimes of main-sequence stars. In a hierarchical triple star, however, the perturbing effect of the distant third star may decircularize the inner orbit significantly on a time-scale of the order of days (as in λ Tau) or centuries (as in β Per). If the inner pair is observed to be semidetached, however, it is plausible to assume that the eccentricity is small. This may be because tidal friction is operating on a comparably short time-scale, and so it is in principle amenable to observation. We attempt to determine a lower limit to the strength of tidal friction in λ Tau and β Per, on the basis of this consideration. Tidal friction will also lead to a secular transfer of angular momentum from the inner orbit to the outer orbit. Too rapid a transfer may lead to orbital shrinkage that is fast compared with the nuclear time-scales of the inner systems, and this can also be ruled out on observational grounds. Thus we may be able to set an upper as well as a lower limit to the strength of tidal friction, on the basis of observations. In a young hierarchical triple, provided that the orbits are fairly nearly orthogonal, tidal friction can serve to reduce the inner orbital period from months to days within a fairly short period of time, of order P ²_out/ P _in. This may be a significant mechanism for producing young short-period binaries.     

The equilibrium of rubble-pile satellites: The Darwin and Roche ellipsoids for gravitationally held granular aggregates

Many new small moons of the giant planets have been discovered recently. In parallel, satellites of several asteroids, e.g., Ida, have been found. Strikingly, a majority of these new-found planetary moons are estimated to have very low densities, which, along with their hypothesized accretionary origins, suggests a rubble internal structure. This, coupled to the fact that many asteroids are also thought to be particle aggregates held together principally by self-gravity, motivates the present investigation into the possible ellipsoidal shapes that a rubble-pile satellite may achieve as it orbits an aspherical primary. Conversely, knowledge of the shape will constrain the granular aggregate's orbit—the closer it gets to a primary, both primary's tidal effect and the satellite's spin are greater. We will assume that the primary body is sufficiently massive so as not to be influenced by the satellite. However, we will incorporate the primary's possible ellipsoidal shape, e.g., flattening at its poles in the case of a planet, and the proloidal shape of asteroids. In this, the present investigation is an extension of the first classical Darwin problem to granular aggregates. General equations defining an ellipsoidal rubble pile's equilibrium about an ellipsoidal primary are developed. They are then utilized to scrutinize the possible granular nature of small inner moons of the giant planets. It is found that most satellites satisfy constraints necessary to exist as equilibrated granular aggregates. Objects like Naiad, Metis and Adrastea appear to violate these limits, but in doing so, provide clues to their internal density and/or structure. We also recover the Roche limit for a granular satellite of a spherical primary, and employ it to study the martian satellites, Phobos and Deimos, as well as to make contact with earlier work of Davidsson [Davidsson, B., 2001. Icarus 149, 375–383]. The satellite's interior will be modeled as a rigid-plastic, cohesion-less material with a Drucker–Prager yield criterion. This rheology is a reasonable first model for rubble piles. We will employ an approximate volume-averaging procedure that is based on the classical method of moments, and is an extension of the virial method [Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium. Yale Univ. Press, New Haven] to granular solid bodies.     

Tidal dissipation within large icy satellites: Applications to Europa and Titan

This paper describes a new approach based on variational principles to calculate the radial distribution of tidal energy dissipation in any satellite. The advantage of the model with respect to classical solutions, is that it relates in a straightforward way the radial distribution of the time-averaged dissipation rate to its sensitivity to the corresponding distribution of viscoelastic parameters. This method is applied to Io-, Europa-, and Titan-like interiors, and it is tested against the results obtained by two classical methods by determining global dissipation as well as radial and lateral distributions within satellite interiors. By exploring systematically the different parameters defining the interior models, we demonstrate that the presence of a deep ocean below an outer ice layer strongly influences the tidal dissipation distribution in both the outer ice layer and in the innermost part of the satellite. On the one hand, the ocean by imposing a large radial displacement at the base of the outer ice I layer, controls the distribution of tidal strain rate within the outer layer, making the tidal strain rate field very weakly sensitive to the viscosity variations. Conversely, in the high-pressure ice layer below the ocean, both tidal strain rate and dissipation are very sensitive to any variation of the ice viscosity. On the other hand, for identical structures of the mantle and of the core, the presence of a subsurface ocean reduces the strength of dissipation in the silicate mantle. The existence of a liquid layer within Europa makes models of the silicate mantle less dissipative than the predictions for Io.     

The sun and exoplanets: The solitude of man

Solar pulsations with a period of P ₀ = 9600.606(12) were discovered in 1974. A more recent discovery is that planetary distances in the solar system are subject to spatial resonance with the parameter L ₀ ?? cP ₀ ?? 9600 ls and that the P ₀ pulsation itself has cosmological significance (coherent cosmic oscillation, or the pace of absolute time of the universe; c is the speed of light). As of June 2011, 552 extrasolar planets have been discovered. Statistical analysis shows that the distribution of the semimajor axes of alien planets does not have L ₀ resonance. Moreover, it appears to have no resonance at all. This frustrates the 20th-century hopes for the existence of extraterrestrial civilizations and possible contact with them. They are simply not there. This explanation of the Fermi paradox, or the Great Silence, appears to rest on the triumph of the anthropic principle, which has been successfully implemented by nature within our planetary system. This leads to a vision whereby the cosmos seems to be created specially for us. The scale L ₀ indicates that the sun is a special quantum object, where L ₀ is a wave function parameter that is not subject to the rational principles of the classical world, but rather follows a peculiar, quantum logic.     

The search for exoplanets and space interferometry

The closing years of the 20th century have allowed us, for the first time, to seriously discuss interferometric instruments deployed in space. With the express purpose of achieving unprecedented spatial resolution, these missions will lead to new astrophysics. Especially—and most challenging—we expect to be able to carry out the first studies of terrestrial exoplanets. The detection and study of the latter promises to usher in a new era in science and will affect a broad spectrum of science and technology. For the first set of interferometric missions—the precursor missions such as SMART and ST-3 and the astrometric SIM, the time line for implementation is such that it could be about 5-10 years until we receive the first results from them. In this review, we describe the impact of interferometry from space on the topic of terrestrial exoplanets. We also briefly review the state of the art of the study of exoplanets as well as discuss the potential impact of several different techniques for their study.     

The third-order theory of artificial satellites of the earth: A first exploration

We propose a mixed analytical and numerical method as a practical means of solving the perturbation equations based on Vinti's intermediate orbit. Hori's averaging method is used to define mean elements σ^* and short-period perturbations Δσ_s. Δσ_s accurate to the third-orbit, are found by Fourier analysis, and dσ*/dt, accurate to the fourth order, from numerical averaging. The present procedure can be used to reduce satellite laser observations with a high precision.     

Contributions of the Pulkovo and Kharkiv Scientific Schools to the search for exoplanets and low-mass dark satellites of stars

This article is devoted to the Pulkovo astronomer, Prof. Aleksandr Nikolaevich Deich (Deutsch) (1899-1986), on the 110-th anniversary of his birth. Deich is known as the founder of the Pulkovo program for observing stars with invisible companions, as well as for his research on the star 61 Cyg, which was suspected, in his time, of having invisible companions with the masses of planets. Astrometric observations on the long focus astrograph and searches for exoplanets of nearby stars are reviewed. Modern methods of searching for exoplanets are summarized briefly. Instrument designs proposed by astronomers at Kharkiv (Scientific Research Institute of Astronomy at Kharkiv National University, NIIA KhNU) and Kazan (Institute of Astronomy, Kazan State University, AO KGU) for use in the search for low-mass dark components of stars are discussed. Examples are given of confirmations of invisible companions of stars which were first discovered by observation. A number of theoretical results on this topic from Kharkiv National University (Scientific Research Institute of Astronomy at Kharkiv and the Dept. of Astronomy) are noted.     

Tidal disruptions: A continuum theory for solid bodies

Although the theory of Roche 1847 for the tidal disruption limits of orbiting satellites assumes a fluid body, a length to diameter of exactly 2.07:1, and a particular body orientation, the theory is commonly applied to the satellites of the Solar System and to small asteroids and comets passing nearby a planet. Clearly these bodies are neither fluid nor generally are that elongated, so a more appropriate theory is needed. Here we present exact analytical results for the distortion and disruption limits of solid spinning ellipsoidal bodies subjected to tidal forces, using the Drucker-Prager strength model with zero cohesion. It is the appropriate model for dry granular materials such as sands and rocks, for rubble-pile asteroids and comets, and for larger satellites, asteroids and comets where the cohesion can be ignored. This study uses the same approach as the studies of spin limits for solid ellipsoidal bodies given in [Holsapple, K.A., 2001. Icarus 154, 432-448; Holsapple, K.A., 2004. Icarus 172, 272-303]. It is a static theory that predicts conditions for breakup and predicts the nature of the deformations at the limit state, but does not track the dynamics of the body as it comes apart. The strength is characterized by a single material parameter associated with an angle of friction, which can range from zero to 90°. The case with zero friction angle has no shear strength whatsoever, so it is then the model of a fluid or gas. The case of 90° represents a material that cannot fail in shear, but still has zero tensile strength. Typical dry soils have angles of friction of 30°-40°. Since the static fluid case is included in the theory as a special case, the classical results of Roche [Roche, E.A., 1847. Acad. Sci. Lett. Montpelier. Mem. Section Sci. 1, 243-262] and Jeans [Jeans, J.H., 1917. Mem. R. Astron. Soc. London 62, 1-48] are included and re-derived in their entirety; but the general solid case has much more variety and applicability. We consider both the spin-locked case, appropriate for most satellites of the Solar System; and the zero spin case, a possible case for a passing stray body. Detailed plots of many special cases are presented, in terms of shape, orientation and mass densities. A very typical result gives a closest approach d=1.5(ρ/ρP)^1/3R in terms of the planet radius R, and the satellite and planet mass densities ρ and ρP. We also use the theory to distinguish between conditions allowing global shape changes leading to new equilibrium states, or those leading to complete disruption. We apply the theory to the potentially hazardous Asteroid 99942 Apophis due to pass very near the Earth in 2029, and conclude it is extremely unlikely to experience any tidal readjustments during its passage. The states of many of the satellites of the Solar System are compared to the theory, and we find that all are well within their tidal disruption limits for expected values of the internal friction.     

Libration of Laplace's argument in the Galilean satellites theory

In the Galilean satellites motion, the Laplace argument ₁3₂+2₃ librates around the value . The amplitude of libration is very small so that the classical theories have not been set up to take into account large librations. On the other hand large librations have to be considered when we describe possible scenarii of capture into resonance by tidal effects. The aim of this paper is to present a new way of applying Hamiltonian perturbation methods to the problem of the Galilean satellites in such a way that the theory is valid for large librations. Preliminary results from such a theory are discussed.     

Tidal dissipation,orbital evolution,and the nature of Saturn's inner satellites

Estimates of tidal damping times of the orbital eccentricities of Saturn's inner satellites place constraints on some satellite rigidities and dissipation functions Q. These constraints favor rock-like rather than ice-like properties for Mimas and probably Dione. Photometric and other observational data are consistent with relatively higher densities for these two satellites, but require lower densities for Tethys, Enceladus, and Rhea. This leads to a nonmonotonic density distribution for Saturn's inner satellites, apparently determined by different mass fractions of rocky materials. In spite of the consequences of tidal dissipation for the orbital eccentricity decay and implications for satellite compositions, tidal heating is not an important contributor to the thermal history of any Saturnian satellite.     

The discovery of Segue 2: a prototype of the population of satellites of satellites

We announce the discovery of a new Milky Way satellite Segue 2 found in the data of the Sloan Extension for Galactic Understanding and Exploration (SEGUE). We followed this up with deeper imaging and spectroscopy on the Multiple Mirror Telescope (MMT). From this, we derive a luminosity of   M _v =−2.5  , a half-light radius of 34 pc and a systemic velocity of  ∼−40 km s⁻¹  . Our data also provide evidence for a stream around Segue 2 at a similar heliocentric velocity, and the SEGUE data show that it is also present in neighbouring fields. We resolve the velocity dispersion of Segue 2 as 3.4 km s⁻¹ and the possible stream as  ∼7 km s⁻¹  . This object shows points of comparison with other recent discoveries, Segue 1, Boo II and Coma. We speculate that all four objects may be representatives of a population of satellites of satellites – survivors of accretion events that destroyed their larger but less dense parents. They are likely to have formed at redshifts   z > 10  and are good candidates for fossils of the reionization epoch.