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 friction in close-in satellites and exoplanets: The Darwin theory re-visited

This report is a review of Darwin’s classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) capture into the 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.     

Exoplanet status report: Observation, characterization and evolution of exoplanets and their host stars

After the discovery of more than 400 planets beyond our Solar System, the characterization of exoplanets as well as their host stars can be considered as one of the fastest growing fields in space science during the past decade. The characterization of exoplanets can only be carried out in a well coordinated interdisciplinary way which connects planetary science, solar/stellar physics and astrophysics. We present a status report on the characterization of exoplanets and their host stars by reviewing the relevant space- and ground-based projects. One finds that the previous strategy changed from space mission concepts which were designed to search, find and characterize Earth-like rocky exoplanets to: A statistical study of planetary objects in order to get information about their abundance, an identification of potential target and finally its analysis. Spectral analysis of exoplanets is mandatory, particularly to identify bio-signatures on Earth-like planets. Direct characterization of exoplanets should be done by spectroscopy, both in the visible and in the infrared spectral range. The way leading to the direct detection and characterization of exoplanets is then paved by several questions, either concerning the pre-required science or the associated observational strategy.     

Precise determination of fundamental parameters of six exoplanet host stars and their planets

The aim of this paper is to determinate the fundamental parameters of six exoplanet host(EH) stars and their planets. Because techniques for detecting exoplanets yield properties of the planet only as a function of the properties of the host star, we must accurately determine the parameters of the EH stars first. For this reason, we constructed a grid of stellar models including diffusion and rotation-induced extra-mixing with given ranges of input parameters(i.e. mass, metallicity and initial rotation rate). In addition to the commonly used observational constraints such as the effective temperature Teff, luminosity L and metallicity [Fe/H], we added two observational constraints, the lithium abundance log N(Li) and the rotational period Prot.These two additional observed parameters can set further constraints on the model due to their correlations with mass, age and other stellar properties. Hence, our estimations of the fundamental parameters for these EH stars and their planets have a higher precision than previous works. Therefore, the combination of rotational period and lithium helps us to obtain more accurate parameters for stars, leading to an improvement in knowledge about the physical state of EH stars and their planets.     

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.     

A search for wide visual companions of exoplanet host stars: The Calar Alto Survey

We have carried out a search for co‐moving stellar and substellar companions around 18 exoplanet host stars with the infrared camera MAGIC at the 2.2 m Calar Alto telescope, by comparing our images with images from the all sky surveys 2MASS, POSS I and II. Four stars of the sample namely HD80606, 55 Cnc, HD46375 and BD–10°3166, are listed as binaries in the Washington Visual Double Star Catalogue (WDS). The binary nature of HD80606, 55 Cnc, and HD46375 is confirmed with both astrometry as well as photometry, thereby the proper motion of the companion of HD46375 was determined here for the first time.We derived the companion masses as well as the longterm stability regions for additional companions in these three binary systems. We can rule out further stellar companions around all stars in the sample with projected separations between 270AU and 2500AU, being sensitive to substellar companions with masses down to ∼60 M_Jup (S /N = 3). Furthermore we present evidence that the two components of the WDS binary BD–10°3166 are unrelated stars, i.e this system is a visual pair. The spectrophotometric distance of the primary (a K0 dwarf) is ∼67 pc, whereas the presumable secondary BD–10°3166B (a M4 to M5 dwarf) is located at a distance of 13 pc in the foreground. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The dynamics of tadpole and horseshoe orbits: II. The coorbital satellites of saturn

The recently discovered coorbital satellites of Saturn, 1980S1 and 1980S3, are shown to be librating in horseshoe orbits. By considering the effects of tangential forces on the semimajor axes of the satellite orbits, we derive an accurate relation between the sum of the satellite masses and (a) their minimum angular separation, (b) the variation of their angular separation with time and (c) the libration period. Observations of (b) and (c) are the most practical methods of determining the satellite masses. The orbits of the coorbital satellites of Dione and Tethys are discussed. We demonstrate the possibility of calculating a new value for the mass of Dione and we show that one of the coorbital satellites of Tethys could be moving in a horseshoe orbit even though another satellite is librating in a tadpole orbit about the leading Lagrangian equilibrium point L₄. The origin of coorbital satellites and the stability of their orbits are discussed.     

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.     

Spectroscopy of stars in ground-based ultraviolet. II: Atmospheres

Some of the tasks of spectroscopic studies of stellar atmospheres in the ground-based ultraviolet are reviewed in historical perspective. Examples used include fragments of spectra that we obtained with the 6-m telescope of the Special Astrophysical Observatory of the Russian Academy of Sciences     

Intrinsic properties of carbon stars. II. Spectra,colours, and HR diagram of cool carbon stars

On the basis of the effective temperature scale proposed previously for cool carbon stars (Paper I), other intrinsic properties of them are examined in detail. It is shown that the major spectroscopic properties of cool carbon stars, including those of molecular bands due to polyatomic species (SiC₂, HCN, C₂H₂ etc.), can most consistently be understood on the basis of our new effective temperature scale and the theoretical prediction of chemical equilibrium. Various photometric indices of cool carbon stars also appear to be well correlated with the new effective temperatures. Furthermore, as effective temperatures of some 30 carbon stars are now obtained, the calibration of any photometric index is straightforward, and some examples of such a calibration are given. In general, colour index-effective temperature calibrations for carbon stars are quite different from those for K-M giant stars. It is found that the intrinsic (R —I)₀ colour is nearly the same for N-irregular variables in spite of a considerable spread in effective temperatures, and this fact is used to estimate the interstellar reddening of carbon stars. An observational HR diagram of red giant stars, including carbon stars as well as K-M giant stars, is obtained on the basis of our colour index-effective temperature calibrations and the best estimations of luminosities. It is shown that carbon stars and M giant stars are sharply divided in the HR diagram by a nearly vertical line at aboutT _eff = 3200 K (logT _eff = 3.50) and the carbon stars occupy the upper right region of M giant stars (except for some high luminosity, high temperature J-type stars in the Magellanic Clouds; also Mira variables are not considered). Such an observational HR diagram of red giant stars shows rather a poor agreement with the current stellar evolution models. Especially, a more efficient mixing process in red giant stars, as compared with those ever proposed, is required to explain the formation of carbon stars.     

Rotational dynamics of planetary satellites: A survey of regular and chaotic behavior

The problem of observability of chaotic regimes in the rotation of planetary satellites is studied. The analysis is based on the inertial and orbital data available for all satellites discovered up to now. The Lyapunov spectra of the spatial chaotic rotation and the full range of variation of the spin rate are computed numerically by integrating the equations of the rotational motion; the initial data are taken inside the main chaotic layer near the separatrices of synchronous resonance in phase space. The model of a triaxial satellite in a fixed elliptic orbit is adopted. A short Lyapunov time along with a large range of variation of the spin rate are used as criteria for observability of the chaotic motion. Independently, analysis of stability of the synchronous state with respect to tilting the axis of rotation provides a test for the physical opportunity for a satellite to rotate chaotically. Finally, a calculation of the times of despinning due to tidal evolution shows whether a satellite's spin could evolve close to the synchronous state. Apart from Hyperion, already known to rotate chaotically, only Prometheus and Pandora, the 16th and 17th satellites of Saturn, pass all these four tests.     

Main-sequence magnetic CP stars: II. Physical parameters and chemical composition of the atmosphere

This paper continues a series of reviews dedicated to magnetic CP stars. The occurrence frequency of CP stars among B5–F0-type main-sequence stars is shown to be equal to about 15–20%. The problems of identification and classification of these objects are addressed. We prefer the classification of Preston, which subdivides chemically peculiar stars into the following groups: Am, λ Boo, Ap/Bp, Hg-Mn, He-weak, and He-strong stars. The main characteristic features of objects of each group are briefly analyzed. The rotation velocities of CP stars are shown to be about three times lower than those of normal stars of the same spectral types (except for λ Boo and He-strong objects). The rotation periods of CP stars range from 0.5 to 100 days, however, there is also a small group of objects with especially long (up to several tens of years) variability periods. All kinds of peculiar stars can be found in visual binaries, with Am-and Hg-Mn-type stars occurring mostly in short-period binaries with P < 10 days, and the binary rate of these stars is close to normal. The percentage of binaries among magnetic stars (20%) is lower than among normal stars. A rather large fraction of CP1-and CP2-type stars was found to occur in young clusters (with ages smaller than 10⁷ years). Photometric and spectral variability of peculiar stars of various types is discussed, and it is shown that only objects possessing magnetic fields exhibit light and spectral variations. The chemical composition of the atmospheres of CP stars of various types is considered. The abundances of various elements are usually determined by comparing the line profiles in the observed spectrum with those of the synthetic spectra computed for various model atmospheres. Different mechanisms are shown to contribute to chemical inhomogeneity at the star’s surface, and the hypothesis of selective diffusion of atoms in a stable atmosphere is developed. Attention is also paid to the problems of the determination of local chemical composition including the stratification of elements. Some of the coolest SrCrEu peculiar stars are found to exhibit fast light variations with periods ranging from 6 to 15 min. These variations are unassociated with rotation, but are due to nonradial pulsations. The final part of the the review considers the fundamental parameters of CP stars. The effective temperatures, luminosities, radii, and masses of these objects are shown to agree with the corresponding physical parameters of normal main-sequence stars of the same spectral types.     

An averaging method to study the motion of lunar artificial satellites II: Averaging and Applications

On this, the second part of a two part study (Steichen, 1998) we further develop a semi-analytical theory for a lunar artificial satellite. This theory is obtained by averaging analytically the Hamiltonian function over period up to a month. The averaged equations are then numerically integrated. The solution is free from singularities at e = 0 and I = 0 and is not expanded in powers of these variables. In the last section, the analytic work is applied to characteristic examples to validate the method used. This revised version was published online in July 2006 with corrections to the Cover Date.     

Atmospheric tides and the rotation of Venus II. Spin evolution

Tides in the atmosphere of Venus may help to stabilize its slow retrograde rotation. The frequency dependence of the body tides also affects its rotational stability. However, the obliquity is probably maintained near 180° by friction between the core and mantle of Venus. In any case, it appears most likely that Venus originated with an obliquity greater than 90°.     

EChO spectra and stellar activity II. The case of dM stars

EChO is a dedicated mission to investigate exoplanetary atmospheres. When extracting the planetary signal, one has to take care of the variability of the hosting star, which introduces spectral distortion that can be mistaken as planetary signal. Magneticvariability has to be taken into account in particular for M stars. To this purpose, assuming a one spot dominant model for the stellar photosphere, we develop a mixed observational-theoretical tool to extract the spot’s parameters from the observed optical spectrum. This method relies on a robust library of spectral M templates, which we derive using the observed spectra of quiet M dwarfs in the SDSS database. Our procedure allows to correct the observed spectra for photospheric activity in most of the analyzed cases, reducing the spectral distortion down to the noise levels. Ongoing refinements of the template library and the algorithm will improve the efficiency of our algorithm.     

Magnetic fields of chemically peculiar stars. II: Magnetic fields and rotation of stars with strong and weak anomalies in the continuum energy distribution

We make a comparative analysis of magnetic fields and rotation parameters of magnetic CP stars with strong and weak anomalies in the spectral energy distribution. Stars with strong depressions in the continuum at 5200 Å are shown to have significantly stronger fields (the mean longitudinal component of the fields of these stars is 〈B _e〉 = 1341 ± 98 G) compared to objects with weaker depressions (〈B _e〉 = 645 ± 58 G). Stars with stronger depressions are also found to occur more commonly among slow rotators. Their rotation periods are, on the average, about 10 days long, three times longer than these of stars with weak depressions (about three days). This fact is indicative of a decrease of the degree of anomality of the magnetic stars continuum spectrum with increasing rotational velocity. Yet another proof has been obtained suggesting that slow rotation is one of the crucial factors contributing to the development of the phenomenon of magnetic chemically peculiar stars.Magnetic CP stars with weak depressions at 5200 Å are intermediate objects between stars with strong depressions and normal A- and B-type stars both in terms of field strength and rotational velocity.