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Zacharias N.,Us Naval Observatory
Astronomical Journal | Year: 2010

The third US Naval Observatory CCD Astrograph Catalog, UCAC3, was released at the IAU General Assembly on 2009 August 10. It is a highly accurate, all-sky astrometric catalog of about 100 million stars in the R = 8-16 mag range. Recent epoch observations are based on over 270,000 CCD exposures, which have been re-processed for the UCAC3 release applying traditional and new techniques. Challenges in the data have been high dark current and asymmetric image profiles due to the poor charge transfer efficiency of the detector. Non-Gaussian image profile functions were explored and correlations are found for profile fit parameters with properties of the CCD frames. These were utilized to constrain the image profile fit models and adequately describe the observed point-spread function of stellar images with a minimum number of free parameters. Using an appropriate model function, blended images of double stars could be fit successfully. UCAC3 positions are derived from two-dimensional image profile fits with a five-parameter, symmetric Lorentz profile model. Internal precisions of about 5 mas per coordinate and single exposure are found, which are degraded by the atmosphere to about 10 mas. However, systematic errors exceeding 100 mas are present in the x, y data which have been corrected in the astrometric reductions following the x, y data reduction step described here. © 2010. The American Astronomical Society. All rights reserved.


Makarov V.V.,Us Naval Observatory | Efroimsky M.,Us Naval Observatory
Astrophysical Journal | Year: 2014

In Efroimsky & Makarov (Paper I), we derived from the first principles a formula for the tidal heating rate in a homogeneous sphere, compared it with the previously used formulae, and noted the differences. Now we present case studies: Mercury, Kepler-10 b, and a triaxial Io. A sharp frequency dependence of k2/Q near spin-orbit resonances yields a sharp dependence of k2/Q (and, therefore, of tidal heating) upon the spin rate. Thereby physical libration plays a major role in tidal heating of synchronously rotating planets. The magnitude of libration in the spin rate being defined by the planet's triaxiality, the latter becomes a factor determining the dissipation rate. Other parameters equal, a strongly triaxial synchronized body generates more heat than a similar body of a more symmetrical shape. After an initially triaxial object melts and loses its triaxiality, dissipation becomes less intensive; the body can solidify, with the tidal bulge becoming a new figure with triaxiality lower than the original. We derive approximate expressions for the dissipation rate in a Maxwell planet with the Maxwell time longer than the inverse tidal frequency. The expressions derived pertain to the 1:1 and 3:2 resonances and a nonresonant case; so they are applicable to most close-in super-Earths detected. In these planets, the heating outside synchronism is weakly dependent on the eccentricity and obliquity, provided both these parameters's values are moderate. According to our calculation, Kepler-10 b could hardly survive the intensive tidal heating without being synchronized, circularized, and reshaped through a complete or partial melt-down. © 2014. The American Astronomical Society. All rights reserved.


Efroimsky M.,Us Naval Observatory | Makarov V.V.,Us Naval Observatory
Astrophysical Journal | Year: 2014

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 and Platzman. 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. However, the two turn out to differ, as in our expression the contributions from all Fourier modes are positive-definite, which is not the case with the formula from Peale & Cassen. Examples of the application of our expression for the tidal damping rate are provided in the work by Makarov & Efroimsky (Paper II) published back to back with the current paper. © 2014. The American Astronomical Society. All rights reserved.


Efroimsky M.,Us Naval Observatory
Astrophysical Journal | Year: 2012

While the seismic quality factor and phase lag are defined solely by the bulk properties of the mantle, their tidal counterparts are determined by both the bulk properties and the size effect (self-gravitation of a body as a whole). For a qualitative estimate, we model the body with a homogeneous sphere, and express the tidal phase lag through the lag in a sample of material. Although simplistic, our model is sufficient to understand that the lags are not identical. The difference emerges because self-gravitation pulls the tidal bulge down. At low frequencies, this reduces strain and the damping rate, making tidal damping less efficient in larger objects. At higher frequencies, competition between self-gravitation and rheology becomes more complex, though for sufficiently large super-Earths the same rule applies: the larger the planet, the weaker the tidal dissipation in it. Being negligible for small terrestrial planets and moons, the difference between the seismic and tidal lagging (and likewise between the seismic and tidal damping) becomes very considerable for large exoplanets (super-Earths). In those, it is much lower than what one might expect from using a seismic quality factor. The tidal damping rate deviates from the seismic damping rate, especially in the zero-frequency limit, and this difference takes place for bodies of any size. So the equal in magnitude but opposite in sign tidal torques, exerted on one another by the primary and the secondary, have their orbital averages going smoothly through zero as the secondary crosses the synchronous orbit. We describe the mantle rheology with the Andrade model, allowing it to lean toward the Maxwell model at the lowest frequencies. To implement this additional flexibility, we reformulate the Andrade model by endowing it with a free parameter ζ which is the ratio of the anelastic timescale to the viscoelastic Maxwell time of the mantle. Some uncertainty in this parameter's frequency dependence does not influence our principal conclusions. © 2012 The American Astronomical Society. All rights reserved.


Efroimsky M.,Us Naval Observatory
Astronomical Journal | Year: 2015

This is a pilot paper serving as a launching pad for study of orbital and spin evolution of binary asteroids. The rate of tidal evolution of asteroidal binaries is defined by the dynamical Love numbers kl divided by quality factors Q. Common in the literature is the (oftentimes illegitimate) approximation of the dynamical Love numbers with their static counterparts. Since the static Love numbers are, approximately, proportional to the inverse rigidity, this renders a popular fallacy that the tidal evolution rate is determined by the product of the rigidity by the quality factor: . In reality, the dynamical Love numbers depend on the tidal frequency and all rheological parameters of the tidally perturbed body (not just rigidity). We demonstrate that in asteroidal binaries the rigidity of their components plays virtually no role in tidal friction and tidal lagging, and thereby has almost no influence on the intensity of tidal interactions (tidal torques, tidal dissipation, tidally induced changes of the orbit). A key quantity that overwhelmingly determines the tidal evolution is a product of the effective viscosity by the tidal frequency . The functional form of the torque's dependence on this product depends on who wins in the competition between viscosity and self-gravitation. Hence a quantitative criterion, to distinguish between two regimes. For higher values of , we get , for lower values we obtain . Our study rests on an assumption that asteroids can be treated as Maxwell bodies. Applicable to rigid rocks at low frequencies, this approximation is used here also for rubble piles, due to the lack of a better model. In the future, as we learn more about mechanics of granular mixtures in a weak gravity field, we may have to amend the tidal theory with other rheological parameters, ones that do not show up in the description of viscoelastic bodies. This line of study provides a tool to exploring the orbital history of asteroidal pairs, as well as of their final spin states. © 2015. The American Astronomical Society. All rights reserved..


Makarov V.V.,Us Naval Observatory
Monthly Notices of the Royal Astronomical Society: Letters | Year: 2012

The relative motion of stars and other celestial objects in very wide pairs, separated by distances of the order of 1 pc, is strongly influenced by the tidal gravitational potential of the Galaxy. The Coriolis component of the horizontal tidal force in the rotating reference frame tends to disrupt such marginally bound pairs. However, even extremely wide pairs of bodies can be bound over intervals of time comparable to the Hubble time, under appropriate initial conditions. Here we show that for arbitrary chosen initial coordinates of a pair of stars, there exists a volume of the space of initial velocity components where the orbits remain bound in the planar tidal field for longer than 10 Gyr, even though the initial separation is well outside the Jacobi radius. The boundary of this phase space of stable orbits is fractal, and the motion at the boundary conditions is clearly chaotic. We found that the pairs may remain confined for several Gyr, and then suddenly disintegrate due to a particularly close rendezvous. By reversing such long-term stable orbits, we find that entrapment of unrelated stars into wide pairs is possible, but should be quite rare. Careful analysis of precision astrometry surveys revealed that extremely wide pairs of stars are present in significant numbers in the Galaxy. These results are expected to help in discriminating the cases of genuine binarity and chance entrapment, and to make inroads in testing the limits of Newtonian gravitation. © 2011 RAS No claim to original US government works. © 2011 RAS.


Efroimsky M.,Us Naval Observatory
Celestial Mechanics and Dynamical Astronomy | Year: 2012

Spin-orbit coupling can be described in two approaches. The first method, known as the "MacDonald torque", is often combined with a convenient assumption that the quality factor Q is frequency-independent. This makes the method inconsistent, because derivation of the expression for the MacDonald torque tacitly fixes the rheology of the mantle by making Q scale as the inverse tidal frequency. Spin-orbit coupling can be treated also in an approach called "the Darwin torque". While this theory is general enough to accommodate an arbitrary frequency-dependence of Q, this advantage has not yet been fully exploited in the literature, where Q is often assumed constant or is set to scale as inverse tidal frequency, the latter assertion making the Darwin torque equivalent to a corrected version of the MacDonald torque. However neither a constant nor an inverse-frequency Q reflect the properties of realistic mantles and crusts, because the actual frequency-dependence is more complex. Hence it is necessary to enrich the theory of spin-orbit interaction with the right frequency-dependence. We accomplish this programme for the Darwin-torque-based model near resonances. We derive the frequency-dependence of the tidal torque from the first principles of solid-state mechanics, i. e., from the expression for the mantle's compliance in the time domain. We also explain that the tidal torque includes not only the customary, secular part, but also an oscillating part. We demonstrate that the lmpq term of the Darwin-Kaula expansion for the tidal torque smoothly passes zero, when the secondary traverses the lmpq resonance (e. g., the principal tidal torque smoothly goes through nil as the secondary crosses the synchronous orbit). Thus, we prepare a foundation for modeling entrapment of a despinning primary into a resonance with its secondary. The roles of the primary and secondary may be played, e. g., by Mercury and the Sun, correspondingly, or by an icy moon and a Jovian planet. We also offer a possible explanation for the "improper" frequency-dependence of the tidal dissipation rate in the Moon, discovered by LLR. © 2012 Springer Science+Business Media B.V. (outside the USA).


Makarov V.V.,Us Naval Observatory
Monthly Notices of the Royal Astronomical Society: Letters | Year: 2013

If the Moon's spin evolved from faster prograde rates, it could have been captured into a higher spin-orbit resonance than the current 1:1 resonance. At the current value of orbital eccentricity, the probability of capture into the 3:2 resonance is as high as 0.6, but it strongly depends on the temperature and average viscosity of theMoon's interior. A warmer, less viscous Moon on a higher eccentricity orbit is even more easily captured into supersynchronous resonances.We discuss two likely scenarios for the present spin-orbit state: a cold Moon on a low-eccentricity orbit and a retrograde initial rotation. © 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.


Makarov V.V.,Us Naval Observatory | Efroimsky M.,Us Naval Observatory
Astrophysical Journal | Year: 2013

We re-examine the popular belief that a telluric planet or a satellite on an eccentric orbit can, outside a spin-orbit resonance, be captured in a quasi-static tidal equilibrium called pseudosynchronous rotation. The existence of such configurations was deduced from oversimplified tidal models assuming either a constant tidal torque or a torque linear in the tidal frequency. A more accurate treatment requires that the torque be decomposed into the Darwin-Kaula series over the tidal modes, and that this decomposition be combined with a realistic choice of rheological properties of the mantle, which we choose to be a combination of the Andrade model at ordinary frequencies and the Maxwell model at low frequencies. This development demonstrates that there exist no stable equilibrium states for solid planets and moons, other than spin-orbit resonances. © Copyright is not claimed for this article. All rights reserved.


Efroimsky M.,Us Naval Observatory | Makarov V.V.,Us Naval Observatory
Astrophysical Journal | Year: 2013

Tidal torques play a key role in rotational dynamics of celestial bodies. They govern these bodies' tidal despinning and also participate in the subtle process of entrapment of these bodies into spin-orbit resonances. This makes tidal torques directly relevant to the studies of habitability of planets and their moons. Our work begins with an explanation of how friction and lagging should be built into the theory of bodily tides. Although much of this material can be found in various publications, a short but self-consistent summary on the topic has been lacking in the hitherto literature, and we are filling the gap. After these preparations, we address a popular concise formula for the tidal torque, which is often used in the literature, for planets or stars. We explain why the derivation of this expression, offered in the paper by Goldreich and in the books by Kaula (Equation (4.5.29)) and Murray & Dermott (Equation (4.159)), implicitly sets the time lag to be frequency independent. Accordingly, the ensuing expression for the torque can be applied only to bodies having a very special (and very hypothetical) rheology which makes the time lag frequency independent, i.e., the same for all Fourier modes in the spectrum of tide. This expression for the torque should not be used for bodies of other rheologies. Specifically, the expression cannot be combined with an extra assertion of the geometric lag being constant, because at finite eccentricities the said assumption is incompatible with the constant-time-lag condition. © Copyright is not claimed for this article. All rights reserved.

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