arXiv subjects

Michael Efroimsky

Publications and source records attributed to Michael Efroimsky.

53 records · Page 3Linked to original sources

arXiv2009

The Physics of Bodily Tides in Terrestrial Planets, and the Appropriate Scales of Dynamical Evolution

Any model of tides is based on a specific hypothesis of how lagging depends on the tidal-flexure frequency. For example, Gerstenkorn (1955), MacDonald (1964), and Kaula (1964) assumed constancy of the geometric lag angle, while Singer (1968) and Mignard (1979, 1980) asserted constancy of the time lag. Thus, each of these two models was based on a certain law of scaling of the geometric lag. The actual dependence of the geometric lag on the frequency is more complicated and is determined by the rheology of the planet. Besides, each particular functional form of this dependence will unambiguously fix the appropriate form of the frequency dependence of the tidal quality factor, Q. Since at present we know the shape of the dependence of Q upon the frequency, we can reverse our line of reasoning and single out the appropriate actual frequency-dependence of the angular lag. This dependence turns out to be different from those employed hitherto, and it entails considerable alterations in the time scales of the tide-generated dynamical evolution. Phobos' fall on Mars is an example we consider.

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arXiv2007

Long-term evolution of orbits about a precessing oblate planet. 3. A semianalytical and a purely numerical approach

Construction of a theory of orbits about a precessing oblate planet, in terms of osculating elements defined in a frame of the equator of date, was started in Efroimsky and Goldreich (2004) and Efroimsky (2005, 2006). We now combine that analytical machinery with numerics. The resulting semianalytical theory is then applied to Deimos over long time scales. In parallel, we carry out a purely numerical integration in an inertial Cartesian frame. The results agree to within a small margin, for over 10 Myr, demonstrating the applicability of our semianalytical model over long timescales. This will enable us to employ it at the further steps of the project, enriching the model with the tides, the pull of the Sun, and the planet's triaxiality. Another goal of our work was to check if the equinoctial precession predicted for a rigid Mars could have been sufficient to repel the orbits away from the equator. We show that for low initial inclinations, the orbit inclination reckoned from the precessing equator of date is subject only to small variations. This is an extension, to non-uniform precession given by the Colombo model, of an old result obtained by Goldreich (1965) for the case of uniform precession and a low initial inclination. However, near-polar initial inclinations may exhibit considerable variations for up to +/- 10 deg in magnitude. Nevertheless, the analysis confirms that an oblate planet can, indeed, afford large variations of the equinoctial precession over hundreds of millions of years, without repelling its near-equatorial satellites away from the equator of date: the satellite inclination oscillates but does not show a secular increase. Nor does it show secular decrease, a fact that is relevant to the discussion of the possibility of high-inclination capture of Phobos and Deimos.

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arXiv2007

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

The Hamiltonian theory of Earth rotation, known as the Kinoshita-Souchay theory, operates with nonosculating Andoyer elements. This situation parallels a similar phenomenon that often happens (but seldom gets noticed) in orbital dynamics, when the standard Lagrange-type or Delaunay-type planetary equations unexpectedly render nonosculating orbital elements. In orbital mechanics, osculation loss happens when a velocity-dependent perturbation is plugged into the standard planetary equations. In attitude mechanics, osculation is lost when an angular-velocity-dependent disturbance is plugged in the standard dynamical equations for the Andoyer elements. We encounter exactly this situation in the theory of Earth rotation, because this theory contains an angular-velocity-dependent perturbation (the switch from an inertial frame to that associated with the precessing ecliptic of date). While the osculation loss does not influence the predictions for the figure axis of the planet, it considerably alters the predictions for the instantaneous spin-axis' orientation. We explore this issue in great detail.

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arXiv2007

The Serret-Andoyer Formalism in Rigid-Body Dynamics: I. Symmetries and Perturbations

This paper reviews the Serret-Andoyer (SA) canonical formalism in rigid-body dynamics and presents some new results. As is well known, the problem of unsupported and unperturbed rigid rotator can be reduced. The availability of this reduction is offered by the underlying symmetry, which stems from conservation of the angular momentum and rotational kinetic energy. When a perturbation is turned on, these quantities are no longer preserved. Nonetheless, the language of reduced description remains extremely instrumental even in the perturbed case. We describe the canonical reduction performed by the Serret-Andoyer (SA) method, and discuss its applications to attitude dynamics and to the theory of planetary rotation. Specifically, we consider the case of angular-velocity-dependent torques, and discuss the variation-of-parameters-inherent antinomy between canonicity and osculation. Finally, we address the transformation of the Andoyer variables into the action-angle ones, using the method of Sadov.

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arXiv2007

Initial Results from the USNO Dispersed Fourier Transform Spectrograph

We have designed and constructed a ``dispersed Fourier Transform Spectrometer'' (dFTS), consisting of a conventional FTS followed by a grating spectrometer. By combining these two devices, we negate a substantial fraction of the sensitivity disadvantage of a conventional FTS for high resolution, broadband, optical spectroscopy, while preserving many of the advantages inherent to interferometric spectrometers. In addition, we have implemented a simple and inexpensive laser metrology system, which enables very precise calibration of the interferometer wavelength scale. The fusion of interferometric and dispersive technologies with a laser metrology system yields an instrument well-suited to stellar spectroscopy, velocimetry, and extrasolar planet detection, which is competitive with existing high-resolution, high accuracy stellar spectrometers. In this paper, we describe the design of our prototype dFTS, explain the algorithm we use to efficiently reconstruct a broadband spectrum from a sequence of narrowband interferograms, and present initial observations and resulting velocimetry of stellar targets.

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arXiv2006

Long-term evolution of orbits about a precessing oblate planet. 2. The case of variable precession

We continue the study undertaken in Efroimsky (2005a) where we explored the influence of spin-axis variations of an oblate planet on satellite orbits. Near-equatorial satellites had long been believed to keep up with the oblate primary's equator in the cause of its spin-axis variations. As demonstrated by Efroimsky and Goldreich (2004), this opinion had stemmed from an inexact interpretation of a correct result by Goldreich (1965). Though Goldreich (1965) mentioned that his result (preservation of the initial inclination, up to small oscillations about the moving equatorial plane) was obtained for non-osculating inclination, his admonition has been persistently ignored for forty years. It was explained in Efroimsky and Goldreich (2004) that the equator precession influences the osculating inclination of a satellite orbit already in the first order over the perturbation caused by a transition from an inertial to an equatorial coordinate system. It was later shown in Efroimsky (2005a) that the secular part of the inclination is affected only in the second order. This fact, anticipated by Goldreich (1965), remains valid for a constant rate of the precession. It turns out that non-uniform variations of the planetary spin state generate changes in the osculating elements, that are linear in the planetary equator's total precession rate, rate that includes the equinoctial precession, nutation, the Chandler wobble, and the polar wander. We work out a formalism which will help us to determine if these factors cause a drift of a satellite orbit away from the evolving planetary equator.

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arXiv2006

On the theory of canonical perturbations and its application to Earth rotation

Both orbital and rotational dynamics employ the method of variation of parameters. We express, in a non-perturbed setting, the coordinates (Cartesian, in the orbital case, or Eulerian in the rotation case) via the time and six adjustable constants called elements (orbital elements or rotational elements). If, under disturbance, we use this expression as ansatz and endow the "constants" with time dependence, then the perturbed velocity (Cartesian or angular) will consist of a partial derivative with respect to time and a so-called convective term, one that includes the time derivatives of the variable "constants." Out of sheer convenience, the so-called Lagrange constraint is often imposed. It nullifies the convective term and, thereby, guarantees that the functional dependence of the velocity upon the time and "constants" stays, under perturbation, the same as it used to be in the undisturbed setting. When the dynamical equations, written in terms of the "constants," are demanded to be symplectic (and the "constants" make conjugated pairs $ Q, P$), these "constants" are called Delaunay elements, in the orbital case, or Andoyer elements, in the rotational case. The Andoyer and Delaunay sets of elements share a feature not visible with a naked eye: in certain cases, the standard equations render these elements non-osculating. Hence, even though the Andoyer variables in the Kinoshita-Souchay theory are introduced in a precessing frame of the Earth orbit, they nevertheless return the angular velocity relative to an inertial frame.

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arXiv2006

Gauge Freedom in Orbital Mechanics

In orbital and attitude dynamics the coordinates and the Euler angles are expressed as functions of the time and six constants called elements. Under disturbance, the constants are endowed with time dependence. The Lagrange constraint is then imposed to guarantee that the functional dependence of the perturbed velocity on the time and constants stays the same as in the undisturbed case. Constants obeying this condition are called osculating elements. The constants chosen to be canonical are called Delaunay elements, in the orbital case, or Andoyer elements, in the spin case. (As some Andoyer elements are time dependent even in the free-spin case, the role of constants is played by their initial values.) The Andoyer and Delaunay sets of elements share a feature not readily apparent: in certain cases the standard equations render them non-osculating. In orbital mechanics, elements furnished by the standard planetary equations are non-osculating when perturbations depend on velocities. To preserve osculation, the equations must be amended with extra terms that are not parts of the disturbing function. In the case of Delaunay parameterisation, these terms destroy canonicity. So under velocity-dependent disturbances, osculation and canonicity are incompatible. (Efroimsky and Goldreich 2003, 2004) Similarly, the Andoyer elements turn out to be non-osculating under angular-velocity-dependent perturbation. Amendment of only the Hamiltonian makes the equations render nonosculating elements. To make them osculating, more terms must enter the equations (and the equations will no longer be canonical). In practical calculations, is often convenient to deliberately deviate from osculation by substituting the Lagrange constraint with a condition that gives birth to a family of nonosculating elements.

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arXiv2003

Gauge Symmetry of the N-body Problem in the Hamilton-Jacobi Approach

In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton-Jacobi method: one begins with the 2-body Hamilton equations, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalise the approach to the N-body case. We re-examine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the so-called Lagrange constraint. It is the condition of the orbital elements being osculating, i.e., of the instantaneous ellipse (or hyperbola) being always tangential to the physical velocity. Imposure of any condition different from the Lagrange constraint (but compatible with the equations of motion) is legitimate and will not alter the physical trajectory or velocity (though will alter the mathematical form of the planetary equations). This freedom of nomination of the supplementary constraint reveals a gauge-type internal symmetry of the celestial-mechanics equations and has consequences for the stability of numerical integrators. Another important aspect of this freedom is that any gauge different from that of Lagrange makes the Delaunay system non-canonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a "generalised Lagrange gauge" wherein the Delaunay system is symplectic. This gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity-independent.

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arXiv2003

Gauge Freedom in the N-body problem of Celestial Mechanics

We summarise research reported in (Efroimsky 2002, 2003; Efroimsky and Goldreich 2003a,b) and develop its application to planetary equations in non-inertial frames. We provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.

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arXiv2002

Complex rotation with internal dissipation. Applications to cosmic-dust alignment and to wobbling comets and asteroids

Neutron stars, asteroids, comets, cosmic-dust granules, spacecraft, as well as whatever other freely spinning body dissipate energy when they rotate about any axis different from principal. We discuss the internal-dissipation-caused relaxation of a freely precessing rotator towards its minimal-energy mode (mode that corresponds to the spin about the maximal-inertia axis). While the body nutates at some rate, the internal stresses and strains within the body oscillate at frequencies both higher and lower than this rate. The internal dissipation takes place mostly the second and higher harmonics. We discuss the application of our findings to asteroids. Regarding the comets, estimates show that the currently available angular resolution of spacecraft-based instruments makes it possible to observe wobble damping within year- or maybe even month-long spans of time. We also discuss cosmic-dust astrophysics; in particular, the role played by precession damping in the dust alignment. We show that this damping provides coupling of the grain's rotational and vibrational degrees of freedom; this entails occasional flipping of dust grains due to thermal fluctuations. During such a flip, grain preserves its angular momentum, but the direction of torques arising from H2 formation reverses. As a result, flipping grain will not rotate fast in spite of the action of uncompensated H2 formation torques. The grains get ``thermally trapped,'' and their alignment is marginal.

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arXiv2002

Mechanical Alignment of Suprathermal Paramagnetic Cosmic-Dust Granules: the Cross-Section Mechanism

We develop a comprehensive quantitative description of the cross-section mechanism discovered several years ago by Lazarian. This is one of the processes that determine grain orientation in clouds of suprathermal cosmic dust. The cross-section mechanism manifests itself when an ensemble of suprathermal paramagnetic granules is placed in a magnetic field and is subject to ultrasonic gas bombardment. The mechanism yields dust alignment whose efficiency depends upon two factors: the geometric shape of the granules, and the angle Phi between the magnetic line and the gas flow. We calculate the quantitative measure of this alignment, and study its dependence upon the said factors. It turns out that, irrelevant of the grain shape, the action of a flux does not lead to alignment if Phi = arccos(1/sqrt{3}).

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arXiv2002

Euler, Jacobi, and Missions to Comets and Asteroids

Whenever a freely spinning body is found in a complex rotational state, this means that either the body is a recent victim of an impact or a tidal interaction, or is a fragment of a recently disrupted progenitor. Another factor (relevant for comets) is outgassing. Due to impacts, tidal forces and outgassing, the asteroidal and cometary precession must be a generic phenomenon: while some rotators are in the state of visible tumbling, a much larger amount of objects must be performing narrow-cone precession not so easily observable from the Earth. The internal dissipation in a freely precessing top leads to relaxation (gradual damping of the precession) and sometimes to spontaneous changes in the rotation axis. Recently developed theory of dissipative precession of a rigid body reveals that this is a highly nonlinear process: while the body is precessing at an angular rate $ ω$, the precession-caused stresses and strains in the body contain components oscillating at other frequencies. Dependent upon the spin state, those frequencies may be higher or, most remarkably, lower than the precession rate. In many states dissipation at the harmonics is comparable to or even exceeds that at the principal frequency. For this and other reasons, in many spin states the damping of asteroidal and cometary wobble happens faster, by several orders, than believed previously. This makes it possible to measure the precession-damping rate. The narrowing of the precession cone through the period of about a year can be registered by the currently available spacecraft-based observational means. However, in the near-separatrix spin states a precessing rotator can considerably slow down its relaxation.

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arXiv2001

Relaxation of Wobbling Asteroids and Comets. Theoretical Problems. Perspectives of Experimental Observation

A body dissipates energy when it freely rotates about any axis different from principal. This entails relaxation, i.e., decrease of the rotational energy, with the angular momentum preserved. The spin about the major-inertia axis corresponds to the minimal kinetic energy, for a fixed angular momentum. Thence one may expect comets and asteroids (as well as spacecraft or cosmic-dust granules) stay in this, so-called principal, state of rotation, unless they are forced out of this state by a collision, or a tidal interaction, or cometary jetting, or by whatever other reason. As is well known, comet P/Halley, asteroid 4179 Toutatis, and some other small bodies exhibit very complex rotational motions attributed to these objects being in non-principal states of spin. Most probably, the asteroid and cometary wobble is quite a generic phenomenon. The theory of wobble with internal dissipation has not been fully developed as yet. In this article we demonstrate that in some spin states the effectiveness of the inelastic-dissipation process is several orders of magnitude higher than believed previously, and can be measured, by the presently available observational instruments, within approximately a year span. We also show that in some other spin states both the precession and precession-relaxation processes slow down considerably. (We call it near-separatrix lingering effect.) Such spin states may evolve so slowly that they can mimic the principal-rotation state.

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arXiv2001

Inelastic Dissipation in Wobbling Asteroids and Comets

Asteroids and comets dissipate energy when they rotate about the axis different from the axis of the maximal moment of inertia. We show that the most efficient internal relaxation happens at the double frequency of body's tumbling. Therefore the earlier estimates that ignore double frequency input underestimate the internal relaxation in asteroids and comets. We show that the Earth seismological data may poorly represent acoustic properties of asteroids and comet as internal relaxation increases in the presence of moisture. At the same time owing to non-linearlity of inelastic relaxation small angle nutations can persist for very long time spans, but our ability to detect such precessions is limited by the resolution of the radar-generated images. Wobbling may provide valuable information on the composition and structure of asteroids and on their recent history of external impacts.

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arXiv2001

Precession of a Freely Rotating Rigid Body. Inelastic Relaxation in the Vicinity of Poles

When a solid body is freely rotating at an angular velocity ${\bf Ω}$, the ellipsoid of constant angular momentum, in the space $Ω_1, Ω_2, Ω_3$, has poles corresponding to spinning about the minimal-inertia and maximal-inertia axes. The first pole may be considered stable if we neglect the inner dissipation, but becomes unstable if the dissipation is taken into account. This happens because the bodies dissipate energy when they rotate about any axis different from principal. In the case of an oblate symmetrical body, the angular velocity describes a circular cone about the vector of (conserved) angular momentum. In the course of relaxation, the angle of this cone decreases, so that both the angular velocity and the maximal-inertia axis of the body align along the angular momentum. The generic case of an asymmetric body is far more involved. Even the symmetrical prolate body exhibits a sophisticated behaviour, because an infinitesimally small deviation of the body's shape from a rotational symmetry (i.e., a small difference between the largest and second largest moments of inertia) yields libration: the precession trajectory is not a circle but an ellipse. In this article we show that often the most effective internal dissipation takes place at twice the frequency of the body's precession. Applications to precessing asteroids, cosmic-dust alignment, and rotating satellites are discussed.

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arXiv1996

Cross-Section Alignment of Oblate Grains

This paper provides a quantitative account of a recently introduced mechanism of mechanical alignment of suprathermally rotating grains. These rapidly rotating grains are essentially not susceptible to random torques arising from gas-grain collisions, as the timescales for such torques to have significant effect are orders of magnitude greater than the mean time between crossovers. Such grains can be aligned by gaseous torques during the short periods of crossovers and/or due to the difference in the rate at which atoms arrive at grain surface. The latter is a result of the difference in orientation of a grain in respect to the supersonic flow. This process, which we call cross-section alignment, is the subject of our present paper. We derive expressions for the measure of cross-section alignment for oblate grains and study how this measure depends upon the angle between the interstellar magnetic field and the gaseous flow and upon the grain shape.

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