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.Comment: Talk at the annual Princeton conference ``New Trends in Astrodynamics" 2005 http://www.math.princeton.edu/astrocon

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}).Comment: 9 figure

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.Comment: 24 pages, Post Script file. To appear in The Astrophysical Journal, Vol. 466, p. 274 - 281, July 199

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.Comment: arXiv admin note: substantial text overlap with arXiv:astro-ph/060552