89 research outputs found
Justification of the two-bulge method in the theory of bodily tides
Mathematical modeling of bodily tides can be carried out in various ways.
Most straightforward is the method of complex amplitudes, which is often used
in the planetary science. Another method, employed both in planetary science
and astrophysics, is based on decomposition of each harmonic of the tide into
two bulges oriented orthogonally to one another.
We prove that the two methods are equivalent. Specifically, we demonstrate
that the two-bulge method is not a separate approximation, but ensues directly
from the Fourier expansion of a linear tidal theory equipped with an arbitrary
rheological model involving a departure from elasticity. To this end, we use
the most general mathematical formalism applicable to linear bodily tides. To
express the tidal amendment to the potential of the perturbed primary, we act
on the tide-raising potential of the perturbing secondary with a convolution
operator. This enables us to interconnect a complex Fourier component of the
tidally generated potential of the perturbed primary with the appropriate
complex Fourier component of the tide-raising potential of the secondary. Then
we demonstrate how this interrelation entails the two-bulge description.
While less economical mathematically, the two-bulge approach has a good
illustrative power, and may be employed on a par with a more concise method of
complex amplitudes. At the same time, there exist situations where the
two-bulge method becomes more practical for technical calculations.Comment: Astronomy & Astrophysics, in pres
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.Comment: 20 pages, 1 figur
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.Comment: Submitted to the "Astronomy and Astrophysics
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
The Efroimsky formalism adapted to high-frequency perturbations
The Efroimsky perturbation scheme for consistent treatment of gravitational
waves and their influence on the background is summarized and compared with
classical Isaacson's high-frequency approach. We demonstrate that the Efroimsky
method in its present form is not compatible with the Isaacson limit of
high-frequency gravitational waves, and we propose its natural generalization
to resolve this drawback.Comment: 7 pages, to appear in Class. Quantum Gra
Gauge Invariant Effective Stress-Energy Tensors for Gravitational Waves
It is shown that if a generalized definition of gauge invariance is used,
gauge invariant effective stress-energy tensors for gravitational waves and
other gravitational perturbations can be defined in a much larger variety of
circumstances than has previously been possible. In particular it is no longer
necessary to average the stress-energy tensor over a region of spacetime which
is larger in scale than the wavelengths of the waves and it is no longer
necessary to restrict attention to high frequency gravitational waves.Comment: 11 pages, RevTe
Improved shaping approach to the preliminary design of low-thrust trajectories
This paper presents a general framework for the development of shape-based approaches to low-thrust trajectory design. A novel shaping method, based on a three-dimensional description of the trajectory in spherical coordinates, is developed within this general framework. Both the exponential sinusoid and the inverse polynomial shaping are demonstrated to be particular two-dimensional cases of the spherical one. The pseudoequinoctial shaping is revisited within the new framework, and the nonosculating nature of the pseudoequinoctial elements is analyzed. A two step approach is introduced to solve the time of flight constraint, related to the design of low-thrust arcs with boundary constraints for both spherical and pseudoequinoctial shaping. The solution derived from the shaping approach is improved with a feedback linear-quadratic controller and compared against a direct collocation method based on finite elements in time. The new shaping approach and the combination of shaping and linear-quadratic controller are tested on three case studies: a mission to Mars, a mission to asteroid 1989ML, a mission to comet Tempel-1, and a mission to Neptune
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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