11 research outputs found
Orbit Determination with the two-body Integrals
We investigate a method to compute a finite set of preliminary orbits for
solar system bodies using the first integrals of the Kepler problem. This
method is thought for the applications to the modern sets of astrometric
observations, where often the information contained in the observations allows
only to compute, by interpolation, two angular positions of the observed body
and their time derivatives at a given epoch; we call this set of data
attributable. Given two attributables of the same body at two different epochs
we can use the energy and angular momentum integrals of the two-body problem to
write a system of polynomial equations for the topocentric distance and the
radial velocity at the two epochs. We define two different algorithms for the
computation of the solutions, based on different ways to perform elimination of
variables and obtain a univariate polynomial. Moreover we use the redundancy of
the data to test the hypothesis that two attributables belong to the same body
(linkage problem). It is also possible to compute a covariance matrix,
describing the uncertainty of the preliminary orbits which results from the
observation error statistics. The performance of this method has been
investigated by using a large set of simulated observations of the Pan-STARRS
project.Comment: 23 pages, 1 figur
Orbit Determination with the two-body Integrals. II
The first integrals of the Kepler problem are used to compute preliminary
orbits starting from two short observed arcs of a celestial body, which may be
obtained either by optical or radar observations. We write polynomial equations
for this problem, that we can solve using the powerful tools of computational
Algebra. An algorithm to decide if the linkage of two short arcs is successful,
i.e. if they belong to the same observed body, is proposed and tested
numerically. In this paper we continue the research started in [Gronchi,
Dimare, Milani, 'Orbit determination with the two-body intergrals', CMDA (2010)
107/3, 299-318], where the angular momentum and the energy integrals were used.
A suitable component of the Laplace-Lenz vector in place of the energy turns
out to be convenient, in fact the degree of the resulting system is reduced to
less than half.Comment: 15 pages, 4 figure
Tidal torques. A critical review of some techniques
We point out that the MacDonald formula for body-tide torques is valid only
in the zeroth order of e/Q, while its time-average is valid in the first order.
So the formula cannot be used for analysis in higher orders of e/Q. This
necessitates corrections in the theory of tidal despinning and libration
damping.
We prove that when the inclination is low and phase lags are linear in
frequency, the Kaula series is equivalent to a corrected version of the
MacDonald method. The correction to MacDonald's approach would be to set the
phase lag of the integral bulge proportional to the instantaneous frequency.
The equivalence of descriptions gets violated by a nonlinear
frequency-dependence of the lag.
We explain that both the MacDonald- and Darwin-torque-based derivations of
the popular formula for the tidal despinning rate are limited to low
inclinations and to the phase lags being linear in frequency. The
Darwin-torque-based derivation, though, is general enough to accommodate both a
finite inclination and the actual rheology.
Although rheologies with Q scaling as the frequency to a positive power make
the torque diverge at a zero frequency, this reveals not the impossible nature
of the rheology, but a flaw in mathematics, i.e., a common misassumption that
damping merely provides lags to the terms of the Fourier series for the tidal
potential. A hydrodynamical treatment (Darwin 1879) had demonstrated that the
magnitudes of the terms, too, get changed. Reinstating of this detail tames the
infinities and rehabilitates the "impossible" scaling law (which happens to be
the actual law the terrestrial planets obey at low frequencies).Comment: arXiv admin note: sections 4 and 9 of this paper contain substantial
text overlap with arXiv:0712.105