1,820 research outputs found
Multidimensional Matrix Inversions and Elliptic Hypergeometric Series on Root Systems
Multidimensional matrix inversions provide a powerful tool for studying
multiple hypergeometric series. In order to extend this technique to elliptic
hypergeometric series, we present three new multidimensional matrix inversions.
As applications, we obtain a new elliptic Jackson summation, as well as
several quadratic, cubic and quartic summation formulas
From order to chaos in Earth satellite orbits
We consider Earth satellite orbits in the range of semi-major axes where the
perturbing effects of Earth's oblateness and lunisolar gravity are of
comparable order. This range covers the medium-Earth orbits (MEO) of the Global
Navigation Satellite Systems and the geosynchronous orbits (GEO) of the
communication satellites. We recall a secular and quadrupolar model, based on
the Milankovitch vector formulation of perturbation theory, which governs the
long-term orbital evolution subject to the predominant gravitational
interactions. We study the global dynamics of this two-and-a-half
degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator
(FLI), used in a statistical sense. Specifically, we characterize the degree of
chaoticity of the action space using angle-averaged normalized FLI maps,
thereby overcoming the angle dependencies of the conventional stability maps.
Emphasis is placed upon the phase-space structures near secular resonances,
which are of first importance to the space debris community. We confirm and
quantify the transition from order to chaos in MEO, stemming from the critical
inclinations, and find that highly inclined GEO orbits are particularly
unstable. Despite their reputed normality, Earth satellite orbits can possess
an extraordinarily rich spectrum of dynamical behaviors, and, from a
mathematical perspective, have all the complications that make them very
interesting candidates for testing the modern tools of chaos theory.Comment: 30 pages, 9 figures. Accepted for publication in the Astronomical
Journa
Medium Earth Orbit dynamical survey and its use in passive debris removal
The Medium Earth Orbit (MEO) region hosts satellites for navigation,
communication, and geodetic/space environmental science, among which are the
Global Navigation Satellites Systems (GNSS). Safe and efficient removal of
debris from MEO is problematic due to the high cost for maneuvers needed to
directly reach the Earth (reentry orbits) and the relatively crowded GNSS
neighborhood (graveyard orbits). Recent studies have highlighted the
complicated secular dynamics in the MEO region, but also the possibility of
exploiting these dynamics, for designing removal strategies. In this paper, we
present our numerical exploration of the long-term dynamics in MEO, performed
with the purpose of unveiling the set of reentry and graveyard solutions that
could be reached with maneuvers of reasonable DV cost. We simulated the
dynamics over 120-200 years for an extended grid of millions of fictitious MEO
satellites that covered all inclinations from 0 to 90deg, using non-averaged
equations of motion and a suitable dynamical model that accounted for the
principal geopotential terms, 3rd-body perturbations and solar radiation
pressure (SRP). We found a sizeable set of usable solutions with reentry times
that exceed ~40years, mainly around three specific inclination values: 46deg,
56deg, and 68deg; a result compatible with our understanding of MEO secular
dynamics. For DV <= 300 m/s (i.e., achieved if you start from a typical GNSS
orbit and target a disposal orbit with e<0.3), reentry times from GNSS
altitudes exceed ~70 years, while low-cost (DV ~= 5-35 m/s) graveyard orbits,
stable for at lest 200 years, are found for eccentricities up to e~0.018. This
investigation was carried out in the framework of the EC-funded "ReDSHIFT"
project.Comment: 39 pages, 23 figure
Non-averaged regularized formulations as an alternative to semi-analytical orbit propagation methods
This paper is concerned with the comparison of semi-analytical and
non-averaged propagation methods for Earth satellite orbits. We analyse the
total integration error for semi-analytical methods and propose a novel
decomposition into dynamical, model truncation, short-periodic, and numerical
error components. The first three are attributable to distinct approximations
required by the method of averaging, which fundamentally limit the attainable
accuracy. In contrast, numerical error, the only component present in
non-averaged methods, can be significantly mitigated by employing adaptive
numerical algorithms and regularized formulations of the equations of motion.
We present a collection of non-averaged methods based on the integration of
existing regularized formulations of the equations of motion through an
adaptive solver. We implemented the collection in the orbit propagation code
THALASSA, which we make publicly available, and we compared the non-averaged
methods to the semi-analytical method implemented in the orbit propagation tool
STELA through numerical tests involving long-term propagations (on the order of
decades) of LEO, GTO, and high-altitude HEO orbits. For the test cases
considered, regularized non-averaged methods were found to be up to two times
slower than semi-analytical for the LEO orbit, to have comparable speed for the
GTO, and to be ten times as fast for the HEO (for the same accuracy). We show
for the first time that efficient implementations of non-averaged regularized
formulations of the equations of motion, and especially of non-singular element
methods, are attractive candidates for the long-term study of high-altitude and
highly elliptical Earth satellite orbits.Comment: 33 pages, 10 figures, 7 tables. Part of the CMDA Topical Collection
on "50 years of Celestial Mechanics and Dynamical Astronomy". Comments and
feedback are encourage
Universal Amplitude Ratios in the Ising Model in Three Dimensions
We use a high-precision Monte Carlo simulation to determine the universal
specific-heat amplitude ratio A+/A- in the three-dimensional Ising model via
the impact angle \phi of complex temperature zeros. We also measure the
correlation-length critical exponent \nu from finite-size scaling, and the
specific-heat exponent \alpha through hyperscaling. Extrapolations to the
thermodynamic limit yield \phi = 59.2(1.0) degrees, A+/A- = 0.56(3), \nu =
0.63048(32) and \alpha = 0.1086(10). These results are compatible with some
previous estimates from a variety of sources and rule out recently conjectured
exact values.Comment: 17 pages, 5 figure
The dynamical structure of the MEO region: long-term stability, chaos, and transport
It has long been suspected that the Global Navigation Satellite Systems exist
in a background of complex resonances and chaotic motion; yet, the precise
dynamical character of these phenomena remains elusive. Recent studies have
shown that the occurrence and nature of the resonances driving these dynamics
depend chiefly on the frequencies of nodal and apsidal precession and the rate
of regression of the Moon's nodes. Woven throughout the inclination and
eccentricity phase space is an exceedingly complicated web-like structure of
lunisolar secular resonances, which become particularly dense near the
inclinations of the navigation satellite orbits. A clear picture of the
physical significance of these resonances is of considerable practical interest
for the design of disposal strategies for the four constellations. Here we
present analytical and semi-analytical models that accurately reflect the true
nature of the resonant interactions, and trace the topological organization of
the manifolds on which the chaotic motions take place. We present an atlas of
FLI stability maps, showing the extent of the chaotic regions of the phase
space, computed through a hierarchy of more realistic, and more complicated,
models, and compare the chaotic zones in these charts with the analytical
estimation of the width of the chaotic layers from the heuristic Chirikov
resonance-overlap criterion. As the semi-major axis of the satellite is
receding, we observe a transition from stable Nekhoroshev-like structures at
three Earth radii, where regular orbits dominate, to a Chirikov regime where
resonances overlap at five Earth radii. From a numerical estimation of the
Lyapunov times, we find that many of the inclined, nearly circular orbits of
the navigation satellites are strongly chaotic and that their dynamics are
unpredictable on decadal timescales.Comment: Submitted to Celestial Mechanics and Dynamical Astronomy. Comments
are greatly appreciated. 28 pages, 15 figure
Spinful bosons in an optical lattice
We analyze the behavior of cold spin-1 particles with antiferromagnetic
interactions in a one-dimensional optical lattice using density matrix
renormalization group calculations. Correlation functions and the dimerization
are shown and we also present results for the energy gap between ground state
and the spin excited states. We confirm the anticipated phase diagram, with
Mott-insulating regions of alternating dimerized S=1 chains for odd particle
density versus on-site singlets for even density. We find no evidence for any
additional ordered phases in the physically accessible region, however for
sufficiently large spin interaction, on-site singlet pairs dominate leading,
for odd density, to a breakdown of the Mott insulator or, for even density, a
real-space singlet superfluid.Comment: Minor revisions and clarification
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