78 research outputs found
Microscopic dynamics underlying the anomalous diffusion
The time dependent Tsallis statistical distribution describing anomalous
diffusion is usually obtained in the literature as the solution of a non-linear
Fokker-Planck (FP) equation [A.R. Plastino and A. Plastino, Physica A, 222, 347
(1995)]. The scope of the present paper is twofold. Firstly we show that this
distribution can be obtained also as solution of the non-linear porous media
equation. Secondly we prove that the time dependent Tsallis distribution can be
obtained also as solution of a linear FP equation [G. Kaniadakis and P.
Quarati, Physica A, 237, 229 (1997)] with coefficients depending on the
velocity, that describes a generalized Brownian motion. This linear FP equation
is shown to arise from a microscopic dynamics governed by a standard Langevin
equation in presence of multiplicative noise.Comment: 4 pag. - no figures. To appear on Phys. Rev. E 62, September 200
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
Mars and frame-dragging: study for a dedicated mission
In this paper we preliminarily explore the possibility of designing a
dedicated satellite-based mission to measure the general relativistic
gravitomagnetic Lense-Thirring effect in the gravitational field of Mars. The
focus is on the systematic error induced by the multipolar expansion of the
areopotential and on possible strategies to reduce it. It turns out that the
major sources of bias are the Mars'equatorial radius R and the even zonal
harmonics J_L, L = 2,4,6... of the areopotential. An optimal solution, in
principle, consists of using two probes at high-altitudes (a\approx 9500-9600
km) and different inclinations, and suitably combining their nodes in order to
entirely cancel out the bias due to \delta R. The remaining uncancelled
mismodelled terms due to \delta J_L, L = 2,4,6,... would induce a bias \lesssim
1%, according to the present-day MGS95J gravity model, over a wide range of
admissible values of the inclinations. The Lense-Thirring out-of-plane shifts
of the two probes would amount to about 10 cm yr^-1.Comment: LaTex2e, 16 pages, 5 figures, no tables. To appear in General
Relativity and Gravitatio
Will the recently approved LARES mission be able to measure the Lense-Thirring effect at 1%?
After the approval by the Italian Space Agency of the LARES satellite, which
should be launched at the end of 2009 with a VEGA rocket and whose claimed goal
is a about 1% measurement of the general relativistic gravitomagnetic
Lense-Thirring effect in the gravitational field of the spinning Earth, it is
of the utmost importance to reliably assess the total realistic accuracy that
can be reached by such a mission. The observable is a linear combination of the
nodes of the existing LAGEOS and LAGEOS II satellites and of LARES able to
cancel out the impact of the first two even zonal harmonic coefficients of the
multipolar expansion of the classical part of the terrestrial gravitational
potential representing a major source of systematic error. While LAGEOS and
LAGEOS II fly at altitudes of about 6000 km, LARES will be placed at an
altitude of 1450 km. Thus, it will be sensitive to much more even zonals than
LAGEOS and LAGEOS II. Their corrupting impact \delta\mu has been evaluated by
using the standard Kaula's approach up to degree L=70 along with the sigmas of
the covariance matrices of eight different global gravity solutions
(EIGEN-GRACE02S, EIGEN-CG03C, GGM02S, GGM03S, JEM01-RL03B, ITG-Grace02s,
ITG-Grace03, EGM2008) obtained by five institutions (GFZ, CSR, JPL, IGG, NGA)
with different techniques from long data sets of the dedicated GRACE mission.
It turns out \delta\mu about 100-1000% of the Lense-Thirring effect. An
improvement of 2-3 orders of magnitude in the determination of the high degree
even zonals would be required to constrain the bias to about 1-10%.Comment: Latex, 15 pages, 1 table, no figures. Final version matching the
published one in General Relativity and Gravitation (GRG
Tidal friction in close-in satellites and exoplanets. The Darwin theory re-visited
This report is a review of Darwin's classical theory of bodily tides in which
we present the analytical expressions for the orbital and rotational evolution
of the bodies and for the energy dissipation rates due to their tidal
interaction. General formulas are given which do not depend on any assumption
linking the tidal lags to the frequencies of the corresponding tidal waves
(except that equal frequency harmonics are assumed to span equal lags).
Emphasis is given to the cases of companions having reached one of the two
possible final states: (1) the super-synchronous stationary rotation resulting
from the vanishing of the average tidal torque; (2) the capture into a 1:1
spin-orbit resonance (true synchronization). In these cases, the energy
dissipation is controlled by the tidal harmonic with period equal to the
orbital period (instead of the semi-diurnal tide) and the singularity due to
the vanishing of the geometric phase lag does not exist. It is also shown that
the true synchronization with non-zero eccentricity is only possible if an
extra torque exists opposite to the tidal torque. The theory is developed
assuming that this additional torque is produced by an equatorial permanent
asymmetry in the companion. The results are model-dependent and the theory is
developed only to the second degree in eccentricity and inclination
(obliquity). It can easily be extended to higher orders, but formal accuracy
will not be a real improvement as long as the physics of the processes leading
to tidal lags is not better known.Comment: 30 pages, 7 figures, corrected typo
Bodily tides near spin-orbit resonances
Spin-orbit coupling can be described in two approaches. The method known as
"the MacDonald torque" is often combined with an assumption that the quality
factor Q is frequency-independent. This makes the method inconsistent, because
the MacDonald theory tacitly fixes the rheology by making Q scale as the
inverse tidal frequency.
Spin-orbit coupling can be treated also in an approach called "the Darwin
torque". While this theory is general enough to accommodate an arbitrary
frequency-dependence of Q, this advantage has not yet been exploited in the
literature, where Q is assumed constant or is set to scale as inverse tidal
frequency, the latter assertion making the Darwin torque equivalent to a
corrected version of the MacDonald torque.
However neither a constant nor an inverse-frequency Q reflect the properties
of realistic mantles and crusts, because the actual frequency-dependence is
more complex. Hence the necessity to enrich the theory of spin-orbit
interaction with the right frequency-dependence. We accomplish this programme
for the Darwin-torque-based model near resonances. We derive the
frequency-dependence of the tidal torque from the first principles, i.e., from
the expression for the mantle's compliance in the time domain. We also explain
that the tidal torque includes not only the secular part, but also an
oscillating part.
We demonstrate that the lmpq term of the Darwin-Kaula expansion for the tidal
torque smoothly goes through zero, when the secondary traverses the lmpq
resonance (e.g., the principal tidal torque smoothly goes through nil as the
secondary crosses the synchronous orbit).
We also offer a possible explanation for the unexpected frequency-dependence
of the tidal dissipation rate in the Moon, discovered by LLR
Phenomenology of the Lense-Thirring effect in the Solar System
Recent years have seen increasing efforts to directly measure some aspects of
the general relativistic gravitomagnetic interaction in several astronomical
scenarios in the solar system. After briefly overviewing the concept of
gravitomagnetism from a theoretical point of view, we review the performed or
proposed attempts to detect the Lense-Thirring effect affecting the orbital
motions of natural and artificial bodies in the gravitational fields of the
Sun, Earth, Mars and Jupiter. In particular, we will focus on the evaluation of
the impact of several sources of systematic uncertainties of dynamical origin
to realistically elucidate the present and future perspectives in directly
measuring such an elusive relativistic effect.Comment: LaTex, 51 pages, 14 figures, 22 tables. Invited review, to appear in
Astrophysics and Space Science (ApSS). Some uncited references in the text
now correctly quoted. One reference added. A footnote adde
A resonant-term-based model including a nascent disk, precession, and oblateness: application to GJ 876
Investigations of two resonant planets orbiting a star or two resonant
satellites orbiting a planet often rely on a few resonant and secular terms in
order to obtain a representative quantitative description of the system's
dynamical evolution. We present a semianalytic model which traces the orbital
evolution of any two resonant bodies in a first- through fourth-order
eccentricity or inclination-based resonance dominated by the resonant and
secular arguments of the user's choosing. By considering the variation of
libration width with different orbital parameters, we identify regions of phase
space which give rise to different resonant ''depths,'' and propose methods to
model libration profiles. We apply the model to the GJ 876 extrasolar planetary
system, quantify the relative importance of the relevant resonant and secular
contributions, and thereby assess the goodness of the common approximation of
representing the system by just the presumably dominant terms. We highlight the
danger in using ''order'' as the metric for accuracy in the orbital solution by
revealing the unnatural libration centers produced by the second-order, but not
first-order, solution, and by demonstrating that the true orbital solution lies
somewhere ''in-between'' the third- and fourth-order solutions. We also present
formulas used to incorporate perturbations from central-body oblateness and
precession, and a protoplanetary or protosatellite thin disk with gaps, into a
resonant system. We quantify these contributions to the GJ 876 system, and
thereby highlight the conditions which must exist for multi-planet exosystems
to be significantly influenced by such factors. We find that massive enough
disks may convert resonant libration into circulation; such disk-induced
signatures may provide constraints for future studies of exoplanet systems.Comment: 39 pages of body text, 21 figures, 5 tables, 1 appendix, accepted for
publication in Celestial Mechanics and Dynamical Astronom
An Assessment of the Systematic Uncertainty in Present and Future Tests of the Lense-Thirring Effect with Satellite Laser Ranging
We deal with the attempts to measure the Lense-Thirring effect with the
Satellite Laser Ranging (SLR) technique applied to the existing LAGEOS and
LAGEOS II terrestrial satellites and to the recently approved LARES
spacecraft.The first issue addressed here is: are the so far published
evaluations of the systematic uncertainty induced by the bad knowledge of the
even zonal harmonic coefficients J_L of the multipolar expansion of the Earth's
geopotential reliable and realistic?
Our answer is negative. Indeed, if the differences Delta J_L among the even
zonals estimated in different Earth's gravity field global solutions from the
dedicated GRACE mission are assumed for the uncertainties delta J_L instead of
using their covariance sigmas sigma_JL, it turns out that the systematic
uncertainty \delta\mu in the Lense-Thirring test with the nodes Omega of LAGEOS
and LAGEOS II may be up to 3 to 4 times larger than in the evaluations so far
published () based on the use of the sigmas of one model at a time
separately. The second issue consists of the possibility of using a different
approach in extracting the relativistic signature of interest from the
LAGEOS-type data. The third issue is the possibility of reaching a realistic
total accuracy of 1% with LAGEOS, LAGEOS II and LARES, which should be launched
in November 2009 with a VEGA rocket. While LAGEOS and LAGEOS II fly at
altitudes of about 6000 km, LARES will be likely placed at an altitude of 1450
km. Thus, it will be sensitive to much more even zonals than LAGEOS and LAGEOS
II. Their corrupting impact has been evaluated with the standard Kaula's
approach up to degree L=60 by using Delta J_L and sigma_JL; it turns out that
it may be as large as some tens percent.Comment: LaTex, 19 pages, 1 figure, 12 tables. Invited and refereed
contribution to The ISSI Workshop, 6-10 October 2008, on The Nature of
Gravity Confronting Theory and Experiment in Space To appear in Space Science
Review
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