209 research outputs found
Towards an interpretation of MOND as a modification of inertia
We explore the possibility that Milgrom's Modified Newtonian Dynamics (MOND)
is a manifestation of the modification of inertia at small accelerations.
Consistent with the Tully-Fisher relation, dynamics in the small acceleration
domain may originate from a quartic (cubic) velocity-dependence of energy
(momentum) whereas gravitational potentials remain linear with respect to mass.
The natural framework for this interpretation is Finsler geometry. The simplest
static isotropic Finsler metric of a gravitating mass that incorporates the
Tully-Fisher relation at small acceleration is associated with a spacetime
interval that is either a homogeneous quartic root of polynomials of local
displacements or a simple root of a rational fraction thereof. We determine the
low energy gravitational equation and find that Finsler spacetimes that produce
a Tully-Fisher relation require that the gravitational potential be modified.
For an isolated mass, Newton's potential is replaced by where is MOND's acceleration scale and is a yet
undetermined distance scale. Orbital energy is linear with respect to mass but
angular momentum is proportional to . Asymptotic light deflection
resulting from time curvature is similar to that of a singular isothermal
sphere implying that space curvature must be the main source of deflection in
static Finsler spacetimes possibly through the presence of the distance scale
that appears in the asymptotic form of the gravitational potential. The
quartic nature of the Finsler metric hints at the existence of an underlying
area-metric that describes the effective structure of spacetime.Comment: Revised version, 9 pages, 1 figure. Accepted for publication in
Monthly Notices of the Royal Astronomical Societ
The Accelerated Kepler Problem
The accelerated Kepler problem is obtained by adding a constant acceleration
to the classical two-body Kepler problem. This setting models the dynamics of a
jet-sustaining accretion disk and its content of forming planets as the disk
loses linear momentum through the asymmetric jet-counterjet system it powers.
The dynamics of the accelerated Kepler problem is analyzed using physical as
well as parabolic coordinates. The latter naturally separate the problem's
Hamiltonian into two unidimensional Hamiltonians. In particular, we identify
the origin of the secular resonance in the accelerated Kepler problem and
determine analytically the radius of stability boundary of initially circular
orbits that are of particular interest to the problem of radial migration in
binary systems as well as to the truncation of accretion disks through stellar
jet acceleration.Comment: 16 pages, 9 figures, in press at Celestial Mechanics and Dynamical
Astronom
Origin theories for the eccentricities of extrasolar planets
Half the known extrasolar planets have orbital eccentricities in excess of
0.3. Such large eccentricities are surprising as it is thought that planets
form in a protoplanetary disk on nearly circular orbits much like the current
states of the solar system planets. Possible explanations for the large
planetary eccentricities include the perturbations that accompany planet-planet
scattering, the tidal interaction between the gas disk and the planets, Kozai's
secular eccentricity cycles, the eccentricity excitation during planetary pair
migration in mean motion resonance, the perturbations by stellar encounters,
stellar-like relaxation that occurs if planets formed through gravitational
instability, and the relative acceleration by the stellar jet system of the
host star with respect to the companion. In this chapter, we comment on the
relevance and characteristics of the various eccentricity origin theories.Comment: 23 pages, 8 figures. Review lecture at the 2006 Aussois Winter School
"Open Problems in Celestial Mechanics". To appear in Lecture Notes in
Physics, Springe
On dynamical friction in a gaseous medium with a boundary
Dynamical friction arises from the interaction of a perturber and the
gravitational wake it excites in the ambient medium. We study the effects of
the presence of a boundary on dynamical friction by studying analytically the
interaction of perturber with uniform rectilinear motion in a uniform
homogeneous medium with a reflecting planar boundary. Wake reflection at a
medium's boundary may occur at the edges of truncated disks perturbed by
planetary or stellar companions as well as in numerical simulations of
planet-disk interaction with no-outflow boundary conditions. In this paper, we
show that the presence of the boundary modifies the behaviour of dynamical
friction significantly. We find that perturbers are invariably pushed away from
the boundary and reach a terminal subsonic velocity near Mach 0.37 regardless
of initial velocity. Dynamical friction may even be reversed for Mach numbers
less than 0.37 thereby accelerating instead of decelerating the perturber.
Perturbers moving parallel to the boundary feel additional friction orthogonal
to the direction of motion that is much stronger than the standard friction
along the direction of motion. These results indicate that the common use of
the standard Chandrasekhar formula as a short hand estimate of dynamical
friction may be inadequate as observed in various numerical simulations.Comment: Revised version, 28 pages, 10 figures, Accepted for publication in
Astrophysics & Space Scienc
The disturbing function for polar Centaurs and transneptunian objects
The classical disturbing function of the three-body problem is based on an
expansion of the gravitational interaction in the vicinity of nearly coplanar
orbits. Consequently, it is not suitable for the identification and study of
resonances of the Centaurs and transneptunian objects on nearly polar orbits
with the solar system planets. Here, we provide a series expansion algorithm of
the gravitational interaction in the vicinity of polar orbits and produce
explicitly the disturbing function to fourth order in eccentricity and
inclination cosine. The properties of the polar series differ significantly
from those of the classical disturbing function: the polar series can model any
resonance as the expansion order is not related to the resonance order. The
powers of eccentricity and inclination of the force amplitude of a :
resonance do not depend on the value of the resonance order but only on
its parity. Thus all even resonance order eccentricity amplitudes are and odd ones to lowest order in eccentricity . With the new
findings on the structure of the polar disturbing function and the possible
resonant critical arguments, we illustrate the dynamics of the polar resonances
1:3, 3:1, 2:9 and 7:9 where transneptunian object 471325 could currently be
locked.Comment: 18 pages, 9 figures, 7 tables. Accepted for publication in Monthly
Notices of the Royal Astronomical Societ
Coorbital capture at arbitrary inclination
The process of capture in the coorbital region of a solar system planet is
studied. Absolute capture likelihood in the 1:1 resonance is determined by
randomly constructed statistical ensembles numbering of
massless asteroids that are set to migrate radially from the outer to the inner
boundaries of the coorbital region of a Jupiter-mass planet. Orbital states
include coorbital capture, ejection, collisions with the Sun and the planet and
free-crossing of the coorbital region. The relative efficiency of retrograde
capture with respect to prograde capture is confirmed as an intrinsic property
of the coorbital resonance. Half the asteroids cross the coorbital region
regardless of eccentricity and for any inclination less than . We
also find that the recently discovered retrograde coorbital of Jupiter,
asteroid 2015 BZ509, lies almost exactly at the capture efficiency peak
associated with its orbital parameters.Comment: 8 pages. 2 figures. Submitted to Journal of Computational and Applied
Mathematic
Dynamical friction for accelerated motion in a gaseous medium
Dynamical friction arises from the interaction of a perturber and the
gravitational wake it excites in the ambient medium. This interaction is
usually derived assuming that the perturber has a constant velocity. In
realistic situations, motion is accelerated as for instance by dynamical
friction itself. Here, we study the effect of acceleration on the dynamical
friction force. We characterize the density enhancement associated with a
constantly accelerating perturber with rectilinear motion in an infinite
homogeneous gaseous medium and show that dynamical friction is not a local
force and that its amplitude may depend on the perturber's initial velocity.
The force on an accelerating perturber is maximal between Mach 1 and Mach 2,
where it is smaller than the corresponding uniform motion friction. In the
limit where the perturber's size is much smaller than the distance needed to
change the Mach number by unity through acceleration, a subsonic perturber
feels a force similar to uniform motion friction only if its past history does
not include supersonic episodes. Once an accelerating perturber reaches large
supersonic speeds, accelerated motion friction is marginally stronger than
uniform motion friction. The force on a decelerating supersonic perturber is
weaker than uniform motion friction as the velocity decreases to a few times
the sound speed. Dynamical friction on a decelerating subsonic perturber with
an initial Mach number larger than 2 is much larger than uniform motion
friction and tends to a finite value as the velocity vanishes in contrast to
uniform motion friction.Comment: Published in MNRAS. Revised version (minor typos corrected
Retrograde resonance in the planar three-body problem
We continue the investigation of the dynamics of retrograde resonances
initiated in Morais & Giuppone (2012). After deriving a procedure to deduce the
retrograde resonance terms from the standard expansion of the three-dimensional
disturbing function, we concentrate on the planar problem and construct
surfaces of section that explore phase-space in the vicinity of the main
retrograde resonances (2/-1, 1/-1 and 1/-2). In the case of the 1/-1 resonance
for which the standard expansion is not adequate to describe the dynamics, we
develop a semi-analytic model based on numerical averaging of the unexpanded
disturbing function, and show that the predicted libration modes are in
agreement with the behavior seen in the surfaces of section.Comment: Celestial Mechanics and Dynamical Astronomy, in pres
A numerical investigation of coorbital stability and libration in three dimensions
Motivated by the dynamics of resonance capture, we study numerically the
coorbital resonance for inclination180 >=I>=0 in the circular restricted
three-body problem. We examine the similarities and differences between planar
and three dimensional coorbital resonance capture and seek their origin in the
stability of coorbital motion at arbitrary inclination. After we present
stability maps of the planar prograde and retrograde coorbital resonances, we
characterize the new coorbital modes in three dimensions. We see that
retrograde mode I (R1) and mode II (R2) persist as we change the relative
inclination, while retrograde mode III (R3) seems to exist only in the planar
problem. A new coorbital mode (R4) appears in 3D which is a retrograde analogue
to an horseshoe-orbit. The Kozai-Lidov resonance is active for retrograde
orbits as well as prograde orbits and plays a key role in coorbital resonance
capture. Stable coorbital modes exist at all inclinations, including retrograde
and polar obits. This result confirms the robustness the coorbital resonance at
large inclination and encourages the search for retrograde coorbital companions
of the solar system's planets.Comment: accepted for publication in Celestial Mechanics and Dynamical
Astronom
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