78,692 research outputs found

### First order resonance overlap and the stability of close two planet systems

Motivated by the population of multi-planet systems with orbital period
ratios 1<P2/P1<2, we study the long-term stability of packed two planet
systems. The Hamiltonian for two massive planets on nearly circular and nearly
coplanar orbits near a first order mean motion resonance can be reduced to a
one degree of freedom problem (Sessin & Ferraz Mello (1984), Wisdom (1986),
Henrard et al. (1986)). Using this analytically tractable Hamiltonian, we apply
the resonance overlap criterion to predict the onset of large scale chaotic
motion in close two planet systems. The reduced Hamiltonian has only a weak
dependence on the planetary mass ratio, and hence the overlap criterion is
independent of the planetary mass ratio at lowest order. Numerical integrations
confirm that the planetary mass ratio has little effect on the structure of the
chaotic phase space for close orbits in the low eccentricity (e <~0.1) regime.
We show numerically that orbits in the chaotic web produced primarily by first
order resonance overlap eventually experience large scale erratic variation in
semimajor axes and are Lagrange unstable. This is also true of the orbits in
this overlap region which are Hill stable. As a result, we can use the first
order resonance overlap criterion as an effective stability criterion for pairs
of observed planets. We show that for low mass (<~10 M_Earth) planetary systems
with initially circular orbits the period ratio at which complete overlap
occurs and widespread chaos results lies in a region of parameter space which
is Hill stable. Our work indicates that a resonance overlap criterion which
would apply for initially eccentric orbits needs to take into account second
order resonances. Finally, we address the connection found in previous work
between the Hill stability criterion and numerically determined Lagrange
instability boundaries in the context of resonance overlap.Comment: Accepted for publication in Ap

### Quantal Overlapping Resonance Criterion in the Pullen Edmonds Model

In order to highlight the onset of chaos in the Pullen-Edmonds model a
quantal analog of the resonance overlap criterion has been examined. A quite
good agreement between analytical and numerical results is obtained.Comment: 12 pages, LATEX, 2 figures available upon request to the Authors,
submitted to Mod. Phys. Lett.

### Chaos Suppression in the SU(2) Yang--Mills--Higgs System

We study the classical chaos--order transition in the spatially homogenous
SU(2) Yang--Mills--Higgs system by using a quantal analog of Chirikov's
resonance overlap criterion. We obtain an analytical estimation of the range of
parameters for which there is chaos suppression.Comment: LaTex, 10 pages, to be published in Phys. Rev.

### The Resonance Overlap and Hill Stability Criteria Revisited

We review the orbital stability of the planar circular restricted three-body
problem, in the case of massless particles initially located between both
massive bodies. We present new estimates of the resonance overlap criterion and
the Hill stability limit, and compare their predictions with detailed dynamical
maps constructed with N-body simulations. We show that the boundary between
(Hill) stable and unstable orbits is not smooth but characterized by a rich
structure generated by the superposition of different mean-motion resonances
which does not allow for a simple global expression for stability.
We propose that, for a given perturbing mass $m_1$ and initial eccentricity
$e$, there are actually two critical values of the semimajor axis. All values
$a a_{\rm unstable}$ are
unstable in the Hill sense. The first limit is given by the Hill-stability
criterion and is a function of the eccentricity. The second limit is virtually
insensitive to the initial eccentricity, and closely resembles a new resonance
overlap condition (for circular orbits) developed in terms of the intersection
between first and second-order mean-motion resonances.Comment: 33 pages, 14 figures, accepte

### A semi-empirical stability criterion for real planetary systems

We test a crossing orbit stability criterion for eccentric planetary systems,
based on Wisdom's criterion of first order mean motion resonance overlap
(Wisdom, 1980).
We show that this criterion fits the stability regions in real exoplanet
systems quite well. In addition, we show that elliptical orbits can remain
stable even for regions where the apocenter distance of the inner orbit is
larger than the pericenter distance of the outer orbit, as long as the initial
orbits are aligned.
The analytical expressions provided here can be used to put rapid constraints
on the stability zones of multi-planetary systems. As a byproduct of this
research, we further show that the amplitude variations of the eccentricity can
be used as a fast-computing stability indicator.Comment: 11 pages, 11 figures. MNRAS accepte

### Drastic facilitation of the onset of global chaos in a periodically driven Hamiltonian system due to an extremum in the dependence of eigenfrequency on energy

The Chirikov resonance-overlap criterion predicts the onset of global chaos
if nonlinear resonances overlap in energy, which is conventionally assumed to
require a non-small magnitude of perturbation. We show that, for a
time-periodic perturbation, the onset of global chaos may occur at unusually
{\it small} magnitudes of perturbation if the unperturbed system possesses more
than one separatrix. The relevant scenario is the combination of the overlap in
the phase space between resonances of the same order and their overlap in
energy with chaotic layers associated with separatrices of the unperturbed
system. One of the most important manifestations of this effect is a drastic
increase of the energy range involved into the unbounded chaotic transport in
spatially periodic system driven by a rather {\it weak} time-periodic force,
provided the driving frequency approaches the extremal eigenfrequency or its
harmonics. We develop the asymptotic theory and verify it in simulations.Comment: 5 pages, 4 figures, LaTeX, to appear PR

### Stability Boundaries for Resonant Migrating Planet Pairs

Convergent migration allows pairs of planet to become trapped into mean
motion resonances. Once in resonance, the planets' eccentricities grow to an
equilibrium value that depends on the ratio of migration time scale to the
eccentricity damping timescale, $K=\tau_a/\tau_e$, with higher values of
equilibrium eccentricity for lower values of $K$. For low equilibrium
eccentricities, $e_{eq}\propto K^{-1/2}$. The stability of a planet pair
depends on eccentricity so the system can become unstable before it reaches its
equilibrium eccentricity. Using a resonant overlap criterion that takes into
account the role of first and second order resonances and depends on
eccentricity, we find a function $K_{min}(\mu_p, j)$ that defines the lowest
value for $K$, as a function of the ratio of total planet mass to stellar mass
($\mu_p$) and the period ratio of the resonance defined as $P_1/P_2=j/(j+k)$,
that allows two convergently migrating planets to remain stable in resonance at
their equilibrium eccentricities. We scaled the functions $K_{min}$ for each
resonance of the same order into a single function $K_c$. The function $K_{c}$
for planet pairs in first order resonances is linear with increasing planet
mass and quadratic for pairs in second order resonances with a coefficient
depending on the relative migration rate and strongly on the planet to planet
mass ratio. The linear relation continues until the mass approaches a critical
mass defined by the 2/7 resonance overlap instability law and $K_c \to \infty$.
We compared our analytic boundary with an observed sample of resonant two
planet systems. All but one of the first order resonant planet pair systems
found by radial velocity measurements are well inside the stability region
estimated by this model. We calculated $K_c$ for Kepler systems without
well-constrained eccentricities and found only weak constraints on $K$.Comment: 11 pages, 7 figure

### Three Body Resonance Overlap in Closely Spaced Multiple Planet Systems

We compute the strengths of zero-th order (in eccentricity) three-body
resonances for a co-planar and low eccentricity multiple planet system. In a
numerical integration we illustrate that slowly moving Laplace angles are
matched by variations in semi-major axes among three bodies with the outer two
bodies moving in the same direction and the inner one moving in the opposite
direction, as would be expected from the two quantities that are conserved in
the three-body resonance. A resonance overlap criterion is derived for the
closely and equally spaced, equal mass system with three-body resonances
overlapping when interplanetary separation is less than an order unity factor
times the planet mass to the one quarter power. We find that three-body
resonances are sufficiently dense to account for wander in semi-major axis seen
in numerical integrations of closely spaced systems and they are likely the
cause of instability of these systems. For interplanetary separations outside
the overlap region, stability timescales significantly increase. Crudely
estimated diffusion coefficients in eccentricity and semi-major axis depend on
a high power of planet mass and interplanetary spacing. An exponential
dependence previously fit to stability or crossing timescales is likely due to
the limited range of parameters and times possible in integration and the
strong power law dependence of the diffusion rates on these quantities.Comment: submitted to MNRA

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