1,270 research outputs found
An Order Theoretic Approach to Net Substitution Effects
We revisit the analysis of discrete comparative statics effects in the classical consumer expenditure minimization framework, using techniques that exploit the order and lattice properties of the problem, without reference to topological properties. It is shown that these comparative statics effects give rise to classes of partial orders, which in turn induce lattice structures that define the critical points of comparability (for the behavior of the utility function), meets and joins, which are used to derive sufficient conditions, from the quasi-supermodular class of properties, for a good(s) to be a net substitute or complement of another. Examples demonstrate the analysis.
Resonant periodic orbits in the exoplanetary systems
The planetary dynamics of , , , and mean motion
resonances is studied by using the model of the general three body problem in a
rotating frame and by determining families of periodic orbits for each
resonance. Both planar and spatial cases are examined. In the spatial problem,
families of periodic orbits are obtained after analytical continuation of
vertical critical orbits. The linear stability of orbits is also examined.
Concerning initial conditions nearby stable periodic orbits, we obtain
long-term planetary stability, while unstable orbits are associated with
chaotic evolution that destabilizes the planetary system. Stable periodic
orbits are of particular importance in planetary dynamics, since they can host
real planetary systems. We found stable orbits up to of mutual
planetary inclination, but in most families, the stability does not exceed
-, depending on the planetary mass ratio. Most of these
orbits are very eccentric. Stable inclined circular orbits or orbits of low
eccentricity were found in the and resonance, respectively.Comment: Accepted for publication in Astrophysics and Space Science. Link to
the published article on Springer's website was inserte
Origin and continuation of 3/2, 5/2, 3/1, 4/1 and 5/1 resonant periodic orbits in the circular and elliptic restricted three-body problem
We consider a planetary system consisting of two primaries, namely a star and
a giant planet, and a massless secondary, say a terrestrial planet or an
asteroid, which moves under their gravitational attraction. We study the
dynamics of this system in the framework of the circular and elliptic
restricted TBP, when the motion of the giant planet describes circular and
elliptic orbits, respectively. Originating from the circular family, families
of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion
resonances are continued in the circular and the elliptic problems. New
bifurcation points from the circular to the elliptic problem are found for each
of the above resonances and thus, new families, continued from these points are
herein presented. Stable segments of periodic orbits were found at high
eccentricity values of the already known families considered as whole unstable
previously. Moreover, new isolated (not continued from bifurcation points)
families are computed in the elliptic restricted problem. The majority of the
new families mainly consist of stable periodic orbits at high eccentricities.
The families of the 5/1 resonance are investigated for the first time in the
restricted three-body problems. We highlight the effect of stable periodic
orbits on the formation of stable regions in their vicinity and unveil the
boundaries of such domains in phase space by computing maps of dynamical
stability. The long-term stable evolution of the terrestrial planets or
asteroids is dependent on the existence of regular domains in their dynamical
neighbourhood in phase space, which could host them for long time spans. This
study, besides other celestial architectures that can be efficiently modelled
by the circular and elliptic restricted problems, is particularly appropriate
for the discovery of terrestrial companions among the single-giant planet
systems discovered so far.Comment: Accepted for publication in Celestial Mechanics and Dynamical
Astronom
Puzzling out the coexistence of terrestrial planets and giant exoplanets. The 2/1 resonant periodic orbits
Hundreds of giant planets have been discovered so far and the quest of
exo-Earths in giant planet systems has become intriguing. In this work, we aim
to address the question of the possible long-term coexistence of a terrestrial
companion on an orbit interior to a giant planet, and explore the extent of the
stability regions for both non-resonant and resonant configurations. Our study
focuses on the restricted three-body problem, where an inner terrestrial planet
(massless body) moves under the gravitational attraction of a star and an outer
massive planet on a circular or elliptic orbit. Using the Detrended Fast
Lyapunov Indicator as a chaotic indicator, we constructed maps of dynamical
stability by varying both the eccentricity of the outer giant planet and the
semi-major axis of the inner terrestrial planet, and identify the boundaries of
the stability domains. Guided by the computation of families of periodic
orbits, the phase space is unravelled by meticulously chosen stable periodic
orbits, which buttress the stability domains. We provide all possible stability
domains for coplanar symmetric configurations and show that a terrestrial
planet, either in mean-motion resonance or not, can coexist with a giant
planet, when the latter moves on either a circular or an (even highly)
eccentric orbit. New families of symmetric and asymmetric periodic orbits are
presented for the 2/1 resonance. It is shown that an inner terrestrial planet
can survive long time spans with a giant eccentric outer planet on resonant
symmetric orbits, even when both orbits are highly eccentric. For 22 detected
single-planet systems consisting of a giant planet with high eccentricity, we
discuss the possible existence of a terrestrial planet. This study is
particularly suitable for the research of companions among the detected systems
with giant planets, and could assist with refining observational data.Comment: Accepted for publication in A&
Continuation and stability deduction of resonant periodic orbits in three dimensional systems
In dynamical systems of few degrees of freedom, periodic solutions consist
the backbone of the phase space and the determination and computation of their
stability is crucial for understanding the global dynamics. In this paper we
study the classical three body problem in three dimensions and use its dynamics
to assess the long-term evolution of extrasolar systems. We compute periodic
orbits, which correspond to exact resonant motion, and determine their linear
stability. By computing maps of dynamical stability we show that stable
periodic orbits are surrounded in phase space with regular motion even in
systems with more than two degrees of freedom, while chaos is apparent close to
unstable ones. Therefore, families of stable periodic orbits, indeed, consist
backbones of the stability domains in phase space.Comment: Proceedings of the 6th International Conference on Numerical Analysis
(NumAn 2014). Published by the Applied Mathematics and Computers Lab,
Technical University of Crete (AMCL/TUC), Greec
Vertical instability and inclination excitation during planetary migration
We consider a two-planet system, which migrates under the influence of
dissipative forces that mimic the effects of gas-driven (Type II) migration. It
has been shown that, in the planar case, migration leads to resonant capture
after an evolution that forces the system to follow families of periodic
orbits. Starting with planets that differ slightly from a coplanar
configuration, capture can, also, occur and, additionally, excitation of
planetary inclinations has been observed in some cases. We show that excitation
of inclinations occurs, when the planar families of periodic orbits, which are
followed during the initial stages of planetary migration, become vertically
unstable. At these points, {\em vertical critical orbits} may give rise to
generating stable families of periodic orbits, which drive the evolution
of the migrating planets to non-coplanar motion. We have computed and present
here the vertical critical orbits of the and resonances, for
various values of the planetary mass ratio. Moreover, we determine the limiting
values of eccentricity for which the "inclination resonance" occurs.Comment: Accepted for publication in Celestial Mechanics and Dynamical
Astronom
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