2,807 research outputs found
The Instability Transition for the Restricted 3-Body Problem. III. The Lyapunov Exponent Criterion
We establish a criterion for the stability of planetary orbits in stellar
binary systems by using Lyapunov exponents and power spectra for the special
case of the circular restricted 3-body problem (CR3BP). The centerpiece of our
method is the concept of Lyapunov exponents, which are incorporated into the
analysis of orbital stability by integrating the Jacobian of the CR3BP and
orthogonalizing the tangent vectors via a well-established algorithm originally
developed by Wolf et al. The criterion for orbital stability based on the
Lyapunov exponents is independently verified by using power spectra. The
obtained results are compared to results presented in the two previous papers
of this series. It is shown that the maximum Lyapunov exponent can be used as
an indicator for chaotic behaviour of planetary orbits, which is consistent
with previous applications of this method, particularly studies for the Solar
System. The chaotic behaviour corresponds to either orbital stability or
instability, and it depends solely on the mass ratio of the binary components
and the initial distance ratio of the planet relative to the stellar separation
distance. Our theoretical results allow us to link the study of planetary
orbital stability to chaos theory noting that there is a large array of
literature on the properties and significance of Lyapunov exponents. Although
our results are given for the special case of the CR3BP, we expect that it may
be possible to augment the proposed Lyapunov exponent criterion to studies of
planets in generalized stellar binary systems, which is strongly motivated by
existing observational results as well as results expected from ongoing and
future planet search missions.Comment: 10 pages, 8 figures, 3 tables; accepted by Astronomy and Astrophysic
Where are the Uranus Trojans?
The area of stable motion for fictitious Trojan asteroids around Uranus'
equilateral equilibrium points is investigated with respect to the inclination
of the asteroid's orbit to determine the size of the regions and their shape.
For this task we used the results of extensive numerical integrations of orbits
for a grid of initial conditions around the points L4 and L5, and analyzed the
stability of the individual orbits. Our basic dynamical model was the Outer
Solar System (Jupiter, Saturn, Uranus and Neptune). We integrated the equations
of motion of fictitious Trojans in the vicinity of the stable equilibrium
points for selected orbits up to the age of the Solar system of 5 billion
years. One experiment has been undertaken for cuts through the Lagrange points
for fixed values of the inclinations, while the semimajor axes were varied. The
extension of the stable region with respect to the initial semimajor axis lies
between 19.05 < a < 19.3 AU but depends on the initial inclination. In another
run the inclination of the asteroids' orbit was varied in the range 0 < i < 60
and the semimajor axes were fixed. It turned out that only four 'windows' of
stable orbits survive: these are the orbits for the initial inclinations 0 < i
< 7, 9 < i < 13, 31 < i < 36 and 38 < i < 50. We postulate the existence of at
least some Trojans around the Uranus Lagrange points for the stability window
at small and also high inclinations.Comment: 15 pages, 12 figures, submitted to CMD
A simple mechanistic model of sprout spacing in tumour-associated angiogenesis
This paper develops a simple mathematical model of the siting of capillary sprouts on an existing blood vessel during the initiation of tumour-induced angiogenesis. The model represents an inceptive attempt to address the question of how unchecked sprouting of the parent vessel is avoided at the initiation of angiogenesis, based on the idea that feedback regulation processes play the dominant role. No chemical interaction between the proangiogenic and antiangiogenic factors is assumed. The model is based on corneal pocket experiments, and provides a mathematical analysis of the initial spacing of angiogenic sprouts
Planets in habitable zones: A study of the binary Gamma Cephei
The recently discovered planetary system in the binary GamCep was studied
concerning its dynamical evolution. We confirm that the orbital parameters
found by the observers are in a stable configuration. The primary aim of this
study was to find stable planetary orbits in a habitable region in this system,
which consists of a double star (a=21.36 AU) and a relatively close (a=2.15 AU)
massive (1.7 Mjup sin i) planet. We did straightforward numerical integrations
of the equations of motion in different dynamical models and determined the
stability regions for a fictitious massless planet in the interval of the
semimajor axis 0.5 AU < a < 1.85 AU around the more massive primary. To confirm
the results we used the Fast Lyapunov Indicators (FLI) in separate
computations, which are a common tool for determining the chaoticity of an
orbit. Both results are in good agreement and unveiled a small island of stable
motions close to 1 AU up to an inclination of about 15 deg (which corresponds
to the 3:1 mean motion resonance between the two planets). Additionally we
computed the orbits of earthlike planets (up to 90 earthmasses) in the small
stable island and found out, that there exists a small window of stable orbits
on the inner edge of the habitable zone in GamCep even for massive planets.Comment: 4 pages, 5 figures, changed 2 references made minor changes due to
referees advic
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