573 research outputs found
The non-integrability of the Zipoy-Voorhees metric
The low frequency gravitational wave detectors like eLISA/NGO will give us
the opportunity to test whether the supermassive compact objects lying at the
centers of galaxies are indeed Kerr black holes. A way to do such a test is to
compare the gravitational wave signals with templates of perturbed black hole
spacetimes, the so-called bumpy black hole spacetimes. The Zipoy-Voorhees (ZV)
spacetime (known also as the spacetime) can be included in the bumpy
black hole family, because it can be considered as a perturbation of the
Schwarzschild spacetime background. Several authors have suggested that the ZV
metric corresponds to an integrable system. Contrary to this integrability
conjecture, in the present article it is shown by numerical examples that in
general ZV belongs to the family of non-integrable systems.Comment: 10 pages, 13 figure
Periodic Orbits and Escapes in Dynamical Systems
We study the periodic orbits and the escapes in two different dynamical
systems, namely (1) a classical system of two coupled oscillators, and (2) the
Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a
general relativistic system). We find their simple periodic orbits, their
characteristics and their stability. Then we find their ordered and chaotic
domains. As the energy goes beyond the escape energy, most chaotic orbits
escape. In the first case we consider escapes to infinity, while in the second
case we emphasize escapes to the central "bumpy" black hole. When the energy
reaches its escape value a particular family of periodic orbits reaches an
infinite period and then the family disappears (the orbit escapes). As this
family approaches termination it undergoes an infinity of equal period and
double period bifurcations at transitions from stability to instability and
vice versa. The bifurcating families continue to exist beyond the escape
energy. We study the forms of the phase space for various energies, and the
statistics of the chaotic and escaping orbits. The proportion of these orbits
increases abruptly as the energy goes beyond the escape energy.Comment: 28 pages, 23 figures, accepted in "Celestial Mechanics and Dynamical
Astronomy
How to observe a non-Kerr spacetime
We present a generic criterion which can be used in gravitational-wave data
analysis to distinguish an extreme-mass-ratio inspiral into a Kerr background
spacetime from one into a non-Kerr background spacetime. The criterion exploits
the fact that when an integrable system, such as the system that describes
geodesic orbits in a Kerr spacetime, is perturbed, the tori in phase space
which initially corresponded to resonances disintegrate so as to form the so
called Birkhoff chains on a surface of section, according to the
Poincar\'{e}-Birkhoff theorem. The KAM curves of these islands in such a chain
share the same ratio of frequencies, even though the frequencies themselves
vary from one KAM curve to another inside an island. On the other hand, the KAM
curves, which do not lie in a Birkhoff chain, do not share this characteristic
property. Such a temporal constancy of the ratio of frequencies during the
evolution of the gravitational-wave signal will signal a non-Kerr spacetime
which could then be further explored.Comment: 4 pages, 2 figure
Analytical description of the structure of chaos
We consider analytical formulae that describe the chaotic regions around the
main periodic orbit of the H\'{e}non map. Following our previous
paper (Efthymiopoulos, Contopoulos, Katsanikas ) we introduce new
variables in which the product (constant) gives
hyperbolic invariant curves. These hyperbolae are mapped by a canonical
transformation to the plane , giving "Moser invariant curves". We
find that the series are convergent up to a maximum value of
. We give estimates of the errors due to the finite truncation of
the series and discuss how these errors affect the applicability of analytical
computations. For values of the basic parameter of the H\'{e}non map
smaller than a critical value, there is an island of stability, around a stable
periodic orbit , containing KAM invariant curves. The Moser curves for are completely outside the last KAM curve around , the curves
with intersect the last KAM curve and the curves with are completely inside the last KAM curve. All orbits in
the chaotic region around the periodic orbit , although they seem
random, belong to Moser invariant curves, which, therefore define a "structure
of chaos". Orbits starting close and outside the last KAM curve remain close to
it for a stickiness time that is estimated analytically using the series
. We finally calculate the periodic orbits that accumulate close to the
homoclinic points, i.e. the points of intersection of the asymptotic curves
from , exploiting a method based on the self-intersections of the
invariant Moser curves. We find that all the computed periodic orbits are
generated from the stable orbit for smaller values of the H\'{e}non
parameter , i.e. they are all regular periodic orbits.Comment: 22 pages, 9 figure
Resonant normal form and asymptotic normal form behavior in magnetic bottle Hamiltonians
We consider normal forms in `magnetic bottle' type Hamiltonians of the form
(second
frequency equal to zero in the lowest order). Our main results are:
i) a novel method to construct the normal form in cases of resonance, and ii) a
study of the asymptotic behavior of both the non-resonant and the resonant
series. We find that, if we truncate the normal form series at order , the
series remainder in both constructions decreases with increasing down to a
minimum, and then it increases with . The computed minimum remainder turns
to be exponentially small in , where is the
mirror oscillation energy, while the optimal order scales as an inverse power
of . We estimate numerically the exponents associated with the
optimal order and the remainder's exponential asymptotic behavior. In the
resonant case, our novel method allows to compute a `quasi-integral' (i.e.
truncated formal integral) valid both for each particular resonance as well as
away from all resonances. We applied these results to a specific magnetic
bottle Hamiltonian. The non resonant normal form yields theorerical invariant
curves on a surface of section which fit well the empirical curves away from
resonances. On the other hand the resonant normal form fits very well both the
invariant curves inside the islands of a particular resonance as well as the
non-resonant invariant curves. Finally, we discuss how normal forms allow to
compute a critical threshold for the onset of global chaos in the magnetic
bottle.Comment: 20 pages, 7 figure
Invariant manifolds and the response of spiral arms in barred galaxies
The unstable invariant manifolds of the short-period family of periodic
orbits around the unstable Lagrangian points and of a barred galaxy
define loci in the configuration space which take the form of a trailing spiral
pattern. In the present paper we investigate this association in the case of
the self-consistent models of Kaufmann & Contopoulos (1996) which provide an
approximation of real barred-spiral galaxies. We also examine the relation of
`response' models of barred-spiral galaxies with the theory of the invariant
manifolds. Our main results are the following: The invariant manifolds yield
the correct form of the imposed spiral pattern provided that their calculation
is done with the spiral potential term turned on. We provide a theoretical
model explaining the form of the invariant manifolds that supports the spiral
structure. The azimuthal displacement of the Lagrangian points with respect to
the bar's major axis is a crucial parameter in this modeling. When this is
taken into account, the manifolds necessarily develop in a spiral-like domain
of the configuration space, delimited from below by the boundary of a
banana-like non-permitted domain, and from above either by rotational KAM tori
or by cantori forming a stickiness zone. We construct `spiral response' models
on the basis of the theory of the invariant manifolds and examine the
connection of the latter to the `response' models (Patsis 2006) used to fit
real barred-spiral galaxies, explaining how are the manifolds related to a
number of morphological features seen in such models.Comment: 16 Page
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