313 research outputs found
The Carter Constant for Inclined Orbits About a Massive Kerr Black Hole: I. circular orbits
In an extreme binary black hole system, an orbit will increase its angle of
inclination (i) as it evolves in Kerr spacetime. We focus our attention on the
behaviour of the Carter constant (Q) for near-polar orbits; and develop an
analysis that is independent of and complements radiation reaction models. For
a Schwarzschild black hole, the polar orbits represent the abutment between the
prograde and retrograde orbits at which Q is at its maximum value for given
values of latus rectum (l) and eccentricity (e). The introduction of spin (S =
|J|/M2) to the massive black hole causes this boundary, or abutment, to be
moved towards greater orbital inclination; thus it no longer cleanly separates
prograde and retrograde orbits. To characterise the abutment of a Kerr black
hole (KBH), we first investigated the last stable orbit (LSO) of a
test-particle about a KBH, and then extended this work to general orbits. To
develop a better understanding of the evolution of Q we developed analytical
formulae for Q in terms of l, e, and S to describe elliptical orbits at the
abutment, polar orbits, and last stable orbits (LSO). By knowing the analytical
form of dQ/dl at the abutment, we were able to test a 2PN flux equation for Q.
We also used these formulae to numerically calculate the di/dl of hypothetical
circular orbits that evolve along the abutment. From these values we have
determined that di/dl = -(122.7S - 36S^3)l^-11/2 -(63/2 S + 35/4 S^3) l^-9/2
-15/2 S l^-7/2 -9/2 S l^-5/2. Thus the abutment becomes an important analytical
and numerical laboratory for studying the evolution of Q and i in Kerr
spacetime and for testing current and future radiation back-reaction models for
near-polar retrograde orbits.Comment: 51 pages, 8 figures, accepted by Classical and Quantum Gravity on
September 22nd, 201
A Study of Elliptical Last Stable Orbits About a Massive Kerr Black Hole
The last stable orbit (LSO) of a compact object (CO) is an important boundary
condition when performing numerical analysis of orbit evolution. Although the
LSO is already well understood for the case where a test-particle is in an
elliptical orbit around a Schwarzschild black hole (SBH) and for the case of a
circular orbit about a Kerr black hole (KBH) of normalised spin, S (|J|/M^2,
where J is the spin angular momentum of the KBH); it is worthwhile to extend
our knowledge to include elliptical orbits about a KBH. This extension helps to
lay the foundation for a better understanding of gravitational wave (GW)
emission. The mathematical developments described in this work sprang from the
use of an effective potential (V) derived from the Kerr metric, which
encapsulates the Lense-Thirring precession. That allowed us to develop a new
form of analytical expression to calculate the LSO Radius for circular orbits
(R_LSO) of arbitrary KBH spin. We were then able to construct a numerical
method to calculate the latus rectum (l_LSO) for an elliptical LSO.
Abstract Formulae for E^2 (square of normalised orbital energy) and L^2
(square of normalised orbital angular momentum) in terms of eccentricity, e,
and latus rectum, l, were previously developed by others for elliptical orbits
around an SBH and then extended to the KBH case; we used these results to
generalise our analytical l_LSO equations to elliptical orbits. LSO data
calculated from our analytical equations and numerical procedures, and those
previously published, are then compared and found to be in excellent agreement.Comment: 42 pages, 9 figures, accepted for publication in Classical and
Quantum Gravit
Central Limit Theorem and recurrence for random walks in bistochastic random environments
We prove the annealed Central Limit Theorem for random walks in bistochastic
random environments on with zero local drift. The proof is based on a
"dynamicist's interpretation" of the system, and requires a much weaker
condition than the customary uniform ellipticity. Moreover, recurrence is
derived for .Comment: 13 pages; to appear in the special issue of J. Math. Phys. on
"Statistical Mechanics on Random Structures
Passive tracer in a flow corresponding to a two dimensional stochastic Navier Stokes equations
In this paper we prove the law of large numbers and central limit theorem for
trajectories of a particle carried by a two dimensional Eulerian velocity
field. The field is given by a solution of a stochastic Navier--Stokes system
with a non-degenerate noise. The spectral gap property, with respect to
Wasserstein metric, for such a system has been shown in [9]. In the present
paper we show that a similar property holds for the environment process
corresponding to the Lagrangian observations of the velocity. In consequence we
conclude the law of large numbers and the central limit theorem for the tracer.
The proof of the central limit theorem relies on the martingale approximation
of the trajectory process
Modelling expected physical impacts and human casualties from explosive volcanic eruptions
A multi-hazard, multi-vulnerability impact model has been developed for application to European volcanoes that could significantly damage human settlements. This impact model is based on volcanological analyses of the potential hazards and hazard intensities coupled with engineering analyses of the vulnerability to these hazards of residential buildings in four European locations threatened by explosive volcanic eruptions. For a given case study site, inputs to the model are population data, building characteristics, volcano scenarios as a series of hazard intensities, and scenarios such as the time of eruption or the percentage of the population which has been evacuated. Outputs are the rates of fatalities, seriously injured casualties, and destroyed buildings for a given scenario. These results are displayed in a GIS, thereby presenting risk maps which are easy to use for presenting to public officials, the media, and the public. Technical limitations of the model are discussed and future planned developments are considered. This work contributes to the EU-funded project EXPLORIS (Explosive Eruption Risk and Decision Support for EU Populations Threatened by Volcanoes, EVR1-2001-00047). </p><p style='line-height: 20px;'> </p
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