242 research outputs found
New views of the solar wind with the Lambert W function
This paper presents closed-form analytic solutions to two illustrative
problems in solar physics that have been considered not solvable in this way
previously. Both the outflow speed and the mass loss rate of the solar wind of
plasma particles ejected by the Sun are derived analytically for certain
illustrative approximations. The calculated radial dependence of the flow speed
applies to both Parker's isothermal solar wind equation and Bondi's equation of
spherical accretion. These problems involve the solution of transcendental
equations containing products of variables and their logarithms. Such equations
appear in many fields of physics and are solvable by use of the Lambert W
function, which is briefly described. This paper is an example of how new
functions can be applied to existing problems.Comment: 16 pages (revtex4), 3 figures, American J. Phys., in press (2004
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
Lamm, Valluri, Jentschura and Weniger comment on "A Convergent Series for the QED Effective Action" by Cho and Pak [Phys. Rev. Lett. vol. 86, pp. 1947-1950 (2001)]
Complete results were obtained by us in [Can. J. Phys. 71, 389 (1993)] for
convergent series representations of both the real and the imaginary part of
the QED effective action; these derivations were based on correct intermediate
steps. In this comment, we argue that the physical significance of the
"logarithmic correction term" found by Cho and Pak in [Phys. Rev. Lett. 86,
1947 (2001)] in comparison to the usual expression for the QED effective action
remains to be demonstrated. Further information on related subjects can be
found in Appendix A of hep-ph/0308223 and in hep-th/0210240.Comment: 1 page, RevTeX; only "meta-data" update
The Analysis of Large Order Bessel Functions in Gravitational Wave Signals from Pulsars
In this work, we present the analytic treatment of the large order Bessel
functions that arise in the Fourier Transform (FT) of the Gravitational Wave
(GW) signal from a pulsar. We outline several strategies which employ
asymptotic expansions in evaluation of such Bessel functions which also happen
to have large argument. Large order Bessel functions also arise in the
Peters-Mathews model of binary inspiralling stars emitting GW and several
problems in potential scattering theory. Other applications also arise in a
variety of problems in Applied Mathematics as well as in the Natural Sciences
and present a challenge for High Performance Computing(HPC).Comment: 8 pages, Uses IEEE style files: Ieee.cls, Ieee.clo and floatsty.sty.
Accepted for publication in High Performance Computing Symposium, May 15-18
(HPCS 2005) Guelph, Ontario, Canad
A Study of the Gravitational Wave Pulsar Signal with orbital and spindown Effects
In this work we present analytic and numerical treatments of the
gravitational wave signal from a pulsar which includes spindown. We consider
phase corrections to a received monochromatic signal due to rotational and
elliptical orbital motion of the Earth, as well as perturbations due to Jupiter
and the Moon. We discuss the Fourier transform of such a signal, which is
expressed in terms of well known special functions and lends itself to a
tractable numerical analysis.Comment: 8 pages, 8 figures. Write-up of talk given at Theory Canada I, June
2005, University of British Columbi
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
A study of the gravitational wave form from pulsars II
We present analytical and numerical studies of the Fourier transform (FT) of
the gravitational wave (GW) signal from a pulsar, taking into account the
rotation and orbital motion of the Earth. We also briefly discuss the
Zak-Gelfand Integral Transform. The Zak-Gelfand Integral Transform that arises
in our analytic approach has also been useful for Schrodinger operators in
periodic potentials in condensed matter physics (Bloch wave functions).Comment: 6 pages, Sparkler talk given at the Amaldi Conference on
Gravitational waves, July 10th, 2001. Submitted to Classical and Quantum
Gravit
A Study of the Orbits of the Logarithmic Potential for Galaxies
The logarithmic potential is of great interest and relevance in the study of
the dynamics of galaxies. Some small corrections to the work of Contopoulos &
Seimenis (1990) who used the method of Prendergast (1982) to find periodic
orbits and bifurcations within such a potential are presented. The solution of
the orbital radial equation for the purely radial logarithmic potential is then
considered using the p-ellipse (precessing ellipse) method pioneered by Struck
(2006). This differential orbital equation is a special case of the generalized
Burgers equation. The apsidal angle is also determined, both numerically as
well as analytically by means of the Lambert W and the Polylogarithm functions.
The use of these functions in computing the gravitational lensing produced by
logarithmic potentials is discussed.Comment: 12 pages, 4 figures. Accepted by MNRAS Sept 6 201
A unified framework for the orbital structure of bars and triaxial ellipsoids
We examine a large random sample of orbits in two self-consistent simulations of N-body bars. Orbits in these bars are classified both visually and with a new automated orbit classification method based on frequency analysis. The well-known prograde x1 orbit family originates from the same parent orbit as the box orbits in stationary and rotating triaxial ellipsoids. However, only a small fraction of bar orbits (~4%) have predominately prograde motion like their periodic parent orbit. Most bar orbits arising from the x1 orbit have little net angular momentum in the bar frame, making them equivalent to box orbits in rotating triaxial potentials. In these simulations a small fraction of bar orbits (~7%) are long-axis tubes that behave exactly like those in triaxial ellipsoids: they are tipped about the intermediate axis owing to the Coriolis force, with the sense of tipping determined by the sign of their angular momentum about the long axis. No orbits parented by prograde periodic x2 orbits are found in the pure bar model, but a tiny population (~2%) of short-axis tube orbits parented by retrograde x4 orbits are found. When a central point mass representing a supermassive black hole (SMBH) is grown adiabatically at the center of the bar, those orbits that lie in the immediate vicinity of the SMBH are transformed into precessing Keplerian orbits that belong to the same major families (short-axis tubes, long-axis tubes and boxes) occupying the bar at larger radii. During the growth of an SMBH, the inflow of mass and outward transport of angular momentum transform some x1 and long-axis tube orbits into prograde short-axis tubes. This study has important implications for future attempts to constrain the masses of SMBHs in barred galaxies using orbit-based methods like the Schwarzschild orbit superposition scheme and for understanding the observed features in barred galaxies
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