Physics and Astronomy Publications

A Study of the Orbits of the Logarithmic Potential for Galaxies

Sree Ram Valluri et al. — Fri, 15 Feb 2013 06:35:30 PST

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 bifurca- tions 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 (pre- cessing ellipse) method pioneered by Struck (2006). This differential orbital equation is a special case of the generalized Burgers equation. The apsidal angle is also deter- mined, 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.

Some applications of the Lambert W function to physics

Sree R. Valluri et al. — Fri, 15 Feb 2013 06:35:28 PST

Two standard physics problems are solved in terms of the Lambert W function, in order to show the applicability of this recently defined function to physics. Other applications of the function are cited, but not described. The problems solved concern Wien’s displacement law and the fringing fields of a capacitor, the latter problem being representative of some problems solved using conformal transformations. The physical content of the solutions remains unchanged, but they gain a new elegance and convenience.

The Carter constant for inclined orbits about a massive Kerr black hole: near-circular, near-polar orbits

Sree Ram Valluri et al. — Thu, 29 Nov 2012 11:45:31 PST

In an extreme mass-ratio binary black hole system, a non-equatorial orbit will list (i.e. increase its angle of inclination, iota) as it evolves in Kerr spacetime. The abutment, a set of evolving, near-polar, retrograde orbits, for which the instantaneous Carter constant (Q) is at its maximum value (Q_X) for given values of latus rectum (˜l) and eccentricity (e), has been introduced as a laboratory in which the consistency of dQ/dt with corresponding evolution equations for d˜l/dt and de/dt might be tested independently of a specific radiation back-reaction model. To demonstrate the use of the abutment as such a laboratory, a derivation of dQ/dt, based only on published formulae for d˜l/dt and de/dt, was performed for elliptical orbits on the abutment. The resulting expression for dQ/dt matched the published result to the second order in e. We believe the abutment is a potentially useful tool for improving the accuracy of evolution equations to higher orders of e and˜l−1.

The Polylogarithm and the Lambert W Functions in Thermoelectrics

Muralikrishna Molli et al. — Tue, 29 Nov 2011 23:10:04 PST

In this work, we determine the conditions for the extremum of the figure of merit, theta², in a degenerate semiconductor for thermoelectric (TE) applications. We study the variation of the function theta² with respect to the reduced chemical potential mu* using relations involving polylogarithms of both integral and nonintegral orders. We present the relevant equations for the thermopower, thermal, and electrical conductivities that result in optimizing theta² and obtaining the extremum equations. We discuss the different cases that arise for various values of r, which depends on the type of carrier scattering mechanism present in the semiconductor. We also present the important extremum conditions for theta² obtained by extremizing the TE power factor and the thermal conductivity separately. In this case, simple functional equations, which lead to solutions in terms of the Lambert W function, result. We also present some solutions for the zeros of the polylogarithms. Our analysis allows for the possibility of considering the reduced chemical potential and the index r of the polylogarithm as complex variables.

A Study of Elliptical Last Stable Orbits about a Massive Kerr Black Hole

P. G. Komorowski et al. — Sun, 10 Jul 2011 17:32:44 PDT

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 normalized spin, (|J|/M², 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 derived from the Kerr metric, which encapsulates the Lense–Thirring precession. This 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 for an elliptical LSO. Formulae for (square of normalized orbital energy) and (square of normalized orbital angular momentum) in terms of eccentricity, ε, and latus rectum, , were previously developed by others for elliptical orbits around an SBH and then extended to the KBH case; we used these results to generalize our analytical 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.

QED Effective Action Revisited

U. D. Jentschura et al. — Sun, 10 Jul 2011 16:45:28 PDT

The derivation of a convergent series representation for the quantum electrodynamic effective action obtained by two of us (S.R.V. and D.R.L.) in [Can. J. Phys. 71, 389 (1993)] is reexamined. We present more details of our original derivation. Moreover, we discuss the relation of the electric-magnetic duality to the integral representation for the effective action, and we consider the application of nonlinear convergence acceleration techniques that permit the efficient and reliable numerical evaluation of the quantum correction to the Maxwell Lagrangian.

Comment on “A Convergent Series for the QED Effective Action”

D. R. Lamm et al. — Sun, 10 Jul 2011 16:45:26 PDT

A Study of the Gravitational Wave Form from Pulsars

S. R. Valluri et al. — Sun, 10 Jul 2011 16:45:25 PDT

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 and a special class of the generalized hypergeometric function of potential relevance. The Zak–Gelfand integral transform that arises in our analytic approach has also been useful for Schrödinger operators in periodic potentials in condensed matter physics (Bloch wavefunctions) and holds promise for the study of periodic GW signals for long integration times.

A Study of the de Broglie Gravitational Waves

Antonio Feoli et al. — Sun, 10 Jul 2011 16:17:13 PDT

We study some interesting properties of the de Broglie gravitational waves. In particular, we investigate the properties of the polarization and the energy momentum tensor and the geodesic deviation associated with these waves. We observe that the polarization tensor has both transverse and longitudinal components and depends on the wave number. We find a new effect which does not occur in the standard gravitational waves. Our waves are responsible of a longitudinal shift of test particles placed along the direction of propagation. The amplitude of the shift decreases when the velocity of the source becomes closer to the speed of light; so slow massive particles must be used for an experimental test of the theory.

A Numerical Method for One-dimensional Action Functionals of Photonic Band-gap Structures

F. Xie et al. — Sun, 10 Jul 2011 16:17:12 PDT

Photonic band gaps (PBG), photonic analogues of electronic semiconductor band gaps, have attracted much attention recently because of numerous potential applications in communications and computing. Aközbek and John (Phys. Rev. E, 57, 2287 (1998).) developed a variational model of such band gaps, using action functionals, where solitary waves are expanded in terms of a finite orthonormal basis. These expansions to finite order N converged to solitary waves. The nonlinear polynomial equations for the coefficients in the expansions, have nonunique solutions. Our paper, makes a study of the multiplicity of the solutions for one-dimensional photonic band-gap structures. It is found that the nonuniqueness grows dramatically with the order of the expansion N. We use homotopy, which continuously deforms the solutions of exactly solvable systems, into the solutions of the systems to be solved with new results in numeric algebraic geometry, such that all solutions are determined. We used Maple 7 to obtain the polynomial equations for the variational coefficients, extending Aközbek and John's approach. A homotopy-based package PHCpack was used to solve the systems for N ≤ 4 and a linearization-extrapolation method was developed to find real solutions for N ≥ 5. The results are compared with the exact soliton solutions and their convergence behavior is discussed. The interplay of geometrical, topological and variational methods is seen in these interesting physical band-gap structures.

Les écarts de bande photoniques (PBG), analogues au écarts de bande électroniques dans les semi-conducteurs, attirent beaucoup l'attention, à cause de leur utilisation potentielle pour les communications et les ordinateurs. Aközbek et John (Phys. Rev. E, 57, 2287 (1998).) ont développé un modèle variationnel pour un tel système, utilisant des fonctionnelles d'action où des ondes solitaires sont exprimées sur une base orthonormale finie. Ces expansions à un ordre fini N convergent vers des ondes solitaires. Nous étudions ici la multiplicité des solutions pour des structures d'écart de bande photonique en 1-D. Nous trouvons que la non-unicité croît dramatiquement avec N. Nous utilisons l'homotopie qui déforme continuellement les solutions de systèmes avec solution exacte, en solutions à être résolues par la nouvelle méthode mixte numérique, algébrique et géométrique, de telle sorte que toutes les solutions sont déterminées. Nous utilisons Maple 7 pour obtenir l'équation polynomiale pour les coefficients variationnels, ce qui constitue une généralisation de la méthode de Aközbek et John. Nous utilisons le module de calculs homotopiques PHCpack pour les systèmes avec N ≤ 4 et une méthode d'extrapolation-linéarisation pour N ≥ 5. Nous comparons nos résultats avec les résultats solitoniques exacts et nous analysons leur convergence. On peut voir le rôle croisé des différentes méthodes géométrique, topologiue et variationnelle dans ces intéressants écarts de bande.

An Extension of the Pais Variational Phase Shift Approximation

W. J. Romo et al. — Sun, 10 Jul 2011 15:52:56 PDT

The Pais variational phase shift approximation has been modified so that it can be applied to systems in which either long range interactions are present or potential resonances exist, or both. Numerical calculations arising from the modified Pais formalism are relatively simple and lead to accurate results even for the lowest partial waves. Short-range and Coulomb potentials as well as their combinations have been considered in the applications of the extended Pais method. The formalism also enables the calculation of partial derivatives of the phase shift with respect to physical quantities such as linear and orbital angular momentum with only minor additional effort. It has possible application to both electron scattering from ions, heavy-ion scattering in nuclear physics and resonance scattering. The generalized Coulomb potential (GCP) has supersymmetry and shape invariance in the r and θ dimensions, and the energy eigenvalues, the Pais variational phase shifts and corresponding wavefunctions can also be obtained by the method of supersymmetric (SUSYQM) quantum mechanics, and shape invariance. The partial derivatives of the Pais phase shifts are also relevant for orbit and glory scattering, and rainbows in collision theory and for the case of scalar waves scattering in a gravitational field by rotating black holes.

An Extension of Newton's Apsidal Precession Theorem

S. R. Valluri et al. — Sun, 10 Jul 2011 15:52:54 PDT

Newton's apsidal precession theorem in Proposition 45 of Book I of the ‘Principia’ has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the applications of this theorem. We have examined the precession of orbits under varying force laws as measured by the apsidal angle θ(N, e), where N is the index for the centripetal force law, for varying eccentricity e. The paper derives a general function for the apsidal angle, dependent only on e and N as the potential is spherically symmetric. Further, we explore approximate ways of the solution of this equation, in the neighbourhood of N= 2 which happens to be the case of greatest historical interest. Exact solutions are derived where they are possible. The first derivatives ∂θ/∂N and ∂θ/∂h[where h(N, e) is the angular momentum] are analytically expressed in the neighbourhood of N= 2 (case of the inverse square law). The value of ∂θ/∂N is computed numerically as well for 1 ≤N < 3. The resulting integrals are interesting improper integrals with singularities at both limits. Some of the integrals, especially for N= 2, can be given in closed form in terms of generalized hypergeometric functions which are reducible in terms of algebraic and logarithmic functions. No evidence was found for isolated cases of zero precession as e was increased. The N= 1 case of the logarithmic potential is also briefly discussed in view of its interest for the dynamics of eccentric orbits and its relevance to realistic galaxy models. The possibility of apsidal precession was also examined for a few cases of high-eccentricity asteroids and extrasolar planets. We find that these systems may provide interesting new laboratories for studies of gravity.

Chebyshev Derived Spindown Parameters for Gravitational Wave Signals from Pulsars

S. R. Valluri et al. — Sun, 10 Jul 2011 15:32:24 PDT

The master equation described by Badri Krishnan et al. (Phys Rev. D, 70, 082001 (2004)) for the time-frequency pattern using the F-statistic is studied in the context of Chebyshev-polynomial modified spindown parameters for the case of gravitational wave pulsar signals. The Chebyshev-polynomial method enables an analytic and numeric evaluation of the Fourier transform (FT) for both the non-demodulated and F-statistic demodulated FT.

Nous étudions l’équation maîtresse décrite par Badri Krishnan et al. (Phys Rev. D, 70, 082001 (2004)) pour le patron en temps-fréquence via la statistique F dans le contexte de paramètres modifiés de spin des polynômes de Chebyshev dans l’étude des ondes gravitationnelles (GW) générées par un pulsar. La méthode des polynômes de Chebyshev permet d’évaluer analytiquement et numériquement la transformée de Fourier (FT) à la fois pour le cas sans démodulation et avec démodulation à statistique F(DeFT).

A Study of the Gravitational Wave Pulsar Signal with Orbital and Spindown Effects

S. R. Valluri et al. — Sun, 10 Jul 2011 15:32:23 PDT

In this work, we present an analytic and a preliminary numerical analysis of the gravitational wave signal from a pulsar that includes simple spindown effects. We estimate the phase corrections to a monochromatic source signal due to rotational and elliptical orbital motion of the Earth, and perturbations due to Jupiter and the Moon. We briefly discuss the Fourier transform of such a signal, expressed in terms of well-known special functions, and its applications.

Nous présentons ici une analyse analyrique et une nouvelle analyse numérique d'un signal d'onde gravitationnelle d'un pulsar, incluant la décroissance rotationnelle. Nous estimons les corrections de phase du signal monochromatiques dues aux mouvements rotationnel et orbital elliptique de la Terre, ainsi qu'aux perturbations par Jupiter et la Lune. Nous discutons brièvement la transformée de Fourier d'un tel signal, exprimée en terme de fonctions bien connues et ses applications.

The Gravitational Wave Pulsar Signal with Jovian and Lunar Perturbations and Orbital Eccentricity Corrections

S. R. Valluri et al. — Fri, 08 Jul 2011 18:30:46 PDT

In this work, we present the analytic treatment of the Fourier transform (FT) of the gravitational wave (GW) signal from a pulsar in a parametrized model discussed by Brady et al (1998 Phys. Rev. D 57 2101). The formalism lends itself to a development of the FT that accounts for the corrections due to the orbital eccentricity, and the Jovian and lunar perturbations in terms of well-known special functions.

Some Applications of the Lambert W Function to Physics

S. R. Valluri et al. — Thu, 26 May 2011 17:25:35 PDT

Two standard physics problems are solved in terms of the Lambert W function, to show the applicability of this recently defined function to physics. Other applications of the function are cited, but not described. The problems solved concern Wien's displacement law and the fringing fields of a capacitor, the latter problem being representative of some problems solved using conformal transformations. The physical content of the solutions remains unchanged, but they gain a new elegance and convenience.

Gravitational Wave Signal Templates, Pattern Recognition, and Reciprocal Eulerian Gamma Functions

Marc Eric Normandin et al. — Thu, 26 May 2011 17:25:34 PDT

The direct detection of Gravitational Waves (GWs) is one of the most challenging problems in experimental gravitation today. It necessitates the use of highly advanced large laser interferometers such as LIGO, VIRGO, LISA, TAMA 300, GEO 600 and AIGO. The analysis of the data from such instruments requires and combines the expertise from a multitude of scientific disciplines. The verification of a detected signal demands an effective way to distinguish the source signal from the background noise. Such a study is required for an all-sky search to determine the @f and @q angles on the sky of gravitational wave sources and their frequencies. In this paper, we present analytical solutions and associated numerical approximations for the inner products employed in matched filtering a GW signal using templates. An exact closed-form expression for the inner products is rigourously derived using the special functions of mathematical physics. The inner products involve reciprocal Eulerian gamma functions, which occur in the study of many diverse phenomena. The spectral noise density of the VIRGO GW detector is shown to be amenable to our analysis. Spectral noise densities like those for LIGO and GEO 600, although different and in a slightly more restricted frequency band, are likewise amenable. We study numerical computation of the inner products, estimate the computational time of the solution on serial and parallel computers, and show the efficiency of the resulting algorithms. The fitting factor that indicates the goodness of fit between a signal and a template is given in closed-form and computed numerically. The numerical plots display an approximate symmetry in the template @f and @q domain.

D-dimensional Bose Gases and the Lambert W Function

J. Tanguay et al. — Wed, 25 May 2011 21:30:11 PDT

The applications of the Lambert W function (also known as the W function) to D-dimensional Bose gases are presented. We introduce two sets of families of logarithmic transcendental equations that occur frequently in thermodynamics and statistical mechanics and present their solution in terms of the W function. The low temperature T behavior of free ideal Bose gases is considered in three and four dimensions. It is shown that near condensation in four dimensions, the chemical potential μ and pressure P can be expressed in terms of T through the W function. The low T behavior of one- and two-dimensional ideal Bose gases in a harmonic trap is studied. In 1D, the W function is used to express the condensate temperature, T_C, in terms of the number of particles N; in 2D, it is used to express μ in terms of T. In the low T limit of the 1D hard-core and the 3D Bose gas, T can be expressed in terms of P and μ through the W function. Our analysis allows for the possibility to consider μ, T, and P as complex variables. The importance of the underlying logarithmic structure in ideal quantum gases is seen in the polylogarithmic and W function expressions relating thermodynamic variables such as μ, T, and P.

The Lambert W Function and Quantum Statistics

Sree Ram Valluri et al. — Sun, 05 Dec 2010 17:12:26 PST

We present some applications of the Lambert W function (W function) to the formalism of quantum statistics (QS). We consider the problem of finding extrema in terms of energy for a general QS distribution, which involves the solution of a transcendental equation in terms of the W function. We then present some applications of this formula including Bose–Einstein systems in d dimensions, Maxwell–Boltzmann systems, and black body radiation. We also show that for the appropriate parameter values, this formula reduces to an analytic expression in connection with Wien’s displacement law that was found in a previous study. In addition, we show that for Maxwell–Boltzmann and Bose–Einstein systems, the W function allows us to express the temperature of the system as a function of the thermodynamically relevant chemical potential, the particle density, and other parameters. Finally, we explore an indirect relationship of the W function to the polylogarithm function and to the Lambert transform.

The Carter Constant for Inclined Orbits about a Massive Kerr Black Hole: I. Circular Orbits

Peter G. Komorowski et al. — Wed, 24 Nov 2010 15:11:58 PST

In an extreme binary black hole system, an orbit will increase its angle of inclination (ι) 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-reactionmodels. For a Schwarzschild black hole, the polar orbits represent the abutment between the prograde and retrograde orbits at whichQis at its maximum value for given values of the latus rectum (˜l ) and the eccentricity (e). The introduction of spin (S˜ = |J|/M2) to themassive 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 characterize 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 LSOs. By knowing the analytical form of ∂Q/∂˜l at the abutment, we were able to test a 2PN flux equation for Q. We also used these formulae to numerically calculate the ∂ι/∂˜l of hypothetical circular orbits that evolve along the abutment. From these values we have determined that ∂ι/∂˜l = −(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. By taking the limit of this equation for ˜l → ∞, and comparing it with the published result for the weak-field radiation reaction, we found the upper limit on |∂ι/∂˜l| for the full range of ˜l up to the LSO. Although we know the value of ∂Q/∂˜l at the abutment, we find that the second and higher derivatives of Q with respect to ˜l exert an influence on ∂ι/∂˜l . Thus the abutment becomes an important analytical and numerical laboratory for studying the evolution of Q and ι in Kerr spacetime and for testing current and future radiation-backreaction models for near-polar retrograde orbits.