Numerical Optical Centroid Measurements

Optical imaging methods are typically restricted to a resolution of order of the probing light wavelength $\lambda_p$ by the Rayleigh diffraction limit. This limit can be circumvented by making use of multiphoton detection of correlated $N$-photon states, having an effective wavelength $\lambda_p/N$. But the required $N$-photon detection usually renders these schemes impractical. To overcome this limitation, recently, so-called optical centroid measurements (OCM) have been proposed which replace the multi-photon detectors by an array of single-photon detectors. Complementary to the existing approximate analytical results, we explore the approach using numerical experiments by sampling and analyzing detection events from the initial state wave function. This allows us to quantitatively study the approach also beyond the constraints set by the approximate analytical treatment, to compare different detection strategies, and to analyze other classes of input states.Comment: 15 pages, 18 figure

Reliability of an experimental method to analyse the impact point on a golf ball during putting

This study aimed to examine the reliability of an experimental method identifying the location of the impact point on a golf ball during putting. Forty trials were completed using a mechanical putting robot set to reproduce a putt of 3.2 m, with four different putter-ball combinations. After locating the centre of the dimple pattern (centroid) the following variables were tested; distance of the impact point from the centroid, angle of the impact point from the centroid and distance of the impact point from the centroid derived from the X, Y coordinates. Good to excellent reliability was demonstrated in all impact variables reflected in very strong relative (ICC = 0.98–1.00) and absolute reliability (SEM% = 0.9–4.3%). The highest SEM% observed was 7% for the angle of the impact point from the centroid. In conclusion, the experimental method was shown to be reliable at locating the centroid location of a golf ball, therefore allowing for the identification of the point of impact with the putter head and is suitable for use in subsequent studies

The Schr\"odinger formulation of the Feynman path centroid density

We present an analysis of the Feynman path centroid density that provides new insight into the correspondence between the path integral and the Schr\"odinger formulations of statistical mechanics. The path centroid density is a central concept for several approximations (centroid molecular dynamics, quantum transition state theory, and pure quantum self-consistent harmonic approximation) that are used in path integral studies of thermodynamic and dynamical properties of quantum particles. The centroid density is related to the quasi-static response of the equilibrium system to an external force. The path centroid dispersion is the canonical correlation of the position operator, that measures the linear change in the mean position of a quantum particle upon the application of a constant external force. At low temperatures, this quantity provides an approximation to the excitation energy of the quantum system. In the zero temperature limit, the particle's probability density obtained by fixed centroid path integrals corresponds to the probability density of minimum energy wave packets, whose average energy define the Feynman effective classical potential.Comment: 29 pages, 2 figures, 1 Table, J. Chem. Phys. (in press

Centroid and moments of an area using a digitizer

The centroid and moments of an area program provides the centroid, moments of inertia, product of inertia, radii of gyration, and area of any closed planar geometric figure. The figure must be available in graphic form and is digitized once with chart digitizer (graphic tablet). The digitizer origin may be set anywhere on the digitizer table. After digitizing, fifteen quantities are calculated and displayed: (1) area (2) moment of inertia of area with respect to digitizer x-axis, (3) moment of inertia of area with respect to digitizer y-axis, (4) product of inertia of area with respect to digitizer axes, (5) first moment of x for digitizer axes, (6) first moment of y for digitizer axes, (7) x coordinate of centroid, (8) y coordinate of centroid, (9) moment of area inertia of with respect to x axis through centroid, (10) moment of inertia of area with respect to y axis through centroid, (11) product inertia of area with respect to x and y axes through centroid, (12) polar moment of inertia of area around centroid, (13) radius of gyration about digitizer x axis, (14) radius of gyration about digitizer y-axis; and (15) variance in the x-direction

Effects of a synthetic jet acting on a separated flow over a hump

The effects of an oscillatory zero-net-mass-flux jet (i.e. synthetic jet) acting on a separated flow over a hump are investigated in terms of two actuation parameters – actuator position and forcing frequency. By considering the vorticity flux balance and introducing a centroid of vorticity production over the hump surface, lift and drag acting on the hump can be expressed as a function of this centroid and the rate of vorticity production. To study the parametric dependence of lift and drag, direct numerical simulation (DNS) is performed by solving compressible, unsteady, laminar flows over a half-cylindrical hump in two dimensions. The DNS results show that periodic actuation significantly reduces the rate of vorticity production at the wall and shifts the centroid upstream so that the drag is reduced and the lift is increased, respectively. When the actuation parameters are varied, it is found that the lift is governed by the horizontal coordinate of the vorticity-production centroid, while the drag is determined by both the vertical coordinate of the centroid and the rate of vorticity production over the hump. This paper explains by using ideal flow models that the vorticity-production centroid is controlled by two factors: one is the actuator position at which clockwise vorticity is generated, and the other is the point where the separation vortex is pinched off, thereby the clockwise vorticity being absorbed

Modification of Projected Velocity Power Spectra by Density Inhomogeneities in Compressible Supersonic Turbulence

(Modified) The scaling of velocity fluctuation, dv, as a function of spatial scale L in molecular clouds can be measured from size-linewidth relations, principal component analysis, or line centroid variation. Differing values of the power law index of the scaling relation dv = L^(g3D) in 3D are given by these different methods: the first two give g3D=0.5, while line centroid analysis gives g3D=0. This discrepancy has previously not been fully appreciated, as the variation of projected velocity line centroid fluctuations (dv_{lc} = L^(g2D)) is indeed described, in 2D, by g2D=0.5. However, if projection smoothing is accounted for, this implies that g3D=0. We suggest that a resolution of this discrepancy can be achieved by accounting for the effect of density inhomogeneity on the observed g2D obtained from velocity line centroid analysis. Numerical simulations of compressible turbulence are used to show that the effect of density inhomogeneity statistically reverses the effect of projection smoothing in the case of driven turbulence so that velocity line centroid analysis does indeed predict that g2D=g3D=0.5. Using our numerical results we can restore consistency between line centroid analysis, principal component analysis and size-linewidth relations, and we derive g3D=0.5, corresponding to shock-dominated (Burgers) turbulence. We find that this consistency requires that molecular clouds are continually driven on large scales or are only recently formed.Comment: 28 pages total, 20 figures, accepted for publication in Ap

Observational Evidence for the Effect of Amplification Bias in Gravitational Microlensing Experiments

Recently Alard\markcite{alard1996} proposed to detect the shift of a star's image centroid, $\delta x$, as a method to identify the lensed source among blended stars. Goldberg & Wo\'zniak\markcite{goldberg1997} actually applied this method to the OGLE-1 database and found that 7 out of 15 events showed significant centroid shifts of $\delta x \gtrsim 0.2$ arcsec. The amount of centroid shift has been estimated theoretically by Goldberg.\markcite{goldberg1997} However, he treated the problem in general and did not apply it to a particular survey or field, and thus based his estimates on simple toy model luminosity functions (i.e., power laws). In this paper, we construct the expected distribution of $\delta x$ for Galactic bulge events by using the precise stellar LF observed by Holtzman et al.\markcite{holtzman1998} using HST. Their LF is complete up to $M_I\sim 9.0$ ($M_V\sim 12$), corresponding to faint M-type stars. In our analysis we find that regular blending cannot produce a large fraction of events with measurable centroid shifts. By contrast, a significant fraction of events would have measurable centroid shifts if they are affected by amplification-bias blending. Therefore, Goldberg & Wo\'zniak's measurements of large centroid shifts for a large fraction of microlensing events confirms the prediction of Han and Alard that a large fraction of Galactic bulge events are affected by amplification-bias blending.Comment: total 15 pages, including 6 figures, and no Table, submitted to ApJ on Apr 26 1998, email [email protected]

Astrometric Image Centroid Displacements due to Gravitational Microlensing by the Ellis Wormhole

Continuing work initiated in an earlier publication (Abe, ApJ, 725 (2010) 787), we study the gravitational microlensing effects of the Ellis wormhole in the weak-field limit. First, we find a suitable coordinate transformation, such that the lens equation and analytic expressions of the lensed image positions can become much simpler than the previous ones. Second, we prove that two images always appear for the weak-field lens by the Ellis wormhole. By using these analytic results, we discuss astrometric image centroid displacements due to gravitational microlensing by the Ellis wormhole. The astrometric image centroid trajectory by the Ellis wormhole is different from the standard one by a spherical lensing object that is expressed by the Schwarzschild metric. The anomalous shift of the image centroid by the Ellis wormhole lens is smaller than that by the Schwarzschild lens, provided that the impact parameter and the Einstein ring radius are the same. Therefore, the lensed image centroid by the Ellis wormhole moves slower. Such a difference, though it is very small, will be in principle applicable for detecting or constraining the Ellis wormhole by using future high-precision astrometry observations. In particular, the image centroid position gives us an additional information, so that the parameter degeneracy existing in photometric microlensing can be partially broken. The anomalous shift reaches the order of a few micro arcsec. if our galaxy hosts a wormhole with throat radius larger than $10^5$ km. When the source moves tangentially to the Einstein ring for instance, the maximum position shift of the image centroid by the Ellis wormhole is 0.18 normalized by the Einstein ring radius. For the same source trajectory, the maximum difference between the centroid displacement by the Ellis wormhole lens and that by the Schwarzschild one is -0.16 in the units of the Einstein radius.Comment: 29 pages, 6 figures, 2 tables, accepted by Ap