Analytical Methods in Gravitational Microlensing

We discuss analytical results dealing with photometric and astrometric gravitational microlensing. The first two sections concern approximation methods that allow us to get solutions of the general lens equation near fold caustics and cusp points up to any prescribed accuracy. Two methods of finding approximate solutions near the fold are worked out. The results are applied to derive new corrections to total amplifications of critical source images. Analytic expressions are obtained in case of the Gaussian, power-law, and limb-darkening extended source models; here we present the first nonzero corrections to the well-known linear caustic approximation. Possibilities to distinguish different source models in observations are discussed on the basis of statistical simulations of microlensed light curves. In the next section, we discuss astrometric microlensing effects in various cases of extended sources and extended lenses, including a simple model of weak statistical microlensing by extended dark matter clumps. Random walks of a distant source image microlensed by stochastic masses are estimated. We note that the bulk motion of foreground stars induces a small apparent rotation of the extragalactic reference frame. Compact analytical relations describing the statistics of such motions are presented.

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Simulations of the gravitational microlensing: Extended source models and impact of binary stars

Journal of Physical Studies ◽

10.30970/jps.16.3904 ◽

2012 ◽

Vol 16 (3) ◽

Cited By ~ 1

Author(s):

V. M. Sliusar ◽

V. I. Zhdanov ◽

A. N. Alexandrov

Keyword(s):

Binary Stars ◽

Extended Source ◽

Gravitational Microlensing ◽

Source Models

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MACHO Alert 95‐30: First Real‐Time Observation of Extended Source Effects in Gravitational Microlensing

The Astrophysical Journal ◽

10.1086/304974 ◽

1997 ◽

Vol 491 (2) ◽

pp. 436-450 ◽

Cited By ~ 119

Author(s):

C. Alcock ◽

W. H. Allen ◽

R. A. Allsman ◽

D. Alves ◽

T. S. Axelrod ◽

...

Keyword(s):

Real Time ◽

Extended Source ◽

Source Effects ◽

Gravitational Microlensing ◽

Time Observation ◽

Real Time Observation

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Semi-Analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian Expansion of Distance Operators W= RC1-nRD1-M, RC1-Nr12-M, R12-Nr13-M

10.26434/chemrxiv.9638879 ◽

2019 ◽

Author(s):

Sandor Kristyan

Keyword(s):

Gaussian Function ◽

Approximate Solutions ◽

Equation System ◽

Least Square ◽

Position Vector ◽

Least Square Fit ◽

Analytic Expressions ◽

The One ◽

Analytical Expressions ◽

Coulomb Integrals

The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)…rho(rk) W(r1,…,rk) dr1…drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases – mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u≠1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrödinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.

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Quasi-Band-Limited Coronagraph for Extended Sources

Journal of Astronomical Instrumentation ◽

10.1142/s2251171721500021 ◽

2021 ◽

Vol 10 (01) ◽

pp. 2150002

Author(s):

Igor Loutsenko ◽

Oksana Yermolayeva

Keyword(s):

High Resolution ◽

High Ratio ◽

Gravitational Lens ◽

High Resolution Imaging ◽

The Sun ◽

Extended Source ◽

Amplitude Image ◽

Extended Sources ◽

Resolution Imaging ◽

Band Limited

We propose a class of graded coronagraphic “amplitude” image masks for a high throughput Lyot-type coronagraph that transmits light from an annular region around an extended source and suppresses light, with extremely high ratio, from elsewhere. The interior radius of the region is comparable with its exterior radius. The masks are designed using an idea inspired by approach due M. J. Kuchner and W. A. Traub (“band-limited” masks) and approach to optimal apodization by D. Slepian. One potential application of our masks is direct high-resolution imaging of exo-planets with the help of the Solar Gravitational Lens, where apparent radius of the “Einstein ring” image of a planet is of the order of an arc-second and is comparable with the apparent radius of the sun and solar corona.

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Measuring Stellar Limb Darkening by Gravitational Microlensing

The Astrophysical Journal ◽

10.1086/376787 ◽

2003 ◽

Vol 594 (1) ◽

pp. 464-482 ◽

Cited By ~ 21

Author(s):

David Heyrovsky

Keyword(s):

Gravitational Microlensing ◽

Limb Darkening

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Gravitational microlensing of an elliptical source near a fold caustic

Bulletin of Taras Shevchenko National University of Kyiv Astronomy ◽

10.17721/btsnua.2018.57.10-15 ◽

2018 ◽

pp. 10-15

Author(s):

A. Alexandrov ◽

V. Zhdanov ◽

A. Kuybarov

Keyword(s):

Amplification Factor ◽

Brightness Distribution ◽

Gravitational Lens ◽

Variable Component ◽

Extended Source ◽

Elliptical Shape ◽

Gravitational Microlensing ◽

Gaussian Source ◽

Analogous Formula ◽

Circular Source

We consider the amplification factor for the luminosity of an extended source near the fold caustic of the gravitational lens. It is assumed that the source has elliptical shape, and the brightness distribution along the radial directions is Gaussian. During the microlensing event the total brightness of all microimages is observed, which changes when the source moves relative to the caustic. The main contribution to the variable component is given by the so-called critical images that arise/disappear at the intersection of the caustic by the source. In the present paper we obtained an analogous formula for elliptical Gaussian source. The formula involves a dependence on the coordinates of the source centre, its geometric dimensions, and its orientation relative to the caustic. We show that in the linear caustic approximation the amplification of the circular and elliptical sources is described by the same (rescaled) formula. However, in the next approximations the differences are significant. We compare analytical calculations of the amplification curves for different orientations of an elliptical source and for a circular source with the same luminosity for the model example.

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The MWA long baseline Epoch of reionisation survey—I. Improved source catalogue for the EoR 0 field

Publications of the Astronomical Society of Australia ◽

10.1017/pasa.2021.50 ◽

2021 ◽

Vol 38 ◽

Author(s):

C. R. Lynch ◽

T. J. Galvin ◽

J. L. B. Line ◽

C. H. Jordan ◽

C. M. Trott ◽

...

Keyword(s):

Angular Resolution ◽

Low Frequency ◽

Calibration Model ◽

Spectral Measurements ◽

Long Baseline ◽

Source Models ◽

Extended Sources ◽

Murchison Widefield Array ◽

Foreground Calibration ◽

Good Agreement

Abstract One of the principal systematic constraints on the Epoch of Reionisation (EoR) experiment is the accuracy of the foreground calibration model. Recent results have shown that highly accurate models of extended foreground sources, and including models for sources in both the primary beam and its sidelobes, are necessary for reducing foreground power. To improve the accuracy of the source models for the EoR fields observed by the Murchison Widefield Array (MWA), we conducted the MWA Long Baseline Epoch of Reionisation Survey (LoBES). This survey consists of multi-frequency observations of the main MWA EoR fields and their eight neighbouring fields using the MWA Phase II extended array. We present the results of the first half of this survey centred on the MWA EoR0 observing field (centred at RA (J2000) $0^\mathrm{h}$ , Dec (J2000) $-27^{\circ}$ ). This half of the survey covers an area of 3 069 degrees $^2$ , with an average rms of 2.1 mJy beam–1. The resulting catalogue contains a total of 80 824 sources, with 16 separate spectral measurements between 100 and 230 MHz, and spectral modelling for 78 $\%$ of these sources. Over this region we estimate that the catalogue is 90 $\%$ complete at 32 mJy, and 70 $\%$ complete at 10.5 mJy. The overall normalised source counts are found to be in good agreement with previous low-frequency surveys at similar sensitivities. Testing the performance of the new source models we measure lower residual rms values for peeled sources, particularly for extended sources, in a set of MWA Phase I data. The 2-dimensional power spectrum of these data residuals also show improvement on small angular scales—consistent with the better angular resolution of the LoBES catalogue. It is clear that the LoBES sky models improve upon the current sky model used by the Australian MWA EoR group for the EoR0 field.

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Modelling of Planetary Gravitational Microlensing Events

10.26686/wgtn.17004814.v1 ◽

2021 ◽

Author(s):

◽

Michael L. J. Miller

Keyword(s):

Extrasolar Planets ◽

Photometric Data ◽

Analytic Method ◽

Numerical Techniques ◽

Root Finding ◽

Gravitational Microlensing ◽

Single Lens ◽

Limb Darkening ◽

Polynomial Root Finding ◽

Speed And Accuracy

<p>This thesis describes and develops procedures for the generation of theoretical lightcurves that can be used to model gravitational microlensing events that involve multiple lenses. Of particular interest are the cases involving a single lens star with one or more orbiting planets, as this has proven to be an effective way of detecting extrasolar planets. Although there is an analytical expression for microlensing lightcurves produced by single lensing body, the generation of model lightcurves for more than one lensing body requires the use of numerical techniques. The method developed here, known as the semi-analytic method, involves the analytical rearrangement of the relatively simple ‘lens equation’ to produce a high-order complex lens polynomial. Root-finding algorithms are then used to obtain the roots of this ‘lens polynomial’ in order to locate the positions of the images and calculate their magnifications. By running example microlensing events through the root-finding algorithms, both the speed and accuracy of the Laguerre and Jenkins-Traub algorithms were investigated. It was discovered that, in order to correctly identify the image positions, a method involving solutions of several ‘lens polynomials’ corresponding to different coordinate origins needed to be invoked. Multipole and polygon approximations were also developed to include finite source and limb darkening effects. The semi-analytical method and the appropriate numerical techniques were incorporated into a C++ modelling code at VUW (Victoria University of Wellington) known as mlens2. The effectiveness of the semi-analytic method was demonstrated using mlens2 to generate theoretical lightcurves for the microlensing events MOA-2009-BLG-319 and OGLE-2006-BLG-109. By comparing these theoretical lightcurves with the observed photometric data and the published models, it was demonstrated that the semi-analytic method described in this thesis is a robust and efficient method for discovering extrasolar planets.</p>

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