scholarly journals Modelling of Planetary Gravitational Microlensing Events

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>

scholarly journals Modelling of Planetary Gravitational Microlensing Events

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>

scholarly journals DoA Estimation Using Neural Tangent Kernel under Electromagnetic Mutual Coupling

Electronics ◽  
2021 ◽  
Vol 10 (9) ◽  
pp. 1057
Author(s):  
Qifeng Wang ◽  
Xiaolin Hu ◽  
Xiaobao Deng ◽  
Nicholas E. Buris
Keyword(s):  
Mutual Coupling ◽  
Deep Neural Network ◽  
Doa Estimation ◽  
Antenna Element ◽  
Root Finding ◽  
Multiple Signal ◽  
Estimation Problems ◽  
Limiting Case ◽  
Signal Doa Estimation ◽  
Polynomial Root Finding

Antenna element mutual coupling degrades the performance of Direction of Arrival (DoA) estimation significantly. In this paper, a novel machine learning-based method via Neural Tangent Kernel (NTK) is employed to address the DoA estimation problem under the effect of electromagnetic mutual coupling. NTK originates from Deep Neural Network (DNN) considerations, based on the limiting case of an infinite number of neurons in each layer, which ultimately leads to very efficient estimators. With the help of the Polynomial Root Finding (PRF) technique, an advanced method, NTK-PRF, is proposed. The method adapts well to multiple-signal scenarios when sources are far apart. Numerical simulations are carried out to demonstrate that this NTK-PRF approach can handle, accurately and very efficiently, multiple-signal DoA estimation problems with realistic mutual coupling.

scholarly journals Efficient high degree polynomial root finding using GPU

2017 ◽  
Vol 18 ◽  
pp. 46-56 ◽  
Author(s):  
Kahina Ghidouche ◽  
Abderrahmane Sider ◽  
Raphaël Couturier ◽  
Christophe Guyeux
Keyword(s):  
Degree Polynomial ◽  
Root Finding ◽  
High Degree ◽  
Polynomial Root Finding

scholarly journals Blended Root Finding Algorithm Outperforms Bisection and Regula Falsi Algorithms

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1118
Author(s):  
Sabharwal
Keyword(s):  
Computational Complexity ◽  
Empirical Evidence ◽  
Analytic Solution ◽  
Fundamental Problem ◽  
Numerical Techniques ◽  
Physical Sciences ◽  
Root Finding ◽  
Numerical Computing ◽  
Newton Raphson ◽  
Raphson Algorithm

Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms.

scholarly journals Exomoon indicators in high-precision transit light curves

2020 ◽  
Vol 638 ◽  
pp. A43
Author(s):  
Kai Rodenbeck ◽  
René Heller ◽  
Laurent Gizon
Keyword(s):  
Extrasolar Planets ◽  
Light Curves ◽  
Low Density ◽  
Instrumental Effects ◽  
The Moon ◽  
Hill Stability ◽  
Follow Up Studies ◽  
Transit Model ◽  
Limb Darkening

Context. While the Solar System contains about 20 times more moons than planets, no moon has been confirmed around any of the thousands of extrasolar planets discovered so far. Considering the large computational load required for the statistical vetting of exomoon candidates in a star–planet–moon framework, tools for an uncomplicated identification of the most promising exomoon candidates could be beneficial to streamline follow-up studies. Aims. Here we study three exomoon indicators that emerge if well-established planet-only models are fitted to a planet–moon transit light curve: transit timing variations (TTVs), transit duration variations (TDVs), and apparent planetary transit radius variations (TRVs). We re-evaluate under realistic conditions the previously proposed exomoon signatures in the TTV and TDV series. Methods. We simulated light curves of a transiting exoplanet with a single moon, taking into account stellar limb darkening, orbital inclinations, planet–moon occultations, and noise from both stellar granulation and instrumental effects. These model light curves were then fitted with a planet-only transit model whilst pretending there were no moon, and we explored the resulting TTV, TDV, and TRV series for evidence of the moon. Results. The previously described ellipse in the TTV-TDV diagram of an exoplanet with a moon emerges only for high-density moons. However, low-density moons distort the sinusoidal shapes of the TTV and the TDV series due to their photometric contribution to the combined planet–moon transit. Sufficiently large moons can nevertheless produce periodic apparent TRVs of their host planets that could be observable. We find that Kepler and PLATO have similar performances in detecting the exomoon-induced TRV effect around simulated bright (mV = 8) stars. Although these stars are rare in the Kepler sample, they will be abundant in the PLATO sample. Moreover, PLATO’s higher cadence yields a stronger TTV signal. We detect substantial TRVs of the Saturn-sized planet Kepler-856 b although an exomoon could only ensure Hill stability in a very narrow orbital range. Conclusions. The periodogram of the sequence of transit radius measurements can indicate the presence of a moon. The TTV and TDV series of exoplanets with moons could be more complex than previously assumed. We propose that TRVs could be a more promising means to identify exomoons in large exoplanet surveys.

scholarly journals Extrasolar planet detections with gravitational microlensing

2007 ◽  
Vol 3 (S249) ◽  
pp. 25-30
Author(s):  
Shude Mao ◽  
Eamonn Kerins ◽  
Nicholas J. Rattenbury
Keyword(s):  
Light Curve ◽  
Real Time ◽  
Extrasolar Planets ◽  
Light Curves ◽  
Extrasolar Planet ◽  
Galactic Centre ◽  
Wide Field ◽  
High Magnification ◽  
Gravitational Microlensing

AbstractMicrolensing light curves due to single stars are symmetric and typically last for a month. So far about 4000 microlensing events have been discovered in real-time, the vast majority toward the Galactic centre. The presence of planets around the primary lenses induces deviations in the usual light curve which lasts from hours (for an Earth-mass [M⊕] planet) to days (for a Jupiter-mass [Mj] planet). Currently the survey teams, OGLE and MOA, discover and announce microlensing events in real-time, and follow-up teams (together with the survey teams) monitor selected events intensively (usually with high magnification) in order to identify anomalies caused by planets. So far four extrasolar planets have been discovered using the microlensing technique, with half a dozen new planet candidates identified in 2007 (yet to be published). Future possibilities include a network of wide-field 2m-class telescopes from the ground (which can combine survey and follow-up in the same setup) and a 1m-class survey telescope from space.

Polynomial root finding technique for joint DOA DOD estimation in bistatic MIMO radar

2010 ◽  
Vol 90 (9) ◽  
pp. 2723-2730 ◽  
Author(s):  
Mohamed Laid Bencheikh ◽  
Yide Wang ◽  
Hongyang He
Keyword(s):  
Mimo Radar ◽  
Bistatic Mimo Radar ◽  
Root Finding ◽  
Polynomial Root Finding
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