Finite-source Effects Research Articles

Finite source effect on the polarization degree induced by a single microlens

We investigate the effect of a single microlens on Stokes parameters. Semi-analytical formulae of the microlensed Stokes parameters are derived. The formulae not only reduce the double integrals in the estimations of those quantities but can also be approximated to a useful form in the bypass case. By using our formulation, we show that a combination of polarimetric data with photometric data enables us to estimate not only the finite source effect but also the direction of the microlens motion.

Stochastic distributions of lens and source properties for observed galactic microlensing events

A comprehensive new approach is presented for deriving probability densities of physical properties characterizing the lens and source that constitute an observed galactic microlensing event. While previously encountered problems are overcome, constraints from event anomalies and model parameter uncertainties can be incorporated into the estimates. Probability densities for given events need to be carefully distinguished from the statistical distribution of the same parameters among the underlying population from which the actual lenses and sources are drawn. Using given model distributions of the mass spectrum, the mass density, and the velocity distribution of Galactic disc and bulge constituents, probability densities of lens mass, distance, and the effective lens–source velocities are derived, where the effect on the distribution that arises from additional observations of annual parallax or finite-source effects, or the absence of significant effects, is shown. The presented formalism can also be used to calculate probabilities for the lens to belong to one or another population and to estimate parameters that characterize anomalies. Finally, it is shown how detection efficiency maps for binary-lens companions in the physical parameters, such as companion mass and orbital semimajor axis, arise from values determined for the mass ratio and dimensionless projected separation parameter, including the deprojection of the orbital motion for elliptical orbits. Compared to the naive estimate based on ‘typical values’, the detection efficiency for low-mass companions is increased by mixing in higher detection efficiencies for smaller mass ratios (i.e. smaller masses of the primary).

Finite source effects in strong lensing: implications for the substructure mass scale

Flux ratio ‘anomalies’ in quadruply imaged gravitational lenses can be explained with galactic substructure of the sort predicted by ΛCDM, but the strength and uniqueness of that hypothesis need to be further assessed. A good way to do that is to use the physical scale associated with the size of the source quasar, and its dependence on wavelength. We develop a toy model to study finite source effects in substructure lensing. Treating substructure as a singular isothermal sphere allows us to compute the images of a finite source analytically, and then to explore how the image configurations and magnifications depend on source position and size. Although simplified, our model yields instructive general principles: image positions and magnifications are basically independent of source size until the source is large enough to intersect a substructure caustic; even sources that are much larger than the substructure Einstein radius can be perturbed at a detectable level; and, most importantly, there is a tremendous amount to be learned from comparing image positions and magnifications at wavelengths that correspond to different source sizes. In a separate analysis, we carefully study four observed radio lenses to determine which of the images are anomalous. In B0712+472, the evidence for a radio flux ratio anomaly is marginal, but if the anomaly is real then image C is probably the culprit. In B1422+231, the anomaly is in image A. Interestingly, B2045+265 and B1555+375 both appear to have two anomalous images. Coincidentally, in each system one of the anomalies is in image C, and the other is in either image A or image B (both possibilities lead to acceptable models). It remains to be seen whether ΛCDM predicts enough substructure to explain multiple anomalies in multiple lenses. When we finally join our modelling results and substructure theory, we obtain lower bounds on the masses of the substructures responsible for the observed anomalies. The mass bounds are broadly consistent with expectations for ΛCDM. Perhaps more importantly, we outline various systematic effects in the mass bounds; poor knowledge of whether the substructure lies within the main lens galaxy or elsewhere along the line of sight appears to be the dominant systematic.

Systematic Analysis of 22 Microlensing Parallax Candidates

We attempt to identify all microlensing parallax events for which the parallax fit improves \Delta\chi^2 > 100 relative to a standard microlensing model. We outline a procedure to identify three types of discrete degeneracies (including a new one that we dub the ``ecliptic degeneracy'') and find many new degenerate solutions in 16 previously published and 6 unpublished events. Only four events have one unique solution and the other 18 events have a total of 44 solutions. Our sample includes three previously identified black-hole (BH) candidates. We consider the newly discovered degenerate solutions and determine the relative likelihood that each of these is a BH. We find the lens of event MACHO-99-BLG-22 is a strong BH candidate (78%), event MACHO-96-BLG-5 is a marginal BH candidate (37%), and MACHO-98-BLG-6 is a weak BH candidate (2.2%). The lens of event OGLE-2003-BLG-84 may be a Jupiter-mass free-floating planet candidate based on a weak 3 sigma detection of finite-source effects. We find that event MACHO-179-A is a brown dwarf candidate within ~100 pc of the Sun, mostly due to its very small projected Einstein radius, \tilde r_E = 0.23+-0.05 AU. As expected, these microlensing parallax events are biased toward lenses that are heavier and closer than average. These events were examined for xallarap (or binary-source motion), which can mimic parallax. We find that 23% of these events are strongly affected by xallarap.

Real-time Earthquake Location Using Kirchhoff Reconstruction

Real-time location of earthquakes can be achieved by using direct imaging of the recorded wave field based on a Kirchhoff reconstruction method similar to that used in the migration of seismic reflection data. The standard method of event location requires the wave arrival at each sensor to be picked and associated with an event. By using direct imaging, the event is identified once in the imaged wave field. The computation is independent of the level of seismic activity and can be carried out on a typical desktop computer. The procedure has been successfully demonstrated in two and three dimensions using data from the Southern California Seismic Network (Trinet). At higher resolutions, the reconstruction method can identify finite source effects. Further work considers extending the method by implementing full elastic theory and solving for moment tensors at all locations in the mesh.

Numerical Modeling of the 13 December 1990 M 5.8 East Sicily Earthquake at the Catania Accelerometric Station

The record of Catania enea–enel accelerometric station, with its anomalously high amplitude, represents an outlier for the dataset of seismograms recorded by the Italian accelerometric network during the 13 December 1990 M 5.8 East Sicily earthquake. Using numerical modeling tools, we corroborate the interpretation of this anomaly as an effect of the crustal structure or site response, rather than as a finite source effect. In the first part of this study, we compare the recorded seismograms to those computed by two numerical methods—the 2D Spectral Element Method (spem), a technique which solves the 2D full-wave propagation through a complex geological structure; and the Wavenumber Integration Method (wim), which solves the 3D full-wave propagation in a horizontally-layered structure. The comparison shows that the recorded waveforms are reproduced accurately using a laterally heterogeneous structure, whereas none of the representations in terms of horizontal plane layers, which simplify the same structure, provides satisfactory results. Furthermore, we compare the horizontal-to-vertical spectral ratios (hvsr) obtained from (1) the earthquake record, (2) the seismogram simulated by the spem, and (3) the seismic noise recorded at the same site. Again, the overall agreement is very good. This study has several outcomes. Firstly, that the numerical modeling approach based on the spem, and used in a previous study to simulate ground shaking for a destructive scenario earthquake, provides reliable results. Secondly, that the 2D model used to represent the crustal structure beneath this area is realistic. Indeed, simplified 1D models may not be an adequate means to reproduce realistic seismograms and predict the ground motion in the frequency band of interest for seismic hazard (i.e., 0.5–10 Hz). Finally, the high amplitude displayed by the Catania station during the 1990 earthquake is explained as a combined effect of site and structure-path, while finite source models appear unnecessary. In general, this study emphasizes the importance of methods that accurately model the wavefield propagation through realistic geologic structures for predicting ground motion.

OGLE‐2003‐BLG‐238: Microlensing Mass Estimate of an Isolated Star

Microlensing is the only known direct method to measure the masses of stars that lack visible companions. In terms of microlensing observables, the mass is given by M = (c2/4G)EθE and so requires the measurement of both the angular Einstein radius θE and the projected Einstein radius E. Simultaneous measurement of these two parameters is extremely rare. Here we analyze OGLE-2003-BLG-238, a spectacularly bright (Imin = 10.3), high-magnification (Amax = 170) microlensing event. Pronounced finite-source effects permit a measurement of θE = 650 μas. Although the timescale of the event is only tE = 38 days, one can still obtain weak constraints on the microlens parallax: 4.4 AU < E < 18 AU at the 1 σ level. Together these two parameter measurements yield a range for the lens mass of 0.36 M☉ < M < 1.48 M☉. As was the case for MACHO-LMC-5, the only other single star (apart from the Sun) whose mass has been determined from its gravitational effects, this estimate is rather crude. It does, however, demonstrate the viability of the technique. We also discuss future prospects for single-lens mass measurements.

OGLE‐2003‐BLG‐262: Finite‐Source Effects from a Point‐Mass Lens

We analyze OGLE-2003-BLG-262, a relatively short, t_E=12.5+-0.1day, microlensing event generated by a point-mass lens transiting the face of a K giant source in the Galactic bulge. We use the resulting finite-source effects to measure the angular Einstein radius, theta_E=195+-17muas, and so constrain the lens mass to the full-width half-maximum interval 0.08 < M/M_sun < 0.54. The lens-source relative proper motion is mu_rel = 27+-2 km/s/kpc. Both values are typical of what is expected for lenses detected toward the bulge. Despite the short duration of the event, we detect marginal evidence for a "parallax asymmetry", but argue that this is more likely to be induced by acceleration of the source, a binary lens, or possibly by statistical fluctuations. Although OGLE-2003-BLG-262 is only the second published event to date in which the lens transits the source, such events will become more common with the new OGLE-III survey in place. We therefore give a detailed account of the analysis of this event to facilitate the study of future events of this type.

A Brown Dwarf Microlens Candidate from the Second Phase of the Optical Gravitational Lensing Experiment

We describe a unique mass determination for a microlensing event from the second phase of the Optical Gravitational Lensing Experiment. The event, sc26_2218, which is very bright (baseline magnitude I = 15.10), appears to exhibit both and finite-source effects. The effect allows us to determine the projected Einstein radius on the observer plane (E ≈ 3.8 AU), while the finite-source effect allows us to determine the ratio of the angular source size and the angular Einstein radius. As the angular size of the star can be estimated using its color and magnitude, we can hence determine the angular Einstein radius θE ≈ 0.1 mas. By combining E and θE, we can determine the lens mass M ≈ 0.050M☉, independent of the source distance. The lens is therefore a brown dwarf candidate, located at a distance of ~6.5 kpc. However, the parallax signature is weak, and so we cannot completely discount the possibility that these signatures originate from binary rotation of the source (which would prevent any estimate of the lens mass) rather than parallax. However, this can be tested by future spectroscopic observations. This event highlights the scientific returns for intense monitoring of bright microlensing events, since the and finite-source effects can be more easily identified because of their high signal-to-noise ratios.

On the Feasibility of Detecting Satellites of Extrasolar Planets via Microlensing

Although many methods of detecting extra-solar planets have been proposed and successful implementation of some of these methods enabled a rapidly increasing number of exoplanet detections, little has been discussed about the method of detecting satellites around exoplanets. In this paper, we test the feasibility of detecting satellites of exoplanets via microlensing. For this purpose, we investigate the effect of satellites in the magnification pattern near the region of the planet-induced perturbations by performing realistic simulations of Galactic bulge microlensing events. From this investigation, we find that although satellites can often cause alterations of magnification patterns, detecting satellite signals in lensing light curves will be very difficult because the signals are seriously smeared out by the severe finite source effect even for events involved with source stars with small angular radii.

First Microlens Mass Measurement: PLANET Photometry of EROS BLG‐2000‐5

We analyze PLANET photometric observations of the caustic-crossing binary lens microlensing event, EROS BLG-2000-5, and find that modeling the observed light curve requires incorporation of the microlens parallax and the binary orbital motion. The projected Einstein radius (E = 3.61 ± 0.11 AU) is derived from the measurement of the microlens parallax, and we are also able to infer the angular Einstein radius (θE = 1.38 ± 0.12 mas) from the finite source effect on the light curve, combined with an estimate of the angular size of the source given by the source position in a color-magnitude diagram. The lens mass, M = 0.612 ± 0.057 M☉, is found by combining these two quantities. This is the first time that parallax effects are detected for a caustic-crossing event and also the first time that the lens mass degeneracy has been completely broken through photometric monitoring alone. The combination of E and θE also allows us to conclude that the lens lies in the near side of the disk, within 2.6 kpc of the Sun, while the radial velocity measurement indicates that the source is a Galactic bulge giant.

Microlens Mass Measurement Using Triple-Peak Events

We show that one can measure the effects of microlens parallax for binary microlensing events with three well-measured peaks -- two caustic crossings plus a cusp approach, and hence derive the projected Einstein radius. Since the angular Einstein radius is measurable from finite source effects for almost any well-observed caustic crossing, triple-peak events can yield the determination of the lens mass. We note that, to a certain extent, rotation of the binary can mimic the effects of parallax, but it should often be possible to disentangle parallax from rotation by making use of the late-time light curve.

Another channel to detect close-in binary companions via gravitational microlensing

Gaudi & Gould showed that close companions of remote binary systems can be efficiently detected by using gravitational microlensing via the deviations in the lensing light curves induced by the existence of the lens companions. In this paper, we introduce another channel to detect faint close-in binary companions by using microlensing. This method utilizes a caustic-crossing binary lens event with a source also composed of binary stars, where the companion is a faint star. Detection of the companion is possible because the flux of the companion can be highly amplified when it crosses the lens caustic. The detection is facilitated since the companion is more amplified than the primary because it, in general, has a smaller size than the primary, and thus experiences less finite source effect. The method is an extension of the previous one suggested to detect close-in giant planets by Graff & Gaudi and Lewis & Ibata and further developed by Ashton & Lewis. From the simulations of realistic Galactic bulge events, we find that companions of K-type main-sequence or brighter stars can be efficiently detected from the current type of microlensing follow-up observations by using the proposed method. We also find that compared with the method of detecting lens companions for which the efficiency drops significantly for binaries with separations ≲0.2 of the angular Einstein ring radius, θE, the proposed method has an important advantage of being able to detect companions with substantially smaller separations down to ∼.

The calculation and measurement of synchrotron radiation flux: finite emittance effects

A finite emittance model (CLSxBW) for calculating the flux of synchrotron radiation is described. CLSxBW accounts explicitly for the finite source size and divergence of the electron beam in the storage ring. Finite source effects are found to be very critical for the proper calculation of the linear polarization of the synchrotron radiation. Intercomparisons of CLSxBW with S chwinger (Bessy computer code) show agreement at 1 and 10 ppm for rectangular and circular detector apertures, respectively. Comparison of emitted power calculated with CLSxBW and power measurements on the white light beamline in BessyII using a broadband detector show excellent quantitative agreement.

A Suggestion for Fitting Ground-motion Attenuation near an Extended Earthquake Source

Peak accelerations recorded near the causative faults of recent large earthquakes have been smaller than expected from published attenuation relations (Somerville, 2000; Anderson et al., 2000). This discrepancy provides a motivation to find new empirical fits of peak ground motion to functional forms that account for the extended nature of the source at locations near the fault (Ohno et al., 1993; Ambraseys and Srbulov, 1998). It has long been proposed (Housner, 1965; Brune, 1970; Hanks and Johnson, 1976) that peak acceleration and short-period velocity spectral response approach constant values at near-fault locations due to the finite-source effect and that the value may be independent of magnitude (for larger magnitudes). Joyner and Boore (1981) introduced a functional form for attenuation of peak ground motion in which the dependencies on distance and magnitude are separable. They account for the finite extent of the source by measuring distance from the closest point of the surface-projection of the fault. Their formulation is appropriate if only a single point on the fault contributes to the peak motion at any observation point. In this note I take the point of view that a considerable portion of the rupture surface contributes to peak ground motion at a distant point. Peak motion is larger for larger magnitudes, in this view, because larger areas of the fault contribute. The dependence found on magnitude at distant points will not apply at near points, where a smaller fault area contributes to the peak motion. Boore, Joyner, and Fumal (1997) (BJF97) continue to assume that magnitude and distance dependence are separable, and they show (Boore et al., 1994) that their data are consistent with that assumption. In contrast to BJF97, attenuation relations of Abrahamson and Silva (1997), Campbell (1997), and Sadigh et al. (1997) are not separable but …

Constraining the Location of Microlensing Objects by Using the Finite Source Effect in EAGLE Events

We propose a new method to constrain the location of microlensing objects using EAGLE (Extremely Amplified Gravitational LEnsing) events. We have estimated the rate of EAGLE events by taking the finite-source effect in to account. We found that the EAGLE event rate for using a 1-m class telescope w hose limiting magnitude is $V \sim 21$ is the same as or higher than that of the ordinary microlensing events which have been found to date. We have also found that the fraction of transit EAGLE events is large enough to detect: between $4 \sim 80 % $ depending on the lens location. Since the lens proper motion can be measured for a transit event, one can distinguish whether the lens is a MACHO (MAssive Compact Halo Object) in our hal o or one of the known stars in the Large Magellanic Cloud (LMC) from the proper motion measurement for each transit EAGLE event. Moreover, we show that the fraction of transit EAGLEs in all EAGLE events signif icantly depends on the lensing locations: the transit EAGLE fraction for the sel f-lensing case is $2 \sim 15$ times larger than that for halo MACHOs. Thus, one can constrain the location of lens objects by the statistics of the tr ansit events fraction. We show that we can reasonably expect $0 \sim 6$ transit events out of 21 EAGLE events in 3 years. We can also constrain the lens population properties at a gre ater than 99% confidence level depending on the number of transit events de tected. We also present the duration of EAGLE events, and show how an hourly ob servational mode is more suitable for an EAGLE event search program.

Optimization of spherical deflecting analyzer with finite-size effects

Spherical analyzers have been widely used in electron spectroscopy due to their high energy resolution. In this article, we present a way of optimizing the performance of a 90° spherical deflecting analyzer (SDA-90) considering finite-source effects, using an imaging matrix formalism. Results are shown for both symmetrically and asymmetrically charged cases. The asymmetrical case is found to be superior both in transmission and resolution. This unexpected advantage can be explained by the rotation of the image caused by the broken symmetry along the principle trajectory in the analyzer using an asymmetrical potential configuration. The net effect is equivalent to a reduction in the lateral magnification and a gain in energy dispersion. Both factors improve the performance of the analyzer. The fringe fields, carefully controlled by two Herzog lenses, are very important in the optimization. A telefocus electron gun is used to check the numerical result, and to investigate the transverse focusing behavior.

Nonlinear sediment response during the 1994 Northridge earthquake: Observations and finite source simulations

We have addressed the long‐standing question regarding nonlinear sediment response in the Los Angeles region by testing whether sediment amplification was similar between the Northridge earthquake and its aftershocks. Comparing the weak‐ and strong‐motion site response at 15 sediment sites, we find that amplification factors were significantly less for the main shock implying systematic nonlinearity. The difference is largest between 2 and 4 Hz (a factor of 2), and is significant at the 99% confidence level between 0.8 and 5.5 Hz. The inference of nonlinearity is robust with respect to the removal of possibly anomalous sediment sites and how the reference‐site motion is defined. Furthermore, theoretical ground‐motion simulations show no evidence of any bias from finite source effects during the main shock. Nonlinearity is also suggested by the fact that the four sediment sites that contain a clear fundamental resonance for the weak motion exhibit a conspicuous absence of the peak in the strong motion. Although we have taken the first step of establishing the presence of nonlinearity, it remains to define the physics of nonlinear response and to test the methodologies presently applied routinely in engineering practice. The inference of nonlinearity implies that care must be exercised in using sediment site data to study large earthquakes or predict strong ground motion.

Extreme Microlensing toward the Galactic Bulge

Extreme microlensing events, defined as events with maximum magnification Amax ≳ 200, are a potentially powerful probe of the mass spectrum and spatial distribution of objects along lines of sight toward the Galactic bulge. About 30 yr-1 such events are expected for main-sequence sources with I0 < 19. For many of these it is possible to measure both a "proper motion" and a "parallax," that together would yield individual mass, distance, and transverse-speed determinations of the lensing object. The proper motion is determined from finite-source effects when the lens transits, or nearly transits, the source. The parallax is determined by observing the difference in the light curve as seen from two Earth observatories separated by about 1 Earth radius, R⊕. The size of the parallax effect is ~Amax R⊕/e, where e is the projected Einstein radius and can be of order 1%. Detection of candidate events requires a pixel-lensing search of the entire bulge once per day, preferably by at least two observatories on different continents. Follow-up observation must be carried out using optical/infrared photometry, with short (e.g., 1 minute) exposures on small ( ≳ 1 m) telescopes. Extreme microlensing observations toward the Large Magellanic Cloud do not appear feasible at the present time.

Einstein Radii from Binary‐Source Lensing Events

We show that the Einstein ring radius and transverse speed of a lens projected on the source plane, E and , can be determined from the light curve of a binary-source event, followed by the spectroscopic determination of the orbital elements of the source stars. The determination makes use of the same principle that allows one to measure the Einstein ring radii from finite-source effects. For the case when the orbital period of the source stars is much longer than the Einstein timescale, P tE, there exists a single twofold degeneracy in determining E. However, when P tE the degeneracy can often be broken by making use of the binary-source system's orbital motion. For ~8% of all lensing events with sources V < 20 seen toward the Large Magellanic Cloud (LMC), one can unambiguously determine whether the lenses are Galactic or whether they lie in the LMC itself. In addition, we propose to include eclipsing binaries as sources for gravitational lensing experiments.