Effects of a deformation of a star on the gravitational lensing

We study analytically a gravitational lens due to a deformed star, which is modelled by using a monopole and a quadrupole moment. Positions of the images are discussed for a source on the principal axis. We present explicit expressions for the lens equation for this gravitational lens as a single real 10th-order algebraic equation. Furthermore, we compute an expression for the caustics as a discriminant for the polynomial. Another simple parametric representation of the caustics is also presented in a more tractable form. A simple expression for the critical curves is obtained to clarify a topological feature of the critical curves; the curves are simply connected if and only if the distortion is sufficiently large.     

The Chang–Refsdal lens revisited

This paper provides a complete theoretical treatment of the point-mass lens perturbed by constant external shear, often called the Chang–Refsdal lens. We show that simple invariants exist for the products of the (complex) positions of the four images, as well as moment sums of their signed magnifications. The image topographies and equations of the caustics and critical curves are also studied. We derive the fully analytic expressions for pre-caustics, which are the loci of non-critical points that map to the caustics under the lens mapping. They constitute boundaries of the region in the image domain that maps on to the interior of the caustics. The areas under the critical curves, caustics and pre-caustics are all evaluated, which enables us to calculate the mean magnification of the source within the caustics. Additionally, the exact analytic expression for the magnification distribution for the source in the triangular caustics is derived, as well as a useful approximate expression. Finally, we find that the Chang–Refsdal lens with additional convergence greater than unity (the 'overfocusing case') can exhibit third-order critical behaviour, if the 'reduced shear' is exactly equal to     , and that the number of images for N -point masses with non-zero constant shear cannot be greater than  5 N − 1  .     

Search for exotic matter from gravitational microlensing observations of stars

We consider small-scale spheroidal clusters of weakly interacting massive particles in our Galaxy as non-compact gravitational microlenses and predict the appearance of caustics in the plane of a lensed source. The crossing of these caustics by a lensed star can produce a large variety of light curves, including some observed in actual microlensing events that have been interpreted as manifestations of binary gravitational lenses. We consider also observable effects during the gravitational microlensing of stars of non-zero angular size with a given brightness distribution across their disks by such an exotic objects as natural wormholes and objects whose space-time environment is described with the NUT metric. We demonstrate that, under certain conditions, the microlensing light curves, chromatic and polarizational effects due to the properties of the lens and the star disk brightness distributions can differ considerably from those observed for a Schwarzschild gravitational lens, so that their analysis can facilitate the identification of such objects.     

A genetic algorithm for the non-parametric inversion of strong lensing systems

We present a non-parametric technique to infer the projected mass distribution of a gravitational lens system with multiple strong-lensed images. The technique involves a dynamic grid in the lens plane on which the mass distribution of the lens is approximated by a sum of basis functions, one per grid cell. We used the projected mass densities of Plummer spheres as basis functions. A genetic algorithm then determines the mass distribution of the lens by forcing images of a single source, projected back on to the source plane, to coincide as well as possible. Averaging several tens of solutions removes the random fluctuations that are introduced by the reproduction process of genomes in the genetic algorithm and highlights those features common to all solutions. Given the positions of the images and the redshifts of the sources and the lens, we show that the mass of a gravitational lens can be retrieved with an accuracy of a few percent and that, if the sources sufficiently cover the caustics, the mass distribution of the gravitational lens can also be reliably retrieved. A major advantage of the algorithm is that it makes full use of the information contained in the radial images, unlike methods that minimize the residuals of the lens equation, and is thus able to accurately reconstruct also the inner parts of the lens.     

Chromaticity of gravitational microlensing events

In this paper, we investigate the colour changes of gravitational microlensing events caused by the two different mechanisms of differential amplification for a limb-darkened extended source and blending. From this investigation, we find that the colour changes of limb-darkened extended source events (colour curves) have dramatically different characteristics depending on whether the lens transits the source star or not. We show that for a source transit event, the lens proper motion can be determined by simply measuring the turning time of the colour curve instead of fitting the overall colour or light curves. We also find that even for a very small fraction of blended light, the colour changes induced by blending are equivalent to those induced by limb darkening, causing serious distortion in the observed colour curve. Therefore, to obtain useful information about the lens and source star from the colour curve of an event, it will be essential to correct for blending. We discuss various methods of blending correction .     

On the astrometric behaviour of binary microlensing events

Despite the suspected binarity for a significant fraction of Galactic lenses, the current photometric surveys detected binary microlensing events only for a small fraction of the total events. The detection efficiency is especially low for non-caustic crossing events, which comprise the majority of the binary lensing events, as a result of the absence of distinctive features in their light curves combined with small deviations from the standard light curve of a single point-mass event. In addition, even if they are detected, it will be difficult to determine the solution of the binary lens parameters owing to the severe degeneracy problem. In this paper, we investigate the properties of binary lensing event expected when they are astrometrically observed by using high-precision interferometers. For this, we construct vector field maps of excess centroid shifts, which represent the deviations of the binary lensing centroid shifts from those of a single lensing event as a function of source position. From the analysis of the maps, we find that the excess centroid shifts are substantial in a considerably large area around caustics. In addition, they have characteristic sizes and directions depending strongly on the source positions with respect to the caustics and the resulting trajectories of the light centroid (astrometric trajectories) have distinctive features, which can be distinguished from the deviations caused by other reasons. We classify the types of the deviations and investigate where they occur. Because of the strong dependence of the centroid shifts on the lens system geometry combined with the distinctive features in the observed astrometric trajectories, astrometric binary lensing observations will provide an important tool that can probe the properties of the Galactic binary lens population.     

The influence of central black holes on gravitational lenses

Recent observations indicate that many if not all galaxies host massive central black holes. In this paper we explore the influence of black holes on the lensing properties. We model the lens as an isothermal ellipsoid with a finite core radius plus a central black hole. We show that the presence of the black hole substantially changes the critical curves and caustics. If the black hole mass is above a critical value, then it will completely suppress the central images for all source positions. Realistic central black holes are likely to have masses below this critical value. Even in such subcritical cases, the black hole can suppress the central image when the source is inside a zone of influence, which depends on the core radius and black hole mass. In the subcritical cases, an additional image may be created by the black hole in some regions, which for some radio lenses may be detectable with high-resolution and large dynamic range VLBI maps. The presence of central black holes should also be taken into account when one constrains the core radius from the lack of central images in gravitational lenses.     

Effects of sherically-symmetric gravitational lenses produced by galaxies and clusters

Images are considered which are produced by a spherically symmetric gravitational lens from a source of small angular size. Simple analytic expressions are derived to describe the position, amplification, shape distortion and time delay for these images provided the mass distribution of the lens corresponds to the King model. Possible types of images are analysed (single and triple images, as well as degenerate double ones at caustics) which may be formed at different positions of the observer relative to the light source and lens. The focal length of galaxies of different types is evaluated on the basis of the obtained results and the conditions are determined at which these galaxies may considerably affect the images of distant sources.     

Pixel lensing as a way to detect extrasolar planets in M31

We study the possibility to detect extrasolar planets in M31 through pixel-lensing observations. Using a Monte Carlo approach, we select the physical parameters of the binary lens system, a star hosting a planet, and we calculate the pixel-lensing light curve taking into account the finite source effects. Indeed, their inclusion is crucial since the sources in M31 microlensing events are mainly giant stars. Light curves with detectable planetary features are selected by looking for significant deviations from the corresponding Paczyński shapes. We find that the time-scale of planetary deviations in light curves increase (up to 3–4 d) as the source size increases. This means that only few exposures per day, depending also on the required accuracy, may be sufficient to reveal in the light curve a planetary companion. Although the mean planet mass for the selected events is about     , even small mass planets  ( M _P < 20 M_⊕)  can cause significant deviations, at least in the observations with large telescopes. However, even in the former case, the probability to find detectable planetary features in pixel-lensing light curves is at most a few per cent of the detectable events, and therefore many events have to be collected in order to detect an extrasolar planet in M31. Our analysis also supports the claim that the anomaly found in the candidate event PA-99-N2 towards M31 can be explained by a companion object orbiting the lens star.     

Gravitational lens under perturbations: symmetry of perturbing potentials with invariant caustics

When the gravitational lensing potential can be approximated by that of a circularly symmetric system affected by weak perturbations, it is found that the shape of the resulting (tangential) caustics is entirely specified by the local azimuthal behaviour of the affecting perturbations. This provides a common mathematical groundwork for understanding problems such as the close–wide  ( d ↔ d ⁻¹)  separation degeneracy of binary lens microlensing light curves and the shear–ellipticity degeneracy of quadruple image lens modelling.     

Effects of microlensing on parameters of the images seen near the critical curves of gravitational lens galaxies

We investigate the effect of microlensing on parameters of the images of distant sources seen near the critical curves of complex gravitational lenses, which are represented as a sum of compact structures—microlenses (stars, star-like or planet-like bodies) and diffusely distributed matter (dust and gas clouds etc.). The observation of merging, cross-shaped, annular, or arc-shaped source images is an indication that the images are close to the critical curves of gravitational lenses. Our analysis and numerical solution have allowed us to determine the structures of the critical curves and caustics formed by macro-and microlenses, as well as to estimate the characteristic perturbations introduced by microlenses at their various positions relative to the critical curve of a regular gravitational lens. We show that, the closer are the microlenses to the critical curve, the larger is the discrepancy between our results and those obtained previously with standard (linearized) allowance for the effect of a regular gravitational lens.     

Where are the binary source gravitational microlensing events? II

Despite the same multiplicity of lenses and sources, the frequency of detection of binary source events is relatively very low compared with that of binary lens events. Dominik pointed out that the rarity of binary source events is caused mainly by the large difference in amplification between the component stars. In this paper, we determine that the fraction of events with similar source star amplifications is as large as ∼8 per cent, and thus show that the very low detection rate for binary source events cannot be explained by this effect alone. By carrying out realistic simulations of binary source events, we find that a significant number of binary source events are additionally missed from detection for various other reasons. First, if the flux ratio between the component stars is very large, the light curve of the bright star is hardly affected by the light from the faint star. Secondly, if the separation is too small, the binary source stars behave like a single star, making it difficult to separate the binary source event from a single source event. Finally, although the probability of detecting binary source events increases as the source separation increases, some fraction of binary source events will still be missed because the light curves of these events will mimic those of single source events with longer time-scales and larger values of the impact parameter.     

Planetary microlensing signals from the orbital motion of the source star around the common barycentre

With several detections, the technique of gravitational microlensing has proven useful for studying planets that orbit stars at Galactic distances, and it can even be applied to detect planets in neighbouring galaxies. So far, planet detections by microlensing have been considered to result from a change in the bending of light and the resulting magnification caused by a planet around the foreground lens star. However, in complete analogy to the annual parallax effect caused by the revolution of the Earth around the Sun, the motion of the source star around the common barycentre with an orbiting planet can also lead to observable deviations in microlensing light curves that can provide evidence for the unseen companion. We discuss this effect in some detail and study the prospects of microlensing observations for revealing planets through this alternative detection channel. Given that small distances between lens and source star are favoured, and that the effect becomes nearly independent of the source distance, planets would remain detectable even if their host star is located outside the Milky Way with a sufficiently good photometry (exceeding present-day technology) being possible. From synthetic light curves arising from a Monte Carlo simulation, we find that the chances for such detections are not overwhelming and appear practically limited to the most massive planets (at least with current observational set-ups), but they are large enough for leaving the possibility that one or the other signal has already been observed. However, it may remain undetermined whether the planet actually orbits the source star or rather the lens star, which leaves us with an ambiguity not only with respect to its location, but also to its properties.     

Colour change measurements of gravitational microlensing events by using the difference image analysis method

Detecting colour changes of a gravitational microlensing event induced by the limb-darkened extended source effect is important for obtaining useful information about both the lens and the source star. However, precise measurements of the colour changes are hampered by blending, which also causes colour changes of the event. In this paper, we show that although the colour change measured from the subtracted image by using the recently developed photometric method of the 'difference image analysis' (DIA) differs from the colour change measured by using the conventional method based on the extraction of the individual source stars' point spread functions, the curve of the colour changes (colour curve) constructed by using the DIA method enables one to obtain the same information about the lens and source star, but with significantly reduced uncertainties due to the absence of blending. We investigate the patterns of the DIA colour curves for both single lens and binary lens events by constructing colour change maps.     

Theoretical light curves of Type II-P supernovae and applications to cosmology

Based on an extensive grid of stellar models between 13 and  25 M_⊙  and a wide range of metallicities, we have studied the light curves of core collapse supernovae, their application to cosmology and their evolutionary effects with redshift. The direct link between the hydrodynamics and radiation transport allows us to calculate monochromatic light curves.
With decreasing metallicity, Z , and increasing mass, progenitors tend to explode as compact blue supergiants (BSG) and produce subluminous supernovae that are approximately 1.5 mag dimmer than normal Type II supernovae (SNe II) with red supergiant (RSG) progenitors. Progenitors with small masses tend to explode as RSGs even at low Z . The consequence for testing the chemical evolution is obvious, namely a strong bias when using the statistics of core collapse supernovae to determine the history of star formation.
Our study is limited in scope with respect to the explosion energies and the production of radioactive Ni. Within the class of extreme SNe II-P supernovae, the light curves are rather insensitive with respect to the progenitor mass and explosion energy compared with analytic models based on parametrized stellar structures. We expect a wider range of brightness due to variations in ⁵⁶Ni because radioactive energy is a significant source of luminosity. However, the overall insensitivity of light curves may allow their use as quasi-standard candles for distance determination.     

Repeating microlensing events in the OGLE data

Microlensing events are usually selected among single-peaked non-repeating light curves in order to avoid confusion with variable stars. However, a microlensing event may exhibit a second microlensing brightening episode when the source or/and the lens is a binary system. A careful analysis of these repeating events provides an independent way to study the statistics of wide binary stars and to detect extrasolar planets. Previous theoretical studies predicted that 0.5–2 per cent of events should repeat due to wide binary lenses. We present a systematic search for such events in about 4000 light curves of microlensing candidates detected by the Optical Gravitational Lensing Experiment (OGLE) towards the Galactic bulge from 1992 to 2007. The search reveals a total of 19 repeating candidates, with six clearly due to a wide binary lens. As a by-product, we find that 64 events (∼2 per cent of the total OGLE-III sample) have been misclassified as microlensing; these misclassified events are mostly nova or other types of eruptive stars. The number and importance of repeating events will increase considerably when the next-generation wide-field microlensing experiments become fully operational in the future.     

Pseudo-infinite guide stars for multi-conjugated adaptive optics on extremely large telescopes

We introduce a novel concept to sense the wavefront for adaptive optics purposes in astronomy using a conventional laser beacon. The concept we describe involves treating the light scattered in the mesospheric sodium layer as if it comes from multiple rings located at infinity. Such a concept resembles an inverse Bessel beam and is particularly suitable for multi-conjugated adaptive optics on extremely large telescopes. In fact, as the sensing process uses light apparently coming from infinity, some problems linked to the finite distance and vertical extent of the guide source are solved. Since such a technique is able to sense a wavefront solely in the radial direction, we propose furthermore a novel wavefront sensor by combining the inverse Bessel beam approach with the recently introduced z -invariant technique for a pseudo-infinite guide star sensor.     

The effect of a binary source companion on the astrometric microlensing behaviour

If gravitational microlensing occurs in a binary source system, both source components are magnified, and the resulting light curve deviates from the standard one of a single source event. However, in most cases only one source component is highly magnified and the other component (the companion) can be treated as a simple blending source: this is a blending approximation. In this paper we show that, unlike the light curves, the astrometric curves, representing the trajectories of the source image centroid, of an important fraction of binary source events will not be sufficiently well-modelled by the blending effect alone. This is because the centroid shift induced by the source companion endures to considerable distances from the lens. Therefore, in determining the lens parameters from astrometric curves to be measured by future high-precision astrometric instruments, it will be important to take the full effect of the source companion into consideration.     

Perturbation theory of N point-mass gravitational lens systems without symmetry: small mass-ratio approximation

This paper makes the first systematic attempt to determine using perturbation theory the positions of images by gravitational lensing due to arbitrary number of coplanar masses without any symmetry on a plane, as a function of lens and source parameters. We present a method of Taylor-series expansion to solve the lens equation under a small mass-ratio approximation. First, we investigate perturbative structures of a single-complex-variable polynomial, which has been commonly used. Perturbative roots are found. Some roots represent positions of lensed images, while the others are unphysical because they do not satisfy the lens equation. This is consistent with a fact that the degree of the polynomial, namely the number of zeros, exceeds the maximum number of lensed images if   N = 3  (or more). The theorem never tells which roots are physical (or unphysical). In this paper, unphysical ones are identified. Secondly, to avoid unphysical roots, we re-examine the lens equation. The advantage of our method is that it allows a systematic iterative analysis. We determine image positions for binary lens systems up to the third order in mass ratios and for arbitrary N point masses up to the second order. This clarifies the dependence on parameters. Thirdly, the number of the images that admit a small mass-ratio limit is less than the maximum number. It is suggested that positions of extra images could not be expressed as Maclaurin series in mass ratios. Magnifications are finally discussed.     

Identification of bright lenses from the astrometric observations of gravitational microlensing events

When a source star is gravitationally microlensed by a dark lens, the centroid of the source star image is displaced relative to the position of the unlensed source star, with an elliptical trajectory. Recently, routine astrometric follow-up measurements of these source star image centroid shifts by using high-precision interferometers have been proposed to measure the lens proper motion, which can resolve the lens parameter degeneracy in the photometrically determined Einstein time-scale. When an event is caused by a bright lens, on the other hand, the astrometric shift is affected by the light from the lens, but one cannot identify the existence of the bright lens from the observed trajectory because the resulting trajectory of the bright lens event is also an ellipse. As results, lensing parameters determined from the trajectory differ from those of a dark lens event, causing an incorrect identification of the lens population. In this paper, we show that although the shape and size of the astrometric centroid shift trajectory are changed because of the bright lens, the angular speed of centroid shifts around the apparent position of the unlensed source star is not affected by the lens brightness. Therefore, one can identify the existence of a bright lens and determine its brightness by comparing the lens parameters determined from the 'angular speed curve' with those determined from the trajectory of observed centroid shifts. Once the lens brightness is determined, one can correct for the lens proper motion. As the proposed method provides information about both the lens brightness (dark or bright) and the corrected values of the physical parameters of the lens, one can constrain the nature of massive compact halo objects (MACHOs) significantly better.