Interferometric visibility and closure phase of microlensing events with finite source size

Interferometers from the ground and space will be able to resolve the two images in a microlensing event. This will at least partially lift the inherent degeneracy between physical parameters in microlensing events. To increase the signal-to-noise ratio, intrinsically bright events with large magnifications will be preferentially selected as targets. These events may be influenced by finite source size effects both photometrically and astrometrically. Using observed finite source size events as examples, we show that the fringe visibility can be affected by ∼5–10 per cent, and the closure phase by a few degrees – readily detectable by ground and space interferometers. Such detections will offer unique information about the lens–source trajectory relative to the baseline of the interferometers. Combined with photometric finite source size effects, interferometry offers a way to measure the angular sizes of the source and the Einstein radius accurately. Limb-darkening changes the visibility by a small amount compared with a source with uniform surface brightness, marginally detectable with ground-based instruments. We discuss the implications of our results for the plans to make interferometric observations of future microlensing events.     

A measurement of the transverse velocity of Q2237+0305

Determination of microlensing parameters in the gravitationally lensed quasar Q2237+0305 from the statistics of high-magnification events will require monitoring for more than 100 years (Wambsganss, Paczynski & Schneider). However, we show that the effective transverse velocity of the lensing galaxy can be determined on a more realistic time-scale through consideration of the distribution of light-curve derivatives. The 10 years of existing monitoring data for Q2237+0305 are analysed. These data display strong evidence for microlensing that is not associated with a high-magnification event. An upper limit of v _t<500 km s⁻¹ is obtained for the galactic transverse velocity, which is smaller than previously assumed values. The analysis suggests that the observed microlensing variation may be predominantly due to stellar proper motions. The statistical significance of the results obtained from our method will be increased by the addition of data points from current and future monitoring campaigns. However, reduced photometric errors will be more valuable than an increased sampling rate.     

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, θ _E, the proposed method has an important advantage of being able to detect companions with substantially smaller separations down to ∼     .     

Parallax microlensing events in the OGLE II data base toward the Galactic bulge

We present a systematic search for parallax microlensing events among a total of 512 microlensing candidates in the OGLE II data base for the  1997–1999  seasons. We fit each microlensing candidate with both the standard microlensing model and a parallax model that accounts for the Earth's motion around the Sun. We then search for the parallax signature by comparing the χ ² of the standard and parallax models. For the events which show a significant improvement, we further use the 'duration' of the event and the signal-to-noise ratio as criteria to separate true parallax events from other noisy microlensing events. We have discovered one convincing new candidate, sc33_4505, and seven other marginal cases. The convincing candidate (sc33_4505) is caused by a slow-moving, and likely low-mass, object, similar to other known parallax events. We found that irregular sampling and gaps between observing seasons hamper the recovery of parallax events. We have also searched for long-duration events that do not show parallax signatures. The lack of parallax effects in a microlensing event puts a lower limit on the Einstein radius projected on to the observer plane, which in turn imposes a lower limit on the lens mass divided by the relative lens–source parallax. Most of the constraints are however quite weak.     

On the probability of microlensing in gamma-ray burst afterglows

The declining light curve of the optical afterglow of gamma-ray burst (GRB) GRB000301C showed rapid variability with one particularly bright feature at about t − t ₀=3.8 d. This event was interpreted as gravitational microlensing by Garnavich, Loeb & Stanek and subsequently used to derive constraints on the structure of the GRB optical afterglow. In this paper, we use these structural parameters to calculate the probability of such a microlensing event in a realistic scenario, where all compact objects in the universe are associated with observable galaxies. For GRB000301C at a redshift of z =2.04, the a posteriori probability for a microlensing event with an amplitude of Δ m 0.95 mag (as observed) is 0.7 per cent (2.7 per cent) for the most plausible scenario of a flat Λ-dominated Friedmann–Robertson–Walker (FRW) universe with Ω_m=0.3 and a fraction f ∗=0.2 (1.0) of dark matter in the form of compact objects. If we lower the magnification threshold to Δ m 0.10 mag, the probabilities for microlensing events of GRB afterglows increase to 17 per cent (57 per cent). We emphasize that this low probability for a microlensing signature of almost 1 mag does not exclude that the observed event in the afterglow light curve of GRB000301C was caused by microlensing, especially in light of the fact that a galaxy was found within 2 arcsec from the GRB. In that case, however, a more robust upper limit on the a posteriori probability of ≈5 per cent is found. It does show, however, that it will not be easy to create a large sample of strong GRB afterglow microlensing events for statistical studies of their physical conditions on microarcsec scales.     

Microlensing by cosmic strings

We consider the signature and detectability of gravitational microlensing of distant quasars by cosmic strings. Because of the simple image configuration such events will have a characteristic lightcurve, in which a source would appear to brighten by exactly a factor of 2, before reverting to its original apparent brightness. We calculate the optical depth and event rate, and conclude that current predictions and limits on the total length of strings on the sky imply optical depths of ≲ 10⁻⁸ and event rates of fewer than one event per 10⁹ sources per year. Disregarding those predictions but replacing them with limits on the density of cosmic strings from the cosmic microwave background fluctuation spectrum, leaves only a small region of parameter space (in which the sky contains about 3 × 10⁵ strings with deficit angle of the order of 0.3 milli-seconds) for which a microlensing survey of exposure 10⁷ source years, spanning a 20–40-year period, might reveal the presence of cosmic strings.     

Short time-scale fluctuations in the difference light curves of QSO 0957+561A,B: microlensing or noise?

From optical R -band data of the double quasar QSO 0957+561A,B, we made two new difference light curves (with an overlap of about 330 d between the time-shifted light curve for the A image and the magnitude-shifted light curve for the B image). We observed noisy behaviours around the zero line and no short time-scale events (with a duration of months), where the term 'event' refers to a prominent feature that may be a result of microlensing or another source of variability. Only one event lasting two weeks and rising −33 mmag was found . Measured constraints on the possible microlensing variability can be used to obtain information on the granularity of the dark matter in the main lensing galaxy and the size of the source. In addition, one can also test the ability of the observational noise to cause the rms averages and the local features of the difference signals. We focused on this last issue. The combined photometries were related to a process consisting of an intrinsic signal plus a Gaussian observational noise. The intrinsic signal has been assumed to be either a smooth function (polynomial), a smooth function plus a stationary noise process, or a correlated stationary process . Using these three pictures without microlensing, we derived some models totally consistent with the observations. We finally discussed the sensitivity of our telescope (at Teide Observatory) to several classes of microlensing variability.     

Rotating haloes and the microlensing MACHO mass estimate

We investigate the implications of a bulk rotational component of the Galactic halo velocity distribution for MACHO mass estimates. We find that for a rotating halo to yield a MACHO mass estimate significantly below that of the standard spherical case, its microlensing must be highly concentrated close to the Sun. We examine two classes of models fitting this criterion: a highly flattened 1/ r ² halo, and a spheroid-like population the density of which falls off as 1/ r ^3.5. The highly flattened 1/ r ² models can decrease the implied average MACHO mass only marginally, and the spheroid models not at all. Generally, rotational models cannot bring the MACHO mass implied by the current microlensing data down to the substellar range.     

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.     

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 baseline flux determination of microlensing events detectable with the difference image analysis method

To improve photometric precision by removing the blending effect, a newly developed technique of difference image analysis (DIA) has been adopted by several gravitational microlensing experiment groups. However, the principal problem of the DIA method is that, by its nature, it has difficulties in measuring the baseline flux F ₀ of a source star, causing a degeneracy problem in determining the lensing parameters of an event. Therefore, it is often believed that the DIA method is not as powerful as the classical method based on PSF photometry for determining the Einstein time-scales t _E of events.
In this paper, we demonstrate that the degeneracy problem in microlensing events, detectable from searches using the DIA method, is not as serious as is often thought. This is because a substantial fraction of events will be high amplification events for which the deviations of the amplification curves, constructed with the wrong baseline fluxes from their corresponding best-fit standard amplification curves, will be considerable, even for a small amount of the fractional baseline flux deviation Δ F ₀ F ₀. With a model luminosity function of source stars and under realistic observational conditions, we find that ∼30 per cent of detectable Galactic bulge events are expected to have high amplifications and their baseline fluxes can be determined with uncertainties Δ F ₀ F ₀≤0.5.     

Predicting the second caustic crossing in binary microlensing events

We fit binary lens models to the data covering the initial part of real microlensing events in an attempt to predict the time of the second caustic crossing. We use approximations during the initial search through the parameter space for light curves that roughly match the observed ones. Exact methods for calculating the lens magnification of an extended source are used when we refine our best initial models. Our calculations show that the reliable prediction of the second crossing can only be made very late, when the light curve has risen appreciably after the minimum between the two caustic crossings. The best observational strategy is therefore to sample as frequently as possible once the light curve starts to rise after the minimum.     

Microlensing of an extended source by a power-law mass distribution

Microlensing promises to be a powerful tool for studying distant galaxies and quasars. As the data and models improve, there are systematic effects that need to be explored. Quasar continuum and broad-line regions may respond differently to microlensing due to their different sizes; to understand this effect, we study microlensing of finite sources by a mass function of stars. We find that microlensing is insensitive to the slope of the mass function but does depend on the mass range. For negative-parity images, diluting the stellar population with dark matter increases the magnification dispersion for small sources and decreases it for large sources. This implies that the quasar continuum and broad-line regions may experience very different microlensing in negative-parity lensed images. We confirm earlier conclusions that the surface brightness profile and geometry of the source have little effect on microlensing. Finally, we consider non-circular sources. We show that elliptical sources that are aligned with the direction of shear have larger magnification dispersions than sources with perpendicular alignment, an effect that becomes more prominent as the ellipticity increases. Elongated sources can lead to more rapid variability than circular sources, which raises the prospect of using microlensing to probe source shape.     

Constraints on σ8 from galaxy clustering in N-body simulations and semi-analytic models

We generate mock galaxy catalogues for a grid of different cosmologies, using rescaled N -body simulations in tandem with a semi-analytic model run using consistent parameters. Because we predict the galaxy bias, rather than fitting it as a nuisance parameter, we obtain an almost pure constraint on σ₈ by comparing the projected two-point correlation function we obtain to that from the Sloan Digital Sky Survey (SDSS). A systematic error arises because different semi-analytic modelling assumptions allow us to fit the r -band luminosity function equally well. Combining our estimate of the error from this source with the statistical error, we find  σ₈= 0.97 ± 0.06  . We obtain consistent results if we use galaxy samples with a different magnitude threshold, or if we select galaxies by b _J-band rather than r -band luminosity and compare to data from the 2dF Galaxy Redshift Survey (2dFGRS). Our estimate for σ₈ is higher than that obtained for other analyses of galaxy data alone, and we attempt to find the source of this difference. We note that in any case, galaxy clustering data provide a very stringent constraint on galaxy formation models.     

The microlensing rate and mass function versus dynamics of the Galactic bar

With the steady increase of the sample size of observed microlenses towards the central regions of the Galaxy, the main source of the uncertainty in the lens mass will shift from the simple Poisson noise to the intrinsic non-uniqueness of our dynamical models of the inner Galaxy, particularly the Galactic bar. We use a set of simple self-consistent bar models to investigate how the microlensing event rate varies as a function of axis ratio, bar angle and velocity distribution. The non-uniqueness of the velocity distribution of the bar model adds a significant uncertainty (by about a factor of 1.5) to any prediction of the lens mass. Kinematic data and self-consistent models are critical to lift the non-uniqueness. We discuss the implications of these results for the interpretation of microlensing observations of the Galactic bulge. In particular we show that Freeman bar models scaled to the mass of the Galactic bulge/bar imply a typical lens mass of around 0.8 M⊙, a factor of 3–5 times larger than the value from other models.     

Detection of stellar spots from the observations of caustic-crossing binary-lens gravitational microlensing events

Recently, Heyrovský & Sasselov investigated the sensitivity of single-lens gravitational microlensing event light curves to spots and found that, during source transit, spots can cause deviations in amplification larger than 2 per cent, and thus be detectable. In this paper, we explore the feasibility of spot detection from the observations of binary-lens microlensing events instead of single-lens events. For this we investigate the sensitivity of binary-lens event light curves to spots and compare it with that of single-lens events. From this investigation, we find that during caustic crossings the fractional amplification deviations of light curves from those of spotless source events are equivalent to those of single-lens events, implying that spots can also be detected with a similar photometric precision to that required for spot detection by observing single-lens events. We discuss the relative advantages of observing binary-lens events over the observations of single-lens events in detecting stellar spots.     

On the intrinsic bias in detecting caustic crossings between Galactic halo and self-lensing events in the Magellanic Clouds

In this paper, we investigate the intrinsic bias in detecting caustic crossings between the Galactic halo and self-lensing gravitational microlensing events in the Magellanic Clouds. For this, we determine the region for optimal caustic-crossing detection in the parameter space of the physical binary separations, ℓ, and the total binary lens mass, M , and find that the optimal regions for both populations of events are similar to each other. In particular, if the Galactic halo is composed of lenses with the claimed average mass of 〈 M 〉∼0.5 M_⊙, the optimal binary separation range of Galactic halo events of 3.5 au≲ℓ≲14 au matches well with that of a Magellanic Cloud self-lensing event caused by a binary lens with a total mass of M ∼1 M_⊙; well within the mass range of the most probable lens population of stars in the Magellanic Clouds. Therefore, our computation implies that if the binary fractions and the distributions of binary separations of the two populations of lenses are not significantly different from each other, there is no strong detection bias against Galactic halo caustic-crossing events.     

Helical fields and filamentary molecular clouds – I

We study the equilibrium of pressure truncated, filamentary molecular clouds that are threaded by rather general helical magnetic fields. We first apply the virial theorem to filamentary molecular clouds, including the effects of non-thermal motions and the turbulent pressure of the surrounding ISM. When compared with the data, we find that many filamentary clouds have a mass per unit length that is significantly reduced by the effects of external pressure, and that toroidal fields play a significant role in squeezing such clouds.
We also develop exact numerical MHD models of filamentary molecular clouds with more general helical field configurations than have previously been considered. We examine the effects of the equation of state by comparing 'isothermal' filaments, with constant total (thermal plus turbulent) velocity dispersion, with equilibria constructed using a logatropic equation of state.
Our theoretical models involve three parameters: two to describe the mass loading of the toroidal and poloidal fields, and a third that describes the radial concentration of the filament. We thoroughly explore our parameter space to determine which choices of parameters result in models that agree with the available observational constraints. We find that both equations of state result in equilibria that agree with the observational results. Moreover, we find that models with helical fields have more realistic density profiles than either unmagnetized models or those with purely poloidal fields; we find that most isothermal models have density distributions that fall off as r ^−1.8 to r ⁻², while logatropes have density profiles that range from r ⁻¹ to r ^−1.8. We find that purely poloidal fields produce filaments with steep radial density gradients that are not allowed by the observations.     

A comprehensive set of simulations of high-velocity collisions between main-sequence stars

We report on a very large set of simulations of collisions between two main-sequence (MS) stars. These computations were carried out with the smoothed particle hydrodynamics method. Realistic stellar structure models for evolved MS stars were used. In order to sample an extended domain of initial parameters space (masses of the stars, relative velocity and impact parameter), more than 14 000 simulations were carried out. We considered stellar masses ranging between 0.1 and  75 M_⊙  and relative velocities up to a few thousand km s⁻¹. To limit the computational burden, a resolution of 1000–32 000 particles per star was used. The primary goal of this study was to build a complete data base from which the result of any collision can be interpolated. This allows us to incorporate the effects of stellar collisions with an unprecedented level of realism into dynamical simulations of galactic nuclei and other dense stellar clusters. We make the data describing the initial condition and outcome (mass and energy loss, angle of deflection) of all our simulations available on the Internet. We find that the outcome of collisions depends sensitively on the stellar structure and that, in most cases, using polytropic models is inappropriate. Published fitting formulae for the collision outcomes, established from a limited set of collisions, prove of limited use because they do not allow robust extrapolation to other stellar structures or relative velocities.     

Globular clusters as microlensing targets

We investigate the possibility of using globular clusters as targets for microlensing searches. Such searches will be challenging and require more powerful telescopes than now employed, but are feasible in the near future. Although expected event rates are low, we show that the wide variety of lines of sight to globular clusters greatly enhances the ability to distinguish between halo models using microlensing observations as compared with LMC/SMC observations alone. In particular, the halo core radius and power-law exponent can be determined with good accuracy.