Observational Evidence for the Effect of Amplification Bias in Gravitational Microlensing Experiments

Recently Alard\markcite{alard1996} proposed to detect the shift of a star's image centroid, $\delta x$, as a method to identify the lensed source among blended stars. Goldberg & Wo\'zniak\markcite{goldberg1997} actually applied this method to the OGLE-1 database and found that 7 out of 15 events showed significant centroid shifts of $\delta x \gtrsim 0.2$ arcsec. The amount of centroid shift has been estimated theoretically by Goldberg.\markcite{goldberg1997} However, he treated the problem in general and did not apply it to a particular survey or field, and thus based his estimates on simple toy model luminosity functions (i.e., power laws). In this paper, we construct the expected distribution of $\delta x$ for Galactic bulge events by using the precise stellar LF observed by Holtzman et al.\markcite{holtzman1998} using HST. Their LF is complete up to $M_I\sim 9.0$ ($M_V\sim 12$), corresponding to faint M-type stars. In our analysis we find that regular blending cannot produce a large fraction of events with measurable centroid shifts. By contrast, a significant fraction of events would have measurable centroid shifts if they are affected by amplification-bias blending. Therefore, Goldberg & Wo\'zniak's measurements of large centroid shifts for a large fraction of microlensing events confirms the prediction of Han and Alard that a large fraction of Galactic bulge events are affected by amplification-bias blending.Comment: total 15 pages, including 6 figures, and no Table, submitted to ApJ on Apr 26 1998, email [email protected]

A method for optimal image subtraction

We present a new method designed for optimal subtraction of two images with different seeing. Using image subtraction appears to be essential for the full analysis of the microlensing survey images, however a perfect subtraction of two images is not easy as it requires the derivation of an extremely accurate convolution kernel. Some empirical attempts to find the kernel have used the Fourier transform of bright stars, but solving the statistical problem of finding the best kernel solution has never really been tackled. We demonstrate that it is possible to derive an optimal kernel solution from a simple least square analysis using all the pixels of both images, and also show that it is possible to fit the differential background variation at the same time. We also show that PSF variations can also be easily handled by the method. To demonstrate the practical efficiency of the method, we analyzed some images from a Galactic Bulge field monitored by the OGLE II project. We find that the residuals in the subtracted images are very close to the photon noise expectations. We also present some light curves of variable stars, and show that, despite high crowding levels, we get an error distribution close to that expected from photon noise alone. We thus demonstrate that nearly optimal differential photometry can be achieved even in very crowded fields. We suggest that this algorithm might be particularly important for microlensing surveys, where the photometric accuracy and completeness levels could be very significantly improved by using this method.Comment: 8,pages, 4 Postscript figures, emulateapj.sty include

Using Astrometry to Deblend Microlensing Events

We discuss the prospect of deblending microlensing events by observing astrometric shifts of the lensed stars. Since microlensing searches are generally performed in very crowded fields, it is expected that stars will be confusion limited rather than limited by photon statistics. By performing simulations of events in crowded fields, we find that if we assume a dark lens and that the lensed star obeys a power law luminosity function, $n(L)\propto L^{-\beta}$, over half the simulated events show a measurable astrometric shift. Our simulations included 20000 stars in a $256\times 256$ Nyquist sampled CCD frame. For $\beta=2$, we found that 58% of the events were significantly blended $(F_{\ast}/F_{tot}\leq 0.9)$, and of those, 73% had a large astrometric shift $(\geq 0.5 pixels)$. Likewise, for $\beta=3$, we found that 85% of the events were significantly blended, and that 85% of those had large shifts. Moreover, the shift is weakly correlated to the degree of blending, suggesting that it may be possible not only to detect the existence of a blend, but also to deblend events statistically using shift information.Comment: 24 pages, 7 postscript Figure

Caustic Crossing Microlensing Event by Binary MACHOs and Time Scale Bias

Caustic crossing microlensing events provide us a unique opportunity to measure the relative proper motion of the lens to the source, and so those caused by binary MACHOs are of great importance for understanding the structure of the Galactic halo and the nature of MACHOs. The microlensing event 98-SMC-01, occurred in June 1998, is the first event for which the proper motion is ever measured through the caustic crossing, and this event may be caused by binary MACHOs as we argue in this Letter. Motivated by the possible existence of binary MACHOs, we have performed the Monte Carlo simulations of caustic crossing events by binary MACHOs and investigated the properties and detectability of the events. Our calculation shows that typical caustic crossing events have the interval between two caustic crossings ($t_{\rm cc}$) of about 5 days. We argue that with the current strategy of binary event search the proper motions of these typical events are not measurable because of the short time scale. Therefore the proper motion distribution measured from caustic crossing events suffers significantly from {`}time scale bias{'}, which is a bias toward finding long time scale events and hence slowly moving lenses. We predict there are two times more short time scale events ($t_{\rm cc}\le 10$ days) than long time scale events ($t_{\rm cc}\ge 10$ days), and propose an hourly monitoring observation instead of the nightly monitoring currently undertaken to detect caustic crossing events by binary MACHOs more efficiently.Comment: 8 pages and 3 figures, accepted for publication in ApJ Letter

On the Nature and Location of the Microlenses

This paper uses the caustic crossing events in the microlens data sets to explore the nature and location of the lenses. We conclude that the large majority of lenses, whether they are luminous or dark, are likely to be binaries. Further, we demonstrate that blending is an important feature of all the data sets. An additional interpretation suggested by the data, that the caustic crossing events along the directions to the Magellanic Clouds are due to lenses located in the Clouds, implies that most of the LMC/SMC events to date are due to lenses in the Magellanic Clouds. All of these conclusions can be tested. If they are correct, a large fraction of lenses along the direction to the LMC may be ordinary stellar binary systems, just as are the majority of the lenses along the direction to the Bulge. Thus, a better understanding of the larger-than-anticipated value derived for the Bulge optical depth may allow us to better interpret the large value derived for the optical depth to the LMC. Indeed, binarity and blending in the data sets may illuminate connections among several other puzzles: the dearth of binary-source light curves, the dearth of non-caustic-crossing perturbed binary-lens events, and the dearth of obviously blended point-lens events.Comment: 15 pages, 2 figures. Submitted to the Astrophysical Journal Letters, 4 January 199

The hydrogen isotopic composition of fossil micrometeorites: Implications for the origin of water on Earth.

Accepted versio

The Age and Structure of the Galactic Bulge from Mira Variables

We report periods and JHKL observations for 648 oxygen-rich Mira variables found in two outer bulge fields at b=-7 degrees and l=+/-8 degrees and combine these with data on 8057 inner bulge Miras from the OGLE, Macho and 2MASS surveys, which are concentrated closer to the Galactic centre. Distance moduli are estimated for all these stars. Evidence is given showing that the bulge structure is a function of age. The longer period Miras (log P > 2.6, age about 5 Gyr and younger) show clear evidence of a bar structure inclined to the line of sight in both the inner and outer regions. The distribution of the shorter period (metal-rich globular cluster age) Miras, appears spheroidal in the outer bulge. In the inner region these old stars are also distributed differently from the younger ones and possibly suggest a more complex structure. These data suggest a distance to the Galactic centre, R0, of 8.9 kpc with an estimated uncertainty of 0.4 kpc. The possible effect of helium enrichment on our conclusions is discussed.Comment: Accepted for MNRAS, 12 pages, 12 figure

Difference image photometry with bright variable backgrounds

Over the last two decades the Andromeda Galaxy (M31) has been something of a test-bed for methods aimed at obtaining accurate time-domain relative photometry within highly crowded fields. Difference imaging methods, originally pioneered towards M31, have evolved into sophisticated methods, such as the Optimal Image Subtraction (OIS) method of Alard & Lupton (1998), that today are most widely used to survey variable stars, transients and microlensing events in our own Galaxy. We show that modern difference image (DIA) algorithms such as OIS, whilst spectacularly successful towards the Milky Way bulge, may perform badly towards high surface brightness targets such as the M31 bulge. Poor results can occur in the presence of common systematics which add spurious flux contributions to images, such as internal reflections, scattered light or fringing. Using data from the Angstrom Project microlensing survey of the M31 bulge, we show that very good results are usually obtainable by first performing careful photometric alignment prior to using OIS to perform point-spread function (PSF) matching. This separation of background matching and PSF matching, a common feature of earlier M31 photometry techniques, allows us to take full advantage of the powerful PSF matching flexibility offered by OIS towards high surface brightness targets. We find that difference images produced this way have noise distributions close to Gaussian, showing significant improvement upon results achieved using OIS alone. We show that with this correction light-curves of variable stars and transients can be recovered to within ~10 arcseconds of the M31 nucleus. Our method is simple to implement and is quick enough to be incorporated within real-time DIA pipelines. (Abridged)Comment: 12 pages. Accepted for publication in MNRAS. Includes an expanded discussion of DIA testing and results, including additional lightcurve example

Microlensing toward crowded fields: Theory and applications to M31

We present a comprehensive treatment of the pixel-lensing theory and apply it to lensing experiments and their results toward M31. Using distribution functions for the distances, velocities, masses, and luminosities of stars, we derive lensing event rates as a function of the event observables. In contrast to the microlensing regime, in the pixel-lensing regime (crowded or unresolved sources) the observables are the maximum excess flux of the source above a background and the full width at half-maximum (FWHM) time of the event. To calculate lensing event distribution functions depending on these observables for the specific case of M31, we use data from the literature to construct a model of M31, reproducing consistently photometry, kinematics and stellar population. We predict the halo- and self-lensing event rates for bulge and disk stars in M31 and treat events with and without finite source signatures separately. We use the M31 photon noise profile and obtain the event rates as a function of position, field of view, and S/N threshold at maximum magnification. We calculate the expected rates for WeCAPP and for a potential Advanced Camera for Surveys (ACS) lensing campaign. The detection of two events with a peak signal-to-noise ratio larger than 10 and a timescale larger than 1 day in the WeCAPP 2000/2001 data is in good agreement with our theoretical calculations. We investigate the luminosity function of lensed stars for noise characteristics of WeCAPP and ACS. For the pixel-lensing regime, we derive the probability distribution for the lens masses in M31 as a function of the FWHM timescale, flux excess and color, including the errors of these observables.Comment: 45 pages, 27 figures LaTeX; corrected typos; published in the Astrophysical Journal Supplemen