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

The Sagittarius dwarf galaxy as a microlensing target

We estimate the optical depth, time-scale distribution and fraction of microlensing events originating from sources in the Sagittarius dwarf galaxy (Sgr) lensed by deflectors in the Milky Way. These events have a time-scale longer by a factor ~1.3 than the MW/MW events and occur mainly on sources fainter than V~21 mag below Sgr's turn off. The fraction of events involving a source in Sgr depends on the location and extinction of the field and on the limiting magnitude of the survey. The contribution of the MW/Sgr events is negligible (<1%) at very low latitudes (|b|<2 deg.) but increases continuously towards higher |b| and becomes dominant near the highest density region of the dwarf galaxy. Sgr is present within the fields of current microlensing surveys and any optical depth map inferred from observations will become biased by the presence of Sgr towards higher |b| where the contribution of MW/Sgr events is significant. Systematic spectroscopic measurements on the sources of all the microlensing events may allow detection of this kind of event for which the degeneracy on the lens mass can be significantly reduced.Comment: 9 pages, 8 figures. Accepted for publication in A&A Main Journa

Variable Stars in the Globular Cluster M5. Application of the Image Subtraction Method

We present $V$-band light curves of 61 variables from the core of the globular cluster M5 obtained using a newly developed image subtraction method (ISM). Four of these variables were previously unknown. Only 26 variables were found in the same field using photometry obtained with DoPHOT software. Fourier parameters of the ISM light curves have relative errors up to 20 times smaller than parameters measured from DoPHOT photometry. We conclude that the new method is very promising for searching for variable stars in the cores of the globular clusters and gives very accurate relative photometry with quality comparable to photometry obtained by HST. We also show that the variable V104 is not an eclipsing star as has been suggested, but is an RRc star showing non-radial pulsations.Comment: submitted to MNRAS, 9 pages, 4 figure

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]

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

MACHO Mass Determination Based on Space Telescope Observation

We investigate the possibility of lens mass determination for a caustic crossing microlensing event based on a space telescope observation. We demonstrate that the parallax due to the orbital motion of a space telescope causes a periodic fluctuation of the light curve, from which the lens distance can be derived. Since the proper motion of the lens relative to the source is also measurable for a caustic crossing event, one can find a full solution for microlensing properties of the event, including the lens mass. To determine the lens mass with sufficient accuracy, the light curve near the caustic crossing should be observed within uncertainty of $\sim$ 1%. We argue that the Hubble Space Telescope observation of the caustic crossing supplied with ground-based observations of the full light curve will enable us to determine the mass of MACHOs, which is crucial for understanding the nature of MACHOs.Comment: 9 pages + 3 figures, accepted for publication in ApJ Letter

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.Comment: 18 pages, 4 figures, accepted for publication in Ap

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

Einstein Radii from Binary Lensing Events

We show that the Einstein ring radius and transverse speed of a lens projected on the source plane, $\hat{r}_{\rm e}$ and $\hat{v}$, 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 time scale, $P\gg t_{\rm e}$, there exists a single two-fold degeneracy in determining $\hat{r}_{\rm e}$. However, when $P \lesssim t_{\rm e}$ the degeneracy can often be broken by making use of the binary-source system's orbital motion. %Once $\hat{r}_{\rm e}$, and thus $\hat{v}$ are determined, one can %distinguish self-lensing events in the Large Magellanic Cloud %from Galactic halo events. For an identifiable 8\% of all lensing events seen toward the Large Magellanic Cloud (LMC), one can unambiguously determine whether the lenses are Galactic, or whether they lie in the LMC itself. The required observations can be made after the event is over and could be carried out for the $\sim 8$ events seen by Alcock et al.\ and Aubourg et al.. In addition, we propose to include eclipsing binaries as sources for gravitational lensing experiments.Comment: 18 pages, revised version, submitted to Ap