I show that when the observables (πE, tE, θE, πs, μs) are well measured up to a discrete degeneracy in the microlensing parallax vector πE, the relative likelihood of the di erent solutions can be written in closed form Pi = KHiBi, where Hi is the number of stars (potential lenses) having the mass and kinematics of the inferred parameters of solution i and Bi is an additional factor that is formally derived from the Jacobian of the transformation from Galactic to microlensing parameters. Here tE is the Einstein timescale, θE is the angular Einstein radius, and (πs;μs) are the (parallax, proper motion) of the microlensed source. The Jacobian term Bi constitutes an explicit evaluation of the \Rich Argument", i.e., that there is an extra geometric factor disfavoring large-parallax solutions in addition to the reduced frequency of lenses given by Hi. I also discuss how this analytic expression degrades in the presence of finite errors in the measured observables.

I investigate the origin of arc degeneracies in satellite microlens parallax π E measurements with only late time data, e.g., t > t 0 + t E as seen from the satellite. I show that these are due to partial overlap of a series of osculating, exactly circular, degeneracies in the π E plane, each from a single measurement. In events with somewhat earlier data, these long arcs break up into two arclets, or (with even earlier data) two points, because these earlier measurements give rise to intersecting rather than osculating circles. The two arclets (or points) then constitute one pair of degeneracies in the well-known four-fold degeneracy of space-based microlens parallax. Using this framework of intersecting circles, I show that next-generation microlens satellite experiments could yield good π E determinations with only about five measurements per event, i.e., about 30 observations per day to monitor 1500 events per year. This could plausibly be done with a small (hence cheap, in the spirit of Gould & Yee 2012) satellite telescope, e.g., 20 cm.

Microlensing can be seen as a version of strong gravitation lensing where the separation angle of the image formed by light de ection by a massive object is too small to be seen by a ground based optical telescope. As a result, what can be observed is the change in light intensity as function of time; the light curve. Conventionally, the intensity of the source is expressed in magnitudes, which uses a logarithmic function of the apparent ux, known as the Pogson formulae. In this work, we compare the magnitudes from the Pogson formulae with magnitudes from the Asinh formulae (Lupton et al. 1999). We found for small uxes, Asinh magnitudes give smaller deviations, about 0.01 magnitudes smalller than Pogson magnitudes. This result is expected to give signicant improvement in detection level of microlensing light curves.

In current microlensing planet searches that are being carried out in a survey/follow-up mode, the most important targets for follow-up observations are lensing events with high magnifications resulting from the very close approach of background source stars to the lens. In this paper, we investigate the dependence of the sensitivity to planets on detailed properties of high-magnification events. From this, it is found that the sensitivity does not monotonically increase as the impact parameter between the lens and the source trajectory decreases. Instead, it is roughly the same for events with impact parameters less than a certain threshold value. It is also found that events involving main-sequence source stars are sensitive to planets in a much wider range of separation and mass ratio, than those events involved with giant source stars. Based on these results, we propose observational strategies for maximal planet detections considering the types of telescopes available for follow-up observations.

Gaudi, Naber & Sackett pointed out that if an event is caused by a lens system containing more than two planets, all planets will affect the central region of the magnification pattern, and thus the existence of the multiple planets can be inferred by detecting additionally deformed anomalies from intensive monitoring of high magnification microlensing events. Unfortunately, this method has important limitations in identifying the existence of multiple planets and determining their parameters (the mass ratio and the instantaneous projected separation) due to the degeneracy of the resulting light curve anomalies from those induced by a single planet and the complexity of multiple planet lensing models. In this paper, we propose a new channel to search for multiple planets via microlensing. The method is based on the fact that the lensing light curve anomalies induced by multiple planets are well approximated by the superposition of those of the single planet systems where the individual planet-primary pairs act as independent lens systems. Then, if the source trajectory passes both of the outer deviation regions induced by the individual planets, one can unambiguously identify the existence of the multiple planets. We illustrate that the probability of successively detecting light curve anomalies induced by two Jovian-mass planets located in the lensing zone through this channel will be substantial. Since the individual anomalies can be well described by much simpler single planet lensing models, the proposed method has an important advantage of allowing one to accurately determine the parameters of the individual planets.

Since the first proposal by Paczynski, great efforts to detect Galactic dark matter by detecting light variations of stars located in the Magellanic Clouds and Galactic bulge caused by gravitational microlensing have been and are being carried out and more than 1,000 events have been successfully detected. In this paper, we review the progress in the theoretical and experimental progresses in microlensing. We begin with basics of microlensing and summarize the results obtained from the last 8 year observations along with the implications of the results. We also discuss the usefulness of microlensing in other fields of astronomy such as the stellar atmosphere, Galactic binary systems, and extra-solar planets. We finally discuss the problems of the current experiments and the new types of observations that can overcome these problems.

As an efficient method to detect blending of general gravitational microlensing events, it is proposed to measure the shift of source star image centroid caused by microlensing. The conventional method to detect blending by this method is measuring the difference between the positions of the source star image point spread function measured on the images taken before and during the event (the PSF centroid shift, δθc,PSF). In this paper, we investigate the difference between the centroid positions measured on the reference and the subtracted images obtained by using the difference image analysis method (DIA centroid shift, δθc.DIA), and evaluate its relative usefulness in detecting blending over the conventional method based on δθc,PSF measurements. From this investigation, we find that the DIA centroid shift of an event is always larger than the PSF centroid shift. We also find that while δθc,PSF becomes smaller as the event amplification decreases, δθc.DIA remains constant regardless of the amplification. In addition, while δθc,DIA linearly increases with the increasing value of the blended light fraction, δθc,PSF peaks at a certain value of the blended light fraction and then eventually decreases as the fraction further increases. Therefore, measurements of δθc,DIA instead of δθc,PSF will be an even more efficient method to detect the blending effect of especially of highly blended events, for which the uncertainties in the determined time scales are high, as well as of low amplification events, for which the current method is highly inefficient.

The lens mass determined from the photometrically obtained Einstein time scale suffers from large uncertainty due to the lens parameter degeneracy. The uncertainty can be substantially reduced if the mass is determined from the lens proper motion obtained from astrometric measurements of the source image centroid shifts, δθc δθc , by using high precision interferometers from space-based platform such as the Space Interferometry Mission (SIM), and ground-based interferometers soon available on several 8-10m class telescopes. However, for the complete resolution of the lens parameter degeneracy it is required to determine the lens parallax by measuring the parallax-induced deviations in the centroid shifts trajectory, Δδθc Δδθc aloe. In this paper, we investigate the detectabilities of δθc δθc and Δδθc Δδθc by determining the distributions of the maximum centroid shifts, f(δθc,max) f(δθc,max) , and the average maximum deviations, (<Δδc,max>) (<Δδc,max>) , for different types of Galactic microlensing events caused by various masses. From this investigation, we find that as long as source stars are bright enough for astrometric observations it is expected that f(δθc) f(δθc) for most events caused by lenses with masses greater than 0.1 M⨀ M⨀ regardless of the event types can be easily detected from observations by using not only the SIM (with a detection threshold but also the δθth\~3μas) δθth\~3μas) but also the ground-based interferometers (withδθth\~3μas) (withδθth\~3μas) . However, from ground-based observations, it will be difficult to detect Δδθc Δδθc for most Galactic bulge self-lensing events, and the detection will be restricted only for small fractions of disk-bulge and halo-LMC events for which the deviations are relatively large. From observations by using the SIM, on the other hand, detecting Δδθc Δδθc will be possible for majority of disk and halo events and for a substantial fraction of bulge self-lensing events. For the complete resolution of the lens parameter degeneracy, therefore, SIM observations will be essential.

Current searches for gravitational microlensing events are being carried out only by a photometric method. In this review paper, we demonstrate that the nature of Galactic lenses can be significantly better constrained with the additional astrometric observations of microlensng events. First, by astromerically observing lensing events, one can resolve the lens parameter degeneracy, and thus the lens mass can be determined with improved precision. Second, by being free from the blending problem, astrometric observations of lensing events will allow one to improve the uncertainties in the determined Einstein time scales. Third, the lens brightness, which could not be measured photometrically, can be measured from the astrometric observations of lensing events, and thus the nature of lens matter can be better constrained. Finally, with the help of astrometric followup observations of a binary-lens event, one can uniquely determine the solution of lens parameters, allowing one to obtain important astronomical information about the source star and the lens itself.