Possible Solution of the long-standing discrepancy in the Microlensing Optical Depth Toward the Galactic Bulge by correcting the stellar number count

We find that significant incompleteness in stellar number counts results in a significant overestimate of the microlensing optical depth $\tau$ and event rate per star per year $\Gamma$ toward the Galactic bulge from the first two years of the MOA-II survey. We find that the completeness in Red Clump Giant (RCG) counts $f_{\rm RC}$ decreases proportional to the galactic latitude $b$, as $f_{\rm RC}=(0.63\pm0.11)-(0.052\pm0.028)\times b$, ranging between 1 and 0.7 at $b=-6^\circ\sim-1.5^\circ$. The previous measurements using all sources by Difference Image Analysis (DIA) by MACHO and MOA-I suffer the same bias. On the other hand, the measurements using a RCG sample by OGLE-II, MACHO and EROS were free from this bias because they selected only the events associated with the resolved stars. Thus, the incompleteness both in the number of events and stellar number count cancel out. We estimate $\tau$ and $\Gamma$ by correcting this incompleteness. In the central fields with $|l|<5^\circ$, we find $\Gamma=[18.74\pm0.91]\times10^{-6}\exp[(0.53\pm0.05)(3-|b|)]$ star$^{-1}$ yr$^{-1}$ and $\tau_{200}=[1.84\pm0.14]\times10^{-6}\exp[(0.44\pm0.07)(3-|b|)]$ for the 427 events with $t_{\rm E}\leq200\,$days using all sources brighter than $I_s\leq20$ mag. Our revised all-source $\tau$ measurements are about 2-$\sigma$ smaller than the other all-source measurements and are consistent with the RCG measurements within 1-$\sigma$. We conclude that the long-standing problem on discrepancy between the high $\tau$ with all-source samples by DIA and low $\tau$ with RCG samples can probably be explained by the incompleteness of the stellar number count. A model fit to these measurements predicts $\Gamma=4.60\pm0.25\times10^{-5}$ star$^{-1}$ yr$^{-1}$ at $|b|\sim-1^\circ.4$ and $-2^\circ.25<l<3^\circ.75$ for sources with $I<20$, where the future space mission WFIRST will observe.Comment: 39 pages, 15 figures, 5 tables, accepted for publication in ApJ. arXiv admin note: substantial text overlap with arXiv:1305.018

Microlensing Characterization of Wide-Separation Planets

With their excellent photometric precision and dramatic increase in monitoring frequency, future microlensing survey experiments are expected to be sensitive to very short time-scale, isolated events caused by free-floating and wide-separation planets with mass as low as a few lunar masses. We estimate the probability of measuring the Einstein radius \theta_E for bound and free-floating planets. We carry out detailed simulations of the planetary events expected in next-generation surveys and estimate the resulting uncertainty in \theta_E for these events. We show that, for main-sequence sources and Jupiter-mass planets, the caustic structure of wide-separation planets with projected separations of < 20 AU substantially increases the probability of measuring the dimensionless source size and thus determining \theta_E compared to the case of unbound planets. In this limit where the source is much smaller than the caustic, the effective cross-section to measure \theta_E to 10% is ~25% larger than the full width of the caustic. Measurement of the lens parallax is possible for low-mass planetary events by combined observations from the ground and a satellite located in an L2 orbit; this would complete the mass measurements for such wide-separation planets. Finally, short-duration events caused by bound planets can be routinely distinguished from those caused by free-floating planets for planet-star separations < 20 AU from either the deviations due to the planetary caustic or (more often) the low-amplitude bump from the magnification due to the parent star.Comment: 10 pages including 7 figures. ApJ, in pres

Short-duration lensing events: II. Expectations and Protocols

Ongoing microlensing observations by OGLE and MOA regularly identify and conduct high-cadence sampling of lensing events with Einstein diameter crossing time, tau_E, of 16 or fewer days. Events with estimated values of tau_E of one to two days have been detected. Short duration events tend to be generated by low-mass lenses or by lenses with high transverse velocities. We compute the expected rates, demonstrate the expected ranges of parameters for lenses of different mass, and develop a protocol for observing and modeling short-duration events. Relatively minor additions to the procedures presently used will increase the rate of planet discovery, and also discover or place limits on the population of high-speed dim stars and stellar remnants in the vicinity of the Sun.Comment: 17 pages; 3 figures; submitted to ApJ 3 July 200

Interpreting the M22 Spike Events

Recently Sahu et al., using the Hubble Space Telescope to monitor stars in the direction of the old globular cluster M22, detected six events in which otherwise constant stars brightened by ~50% during a time of <1 day. They tentatively interpret these unresolved events as due to microlensing of background bulge stars by free-floating planets in M22. I show that if these spike events are due to microlensing, the lensing objects are unlikely to be associated with M22, and unlikely to be part of a smoothly distributed Galactic population. Thus either there happens to be a massive, dark cluster of planets along our line-of-sight to M22, or the spike events are not due to microlensing. The lensing planets cannot be bound to stars in the core of M22: if they were closer than 8 AU, the lensing influence of the parent star would have been detectable. Moreover, in the core of M22, all planets with separations > 1 AU would have been ionized by random stellar encounters. Most unbound planets would have escaped the core via evaporation which preferentially affects such low-mass objects. Bound or free-floating planets can exist in the outer halo of M22; however, for reasonable assumptions, the maximum optical depth to such a population falls short of the observed optical depth, tau ~ 3x10^{-6}, by a factor of 5-10. Therefore, if real, these events represent the detection of a significant free-floating Galactic planet population. The optical depth to these planets is comparable to and mutually exclusive from the optical depth to resolved events measured by microlensing survey collaborations toward the bulge, and thus implies a similar additional mass of lensing objects. Such a population is difficult to reconcile with both theory and observations.Comment: Minor changes. 12 pages, 4 figures, 2 tables. Accepted to ApJ. To appear in Feb 10, 2002 issue (v566

Short-duration lensing events: I. wide-orbit planets? free-floating low-mass objects? or high-velocity stars?

Short duration lensing events tend to be generated by low-mass lenses or by lenses with high transverse velocities. Furthermore, for any given lens mass and speed, events of short duration are preferentially caused by nearby lenses (mesolenses) that can be studied in detail, or else by lenses so close to the source star that finite-source-size effects may be detected, yielding information about both the Einstein ring radius and the surface of the lensed star. Planets causing short-duration events may be in orbits with any orientation, and may have semimajor axes smaller than an AU, or they may reach the outer limits of their planetary systems, in the region corresponding to the Solar System's Oort Cloud. They can have masses larger than Jupiter's or smaller than Pluto's. Lensing therefore has a unique potential to expand our understanding of planetary systems. A particular advantage of lensing is that it can provide precision measurements of system parameters, including the masses of and projected separation between star and planet. We demonstrate how the parameters can be extracted and show that a great deal can be learned. For example, it is remarkable that the gravitational mass of nearby free-floating planet-mass lenses can be measured by complementing observations of a photometric event with deep images that detect the planet itself. A fraction of short events may be caused by high-velocity stars located within a kpc. Many high-velocity lenses are likely to be neutron stars that received large natal kicks. Other high-speed stars may be members of the halo population. Still others may be hypervelocity stars that have been ejected from the Galactic Center, or runaway stars escaped from close binaries, possibly including the progenitor binaries of Type Ia supernovae.Comment: 17 pages; 2 figures; submitted to ApJ 3 July 200

A Simple Model for r-Process Scatter and Halo Evolution

Recent observations of heavy elements produced by rapid neutron capture (r-process) in the halo have shown a striking and unexpected behavior: within a single star, the relative abundances of r-process elements heavier than Eu are the same as the same as those of solar system matter, while across stars with similar metallicity Fe/H, the r/Fe ratio varies over two orders of magnitude. In this paper we present a simple analytic model which describes a star's abundances in terms of its ``ancestry,'' i.e., the number of nucleosynthesis events (e.g., supernova explosions) which contributed to the star's composition. This model leads to a very simple analytic expression for the abundance scatter versus Fe/H, which is in good agreement with the data and with more sophisticated numerical models. We investigate two classes of scenarios for r-process nucleosynthesis, one in which r-process synthesis events occur in only \sim 4% of supernovae but iron synthesis is ubiquitous, and one in which iron nucleosynthesis occurs in only about 9% of supernovae. (the Wasserburg- Qian model). We find that the predictions in these scenarios are similar for [Fe/H] \ga -2.5, but that these models can be readily distinguished observationally by measuring the dispersion in r/Fe at [Fe/H] \la -3.Comment: AASTeX, 21 pages, includes 4 figure