«Rates for Parallax-Shifted Microlensing Events from Ground-Based Observations of the Galactic Bulge Ari Buchalter∗1 and Marc Kamionkowski†2 ∗ ...»
Rates for Parallax-Shifted Microlensing Events from
Ground-Based Observations of the Galactic Bulge
Ari Buchalter∗1 and Marc Kamionkowski†2
Department of Astronomy, Columbia University, New York, NY 10027
Department of Physics, Columbia University, New York, NY 10027
The parallax eﬀect in ground-based microlensing (ML) observations consists
of a distortion to the standard ML light curve arising from the Earth’s orbital
motion. This can be used to partially remove the degeneracy among the system parameters in the event timescale, t0. In most cases, the resolution in current ML surveys is not accurate enough to observe this eﬀect, but parallax could conceivably be detected with frequent followup observations of ML events in progress, providing the photometric errors are small enough. We calculate the expected fraction of ML events where the shape distortions will be observable by such followup observations, adopting Galactic models for the lens and source distributions which are consistent with observed microlensing timescale distributions. We study the dependence of the rates for parallax-shifted events on the frequency of followup observations and on the precision of the photometry.
For example, we ﬁnd that for hourly observations with typical photometric errors of 0.01 mag, 6% of events where the lens is in the bulge, and 31% of events where the lens is in the disk, (or ≈ 10% of events overall) will give rise to a measurable parallax shift at the 95% conﬁdence level. These fractions may be increased by improved photometric accuracy and increased sampling frequency.
While long-duration events are favored, the surveys would be eﬀective in picking out such distortions in events with timescales as low as t0 ≈ 20 days. We study the dependence of these fractions on the assumed disk mass function, and ﬁnd firstname.lastname@example.org email@example.com –2– that a higher parallax incidence is favored by mass functions with higher mean masses. Parallax measurements yield the reduced transverse speed, v, which ˜ gives both the relative transverse speed and lens mass as a function of distance.
We give examples of the accuracies with which v may be measured in typical ˜ parallax events. Fitting ML light curves which may be shape-distorted (e.g., by parallax, blending, etc.) with only the 3 standard ML parameters can result in inferred values for these quantities which are signiﬁcantly in error. Using our model, we study the eﬀects of such systematic errors and ﬁnd that, due primarily to blending, the inferred timescales from such ﬁts, for events with disk lenses, tend to shift the event duration distribution by ≈ 10% towards shorter t0. Events where the lens resides in the bulge are essentially unaﬀected. In both cases, the impact-parameter distribution is depressed slightly at both the low and high ends.
Subject headings: Galaxy: general – Galaxy: structure — gravitational lensing
The recent observations by the MACHO (Alcock et al. 1995a, Alcock et al. 1996), EROS (Aubourg et al. 1993), OGLE (Udalski et al. 1994a), and DUO (Alard et al. 1995) collaborations of gravitational microlensing (ML) of stars in the LMC and Galactic bulge have generated tremendous excitement in astrophysics. These surveys provide a new probe of Galactic structure and low-mass stellar populations. ML observations of the LMC allow measurements of the dark-matter content of the Galactic halo (Bennett et al. 1996), placing important constraints on dark-matter theories. Also intriguing have been observations of ML events towards the Galactic center, which probe the inner disk and bulge of our Galaxy.
The roughly 100 events observed to date imply an optical depth towards the bulge of (3.3 ± 1.2) ×10−6, three times that predicted by theoretical estimates (Griest et al. 1991), and reveal the need for a better understanding of Galactic structure.
One explanation for the observed excess of bulge events would be the presence of a hitherto undiscovered population of compact, sub-stellar objects, implying the existence of more mass in the disk than previously believed (Alcock et al. 1994; Gould, Miralda-Escude, & Bahcall 1994). This would require an upturn in the stellar mass function (MF) below the hydrogen-burning limit. Another possibility is that the lenses comprise an ordinary stellar population, and the enhancement of ML events is due to non-axisymmetric structure, such as a bar, in the Galactic bulge. The optical depth calculated from a self-consistent bar –3–
violating the assumption of achromaticity, and distorting the shape, [c.f. Eq. (1)] of the light curve (Kamionkowski 1995; Buchalter, Kamionkowski, & Rich 1995, hereafter BKR).
Ground-based observations of this color-shift eﬀect in two or more wavebands can remove entirely the degeneracy in t0. If the lenses are not completely dark, then color shifting should, in principle, aﬀect every event and the observed incidence of this eﬀect is simply a function of resolution of the data (namely sampling frequency and level of photometric error). The MACHO collaboration has already observed another type of distortion, where the time symmetry of the light curve is broken by the Earth’s orbital motion (Alcock et al. 1995b). This parallax eﬀect allows one to compare the projected Einstein ring with the size of the Earth’s orbit and thereby obtain an additional constraint relating Ml, v, and Dol, eﬀectively removing one degree of degeneracy. Strictly speaking, this eﬀect is also present in every event, but is not expected to be frequently observed by any of the existing ML surveys; the light curves are sampled too infrequently and the photometric errors are too large. The single conﬁrmed parallax event detected to date was fortuitous in that it was both long enough (t0 ≈ 110 days) for the light curve to be heavily sampled, and well-situated during the MACHO observing season so that the asymmetry could be ﬁt along the entire light curve. In addition, the eﬀect is more dramatic for longer events, during which the Earth can move through an appreciable fraction of its orbit. It is only for these rarer long events that a parallax shift may be observed by the low-resolution ML surveys. However, with the early-warning alert systems developed by both the MACHO and OGLE groups (Stubbs et al. 1994; Udalski et al. 1994b), it is conceivable that a program of followup observations with frequent and precise measurements could measure the light curves with suﬃcient accuracy to detect the parallax eﬀect in a larger fraction of events. The PLANET (Albrow et al. 1995) and GMAN (Pratt et al. 1996) collaborations are currently performing such observations, with the primary purpose of detecting planets around lenses, and Tytler et al. (1996) are proposing another such search. Planetary masses would give rise to smaller event timescales and thus produce narrow spikes on the lensing light curve which likely go undetected in current surveys. These spikes could be resolved with observations by dedicated telescopes performing rapid sampling of events in progress with high photometric precision. Such ground-based surveys are well-suited to pick out the parallax eﬀect for shorter-term events, as well as distortions due to unlensed light from the lens.
In this paper, we determine the fraction of ML events toward the Galactic bulge which should show a measurable parallax shift in such followup programs. We calculate the fractions which will arise if the lenses are all in the bulge and if the lenses are all in the disk, using realistic models for the lens distributions which are consistent with the currently observed ML timescale distribution. A Monte Carlo technique is used to simulate –5– events and generate parallax-distorted light curves for each. It is then determined for each event whether the distorted light curve can be distinguished at the 95% conﬁdence level from a standard ML light curve. The calculation is performed for several values of the sampling frequency and for several values of photometric accuracies that may be attainable.
Our results indicate that, depending on the survey sensitivity, the expected fraction of parallax-shifted events (FP SE ) ranges from 0.06 to 0.31 for bulge self-lensing events, 0.31 to
0.77 for disk-lensing-bulge events, and 0.49 to 0.83 for (rare) disk self-lensing events, with current-generation followup surveys favoring the lower values. Since the parallax eﬀect is particularly important for the case of disk lenses, we also examine the dependence of this eﬀect on the adopted mass function (MF) for the disk, and ﬁnd higher incidences from MFs with higher mean masses. We further calculate the dependence of FP SE on event timescale, ﬁnding that the intensive followup programs should measure a substantial contribution to FP SE from events with t0 as low as 20 days, and we also examine what may be learned from a typical PSE. We emphasize that, like color-shift analysis, parallax analysis can be directly applied to all events, so that information about the mass, velocity and spatial distributions of the lenses becomes available on an event by event—rather than on a statistical—basis.
Although parallax and blending eﬀects may be present to some degree in a large fraction of events, they are expected to be small in most cases, so that a standard three-parameter ﬁt to a slightly distorted light curve may be deemed adequate. However, ﬁtting the standard ML ampliﬁcation function to a light curve distorted by these eﬀects can result in systematic errors in the inferred ﬁt parameters, most notably t0 and umin. These inaccuracies may lead to a systematic miscalculation of the duration and ampliﬁcation distributions, and of the overall optical depth. Thus, we perform another simulation to generate shape-distorted light curves including both parallax and color-shift eﬀects (i.e., blending due to the lens).
A standard three-parameter ﬁt to these light curves is then applied, and the inferred parameters are used to compare the resulting ampliﬁcation and event duration distributions with their actual values. We ﬁnd for the latter that the errors incurred by three-parameter ﬁts are negligible for bulge lenses, but may be quite large (≥ 10%) if the lenses are in the disk, particularly if the disk MF declines at the low-mass end. We also ﬁnd that errors in the inferred minimum impact parameter lead to a non-uniform distribution in peak ampliﬁcation.
The plan of the paper is as follows: In Section 2, we review the characteristics of parallax-distorted events. In Section 3, we summarize the models and techniques used in the calculation and present the results of our calculations of expected fraction of PSEs. In Section 4, we examine the impact of parameterizing shape-distorted light curves with the standard ML ﬁt on the inferred ML parameters, and in Section 5 we make some concluding remarks.
2. PARALLAX-SHIFTED MICROLENSING EVENTS
To parameterize the parallax eﬀect, consider ﬁrst a standard ML event as viewed at the position of the Sun. In that case, we regain the constant-velocity condition, so that the scaled distance from the lens to the line of sight, u(t), is properly described by Eq. (1), where umin now represents the minimum distance from the lens to the Sun-source line of sight. We can modify the expression for u(t) to incorporate the Earth’s orbital motion by projecting to the position of the lens the motion of the observer (Earth) along the ecliptic.
Figure 1 depicts a simpliﬁed version of the geometry where we have assumed, for the sake of illustration, that the ecliptic is perpendicular to the Sun-source line. In this case, we see from Figure 1(a) that the projection of the Earth’s orbit traces out a circle with a radius (scaled in units of Re ) given by
where tc is the time when the Earth is closest to the Sun-source line, Ω0 = 2π yr−1, tp is the time of perihelion and = 0.017 is the eccentricity of Earth’s orbit. We include the dependence on in our expression for Ω since it may contribute appreciably near t = tc, though we omit it in the expression for α, where it is always negligible. The position of the lens is given by Figure 1(b) as
Fig. 1.— (a) Simpliﬁed geometry of the parallax cone, assuming the ecliptic is perpendicular to the Sun-source line. The Earth’s orbit, projected to the position of the lens, traces out a circle of radius a. (b) Cross section of the parallax cone at the distance of the lens. The lens velocity is denoted by v. Note that umin is no longer the minimum distance from the lens to the observer-source line of sight, but rather to the Sun-source line of sight.
so that a parallax-shifted light curve is parameterized by the 5 quantities tmax, t0, umin, α, and φ. In our simulations, we include an additional parameter to account for the presence of unlensed light (either due to the lens or a chance blended star) in the light curve.
Figure 2 illustrates an example of a ML light curve distorted by the parallax eﬀect.
The solid curve is a simulated parallax-shifted event (PSE), arising from a bulge self-lensing conﬁguration with t0 = 100 days and α = 0.13. The dashed curve is a standard ML curve with the same peak height (Amax), peak time (tmax), and FWHM, and has a timescale of t0 = 97 days. It is important to note that since the two curves have diﬀerent shapes, as indicated by the residuals in the lower panel, a ﬁt to a PSE which involves only the parameters tmax, t0, and umin can generate inferred values of these quantities which are signiﬁcantly in error. This is further evidenced by the MACHO collaboration analysis of their PSE, in which they ﬁnd the best-ﬁt standard curve to have t0 = 141 days and umin = 0.101, and the best-ﬁt parallax curve to have t0 = 111 days and umin = 0.159 (Alcock et al. 1995b). Failing to account for distortions in observed ML light curves can lead to errors in the inferred timescales and thus in the inferred optical depth as well.
3. EXPECTED FRACTIONS OF PARALLAX-SHIFTED EVENTS