A

where u = b/RE is the dimensionless lens-source separation. The total magnification of both images is given by u2

For a lens of M = 1M©, that is half-way to a source in the Galactic center (at DS = 8kpc), we find where x = DL/DS, so RE is similar to the orbital radius of planets in our own Solar System. This also implies that 0E ~ 1 mas. Since the image separation is of order ~ 0E, this implies that images will not generally be resolved with virtually all planned and future astronomical instruments (with a few exceptions (Delplancke et al., 2001)). On the other hand, if we assume a typical Galactic velocity of v± = 100 km/sec for the relative velocity between the lens star and the line-of-sight to the source, then the typical Einstein radius crossing time for a lens in the Galactic disk and a bulge source is tE = Re/v± ~ 2 months. Thus, the main observational effect for lensing by stars within the Milky Way is the time varying magnification instead of the image separation, and this is why it is referred to as microlensing instead of lensing.

The microlensing light curve is generally described by eq. 3.6 with the lens-source separation given by assuming that the relative motion between the lens and the observer-source line-of-sight. Thus, a single-lens microlensing light curve is described by three parameters, the time of peak magnification, t0, the Einstein radius crossing time or width, tE, and the minimum separation, u0, which determines the peak magnification. u0 is the only parameter that affects the intrinsic light curve shape, as shown in Fig. 3.2, but only tE constraints the physically interesting parameters of the event: the lens mass, M, the lens distance, DL, and the relative velocity, v±.

The first microlensing events were discovered in 1993 towards the Large Magellanic Cloud (LMC) by the MACHO Project (Alcock et al., 1993) and towards the Galactic bulge by the OGLE Collaboration. (Udalski et al., 1993). The early emphasis of microlensing surveys was the search for dark matter in the Milky Way's halo (Paczynski, 1986), but this issue has been largely resolved with the demonstration that the excess microlensing seen toward the LMC by the MACHO group (Alcock et al., 2000b; Bennett, 2005) requires at most 20% of the Milky Way's dark matter in the form of stellar mass objects, while the results of the EROS group (Tisserand et al., 2007) suggest that much of this microlensing excess may be caused by stars associated with the LMC itself (Sahu, 1994), perhaps in the LMC halo (Wu, 1994).

With the dark matter microlensing question mostly resolved, the prime focus of microlensing observations has shifted to the detection of extrasolar planets. Microlensing was first suggested as a method to find planets by Liebes (1964), but as Mao & Paczynski (1991) pointed out, this requires a consideration of multiple lens systems.

Fig. 3.2. Example microlensing light curves for a point source and a single lens that moves with a constant lens velocity with respect to the observer-source line-of-sight. Light curves with uo = 0.05, 0.1, 0.2, 0.4, and 0.8 are shown.

Fig. 3.2. Example microlensing light curves for a point source and a single lens that moves with a constant lens velocity with respect to the observer-source line-of-sight. Light curves with uo = 0.05, 0.1, 0.2, 0.4, and 0.8 are shown.

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