Planetary Parameters from Microlensing Events

The determination of the properties of the lens systems that are detected in mi-crolensing events is often a serious challenge. The simple form of the microlensing

Detection Threshold = 4% deviation


Fig. 3.5. The planetary deviation detection probability is plotted for different values of the planetary mass ratio, e, and the stellar radii, p (in units of RE. A planet is considered to be "detected" if the lightcurve deviates from the standard point lens lightcurve by more than 4% for a duration of more than tE/200.

light curves shown in Fig. 3.2 is an advantage when trying to identify microlensing events, but as I mentioned in Sect. 3.2.1, in a single lens event it can also be a drawback when trying to interpret observed microlensing events. For most single lens events, it is only the tE parameter that constrains the physically interesting parameters of the event: the lens mass, M, the lens distance, DL, and the relative velocity, v±. The single lens parameters u0 and t0 don't constrain lens system parameters that are of much interest.

In addition to the parameters needed to describe a single lens event, a planetary microlensing event must have three additional binary lens parameters: the planetary mass ratio, q = e/(1 — e), the star-planet separation, d, (which is in units of RE), and the angle between the star-planet axis and the trajectory of the source with respect to the lens system, 0. So, two of these new parameters, q and d, directly constrain planetary parameters of interest, although d is normalized to RE, which may not be known. Most planetary light curves, at least those for low-mass planets, also have caustic crossings or a close approach to a cusp that reveal light curve features due to the finite size of the source star. This enables the source radius crossing time, t*, to be measured.

The determination of the star-planet separation and the planetary mass fraction is usually quite straightforward from the microlensing light curve. For events at moderate magnification, due to the planetary caustic, the separation can be determined by the magnification predicted by the single lens model that describes the event outside the region of the planetary deviation following eq. 3.15. This still leaves an ambiguity between the d < 1 and d > 1 solutions, but this is easily resolved by the drastically different magnification patterns in the vicinity of major image and minor image caustics, as shown in Fig. 3.3. The planetary mass fraction, q, can generally be determined by the duration of the planetary perturbation. In some cases, if the time scale of the deviation is similar to or smaller than t*, both q and t* determine the deviation time scale, but good light curve coverage with moderately precise photometry allow both q and t* to be determined (Gaudi & Gould, 1997).

The situation is somewhat different for high magnification, stellar caustic deviation events. Dominik (1999) pointed out an approximate degeneracy in the properties of the stellar caustic under the transformation d ^ 1/d, which means that there may be a d ^ 1/d ambiguity in the modeling of stellar caustic planetary events. This is apparent from the magnification patterns shown in Fig. 3.3. For a source trajectory nearly parallel to the lens axis or for d ~ 1, this degeneracy breaks down, so the ambiguity disappears. With precise photometry it is usually possible to distinguish between the d < 1 and d > 1 solutions, and this has been the case for all events observed to date.

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