Finite Source Effects

Microlensing is arguably the exoplanet search technique that is most sensitive to low-mass planets, and the lower limit in sensitivity of the microlensing method is set by the finite angular size of the source stars. Roughly speaking, when the angular radius of the planetary Einstein ring, sjep0E, is much smaller than the source star angular radius, 0*, we expect that planetary signal to be washed out. But this is only a crude, order-of-magnitude estimate, and a full finite source solution to the lens equation, 3.9, is required to determine the precise limits on the microlensing planet detection method set by the finite angular size of the sources.

Full finite source planetary microlensing light curves were first calculated by Bennett & Rhie (1996) using the methods described in Sect. 3.2.2. Results of these calculations are reproduced in Figs. 3.4 and 3.5. Figure 3.4 shows a series of planetary light curves with planetary mass fractions of e = 10~4 and 10~5. For a typical lens star mass of ~ 0.3M©, these correspond to 1M® and 10M®, respectively. The finite source light curves are characterized by the source star radius in Einstein ring units: p = 0„/0E. The p values shown in Fig. 3.4 are 0.003, 0.006, 0.013, and 0.03, and these span the expected range of p for a low mass planetary host star in the Galactic bulge with a source star ranging in radius from 1R© to 10R©, which is a typical radius for a "red clump" K-giant in the bulge. A number of the planet detections to date actually have p values in the 0.4-1 x10~3 range because the lens stars reside in the bulge and have a larger than average mass. (Both of these imply a larger 0E.)

Several general trends are apparent from Fig. 3.4. First, the planetary deviations are easily detectable for p = 0.003, but the signals are much weaker for p = 0.03. This implies that 1M® planets are easily detected with main sequence source stars, but planets of 10M© are close to the lower limit of detectability for giant source stars. Another notable feature of these light curves is that the planetary signals with d = 0.8 are more easily washed out by the finite source effects than those with

t/tE t/tE

Fig. 3.4. Microlensing lightcurves which show planetary deviations are plotted for a mass ratio of e = 10-4 & 10-5 and separations of d = 1.3 & 0.8. The main plots are for a stellar radius of p = 0.003 while the insets show light curves for radii of 0.006, 0.013, and 0.03 as well. The dashed curves are the unperturbed single lens lightcurves. For each of these lightcurves, the source trajectory is at an angle of sin-1 0.6 with respect to the star-planet axis. The impact parameter u0 = 0.27 for the d = 0.8 plots and u0 = 0.32 for the d = 1.3 plots.

Fig. 3.4. Microlensing lightcurves which show planetary deviations are plotted for a mass ratio of e = 10-4 & 10-5 and separations of d = 1.3 & 0.8. The main plots are for a stellar radius of p = 0.003 while the insets show light curves for radii of 0.006, 0.013, and 0.03 as well. The dashed curves are the unperturbed single lens lightcurves. For each of these lightcurves, the source trajectory is at an angle of sin-1 0.6 with respect to the star-planet axis. The impact parameter u0 = 0.27 for the d = 0.8 plots and u0 = 0.32 for the d = 1.3 plots.

d = 1.3. This is a consequence of the large magnification deficit seen between the two planetary caustics for a minor image perturbation, as shown in the left hand panel of Fig. 3.3. When a large finite source effectively averages over the vicinity of a minor image planetary caustic, the positive and negative deviations effectively cancel each other out (Bennett & Rhie, 1996; Gould & Gaucherel, 1997). In contrast, the major image planetary caustic magnification deviation pattern is mostly positive, so the finite source effect merely smoothes it out.

Figure 3.5 shows how the planet detection probability varies as a function of d for the same e and p values used for Fig. 3.4. The greater tolerance of deviations with d > 1 to finite source effects is apparent for e = 10~4, p = 0.03 and e = 10~5, p = 0.013. The behavior of Fig. 3.5 near d = 1 is even more interesting. For e = 10~4 and p = 0.003 or 0.006, the detection probability reaches a maximum at d « 1, but for p = 0.013, the probability has a local minimum at d =1, and for p = 0.03 or any of the p values with e = 10~5, the detection probability = 0 at d = 1. This

Detection Threshold = 4% deviation

d/RE

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.

is due to the fact that for d « 1 the planetary and stellar caustics merge for form a relatively large single caustic that is extended along the lens axis. This caustic is large, but relatively weak, and it has associated positive and negative deviation regions that tend to cancel when averaged over by a moderately large finite source.

However, some features of Fig. 3.5 are dependent on the somewhat arbitrary choice of the event detection threshold, and the sensitivity of a real observing strategy can differ from this. In fact, the planetary deviation detected in event OGLE-2005-BLG-169 (Gould et al., 2006) would not have passed the selection criteria for Fig. 3.5, but the planet is nevertheless detected with a strong signal. The reason for this is that it was identified as a very high magnification event with a very high sensitivity to planets, and for this reason it was observed much more frequently than most events with potential planetary signals. The additional observations provided enough additional signal to allow the definitive detection of a relatively low-amplitude signal.

As a practical matter, finite source effects imply a lower planetary mass limit of Mp > 5M© for giant source stars in the bulge, and a limit of Mp > 0.05M© for bulge main sequence stars. Thus, searches for terrestrial exoplanets must focus on main sequence source stars.

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