Info

M/M0

Fig. 3.6. The predicted fractional brightness, ftens = Fl/(Fs + Fl), of the OGLE-2003-BLG-169 lens is plotted in the top panel as a function of mass in the BVIJH passbands. The predicted offsets of the centroids of the blended source+lens images in different passbands are shown in the bottom panel, assuming that the images are taken 2.4 years after peak magnification.

M/Ms

Centroid Offsets

Centroid Offsets

M/M0

Fig. 3.6. The predicted fractional brightness, ftens = Fl/(Fs + Fl), of the OGLE-2003-BLG-169 lens is plotted in the top panel as a function of mass in the BVIJH passbands. The predicted offsets of the centroids of the blended source+lens images in different passbands are shown in the bottom panel, assuming that the images are taken 2.4 years after peak magnification.

determined with an image that has sufficient angular resolution to resolve the source and lens stars from the unrelated stars in the field. This generally requires space-based imaging with the Hubble Space Telescope (HST), or possibly ground-based adaptive optics imaging because of the extreme crowding in the Galactic bulge fields where microlensing events are most easily found. (The lens-source relative proper motion has typical value ~ 5mas/yr, so the lens and source are not typically resolved from each other until a decade or more after the event.) If the combined lens-plus-source image is significantly brighter than the brightness of the source from the microlensing fit, then the difference determines the brightness of the lens. This then allows the mass of the planetary host (lens) star to be determined using a main sequence star mass-luminosity relation (Bennett et al., 2007a).

The top panel of Fig. 3.6 shows the predicted brightness of the lens for the OGLE-2005-BLG-169 event in the BVIJH passbands. This indicates that the lens star will easily be detected if it is a main sequence star, since even a 0.08M© lens star will contribute > 40% of the H-band flux and > 10% of the I-band flux. This case is more favorable than most because of a relatively large 0E value, but in most cases, the lens star will be detectable in the H-band unless it is a late M-dwarf located in the bulge. However, for Galactic disk lenses at a certain range of distances (corresponding to 0.2M© < M < 0.4 for OGLE-2005-BLG-169 in the IJH-bands)

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Fig. 3.7. The top-left panel shows the fraction of the source+lens flux for event OGLE-2003-BLG-235/MOA-2003-BLG-53 that is predicted to come from the lens in the HST-I, V, and B passbands as a function of lens mass. The bottom-left panel shows the predicted color-dependent centroid shifts as a function of mass for 1.78 years of relative proper motion at = 3.3mas/yr. The measured values of flens in the I-band and the color dependent centroid shifts and error bars are indicated with their error bars. These are plotted at an arbitrary value for the stellar mass (M*). The centroids of the source+lens star blended images in the individual HST/ACS/HRC images are shown in the right panel as red circles (I), green squares (V), and blue triangles (B). The crossed error bars are the average centroid in each passband.

the mass-distance relation, eq. 3.17, combines with the mass-luminosity relation to yield a nearly flat mass-brightness relation for the planetary host star. In these cases, it is useful to have images in shorter wavelength bands, such as V and B because this cancelation does generally not occur in the optical and infrared passbands for the same range of lens star masses.

High resolution images in multiple colors also allow an independent method for estimating the lens star brightness, as shown in the bottom panel of Fig. 3.6 and the bottom-left panel of Fig. 3.7. Because the lens and source stars usually have different colors, the centroid of the blended source+lens image will usually be color dependent. So, an additional constraint on the lens star is obtained by measuring the centroid offset between the centroids of the blended source+lens in different passbands. As indicated in Fig. 3.7, this effect was marginally detected for the first planet detected by microlensing (Bennett et al., 2006) with HST images taken only 1.8 years after peak magnification. Also, because this color dependent centroid shift depends on the relative lens -source proper motion, ^rei, it can be used to help determine 0E for planetary events with no finite source effects, and hence, no measurement of t*.

Simulated HST images:

Simulated HST images:

Fig. 3.8. Simulated image stacks of multiple dithered exposures of the OGLE-2005-BLG-169 source and lens star 2.4 years after peak magnification using the HST/ACS High Resolution Camera (HRC) in the F814W filter band. The top row of images assumed a host star mass of Mt = 0.08M©, the middle row assumes Mt = 0.35M©, and the bottom row assumes Mt = 0.63M©. In each row, the image on the left shows the raw image stack sampled at one half the native HRC (28 mas) pixel size. The central column shows the residuals after subtraction of the best fit PSF model, showing the blended image elongation along the c-axis due to the lens-source separation. The right hand column shows these residuals rebinned to the 28 mas native pixel scale.

Fig. 3.8. Simulated image stacks of multiple dithered exposures of the OGLE-2005-BLG-169 source and lens star 2.4 years after peak magnification using the HST/ACS High Resolution Camera (HRC) in the F814W filter band. The top row of images assumed a host star mass of Mt = 0.08M©, the middle row assumes Mt = 0.35M©, and the bottom row assumes Mt = 0.63M©. In each row, the image on the left shows the raw image stack sampled at one half the native HRC (28 mas) pixel size. The central column shows the residuals after subtraction of the best fit PSF model, showing the blended image elongation along the c-axis due to the lens-source separation. The right hand column shows these residuals rebinned to the 28 mas native pixel scale.

The stable point-spread function (PSF) of space-based telescopes, such as HST, allows the measurement of the image elongation due to the growing separation of the lens and source stars after the microlensing event. Simulations of this effect for the OGLE-2005-BLG-169 event are shown in Fig. 3.8 for three different cases: Ml = 0.08M©, Ml = 0.35M©, and ML = 0.63M©. This event has a higher relative proper motion than most events, but this simulation assumes images taken only 2.4 years after peak magnification, so for other events, it may be necessary to obtain the follow-up space-based images ~ 4 years after peak magnification.

When yU,rei can be measured from image elongation and/or the color dependent centroid shift, then the angular Einstein radius can be determined via

so that mass-distance relation can be determined even when t* cannot be measured.

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