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What are microlensing anomalies and what do they teach us?

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The simple microlensing light curves that are caused by a point-like foreground object lensing a point-like background source of light can be described by four parameters: the normal brightness of the background star, the time at which peak brightness occurs, the value of the peak brightness, and the duration of the event. Only the last of these gives any information about the lens itself through the equation:

In this equation, M is the mass of lens, D_S is the distance between the source and the observer, D_L is the distance between the lens and the observer, and v_L is the velocity of the lens across the sky. (G is the Gravitation constant, and c is the speed of light.) Since the source star can be seen and studied, its distance D_S can generally be determined to 20% accuracy. The lens is generally unseen, and thus its mass, distance and velocity are unknown. By measuring the duration of the event, the equation above can be used to learn how the mass, distance and velocity of the lens are related, but the lens parameters cannot be determined independently. This leads to what are called degeneracies: a lens moving very quickly can be confused with a low-mass lens and nearby lens can be confused with one that is further away (Fig. 7).

Fig. 7 --- The limited information available for a simple microlensing light curve creates a degeneracy between the lens mass, lens distance, and lens velocity. Inside the shaded blue region are possible values for the lens mass and lens distance for an event lasting 40 days, assuming that the lens velocity lies between 100 and 300 km/s, reasonable values for stars in our Galaxy. (The source star is assumed to be in the Galactic Bulge near the center of the Galaxy, a distance of 8 kiloparsec or about 25 lightyears from us.) Events that last about 40 days are probably caused by lenses that have the mass of typical stars: 0.1 times the Sun's mass or higher. Click on the figure for a zoom.

In order to break these degeneracies to learn more about the lenses, more information is needed, information that is sometimes provided by microlensing anomalies. Microlensing anomalies are deviations from the standard microlensing light curve and are caused by the extended size of a source or lens, or by complicated (non-linear) motion between the observer, source or lens while the event is underway. Some microlensing survey teams can now detect and immediately alert the community via electronic mail whenever a microlensing event is occurring (see, for example, the MACHO team alert web site at http://darkstar.astro.washington.edu). This allows other groups of astronomers such as the GMAN (Global Microlensing Alert Network) and the Groningen-based PLANET (Probing Lensing Anomalies NETwork) to do more detailed studies of these light curves to study microlensing anomalies. They can then also issue electronic alerts whenever a microlensing event exhibits odd behavior characteristic of an anomaly.

Suppose, for example, that the source star is so large that as it passes behind the lens, different parts of the source star are magnified differently. First the outer atmosphere of the leading edge, or limb, of the star will be magnified, then the central regions of the stellar disk, and finally the trailing limb. Since only a portion of the star is magnified at any given time, the peak brightness of the light curve will not be as large as it would for a small point-like source star; the light curve will be anomalous. The difference or anomaly will only be apparent as the star is passing just behind the lens. The astronomer can measure how long the anomaly lasts. By comparing this to the known size of the background star (found by other methods), the astronomer can then determine how long it took for the lens to cross over the source star and thus the angular speed of the lens. This has already been done for a handful of cases.

Another example of anomaly is the slight wave on top of a normal microlensing light curve caused by the circular motion of the Earth (and therefore the observer) around the Sun. If a microlensing light curve lasts long enough so that the Earth moves far enough along the arc of its orbit, this anomaly can be measured and used to infer motion of the lens relative to the (known) Earth motion. This anomaly has been measured in a handful of microlensing events by the MACHO collaboration.

As we have discussed, simple microlensing is achromatic, that is it looks the same in any filter or color of light. But if the source star is large enough so that portions of the source star are magnified in sequence, the peak of the light curve may be bluer than the wings since the center of a star is bluer (hotter) than the outer atmosphere of its limb. This anomaly has been measured in one event by the PLANET collaboration and used to characterize the atmosphere of a giant star about 8 kiloparsecs from us, buried in the bulge at the center of the Milky Way.

Probably the most common type of anomaly, occurring in about 10% of all microlensing events to date, is that caused by a double, or binary, lens. If two lenses are far enough apart (many times their own individual Einstein ring radii), then they will behave like separate lenses, and the light curve will exhibit either one or two simple bumps (depending on whether or not the source passes close behind only one or both lenses). On the other hand, if the lenses are fairly close to one another, then the magnification pattern is altered so that at certain positions on the sky their individual lensing strengths can reinforce each other tremendously. These positions are called "caustics'' and if a source passes behind one of them its magnification can reach enormous values. A close binary lens passing in front of a background source will therefore create generate sharp peaks on top of an otherwise fairly smooth light curve. The first such binary lens light curve was reported by the OGLE team (Fig. 8).

Fig. 8 --- The first binary lens light curve was reported by the OGLE team. The two lens have similar mass and reinforce one another to produce the sharp peaks spaced by about 50 days. Before and after the microlensing, the source star has maintained constant brightness. Click on the figure for a zoom. (Courtesy OGLE team.)

The sharp peaks in source brightness are similar to the sharp, bright features that appear on a tablecloth when candlelight passes through a wine glass. If the wine glass is moved, the bright patterns move over the tablecloth. When a complex astronomical lens like a double star (wine glass) moves between the source (candle) and an observer (fixed position on the table cloth), the telescope records the pattern of changing light, which will have sharp peaks whenever the source passes behind the caustic structure (Fig. 9).

Fig. 9 --- Left: An equal mass double lens (two black dots) creates a caustic structure --- regions where the combined effect of both lenses is enormous --- shown here as the solid black line between the two lenses. Right: The light curves of sources passing behind this caustic structure will exhibit rapid increases of brightness at the moment of crossing. The exact light curve shape depends on the path (shown as the colored lines in the left-hand figure) that the source takes as well as on the mass ratio of the lens and their angular separation. Click on the figures for a zoom. (Adapted from Paczynski 1996.)

By carefully measuring the anomalous structure in a binary lens light curve, the ratio of the mass of one lens to the mass of its partner can be measured. One can also measure the angular distance between the lenses at the time the anomaly occurred as a fraction of their Einstein ring radii. Microlensing thus gives astronomers a way to determine which types of binary stars are common in our Galaxy.

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