4.7 Galactic microlensing

It has been known for more than two decades that halos of galaxies must contain some unknown kind of dark matter. Many different particles/objects had been suggested as constituents of this halo dark matter. The candidates can be divided into the two broad categories “elementary particles” and “astronomical bodies”. A conservative candidate for this dark matter are brown dwarfs, objects with masses less than 0.08M ⊙ so that the central temperature is not high enough to start helium fusion. These objects are certain to exist, we just do not know how many there are.

In 1986 Paczyński [136Jump To The Next Citation Point] suggested a method to test observationally whether the Milky Way halo is made of such brown dwarfs (or other astronomical objects in roughly this mass range). Subsequently this type of dark matter candidate was labelled “Macho” for MAssive Compact Halo Object [69]. If one could continuously observe the brightness of stars of our neighbouring galaxy Large Magellanic Cloud (LMC) one should see typical fluctuations in some of these stars due to the fact that every now and then one of these compact halo objects passes in front of the star and magnifies its brightness. The only problem with this experiment is the low probability for such an event: Only about one out of three million LMC stars would be significantly magnified at any given time.

The underlying scenario is very simple: Due to the relative motion of observer, lensing Macho and source star the projected impact parameter between lens and source changes with time and produces a time dependent magnification. If the impact parameter is smaller than an Einstein radius then the magnification is μmin > 1.34 (cf. Equation (22View Equation)).

For an extended source such a sequence is illustrated in Figure 20View Image for five instants of time. The separation of the two images is of order two Einstein radii when they are of comparable magnification, which corresponds to only about a milliarcsecond. Hence the two images cannot be resolved individually, we can only observe the brightness of the combined image pair. This is illustrated in Figures 21View Image and 22View Image which show the relative tracks and the respective light curves for five values of the minimum impact parameter umin.

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Figure 20: Five snapshots of a gravitational lens situation: From left to right the alignment between lens and source gets better and better, until it is perfect in the rightmost panel. This results in the image of an “Einstein ring”.
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Figure 21: Five relative tracks between background star and foreground lens (indicated as the central star) parametrized by the impact parameter umin. The dashed line indicates the Einstein ring for the lens (after [136Jump To The Next Citation Point]).
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Figure 22: Five microlensing lightcurves for the tracks indicated in Figure 21View Image, parametrized by the impact parameter umin. The verical axes is the magnification in astronomical magnitudes relative to the unlensed case, the horizontal axis displays the time in “normalized” units (after [136]).

Quantitatively, the total magnification μ = μ1 + μ2 of the two images (cf. Equation (22View Equation)) entirely depends on the impact parameter u(t) = r(t)∕RE between the lensed star and the lensing object, measured in the lens plane (here RE is the Einstein radius of the lens, i.e. the radius at which a circular image appears for perfect alignment between source, lens and observer, cf. Figure 20View Image, rightmost panel):

u2 + 2 μ (u) = -∘---------. (43 ) u (u2 + 4)
The time scale of such a “microlensing event” is defined as the time it takes the source to cross the Einstein radius:
∘ ----∘ -------∘-------- RE M DL DL ( v⊥ )− 1 t0 = --- ≈ 0.214 yr ---- ------- 1 − --- ---------−1 . (44 ) v⊥ M ⊙ 10 kpc DS 200 km s
Here v⊥ is the (relative) transverse velocity of the lens. We parameterized the time scale by “typical” numbers for the distances of lensed and lensing star and the relative transverse velocity. Note also that here we used the simple relation D = D − D LS S L (which is not valid for cosmological distances).

Note that from Equation (44View Equation) it is obvious that it is not possible to determine the mass of the lens from one individual microlensing event. The duration of an event is determined by three unknown parameters: the mass of the lens, the transverse velocity and the distances of lens and source. It is impossible to disentangle these for individual events. Only with a model for the spatial and velocity distribution of the lensing objects and comparison with “simulated microlensing events” it is possible to obtain information about the masses of the lensing objects and their density.

What seemed to be an impossible task at the time – namely determine the brightness of millions of stars on an almost nightly basis – was turned into three big observational campaigns within few years (MACHO, EROS, OGLE experiments). These groups looked at millions of stars in the LMC and towards the bulge of the Milky Way, and their first results appeared essentially simultaneously in the fall of 1993 [511186]. In the meantime more groups have joined this effort, some of them with special emphases: e.g. on covering ongoing microlensing events (PLANET, DUO), or on extending the microlensing search to unresolved stars (“pixel lensing”) in the Andromeda galaxy (AGAPE) [4467], or to cover the Magellanic Clouds completely around the year (MOA). Here is a list of groups currently active in the search for microlensing signatures of compact objects in the halo of the Milky Way or elsewhere:

The observations towards the Large Magellanic Cloud show that there are fewer microlensing events than one would expect if the halo of the Milky Way was made entirely of these compact objects. The latest published results from the microlensing experiments that monitor stars in the LMC indicate that the optical depths toward the LMC is about τ ≈ 3 × 10 −7. The observations are consistent with 50% of the Milky Way halo made of compact objects with most likely masses of +0.3 0.5−0.2M ⊙ [6]. But the number of observed events is still small (in this analysis eight events were used) and hence the uncertainties are large; in fact, it cannot even be excluded that none of the observed events is due to an unknown halo population [64].

The same type of experiment (searching for microlensing events) is being performed in the direction of the galactic bulge as well, the central part of the Milky Way. By now more than 200 microlensing events have been detected in this direction (for an example see Figure 23View Image). Among them are a number of “binary lens”-events (which have a very typical signature of at least two caustic crossings, cf. Figure 24View Image). This is about three times as many microlensing events as were expected/predicted. Several groups try to explain this “over-abundance” of events to have a new look at the stellar content and the dynamics of the bar/bulge of the Galaxy. The latest published results can be found in [7].

With these microlensing experiments gravitational lensing has established itself as a new tool to study the structure of the Milky Way. This type of microlensing also holds some promise for the future. It can be used, e.g. to study the frequency of binary stars. One of the most interesting possibilities is to detect planets around other stars by extending the sensitivity of the binary lenses to smaller and smaller companion masses [115195].

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Figure 23: Observed Lightcurve of a microlensing event towards the bulge of the galaxy, event OGLE #6 [188]: The I-band magnitude is plotted as a function of time (Julian days). In the top panel, the constant V-I color of the star is shown. The maximum magnification is μ = 6.9 (or 2.1 mag), the duration of the event is 8.4 days. The star has constant brightness in the following year (Credits: Andrzej Udalski.)
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Figure 24: Lightcurve of a binary microlensing event towards the bulge of the galaxy, event OGLE #7 [187]: The I-band the magnitude is plotted over time (Julian days). In the top panel the constant V-I-color of the star is shown. The maximum magnification is more than 2.5 mag higher than the unlensed brightness. The duration of the event is about 80 days. The two insets at the left part show a zoom of the two peaks. The star had constant brightness in the year preceding the microlensing event (1992). A model for this event finds a mass ratio of 1.02 between the two lensing stars, and a separation of 1.14 Einstein radii. (Credits: Andrzej Udalski.)

For a recent comprehensive presentation of galactic microlensing and beyond see [137]. Various aspects of microlensing in the local group are reviewed in detail. Another review article on the basics and the results of galactic microlensing can be found in [156].

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