Micro lenses located in distant galaxies are massive, compact astronomical objects with a gravitational force field of a sufficient strength to bend and refocus the light of a very distant source of light into multiple images separated by about a microarcsecond. (There are 3600 seconds of arc in one degree; a microarcsecond is 1 one-millionth of a second of arc.) In order to generate this image splitting, the lenses must have masses comparable to the mass of a star. Stars in our own Galaxy, the Milky Way, can also act as lenses by creating multiple images of background stars located at larger distances in the Milky Way or in nearby galaxies. Because the bending angles depend not only on the mass of the lens, but also on the distance to the lens (DL) and background source (DS), (see Fig. 1), stars in our own Galaxy create multiple images separated by milliarcseconds (1 one-thousandth of a second of arc), not microarcseconds. Nevertheless astronomers still call them microlenses. As we shall see, most of the microlenses that have been discovered to date reside in our own Galaxy.
Fig. 1 --- Side view: A massive lens (L) bends the light of a distant luminous
source (S) by an angle
so that in principle the observer (O) sees
two images (I1 and I2).
The amount of the bending depends
on the mass M of the lens and the closest distance r
of the light
rays to the massive lens. The greater the mass of the lens or
the closer the light approaches the lens, the further apart the two
images will be split. Click on figure for a zoom.
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Suppose an astronomer could point a telescope capable of resolving details smaller than milliarcseconds in the direction of the sky where a microlens happens to be passing almost directly in front of a background luminous source star. She would not see the source, and perhaps not even the lens (unless it is very bright). Instead she would see two distorted images of the background source star (Fig. 2). If, from the perspective of the astronomer, the lens passes directly in front of the source, these images would be so distorted as to smear into an exceedingly bright ``Einstein ring'' centered on the position of the lens. In all other cases, the astronomer would observe two images, one inside the position of the Einstein ring and one outside. The combined brightness of the two images would exceed that of the background star when it is not lensed.
Fig. 2 --- Sky view: An astronomer with a futuristic telescope looking at the sky may have this view: two
distorted images (blue bananas), a brighter one (I2) that is closer to the
true position of the background source star and a fainter image (I1)
that is located on the other side. The observer will see nothing
at the true position of the source (blue circle), and may also not see the microlens in the center,
depending on its brightness. The dashed line marks the position of
the Einstein Ring for this lens. The red lines mark the angular
distance on the sky
between the lens and the source,
and the angular size
of the Einstein ring radius. Click on figure for a zoom.
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Unfortunately, telescopes do not yet exist that are capable of
resolving optical details as small as the images created by Galactic
microlenses.
The angular distance
between two images of a distant star being lensed by
a nearer Galactic star with the mass of our Sun is about the size of
a stuiver (Dutch nickel) in Groningen, Holland as it would appear to a tourist in Florence Italy!
Consequently, with current telescopes the observer sees neither
the splitting nor the distortion of the two images, but one
image (a blurring of the two microimages) that is brighter than
the unlensed source. The amount A by which the blurred
combined image is brighter depends on the angular separation
between the background source and the lens compared to the angular
size
of the Einstein ring:
/.
So, when the background source star lies almost behind the lens,
will be small,
so that u will be small, and A the amplification will be high:
the background source will appear much brighter than it actually is.
The difficulty is that
the observer does not know what the intrinsic (unlensed) brightness of the
source star should be, so how can she be sure that the bright image she sees
is the result of a faint background star being microlensed rather than
just an ordinarily bright (unlensed) star?
She must be patient. Stars and other objects in the Milky Way
are in constant motion. Relative motion between the observer,
lens, and source causes the angular distance
between
the lens and background source to change with time. This in turn changes the relative
positions (which can't be measured) and
the combined brightness (which can be measured) of the two images.
So, as the source first moves nearly behind the
lens and then recedes from it, the combined brightness of the
images will increase and then decrease (Fig. 3).
| Fig. 3 --- As the source (open blue circle) moves behind the lens with a speed vL across the sky, the two images (distorted blue patches) move in such a way that the line joining them on the sky always passes through both the lens and the real position of the source. The brightness of the two images, which is proportional to the area of the distorted blue patches, changes as the background source moves across the sky, reaching a combined maximum when the angular separation between the lens and source is the smallest. As the source recedes from the lens, the inner image fades and the outer image approaches a position and brightness that coincides with the unlensed source. (Adapted from Paczynski 1996.) This means that an astronomer monitoring the background source star will see its brightness increase and then decrease in a symmetrical manner. Click on figure for a zoom. |
The amount by which the brightness of the background star will
appear to change depends on the angular separation of the source
and lens at closest approach. If the source just grazes the
Einstein ring radius
(about 1 milliarcsecond) for the lens,
the maximum increase in source brightness will be 34%.
If instead the source passes as close as one-tenth of the Einstein radius,
it will experience a ten-fold increase in its normal
brightness (Fig. 4).
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Fig. 4 --- Background source stars may pass behind the lens
with different minimum angular separations. Those that pass
closest to the lens (in projection on the sky) will experience the
greatest magnification in their brightness. Sources that
pass an angular distance from the lens equal to the Einstein
ring radius
will be magnified by a factor 1.34. The light curves (brightness of
the sources as a function of time) are shown on the right for each
of the possible paths shown on the left.
Since each of the random possibilities for source position
shown above is equally likely, each of the resulting light curves
show below have equal probability. Click on figures for a zoom.
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The length of time over which this takes place and thus the duration of the microlensing event depends on the mass of the lens, the speed of the source across the sky, and distances of the observer to the lens and source. Faster motion and smaller lens masses will produce shorter microlensing ``events.'' For stellar mass lenses traveling at speeds typical for stars in the Milky Way, microlensing events should last a few weeks to a few months. Since the focusing of the light rays is independent of wavelength, microlensing light curves are ``achromatic,'' that is, they have the same shape regardless of the filter in the telescope camera.