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Let us now address the issue of 
gravitational microlensing. 

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This is a lensing of stars by other 
stars. 

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As you may recall, one possibility for 
the constituentce of baryonic dark matter 

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or maybe all of the dark matter was the 
in the form of some under luminous 

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substellar objects, which could be brown 
dwarfs or planets or even little black 

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holes or neutron stars, 
and gravitational microlensing offers a 

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way to test that. 
So if any such object passes along the 

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line of sight of some background star, it 
will cause gravitational magnification 

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and there, there will be first horizon 
brightness as it approaches the critical 

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radius and [UNKNOWN] so it will be a 
symmetric light curve, 

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it will also be same at all wavelengths 
because gravitational lensing is 

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achromatic. 
How often will this happen depends 

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obviously on a cross section and cross 
sections for stars that are very small, 

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they're down, because they're so very 
small compared to interstellar distances. 

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So in principle for any given star, it'll 
be extremely rare, and therefore, you'd 

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have to monitor a large number of stars 
in order to be able to catch one of 

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these. 
Here's some of the model light curves for 

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microlensing event. 
The peak magnification will depend on how 

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well aligned the passage is. 
The, the closer the better, of course. 

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And, how long does the event last will 
depend on the velocity of the lens 

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relative to the line of sight. 
Faster moving ones will cause shorter 

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events, obviously. 
In order to make this practical, one had 

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to monitor many millions of stars, 
and one good solution was to look towards 

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the large Magellanic Cloud, 
which is about 50 kiloparsecs away, where 

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many, many stars can be in the same field 
of view and then monitor for a whil 

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looking for such events. 
This was done by a team called MACHO, for 

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Massive Compact Halo Object, and also 
another team called OGLE, and since then, 

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many others. 
So there is a probability of seeing 

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lensing happen to any given star is about 
one part in 10 million per year, then you 

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better look at tens of millions of stars. 
Another possible direction is towards the 

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galactic bulge where a large number of 
galactic stars will be the same. 

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So MACHO and OGLE were just two 
experiments that started the whole thing. 

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MACHO monitored almost 12 millions stars 
for several years and OGLE continues to 

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monitor about 33 million stars. 
Many others have been started since then, 

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and now, this is a really 
well-established field. 

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Shown here on the right, is the first 
MACHO event detected, detected in 

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microlensing, 
and the two panels show light curving to 

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different filters. 
This was an important discriminator 

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against variable stars, because variable 
stars will tend to vary differently in 

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different filters, whereas gravitational 
lensing is unique in the sense that it 

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will work exactly the same for all 
wavelengths. 

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And on the left, we see many more 
different events, 

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to date there have been probably 
thousands and thousands of microlensing 

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discovered by different groups. 
Recall that Einstein radius essentially 

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defines the cross section for strong 
lensing. 

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And in this case, it would be as a 
critical impact parameter for lens 

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passing along the line or close to the 
line of sight. 

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Because mass of the lens and there is 
under square root in Einstein radius in 

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the area of the cross section is a square 
of that, it will be directly proportional 

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to the mass of the lens, 
thus, we can have an insight into the 

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masses of the lenses, 
and if we look at the whole population of 

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stars, then we can just add them up. 
So the net total fraction of all stars 

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that are being lensed per unit time is 
giving you effectively an optical depth 

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due to gravitational microlensing. 
The velocities of lenses are indicative 

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of the velocity dispersion of their 
populations, say halo, but they do not by 

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themselves say anything about lenses, 
and also, the implification says exactly 

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nothing about any given lens. 
Now, since their distances involved in 

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the expression of the Einstein radius, 
we have to know where the lenses are, and 

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so, the interpretation of the result 
depends critically on where do you assume 

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the lensing is happening? Are there stars 
near us? Are there stars in a giant cloud 

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or somewhere in between, like in a 
galactic halo? 

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If you know or assume, somehow, 
velocities of the lenses, 

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say we know that the velocity of 
dispersion of halo stars is couple 

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hundred kilometers per second, then you 
can infer masses of lenses from the 

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durations of the events, 
and typically, those are measured in days 

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or tens of days. 
The shorter the event, the smaller the 

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mass, so for an, an individual event 
there are only two things to measure. 

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We measure duration and the amplitude. 
The amplitude is simply telling you how 

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well aligned you are, it doesn't say 
anything about lenses. 

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The duration tells you about either their 
velocity dispersion or the individual 

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masses, 
and the total number of lenses you see is 

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telling you something about their 
density. 

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So here are the original MACHO results, 
they're expressed in probability contours 

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as a function of the lens mass and a 
fraction of the mass of the galactic halo 

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that could be attributed to them, 
assuming they really are in the galactic 

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halo. 
The surprising result here was the 

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typical mass was supposed to be about 
half solar mass. 

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This was unexpected because the only 
thing that could be like that would be 

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white dwarfs, which are evolutionary 
remnants of more massive stars, 

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but that means that there must have been 
some huge population of projectors of 

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those with other consequences which are 
not seen. 

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And so most likely the solution of this 
is that lenses were actually not in 

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galactic halo. 
They were either in our disc of our 

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galaxy or imaginary clouds themselves. 
But even if they were in a galactic halo, 

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the sheer frequency of them eliminated 
the possibility that all of the dark 

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matter is in form of MACHOs, regardless 
of what matters are brown dwarfs or black 

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holes or anything else. 
This was still a very powerful result. 

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Sometimes you can't find where the 
lenses. One it sufficiently close by that 

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you can use parallax and the other one is 
there is a binary, because then you can 

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use geometry of the binary star to infer 
how far it was. 

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A more recent twist on this is that this 
kind of measurement is used to discover 

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planets around other stars. 
A complimentery method to those, say 

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like, say Kepler Satellite that uses 
eclipses or kinematical measurements with 

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radial velocities. 
Next time, we will talk about dark 

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energy, 
probably the single most understanding 

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problem of physical cosmology today.