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Dark Matter in our Galactic
Halo
Dark Matter in our Galactic Halo
The rotation curve of the disk of our galaxies implies that our Galaxy contains more mass
than just the visible stars and gas. Extra mass – the dark matter – is normally assumed to
reside in an extended, roughly spherical halo around the Galaxy. But what is this dark
matter? Possibilities include:
• molecular hydrogen gas clouds
• very low mass stars/brown dwarfs
• stellar remnants: white dwarfs, neutron stars, black holes
These possibilities are all baryonic forms of dark matter. This means they are made up of
ordinary matter (protons and neutrons). Other possibilities include:
• primordial black holes
• elementary particles, probably currently unknown (e.g. WIMPS)
These are non-baryonic forms of dark matter. On the largest scales in the Universe, there is
strong evidence that the dark matter must be non-baryonic. This is because the abundance
of light elements (hydrogen, helium, and lithium) formed in the Big Bang depend on how
many baryons (protons + neutrons) there were. Measurements of the light element
abundances allow measurement of the number of baryons. Observations of dark matter in
galaxy clusters suggest that there is too much dark matter for it all to be baryonic; most of it
must be non-baryonic.
Gravitational Lensing
On galaxy scales, there is no similar argument. How can we try to understand the nature of
this dark matter? One such powerful tool is that provided by gravitational lensing.
Gravitational Lensing
q
r
M
x

Simply stated, “gravitational lensing” is the bending or deflection of light by massive
bodies. Hence images of background objects are distorted if there is a massive foreground
object along the line of sight. Bending of light is similar to the deflection of massive
particles as they move past a mass. Let us consider a photon moving past a point mass M,
with an “impact parameter”  as shown in the Figure below. From classical Newtonian
gravity, the photon will undergo an acceleration perpendicular to the direction of its
motion. This deflection can be calculated as a function of the point mass (M) and the
distance the photon passes at ().
Gravitational Lensing
The amount of this deflection is simply:
dv GM
 2 sin q
dt
r
If we substitute dx = cdt, then we get:
v 

GM
c
1


 x 2   2 x 2   2

GM
c
 x

2

 2


3 / 2

1/ 2
dx
dx
The integral above is analytic and works out to 2/2, so:
v 
2GM
c
and the Newtonian deflection angle is then (in the limit of small deflection):

v 2GM

c
c 2
Gravitational Lensing
In the general relativistic case, however, gravity
affects both the spatial and time component of a
photon’s path, so the actual bending is twice this
value. Thus, we define the deflection angle,
otherwise known as the Einstein angle, as:
4GM
c 2
The standard geometry of a gravitational lens is
depicted in the figure. There are three main
components: the observer, the lens or lensing object,
and the source. The observer is positioned at point O
and the source at point S. The important distances
and angles in this configuration are:

Dd =
Ds =
D
 ds =
 =
q =
 =

 =
distance from the observer to the lens
distance from the observer to the light source
distance from the lens to the source
true angle between the lens and the source
observed angle between the lens and the source
distance from the lens to a passing light ray
the Einstein angle of deflection
S’
Gravitational Lensing
Due to the deflection of light by the lensing object (the mass M), an image of the source is
seen by the observer at point I at the angle q, and the source is interpreted to be at point S.
The amount of deflection is a function of the impact parameter and mass M.
Before we derive relationships from this lens geometry, a few assumptions need to be
made. First, it is assumed that the distances from the observer to the lens and from the lens
to the source are much larger than the dimensions of the lensing mass itself. Therefore, the
deflection of light is assumed to be instantaneous upon encountering the lens. Additionally,
the lens is assumed to be two dimensional, in a plane perpendicular to the line of sight
between the observer and the source. These assumptions allow for the “thins lens”
approximation. Also, the fact that the distances are large and the angles small, allows for
the small angle approximation. On a last technical note, all of the angles are truly defined
as vectors, but for the assumption of a two dimensional mass distribution.
From ∆OSI and the law of sines,
sin 180    sin q   

Ds
Dds
Since all the angles are small, sin(q - )  q - , and sin(180 - ) = sin  . So


 q 
Dds 

(we will neglect the vector signs from
Ds
now on based on our initial assumptions)
Gravitational Lensing
Since we have already seen that:

4GM
c 2
and from simple geometry for small angles: q = /Dd
 4GM Dds  1
 
2
 c Dd Ds  q
So we now have a relationship between  and q. To simplify this expression, let us define a
characteristic bending angle o that is a function only of the mass of the lens and the
distances involved:
1/ 2
 4GM Dds 

 0   2
 c Dd Ds 
  q  
So that:
02
 q 
q
and
q 2  q  0 2  0
Whose solution is:
1
q     4 0 2   2 
2

Gravitational Lensing
This final equation defines the location of the images as a function of the true position of
the source with respect to the lens and o. Notice that the equation is quadratic and
therefore has more than one solution, implying there will be multiple images for a given .
The value of o defines a characteristic bending angle for a given lens. Notice that if  = 0,
namely when the source, lens, and observer all lie on the same optic axis, q = o, which is
often referred to as the Einstein radius qE . For a source exactly behind the lens, the source
would appear as an “Einstein ring” on the sky, with radius qE. For  > 0, we get two
images, one inside and one outside the Einstein ring radius.
Lensing Regimes
Conceptually, the simplest situation for gravitational lensing is when the lens is massive
enough to produce a large angle of deflection. The case where we can resolve multiple
images of the background source is called “strong lensing”.
Strong lensing normally requires a lens as massive as a galaxy (which BTW are extended
and therefore slightly more complicated than the simple point mass we considered).
Consider instead a solar mass star half way between us and the Galactic bulge. Here,
Ds = 8 kpc
Dd = 4 kpc
Dds = 4 kpc
Then, using:
1/ 2
 4GM Dds 

 0   2
c
D
D
d s 

to calculate the Einstein ring radius, we get that :
qE = 510-8 radians = 10-3 arcseconds.
Lensing Regimes
This is really tiny. We cannot resolve multiple images when the separation is only 1
miliarcsecond. This regime of lensing is called “microlensing”. For experiments trying to
detect stellar-mass lenses in the Galactic halo using stars in the Large Magellanic cloud
(LMC) which is 50 kpc away, you can show that the angular separation of the lensed
objects would be sub-milliarcseconds. So how would we try to use this technique to detect
a large population of dark massive point like objects in our Galactic halo? Well, all is not
lost, since even though we cannot resolve the multiple images of background sources
produced by these dark point masses, lensing produces magnification so the relative motion
between the lens and the background source would cause a change in brightness as the lens
passes in front of the source.
To see that at least one image will be magnified, consider a polar coordinate system with
the lens at the center, and consider light passing through a differential area, dA. Due to the
gravitational lens, the angles are distorted, so the observer sees this light as squeezed into a
different area element dA. You can think of this as the lens “focusing” the light from area
dA to area dA. The magnification is then simply the ratio of the two areas:


dA q  q

dA
  
q q
 
Lensing Regimes
r  D

dr  Dd 
r
D

d 

1
dr
D
q
r
D

dq 
1
dr 
D
dA   r   dr 
dA   r   dr
 D 2q dq
 D 2  d 

r
D
q

dA q  q


dA
  

q q
 
Lensing Regimes
dq 1 

 1
2
2
d 2 
4

o 

which means that the magnification is:
But:
1 


4  4 2   2
o



1/ 2
4

2
o




2


1/ 2

 2


This equation yields two magnifications, one for each of the two solutions to the quadratic
form of the lens equation for q. The figure above shows the dependence of the
magnification on  and o. You can see that as   , +  1 and -  0; this indicates
there is no lensing. As you can see from the figure above, when   o, there is
considerable lensing.
Lensing Regimes
There are a few other useful relations. Since the magnification of a lens is given by:
1/ 2
2

4 o   2
1 



 2
1
/
2

4  4 2   2

o


The brightness ratio of two images is:




which simplifies to:

   4 o 2   2


   4 o 2   2



  

1/ 2
  
1/ 2
 q 
     1 
  q 2 
2
2
If  = o (for the source to be within the Einstein ring), then the magnification of each
image would be:

1 1
    5  2 
4 5

The total magnification of the lens would be
tot = + + - = 1.34 (since both images would be landing on top of each other)
Lensing Regimes
Thus, when an object is microlensed, it appears to brighten by at least 34%. This
magnification corresponds to a brightening of 0.32 magnitudes which is easily detectable.
So, as a dark mass passes along the line of sight between us and an observed star in the
LMC, for example, the star would appear to brighten and then fade as the alignment is lost.
There are several experiments underway that since ~1990 are trying to detect microlensing
events by monitoring millions of stars in the LMC and SMC. An illustration of this idea is
shown below.
(Diagram from Keener 1998)
Lensing Regimes
Suppose that between us and the LMC there are a large number of dark, compact objects.
At any one time, we will see a clear lensing event (i.e., the background star will be
magnified to a detectable level) if the line of sight passes through the Einstein ring of one
of the lenses. The area of this ring on the sky is qE2 where qE is o which is given by:
1/ 2
 4GM Dds 

 0   2
 c Dd Ds 
The probability that this lens will magnify a given source is:
 D 
2
P  q E   ds M
 Dd Ds 
which as you can see is directly proportional to the mass of the lens. The same would be
true for a population of lenses with some total mass. So we can conclude that measuring the
fraction of stars that are being lensed at any one time measures the total mass in lenses,
independent of their individual masses. But, we need to know where the lenses are to get
the right mass estimate.
Microlensing Experiments
Several experiments have searched for microlensing events:
• toward the Galactic Bulge (lenses are disk or bulge stars)
• toward the Magellanic Clouds (lenses could be stars in the LMC / SMC, or halo
objects)
MACHO (Massive Compact Halo Object):
• observed 11.9 million stars in the Large Magellanic Cloud for a total of 5.7 years.
OGLE (Optical Gravitational Lensing Experiment):
• ongoing experiment
• presently monitor 33 millions stars in the LMC, plus 170 million stars in the Galactic
Bulge.
Microlensing Observables
Suppose that between us and the Magellanic Clouds there are a large number of dark,
compact objects.
Source stars
in the LMC
Unseen lenses in
the Galactic halo
At any one time, we will see a clear lensing event (i.e. the background star will be
magnified) if the line of sight passes through the Einstein ring of one of the lenses.
Previously derived the angular radius of the Einstein ring on the sky qE. Area is qE2.
Microlensing Observables
2 GMd LS
qE 
c dLdS
Single lens of mass M, at distance dL. Observer - source
distance is dS, lens - source distance is dLS (= dS - dL)
Probability that this lens will magnify a given source is:
 d 
2
P  q E   LS M
directly proportional to the mass of the lens
 dLdS 
Same is obviously true for a population of lenses, with total mass Mpop - just add up the
individual probabilities. Conclude:
• measuring the fraction of stars that are being lensed at any one time measures the
total mass in lenses, independent of their individual masses
• geometric factors remain - we need to know where the lenses are to get the right
mass estimate.
Microlensing Observables
No way to determine from a single image whether a given star is being magnified by
lensing. Need a series of images to see star brighten then fade as the alignment changes:
Position of Einstein
ring when event
starts
Line of sight
…when event ends
Motion of lens
Lensing time scale: equals the physical distance across the Einstein ring divided by the
relative velocity of the lens:

2d Lq E
L
Microlensing Observables

4 GMd L d LS
 Lc
dS
Time scale is proportional to the square root of the individual
lens masses.
Put in numbers appropriate for disk stars lensing stars in the Galactic bulge:
• dS = 8 kpc, dL = dLS = 4 kpc
• M = 0.3 Msun
• vL = 200 km s-1
  40
M
days
0.3M sun
Weak dependence on mass is very convenient observationally, if we observe every night
can detect:
• events with  ~ 1 day: M < Jupiter mass (10-3 Msun)
• events with  ~ 1 year: M ~ 25 Solar masses (e.g. stellar mass black holes)
• + everything in between …
Microlensing Observables
MACHO project detected 13-17 microlensing events toward the LMC in just under 6 years
of operation, compared to only 2-4 expected on the basis of known stellar populations.
Lensing events are expected to be achromatic (same light curve in different wavebands),
which helps distinguish them from variable stars.
Microlensing Observables
For each event, there are only two observables:
• duration  - if we know the location of the lens along the line of sight this gives the
lens mass directly
• peak amplification A: this is related to how close the line of sight passes to the
center of the Einstein ring
u
b
A
Define
b
d Lq E
u2  2
u u2  4
(we won’t try to
prove this formula)
Note: amplification tells us nothing useful about the lens.
Additionally, observing many events gives an estimate of the probability that a given
source star will be lenses at any one time (often called the optical depth to microlensing).
This measures the total mass of all the lenses, if their location is known.
Microlensing Observables
Based on the number and duration of MACHO events:
If the lenses are objects in the Galactic Halo
• 20% of the mass of the Galactic halo (inferred from the Galactic rotation curve) is
in the form of compact objects
• Typical mass is between 0.15 Msun and 0.9 Msun
• Idea that all the mass in the halo is MACHOs is definitely ruled out
One interpretation of these results is that the halo contains a much larger population of
white dwarf stars than suspected.
Other authors suggest the lenses may not be our halo at all, but rather reside in the
Magellanic Clouds. If correct, implies that none of the halo is in the form of planetary mass
to ~10 Msun compact objects.
Microlensing Observables
Ambiguity in the distance to the lenses is the main problem. Can be resolved in a few
special cases:
a) If distortions to the light curve caused by the motion of the Earth around the Sun
can be detected (`parallax events’)
b) If the lens is part of a binary system. Light curves produced by binary lenses are
much more complicated, but often contain sharp spikes (`caustic crossings’) and
multiple maxima. Provide more information about the event.
One event seen toward the Small Magellanic Cloud was a binary event, and it is known to
lie close to the SMC.
My guess is therefore that the majority of the lenses are not in the Galactic halo, which is
probably made up of elementary particle dark matter instead…
Microlensing Observables
Observed binary lensing event.
Note: a star with an orbiting planet is just
a special case of a binary system with a
large difference in masses.
Much more numerous events toward the
Bulge are being monitored for signs of
any planets, so far without any definite
detections…