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Worksheet 4 - Advanced Worksheet |
A. Gravitational Lensing
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Some of the most convincing evidence for dark matter comes from a phenomenon known as
gravitational lensing. This was first predicted by Einstein in his theory of relativity. The theory
predicts that large masses in outer space, such as clusters of galaxies, bend light that travels
near them. So as the light from a distant star passes by a large mass its path is distorted by
gravity. Gravitational lensing was first observed experimentally in 1919 when physicist Arthur
Eddington observed light from a distant star being bent by the Sun.
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The angle  (measured in radians) by which light from a distant star or
galaxy is bent by a mass M is given by the following formula
where G = 6.67 x 10-11 Nm2/kg2, d is the closest the light comes to the
centre of the object, and c is the speed of light.
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1. |
Given that the mass of the Sun is 1.99 x 1030 kg and its radius is
6.96 x 108 m, calculate the angle of deflection
for light
from a distant
star that passes very close to the Sun's surface.
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2. |
A ray of light that passes within a distance of 16 million light
years from the centre of a cluster of galaxies is bent
by an
angle of
2.0x 10-5 radians. Use gravitational lensing to calculate the mass
of the cluster.
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3. |
Order the following three scenarios according to the angle of
deviation (from highest to lowest) for light that just
passes by the
edges of the clusters.
a) A cluster of galaxies with a mass of 1014 times the mass of the
Sun and a radius of 107 light years.
b) A cluster of galaxies with a mass of 5 x 1014 times the mass of the
Sun and a radius of 3 x 106 light years.
c) A cluster of galaxies with a mass of 2 x 1014 times the mass of the
Sun and a radius of 4 x 106 light years.
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B. "Seeing" Dark Matter on Earth: WIMP Collisions
One of the many experiments currently underway on Earth in the search for dark matter is located
in rural Minnesota, U.S.A. It is 700 m underground in an abandoned mine and is called the
Cryogenic Dark Matter Search (CDMS). The experiment involves a number of
250 g crystals of
germanium cooled down to just above absolute zero (-273° C) and is designed to detect dark
matter if it is made of weakly interacting massive particles (WIMPs). To date, the experiment has
not detected any WIMPs.
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4.
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If dark matter is made of WIMPs then billions of these particles
from space are raining down on Earth each second.
Although they
typically pass through solid objects as if they are not there, there
is a very small chance that a WIMP
will collide with a nucleus of
an atom within any material it happens to pass through.
So, at CDMS there is a very small probability that a WIMP will
collide with the nucleus of a germanium atom in the
detector.
This collision would be elastic, and is illustrated below.
You have been hired as a consultant by CDMS and some of the
physicists ask you for help with the following problem:
Suppose a WIMP has a mass of 1.07 x 10-25 kg and an initial
speed of 230 km/s. It collides with the nucleus of a
stationary
germanium atom with a mass of 1.19 x 10-25 kg. The germanium
atom is deflected with an energy of
10 keV (1 eV = 1.60 x 1019 J). The physicists would like to know in which direction the
germanium atom travels
after the collision.
Find the answer to this problem and write a clear, detailed
explanation of how you arrived at it so that you can send
it to the CDMS physicists.
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C. |
Density of Dark Matter (Challenging)
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5. |
The total mass of dark matter, Mdark, within a galaxy increases
linearly with distance r from the centre of the
galaxy, i.e.,
Assuming that dark matter is distributed in a spherically
symmetric fashion, use this fact about the mass of dark
matter to
write a proportionality statement (e.g., y  x ) for the relationship
between the density of dark matter
and the distance from the
centre of a galaxy.
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