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ASTR415 Spring 2009 TERM PROJECT
Due May 7th, 2009
YOUR FIRST COSMOLOGICAL SIMULATION OF A BLACK HOLE!
(also known as secondary infall problem)
Write a report and prepare a class presentation on the results of your
numerical simulations of dark matter accretion around a black hole at
rest in an expanding Universe. This problem is quite difficult for an
undergraduate level course: do not worry too much if your results do
not agree with the expected solution!
In order to solve this problem you need to generalize the integration
package you wrote for PS5 so that it can solve the
-body problem in
an expanding Universe assuming spherical geometry and fixed time steps.
The following ODEs should be integrated in order to solve the
problem (see also my handwritten notes):
This equations describe the motion of spherical shells of dark matter
around the black hole in comoving coordinates:
is the
peculiar velocity of a shell at comiving distance
from the black
hole and
is the difference
between the mass within
at time
and the original mass at time
when the Universe was homogeneous (constant density). To
convert the distances in physical units
use the formula:
, where the redshift
depends on time as
and where
Gyr is the age of the
Universe. The Hubble parameter in a flat matter dominated Universe is:
, where
km/s.
Alternatively you can integrate the equation as a function of redshift
rather than time
(it may be easier). Using the
relationship
you get the following equation:
Assume initial conditions as follows: start the simulation at redshift
and assume that the dark matter has constant density
g/cm
. Remember to place a black hole
of mass
M
at
. If I forgot to provide enough
explanations to solve the problem feel free to ask me. Have fun!
Hints: The dark matter should create a spherical halo around the
black hole with mass
and have a total mass
within the halo radius that increases with decreasing redshift as
.
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Massimo Ricotti
2009-04-16