ASTR340
The Origins of the Universe
(Spring 2004; Papadopoulos)
FINAL EXAMINATION
[10.30am-12.30pm, Monday, 17th May 2004]
Name:
Student Number:
Please read and sign the following honor pledge:
“I pledge on my honor that I have not given or received any unauthorized
assistance on this examination.”
signed
Instructions for this examination:
1. Answer all questions
2. This is a closed-book exam: no notes are allowed.
3. Please write your answers directly on this exam paper in the gaps
after each question. Do not feel that you have to use all of the
available space.
4. If you need more space than provided, write the continuation of
your answer on the back or on a separate sheet of paper. Be sure
to write your name and the question number on every additional
sheet you use.
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SECTION A (35 points)
Problems 1-5, four points each. Problems 6-8, five points each.
Problem 1
An inertial frame of reference is one in which (a) an object has no inertia; (b)
every object is either stationary or moving with constant velocity; (c) Newton’s first law
of motion is valid; (d) no object can rotate.
b
Problem 2
If the gravitational attraction between the Earth and the Moon were nullified at
the time of new Moon without making any other change in the solar system, the Moon
would (a) gradually escape from the solar system; (b) strike the Earth; (c) move inward
and strike the Sun; (d) follow a slightly eccentric orbit around the Sun similar to that of
the earth.
d
Problem 3
If the Sun’s rotation were stopped, (a) the orbits of planets would be changed
markedly; (b) the orbits of planets would remain the same; (c) the pattern of the seasons
on Earth would be changed; (d) tides on the Earth would cease.
b
Problem 4:
Newton’s law of gravity says that force is proportional to what power of the distance (d)
separating the two bodies?
a) d2
b) d1
c) d-1
d) d-2 X
e) d-3
Problem 5:
In a white dwarf the force exerted by its mass is balanced by
a) Gaseous pressure
b) Electron degeneracy X
c) Electromagnetic force
d) Neutron degeneracy
e) Strong force.
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Problem 6:
State the Cosmological Principles. Briefly describe one piece of observational
evidence that supports the validity of one of these Principles.
The Universe is Homogeneous and Isotropic.
Isotropy is supported by the measurements of the Cosmic Background Radiation
(CRB). The CBR has been measured more recently by the COBE explorer and found
to be isotropic to better than 10-6 and representative of the spectrum of Blackbody
radiation at 2.735 K with an uncertainty of .01 K.
Problem 7:
Suppose the Sun were to suddenly and magically turn into a black hole (without
changing its mass in the process). What would happen to the orbit of the Earth? In
other words, would the orbit of the Earth spiral into the black hole, fly into space, or
remain unchanged? Briefly justify your answer.
The orbit of the earth will be unaffected, since the gravitational force depends only on
the total mass of the Sun and the distance of the earth from the center of the sun and
both are unchanged.
Problem 8:
In the space below, sketch a graph of how the scale factor changes as a function of
cosmic time for the three possible “standard models” of the universe. Remember to
label the axes of the graph, as well as the three cases you plot. Neglect the effects of
dark energy (i.e. =0)
See Figure 11.2 of book
R
k=-1
k=0
k=+1
t
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SECTION B (65 points)
Problem 9. This problem concerns black holes
1. [5 points] Observations with radio and X-ray telescopes reveal that there is some
amount of radiation being emitted from the vicinity of this black hole. What do
we believe gives rise to this energy source? Describe the process.
The process is known as accretion and the observed body is an accretion disk. It is a disk
of gas that accumulates around the center of strong gravitational attraction such it exists
in the ergosphere of a rotating black hole. As the gas spirals into the black hole is
frictionally heated to high temperatures with characteristic emission in the X-ray range.
Most often this process occurs in binary systems one of which is a black hole that siphons
gas from the other.
2.
[10 points] Suppose you are in a spacecraft hovering at some distance above this
black hole. You tie a clock to a rope and start lowering the clock towards the
black hole. Describe how the rate at which you see the clock “tick” changes as it
is lowered towards the black hole. Be sure to highlight the significance of the
event horizon in your answer. Try to use a plot that shows the change in the
duration of time length t in relation to my tick labeled as to as a function of the
distance R from the center of the hole.
As the clock is lowered into the black hole I will see the signals arriving at pre-described
intervals to start arriving at longer and longer periods due to the gravitational redshift.
The ticks will continue to slow down as the clock approaches the event horizon, but I will
not see it crossing the event horizon. The time will seem to stop for the clock as it
approaches the event horizon and radio beacon signal of the clock is redshifted away. At
some point a last, highly redshifted signal arrives and nothing afterwards.
From invariance of the Schwarzschild metric
(1  RS / R)c 2 t 2 ( R)  c 2 t o2
and
t ( R) 
t o
1  RS / R
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t(R)/to
1
1
R/RS
GRADING – 5 points for correct description. 3 points for use of formula as time or
frequency redshift. 2 points for graph ( Give 5 points for graph even without
formula).
3. [5 points] You notice that, even though you are lowering the clock straight into
the black hole, there is an apparent force that is trying to make the clock orbit
around the black hole. This effect becomes stronger and stronger as the clock
gets closer and closer to the black hole. When the clock gets sufficiently close
(but still somewhat outside of the event horizon), this effect becomes irresistible.
Despite your best efforts, the rope breaks and the clock spirals into the black hole.
What is going on?
The hole is a rotating or Kerr black hole. As soon as the clock crosses the static surface
and enters the ergosphere it cannot remain motionless. The black hole compels the clock
to participate in its rotation. Inside the static surface even light even light is dragged
around the black hole. While the light cone is constantly tipped as we approach and enter
the static surface we still receive the signals.
Problem 10: (20 points) Describe all the types of redshifts that are measured in the
Universe. For each one
a. Describe the physical origin
b. Give an example of its occurrence
c. Show the equation that describes it (Make sure to define
the symbols in the formulas)
Definition of redshift factor z=rec-o)/o
I. Doppler shift due to motion of source with respect to observer. Hydrogen lines
received on the earth from the Sun. Whistle of an approaching or receding train. For
non-relativistic speeds z=V/c where V is the radial velocity.
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2. Gravitational Redshift: It occurs in regions that the strength of gravity near the
source is larger than the in the observer’s region. For example reaching us from
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an accretion disk. z 
 1 . Rs is the Schwarzschild’s radius.
1  Rs / R
3. Cosmological Redshift: It is due to the expansion of the Universe. It only occurs
between galaxies but not inside a galaxy. For example radiation we receive from
Quasars. z=Ro/R-1, where Ro is the value of the scaling factor at the time of
reception and R at the time of emission.
Grading: 7 points for each correct answer. In each answer 4 points for correct
description, 2 points for example and point for formula.

Problem 11: (15 points) Consider the Universe model shown in the figure below. At
time t1 a star in a galaxy X emits the line of an element with wavelength measured in
the laboratory of .1 micron. What will be the wavelength of the line for
i. An observer that receives it on the surface of a planet located in
another galaxy Y at time t2.
ii. An observer that receives it at the same time t2 but at the center of
the galaxy Y at a distance R=3/4 RS from the center of black hole.
iii. For an observer that receives it in a galaxy Z at time t3 when the
Universe is expanding at a rate R/t=.1 c.
R/Ro
Rt=.1c
2.1
2
1
t1
t
t
t3
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The wavelength will be given by
rec=(z+1)o
In the first case the value of z is z1=R(t2)/R(t1)-1=2-1=1 Therefore rec=2o=.2 m
In the second case in addition to the cosmological redshift we have to add the
Gravitational” redshift” since the observer is located near a black hole. Notice,
however, that in fact this will be blue shift, since the photon is coming and accelerated
towards the black hole. As a result the total value of z will be the value of z1 above minus
the value of z caused by the black hole which is
z 2  1 / 1  RS / R  1  1 .
As a result of the two exactly opposite effects the observer will receive un-shifted
radiation of .1 m.
In the last casethere will be a cosmological shift and a Doppler shift since the Universe is
expanding at .1c. It is sufficient to use the non-relativistic formula so that the Doppler
shift will have a value of z3=.1, while the cosmological redshift in this case will be
is z1=R(t3)/R(t1)-1=2.1-1=1.1. The total value of z will be 1.1+.1=1.2 and the received
radiation will have a wavelength of 1.2 m.

Problem 12: (10 points) Describe the Olbers’ paradox and its resolution.
Olbers in 1983 questioned the fact that if the Universe is infinite the night sky should
bright, as bright a s the surface of a star, since the line of sight of an observer will
always terminate on the surface of a star and the 1/R2 reduction of the light intensity
with distance will be compensated by the fact that the solid angle that we receive
radiation increases with distance as R2. Since the extinction due to the interstellar
medium is small the night sky should be very bright.
The paradox is resolved in the expanding Universe model, since redshifting brings the
radiation to well below the visible band as well as by the fact than even in an infinite
Universe light from very distance objects has not yet reached us due to the finite
speed of light.
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