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Physics 55 Final Exam
Professor Henry Greenside
Thursday, December 15, 2005
This 3-hour exam is closed book and no electronic aids are allowed except for a scientific calculator. If you
have any questions during the exam, please do not hesitate to ask me.
Before beginning, please look the exam over to get a sense of how many points each section is worth.
The following data and equations may be helpful:
E=
1
mv 2 ,
2
E = mgh,
sum mv before = sum mv after,
p2 = a 3 ,
L
,
4πd2
d (in pc) =
z=
v
∆λ
= ,
λ
c
0
r
t =t
n=
1−
v 2
c
N
1
= 3,
V
d
G ≈ 7 × 10−11
m3
,
kg s2
,
L =L
r
c ≈ 3 × 108
1−
v 2
c
rv 2
.
G
GM
g= 2 ,
d
v=
,
m0 = q
1 ly ≈ 1013 km,
2GM
.
r
Rstar
2d
2 ,
1 − vc
kg m2
,
s
2, 900, 000
.
T (kelvin)
1/2
v=
A = 4πr 2 ,
1 pc ≈ 3 ly,
1
m
C = 2πr,
h ≈ 7 × 10−34
r
λmax (in nm) =
Tplanet = (1 − r)1/4
180 s
,
π d
m
,
s
M=
L = A · σT 4 ,
v = H0 d,
α (in deg) =
1 AU ≈ 108 km,
4π 2
a3 ,
G(M1 + M2 )
1
,
p (in ”)
λf = c,
0
sum mvr before = sum mvr after.
M1 M2
F =G
,
d2
1
a = Ftotal ,
m
d = vt,
b=
p2 =
E = mc2 .
E = hf,
Tstar .
v1 + v 2
.
1 + v1 2v2
c
V =
4 3
πr .
3
σ ≈ 6 × 10−8
1 nm = 10−9 m.
m2
watt
.
Kelvin4
True or False Questions (1 point each)
Please circle “T”or “F” to indicate respectively whether each statement is true or false.
1. T / F A good place to look for a pulsar is at the center of a planetary nebula.
2. T / F In the year 8,000 AD, Earth could receive a signal from aliens in some star system saying that
they had found and deciphered the Pioneer 10 plaque designed by Carl Sagan and other scientists.
3. T / F Light can be accelerated.
4. T / F All galaxies have redshifts when observed through a telescope on Earth.
5. T / F According to Einstein’s theory of special relativity, the hotter a cannon ball, the larger the
force that is needed to accelerate the cannon ball.
6. T / F All white dwarfs look white.
7. T / F WIMPs are currently believed to be a more likely explanation for dark matter than MACHOs.
8. T / F Most of the helium in the universe was created by the fusion of hydrogen in the cores of stars.
9. T / F All stars of the celestial sphere can be observed over the course of a year in an equatorial local
sky.
10. T / F Because of the expansion of the universe, you are bigger today than you were yesterday.
11. T / F When a white dwarf becomes a supernova, a lot of carbon is spewed out into space.
12. T / F It is not possible to create a proton-antiproton pair from a photon that was produced by the
annihilation of an electron with a positron.
13. T / F Given that the star Mintaka, in the right side of Orion’s belt, lies almost exactly on the
celestial equator, the star Rigel must have a negative declination.
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14. T / F If the strong interaction were somehow to become much weaker than the electromagnetic
interaction with all other properties being the same (so protons and neutrons still attract each other
but only at extremely short distances), then the only atoms in the universe would be isotopes of
hydrogen.
15. T / F Our solar system is filled with dark matter.
16. T / F The star-gas-star cycle of the Milky Way—in which stars are born, stars spew their contents
into the galaxy when they die, and new stars are later born—will not continue forever.
17. T / F Over the next billion years, the surface temperature of the Sun will slowly increase.
18. T / F Collisions of galaxies were much more common six billion years ago.
19. T / F It is possible for the solar day of a planet to be longer in time than its sidereal year.
20. T / F Near stars, space has a spherical geometry (as opposed to flat or saddle-shaped geometry).
21. T / F If the Earth had an exactly circular orbit, there would not be any seasons.
Multiple Choice Questions (4 points each)
Circle the letter that best answers each question.
1. Which of the following is closest in size to a white dwarf?
(a) A Duke football.
(b) Duke’s Wallace Wade football stadium.
(c) The city of Durham.
(d) The Earth.
(e) Jupiter.
2. A main sequence star of what spectral type will eventually fuse nuclei in more than two shells at the
same time?
(a) Type G.
(b) Type K.
(c) Type M.
(d) Type O.
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3. A certain remote object in the sky is visible with a radio telescope but not visible with an optical
telescope. A recording of the object’s apparent brightness as a function of time shows preciselyrepeating brief flashes about once a second. This object is most likely
(a) a black hole.
(b) the accretion disk around a black hole.
(c) a message from an advanced civilization.
(d) a neutron star.
(e) a white dwarf.
4. As a certain main sequence star approaches the Earth,
(a) the absolute magnitude M of this star will increase.
(b) the absolute magnitude M of this star will decrease.
(c) the apparent magnitude m of this star will increase.
(d) the apparent magnitude m of this star will decrease.
5. Hubble’s law in the form d = (1/H0 )v allows one to deduce a distance d from the speed v of a remote
galaxy. What is the meaning of the distance d?
(a) It is the distance between the galaxy and Earth right now.
(b) It is the distance between the galaxy and Earth when the light first started leaving the galaxy.
(c) It is some other distance.
6. For the last three nights leading up to today’s exam, the phase of the Moon was closest to
(a) new.
(b) waxing crescent.
(c) first quarter.
(d) full.
7. A rocket travels past you close to the speed of light. Which one of the following statements is false?
(a) Someone on the rocket observes your watch to run slowly.
(b) You measure the length of the rocket to be shorter than its length when motionless.
(c) You measure the number of atoms in the rocket to be greater than the number of atoms in the
rocket when it is motionless.
(d) You measure the mass of the rocket to be larger than its mass when sitting still.
8. A certain B-type star has a luminosity equal to 0.02 times the Sun’s luminosity. Given that the Sun is
a G-type star,
(a) the B-type star is the more massive star.
(b) our Sun is the more massive star.
(c) there is insufficient information to deduce which star is more massive.
9. If a sundial on your boat indicates that it is 1 P.M. and your universal time clock (the time in Greenwich
England) reads 3 P.M., then
(a) you are west of Greenwich (i.e., you are west of the prime meridian).
(b) you are east of Greenwich.
(c) there is not enough information to determine your longitude.
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10. The figure below shows how the sky would look on a certain day if you could see stars during the day.
At noon, the Sun appears near the stars of the constellation Gemini as shown. Circle the constellation
in which the Sun will appear near sunset.
11. The best explanation for why the spectra of extremely remote galaxies are red shifted is that
(a) the photons from stars in those galaxies undergo a gravitational red shift as the photons leave the
large gravitational field of the galaxies.
(b) the galaxies are moving away from Earth at high speed and so are Doppler shifted.
(c) the wavelengths of the photons on their way to Earth are stretched as the space between the
remote galaxy and Earth expands.
12. Applying the orbital velocity law Mr = rv 2 /G to a star orbiting 40,000 ly from the center of the Milky
Way allows us to determine
(a) the mass of the Milky Way that lies within 40,000 ly from the galactic center.
(b) the total mass of the Milky Way galaxy.
(c) the mass of the black hole that is believed to reside in the center of the galaxy.
(d) the percentage of the galaxy’s mass made of dark matter.
13. Which one of the following can not be deduced by looking at only the output of a spectroscope
connected to a telescope?
(a) the chemical composition of a star.
(b) the rotation rate of a star.
(c) the surface temperature of a star.
(d) the mass of the star.
(e) the presence of a binary companion.
14. You are in a hot dense place, You are surrounded by protons and neutrons, some rapidly fusing into
helium, You notice that your surroundings are cooling (good, because it is really hot) and the density
is rapidly dropping. Within about three minutes, the fusion reactions stop. Where are you?
(a) In the center of a star like our Sun.
(b) In the early universe during the era of nucleosynthesis.
(c) In the Harris nuclear power plant south of Raleigh.
(d) In the center of a star much smaller than our Sun.
(e) In the center of a massive star that is about to become a supernova.
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15. Astronauts inside the Space Shuttle float around as it orbits the Earth because
(a) there is no gravity in space.
(b) they are falling in the same way as the Space Shuttle.
(c) they are above the Earth’s atmosphere.
(d) there is less gravity inside the Space Shuttle.
16. An A-type main sequence star S is moving with speed v1 toward a spectrometer E on Earth. As shown
in Eq. (1) below, between the star and Earth are two clouds C1 and C2 of cold gases such that cloud C1
is motionless with respect to Earth while cloud C2 is moving toward Earth with speed v2 > v1 .
v
S →1
v2
−→
C2
C1
E
(1)
When light from a thermal emitter passes through a stationary laboratory flask of gas C 1 , an absorption
line is observed at wavelength λ1 , and similarly a stationary flask of gas C2 causes an absorption line
to appear at wavelength λ2 much larger than λ1 . If λ01 and λ02 are the wavelengths of the absorption
lines observed on Earth in the scenario Eq. (1), then we can conclude that
(a) λ01 = λ1 and λ02 = λ2 .
(b) λ01 > λ1 and λ02 > λ2 .
(c) λ01 < λ1 and λ02 > λ2 .
(d) λ01 > λ1 and λ02 < λ2 .
(e) something else is observed.
17. It is surprising to discover a binary star system in which one member is a 15MSun main-sequence star
and other member is a 10MSun red giant because
(a) it doesn’t make sense to find a red giant in a binary star system.
(b) the two stars in a binary system should be at the same point in stellar evolution so both should
be main sequence stars or both be red giants.
(c) the two stars should be the same age, so the more massive star should be the red giant.
(d) a star with mass 15MSun is too massive to be a main-sequence star.
18. Towards the end of the life of a high-mass star, the star’s core has multiple concentric shells in which
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the nuclei neon (20
10 Ne), magnesium (12 Mg), and silicon (14 Si) are fusing. In which order do these shells
occur, from innermost to outermost?
(a) Ne, Mg, Si.
(b) Si, Mg, Ne.
(c) Mg, Si, Ne.
(d) Mg, Ne, Si.
(e) some other order.
19. Suppose that we discovered that the average distance of the Earth to the Sun was unfortunately in
error and should be 1% greater than its current value. How would this affect our estimate of the
distance to the Andromeda galaxy?
(a) It would mean that the distance to the Andromeda galaxy is 1% greater than we thought.
(b) It would mean that the distance to the Andromeda galaxy is 1% smaller than we thought.
(c) It would not have any effect on our estimate of the distance to the Andromeda galaxy.
(d) It would mean that all the other standard candles can not be trusted.
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20. Which one of the following conservation laws would one use to deduce the speed of a planet at aphelion
if the speed of the planet is known at perihelion? Assume that the distances of the planet to the Sun
at aphelion and at perihelion are known.
(a) Conservation of energy.
(b) Conservation of mass.
(c) Conservation of momentum,
(d) Conservation of angular momentum,
(e) Conservation of baryon number.
21. A CCD (charge-coupled diode) is an electronic device that, when connected to a telescope, can be used
to measure directly the value of a star’s
(a) spectrum.
(b) luminosity.
(c) apparent brightness.
(d) surface temperature.
(e) magnetic field strength.
22. The best fuel for an interstellar rocket would be
(a) chemicals like hydrazine and an oxidizer like nitrogen tetroxide.
(b) uranium for use in a fission reactor.
(c) deuterium and helium-3 for use in a fusion reactor.
(d) a tank of electrons and a tank of positrons.
23. Within a few days of the winter solstice, a star with which RA will be impossible to see in the night
sky?
(a) 0h .
(b) 6h .
(c) 12h .
(d) 18h .
24. Which event marks the beginning of a massive-star supernova?
(a) The initiation of the carbon-nitrogen-oxygen (CNO) cycle of proton fusion.
(b) The rapid collapse of an iron core into a compact ball of neutrons.
(c) The onset of helium burning after a helium flash.
(d) The onset of neon burning in a shell around the core.
25. You learned in this course that a clock runs more quickly toward the front of an accelerating rocket
compared to an identical clock at the back of the rocket. (The front of the rocket is the direction of
acceleration.) If the rocket is now made substantially longer,
(a) the clock at the front will tick even more quickly compared to the clock at the back.
(b) the clock at the front will tick at a rate closer to the rate of the clock at the back.
(c) changing the length of the rocket will have no effect on how rapidly one clock ticks compared to
the other.
7
26. Constellations that are visible in the part of the sky near the horizon where the Sun is about to rise
(a) are constellations that will be near your meridian at midnight later in the year.
(b) are constellations that were near your meridian at midnight earlier in the year.
(c) are constellations that will remain visible throughout the year.
27. The gold atoms (197
79 Au) in your jewelry were created
(a) during the Big Bang.
(b) in the core of small-mass stars
(c) in the core of intermediate-mass stars
(d) in the core of high-mass stars
(e) during the supernova of a high-mass star
28. Many ancient cultures believed that Algol was an evil star because
(a) Algol was in a region of the sky where many meteor showers were observed.
(b) Algol wanders around the celestial sphere like a planet but in the wrong direction.
(c) Algol varies in brightness and in color every few days, for a few hours at a time.
(d) Algol sometimes looked like a single star, sometimes like a double star.
29. Which of the following objects in our solar system are currently in locations other than where they
initially formed?
(a) the rocky planets.
(b) the asteroids.
(c) the gas giants.
(d) the members of the Kuiper belt.
(e) the members of the Oort cloud.
30. Which of the following statements about nuclear burning in a massive star is not true?
(a) Each new stage of fusion requires a higher temperature than the previous stages.
(b) At each new stage of fusion, the core shrinks further.
(c) Each stage of fusion creates a nucleus with a greater atomic mass number.
(d) Each new stage of fusion lasts longer than the previous stages.
31. Which one of the following statements is not true regarding why a tokamak is an attractive solution
for solving the Earth’s energy crisis.
(a) The only waste products are He nuclei, neutrons, and lots of energy.
(b) A tokamak is simple to build and maintain compared to a fission reactor.
(c) The deuterium used by a tokamak is abundantly available from seawater.
(d) There is no chance for nuclear proliferation.
32. Stars A and B have parallax angles of 0.08” and 0.03” respectively as measured from Earth. Which
star would humans likely first visit upon developing an interstellar rocket?
(a) Star A.
(b) Star B.
(c) Each would be equally likely to visit.
8
33. On a certain night at midnight, you see the Moon and a certain star on your meridian. Then the
following night at midnight, you will see that
(a) the Moon and star are still on your meridian.
(b) the Moon and star have already crossed your meridian.
(c) the Moon and star have not yet crossed your meridian.
(d) the Moon has crossed your meridian while the star has not yet crossed your meridian.
(e) the Moon has not yet crossed your meridian while the star has already crossed your meridian,
34. For an astronaut in orbit around the Earth during a space walk, which of the following actions would
increase the tidal force acting on the astronaut?
(a) Move the astronaut further away from the Earth.
(b) Turn the Earth into a black hole of identical mass (without changing the distance to the astronaut).
(c) Decrease the mass of the astronaut (without changing the size of the astronaut).
(d) Increase the mass of the Earth.
35. Why is the Hyades open cluster important for building up a catalog of accurate luminosities of mainsequence stars?
(a) The Hyades has many bright stars that are easy to observe.
(b) The Hyades is close enough to measure the distance of each star accurately via parallax.
(c) The Hyades contains many Cepheid variable stars.
(d) Over many decades, the apparent brightnesses of most of the Hyades stars have been measured.
36. What is the primary practical difficulty that limits the use of Hubble’s law for measuring distances?
(a) Redshifts of galaxies are hard to measure.
(b) The recession velocities of galaxies are so large that they are hard to measure.
(c) We do not know Hubble’s constant H0 accurately.
(d) We do not know how to take into account the motion of the Earth relative to the Milky Way.
37. Since the deuterium created in the cores of stars is completely consumed by nuclear reactions, the
deuterium in seawater and elsewhere are primordial nuclei left over from the Big Bang. Measuring the
amount of deuterium in the universe is important because it allows us to estimate
(a) the temperature of the universe at the end of the age of nuclei.
(b) the amount of ordinary matter made of protons and neutrons.
(c) the expansion rate of the universe.
(d) the age of the universe.
38. Senator Hillary Clinton believes that the Hubble constant H0 has the value 70 (km/s)/Mpc while
President Bush believes its value to be 50 (km/s)/Mpc. Which statement therefore follows?
(a) The Senator believes that the universe is expanding while the President does not.
(b) The Senator believes that the Andromeda galaxy (a member of our Local Group) is moving away
faster than the President believes.
(c) The Senator believes that the universe is expanding more rapidly than what the President believes.
(d) The Senator believes that the universe is expanding less rapidly than what the President believes.
(e) The Senator believes that the universe will someday stop expanding while the President believes
that the universe will expand forever.
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Open Questions
For problems that require numerical answers, give your answer to one significant unit unless otherwise
indicated.
You must justify your answers to get credit.
If I can not easily read and understand your answer, you will lose credit.
If you don’t have enough room for your answers, you can use additional blank pages to write down your
answers. Please make sure to write your name on all additional pages and also write down clearly the
problem number.
1. Next to each of the following characteristic of a star, indicate with the letters “i”, “d”, or “s” whether
the characteristic respectively increases, d ecreases, or stays the same after the hydrogen fuel in the
core runs out, the core is contracting, and hydrogen shell burning begins. (No explanation is needed
for this problem.)
(a) (2 points) The star’s radius.
(b) (2 points) The surface temperature of the star.
(c) (2 points) The star’s luminosity.
(d) (2 points) The temperature at the center of the star.
(e) (2 points) The peak wavelength of light emitted from the surface of the star.
2. (6 points) A certain star crossing the meridian in the local sky of Stockholm, Sweden, has the smallest
possible declination that a star can have and still be visible from Stockholm. What are the altitude
and direction of this same star as it crosses the meridian of the local sky in Durham?
Note: The latitudes of Stockholm and Durham are respectively 59◦ N and 36◦ N.
10
3. (18 points) After graduating from Duke, you become involved with a science outreach program for
Durham middle schools and you organize a green-laser tour of the Durham night sky. In the space
below:
(a) mention three of your favorite celestial objects that you would point out to these students
(b) for each object, indicate with appropriate sketches and some explanation how your students can
locate these objects with respect to some easily identified constellations. You need to include a
least two or more constellations in your sketches, to give a sense of scale and also to make it easier
to locate a region of the celestial sphere.
(c) most importantly, explain why each of your objects is interesting from a scientific point of view.
Your list of objects can not include the Moon, the stars Polaris or Algol, or any of the planets. Do not
choose constellations unless there is a scientific insight they convey, e.g., do not choose the Big Dipper
just because it is easily identified and because it stars can be used to find neighboring asterisms. Do
not mention cultural facts such as what the Greeks or Mayans thought of your objects.
For example, if you were discussing the Moon, you could scientific facts such as: the Moon was believed
to have formed during an immense collision of Earth with a Mars-like object; Earth is the only rocky
planet with a sizeable Moon; the Moon’s craters reflect the heavy bombardment of all solar system
masses by planetesimals during the formation of the solar system; our Moon is actually one of the
biggest Moons in the solar system; the Moon does not have a magnetic field and so is not protected by
the solar wind; the Moon does not have an atmosphere because its surface is too hot and its gravity
is too weak to hold onto most gases; tidal forces have caused the Moon to become synchronized with
the Earth such that the Moon rotates once per orbit period, so the same side always faces the Earth.
Inappropriate facts would be that North Americans see a face in the Moon while Europeans see a
rabbit, or that many cultures use the Moon as the basis of their calendars.
Your grade for this problem will depend especially on your ability to impress middle school children
with the riches of the night sky.
11
12
4. (6 points) Describe briefly the two key pieces of evidence for the Big Bang and what experimental
equipment was needed to obtain this evidence.
5. (5 points) Supernova remnant Simeis 147 is known to be 3,000 ly away and has a diameter of about
150 ly. What is the angular width of this supernova remnant in the sky, in multiples of your little
finger held at arm’s length?
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6. An interstellar rocket leaves the Earth with constant speed 0.8c to travel to a star 200 ly away.
(a) (3 points) How long in years does it take the rocket to reach the star as measured by someone
on Earth?
(b) (3 points) How long in years does it take the rocket to reach the star as measured by someone
on the rocket?
(c) (3 points) What is the distance traveled (in light-years) according to someone on the rocket?
(d) (3 points) What is the speed of the star toward the rocket according to someone on the rocket?
Express your answer as fraction of the speed of light and discuss briefly whether your answer
makes sense.
14
7. (5 points) About five billion years from now, the Sun will evolve into a red giant with a radius of
about 100 RSun ≈ 7 × 107 km (just beyond Mercury’s orbit) and with a surface temperature of about
3000 K. Ignoring any possible greenhouse effect and assuming the reflectivity of the Earth continues
to have the value 0.36, what will be the new steady state surface temperature of the Earth? Will the
oceans still remain in liquid form?
8. (5 points) By applying Einstein’s principle of equivalence to a rocket that is accelerating with a
constant acceleration and to the same rocket sitting still on Earth’s surface, explain why objects of
greatly unequal mass that are dropped in a vacuum at the same time on Earth will hit the ground at
the same time.
15
9. (6 points) Using at least a few sentences for each, explain briefly two ways that Einstein’s theory of
general relativity has been confirmed experimentally.
10. (6 points) Using at least a few sentences for each, explain briefly two examples of how general
relativity has proved to be essential and invaluable to astronomers, to understand what would otherwise
be incomprehensible observations.
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11. Draw a schematic figure illustrating Hubble’s law. You should give the variables associated with the
horizontal and vertical axes (2 points), their physical units (2 points), the numerical ranges of the
horizontal and vertical axes (4 points), and indicate the approximate functional dependence showing
how the vertical variable varies with the horizontal variable (1 point).
12. (5 points) Explain briefly how Hubble obtained the values for the data points that led to his great
discovery.
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13. (5 points) Assume that Hubble had instead carried out his measurements for a static infinite universe
of the sort that Isaac Newton and many others believed in before the discovery of the Big Bang. In
such a universe, galaxies everywhere move about randomly like atoms in a gas, with relative speeds
that are much smaller than c, and the size and overall properties of the universe do not change over
time.
Using the same variables and units as Hubble’s original plot, draw schematically how Hubble’s plot
would appear for a steady-state universe.
14. (5 points) Finally, assume that Hubble had carried out his measurements on a collapsing universe.
This would correspond to a raisin cake (or balloon with buttons on it) that is shrinking over time.
Assuming a steady rate of contraction and using the same variables and units as Hubble’s original plot,
draw schematically how Hubble’s plot would appear for a contracting universe.
18