PHYS 175 Final Exam April 21, 2021
1. Neutron Star - Gravitational Binding
Consider a neutron star of mass M and radius R. IN THIS PROBLEM YOU WILL IG-
NORE ALL NUMERICAL FACTORS. All equal signs should be replaced by “∼” mean-
ing “is of the order of magnitude of.” Use the following symbols: mp ≡ mass of proton
(which is the same as the mass of a neutron for the purposes of this problem), ρnucl ≡
nuclear mass density, h ≡ Planck’s constant, G ≡ Newton’s gravitational constant, and Nn ≡
number of neutrons in the nuetron star.
(a) 3 points In one or two sentences, concisely explain why atomic nuclei heavier than about
Uranium cannot be bound by the strong nuclear force.
(b) 2 points According to the de Broglie relation, what is the minimum momentum of neu-
trons in a neutron star? Express your answer in terms of h and R. In one short sentence,
concisely explain why this is the minimum.
(c) 5 points When we applied the Pauli Exclusion Principle to a white dwarf we considered
first the one-dimensional case and indicated the allowed momenta of the electrons as points
on a one-dimensional px axis, which allowed us to determine pmax. We then extended this
argument to the three-dimensional case by considering a three-dimensional “lattice” of
allowed momenta in px, py, pz space. Draw a similar “lattice diagram” to indicate the
allowed values of neutron momentum in px, py, pz space. According to the Pauli exclusion
principle, by what factor is the maximum momentum of the neutrons greater than the
minimum momentum? Briefly explain the origin of this factor, using your diagram to
support your explanation.
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PHYS 175 Final Exam April 21, 2021
(d) 2 points What is the maximum velocity of the neutrons? Express your answer in terms
of h, M , R, and mp.
(e) 2 points What is the escape velocity of neutrons from the surface of the neutron star?
How should this escape velocity be related to the maximum velocity of the neutrons you
calculated in the previous part, in order that the neutrons be gravitationally bound?
(f) 4 points Using your result from the previous part, show a derivation for the minimum
neutron star mass required for the neutrons to be gravitationally bound. Express your
answer in terms of mp, ρnucl, h, and G.
(g) 3 points What is the fundamental difference between the strong nuclear force and the
gravitational force that allows a neutron star (a “giant atomic nucleus,” many orders of
magnitude larger than a Uranium nucleus) to exist? Based on your result in the previous
part, if G were larger, how would this affect the minimum neutron star mass? In one
sentence, give a concise intuitive reason why.
END OF QUESTION 1
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PHYS 175 Final Exam April 21, 2021
2. Neutron Star - Magnetic Field
Maxwell’s classical theory of electromagnetism is linear, meaning light does not interact with
light. However, in Quantum Electrodynamics (QED), photons can interact with each other,
for example, via the Feynman diagram shown below. This diagram describes two photons
interacting via the intermediate creation and annihilation of virtual electron-positron pairs.
(The solid lines with arrows indicate electrons; reversing an arrow turns it into a positron.)
(a) 2 points In order for such photon-photon interactions to occur, the photons must have
an energy at least sufficient to create an electron-positron pair at rest. What is this mini-
mum required photon energy, Eγ , and the corresponding photon frequency, f? (Symbolic
answers, not numerical.)
(b) 3 points Suppose we have a uniform magnetic field of strength B in the z-direction.
If an electron is given an initial speed v in the xy-plane, calculate the frequency, f , of
its subsequent circular motion. Use non-relativistic physics and express your answer in
terms of e (the magnitude of the charge on the electron), m (the mass of the electron),
and B, and any numerical factors that arise. Note that an electron in circular motion is
accelerating, and so will emit photons (at the same frequency as the circular motion).
(c) 3 points Combine the previous two parts to determine the minimum strength of magnetic
field, B, required for electrons (or positrons) moving in this magnetic field to emit virtual
photons with sufficient energy to interact with each other. Let’s call this critical value of the
strength of the magnetic field BQED. Work out BQED both symbolically and numerically
(in Tesla).
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PHYS 175 Final Exam April 21, 2021
(d) 3 points Convert the numerical value of the magnetic field strength, BQED, that you
calculated in the previous part, into an equivalent mass density (a numerical answer in
kg m−3) and compare this with the mass density of lead (11, 400 kg m−3).
(e) 4 points Suppose we start with an ordinary star with radius Rstar and surface magnetic
field strength ∼ 1 Gauss (convert this to Tesla). Using the argument discussed in the notes
and lectures, what would Rstar need to be in order that, if it collapsed to a neutron star
of radius ∼ 13 km, its surface magnetic field strength would be equal to BQED? Express
Rstar in terms of the radius of the Sun, R. The largest stars in the universe have a radius
of about 1800R. Is it plausible that there exist neutron stars with a surface magnetic
field strength on the order of BQED or larger?
END OF QUESTION 2
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PHYS 175 Final Exam April 21, 2021
3. Gravitational Lensing
In the diagram below, point A is the background object (galaxy or quasar) being lensed, point
B is the Earth, and point C is the foreground lensing object (galaxy or cluster of galaxies) of
mass M .
(a) 5 points Using Einstein’s deflection of light formula discussed in the notes and the video
lectures, derive a formula for the mass of the foreground lensing object in terms of α, d,
D, G, and c. Use the small angle approximation and show all steps in your derivation.
(b) 4 points The image below shows the first example of gravitational lensing ever de-
tected. It shows two images (the two bright spots with diffraction spikes) of a single
background object (a quasar), separated by about 6 arcseconds. The quasar is about
8.7 billion light years from Earth. The foreground lensing object is a cluster of galaxies
about 3.7 billion light years from Earth. (The fuzzy blob with a bright center located
approximately on the line joining the two images of the quasar, closer to the bottom-right
quasar image, is one of the elliptical galaxies that is part of the cluster of galaxies). Make
the simplifying assumption that the center of mass of the galaxy cluster is located exactly
at the midpoint of the line joining the two quasar images. Using this simplifying assump-
tion, estimate the mass of the galaxy cluster and express your answer in terms of solar
masses.
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PHYS 175 Final Exam April 21, 2021
END OF QUESTION 3
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PHYS 175 Final Exam April 21, 2021
4. Sirius A and B
Sirius A is the brightest star in the night sky. It is a main sequence star with mass 2.02 M.
Sirius B is a white dwarf companion to Sirius A. The semimajor axis and period of this binary
star system is 19.5 AU and 50.0 years, respectively. The distance to the system, from the Earth,
is 8.60 light years, and the flux of light from Sirius B as seen on Earth is 1.20× 10−10 W m−2.
(a) 4 points Use Newton’s generalization of Kepler’s third law to estimate the mass of Sir-
ius B (express your answer in terms of M).
(b) 4 points Sirius B has a very thin, dense, and hot hydrogen atmosphere. The highly
ionized atmosphere emits a wide range of light, including light corresponding to the Hα
spectral line of hydrogen. In the lab, the Hα spectral line has a wavelength of 656.28 nm.
The observed wavelength of this light from Sirius B is 656.44 nm. Use the gravitational
time dilation formula to estimate the radius of Sirius B (express your answer in terms of
R⊕, the radius of the Earth).
(c) 2 points From the information given in the problem introduction, estimate the luminosity
of Sirius B (express your answer in terms of L).
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PHYS 175 Final Exam April 21, 2021
(d) 2 points Estimate the surface temperature of Sirius B.
(e) 2 points Estimate the wavelength of the peak intensity thermal radiation emitted by
Sirius B (express your answer in nm). Is this infrared, visible, or ultraviolet radiation?
END OF QUESTION 4
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PHYS 175 Final Exam April 21, 2021
5. Dark Energy and Inflation
(a) 1 point Consider a gas with pressure P inside a cylinder of volume V and cross-sectional
area A, with a moveable piston at one end. Let the gas expand by moving the piston
through a distance ∆x. Derive an expression for the work done by the gas. Express your
answer in terms of ∆V , the corresponding increase in volume occupied by the gas.
(b) 1 point Considering part (a), what is the change in the amount of energy stored in the
gas during this expansion? (Assume no thermal energy enters or leaves the gas during its
expansion.)
(c) 3 points If Dark Energy is a kind of “gas” with a mass density ρ (energy density ρc2)
that is uniform in space and constant in time, what is the change in the amount of energy
stored in the gas during this expansion?
(d) 1 point Combine your results from parts (b) and (c): how is the pressure of Dark Energy
related to its mass density?
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PHYS 175 Final Exam April 21, 2021
(e) 2 points Write down the general relativistic equation for cosmic acceleration derived in
class (a¨/a = . . .), where a = a(t) is the cosmological scale factor. Taking ρ and P to refer
to Dark Energy, and using your result from part (d), express a¨/a in terms of ρ.
(f) 3 points What is the general solution to an equation of the form a¨/a = T−2, where T is
a constant? Differentiate your general solution to verify that this is correct.
(g) 1 point During the present epoch, the mass density of Dark Energy is equivalent to about
5 protons m−3. What is the corresponding time scale, T , expressed in billions of years?
(h) 1 point During the inflation epoch, the mass density of the inflaton vacuum was believed
to be about 1080 kg m−3. What is the corresponding time scale, T , expressed in seconds?
(i) 3 points If inflation happened in an interval of time equal to 100T , by what factor would
the scale factor of space have expanded? What is the time interval, 100T in seconds?
END OF QUESTION 5
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PHYS 175 Final Exam April 21, 2021
6. Solar System Orbits, Galactic Orbits, and Dark Matter
(a) 4 points In a spiral galaxy (seen edge-on in the picture below), most of the ordinary
matter is in the central “bulge,” just as most of the matter in the Solar System is in the
Sun. Thus, we might expect that the stars, gas, and dust in such a galaxy orbit this
central bulge much like planets orbit the Sun.
Assuming the mass of the central bulge is M , use Newtonian gravity to determine the
speed of a star, v(r), in its circular orbit at radius r. A graph of this result for planets
orbiting the Sun is shown below. If the mass of the Sun is kept the same, but the orbital
radius of the planet is increased by a factor of 4, by what factor should the orbital speed
of the planet be reduced? If the orbital radius of the planet is kept the same, but the mass
of the Sun is increased by a factor of 4, by what factor is the orbital speed of the planet
increased?
(b) 5 points When Vera Rubin measured the speeds of stars (and blobs of gas and dust)
orbiting the central bulge of the Andromeda galaxy, she obtained a graph like that shown
below (curve B), a result that is qualitatively completely different from the expected graph
(curve A). Arguably, the most natural way to interpret this huge discrepancy is to imagine
that the galaxy is immersed in an unseen form of matter called Dark Matter. If we assume,
for simplicity, that this Dark Matter is spherically symmetrically distributed about the
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PHYS 175 Final Exam April 21, 2021
center of the galaxy, then the orbital speed of a star (or blob of gas or dust) would be
determined by the gravitational force it experiences from the total matter contained within
the sphere of space bounded by the object’s orbit (think of the object’s orbit as the equator
of this sphere).
For example, in the Triangulum galaxy, the most distant stars (or clouds of gas or
dust) have an orbital radius of about 40, 000 light years, and an orbital speed of about
120 km s−1. Estimate the total amount of mass (in ordinary matter plus Dark Matter)
contained in a sphere of radius 40, 000 light years centered on the galaxy. Express your
answer in terms of M. The total mass of ordinary matter in the Triangulum galaxy,
within this radius, has been measured to be about 7 billion Suns. What is the ratio of
the mass of Dark Matter to the mass of ordinary matter in this region of the Triangulum
galaxy?
END OF QUESTION 6
END OF FINAL
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