Astronomy 241: Review Questions #3
Distributed: December 5, 2013
Review the questions below, and be prepared to discuss them in class. For each question, list
(a) the general topic, and (b) the key laws and equations needed to find an answer; bring this
list to class on Dec. 10. Modified versions of some of these questions will be used in the final
exam on Dec. 19.
1. An electron at rest is struck by an X-ray photon. The photon has energy Eγ = 1 KeV !
1.60 × 10−16 kg m2 s−2 and momentum pγ = Eγ /c, where c = 3 × 108 m s−1 is the speed of
light. Use conservation of energy and momentum to show that the electron cannot simply
absorb the photon. (Hint: you may safely assume the electron is non-relativistic, and use
classical expressions for its kinetic energy and momentum.)
2. At a distance of r = 1 AU from the Sun, the solar wind has a typical velocity v !
400 km s−1 and a typical density ρ ! 1.2 × 10−20 kg m−3 .
(a) Estimate the rate Ṁ at which the Sun is losing mass due to the solar wind.
(b) How long would it take the Sun to vanish if it continued losing mass at this rate?
(c) Estimate the amount of energy per unit time required to sustain this wind. This is the
Sun’s “wind luminosity”, Lwind . Compare your result to the Sun’s “photon luminosity”,
L" ! 3.8 × 1026 kg m2 s−3 .
3. Assume that a large solar flare occurs in a region where the magnetic field strength is
B ! 0.03 kg s−1 C−1 , and that it releases 1025 kg m2 s−2 .
(a) What was the magnetic energy density in the region before the eruption began?
(b) What is the minimum volume required to supply the magnetic energy needed to power
the flare? Assuming for simplicity the volume supplying this energy is a sphere, calculate
its radius.
(c) Assuming a coronal density ρ ! mp · 1015 m−3 (eg, Fletcher, L. & Hudson, H.S. 2008,
Ap.J. 675, 1645), how long would it take an Alfven wave to travel across this spherical
volume?
4. The solar corona is in approximate hydrostatic equilibrium out to a radius of several R" .
Over this range, the gravitational acceleration g(r) varies significantly, so the simple scaleheight calculation we used for the photosphere and chromosphere is not valid, and another
solution is needed. For simplicity, assume the the corona is isothermal (T ! 1.0 × 106 K)
and composed of completely ionized hydrogen (µ ! 0.5) throughout.
(a) Use hydrostatic equilibrium and the ideal gas law to set up a differential equation for
the pressure, P (r).
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(b) Show that the solution to this equation is
P (r) = P∗ exp
!
GM" µmH
kB T r
"
(1)
where P∗ is a parameter with units of pressure.
5. At radius r0 = 1.4R" ! 9.8 × 108 m the corona’s temperature is T ! 1.0 × 106 K. For
simplicity, assume the corona is composed of ionized hydrogen, with µ = 0.5.
(a) Directly evaluate the scale height
H=
kT
µmH g0
(2)
where g0 is the Sun’s gravitational acceleration at radius r0 .
(b) The hydrostatic isothermal solution yields
P (r) = P∗ exp
!
GM" µmH
kB T r
"
(3)
where (GM" µmH )/(kB T ) ! 8 × 109 m. Evaluate the scale height using the relation
1 dP
1
=
.
H
P dr
(4)
Compare this result with the one from (a).
6. In class I compared the rms thermal velocity in the solar corona with the escape velocity, and argued that gas must be constantly escaping from the upper layers of the corona.
Another way to see this is to apply the same rule we previously1 used to determine if a
planet’s atmosphere will escape: vrms > 16 vesc . Use this criterion to determine the “escape
temperature” (ie, the minimum temperature at which vrms > 16 vesc ) as a function of radius
r (measured from the center of the Sun). Compare this temperature to the characteristic
temperature of the solar corona, Tcorona ! 1.0 × 106 K.
7. Earth’s magnetosphere extends to the radius where the solar wind’s energy density
uw = ρv 2 /2 is approximately equal to the energy density um = B 2 /2µ0 of Earth’s magnetic
field. This radius is about 10RE . In contrast, Jupiter’s magnetosphere is much larger; if it
could be seen from Earth, it would appear almost 0.5◦ in diameter.
(a) Assuming a typical Earth-Jupiter distance of 5 AU, what is the radius of Jupiter’s magnetosphere?
(b) How large would the Earth’s magnetosphere be if the Earth was 5 AU from the Sun?
(c) Jupiter’s magnetic field is ∼ 14 times stronger than Earth’s. Does this explain the radius
of Jupiter’s magnetopause?
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Back in September!
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8. If protons were unable to “tunnel” through the Coulomb barrier, the Sun’s internal
temperature would have to reach Tc ! 1.1 × 1010 K before the p + p → 2 H + e+ + νe reaction
could get going. Estimate the radius the Sun would contract to in order to reach this
temperature.
9. Kepler’s laws were deduced empirically from a detailed analysis of planetary positions.
Subsequently, these laws were derived from Newton’s laws of motion and gravity. In the
process, Kepler’s laws were generalized to include (i) bodies of comparable masses m1 , m2 ,
and (ii) unbound orbital motion.
(a) State Kepler’s three laws for the motion of planets around the Sun in their original
empirical forms.
(b) State the Newtonian generalizations of Kepler’s three laws.
10. A satellite of mass m is on a circular orbit of radius r around a planet of mass M % m.
Suppose we increase the satellite’s speed by a factor of α ≥ 1 (ie, $vfinal = α$vinitial ).
(a) Compute the major axis, minor axis, pericenter distance, and apocenter distance of the
new orbit.
√
(b) Note that that rapo → ∞ as α → 2. Explain this result by applying the virial theorem
(2K + U = 0) to the initial orbit.
11. Callisto rotates slowly and has albedo A ! 0.2.
(a) What is Callisto’s surface temperature at the subsolar point (that is, the point on Callisto
directly facing the Sun)?
(b) At this temperature, could Callisto retain an atmosphere of He? Of N2 ? Of Ar?
12. A rocky object orbits the Sun on a highly elliptical orbit with an apocentric distance of
many tens of AU. On one of its passages through the inner solar system, it hits the Moon.
(a) Given that the Earth orbits the Sun at ∼ 30 km s−1 , and the Moon orbits the Earth at
∼ 1 km s−1 , what is the maximum impact velocity the object can have?
(b) Suppose the impact forms a crater 90 km in diameter and 5 km deep. What is the mass
of the rock ejected?
(c) Suppose the object producing this crater is 1 km in radius. What is the average velocity
of the ejected material? (Hint: use a kinetic energy argument).
(d) Compare this velocity to the Moon’s escape velocity. Is it likely that a substantial
fraction of the ejected material can escape the Moon’s gravity?
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13. A number of planetary systems contain “hot Jupiters” – Jovian planets which orbit
their parent stars at distances r < 0.1 AU.
(a) How close could Saturn orbit the Sun without being tidally disrupted?
(b) How close could Saturn orbit the Sun and retain its A ring (neglecting solar heating)?
(c) Allowing for the effects of solar heating, what would actually happen to Saturn’s rings?
14. Describe the internal heating mechanisms in terrestrial planets, giant planets, and
(some) moons of giant planets. How is energy transported inside these objects?
15. Saturn’s rings orbit in a plane aligned with the planet’s equator. One of Saturn’s
moons, Mimas, orbits at a distance rmi ! 1.85 × 108 m in a plane which is slightly inclined
with respect to the rings.
(a) Mimas attracts particles in the rings, creating a variety of features visible on close-up
images. These features, however, are not seen throughout the rings; instead, they only
appear at certain definite radii. Why?
(b) One such feature appears in the rings at a radius of ∼ 0.731rmi . How would this feature
be labeled?
(c) On even closer inspection, this feature has two parts. One part is created by particles
moving slightly above and below the ring plane, while the other is created by particles moving
radially while staying in the ring plane. Moreover, the two parts appear at slightly different
radii. Why?
16. Consider a comet approaching the Sun on an orbit with rapo = 15 AU and rperi = 0.5 AU.
(a) Water ice sublimates rapidly at T ≥ 190 K. If the comet’s nucleus has an albedo of
A = 0.1, how far from the Sun does the comet become active?
(b) At pericenter, solar heating is almost entirely balanced by sublimation of H2 O. If Q
is the outgassing rate (units are molecules s−1 ), the cooling term is Hs Q/NA , where Hs !
5 × 104 kg m2 s−2 mole−1 is the latent heat of sublimation and NA = 6.02 × 1023 is Avogadro’s
number. If the nucleus is ∼ 10 km in diameter, what is Q?
(c) The comet’s ion tail is swept away from the Sun by the solar wind, which has a typical
velocity of 400 km s−1 . Observations show that the ion tail does not point exactly outward
from the Sun; as a result of the comet’s motion, it makes an angle of φ with respect to a line
drawn outward from the Sun through the nucleus of the comet. Find φ.
17. The late and lamented Comet ISON reached a pericenter distance of rperi ! 0.0124 AU !
2.66 R" (measured from the center of the Sun) and promptly disintegrated.
(a) At this distance, water ice sublimates rapidly. Consider the fate of a icy nucleus with
a radius of r = 1 km. The cooling term is Hs Q/NA , where Hs ! 5 × 104 kg m2 s−2 mole−1
is the latent sublimation heat of ice, NA = 6.02 × 1023 is Avogadro’s number, and Q is the
outgassing rate (in molecules s−1 ). Estimate (i) the rate of sublimation (in kg s−1 ) and (ii)
the time required to sublimate the entier nucleus.
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(b) It’s plausible that ISON also contained some rocky material. Estimate the black-body
equilibrium temperature of a chunk of rock attains at ISON’s minimum distance rperi . Compare your result to the vaporization temperatures of typical rocks; what do you expect will
happen to rocky fragments?
18. Briefly describe the two main types of planets found in our solar system, and describe
one of the two current explanations for this planetary dichotomy.
19. The Sun is powered by a sequence of nuclear reactions which convert hydrogen to
helium.
(a) Write down the four steps of the PP I chain.
(b) Which step is the slowest, and why is it so slow?
(c) What is the minimum temperature required for the PP I chain?
(d) Why is the reaction rate so sensitive to temperature?
(e) How did the Sun get hot enough for this reaction to begin?
20. A typical model of the Sun is based on four key differential equations.
(a) Write down the left-hand sides of these equations, and briefly describe the four functions
they define.
(b) Three additional equations must be specified to complete the solar model. What physical
quantities do these relations specify?
(c) Two of the equations in part (a) also appear in models of giant planets such as Jupiter.
Write down these equations in full.
21. Explain how the Sun maintains a constant rate of nuclear energy release. What happens
if the Sun’s core begins to produce too much energy? What would happen to the core
temperature Tc if nuclear reactions ceased?
22. Consider a highly simplified model of the greenhouse effect, in which the greenhouse
gas forms a thin layer which is completely transparent to solar radiation, and completely
opaque to the infrared radiation emitted by the planet’s surface. Assume that the top and
bottom surface areas of the layer are both equal to the planet’s surface area.
(a) Let F be the flux of solar radiation, Rp be the planet’s radius, and a the planet’s albedo.
What is the planet’s heat input Wabs (units: kg m2 s−3 = W) due to solar radiation?
(b) Assume that the greenhouse gas layer radiates like a blackbody with temperature Tg .
What is Tg ?
(c) Finally, show that the surface temperature of the planet is Tp = 21/4 Tg .
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23. The greenhouse skeptic argues “The ‘greenhouse gasses’ in the Earth’s atmosphere are
cooler than the surface of the Earth. The Second Law of Thermodynamics says heat can
never flow from cold to hot. Thus it’s impossible for greenhouse gasses to warm the Earth.”
Your response?
24. On 10 August, 1972, a meteor with an estimated mass of m ! 105 kg skimmed the
Earth’s atmosphere but did not hit the Earth. At closest approach, it was ∼ 58 km above
sea level. Its orbit is shown in Fig. 1. Ignore atmospheric drag in answering parts (a), (b),
and (c).
(a) Eyewitness reports, movies, and sensor data indicate that the meteor’s speed at closest
approach was vp ! 15 km s−1 . Assuming the meteor had the mass m given above, what was
its orbital energy E and angular momentum L?
(b) What was its initial velocity v0 before it was accelerated by Earth’s gravity?
(c) What was the impact parameter b of its initial orbit?
(d) Atmospheric drag slowed the meteor by ∆v ! 0.8 km s−1 , but it was still moving too fast
to be captured by Earth’s gravity. What ∆v would be required for capture?
(e) If the meteor had been captured into an orbit around the Earth, what would its next
encounter with the Earth have been like?
25. Conservation of angular momentum can be used to test the idea that the Pluto-Charon
system formed as a result of a collision between two “dwarf planet” progenitors. We know
that Pluto and Charon are on circular orbits about their mutual center of mass, with an
orbital period P = 5.52 × 105 s and semi-major axis a = 1.94 × 107 m. We also know
that Pluto has mass MP = 1.3 × 1022 kg and radius RP = 1.15 × 106 m, Charon has mass
MC = 1.4 × 1021 kg and radius RC = 6.0 × 105 m, and that both objects rotate once per
orbital period.
(a) Calculate the orbital angular momentum LP and LC and rotational angular momentum
SP and SC of Pluto and Charon. You may assume both objects are homogeneous spheres,
with moments of inertia I = 25 MR2 . List these four angular momenta in order from largest
Figure 1: Orbit of the Great Daylight Fireball of 1972. Before being accelerated by the
Earth’s gravity, its initial velocity was v0 ; if it had continued in a straight line, it would have
passed the Earth at minimum distance b.
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to smallest, and sum them to get the total angular momentum JPC of the Pluto-Charon
system.
(b) Assume that the two progenitors which collided to form Pluto and Charon were identical,
with masses M = 12 (MP + MC ) and internal densities ρ = 2000 kg m−3 . Calculate the radii
R of these objects.
(c) Now assume the two progenitors fell together on a parabolic orbit (total orbital energy
E = 0), and just grazed each other at closest approach (rperi = 2R). Calculate the velocity
vperi of each object with respect to their mutual center of mass, and use the result to calculate
the total angular momentum Lorb of their pre-encounter orbit.
(d) Compare the current angular momentum JPC of the Pluto-Charon system to the orbital
angular momentum Lorb of the progenitors. Does it seem plausible that the Pluto-Charon
system could have formed in this way? (Note: tidal interactions insure that even a grazing
encounter like the one considered here will result in the progenitors merging quite rapidly.)
26. The concepts of mean free path & and cross section σ can be extended to include
not just simple physical collisions but other kinds of interactions between particles as well.
In the center of the Sun, a proton has several different mean free path lengths. For this
problem, assume the Sun’s central density and temperature are ρc ! 1.5 × 105 kg m−3 and
Tc ! 1.6 × 107 K, and assume the Sun is composed of pure hydrogen.
(a) The radius of a proton is Rp ! 10−15 m. Two protons physically collide with each other
when the distance between them is rgeo = 2Rp . Assume that protons move on straight lines,
(ie, neglect electromagnetic forces); what are the cross section σgeo and mean free path &geo
for physical (ie, geometric) collisions?
(b) In the Sun, electromagnetic forces deflect protons long before they can approach to within
rgeo . The electromagnetic interaction energy for a pair of protons is
Uem (r) =
1 e2
,
4π)0 r
(5)
where e ! 1.60 × 10−19 C is the elementary charge and )0 ! 8.9 × 10−12 kg−1 m−3 s2 C2 is the
permittivity of free space. Assuming protons have average energy 32 kTc , find the distance
rem at which two protons are significantly deflected by electromagnetic forces.
(c) Using your result for part (b), how far does a typical proton in the Sun’s core travel
between significant deflections? This is the mean free path &em for electromagnetic encounters
between protons. What is the corresponding cross section σem ?
(d) Finally, a typical proton in the Sun’s core survives for tnuc ! 3 × 1017 s before becoming
involved in a nuclear reaction. Assuming the proton moves at the thermal velocity, how far
does it travel in this time? This is the mean free path &nuc for nuclear reactions between
protons. What is the corresponding cross section σnuc ?
27. Inside the Sun, energy is transported outward by two key mechanisms, radiation and
convection. Radiative transport implies that the temperature gradient is
dT
L
3 κρ
,
=−
3
dr
16 σSB T 4πr 2
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(6)
where T is temperature, r is radius, κ is opacity, ρ is density, σSB = 5.67 × 10−8 kg s−3 K−4 is
the Stefan-Boltzmann constant, and L is the Sun’s luminosity. Convective transport implies
that the temperature gradient is
#
dT
1
= 1−
dr
γ
$
T dP
,
P dr
(7)
where γ = 53 is the ratio of specific heats, and P is pressure.
(a) At radius R1 = 0.714R" , where the Sun’s radiative zone ends and its convective zone
begins, these two expressions yield (almost) identical values for the temperature gradient,
dT /dr. Why?
(b) The total mass within radius R1 is M1 = 0.976M" . (That’s right: nearly 98% of the
Sun’s mass lies within the radiative zone!) Evaluate the gravitational acceleration g1 at this
radius.
(c) At radius R1 , the temperature is T1 = 2.18 × 106 K and the mean molecular weight is
µ = 0.60. Evaluate the pressure scale height
H1 =
kT1
,
µmH g1
(8)
where k = 1.38 × 10−23 kg m2 s−2 K−1 is Boltzmann’s constant. How does the scale H1
compare to the Sun’s radius?
(d) At radii near R1 , the pressure is
P (r) ! P1 e(R1 −r)/H1 ,
(9)
where P1 = 5.59 × 1012 kg m−1 s−2 is the pressure at R1 . Use Eq. (7) above to evaluate the
convective temperature gradient, (dT /dr)con .
(e) At radius R1 , the density is ρ = 187 kg m−3 and the opacity is κ = 2.0 kg−1 m2 . Use
Eq. (6) above to evaluate the radiative temperature gradient, (dT /dr)rad.
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