Download Astronomy 241: Problem Set #3 due September 11, 2012 Solve the

Astronomy 241: Problem Set #3
due September 11, 2012
Solve the problems listed below, and write up your answers clearly and completely. Do not
turn in rough work – instead, make a clean copy after checking your calculations. Use English
sentences and phrases to explain your solution and describe key equations. Show your work!
1. Introductory astronomy textbooks claim that the Moon always shows the same face to the
Earth. This is not quite true; the Moon exhibits a rocking motion known as as “libration”
(see http://www.ifa.hawaii.edu/∼barnes/ast241/moon.mpg). Libration has two main
components. The first component arises because the Moon’s rotation axis is not exactly
parallel to its orbital axis; as it rotates and revolves around the Earth, its north and south
poles are alternately exposed to our view (hence the up-and-down motion seen in the video).
The second component of libration arises because the Moon’s orbit is elliptical, not circular
(this produces the left-and-right motion seen in the video).
(a) Qualitatively explain the second component of the Moon’s libration. Assume (1) that
the Moon rotates with constant angular velocity and a rotation period equal to its sidereal
orbital period of P = 27.3 day, and (2) that the Moon’s elliptical orbit obeys Kepler’s laws
precisely. (Hint: sketch the Moon’s orbit around the Earth, exaggerating its eccentricity.
Then imagine the Moon has a mountain which points directly toward the Earth when the
Moon is at pericenter; which way will it point at other times?)
(b) From the video, how many degrees left-and-right does the Moon appear to swing as it
orbits the Earth? Make a rough measurement by tracking a specific feature on the Moon
over one cycle, ignoring up-and-down motion.
(c) Given that the Moon’s orbit has an eccentricity e ∼ 0.06, estimate how many degrees
left-and-right you expect the Moon to swing. Is this consistent with what you see in the
video? (Note the word estimate. If you’re working out integrals, you’re being too precise!)
2. Given that (1) the Sun is ∼ 400 times further away from the Moon than the Earth
is, and (2) the Sun’s mass is ∼ 3 × 105 times the Earth’s mass, which produces a greater
gravitational force on the Moon, the Earth or the Sun? Compute the ratio of these forces (a
precision of ∼ 10% is adequate). Are you surprised by your result?
3. Recall that (1) the Sun is (typically) ∼ 400 times further away than the Moon, and (2)
the Sun’s mass is ∼ 2.5 × 107 times the Moon’s mass.
(a) Which produces a greater tidal force on the Earth, the Sun or the Moon? Compute the
ratio of these forces.
(b) Diagram the tidal forces due to the Moon and the Sun on the Earth when the Moon is
(i) New, (ii) at First Quarter, (iii) Full, and (iii) at Third Quarter (see Fig. 19.4(b)). To
keep things straight, use one color for the Moon’s tidal forces, and another for the Sun’s.
Then diagram the total tidal force. How would you expect tide heights to depend on the
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phase of the Moon?
(c) The Earth’s orbit around the Sun has eccentricity eE # 0.017. How many times stronger
is the Sun’s tidal force when the Earth is at perihelion compared to aphelion?
(d) The Moon’s orbit around the Earth has eccentricity eM # 0.055. How many times
stronger is the Moon’s tidal force when it’s at pericenter compared to apocenter?
4. What would the ocean tides be like if the Earth was fixed in inertial space, and therefore
prevented from accelerating even though subject to the Moon’s gravitational field?
(a) Describe the tides qualitatively.
(b) Estimate their height in meters.
5. Suppose we could double the Moon’s mass without changing its present orbit.
(a) How much would the height of tides on Earth change as a result?
(b) How would the gradual expansion of the Moon’s orbit due to tidal friction change as a
result?
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