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Our Universe
Coursework 1
Due in on Wednesday of week 2 at 16:00
Exercise class question - not to be handed in
This question concerns the precession of the equinoxes, the phenomenon where the direction pointed
to by the Earth’s rotation axis changes continuously. This effect arises because of the gravity of the
Sun and the Moon acting on the spinning Earth. It only occurs because the Earth is non-spherical.
The rotation of the Earth causes it to bulge at the equator, and the resulting shape is an oblate
spheroid. The precession of the Earth is very similar in concept to the precession of a gyroscope
or a spinning top that you may be more familiar with.
Figure 1: Spinning top
1. We begin by considering a spinning top that consists of a thin disc mounted on an essentially
massless spindle, as shown in Figure 1. The angular momentum is denoted by the vector J, and its
rate of change is denoted by J̇. Gravity attempts to tilt the top by exerting a force of magnitude
F = mg at the centre of mass, where m is the mass of the top. This gives rise to a torque J̇ = r×F.
In which direction does the spin axis of the top begin to precess? Show that the precession torque
is given by J˙ = mgr sin θ.
2. The rate of change of a vector J when rotated about an arbitrary axis is J̇ = ω × J, where
ω is the rotation vector whose magnitude is the rate of rotation measured in radians per second.
Obtain an expression for the precession rate of the spinning top.
Figure 2: Left panel: Exaggerated depiction of oblate, spinning Earth and Sun during northern hemisphere
summer. Right panel: Earth and Sun during northern winter.
3. Figure 2 shows the gravitational force of the Sun acting on a point on the Earth’s surface during
both the summer and winter solstices. Decompose this force into one that acts along the direction
joining the centres of the Earth and the Sun, and one that is perpendicular to this direction. The
component that acts along the line of centres just contributes to the orbital motion of the Earth
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around the Sun, the perpendicular component is responsible for the precession. Write down an
expression for the torque that drives the precession of the Earth.
4. Now consider the direction in which this precessional torque acts during both the summer and
winter solstices (hint: it should either be into or out of the page in each case). Using your results,
argue that the torque acting, when integrated over the orbit of the Earth around the Sun, gives
rise to a well defined direction of precession.
5. Now that you have determined the direction of precession, explain why the tropical year is
shorter than the sidereal year. The tropical year is the time between successive vernal equinoxes,
which are defined to occur when the spin axis of the Earth is at right angles to the line that joins
the centres of the Earth and the Sun. Draw a diagram to help visualise what is going on.
Please note that all questions below should be handed in for assessment
Question 1
Astronomical distance scales
1. Calculate the number of metres in one light year.
2. The Astronomical Unit (AU) is the mean distance between the Sun and the Earth, and
1 AU = 1.5 × 1011 m. Pluto orbits the Sun at a distance of 40 AU. Calculate the light travel
time between the Sun and Pluto.
3. The Voyager I spacecraft is the furthest human-built spacecraft from Earth, and is currently
at a distance of 125 AU travelling at 17 km/s. Estimate the year of its launch and compare
your answer with the actual launch date. Estimate how long it would take a spacecraft
travelling with Voyager I’s speed to:
a) Reach Proxima Centauri, the nearest star to the Sun (distance ≈ 4.3 lightyears)
b) Travel across the diameter of the Milky Way galaxy (D ∼ 105 lightyears)
c) Travel to the Andromeda galaxy (distance approximately 2 × 106 lightyears)
4. A 30 year-old astronaut launches from Earth and travels to Proxima Centauri and back at
a speed equal to 99.5 % of the speed of light, leaving a twin sister at home. How old will
the astronaut be upon their return to Earth as determined by an observer who remained on
Earth during the voyage? How old will the twin sister be? [Hint: special relativity.]
Question 2
With the aid of a diagram explain the following:
1. Why an astronomer who observes stars at a fixed point in the sky (e.g. the zenith) during a
night only observes stars whose declination values are all the same but whose right ascension
values are different.
2. Why a lunar eclipse does not occur every month.
3. The occurence of annular eclipses.
4. The difference between a synodic and a sidereal month.
Question 3
With the aid of a diagram, explain briefly why precession of the equinoxes occurs. What is the
period associated with the precession of the equinoxes? Describe two astronomical consequences of
this precession.
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