Download Set 2

Our Universe
Coursework 2
Due in on Tuesday of week 4 at 16:00
Exercise class question - not to be handed in
In 1786 James Bradley, the 3rd Astronomer Royal, noticed that the apparent positions of stars on
the night sky shift slightly relative to their real positions as the Earth orbits the Sun. This effect
is due to the combined effect of the finite speed of light and the motion of the Earth around its
orbit, and is known as stellar aberration.
1. Using an appropriate diagram, explain why stellar aberration occurs and estimate the maximum shift of stellar positions on the sky that arises because of this effect. Quote your answer
in arcseconds.
2. Consider a distant star whose position in physical space places it directly over the Sun in
a direction that is perpendicular to the plane of the Earth’s orbit. Assume that the star is
essentially at infinity so that the light rays from the star travel as parallel lines perpendicular
to the Earth’s orbit plane. A space-craft that orbits the Sun, following the Earth’s orbit,
continuously monitors the position of the star on the sky for the duration of one year. Sketch
the apparent motion of the star as recorded by the space-craft. You should assume that the
only motion exhibited by the space-craft is circular orbit around the Sun.
Please note that all questions below should be handed in for assessment
Homework question 1
Using a diagram explain how the apparent retrograde motion of Mars in the night sky can be
explained by a heliocentric model of the Solar System in which Mars and the Earth undergo circular
orbit around the Sun. Which observations of Venus undertaken by Galileo prove that Venus orbits
the Sun on an inferior orbit. Using a diagram explain why these observations are incompatible
with a geocentric model.
Homework question 2
Copernicus devised a method for measuring the distance of a superior planet (such as Jupiter) from
the Sun that involves measuring the time interval from the moment that the Earth and Jupiter
are in opposition until the moment when they are next in quadrature. Earth and Jupiter are
in quadrature when the Sun-Earth-Jupiter form a right angle. Using trigonometry, explain how
Copernicus’ method works. You should assume that the Earth and Jupiter are on circular orbits
with sidereal periods E and P , respectively, such that their orbital angular velocities equal 360o /E
and 360o /P . You should note that Copernicus’ method only calculates the Jupiter-Sun distance in
terms of the Earth-Sun distance.
Hint: draw a sketch that shows Earth and Jupiter in opposition, and another one showing them
in quadrature (remembering that Jupiter orbits much more slowly than the Earth). Think about
how you can calculate the Earth-Sun-Jupiter angle from the time interval between opposition and
quadrature.
Assuming that Jupiter was at opposition at midnight on 1st December 2012 and will be at
quadrature on 26th February 2013 (time to quadrature is 87 days), calculate the distance to Jupiter
in terms of the Earth-Sun distance. You should assume that Earth’s sidereal period is 1 year (365.25
days) and Jupiter’s sidereal period is 11.86 years.
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Homework question 3
The satellite Hipparchos was able to measure angular distances on the sky to an accuracy of 0.001
arcseconds. What is the largest stellar distance that could be measured using parallax with a
baseline equal to twice the radius of the Earth’s orbit around the Sun ? Express your answer in
parsecs and light years.
Homework question 4
Observations of Saturn at opposition indicate the existence of an orbiting satellite. The maximum
observed angular separation of the satellite from Saturn is measured to be 3.26 arcminutes, and
the orbital period of the satellite around Saturn is observed to be 16 days. Assuming that Saturn
orbits the Sun on a circular orbit at a distance 9.6 AU, and the Earth undergoes circular orbit with
a distance of 1 AU, estimate the combined mass of Saturn and the satellite. The following data
may be useful: 1 AU = 1.5 × 1011 m; G = 6.67 × 10−11 N m2 kg−2 . Note that the measurement of
the angular distance of the satellite from Saturn is not a parallax measurement, but arises purely
because of the orbital motion of the satellite around Saturn.
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