Download Coursework 2 File

Our Universe
Coursework 2
Due in on Wednesday of week 3 at 16:00
Useful Information
• 1 AU = 1.5 × 1011 m
• Gravitational constant G = 6.67 × 10−11 N m2 / kg2
• Solar mass 1.98 × 1030 kg
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 spacecraft 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 spacecraft. You should assume that the
only motion exhibited by the spacecraft is circular orbit around the Sun.
Please note that all questions below should be handed in for assessment
Homework question 1
In around 280 B.C. Aristarchus of Samos devised a method of determining the relative size of the
Moon and the Earth using a total lunar eclipse. Consider a simplified geometry for a total lunar
eclipse in which the Sun acts as a point source of light at infinity. The trajectory of the Moon
behind the Earth is a straight line, running perpendicular to the rays of the Sun, along which the
Moon travels at constant velocity. With the aid of a diagram, explain how the ratio of the Moon
and Earth diameters can be estimated from the following two measurements: (i) the time from the
moment the Moon just begins to enter eclipse to the moment when it just reaches totality; (ii) the
time for which the Moon is in totality, completely obscured by the shadow of the Earth.
Aristarchus estimated that the Earth has a diameter three times larger than the Moon. In lecture
3 we learned that Aristarchus estimated that the distance between the Earth and the Sun is
approximately 20 times larger than the distance between the Earth and the Moon. Using the fact
that during a total solar eclipse the Sun and Moon appear to have the same size, Aristarchus was
able to estimate the relative diameters of the Sun and the Earth. What was his estimate? Can you
offer an explanation of why Aristarchus was the first person in recorded history to suggest that the
Earth orbits around the Sun, rather than the other way round?
Homework question 2
(i). Define what is meant by the sidereal period for a single planet.
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(ii). Define what is meant by the synodic period for two planets – the Earth and a superior planet.
(iii). By assuming that the planets in the Solar System are on circular orbits, Copernicus devised
a method for calculating the sidereal period for either an inferior or superior planet based on
knowledge of the Earth’s sidereal period and observations of the synodic period. In lectures we
derived the relation between these periods for an inferior planet such as Venus or Mercury. Derive
the corresponding expression for a superior planet using the same reasoning as provided in lectures.
Jupiter has a synodic period of 398.9 days. Calculate its sidereal period.
(iv). 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 lines joining the Sun-Earth and 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.
(v). Jupiter was at opposition at midnight on 5th January 2014 and was at quadrature on 1st April
2014 (time to quadrature is 85 days). Calculate the distance to Jupiter in terms of the Earth-Sun
distance using your knowledge of the Earth’s and Jupiter’s sidereal periods.
Homework question 3
(i). 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.
(ii). The recently launched satellite Gaia will be able to measure angular distances to an accuracy of
20 × 10−6 arcseconds. What is the largest stellar distance that it will be able to measure (measured
in parsecs and light years)?
* Homework question 4
Halley’s comet orbits the Sun with a semimajor axis of a = 17.8 AU and an eccentricity e = 0.967.
(i). Calculate the orbital period in years (assuming its mass is very small compared to that of the
Sun). Note that you will need information contained in lecture 6 to answer this question.
(ii). Calculate the distance of its closest approach to the Sun (perihelion) in AU.
(iii). Calculate its furthest distance from the Sun (aphelion) in AU.
(iv). The angular p
momentum of a body of mass mb orbiting a star of mass M∗ is given by the
expression J = mb GM∗ a(1 − e2 ). Calculate the velocity of Halley’s comet when it is at perihelion
and aphelion, expressing your answer in metres per second and kilometres per second.
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