* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Coursework 2 File
Astrobiology wikipedia , lookup
Definition of planet wikipedia , lookup
Tropical year wikipedia , lookup
History of Solar System formation and evolution hypotheses wikipedia , lookup
Lunar theory wikipedia , lookup
Rare Earth hypothesis wikipedia , lookup
Aquarius (constellation) wikipedia , lookup
Planetary habitability wikipedia , lookup
Formation and evolution of the Solar System wikipedia , lookup
Extraterrestrial life wikipedia , lookup
Copernican heliocentrism wikipedia , lookup
Satellite system (astronomy) wikipedia , lookup
Late Heavy Bombardment wikipedia , lookup
Geocentric model wikipedia , lookup
Comparative planetary science wikipedia , lookup
Astronomical unit wikipedia , lookup
Dialogue Concerning the Two Chief World Systems wikipedia , lookup
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. 1 (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. 2