AST221_2023F_Assignment1

Assignment 1 AST221H Due: Sept. 29, 2023 You may work together with other students to solve these problem sets, but all solutions must be written and submitted independently. Show your work clearly for full marks, including intermediate steps, and diagrams if necessary! Answers that are not fully explained will not get full marks. Check the syllabus for reading recommendations. Include units! Submit your assignment as a single .pdf file following the instructions on Quercus. This includes the plot in Question 4! Problem 1: Parallax a (3 points) If parallax can be measured with an accuracy of 0.01 arcseconds, and the mean density of stars in the Solar neighbourhood is 0.1 pc − 3 , how many stars can have their distances measured via parallax? Clearly state any assumptions you make to solve this problem. b (2 points) Parallax doesn’t have to be measured using the orbit of Earth as a baseline. In 1672, an international effort was made to measure the parallax angle of Mars at opposition, when it was on the opposite side of the Earth from the Sun, and thus closest to Earth, using two widely-separated observers on Earth. Consider two observers at the same longitude but one at latitude of 45 degrees North and the other at 45 degrees South. Work out the physical separation s between the observers given the radius of Earth is R E ≈ 6400 km. c (2 points) If the parallax angle measured was 25 arcseconds, using your previous result, what was the distance to Mars? Give your answer in both km and au (astronomical units). d (3 points) If we want to measure the distance to Mars with a precision of 10%, how closely must the clocks used by the two observers be synchronized? Ignore the rota- tion of the Earth, and assume the average orbital velocities of Earth and Mars are 29.79 km s − 1 and 24.13 km s − 1 , respectively. Explain your answer. Problem 2: The Alpha Centauri system a (3 points) The low-mass star, Proxima Centauri, is the closest known star to the Sun, and has a parallax of 0.77 ′′ and an apparent magnitude m = 11 . 05. What is Proxima Centauri’s distance in light years? How much brighter or fainter in luminosity is the star Proxima Centauri compared to the Sun? b (2 points) Given your result in a), how close to Proxima Centauri would a planet need to orbit to receive the same amount of flux as the Earth does from the Sun? What would the period be for such a planet? 1