A Problem book in Astronomy and Astrophysics Compilation of problems from International Olympiad in Astronomy and Astrophysics (2007-2012) Edited by: Aniket Sule on behalf of IOAA international board July 23, 2013 ii Copyright ©2007-2012 International Olympiad on Astronomy and Astrophysics (IOAA). All rights reserved. This compilation can be redistributed or translated freely for educational purpose in a non-commercial manner, with customary acknowledgement of IOAA. Original Problems by: academic committees of IOAAs held at: – Thailand (2007) – Indonesia (2008) – Iran (2009) – China (2010) – Poland (2011) – Brazil (2012) Compiled and Edited by: Aniket Sule Homi Bhabha Centre for Science Education Tata Institute of Fundamental Research V. N. Purav Road, Mankhurd, Mumbai, 400088, INDIA Contact: [email protected] Thank you! Editor would like to thank International Board of International Olympiad on Astronomy and Astrophysics (IOAA) and particularly, president of the board, Prof. Chatief Kunjaya, and general secretary, Prof. Gregorz Stachowaski, for entrusting this task to him. We acknowledge the hard work done by respec- tive year’s problem setters drawn from the host countries in designing the problems and fact that all the host countries of IOAA graciously agreed to permit use of the problems, for this book. All the members of the interna- tional board IOAA are thanked for their support and suggestions. All IOAA participants are thanked for their ingenious solutions for the problems some of which you will see in this book. iv Thank you! A Note about Problems You will find a code in bracket after each problem e.g. (I07 - T20 - C). The first number simply gives the year in which this problem was posed I07 means IOAA2007. Second one gives information about the test and question number. T stands for theory, D for data analysis, O for observation and G for group tasks. In theory numbers greater than 15 denote long questions. The last letter denotes difficulty level of the question as perceived by the editor. A being the simplest and D the most difficult. The prefix “EA” before a line in a problem indicates that this was not part of the original question but its an addition by editor. General Marking Principle 1. Giving correct answer without detailed cal- Deduct 50% of the marks culation for that part 2. Minor mistakes in the calculations e.g. Deduct 20% of the marks wrong signs, symbols, substitutions for that part 3. Units missing from final answers Deduct 10% of the marks for that part 4. Too few or too many significant digits in Deduct 10% of the marks the final answer for that part 5. Using incorrect physical concept (despite No points given correct answers) 6. Error propagated from earlier parts: minor Full points (i.e. no deduc- errors tions) 7. Error propagated from earlier parts: major Deduct 20% of the marks errors for the final answer Preface International Olympiads are high school competitions where each country sends a team of students accompanied by mentors and the students compete individually in a set of tests and awarded medals based on their performance. Currently, such Olympiads exist for Physics, Chemistry, Biology, Mathemat- ics, Astronomy and Astrophysics, Informatics, Earth Sciences, Linguistics etc. There are also special Olympiads like International Junior Science Olympiads aimed at middle school and early high school students. These International Olympiads have also given birth to smaller, regional events like Asian Physics Olympiad, Latin American Astronomy and Astrophysics Olympiad and Eu- ropean Girls’ Mathematics Olympiad to name a few. International Olympiad on Astronomy and Astrophysics (IOAA) started in 2007. We have had 6 Olympiads so far. They were hosted by Thailand (2007), Indonesia (2008), Iran (2009), China (2010), Poland (2011) and Brazil (2012). The next two Olympiads will be hosted by Greece (2013) and Romania (2014). Currently roughly 30 teams participate in IOAA. Each team consists of upto five students chosen through national level selection process in the respective country and it is accompanied by two mentors. There are three rounds of tests. Theoretical problems, data analysis problems and night sky observation test. Theoretical round typically has 15 short questions and 2-3 long questions to be solved in five hours. Data analysis usually has two problems to be solved in four hours and night sky observation round has 4-5 tasks with typical time upto 5 minutes for each task. The scores for three rounds for each participant are totalled with 50% weigh- tage to theory and 25% each for the other two. Post totalling, the scores are renormalised by the “average score of the top three participants”. As per these renormalised scores, participants scoring more than 90%, 78%, 65% and 50% qualify for Gold medals, Silver Medals, Bronze Medals and Honourable Mention certificates respectively. Additional special certificates are awarded for outstanding performances in one or more tests. From 2011, there is an additional team competition, where all members of the team work together vi Preface on one or more problems. This compilation includes problems from the first six IOAAs. The problems in the respective years were originally designed by the respective host countries. A sketch of possible solution was also provided. Post-IOAA, these problems were also made available on the respective IOAA websites. This compilation, however, aspires to present the problem set in a pedagogically more useful way. The problems are divided as per concepts involved and also graded as per difficulty level. At some places, additional sub-questions are added to stimulate thought process. Additional notes are added in solutions to make the solutions self explanatory. IOAA statutes demand that original solutions should not involve calculus. However, some times same problem can be solved more elegantly using calculus. Sometimes, a problem can be solved in more than one elegant ways. In such cases, both solutions are included. In some cases numerical values from original problems are tweaked slightly to make the problems more realistic. If a problem involves multiple concepts, it is included in the chapter where last of the concepts involved is covered. Nu- merical values use SI system of units wherever applicable. It is implicitly expected that students use proper number of significant digits and at least have a rough estimate of amount of error in their answer. The problems and solutions remain property of IOAA and any reproduction in any language should carry proper acknowledgements. The current syllabus of IOAA is appended at the start of this book. As one can see, except the “non-calculus based solutions” restriction, the syllabus more or less covers same range of topics as covered in any typical under- graduate astrophysics course in any university. Thus, we hope that this book can serve as a useful problem set for those courses too. Needless to say, po- tential Astronomy and Astrophysics Olympiad participants would find this compilation extremely handy for their preparation. Happy solving! Aniket Sule Table of Constants The Sun Mass M = 1.9891 1030 Kg ⊙ Radius R = 6.955 ×108 m ⊙ × Luminosity L = 3.826 1026 W ⊙ Apperant magnitude at mid-day m = -26.72× ⊙ Absolute V-band magnitude MV⊙ = 4.82 Absolute bolometric magnitude Mbol = 4.72 Apperant angular diameter θ = 30’ ⊙ Temperature on the surface T = 5778K ⊙ SolarConstant(atEarth) S = 1366W/m2 The Earth Mass M = 5.9736 1024 Kg ⊕ Radius R = 6.3708 × 106 m ⊕ × Mean density ρ = 5515 Kg/m3 ⊕ Gravitational acceleration on the g = 9.81 m/s2 surface Inclination of the axis ǫ = 23◦26′ Albedo α = 0.39 ⊕ The Moon 22 Mass MM = 7.4377 10 Kg Radius R = 1.7374 × 106 m M × Mean distance from Earth d = 3.78 108 m M × Synodic period Psy = 29.5306 days Apperant magnitude (full moon) mmoon = -12.74 Albedo α = 0.14 Inclination of the lunar orbit w.r.t. = 5.14◦ the ecliptic The Venus Radius RV = 0.949 R ⊕ Orbital semi-major axis aV enus = 0.732 A.U. Orbital period TV enus = 224.70 days viii Table of Constants Albedo α = 0.87 ⊕ The Mars Mass M = 6.421 1023kg Mars × Radius RMars = 3393km Orbital semi-major axis aMars = 1.52 A.U. Rotational period TMars = 24.623h Orbital radius of Phobos aPh = 9380km The Jupiter Mass M = 1.898 1027 Kg J × Orbital semi-major axis aJup = 5.204 A.U. Number of arcseconds in a rad. = 206265” 1 sidereal day = 23h 56m 4s.1 1 tropical year = 365.2564 solar days = 3.1557 107 s 1 sidereal year = 365.2422× solar days 1 Astronomical Unit (A.U.) a = 1.4960 1011 m ⊕−⊙ 1 lightyear (ly) = 9.46 ×1015 m = 6.324× 104 A. U. × 1parsec(pc) =3.0856 1016 m = 3.262 ly× Distance to the galactic centre dGC = 8.3 0.3 kpc Speed of Light c = 2.99792458± 108 m/s ×11 2 2 Universal Gravitational Constant G = 6.6726 10− Nm kg− × 34 Planck Constant h = 6.62 10− J s × · Hubble Constant H0 = (67.80 0.77) Km/s/Mpc ± 9 Age of the Universe t0 = 13.77 10 years × 8 2 4 Stephan’s Constant σ = 5.67 10− W m− K− × 23 2 Boltzmann Constant κB = 1.38 10− JK− × 3 Wien’s displacement law λmT = 2.898 10− m K × 19 · Charge of a electron qe = 1.602 10− C × 31 Mass of a electron me = 9.1 10− kg × 2 Mass of a proton mp = 938.27 MeV/c 2 Mass of a neutron mn = 939.56 MeV/c 2 Mass of a Deuterium atom mD = 1875.60 MeV/c 2 Mass of a Helium-3 atom mHe3 = 2808.30 MeV/c Mass of a Helium-4 atom mHe = 4.002603 a.m.u.