AN INTRODUCTION TO ASTROPHYSICS AND COSMOLOGY BY ANDREW NORTON – 2016 Chapter 1 Manipulating numbers and symbols 7 Introduction 7 1.1 Algebra and physical quantities 7 1.1.1 Manipulating algebraic expressions 7 1.1.2 Rearranging algebraic equations 9 1.1.3 Solving simultaneous equations 10 1.2 Powers, roots and reciprocals 10 1.2.1 Combining powers 11 1.2.2 Solving polynomial equations 12 1.3 Imaginary numbers 13 1.4 Unit (dimensional) analysis 14 1.5 Function notation 14 1.6 Powers of ten and scientific notation 15 1.7 Significant figures 16 1.8 Experimental uncertainties 17 1.8.1 Types of uncertainty 17 1.8.2 Estimating random uncertainties 19 1.8.3 Uncertainties when counting randomly occurring events 20 1.8.4 The uncertainty in a mean value 21 1.8.5 Combining uncertainties in a single quantity 21 1.9 Logarithms and logarithmic functions 22 1.10 Graphs 24 1.10.1 Straight-line graphs 25 1.10.2 Making curved graphs straight 27 1.11 Angular measure 28 1.12 Trigonometry 29 1.12.1 Trigonometric ratios 29 1.12.2 The sine rule and cosine rule 30 1.12.3 Trigonometric functions 31 1.12.4 Inverse trigonometric functions 33 1.13 Vectors 34 1.13.1 Vector components 34 1.13.2 Addition and subtraction of vectors 35 1 (Please supply running head) 1.13.3 Position and displacement vectors 36 1.13.4 Unit vectors 36 1.13.5 The scalar product 37 1.13.6 The vector product 37 1.14 Coordinates 38 1.15 Scalar and vector fields 40 1.16 Matrices 42 1.16.1 Combining matrices 42 1.16.2 Special types of matrices 44 1.16.3 Transposing matrices 45 1.16.4 The determinant of a matrix 45 1.16.5 Adjoint and reciprocal matrices 46 Chapter 2 Stars and planets 51 Introduction 51 2.1 Measuring stars and planets 51 2.2 Units in astrophysics 53 2.3 Positions, distances and velocities 54 2.3.1 Observing the positions of stars 54 2.3.2 Measuring the velocities of stars 58 2.4 Spectra and temperatures 59 2.5 Luminosities and fluxes 63 2.6 Astronomical magnitudes 63 2.7 Colours and extinction 66 2.8 The Hertzsprung–Russell diagram 67 2.9 Masses of stars 69 2.10 Life cycles of stars 76 2.11 Stellar end-points 81 2.12 Planetary structure 81 2.12.1 Terrestrial planets 81 2.12.2 Giant planets 83 2.13 Extrasolar planets and how to find them 86 2.14 Astronomical telescopes 88 2.14.1 Telescope characteristics 89 2.14.2 Telescopes in other parts of the electromagnetic spectrum 92 2 Chapter 3 Galaxies and the Universe 97 Introduction 97 3.1 The Milky Way – our galaxy 97 3.2 Other galaxies 98 3.2.1 Classification of galaxies 98 3.2.2 Origin and evolution of galaxies 101 3.2.3 Measuring galaxy properties 101 3.3 The distances to other galaxies 102 3.4 Active galaxies 105 3.4.1 The spectra of active galaxies 105 3.4.2 Types of active galaxy 110 3.5 The spatial distribution of galaxies 113 3.6 The structure of the Universe 115 3.7 The evolution of the Universe 120 3.8 Observational cosmology 123 3.9 Cosmological questions 125 Chapter 4 Calculus 129 Introduction 129 4.1 Differentiation and curved graphs 129 4.2 Differentiation of known functions 131 4.3 The exponential function 133 4.4 The chain rule 135 4.5 Logarithmic differentiation 138 4.6 Expansions 139 4.7 Partial differentiation 142 4.8 Differentiation and vectors 143 4.9 Differential equations 144 4.10 Integration and curved graphs 146 4.11 Integration of known functions 147 4.12 Integration by substitution 149 4.13 Integration by parts 152 4.14 Multiple integrals 153 3 (Please supply running head) Chapter 5 Physics 159 Introduction 159 5.1 Describing motion 159 5.1.1 Motion in one dimension 159 5.1.2 Motion in two or three dimensions 161 5.1.3 Periodic motion 162 5.2 Newton’s laws 163 5.2.1 Newton’s laws of motion 164 5.2.2 Newton’s law of gravitation 165 5.3 Relativistic motion 166 5.4 Predicting motion 168 5.4.1 Work, energy, power and momentum 168 5.4.2 Relativistic mechanics 170 5.5 Rotational motion 171 5.6 Properties of gases 174 5.7 Atoms and energy levels 178 5.7.1 Atomic structure 178 5.7.2 Photons and energy levels 180 5.8 Quantum physics 184 5.8.1 Wave mechanics 185 5.8.2 Quantum mechanics in atoms 189 5.9 Quantum physics of matter 191 5.9.1 Quantum gases 191 5.9.2 Nuclear physics 195 5.9.3 Particle physics 200 5.10 Electromagnetism 202 5.10.1 Electricity and magnetism 202 5.10.2 Electromagnetic waves 204 5.10.3 Spectra 207 5.10.4 Opacity and optical depth 211 Solutions 221 4 Introduction In order to successfully study one or both of the Open University’s Level 3 modules, S382 Astrophysics or S383 The Relativistic Universe, you should already be familiar with various topics in cosmology, astronomy, planetary science, physics and mathematics. The level of skills, knowledge and understanding that we expect you to have when you embark on either of these modules is equivalent to the end-points of the OU’s Level 2 modules: S282 Astronomy, S283 Planetary Science and the Search for Life, S217 Physics: from Classical to Quantum and either MST224 Mathematical Methods or MST210 Mathematical Methods, Models and Modelling. To ascertain whether or not you meet the required level before embarking on S382 and/or S383 you should work through the document entitled Are You Ready For S382 or S383? which is available from the Faculty website. If, as a result of attempting the questions in that document, you realise that you need to revise your skills, knowledge and understanding in certain areas of mathematics, physics, cosmology, astronomy and planetary science, then you should study the relevant chapters of this document carefully. There are five main chapters to this document – one each to introduce the astronomy and planetary science, the cosmology and the physics background, plus two chapters of mathematics. It is important to note that, because most of this document revisits concepts and phenomena that are covered in detail in Level 2 Open University modules, the treatment here is much less rigorous than in the modules themselves. For the most part, the subjects covered here are merely presented to you rather than developed gradually through detailed argument. This is to enable you to get rapidly ‘to the point’ and appreciate the key information you need in order to understand what follows, and to allow you to progress quickly to the main substance of the Level 3 modules . 5 Introduction Acknowledgements The material in this document has been drawn from the physics, maths and astronomy that is taught in various other OU modules, including S282, S283, SXR208/SXPA288, S207/S217, S103/S104 and S151. The authors of the relevant parts of those courses: David Adams, John Bolton, David Broadhurst, Jocelyn Bell Burnell, Derek Capper, Alan Cayless, Andrew Conway, Alan Cooper, Dan Dubin, Alan Durrant, Tony Evans, Stuart Freake, Iain Gilmour, Simon Green, Iain Halliday, Carole Haswell, Keith Higgins, Keith Hodgkinson, Anthony Jones, Barrie Jones, Mark Jones, Sally Jordan, Ulrich Kolb, Robert Lambourne, Ray Mackintosh, Lowry McComb, Joy Manners, David Martin, Pat Murphy, Andrew Norton, Lesley Onuora, John Perring, Michael de Podesta, Shelagh Ross, David Rothery, Sean Ryan, Ian Saunders, Mark Sephton, Richard Skelding, Tony Sudbery, Elizabeth Swinbank, John Zarnecki and Stan Zochowski are gratefully acknowledged, along with the other members of the teams responsible for those modules. Grateful acknowledgement is also offered to Carolin Crawford for critically reading and Amanda Smith for proof-reading this document, although any remaining errors are the responsibility of the editor. Grateful acknowledgement is made to the following sources of figures: Figure 1.4 (Photograph of Jupiter): NASA/Science Photo Library; Figure 1.4 (Photograph of the Earth): NASA; Figure 1.4 (Photograph of a galaxy): The Regents, University of Hawaii; Figure 2.2: Till Credner, Allthesky.com; Figure 2.29: Observatoire de Paris; Figure 2.31: S Korzennik, Harvard University Smithsonian Center for Astrophysics; Figure 3.10: Lee, J. C. etal. (2002) ‘The shape of the relativistic Iron Kα Line from MCG 6-30-15 measured with the Chandra high energy transmission grating spectrometer and the Rossi X-Ray timing explorer’, Astrophysical Journal, Vol 570. ⃝c The American Astrophysical Society; Figure 3.13: Le Grand Atlas de l’Astronomie 1983. Encyclopaedia Universalis; Figure 3.14: W. N. Colley and E. Turner (Princeton University), J. A. Tyson (Bell Labs, Lucent Technologies) and NASA; Figure 3.17: Adapted from Landsberg, P. T. and Evans, D. A. Mathematical Cosmology. 1977, Oxford University Press; Figure 3.21: Adapted from Schwarzschild, B. (1998) ‘Very Distant Supernovae suggest that the cosmic expansion is speeding up’, Physics Today, June 1998. American Institute of Physics; Figures 3.22 and 3.23: Bennett, C. L. et al. ‘First Year Wilkinson Microwave Anisitrophy Probe (WMAP) Observations’, Astrophysical Journal Supplement Series, Volume 148, Issue 1. Every effort has been made to contact copyright holders. If any have been inadvertently overlooked we will be pleased to make the necessary arrangements at the first opportunity. 6 Chapter 1 Manipulating numbers and symbols Introduction In this chapter, we will concentrate on the various rules for manipulating numbers and algebraic symbols, including how to manipulate equations containing fractions, powers, logarithms and trigonometric functions, and how to deal with vectors and matrices.