A Web of Worlds presents The Ultimate Cheat Sheet for Astrophysics Students Lachlan Marnoch www.webofworlds.net v 1.0 July 28, 2018 Contents 1 Physics 5 1.1 Motion...........................................5 1.1.1 Velocity......................................5 1.1.2 Acceleration....................................5 1.1.3 Newton's Laws..................................5 1.1.4 Momentum....................................5 1.1.5 Centripetal Force.................................5 1.1.6 Kinetic Energy..................................5 1.1.7 Projectile Motion.................................6 1.1.8 Rotation......................................6 1.1.9 Euler-Lagrange and the Hamiltonian......................7 1.2 Oscillations........................................7 1.2.1 Springs......................................7 1.3 Materials.........................................8 1.3.1 Density......................................8 1.4 Energy...........................................8 1.4.1 Work.......................................8 1.5 Forces...........................................8 1.5.1 Buoyancy (Archimedes' Principle).......................8 1.5.2 Friction......................................8 1.6 Waves...........................................8 1.6.1 Wavelength....................................8 1.6.2 Angular Frequency................................8 1.7 Newtonian Gravity....................................8 1.7.1 Force of Gravity.................................8 1.7.2 Gravitational Potential (potential energy per unit mass)...........8 1.7.3 Gravitational field................................8 1.7.4 Gravitational Potential Energy.........................8 1.7.5 Kepler's Third Law................................9 1.8 Electromagnetism.....................................9 1.8.1 Notation......................................9 1.8.2 Maxwell's Equations...............................9 1.8.3 Lorentz Force...................................9 1.8.4 Electric Field...................................9 1.8.5 Dipole moment.................................. 10 1.8.6 Electric potential................................. 10 1.8.7 Electric potential difference........................... 10 1.8.8 Electric potential energy............................. 10 1.8.9 Charge densities................................. 10 1.8.10 Current densities................................. 10 1.8.11 Circuits...................................... 11 1.8.12 Capacitors.................................... 11 1.8.13 Magnetic fields.................................. 12 1.8.14 Inductors..................................... 12 1.8.15 Materials..................................... 12 1.9 Special Relativity..................................... 13 1 1.9.1 Interval...................................... 13 1.9.2 Four-vectors.................................... 14 1.9.3 Frames of Reference............................... 15 1.9.4 Proper Velocity.................................. 15 1.10 General Relativity.................................... 15 1.10.1 Metrics...................................... 15 1.10.2 Rindler coordinates................................ 16 1.10.3 Einstein notation................................. 16 1.10.4 Christoffel symbols................................ 16 1.10.5 Covariant derivatives............................... 17 1.10.6 Riemann curvature tensor............................ 17 1.10.7 Ricci curvature tensor.............................. 17 1.10.8 Einstein's equations............................... 17 1.11 Thermodynamics..................................... 17 1.11.1 Ideal Gases.................................... 17 1.11.2 Microstates.................................... 17 1.11.3 Entropy...................................... 18 1.11.4 Black bodies................................... 18 1.12 Quantum Mechanics................................... 18 1.12.1 The Uncertainty Principle............................ 18 1.12.2 Bras and Kets.................................. 18 1.12.3 Rules for an Inner Product........................... 18 1.12.4 The Born Rule.................................. 18 1.12.5 Expectation.................................... 18 1.12.6 Variance...................................... 19 1.12.7 Standard Deviation............................... 19 1.12.8 Trace....................................... 19 1.12.9 Partial Trace................................... 19 1.12.10 The Schr¨odingerEquation............................ 19 1.12.11 Heisenberg equation of motion......................... 19 1.12.12 Operators..................................... 19 2 Astrophysics & Astronomy 22 2.1 Astrometry........................................ 22 2.1.1 Redshift...................................... 22 2.1.2 Apparent magnitude............................... 22 2.1.3 Absolute magnitude............................... 22 2.1.4 Relative magnitudes............................... 22 2.1.5 Flux-magnitude relationship........................... 22 2.1.6 Color....................................... 22 2.1.7 Metallicity.................................... 22 2.2 Stars............................................ 22 2.2.1 Stellar Structure Equations........................... 22 2.2.2 Timescales.................................... 23 2.2.3 Gravitational potential energy......................... 23 2.2.4 Eddington Limit (hydrostatic equilibrium)................... 23 2.2.5 Mass-Luminosity Relationship......................... 23 2.3 Galaxies.......................................... 23 2.3.1 Hubble Eliptical Galaxy Classification..................... 23 2.3.2 S´ersicProfile................................... 23 2.3.3 Density of stars in the Milky Way Galaxy................... 24 2.4 Black Holes........................................ 24 2.4.1 Schwarszchild Radius.............................. 24 2.5 Instrumentation...................................... 24 2.5.1 Lensmaker's equation.............................. 24 2.5.2 Focal ratio / Focal number........................... 24 2.5.3 Field of view................................... 24 2.5.4 Resolution Limits................................. 24 2 2.5.5 Nyquist sampling................................. 24 2.5.6 Plate scale.................................... 24 2.5.7 Fitting error................................... 24 2.5.8 Adaptive optics error............................... 25 2.5.9 Signal-to-noise ratio............................... 25 2.5.10 Atmospheric Extinction............................. 25 2.5.11 Rocket science.................................. 25 3 Mathematics 26 3.1 Notation.......................................... 26 3.2 Algebra.......................................... 26 3.2.1 Factorisation................................... 26 3.2.2 Absolute Value.................................. 26 3.2.3 Quadratics.................................... 26 3.2.4 Logarithms.................................... 27 3.2.5 Vectors...................................... 27 3.2.6 Factorials..................................... 27 3.2.7 Inner product definition............................. 28 3.2.8 Complex Numbers................................ 28 3.2.9 Power Series................................... 28 3.2.10 Matrix Operations................................ 28 3.2.11 Matrix Types................................... 29 3.2.12 Change of Basis Unitary............................. 32 3.2.13 Commutator................................... 32 3.2.14 Anticommutator................................. 32 3.2.15 Cauchy-Schwarz Inequality........................... 32 3.3 Geometry......................................... 32 3.3.1 Pythagorean theorem.............................. 32 3.3.2 Properties of shapes............................... 32 3.3.3 Properties of solids................................ 32 3.3.4 Circular formulae................................. 33 3.3.5 Useful Functions................................. 33 3.3.6 Coordinates.................................... 34 3.3.7 Hyperbolic Functions.............................. 34 3.4 Trigonometry....................................... 35 3.4.1 Identities..................................... 35 3.5 Calculus.......................................... 39 3.5.1 Limits....................................... 39 3.5.2 Properties..................................... 39 3.5.3 Differentiation.................................. 39 3.5.4 Partial Differentiation.............................. 41 3.5.5 The Differential.................................. 41 3.5.6 Line Element................................... 41 3.5.7 Integration.................................... 41 3.5.8 Vector Calculus.................................. 43 3.5.9 Dirac Delta Function............................... 44 3.5.10 Approximations.................................. 44 4 Statistics 45 4.1 Variance.......................................... 45 4.2 Standard Deviation.................................... 45 A Values 46 A.1 Physics.......................................... 46 A.1.1 Physical Constants................................ 46 A.1.2 Useful Quantities................................. 47 A.2 Astronomy........................................ 47 A.2.1 Useful Quantities................................. 47 A.3 Mathematics....................................... 48 3 B Units of Measurement 49 B.1 Natural Units....................................... 49 B.2 SI System......................................... 49 B.2.1 Base Units.................................... 49 B.2.2 Derived Units................................... 50 B.3 CGS (centimetres-grams-seconds)...........................