These lecture notes are constantly being updated, extended and improved This is Version F2021:v0.1 Last Modified: Aug 14, 2021 2 Course Outline Astronomy is the study of everything in our Universe outside of the Earth’s atmo- sphere, including the Universe as a whole (‘cosmology’). As such, it involves all of physics: quantum mechanics, nuclear and particle physics, classical mechanics, special and general relativity, electromagnetism, statistical physics, hydrodynamics, plasma physics, and even solid state physics. As we will see, though, with the excep- tion of rocky planets and asteroids, all objects in the Universe can be characterized as some kind of fluid. In addition, almost all information we receive from these ob- jects reaches us in the form of radiation (photons)1. Consequently, in this course on physical processes in astronomy we will focus almost exclusively on fluid dynam- ics, including neutral fluids, charged fluids (aka plasmas), and collisionless fluids, and radiative processes. We start in Part I with standard hydrodynamics, which applies mainly to neutral, collisional fluids. We derive the fluid equations from kinetic theory: starting with the Liouville theorem we derive the Boltzmann equation, from which in turn we derive the continuity, momentum and energy equations. Next we discuss a variety of different flows; vorticity, incompressible barotropic flow, viscous flow, accretion flow, and turbulent flow, before addressing fluid instabilities and shocks. Next, in Part II, we briefly focus on collisionless dynamics, highlighting the subtle differences between the Jeans equations and the Navier-Stokes equations. We also give a brief account of orbit theory, and discuss the Virial theorem. In Part III we turn our attention to plasmas. We discuss how the relevant equations (Vlasov and Lenard-Balescu) derive from kinetic theory, and then discuss plasma orbit theory, magnetohydrodynamics (MHD) and several applications. Finally, in Part IV we discuss radiative processes and the interaction of light with matter (scattering & absorption). We end with a brief discussion of the cooling function, which describes how baryonic matter can dissipate its internal energy, leading to the formation of galaxies, stars and planets. It is assumed that the student is familiar with vector calculus, with curvi-linear coordinate systems, and with some radiative essential. A brief overview of these topics, highlighting the most important essentials, is provided in Appendices A-E and K. The other appendices present detailed background information that is provided for the interested student, but which is not considered part of the course material. 1Other astrophysical messengers include neutrinos, cosmic rays and gravitational waves 3 CONTENTS Part I: Fluid Dynamics 1: Introduction to Fluids and Plasmas ......................... ............9 2: Dynamical Treatments of Fluids ........................... ............16 3: Hydrodynamic Equations for Ideal Fluids .................... ..........25 4: Viscosity, Conductivity & The Stress Tensor . ..........28 5: Hydrodynamic Equations for Non-Ideal Fluids . .........34 6: Equations of State .................................... .................38 7: Kinetic Theory: from Liouville to Boltzmann . ......45 8: Kinetic Theory: from Boltzmann to Navier-Stokes . ........60 9: Vorticity & Circulation . .............71 10: Hydrostatics and Steady Flows ........................... .............79 11: Viscous Flow and Accretion Flow ............................ .........91 12: Turbulence .......................................... ................102 13: Sound Waves ......................................... ...............110 14: Shocks ............................................. ..................118 15: Fluid Instabilities . .............127 Part II: Collisionless Dynamics 16: Jeans equations & Dynamical Modeling ....................... .......138 17: Orbit Theory ......................................... ...............147 18: Virial Theorem & Gravothermal Catastrophe ................ ........151 19: Collisions & Encounters of Collisionless Systems . .... 156 Part III: Plasma Physics 20: Characteristic time & length scales . ..........167 21: Plasma Orbit Theory .................................... ............178 22: Plasma Kinetic Theory ................................... ...........186 23: Vlasov Equation & Two-Fluid Model .......................... ...... 192 24: Magnetohydrodynamics ................................ ..............199 Part IV: Radiative Processes 25: The Interaction of Light with Matter. I - Scattering ........... .......212 26: The Interaction of Light with Matter. II- Absorption .......... .......222 27: The Interaction of Light with Matter. III - Extinction . ......225 28: Radiative Transfer .................................... ...............229 29: Continuum Emission Processes ............................ ...........237 30: The Cooling Function.................................... ............249 4 APPENDICES Appendix A: Vector Calculus....................................... .........251 Appendix B: Conservative Vector Fields .............................. .......256 Appendix C: Integral Theorems................................... ...........257 Appendix D: Curvi-Linear Coordinate Systems ......................... .....258 Appendix E: The Levi-Civita Symbol .................................. .....268 Appendix F: The Viscous Stress Tensor .............................. .......269 Appendix G: Poisson Brackets ..................................... .........272 Appendix H: The BBGKY Hierarchy ................................... .... 274 Appendix I: Derivation of the Energy equation ......................... .....279 Appendix J: The Chemical Potential ................................. .......283 Appendix K: Radiation Essentials .................................... .......286 Appendix L: Velocity Moments of the Collision Integral . ..292 5 LITERATURE The material covered and presented in these lecture notes has relied heavily on a number of excellent textbooks listed below. The Physics of Fluids and Plasmas • by A. Choudhuri (ISBN-0-521-55543) The Physics of Astrophysics–I. Radiation • by F. Shu (ISBN-0-935702-64-4) The Physics of Astrophysics–II. Gas Dynamics • by F. Shu (ISBN-0-935702-65-2) Modern Fluid Dynamics for Physics and Astrophysics • by O. Regev, O. Umurhan & P. Yecko (ISBN-978-1-4939-3163-7) Principles of Astrophysical Fluid Dynamics • by C. Clarke & B.Carswell (ISBN-978-0-470-01306-9) Introduction to Plasma Theory • by D. Nicholson (ISBN-978-0-894-6467705) Introduction to Modern Magnetohydrodynamics • by S. Galtier (ISBN-978-1-316-69247-9) Astrophysics: Decoding the Cosmos • by J. Irwin (ISBN-978-0-470-01306-9) Theoretical Astrophysics • by M. Bartelmann (ISBN-978-3-527-41004-0) Radiative Processes in Astrophysics • by G Rybicki & A Lightman (ISBN-978-0-471-82759-7) Galactic Dynamics • by J. Binney & S. Tremaine (ISBN-978-0-691-13027-9) Modern Classical Physics • by K. Thorne & R. Blandford (ISBN-978-0-691-15902-7) 6 7 Part I: Fluid Dynamics Almost everything we encounter in the Universe, from gas planets to stars, and from the interstellar medium to galaxies, can be categorized as some kind of fluid. Hence, understanding astrophysical processes requires a solid understanding of fluid dynamics. The following chapters present fairly detailed description of the dynamics of fluids with an application to astrophysics. Fluid dynamics is a rich topic, and one could easily devote an entire course to it. The following chapters therefore only scratch the surface of this rich topic. Readers who want to get more indepth information are referred to the following excellent textbooks - The Physics of Fluids and Plasmas by A. Choudhuri - Modern Fluid Dynamics for Physics and Astrophysics by O.Regev et al. - The Physics of Astrophysics II. Gas Dynamics by F. Shu - Principles of Astrophysical Fluid Dynamics by C. Clarke & B. Carswell - Modern Classical Physics by K.Thorne & R. Blandford 8 CHAPTER 1 Introduction to Fluids & Plasmas What is a fluid? A fluid is a substance that can flow, has no fixed shape, and offers little resistance to an external stress In a fluid the constituent particles (atoms, ions, molecules, stars) can ‘freely’ • move past one another. Fluids take on the shape of their container (if not self-gravitating). • A fluid changes its shape at a steady rate when acted upon by a stress force. • What is a plasma? A plasma is a fluid in which (some of) the consistituent particles are electrically charged, such that the interparticle force (Coulomb force) is long-range in nature. Fluid Demographics: All fluids are made up of large numbers of constituent particles, which can be molecules, atoms, ions, dark matter particles or even stars. Different types of fluids mainly differ in the nature of their interparticle forces. Examples of inter-particle forces are the Coulomb force (among charged particles in a plasma), vanderWaals forces (among molecules in a neutral fluid) and gravity (among the stars in a galaxy). Fluids can be both collisional or collisionless, where we define a collision as an interaction between constituent particles that causes the trajectory of at least one of these particles to be deflected ‘noticeably’. Collisions among particles drive the system towards thermodynamic equilibrium (at least locally) and the velocity distribution towards a Maxwell-Boltzmann distribution. In neutral fluids the particles only interact with each other on very small scales. Typically the inter-particle