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Here 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’

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Here 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’ These lecture notes describe the material covered during the Spring 2020 semester of the course Astrophysical Flows at Yale University One goes deep into the flow Becomes busy flowing to it Lost in the flow What happens Is that the one forgets What he is flowing for from ”The River Flows” by Sachin Subedi 2 CONTENTS 1: Introduction to Fluids and Plasmas ......................... ................7 2: Dynamical Treatments of Fluids ........................... ................14 3: Hydrodynamic Equations for Ideal Fluid ..................... ..............23 4: Viscosity, Conductivity & The Stress Tensor . ..............26 5: Hydrodynamic Equations for Non-Ideal Fluid . .............32 6: Kinetic Theory I: from Liouville to Boltzmann . ..........36 7: Kinetic Theory II: from Boltzmann to Navier-Stokes . ............52 8: Vorticity & Circulation . .................64 9: Hydrostatics and Steady Flows ........................... .................72 10: Viscous Flow and Accretion Flow ............................ ..............84 11: Turbulence .......................................... ......................95 12: Sound Waves ......................................... ....................103 13: Shocks ............................................. ......................111 14: Fluid Instabilities . ..................119 15: Collisionless Dynamics: CBE & Jeans equations . ........129 16: Collisions & Encounters of Collisionless Systems . .........146 17: Solving PDEs with Finite Difference Methods . .........157 18: Consistency, Stability, and Convergence . ................173 19: Reconstruction and Slope Limiters ......................... ...............180 20: Burgers’ Equation & Method of Characteristics.............. ..............190 21: The Riemann Problem & Godunov Schemes . ..........198 22: Plasma Characteristics ................................. ..................212 23: Plasma Orbit Theory .................................... .................223 24: Plasma Kinetic Theory ................................... ................231 25: Vlasov Equation & Two-Fluid Model .......................... ...........237 26: Magnetohydrodynamics ................................ ..................244 3 APPENDICES Appendix A: Vector Calculus....................................... .........257 Appendix B: Conservative Vector Fields .............................. .......262 Appendix C: Integral Theorems................................... ...........263 Appendix D: Curvi-Linear Coordinate Systems ......................... .....264 Appendix E: Differential Equations .................................. ........274 Appendix F: The Levi-Civita Symbol .................................. ..... 280 Appendix G: The Viscous Stress Tensor .............................. .......281 Appendix H: Equations of State .................................... ........ 284 Appendix I: Poisson Brackets ..................................... ..........291 Appendix J: The BBGKY Hierarchy ................................... .....293 Appendix K: Derivation of the Energy equation ......................... ....298 Appendix L: The Chemical Potential ................................. .......302 4 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–II. Gas Dynamics • by F. Shu (ISBN-0-935702-65-2) Introduction to Plasma Theory • by D. Nicholson (ISBN-978-0-894-6467705) The Physics of Plasmas • by T. Boyd & J. Sanderson (ISBN-978-0-521-45912-9) Modern Fluid Dynamics for Physics and Astrophysics • by O. Regev, O. Umurhan & P. Yecko (ISBN-978-1-4939-3163-7) Theoretical Astrophysics • by M. Bartelmann (ISBN-978-3-527-41004-0) Principles of Astrophysical Fluid Dynamics • by C. Clarke & B.Carswell (ISBN-978-0-470-01306-9) Introduction to Modern Magnetohydrodynamics • by S. Galtier (ISBN-978-1-316-69247-9) Modern Classical Physics • by K. Thorne & R. Blandford (ISBN-978-0-691-15902-7) Galactic Dynamics • by J. Binney & S. Tremaine (ISBN-978-0-691-13027-9) Galaxy Formation & Evolution • by H. Mo, F. van den Bosch & S. White (ISBN-978-0-521-85793-2) 5 6 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. • 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 force is a vanderWaals force, which drops off very rapidly. Put differently, the typical cross section for interactions is the size of the particles (i.e., the Bohr radius for atoms), which is very small. Hence, to good approximation 7 Figure 1: Examples of particle trajectories in (a) a collisional, neutral fluid, (b) a plasma, and (c) a self-gravitating collisionless, neutral fluid. Note how different the dynamics are. particles in a neutral fluid move in straight lines in between highly-localized, large- angle scattering events (‘collisions’). An example of such a particle trajectory is shown Fig. 1a. Unless the fluid is extremely dilute, most neutral fluids are collisional, meaning that the mean free path of the particles is short compared to the physical scales of interest. In astrophysics, though, there are cases where this is not necessarily the case. In such cases, the standard equations of fluid dynamics may not be valid! 2 In a fully ionized plasma the particles exert Coulomb forces (F~ r− ) on each other. ∝ Because these are long-range forces, the velocity of a charged particle changes more likely due to a succession of many small deflections rather than due to one large one. As a consequence, particles trajectories in a highly ionized plasma (see Fig. 1b) are very different from those in a neutral fluid. In a weakly ionized plasma most interactions/collisions are among neutrals or be- tween neutrals and charged particles. These interactions are short range, and a weaky ionized plasma therefore behaves very much like a neutral fluid. In astrophysics we often encounter fluids in which the mean, dominant interparticle force is gravity. We shall refer to such fluids are N-body systems. Examples are dark matter halos (if dark matter consists of WIMPs or axions) and galaxies (stars act like neutral particles exerting gravitational forces on each other). Since gravity is a long-range force, each particle feels the force from all other particles. Consider the gravitational force F~i at a position ~xi from all particles in a relaxed, equilibrium system. We can then write that F~ (t)= F~ + δF~ (t) i h ii i 8 Here F~ i is the time (or ensemble) averaged force at i and δF~i(t) is the instantaneous deviationh i due to the discrete nature of the particles that make up the system. As N then δF~ 0 and the system is said to be collisionless; its dynamics → ∞ i → are governed by the collective force from all particles rather than by collisions with individual particles. As you learn in Galactic Dynamics, the relaxation time of a gravitational N-body system, defined as the time scale on which collisions (i.e., the impact of the δF~ above) cause the energies of particles to change considerably, is N t t relax ≃ 8 ln N cross where tcross is the crossing time (comparable to the dynamical time) of the system. 10 50 60 Typically N 10 (number of stars in a galaxy) or 10 − (number of dark matter ∼ particles in a halo), and tcross is roughly between 1 and 10 percent of the Hubble time (108 to 109 yr). Hence, the relaxation time is many times the age of the Universe, and these N-body systems are, for all practical purposes, collisionless. As a consequence, the particle trajectories are (smooth) orbits (see Fig. 1c), and understanding galactic dynamics requires therefore a solid understanding of orbits. Put differently, ‘orbits are the building blocks on galaxies’. Collisional vs. Collisionless Plasmas: If the collisionality of a gravitational system just depends on N, doesn’t that mean that plasmas are also collisionless? After all, the interparticle force in a plasma is the Coulomb force, which has the same long-range 1/r2 nature as gravity. And the number of particles N of a typical plasma is huge ( 1010) while the dynamical time can be large as well (this obviously depends on the length≫ scales considered,
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