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Introduction to PLASMA ASTROPHYSICS (Selected 10 lectures)
Boris V. Somov Astronomical Institute and Faculty of Physics Moscow State University

Pushino-na-Oke, 2008


Contents
Ab out These Lectures 1 Particles and Fields: Exact Self-Consistent Description 1.1 Liouville's theorem . . . . . . . . . . . . . . . . . . . 1.1.1 Continuity in phase space . . . . . . . . . 1.1.2 The character of particle interactions 1.1.3 The Lorentz force, gravity . . . . . . . . . 1.1.4 Collisional friction . . . . . . . . . . . . . . . 1.1.5 The exact distribution function . . . . 1.2 Charged particles in the electromagnetic field . . . . . . . . . 1.2.1 General formulation of the problem . 1.2.2 The continuity equation for electric charge . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Initial equations and initial conditions 1.2.4 Astrophysical plasma applications . . 1.3 Gravitational systems . . . . . . . . . . . . . . . . . . . . . . 1.4 Practice: Exercises and Answers . . . . . . . . . . . . . . . . 2 Statistical Description of Interacting Particle Systems 2.1 The averaging of Liouville's equation . . . . . . . . . . . . . 2.1.1 Averaging over phase space . . . . . . . 2.1.2 Two statistical p ostulates . . . . . . . . 2.1.3 A statistical mechanism of mixing . 2.1.4 Derivation of a general kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 A collisional integral and correlation functions . . . . . . . . 2.2.1 Binary interactions . . . . . . . . . . . . . . 2.2.2 Binary correlation . . . . . . . . . . . . . . 1 5 9 9 9 12 15 16 18 22 22 24 25 26 28 29 33 33 33 35 37 40 43 43 46

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2 2.2.3 2.3 2.4

CONTENTS

. Equations for correlation functions . . . . . . . . . . . . . . . Practice: Exercises and Answers . . . . . . . . . . . . . . . .

The collisional integral and binary correlation . . . . . . . . . . . . . . . . . . . . .

48 51 54

3 Weakly-Coupled Systems with Binary Collisions 55 3.1 Approximations for binary collisions . . . . . . . . . . . . . . 55 3.1.1 The small parameter of kinetic theory 55 3.1.2 The Vlasov kinetic equation . . . . . . . 58 3.1.3 The Landau collisional integral . . . . . 59 3.1.4 The Fokker-Planck equation . . . . . . . 61 3.2 Correlations and Debye-Huckel shielding . . . . . . . . . . . . 64 Е 3.2.1 The Maxwellian distribution function 64 3.2.2 The averaged force and electric neutrality . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.3 Pair correlations and the Debye-Huckel Е radius . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3 Gravitational systems . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Comments on numerical simulations . . . . . . . . . . . . . . 71 3.5 Practice: Exercises and Answers . . . . . . . . . . . . . . . . 73 4 Macroscopic Description of Astrophysical Plasma 4.1 Summary of microscopic description . . . . . . . . . . . . . . 4.2 Definition of macroscopic quantities . . . . . . . . . . . . . . 4.3 Macroscopic transfer equations . . . . . . . . . . . . . . . . . 4.3.1 Equation for the zeroth moment . . . . 4.3.2 The momentum conservation law . . . 4.4 The energy conservation law . . . . . . . . . . . 4.4.1 The second moment equation . . . . . . 4.4.2 The case of thermo dynamic equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 The general case of anisotropic plasma 4.5 General properties of transfer equations . . . . . . . . . . . . 4.5.1 Divergent and hydro dynamic forms . 4.5.2 Status of the conservation laws . . . . . 4.6 Equation of state and transfer coefficients . . . . . . . . . . . 4.7 Gravitational systems . . . . . . . . . . . . . . . . . . . . . . 77 77 78 80 80 82 86 86 89 90 92 92 94 95 98


CONTENTS 5 The 5.1 5.2 5.3 5.4 Generalized Ohm's Law in Plasma The classic Ohm's law . . . . . . . . . . . . . . . . . . . . . Derivation of basic equations . . . . . . . . . . . . . . . . . The general solution . . . . . . . . . . . . . . . . . . . . . . The conductivity of magnetized plasma . . . . . . . . . . . 5.4.1 Two limiting cases . . . . . . . . . . . . . . 5.4.2 The physical interpretation . . . . . . . Currents and charges in plasma . . . . . . . . . . . . . . . . 5.5.1 Collisional and collisionless plasmas 5.5.2 Volume charge and quasi-neutrality Practice: Exercises and Answers . . . . . . . . . . . . . . .

3 101 101 102 106 107 107 108 111 111 115 117 119 119 119 120 123 126 126 128 131 133 134 135 135 139 141 143 146

5.5

5.6

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6 Single-Fluid Mo dels for Astrophysical Plasma 6.1 Derivation of the single-fluid equations . . . . . . . . . . . . 6.1.1 The continuity equation . . . . . . . . . . 6.1.2 The momentum conservation law . . 6.1.3 The energy conservation law . . . . . . 6.2 Basic assumptions and the MHD equations . . . . . . . . . 6.2.1 Old simplifying assumptions . . . . . . 6.2.2 New simplifying assumptions . . . . . 6.2.3 Non-relativistic MHD . . . . . . . . . . . 6.2.4 Energy conservation . . . . . . . . . . . . . 6.2.5 Relativistic magnetohydro dynamics 6.3 Magnetic flux conservation. Ideal MHD . . . . . . . . . . . 6.3.1 Integral and differential forms of the law . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 The ideal MHD . . . . . . . . . . . . . . . . 6.3.3 The `frozen field' theorem . . . . . . . . 6.4 Magnetic reconnection . . . . . . . . . . . . . . . . . . . . . 6.5 Practice: Exercises and Answers . . . . . . . . . . . . . . . 7 MHD in Astrophysics 7.1 The main approximations in ideal MHD . . . . . . . . . . . 7.1.1 Dimensionless equations . . . . . . . . . 7.1.2 Weak magnetic fields in astrophysical plasma . . . . . . . . . . . . . . . . . . . . . 7.1.3 Strong magnetic fields in plasma . . 7.2 Accretion disks of stars . . . . . . . . . . . . . . . . . . . . 7.2.1 Angular momentum transfer . . . . . .

149 . 149 . 149 . . . . 152 153 157 157


4

CONTENTS 7.2.2 Accretion in cataclysmic variables 7.2.3 Accretion disks near black holes . . 7.2.4 Flares in accretion disk coronae . . Astrophysical jets . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Jets near black holes . . . . . . . . . . . 7.3.2 Relativistic jets from disk coronae Practice: Exercises and Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 160 161 162 162 165 166 169 169 172 173 175 181 183 183 188 191

7.3

7.4

8 Plasma Flows in a Strong Magnetic Field 8.1 The general formulation of a problem . . . . . . . . . . . . 8.2 The formalism of 2D problems . . . . . . . . . . . . . . . . 8.2.1 The first typ e of problems . . . . . . . . 8.2.2 The second typ e of MHD problems 8.3 The existence of continuous flows . . . . . . . . . . . . . . . 8.4 Flows in a time-dependent dipole field . . . . . . . . . . . . 8.4.1 Plane magnetic dip ole fields . . . . . . 8.4.2 Axial-symmetric dip ole fields . . . . . 8.5 Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Ab out These Lectures
If you want to learn the most fundamental things about plasma astrophysics with the least amount of time and effort - and who doesn't? - this text is for you. The textbook is addressed to students without a background in plasma physics. It grew from the lectures given at the Moscow Institute of Physics and Technics (the `fiz-tekh') since 1977. A similar full-year course was offered to the students of the Astronomical Division in the Faculty of Physi