Theoretical Astrophysics E-Book

Matthias Bartelmann

• Herausgeber: Wiley-VCH
• Kategorie: Lebensstil
• Sprache: Englisch
• Veröffentlichungsjahr: 2013

Seitenzahl: 405

Beschreibung

Beginning from first principles and adopting a modular structure, this book develops the fundamental physical methods needed to describe and understand a wide range of seemingly very diverse astrophysical phenomena and processes. For example, the discussion of radiation processes including their spectra is based on Larmor's equation and extended by the photon picture and the internal dynamics of radiating quantum systems, leading to the shapes of spectral lines and the ideas of radiation transport. Hydrodynamics begins with the concept of phase-space distribution functions and Boltzmann's equation and develops ideal, viscous and magneto-hydrodynamics all from the vanishing divergence of an energy-momentum tensor, opening a natural extension towards relativistic hydrodynamics. Linear stability analysis is introduced and used as a common and versatile tool throughout the book. Aimed at students at graduate level, lecturers teaching courses in Theoretical Astrophysics or advanced topics in modern astronomy, this book with its abundant examples and exercises also serves as a reference and an entry point for more advanced researchers wanting to update their knowledge of the physical processes that govern the behavior and evolution of astronomical objects.

Leseprobe

Contents

Preface

Acknowledgements

1 Theoretical Foundations

1.1 Units

1.2 Lorentz Invariance

1.3 Electromagnetism

1.4 Elementary Kinetic Theory

Further Reading

2 Radiation Processes

2.1 Thomson Scattering

2.2 Spectra

2.3 Synchrotron Radiation

2.4 Bremsstrahlung

2.5 Radiation Damping

2.6 Compton Scattering

2.7 Radiative Quantum Transitions

2.8 Shapes of Spectral Lines

2.9 Radiation Quantities

2.10 The Planck Spectrum and Einstein Coefficients

2.11 Absorption and Emission

Further Reading

3 Hydrodynamics

3.1 The Equations of Ideal Hydrodynamics

3.2 Relativistic Hydrodynamics

3.3 Viscous Hydrodynamics

3.4 Flows under Specific Circumstances

3.5 ShockWaves

3.6 Instabilities

Further Reading

4 Fundamentals of Plasma Physics and Magneto-Hydrodynamics

4.1 Collision-Less Plasmas

4.2 Electromagnetic Waves in Media

4.3 Dispersion Relations

4.4 Electromagnetic Waves in Thermal Plasmas

4.5 The Magneto-Hydrodynamic Equations

4.6 Generation of Magnetic Fields

4.7 Ambipolar Diffusion

4.8 Waves in Magnetised Cold Plasmas

4.9 Hydromagnetic Waves

Further Reading

5 Stellar Dynamics

5.1 The Jeans Equations and Jeans′Theorem

5.2 Equilibrium and Stability

5.3 Dynamical Friction

Further Reading

6 Brief Summary and Concluding Remarks

Index

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The Author

Prof. Dr. Matthias Bartelmann

Universität Heidelberg

Inst. Theoretische Astrophysik

Albert-Überle-Str. 2

69120 Heidelberg

Germany

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Preface

This book is not in any sense complete or exhaustive, and it is not meant to be. Its subject, theoretical astrophysics, is vast and cannot possibly be comprehensively covered in a single volume.

This book has a rather different purpose. It is intended as a textbook for students who have a reasonably complete knowledge of the material usually taught in the introductory courses on theoretical physics: classical mechanics, electrodynamics, quantum mechanics, and thermodynamics. Building upon this assumed foundation, this book adds material typically not covered by the introductory lectures, but required for research work in theoretical astrophysics. It may also be useful as a resource for researchers. Arguably the most important extensions are radiation processes, hydrodynamics, plasma physics and magneto-hydrodynamics, and stellar dynamics.

This book provides introductions to these four areas. It is structured into four main chapters and an initial chapter summarising some essential theoretical concepts which the following chapters build upon.

The chapter on radiation processes begins with the Larmor equation from electrodynamics and derives Thomson scattering and a general approach to calculating spectra from it, which is then applied to synchrotron radiation and bremsstrahlung. Up to this point, electromagnetic radiation is described as a classical wave that does not exchange momentum with the charges it originates from or interacts with. The back-reaction of radiation on the radiating charge is then discussed before Compton scattering is introduced, and with it, the photon picture of electromagnetic radiation. The internal structure of radiating systems such as atoms follows, leading to the calculation of cross sections for the interaction of quantummechanical systems with radiation and of the shapes of spectral lines. Finally, radiation is described as an ensemble of photons. Specific intensity, emissivity and opacity, the Planck spectrum and radiation transport are introduced therein.

The chapter on hydrodynamics begins with a derivation of the ideal hydrodynamical equations from elementary kinetic theory. It is emphasised that these equations express the (local) conservation of the energy-momentum tensor. This opens the door for relativistic hydrodynamics as well as various extensions, for example, viscous hydrodynamics and magneto-hydrodynamics. The assumption of an infinitely small mean-free path from ideal hydrodynamics is then relaxed, leading to viscous hydrodynamics. Inviscid and viscous flows are considered under certain simplifying conditions. The formation of shocks and the Sedov solution follow before the discussion of several fluid instabilities concludes the chapter. The discussion in this chapter emphasises the roots of hydrodynamics in the conservation equation for the energy-momentum tensor, the common origin of non-ideal hydrodynamical effects in particle transport, the importance of integrated statements such as Kelvin’s theorem, the Bernoulli equation and the Rankine–Hugoniot conditions, and the general approach to linear perturbation or stability analysis.

The chapter on plasma physics begins with the introduction of the plasma parameters and proceeds to the propagation of electromagnetic waves through a plasma. Dispersion relations are generally derived for transverse and longitudinal waves, touching the phenomenon of Landau damping, and specified for thermal plasmas. The equations of magneto-hydrodynamics are introduced next, emphasising their common ground with hydrodynamics in the vanishing divergence of an energy-momentum tensor. The generation of magnetic fields is briefly discussed, followed by ambipolar diffusion as an example for non-ideal coupling between the plasma charges and the fluid particles. The propagation of electromagnetic waves through cold, magnetised plasmas is studied next, and the chapter concludes with a linear stability analysis, revealing the variety of hydromagnetic and Alfvén modes.

The chapter on stellar dynamics begins with deriving Jeans’ equations in parallel to the hydrodynamical equations, emphasising the importance of anisotropic pressure. Stability criteria for stellar-dynamical systems are then derived, leading to the Jeans and Toomre criteria. Finally, the phenomenon of dynamical friction is introduced and discussed, ending with Chandrasekhar’s formula for the friction force.

Preparing for this selection of subjects, the initial chapter briefly summarises special relativity and relativistic electrodynamics as well as elementary kinetic theory to lay the foundation for the discussions in the following main chapters.

In all chapters, the attempt was made to trace these four areas of theoretical astrophysics back to their origins in the fundamental concepts of theoretical physics. Rather than discussing many examples and trying to cover as many astronomical and astrophysical phenomena as possible, the goal of this book is to reveal the roots of the common approaches in theoretical astrophysics, the choices and assumptions made and the methodical similarities appearing throughout. The book does not aim at explaining the richness of astrophysical phenomena, but at enabling the reader to understand and apply the rich toolbox of theoretical astrophysics by her, or himself. In this spirit, the notorious phrases “one can show” or “as can be shown” do not appear in this book. Every subject discussed is derived from first principles, which is considerably more important to the author than completeness.

This book grew from a one-semester course in theoretical astrophysics developed and regularly taught at the University of Heidelberg. The course was comprised of four hours of lecture and a two-hour tutorial per week. The amount of material collected here is probably at the upper end of what can be covered in a single term of 15 weeks. If it needs to be pruned, the general idea of the course should not be given up: to reveal the foundations of theoretical astrophysics including its important general assumptions, and to identify the common methodical approaches.

By far most, if not all, of the material summarised and compiled in this book is not new. Its intention lies in the foundation and the arrangement of matters, which may be best seen from a common and unifying perspective. It is not at all possible to give full reference to the original derivations and presentations, nor to mention any specific research results. This is therefore not even attempted. Rather, we give a list of more specialised textbooks and refer to them for further reading on individual subjects.

Acknowledgements

Since this book emerged from a course in theoretical astrophysics, many students were exposed to it. I am most grateful to the plethora of comments and suggestions, and the constructive criticism I received, which helped to clarify and sharpen the arguments, to prune derivations of unnecessary loops and details, and to remove errors. In particular, I want to mention Santiago Casas, Martina Schwind, Elena Sellentin and Benjamin Wallisch, who all read and commented on large parts or all of the book.

The perpetual discussions with themembers of the cosmology group at the Institut für Theoretische Astrophysik of Heidelberg University were extremely helpful and clarifying. I wish to thank you all for your great help and support! In particular, I should mention Christian Angrick and Philipp Merkel for designing many problems, working out solutions, unfailingly criticising weaknesses and improving the line of reasoning.

The idea for a course like this dates back to the discussions I had with my friend and colleague Achim Weiß, surely more than a decade ago. I have profited greatly over the years from his unconventional and stimulating view on our common field of research. Many colleagues inspired and impressed me with their broad and comprehensive view of physics, their clarity of thinking and their attitude both towards research and teaching. I wish to expressly name Jürgen Ehlers, Wolfgang Hillebrandt, Jens Niemeyer, Björn Schäfer, Peter Schneider, Norbert Straumann, Christof Wetterich and Simon White, who shaped and influenced this book in many ways, mostly without knowing it.

Large parts of this book were written at personally difficult times. For the support and encouragement I received during this time, I am deeply grateful to a fine group of friends and colleagues: Thank you all for your kindness and attention, and for the precious, lovely hours!

1

Theoretical Foundations

1.1 Units

1.1.1 Lengths, Masses, Times, and Temperatures

We use Gaussian centimetre-gram-second (cgs) units throughout. Lengths are measured in centimetres, masses in grams and time in seconds. The derived units of force, energy and power are listed in Table 1.1. Temperatures are unvariedly measured in Kelvin (K).

The main reason for using these rather than SI units is they allow electromagnetic relations to be expressed in a much easier way, as we shall now discuss.

1.1.2 Charges and Electromagnetic Fields

The unit of charge is chosen such that the Coulomb force between two charges q separated by the distance r is

(1.1)

Table 1.1 The units of force, energy and power are listed here in the cgs system together with their relations to SI units.

Table 1.2 This table lists the units of charge, electric and magnetic field in the Gaussian cgs system, their physical dimensions, and alternative units.

With this choice, the dielectric constant of the vacuum, , becomes dimensionless and unity. Electric and magnetic fields are defined to have the same unit. This is most sensible in view of the fact that they are both related, and can be converted into each other, by Lorentz transforms. Their unit is chosen such that the force caused by an electric field E on a charge q is

(1.2)

This implies that charge, electric and magnetic fields must have the units given in Table 1.2. The squared electric or magnetic field strengths then have the dimension of an energy density.

By definition, the units of charge in the SI and the Gaussian cgs systems are related by

(1.3)

Electrostatic potential differences, or electrostatic potential energy changes per unit charge, are measured in volts in SI units. Consequently, we must have

(1.4)

The energy gained by a unit charge moving through an electrostatic potential difference of 1Volt, defined as the electron volt, must then be

(1.5)

1.1.3 Natural Constants

The most frequently used natural constants in cgs units are tabulated in Table 1.3.

1.2 Lorentz Invariance

This section summarises the concepts of special relativity and their consequences for the structure of space-time and for the dynamics of a particle. Its most important resultsare the relativistic time dilation (1.32) and the Lorentz contraction (1.36), the addition theorem for velocities (1.38) and the transformation of angles (1.41), the combination of energy and momentum into the momentum four vector (1.59) and the relativistic relations (1.62) and (1.63) between energy, momentum and velocity.

Table 1.3 The most frequently used natural constants are tabulated here with their common symbols and their values in cgs units. The values are taken from the Particle Data Group (http://pdg.lbl.gov/, last accessed 25 September 2012).

Quantity

Symbol

Value in cgs units

Light speed

c

2.9979 × 10

10

Elementary charge

e

4.8032 × 10

–10

Electron mass

m

e

9.1094 × 10

–28

Proton mass

m

p

1.6726 × 10

–24

Boltzmann’s constant

k

B

1.3806 × 10

–16

Newton’s constant

G

6.6738 × 10

–8

Planck’s constant

ħ

1.0546 × 10

–27

Perhaps it is helpful to begin with the statement that classical physics aims to quantify the behaviour of physical entities in space with time. Point mechanics, for example, studies the trajectories of particles with negligible extension. A trajectory can be quantified by a vector-valued function which assigns a spatial vector to any instant t from a finite or infinite time interval. Field theory describes forces as the effect of fields, which are functions of space and time obeying their own dynamics. Immediately, we are led to the question of how we want to identify points in space and instants in time in a quantifiable manner.

This is achieved by a reference frame or a coordinate system. In Newtonian physics, space and time were both assumed to be absolute. A rigid reference frame was assumed to exist which identified each point in space by a triple of real-valued, spatial coordinates, and by a real number for the time. Having formulated the laws of physics in this absolute frame, the immediate further question arises as to how other frames of reference, or coordinate systems, could be chosen such that those laws would remain valid without changing their form. The answer of Newtonian physics was that the laws of physics are the same in all so-called inertial frames. In slightly different words, the laws of physics were claimed to be invariant under all transformations, leading from one inertial frame to another.