Theoretical Astrophysics

An Introduction

Matthias Bartelmann

• Publisher: Wiley-VCH
• Year: 2013
• Print ISBN: 9783527410040
• eBook ISBN: 9783527669776
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Astronomy & Astrophysics

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.

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 results are 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	me	9.1094 × 10^–28
Proton mass	mp	1.6726 × 10^–24
Boltzmann’s constant	kB	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 refer...

Table of contents

• Cover
• Contents
• Related Titles
• Title Page
• The Author
• Copyright
• Preface
• Acknowledgements
• 1 Theoretical Foundations
• 2 Radiation Processes
• 3 Hydrodynamics
• 4 Fundamentals of Plasma Physics and Magneto-Hydrodynamics
• 5 Stellar Dynamics
• 6 Brief Summary and Concluding Remarks
• Index