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Jan Trulsen

The Institute of Theoretical AstrophysicsUniversity of Oslo

Winter 2009

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Foreword

The following compendium constitutes the curriculum for the course AST 3210, RadiationI as taught at the Institute of Theoretical Astrophysics at the University of Oslo for thespring semester 2009. The compendium is in a preliminary form, figures and the selectionof problems in particular are still incomplete. Suggestions for improvement of the text and

reports of misprints are most welcomed.The course aims at providing quantitative training in basic physics concepts constituting

the foundation for any advanced study of astrophysical phenomena. The course starts with abrief review of Maxwell equations and electromagnetic waves, then proceeds to a discussion ofelectromagnetic radiation emitted by accelerated charged particles. Next, an introduction toquantum mechanics at an intermediate level is given to lay the foundation for an understand-ing of atomic and molecular spectra. An introduction to basic statistical concepts, statisticalphysics and thermodynamics including the Saha equation is included, mostly for referencepurposes. Finally, the course includes an introduction to the topic of radiation transport.The latter topic deals with the question of how radiation is generated in, interacts with, andis transported through material media. It has been necessary to limit the major part of the

discussion of the latter topic to the local thermal equilibrium (LTE) approximation.For reference purposes the compendium also contains an introduction to fluid mechan-

ics including magnetohydrodynamics (MHD) and dimensional analysis, in addition to anappendix outlining useful vector calculus results.

The different topics covered, are discussed as topics of physics. The selection of topics,examples and problems have been made on the basis of their astrophysical relevance. Themore detailed discussion of astrophysical phenomena is deferred to subsequent and morespecialized courses in astrophysics.

The course is best suited for students who have completed at least one year training in bothmathematics and in physics. She is expected to have working knowledge of linear algebra,including the eigenvalue problem, calculus including differential equations and the Gauss and

Stoke integral theorems. Likewise he is expected to be acquainted with basic properties ofMaxwells equations, Newtonian mechanics and also to have passed a first courses in quantummechanics and statistical physics.

In the compendium emphasis is put on the derivation of import results from the basicequations, sometimes based on simplified models. The focus of the course is, never the less,on applying these results for the solution of relevant astrophysical questions. This is reflectedin the fact that the course makes use of open book exams.

The compendium is supplied with a number of problems to be solved, some simple, othersmore demanding. Most of these problems can be and should be solved analytically. Otherswill require the use of computers for numerical or graphical reasons. The student is thereforeexpected to be familiar with a suitable programming language such as MATLAB, Python,

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ii

IDL or others.SI units are used throughout the compendium. This modern unit system is by now

totally dominating the science literature. The student should, however, be warned that one

may still find astrophysical texts, particularly within the field of radiation transport, clingingto obsolete unit systems.

No explicit reference to literature is made in the compendium. Instead the student isdirected to standard textbooks for further reading. Much additional information can of coursebe found on the web, for instance on Wikipedia, and in the professional literature. The studentis encouraged to familiarize herself and make regular use of these resources. Below is collectionof textbooks that have been consulted during the writing of the compendium:

J. D. Jackson: Classical Electrodynamics, John Wiley, ISBN 0-471-30932-X D. J. Griffiths: Introduction to Quantum Mechanics, Pearson Prentice Hall, ISBN 0-

13-191175-9

C. Kittel & H. Kroemer: Thermal Physics, Freeman, ISBN 0-77167-1088-9 B. H. Bransden & C. J. Joachain: Physics of atoms and molecules, Longman Scientific

and Technical, ISBN 0-582-44401-2

D. Mihalas & B. Weibel-Mihalas: Foundations of Radiation Hydrodynamics, Dover,ISBN 0-486-40925-2

Oslo, Winter 2009 Jan Trulsen

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Contents

1 Electromagnetic Waves 3

1.1 Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 The Poynting Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Electromagnetic Waves in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Complex field notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 The Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Wave Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.6 Non-Monochromatic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6.1 Wave coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.6.2 Partial polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6.3 Spatial-temporal Fourier representation . . . . . . . . . . . . . . . . . 19

1.6.4 Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7 Specific Intensity of Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.8 Interaction of Waves and Matter . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.8.1 Non-magnetized Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . 241.8.2 Magnetized Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.9 The Ray Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.10 The Radiative Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . 34

2.1 Electromagnetic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Radiation from Point Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3 Radiated Energy and Power Spectrum . . . . . . . . . . . . . . . . . . . . . . 42

2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4.1 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4.2 Cyclotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.4.3 Thomson scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Spectra of OneElectron Atoms 57

3.1 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2 The One-Electron Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Physical Interpretation of Quantum Numbers . . . . . . . . . . . . . . . . . . 69

3.4 The Isotope Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 An External Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.6 The Electron Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.7 Total Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

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3.8 Spectroscopic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.9 Transition Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.10 Selection Rules and Atomic Lifetimes . . . . . . . . . . . . . . . . . . . . . . . 88

3.11 Spectral Line Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.11.1 Natural line profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.11.2 Collisional or pressure broadening . . . . . . . . . . . . . . . . . . . . 933.11.3 Thermal Doppler broadening . . . . . . . . . . . . . . . . . . . . . . . 94

3.11.4 The Voigt line profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.12 Fine Structure Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.13 T he Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.13.1 The weak field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.13.2 The strong field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.13.3 Intermediate magnetic field strength . . . . . . . . . . . . . . . . . . . 105

3.13.4 Physical visualization of angular momentum coupling . . . . . . . . . 106

3.13.5 Polarization and directional effects . . . . . . . . . . . . . . . . . . . . 107

3.14 T he Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.15 Nuclear Spin and Hyperfine Effects . . . . . . . . . . . . . . . . . . . . . . . . 111

4 Spectra of Many-Electron Atoms 115

4.1 The Pauli Exclusion Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2 The Central Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3 Angular Momenta and their Summation . . . . . . . . . . . . . . . . . . . . . 119

4.4 Electron Correlation and Fine Structure Effects . . . . . . . . . . . . . . . . . 121

4.5 Spectroscopic Notation and the Periodic System . . . . . . . . . . . . . . . . 126

4.6 Summary of Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.7 Alkali Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.8 Helium and the Alkaline Earths . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.9 Effects of External Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.9.1 The Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.9.2 The Stark effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Molecular Spectra 139

5.1 The Diatomic Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.2 Molecular Vibration and Rotation . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3 Selection Rules for Vibrational-Rotational Transitions . . . . . . . . . . . . . 146

5.4 Generalized Oscillator-Rotor Models . . . . . . . . . . . . . . . . . . . . . . . 148

5.5 Electronic-Vibrational-Rotational Spectra . . . . . . . . . . . . . . . . . . . . 151

5.6 Comments on Polyatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . 153

5.7 Coupling of Angular Momenta in Molecules . . . . . . . . . . . . . . . . . . . 154

6 Thermal and Statistical Physics 157

6.1 Probability Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2 Entropy and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.3 The Boltzmann Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.4 Particles in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.5 The Maxwell Velocity Distribution . . . . . . . . . . . . . . . . . . . . . . . . 166

6.6 The Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

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8.1 Emissivity and Extinction Coefficient . . . . . . . . . . . . . . . . . . . . . . . 2478.1.1 Radiation and Matter in Thermal Equilibrium . . . . . . . . . . . . . 247

8.1.2 Local Thermal Equilibrium (LTE) . . . . . . . . . . . . . . . . . . . . 2488.1.3 Contributions to Emissivity and Extinction Coefficient . . . . . . . . . 248

8.2 Plane-parallel Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2518.2.1 Interior approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528.2.2 Surface approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

8.3 Line Formation for Optically Thick Medium . . . . . . . . . . . . . . . . . . . 254

9 Dimensional analysis 257

9.1 The -Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2579.2 Simple Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

A Vector Calculus and the operator 265A.1 The grad, div and curl Operators . . . . . . . . . . . . . . . . . . . . . . . . . 265A.2 Orthogonal Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . 267

A.2.1 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 270A.2.2 Cylinder coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271A.2.3 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

A.3 Introduction of the operator . . . . . . . . . . . . . . . . . . . . . . . . . . 271A.4 General operator Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 272A.5 Integral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273A.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275A.7 The Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

A.8 The Dirac function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

B Solutions to Selected Problems 283

C Physical Constants 285

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List of Symbols

aB Bohr radiusc speed of light (in vacuum)e elementary charge

e polarization vector

g Lande factorg multiplicity

g gravitaional accelerationh Plancks constant = h/2

j angular momentum quantum numberj electric current densityk wave number

k wave vectork spring constant quantum number

m particle massmE electric dipole momentmj azimuthal angular momentum quantum numberm quantum numbermL orbital magnetic dipole momentms azimuthal spin quantum numbermh mass of hydrogen atommS spin magnetic dipole momentn principal quantum numbernQ quantum consentration

p degree of polarization

p momentumpF Fermi momentumr position vectors spin quantum numbert timeu energy densityu specific internal energyuB magnetic field energy densityuE electric field energy densityv vibrational quantum numberv flow velocity

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vph phase speedvgr group velocitywfi transition rate

A directed surface areaB rotational constantB magnetic field (magnetic flux density)B Plancks radiation functionCs sound speedD viscous dissipation rateDe dissociation energy

E electric fieldF forceF Helmholtz free energyFc Coriolis force per unit volume

Fem Lorentz force per unit volumeF viscous force per unit volumeG gravitational constantI electric currentI Stokes parameterIe moment of inertiaH HamiltonianJ total angular quantum number

J total angular momentumJz azimuthal total angular momentumL total orbital angular quantum number

L associate Laguerre polynomialL orbital angular momentumLz azimuthal component of orbital angular momentumM Mach numberMJ total azimuthal angular quantum numberML total azimuthal orbital angular quantum numberMN azimuthal nuclear orbital angular quantum numberMS total azimuthal spin angular quantum numberN number of particlesN nuclear orbital angular quantum numberNz azimuthal component of nuclear orbital angular momentum

N nuclear orbital angular momentumN refractive indexNr ray refractive indexP pressureP Poyntings vectorP spectral power densityPm associate Legendre polynomialQ electric chargeQ Stokes parameterR radius of curvatureRe equilibrium internuclear distance

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CONTENTS ix

Rn Laguerre functionRH Rydberg constantS total spin angular quantum number

S spin angular momentumSz azimuthal component of spin angular momentumT temperatureU potential energyU Stokes parameterV Stokes parameterV volumeVA Alfven speedW energyZ charge numberZ partition function

Z partition function fine structure constant phase shift0 permitivity of vacuum coefficient of viscosity Boltzmann constant coefficient of heat conduction wavelength chemical potential mean molecular weight reduced mass

0 permeability of vacuum frequency0 vibrational frequency phase azimuthal anglee electric charge densitym mass density repetence, = 1/ electric conductivity fundamental entropy StefanBoltzmann constant

fundamental temperature polar angle angular frequencyL angular Larmor frequency vorticity circulation magnetic flux rate of strain increasec centripetal potentialg gravitational potential velocity potential

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wave function angular velocity

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mtex

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Chapter 1

Electromagnetic Waves

Any study of distant astronomical objects is totally dependent on the information that maybe gathered on these objects. Apart from sending spacecrafts to perform actual in situ mea-surements, four different information channels are available to us. We may collect informationthrough the electromagnetic radiation reaching us from these distant objects. We may studycosmic rays, material particles moving with velocities close to the speed of light and originat-ing in different parts of the universe. We may study the flux of neutrinos and anti-neutrinos,that is, near mass-less particles moving with the speed of light and created by nuclear reac-tions in the interior of stars. Finally, we may try to infer information from the presence ofgravitational waves.

Gravitational waves have been predicted since the time of birth of the theory of generalrelativity. The time history of the Hulse-Taylor binary pulsar provides indirect evidence fortheir existence, but we will have to wait well into this century before gravitational wavespossibly become an important information channel of astrophysical phenomena.

Neutrinos are produced through several nuclear reactions, for instance, the reaction

1H + 1H 2H + e+ + ,which plays a central role in the energy production in the interior of any star. The neutrinoflux is difficult to detect because these particles hardly interact with matter, in fact, theneutrino flux passes through the solid Earth almost unobstructed. One successful detectionscheme has been to make use of the reaction

37Cl + 37Ar + e

in large chlorine-filled tanks located in deep underground pits, well shielded from the per-turbing influence of cosmic rays. An outstanding problem in astrophysics for many years wasthat the neutrino flux detected in this way was a factor 2 - 3 less than theoretically expected.We now have an explanation of this puzzle.

Cosmic rays have been observed over many years through the tracks left in photographicemulsions placed behind leaden shields of varying thicknesses. Cosmic rays are produced byseveral astrophysical processes. Supernova remnants are one of the more important sources.Cosmic ray observations provide important constraints for any understanding of processesresponsible for the acceleration of these high-energy particles.

The by far most important information channel on astronomical objects is, however, stillthe electromagnetic radiation with frequencies ranging from radio-waves through the infrared,

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4 CHAPTER 1. ELECTROMAGNETIC WAVES

visible and ultraviolet parts of the spectrum to energetic X-rays and -rays. Technologicaldevelopments have over recent years significantly improved our ability to observe this spec-trum both from ground-based observatories, but also from space-born platforms without the

perturbing effects of our atmosphere. It is therefore only natural that our discussion of fun-damental topics in astrophysics starts with a short review of basic parts of electromagnetictheory.

1.1 Electric and Magnetic Fields

The Maxwell equations constitute the basis for any discussion of electromagnetic phenomena.The reader is assumed to be acquainted with the physical content of these equations already.We shall therefore only briefly review some of their properties. In the simplest version theseequations take the form

E= 0

(1.1)

E= Bt

(1.2)

B = 0 (1.3) B = 0

j + 0

E

t

. (1.4)

Here E is the electric field intensity and B the magnetic flux density. For convenience, EandB will be referred to as the electric and magnetic fields in the following. The electric chargeand current densities and j as well as the fields E and B are all functions of space r and

time t. The constants 0 and 0 are the permittivityand permeability of vacuum. In SI-units,the latter coefficient is defined as 0 4 107 H/m while 0 = 1/(0c2) where c is thespeed of light in vacuum, c = 2.997925 108 m/s. Basic properties of the divergence and curloperators, div = and curl = , acting on the electric and magnetic fields are reviewedin appendix A. The explicit forms of the divergence and curl operators depend on the choiceof coordinate system. For Cartesian, cylindrical and spherical coordinates, as well as for anarbitrary orthonormal curvilinear coordinate system, these forms are listed in appendix A.

Maxwell equations in the form (1.1) - (1.4) are sometimes referred to as the vacuum versionof the Maxwell equations. This means that all sources of charge and current densities areincluded in and j. In material media, it is often convenient to restrict and j to the freecharge and conduction current densities. Polarization charge and current densities and the

magnetization current density are then included through the permittivity and permeabilityof the medium. We shall return to this aspect in a later chapter.

The Maxwell equations express the fact that the charge and current densities and j arethe sources for the electric and magnetic fields E and B. Gauss law for electric fields (1.1)states that electric field lines originate in electric charges. This is seen by making use of thedefinition (A.3) of the divergence operator as applied to the electric field E,

E limV0

1

V

A

d2AE. (1.5)

where d2A is an outward pointing differential element of the closed surface A bounding theinfinitesimal volume V. The geometry is illustrated in figure 1.1a. According to (1.1), this

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1.1. ELECTRIC AND MAGNETIC FIELDS 5

a)

d

Q

V

b)C

I

d

En

l

B

A

A

A

Figure 1.1: Illustrating Gauss and Amperes laws

means that the total flux of the electric field,

A d

2A E, out of any volume element V, isequal to the total amount of electric charge Q = V d3r inside that volume, except for thefactor of proportionality 10 ,

0.

The corresponding Gauss law for magnetic fields (1.3), expresses the fact that magneticcharges do not exist. This means that the magnetic field has a solenoidalcharacter: a magneticfield line never ends, or equivalently, the net magnetic flux

A d

2A B leaving any volumeelement V bounded by the surface A vanishes,

For slowly time-varying phenomena the last term of the Maxwell law (1.4) may often beneglected, |0E/t ||j |. The simplified version of the Maxwell law,

B = 0j, (1.6)

is referred to as the Ampere law. The Ampere law may be given a simple geometric interpre-tation by making use of the definition (A.4) of the curl operator as applied to the magneticfield B. The component of the curl of the vector field B along a direction n is given by

n B limA0

1

AC

d B. (1.7)

where the line integral on the right hand side is taken along the closed contour C boundingthe open surface A= A n. The positive direction of the contour C and the direction n ofthe surface Aare related through the well-known right hand rule as indicated in figure 1.1b.Thus, Amperes law states that the circulation

C d B of the magnetic field around the

perimeter C of any open surface A, equals the total electric current I = A d2A j passingthrough that surface, except for the constant of proportionality 0,

Cd B = 0I.

For slowly time-varying (low-frequency) phenomena, the Faraday law (1.2) represents themain coupling between electric and magnetic fields. A time varying magnetic flux A d2AB

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6 CHAPTER 1. ELECTROMAGNETIC WAVES

inside a given electric circuit C bounding an area A results in an induced electromotive force1

C d E,

C d E= ddt A d2AB.This in turn may drive electric currents which couple back to the magnetic field. This classof phenomena includes for instance the dynamics of stellar atmospheres, the generation ofstellar and large scale inter-stellar magnetic fields. We shall return to this class of phenomenain later chapters.

In the high-frequency limit, the displacement current, the 0E/t-term of the Maxwelllaw (1.4), contributes importantly to the coupling between the electric and magnetic fields.This term accounts for the existence of electromagnetic waves and therefore for how electricand magnetic fields may decouple from their proper sources and j and propagate away.Before we turn to a review of basic properties of these waves, we shall need another importantproperty of electromagnetic fields.

Quiz 1.1 : Write the explicit form of Maxwell equations (1.1) - (1.4) in Cartesian andspherical coordinates.

Quiz 1.2 : Make use of the Gauss and Stoke integral theorems, (A.27) and (A.28), torecast Maxwell equations (1.1) - (1.4) into the equivalent integral form

0(1.8)

Cd E= d

dt

A

d2A B (1.9)

A d2AB = 0 (1.10)C

d B = 0

I + 0d

dt

A

d2A E

. (1.11)

Give the proper definitions of all quantities involved.

Quiz 1.3 : Show that the equation

/t + j = 0 (1.12)follows from Maxwells equations. What is the corresponding integral form of thisequation? Give a physical interpretation of the terms appearing in the equation.

Quiz 1.4 : Make use of the Maxwell equations in integral form toa) show that the electric field from a point charge Q at the origin is given by

E(r) =Q

40 r2r

b) show that the magnetic field from a constant current I flowing along the positivez-axis is given by

B(r) =0 I

2

1The traditional notation is rather unfortunate, the electromotive force is no force but an induced electricvoltage!

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1.2. THE POYNTING THEOREM 7

c) find the current I flowing in a plane circular circuit of radius a and resistanceR during the time interval T in which a uniform magnetic field perpendicular to theplane of the circuit increases linearly from 0 to B. Determine the direction of the

current.Interprete the symbols used in these results!

1.2 The Poynting Theorem

To create electric or magnetic fields, energy is required. This energy may be recovered whenthe fields are destroyed. Therefore, the fields represent potential energy. The electromag-netic fields are also able to transport this potential energy from one location to another.Explicit expressions for the energy density and the energy flux associated with the electro-magnetic fields are readily derived from Maxwell equations. Scalar multiplications of (1.2)

and (1.4) with B/0 and E/0 and a subsequent addition of terms lead to the result 1

0(E B) =

t(

02E2 +

1

20B2) j E. (1.13)

In deriving the left hand term we made use of the vector identity (A.25). Equation (1.13)is known as the Poynting theorem and allows for the following interpretation. The last termj E is recognized as the work performed by the current source j on the fields per unitvolume and time. With the geometric interpretation of the divergence operator in mind, theleft hand term must represent the outflow of energy per unit volume and time, and thereforethat the Poynting vector,

P =1

0

(EB), (1.14)expresses the energy flux carried by the fields. If the work done by the source j on the field Edoes not balance the energy outflow, the energy density in the fields must change with time.Thus, we interpret

u = uE+ uB =02E2 +

1

20B2 (1.15)

as the sum of electric and magnetic field energy densities.

Quiz 1.5 : Show for any volume V with boundary surface A thatd

dt Vu d3r +

AP d2A=

Vj Ed3r (1.16)

where P and u are defined by (1.14) and (1.15). Interpret the physical meaning ofeach term.

1.3 Electromagnetic Waves in Vacuum

The Maxwell equations may be written as a vector equation in, for instance, the electric fieldalone. Taking the curl of (1.2) and substituting the expression for B from (1.4) leads tothe wave equation for the electric field in vacuum

2E

1

c2

2

t2E= 0. (1.17)

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8 CHAPTER 1. ELECTROMAGNETIC WAVES

Physically allowable electric fields in vacuum must satisfy (1.17) together with the divergencecondition E= 0. For instance, any field of the form

E= xf(z ct)where f is an arbitrary, twice differentiable function will be allowed.

In the following we shall often be satisfied with a simpler class of solutions. Direct sub-stitution will show that the electromagnetic fields

E(r, t) = E0 cos(k r t)B(r, t) = B0 cos(k r t) (1.18)

constitute a solution of Maxwell equations (1.1) - (1.4) in a source-free region of space, =j = 0, provided the conditions

= kc (1.19)and

k E0 = B0 and k E0 = 0 (1.20)are satisfied. Here k and are the wave vector and angular frequency of the wave, and E0and B0 are the amplitude vectors of the electric and magnetic fields. The dispersion relation(1.19) shows that a definite relationship exists between the frequency of the wave and the wavenumber k = |k |. The condition (1.20) means that the three vectors E, B and k constitute aright-handed orthogonal set of vectors.

Figure 1.2: Field distribution in a plane wave

In figure 1.2 the distribution of electric and magnetic fields in the harmonic wave (1.18)along the propagation direction k at a fixed time t is given. The argument

= k r t (1.21)

in (1.18) is the phase of the wave. The wave is called a plane wave because the phase at anygiven time t is constant in the planes k r = constant. We may also find wave solutions withconstant phase surfaces in the form of cylinders or spheres. These waves are correspondinglycalled cylindricalor spherical waves. To remain at a constant phase of the wave, an observermust move in the direction of the wave vector k with the phase speed

vph = /k. (1.22)

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1.3. ELECTROMAGNETIC WAVES IN VACUUM 9

With the dispersion relation (1.19), the phase speed is equal to the speed of light c in vacuum.The energy density and energy flux in the wave, as given by (1.15) and (1.14), oscillate in

space and time. Physically more meaningful quantities are found by forming time averages f

of the relevant quantities f(t),

f 1T

T0

f(t) dt,

where T represents the period of the time varying quantity. For the time averaged energydensity of the plane wave (1.18) we find

u =

02E20 +

1

20B20

cos2 =

04E20 +

1

40B20 =

02E20 . (1.23)

In (1.23) we made use of the conditions (1.19) and (1.20). We note that the electric andthe magnetic fields in the wave contribute equally to the time averaged energy density in the

wave.The time averaged energy flux in the wave is

P =1

0E0 B0cos2 = 0

2E20 c k = uc k. (1.24)

This means that the electromagnetic wave transports its average energy density u with thespeed of light c in the direction of the wave vector k, that is, the phase velocity and the energypropagation velocity in the wave are identical. This conclusion is valid for electromagneticwaves in vacuum. For electromagnetic waves in material media these two velocities, the latterrepresented by the group velocity

vg =

(k)

k , (1.25)

generally differ in both magnitude and direction.

Quiz 1.6 : Define wavelength , period T and frequency in terms of the wave numberk and the angular frequency . Rewrite the dispersion relation (1.19) in terms of and .

Quiz 1.7 : Verify that

cos2(k r t) = 2

2/

0cos2(k r t) dt = 1

2

and therefore that (1.23) and (1.24) are correct.

Quiz 1.8 : Show that

E(r, t) = c cos(kr t)/rB(r, t) = cos(kr t)/r (1.26)

is an asymptotic solution of (1.1) - (1.4) for large r, provided = j = 0 and thedispersion relation (1.19) is satisfied. Here and together with r are unit vectorsin a spherical coordinate system. How would you classify this solution?

[Hint: Asymptotic means that the expressions (1.26) satisfy (1.1) - (1.4) to any givendegree of accuracy for large enough values of r.]

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10 CHAPTER 1. ELECTROMAGNETIC WAVES

Quiz 1.9 : Two plane waves with identical amplitude vectors, but with different wavenumbers k = k0 k, | k | k0, both propagate along k0. Make use of identitiesfor trigonometric functions to show that the resulting wave field consists of a rapidly

varying harmonic part with phase function 0 = k0 r 0t multiplied with a slowlyvarying envelope

cos(k r t).Verify that the envelope moves with the group velocity

vg =(k)

k|k=k0 .

1.3.1 Complex field notation

The electric and magnetic fields are both real quantities. Since the Maxwell equations arelinear equations in E and B with real coefficients it is often mathematically convenient tointroduce complex notation for these fields. In this notation the wave solution (1.18) is written

E(r, t) = E0 exp(k r t)B(r, t) = B0 exp(k r t), (1.27)

where =1 is the imaginary unit and E0, B0, k and have to satisfy conditions (1.19)

and (1.20). Due to the de Moivre identity

exp() = cos + sin , (1.28)

the physically relevant fields are recovered by taking the real part of the corresponding com-plex fields.

A convenient feature of the complex notation and in fact the main reasons for itsintroduction is that taking derivatives with respect to r and t of the complex exponentialfunction is seen to be equivalent to simple algebraic operations,

k and t

(1.29)

that is, any -operator may be replaced with the vector k, any time derivative /t withthe scalar factor . We shall make repeated use of complex fields in the following.

The electric field amplitude vector E0

in the complex notation (1.27) represents not onlythe (real) amplitude E0 of the wave but also the (complex) polarization vector of the wave,

E0 = E0. (1.30)

The polarization vector has unit length, = 1. We shall return to a discussion of thephysical importance of the polarization vector in section 1.5.

Useful as the complex field notation may be, it still has its limitations. The complex fieldscannot be used directly when calculating non-linear field quantities. It is necessary to findthe real (physical) part of the fields before substitution into the non-linear expression. Onenotable exception to this rule, however, exists. Time averaged quadratic field quantities maybe calculated as one half the real part of the product of the two complex fields with one of

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1.4. THE ELECTROMAGNETIC SPECTRUM 11

the field factors complex conjugated. Thus, the time averaged energy density and energy fluxin the electromagnetic wave are calculated in terms of the complex fields E and B as

u = 12

Re02EE + 1

20B B = 0

4|E0 |2 + 1

40|B0 |2= 0

2|E0 |2 (1.31)

and

P =1

2Re

1

0E B

=

02

|E0 |2 c k. (1.32)

These results are seen to agree with our previous results (1.23) and (1.24).

Quiz 1.10: Write the spherical wave solution (1.26) in complex notation.

Quiz 1.11: Prove that

fphys(t)gphys(t) =12

Re fcmpl(t)gcmpl(t)where fcmpl(t) exp(t) and fphys(t) = Re (fcmpl(t)) and similarly for g. What isthe corresponding value for fcmpl(t)gcmpl(t)?

1.4 The Electromagnetic Spectrum

Electromagnetic waves of astrophysical interest extend from radio waves through the infrared,visible and ultraviolet spectral ranges to X-rays and hard -rays. The frequencies and wave-lengths corresponding to these classifications are illustrated in figure 1.3.

Figure 1.3: The electromagnetic spectrum

The Earths ionosphere represents a limiting factor at low frequencies. Only waves withfrequencies exceeding a minimum value depending on the maximum value of the electrondensity in the ionosphere and the angle of incidence are able to penetrate. This minimumfrequency is normally in the 5 - 10 MHz range. Thus, the so-called kilo-metric radiation, 1 km, originating in the Jovian magnetosphere was first found when satellites madeobservations outside the ionosphere possible. Similarly, atmospheric gases are responsible forabsorption bands at different frequencies. H2O-vapor and CO2 give rise to absorption bandsin the infrared range. The ozone-layer is an effective absorbing agent in the ultraviolet.

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12 CHAPTER 1. ELECTROMAGNETIC WAVES

1.5 Wave Polarization

At a given position r, the electric field vector of (1.18) oscillates along the fixed E0-direction

as a function of time. The wave is therefore said to be linearly or plane polarized. In thecomplex notation (1.27) this case corresponds to a real polarization vector, for instance, = xfor k = z. This is, however, not the most general case. If the polarization vector is chosen as = (x y)/2 for k = z, it is easily seen that the tip of the (physical) electric field vectorat a given position will perform a circular motion as a function of time.

In the more general plane wave solution the (physical) electric field can be written in theform

E(r, t) = E0( p cos cos + q sin sin ) (1.33)

where is a constant, [/2, /2], where the constant unit vectors p and q satisfy therequirement

pq = k,

and where the phase function is given by (1.21). In this case the tip of the electric fieldvector at any given position and as a function of time will trace an ellipse with semi-majorand semi-minor axis, a = E0 cos and b = E0 | sin | oriented along p and q. The geometryis illustrated in figure 1.4.

The ellipticityof the ellipse is given by tan . With tan = 0 the ellipse reduces to a lineand the linearly polarized wave is recovered. For | tan |= 1 the ellipse reduces to a circle. Thewave is then circularly polarized. For any other value of the wave is elliptically polarized.If < 0 the angle in figure 1.4 increases with time and the tip of the electric field vectortherefore traces the ellipse (circle) in a clockwise direction looking along the wave vector k.The wave is then said to be right-handedelliptically (circularly) polarized. With the opposite

sign for the wave is left-handedpolarized.2

Figure 1.4: Polarization ellipse

The polarization of an electromagnetic wave is often conveniently described in terms ofthe Stoke parameters. With the z-axis along the wave vector k and for an arbitrarily chosen

2Be aware that in the literature left/right handed polarization is sometimes defined as looking along k,that is, looking toward the infalling radiation! Note also that the connection between the sign of and right-or left-handed polarization assumes the phase function to be defined as = k r t. Different definitions of are in use in the litterature.

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1.5. WAVE POLARIZATION 13

set of orthogonal x- and y-axis, these parameters are defined by

I = 2 (E2x(t) + E2y(t))

Q = 2 (E2x(t) E2y(t))U = 4 Ex(t)Ey(t)

V = 4 Ex(t)Ey(t /2).(1.34)

The over-bar in (1.34) again represents time average. In the last expression a time delaycorresponding to a phase shift of /2 is introduced in the y-component before the averagingis performed. With the electric field given in terms of (1.33), the Stoke parameters reduce to

I = E20Q = E20 cos2cos2U = E20 cos2sin2

V = E20 sin2

(1.35)

and therefore alsoI2 = Q2 + U2 + V2. (1.36)

The Stoke parameters all have the same physical dimension. For an electromagnetic wavein vacuum, I is proportional to the energy density or the energy flux associated with thewave. We notice that the ratio of the two axis of the polarization ellipse, the ellipticity tan ,may be determined from the ratio of V and I,

sin2 =V

I

while the orientation of the polarization ellipse with respect to the chosen x- and y-axis

follows fromtan2 =

U

Q.

Quiz 1.12: Verify that (1.27) with k = z and = (x y)/2 represents circularlypolarized waves.

Quiz 1.13: From the Stoke parameters, how do you determine if the arriving wave isleft- or right-handed polarized?

Quiz 1.14: The Stoke parameters I, Q, U and V refer to a particular choice of ori-entation (x, y) of the measuring instrument. We may indicate this by marking the

Stoke parameters with this particular orientation, Ix,y and so on. Show that if themeasuring instrument is rotated an angle /4 to the new orientation (, ) then

I, = Ix,y, V, = Vx,y, Q, = Ux,y and U, = Qx,y.

Quiz 1.15: Find the corresponding magnetic field B that will make (1.33) part of asolution of Maxwell equations (1.1) - (1.4) in vacuum. What relation must exist be-tween and k? What is the amplitude of the electric field in the wave? Are you ableto classify this wave as plane, cylindrical or spherical? How would you characterizethe polarization of the wave? Discuss the following statement: The superpositionof two plane polarized plane waves with identical wave vectors k generally results inan elliptically polarized plane wave.

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14 CHAPTER 1. ELECTROMAGNETIC WAVES

Quiz 1.16: With the z-axis along k and given orthogonal x- and y-axis (see figure 1.4)show that the Ex and Ey components of the electric field (1.33) can be written

Ex = E0x cos( + x)

Ey = E0y cos( + y).

Show that the amplitudes E0x and E0y and the relative phase shift = y x maybe expressed uniquely in terms of and the angle as

E20x = E20 (cos

2 cos2 + sin2 sin2 )

E20y = E20 (cos

2 sin2 + sin2 cos2 )

tan = tan2/ sin2.

What are the Stoke parameters I, Q, U and V in terms of E0x, E0y and ?

1.6 Non-Monochromatic Waves

The solutions of Maxwell equations studied in the previous sections were all examples ofmonochromatic waves. To produce a strictly monochromatic wave, rather stringent conditionshave to be met by the wave source. In particular, the source will have to maintain a strictlyharmonic character over (ideally) infinite times. The two best known examples of sourcesapproximately satisfying this condition are the non-modulated electronic oscillator and thelaser. In this section we will consider three different aspects related to the non-monochromatic

character of electromagnetic radiation: finite wave coherence, partial polarization and powerspectra.

1.6.1 Wave coherence

The monochromatic wave may be characterized as having infinite phase memory, the wavephase varying strictly linearly with time. Naturally occurring electromagnetic radiationhave finite phase memory. After a coherence time , the wave tends to forget its previousphase. As an example we visualize the wave as consisting of a superposition of finite timesegments, each of duration during which the electric and magnetic fields vary according to(1.18), but where the phase experiences an arbitrary jump between each successive segment.

This is illustrated in figure 1.5. The segment length is the coherence length = c.

Figure 1.5: Wave with finite coherence time

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1.6. NON-MONOCHROMATIC WAVES 15

Any harmonically oscillating time segment,

E(t) = cos(0t) 2 t < 20 otherwise,

(1.37)

may be represented as a superposition of harmonic oscillations of different frequencies.

Let f(t) be a square integrable function, |f(t) |2 dt < .

The Fourier transform and its inverse are defined as

f() = F[f(t)] 12

f(t) exp(t) dt (1.38)

f(t) =

F1[f()] =

f()exp(

t) d. (1.39)

If f(t) is real, then f() = f().

The following relations hold

F[ ddt

f(t)] = f() (1.40)

F[f(at + b)] = 1|a | exp(

ab)f

a

. (1.41)

Some generalized Fourier transform pairs are

F[1] = ()F[sgn(t)] = ()1. (1.42)

Parseval theorem: If f(t) and g(t) are real, f = F[f] and g = F[g], then

f(t)g(t) dt = 2

f()g() d

= 4

0

Re(f()g()) d.(1.43)

Table 1.1: The Fourier transform

In fact, Fourier transform theory tells that any square integrable function f(t), that is afunction satisfying

|f(t) |2 dt < , may be written in the form

f(t) =

f() exp(t) d (1.44)

where the complex amplitude f() is given by

f() =1

2

f(t) exp(t) dt. (1.45)

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16 CHAPTER 1. ELECTROMAGNETIC WAVES

The two relations (1.44) and (1.45) form a Fourier transform pair. Fourier transforms mayalso be generalized to include non square integrable functions. A summary of some importantproperties of the Fourier transform is given in table 1.1. For reference purposes we also note

that if f(t) is any periodic function, f(t + T) = f(t), then it may be expanded in a Fourierseries, consisting of harmonically varying terms in the fundamental angular frequency 2 /Tand its harmonics. A summary of Fourier series is given in table 1.2.

Let f(t) be a periodic function, f(t + T) = f(t).The function f(t) may be expanded in the trigonometric series

f(t) =

n=fn exp( 2n

Tt) (1.46)

withfn =

1

T

T/2T/2

f(t) exp(2n

Tt) dt (1.47)

If f(t) is a real function, then fn = fn.If g(t) = f(t + b) then

gn = exp( 2nT

b) fn. (1.48)

Parseval theorem: For real periodic functions f(t) and g(t), both ofperiod T and with f g = 0, then

1

T

T/2T/2

f(t)g(t) dt =

n=fngn

=

d

n=1

2Re(fngn) ( 2n

T).

(1.49)

Table 1.2: The Fourier series

After this intermezzo on mathematical methods let us return to our discussion on wavecoherence. Substituting the expression (1.37) for E(t) with

cos 0t =1

2(exp(0t) + exp(0t))

into (1.45) and performing the integral we find

E() =

4

sinc ( + 0)

2+ sinc ( 0)

2

(1.50)

where the sinc-function is defined as

sinc x sin xx

. (1.51)

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1.6. NON-MONOCHROMATIC WAVES 17

Figure 1.6: Amplitude distribution for a finite harmonic segment

The result (1.50) is plotted in figure 1.6 for a case where 0 1, that is, when there isnegligible cross-talk between the two sinc-functions. The two wings of the plot have typical

half-widths (from line center to first zero-crossing)

=2

. (1.52)

According to (1.44) the finite harmonic segment may be considered made up from a superpo-sition of (infinite) harmonic oscillations with frequencies within an interval with half-width varying inversely with the duration .

Formula (1.50) only takes into account the contribution to the complex amplitude E()from the time segment /2 t < /2. Adjacent time segments will give similar contribu-tions. In fact, if a constant phase is added to the argument of the cos-function in (1.37),the two terms in (1.50) get additional complex conjugate phase factors exp(

) and such

that E() = E() in accordance with table 1.1. The contribution from the different timesegments will contain the identical sinc functions multiplied with random phase factors of theform exp(). The frequency dependence of the amplitude function E() will, therefore, beidentical for a single time segment and a sequence of such segments. We may conclude thatany wave with finite coherence time can be considered as a non-monochromatic wave, witha characteristic frequency half-width given by (1.52).

For a simple estimation of the coherence time of a given radiation field, Youngs two-slitinterference experiment may be used (see quiz 1.18). The experiment is illustrated schemati-cally in figure 1.7. A plane wave, illustrated by selected phase fronts, is falling perpendicularlyon a screen with two parallel slits a distance d apart. The resulting interference pattern isfocused by a lens L on a second screen S one focal length f away. A series of intensity maximawill be observed on the screen. The intensity of the n.th interference fringe, at the locationcorresponding to an optical path difference n for the rays emerging from the two slits, fallsbelow the diffraction limit (dash-dotted) due to the slit width b as n c. A count ofthe number of visible fringes is thus a direct measure of the time that the radiation fieldremembers its own phase, that is, the coherence time of the radiation field.

Quiz 1.17: What is the width of the wavelength range corresponding to (1.52)?

Quiz 1.18: Light from a distant source with finite coherence time falls perpendicularlyon a screen containing two parallel slits of width b and a distance d apart. The lightis subsequently focused by a lens on another screen. A series of interference maxima

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18 CHAPTER 1. ELECTROMAGNETIC WAVES

Figure 1.7: Youngs two slit interference experiment

(fringes) of increasing order n will be seen. What is the maximum order n that youexpect to see? Which of the quantities , b and d will be involved? Estimate thereduction factor for the interference fringe of order n due to the finite coherence time.

Quiz 1.19: For a Gaussian shaped pulse of characteristic width t = , that is,

f(t) = exp

t

2

22

< t < ,

show that the corresponding width of the amplitude spectrum f() is = (t)1.

1.6.2 Partial polarizationApart from the finite phase memory discussed above, the polarization of the wave may changefrom one segment to the next. This phenomenon is conveniently described in terms of theStoke parameters (1.35). The time averaging in (1.35) will now have to be extended overmany coherence times over which the parameters and take different values for adjacentsegments. For simplicity we assume E0 to remain constant. The result is then

I = E20Q = E20 cos2 cos2

U = E20 cos2 sin2

V = E20 sin2 ,

(1.53)

where the time averaging is taken over a time interval N for N large. Making use of theSchwartz inequality for integrals,b

af(t)g(t) dt

2ba

f2(t) dt

ba

g2(t) dt, (1.54)

it will now be apparent that for instance

U2 = E40 cos2 sin22 E40 cos2 2 sin2 2 ,

and therefore thatI2

Q2 + U2 + V2. (1.55)

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1.6. NON-MONOCHROMATIC WAVES 19

The degree of polarization,

p

Q2 + U2 + V2

I, (1.56)

describes the degree to which the wave remembers its own polarization. With p = 1 the waveis completely polarized, with p = 0 the wave has random polarization. In the first case thepolarization is constant between successive segments. In the second case the wave chooses itspolarization in each segment completely at random.

Two waves with finite coherence times, propagating in the same direction with the samefrequency and with electric fields E(1) and E(2) are said to be incoherent if the time averagedproduct of any two components of the two fields vanishes,

E(1)x (t)E

(2)y (t) = 0, .

The Stoke parameters for incoherent waves are additive. That is, if we define the Stokeparameters for the two waves (I(1), Q(1), U(1), V(1)) and (I(2), Q(2), U(2), V(2)), then the cor-responding parameters for the resultant wave is

(I,Q ,U ,V) = (I(1), Q(1), U(1), V(1)) + (I(2), Q(2), U(2), V(2)). (1.57)

This means that any partially polarizedwave, 0 < p < 1, may be considered as a superpositionof two incoherent waves, one completely polarized and one with random polarization,

I(1) =

Q2 + U2 + V2, Q(1) = Q, U(1) = U, V(1) = V

and

I(2) = I I(1), Q(2) = U(2) = V(2) = 0.The degree of polarization of the radiation arriving from the universe is important for the

interpretation of the physical conditions responsible for the emitted wave. Our knowledgeof stellar, interstellar and galactic magnetic fields and conditions in interstellar dust cloudsdepends to a large extent on this type of information.

1.6.3 Spatial-temporal Fourier representation

Any square integrable spatial function f(r) may be expressed in terms of its spatial Fourieramplitude f(k) as

f(r) = f(k)exp(k r) d3k (1.58)f(k) =

1

(2)3

f(r) exp(k r) d3r. (1.59)

The integrals over r and k here extends over the entire r and space. Note that we for laterconvenience chose the opposite sign convention in (1.58)-(1.59) compared to (1.38)-(1.39).Substituting expressions for the electromagnetic fields Eand B and the corresponding sourcedensities j and in terms of their spatial-temporal Fourier amplitudes, for instance,

E(r, t) = E(k, )exp((k r t)) d3k d, (1.60)

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20 CHAPTER 1. ELECTROMAGNETIC WAVES

in the Maxwell equations leads to

k

E

/0

k E Bk B

k B 0(j 0E)

exp((k r t)) d3k d = 0. (1.61)For this set of equations to be satisfied identically for any allowed r and t, the integrandsmust vanish. For electromagnetic waves in vacuum, j = = 0, this leads to a set of algebraicequations identical to those previously studied for plane waves in the complex notation (seesection 1.3.1), and requires E and B to vanish identically, except for combinations ofk and satisfying the dispersion relation for electromagnetic waves in vacuum, = kc.

This result means that any (poly-chromatic) wave field may be constructed as a superposi-tion of plane, harmonic waves. For instance, for any plane wave propagating in the z-direction

the Fourier amplitude of the electric field will be of the form

E(k, ) = E()(kx)(ky)(kz /c), E() z = 0, (1.62)

and therefore

E(r, t) =

E() exp(

c(z ct)) d (1.63)

since the Dirac -functions allow for immediate integration over the wave vector k. With

B(k, ) =k

E(k, ), (1.64)

the corresponding magnetic field is

B(r, t) =1

cz

E() exp(

c(z ct)) d. (1.65)

It should be born in mind that the details of a computed amplitude spectrum E() for agiven location r depends not only on the properties of the physical wave field E(t) present,but also on the duration T of the observational time series available. The observationalamplitude spectrum is a combination of properties inherent in the physical signal present anda windowing effect.

1.6.4 Power spectrum

From the Parseval theorem (1.43) given in table 1.1, the average value of the product of tworeal functions f(t) and g(t) over the time interval (T /2, T /2) can be expressed as

f(t)g(t) =1

T

T/2T/2

f(t)g(t) dt =4

T

0

Re(fT()gT()) d (1.66)

where f and g are the Fourier transforms of f and g over the time interval of length T. Thetheorem (1.66) may be used to express the average value of the power density carried by anelectromagnetic wave, in terms of the Fourier transforms of the electric and magnetic fields

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1.6. NON-MONOCHROMATIC WAVES 21

in the wave, ET(r, ) and BT(r, ). Thus, the average value of the Poynting vector (1.14)at position r may be written

P(r) 0P(r, ) d

0P(r, ) d, (1.67)

with spectral power density

P(r, ) =4

T

1

0ReET(r, ) BT(r, )

(1.68)

and the corresponding expression for P(r, ). In the following we will be interchangingfreely between spectral quantities n terms of angular frequency, [] = rad/s, and frequency,[] = Hz. A simple relation exists between these quantities. With d = 2d it thus followsthat P(r, ) = 2P(r, = 2). n P(r, ) represents the energy transported by theelectromagnetic fields per unit time interval per unit frequency interval through a unit area

located at position r and oriented in the direction n, that is, [P] = W m2Hz1.From (1.68) we see that each frequency component in the radiation field contributes to

the total power density only at that frequency. The contribution P(r, ) d to the powerdensity from an infinitesimal frequency interval (, + d) only depends on the amplitudesof the electric and magnetic fields in that frequency interval. We also note that with thespectral power density in the form (1.67) only positive frequencies contribute to the totalpower density.

As an example let us return to the case of a general plane wave propagating along thez-axis as described by (1.63) and (1.64). Substitution into (1.68) leads to the result

P =4

T

c0|ET()

|2 z, (1.69)

independent of position r. We remember that the length T of the observational time windowenters the result not only as an explicit factor, but also through the actual form of theamplitude spectrum ET().

Quiz 1.20: A plane-polarized, monochromatic plane wave propagating along the z-axiswith electric field Ex(z, t) = E0cos(kzt) is sampled over a time interval T. What isthe observed power spectrum z P? What is the full width at half height FWHHof the power spectrum?

Quiz 1.21: Two identical waves as described in quiz 1.20 but with slightly different

frequencies, 1 and 2 with | 2 1 | /1 0.01, falling on an ideal detector aresampled over a time interval T. What is the power spectrum of the combined wave?Will your result be different whether the power spectrum is calculated by first addingthe electric field of the two individual waves or by first calculating the power spectrumof the individual waves and then adding? What time interval T is needed in order toidentify the two frequencies 1 and 2 in the combined spectrum?

Quiz 1.22: For a plane wave with an electric field given as a series of harmonic segmentsas discussed in section 1.6.1, what is the corresponding spectral power density P?For simplicity, choose a time interval for the Fourier transform of length T = N where N is an integer number. Compare with the corresponding result for one har-monic segment of length T = N .

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22 CHAPTER 1. ELECTROMAGNETIC WAVES

The energy transported through a given area d2A= d2

An generally has contributions from

individual electromagnetic waves propagating in different directions s. It will be convenientto work with quantities that explicitly reveal this directional dependence. We thus define

n P(r, )

I(r, s, ) n sd2 (1.70)

where d2 represents the differential solid angle element centered on s as illustrated in figure1.8. I or the corresponding I is the specific intensity of the radiation field. I representsthe energy transported per unit time through a unit area oriented in direction n per unitfrequency interval and per unit solid angle centered on the direction s, that is [I] = W m2Hz1 sr1.

If the specific intensity is independent of the direction s, the radiation field is said to be

isotropic. If the intensity is independent of position r the radiation field is homogeneous.

Figure 1.8: Geometry for specific intensity

From the specific intensity several useful quantities may be derived. The mean intensityis defined as the intensity averaged over all directions,

J 14

I d2 = 1

4

20

11

I dcos d. (1.71)

For an isotropic radiation field we have J =I.The net energy fluxF is defined as

F(r, n)

I cos d2 =2

0

11

I cos dcos d. (1.72)

F(r, n) represents the net flow of energy per unit frequency interval and per unit timethrough a unit area with normal n located at r. Flow of energy in direction n contributespositively, flow of energy in the opposite direction contributes negatively. It is often convenientto write the net flow of energy as the difference between the unidirectional energy flux in eachdirection

F =

F+

F =

2

0 1

0 I cos dcos d +

2

0 0

1I cos dcos d. (1.73)

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1.7. SPECIFIC INTENSITY OF RADIATION 23

F+ represents the net energy flux per unit area in the +n-direction per unit specified fre-quency interval, F the corresponding net flux in the opposite direction. For an isotropicradiation field we have

F+ =

F =

I and therefore

F = 0.

All quantities discussed above are spectral quantities measured per unit angular frequency or per unit frequency interval. From the spectral quantities the corresponding totalquantities are derived by integrating over (angular) freguency, for instance the total energyflux

F+ =

dF+ =

dF+ .

We have so far expressed quantities like spectral power density or specific intensity interms of the electromagnetic wave fields Eand B. According to the correspondence principleit is often advantageous to express these quantities in terms of the alternative corpuscularview, that is, that the energy transported by the radiation is carried by photons of the proper

frequency. To this end the photon probability density frs or alternatively frs are needed.frs represents the number of photons per unit volume, per unit frequency interval andtraveling per unit solid angle in the specified direction s. Each photon with energy = hand traveling (in vacuum) with the speed of light c in direction s contributes to the specificintensity I in that direction. We note that [frs] = m3 Hz1 sr1. In terms of thesequantities the specific intensity is given as

I c frs or I h c frs. (1.74)

We end this initial introduction to the specific intensity with a simple, yet illustrativeexample. Consider the radiation field in vacuum from an extended source, a star with aprojected area

Aas indicated in figure 1.9. We assume that at a distance r1 the radiation

from the star fills the solid angle 1 = A/r21 uniformly. The energy flux through a radiallydirected unit area is then given by

F(r1) =I(r1) 1,whereI(r1) is the specific intensity at radius r1. For this result we did assume Omega 1and therefore that we could assume cos 1. The total energy flux through a spherical shellof radius r1 is 4r

21F(r1) = 4AI(r1). The corresponding flux through a spherical shell

of radius r2, where the specific intensity is I(r2), is given by 4r22F(r2) = 4AI(r2). Ifno energy is created or lost between radii r1 and r2, these quantities must be equal. Thismeans that we have arrived at the conclusion that the specific intensity is independent of thedistance from the star,

I(r2) =I(r1). (1.75)At first sight this result may come as a surprise: the specific intensity of radiation from theSun is the same at the distance of the pluton Pluto as it is at the planet Earth!

Quiz 1.23: Comment on the claim: The specific intensity resulting from a radiatingpoint source is singular.

Quiz 1.24: Two identical bolometers are directed toward the Sun but placed at thedistances of the Earth and Pluto. Will they measure identical energy deposition byradiation from the Sun? Discuss. How is a constant specific intensity with distanceconsistent with the expectation of a r2 dependent energy flux? [An ideal bolometermeasures the total energy deposition from infalling radiation.]

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24 CHAPTER 1. ELECTROMAGNETIC WAVES

Figure 1.9: Intensity along rays

1.8 Interaction of Waves and Matter

We have so far considered properties of electromagnetic waves in vacuum. In a mediumadditional effects will be important. The electric field E of an electromagnetic wave presentwill induce electric currents in the medium. These currents will act back to determine the

properties of the wave. The simple dispersion relation = kc valid for electromagnetic wavesin vacuum will have to be replaced. The phase and group velocities of the wave will generallytake different values, possibly even pointing in different directions. Waves may suffer refractiveeffects in spatially varying media, they may suffer absorption, and the medium may act togenerate additional wave energy. The latter two effects constitute the main subjects for thesubsequent chapters. Here we first consider refractive effects.

1.8.1 Non-magnetized Plasma

The common astrophysical medium is an ionized gas, or plasma, with free electron densityne(r). The electric field E of an electromagnetic wave will set these electrons into oscillating

motion and thus generate an electric current density jp in the medium. Ions will also con-tribute to the total induced currents. But because of their much larger mass their effects willbe much smaller. For a simplest model we assume the ions to be immobile. The electronswill be assumed mobile but cold. The latter condition means that we are neglecting effectsfrom thermal motions of the electrons. As long at the thermal speed of the electrons is muchless than the phase speed of the waves, this is normally a useful approximation.

For a quantitative analysis we assume the electromagnetic wave to be harmonically varyingin space and time with angular frequency and wave vector and k,

E= E0 exp(k r t).This electric field will give rise to an oscillating velocity v for electrons at r given by

mv = eE. (1.76)Here effects due to the Lorentz force from the magnetic field B associated with the wave wereneglected. We also assumed no external constant magnetic field B0 to be present. With neelectrons per unit volume the resulting induced electric current density is

jp = enev = 2p

0E= 0(K 1)E (1.77)

where

K = 1

2p

2

(1.78)

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1.8. INTERACTION OF WAVES AND MATTER 25

is the dielectric coefficient of the medium. The plasma frequency p is defined by

2p

nee2

m0. (1.79)

We notice that jp is parallel but 90 out of phase with E. This means that the wave will

not suffer damping due to Ohmic heating of the medium. The induced current jp will beaccompanied by the charge density p = k jp/, which when substituted into Gauss law(1.1) gives rise to the condition

Kk E= 0. (1.80)For = p this requires k E= 0, that is, the wave will be a transverse wave.

Substitution of the induced current density (1.77) into (1.2) leads to the wave equation

N2k (k E) + KE= 0 (1.81)where

N2 c2k22

(1.82)

defines the refractive indexNof the medium. The wave equation (1.81) allows for non-trivialtransverse solutions (solutions with k E= 0 and E= 0!) only ifk and together satisfythe dispersion relation

N2 = K or equivalently 2 = 2p + c2k2. (1.83)

Figure 1.10: Refractive index of the non-magnetized plasma

The properties of transverse electromagnetic waves in a plasma differ significantly from thecorresponding property of electromagnetic waves in vacuum. The waves are now dispersive,that is, the phase velocity of the wave varies with frequency (or wave number k). Forfrequencies < p the refractive index N takes imaginary values. This means that theelectric field of the wave will be decaying exponentially with distance. Such a wave is calledan evanescent wave. Propagating electromagnetic waves only exist for frequencies > pwhere refractive index N is real and less than unity. This means that the phase velocitymagnitude vph = /k of the wave exceeds the velocity of light in vacuum. The magnitude ofthe group velocity vg = /k is, however, less than c and tends to zero as the frequencyapproaches the plasma frequency. The result is plotted in figure 1.10. The quantities vph, cand vg are graphically represented by the tangence of the indicated angles.

Quiz 1.25: Calculate the group velocity vg resulting from the dispersion relation (1.83)and show that vphvg = c

2 for electromagnetic waves propagating in an ionized non-magnetized plasma.

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26 CHAPTER 1. ELECTROMAGNETIC WAVES

1.8.2 Magnetized Plasma

The result of the previous subsection was an isotropic dispersion relation, = (k). That is,

the frequency depends on the magnitude of the wave vector, but is independent of its direction.This means that the group velocity vg = /k and the wave vector k are parallel. Thisresult will be different if a static magnetic field B0 = B0z is permeating the ionized gas.

With the magnetic field present the oscillating velocity of the electrons (1.76) must bemodified by including the Lorentz force from the static magnetic field B0

mv = e(E+ v B0). (1.84)The induced current density is now given by

jp = enev = 0 (K 1) E (1.85)where the dielectric tensor K is a tensor of rank 2 (a 3

3 matrix for any given choice of

coordinate system) with elements

Kxx = Kyy = S = 1 +2

1 2 (1.86)

Kxy = Kyx = D = 2

1 2 (1.87)Kzz = P = 1 2 (1.88)

Kxz = Kyz = Kzx = Kzy = 0 (1.89)

with

p/,

b/, and

b eB0

m. (1.90)

The induced current jp is no longer parallel with E. The dielectric tensor is Hermitean(Kij = Kji), the wave will therefore still not suffer damping due to Ohmic heating of themedium (see quiz 1.26). Gauss law (1.1) leads to the condition

k K E= 0, (1.91)and we can no longer assume the wave to be transverse (k E= 0).

The wave equation this time reads

N2k (k E) + K E= 0. (1.92)

Without loss of generality we may choosek to lie in the xz-plane,

k = (sin , 0, cos ). Thelinear, homogeneous set of equations (1.92) may then be written in matrix form N2 cos2 + S D N2 sin cos D N2 + S 0

N2 sin cos 0 N2 sin2 + P

ExEyEz

= 0. (1.93)Non-trivial solutions for E(E= 0) will only exist if this set of equations is linearly dependent,that is, if the system determinant vanishes,

det

N2 cos2 + S D N2 sin cos

D N2 + S 0

N2 sin cos 0

N2 sin2 + P

= AN4 BN2 + C = 0 (1.94)

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1.8. INTERACTION OF WAVES AND MATTER 27

with

A = Ssin2 + P cos2

B = (S2 D2)sin2 + SP(1 + cos2 ) (1.95)C = (S2 D2)P.

Equation (1.94) is the dispersion relation for electromagnetic waves in our cold, magnetizedplasma. It is a quadratic polynomial equation with two solutions for N2 as functions of and , solutions that may be written in the form = (k, ). These solutions correspond todifferent wave modes with wave polarization determined by substituting the proper valueforN2 for a given value of in (1.92) and solving for E. In figure 1.11 the different solutionsfor N2 are plotted as functions of for = 0 (green) and = 90 (red) for two cases: a)p > b and b) p < b. Symbols o, x, z and w stand for ordinary, extraordinary, z andwhistler modes. The o and x modes are of special importance in radio astronomy. These are

the only modes that may escape from an astrophysical radio source. Propagation away fromthe source towards decreasing p and b is equivalent to moving to the right in figure 1.11.The o and x modes meet no obstacle in this respect, whereas the z and whistler modes willencounter a resonance (N2 ) and be absorbed. Contrary, when propagating towardsincreasing b and p the o and x modes will eventually meet a cutoff (N2 0) where theywill be refracted or reflected.

Figure 1.11: Refractive index for magnetized plasma

With a dispersion relation for the magnetized plasma of the general form = (k, ) thecorresponding group velocity vg is no longer parallel with the wave vector k, in fact,

vg =

k | k +1

k

|k =

k | k lnN | . (1.96)and therefore also

vg =

k |

1 +

lnN

|2

(1.97)

For this result we made use of the mathematical identity (1.99). The group velocity vg makesan angle with the magnetic field B0 where

cos =cos + lnN | sin

1 + lnN

|

2. (1.98)

40/304

28 CHAPTER 1. ELECTROMAGNETIC WAVES

We note that vg, k and B0 are coplanar.

Figure 1.12: Refractive index a) and phase velocity b) surfaces with graphical construction

of the group velocity direction for an axisymmetric medium

The situation is illustrated in figure 1.12. In a) the red curve represents the functionN() = ck()/ for a given . This curve will when rotated around the B0 axis generatewhat is known as the refractive index surface for the chosen wave mode. From the graphicalconstruction one readily notes that the group velocity will always be perpendicular to therefractive index surface. An alternative construction, where the refractive index surface isreplaced by the phase velocity surface (inverse refractive index surface) 1/N() for a given ,is shown in b). The normal n to this surface is parallel with

k

ln 1/N |

= k +lnN

|

while the group velocity vg is still given by (1.96). Thus, the wave vector k bisects the anglebetween the phase velocity surface normal n and the group velocity vg.

Figure 1.13: Phase velocity and inverse ray refractive index surfaces for three frequencies > b > p

In the upper panel of figure 1.13 cuts through the phase velocity surface corresponding tothe situation in figure 1.11b (b > p) for the three frequencies denoted 1, 2 and 3 in figure

41/304

1.8. INTERACTION OF WAVES AND MATTER 29

1.11b are given. The magnetic field is parallel with the vertical axis. The dotted circles (red)represents an electromagnetic wave in vacuum. At the lower frequency (1) the ordinary andthe z modes exist, the latter one only for angles between the magnetic field B0 and the

wave vector k exceeding a minimum value. For the middle frequency (2) only the ordinarymode is allowed. At the higher frequency (3) the ordinary and the extraordinary modesboth propagate. With still increasing frequencies the phase velocity surfaces for both modesapproach that of electromagnetic waves in vacuum. For later reference, in the lower panel offigure 1.13 corresponding cuts through the inverse ray refractive index are shown.

In an isotropic plasma electromagnetic waves are transverse and they may take any polar-ization, much like electromagnetic waves in vacuum. In the magnetized plasma the situationis different. Each mode takes a particular polarization as required by the wave equation, andthe waves are not generally transverse. For a short summary we restrict our discussion to > (p, b). For = 0

the ordinary and extraordinary modes are both transverse and cir-

cularly polarized, but with opposite handedness. For = 90 the ordinary mode is transverseand plan polarized with the electric field parallel with B0. The corresponding extraordinarymode, however, is non-transverse with the non-vanishing components of the electric fieldsatisfying Ex/Ey = D/S.

Refraction is but one of many effects that will determine the propagation of radiationfields in the presence of matter. The induced acceleration of free electrons by the electricfield of the wave will also lead to generation of radiation traveling in different directions(Thompson scattering). Photons may suffer collisions with electrons and be deflected witha corresponding change in energy (Compton scattering). More importantly, photons may beabsorbed by atoms or molecules, the excess energy later to be re-emitted in arbitrary direction

and possibly at different frequencies in a spontaneous emission transition or as a copy of atriggering photon in a stimulated emission process. A study of radiative transport aims atexplaining and predicting the cumulative effect of all these different processes.

Quiz 1.26: With jp as given by (1.85) show that the corresponding Ohmic heating rate12 Re(E

jp) = 0 whatever the choice of the electric field E.Quiz 1.27: Show that cutoff and resonance in a magnetized plasma occur where the

conditions (S2 D2)P = 0 and tan2 = P/S are satisfied, respectively. Determinethe corresponding values of for given p and b.

Quiz 1.28: For o and x modes propagating parallel with the magnetic field ( = 0)

in a plasma determine the corresponding wave polarizations . Argue that a planepolarized wave can be formed from a superposition of o and x modes. What happensto the polarization plane as the wave propagates along the magnetic field. Why is theeffect called Faraday rotation? [Hint: Note that the refractive index and thereforealso the wave number k for the two modes will be different.]

Quiz 1.29: Determine the wave polarizations for o and x modes propagating perpen-dicular to the magnetic field in a plasma.

Quiz 1.30: For three variables x, y and z, functionally connected, z = z(x, y), showthat

z

x |y

x

y |z

y

z |x =

1. (1.99)

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30 CHAPTER 1. ELECTROMAGNETIC WAVES

[Hint: Note that dz = zx |y dx + zy |x dy, write the corresponding expression for dyand then eliminate dy.]

1.9 The Ray Equations

We now turn to the more quantitative discussion of refractive effects for the transport ofradiation in media, for the time being neglecting absorption and emission processes. Welimit our discussion to radiation transport in a time stationary medium. The medium will beallowed to be inhomogeneous, but the length scales for spatial variations in the medium willbe assumed to be long compared to the wave lengths of interest.

Electromagnetic waves in such a medium are governed by a local dispersion relation

= (r,k). (1.100)

As the wave propagates through the medium, the frequency of the wave remains constant,while the wave vector k changes with position r. The corresponding changes in wave vectorand position are found by forming the total time derivative of the dispersion relation (1.100)

0 =d

dt=

k

dk

dt+

r

dr

dt,

that is,

dr

dt=

k(1.101)

dkdt

= r

. (1.102)

We must here interpret r and k as the instantaneous central location and wave vector of apropagating wave packet. In particular, we have identified the propagation velocity of thewave packet with the group velocity vg = /k, that is, the velocity at which energy istransported in the wave. The result (1.101)-(1.102) is referred to as the ray equations andforms the basis of geometrical optics.3

Figure 1.14: Incompressible flow through the ray equations

3Note that with momentum p = k and Hamiltoniam H = the ray equations are identical in form withthe Hamilton equations (3.5)-(3.6).

43/304

1.9. THE RAY EQUATIONS 31

A remarkable property of the ray equations is that they represent an incompressible flowin the six dimensional phase space (r,k). That is, the divergence of the six dimensional phasespace velocity (dr/dt, dk/dt) = (/k,

/r) vanishes identically,

r dr

dt+

k dk

dt 0. (1.103)

Physically, this result means that any 6-dimensional volume element d3r d3k in (r,k)-spaceremains constant as transformed by the ray equations. Mathematically this may be reformu-lated in the following way: any infinitesimal six dimensional volume element d3r d3k centeredat (r,k) at time t transforms into the volume element d3r d3k centered at (r,k) at time t

and such that

d3r d3k Jd3r d3k = d3r d3k (1.104)since the Jacobian

J (r,k)(r,k)

= 1, (1.105)

due to the property (1.103) (see quiz 1.31). This incompressibility property is illustratedschematically in figure 1.14. The volume of the shaded 6-dimensional (r,k)-element remainsconstant as it is transformed in time by the ray equations.

To study an effect of the constraint (1.104), let us consider a locally axisymmetric medium,that is, a medium with a dispersion relation for a chosen wave mode of the form,

= (r,k) = (r, k , ). (1.106)

This is in accordance with the properties of the magnetized plasma of section 1.8.2. Theangular wave frequency depends on the direction of the wave vector k only through theangle between k and the local magnetic field B0. Let us further consider waves with wavevectors in d3k = k2dkd2k centered on k with d

2k = dcos d. It is convenient to expressthe k-space volume element in terms of the corresponding angular frequency and cos where is the angle between the group velocity and the static magnetic field. Performingthese transformations and making use of (1.97) we find

d3k = k2 dk d2k =2

c2N2

k |

cos cos |

d d2 =2

c2N2r

vgd d2, (1.107)

whereN2r N2

1 +

lnN

|2 cos cos | (1.108)

defines the ray refractive index Nr and d2 is the solid angle element centered on the raydirection vg corresponding to the chosen wave vector k. We notice that for an isotropicmedium the ray refractive index reduces to the refractive index, Nr =N.

The energy carried by waves that passes through an arbitrary surface element d2A atposition r with wave vectors within the chosen d3k element during the time interval dt willbe located within the volume element

d3r = d2A

vg dt. (1.109)

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32 CHAPTER 1. ELECTROMAGNETIC WAVES

Figure 1.15: Energy transport in anisotropic medium

The geometry is illustrated in figure 1.15. The curved line represents a ray of radiation with

local direction of propagation s = vg. The wave energy for wave vectors k with directionswithin d2k and located within the shaded cylindrical volume will pass through the surfaceelement d2A= d2A n in direction s within d2 during the time interval dt. But, accordingto (1.104) this means that

d3r d3k =2

c2N2r d2A sdt d d2 = constant along the ray. (1.110)

If the radiation travels unscattered and without energy gain or loss through the back-ground medium, then the amount of energy that is transported along a ray in direction sthrough the given surface element d2A at position r, within the solid angle element d2centered on s, in the frequency interval d during the time interval dt, must also remain

constant as transformed along the ray, that is,

I d2A sdt d d2 = constant along the ray. (1.111)

If we now compare (1.110) and (1.111) it then follows that also the ratio

IN2r

= constant along the ray. (1.112)

If the anisotropy of the medium can be neglected, that is, the dispersion relation maybe approximated as = (r, k), considerable simplifications result. For this case the rayrefractive index equals the refractive index, clNr =

Nand the vectors k and vg are colinear.

In dilute media like stellar atmospheres and at high enough frequencies the refractive indexN is close to unity. The factorN2r in (1.112) is therefore often omitted in the literature. Thisomission is equivalent to the expectation that each ray travels along straight lines. Otherwise,in inhomogeneous media whereNr andNare functions of position r, the rays will be curvedlines, that is, the waves will be refracted or reflected by the medium.

A simple example will be illustrative.

Consider a plane-stratified ionized medium for which the electron density decreases withincreasing z as illustrated in figure 1.16b. Let a ray of electromagnetic radiation fill the smallsolid angle element 1 =

21 centered on the positive z-axis at level z1 where the refractive

index is

N1. As the ray propagates to level z2 where the refractive index has changed to the

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1.9. THE RAY EQUATIONS 33

Figure 1.16: Ray refraction in plane-stratified ionized medium

valueN2, the ray will be subject to Snells law, requiring that the relationN1 sin 1 =N2 sin 2is satisfied. For small this can be approximated by

N1 1 =N2 2. (1.113)This means that off-vertical rays will be slightly refracted as illustrated in figure 1.16c. Therefractional effect is important when the wave frequency is comparable t

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