0521770017 cambridge university press an introduction to radiative transfer dec 2001

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An Introduction to Radiative Transfer
Methods and applications in astrophysics
Astrophysicists have developed several very different methodologies for solving
the radiative transfer equation. An Introduction to Radiative Transfer presents
these techniques as applied to stellar atmospheres, planetary nebulae,
supernovae and other objects with similar geometrical and physical conditions.
Accurate methods, fast methods, probabilistic methods and approximate
methods are all explained, including the latest and most advanced techniques.
The book includes the different methods used for computing line profiles,
polarization due to resonance line scattering, polarization in magnetic media and
similar phenomena. Exercises at the end of each chapter enable these methods to
be put into practice, and enhance understanding of the subject. This textbook
will be of great value to graduates, postgraduates and researchers in
astrophysics.
A NNAMANENI P ERAIAH obtained his doctorate in radiative transfer from
Oxford University. He was formerly a Senior Professor at the Indian Institute of
Astrophysics, Bangalore, India. He has held positions in India, Canada,
Germany and the Netherlands. His research interests include developing
solutions to the radiative transfer equation in stellar atmospheres and line
formation in expanding atmospheres with different physical and geometrical
conditions.



An Introduction to
Radiative Transfer
Methods and applications

in astrophysics
Annamaneni Peraiah


         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
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© Cambridge University Press 2004
First published in printed format 2001
ISBN 0-511-03401-6 eBook (Adobe Reader)
ISBN 0-521-77001-7 hardback
ISBN 0-521-77989-8 paperback


Contents

Preface xi

Chapter 1 Definitions of fundamental quantities of the radiation field 1
1.1

Specific intensity 1

1.2

Net flux 2

1.2.1

Specific luminosity 4

1.3

Density of radiation and mean intensity 5

1.4

Radiation pressure 7

1.5

Moments of the radiation field 8

1.6

Pressure tensor 8

1.7

Extinction coefficient: true absorption and scattering 9

1.8

Emission coefficient 10

1.9

The source function 12

1.10

Local thermodynamic equilibrium 12

1.11

Non-LTE conditions in stellar atmospheres 13

1.12

Line source function for a two-level atom 15

1.13

Redistribution functions 16

1.14

Variable Eddington factor 25
Exercises 25
References 27

Chapter 2 The equation of radiative transfer 29
2.1

General derivation of the radiative transfer equation 29

2.2

The time-independent transfer equation in spherical symmetry 30

2.3

Cylindrical symmetry 32

v


Contents

vi

2.4

The transfer equation in three-dimensional geometries 33

2.5

Optical depth 38

2.6

Source function in the transfer equation 39

2.7

Boundary conditions 40

2.8

Media with only either absorption or emission 41

2.9

Formal solution of the transfer equation 42

2.10

Scattering atmospheres 44

2.11

The K -integral 46
, , X operators 47

2.12

Schwarzschild–Milne equations and

2.13

Eddington–Barbier relation 51

2.14

Moments of the transfer equation 52

2.15

Condition of radiative equilibrium

2.16

The diffusion approximations 53

2.17

The grey approximation 55

2.18

Eddington’s approximation 56

53

Exercises 58
References 63

Chapter 3 Methods of solution of the transfer equation 64
3.1

Chandrasekhar’s solution

64

3.2

The H -function 70

3.2.1

The first approximation 72

3.2.2

The second approximation 73

3.3

Radiative equilibrium of a planetary nebula 74

3.4

Incident radiation from an outside source 75

3.5

Diffuse reflection when ω = 1 (conservative case) 78

3.6

Iteration of the integral equation 79

3.7

Integral equation method. Solution by linear equations 82
Exercises 83
References 86

Chapter 4 Two-point boundary problems 88
4.1

Boundary conditions 88

4.2

Differential equation method. Riccati transformation 90

4.3

Feautrier method for plane parallel and stationary media 92

4.4

Boundary conditions 93

4.5

The difference equation 94

4.6

Rybicki method 99

4.7

Solution in spherically symmetric media 101


Contents

vii

4.8

Ray-by-ray treatment of Schmid-Burgk 106

4.9

Discrete space representation 108
Exercises 109
References 110

Chapter 5 Principle of invariance 112
5.1

Glass plates theory 112

5.2

The principle of invariance 116

5.3

Diffuse reflection and transmission 117

5.4

The invariance of the law of diffuse reflection

5.5

Evaluation of the scattering function 120

5.6

An equation connecting I (0, µ) and S0 (µ, µ ) 123

5.7

The integral for S with p(cos ) =

5.8

The principle of invariance in a finite medium 126

5.9

Integral equations for the scattering and transmission functions 130

5.10

The X - and the Y -functions 133

5.11

Non-uniqueness of the solution in the conservative case 135

5.12

Particle counting method 137

5.13

The exit function 139

119

(1 + x cos ) 125

Exercises 143
References 144

Chapter 6 Discrete space theory 146
6.1

Introduction 146

6.2

The rod model 147

6.3

The interaction principle for the rod 148

6.4

Multiple rods: star products 150

6.5

The interaction principle for a slab 152

6.6

The star product for the slab 154

6.7

Emergent radiation 157

6.8

The internal radiation field 158

6.9

Reflecting surface 163

6.10

Monochromatic equation of transfer 163

6.11

Non-negativity and flux conservation in cell matrices 168

6.12

Solution of the spherically symmetric equation 171

6.13

Solution of line transfer in spherical symmetry 179

6.14

Integral operator method 185
Exercises 190
References 191


Contents

viii

Chapter 7 Transfer equation in moving media: the observer frame 193
7.1

Introduction 193

7.2

Observer’s frame in plane parallel geometry 194

7.3

Wave motion in the observer’s frame 199

7.4

Observer’s frame and spherical symmetry 201

7.4.1

Ray-by-ray method 201

7.4.2

Observer’s frame and discrete space theory 205

7.4.3

Integral form due to Averett and Loeser 209
Exercises 215
References 215

Chapter 8 Radiative transfer equation in the comoving frame 217
8.1

Introduction 217

8.2

Transfer equation in the comoving frame 218

8.3

Impact parameter method 220

8.4

Application of discrete space theory to the comoving frame 225

8.5

Lorentz transformation and aberration and advection 238

8.6

The equation of transfer in the comoving frame 244

8.7

Aberration and advection with monochromatic radiation 247

8.8

Line formation with aberration and advection 251

8.9

Method of adaptive mesh 254
Exercises 261

References 262

Chapter 9 Escape probability methods 264
9.1

Surfaces of constant radial velocity 264

9.2

Sobolev method of escape probability 266

9.3

Generalized Sobolev method 275

9.4

Core-saturation method of Rybicki (1972) 282

9.5

Scharmer’s method

9.6

Probabilistic equations for line source function 297

9.6.1

Empirical basis for probabilistic formulations 297

9.6.2

Exact equation for S/B 300

9.6.3

Approximate probabilistic equations 301

9.7

Probabilistic radiative transfer 303

9.8

Mean escape probability for resonance lines 310

9.9

Probability of quantum exit 312

9.9.1

The resolvents and Milne equations 319

287


Contents

ix

Exercises 324
References 326

Chapter 10 Operator perturbation methods 330
10.1

Introduction 330

10.2

Non-local perturbation technique of Cannon 331

10.3

Multi-level calculations using the approximate lambda operator 338

10.4

Complete linearization method 345

10.5

Approximate lambda operator (ALO)

10.6

Characteristic rays and ALO-ALI techniques 353

348

Exercises 359
References 359

Chapter 11 Polarization 362
11.1

Elliptically polarized beam 363

11.2

Rayleigh scattering 365

11.3

Rotation of the axes and Stokes parameters 367

11.4

Transfer equation for I (θ, φ) 368

11.5

Polarization under the assumption of axial symmetry

11.6

Polarization in spherically symmetric media 376

11.7

Rayleigh scattering and scattering using planetary atmospheres 387

11.8

Resonance line polarization 397

373

Exercises 412
References 413

Chapter 12 Polarization in magnetic media 416
12.1

Polarized light in terms of I , Q, U , V 416

12.2

Transfer equation for the Stokes vector 418

12.3

Solution of the vector transfer equation with the Milne–Eddington
approximation 421

12.4

Zeeman line transfer: the Feautrier method 423

12.5

Lambda operator method for Zeeman line transfer 426

12.6

Solution of the transfer equation for polarized radiation 428

12.7

Polarization approximate lambda iteration (PALI) methods 433
Exercises 438
References 439

Chapter 13 Multi-dimensional radiative transfer 441
13.1

Introduction 441


Contents

x

13.2

Reflection effect in binary stars 442

13.3

Two-dimensional transfer and discrete space theory 449

13.4

Three-dimensional radiative transfer 452

13.5

Time dependent radiative transfer 455

13.6

Radiative transfer, entropy and local potentials 460

13.7

Radiative transfer in masers 466
Exercises 466
References 467
Symbol index 469
Index 477


Preface

Astrophysicists analyse the light coming from stellar atmosphere-like objects with
widely differing physical conditions using the solution of the equation of radiative
transfer as a tool. A method of obtaining the solution of the transfer equation
developed to suit a given physical condition need not necessarily be useful in a

situation with different physical conditions. Furthermore, each individual has his/her
preferences to a particular type of methodology. These factors necessitated the
development of several widely differing methods of solving the transfer equation.
In the second half of the twentieth century several books were written on the
subject of radiative transfer: one each by Chandrasekhar, Kourganoff and Sobolev,
two books by Mihalas, two by Kalkofen and more recently two books by Sen and
Wilson. These books, which describe the developments of the transfer theory, will
remain milestones. They will be of great value to the researcher in this field. A
beginner needs to understand the basic concepts and the initial development of the
subject to proceed to use the latest advances. It is felt that it is necessary to have
a book on radiative transfer which presents a comprehensive view of the subject
as applied in astrophysics or more particularly in stellar atmospheres and objects
with similar geometrical and physical conditions. This book serves such a purpose.
Several methods are presented in the book so that the students of radiative transfer
can familiarise themselves with the techniques old and new.
It became a daunting task to include all the existing techniques in the book as
there is a restriction on its size. This resulted in leaving out a few methods that
are of equal interest as those that appear in the book. I apologize to the authors of
these methods in advance. The subject matter of the book assumes of the student a
knowledge of basic mathematics and physics at the undergraduate level. This book

xi


xii

Preface

is intended to be included in the advanced course work of undergraduate students,
and the course work of graduate students. Several exercises have been included at

the end of each chapter for practising the concepts described in the chapter. These
problems are straightforward and can be solved by direct application of the theory.
Some of them involve just supplying the intermediate steps in the derivations of the
chapter.
The material in the book is largely drawn from the books mentioned earlier and
from various other references cited at the end of each chapter. If there are any errors
these are mine and I shall be grateful if these are brought to my attention. Any
suggestions for improvements and corrections are welcome.
It is a pleasure to thank Dr W. Kalkofen for a brief discussion on the subject
matter of the book. I am grateful to Professor K. K. Sen for not only giving a few
tips on writing books but also for going through the first draft and pointing out
several typographical errors and adding a few conceptual points. This book would
not have been possible without the active help from Mr Baba Anthony Varghese
who very patiently typed the text. His phenomenal computer expertise enabled the
book to rapidly and easily take its present form. It is pleasure to thank him for all
this. I thank Drs A. Vagiswari and Christina Louis for their magnanimous and kind
help in securing me any reference that I needed. Further, I thank Mr M. Srinivasa
Rao, Mr S. Muthukrishnan and Mrs Pramila Kaveriappa for helping me in various
ways during the writing of the book.
There is one person whose memory always lingers on in my mind – that of
Professor M. K. Vainu Bappu. From him I have learnt several aspects not only of
science but also of life. I fondly cherish the memory of my association with him.
I am grateful to my wife Jayalakshmi and my children Rajani (Vaidhyanathan),
Chandra (Edith) and Usha (Madhusudan) – spouses in brackets – for the love and
affection shown to me.
Finally I thank the staff of Cambridge University Press who have been connected
with the publication of the book, especially Dr Simon Mitton and Miss Jacqueline
Garget for clearing my doubts from time to time and Ms Maureen Storey, who very
patiently went through the manuscript and suggested several corrections.

Bangalore
October 2000

Annamaneni Peraiah


Chapter 1
Definitions of fundamental quantities of the
radiation field

Specific intensity

1.1

This is the most fundamental quantity of the radiation field. We shall be dealing with
this quantity throughout this book.
Let d E ν be the amount of radiant energy in the frequency interval (ν, ν + dν)
transported across an element of area ds and in the element of solid angle dω during
the time interval dt. This energy is given by
d E ν = Iν cos θ dν dσ dω dt,

(1.1.1)

where θ is the angle that the beam of radiation makes with the outward normal to
the area ds, and Iν is the specific intensity or simply intensity (see figure 1.1).
The dimensions of the intensity are, in CGS units, erg cm−2 s−1 hz−1 ster−1 . The
intensity changes in space, direction, time and frequency in a medium that absorbs

Ω

Figure 1.1 Schematic
diagram which shows how
the specific intensity is
defined.

dω
P

θ

ds

1

Normal to ds = n


1 Definitions of fundamental quantities of the radiation field

2

and emits radiation. Iν can be written as
Iν = Iν (r, , t),

where r is the position vector and
be written as
Iν = Iν (x, y, z; α, β, γ ; t),

(1.1.2)

is the direction. In Cartesian coordinates it can

(1.1.3)

where x, y, z are the Cartesian coordinate axes and α, β, γ are the direction cosines.
If the medium is stratified in plane parallel layers, then
Iν = Iν (z, θ, ϕ; t),

(1.1.4)

where z is the height in the direction normal to the plane of stratification and θ and
ϕ are the polar and azimuthal angles respectively. If Iν is independent of ϕ, then we
have a radiation field with axial symmetry about the z-axis. Instead of z, we may
choose symmetry around the x-axis.
In spherical symmetry, Iν is
Iν = Iν (r, θ; t),

(1.1.5)

where r is the radius of the sphere and θ is the angle made by the direction of the
ray with the radius vector.
The radiation field is said to be isotropic at a point, if the intensity is independent
of direction at that point and then
Iν = Iν (r, t).

(1.1.6)

If the intensity is independent of the spatial coordinates and direction, the radiation
field is said to be homogeneous and isotropic. If the intensity Iν is integrated over
all the frequencies, it is called the integrated intensity I and is given by

∞

I =
0

Iν dν.

(1.1.7)

There are other parameters that characterize the state of polarization in a radiation
field. These are studied in chapters 11 and 12.

1.2

Net flux

The flux Fν is the amount of radiant energy transferred across a unit area in unit
time in unit frequency interval. The amount of radiant energy in the area ds in the
direction θ (see figure 1.1) to the normal, in the solid angle dω, in time dt and in


1.2 Net flux

3

the frequency interval (ν, ν + dν) is equal to Iν cos θ dω dν ds dt. The net flow in
all directions is
Iν cos θ dω,

dν ds dt

or
Fν =

Iν cos θ dω.

(1.2.1)

The integration is over all solid angles. This is the net flux and is the rate of flow of
radiant energy per unit area per unit frequency.
In polar coordinates, where the outward normal is in the z-direction, we have
dω = sin θ dθ dϕ,

(1.2.2)

where ϕ is the azimuthal angle. The net flux Fν then becomes
Fν =

π

2π
0

0

Iν cos θ sin θ dϕ dθ.

(1.2.3)

The dimensions of flux are erg cm−2 s−1 hz−1 . Equation (1.2.3) can also be written

as
Fν =

π/2

2π

dϕ
0

0

Iν cos θ sin θ dθ +

= Fν (+) − Fν (−),

π

2π

dϕ
0

π/2

Iν cos θ sin θ dθ
(1.2.4)

where
Fν (+) =

π/2

2π
0

0

Iν cos θ sin θ dθ dϕ

(1.2.5)

Iν cos θ sin θ dθ dϕ.

(1.2.6)

and
Fν (−) =

π/2

2π
0

π

The physical meaning of equation (1.2.4) is as follows: Fν (+) represents the
radiation illuminating the area from one side and Fν (−) represents the radiation
illuminating the area from another side. Therefore Fν , the flux of radiation transported through the area, is the difference between these illuminations of the area.
The flux depends on the direction of the normal to the area. The dependence of the

flux on direction shows that flux is of vector character. In the Cartesian coordinate
system, let the angles made by the direction of radiation with the axes x, y and z
be α1 , β1 and γ1 respectively, then the flux or radiation along the coordinate axes is
given by
Fν (x) =

Iν cos α1 dω,

(1.2.7)


1 Definitions of fundamental quantities of the radiation field

4

Fν (y) =

Iν cos β1 dω,

(1.2.8)

Fν (z) =

Iν cos γ1 dω.

(1.2.9)

Furthermore, if α2 , β2 and γ2 are the angles made by the coordinate axes and the
normal to the area and θ is the angle between the normal and the direction of the
radiation, then

cos θ = cos α1 cos α2 + cos β1 cos β2 + cos γ1 cos γ2 .

(1.2.10)

Substituting equation (1.2.10) into equation (1.2.1), we get
Fν = cos α2 Fν (x) + cos β2 Fν (y) + cos γ2 Fν (z).

(1.2.11)

The integrated flux over frequency is
∞

F=
0

Fν dν.

(1.2.12)

If the radiation field is symmetric with respect to the coordinate axes, then the net
flux across the surface oriented perpendicular to that axis is zero as the oppositely
directed rays cancel each other. In a homogeneous planar geometry, Fν (x) and Fν (y)
are zeros and only Fν (z) exists. In such a situation, we have
Fν (z, t) = 2π

+1
−1

I (z, µ, t)µ dµ,

(1.2.13)

where µ = cos θ .
The astrophysical flux FAν (z, t) normally absorbs the π on the RHS of equation
(1.2.13) and is written as
FAν (z, t) = 2

+1
−1

I (z, µ, t)µ dµ

(1.2.14)

and the Eddington flux FEν is defined as
FEν (z, t) =

1.2.1

1
2

+1
−1

I (z, µ, t)µ dµ.

(1.2.15)

Specific luminosity

The specific luminosity was suggested by Rybicki (1969) and Kandel (1973). We
define it following Collins (1973).
From figure 1.2, we define the specific luminosity L(ψ, ξ ) in terms of the
orientation variables ψ and ξ as
L(ψ, ξ ) = 4π

I (θ, φ)n(θ,
ˆ φ) · o(θ,
ˆ φ) d A(θ, φ),

(1.2.16)

A

where n(θ,
ˆ φ) and o(θ,
ˆ φ) are position dependent unit vectors normal to the surface
and in the direction of the observer respectively. The area A over which the specific


1.3 Density of radiation and mean intensity

5

intensity I (θ, φ) is to be integrated is the ‘observable’ surface and is defined by the
orientation angles ψ and ξ . It is obvious from equation (1.2.16) that L(ψ, ξ ) is a
function of the orientation of the object with respect to the observer and is measured
per unit solid angle; the total luminosity L is given in terms of L(ψ, ξ ) as
L=

1
4π

L(ψ, ξ ) d (ψ, ξ ).

(1.2.17)

4π

Density of radiation and mean intensity

1.3

Let V and be two regions (see figure 1.3) the latter being larger than the former in
linear dimensions but sufficiently small for a pencil not to have its intensity changed
appreciably in transit. The radiation travelling through V must have crossed the
region
through some element; let d be such an element with normal N. The

Z
To
Observer
o n
ξ

θ
Y

ψ

Figure 1.2 The angles θ
and φ are the angular
coordinates of a point on the
stellar surface, and therefore
represent a local structure.
The angles ψ and ξ
represent the orientation of
the stellar body (from
Collins (1973), with
permission).

φ

X

N

dσ n
Ω

r

dΣ

dω
V P
Σ

Figure 1.3 Schematic

diagram to define density of
radiation.


1 Definitions of fundamental quantities of the radiation field

6

energy passing through d
unit time is
Iν ( , N) d

which also passes through dσ with normal n on V per

dω dν,

(1.3.1)

· n) dσ/r 2 .

(1.3.2)

where
dω = (

If l is the length travelled by the pencil in V , then an amount of energy
Iν (

· n)(

· N) dσ d dν l
(1.3.3)
c
r2
will have travelled through the element in time l/c, where c is the velocity of light.
The solid angle dω subtended by d at P is ( · N) d /r 2 and the volume
intercepted in V by the pencil is given by
d V = l(

· n) dσ.

(1.3.4)

This amount of energy is given by
1
Iν dν d V dω.
(1.3.5)
c
Therefore, the contribution to the energy per unit volume per unit frequency range
(in the interval ν, ν + dν) coming from the solid angle dω about the direction is
Iν dω/c and the energy density is defined as
Uν =

1
c

Iν dω.

(1.3.6)

The average intensity or mean intensity Jν is
Jν =

1
4π

Iν dω,

(1.3.7)

so that
4π
Jν .
c
For an axially symmetric radiation field, Jν is given by
Uν =

Jν =

1
2

=

1
2

π
0

Iν sin θ dθ

+1
−1

(1.3.8)

I (µ) dµ.

(1.3.9)

The integrated energy density U is
∞

U=
0

Uν dν =

1
c

I dω.

(1.3.10)

The dimensions of energy density are erg cm−3 hz−1 and those of the integrated
energy density are erg cm−3 . The dimensions of the mean intensity are erg cm−2
s−1 hz−1 .

1.4 Radiation pressure

1.4

7

Radiation pressure

A quantum of energy hν will have a momentum of hν/c, where c is the velocity of
light in the direction of propagation. The pressure of radiation at the point P (see
figure 1.1) is calculated from the net rate of transfer of momentum normal to an area
ds, which contains the point P. The amount of radiant energy in the frequency range
(ν, ν + dν) incident on ds making an angle θ with the normal to ds traversing the
solid angle dω in time dt is
Iν cos θ dω dν ds dt.

(1.4.1)

The momentum associated with this energy in the direction Iν is
1
Iν cos θ dω dν ds dt.
c

(1.4.2)

Therefore the normal component of the momentum transferred across ds by the
radiation is
1
dσ dt Iν cos2 θ dω dt.

c

(1.4.3)

The net transfer of momentum across ds by the radiation in the frequency interval
(ν, ν + dν) is
dσ dt
c

Iν cos2 θ dω dν,

(1.4.4)

where the integration is over the whole sphere. The pressure at the point P is the net
rate of transfer of momentum normal to the element of the surface area containing
P in the unit area; the pressure pr (ν) dν can be written in the frequency interval as
pr (ν) =

1
c

π

2π
0

Iν cos2 θ sin θ dθ dϕ.

0

(1.4.5)

If the radiation field is isotropic, then
pr (ν) =

2π
Iν
c

π

µ2 dµ =

0

4π
Iν
3c

(µ = cos θ)

(1.4.6)

or in terms of energy density Uν
pr (ν) =

1
Uν .
3

(1.4.7)

The radiation pressure integrated over all frequencies is
∞

pr =

pr (ν) dν

(1.4.8)

I cos2 θ dω,

(1.4.9)

0

or
pr =

1
c


1 Definitions of fundamental quantities of the radiation field

8

where I is the integrated intensity. Furthermore
pr =

1
U.
3

(1.4.10)

It can be seen that the dimensions of radiation pressure are the same as those of
energy density, that is, erg cm−3 hz −1 and the integrated radiation pressure has the
dimensions of erg cm−3 .

1.5

Moments of the radiation field

Moments are defined in such a way that the nth moment over the radiation field is
given by
Mn (z, n) =

1
2

+1
−1

Iν (z, µ)µn dµ.

(1.5.1)

Following Eddington, we can have the zeroth, first and second moments as:

1. Zeroth moment (mean intensity):
Jν (z) =

1
2

+1
−1

I (z, µ) dµ.

(1.5.2)

2. First moment (Eddington flux):
Hν (z) =

1
2

+1
−1

I (z, µ)µ dµ.

(1.5.3)

3. Second moment (the so called K -integral):
K ν (z) =

1.6

1
2

+1
−1

I (z, µ)µ2 dµ.

(1.5.4)

Pressure tensor

The rate of transfer of the x-component of the momentum across the element of
surface normal to the x-direction by radiation in the solid angle dw per unit area in
the direction whose direction cosines are l, m, n is
1
I l dω l,
c

(1.6.1)

where I is the integrated radiation. If monochromatic radiation is considered, then
I should be replaced by Iν dν. The total rate of x-momentum transfer across the
element per unit area is pr (x x):


1.7 Extinction coefficient: true absorption and scattering

pr (x x) =

1
c

9

I l 2 dω.

(1.6.2)

Similarly the y- and z-components are given by
pr (x y) =

1
c

I lm dω

and

pr (x z) =

1
c

I ln dω.

(1.6.3)

The quantities pr (yx), pr (yy), pr (yz), pr (zx), pr (zy) and pr (zz) are similarly

defined for elements of the surfaces normal to the y- and z-directions. These nine
quantities constitute the ‘stress tensor’.
One can see that pr (x y) = pr (yx), pr (x z) = pr (zx) and pr (yz) = pr (zy) or
that the tensor is symmetrical. The mean pressure p¯ is defined by
p¯ =

1
[ pr (x x) + pr (yy) + pr (zz)],
3

p¯ =

1
3c

(1.6.4)

and
Iω =

1
U,
3

(1.6.5)

as l 2 + m 2 + n 2 = 1.
In the case of an isotropic radiation field
p¯ = pr (x x) = pr (yy) = pr (zz) =

and
pr (x y) = pr (yx) = 0,
pr (x z) = pr (zx) = 0,
pr (yz) = pr (x y) = 0.

1.7

1
U,
3

(1.6.6)








(1.6.7)

Extinction coefficient: true absorption and scattering

A pencil of radiation of intensity Iν is attenuated while passing through matter of
thickness ds and its intensity becomes Iν + d Iν , where
d Iν = −Iν κν ds.

(1.7.1)

The quantity κν is called the mass extinction coefficient or the mass absorption
coefficient. κν comprises two important processes: (1) true absorption and (2) scattering. Therefore we can write
κν = κνa + σν ,

(1.7.2)

where κνa and σν are the absorption and scattering coefficients respectively. Absorption is the removal of radiation from the pencil of the beam by a process


1 Definitions of fundamental quantities of the radiation field

10

which involves changing the internal degrees of freedom of an atom or a molecule.
Examples of these processes are: (1) photoionization or bound–free absorption by
which the photon is absorbed and the excess energy, if any, goes into the kinetic
energy of the electron thermalizing the medium; (2) the absorption of a photon by a
freely moving electron that changes its kinetic energy which is known as free–free
absorption; (3) the absorption of a photon by an atom leading to excitation from
one bound state to another bound state, which is called bound–bound absorption
or photoexcitation; (4) the collision of an atom in a photoexcited state which will
contribute to the thermal pool; (5) the photoexcitation of an atom which ultimately
leads to fluorescence; (6) negative hydrogen absorption, etc. The reversal of the
above processes may contribute to the emission coefficient (see section 1.8).
The coefficient κνa depends on the thermodynamic state of the matter at (pressure
p, temperature T , chemical abundances αi ) any given point in the medium. At the
point r the coefficient is given by
κνa (r, T ) = κνa [ p(r, T ), T (r ), αi (r, T ), . . . , ακ (r, T )],

(1.7.3)

when there is local thermodynamic equilibrium (LTE). This kind of situation does
not exist in reality and one needs to determine the κνa in a non-LTE situation. In static
media κνa is isotropic while in moving media it is angle and frequency dependent due
to Doppler shifts.
Another process by which energy is lost from the beam is the scattering of
radiation which is represented by the mass scattering coefficient κνs . Scattering
changes not only the photon’s direction but also its energy. If we define the albedo
for single scattering as ων , then
ων =

σν
,
κν

(1.7.4)

is the ratio of scattering to the extinction coefficients.
The extinction coefficient is the product of the atomic absorption coefficients or
scattering coefficients (cm2 ) and the number density of the absorbing or scattering
particles (cm−3 ). The dimension of κν is cm−1 and 1/κν gives the photon mean free
path which is the distance over which a photon travels before it is removed from the
pencil of the beam of radiation.

1.8

Emission coefficient

Let an element of mass with a volume element d V emit an amount of energy d E ν
into an element of solid angle dω centred around in the frequency interval ν to

ν + dν and time interval t to t + dt. Then
d E ν = jν d V dω dν dt,

(1.8.1)


1.8 Emission coefficient

11

where jν is called the macroscopic emission coefficient or emissivity. The emissivity has dimensions erg cm−3 sr−1 hz−1 s−1 . Emission is the combination of the
reverse of the physical processes that cause true absorption. These processes are:
(a) radiative recombination: when a free electron occupies a bound state creating
a photon whose energy is the sum of the kinetic energy of the electron and the
binding energy; (b) bremsstrahlung: a free electron moving in one hyperbolic orbit
emits a photon by moving into a different hyperbolic orbit of lower energy; (c)
photo de-excitation or collisional de-excitation: a bound electron changes to another
bound state by emitting a photon through collision; (d) collisional recombination: a
photoexcited atom contributes photon energy by collisional ionization; the reverse
of this is called (three-body) collisional recombination; and (e) fluorescence: if a
photon is absorbed by an atom and it is excited from bound state p to another bound
state r , decays to an intermediate bound state q and then to the original state p,
this process is called fluorescence. The energy from the original absorbed photon is
re-emitted in two photons each of different energy.
A true picture of the occupation numbers is obtained only when the statistical
equilibrium equation, which describes all necessary processes that are to be taken
into account, is written. When LTE exists, the emission coefficient is given by
jνa (LTE) = κνa Bν (T ),

(1.8.2)

where Bν (T ) is the Planck function:
Bν (T ) =

hν
2hν 3
exp
kT
c2

−1

−1

.

(1.8.3)

Equation (1.8.2) is known as Kirchhoff–Planck relation. In a non-LTE situation one
has to consider stimulated emission due to the presence of the radiation field and
spontaneous emission and the Einstein transition coefficients involved.
Emission of radiation can also be from the scattered photons. One can write
jνs (r, ) =

1
4π

σνs , (r, t) p(ν, ; ν ,

; r, t)Iν (r,

, t) dν dω . (1.8.4)

The phase function p can be normalized in such a way that
p(ν ,

; ν, , ;r, t) dν dω = 4π.

(1.8.5)

This is the manifestation of the conservation of radiation flux, that is, the emitted
radiation balances that removed from the beam.
Equation (1.8.2) should be corrected for the stimulated scattering by multiplying
it by the correction factor
1+

c2
Iν (r, , t) .
2hν 3

(1.8.6)