Tetsuo Sasao and André B. Fletcher
Introduction to VLBI Systems
Chapter 1
Lecture Notes for KVN Students
Based on Ajou University Lecture Notes(to be further edited)
Version 2.
Issued on February 28, 2005.
Revised on March 18, 2005.
Revised on February 4, 2006.

Basic Knowledge ofRadio Astronomy
Contents
1 ‘Windows’ for Ground–based Astronomy

3

2 Various Classifications of Astronomical Radio Sources

4

3 Spectra of Typical Continuum Radio Sources

4

4 Spectralline Radio Sources

5

5 Designations of Astronomical Radio Sources

8

6 Designations of Frequency Bands

8

7 Basic Quantities in Radio Astronomy
7.1 Intensity (Specific/Monochromatic Intensity) Iν or Brightness
Bν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Constancy of the Monochromatic Intensity Iν . . . . . . . . .
7.3 ‘Spectral Flux Density’ or ‘Flux Density’ or ‘Flux’ Sν . . . . .
7.4 Power/Energy/Radiation Flux Density S . . . . . . . . . . . .
7.5 Spectral Energy Density per Unit Solid Angle uν . . . . . . .
7.6 Spectral Energy Density Uν . . . . . . . . . . . . . . . . . . .

9

1

9
10
11
12
12
13

8 Emission and Absorption of Electromagnetic Radiation
8.1 Elementary Quantum Theory of Radiation (A. Einstein, 1916)
8.2 Einstein Coefficients . . . . . . . . . . . . . . . . . . . . . . .
8.3 Number Density of Photons . . . . . . . . . . . . . . . . . . .

13
13
17
17

9 Blackbody Radiation
9.1 Two Extreme Cases of the Planck Spectrum . . . . . . . . . .
9.2 Wien’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Spectral Indices of Thermal Continuum Radio Sources . . . .
9.4 Stars are Faint and Gas Clouds are Bright in the Radio Sky .
9.5 Stefan–Boltzmann Law . . . . . . . . . . . . . . . . . . . . . .
9.6 Total Blackbody Radiation from a Star or a Gas Cloud . . . .
9.7 Universality of the Relationship among Einstein’s Coefficients
9.8 Another Important Quantity in Radio Astronomy . . . . . . .

18
20
22
22
22
24
24
25
26

10 Radiative Transfer
26
10.1 Phenomenological Derivation of the Radiative Transfer Equation 26
10.2 Derivation of the Radiative Transfer Equation from Einstein’s
Elementary Quantum Theory of Radiation . . . . . . . . . . . 27
10.3 The Simplest Solutions of Radiative Transfer Equation . . . . 28
10.4 What is LTE? . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
10.5 Spectrum of the Orion Nebula . . . . . . . . . . . . . . . . . . 33
11 Synchrotron Radiation34
11.1 Non–Relativistic Case . . . . . . . . . . . . . . . . . . . . . . 34
11.2 Relativistic Case . . . . . . . . . . . . . . . . . . . . . . . . . 36

2

1‘Windows’ for Ground–based Astronomy

Figure 1 shows how the Earth’s atmosphere transmits electromagnetic waves
from the Universe. The curve over the hatched area in this figure corresponds
to the height in the atmosphere at which the radiation is attenuated by afactor of 1/2.

Figure 1: ‘Windows’ for ground-based astronomy (from Rohlfs & Wilson,
2001).
This figure shows that the Earth’s atmosphere is largely transparent in
the radio (15 MHz – 300 GHz) and visible light (360 THz – 830 THz) regionsof the electromagnetic spectrum. These two regions are the main ‘windows’
for ground-based astronomical observations. VLBI (Very Long Baseline Interferometry) is one of the techniques used to observe astronomical radiosources through the ‘radio window’.

3

2

Various Classifications of Astronomical Radio Sources

There are several ways to classify astronomical radio sources. Typical classifications are:•

– Thermal source
Source emitting via a thermal mechanism (e.g. blackbody radiation),– Non-thermal source
Source emitting via a non-thermal mechanism (e.g. synchrotron
radiation, inverse Compton scattering, annihilation radiation, maser
emission, etc.),•– Continuum source
Source emitting over a broad range of frequencies,– Spectralline source
Source emitting in narrow lines at specific frequencies,and•– Galactic source
Source inside our Milky Way Galaxy,– Extragalactic source
Source outside our Galaxy.

3

Spectra of Typical Continuum Radio Sources

Figure 2 shows typical spectra of continuum radio sources.
These spectra are characterized by a quantity called ‘spectral index’ α, defined by the formula: Sν ∝ ν −α , where Sν is a quantity called the ‘flux
density’ [unit: Jy (Jansky) ≡ 10−26 W m−2 Hz −1 ], which is a measure of thestrength of the radiation from a source.
Thermal and non–thermal sources usually show different spectral indices,namely:α∼= −2 for thermal sources,andα ≥ 0 for non-thermal sources.

4

Figure 2: Spectra of galactic and extragalactic continuum radio sources (fromKraus, 1986).

4

Spectralline Radio Sources

An example of a thermal spectralline source showing many molecular lines
is given in Figure 3.
Examples of spectra of non-thermal maser line sources obtained in test
observations for the VERA (VLBI Exploration of Radio Astrometry) arrayin Japan are shown in Figures 4, 5, and 6.

5

Figure 3: Thermal molecular lines observed in the Orion-KL star–forming
region (Kaifu et al., 1985).

Figure 4: H2 O maser lines in Cep A (left), and RT Vir (right) (courtesy of
VERA group, Japan, 2002).

6

Figure 5: SiO maser lines in Orion-KL (left), and R Cas (right) observed inpointing tests for a VERA antenna (courtesy of VERA group, Japan, 2002).

Figure 6: H2 O maser lines in W49 N and OH43.8, observed simultaneouslywith a VERA dual-beam antenna (courtesy of VERA group, Japan, 2002).

7

5

Designations of Astronomical Radio Sources

There are many designation systems for naming radio sources. Examples
are given in the following. As a result, radio sources are often identified byseveral different names.
• IAU 1974 system of designation (‘IAU name’, see, e.g., Explanatory
Supplement to American Astronomical Almanac)– identifies a radio source by a six, seven, or eight-digit number,such as 0134+329, which tells us that the right ascension of thesource is 01h 34m , and its declination is +32.◦ 9 (usually in the
B1950 equinox system).
– sometimes source types or catalog acronyms are added, e.g., pulsars are called ‘PSR0950+08’, and sources from the Parkes Sky
Survey as ‘PKS1322-42’.
• 3C-name, 4C-name– identifies an extragalactic radio source by a serial number in the
Third Cambridge Catalog, and by declination in the Fourth Cambridge Survey Catalog.
– examples are 3C84, 3C273, 3C345, 4C39.25, etc.
• W-name– is based on Westerhout’s (1958) catalog of HII regions.
– examples are W49 N, W3(OH), W51 M, etc.

6

Designations of Frequency Bands

The basic unit of radio frequency is Hz (Hertz: cycle per second). In orderto describe high frequency, typically used in radio astronomical observations
or communications, we frequently use the following secondary units: kHz(kilohertz: 103 Hz), MHz (megahertz: 106 Hz), GHz (gigahertz: 109 Hz),and THz (terahertz: 1012 Hz).
Radio frequency bands used in astronomical observations, as well as incommunications, are designated by one or a few alphabetic characters. Twosystems of designations are shown in Figure 7.
In radio astronomy, IEEE STD-521-1976 designations have been used.
Examples are:8

Figure 7: Frequency designations.
• L–band : GPS satellite, observations of OH masers,• S– and X–bands : geodetic VLBI observations,• K–band : observations of H2 O masers, and• Q–band : observations of SiO masers.

7

Basic Quantities in Radio Astronomy

The following quantities are commonly used in radio astronomy to characterize radio waves coming from celestial bodies.

7.1

Intensity (Specific/Monochromatic Intensity) Iν or
Brightness Bν

The intensity Iν (or brightness Bν ) is the quantity of electromagnetic radiation energy incoming from a certain direction in the sky, per unit solid angle,
per unit time, per unit area perpendicular to this direction, and per unit
frequency bandwidth with center frequency ν. The SI unit of this specific(or monochromatic) intensity is thus: W m−2 Hz−1 sr−1 (Figure 8).
In terms of the monochromatic intensity Iν , the power dW of radiation
coming from a direction s within a solid angle dΩ, through a cross-section ofarea dσ with a normal inclined by an angle θ from the direction s, within afrequency bandwidth dν centered at ν, is given by:
dW = Iν (s) cos θ dΩ dσ dν.
9

(1)

s

dΩ
θ

dσ

Figure 8: Definition of Specific/Monochromatic Intensity Iν (or Brightness
Bν ).

7.2

Constancy of the Monochromatic Intensity Iν

The monochromatic intensity Iν remains constant along a ray path, irrespective of the distance from the emitting source, as long as no absorption,
emission, or scattering occurs along the path. An explanation of this constancy may be given as follows.
W2

σ2
Ω2

W1

σ1
Ω

r2

1

r1

Figure 9: Constancy of the monochromatic intensity Iν along the ray path.
Let us assume a spherically symmetric distribution of radiation from anemitting source at the center, without any intervening absorption, emission,or scattering, as shown in Figure 9. Let a cone-like ray tube of constant10

opening angle intersect a sphere of radius r, forming a cross–section of area
σ, and let the source be seen to subtend a solid angle Ω at this radius. The
power of the radiation with bandwidth ∆ν passing through this cross–section
is W = Iν Ω σ∆ν, which must be constant at any radius along the tube, aslong as the radius is much larger than the source size. Now, the solid angle ofthe source Ω and the area of the cross–section σ vary as Ω ∝ r −2 and σ ∝ r 2 ,respectively. Therefore, the monochromatic intensity Iν = W/(Ω σ ∆ν)must be constant along the path.

7.3‘Spectral Flux Density’ or ‘Flux Density’ or ‘Flux’Sν

The spectral flux density Sν is the quantity of radiation energy incoming
through a cross section of unit area, per unit frequency bandwidth, andper unit time. A special unit called ‘Jansky (Jy)’ is widely used in radio
astronomy for the spectral flux density. This unit is defined as: 1 Jy = 10−26
W m−2 Hz−1 .
s

θ

∆σ

Figure 10: Definition of spectral flux density Sν .
The spectral flux density Sν is related to the intensity Iν by an integral
over a solid angle Ω:
Sν =

ZZ

Iν (s) cos θ dΩ,

Ω

=

ZZ

Iν (θ, φ) cos θ sin θ dθ dφ,

Ω

11

(2)

where Ω could be, for example, a solid angle subtended by a radio source(see Figure 10).

7.4

Power/Energy/Radiation Flux Density S

The power/energy/radiation flux density S is the quantity of radiation energy, over the whole frequency range, incoming through a cross section ofunit area, per unit time. Therefore,Z∞S=Sν dν.

(3)0

When we are interested in the “received” power flux density only, we
restrict the range of integration to the observing bandwidth ∆ν, i.e.,Z

S=Sν dν.

(4)∆νThe unit of power flux density is: W m−2 .

7.5

Spectral Energy Density per Unit Solid Angle uν

The spectral energy density per unit solid angle, uν (s), is the volume densityof the radiation energy incident from a certain direction s, per unit solid
angle, and per unit frequency bandwidth. The unit is J m−3 Hz−1 sr−1 .
s

dσ

c dt

Figure 11: Radiation energy per unit solid angle in a tube.
Let us consider a cylindrical tube with a cross section of area dσ perpendicular to the ray propagation direction, and with a length cdt, which is the
distance travelled by the radiation during a time interval dt at light speed c( = 2.998 × 108 m s−1 ) (see Figure 11). The radiation energy dUν (J Hz−1
sr−1 ) per unit solid angle, and per unit frequency bandwidth, contained in
the cylinder may be expressed either in terms of the spectral energy density
per unit solid angle, uν (s), or in terms of the intensity Iν (s), as:dUν = uν (s) cdt dσ,dUν = Iν (s) dt dσ.
12(5)Therefore, these quantities are related to each other by:Iν (s) = c uν (s).

7.6

(6)

Spectral Energy Density Uν

The spectral energy density Uν is the volume density of the energy of theradiation, per unit frequency bandwidth, incoming from all directions (Figure
12).

Figure 12: Spectral energy density: energy of radiation coming from all
directions, contained in a unit volume.
Therefore,Uν =Iuν (s)dΩ =1cIIν (s)dΩ.

(7)

The unit is J m−3 Hz−1 .

8Emission and Absorption of Electromagnetic Radiation

8.1

Elementary Quantum Theory of Radiation (A. Einstein, 1916)

Let us consider a gaseous medium consisting of a large number of particles(atoms or molecules) of the same species. Let us also consider that eachparticle exists in one of the quantum states Z1 , Z2 , ..., with energy levels
E1 , E2 , ... . If Em < En for a pair of states Zm and Zn , a particle canabsorb an amount of radiation energy En − Em in a transition from Zm to
Zn . Also, a particle can emit the same amount of energy in a transition from13

Zn to Zm (Figure 13). The frequency of the absorbed or emitted radiation
is determined by the equation:hνmn = En − Em ,(8)

where h is the Planck constant (h = 6.626 × 10−34 J s).

E=En

Zn

hνmn=En-Emhνmn=En-Em

Zm

E=Em

Z3
Z2
Z1

E=E3
E=E2
E=E1

Figure 13: Energy levels and transitions with emission (left–hand arrow) orabsorption (right–hand arrow) of radiation.
The energy levels here may be distributed continuously (continuum emission) or discretely (line emission). Note that even in the discrete level case,the frequency is spread over a finite ‘line width’, due to the Doppler shifts in
randomly moving gaseous media in the universe.
Three kinds of transitions may occur between these two states (Figure14), as follows:(1) Spontaneous emission
Z n → Zm
The spontaneous emission emerges due to a transition which occurs ‘byitself’, without any external influence (Figure 15). The probability dfsp forthe spontaneous emission to occur within a small solid angle dΩ towards a
direction -s, within a small frequency bandwidth dν around the frequency
νmn = (En − Em ) / h, and during a small time interval dt, must be proportional to dν dΩ dt. Therefore, the probability can be expressed through a
14

E=En

Zn

Zm

Spontaneous Emission

Absorption

Induced Emission

E=Em

Figure 14: The three possible transitions between two energy states.
certain coefficient αnm as:
dfsp = αnm dν dΩ dt.

(9)E= E n

E=Em

-s

dΩ

Figure 15: Spontaneous emission of radiation.
(2) Absorption
Z m → Zn
Some of the radiation passing through the gaseous medium may be absorbed in a transition from Zm to Zn , at a frequency around νmn = (En −Em ) / h (Figure 16). The absorption probability dfab for the radiation incident from a direction within a small solid angle dΩ around s, and in a small
frequency bandwidth dν, during a short time interval dt, must be proportional to dν dΩ dt and to the spectral energy density per unit solid angle,uν (s), of the radiation itself. Therefore, introducing a proportionality coeffincient βm
, we have:ndfab = βmuν (s) dν dΩ dt.
(10)(3) Induced (or stimulated) emission
Z n → Zm
Radiation with a frequency ν incident from a certain direction s may
induce (or ‘stimulate’) transitions of gas particles from higher to lower energy
15

dΩ

s

E=E n
u ν (s)
E=Em

Figure 16: Absorption of radiation.
levels, such that the newly emitted radiation has the same frequency ν, andthe same direction of propagation -s, as the incident radiation (Figure 17).
The probability dfst for the induced (or stimulated) emission to occur in adΩs

E= E nu ν (s)

E= Em

Figure 17: Induced (or stimulated) emission of radiation.
direction within a small solid angle dΩ around -s, and in a small frequency
bandwidth dν around νmn = (En − Em ) / h, during a short time interval
dt, must be proportional to dν dΩ dt and to the spectral energy density perunit solid angle uν (s) of the incident radiation. Therefore, introducing a
proportionality coefficient βnm , we have:
dfst = βnm uν (s) dν dΩ dt.
16

(11)

8.2

Einstein Coefficientsn
The three coefficients αnm , βm
, and βnm , describing the probabilities of thethree possible transitions:dfsp = αnm dν dΩ dt,ndfab = βmuν (s) dν dΩ dt,mdfst = βn uν (s) dν dΩ dt,(12)

are called the “Differential Einstein Coefficients”. They are usually isotropic,
i.e. they do not depend on the direction of propagation of radiation.
Einstein originally described the transition probabilities in the following
form, for the case of an isotropic radiation field, and with the particles at
rest:
dWsp = Amn dt,ndWab = Bm
Uν dt,
mdWst = Bn Uν dt,(13)

where Uν = uν (s) dΩ is the spectral energy density (in units of J m−3 Hz−1 ).
nm
The coefficients Am
n , Bm , and Bn , are called the “Einstein Coefficients”. Insuch a case, the spectral lines due to the transitions between discrete energy
levels must be monochromatic lines having no Doppler broadening, since theparticles are at rest. If we introduce f (ν) as the probability distribution of
frequency within a spectral line in a real interstellar medium in motion, thetwo sets of coefficients are related to each other, as follows:HAmn f (ν),4πn= Bm
f (ν),
m= Bn f (ν).

αnm =
n
βm
βnm

8.3

(14)

Number Density of Photons

Let the number densities of particles (number of particles per unit volume) in
the states Zm and Zn be nm and nn , respectively. Then the number density ofphotons emitted by the Zn → Zm transition into the solid angle dΩ aroundthe direction -s, within the bandwidth dν, during the time interval dt, isequal to:nn (dfsp + dfst ) = nn (αnm + βnm uν ) dν dΩ dt.

17

nn

Z n Ennm

Z m EmFigure 18: Number densities of particles in the two energy states.
On the other hand, the number density of photons in the same solid angledΩ, and the same bandwidth dν, absorbed by the Zm → Zn transition during
the time interval dt, is equal to:nuν dν dΩ dt.
nm dfab = nm βm

The differential changes in the number density of photons per unit solid
angle around -s, and per unit bandwidth around νmn = (En − Em )/h, in thecourse of the absorption and emission is now given by:ndnp (s, ν) dν dΩ = {−nm βmuν + nn [αnm + βnm uν (s)]}dν dΩ dt.

Therefore, the time variation of the number density of photons is describedby the following equation (Figure 19):dnp (s, ν)n= (nn βnm − nm βm)uν (s) + nn αnm .
dt

nn

s
np (s, ν) at t

nm

(15)

dΩ
np (s, ν) at t + dt

Figure 19: Time variation of the number density of photons.

9Blackbody Radiation

We now consider the case when the radiation and matter are in thermodynamic equilibrium. Then, we have the following:18

1. Stationary conditions:
The number of transitions Zm → Zn must be equal to the number ofopposite transitions Zn → Zm . Requiring this stationarity in equation(15), we have:nn m βmuν (s) = nn [αnm + βnm uν (s)].
(16)
2. Boltzmann distribution:
The probability Pn for a particle to be in a state Zn , with energy level
En , is given by the formula:
En

Pn = gn e− kT ,(17)

where k = 1.381 × 10−23 J K−1 is the Boltzmann constant, T is theabsolute temperature of the medium in Kelvin (K), and gn is the statistical weight (reflecting in particular the degree of degeneracy) of thestate Zn .
If we denote the number density of all particles as n, the ratio nn /n must
be equal to the probability Pn (at least in the statistical sense). Therefore,
we haveEmnm= gm e− kT ,
(18)
n
and
En
nn
(19)
= gn e− kT .
nInserting equations (18) and (19) into equation (16), we obtain
Em

En

n
e− kT gm βmuν (s) = e− kT gn [αnm + βnm uν (s)].

(20)

Einstein discussed the implications of this equation, as follows:
1. The energy density per unit solid angle of the thermal radiation musttend to infinity (uν (s) → ∞) when the temperature of the medium
tends to infinity (T → ∞) in equation (20). Hence, we obtainng m βm= gn βnm .

(21)Consequently, equation (20) becomes(e

En −EmkT− 1)βnm uν (s) = αnm .

(22)

If we take into account the relation hν = En −Em (hereafter, we denoteνmn = ν for simplicity), the energy density per unit solid angle can be
expressed asαm1uν = nm hν.
(23)βn e kT − 1
19Henceforth, we omit the s dependence in uν , since the RHS of equation(23) does not depend on any specific direction, as expected from theisotropic nature of thermal radiation.
2. The energy density per unit solid angle uν must follow the well-known
Rayleigh–Jeans radiation law:2ν 2uν = 3 kT,c(24)

in the classical limit hν  kT . Therefore, we must require in equation
(23)
2hν 3
αnm
.
(25)
=
βnm
c3
The above discussions thus lead to Planck’s formula of blackbody radiation:
uν =

2hν 3
1
.
hν
3
c e kT − 1(26)

For the intensity Iν , we obtain
Iν = cuν =

2hν 3
1
,
hν
2
c e kT − 1(27)(see Figure 20). For the energy density Uν , we have
Uν =

9.1

I

uν dΩ = 4πuν =18πhν 3.
hν
3c e kT − 1(28)

Two Extreme Cases of the Planck Spectrumhνhν• Rayleigh–Jeans region (hν  kT , and hence e kT ' 1 + kT
):

Iν =

2ν 2
kT.
c2

(29)

Note that thermal radiation in the radio frequency range is mostly in
the Rayleigh–Jeans region.
• Wien region (hν  kT ):Iν =2hν 3 − hνe kT .
c2

20

(30)

104

102

1

10−2

Wavelength λ (m)
10−4
10−6

10−8

10−10

10−12

10−14

1015
1010

Intensity Iν (W Hz−1 m−2 rad−2)

105

1
10−5
10−10
10−15
10−20
T=1K 10K 102K 103K 104K 105K106K107K 108K109K1010K

104

106

108

1010

1012
1014
1016
Frequency ν (Hz)

1018

1020

1022

10−25

Figure 20: Planck spectrum of black body radiation. Each curve corresponds
to a certain absolute thermodynamic temperature value.

21

9.2

Wien’s Law

The peak frequency of the Planck spectrum at a given temperature T is
νmax (in Hz) = 5.8789 × 1010 T (in K).

(31)

One can easily derive this law by equating to zero the derivative of theintensity Iν with respect to frequency ν, in equation (27).
This, Wien’s law, explains why stars with higher temperatures appear
bluer, and those with lower temperatures appear redder. Conversely, astronomers infer one of the most important physical parameters of stars, thesurface temperature, from measuring their color.
Can we observe the peak of the thermal blackbody spectrum inthe radio region?
• No hope for thermal radiation from a stellar surface, or from a fairly
warm interstellar cloud.
• For submillimeter waves at around 500 GHz (which are still regarded
as radio waves), we can see the peak if the temperature of the medium
is below 10 K (T ≤ 10 K).
• For example, the cosmic background radiation with T ' 2.7 K has its
peak at around 170 GHz.

9.3

Spectral Indices of Thermal Continuum Radio Sources

We now understand why the spetral indicies α (Sν ∝ ν −α ) of thermal continuum radio sources like the Moon, the quiet Sun, and the left half portion of
the spectrum of Orion nebula, are all close to −2 (α ' −2). This is because2
thermal continuum sources mostly show Rayleigh–Jeans spectra, Iν = 2νc2 kT ,in the radio region.

9.4

Stars are Faint and Gas Clouds are Bright in theRadio Sky

According to the Planck spectrum, the intensity of a hotter black body isalways stronger than the intensity of a colder body, for any frequency range.
On the other hand, the flux density Sν , which we directly detect with ourradio telescopes, is proportional to the product of the intensity Iν and thesolid angle of the radio source Ω (Sν ∝ Iν Ω). While the surface temperatures
22

Figure 21: Spectra of continuum radio sources (Kraus, 1986; left), and an
optical image of the Moon (right). Do we see such a halfmoon image in radio
waves?

23

of ordinary stars are fairly high (several thousands to several tens of thousand
Kelvin), their solid angles, which are inversely proportional to their squared
distances, are very small. For example, if we were to put the Sun at the
distance of the nearest star, which is about 3 light years, or 1 parsec, theSun’s angular diameter would be only ' 0.01 arcsecond! As a result, theflux density 5 × 106 Jy of the quiet Sun at 10 GHz (see Figure 2) would be
reduced down to 125 µJy at the distance of the nearest star. On the contrary,
the interstellar gas clouds, as cold as tens to hundreds of Kelvin, still clearlyshine in the radio sky, because their angular diameters are as large as minutes
to degrees of arc. For example, Figure 2 shows that the flux density of theOrion Nebula is about 500 Jy at around 500 MHz.

9.5

Stefan–Boltzmann Law

The total intensity I(T ) of the thermal radiation from a black body of temperature T , over the entire frequency range, is given byI(T ) =

Z∞Iν dν =

Z∞
0

0

2hν 3
1
1
dν = σT 4 ,hν2c e kT − 1π(32)

where σ is the Stefan–Boltzmann constant:
2π 5 k 4σ== 5.6697 × 10−8 W m−2 K−4 .
15c2 h3
Equation (32) can be derived using the integration formula:Z∞0

Bn
2x2n−1
dx =
,
2πx
e −1
2n

(33)

where Bn is the n-th Bernoulli number, and B2 = 1/30.

9.6

Total Blackbody Radiation from a Star or a Gas
Cloud

The power flux density S? , at a surface of a blackbody, which is the powerover the entire frequency range through a cross section of unit area of thesurface (Figure 22), is given by the equation:
S? =

Z∞

π

Sν dν =00π2

= I(T )Z∞ Z2 Z2π

Z

Z2π

0

0

0

Iν cos θ sin θ dφ dθ dν0cos θ sin θ dφ dθ = π I(T ) = σT 4 .
24

(34)

θ

φ

Figure 22: Thermal radiation from a surface of a black body.
Therefore, the total power F of the blackbody radiation from a sphericalstar with radius R is equal to:
F = 4πR2 S? = 4πR2 σT 4 .
The power flux density of the thermal radiation which we receive from a staror a cloud of solid angle Ω, surface temperature T , and distance r, is fromequation (32):
ΩσT 4
S=ΩI=
,
π
which, for the spherically symmetric case where Ω = πR2 /r 2 , is reduced toF
R2 4 R2,
S = 2 σT = 2 S? =
r
r
4πr 2
as expected.

9.7

Universality of the Relationship among Einstein’s
Coefficients

We derived equations (21) and (25), giving the following relationships among
Einstein’s coefficients:ng m βm= gn βnm ,
2hν 3
αnm
=
,
βnm
c3

or, equivalently,ngm Bm= gn Bnm ,8πhν 3
Am
n
,
=
Bnm
c3

25

by assuming thermodynamic equilibrium. However, the resulting equations
above do not contain any quantity characteristic of thermodynamics. Thismeans that the equations must hold universally, irrespective of whether thecondition of thermodynamic equilibrium is fulfilled. These relationships are
determined by the microscopic interactions between a photon and a particle,
and are not influenced by the environment (temperature, pressure, ... etc.).
We obtained the relationships under the assumption of thermodynamic equilibrium, where it is most easily derived. Analogously, if you find a big hole
in a road in daytime, you would assume that the hole still exists at night,
when you cannot see it well.

9.8

Another Important Quantity in Radio Astronomy

Brightness Temperature
The brightness temperature TB of a source with monochromatic intensity(or surface brightness) Iν , is a quantity which is defined by the equation:
c2
TB =
Iν .
2kν 2

(35)

This is a quantity with the dimension of temperature obtained by a ‘forced’
application of the Rayleigh–Jeans formula to the radiation from any radiosource. If the radiation comes from a sufficiently hot (T  10 K) thermal
source, without noticeable absorption or additional emission along the propagation path, the brightness temperature must correspond to the physicaltemperature of the source. If the source is non–thermal, the brightness temperature has no relevance to any real temperature. For example, for somemaser sources, the brightness temperature could be as high as 1014 K, although the physical temperature values of the ‘masing’ (i.e. maser–emitting)gas clouds are only several hundreds of Kelvin. The word ‘brightness temperature’ is a jargon term which is used only, but quite frequently, in radio
astronomy.

10

Radiative Transfer

10.1

Phenomenological Derivation of the Radiative Transfer Equation

Radiative transfer theory describes how the intensity varies as radiation propagates in an absorbing and/or emitting medium. The equation of radiativetransfer can be phenomenologically derived as follows:26

Let the intensity Iν change by dIν due to absorption and emission as
the radiation passes through an infinitesimal distance dl (Figure 23). ThesIν (s, l)Iν (s, l + dl)
dl

Figure 23: Radiative transfer through a cylinder of infinitesimal length.
variation can be described as a sum of two contributions: one due to theabsorption −κν Iν dl, and one due to the emission ν dl, where κν and ν are
the following frequency–dependent coefficients:
κν
ν

:
:

opacityemissivity

m−1 ,W m−3 Hz−1 sr−1 .

We then obtain the radiative transfer equation in the following form:dIν= −κν Iν + ν .
dl

10.2

(36)

Derivation of the Radiative Transfer Equation fromEinstein’s Elementary Quantum Theory of Radiation

Equation (36) can also be derived from equation (15) obtained in the discussion of Einstein’s elementary quantum theory of radiation:dnp (s, ν)n= (nn βnm − nm βm)uν (s) + nn αnm ,dt
where np is the number density of photons per unit solid angle and per unit
frequency bandwidth, uν is the spectral energy density per unit solid angle,
nm and nn are the number densities of the particles in states Zm and Zn ,nand αnm , βnm and βm
are Einstein’s differential coefficients. In fact, using therelations:uν (s) = hνnp (s, ν),Iν (s) = cuν (s),dl = cdt ,we can transform equation (15) to:dIνhνn= −(nm βm− nn βnm ) Iν + hνnn αnm .
dl
c
27

(37)

Therefore, if we denotenκν = (nm βm− nn βnm ) hνcν = hνnn αnm: absorption and induced emission,: spontaneous emission,(38)

equation (37) is reduced to the radiative transfer equation (36).
Role of the induced emission
In the resulting radiative transfer equation:dIν= −κν Iν + ν ,dl
the opacity κν now contains not only the contribution of simple absorption
but also that of induced emission, as we see in equation (38). According toequation (21) obtained by Einstein:ng m βm= gn βnm ,we can now express the opacity κν asκν = (nm −gmn hνnn )βm
.
gn
c

(39)

If we consider the simple case where the statistical weights of the two states
are equal to each other (gm = gn ), the above equation (39) reduces tonκν = (nm − nn )βm

hν
.
c

(40)

It is worthy to note that the opacity takes on a negative value when thenumber density of the particles in the higher energy level is larger than thatat the lower level (nn > nm ).

10.3

The Simplest Solutions of Radiative Transfer Equation

Let us solve the radiative transfer equation (36):dIν= −κν Iν + ν ,dlunder the following simple conditions.

28

1. When κν = 0,
the solution of equation (36) is just a simple integral:Iν (l) = Iν (0) +Zlν (l0 )dl0 .

(41)

0

2. When ν = 0,
the solution is an exponential function:−Iν (l) = Iν (0)eRlκν (l0 )dl0

0

.

(42)

If the medium is homogeneous, i.e. κν = constant everywhere, thenIν (l) = Iν (0)e−κν l .

(43)

Here,
κν > 0 (positive opacity) implies an exponential decay, which arises
from ordinary absorption, andκν < 0 (negative opacity) implies an exponential growth, which arises
from maser amplification.
Maser Amplification
When the number density nn of particles at a higher energy level En
is, for some reason, larger than nm at a lower energy level Em (sucha situation is called a “population inversion”), the opacity becomes
negative (κν < 0), and the radiation of the frequency corresponding to
the transition between the energy levels is exponentially amplified along
the initial direction of propagation (Figure 24). Since this amplificationis due to induced (or stimulated) emission, it is called a “MASER”(Microwave Amplification of Stimulated Emission of Radiation). The
same mechanism in the visible light region of electromagnetic waves
is called the “LASER” (Light Amplification of Stimulated Emission ofRadiation).
If the thermal equilibrium condition is fulfilled in the gas medium, theBoltzmann distribution always ensures nn < nm , and no maser amplification can occur. Therefore, maser emission is essentially non–thermal.
We need some “pumping mechanism” which realizes the population inversion in order to get the maser mechanism to work. In actual interstellar space, strong infrared radiation from stars, or collisions of gas
molecules, may serve as a pumping mechanism.
29

incident radiation

particle
induced emission

Figure 24: A schematic view of maser amplification. Induced emission causes
new induced emission, initiating an “avalanche” of maser emission photons
all traveling in a common direction parallel to the incident photon.
Usually, masers are strong radio sources. For example, the brightness
temperature of some H2 O masers is as high as 1014 K.
3. In the purely thermal equilibrium case,
we obtain from the detailed balance:−κν Iν + ν = 0,

and therefore Iν =

ν
,
κν

and from the Planck spectrum:
1
2hν 3
,
Iν = Bν (T ) ≡ 2 hνc e kT − 1
where Bν (T ) is defined to be the Planck function. From the above
equations, we have a relationship between the emissivity and opacity,which is called Kirchoff’s law:ν= Bν (T ) .
(44)
κν
4. Local thermodynamic equilibrium (LTE)In a fairly wide range of real circumstances in interstellar gas media, andin laboratory conditions, at a given temperature T , there is the casewhen Kirchoff’s law ν /κν = Bν (T ) holds to a good approximation,
but the radiation intensity Iν is not equal to the Planck function30

Bν (T ) at temperature T . Such a case is called “local thermodynamic
equilibrium”, denoted by “LTE”.
In LTE, the radiative transfer equation reduces todIν= −κν [Iν − Bν ] .
dl

(45)

Then, introducing the “optical depth” τν , which satisfiesdτν = −κν dl,(46)

we can further transform equation (45) todIν= Iν − Bν (T ) .
dτν

(47)

Suppose we have an emitting and absorbing gas medium confined within
a finite linear extent l0 , and we consider the radiation coming from theoutside (l < 0, see Figure 25). The optical depth τν is chosen to beequal to τν (0) at l = 0, and to 0 at l = l0 .
The solutions to equation (47) are:
Background
Radiation

Medium with finite κ ν and ε νdlIν (0)

I ν (l0 )sIν (s, l)Iν (s, l + dl)
d τν

0

l

τν (0)

τνκ =0ν
εν = 0l0τν ( l0 ) = 0

Figure 25: Radiative transfer in LTE and optical depth.
within the medium (0 ≤ l ≤ l0 ):Iν (l) = Iν (0)eτν −τν (0)

+e

τν

τZ
ν (0)
τν

31

0

Bν (T (τ 0 ))e−τ dτ 0 ,(48)

and outside the medium (l0 ≤ l):Iν (l) = Iν (l0 ) = Iν (0)e−τν (0)

+

τZ
ν (0)

0

Bν (T (τ 0 ))e−τ dτ 0 ,

(49)

0

where Iν (0) is the incoming background radiation.
5. LTE and isothermal medium
If the medium is isothermal (T (τ 0 ) = T = const), the solutions are:within the medium (0 ≤ l ≤ l0 ):Iν (l) = Iν (0)eτν −τν (0) + Bν (T )(1 − eτν −τν (0) ),(50)

and outside the medium (l0 ≤ l):Iν (l) = Iν (l0 ) = Iν (0)e−τν (0) + Bν (T )(1 − e−τν (0) ) .

(51)

Note that Iν → Bν (T ) when τν (0) → ∞ in the above equations. Thismeans that thermal radiation becomes blackbody radiation reflecting the temperature of the medium when, and only when,the medium is completely opaque.
If we denote the intensity Iν in terms of the brightness temperature
TB , and adopt the Rayleigh–Jeans approximation for Bν (T ):
Iν =

2ν 2
2ν 2
kT
and
B
(T
)
=
kT ,
B
ν
c2
c2

the equation (51) for the solution of the radiative transfer equation
outside the medium can be described by
TB (l) = TB (0)e−τν (0) + T (1 − e−τν (0) ) .

10.4

(52)

What is LTE?

How can the Kirchoff law ν /κν = Bν (T ) be satisfied, even though the intensity of the radiation is not blackbody?
The opacity and emissivity are expressed in terms of Einstein’s differential
coefficients, as we saw in equation (38)nκν = (nm βm− nn βnm )ν = hνnn αnm .
32

hν
,
c

Taking into account Einstein’s relations among the differential coefficients inequations (21) and (25):ng m βm= gn βnm ,
αnm
2hν 3
,
=
βnm
c3

we can express the emissivity–opacity ratio as
1
ν
2hν 3
= nm g n,κν− 1 c2nn g m(53)

using equation (39). Obviously, the right hand side of equation (53) becomesthe Planck function if the Boltzmann distribution
Pn =

En
nn
= gn e− kT ,n(equation (17)) holds, and hencehν
nm g n= e kT .
nn g mTherefore, we can interpret LTE as a physical situation where the Boltzmann
distribution is established among the particles in a medium due, for example,
to their mutual collisions, but in which the radiation is still not in equilibriumwith the particles.

10.5

Spectrum of the Orion Nebula

Now we are in a position to interpret qualitatively the bend in the spectrumof the Orion Nebula, as shown in Figures 2 and 26. If we neglect the contribution of the background radiation in the solution of the radiative transfer
equation in the isothermal LTE case (equation (51)), the intensity is givenbyIν = Bν (T )(1 − e−τν (0) ).
Therefore, the bending can be explained if the nebula is completely opaque(τν (0)  1) at low frequencies, but not at high frequencies. In the higher
frequency range, the radiation no longer shows the blackbody spectrum. But
this type of radiation is still usually included in the category of thermalradiation, since this is caused by the thermal motion of free electrons in
the plasma gas, and tends to have the blackbody (Planck) spectrum in thecompletely opaque limit.
33

Figure 26: Interpretation of the spectrum of the Orion Nebula. Left panel isfrom Kraus (1986).

11

Synchrotron Radiation

Synchrotron radiation is a typical example of non-thermal radiation causedby the high–speed (“relativistic”) electrons in their accelerated helical motion
in a magnetic field. Historically, synchrotron radiation was first discovered in
1948, as the light emitted by a particle accelerator called the “Synchrotron”.
Synchrotron radiation is dominant in a variety of astronomical objects, including “Active Galactic Nuclei” (AGNs), Supernova Remnants (SNRs), andsolar flares (see Figure 27 as an example).

11.1

Non–Relativistic CaseIn the non–relativistic case, the analog of synchrotron radiation is known as“cyclotron” or “gyro–synchrotron” radiation. In this case, the balance of theLorentz force and the centrifugal force2
m0 v ⊥,
rgives rise to “Larmor Precession” with “gyro–frequency” νG :e(v ⊥ × B) =

νG =

1 eB
v⊥
=
,
2πr
2π m0
34

Figure 27: Many galaxies show activity in their nuclei, involving violent
ejections of high–energy electrons and resultant synchrotron radiation. A
nearby galaxy M87, also known as Virgo A or 3C 274 in radio astronomy, is an
example. Left: Infrared image of M87 by 2 Micron All Sky Survey (2MASS).
Right: Close–up view of a collimated jet by Hubble Space Telescope (HST).
(Figure courtesy of NASA/IPAC Extragalactic Database (NED) operated
by the Jet Propulsion Laboratory, California Institute of Technology, undercontract with the National Aeronautics and Space Administration).
B

Centrifugal force

vP

v

v

T

vT
Lorentz force

acceleration

Radiation in non-relativistic case

Figure 28: “Cyclotron” or “Gyro–synchrotron” radiation35

where m0 , e and v are rest mass, electric charge and velocity of an electron,
while B is the magnetic flux density (Figure 28).
The gyro–frequency νG does not depend on the velocity of the electron.
Therefore, for any electron velocity, cyclotron radiation is emitted as a line atfrequency νG , which is modulated only by the spatial and temporal variationof the magnetic field.

11.2

Relativistic CaseIn the relativistic case, when the electron velocity v approaches light velocityc, the radiation from an accelerated electron is emitted almost exclusively
in the direction of movement of the electron, due to the relativistic beaming
effect (Figure 29). A distant observer can detect this pulse–like radiation
only when the narrow beam is directed along or near the line of sight. Since
the beam direction rotates around the magnetic field at high speed, theresulting high frequency pulses from a single electron produce a continuum–
like spectrum, with a peak frequency νmax :νmax ∝

νG2 .
1 − vc2(54)

The energy distribution of high–energy electrons in active regions, such
as AGNs and SNRs, usually follows a power law:
N (E) ∝ E −γ ,(55)

where E is the energy of the electron, N (E) is the number density of electrons with energy E per unit energy range, and γ is the index of the energy
spectrum. Since the energy of an electron with velocity v:
m0 c 2,E = mc2 = q
2
1 − vc2(56)

is proportional to the square root of its peak frequency νmax given in equation
1/2(54) (i.e. E ∝ νmax), a large number of electrons in a wide range of energy
yield a compound spectrum with energy density Uν , which is roughly givenby1−γUν ∝ ν 2 .
(57)
Since the index γ of the energy spectrum of high–energy electrons, as observed in the cosmic rays, is roughly γ ' 2.4, we can expect that the spectrumof the synchrotron radiation is approximatelyUν ∝ ν −0.7 .
36

(58)

B

accelerationv
Radiation in relativistic case

Figure 29: Synchrotron radiation from a relativistic electron.

37

Therefore, the spectral index (Sν ∝ ν −α ) is around α ' 0.7.
Thus, we can now explain the observed positive spectral indices of thespectra of the non–thermal sources such as Crab Nebula, Cygnus A, 3C273,
the disturbed Sun, etc. (Figure 30).

Figure 30: Spectra of continuum radio sources, including non–thermal
sources with positive spectral indices (Kraus, 1986; left), and the Crab Nebula supernova remnant, a typical synchrotron radiation source.

References
Einstein, A., 1916, Zur Quantentheorie der Strahlung, Mitt. Phys.
Ges. (Zürich), 18, 47–62. (This paper was also published in
Physikalishe Zeitschrift, 1917, 18, 121.)
Kaifu, N., Suzuki, H., Ohishi, M., Morimoto, M., and Miyaji, T.,
1985, NRO Graphic Report No.1 (Nobeyama, Japan).
Kraus, J.D., 1986, Radio Astronomy, 2nd Edition, Cygnus–Quasar
Books (Powell, Ohio, U.S.A.).
Rohlfs, K., and Wilson, T.L., 2000, Tools of Radio Astronomy,
3rd Edition, Springer–Verlag (Germany).
38

Westerhout, G., 1958, A Survey of the Continuous Radiation fromthe Galactic System at a Frequency of 1390 Mc/s, Bull. Astron. Inst. Netherlands, 14, 215.

39