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4 Free–Free Radiation‣ Essential Radio Astronomy
Essential Radio Astronomy
3 Radio Telescopes and Radiometers
5 Synchrotron Radiation
Chapter 4
Free–Free Radiation
4.1 Thermal and Nonthermal Emission
Larmor’s formula (Equation 2.143)
2q
2
P =
v̇
2
(4.1)
3
3c
states that electromagnetic radiation with power P is produced by accelerating (or decelerating;
hence the German name bremsstrahlung meaning “braking radiation”) an electrical charge q .
Free charged particles can be accelerated by electrostatic or magnetic forces, gravitational
acceleration being negligible by comparison. Electrostatic bremsstrahlung is the subject of this
chapter, and its magnetic counterpart magnetobremsstrahlung or “magnetic braking radiation”
(e.g., synchrotron radiation) is covered in Chapter 5.
Thermal emission is produced by a source whose emitting particles are in local
thermodynamic equilibrium (LTE) (Section 2.2.2); otherwise nonthermal emission is produced.
Most astronomical sources of electrostatic bremsstrahlung are thermal because the radiating
electrons have the Maxwellian velocity distribution (Appendix B.8) of particles in LTE. The
relativistic electrons in most astronomical sources of magnetobremsstrahlung have power-law
energy distributions and hence are not in LTE, so synchrotron sources are often called
nonthermal sources. However, electrostatic and magnetic bremsstrahlung are not synonymous
with thermal and nonthermal radiation, respectively. For example, electrons with a relativistic
Maxwellian energy distribution are in LTE and can emit thermal synchrotron radiation.
Remember also that a thermal source does not have a blackbody spectrum if the source opacity
is small and the emission coefficient depends on frequency.
4.2 Hii Regions
The electrostatic force is so much stronger than gravity that free charges in interstellar gas
quickly rearrange themselves so that the negative charges of free electrons in an ionized cloud
neutralize the positive charges of ions on all scales larger than the Debye length λ ≤ 1 m
(Equation 4.46). As an electron (charge −e ≈ −4.8 × 10
statcoulomb) passes by an ion
(charge +Z e for an atom with Z electrons removed), the Coulomb force (Equation 2.134)
causes an acceleration of magnitude
D
−10
|v̇ | =
F
me
https://www.cv.nrao.edu/~sransom/web/Ch4.html
=
Ze
2
me r
2
,
(4.2)
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4 Free–Free Radiation‣ Essential Radio Astronomy
where m ≈ 9.1 × 10
g is the electron mass and r is the distance between the electron and
the ion. The resulting emission is called free–free radiation because the electron is free both
before and after the interaction; it is not captured by the ion. If the ionized interstellar cloud is
reasonably dense, the electrons and ions interact often enough that they quickly come into LTE
at some common temperature, so free–free radiation is usually thermal emission.
−28
e
Interstellar gas is primarily hydrogen and helium, plus trace amounts of heavier elements
such as carbon, nitrogen, oxygen, neon, silicon, and iron. Astronomers call all of these heavier
elements metals, meaning elements that readily form positive ions, even though most are not
metallic in the usual sense of being solid, malleable, ductile, and electrically conducting solids at
room temperature. Much of the interstellar hydrogen is in the form of neutral atoms (called Hi in
astronomical terminology) or diatomic molecules (H ), but some is ionized. The singly ionized
hydrogen atom H is referred to as Hii by astronomers, doubly ionized oxygen O is called
Oiii, triply ionized carbon C
is called Civ, etc.
2
+
++
+++
In 1939 the astronomer Bengt Strömgren realized that the interstellar medium can be
divided into distinct regions in which hydrogen is either (1) mostly atomic or molecular, with
nearly all of the hydrogen atoms in the ground electronic state or (2) almost completely ionized.
Furthermore, the boundaries separating these Hi and Hii regions are very thin. Sometimes the
Hii regions surrounding stars are called Strömgren spheres (Figure 4.1) after his early
theoretical models. What is the microscopic physical basis for this picture?
Figure 4.1: A Strömgren sphere of ionized hydrogen (Hii) with Strömgren radius R
surrounded by a thin shell of partially ionized hydrogen (Hi + Hii) surrounded by neutral
hydrogen (Hi).
S
A hydrogen atom in the ground state has the smallest and most tightly bound electronic
orbit around the nuclear proton that is consistent with a stationary electronic wave function. (See
Section 7.2.1 and Figure 7.1 to review the Bohr model of hydrogen atoms.) The permitted
electronic energy levels are characterized by their principal quantum numbers
n = 1, 2, 3, …, where n = 1 corresponds to the ground state. Although quantum mechanics
forbids an electron in the ground state from radiating according to the classical Larmor formula,
it permits radiative decay from higher levels n = 2, 3, …; and Larmor’s equation fairly
accurately predicts the radiative lifetimes of excited hydrogen atoms. The orbital radius a of an
electron in the nth energy level is a = n a , where a ≈ 5.29 × 10 cm is called the Bohr
radius. Applying Larmor’s equation shows that the radiative lifetime τ is proportional to a and
hence to n . Thus the (incorrect) classical result τ ≈ 5.5 × 10
s for radiation from the
n = 1 ground state can be scaled to estimate the radiative lifetimes of excited states. For
example, the approximate radiative lifetime of the n = 2 state would be
τ ≈ 2 ⋅ 5.5 × 10
s ≈ 3 × 10
s, in reasonable agreement with the accurate quantummechanical result τ ≈ 2 × 10 s. Excited hydrogen atoms spontaneously decay very quickly
to the ground state by emitting radiation so, at any instant, almost all neutral atoms are in the
ground state.
n
−9
2
n
0
0
3
n
−11
6
6
−11
−9
−9
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Hydrogen atoms in the ground state can be ionized by photons with energy E ≥ 13.6 eV
(1 electron Volt ≈ 1.60 × 10
erg). Such energetic photons have frequencies higher than the
Rydberg frequency R c = E/h ≈ 3.29 × 10 Hz (Equation 7.11) and wavelengths shorter
than λ = 912 Å = 912 × 10
m, a far-ultraviolet (UV) wavelength. These Lyman
continuum photons are produced in significant numbers by the Wien tail of blackbody
radiation from stars hotter than T ∼ 3 × 10 K. The rate Q at which a star with spectral
luminosity L produces photons that can ionize hydrogen atoms in the ground state is
−12
15
∞
−10
4
H
ν
∞
QH = ∫
(
Lν
(4.3)
) dν .
hν
R∞ c
If a star emits Q Lyman continuum photons per second, it will photoionize hydrogen atoms
with number density n throughout some volume V surrounding the star. The helium mixed
with the hydrogen can often be ignored because its ionization potential is so high, E ≈ 24.5 eV,
that only exceptionally hot stars can ionize significant amounts of helium.
H
H
The absorption cross section of a neutral hydrogen atom to photons with energies just
above 13.6 eV is large enough, σ ≈ 10
cm , that each ionizing photon is absorbed and
produces a new ion shortly after it passes from the ionized Strömgren sphere into the
surrounding Hi region. The thickness ΔR of the partially ionized shell surrounding a
Strömgren sphere (Figure 4.1) is
−17
2
S
ΔRS ≈ (nH σ)
For example, if the neutral hydrogen density is n
H
ΔRS ≈ (10
3
cm
−3
× 10
−17
−1
= 10
2
cm )
(4.4)
.
3
−1
atoms
≈ 10
14
cm
−3
, then
cm ≪ 1
(4.5)
pc.
Light travels ≈ 10 cm per hour, so an ionizing photon typically survives only about an hour
in such an Hi cloud before being absorbed.
14
Once ionized from Hi into free protons (H ions) and electrons, the Hii region has a much
lower opacity to ionizing photons. Thus a new ionizing star turning on in a uniform-density Hi
cloud will fully ionize a sphere whose Strömgren radius (Figure 4.1) grows with time until
equilibrium between ionization and recombination is reached. This is sometimes called an
ionization bounded Hii region. If the surrounding Hi cloud is small enough that the star can
ionize it completely, the Hii region is said to be matter bounded or density bounded.
+
Inside the Hii region, electrons and protons occasionally collide and recombine at a
volumetric recombination rate ṅ that can be written as
H
(4.6)
ṅH ≈ αH ne np ,
where ṅ is the number of recombinations per unit time per unit volume (e.g., cm
H
αH ≈ 3 × 10
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−13
cm
3
s
−1
−3
s
−1
),
(4.7)
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is the recombination coefficient for hydrogen, and the collision rate per unit volume is
proportional to the product n n of the electron and proton densities. For example, if
n = n
= 10
cm
,
e
3
e
p
−3
p
ṅH ≈ 3 × 10
−13
cm
3
s
−1
× 10
3
cm
−3
× 10
3
cm
−3
≈ 3 × 10
−7
cm
−3
s
−1
(4.8)
.
This example shows that the recombination time
ne
τ ≡
≈ 3.3 × 10
9
s ≈ 10
2
(4.9)
yr
ṅH
is usually much shorter than the > 10 year lifetime of an ionizing star. (A useful relation to
remember is 1 year ≈ 10 s.) The volume V of an ionization-bounded Hii region grows until
the total ionization and recombination rates in the Strömgren sphere are equal. In equilibrium,
6
7.5
4
QH = ṅH V = αH ne np
πR
3
(4.10)
3
S
yields a Strömgren radius
1/3
3QH
RS ≈ (
2
(4.11)
.
)
4παH ne
For example, an O5 star (very hot and luminous) emits Q
second. If n ≈ 10 cm ,
H
3
≈ 6 × 10
49
ionizing photons per
−3
e
RS ≈ [
3 ⋅ 6 × 10
4π ⋅ 3 × 10
−13
cm
3
49
s
s
−1
1/3
−1
(10
3
cm
−3
2
]
≈ 3.6 × 10
18
cm ≈ 1.2 pc.
)
This example illustrates that R ≫ ΔR ; that is, the radius of the fully ionized Strömgren
sphere is much larger than the thickness of its partially ionized skin.
S
S
Two distinct kinds of stars produce most of the Hii regions in our Galaxy:
1. The most massive (M ≥ 15M ) short-lived (lifetimes ≤ 10 yr) main-sequence stars
are big enough (R ∼ 10R ) and hot enough (T ≥ 3 × 10 K) to be very luminous
sources of ionizing UV. Such stars were recently formed by gravitational collapse and
fragmentation of interstellar clouds containing neutral gas and dust grains.
2. Old lower-mass (1 < M /M < 8 ) stars whose main-sequence lifetimes are less than
the age of our Galaxy ( ≈ 10 yr) eventually become red giants and finally white
dwarfs. Young white dwarfs are small (R ∼ 10 R ) but hot enough to ionize the
stellar envelope material that was ejected during the red giant stage, and these ionized
regions are called planetary nebulae because many looked like planets to early
astronomers using small telescopes.
7
⊙
4
⊙
⊙
10
−2
⊙
Most ionizing stars are approximately blackbody emitters and their ionizing photons from
the high-frequency Wien tail have energies only somewhat greater than the E = 13.6 eV
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4 Free–Free Radiation‣ Essential Radio Astronomy
minimum needed to ionize a hydrogen atom from its ground state. Momentum conservation
during ionization ensures that nearly all of the photon energy in excess of 13.6 eV is converted
into kinetic energy of the ejected electron. Collisions between these hot photoelectrons, and
between electrons and ions, thermalize the ionized gas and gradually bring it into local
thermodynamic equilibrium (LTE). Consequently, the thermalized electrons have a Maxwellian
energy distribution. Eventually this heating is balanced by radiative cooling. Collisions of
electrons with “metal” ions can excite low-lying (a few eV) energy states that decay slowly via
forbidden transitions and emit visible photons that may escape from the nebula. Examples of
visible cooling lines include the green lines of Oiii at λ = 4959 Å and 5007 Å, first discovered
in nebulae and named nebulium lines because these forbidden lines hadn’t been observed in the
laboratory and were thought to be from a new element found only in nebulae (just as helium
lines in the solar spectrum were once ascribed to a new element found in the Sun). The Balmer
hydrogen recombination lines Hα at λ = 6563 Å and Hβ at λ = 4861 Å also contribute to the
characteristic colors of Hii regions.
Thermal equilibrium between heating and cooling of Hii regions is usually reached at a
temperature close to T ≈ 10 K [77] that is much higher than the initial temperature
T < 100 K of the neutral interstellar gas. The heated gas expands, reversing any infall onto the
ionizing star and sending shocks into the surrounding cold gas, thereby both inhibiting and
stimulating the subsequent production of stars in the region. Typical Hii regions have sizes ∼ 1
pc, electron number densities ∼ 10 cm , and masses up to 10 M .
4
3
4
−3
⊙
The free–free radio emission from an Hii region is a tracer of the electron temperature,
electron density, and ionized volume. It constrains the production rate Q of ionizing photons
and, with an assumption about the initial mass function (IMF) (the mass distribution of new
stars), the total star-formation rate in an Hii region. The radio data are important for reliable
quantitative estimates of star formation because they do not suffer from extinction by interstellar
dust.
H
Ultra-compact (UC) Hii regions [23] are small (diameter 2R ≤ 0.03 pc ) but dense (
n > 10
cm
) Hii regions ionized by O and B stars so young that they are still optically
obscured by the dusty molecular clouds from which they formed. The dust makes them strong
far-infrared sources, and free–free radio emission penetrates the dust so they can be imaged and
studied at radio wavelengths. UC Hii regions are valuable tracers of the formation and early
evolution of massive stars, and of their interactions with their environment. On a much larger
scale, extragalactic ultra-dense (UD) Hii regions [59] are ionized by young super star clusters
(SSCs) so dense and massive that they may be the progenitors of long-lived globular clusters,
and so luminous that they may disrupt the ISM in dwarf galaxies.
S
4
−3
e
Planetary nebulae are Hii regions surrounding the hot (up to T ∼ 10 K) white-dwarf
cores of low-mass (1 < M < 8 ) stars that have ejected their outer envelopes as stellar winds.
White dwarfs are small, about the size of the Earth, so they are much less luminous than massive
O stars and the ionized nebular masses are only 0.1 < M /M < 1 . Planetary nebulae are
nonetheless fairly luminous indicators of the last stages in the lives of low-mass stars. They are
potentially useful as a record of low-mass star formation throughout the history of our Galaxy.
However, the optical selection of planetary nebulae is affected by dust extinction. Far-infrared
and radio selection may avoid this limitation. Planetary nebulae are not particularly luminous
radio sources, but they are the most numerous compact radio continuum sources in our Galaxy.
5
⊙
⊙
4.3 Free–Free Radio Emission from Hii Regions
Thermal bremsstrahlung from ionized hydrogen is often called free–free emission because it is
produced by free electrons scattering off ions without being captured—the electrons are free
before the interaction and remain free afterward. What are the basic properties of free–free radio
emission from an astrophysical Hii region? Despite all of the simplifications introduced in
Section 4.2, this problem can’t be solved without several additional approximations. A certain
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4 Free–Free Radiation‣ Essential Radio Astronomy
amount of astrophysical “intuition” is required to distinguish important from negligible effects.
For example, the energy lost by an electron when it interacts with an ion is much smaller than
the initial electron energy. Radiation from electron–electron collisions and from ions can be
ignored. Formally divergent integrals over impact parameters can be avoided with physical
limits to the range of integration and still yield reasonably accurate, but not exact, results. Most
astrophysical conditions are so far removed from personal experience (How hot does 10 K
feel? How big is a parsec compared with the height of a tree? How much is M ≈ 2 × 10 g
compared with the mass of a person?) that astrophysical intuition depends on being familiar with
numerical values for the relevant parameters, so it is possible to decide quickly what is
important and what can be neglected. As Linus Pauling said, “The way to get good ideas is to
get lots of ideas and throw the bad ones away.” The following analysis of free–free emission
from an Hii region illustrates approaches and techniques used to solve “messy” astrophysical
problems. For similar but alternative analyses, see the chapters on bremsstrahlung in Rybicki
and Lightman [98] and in Wilson et al. [116].
4
33
⊙
Why should an Hii region emit radio radiation at all? The answer is, because charged
particles are being accelerated electrostatically, and nonrelativistic free accelerated charges
radiate power according to Larmor’s formula (Equation 2.143). Electrostatic interactions among
many kinds of charged particles take place in an Hii region, but most do not emit significant
amounts of radiation. The magnitude of the acceleration v̇ is inversely proportional to the
particle mass m. The lightest ion is the hydrogen ion. Its mass is the proton mass
m
≈ 1.66 × 10
g, which is about 2 × 10 times the electron mass
g. In any electron–ion collision the electron will therefore radiate at least
m ≈ 9.11 × 10
(m /m ) ≈ 4 × 10 as much power as the ion, so all ionic radiation can be neglected.
Interactions between identical particles also do not radiate significantly because the
accelerations of the two particles are equal in magnitude but opposite in direction: v˙ = −v˙ .
Their radiated electric fields are equal in magnitude but opposite in sign, so the net radiated
electric field approaches zero at distances much larger than the collision impact parameter
(Figure 4.2) and the radiation from electron–electron collisions can be ignored. The bottom line
is that only the electron–ion collisions are important, and only the electrons radiate
significantly.
−24
3
p
−28
e
2
p
6
e
1
2
4.3.1 Radio Radiation From a Single Electron–Ion Interaction
Figure 4.2 shows an electron passing by a far more massive ion of charge Z e , where Z = 1 for
a singly ionized atom such as hydrogen. Each electron–ion interaction will generate a single
pulse of radiation. The total energy emitted and the approximate frequency spectrum of the pulse
will be derived in this section. The exact spectrum of an individual pulse is not needed because
the broad distribution of electron energies and impact parameters smears out spectral details in
the total spectrum of an Hii region.
Figure 4.2: A light, fast electron passing by a slow, heavy ion. Low-energy radio photons are
produced by weak scattering in which the velocity vector v ⃗ of the electron changes little. The
distance of closest approach b is called the impact parameter and the interval τ = b/v is
called the collision time.
Radio photons are produced by weak interactions because the energy E = hν of a radio
photon is much smaller than the average kinetic energy ⟨E ⟩ of an electron in an Hii region. The
numerical comparison below is an example of how astrophysical “intuition” can simplify the
electron–ion scattering problem.
e
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The mean electron energy in a plasma of temperature T is
⟨E e ⟩ =
For an Hii region with T
⟨E e ⟩ ≈
∼ 10
4
3kT
(4.12)
.
2
K,
3 ⋅ 1.38 × 10
−16
−1
erg K
⋅ 10
4
K
≈ 2 × 10
2
−12
erg
≈ 1
eV.
(4.13)
(This is another useful conversion factor to remember: 1 eV is the typical particle kinetic energy
associated with the temperature T ≈ 10 K.) By comparison, the energy of a radio photon of
frequency ν = 10 GHz is only
4
E = hν ≈ 6.63 × 10
−27
⋅ 10
erg s
10
Hz
≈ 6.63 × 10
−17
≈ 4 × 10
erg
−5
(4.14)
eV.
The weak interactions that produce radio photons cause the trajectory of the electron to deflect
by only a small angle ( ≪ 1 radian). As shown in Figure 4.2, the electron’s path can be
approximated by a straight line.
During the interaction, the electron will be accelerated electrostatically both parallel to and
perpendicular to its nearly straight path:
F∥ = me v̇ ∥ =
F⊥
= me v̇
⟂ =
−Z e
r
sin ψ =
−Z e
2
sin ψ cos
2
r
Ze
2
ψ
,
(4.15)
2
b
2
2
2
cos ψ =
Ze
2
cos
3
ψ
,
(4.16)
2
b
where cos ψ = b/r and b is the impact parameter of the interaction, the minimum value of the
distance r between the electron and the ion.
For any impact parameter b, these two equations can be solved to show that the maximum
of v̇ is a nonnegligible 38% of the maximum of v̇ ⟂ . Even so, the radio radiation arising from
v̇ is completely negligible. Plotting the variation with time of v̇ and v̇ ⟂ during the interaction
shows pulses with quite different shapes (Figure 4.3).
∥
∥
∥
Figure 4.3: The acceleration of an electron by an ion may be resolved into components
perpendicular (⟂ ) to and parallel (∥) to the electron’s velocity. The perpendicular acceleration
(Equation 4.16) yields a roughly Gaussian pulse whose power spectrum extends to low (radio)
frequencies. The parallel acceleration (Equation 4.15) gives a roughly sinusoidal pulse with
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4 Free–Free Radiation‣ Essential Radio Astronomy
frequencies. The parallel acceleration (Equation 4.15) gives a roughly sinusoidal pulse with
no “DC” component, so the resulting radiation is strongest at higher (infrared) frequencies
and very weak at radio frequencies.
The pulse duration is comparable with the collision time τ ≈ b/v . The v̇ pulse is roughly
a sine wave of angular frequency ω ∼ τ
= v/b , which is much higher than radio frequencies
for all relevant impact parameters b (Equation 4.43). The parallel acceleration produces some
infrared radiation but very little radio radiation. The v̇ ⟂ pulse is a single peak whose frequency
spectrum extends from zero up to ∼ v/b because the Fourier transform of a Gaussian is also a
Gaussian (Appendix B.4), so it is stronger at radio frequencies.
∥
−1
Inserting v̇ ⟂ from Equation 4.16 into Larmor’s formula (Equation 2.143) gives the
instantaneous power emitted by the acceleration perpendicular to the electron velocity:
2
P =
2
e v̇
3
2
⟂
2e
=
3
2
Z
3
c
2
e
4
2
3c
cos
(
3
ψ
2
2
(4.17)
) .
b
me
The total energy W emitted by the pulse is
∞
W = ∫
(4.18)
P dt.
−∞
Because (ΔE ) /E = E /E ≪ 1 for radio photons, the electron velocity is nearly constant.
The interaction diagram in Figure 4.2 shows that
e
e
γ
e
v =
dx
and
x
tan ψ =
dt
(4.19)
,
b
so
v =
b d tan ψ
b sec
=
dt
2
ψ dψ
b dψ
=
dt
cos
2
(4.20)
,
ψ dt
and
dt =
b
dψ
(4.21)
.
v cos 2 ψ
Inserting Equations 4.17 and 4.21 into 4.18 gives
W =
2
Z
3
2
e
2
π/2
6
∫
4
3 c m b
e
−π/2
b cos
v cos
6
2
ψ
ψ
dψ =
4
Z
3
2
e
2
π/2
6
∫
3
3 c m b v
e
cos
4
ψ dψ.
(4.22)
0
The integral in Equation 4.22 is evaluated in Appendix B.7; it is
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4 Free–Free Radiation‣ Essential Radio Astronomy
π/2
∫
cos
4
3π
ψ dψ =
(4.23)
.
16
0
Thus the pulse energy W radiated by a single electron–ion interaction characterized by impact
parameter b and velocity v is
W =
πZ
3
2
e
6
1
(
2
(4.24)
).
3
b v
4c me
This energy is emitted in a single pulse of duration τ ≈ b/v , so the pulse power spectrum
(Appendix A.4) is nearly flat over all frequencies ν < ν
≈ (2πτ )
≈ v/ (2πb) and falls
rapidly at higher frequencies. It is possible to calculate the actual Fourier transform of the pulse
shape (solid curve in Figure 4.4), but doing so would only add an unnecessary complication to
an already complicated calculation. The ranges of velocities v and impact parameters b
characterizing electron–ion interactions in an Hii region are so wide that averaging over all
collision parameters will wash out fine details in the spectrum associated with any particular v
and b.
−1
max
Figure 4.4: The actual power spectrum of the electromagnetic pulse generated by one
electron–ion interaction is nearly flat up to frequency ν ≈ v/ (2πb) , where v is the electron
speed and b is the impact parameter, and declines at higher frequencies. The approximation
P
= 1 for all ν < v/ (2πb) and P
= 0 at higher frequencies (dashed line) is quite good at
radio frequencies ν ≪ v/ (2πb) .
ν
ν
The typical electron speed in a T ∼ 10 K Hii region is v ≈ 7 × 10 cm s and the
minimum impact parameter is b
∼ 10
cm, so ν
≈ 10
Hz, much higher than radio
frequencies. In the approximation that the power spectrum is flat out to ν = ν
and zero at
higher frequencies (dashed line in Figure 4.4), the average energy per unit frequency emitted
during a single interaction is approximately
4
7
−7
−1
14
min
max
max
Wν ≈
W
νmax
= (
πZ
3
2
e
2
6
) (
3
4c me b v
2πb
),
(4.25)
v
which simplifies to
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4 Free–Free Radiation‣ Essential Radio Astronomy
Wν ≈
π
2
2
Z
2
3
e
6
2
(
1
2
2
),
b v
c me
ν < νmax ≈
v
≈ 10
14
(4.26)
Hz.
2πb
4.3.2 Radio Radiation From an Hii Region
The strength and spectrum of radio emission from an Hii region depends on the distributions of
electron velocities v and collision impact parameters b (Figure 4.2). The distribution of v
depends on the electron temperature T . The distribution of b depends on the electron number
density n (cm ) and the ion number density n (cm ).
−3
−3
e
i
In LTE, the average kinetic energies of electrons and ions are equal. The electrons are much
less massive, so their speeds are much higher and the ions can be considered nearly stationary
during an interaction (Figure 4.5).
Figure 4.5: The number of electrons with speeds v to v + dv passing by a stationary ion and
having impact parameters in the range b to b + db during the time interval t equals the
number of electrons with speeds v to v + dv in the cylindrical shell shown here.
The number of electrons passing any ion per unit time with impact parameter b to b + db
and speed range v to v + dv is
ne (2πb db)
v f (v)
(4.27)
dv,
where f (v) is the normalized (∫ f (v) dv = 1 ) speed distribution of the electrons. The number
ṅ (v, b) of such collisions per unit time per unit volume per unit velocity per unit impact
parameter is
c
ṅc (v, b) = (2πb) vf (v)
(4.28)
ne ni .
The spectral power at frequency ν emitted isotropically per unit volume is 4πj , where j is the
emission coefficient defined by Equation 2.26. Thus
ν
∞
4πjν = ∫
Substituting the results for W
Equation 4.29 gives
ν
(v, b)
4πjν
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∞
∫
b=0
ν
Wν (v, b) ṅc (v, b) dv db.
(4.29)
v=0
(Equation 4.26) and ṅ
c
(v, b)
(Equation 4.28) into
(4.30)
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4 Free–Free Radiation‣ Essential Radio Astronomy
∞
∞
= ∫
b=0
3
π Z
=
2
∫
e
2
3
∞
6
e ne ni
6
2
2
) 2πb db ne ni vf (v) dv
c me
∞
f (v)
∫
2
3
2
2c me b v
v=0
2
π Z
(
db
dv ∫
v
v=0
(4.31)
.
b
b=0
Equation 4.31 exposes a problem: the integral
∞
db
∫
(4.32)
b
b=0
diverges logarithmically. There must be finite physical limits b
and b
on the range of the impact parameter b that prevent this divergence:
min
3
4πjν =
π Z
2
∞
6
e ne ni
2
3
c me
bmax
f (v)
∫
max
db
dv ∫
v
v=0
(to be determined)
(4.33)
.
b
bmin
The distribution f (v) of electron speeds in LTE is the nonrelativistic Maxwellian
distribution (see Appendix B.8 for its derivation):
Figure 4.6: The nonrelativistic Maxwellian distribution of particle speeds in LTE
(Equation 4.34), where v
= (3kT /m)
is the rms speed of particles with mass m at
temperature T .
1/2
rms
2
f (v) =
4v
−
−
√π
(
me
3/2
)
2
exp (−
2kT
me v
(4.34)
).
2kT
Equation 4.34 can be used to evaluate the integral over the electron speeds in Equation 4.33:
∞
f (v)
∫
dv =
v
v=0
Substituting x
2
≡ me v / (2kT )
∞
−
−
√π
so dx
dv =
v
(
me
3/2
)
2kT
∞
∫
4
−
−
√π
(
2
v exp(−
me
2kT
)dv.
(4.35)
gives
3/2
)
me v
2kT
v=0
= me v dv/ (kT )
f (v)
∫
v=0
4
∞
∫
x=0
kT
e
−x
dx
(4.36)
me
(4.37)
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4 Free–Free Radiation‣ Essential Radio Astronomy
=
2
−
−
√π
= (
1/2
me
(
)
∞
∫
2kT
2me
e
−x
dx
x=0
1/2
)
(4.38)
.
πkT
In conclusion, the free–free emission coefficient can be written as
2
π Z
jν =
2
6
e ne ni
2
3
(
2me
1/2
)
ln (
πkT
4c me
bmax
(4.39)
).
bmin
The remaining problem is to estimate the minimum and maximum impact parameters b
and b . These estimates don’t have to be very precise because only their logarithms appear in
Equation 4.39.
min
max
To estimate the minimum impact parameter b , notice that the net impulse (change in
momentum) during a single electron–ion interaction
min
∞
me Δv = ∫
(4.40)
F dt
−∞
comes entirely from the perpendicular component F of the electric force because the
contribution from E is antisymmetric about t = 0 (Figure 4.3). Inserting F from
Equation 4.16 into Equation 4.40 gives
⊥
∥
⊥
∞
me Δv = ∫
Ze
2
(
r
−∞
∞
cos ψ
) dt = Z e
2
2
cos
∫
3
2
ψ
dt.
(4.41)
b
−∞
Using Equation 4.21 to change the variable of integration from t to ψ gives
me Δv =
Ze
π/2
2
bv
∫
cos ψ dψ =
2Z e
2
.
(4.42)
bv
−π/2
The maximum possible momentum transfer m Δv during the free–free interaction is twice the
initial momentum m v of the electron, so the impact parameter of a free–free interaction cannot
be smaller than
e
e
bmin ≈
Ze
2
2
.
(4.43)
me v
This result is based on a purely classical treatment of the interaction (see also Jackson [56,
section 13.1 and problem 13.1] for a more detailed discussion). The uncertainty principle (
ΔxΔp ≃ ℏ ) implies an independent quantum-mechanical limit
(4.44)
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4 Free–Free Radiation‣ Essential Radio Astronomy
ℏ
bmin =
,
me v
but this lower limit is generally smaller than the classical limit in Hii regions and hence may be
ignored. This claim can be tested by computing the ratio of the classical to quantum limits for
the rms electron velocity v = (3kT /m ) :
1/2
e
(
Ze
2
2
−1
ℏ
)(
)
=
Ze
me v
me v
2
Ze
=
ℏv
2
ℏ
(
me
1/2
)
(4.45)
.
3kT
In an Hii region with T ≈ 10 K and Z = 1, this ratio ≈ 3, so the classical limit is stronger.
Only in much cooler (T < 10 K) plasmas is the quantum-mechanical limit important.
4
3
There are two effects that might determine the upper limit b
to the impact parameter.
Because electrostatic forces always dominate gravity on small scales, electrons in the vicinity of
a nearly stationary ion are free to rearrange themselves to neutralize, or shield, the ionic charge.
The characteristic scale length of this shielding is called the Debye length. From Jackson [56],
the Debye length is
max
λD ≈ (
1/2
kT
4πne e
2
)
(4.46)
.
The Debye length is quite large in the low-density plasma of a typical Hii region. For example, if
T ≈ 10 K and n ≈ 10
cm
,
4
3
−3
e
1.38 × 10
−16
−1
erg K
⋅ 10
4
λD ≈ [
4π ⋅ 10
3
cm
−3
⋅ (4.8 × 10
−10
1/2
K
statcoulomb)
2
]
≈ 22
cm.
(4.47)
An independent upper limit to the impact parameter is the largest value of b that can generate a
significant amount of power at some relevant radio frequency ν . Recall that the pulse power per
unit bandwidth is small above angular frequency ω ≈ v/b so
bmax ≈
v
ω
=
v
,
2πν
(4.48)
is the maximum impact parameter capable of emitting significant power at frequency ν .
The controlling upper limit b
in any particular situation is the smaller of these two
upper limits. Numerical values for the lower and upper limits in a typical Hii region observed at
ν = 1 GHz are derived in the boxed example, and Equation 4.48 gives the smaller
b
∼ 10
cm.
max
−2
max
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4 Free–Free Radiation‣ Essential Radio Astronomy
Example. Estimate b
and b
for a pure Hii region (Z
at a fairly low frequency ν = 1 GHz = 10 Hz:
min
max
= 1
) with T
≈ 10
4
K observed
9
bmin ≈
Ze
2
e
≈
2
2
3kT
me v
(4.8 × 10
−10
statcoulomb)
2
≈
3 ⋅ 1.38 × 10
bmax ≈
v
≈ (
3kT
2πν
≈ (
−16
−1
erg K
⋅ 10
4
≈ 5.6 × 10
−8
cm,
K
1/2
−1
)
(2πν )
me
3 ⋅ 1.38 × 10
−16
−1
erg K
9.1 × 10
≈ 1.1 × 10
−2
−28
⋅ 10
4
1/2
K
)
(2π × 10
9
s
−1
)
−1
g
cm.
The maximum impact parameter capable of generating power at this frequency is much
smaller than the λ ≈ 22 cm Debye length in an Hii region with electron density
(Equation 4.47), so the Debye length is irrelevant. A T = 10 K electron
n ≈ 10 cm
takes so long to move 22 cm that it would emit at unobservably low frequencies
ν < 1 MHz. The Debye length becomes relevant only in much denser plasmas such as the
solar chromosphere (n ≈ 10 cm ).
D
3
4
−3
e
12
−3
e
Our simple estimate of the ratio
bmax
bmin
≈ (
3kT
1/2
)
−1
(2πν )
(
3kT
me
Ze
2
) ≈ (
3kT
me
3/2
me
)
(4.49)
2
2πZ e ν
is very close to the result of the very detailed derivation in Oster [76]. The ratio (b /b
of order 10 , which is much greater than the fractional velocity range σ /v ≈ 1 in the
Maxwellian velocity distribution (Figure 4.6). Also note that
max
min
)
is
5
v
ln(
bmax
(4.50)
) ∼ 12
bmin
varies slowly with changes in either b
or b , so small uncertainties in these limits have
very little effect on the calculated emission coefficient of an Hii region.
max
min
Because the Hii region is in local thermodynamic equilibrium (LTE) at some temperature
, Kirchhoff’s law (Equation 2.30) immediately yields the absorption coefficient κ
(Equation 2.18) in terms of the emission coefficient and the blackbody brightness B (T ):
T
ν
κ =
jν
Bν (T )
2
≈
jν c
2kT ν
2
(4.51)
in the Rayleigh–Jeans limit. Thus
(4.52)
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4 Free–Free Radiation‣ Essential Radio Astronomy
1
κ =
ν
2
T
[
Z
3/2
2
e
c
6
1
ne ni
−
−
−−
−−
−−
√2π(m k)
e
]
3
π
2
ln (
4
bmax
).
bmin
The limit b
(Equation 4.48) is inversely proportional to frequency so the absorption
coefficient is not exactly proportional to ν . A good numerical approximation is
κ (ν ) ∝ ν
.
max
−2
−2.1
The total opacity τ of an Hii region is the integral of −κ along the line of sight, as
illustrated in Figure 4.7:
Figure 4.7: Astronomers often approximate Hii regions by uniform cylinders whose axis is the
line of sight because this gross oversimplification finesses the radiative-transfer problem. It is
for good reason that astronomers often feature in jokes beginning “Consider a spherical
cow….”
τ = −∫
ne ni
κ ds ∝ ∫
ν
los
2.1
T
2
ne
ds ≈ ∫
3/2
ν
2.1
T
(4.53)
ds.
3/2
At frequencies low enough that τ ≫ 1 , the Hii region becomes opaque, its spectrum approaches
that of a blackbody with brightness temperature approaching the electron temperature (
T
≈ T ∼ 10 K), and its flux density obeys the Rayleigh–Jeans approximation S ∝ ν . At
very high frequencies, τ ≪ 1 , the Hii region is nearly transparent, and
4
2
b
S ∝
2kT ν
2
2
τ (ν ) ∝ ν
−0.1
.
(4.54)
c
On a log-log plot, the overall spectrum of a uniform Hii region looks like Figure 4.8, with the
spectral break corresponding to the frequency at which τ ≈ 1 .
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4 Free–Free Radiation‣ Essential Radio Astronomy
Figure 4.8: The radio spectrum of an Hii region. It is a blackbody at low frequencies, with
slope 2 if a uniform cylinder as shown in Figure 4.7 and < 2 otherwise. At some frequency ν
the optical depth τ = 1 , and at much higher frequencies the spectral slope becomes ≈ −0.1
because the opacity coefficient κ (ν ) ∝ ν
. The source brightness at low frequencies
equals the electron temperature. The brightness at high frequencies is proportional to the
emission measure (Equation 4.57) of the Hii region.
−2.1
The spectral slope on a log-log plot is often called the spectral index and denoted by α ,
whose sign is defined ambiguously:
d log S
α ≡ ±
(4.55)
.
d log ν
Beware the ± sign! Unfortunately both sign conventions are found in the literature, and you
have to look carefully at each paper to find out which one is being used. With the + sign
convention, the low-frequency spectral index of a uniform Hii region would be α = +2 . The −
sign convention was introduced in the early days of radio astronomy because most sources
discovered at low frequencies are stronger at low frequencies than at high frequencies. Thus
α = +0.7 might mean
d log S
(4.56)
= −0.7.
d log ν
The (+) spectral index of any inhomogeneous Hii region will be α ≈ −0.1 well above the
break frequency, but the break will be more gradual and the low-frequency slope will be
somewhat less than +2 just below the break. For example, ionized winds from stars are quite
inhomogeneous. Mass conservation in a constant-velocity, isothermal spherical wind implies
that the electron density is inversely proportional to the square of the distance from the star:
n ∝ r
. The low-frequency spectral index of free–free emission by such a wind is closer to
+0.6 than to +2 .
−2
e
The emission measure (EM) of an Hii region is defined by the integral of n along the line
of sight expressed in astronomically convenient units:
2
e
(4.57)
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4 Free–Free Radiation‣ Essential Radio Astronomy
EM
pc
cm
≡ ∫
−6
(
cm
los
2
ne
) d(
−3
s
).
pc
Because κ is proportional to n n ≈ n , the optical depth τ is proportional to the emission
measure. The emission measure is commonly used to parameterize τ in astronomically
convenient units:
2
e
τ ≈ 3.014 × 10
i
e
−2
(
T
−3/2
)
(
K
−2
ν
)
EM
(
GHz
pc
cm
−6
) ⟨gff ⟩ ,
(4.58)
where the free–free Gaunt factor [18] ⟨g ⟩ is a parameter that absorbs the weak frequency
dependence associated with the logarithmic term in κ :
ff
⟨gff ⟩ ≈ ln[4.955 × 10
−2
−1
ν
(
)
T
] + 1.5 ln(
GHz
(4.59)
).
K
Mezger and Henderson [73] found a very good approximation for the free–free opacity τ that is
easy to evaluate numerically:
τ ≈ 3.28 × 10
−7
−1.35
T
(
10
4
)
−2.1
ν
(
)
GHz
K
EM
(
pc
cm
−6
).
(4.60)
Mezger and Henderson [73, Table 6] lists the errors introduced by this approximation over wide
ranges of temperature and frequency.
Example. The interstellar medium of our Galaxy contains a diffuse ionized component,
some of which is “warm” (T ≈ 10 K) and some is “hot” (T ≈ 10 K). These two phases
are roughly in pressure equilibrium so the hot medium is less dense by a factor of ∼ 10 .
The combination of high T and low n of the hot phase means that only the warm
component contributes significantly to the free–free opacity of the ISM. The warm ionized
gas is largely confined to the disk of our Galaxy, where we reside. There must be some
frequency ν below which this disk becomes opaque and we cannot see out of our Galaxy,
even in the direction perpendicular to the disk. From the observed brightness spectrum in
the direction perpendicular to the disk, Cane [21] found that τ ≈ 1 at ν ≈ 2 MHz. This
result can be inserted into Equation 4.60 to estimate the rms electron density in the warm
ISM:
4
6
2
e
e
1 ≈ 3.28 × 10
2
⟨ne ⟩
1/2
−7
≈ (
⋅ (1)
−1.35
0.002
⋅ 0.002
2.1
3.28 × 10
−4
−2.1
2
⋅ ⟨ne ⟩ ⋅ 1000
pc,
1/2
)
≈ 0.08
cm
−3
.
From the optical depth τ and the electron temperature T it is possible to calculate the
brightness temperature
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4 Free–Free Radiation‣ Essential Radio Astronomy
Tb = T (1 − e
−τ
(4.61)
)
of free–free emission. The line-of-sight structure of an Hii region is not normally known, so it is
common to approximate the geometry of an Hii region by a circular cylinder whose axis lies
along the line of sight, and whose axis length equals its diameter. Suppose further that the
temperature and density are constant throughout this volume. Then it is very easy to estimate
physical parameters of the Hii region (e.g., electron density, temperature, emission measure,
production rate Q of ionizing photons) from the observed radio spectrum, once the distance to
the Hii region is known.
H
A useful approximation relating the production rate of ionizing photons to the free–free
spectral luminosity L at the high frequencies where τ ≪ 1 of an Hii region in ionization
equilibrium is [95]
ν
(
QH
s
−1
) ≈ 6.3 × 10
52
−0.45
T
(
10
4
)
0.1
ν
(
)
GHz
K
Lν
(
10
20
W
−1
).
(4.62)
Hz
Example. Suppose an idealized Hii region with temperature T = 10 K at distance
d = 10 kpc produces the radio spectrum shown in Figure 4.8 where S is in Jy and ν is in
GHz. What constraints on the Hii region are provided by this spectrum?
At low frequencies ν ≪ 1 GHz, the Hii region is optically thick, so it is a blackbody
radiator and obeys the Rayleigh–Jeans approximation from which the source solid angle
can be derived. From Figure 4.8, S ≈ 0.1 Jy at ν = 0.3 GHz (λ = 1 m) so
4
2
2
Ω =
λ S
2kT
(1 m)
⋅ 0.1 × 10
−26
−1
−2
W m
Hz
≈
≈ 3.6 × 10
2 ⋅ 1.38 × 10
−23
−1
J K
⋅ 10
4
−9
sr.
K
Thus the angular diameter of the Hii region is
θ ≈ (
4Ω
1/2
)
≈ 7 × 10
−5
rad
π
and its linear diameter is about 0.7 pc.
At high frequencies ν ≫ 1 GHz, the opacity τ ≪ 1 so the ratio of observed flux density
to blackbody flux density is ≈ τ . At ν = 10 GHz, the observed flux density is
S ≈ 0.8 Jy while the Rayleigh–Jeans extrapolated flux density is
S ≈ 0.1 Jy (
ν
2
)
≈ 110
Jy,
0.3 GHz
so the optical depth at 10 GHz is τ
≈ 0.8/110 ≈ 0.007
.
Free–free emission accounts for about 10% of the 1 GHz continuum luminosity in most
spiral galaxies. It is the strongest component in the frequency range from ν ≈ 30 GHz to
ν ≈ 200 GHz, above which thermal emission from cool dust grains dominates (Figure 2.24).
Free–free absorption flattens the low-frequency spectra of spiral galaxies, and the frequency at
which τ ≈ 1 is higher in galaxies with high star-formation rates, especially if the star formation
is confined to a compact region near the nucleus. If the free–free and synchrotron emission from
a starburst galaxy are roughly cospatial, its radio brightness temperature at frequencies
ν < 100 GHz is [30]
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4 Free–Free Radiation‣ Essential Radio Astronomy
Tb ∼ T [1 − exp(−τ )] [1 + 10(
ν
(4.63)
0.1+α
)
],
GHz
where T ≈ 10 K and α ≈ −0.8 is the spectral index of the synchrotron radiation. Free–free
absorption of the synchrotron radiation limits the maximum brightness temperature to
≤ 10 K at frequencies ν ≥ 1 GHz. This limit can be used to identify the energy source
T
powering a compact radio source at the center of a galaxy: if its brightness temperature is
significantly higher than 10 K, it is powered by an AGN, not a compact starburst.
4
5
b
5
3 Radio Telescopes and RadiometersBibliographyIndex
5 Synchrotron Radiation
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