Essential Radio Astronomy

Chapter 5 Synchrotron Radiation

5.1 Magnetobremsstrahlung

Any accelerated charged particle emits electromagnetic radiation with power specified by Larmor’s formula (Equation 2.143). In astrophysical situations, electromagnetic forces produce the strongest accelerations of charged particles. Acceleration by an electric field accounts for free–free radiation. Acceleration by a magnetic field produces magnetobremsstrahlung, the German word for “magnetic braking radiation.” The lightest charged particles (electrons, and positrons if any are present) are accelerated more than relatively massive protons and heavier ions, so electrons (and possibly positrons) account for virtually all of the radiation observed. The character of magnetobremsstrahlung depends on the speeds of the electrons, so these somewhat different types of radiation are given specific names. Gyro radiation comes from electrons whose velocities are much smaller than the speed of light: vc. Mildly relativistic electrons whose kinetic energies are comparable with their rest mass mec2 emit cyclotron radiation, and ultrarelativistic electrons (kinetic energies mec2) produce synchrotron radiation.

Synchrotron radiation is ubiquitous in astronomy. It accounts for most of the radio emission from active galactic nuclei (AGNs) thought to be powered by supermassive black holes in galaxies and quasars, and it dominates the radio continuum emission from star-forming galaxies like our own at frequencies below ν30 GHz. The magnetosphere of Jupiter is a synchrotron radio source. The optical emission from the Crab Nebula supernova remnant, the optical jet of the radio galaxy M87, and the optical through X-ray emission from many quasars is synchrotron radiation.

The relativistic electrons in nearly all synchrotron sources have power-law energy distributions, so they are not in local thermodynamic equilibrium (LTE). Consequently, synchrotron sources are often called “nonthermal” sources. However, a synchrotron source with a relativistic Maxwellian electron-energy distribution would be a thermal source, so “synchrotron” and “nonthermal” are not completely synonymous.

Even though synchrotron radiation is quite different from free–free emission, notice how many themes from the derivation of free–free source spectra are repeated for synchrotron sources—Larmor’s equation is used to derive the total power and spectrum of radiation by a single electron, the spectrum of an optically thin source is obtained as the superposition of the spectra of individual electrons, the electron energy distribution is broad enough that the spectrum of an individual electron can be approximated by a delta function, Kirchhoff’s law yields the absorption coefficient in terms of the emission coefficient (even though the synchrotron source is not in LTE!), and the simple “cylindrical cow” geometry is used to yield the spectrum of a source that is optically thick at low frequencies.

5.1.1 Gyro Radiation

Larmor’s equation is valid only for gyro radiation from a particle with charge q moving with a small velocity vc. The magnetic force F exerted on the particle by a magnetic field B is

F=q(v×B)c. (5.1)

The magnetic force is perpendicular to the particle velocity, so Fv=0. Consequently the magnetic force does no work on the particle, does not change the particle’s kinetic energy mv2/2, and does not change the component of velocity v parallel to the magnetic field. Because both |v| and v are constant, the magnitude of the velocity component |v| perpendicular to the magnetic field must also be constant. In a uniform magnetic field, the particle moves along the magnetic field line on a helical path with constant linear and angular speeds. In the inertial frame moving with velocity v, the particle orbits in a circle of radius r perpendicular to the magnetic field with the angular velocity ω needed to balance the centripetal and magnetic forces:

m|v˙|=mω2r=qc|v×B|=qcωrB; (5.2)

the orbital angular frequency is

ω=qBmc. (5.3)

Equation 5.3 implies that the orbital frequency is independent of the particle speed so long as vc. The angular gyro frequency ωG is defined by

ωGqBmc. (5.4)

This definition holds for any particle speed, so the gyro frequency equals the actual orbital frequency if and only if vc.

The angular gyro frequency (rad s-1) of an electron is

ωG=eBmec =4.8×10-10statcoulB9.1×10-28g3×1010cms-1 (5.5)
17.6×106rads-1B(gauss). (5.6)

In terms of νGωG/(2π) the electron gyro frequency in MHz is

(νGMHz)=2.8(Bgauss). (5.7)

The typical interstellar magnetic field strength in a normal spiral galaxy like ours is B10μG, so the electron gyro frequency is only νG=2.8MHz10×10-6gauss28Hz. The associated gyro radiation cannot propagate through the ISM because its frequency ν28Hz is less than the plasma frequency (Equation 6.40).

Gyro radiation from nonrelativistic electrons is observable in very strong magnetic fields such as the B1012 gauss magnetic field of a neutron star. For example, the binary X-ray source Hercules X-1 exhibits an X-ray absorption line at photon energy E34 keV (Figure 5.1).

Figure 5.1: Gyro-resonance absorption line near 34 keV [41].

This spectral feature is thought to be a gyro-resonance absorption, in which case the frequency of this absorption line directly measures the magnetic field strength near the Her X-1 neutron star. The observed photon energy corresponds to the frequency

ν=Eh34×103eV1.60×10-12ergeV-16.63×10-27ergs8.2×1018Hz. (5.8)

Equating this frequency to the gyro frequency yields the magnetic field:

B =2πνGmece (5.9)
2π8.2×1018Hz9.1×10-28g3×1010cms-14.8×10-10statcoul (5.10)
2.9×1012gauss. (5.11)

5.2 Synchrotron Power

Cosmic rays are celestial particles (e.g., electrons, protons, and heavier nuclei) with extremely high energies. Cosmic-ray electrons in the interstellar magnetic field emit the synchrotron radiation that accounts for most of the continuum emission from our Galaxy at frequencies below about 30 GHz. Larmor’s formula can be used to calculate the synchrotron power and synchrotron spectrum of a single electron in the inertial frame in which the electron is instantaneously at rest, but the Lorentz transform of special relativity is needed to transform these results to the frame of an observer at rest in the Galaxy.

5.2.1 Lorentz Transforms

Figure 5.2: An “event” viewed by observers in two coordinate frames. The unprimed frame is the rest frame and the primed frame is moving to the right with velocity v.

For any pointlike event, the Lorentz transform (see Appendix C for a derivation) relates the coordinates (x,y,z,t) in the unprimed inertial frame and the coordinates (x,y,z,t) in the primed frame moving with velocity v in the x-direction (Figure 5.2). They are

x=γ(x+vt),y=y,z=z,t=γ(t+βx/c), (5.12)
x=γ(x-vt),y=y,z=z,t=γ(t-βx/c), (5.13)


βv/c (5.14)


γ(1-β2)-1/2 (5.15)

is called the Lorentz factor. The Lorentz transform is linear, so even for finite coordinate differences (Δx,Δy,Δz,Δt) and (Δx,Δy,Δz,Δt) between two events, the differential form of the Lorentz transform is

Δx=γ(Δx+vΔt),Δy=Δy,Δz=Δz,Δt=γ(Δt+βΔx/c), (5.16)
Δx=γ(Δx-vΔt),Δy=Δy,Δz=Δz,Δt=γ(Δt-βΔx/c). (5.17)

5.2.2 Relativistic Masses

The rest mass me of an electron can be converted to an energy by Einstein’s famous mass–energy equation

E=mc2. (5.18)

The energy corresponding to the electron’s rest mass me is

E0=mec2 =9.1×10-28g(3×1010cms-1)2=8.2×10-7erg (5.19)
=8.2×10-7erg1.60×10-12erg(eV)-1=5.1×105eV=0.51MeV. (5.20)

Cosmic-ray electrons having masses m=γmeme (Equation C.28) and total energies EE0=0.51MeV are called ultrarelativistic.

Ultrarelativistic electrons still move on spiral paths along magnetic field lines, but the angular frequencies ωB of their orbits are lower because their inertial masses are multiplied by γ (Appendix C):

ωB=eB(γme)c=ωGγ. (5.21)

The orbital frequency of a cosmic-ray electron with γ=105 in a B10μG interstellar magnetic field is only

νBωB2π28×10-5Hz1cycleperhour. (5.22)

Because vc whenever γ1, the orbital radius r of an ultrarelativistic electron is nearly independent of γ and can be quite large:

rcωB3×1010cms-12π28×10-5Hz1.7×1013cm1AU. (5.23)

Equation 5.21 is not promising for the production of observable synchrotron radiation: the high observed masses m=γme of relativistic electrons reduce their orbital frequencies and accelerations to extremely low values. However, two compensating relativistic effects can explain the strong synchrotron radiation observed at radio frequencies: (1) the total radiated power in the observer’s frame is proportional to γ2 and (2) relativistic beaming turns the low-frequency sinusoidal radiation in the electron frame into a series of extremely sharp pulses containing power at much higher frequencies γ3νB=γ2νG in the observer’s frame. These relativistic corrections are derived in Sections 5.2.3 and 5.3.1.

5.2.3 Synchrotron Power Radiated by a Single Electron

Let primed coordinates describe the inertial frame in which the electron is (temporarily) nearly at rest. Larmor’s equation (Equation 2.143) gives the radiated power in the electron rest frame as

P=2(e)2(a)23c3=2e2(a)23c3 (5.24)

because e=e (electric charge is a relativistic invariant) follows directly from Maxwell’s relativistically correct equations.

The magnetic acceleration a=(ay2+az2)1/2 of the electron in the frame of an observer at rest in the Galaxy can be derived by applying the chain rule for derivatives to the differential Lorentz coordinate transform (Equations 5.16 and 5.17).

aydvydt=dvydtdtdt=1γdvydtdtdt=ayγ2. (5.25)

Similarly, az=az/γ2 so

a=aγ2. (5.26)


P=2e2(a)23c3=2e2a2γ43c3. (5.27)

The next step is to transform from the radiated power P=dE/dt in the electron frame to P=dE/dt, the power measured by an observer at rest in the Galaxy, by applying the chain rule to the mass–energy Equation 5.18:

PdEdt=dEdtdtdt=dEdEdEdtdtdt=γPγ-1=P; (5.28)

that is, the power is a relativistic invariant. Consequently,

P=P=2e2a2γ43c3  (a=0). (5.29)

To calculate a, combine force balance in a circular orbit

advdt=ωBv (5.30)

with Equation 5.21 for ωB to get

a=eBvγmec=eBvsinαγmec, (5.31)

where the constant angle α between the electron velocity v and the magnetic field B is called the pitch angle.

Inserting a from Equation 5.31 into Equation 5.29 gives the power radiated by a single electron moving with pitch angle α:

P=2e23c3γ2e2B2me2c2v2sin2α. (5.32)

This power is usually written in terms of the Thomson cross section of an electron, σT. The Thomson cross section is the classical radiation-scattering cross section of a charged particle. If a plane wave of electromagnetic radiation passes a free charged particle initially at rest, the electric field of that radiation will accelerate the particle, which in turn will radiate power in all directions according to Larmor’s equation. This process is called scattering rather than absorption because the total power in electromagnetic radiation is unchanged—all of the power extracted from the incident plane wave is reradiated at the same frequency but in other directions. It is a straightforward exercise to show that the geometric area that would intercept this amount of incident power from the plane wave is

σT8π3(e2mec2)2. (5.33)


σT=8π3[(4.80×10-10statcoul)29.11×10-28g(3.00×1010cms-1)2]6.65×10-25cm2. (5.34)

The reason for using the Thomson cross section will become clear in the discussion of inverse-Compton scattering of radiation by the same cosmic rays that are producing synchrotron radiation (Section 5.5.1).

It is also conventional to eliminate the B2 in Equation 5.32 in favor of the magnetic energy density

UB=B28π. (5.35)


P=[8π3(e2mec2)2]2(B28π)cγ2v2c2sin2α (5.36)

simplifies to

P=2σTβ2γ2cUBsin2α. (5.37)

The synchrotron power radiated by a single electron depends only on physical constants, the square of the electron kinetic energy (via γ2), the magnetic energy density UB, and the pitch angle α.

The relativistic electrons in radio sources can have lifetimes of thousands to millions of years before losing their ultrarelativistic energies via synchrotron radiation or other processes. During their lifetimes they are scattered repeatedly by magnetic-field fluctuations and charged particles in their environment, and the distribution of their pitch angles gradually becomes random and isotropic. The average synchrotron power P per electron in an ensemble of electrons having the same Lorentz factor γ and isotropically distributed pitch angles α is

P=2σTβ2γ2cUBsin2α, (5.38)

where sin2α is the average over all pitch angles:

sin2α sin2αdΩ𝑑Ω=14πsin2αdΩ (5.39)
=14πϕ=02πα=0πsin2αsinαdαdϕ=14π2π43 (5.40)
=23. (5.41)

Thus the average synchrotron power per relativistic electron in a source with an isotropic pitch-angle distribution is

P=43σTβ2γ2cUB. (5.42)

For all γ1, the factor β2=1-γ-21 can be ignored. Relativistic effects multiply the average radiated power by a factor γ2 compared with the nonrelativistic (γ=1) Larmor equation.

5.3 Synchrotron Spectra

5.3.1 The Synchrotron Spectrum of a Single Electron

Why does synchrotron radiation appear at frequencies much higher than ωB=ωG/γ? First, relativistic aberration beams the dipole pattern of Larmor radiation in the electron frame sharply along the direction of motion in the observer’s frame as v approaches c (Figure 5.3). Relativistic photon beaming follows directly from the relativistic velocity addition equations (Equations C.22 through C.26) that relate the photon velocity component vx in the unprimed frame of the observer to the photon velocity component vx in the primed frame and the velocity v=βc of the primed frame in which the electron is instantaneously at rest:

vxdxdt =dxdtdtdt=γ(dxdt+vdtdt)(dtdt)-1 (5.43)
=γ(vx+v)[γ(1+βcdxdt)]-1, (5.44)
vx=(vx+v)(1+βvxc)-1. (5.45)

In the y-direction,

vydydt=dydtdtdt=dydt(dtdt)-1, (5.46)
vy=vyγ(1+βvxc)-1. (5.47)

Consider the synchrotron photons emitted with speed c at an angle θ from the x-axis. Let vx and vy be the projections of the photon speed onto the x- and y-axes. Then

cosθ=vxc,sinθ=vyc. (5.48)

In the observer’s frame the same photons have

cosθ=vxc,sinθ=vyc. (5.49)

Inserting the velocity Equations 5.45 and 5.47 into the angle Equations 5.48 and 5.49 yields relations connecting θ and θ:

cosθ=(vx+v1+βvx/c)1c=(ccosθ+v1+βccosθ/c)1c=cosθ+β1+βcosθ (5.50)


sinθ=vycγ(1+βvx/c)=sinθγ(1+βcosθ). (5.51)

In the frame moving with the electron, the Larmor equation implies a power pattern proportional to cos2θ with nulls at θ=±π/2. In the observer’s frame, these nulls are offset by much smaller angles

θ=±arcsin(1/γ). (5.52)

An observer at rest sees the radiation confined to a very narrow beam of width 2/γ between nulls, as shown in Figure 5.3. For example, a 10 GeV electron has γ2×104 so 2/γ10-4rad20 arcsec! Although the electron is emitting continuously, the observer sees a short pulse of radiation only from the tiny fraction

22πγ=1πγ (5.53)

of the electron orbit where the electron is moving almost directly toward the observer.

Figure 5.3: Relativistic aberration transforms the dipole power pattern of Larmor radiation in the electron rest frame (dotted curve) into a narrow searchlight beam in the observer’s frame. The solid curve is the transformed pattern for γ=5. The observed angle between the nulls of the forward beam falls to Δθ=2arcsin(1/γ), which approaches Δθ=2/γ in the limit γ1.

The duration Δtp of the observed pulse is even shorter than the time Δt the electron needs to cover 1/(πγ) of its orbit because the electron is moving almost directly toward the observer with a speed approaching c (Figure 5.4) when it is observable. As it travels toward the observer, the electron nearly keeps up with the radiation that it emits:

Δtp =t(endofobservedpulse)-t(startofobservedpulse) (5.54)
=Δxv+(x-Δx)c-xc. (5.55)
Figure 5.4: The beamed radiation from an ultrarelativistic electron is visible only while the electron’s velocity points within ±1/γ of the line of sight (Δθ2/γ). During that time Δt the electron moves a distance Δx=vΔt toward the observer, almost keeping up with the radiation that travels a distance cΔt. As a result, the observed pulse duration is shortened by a factor (1-v/c).

The first term in this equation represents the time taken by the electron to cover the distance Δx, the second is the light travel time from the electron position at the end of the pulse seen by the observer, and the third is the light travel time from the electron position at the beginning of the pulse seen by the observer. Note that the observed pulse duration

Δtp=Δxv-Δxc=Δxv(1-vc)Δxv=Δt (5.56)

is much less than the time Δt the electron needs to move a distance Δx because, in the observer’s frame, the electron nearly keeps up with its own radiation. In the limit vc,

(1-vc)=(1-vc)1+v/c1+v/c=1-v2/c21+v/cγ-22=12γ2 (5.57)


Δtp=Δt2γ2=Δxv12γ2=ΔθωB12γ2. (5.58)

Recall that Δθ2/γ (Figure  5.4) so

Δtp=2γωB2γ2=1γ3ωB=1γ2ωG (5.59)

is the full observed duration of the pulse. Allowing for the motion of the electron parallel to the magnetic field replaces the total magnetic field by its perpendicular component B=Bsinα, yielding

Δtp=1γ2ωGsinα, (5.60)

where α is the pitch angle of the electron. Thus a plot of the power received as a function of time is very spiky. If γ104 and B10μG, the half width of each pulse is Δtp/2<10-10 s and the spacing between pulses is γ/νG>102 s (Figure 5.5).

Figure 5.5: Synchrotron radiation is a very spiky series of widely spaced narrow pulses. The numerical values on this plot of power versus time correspond to an electron with γ104 in a magnetic field B10μG.

The observed synchrotron power spectrum is the Fourier transform of this time series of pulses. The pulse train is the convolution of the individual pulse profile with the shah function (see Figure A.1 or Bracewell [15], a valuable reference book):

III(t/Δt)n=-δ[(t/Δt)]-n), (5.61)

where each delta function δ is an infinitesimally narrow spike at integer t/Δt=n and whose integral is unity. The convolution theorem (Equation A.15) states that the Fourier transform of the pulse train is the product of the Fourier transform of one pulse and the Fourier transform of the shah function.

The Fourier transform of the shah function is also the shah function (Figure A.1), so the similarity theorem (Equation A.11) implies that the Fourier transform of

III(tνGγ) (5.62)

in the time domain is proportional to

III(νγνG), (5.63)

which is a nearly continuous series of spikes in the frequency domain. Adjacent spikes are separated in frequency by only

Δν=νGγ<10-3Hz. (5.64)

Although this is not formally a continuous spectrum, the frequency shifts caused by even tiny fluctuations in electron energy, magnetic field strength, or pitch angle cause frequency shifts much larger than Δν, so the spectrum of synchrotron radiation is effectively continuous.

Figure 5.6: This figure shows four different ways to plot the synchrotron spectrum of a single electron in terms of F(x)xxK5/3(η)𝑑η, where xν/νc is the frequency in units of the critical frequency νc. Although they all plot the same spectrum, they look quite different and emphasize or suppress information in different ways. (1) Simply plotting F(x) versus x on linear axes (lower left panel) completely obscures the spectrum below the peak of F(x) at x0.29. (2) Replotting on logarithmic axes (upper left panel) shows that the low-frequency spectrum has a logarithmic slope of 1/3, but it obscures the fact that most of the power is emitted at frequencies near x1 because F(x) is the spectral power per unit frequency, not per unit log(frequency). (3) The power per unit logx is F(logx)=ln(10)xF(x), which is plotted on logarithmic axes in the upper right panel. It has a slope of 4/3 at low frequencies, making it clearer that most of the power is emitted near x1, as required to justify the approximation (used in the next section) that all of the power is emitted at x=1. (4) The lower right panel plots F(logx) with a linear ordinate but a logarithmic abscissa to expand the low-frequency spectrum lost in the lower left panel. It is clearly consistent with the approximation that all emission is near x=1 but doesn’t clearly show that the low-frequency spectrum is a power law. Note also that the peak of F(logx) is at x1.3, not x0.29. Areas under the curves in the two lower panels are proportional to the power radiated in given frequency ranges. Both lower panels show that about half of the power is emitted at frequencies below the critical frequency and half at higher frequencies.

Thus the synchrotron spectrum of a single electron is fairly flat at low frequencies and tapers off at frequencies above

νmax12Δtpπγ2νGsinαγ2B. (5.65)

It isn’t necessary to know the Fourier transform of the pulse shape precisely to calculate the synchrotron spectra of celestial sources because real sources don’t contain electrons with just one energy and one pitch angle in a uniform magnetic field. The actual energy distribution of cosmic rays in a real radio source is a very broad power law, broad enough to smear out the details of the spectrum from each electron-energy range. Just for the record, the synchrotron power spectrum of a single electron is

P(ν)=3e3Bsinαmec2(ννc)ν/νcK5/3(η)dη, (5.66)

where K5/3 is a modified Bessel function and νc is the critical frequency whose value is

νc=32γ2νGsinα. (5.67)

For the full mathematical derivation of Equations 5.66 and 5.67, see Pacholczyk [77], a valuable reference for details of radiation processes.

The synchrotron power spectrum of a single electron is plotted in Figure 5.6. It has a logarithmic slope

dlogP(ν)dlogν13 (5.68)

at low frequencies, a broad peak near the critical frequency νc, and falls off sharply at higher frequencies. One way to look at νc is

νc=(32sinα)(Emc2)2eB2πmecE2B. (5.69)

That is, the frequency at which each electron emits most strongly is proportional to the square of its energy multiplied by the strength of the perpendicular component of the magnetic field.

5.3.2 Synchrotron Spectra of Optically Thin Radio Sources

If a synchrotron source containing any arbitrary distribution of electron energies is optically thin (τ1), then its spectrum is the superposition of the spectra from individual electrons and its flux density cannot rise faster than ν1/3 at any frequency ν. In other words, the (negative) spectral index α-dlogPν/dlogν (be careful not to confuse this spectral index α with the electron pitch angle α) must always be greater than -1/3. Most astrophysical sources of synchrotron radiation have spectral indices near α0.75 at frequencies where they are optically thin, and their high-frequency spectral indices reflect their electron energy distributions, not the spectra of individual electrons.

The energy distribution of cosmic-ray electrons in most synchrotron sources is roughly a power law:

n(E)dEE-δdE, (5.70)

where n(E)dE is the number of electrons per unit volume with energies E to E+dE. The energy range around γ104 is relevant to the production of radio radiation. Because n(E) is nearly a power law over more than a decade of energy and the critical frequency νc is proportional to E2, the synchrotron spectrum will reflect this power law over a frequency range of at least 102=100. Consequently the detailed spectra of individual electrons can be ignored because they are smeared out in the source spectrum by the broad power-law energy distribution. The source spectrum can be calculated with good accuracy from the approximation that each electron radiates all of its average power (Equation 5.42)

P=-dEdt=43σTβ2γ2cUB (5.71)

at the single frequency

νγ2νG, (5.72)

which is very close to the critical frequency (Equation 5.67). Then the emission coefficient (Equation 2.26) of synchrotron radiation by an ensemble of electrons is

jνdν=-dEdtn(E)dE, (5.73)


E=γmec2(ννG)1/2mec2. (5.74)

Differentiating E(ν) in Equation 5.74 gives

dEmec2ν-1/22νG1/2dν (5.75)


jν(43σTβ2γ2cUB)(E-δ)(mec2ν-1/22νG1/2). (5.76)

Eliminating E in favor of ν/νG and then using νGB yields an expression for jν in terms of ν and B only:

jν(ννG)B2(ννG)-δ/2(ννG)-1/2(νB)B2(νB)-δ/2(νB)-1/2. (5.77)

This simplifies to

jνB(δ+1)/2ν(1-δ)/2. (5.78)

Thus the spectrum of optically thin synchrotron radiation from a power-law distribution n(E)E-δ of electrons is also a power law, and the spectral index α=-dlnS/dlnν depends only on δ:

α=δ-12. (5.79)

In our Galaxy and in many other synchrotron sources, α0.75 near ν1 GHz, so the radio spectra imply δ2.5. This value of δ reflects the initial power-law energy slope δ0 of cosmic rays accelerated in shocks, such as the shocks produced by supernova remnants expanding into the ambient interstellar medium, modified by loss processes that deplete the population of relativistic electrons. For example, the rate at which an electron loses energy to synchrotron radiation is proportional to E2, so higher-energy electrons are depleted more rapidly. The critical frequency is also proportional to E2, so synchrotron losses eventually steepen source spectra at higher frequencies. If relativistic electrons with initial power-law slope δ0 are continuously injected into a synchrotron source, synchrotron losses will eventually steepen that slope to δ=δ0+1 at high energies and the high-frequency spectral index will steepen by Δα=1/2. See Pacholczyk [77] for a more detailed discussion of energy losses and their spectral consequences.

5.3.3 Synchrotron Self-Absorption

The brightness temperatures of synchrotron sources cannot become arbitrarily large at low frequencies because for every emission process there is an associated absorption process. If the emitting particles are in local thermodynamic equilibrium (LTE), they have a Maxwellian energy distribution and the source is thermal. No thermal source can have a brightness temperature greater than the kinetic temperature of the emitting particles. If the energy distribution of relativistic electrons in a synchrotron source were a (relativistic) Maxwellian, the electrons would have a well-defined kinetic temperature, and synchrotron self-absorption would prevent the brightness temperature of the synchrotron radiation from exceeding the kinetic temperature of the emitting electrons. Most astrophysical synchrotron sources are nonthermal sources because the energy distribution of the relativistic electrons is a power law and there is no well-defined electron temperature. However, synchrotron self-absorption occurs for any electron energy distribution, and the low-frequency spectrum of an optically thick synchrotron source is a power law whose slope is α=-dlnS/dlnν=-5/2. That result is derived below.

Electrons with energy E=γmec2 emit most of their synchrotron power near the critical frequency

νcγ2eB2πmec, (5.80)

so the synchrotron emission at frequency ν comes primarily from electrons with Lorentz factors near

γ(2πmecνeB)1/2. (5.81)

In this approximation that only those electrons having one particular energy E contribute to the emission (and hence absorption) at each frequency ν, all other electrons could have a relativistic Maxwellian energy distribution to match without changing the resulting emission and absorption at that frequency. Consequently, a sufficiently bright synchrotron source will be optically thick, and its brightness temperature at any frequency cannot exceed the effective temperature of the electrons emitting at that frequency.

In an ultrarelativistic gas, the ratio of specific heats at constant pressure and at constant volume is cp/cv=4/3, not the nonrelativistic 5/3, so the relation between electron energy E and temperature Te is

E=3kTe,not3kTe2. (5.82)

Thus the effective temperature of relativistic electrons with energy E can be defined as

TeE3k=γmec23k, (5.83)

even if the ensemble of electrons has a nonthermal energy distribution. Using Equation 5.81 to eliminate γ in favor of ν gives the effective temperature of those electrons producing most of the synchrotron radiation at frequency ν:

Te(2πmecνeB)1/2mec23k. (5.84)


(TeK)1.18×106(νHz)1/2(Bgauss)-1/2. (5.85)

For example, the effective temperature of the relativistic electrons emitting synchrotron radiation at ν=0.1GHz=108Hz in a B=100μgauss =10-4 gauss magnetic field is

(TeK)1.18×106(108)1/2(10-4)-1/21012. (5.86)

At a sufficiently low frequency ν, the brightness temperature Tb of any synchrotron source will approach the effective electron temperature Te of electrons emitting at that frequency and the source will become opaque. Equation 2.33 defines brightness temperature in the Rayleigh–Jeans limit:

TbIνc22kν2. (5.87)

Setting TbTe and using Equation 5.85 to eliminate Te in favor of ν and B gives

Iν2kTeν2c2ν1/2ν2B-1/2=ν5/2B-1/2. (5.88)

Thus at low frequencies the spectrum of a synchrotron self-absorbed and spatially homogeneous source is a power law of slope 5/2:

S(ν)ν5/2, (5.89)

independent of the slope δ of the electron-energy spectrum. The flux density of an opaque but truly thermal source (e.g., an Hii region) is proportional to ν2; the extra ν1/2 for synchrotron radiation comes from the fact that Teν1/2 (Equation 5.85).

The full spectrum of a homogeneous cylindrical synchrotron source (Figure 5.7) is [77]

S(νν1)5/2{1-exp[-(νν1)-(δ+4)/2]}, (5.90)

where ν1 is the frequency at which τ=1. Real astrophysical sources are inhomogeneous, so synchrotron self-absorption always yields slopes much lower than 5/2, as illustrated by Figure 5.8.

Figure 5.7: The spectrum of a homogeneous cylindrical synchrotron source in terms of the frequency ν1 at which τ=1. Equation 5.90 shows that it approaches a power law with slope 5/2 at frequencies νν1 and slope (1-δ)/2 for νν1. Real astrophysical sources are inhomogeneous, so their low-frequency spectral slopes are smaller than 5/2 and their spectral peaks are not so sharp.

Substituting TbTe into Equation 5.85 yields an estimate of the magnetic field strength in a self-absorbed source whose brightness temperature has been measured at frequency ν:

(Bgauss)1.4×1012(νHz)(TbK)-2. (5.91)

For example, a self-absorbed radio source with observed brightness temperature Tb1011 K at ν=1 GHz has a magnetic field strength


The spectra of celestial radio sources are more complex because real sources have nonuniform magnetic fields and electron energy distributions in geometrically complex structures. Representative spectra of powerful radio galaxies and quasars are illustrated in Figure 5.8.

Figure 5.8: Representative spectra of radio galaxies and quasars [110]. The radio source 3C 84 in the nearby galaxy NGC 1275 contains a very compact nuclear component that is opaque below about 20 GHz. The radio galaxy 3C 123 is transparent at all plotted frequencies, and energy losses steepen its spectrum above a few GHz. The quasar 3C 48 is synchrotron self-absorbed only below 100 MHz, while the quasar 3C 454.3 contains structures of different sizes that become opaque at different frequencies.

Synchrotron radiation from cosmic-ray electrons accelerated by the supernova remnants of relatively massive (M>8M) and short-lived (T<3×107 yr) stars dominates the radio continuum emission of the nearby starburst galaxy M82 (Figure 8.13) at frequencies ν<30 GHz (see the dot-dash line in Figure 2.24). Thermal emission (dashed line) from Hii regions ionized primarily by even more massive (M>15M) and shorter-lived stars is strongest between about 30 and 200 GHz. At frequencies well below 1 GHz, free–free absorption flattens the overall spectrum.

5.4 Synchrotron Sources

5.4.1 Minimum Energy and Equipartition

The existence of a synchrotron source implies the presence of relativistic electrons with some energy density Ue and a magnetic field whose energy density is UB=B2/(8π). What is the minimum total energy in relativistic particles and magnetic fields required to produce a synchrotron source of a given radio luminosity

L=νminνmaxLν𝑑ν (5.92)

over the frequency range conventionally bounded by νmin=107 Hz and νmax=1011 Hz?

The energy density of relativistic electrons in the energy range Emin to Emax is

Ue=EminEmaxEn(E)𝑑E, (5.93)

where n(E)dE is the number density of electrons in the energy range E to E+dE. Electrons with energy E emit most of the radiation seen at frequency νE2B, so the electron energy corresponding to radiation at frequency ν satisfies

EB-1/2. (5.94)

Thus the ratio of Ue to L can be written in terms of the energy limits:

UeLEminEmaxEn(E)𝑑E-EminEmax(dE/dt)n(E)𝑑E, (5.95)

where the synchrotron power emitted per electron is (-dE/dt)B2E2. For a power-law electron energy distribution n(E)E-δ,

UeLEminEmaxE1-δ𝑑EB2EminEmaxE2-δ𝑑EE2-δ|EminEmaxB2E3-δ|EminEmax. (5.96)

The energy limits Emin and Emax are both proportional to B-1/2 (Equation 5.94) so

UeL(B-1/2)2-δB2(B-1/2)3-δ=B-1+δ/2B2B-3/2+δ/2=B-3/2 (5.97)

Thus the electron energy density needed to produce a given synchrotron luminosity scales as

UeB-3/2, (5.98)

while the magnetic energy density is

UBB2. (5.99)

The “invisible” cosmic-ray protons and heavier ions emit negligible synchrotron power but they still contribute to the total cosmic-ray particle energy. If the ion/electron energy ratio is η, then the total energy density in cosmic rays is UE=(1+η)Ue and the total energy density U of both cosmic rays and magnetic fields is

U=(1+η)Ue+UB. (5.100)

Cosmic rays collected near the Earth have η40, but the value of η in radio galaxies and quasars has not been measured.

The greatly differing dependences of Ue and UB on B means that the total energy density U has a fairly sharp minimum near equipartition, the point at which (1+η)UeUB (Figure 5.9).

Figure 5.9: For a source of a given synchrotron luminosity, the particle energy density UE(1+η)Ue is proportional to B-3/2 and the magnetic energy density UB is proportional to B2. The total energy density U=UE+UB has a fairly sharp minimum near equipartition of the particle and magnetic energy densities (UEUB).

The minimum of the total energy density U occurs at

dUdB=d[(1+η)Ue+UB]dB=0. (5.101)

The logarithmic derivative of the electron energy density UeB-3/2 is

dUedBUe-1=-(32)B-5/2B3/2=-32B, (5.102)


dUedB=-3Ue2B. (5.103)

The logarithmic derivative of the magnetic-field energy density UBB2 is

dUBdBUB-1=2BB2=2B, (5.104)


dUBdB=2UBB. (5.105)

Inserting Equations 5.103 and 5.105 into the minimum-energy Equation 5.101 gives

-3(1+η)Ue2B+2UBB=0. (5.106)

The ratio of cosmic-ray particle energy density to magnetic field energy that minimizes the total energy is

particleenergydensitymagneticfieldenergydensity=(1+η)UeUB=43. (5.107)

This ratio is nearly unity, so minimum energy implies (near) equipartition of energy: the total cosmic-ray energy density (including the energy of the nonradiating ions) (1+η)Ue is nearly equal to the total magnetic energy density UB. It is not known whether most synchrotron sources are in equipartition, but radio astronomers often assume so because

  1. 1.

    it is physically plausible—systems with interacting components often tend toward equipartition;

  2. 2.

    extragalactic radio sources with high luminosities L and large volumes V such as Cyg A have enormous total energy E=UV requirements even near equipartition; the “energy problem” is even worse otherwise;

  3. 3.

    it eliminates an unknown parameter and permits estimates of the relativistic particle energies and the magnetic field strengths of radio sources with measured luminosities and sizes.

Getting the actual numerical values of the particle and magnetic-field energy densities from the synchrotron emission coefficient is a straightforward but tedious algebraic chore (Wilson et al. [116, Section 10.10]). The results (from Pacholczyk [77, p. 171]) are summarized in Equations 5.109 and 5.110. The functions c12 and c13 in these equations absorb the integrations from frequency νmin to νmax and the physical constants in Gaussian CGS units. The values of c12 and c13 are plotted in Figures 5.10 and 5.11.

For a spherical radio source with radius R and magnetic field strength B, the total magnetic energy is

EB=UBV=B28π4πR33=B2R36. (5.108)

The minimum-energy magnetic field strength for a source of radio luminosity L is

Bmin=[4.5(1+η)c12L]2/7R-6/7gauss (5.109)

and the corresponding total energy is

Emin(total)=c13[(1+η)L]4/7R9/7ergs. (5.110)

The synchrotron lifetime of a source is defined as the ratio of total electron energy Ee to the energy loss rate L from synchrotron radiation:

τsEeL. (5.111)

It approximates the lifetime of a synchrotron source if the primary loss mechanism is synchrotron radiation; if other loss mechanisms (e.g., inverse-Compton scattering) are significant, the actual source lifetime will be shortened. The synchrotron lifetime can be written as

τsc12B-3/2. (5.112)
Figure 5.10: Plots of c12 in Gaussian CGS units as a function of (negative) spectral index α-dlogS/dlogν for νmin=106 Hz (dashed curves) and 107 Hz (solid curves) and νmax=1010 Hz, 1011 Hz, and 1012 Hz.
Figure 5.11: Plots of c13 in Gaussian CGS units as a function of (negative) spectral index α-dlogS/dlogν for νmin=106 Hz (dashed curves) and 107 Hz (solid curves) and νmax=1010 Hz, 1011 Hz, and 1012 Hz.

5.4.2 The Eddington Luminosity Limit

The steady-state luminosity of an astronomical object of total mass M is limited by the requirement that the outward radiation pressure cannot exceed the pull of gravity. Otherwise, radiation pressure would expel the outer layers of a star or disrupt accretion onto a compact object such as a black hole or neutron star. If the atmosphere of the star or the infalling material is primarily ionized hydrogen, the free electrons will Thomson-scatter outflowing radiation. The Thomson-scattering cross section σT is given by Equation 5.33. Each electron being pushed away by radiation pressure will drag along one proton (mpme) with it to maintain charge neutrality. Balancing the forces from radiation and gravity on each electron/proton pair at distance r from the accreting object defines the Eddington luminosity:

LE4πr2σTc=GM(mp+me)r2GMmpr2. (5.113)

Both forces are proportional to r-2 so

LE4πGMmpcσT (5.114)

is proportional to the mass M and independent of distance. In CGS units,

LE(ergs-1)=6.28×104M(g). (5.115)

Normalized to “solar” units L3.83×1033 erg s-1 and M1.99×1033 g,

(LEL)6.28×1041.99×1033g3.83×1033ergs-1(MM), (5.116)
(LEL)3.3×104(MM). (5.117)

For example, as the mass of a main-sequence star approaches M100M, its luminosity approaches its Eddington luminosity. Very massive stars often have radiation-driven winds, and stars more massive than 100M may not be stable. The Eddington limit doesn’t apply only to objects surrounded by ionized hydrogen, and analogs to the classical Eddington limit can be derived for other absorbers, such as interstellar dust. The ratio of the absorption cross section to mass for a typical dust grain is 500 times larger than the ratio for ionized hydrogen, so the maximum luminosity-to-mass ratio of a dusty galaxy is 500 times lower than given by Equation 5.117. Thus a supermassive black hole emitting at its Eddington limit for ionized hydrogen in the nucleus of a dusty galaxy may remove the dusty ISM if the galaxy mass is less than 500 times the black-hole mass. Such radiative “feedback” processes may account for the observed mass ratio 500 of galaxy bulges and their central black holes [37].

5.4.3 Application to the Luminous Radio Galaxy Cyg A

Figure 5.12: A high-resolution VLA image of the radio source Cyg A. The bright central component is thought to coincide with a supermassive black hole that accelerates the relativistic electrons along two jets terminating in lobes well outside the host galaxy. Image credit: NRAO/AUI/NSF Investigators: R. Perley, C. Carilli, & J. Dreher.

Cyg A is a luminous double radio source (Figure 5.12) in a peculiar galaxy at a distance d230 Mpc. Its radio lobes have radii R30 kpc, and the total flux density of Cyg A is


To estimate the total radio luminosity of Cyg A, first convert the data from “astronomical” units to Gaussian CGS units:

R =30kpc(103pckpc)(3.09×1018cmpc)9.0×1022cm,
Sν =2000Jy(10-23ergs-1Hz-1cm-2Jy)(ν109Hz)-0.8
d =230Mpc(106pcMpc)(3.09×1018cmpc)7.1×1026cm.

The spectral luminosity of Cyg A is

Lν 4πd2Sν

The total radio luminosity of Cyg A in the frequency range 107 Hz to 1011 Hz is

L =107Hz1011HzLν𝑑ν2.0×1042ergs-1Hz-1(ν0.20.2)|ν=107Hzν=1011Hz

In units of the bolometric solar luminosity L3.83×1033ergs-1, the radio luminosity of Cyg A is


The radio power from Cyg A exceeds the total power produced by all of the stars in our Galaxy.

The energy source for this radio emission is a compact object at the center of the host galaxy. The Eddington limit (Equation 5.117) yields a lower limit to its mass M:


Note that the Eddington mass limit depends only on the instantaneous power emitted by the source, not on the total energy of the source, the source age, or any other indicator of its history.

The magnetic field strength Bmin that minimizes the total energy in the relativistic particles and magnetic fields implied by the luminous synchrotron source can be estimated with Equation 5.109. Approximate Cyg A (Figure 5.12) by two spherical lobes of radius R30 kpc and luminosity L/2 each, where L is the total luminosity of Cyg A:

Bmin [4.5(1+η)c12(L/2)]2/7R-6/7

The ion/electron energy ratio η has not been measured in extragalactic radio sources such as Cyg A. The cosmic rays accelerated by a supermassive black hole might be primarily electrons and positrons. Electrons and positrons have equally large charge/mass ratios, so an electron–positron plasma would have η1. If electrons and protons are accelerated to the same velocity (same γ), then the protons carry mp/me2×103 as much energy but emit almost nothing and η2×103. Fortunately, Bmin(1+η)2/7 is only weakly dependent on η—varying η from 1 to 2×103 changes (1+η)2/7 from about 1 to 9:

Bmin 1.45×10152.1×10-20(1to9)gauss (5.118)
(30to300)×10-6gauss10-4gauss. (5.119)

The minimum total energy (Equation 5.110) of Cyg A is twice the energy of each lobe:

Emin 2(lobes)c13[(1+η)L]4/7R9/7
where (1+η)4/7 is in the range of about 1 to 80;
Emin 4×1044.1×10253.26×1029(1to80)ergs (5.120)
5.4×1059(1to80)ergs5×1060ergs. (5.121)

Such large calculated energies can be confirmed observationally for sources in clusters of galaxies. Figure 8.15 shows that the radio source (red) in the galaxy cluster MS07356+7421 has displaced the X-ray emitting gas (blue) in a large volume. The gas pressure can be derived from the intensity of its the X-ray emission, and the total energy required to displace the gas is the product of the volume and the pressure [71].

This enormous energy implies an independent lower limit to the mass of the central object powering the radio source. Mass cannot be converted to energy with more than 100% efficiency, so the minimum mass needed to produce Emin is

M Eminc25×1060ergs(3×1010cms-1)26×1039g, (5.122)
M 6×1039g(M1.99×1033g)3×106M. (5.123)

This is a very conservative lower limit. Nuclear fusion can convert mass to energy with only 1% efficiency, so M>3×108M would be required for nuclear fusion in stars. Accretion onto a spinning black hole can yield efficiencies up to (1-3-1/2)0.4 in theory, implying M>107M. Many authors assume that mass accreted by astronomical black holes is converted to energy with about 10% efficiency; this yields M>3×107M. The small size of the radio core measured by Very Long Baseline Interferometry (VLBI) and its observed flux variability on timescales of months to years combined with the large minimum masses estimated from the Eddington limit and the total energy of the radio lobes together make it difficult to avoid the conclusion that the compact, massive object powering the radio source is a supermassive black hole (SMBH). The adjective supermassive is used for black holes with M>106M, which is far more massive than the most massive stars, M100M.

A lower limit to the age τ of the radio source Cyg A is the average synchrotron lifetime (Equation 5.111) of the relativistic electrons estimated by taking the ratio of the electron energy to the observed synchrotron luminosity:

τ τsEeLEmin/(1+η)L, (5.124)
τ 5.4×1059erg(1+η)4/71.33×1045ergs-1(1+η)4×1014sη-3/71014s3×106yr. (5.125)

Because each electron radiates energy at a rate proportional to E2 and the critical frequency is proportional to E2, the most energetic electrons emitting at the highest frequencies have the shortest lifetimes. The rapid depletion of high-energy electrons steepens the radio spectrum (Figure 5.13) of Cyg A frequencies higher than ν1GHz.

Figure 5.13: The radio spectrum of Cyg A (and Cas A, Vir A) from Baars et al. [6]. Note the spectral steepening above ν103 MHz.

Suppose that new relativistic electrons are continuously injected with a power-law energy distribution

N(E)E-δ0 (5.126)

into a radio source. After a long time, electrons emitting at frequencies higher than ν will be depleted by radiative losses E2 and these high-energy electrons will eventually reach an energy distribution N(E)E-(δ0+1). Consequently, the (negative) spectral index will be

α0=δ0-12 (5.127)

at low frequencies and approach

α=(δ0+1-1)/2=(α0+1/2) (5.128)

at higher frequencies; that is, the high-frequency spectrum steepens by Δα=1/2.

If the observed frequency ν of the spectral bend is high enough, the implied synchrotron lifetime of electrons with νcν may be less than the time needed for new relativistic electrons to travel from the radio core to the emitting feature in a jet or lobe. This implies in situ acceleration—something outside the radio core (e.g., shocks in the jet) must replenish the supply of relativistic electrons. The radio spectrum of Vir A, the source in the galaxy M87, is straight to at least ν30 GHz (Figure 5.13), and this synchrotron emission extends to optical frequencies, so many of the cosmic rays must be accelerated in the bright shocked regions, not just near the black hole.

5.5 Inverse-Compton Scattering

The ambient radiation field is normally fairly isotropic in the rest frame of a synchrotron source. However, such a radiation field looks extremely anisotropic to each ultrarelativistic (γ1) electron producing the synchrotron radiation. Relativistic aberration (Section 5.3.1) causes nearly all ambient photons to approach within an angle γ-1 rad of head-on (Figure 5.14). Thomson scattering of this highly anisotropic radiation systematically reduces the electron kinetic energy and converts it into inverse-Compton (IC) radiation by upscattering radio photons to become optical or X-ray photons. Inverse-Compton “cooling” of the relativistic electrons also limits the maximum rest-frame brightness temperature of an incoherent synchrotron source to Tb1012 K.

Figure 5.14: For a relativistic electron at rest in the “primed” frame moving with velocity v along the x-axis, the angle of incidence θ of incoming photons will be much less than the corresponding angle θ in the rest frame of the observer (Equations 5.50 and 5.51). This figure shows the aberration of an isotropic radiation field (left) seen in a moving frame with γ=(1-v2/c2)-1/2=5 (right).

5.5.1 IC Power from a Single Electron

To derive the equations describing inverse-Compton scattering, first consider nonrelativistic Thomson scattering in the rest frame of an electron. If the Poynting flux (power per unit area) of a plane wave incident on the electron is

S=c4πE×B=c4π|E|2, (5.129)

the electric field of the incident radiation will accelerate the electron, and the accelerated electron will in turn emit radiation according to Larmor’s equation. The net result is simply to scatter a portion of the incoming radiation, with no net transfer of energy between the radiation and the electron. The scattered radiation has power

P=|S|σT, (5.130)


σT8π3(e2mec2)26.65×10-25cm2 (5.131)

is called the Thomson cross section (Equation 5.33) of an electron. In other words, the electron will extract from the incident radiation the amount of power flowing through the area σT and reradiate that power over the doughnut-shaped pattern given by Larmor’s equation. The scattered power can be rewritten as

P=σTcUrad, (5.132)

where Urad=|S|/c is the energy density of the incident radiation.

Next consider radiation scattering by an ultrarelativistic electron. Equation 5.132 is valid only in the primed frame instantaneously moving with the electron:

P=σTcUrad. (5.133)

This nonrelativistic result needs to be transformed to the unprimed rest frame of an observer. Using the result P=P (Equation 5.28) gives

P=σTcUrad. (5.134)

To transform Urad into Urad, suppose that an electron moving with speed v=vx in the rest frame of the observer is hit successively by two low-energy photons approaching from an angle θ in the observer’s frame (θ in the electron frame) from the x-axis as shown in Figure 5.15. If the coordinates corresponding to the first and second photons hitting the electron (which is always located at x=y=z=0) are

(x1,0,0,t1)and(x2,0,0,t2) (5.135)

in the observer’s frame, then the Lorentz transform Equation 5.12 gives the coordinates of these two events as

(γvt1,0,0,γt1)and(γvt2,0,0,γt2), (5.136)

as shown in Figure 5.15.

Figure 5.15: Two successive photons striking an electron moving to the right. The photons approach at angle θ from the x-axis, as seen in the unprimed observer’s frame.

In the observer’s frame, the time Δt elapsed between the arrival of these two photons at the plane (dashed line in Figure 5.15) normal to the direction of propagation is

Δt =t2+(x2-x1)ccosθ-t1 (5.137)
=γt2+(γvt2-γvt1)ccosθ-γt1 (5.138)
=(t2-t1)[γ(1+βcosθ)], (5.139)

where βv/c. The time between being hit by the two photons in the electron’s frame is Δt=t2-t1 so

Δt=Δt[γ(1+βcosθ)]. (5.140)

The relativistic Doppler equation follows immediately from Equation 5.140. Let Δt be the time between the arrivals of two successive cycles of a wave whose frequency is ν=(Δt)-1 in the observer’s frame and ν=(Δt)-1 in the moving frame. Then

ν-1=(ν)-1[γ(1+βcosθ)] (5.141)


ν=ν[γ(1+βcosθ)]. (5.142)

In the electron’s frame, the frequency ν and energy E=hν of each photon are multiplied by [γ(1+βcosθ)]. Moreover, the rate at which successive photons arrive is multiplied by the same factor. If nγ is the photon number density in the observer’s frame, then nγ=nγ[γ(1+βcosθ)]. In the observer’s frame, the radiation energy density is

Urad=nγhν. (5.143)

In the electron’s frame

Urad=nγhν=nγ[γ(1+βcosθ)]hν[γ(1+βcosθ)]=Urad[γ(1+βcosθ)]2. (5.144)

Thus the transformation between Urad and Urad depends on the angle θ between the direction of the photons in the plane wave and the direction of the electron motion.

For a radiation field of total energy density Urad that is isotropic in the observer’s frame, the energy density per unit solid angle is Urad/4π and the total energy density in the electron frame can obtained by integrating Equation 5.144 over all directions:

Urad=Urad4πϕ=02πθ=0π[γ(1+βcosθ)]2sinθdθdϕ, (5.145)

where ϕ is the azimuthal angle around the x-axis. Thus

Urad=Uradγ22θ=0π(1+βcosθ)2sinθdθ. (5.146)

To evaluate this integral, substitute zcosθ so dz=-sinθdθ and eliminate β in favor of γ:

Urad =Uradγ221-1(1+βz)2(-1)𝑑z=Uradγ2(1+β2/3) (5.147)
=Urad[γ2+γ23-(γ23-γ2β23)]=Urad[4γ23-13γ2(1-β2)]. (5.148)

Recall that γ2(1-β2)=1 so

Urad=Urad4(γ2-1/4)3. (5.149)

Substituting this result for Urad into Equation 5.134 yields

P=43σTcUrad(γ2-1/4) (5.150)

for the total power radiated after inverse-Compton upscattering of low-energy photons. The initial power of these photons was σTcUrad, so the net power added to the radiation field by inverse-Compton scattering is

PIC=43σTcUrad(γ2-14)-σTcUrad=43σTcUrad(γ2-1). (5.151)

Replacing (γ2-1) by β2γ2 gives the final result

PIC=43σTcβ2γ2Urad (5.152)

for the net inverse-Compton power gained by the radiation field and lost by the electron. Dividing by the corresponding synchrotron power (Equation 5.42)

Psyn=43σTcβ2γ2UB (5.153)

reveals the remarkably simple ratio of IC to synchrotron radiation losses:

PICPsyn=UradUB. (5.154)

The IC loss is proportional to the radiation energy density and the synchrotron loss is proportional to the magnetic energy density. Note that synchrotron and inverse-Compton losses have the same electron-energy dependence (dE/dtγ2), so their effects on radio spectra (Equation 5.128) are indistinguishable.

5.5.2 The IC Spectrum of a Single Electron

What is the spectrum of the inverse-Compton radiation? Suppose the ambient radiation field in the observer’s frame contains only photons of frequency ν0, and consider scattering by a single electron moving with ultrarelativistic velocity +v along the x-axis. In the inertial frame moving with the electron, relativistic aberration causes most of the photons to approach nearly head-on. The relativistic Doppler Equation 5.142 gives the frequency ν0 in the electron frame of a photon approaching near the x-axis (θ1); it is

ν0=ν0[γ(1+βcosθ)]ν0[γ(1+β)]. (5.155)

In the electron frame, Thomson scattering produces radiation with the same frequency as the incident radiation: the scattered photons have ν=ν0. In the observer’s frame, relativistic aberration beams the scattered photons in the direction of the electron’s motion, and the frequency ν of radiation scattered nearly along the +x-direction (θ0) is given by the relativistic Doppler formula:

ν=ν[γ(1+βcosθ)]ν[γ(1+β)]ν0[γ(1+β)]2. (5.156)

In the ultrarelativistic limit β1,

νν04γ2. (5.157)

This is the maximum frequency of the upscattered radiation in the observer’s frame.

Oblique collisions (θ>0) result in lower frequencies ν. For an isotropic radiation field in the observer’s frame, the average energy E of scattered photons equals the average scattered power PIC per electron divided by N˙IC, the number of photons scattered per second per electron. This rate is the scattered power divided by the photon energy in the observer’s frame, or

N˙IC=σTcUradhν0. (5.158)


E=hν=PICN˙IC=43σTcβ2γ2Urad(hν0σTcUrad) (5.159)

and the average frequency ν of upscattered photons is

νν0=43γ2. (5.160)

For example, isotropic radio photons at ν0=1 GHz, IC scattered by electrons having γ=104, will be upscattered to an average frequency


corresponding to X-ray radiation. The principal astronomical effect of inverse Compton scattering is to drain energy from cosmic-ray electrons that produce radio radiation and use it to produce X-ray radiation instead.

Because the maximum frequency (Equation 5.157) is only three times the average frequency (Equation 5.160), the IC spectrum must be sharply peaked near the average frequency. The detailed Compton-scattering spectrum resulting from an isotropic single-frequency radiation field has been calculated (Blumenthal and Gould [13]; see also Pacholczyk [77]). It is indeed sharply peaked just below the maximum ν/ν0=4γ2, as shown in Figure 5.16.

Figure 5.16: The inverse-Compton spectrum of electrons with energy γ irradiated by photons of frequency ν0. The log-log plot of power per logarithmic frequency range (right) indicates more clearly just how peaked the spectrum is.

This spectrum is even more peaked than the synchrotron spectrum of monoenergetic electrons. Therefore it is not necessary to use the detailed Compton-scattering spectrum of monoenergetic electrons to calculate the inverse-Compton spectrum of an astrophysical source containing a power-law distribution of relativistic electrons. If the electron-energy distribution is n(E)E-δ, the inverse-Compton spectrum will also be a power law with spectral index

α=δ-12, (5.161)

which is the same spectral index that Equation 5.79 gives for synchrotron radiation emitted by the same power-law distribution of electron energies.

5.5.3 Synchrotron Self-Compton Radiation

Synchrotron self-Compton radiation results from inverse-Compton scattering of synchrotron radiation by the same relativistic electrons that produced the synchrotron radiation. Equation 5.154,

PICPsyn=UradUB, (5.162)

implies that multiplying the density of relativistic electrons by some factor f multiplies both the synchrotron power and its contribution to Urad by f, so the synchrotron self-Compton power is proportional to f2.

The self-Compton radiation also contributes to Urad and leads to significant second-order scattering as the synchrotron self-Compton contribution to Urad approaches the synchrotron contribution in compact sources. This runaway positive feedback is a very sensitive function of the source brightness temperature, so inverse-Compton losses very strongly cool the relativistic electrons if the source brightness temperature exceeds Tb1012 K in the rest frame of the source. Radio sources with brightness temperatures significantly higher than

Tmax1012K (5.163)

in the observer’s frame are either Doppler boosted or not incoherent synchrotron sources (e.g., pulsars are coherent radio sources). The active galaxy Markarian 501 emits strong synchrotron self-Compton radiation and the radio emission approaches this rest-frame brightness limit for incoherent synchrotron radiation. The synchrotron and synchrotron self-Compton spectra of Mrk 501 are shown in Figure 5.17.

Figure 5.17: The synchrotron (peaking near 1019 Hz) and synchrotron self-Compton (peaking near 1027 Hz) spectra of Mrk 501 [60]. The thin curve shows the best-fit synchrotron self-Compton model, the thick points represent the X-ray data, and the γ-ray data are plotted as points with error bars. The ordinate νFν on this plot is proportional to flux density per logarithmic frequency range, so the relative heights of the two peaks indicate their relative contributions to Urad.

5.6 Extragalactic Radio Sources

5.6.1 Relativistic Bulk Motion

The results above apply only to radio-emitting plasmas that are not moving relativistically with respect to the observer. Bright radio-source components (discrete regions of enhanced brightness) are often seen to move with apparent transverse velocities exceeding the speed of light. This illusion of superluminal velocities can occur if the components are moving obliquely toward the observer with relativistic speeds, as shown in Figure 5.18.

Figure 5.18: A source moving with speed v<c at an angle θ<π/2 from the line of sight may appear to be moving faster than c in projection onto the sky because the light travel time is reduced by vtcosθ/c during time t.

Suppose the radio-emitting component is moving toward the observer with constant speed v=βc at an angle θ from the line of sight. Consider two “events” in the moving component, the first occurring a distance r from the observer at time t=0, and the second at time t. Radiation from the first and second events will be received at times

t1=r/candt2=r-vtcosθc+t, (5.164)

respectively. The difference between these times is

t2-t1=t[1-(vcosθ)/c]. (5.165)

The apparent transverse velocity of the moving component is the actual transverse distance covered in time t divided by the apparent time interval (t2-t1):

v(apparent)=vtsinθt2-t1=vtsinθt[1-(vcosθ)/c], (5.166)
β(apparent)=βsinθ1-βcosθ. (5.167)

For every speed β there is an angle θm that maximizes β(apparent). That angle satisfies

β(apparent)θ=0=(1-βcosθm)βcosθm-(βsinθm)2(1-βcosθm)2, (5.168)
βcosθm-β2cos2θm-β2sin2β=βcosθm-β2=0. (5.169)


cosθm=β (5.170)


sinθm=(1-cos2θm)1/2=(1-β2)1/2=γ-1. (5.171)

Inserting cosθ=β and sinθ=γ-1 into Equation 5.167 for β(apparent) yields the highest apparent transverse speed of a source whose actual speed is β:

max[β(apparent)]=β(1-β2)1/21-β2=βγ. (5.172)
Figure 5.19: Apparently superluminal motion of the radio components in the quasar 3C 279 [81].

Figure 5.19 shows five successive high-resolution radio images of the quasar 3C 279. The bright component at the left is taken to be the fixed radio core, and the bright spot at the right appears to have moved 25 light years across the plane of the sky between 1991 and 1998, for an apparently superluminal motion of 25 light years in 7 years: β(apparent)3.6. What is the minimum component speed β consistent with these images? What is the corresponding angle θm between that motion and the line of sight? From the results above,

βγ=β(1-β2)1/2β(apparent), (5.173)


β [β2(apparent)1+β2(apparent)]1/2, (5.174)
β [(25/7)21+(25/7)2]1/20.96. (5.175)

The corresponding θm is given by

cosθm=β0.96, (5.176)
θm0.28rad16. (5.177)

The relativistic Doppler formula (5.142) relates the frequency ν emitted in the component frame to the observed frequency ν. Note that we have replaced θ by (π-θ) radians in the current analysis by calling it the angle between the line of sight and the velocity of an approaching component, so

ν=νγ(1-βcosθ), (5.178)

where θ=0 now corresponds to a radio component moving directly toward the observer. The quantity

δ[γ(1-βcosθ)]-1=νν (5.179)

is called the Doppler factor. If θ=π/2, there is a transverse Doppler shift

δ=νν=γ-1. (5.180)

The transverse Doppler shift has no nonrelativistic counterpart because the source has no component of velocity parallel to the line of sight; it exists only because moving clocks run slower by a factor γ. The Doppler factors associated with a given source speed β range from

δ(2γ)-1 (5.181)

for directly receding (θ=π rad) sources to

δ2γ (5.182)

for directly approaching (θ=0) sources. For example, the ratio ν/ν of the observed frequency to the emitted frequency in 3C 279 (β=0.96, cosθ=β) is

νν =[γ(1-βcosθ)]-1=[γ(1-β2)]-1=γ

The observed flux density S of a relativistically moving component emitting isotropically in its rest frame depends critically on its Doppler factor δ. The exact amount of Doppler boosting caused by relativistic beaming is somewhat model dependent [100] but probably lies in the range

δ2+α<SS0<δ3+α, (5.183)

where S0 would be the observed flux density if the source were stationary and α=-dlogS/dlogν is the (negative) spectral index. If γ5, then 0.1<δ<10 depending on the angle θ. Relativistic components approaching at angles θ<γ-1 can easily be boosted by factors >103 compared with components moving in the sky plane or away from us. For example, the approaching jet of 3C 279 (δ=ν/ν3.6 and α0.7) is Doppler boosted by

δ2+α<SS0 <δ3+α,
3.62.7<SS0 <3.63.7,
32<SS0 <114.

The receding counterjet is dimmed by a comparable factor, so the jet/counterjet observed flux-density ratio of 3C 279 is probably >103.

Doppler boosting strongly favors approaching relativistic jets and components and discriminates against those with θ>γ-1 in flux-limited samples of compact radio sources. Radio quasars aren’t isotropic candles spread throughout the universe; they are beamed flashlights. The brightest aren’t always the most luminous; they are just pointing in our direction. For every flashlight we see, there are many others in the same volume of space that we don’t see simply because they are not pointing at us.

Figure 5.20: Inner jets of the radio galaxy 3C 31 [64]. If the jets are close to the plane of the sky and decelerate from relativistic speeds as they recede from the core, only the inner portions of the jets are Doppler dimmed. Image credit: NRAO/AUI/NSF. The simulation at shows how 3C 31 would appear at different angles θ between the jet and the line of sight. The best fit to the data occurs at θ=52. Image credit: NRAO/AUI/NSF. Investigators: Alan Bridle & Robert Laing.
Figure 5.21: This VLA image of the radio-loud quasar 3C 175 shows the core, an apparently one-sided jet, and two radio lobes with hot spots of comparable flux densities. The jet is intrinsically two sided but relativistic, so Doppler boosting brightens the approaching jet and dims the receding jet. Both lobes and their hot spots are comparably bright and thus are not moving relativistically. Image credit: NRAO/AUI/NSF Investigators: Alan Bridle, David Hough, Colin Lonsdale, Jack Burns, & Robert Laing.

The fact that the two lobes of very extended radio sources such as Cyg A (Figure 5.12) and 3C 348 (Figure 8.14) typically have flux ratios <2 indicates that the lobes are moving outward with speeds vc. The radio jets feeding nearly equal lobes often appear quite unequal, with one jet being very strong and the other undetectable. The jets of very luminous sources often terminate in bright hotspots in the lobes.

Because the jets feed the lobes, the lobe symmetry suggests that the jets are intrinsically similar, but the approaching jet is boosted while the receding counterjet is dimmed. Another feature of many radio jets is gaps near the core. If jets start out relativistic at the core and are inclined by more than θγ-1 from the line of sight, both will be Doppler dimmed. If they proceed with constant θ but gradually decelerate as they move away from the core, one or both may become visible beyond the point where γθ-1.

Extragalactic radio sources with jets and lobes can be divided into two morphological classes: (1) those, like 3C 31 (Figure 5.20), that appear to fade away at large distances from the center and (2) sources with edge-brightened lobes, like 3C 175 (Figure 5.21). Such sources are called FR I and FR II sources, respectively, after Fanaroff and Riley [38], who first made such classifications and noted that FR I sources are usually less luminous than FR II sources, with the dividing line being Lν1024WHz-1 at 1.4 GHz. FR I sources generally have lower equipartition energy densities and hence lower equipartition pressures. The jets of FR I sources are fairly symmetric at distances greater than several kpc from the cores, suggesting that the low-luminosity jets are quickly decelerated to nonrelativistic speeds. The low-energy FR I jets are easily influenced by ambient matter. Low-luminosity galaxies moving through the intracluster medium of a cluster of galaxies frequently have bent head-tail radio morphologies similar to the wake of a moving boat.

5.6.2 Unified Models

The combination of orientation-dependent beaming and obscuration by dust has led to various unified models (Figure 5.22) of active galactic nuclei (AGN).

Figure 5.22: This cartoon shows the main features of a “unified model” for active galactic nuclei [109]. Image Credit: Robert Findlay.

These models attribute some or all of the differences between observationally different objects to the inclinations of their jets relative to the line of sight. If the inclination is small, the base of the approaching jet will be strongly Doppler boosted, and the compact optical broad-line region and inner accretion disk will not be obscured by the larger dusty accretion torus lying in a plane normal to the jet. The observed radio emission will be dominated by a one-sided jet that may be variable in intensity and apparently superluminal. Thermal emission from the inner parts of the accretion disk may be visible as a big blue bump in the optical/UV spectrum, and Doppler-broadened emission lines from the small (<1 pc) broad-line region will not be obscured. If the optical AGN emission is much brighter than the starlight of the host galaxy, the object will be called a quasi-stellar object (QSO), but otherwise a Seyfert I galaxy. In extreme cases, optical synchrotron emission may dominate the big blue bump and emission lines. Objects with lineless power-law optical spectra are often called BL Lac objects after their prototype BL Lacertae, which was originally thought to be a Galactic star (hence the constellation name). If the inclination angle is larger than about 45 degrees, the optical core may be obscured by the dusty torus and highly relativistic radio jets may be Doppler dimmed, and we will see either a double-lobed radio galaxy or a Seyfert II galaxy (a Seyfert galaxy with only the narrow emission lines directly visible). The ongoing debate over unified models is not about whether relativistic beaming and dust obscuration affect the appearance of AGNs, but how much.

5.6.3 Radio Emission from Normal Galaxies

The radio emission from a normal galaxy is not powered by an AGN. The continuum radio emission from normal galaxies is dominated by a combination of

  1. 1.

    free–free emission from Hii regions ionized by massive (M>15M) main-sequence stars, and

  2. 2.

    synchrotron radiation from cosmic-ray electrons, most of which were accelerated in the supernova remnants (SNRs) of massive (M>8M) stars.

Stars more massive than 8M have main-sequence lifetimes τ<3×107 yr, much less than the >1010 yr age of our Galaxy. Also, the synchrotron lifetimes of cosmic-ray electrons in the typical interstellar magnetic field are τ<108 yr. Thus the current radio continuum emission from normal galaxies is an extinction-free tracer of recent star formation, unconfused by emission from older stars. The radio emission is roughly coextensive with the locations of star formation, spanning the stellar disks of most spiral galaxies.

Massive stars form by gravitational collapse in dusty molecular clouds. The dust absorbs most of their visible and ultraviolet radiation, is heated to temperatures of several tens of K, and reemits the input energy at far-infrared (FIR) wavelengths λ100μm. The molecular clouds are not opaque at FIR wavelengths, so FIR luminosity is a good quantitative measure of the current star-formation rate. Remarkably, the radio luminosities of normal galaxies are very tightly correlated with their FIR luminosities.

Figure 5.23: The FIR/radio (1.4 GHz) correlation for normal galaxies [27].

The physical origin of this famous FIR/radio correlation (Figure 5.23) is poorly understood, particularly at low frequencies where most of the radio emission is synchrotron radiation. It is not surprising that the FIR and free–free radio fluxes would be correlated because both are roughly proportional to the ionizing luminosities of massive young stars. However, free–free emission accounts for only a small fraction of the total radio luminosity at low frequencies ν30 GHz. The FIR/radio spectrum of the nearby starburst galaxy M82 (Figure 2.24) is typical.

At ν1 GHz, about 90% of the radio flux is produced by synchrotron radiation, yet the FIR/radio luminosity ratio is confined to a very narrow range. If the star-formation rate (SFR) in a galaxy is fairly constant on timescales longer than 3×107 yr, then the number of young SNRs would be proportional to the present number of massive stars, so it is plausible that the current production rate of cosmic-ray electrons is proportional to the current star-formation rate. However, most of the synchrotron radiation from normal galaxies does not originate in the SNRs themselves, but rather comes from cosmic-ray electrons that have diffused into the interstellar medium (ISM). The power radiated by each electron is proportional to the magnetic energy density UB=B2/(8π) in the ISM. The equipartition fields in normal galaxies range from very low values to B5μG in a typical spiral galaxy like ours, to B100μG in M82 and up to B1000μG in a particularly compact and luminous starburst galaxy such as Arp 220. Thus the power radiated by each cosmic-ray electron must vary by up to several orders of magnitude from one galaxy to another, yet all of these galaxies obey the same FIR/radio correlation.

The calorimeter model [112] was devised to explain how the FIR/radio ratio could be independent of UB. The total radio energy radiated by each electron might be independent of UB if the lifetime of the electron is proportional to UB-1. Thus, a cosmic-ray electron in a strong magnetic field radiates a high power for a short time, while one in a weak magnetic field radiates a lower power for a proportionately longer time. For a given production rate of cosmic-ray electrons, the average synchrotron power will then be independent of UB. The calorimeter model works well to explain the FIR/radio correlation so long as the fraction of the electron energy going into synchrotron radiation is about the same in all normal galaxies. However, there are many other energy-loss channels. One is inverse-Compton scattering off the cosmic microwave background, starlight, FIR radiation, etc. Another is diffusion out of the magnetic field of a galaxy—some electrons escape silently into intergalactic space. Electrons may also lose energy by colliding with atoms in the ISM.

Galaxy–galaxy collisions can trigger intense starbursts (star-formation episodes so intense that they will deplete the available ISM on timescales much shorter than 1010 yr) within several hundred parsecs of the centers of galaxies and produce compact central sources. The radiation energy density in a compact starburst galaxy such as Arp 220 can reach Urad10-6ergcm-3, yet Arp 220 still obeys the FIR/radio correlation. The fact that inverse-Compton cooling doesn’t seriously deplete the population of synchrotron-emitting electrons sets a lower limit to the interstellar magnetic energy density UB=B2/(8π)Urad via Equation 5.154. This limit is B1000μG in Arp 220 [28].

Despite its uncertain theoretical basis, the FIR/radio correlation makes radio continuum emission from normal galaxies a very useful, quantitative, and extinction-free indicator of the rate at which massive stars are being formed. The rate (measured in units of solar masses per year) at which stars with masses M>5M are formed in a galaxy can be estimated from the thermal (free–free) and nonthermal (synchrotron) spectral luminosities by the following equations [24]:

(LTWHz-1)5.5×1020(νGHz)-0.1[SFR(M>5M)Myr-1], (5.184)
(LNTWHz-1)5.3×1021(νGHz)-0.8[SFR(M>5M)Myr-1]. (5.185)

5.6.4 Extragalactic Radio-Source Populations and Cosmological Evolution

Surveys of discrete radio sources have been made over large areas of the sky and at many frequencies ranging from 38 MHz to 857 GHz. The most extensive sky survey is the NRAO VLA Sky Survey (NVSS) [29], which covers the whole sky north of declination (latitude on the celestial sphere) δ=-40 and detected nearly 2×106 sources stronger than S2.3 mJy at 1.4 GHz. Extremely sensitive sky surveys covering much smaller areas have reached flux densities S5μJy. Sources detected by blind surveys covering representative areas of sky give us an unbiased statistical sample of the radio-source population.

The distribution of discrete sources on the sky is extremely isotropic, as shown in Figure 5.24. This isotropy indicates that nearly all radio sources in a flux-limited sample are extragalactic—the center of our Galaxy is barely visible as the curved band at the left of the upper panel. A similar plot of the brightest galaxies selected at optical or near-infrared wavelengths is much clumpier than the radio plots because galaxies cluster on scales 10 Mpc in size. The reason for this difference is that the strongest extragalactic radio sources are much farther away than the optically brightest galaxies. Radio galaxies are at least as clustered as optical galaxies, but the average distance between radio galaxies is so much greater than 10 Mpc that their clustering can be detected only by sensitive statistical tests. Indeed, the distribution of radio sources on the sky is so uniform that the small (<1%) dipole anisotropy in source density caused by Doppler boosting from the Earth’s motion relative to the frame defined by distant galaxies has been detected. The velocity of the Earth deduced from this anisotropy is consistent with the motion deduced from the corresponding anisotropy in the cosmic microwave background radiation produced by the hot big bang [11].

Figure 5.24: These two equal-area plots show the sky distribution of discrete sources stronger than S=100 mJy at 1.4 GHz in the 82% of the sky north of δ=-40 (upper panel) and the sky distribution of sources stronger than S=2.5 mJy at 1.4 GHz in the 1.70% of the sky north of δ=+75 (lower panel).

Only a small fraction (1%) of radio sources in a flux-limited sample are closer than about 100 Mpc. By identifying those sources with bright galaxies and determining their Hubble distances, we can determine the space density of nearby radio sources as a function of radio spectral luminosity; this is called the local luminosity function. The local luminosity function can be further refined by specifying separately the space densities of radio sources powered by AGN and those powered by star-forming galaxies containing Hii regions, SNRs, etc. (Figure 5.25).

Figure 5.25: The 1.4 GHz local luminosity functions of normal star-forming galaxies (filled symbols) and of AGN (open symbols) [30].

In a given volume of space, radio sources in star-forming galaxies outnumber radio galaxies containing AGN by an order of magnitude. However, the rarer AGN produce all of the most luminous sources, so they account for slightly over half of all radio emission produced by discrete sources.

If we assume that the comoving space density of radio sources in the expanding universe is independent of time, we can use the local luminosity function to calculate the total number of radio sources per steradian of sky as a function of flux density. The resulting source counts are usually tabulated in differential form: n(S)dS is the number of sources per sr with flux densities between S and S+dS. In a static Euclidean universe, the flux density of any source at distance r is proportional to r-2, and the volume enclosed by a sphere is proportional to r3, so the integral number N(>S) of sources stronger than any given flux density S should be proportional to S-3/2 and the differential number n(S) per unit flux density should be n(S)S-5/2. Plotting the normalized differential source count n(S)×S5/2 as a function of S should yield a horizontal line in a static Euclidean universe.

The actual plot for sources selected at 1.4 GHz is shown in Figure 5.26. The source counts are not consistent with a static Euclidean universe, so most radio sources cannot be “local” extragalactic sources. In an expanding universe with a constant comoving source density, distant sources will be Doppler dimmed and the normalized source counts should decline monotonically at low flux densities. This is not the case either; the normalized counts have a clear maximum near S500 mJy. This peak indicates that radio sources must be evolving on cosmological timescales; that is, their comoving space density varies with time and was higher at some time in the past. The discovery of radio-source evolution was used as evidence against the steady-state model of the universe [99], years before the discovery of the cosmic microwave background radiation decisively confirmed the hot big-bang model.

Detailed models consistent with the local luminosity function, radio source counts, and redshift distributions of radio sources identified with galaxies and quasars have been constructed to measure the amount of evolution. The results are actually quite simple: cosmological evolution is so strong that most radio sources in flux-limited samples have redshifts near the median z0.8. To radio astronomers, the universe looks like a nearly hollow spherical shell centered on the Earth. This observation does not conflict with the Copernican principle, which states that the Earth is not in a special position at the center of the universe; it requires only that the universe evolve with time. Most radio sources seen today have distances of 5 to 10×109 light years, and their dominance reflects the higher AGN and star-formation activity of 5 to 10 Gyr ago. For sources in a thin shell, there is little correlation between flux density and average distance; rather, flux density is more closely correlated with absolute luminosity as shown by labels on the lower and upper abscissae in Figure 5.26. Consequently, radio-loud AGNs in radio galaxies and quasars (dashed curve in Figure 5.26) account for most sources stronger than S0.1 mJy at 1.4 GHz, and the numerous but less luminous star-forming galaxies (dotted curve in Figure 5.26) dominate the microJy radio-source population.

Figure 5.26: The 1.4 GHz Euclidean-normalized luminosity function ϕ and source count S5/2n(S) [31] are consistent with strong (10×) luminosity evolution of all extragalactic radio sources. The dashed and dotted curves indicate the contributions of AGNs and star-forming galaxies, respectively, to the total source count.