A source moving with speed $v < c$ at an angle $\theta < \pi /2$ 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 $v t \cos\theta / c$ in time $t$.

Suppose the radio-emitting component is moving toward the observer with constant speed $v = \beta c$ at an angle $\theta$ 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

$$t_1 = r/c,$$ and

$$t_2 = {r - vt\cos\theta \over c} + t~,$$ respectively. The apparent transverse velocity of the moving component is the actual transverse distance covered in time $t$ divided by the apparent time interval $(t_2 - t_1)$:

$$v_\bot{\rm (apparent)} = {v t \sin \theta \over t_2 - t_1}$$ $$v_\bot{\rm (apparent)} = { v t \sin \theta \over t (1 - v \cos \theta / c) } $$ $$\bbox[border:3px blue solid,7pt]{\beta_\bot{\rm (apparent)} = {\beta \sin \theta \over 1 - \beta \cos\theta}}\rlap{\quad \rm {(5F1)}}$$

For every speed $\beta$ there is an angle $\theta_{\rm m}$ that maximizes $\beta_\bot{\rm (apparent)}$. That angle satisfies

$${\partial \beta_\bot{\rm (apparent)} \over \partial \theta} = 0 = { (1 - \beta\cos\theta_{\rm m}) \beta \cos \theta_{\rm m} - (\beta \sin\theta_{\rm m}) ^2 \over (1 - \beta \cos \theta_{\rm m})^2 }~.$$

and

Inserting $\cos \theta = \beta$ and $\sin \theta = \gamma^{-1}$ into Equation 5F1 for $\beta_\bot{\rm (apparent)}$ yields the highest apparent transverse speed of a source whose actual speed is $\beta$:



$$\bbox[border:3px blue solid,7pt]{{\rm max}[\beta_\bot{\rm (apparent)}] = {\beta (1 - \beta^2)^{1/2} \over 1 - \beta^2} = \beta \gamma}\rlap{\quad \rm {(5F4)}}$$

$$\beta \cos \theta_{\rm m} - \beta^2 \cos^2 \theta_{\rm m} - \beta^2 \sin^2 \beta = 0$$ $$\beta \cos \theta_{\rm m} - \beta^2 = 0$$ Thus$$\bbox[border:3px blue solid,7pt]{\cos \theta_{\rm m} = \beta}\rlap{\quad \rm {(5F2)}}$$ $$\bbox[border:3px blue solid,7pt]{\sin \theta_{\rm m} = (1 - \cos^2 \theta_{\rm m})^{1/2} = (1 - \beta^2)^{1/2} = \gamma^{-1}}\rlap{\quad \rm {(5F3)}}$$

Example: A sequence of VLBI (Very Long Baseline Interfereometry) observations of the radio jet near the core of the quasar 3C 279 shows an apparently superluminal motion of 25 light years in 7 years: $\beta_\bot{\rm (apparent)} \approx 3.6$.



Apparently superluminal motion of the radio components in 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 on the plane of the sky between 1991 and 1998. Image credit





What is the minimum component speed $\beta$ consistent with these observations? What is the corresponding angle $\theta_{\rm m}$ between that motion and the line of sight?

$$\beta\gamma = {\beta \over (1 - \beta^2)^{1/2}} \geq \beta_\bot{\rm (apparent)}$$ so $$\beta \geq \biggl[{\beta^2_\bot{\rm (apparent)} \over 1 + \beta^2_\bot{\rm (apparent)}}\biggr]^{1/2}$$ $$ \beta \geq \biggl[{(25/7)^2 \over 1 + (25/7)^2}\biggr]^{1/2} \approx 0.96$$ The corresponding $\theta_{\rm m}$ is given by $$\cos \theta_{\rm m} = \beta \approx 0.96$$ $$ \theta_{\rm m} \approx 0.28 {\rm ~rad} \approx 16^\circ ~.$$



The relativistic Doppler formula (Eq. 5E2) relates the frequency $

u'$ emitted in the component frame to the observed frequency $

u$. Note that we have replaced $\theta$ by ($\pi - \theta$) radians in the current analysis by calling it the angle between the line-of-sight and the velocity of an approaching component, so

$$

u = {

u' \over \gamma(1 - \beta \cos\theta)}~,$$ where $\theta = 0$ now corresponds to a radio component moving directly toward the observer. The quantity

$$\delta \equiv [\gamma(1 - \beta \cos \theta)]^{-1} = {

u \over

u'}$$ is called the Doppler factor. If $\theta = \pi /2$, there is a transverse Doppler shift

$$\bbox[border:3px blue solid,7pt]{ \delta = {

u \over

u'} = \gamma^{-1}}\rlap{\quad \rm {(5F5)}}$$ 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 $\gamma$. The Doppler factors associated with a given source speed $\beta$ range from

$$\delta \geq (2 \gamma)^{-1}$$ for directly receding ($\theta = \pi$ rad) sources to

$$\delta \leq 2 \gamma$$ for directly approaching ($\theta = 0$) sources.



Example: What is the ratio $

u /

u'$ of the observed frequency to the emitted frequency in the example above ($\beta = 0.96$, $\cos \theta = \beta$)?

$$ {

u \over

u'} = [\gamma (1 - \beta \cos\theta)]^{-1} =

[\gamma (1 - \beta^2)]^{-1} = \gamma$$ $$ {

u \over

u'} = (1 - \beta^2)^{-1/2} \approx 3.6$$





The observed flux density $S$ of a relativistically moving component emitting isotropically in its rest frame depends critically on its Doppler factor $\delta$. The exact amount of Doppler boosting is somewhat model-dependent (Scheuer, P. A. G., & Readhead, A. C. S. 1979, Nature, 277, 182) but probably lies in the range

$$\bbox[border:3px blue solid,7pt]{\delta^{2 + \alpha} < {S \over S_0} < \delta^{3 + \alpha}~,}\rlap{\quad \rm {(5F6)}}$$ where $S_0$ would be the observed flux density if the source were stationary and $\alpha = -d \log S / d \log

u$ is the (negative) spectral index. If $\gamma \sim 5$, then $0.1 < \delta < 10$ depending on the angle $\theta$. Relativistic components approaching at angles $\theta < \gamma^{-1}$ can easily be boosted by factors $ >10^3$ compared with components moving in the sky plane or away from us.



Doppler boosting strongly favors those relativistic jets and components approaching us and discriminates against those with $\theta > \gamma^{-1}$ 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.



Example: What is the amount of Doppler boosting for the approaching jet of 3C 279 if $\delta =

u /

u' \approx 3.6$ and $\alpha \approx 0.7$?

$$\delta^{2 + \alpha} < {S \over S_0} < \delta^{3 + \alpha}$$ $$3.6^{2.7} < {S \over S_0} < 3.6^{3.7}$$ $$ 32 < {S \over S_0} < 114 $$ The receding counterjet is dimmed by a comparable factor, so the jet/counterjet flux-density flux-density ratio is probably $> 10^3$. This large ratio explains why so many bright jets have no detectable counterjets.





The fact that the two lobes of very extended radio sources like Cyg A typically have flux ratios $< 2$ indicates that the lobes are moving outward with speeds $v \ll c$. 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.





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





Since 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 a gap near the core. If such jets start out relativistic and inclined by more than $\theta \sim \gamma^{-1}$ from the line-of-sight, both will be Doppler dimmed. If they proceed with constant $\theta$ but gradually decelerate as they move away from the core, one or both may become visible beyond the point where $\gamma \sim \theta^{-1}$.





Radio (red) and optical (blue) images of the radio galaxy 3C 31. 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 Here is a simulation of how 3C 31 would appear if observed over a range of angles $\theta$ between the jet and the line of sight. The best fit to the data occurs at $\theta = 52^\circ$.



Extragalactic radio sources with jets and lobes can be divided into two morphological classes: (1) those, like 3C 31, that appear to fade away at large distances from the center and (2) sources with edge-brightened lobes (e.g., 3C 175). Such sources are called FR I and FR II sources, respectively, after Fanaroff and Riley (1974, MNRAS, 167, 31P), who first made such classifications and noted that FR I sources are usually less luminous than FR II sources, with the dividing line being $L_

u \sim 10^{24}$ W Hz$^{-1}$ 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 ralaxies moving through the intracluster medium of a cluster of galaxies frequently have bent or head-tail radio morphologies similar to the wake of a moving boat.





The isolated red dot in this false-color high-resolution image of the radio source 3C 83.1B is the core located in the nucleus of the galaxy NGC 1265, which is moving through the intergalactic medium of the Perseus Cluster at a velocity of about 2000 km s$^{-1}$. Drag by the cluster gas bends the radio source, which would otherwise be straight. Image credit

Unified Models

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



This cartoon shows the main features of a "unified model" for AGN as reviewed by Urry, C. M., & Padovani, P. 1995, PASP, 107, 803.



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 (BLR) 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); 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 $45^\circ$, the optical core may be obscured by the dusty torus and highly relativistic radio jets may be Doppler dimmed, and we will see a 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.

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) free-free emission from HII regions ionized by massive ($M > 15 M_\odot$) main-sequence stars and

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

Stars more massive than $\sim 8 M_\odot$ have main-sequence lifetimes $\tau < 3 \times 10^7$ yr, much less than the age of our Galaxy, which is $> 10^{10}$ yr. Also, the synchrotron lifetimes of cosmic-ray electrons in the typical interstellar magnetic field is $\tau < 10^8$ yr. Thus the current radio continuum emission from normal galaxies is an extinction-free tracer of recent star formation, uncontaminated by older stars. The radio emission is roughly coextensive with the locations of star formation, spanning the stellar disks of most spiral galaxies. Sometimes galaxy-galaxy collisions trigger intense starbursts (star-formation episodes so intense that they will deplete the available ISM on time scales much shorter than $10^{10}$ yr) within several hundred parsecs of the centers of galaxies and produce compact central sources.



Contour plots of the 1.49 GHz continuum emission from a range of nearby galaxies (Condon, J. J. 1992, ARA&A, 30, 575). The horizontal bars are $\approx 3$ kpc long, and the contours are spaced by factors of $2^{1/2}$ in brightness.

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 re-emits the input energy at far-infrared (FIR) wavelengths $\lambda \sim 100\,\mu$m. 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.





The FIR/radio (1.4 GHz) correlation for normal galaxies (Condon, J. J.,

Anderson, M. L., & Helou, G. 1991, ApJ, 376, 95).



The physical origin of this famous FIR/radio correlation is not well 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, since 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 $

u \ll 30$ GHz. The FIR/radio spectrum of the nearby starburst galaxy M82 is typical:



The FIR/radio spectrum of M82 (Condon, J. J. 1992, ARA&A, 30, 575) is typical for normal galaxies.

At $

u \approx 1$ 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 $3 \times 10^7$ 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 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 $U_{\rm B} = B^2 / (8 \pi)$ in the ISM. The equipartition fields in normal galaxies range from very low values to $B \sim 5\,\mu$G in a typical spiral galaxy like ours, to $B \sim 100\,\mu$G in M82 and up to $B \sim 1000\,\mu$G 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 obey the same FIR/radio correlation.

The calorimeter model (Volk, H. J. 1989, A&A, 218, 67) was devised to explain how the FIR/radio ratio could be independent of $U_{\rm B}$. The total radio energy radiated by each electron might be independent of $U_{\rm B}$ if the lifetime of the electron is proportional to $U_{\rm B}^{-1}$. 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 low power for a proportionately longer time. For a given production rate of cosmic-ray electrons, the average power radiated will then be independent of $U_{\rm B}$. 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.

Despite its weak 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 $M > 5 M_\odot$ are formed in a galaxy can be estimated from the thermal (free-free) and nonthermal (synchrotron) spectral luminosities by the following equations (Condon, J. J. 1992, ARA&A, 30, 575):

$$\bbox[border:3px blue solid,7pt]{\biggl({ L_{\rm T} \over {\rm W~Hz}^{-1} } \biggr) \approx 5.5 \times 10^{20} \biggl( {

u \over {\rm GHz}} \biggr)^{-0.1} \biggl[ {SFR(M > 5 M_\odot) \over M_\odot {\rm ~yr}^{-1} } \biggr]}\rlap{\quad \rm {(5F7)}}$$ $$\bbox[border:3px blue solid,7pt]{\biggl({ L_{\rm NT} \over {\rm W~Hz}^{-1} } \biggr) \approx 5.3 \times 10^{21} \biggl( {

u \over {\rm GHz}} \biggr)^{-0.8} \biggl[ {SFR(M > 5 M_\odot) \over M_\odot {\rm ~yr}^{-1} } \biggr]}\rlap{\quad \rm {(5F8)}}$$





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 20 GHz. The most extensive sky survey is the NRAO VLA Sky Survey (NVSS), which covered the whole sky north of declination (latitude on the celestial sphere) $\delta = -40^\circ$ and detected nearly $2 \times 10^6$ sources stronger than $S = 2.5$ mJy at 1.4 GHz. Extremely sensitive sky surveys covering much smaller areas have reached flux densities $S \sim 10\,\mu$Jy. 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.



Equal-area plot showing the sky distribution of discrete sources stronger than 100 mJy at 1.4 GHz.

The distribution of sources stronger than 2.5 mJy at 1.4 GHz within $15^\circ$ of the north celestial pole.

This isotropy indicates that nearly all 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 first plot. 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 $\sim 10$ Mpc in size. The reason for this difference is that the strongest extragalactic 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 much greater than 10 Mpc, so 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 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 (Blake, C., & Wall, J. V. 2002, Nature, 416, 150).

Only a small fraction ($\sim 1$%) of radio sources in a flux-limited sample are nearer than about 100 Mpc. By identifying those sources with nearby galaxies and determining the Hubble distances of those galaxies, we can determine the space density of radio sources as a function of radio spectral luminosity; this is called the luminosity function. The radio luminosity function can be further refined if we specify independently the space densities of radio sources powered AGN and those powered by star-forming galaxies containing HII regions, SNRs, etc.

The 1.4 GHz local luminosity functions of normal star-forming galaxies (filled symbols) and of AGN (open symbols).





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: $n(S) d S$ is the number of sources per sr with flux densities between $S$ and $S + d S$. 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 $r^3$ , so the number of sources stronger than any given flux density should be proportional to $S^{-3/2}$ and the differential number $n(S)$ should be $n(S) \propto S^{-5/2}$. By plotting the normalized source counts $n(S) \times S^{5/2}$ as a function of $S$ , we would get a horizontal line if we were in a static Euclidean universe. The actual plot for sources selected at 1.4 GHz is shown below.





The 1.4 GHz Euclidean-normalized luminosity function $\phi$ and source counts $S^{5/2}n(S)$ are consistent with strong ($\sim 10 \times$) luminosity evolution of all radio sources.



Clearly, the source counts do not reflect 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 much fainter, 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 $S \sim 500$ mJy. This excess of sources indicates that radio sources must be evolving on cosmological time scales; that is, their comoving space density varies with time. The discovery of radio-source evolution was used as evidence against the steady-state model of the universe 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 $\langle z \rangle\sim 0.8$, where

$$\bbox[border:3px blue solid,7pt]{(1 + z) \equiv {\lambda_{\rm observed} \over \lambda_{\rm emitted}}}\rlap{\quad \rm {(5F9)}}$$ is the factor by which lengths growing with the universe (e.g., distances between widely-spaced galaxies, wavelengths of photons) have grown. The universe looks like a nearly hollow shell to radio astronomers. Most radio sources seen today have distances of 5 to $10 \times 10^9$ 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 almost no correlation between flux density and average distance; rather, flux density is more closely correlated with absolute luminosity. Consequently, luminous radio galaxies and quasars account for most radio sources stronger than $S \sim 0.1$ mJy at 1.4 GHz, and the numerous but less luminous star-forming galaxies dominate the microJy radio-source population.



The evolution of the star-formation rate density (the number of solar masses per year of stars produced per cubic Megaparsec) was an order of magnitude higher in the past ($z > 1$) than it is today ($z = 0$). The 1.4 GHz local luminosity function of star-forming galaxies and a simple model for their cosmological evolution yields the black point and solid curve, respectively.



