ARlogo Annu. Rev. Astron. Astrophys. 1984. 22: 319-58
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6.1 Velocity Estimates

Without direct velocity indicators such as emission lines from material in the jet flows, estimates of the average flow velocities vj are indirect and sensitive to the initial assumptions. Most methods assume jet properties to be stationary in time and estimate vj from observables using one or more of the following arguments.

6.1.1   ENERGY FLUX   If synchrotron losses from a lobe of luminosity Llobe are continuously replenished by the energy influx from a jet at an efficiency epsilon, the energy flux supplied to the lobe must be Lj = Llobe / epsilon. The flux Lj is related to other jet parameters by

Equation 1

where Aj, vj, rhoj, gammaj, and hj are the cross-sectional area, flow velocity, density, Lorentz factor, and enthalpy per unit rest mass of the jet at some point along it, respectively. (The transverse structure of the jet is usually ignored when estimating vj.)

6.1.2   MOMENTUM FLUX   For jets which terminate at hot spots, the thrust Tj / h in the rest frame of the hot spot must balance ph AH, where p h and Ah are the minimum pressure and cross-sectional area of the hot spot estimated from its synchrotron parameters. The thrust Tj = Aj rhoj vj2 gammaj2 in the galaxy or QSR frame can be calculated from this by assuming the dynamics of the interaction between the hot spot and the ambient density rhoe to relate vj to the velocity v h of advance of the hot spot. Estimates for Tj can also be made for C-shaped jets in "head-tail" sources if these jets are bent by the ram pressure of an intergalactic density rhoe through which their parent galaxy moves at velocity vg (10, 128). If the radius of curvature of the C-structure is Rc and the scale over which the ram pressure is transmitted to the jet is H, momentum balance requires that Tj = rhoe vg2 Aj Rc / H. The parameter H is the radius Rj of the jet in the "naked" jet-bending model of (10), but it is a scale associated with the ISM of the parent galaxy in the "shielded" model of (128).

6.1.3   MASS FLUX   The mass flux dm / dt = Aj rhoj vj gammaj down the jet must meet either reasonable constraints on the rate of ejection from the "central engine" or constraints from depolarization data on the total mass injected into the lobes over the lifetime of the source. (The latter constraints are generally less stringent, but they could be tightened by high-resolution polarimetry of lobes at frequencies below 1 GHz.)

6.1.4   JET EXPANSION   The Mach number Mj of a free jet where it detached from its confinement (Section 4.1) can be estimated from its synchrotron expansion rate via dPhi / dTheta = (2 / Mj)sec(i), where i is the (assumed) angle between the jet axis and the plane of the sky. Then vj = Mj sqrt[Gamma pj / rhoj], where Gamma is the ratio of principal specific heats in the jet, and a lower limit to pj is obtained from the synchrotron parameters. If the jet is actually confined, Mj is overestimated. If the weak radio galaxy jets are alternately free and confined (Section 4.1.1), Mj is best indicated by the rapid expansions at z approx 1 - 10 kpc, which imply that Mj approx 10 there.

6.1.5   ELIMINATING THE JET DENSITY   Usage of the techniques of Sections 6.1.1 - 6.1.4 alone requires estimates of rhoj, generally from centimeter-wavelength Faraday depolarization data that are hard to obtain and to interpret (87). High signal-to-noise is needed to reduce Ricean bias in the polarized signal (183, 270). The configuration of Bj is unclear (Section 3.2), particularly the scale distribution of its reversals, which may "hide" thermal gas. Some jets are surrounded by emission-line filaments (Section 5.1.2) and magnetoionic media with clumping scales of ~ 1 - 5 kpc (33, 183; Table 1, ref. L2). Differential Faraday rotation across the radio beam by such media may decouple the observed depolarization from rhoj, so even setting limits to rhoj from low-resolution depolarization data without mapping the rotation measure gradients is hazardous, especially at low frequencies. The methods in Section 6.1.1 - 6.1.4 all permit vj -> c if rhoj -> 0, but Table 4 shows how vj can be constrained using combinations of these methods to eliminate rhoj for "cold" (hj << rhoj vj2) jets. As hj / rhoj vj2 = Gamma / Mj2(Gamma - 1) from the gas laws and Section 6.1.4, this is a good assumption for jets with Mj geq 4. These combinations also eliminate Aj, bypassing the (uncertain) relationship between the jet's synchrotron width Phi and its flow radius Rj. They normally yield velocities in the range 1000 < vj < 30,000 km s-1 unless low efficiencies epsilon or high mass fluxes dm/dt are assumed.

Table 4. Parameter combinations for cold Jets
Table 4

6.1.6   SUPERLUMINAL MOTION   A simple model for the observed proper motions of knots in compact sources (57) is that apparent "superluminal" motion at betaapp = vapp/c > 1 arises for features in the approaching side of a high- gammaj jet at a large angle i to the plane of the sky, whereupon vj = vapp / [betaapp sin(i) + cos(i)].

6.1.7   DOPPLER BOOSTING   With the above notation, vj is related to the ratio of intensities calSapp of the approaching and receding sides of an intrinsically symmetric (calS = 1) jet as vj sin(i) = c[calSappdelta - 1] / [calSappdelta + 1], where delta = 1 / (2 + alpha) for a continuous jet with a nu-alpha spectrum (23). Assuming calSapp to be due entirely to Doppler boosting therefore constrains betajsin(i). Note that with the typical value alpha = 0.65 (Section 4.2), calSapp varies as gammaj5.3 if the line of sight is < 1/gammaj radians from the jet axis; also, note that a jet can be "one sided" as in Section 3.1 (calSapp > 4 : 1) if betaj sin(i) > 0.26, which at i = 30° (the median value for randomly oriented sources) requires only that betaj > 0.52.

6.1.8   JET WIGGLING   Many jets wiggle around their mean direction (e.g. 29, 172, 183, 214, 220, 262, 263). Mechanisms for periodic lateral deflections Delta as a function of angle Theta from the core [reviewed in (252)] include (a) orbital motion of the primary collimator around a companion mass in the parent nucleus (9, 151) or a nearby member of the same group or cluster (22, 263); (b) precession of the primary collimator or of a larger-scale recollimating atmosphere due to interaction with another body (9, 104, 123, 143, 205, 284); and (c) growth of helical Kelvin-Helmholtz instabilities at the boundary of a confined jet (Section 6.4). Pure orbital motion leads to C-symmetry between the two sides of a jet, with fixed wiggle amplitude Delta and a period tau0. Pure precession of the source of a free nonrelativistic jet leads to S-symmetry, linear growth of Delta with Theta, and a period taup. The analogue for relativistic jets is more complicated as the S-symmetry is broken by light travel time effects, which might themselves indicate vj if other distortions were absent (63, 104, 143). Helical surface instabilities on a confined expanding jet make wiggles whose amplitudes Delta and wavelengths lambdai both grow with Theta; linear theory has been used to estimate the most rapidly growing wavelength lambdai as a function of jet radius Rj, Mach number Mj, and density contrast rhoj / rhoe (Section 6.4).

Attempts to constrain vj from jet-wiggling data "match" an observed pattern Delta(Theta) to one of these pure forms to find a characteristic wavelength lambda0 and a self-consistent estimate of the characteristic period tau0 or taup. Then, vj or Mj is derived from one of the following: vj = lambda0 / tau0, vj = lambda0 / taup, or lambda0 approx lambdai = Rj F1 (Mj) F2(rhoj / rhoe), where the functions F1 and F2 are provided by (linear) instability theory. These methods are fraught with uncertainties. Well-studied jets rarely match simple orbital or ballistic precessional shapes convincingly (22, 104, 105, 257), so additional poorly constrained parameters (e.g. multiple or eccentric orbits, variation of precession cone angle with time) are invoked. Even goodness of fit to a simple C- or S-shape does not guarantee uniqueness of the model (63, 122). Bending and buoyancy effects (e.g. 119, 235, 283) may also be present and - unless the jet is denser than the ambient medium - lateral motions may excite surface instabilities, whose growth also alters the shape of the jet (12). Linear instability theory may be inadequate to describe any mode that grows sufficiently to become detectable on radio maps.

6.2. The Velocity Dilemma

The above methods give velocities ranging from vj approx 1000 km s-1 in C-shaped jets in head-tail sources (using Section 6.1.2) to approx c (using Section 6.1.6 to interpret one sidedness or Section 6.1.7 to interpret superluminal motion). This uncertainty in vj seriously obstructs progress in elucidating the physics of radio jets.

6.2.1   ARGUMENTS FOR vj approx c ON PARSEC SCALES   Five arguments favor vj approx c on parsec scales in some sources:

  1. The superluminal separations of knots in VLBI "core-jet" structures can be explained if gammaj approx 2.5 - 10 (for H0 = 100) and these structures are within 1 /gammai radians of the line of sight (Section 6.1.6).

  2. The same parameters entail Doppler boosting (Section 6.1.7), which accounts for the one sidedness of these VLBI core-jets.

  3. Similar assumptions (but with higher values of gammaj) may explain the high brightness temperatures implied by the rapid low-frequency variability of some compact radio sources (89, and references therein).

  4. The same assumptions explain the low Compton X-ray fluxes from compact radio sources (e.g. 155).

  5. The small angles to the line of sight (i approx 90°) required by this interpretation of the properties of strong compact cores are consistent with the large apparent bending of the jets in core-dominated sources (Section 3.3).

There is little evidence against vj approx c on parsec scales: 3C 147 has a complex, two-sided parsec-scale structure (194), but two sidedness may be ascribed to bending a one-sided jet across the line of sight, in a suitably small number of cases, without endangering the relativistic-jet picture of compact sources.

6.2.2   ARGUMENTS AGAINST vj approx c ON KILOPARSEC SCALES   The sensitivity of Doppler boosting (Section 6.1.7) to vjsin(i) argues against vj approx c in the C-shaped jets in "narrow head-tail" sources. If these are indeed swept back by ram pressure of the intergalactic medium (Section 6.1.2), vj changes direction along them by as much as 90° (e.g. 171, 214, 257). If vj approx c, they would (a) have large side-to-side asymmetries and (b) brighten or fade dramatically as they bend, in conflict with observation (265). The orientations of dust lanes in some weak radio galaxies also suggest that brightness asymmetry at the bases of their two-sided jets (Section 3.1) is unlikely to be due to Doppler favoritism. The jets are generally > 70° from the dust lanes in projection (134 and Table 1, ref. L1), so they should generally be nearly perpendicular to them in three dimensions. The orientation of the dust lane (266) in M84 thereby suggests that the northern jet, which is the brighter and has the one-sided base (Figure 1), is either receding or very close to the plane of the sky (if it is an outflow). Both this constraint and the fact that it becomes two sided without bending argue that its greater brightness is due to greater power output or greater dissipation on its side of the nucleus, rather than to Doppler boosting. R.A. Laing (personal communication) finds similar results in NGC 3665 and for the possible jet in NGC 612, although in Cen A the peculiar velocities of the optical filaments (Section 5.1.2) argue that the bright radio jet is approaching.

This evidence against vj approx c in sources with Ptot1.4 < 1025 W Hz-1 (FR I structures) leaves open the possibility that vj increases with Ptot, so that the long one-sided jets in powerful sources might be Doppler-boosted flows with gammaj >> 1. Some bent one-sided jets have smooth brightness variations [e.g. 1150 + 497 (Figure 3), 4C 32.69 (191)], which are inconsistent with changing Doppler boosts in high-gammaj jets if they bend because they are confined. Such jets could be ballistic, however, with their shapes arising from wobble (precession?) of the primary collimator; vj would then not follow the bends, but the wiggle pattern would move radially as a whole. Changes in vjsin(i) and in the Doppler boosting (Section 6.1.7) may then be small. We must know whether or not such jets are confined (Section 4.1), and if so where, to decide whether their brightness distributions argue against vj approx c. Doppler boosting models for long one-sided jets also require large angles i, so boosted one-sided jets would be significantly longer in three dimensions than they appear in projection. It is unclear whether this seriously conflicts with vj approx c in these jets, as the existing statistics of QSR source sizes come from samples containing significant numbers of one-sided jets (Section 2.2). Maps with greater dynamic range are needed to assess the degree of one sidedness of these jets (we do not know by how much calS > 4 in most cases), as the average deprojection increases with the average asymmetry.

6.2.3   ARGUMENTS FOR vj approx c ON KILOPARSEC SCALES   Table 1 lists 22 sources with VLBI jets or elongations. Of these, five exhibit superluminal expansion - 3C 120, 3C 179, 3C 273, 3C 279, 3C 345 (57). In all five, the kiloparsec- and parsec-scale jets start on the same side of the core, as in Figure 4 (see references in Table 1). Sixteen others have VLBI elongations and kiloparsec-scale jets, but their proper motions on parsec scales are unknown. In 11 of these (NGC 315, 3C78, 3C84, 0957 + 56, 3C111, M87, Cen A, NGC 6251, 3C371, 3C405, and 3C418), the larger-scale jet starts on the same side as the small, e.g. Figures 2 and 5. Of the remaining five, two (3C 147 and 3C236) have two-sided small-scale structure, two (M84 and 3C 454.3) do not have closure-phase VLBI maps, and 3C 309.1 has complex structure. The correlation between small- and large-scale sidedness argues that one sidedness has the same cause on both scales. It supports the idea that vj can be high enough on kiloparsec scales for Doppler favoritism to be important, if one is convinced by the case for gammaj >> 1 on parsec scales (Section 6.2.1). This case is strongest for the five superluminal sources, but it is not yet impregnable. Since there are no known coreless large-scale jets, either both the cores and the jets are Doppler boosted or the luminosities of intrinsically one-sided jets are coupled to those of the cores; the reason for such coupling over such a wide range of linear scales is unclear if the sidedness is due to asymmetric dissipation. If the kiloparsec-scale sidedness is intrinsic (124, 213, 218, 273, 280), these data require a switching time-scale tau > dj/vj and an alternative model for superluminal expansions. The constraint tau > dj/vj is often hard to reconcile with tau being less than the synchrotron lifetimes in the hot spots (e.g. 113, 265). On balance, the correlation between parsec and kiloparsec sidedness favors gammacore approx gammaj geq 1 in powerful sources.

Other (weaker) arguments for vj approx c on kpc scales are the following:

  1. Relativistic jets need less confinement, since Doppler boosting and projection effects at large angles to the plane of the sky mean that the standard synchrotron calculation may overestimate pj.

  2. It is difficult to brake a jet by entrainment between the parsec and kiloparsec scales without converting much of its energy into heat, unless Mj is low (222). Losing ~ c/vj times the total power of a strong extended radio source by dissipation as "waste heat" near its nucleus is a daunting prospect for source models if vj << c.

6.3. Jet/Hot-Spot Symmetries and the Sidedness Dilemma

The symmetries of the regions where powerful jets end may also offer clues to the reasons for their one sidedness (215). If it is always due to Doppler favoritism, the jetted and unjetted lobes should look similar - unless high-gammaj jets push the hot spots out at vh approx c, in which case the brighter jet should appear to feed the brighter and more distant hot spot if the two sides of the source have the same history (146). (In the extreme case of a "young" high-gammaj source, radiation from the receding side may also not yet have reached us.)

In 34 of the 46 FR II sources in Table 1 with one-sided jets, one lobe has a significantly brighter hot spot than the other on the highest resolution map available. Seventeen of the 34 have fC = Score5 / Stot1.4 > 0.05; the brighter jet points to the brighter hot spot in 16 of these. Unless the jets are "young", either the brighter jet has a higher thrust or the jets and the hot spots in these sources are both Doppler boosted. In the 17 cases with fC < 0.05, the brighter jet points to the brighter hot spot in ten and to the weaker in seven. This is consistent with one sidedness due either to differential dissipation or to Doppler boosting. There is no trend in either group for the jetted hot spot to be more distant, so if boosting is important the hot-spot separations must not reflect travel time differences from simultaneous ejecta. They might instead be determined by the history of the source, e.g. by a wandering or intermittent jet illuminating different parts of a lobe at different times. These trends imply that either (a) jet one sidedness has different causes in FR II sources with different fC or (b) the jets, but not the hot spots, are boosted in sources with fC < 0.05, while both are boosted if the core is strong. The relative brightnesses of hot spots are sensitive to linear resolution, however, so the trends must be checked with more uniform data.

6.4. Stability

Jets are surprisingly stable. They can extend for hundreds of kiloparsecs or bend through geq 90° (in C-shaped "head-tails") without disruption. Early analyses of the stability of confined cylindrical jets to helical, fluting, and pinching perturbations analogous to the Kelvin-Helmholtz instabilities of a vortex sheet (15, 92, 93, 95, 110, 195) suggested that jets are generally unstable to modes with wavelengths of a few jet radii. The growth rates are less for Mj > 1 and for vj approx c, but the stability of observed jets forces re-examination of simplifying assumptions made in these analyses. The stabilizing influence of a surface shear layer on modes with wavelengths less than its scale depth was examined in (92) and (196), and that of jet expansion on long-wavelength modes in (111). Within a thermally confined jet, B|| may stabilize long-wavelength pinching modes (12, 92, 195). The firehose instability can be inhibited by sizable Bj and by linking the inertia of a plasma cocoon around the jet to Bj (12). The stability of magnetically confined jets has yet to be studied thoroughly, although first steps have been taken (12, 16, 61). Progress here is hampered by ignorance of basic MHD parameters in jets: we know little about ion or electron temperatures, field strengths, particle densities, and sound or Alfvén speeds, independent of the assumption of equipartition. Currently favored models of jet production from rotating disks or tori near supermassive objects (e.g. 251) may produce flows with net helicity. The influence of such helicity on jet stability merits attention, as helicity can lead to efficient generation of large-scale Bj by turbulent amplification of small seed fields (72).

Instabilities in real jets may grow algebraically, rather than exponentially. Exponential growth can be stopped in many ways - shock formation when the perturbation velocities become supersonic, shifting of the modes to longer wavelengths as their amplitudes grow, or saturation of the instabilities by in situ particle acceleration (15, 93, 94). Nondestructive instabilities might dominate the observed shapes and relative brightnesses of radio jets and lobes (15, 112, 286): algebraic growth of short-wavelength instabilities may help to keep jets bright (Section 4.3), long-wavelength helical modes to explain jet wiggling (Section 6.1.8), and pinching modes to form knots (Sections 4.1.4 and 4.3). Instability growth may also determine overall source sizes. MHD stability analyses including jet expansion, velocity and density gradients, and realistic Bj configurations and velocity profiles are needed; the analytical difficulties are great, and numerical simulations that do not legislate axisymmetry may be required.

6.5. Unified Models

"Unified" models seek to relate differences between sources with weak and powerful radio cores solely to differences in viewing angle. If the arguments in Section 6.2.1 indeed support gammaj >> 1 and i approx 90° in the jets of core-dominated sources, a randomly oriented sample should contain ~ 2gammaj2 unboosted sources for every boosted one if the jets have narrow cone angles. For <gammaj> approx 5 (169), core-dominated sources would number only a few percent of their parent population in the plane of the sky, which may therefore be a well-known class of object. Proposed parent populations for the core-dominated QSRs are radio-quiet QSOs (223) and QSRs with lobe-dominated extended radio sources (23, 169). The latter proposal is not encouraged by the fact that the lobe-dominated sources have weaker [Fe II] emission and broader lines than core-dominated sources (242, 281). It is hard to see how such differences in the line strengths could be produced by the small aspect differences (Deltai approx 1 / gammaj radians) over which the Doppler-boosting factor varies markedly (115). Although the flux density distribution of strong radio sources in optically selected QSO samples conflicts with the unified models (131, and references therein), VLA studies of the Schmidt-Green QSO sample to a limiting flux density of 250 µJy at 6cm are consistent with them over most of the flux density range. The "excess" of strong sources may be due to a separate population of extended sources, most of whose emission is presumably unbeamed (K. Kellermann et al., in preparation). As the emission lines cannot be beamed, models that beam the optical continuum luminosities of core-dominated QSRs predict the existence of emission-line QSOs without nonthermal continua; these have not been detected.

About half of all core-dominated sources have detectable kiloparsec-scale secondary structure, which is generally one sided, as in Figure 5 (182, 185, 274). If the parent population is to be "radio quiet" (223) or to have normal two-sided lobes (169), both "unified models" must assert that most of this one-sided secondary structure is also boosted. The parent population of most core-dominated sources must then be a class of numerous weak extended radio sources with at least mildly relativistic kiloparsec-scale jets and relatively strong Fe lines.

6.6. A Broader Unified Model

As about half of all radio galaxies and nearly all QSRs have detectable radio cores (131, and references therein), there must be a mixture of boosted and unboosted contributions to the core emission. Furthermore, some kiloparsec-scale jets emerging from weak cores must have nonrelativistic velocities and intrinsic emission asymmetries (Section 6.2.1), while some jets emerging from powerful cores may be relativistic (Section 6.2.3). It may be that gammacore geq gammaj > gammah> in all sources, while all three tend to increase with the actual source power (measured by the luminosity of the most extended radio features). The correlations between Pcore and fC, the occurrence rates of jets (Section 2.3), their sidedness (Section 3.1), their magnetic field configuration (Section 3.2), and the large-scale source structure (FR class) might be assimilated in a broader unified model as follows.

The kiloparsec-scale jets in most weak sources have vj << c, and so appear two sided, with minor asymmetries that are either intrinsic or the result of asymmetric internal dissipation of flow kinetic energy to synchrotron radiation. They expand rapidly, so Bperp dominates over B|| except at their bases. They have low thrusts and so are readily bent, sometimes maintaining B|| layers at their edges by shearing or stretching as they bend. Low Mach numbers allow them to become turbulent, to entrain material and thus to decelerate [all effects that keep them well lit up (Section 4.3)], and to terminate gently without forming hot spots. These characteristics lead to FR I morphology (Figure 1). Weak sources with gammaj = gammah approx 1 but gammacore >> 1 would be strongly core dominated if oriented near the line of sight; their extended low-brightness FR I structure would be detected only on maps with high dynamic range. Such sources could be BL Lac objects with very weak large-scale structure (23, 37, 254). There cannot, however, be large numbers of sources with gammacore >> gammaj, or else we would see many "coreless jets."

The kiloparsec-scale jets in more powerful sources may have higher vj. They may also have higher Mach numbers, leading to narrower cone angles where they are free and to prominent hot spots where they end. They may be more stable, less turbulent, and thus dimmer relative to their lobes, leading to FR II morphology. Higher vj may lead, however, to deeper boundary layers with the intergalactic medium, in which B|| is maintained by shearing (129, 208). The combination of such shearing and good collimation could make the jets that do stay lit up appear B||- dominated (Section 3.2) at low transverse resolution. If vj -> c in the more powerful sources, Doppler boosting may contribute to correlations between jet detectability, fC, and jet/hot-spot symmetries (Section 6.3). The jets and some core emission in powerful sources near the plane of the sky would be beamed away from us, producing "jetless" FR II sources with weak cores, as in the distant 3CR2 radio galaxies (Section 2.2). Similar sources turned toward us would have strong cores and one-sided jets, as in the extended 3CR2 QSRs. The ~ 40 to 50% detection rate of jets in 3CR2 QSRs requires, however, that only mild boosting (gammaj leq 2) is usually involved, and the lack of "coreless jets" again implies that gammacore approx gammaj in general. Intrinsic asymmetries may therefore still be significant in the powerful sources. There are weak relationships between fC and projected linear size, misalignments, and lobe separations among extended QSRs (130); these relationships are consistent with some core boosting in these sources.

Such "unified models" of extragalactic radio sources may ultimately be judged by whether or not the optical and X-ray differences between different source types can be correlated with intrinsic source power and with indicators of the viewing angle.

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