Annu. Rev. Astron. Astrophys. 1984. 22: 319-58 Copyright © 1984 by Annual Reviews. All rights reserved

6. JET LAG-MISSING PHYSICS

6.1 Velocity Estimates

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

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

where A_j, v_j, _j, _j, and h_j 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 v_j.)

6.1.2   MOMENTUM FLUX   For jets which terminate at hot spots, the thrust T_{j / h} in the rest frame of the hot spot must balance p_h A_H, where p _h and A_h are the minimum pressure and cross-sectional area of the hot spot estimated from its synchrotron parameters. The thrust T_j = A_{j j} v_j² _j² 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 _e to relate v_j to the velocity v _h of advance of the hot spot. Estimates for T_j 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 _e through which their parent galaxy moves at velocity v_g (10, 128). If the radius of curvature of the C-structure is R_c and the scale over which the ram pressure is transmitted to the jet is H, momentum balance requires that T_j = _e v_g² A_j R_c / H. The parameter H is the radius R_j 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 = A_{j j} v_{j j} 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 M_j of a free jet where it detached from its confinement (Section 4.1) can be estimated from its synchrotron expansion rate via d / d = (2 / M_j)sec(i), where i is the (assumed) angle between the jet axis and the plane of the sky. Then v_j = M_j sqrt[ p_j / _j], where is the ratio of principal specific heats in the jet, and a lower limit to p_j is obtained from the synchrotron parameters. If the jet is actually confined, M_j is overestimated. If the weak radio galaxy jets are alternately free and confined (Section 4.1.1), M_j is best indicated by the rapid expansions at z 1 - 10 kpc, which imply that M_j 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 _j, 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 B_j 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 _j, so even setting limits to _j 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 v_j -> c if _j -> 0, but Table 4 shows how v_j can be constrained using combinations of these methods to eliminate _j for "cold" (h_j << _j v_j²) jets. As h_j / _j v_j² = / M_j²( - 1) from the gas laws and Section 6.1.4, this is a good assumption for jets with M_j 4. These combinations also eliminate A_j, bypassing the (uncertain) relationship between the jet's synchrotron width and its flow radius R_j. They normally yield velocities in the range 1000 < v_j < 30,000 km s^-1 unless low efficiencies or high mass fluxes dm/dt are assumed.

Table 4. Parameter combinations for cold Jets

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 _app = v_app/c > 1 arises for features in the approaching side of a high- _j jet at a large angle i to the plane of the sky, whereupon v_j = v_app / [_app sin(i) + cos(i)].

6.1.7   DOPPLER BOOSTING   With the above notation, v_j is related to the ratio of intensities _app of the approaching and receding sides of an intrinsically symmetric ( = 1) jet as v_j sin(i) = c[_app - 1] / [_app + 1], where = 1 / (2 + ) for a continuous jet with a ^- spectrum (23). Assuming _app to be due entirely to Doppler boosting therefore constrains _jsin(i). Note that with the typical value = 0.65 (Section 4.2), _app varies as _j^5.3 if the line of sight is < 1/_j radians from the jet axis; also, note that a jet can be "one sided" as in Section 3.1 (_app > 4 : 1) if _j sin(i) > 0.26, which at i = 30° (the median value for randomly oriented sources) requires only that _j > 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 as a function of angle 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 and a period ₀. Pure precession of the source of a free nonrelativistic jet leads to S-symmetry, linear growth of with , and a period _p. The analogue for relativistic jets is more complicated as the S-symmetry is broken by light travel time effects, which might themselves indicate v_j if other distortions were absent (63, 104, 143). Helical surface instabilities on a confined expanding jet make wiggles whose amplitudes and wavelengths _i both grow with ; linear theory has been used to estimate the most rapidly growing wavelength _i as a function of jet radius R_j, Mach number M_j, and density contrast _j / _e (Section 6.4).

Attempts to constrain v_j from jet-wiggling data "match" an observed pattern () to one of these pure forms to find a characteristic wavelength ₀ and a self-consistent estimate of the characteristic period ₀ or _p. Then, v_j or M_j is derived from one of the following: v_j = ₀ / ₀, v_j = ₀ / _p, or _{0 i} = R_j F₁ (M_j) F₂(_j / _e), 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 v_j 1000 km s^-1 in C-shaped jets in head-tail sources (using Section 6.1.2) to c (using Section 6.1.6 to interpret one sidedness or Section 6.1.7 to interpret superluminal motion). This uncertainty in v_j seriously obstructs progress in elucidating the physics of radio jets.

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

1. The superluminal separations of knots in VLBI "core-jet" structures can be explained if _j 2.5 - 10 (for H₀ = 100) and these structures are within 1 /_i 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 _j) 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 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 v_j 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 v_j c ON KILOPARSEC SCALES   The sensitivity of Doppler boosting (Section 6.1.7) to v_jsin(i) argues against v_j 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), v_j changes direction along them by as much as 90° (e.g. 171, 214, 257). If v_j 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 v_j c in sources with P_tot^1.4 < 10²⁵ W Hz^-1 (FR I structures) leaves open the possibility that v_j increases with P_tot, so that the long one-sided jets in powerful sources might be Doppler-boosted flows with _j >> 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-_j 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; v_j would then not follow the bends, but the wiggle pattern would move radially as a whole. Changes in v_jsin(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 v_j 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 v_j 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 > 4 in most cases), as the average deprojection increases with the average asymmetry.

6.2.3   ARGUMENTS FOR v_j 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 v_j can be high enough on kiloparsec scales for Doppler favoritism to be important, if one is convinced by the case for _j >> 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 > d_j/v_j and an alternative model for superluminal expansions. The constraint > d_j/v_j is often hard to reconcile with being less than the synchrotron lifetimes in the hot spots (e.g. 113, 265). On balance, the correlation between parsec and kiloparsec sidedness favors _{core j} 1 in powerful sources.

Other (weaker) arguments for v_j 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 p_j.

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 M_j is low (222). Losing ~ c/v_j 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 v_j << 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-_j jets push the hot spots out at v_h 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-_j 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 f_C = S_core⁵ / S_tot^1.4 > 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 f_C < 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 f_C or (b) the jets, but not the hot spots, are boosted in sources with f_C < 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 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 M_j > 1 and for v_j 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 B_j and by linking the inertia of a plasma cocoon around the jet to B_j (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 B_j 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 B_j 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 _j >> 1 and i 90° in the jets of core-dominated sources, a randomly oriented sample should contain ~ 2_j² unboosted sources for every boosted one if the jets have narrow cone angles. For <_j> 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 (i 1 / _j 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 _{core j} > _h> 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 P_core and f_C, 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 v_j << 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 B 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 _j = _h 1 but _core >> 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 _core >> _j, or else we would see many "coreless jets."

The kiloparsec-scale jets in more powerful sources may have higher v_j. 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 v_j 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 v_j -> c in the more powerful sources, Doppler boosting may contribute to correlations between jet detectability, f_C, 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 (_j 2) is usually involved, and the lack of "coreless jets" again implies that _{core j} in general. Intrinsic asymmetries may therefore still be significant in the powerful sources. There are weak relationships between f_C 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.