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Astron. Astrophys. 325, 57-73 (1997)

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4. Discussion

Although no new high frequency counterpart of any radio hot spot could be found in the present work we are able to extend the scope of Paper I considerably: We improved the spectra of 3C 20 and 3C 33. The upper limits to the K-band flux for the bright hot spots A & D in Cygnus A enable us to derive rather reliable synchrotron spectra which constrain the high frequency cutoff to a factor of 2. In addition, new millimetre and K-band measurements of 3C 123 and Pictor A have sorted out the ambiguities in the overall spectra which were present in our earlier work (Paper I). Likewise, new optical and K-band photometry of the terminal hot spot in the jet of 3C 273 has led to a reliable overall spectrum (Meisenheimer et al. 1996b) which we include for completeness in Table 5 and the present discussion.

Altogether we can base our updated discussion of the overall spectra of radio hot spots on 8 cases: The hot spots in 3C 20 W, 3C 33 S, 3C 111 E, 3C 123 E, 3C 273 A, 3C 405 A, D (Cygnus A) and Pictor A west. Although the improvement in number might not seem significant, the improvement in quality (6 out of the 8 hot spots have much better determined spectra) makes a revision of the discussion presented in Paper I mandatory. Moreover we feel that the detection of optical or near infrared synchrotron radiation from locations well outside prominent hot spots or knots is so striking that it no longer can be ignored (as we did in Paper I).

Therefore, we divide the following discussion into three parts: In the first paragraph (4.1) we will apply the standard Fermi acceleration model for the synchrotron spectra of radio hot spots (along the lines of Paper I). In 4.2 we will discuss the obvious shortcomings of the standard scenario before in the final part (4.3) we will try to summarize what the diversity of hot spot spectra could tell us about the physical processes which make the hot spots shine so brightly in the radio and sometimes even at optical frequencies.

4.1. Fermi acceleration models for radio hot spots

The standard model of particle acceleration in radio hot spots assumes diffusive shock acceleration (first-order Fermi acceleration) in which the relativistic electrons are accelerated by multiple scattering across the velocity jump of a strong shock front. Such a shock (Mach disk) is expected to exist in the working surface of supersonic jets. In Paper I we pursued this standard model in great detail in order to explore whether it leads to a self-consistent description of the particle spectra (as inferred from the observed synchrotron spectrum) and other parameters like size and shape of the hot spot emission region, magnetic field strength and the speed of the inflowing jet plasma. We concluded that self-consistent solutions can be found for all 6 hot spots contained in Paper I and presented a rather detailed set of physical parameters for them (see Table 5 and Figs. 8 and 9 in Paper I). Thus our first objective is to check our former results on the basis of the improved spectra for these 6 hot spots and the inclusion of the two bright hot spots of Cygnus A for which we could determine the spectra. In this section we derive the physical parameters exactly in the same way as outlined in Paper I. Thus the parameters summarized in Table 6 can directly be compared with those of Paper I. 2

In Paper I we identified two distinct types of hot spot spectra:

  • (A) low loss hot spots which are characterized by their low frequency power law of rather flat slope ([FORMULA]) extending out towards high frequencies ([FORMULA] Hz) where the spectra turn down in a high frequency cutoff.
    This is in remarkable contrast to the spectra of
  • (B) high loss hot spots in which the flat spectrum ([FORMULA]) extends only to around [FORMULA] Hz where it bends by [FORMULA] before the cutoff at around [FORMULA] Hz is reached. Such a [FORMULA] break is naturally expected if the electrons which have been accelerated in the shock front lose most of their energy within a hot spot emission region which extends downstream of the shock over several hundred or thousand times the electron mean free path [FORMULA] (Heavens & Meisenheimer 1987).

The low and high loss hot spots are shown in Figs. 3, 4 & 5 by a gray and black shading, respectively.

[FIGURE] Fig. 5. Distribution of parameters concerning the particle acceleration in hot spot: a Maximum energy [FORMULA], b acceleration time scale [FORMULA], c electron mean free path [FORMULA]. Shading as in Fig. 3. The three high loss hot spots with [FORMULA], [FORMULA] yrs, and [FORMULA] pc are not detected in either the K-band or the optical.

Since the spectra of the three low loss hot spots from Paper I (3C 20 west, 3C 33 south, 3C 111 east) have been refined only marginally our previous results hold unchanged. The main difference concerns 3C 33 south, the complex emission region of which we treat here as a single source, rather than a two component object as in Paper I. Therefore our one-dimensional model is not strictly applicable and several parameters in Table 6 are only of qualitative nature (see also Section 4.2).

The most substantial adjustments have to be made for the hot spots in 3C 123 east and Pictor A west since their spectra are changed considerably by new flux measurements: The best fit spectrum of 3C 123 east is now definitely characterized by a high frequency cutoff in the millimetre range which makes the [FORMULA] break rather inconspicuous although our model fits still place the break frequency as low as [FORMULA] GHz. So our classification of 3C 123 E as a high loss hot spot remains valid, despite the fact that [FORMULA] is an order of magnitude below the upper limit in Paper I. The complicated structure of this hot spot has not been disentangled by any better radio maps. Thus the physical parameters in Table 6 are again only qualitative.

Due to better radio data and the correction of the erroneous NIR photometry, the spectrum of Pictor A west has been altered completely: The lack of any low frequency break classifies this hot spot now as low loss type, albeit with a very steep power law slope [FORMULA]. As in 3C 33 south this fact and the largely extended radio and optical emission region makes the application of the standard model rather dubious. Even if we apply the model only to the thin sheet (filament ?) of highest surface brightness at the leading edge of the hot spot we get a large discrepancy between [FORMULA] and [FORMULA] indicating that the model assumptions fail.

Notwithstanding the fact that new NIR photometry of the leading hot spot 3C 273 A of the jet of 3C 273 (Neumann et al. 1996), together with the optical data published in Röser & Meisenheimer (1991) allows us to determine the high frequency shape of its spectrum with much superior accuracy (Meisenheimer et al. 1996b), the essential conclusions of Meisenheimer & Heavens (1986) which were included in Paper I remain unchanged: The [FORMULA] break at GHz-frequencies is confirmed and the high frequency cutoff can be accurately pinned down at [FORMULA] Hz (see Table 5). Moreover, the positional offset between the mean radio emission and the optical peak which is predicted in the model of synchrotron losses in a finite downstream emission region (Meisenheimer & Heavens 1986) seems to be confirmed by a recent comparison between a radio maps and HST images (Röser et al. 1996). So we are rather confident that the model applies to this source and the parameters given in Table 6 are reliable.

Applying our model to the brightest hot spots in Cygnus A which both have well defined low frequency breaks by [FORMULA] and a measurable length L of their downstream emission region leads to an equally self-consistent sets of parameters. We regard this as further evidence that the basic concept of a thin acceleration region (presumably at a strong shock) which is followed downstream by an extended emission region is sensible. It is interesting to note that in all three well defined high loss hot spots we derive a mildly relativistic inflow speed [FORMULA] which is at the lower end of the distribution in Fig. 4b. This might be due to the fact that all these hot spots should be classified as "secondary hot spots" in the sense that they have a more compact precursor in which the jet might be slowed down considerably. The wide spread of [FORMULA] in low loss hot spots might be due to large uncertainties in both the length L and [FORMULA]. The "true" distribution may well be consistent with [FORMULA].

If we ignore the aforementioned difficulties in applying the simple shock acceleration model to some hot spots with complicated structure we still find a rather self-consistent picture: In general, the minimum energy field estimate [FORMULA] and that from the downstream losses [FORMULA] agree within the errors. The (averaged) best guess magnetic field values cluster quite narrowly around [FORMULA] nT (Fig. 4a). This means that the large spread of observed cutoff frequencies [FORMULA] Hz is mainly caused by a comparable spread in maximum energies [FORMULA]. Within the shock acceleration model this has to be assigned to a rather wide distribution of acceleration time scales [FORMULA] which are determined by the mean free path of the electrons [FORMULA] (or the diffusion coefficient). However, it is interesting to note that in both the distributions of the magnetic field strength and that of the maximum energy [FORMULA] the hot spots seem to be overtaken by the brightest knots of optical jets. Maybe the hot spot phenomenon is not as outstanding as hydrodynamic jet simulation suggest and resembles the "normal" knots in the jet flow.

As emphasized in Paper I the most important consistency check of the shock acceleration model considers whether the number of relativistic electrons [FORMULA] (between a "cut-on" energy [FORMULA] and the maximum energy [FORMULA]) is only a (small) fraction of the total number of electrons in the jet flow, and whether the energy injected into relativistic electrons at the shock [FORMULA] can be provided by the kinetic energy flux of the jet [FORMULA]. As the jet's ram pressure has at least to balance the minimum pressure in the hot spot, the estimate for [FORMULA] gives a handle to set a lower limit to the incoming proton and kinetic energy flux. Thus the model sets upper limits to the required fraction of relativistic electrons [FORMULA] (assuming a neutral ep-jet: [FORMULA]) and to the acceleration efficiency [FORMULA]. The upper limits for [FORMULA] and [FORMULA] derived from our best guess parameter set are given at the bottom of Table 6 and visualized in the histograms of Fig. 6a,b. Obviously only one of the two hot spots with the outstanding steep low frequency spectra [FORMULA] (Pictor A west) is in conflict with the requirement [FORMULA] and [FORMULA] (3C 33 south is no more extreme than 3C 20). This conflict could be solved by assuming a higher "cut-on" frequency [FORMULA] (instead of the standard value [FORMULA] we assumed here). But together with the other morphological and spectral evidence pointing away from the standard model we think it is highly unlikely that such a fine tuning could save the simple shock acceleration scenario for the extended hot spots of 3C 33 and Pictor A.

[FIGURE] Fig. 6. Histogram of a the maximum fraction of electrons with relativistic energies [FORMULA], and b maximum required acceleration efficiency [FORMULA]. Shading as in Fig. 3. Note that only one hot spot, Pic A west lies in the forbidden area [FORMULA].

At this point, we should emphasize that the high frequency spectra of both the high- and low-loss hot spots are very well described by our simple one-dimensional model which essentially assumes a constant down-stream magnetic field. On the other hand, one might expect that the complicated flow pattern in the hot spot region (e.g. Kössl et al. 1990) should produce wide variations in the local magnetic field strength and thus "smear out" the cutoff spectrum which we derived from the idealized field geometry. The observed lack of this "smearing out" effect can be understood as follows: In the observed frequency range (just above the cutoff frequency [FORMULA]) the typical spectral index is [FORMULA]. Accordingly, the mean emissivity (per unit volumn) is a strong function of the local magnetic field: [FORMULA] or steeper. Thus those parts of the hot spot with the highest magnetic fields will dominate the spectral shape around [FORMULA]. A significant smoothing of the spectrum could only occur if the total emission region is much larger than the acceleration region (near the shock). However, model calculations which take into account both a wide variety of field geometries (i.e. including fields the strength of which decline rapidly away from the shock) and a proper treatment of the synchrotron losses result in spectra which always show the low-loss or high-loss characteristics without significantly changing the shape around [FORMULA]. Only with a very carefull balance of declining field strength and synchrotron losses it is possible to alter the shape of the high frequency spectrum. But these solutions are only possible in a hot spot region without sharp boundaries. If the volumn which contains the widely varying field is limited the dominance of the regions with the highest magnetic field inevitably determine the spectra.

In summary, we conclude that first-order Fermi acceleration at a strong, non-relativistic shock can well account for the observed properties of 6 out of the 8 hot spots in our sample. The major arguments which support this are:

  • (a) The low frequency spectral indices lie exactly in the range [FORMULA] which is predicted for mildly relativistic shocks (Bell 1978, Ballard & Heavens 1991).
  • (b) The independent magnetic field estimates [FORMULA] and [FORMULA] (from downstream losses) agree for mildly relativistic jet speeds [FORMULA].
  • (c) Both the fraction of relativistic electrons [FORMULA] and the acceleration efficiency [FORMULA] lie comfortably below 1.
  • (d) Emission regions appear longer (along the jet axis) in high loss hot spots while they seem to be thin sheets in low loss hot spots.
  • (e) The only source for which a comparison between the radio and optical morphology is available at the required resolution [FORMULA], 3C 273 A shows exactly the predicted offset between its optical peak ([FORMULA]) and the radio hot spot (at [FORMULA]).

The obvious counter-examples, the hot spots in 3C 33 south and Pictor A west, fail to show more than two of the properties (a) - (e).

4.2. Limitations of the Fermi acceleration model

In the present sample the most obvious conflict with the standard shock acceleration model occurs in the hot spots of 3C 33 south and Pictor A west, both of which show a steep power law index [FORMULA] and an optical emission region clearly extending beyond a sheet or disk which would be expected for shock acceleration where the acceleration region should extend no more than a few mean free paths [FORMULA] (see Table 6) from the major velocity jump. The most striking example for extended synchrotron light is found near the hot spot of Pictor A west, in which a filament stretches out by [FORMULA] kpc on either side of the jet. We think that this evidence for widely distributed particle acceleration has to be seen in the context of the finding that the optical emission from radio jets is also not confined to bright, individual knots but follows the radio morphology very closely (Meisenheimer 1991, Boksenberg et al. 1992, Meisenheimer et al. 1996a).

The detection of the radio jet in 3C 303 together with the clearly extended morphology of the hot spot on our K-band image (Fig. 1(g)) indicates that this object belongs into the same category.

But extended optical emission is not the only evidence for particle acceleration taking place outside hot spots and bright knots. Even in the case of the well established high loss hot spots in 3C 273 and Cygnus A there is a problem: We have determined the break frequency [FORMULA] and thus the maximum energy with which the electrons leave the hot spot downstream. Even if the magnetic field strength in the lobes surrounding the hot spot is comparable to that in the hot spot (which is not supported by minimum energy estimates of the field, see e.g. Carilli et al. 1991) the lobe spectra should not extend beyond [FORMULA] if indeed the particle acceleration is done in the hot spots only. But the lobe spectra of Cygnus A extend well beyond 30 GHz (i.e. [FORMULA]) near the edges of the radio source (see Carilli et al. 1991, Meisenheimer 1996) and the extended radio "halo" around the hot spot in 3C 273 A seems to be visible on a deep K-band image (Neumann, 1995). So either the magnetic field is significantly enhanced over minimum energy in the lobes or additional acceleration is required outside the main shock.

Therefore, we would like to suggest the following generalization for in situ particle acceleration in extragalactic radio sources: There exists an acceleration mechanism in highly magnetized plasma which is not directly coupled to strong shocks but may work everywhere where shear or turbulence in the plasma flow generate strong magnetic fields. Although there is still little observational evidence how this process works in detail we tend to link its occurence to electro-magnetic plasma processes like reconnection and/or the generation of strong electric currents. This acceleration mechanism has to be mainly responsible for the very efficient particle acceleration which is required to explain the continuous emission of synchrotron light in optical jets (M 87, 3C 273, PKS 0521-36 and others). Therefore we will refer to it as "jet-like" acceleration. It generates rather step power-law spectra [FORMULA] and can reach very high energies ([FORMULA]).

At the strong shock (Mach disk) in the working surface of a supersonic jet the "jet-like" acceleration is either substituted or resembles the standard diffusive shook acceleration mechanism (Bell 1978). This leads to the "standard" shock acceleration spectra with slope [FORMULA] observed in most hot spots. However, our observations of "broken" high-loss spectra in 3C 273 A and Cygnus A (hot spots A & D) which indicate an extended downstream emission region dominated by synchrotron losses (without re-acceleration) pose a severe problem to this concept of two distinct acceleration processes: Synchrotron losses can only dominate if the efficiency of the "jet-like" acceleration is un-important or quenched for some range down-stream of a strong shock. One may speculate that this could be caused by a "change of state" when magnetized plasma passes a strong shock. Moreover, further down-stream (e.g. in the lobes) the plasma seems to regain its ability to support jet-like acceleration in order to boost the maximum electron energy well above the value which is inferred from the losses within the hot spot (see also Meisenheimer 1996). We tendatively identify this effect with some kind of "magnetic tension" in jet-like plasmas which is relaxed within strong shocks but can subsequently build up again by the strong velocity shear and turbulence expected near the "contact discontinuity" between radio plasma and shocked outer material. If correct this interpretation would imply that velocity shear or turbulence are the prime movers of "jet-like" acceleration while a more ordered velocity jump or gradient leads to the classical shock acceleration spectrum. The relative importance of shock and jet-like acceleration in hot spots may depend on the distance between the Mach-disk and the contact discontinuity, which seems to be rather transient in numerical simulations of jets (see e.g. Kössl et al. 1990). Somewhat against intuition, the highest electron energies seem to be reached in those hot spots in which the jet-like acceleration dominates (Pictor A (west) and 3C 33 south), making them the optically brightest hot spots detected so far.

4.3. General results

The analysis of our sample of those 8 hot spot spectra, which are accurately determined by this work, strongly suggests that the emission of synchrotron light ([FORMULA] Hz) is only possible if maximum electron energies reached by the acceleration process exceed [FORMULA]. The second parameter that determines the maximum observed frequency - the magnetic field strength [FORMULA] seems to play a minor role since all hot spots in our sample (including those with [FORMULA] Hz) show essentially the same value [FORMULA] nT.

Independent of the acceleration mechanism, the energy [FORMULA] will essentially be set by the balance of synchrotron losses and acceleration gains, i.e. [FORMULA]. From the observed cutoff frequency [FORMULA] and the magnetic field in the acceleration region one can directly calculate [FORMULA] and thus gets [FORMULA] for any underlying acceleration process. It is obvious from Fig. 5b that acceleration time scales of [FORMULA] years are the essential parameter to make a hot spot optically visible.

It is still unclear which physical parameter or process sets the time scale [FORMULA]. But we have presented here some more hints that nature has found at least two ways for effective particle acceleration: One process is identical or closely related to the standard concept of diffusive shock acceleration (Bell 1978). It produces an electron spectrum [FORMULA] with a slope [FORMULA], for which we find evidence in 6 of our 8 hot spots. Under favorable conditions an acceleration times scale of [FORMULA] yrs can be reached (but this seems to be rare since there are tens of radio hot spots which are bright enough in the radio to be optically detectable if [FORMULA] Hz).

The second process which is capable to reach [FORMULA] does not seem to be confined to strong shocks but could work everywhere in highly magnetized plasmas. Its existence is required in order to explain the constant electron spectrum in the jet of M 87 (Meisenheimer et al. 1996a) but it also has to be at work in hot spots with very extended optical emission regions ([FORMULA] kpc). It is characterized by rather steeper electron spectra ([FORMULA]) and high acceleration efficiency ([FORMULA] yrs) up to extremely high energies ([FORMULA] or more).

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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