6 Discussion

   
6.1 The jet's extensions

Before analysing the jet emission proper, we consider the relation of the extensions to the jet. Our new images confirm the outer extension's nature as spiral galaxy. The spectral index map gives us few hints about the nature of the emission mechanism for the inner and southern extensions. We have therefore measured their integral fluxes at 300nm and 620nm (this paper) and at 1.6 $\mu$ (HST NICMOS camera 2 imaging, in prep.). Figure 5 shows the resulting spectral energy distributions. In1, with the steepest $\alpha _{RU}$, is also the brightest in the infrared. The infrared flux points of both extension S and of In2 lie on the continuation of the power law with index -0.4, determined on the optical spectral index map. If this power law were valid up to radio wavelengths, it would predict a flux at $\lambda 6$cm of about 50 $\mu$Jy for the northwestern knot In1, and of about 45 $\mu$Jy for the southern extension S. This is a factor of order 100 below the flux from the jet at this wavelength (Conway et al. 1993), and any emission at this level would not be detected even on our new VLA maps with an rms noise of $\approx$0.8mJy. There is no similarly obvious extrapolation of In1's SED to radio wavelengths. A full comparison will be presented in a future paper. The inner extension shows a polarisation signal on unpublished ground-based maps made by our group and on polarimetric images obtained with the Faint Object Camera on board HST (Thomson et al. 1993). A polarisation signal could be due to scattered quasar light (Röser & Meisenheimer 1991) or, together with possible radio emission, synchrotron radiation. We aim to clarify the nature of the inner extensions by planned polarimetric and spectroscopic observations with the VLT.

Figure 5: Broad-band flux points for the radio-quiet extensions to the jet from HST imaging at 1.6$\mu$, 620nm and 300nm.

   
6.2 The hot spot

The optical counterpart to the radio hot spot (H2) appears very faint on both images, and the spectral index map shows a flip from flat to steep there. Both features are explained by the Meisenheimer & Heavens (1986) hot spot model: the lifetime of electrons emitting in the optical is quenched by the strong magnetic field in the hot spot. The spectral index flip is expected if there is an offset between the emission peaks at different frequencies. This offset is predicted by the theory by Meisenheimer & Heavens (1986) and is clearly seen when comparing optical and radio images (Röser et al. 1997; a detailed comparison of all data will be presented in a future paper). By its spectrum and morphology, the hot spot appears exactly as expected for a high-loss synchrotron emission region downstream of a localized strong shock with first-order Fermi acceleration. Finally, we note that although the term "hot spot'' is historically in use for this part of the jet, the term is ill-defined and may have become inadequate, but we defer a discussion of its adequacy to a future paper.

   
6.3 Main body of the jet

Our HST images (Figs. 1 and 2) show a close coincidence of the jet's morphology over a factor of 2 in frequency (optical and UV). The spectral index map (Fig. 3) shows amazingly smooth variations in the physical conditions over the entire jet. The electrons' synchrotron cooling leaves little imprint on the spectral index along the jet, contrary to expectations.

   
6.3.1 The spectral index at HST resolution

Figure 6: Optical spectral index against red-band brightness for the outer half of the jet in 3C273 (data as in Fig. 4). There is no strict correlation like that observed in the jet of M 87.

The two striking features of our spectral index map are the smooth variation of the spectral index over the entire jet (Fig. 4), and the lack of a strong correlation between red-band brightness and spectral index (Fig. 6) like that observed in the jet of M 87 (Meisenheimer et al. 1996a,b; Perlman et al. 2001). There are large but smooth variations of the spectral index, while the surface brightness remains fairly constant over the jet's projected extent of about 10 $^{\prime \prime }$. Conversely, there are large local variations of surface brightness without strong changes in the spectral index. We do not observe abrupt changes of the spectral index inside the jet like we do in the hot spot. The run of the spectral index is thus consistent with the complete absence of energy losses over scales of many kiloparsecs, in spite of the large observed synchrotron luminosity, which indicates the need for re-acceleration of particles inside the jet. The mismatch between synchrotron cooling scale and extent of the jet even led Guthrie & Napier (1975) to conclude that the jet's emission could not be synchrotron radiation. The same lack of apparent synchrotron cooling is implied by the surprising overall correspondence between $\alpha _{BRI}$ and $\alpha _{RU}$ (Fig. 4) which have been determined at vastly different resolutions (1 $.\!\!^{\prime\prime}$3 and 0 $.\!\!^{\prime\prime}$2, corresponding to 4.2h₆₀kpc and 640h₆₀pc, respectively). The small differences (strongest in regions A and C2/D1) can be explained at least in part by the different beam sizes: the spectral index determined by smoothing the HST images to 1 $.\!\!^{\prime\prime}$3 makes the discrepancy smaller. In addition, the emission region becomes wider towards longer wavelengths, an effect already noticeable in the near-infrared $K^{\prime}$ band (Neumann et al. 1997) and attributed to a "backflow'' (Röser et al. 1996). The presence of such a steep-spectrum, diffuse component more extended than the jet channel might also explain the discrepancy. The general agreement indicates a constancy of the shape of the spectrum even at the scales resolved by HST. To first order, the appearance of the jet at all wavelengths can be explained through such a constant spectral shape consisting of a low-frequency power law and a high-frequency curved cutoff. The assumed spectral shape is shifted systematically in the $\log\nu-\log S_{\nu}$ plane, passing different parts of the curved cutoff through the optical flux point. This leads to similar morphological features at all wavelengths, in accord with observations (Conway et al. 1993; Bahcall et al. 1995). Moving the spectral shape through a constant optical flux point leads to a correlation of steeper optical spectral index with higher radio flux. This correlation reproduces the observed tenfold increase of the radio surface brightness from A to D2/H3 (Conway et al. 1993) and the overall steepening of the optical spectral index. However, this steepening happens on much longer time scales than expected from synchrotron cooling alone. Assuming an electron energy distribution with a fairly sharp cutoff, drastic jumps in $\alpha _{RU}$ would be expected at the locations of the shock fronts if the jet knots were (strong) shocks like the hot spot without extended re-acceleration acting between them. The absence of strong cooling makes it impossible to pinpoint localised sites at which particles are either exclusively accelerated or exclusively undergo strong losses. Any acceleration site must therefore be considerably smaller than the beam size we used ( 640h₆₀^-1pc), and these acceleration sites must be distributed over the entire jet to explain the absence of cooling. This, together with the low-frequency spectral index of $\approx$-0.8, corresponds to the jet-like acceleration mechanism proposed by Meisenheimer et al. (1997). Meisenheimer et al. (1996a) have presented fits of synchrotron continua to the observed SEDs of knots A, B, C, D and the hot spot H at 1 $.\!\!^{\prime\prime}$3 which have also been used by Röser et al. (2000). It is noted that the optical spectral index predicted from these spectra ( $\alpha _{{\rm fit}}$ in Fig. 4) is always steeper than the observed spectral index, that is below $\alpha _{BRI}$ and $\alpha _{RU}$. This indicates that the fitted spectrum is not fully adequate at the highest frequencies. This may again be due to contamination of the near-infrared flux by the same "backflow'' component mentioned above. The contamination would make the IR-optical spectral index (which dominates the run of the spectrum at high frequencies) steeper than the optical spectral index, as observed. Alternatively or additionally, there could be deviations of the optical-UV spectral shape from a standard synchrotron cutoff spectrum which may be interpreted as the first observational hint towards the existence of a second, higher-energy electron population producing the X-ray emission (Röser et al. 2000). Detailed statements about the adequacy of model spectra have to be deferred to a future paper considering the full radio, infrared and optical data set.

   
6.3.2 Can beaming account for the lack of cooling?

Heinz & Begelman (1997) proposed that sub-equipartition magnetic fields combined with mildly relativistic beaming could explain the lack of cooling in the jet of M 87 - which is, however, ten times shorter than that of 3C273. As an alternative to postulating re-acceleration, we consider whether low magnetic field values and beaming could lead to electron lifetimes sufficient to allow electrons to be accelerated at region A to illuminate the entire jet down to the hot spot over a projected extent of 32 h₆₀^-1kpc (the argument will become even more stringent by demanding acceleration in the quasar core). We consider the electron lifetime against synchrotron and inverse Compton cooling off cosmic microwave background photons; the synchrotron self-Compton process is negligible for electrons in the jet (Röser et al. 2000), as is Compton scattering off the host galaxy's star light. The total energy loss rate of an electron with energy E due to synchrotron radiation and inverse Compton scattering, averaged over many pitch-angle scattering events during its lifetime, is

$$% -\frac{{\rm d}E}{{\rm d}t} = \frac{4}{3} \sigma_{{\rm T}} c U_{\rm tot} \beta^2 \left(\frac{E}{m_{\rm e}c^2}\right)^2,$$ (4)

where $U_{\rm tot} = U_{{\rm CBR}}(z) + U_{\rm mag}$ is the sum of the energy densities of the background radiation and magnetic field, respectively, and $\sigma_{{\rm T}}$ is the Thomson cross-section (Longair 1994). We integrate this equation from $E=\infty$ at t=0 to E(t), assuming $\beta = 1$ (appropriate for the highly relativistic electrons required for optical synchrotron radiation) and substitute for the electron's energy $E = \gamma m_{\rm e}c^2$. Inverting yields the maximum time that can have elapsed since an electron was accelerated, given its Lorentz factor $\gamma$ (van der Laan & Perola 1969):

$$% t(\gamma) = \frac{m_{\rm e}c^2}{ \frac{4}{3} \sigma_{{\rm T}} c U_{\rm tot} \gamma}\cdot$$ (5)

This is the "electron lifetime'', inversely proportional to both the energy density in which the electron has been "ageing'', and the electron's own energy. As most of the electron's energy is radiated at the synchrotron characteristic frequency $\propto \gamma^2 B$, we can substitute for $\gamma$ in Eq. (5) in terms of the observing frequency and the magnetic field in the source. Hence, Eq. (5) becomes (in convenient units)

$$% t_{{\rm life}} = \frac{51\,000\,{\rm y}}{B_{-9,{\rm IC}}(z... ...left(\frac{B_{-9,{\rm jet}}}{\nu_{15}} \right)^{\frac{1}{2}},$$ (6)

where $B_{-9,{\rm jet}}$ is the magnetic flux density in nT of the jet field, the background radiation energy density has been expressed in terms of an equivalent magnetic field, $B_{-9,{\rm IC}} = (1+z)^2\times 0.45\,{\rm nT}$, and $\nu_{15}$ is the observing frequency in 10¹⁵Hz (van der Laan & Perola 1969). Note that as the substituted $\gamma \propto B^{-\frac{1}{2}}$, setting $B_{-9,{\rm jet}}=0$ is now meaningless. To be fully adequate for electrons in a relativistic jet at cosmological distances, the equation needs to be modified. Firstly, the frequency local to the source is (1+z) times the observing frequency because of the cosmological redshift. Furthermore, the radiating electron may be embedded in a relativistic flow with bulk Lorentz factor $\Gamma$ with three consequences: relativistic time dilation and Doppler shift, and boost of the background radiation energy density. The relativistic time dilation enhances the electron lifetime in the jet frame by a factor $\Gamma$. The Doppler shift between the emission frequency $\nu_{{\rm int}}$ in the jet frame (equal to the characteristic frequency) and the observation frequency $\nu_{{\rm obs}}$ is given by $\nu_{{\rm int}} = \nu_{{\rm obs}} {\cal D}^{-1}$, where ${\cal D}(\Gamma,\theta) = [\Gamma(1-\beta_{{\rm jet}} \cos \theta)]^{-1}$, the Doppler boosting factor for an angle $\theta$ to the line of sight (see Hughes & Miller 1991, e.g.). A relativistic flow perceives the energy density of the background radiation field boosted up by a factor $\Gamma^2$ (Dermer 1995). Inserting these yields

$$% t_{{\rm life}}({\cal D},z) = \frac{\Gamma {\cal D}^{\frac{... ...\frac{B_{-9,{\rm jet}}}{(1+z)\nu_{15}} \right)^{\frac{1}{2}}.$$ (7)

The electron lifetime attains a maximum value at a certain value of the jet's magnetic field (Fig. 7) (van der Laan & Perola 1969). On either side of the maximum, the lifetime is decreased by larger losses suffered before it is observed: for a higher magnetic field in the jet, the synchrotron cooling is more rapid ( $t \propto 1/U_{\rm tot}$, Eq. (5)).

Figure 7: Maximum age against synchrotron and inverse Compton cooling (off microwave background photons at 3C273's redshift) of an electron radiating at observed UV wavelengths plotted against the jet magnetic field. Solid line, no beaming; dashed line, relativistic Doppler boosting with $\Gamma =10$ and a line-of-sight angle $\theta =45\hbox {$^\circ $ }$.

A lower jet field requires an electron of higher Lorentz factor for emission at the given frequency, which also suffers more rapid losses ( $t \propto 1/\gamma$, Eq. (5)). By differentiation of Eq. (7), the value of the maximum lifetime is

$$% t_{{\rm max}}(B_{{\rm IC}}, {\cal D}, z) = \sqrt{\frac{\cal... ...\frac{3}{2}} \left[(1+z) \nu_{15}\right]^{\frac{1}{2}} }\cdot$$ (8)

Note that the largest possible value for the factor $\sqrt{{\cal D}/\Gamma}$ is $\sqrt {2}$. $t_{{\rm max}}$ is a firm upper limit for the lifetime of a synchrotron-radiating electron from a source at redshift z in a flow of bulk Lorentz factor $\Gamma$, whatever the magnetic field strength in the source. It is deduced only from the fact that synchrotron emission is observed at a certain frequency, and that electrons which can radiate at this frequency suffer drastic energy losses either by synchrotron or by inverse Compton cooling between the time of acceleration and the time of emission. The only additional assumptions are rapid pitch-angle scattering and a homogeneity of conditions throughout the electron's lifetime.

The VLBI jet close to the core has a line-of-sight angle near 10$^\circ$ and a bulk Lorentz factor near 10 (Abraham & Romeo 1999). A line-of-sight angle $\theta \approx 45\hbox{$^\circ$ }$ has been inferred for the flow into the hot spot from independent considerations of the jet's polarisation change there and the hot spot's morphology (Conway & Davis 1994; Meisenheimer et al. 1997). We have plotted the lifetime of an electron responsible for emission from the jet in 3C273 observed at 300nm as function of the jet's magnetic field in Fig. 7 for the extreme cases of no beaming in the optical jet and beaming identical to that in the VLBI jet with $\Gamma=10, \theta = 10\hbox{$^\circ$ }$, and for an intermediate case with $\Gamma=10, \theta=45\hbox{$^\circ$ }$ (though note that $\Gamma=10, \theta = 10\hbox{$^\circ$ }$ is unrealistic as there is a difference in position angle between the VLBI jet at 244$^\circ$ and the arcsecond jet at 222$^\circ$). The equipartition flux densities derived for the jet lie in the range of 15nT (Neumann 1995) up to 67nT for regionA (Röser et al. 2000), leading to maximum ages of 100-800 years, less than the light travel time from one bright region to the next. In the absence of beaming effects, the largest possible lifetime for electrons in 3C273 from Eq. (8) is about 58000y, again short of the required values. The "boosted lifetime'' can be at most $\sqrt {2}$ larger than this. There is thus no combination of $\Gamma, \theta$ which enhances the electron lifetime to the 100000y required for illumination of the entire jet in 3C 273 by UV-radiating electrons. Thus, the invocation of mild or even drastic beaming and/or sub-equipartition fields cannot resolve the discrepancy between the synchrotron loss scale and the extent of the optical jet of 3C273, as has been possible for the jet in M 87 (Heinz & Begelman 1997). As another alternative to invoking quasi-permanent re-acceleration, the existence of a "loss-free channel'' in which electrons can travel down a jet without synchrotron cooling has been proposed by Owen et al. (1989). As an extreme version of this case, we assume that the electron travels along the jet in zero magnetic field and is observed as soon as it enters a filament with magnetic field $B_{{\rm fil}}$. The energy loss between acceleration and synchrotron emission is then only due to inverse Compton scattering. The lifetime in 3C273 is then 130000y $\times \Gamma^{-1} \sqrt{B_{{\rm -9,fil}} {\cal D}/\nu_{15}}$. Again, if the jet flow in 3C273 is highly relativistic, the electrons suffer heavy inverse Compton losses and the lifetime mismatch persists. In any case, it remains to be shown that the "loss-free channel'' is a physically feasible configuration of an MHD jet.