Non-Thermal Models

Inverse Compton and Synchrotron Emission from the Radio and Optical-Emitting Electron Population

It has long been realised that the outward motion of hot spots in FRII radio galaxies is sub-relativistic. The most recent work (Arshakian & Longair 2000) obtains a mean outward velocity of 0.11c $\pm$ 0.013c. While the velocity of a hot spot in any individual object is uncertain, we shall neglect boosting or diminution of the hot spot fluxes by bulk relativistic motion.

The broad-band spectrum (Fig. 8) shows that the X-ray emission of the western hot spot cannot be synchrotron radiation from a high energy extension of the population of electrons responsible for the radio to optical synchrotron radiation. It is, therefore, natural to consider inverse Compton scattering for the X-ray emission. In the bright core of the hot spot, the radiant energy density ( $\epsilon_{rad}$) of the synchrotron radiation dominates that of the microwave background radiation and the galaxy starlight (Tab. 3), so it is appropriate to consider a synchrotron self-Compton model. When the scattering electrons follow a power-law distribution in $\gamma$ (= E/mc$^{2}$), n($\gamma$) = n $_{e_0}\gamma^{-\rm {p}}$ over all $\gamma$, then the inverse Compton spectrum is a power law with a spectral index $\alpha_C = (\rm {p}-1)/2$, the same index as the synchrotron spectrum. The fact that the radio ($\alpha_{rad}$ = 0.740 $\pm$ 0.015) and X-ray ( $\alpha_{X-ray}$ = 1.07 $^{+0.11}_{-0.11}$) spectral indices are different then suggests difficulties with an inverse Compton model. However, in reality, the electron energy spectrum is a power law over only a certain range of $\gamma$ (from $\gamma_{min}$ to $\gamma_{max}$), and the synchrotron spectrum is a power law over only a certain range of $\nu$ (from $\nu_{s,min}$ to $\nu_{s,max}$). These limited ranges yield ``end effects'' in synchrotron self-Compton spectra: the inverse Compton scattered spectra must turn down below $\nu_{c,min} \sim 4\gamma^{2}_{min} \nu_{s,min}$ and above $\nu_{c,max} \sim 4\gamma^{2}_{max} \nu_{s,max}$ (e.g. Blumenthal & Gould 1970; Rybicki & Lightman 1979). Further, the synchrotron self-Compton spectrum is no longer an exact power law between these limits.

In order to provide a more realistic evaluation of inverse Compton and synchrotron self-Compton models, we have performed numerical calculations of spectra in spherical geometries using the computer code of Band & Grindlay (1985, 1986), which was kindly provided by Dan Harris. Given the observed radio - optical spectrum (Fig. 8), we have assumed a power law electron spectrum with $p$ = 2.48 from $\gamma_{min}$ to $\gamma_{max}$. The magnetic field is treated as a variable. We first computed a synchrotron self-Compton spectrum which passes through the Chandra-measured flux (model 1). This was achieved for a magnetic field strength of $3.3\times10^{-5}$ gauss, a factor of 14 below the equipartition field in the radio ``core'' of the hot spot (Tab. 4). However, the predicted spectrum is similar to that of the radio source and does not match the measured X-ray spectrum (Fig. 8).

As an alternative to reducing the field strength, the power radiated in inverse Compton radiation could be increased by increasing the radiation density. The nucleus of Pictor A might emit a narrow, collimated beam of radiation (like those inferred to be present in BL Lac objects) along the axis of the jet. This beam could illuminate the hot spot, but be invisible to us. However, an (isotropic) nuclear luminosity of 1.6 $\times$ 10$^{48}$ erg s$^{-1}$ is needed merely to equal the radiant energy density in synchrotron radiation in the core of the hot spot. The (isotropic) nuclear luminosity would have to be $\simeq$ 1.5 $\times$ 10$^{50}$ erg s$^{-1}$ to provide sufficient radiation for the hot spot to radiate the observed X-ray flux by inverse Compton scattering if the field has its equipartition value. This luminosity is unreasonably large.

An alternative is that the X-rays are a combination of synchrotron and inverse Compton radiation. In model 2, we assumed that the turnover in the synchrotron spectrum at 10$^{13-14}$ Hz is an effect of synchrotron losses on a continuously injected electron spectrum with energy index $p$ = 2.48. The assumed electron spectrum is thus a broken power law (Kardashev 1962):

$$n_e=n_{e_0}\gamma^{-p} \gamma \ll\ \gamma_{\rm break}$$ (3)

$$n_e=n_{e_0}\gamma_{\rm break}\gamma^{-(p+1)} \gamma \gg\ \gamma_{\rm break}$$ (4)

The synchrotron spectrum then has an index $\alpha$ = 0.74 well below the break and $\alpha$ = 0.74 + 0.5 = 1.24 is expected well above it (dotted line in Fig. 8). By adding this spectrum to the predicted synchrotron self-Compton component for a magnetic field of $5.3\times10^{-5}$ gauss (dot-dashed line in Fig. 8), which is again well below equipartition, we obtain the solid line in Fig. 8, which is a good description of the Chandra spectrum. The parameters of model 2 are given in Tab. 4. Because the energy density in magnetic field is $\gtrsim$ an order of magnitude larger than the synchrotron radiation density, the power in the first order synchrotron self-Compton component is less than that in the synchrotron radiation (e. g. Rees 1967). Further, the second order scattered component is even weaker (3 orders of magnitude below the first order radiation in the Chandra band).

Assuming that the electrons are continuously accelerated in the hot spot, we may interpret the turnover frequency as the frequency at which the ``half-life'' to synchrotron losses of the radiating electrons is equal to their escape time from the hot spot. For the field of $5.3\times10^{-5}$ gauss needed to match the X-ray spectrum, electrons radiating at 10$^{14}$ Hz have a synchrotron ``half-life'' of $\simeq$ 10$^{4}$ yrs. For a hot spot radius of $\simeq$ 250 pc, the corresponding streaming velocity of the relativistic electrons is $\simeq$ 0.1c. The problem with model 2 is the fine tuning needed: synchrotron and synchrotron self-Compton emission must be present in the Chandra band with comparable fluxes.

Inverse Compton and Synchrotron Emission from Hypothetical New Electron Populations

In the previous subsection, we showed that the X-ray emission is so strong that the magnetic field in the hot spot must be a factor of $\sim$ 14 below equipartition if the X-rays are produced by inverse Compton scattering from the electrons that generate the observed radio emission. Even then, the observed X-ray spectrum is different to the predicted one (Fig. 8), and so this model (model 1, Tab. 4) may be ruled out. The other process capable of generating the observed X-rays is synchrotron radiation. However, the X-ray emission of the hot spot is not a smooth continuation of the radio - optical synchrotron spectrum. In model 2, we contrived to fit the X-ray spectrum by a combination of synchrotron emission from an extension of the radio-optical spectrum and synchrotron self-Compton from the radio-optical synchrotron-emitting electrons. In view of the unsatisfactory nature of these models, we now consider inverse Compton and synchrotron models involving hypothetical new electron populations.

i) X-rays as inverse Compton radiation

For a successful inverse Compton scattering model, a low energy population of relativistic electrons with an index of the energy spectrum p $\simeq$ 2$\alpha_{X}$ + 1 is needed. The required value of p is thus p$_{{\rm hs}}$ $\simeq$ 3.14 $^{+0.22}_{-0.22}$. Fig. 9 shows a model (model 3, Tab. 4) for the hot spot in which a synchrotron-emitting component with a larger spectral index than that observed in the radio has been added at low radio frequencies. The spectrum of the hot spot has not been measured below 327 MHz, but the extension of this hypothetical spectrum down to lower frequencies does not exceed the measured integrated flux density of Pictor A (the lowest such measurements are at $\sim$ 80 MHz, see PRM). It is envisaged that the population of electrons which emits the observed synchrotron radio emission is in a relatively strong (e.g. equipartition, Tab. 3) magnetic field, so that the synchrotron self- Compton emission from this electron population is negligible, as shown above. However, the observed radio-optical synchrotron radiation, as well as the radio synchrotron radiation of the hypothesised component, is available for scattering by the hypothetical low energy electron population. As can be seen from Fig. 9, the low energy electrons must radiate predominantly through the inverse Compton channel, requiring a very weak magnetic field (model 3, Tab. 4) which is $\sim$ 100 times weaker than the equipartition field in the core of the western hot spot. The calculation shows that the second order Compton scattered component is $\gtrsim$ 2 orders of magnitude weaker than the first order Compton scattered component in the Chandra band. We find that the spectrum of the low energy electrons has to be steep (p = 3.3) as shallower spectra (p $<$ 3.3) cause the inverse Compton spectrum to increase above $10^{17}$ Hz, contrary to our X-ray observations. Taking this model to its extreme, the hypothetical electron population could be in a region free of magnetic field, and thus radiate no radio emission at all. This requirement for a weak or absent magnetic field associated with this population is difficult to understand in view of the finding that the X-ray, optical and radio emitting regions of the western hot spot are co-spatial (Section 3.2.1 and Fig. 3). High resolution radio observations at lower frequencies would provide stronger constraints on this model.

ii) X-rays as synchrotron radiation

Alternatively, the X-ray emission of the hot spot could be synchrotron radiation from a separate population of electrons. In such models (models 4a and b), the energy index of the X-ray emitting electrons would be p$_{{\rm hs}}$ $\simeq$ 3.14 $^{+0.22}_{-0.22}$, similar to the above inverse Compton model. In view of the short synchrotron loss times (a few years for a 5 keV-emitting electron in the equipartition field), the energy index at injection would be p $_{{\rm hs, inj}}$ $\simeq$ 2.14 $^{+0.22}_{-0.22}$ in a steady state, continuous injection model. The synchrotron break frequency must thus be below the X-ray band but above 10$^{11}$ Hz (otherwise the emission would exceed the observed flux at the latter frequency). Models 4b and 4a (Tab. 4), in which we have adopted the equipartition field, represent these two extremes for the synchrotron break frequency, respectively. Thus, in both these models, the extrapolation of the X-ray spectrum to lower frequencies does not exceed the observed radio, infrared or optical flux. Interestingly, the required value of p $_{{\rm hs, inj}}$ is the same to within the errors as the canonical index for particle acceleration by a strong shock (p = 2, e.g. Bell 1978; Blandford & Ostriker 1978). In this picture, electrons would have to be continuously reaccelerated by such shocks on pc scales.