Non-Thermal Models

Some estimated parameters of the jet are given in Tab. 5. It is apparent that the microwave background radiation dominates the radiant energy within the jet, unless the radius of the radio synchrotron-emitting region is much smaller than the observed radius of the X-ray jet. The X-rays from the jet could be either inverse Compton or synchrotron radiation, which we discuss in turn. Evaluation of these models is limited by our ignorance of the jet's radio spectrum. For concreteness, we shall adopt a jet radius of 1 $^{\prime\prime}$ (950 pc) for both the radio and X-ray emission.

i) X-rays as inverse Compton radiation

Adopting $\alpha_{\rm {r,jet}}$ = 0.87, S $_{\rm {jet, 1.4 GHz}}$ = 152 mJy for the part of the jet detected in X-rays (Section 4.3.1), and the equipartition field H$_{\rm {jet}}$ = 2.3 $\times$ 10$^{-5}$ gauss, the predicted inverse Compton scattered X-ray flux falls a factor of $\simeq$ 500 below that observed. In order to match the observed X-ray flux, we need to reduce the magnetic field to H$_{\rm {jet}}$ $\simeq$ 7 $\times$ 10$^{-7}$ gauss, a factor of $\simeq$ 30 below equipartition. Alternatively, as for the hot spot, we could invoke a radiation beam from the nucleus to boost the radiation density in the jet. To obtain the observed X-ray flux from the jet while retaining the equipartition field requires an (isotropic) nuclear luminosity of $\sim$ 2 $\times$ 10$^{48}$ erg s$^{-1}$, which is implausibly high. We could also suppose, as we did for the hot spot, that the observed radio jet contains a stronger (e.g. equipartition) magnetic field and there is, in addition, a population of electrons in a weak or absent field. Because we do not know the radio spectrum of the jet, and have thus not been able to demonstrate that the radio and X-ray spectral indices are different, we cannot prove that the radio- and X-ray-emitting electrons represent different populations, as we could for the hot spot.

The above discussion neglects relativistic boosting or diminution by possible bulk relativistic motion of the jet. There is now a strong case for energy transport at bulk relativistic velocities to the hot spots in FRII sources (e.g. Bridle 1996). For emission which is isotropic in the rest frame and has a power law spectrum, the observed flux density, $F_{\nu}(\nu)$, is related to the flux density in the rest frame, $F^{\prime}_{\nu}(\nu)$, by

$$F_{\nu}(\nu)\ = \delta^{3 + \alpha}F^{\prime}_{\nu}(\nu)$$ (5)

where $\delta = [\Gamma(1\ - \beta\ \cos\theta)]^{-1}$, $\Gamma$ is the Lorentz factor of the bulk flow, $\beta$ is the bulk velocity in units of the speed of light, and $\theta$ is the angle between the velocity vector and the line of sight (e.g. Urry & Padovani 1995). This equation assumes that the emission comes from a discrete, moving source. For a smooth, continuous jet, the exponent 3+$\alpha$ becomes 2+$\alpha$ (Begelman, Blandford & Rees 1984). Equation (5) will describe relativistic boosting or diminution of the jet's synchrotron radiation, as long as that radiation is isotropic in the jet's rest frame.

As noted above (Tab. 5), the radiation density in the jet is dominated by the microwave background radiation, which is isotropic in the observer's frame and anisotropic in the rest frame of the jet. In this case, the principal dependence of the inverse Compton scattered radiation on $\theta$ is given by (Dermer 1995)

$$F_{\nu}(\nu)\ = \delta^{4 + 2\alpha}F^{\prime}_{\nu}(\nu)$$ (6)

where an additional term which depends slowly on $\cos\theta$ has been omitted (Begelman & Sikora 1987; Dermer 1995). Retaining the discrete source model, the ratio of inverse Compton scattered to synchrotron flux is then

$$F_{C,\nu}(\nu)/F_{S,\nu}(\nu) \propto \delta^{1 + \alpha}.$$ (7)

The orientation of the jet in Pictor A w.r.t. the line of sight is unknown. However, various considerations (Section 4.1) indicate that $\theta$ is at least 23$^{\circ }$ and probably larger. For this value of $\theta$, the maximum possible value of $\delta^{1 + \alpha}$ is $\sim$ 5.8 (achieved at $\Gamma$ $\simeq$ 2.6), and this number is smaller for larger $\theta$. We conclude that the factor of 500 discrepancy between the predicted and observed inverse Compton X-rays for an equipartition field cannot be resolved by differential relativistic boosting.

The following simple fact argues against inverse Compton scattering for the jet's X-ray emission: in the radio band, the western lobe dominates the jet, while in X-rays the converse is true. The ratio of the rates of energy loss to inverse Compton scattering and synchrotron radiation is

$${{(\rm dE/dt)_{IC}}\over{(\rm dE/dt)_{synch}}} = {{\epsilon_{rad}}\over{\epsilon_{mag}}}.$$ (8)

The radiant energy density in both the jet and western lobe is dominated by the microwave background radiation. The equipartition magnetic field in the jet is $\simeq$ 2.3 $\times$ 10$^{-5}$ gauss (for r$_{\rm j}$ = 1 $^{\prime\prime}$, see Tab. 5) and that in the lobe is $\sim$ 5 $\times$ 10$^{-6}$ gauss. Thus one expects that the ratio (dE/dt)$_{\rm IC}$/(dE/dt)$_{\rm synch}$ should be larger for the lobe than the jet. Given that the lobe's radio synchrotron radiation overwhelmingly dominates that of the jet, the lobe's inverse Compton emission should dominate the jet by an even larger factor, contrary to observation. However, the jet's inverse Compton emission would be larger if either a) its magnetic field is well below equipartition, or b) a narrow beam of radiation is emitted by the nucleus along the jet, providing a larger $\epsilon_{rad}$ than the microwave background, as discussed above. Both of these possibilities are ad hoc and so we consider the prominence of the jet compared to the lobe in the Chandra image as an argument against inverse Compton scattering.

ii) X-rays as synchrotron radiation

As already noted, the available data are consistent with a single power law with $\alpha_{\rm {rx,jet}}$ = 0.87 from 1.4 GHz to 10 keV. This is, however, an unlikely physical situation for a purely synchrotron spectrum in view of the short energy loss times of the X-ray emitting electrons to synchrotron and inverse Compton radiation. For electrons emitting synchrotron radiation at 1.4 GHz and 1 keV in the equipartition field of 2.3 $\times$ 10$^{-5}$ gauss, the times to lose half their energy are $\simeq$ 4 $\times$ 10$^{6}$ and 300 yrs, respectively. If the energies of the radio-emitting electrons are not significantly reduced by synchrotron and inverse Compton losses, we would expect the radio spectrum to be flatter than the X-ray spectrum; in the simplest case involving continuous injection, $\alpha_{\rm {r,jet}}$ = $\alpha_{\rm {x,jet}}$ - 0.5 $\simeq$ 0.4. It would thus be valuable to measure $\alpha_{\rm {r,jet}}$.

For a synchrotron model in which the electrons are isotropic in the frame of bulk jet motion, the observed ratio of flux in the approaching side of the jet to the receding side is given by

$$R=\frac{F_{\nu_{\rm app}}(\nu)}{F_{\nu_{\rm rec}}(\nu)}=\left( \frac{1+\beta\cos\theta}{1-\beta\cos\theta}\right)^{3+\alpha}.$$ (9)

From X-ray observations $\alpha\approx0.9$ and $R\ge10$ at the brightest part of the X-ray jet. Assuming $\theta$ $>$ 23$^{\circ }$ (section 4.1), then $\beta$ $>$ 0.3. This is approximately three times larger than the inferred average recessional speed of hot spots from the nucleus. If the X-rays are the result of inverse Compton scattering of the microwave background radiation, the exponent 3+$\alpha$ in equation (9) becomes 4+2$\alpha$ (cf. equation 6) and a somewhat smaller lower limit to $\beta$ is obtained.