A MULTI-WAVELENGTH STUDY OF THE JET, LOBES, AND CORE OF THE QUASAR PKS 2101−490

Figure 1 illustrates the radio structure of the source and the naming convention used for the various features in the radio maps. Figure 2 illustrates the polarization characteristics of the jet. We have extracted radio, optical, and x-ray flux densities from the seven major emission regions identified in Figure 1. In this section, we describe the observations in each waveband, as well as the methods used to measure flux densities and sizes for the individual jet knots. Table 1 lists the observational information for all data used in this study.

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PKS 2101−490 was observed with the ATCA at 4.8 GHz and 8.64 GHz in two configurations, 1.5A and 6A, on 2000 May 25 and 2000 September 2, respectively, and in a single configuration (6C) at 17.7 GHz and 20.2 GHz on 2004 May 10. In each case, a full 12 hr synthesis was obtained, recording 128 MHz bandwidth in all four polarization products. Regular scans on the nearby phase calibrator PKS 2106−413 were scheduled throughout the observations, as well as scans on the ATCA flux calibrator PKS 1934−638. Standard calibration and editing procedures were carried out using the MIRIAD data analysis package. Following calibration, the data were exported to DIFMAP and several imaging/self-calibration iterations were performed. The data were both phase and amplitude self-calibrated.

2.2.1. Radio Knot Flux Density and Size Measurements

PKS 2101−490 was observed with the Advanced Camera for Surveys (ACS) Wide Field Camera 1 (WFC1) on the HST in two filters (F475W and F814W) on 2005 March 8. A total exposure time of 2.3 ks was obtained in each filter. A sub-pixel dithering pattern, with CR-SPLIT images at each of three positions along a chip diagonal, was utilized to eliminate bad pixels and allow us to maximize the angular resolution, as the ACS/WFC does not fully sample the PSF at either 4750 or 8000 Å. An ORIENT was chosen such that the jet did not fall within 25° of a diffraction spike. The data were retrieved from the Multi-mission Archive at Space Telescope (MAST) Web site,¹³ however, multiple peaks in the pipeline drizzled image of the quasar core suggested that a re-reduction of the data was required. In addition, the pipeline does not take into account sub-pixel dithering, which was performed in order to recover PSF information from the undersampling of the HST PSF. The re-reduction involved running MULTIDRIZZLE with a smaller PIXFRAC and scale, so that the images could be sub-sampled to 0.0247 arcsec pixels, and checking the alignment of the images with TWEAKSHIFTS. The best reference files indicated in the HST archive were used for the re-reduction. The position of the optical quasar core was aligned with the position of the radio core. This required shifting the optical data by approximately 04.

We used TinyTIM simulations (Krist & Burrows 1994; Suchkov & Krist 1998; Krist & Hook 2004) to obtain subsampled PSF simulations for both bands. For those simulations, we assumed an optical spectrum of the form F_ν∝ν⁻¹, although experience has shown that the PSF shape is not heavily dependent on spectral slope. Rotating the PSF to a north-up position for use with the drizzled images required independently rotating the two axes as the WFC detector's axes are not completely orthogonal on the sky. We normalized the PSF to the total number of counts in a 2'' circular aperture centered on the source. This takes advantage of the fact that charge "bleed" on the ACS is linear and charge is conserved for a saturated source (Gilliland 2004), and enables optimal matching of the PSF's outer portion to what is observed. Because the quasar core was saturated in the individual exposures, this inevitably led to negatives in the central pixels when PSF subtraction was done; however, given the small residuals in other places plus the smooth off-jet isophotes, we believe that the result is reliable. Subtraction of the PSF allowed us to look for optical jet components within 1''–2'' and resulted in a detection of Knot 1 in both bands.

Figure 3 shows the resulting optical maps with radio contours overlaid. In these images, the HST data have been smoothed with a 0.3 arcsec FWHM Gaussian to better show the optical emission from knot K6, which is clearly detected in both filters. It is interesting to note that the F814W image of K6 reveals an elongated, double structure that is not apparent in the F475W image of K6. We further note that the position of the optical emission from K6 is coincident with the radio position to within the uncertainties associated with the optical–radio image alignment. We consider the interpretation of the optical data in Section 5.

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2.3.1. Optical Flux Density Measurements

2.3.2. The Extent of Optical Emission from Knot 6

We estimated the size of the optical emission from K6 as follows. First, the optical images were convolved with 015 FWHM Gaussian. We then produced integrated profiles both parallel and perpendicular to the jet axis for the knot K6, a nearby star, and the TinyTim generated PSF using the ds9 projection capability. From the integrated profiles, we measured the FWHM of each¹⁵ feature to be Θ_K6, Θ_star, and Θ_PSF. The profiles of the star and TinyTim PSF are in good agreement with each other, and indicate that in the smoothed maps, the PSF FWHM is Θ_PSF ≈ Θ_star ≈ 018 for both the F475W and F814W bands. We then calculated the de-convolved (intrinsic) extent of the optical emission associated with the knot K6 as $\Theta _{\rm K6, intrinsic} = \ssty\sqrt{\Theta _{\rm K6}^2 - \Theta _{\rm PSF/Star}^2}$. The measured PSF sizes in the smoothed HST maps correspond to an un-smoothed PSF of 011 in both bands, as expected for ACS/WFC. We note that the profile of K6 along the jet in the F814W band appears double peaked, in contrast to the F475W profile which is single peaked (see Figures 3 and 4).

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The optical knot appears smaller than the associated radio knot; in both bands, the cross-jet width of the optical emission associated with K6 is significantly less than the measured width in the radio band (Table 3). The length of K6 parallel to the jet in the F814W map appears marginally resolved, while the length of K6 in the F814W map is clearly resolved, and appears to consist of two peaks (Figure 4). A discussion of the optical emission from K6 is presented in Section 5.

PKS 2101−490 was observed with the Chandra x-ray observatory on 2004 December 17 (Cycle 6) using the Advanced CCD Imaging Spectrometer (ACIS-S) for a total exposure time of 42 ks (Chandra ObsID 5731). To reduce the effect of pile-up in the quasar core, a 1/4 subarray mode was used with a single CCD, so that the frame time was 0.8 s. The source was positioned close to the read-out edge of the CCD to reduce the effect of charge transfer inefficiency. A new Event 2 data file was made with pixel randomization removed, and the x-ray and radio core positions were aligned, requiring a shift of the x-ray data by approximately 04. The data were restricted to the energy range 0.5–7 keV.

2.4.1. X-ray Flux Density Measurements

The x-ray flux densities for individual knots were obtained by calculating the background-subtracted counts in each knot region and multiplying by the conversion factor. The conversion factor was estimated by fitting an absorbed power-law spectrum to the x-ray counts extracted from the whole jet. The x-ray spectra of the jet and lobe were fitted using the Sherpa software package by minimizing the Cash statistic (related to the log of the likelihood). The instrument response functions (RMF and ARF) were determined from the CALDB calibration database appropriate for the position of the source on the ACIS-S3 chip. In all model fits, the neutral hydrogen column density was fixed at the Galactic value 3.4 × 10²⁰ cm⁻² as determined from the COLDEN¹⁶ column density calculator provided by the Chandra X-ray Center. The uncertainties for the flux density and spectral index were calculated using the "Covariance" routine in Sherpa. The results of this routine are valid provided that the surface of log likelihood is approximately shaped like a paraboloid. The "Interval-Projection" routine in Sherpa was used to verify that this condition was met.

We extracted a total of 138 counts from the jet having energies in the range 0.5–7.0 keV. Fitting a power law gives 1 keV flux density S = 3.3 ± 0.4 nJy and spectral index α_X = 0.85 ± 0.2. This implies that the conversion factor is G ≈ 1.0 μJy count⁻¹ s⁻¹ for a spectral index of α ≈ 0.85. In the counter lobe, we extracted a total of 55 counts in the energy range 0.5–7.0 keV. Following the same procedure as for the jet, we find flux density S = 1.5 ± 0.2 nJy at 1 keV and spectral index α = 1.3 ± 0.3, implying the conversion factor is G = 1.15 μJy count⁻¹ s⁻¹ for a spectral index α ≈ 1.3. These estimates of the conversion factor are consistent with the predictions of the Chandra proposal planning toolkit.

Figure 5(A) is a comparison of x-ray and radio structure in the jet of PKS 2101−490 and Figure 5(B) illustrates the regions used to extract x-ray counts for the knots. The regions shown in Figure 5(B) (except for the hotspot extraction region—this region is discussed further below) have sides ≳ 16. Knots 5, 6, and the hotspot are difficult to separate in the Chandra image. In order to estimate the counts associated with the hotspot and avoid contamination from knot 6, a region encompassing only one side of the hotspot is used (the side furthest from knot 6). Due to the background in the vicinity of the hotspot and the possibility of contamination from knot 6, the few counts within this aperture may not be associated with the hotspot, and therefore the flux density estimate for the hotspot is an upper limit. An aperture correction of two is used when calculating the upper limit, since the flux extraction region encompasses only one side of the hotspot.

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In the Chandra image, knot 1 is blended with the wings of the x-ray core. To place a limit on the x-ray flux density from knot 1, a sector of an annulus centered on the core was used, as shown in Figure 5(B). The background was estimated using the section of the annulus excluding knot 1. The data are consistent with zero counts from knot 1.

2.4.2. The Extent of X-ray Emission from Knot 6

Figure 6 illustrates the radio and x-ray longitudinal jet profiles. The x-ray surface brightness is greatest near the end of the jet. This is in contrast to the trend that is seen in sources such as 3C273 (e.g., Marshall et al. 2001; Sambruna et al. 2001), 0827+243 (Jorstad & Marscher 2004), and 1127–145 (Siemiginowska et al. 2002, 2007), where the x-ray brightness peaks closer to the core and the brightness of the knots decreases with distance from the core (see Georganopoulos & Kazanas 2004, for a discussion of this phenomenon).

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The spectral energy distributions of the jet knots are typical of quasar x-ray jet knots such as those in PKS 0637−752: a single or broken power law is unable to fit the entire radio to x-ray SED, so that two spectral components are required to model the data (see Figure 7). The x-ray spectral index of the entire jet α^7.0  keV_0.5  keV = 0.85 ± 0.2 is consistent with the radio spectral index of the entire jet α_R = 0.81 ± 0.01. The data are therefore consistent with an IC interpretation for the x-ray emission.

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3.2.1. Synchrotron Self-Compton Model

3.2.2. IC/CMB Modeling

In this section, we model the knot x-ray emission in terms of IC scattering of CMB photons in a highly relativistic jet directed close to the line of sight (Tavecchio et al. 2000; Celotti et al. 2001).

Assumed form of the electron energy distribution: in order to apply the analytic one-zone IC/CMB model (Dermer 1995; Harris & Krawczynski 2002; Worrall 2009), we assume a single power-law electron energy distribution N(γ) = K_eγ^−a between a minimum and a maximum Lorentz factor, γ_min and γ_max, with a = 2.6 (corresponding to synchrotron spectral index, α = 0.8). We assume γ_max = 10⁵ for all knots. The assumed value for γ_max allows the synchrotron spectrum to cutoff at a frequency below that of the HST observing frequencies. The value of the high energy cutoff is not well constrained, however, the conclusions drawn from this model are not sensitive to the assumed value of γ_max (see Schwartz et al. 2006a, Appendix A). In contrast to the single power-law electron energy distribution assumed here, in Section 5 we model the synchrotron spectrum of Knot 6 in terms of a broken power law. We note that the results of the IC/CMB model are insensitive to the details of the high-energy end of the electron energy distribution, provided the break in the distribution occurs at γ_break ≫ γ_min. Therefore, the assumption of a single power-law distribution in this section does not affect the conclusions drawn from this model for Knot 6, nor does it alter the conclusions drawn in later sections.

The low energy cutoff, γ_min, is constrained so that the model does not produce optical IC/CMB emission above the HST upper limits or detections. This constraint on γ_min is possible because an extrapolation of the IC/CMB spectrum from x-ray to optical frequencies lies above the optical upper limits, and in the case of Knot 6, is comparable to the detection level but with an incorrect spectral index. In addition to this constraint, γ_min must not be so high that there is no significant x-ray emission produced at 0.5 keV. As we show in the following sections, the IC/CMB model requires a Doppler factor δ ∼ 6 at least in some parts of the jet. Assuming ν^ic/cmb_min ≈ 1.6 × 10¹¹ × δ²γ_min² Hz (see, e.g., Worrall 2009), we are able to constrain the value for γ_min to lie in the approximate range 10 ≲ γ_min ≲ 200. Mueller & Schwartz (2009) find marginal evidence for a value γ_min ≈ 50 in the jet of PKS 0637−752 based on spectral fitting of the jet x-ray spectrum. We therefore adopt γ_min = 50 in the following analysis.

The model and assumptions: we use the standard one-zone IC/CMB model equations (see Worrall 2009) assuming a continuous jet geometry (i.e., S_ν∝δ^{2 + α} for the synchrotron flux density). Many of the jet knots appear elongated along the jet axis. Therefore, since this model requires the jet viewing angle to be small, we take into account projection effects in the calculation of knot volumes. Without independent constraints on the jet viewing angle, we simply assume a representative value in order to make an approximate de-projection. As we show in the following sections, the IC/CMB model requires a Doppler factor of the order of δ ≳ 6 at least in some parts of the jet. This implies that the jet viewing angle must be ≲ 9°. Angles significantly less than 9° are unfavorable since that would imply uncomfortably large source size. For example, if the jet lies closer than 5° to the line of sight, then the total source size must be >2.3 Mpc. Therefore, we assume a jet viewing angle of θ = 9° in making an approximate de-projection of the knot length along the jet axis, and note that this angle corresponds approximately to the maximum jet viewing angle given the derived Doppler factor δ ≳ 6. The de-projected volumes, V, are related to the projected volumes V₀ as V = V₀/sin θ. The projected volumes V₀ are listed in Table 2. For the purposes of the current calculations, and to reduce the number of model parameters, we further assume that the proton contribution to the particle energy density is negligible.

The results of IC/CMB modeling are presented in Table 4. Included in the table is the jet kinetic energy flux, which we calculate based on the derived jet parameters in the case of purely leptonic and electron/proton jets, using Equation (B17) of Appendix B in Schwartz et al. (2006a), with the following simplifying assumptions: the bulk Lorentz factor of the jet Γ ≫ 1, equipartition between particle and magnetic field energy densities, and a tangled magnetic field geometry such that 〈B'²_⊥〉 = (2/3)B². With these assumptions, the expression for kinetic luminosity in the case of a purely leptonic and electron/proton jet, respectively, are (in cgs units)

$$\begin{eqnarray*} L_{{\rm jet}}^{e^{+/-}} &\approx & \pi R^2 \Gamma ^2 c \left(\frac{B^2}{3 \pi } \right) {\rm erg\,s^{-1}} \nonumber \\ L_{{\rm jet}}^{e^{-} p} &\approx & \pi R^2 \Gamma ^2 c \left(\left(\frac{\Gamma - 1}{\Gamma } \right) n_e m_p c^2 + \frac{B^2}{3 \pi } \right) {\rm erg\,s^{-1}.} \nonumber \end{eqnarray*}$$

In the case of an electron/proton jet, we assume one cold proton for each relativistic electron. The electron density, n_e, in cgs units, is calculated from the spectral fits as

$$\begin{equation} n_e \approx \frac{B^2}{8 \pi m_e c^2} \left(\frac{a-2}{a-1} \right) \gamma _{\rm min}^{-1}, \end{equation} \tag{ 1 }$$

where the magnetic field strength B is in Gauss.

Table 4. IC/CMB Modeling Results

Knot ID	$\frac{F_{\rm 1 keV}}{F_{\rm 17.7 \: GHz}}$	δ	Beq	ne	$L^{e^{-}e^{+}}_{\rm jet}$	$L^{e^{-}p}_{\rm jet}$
	(× 10−8)		(μG)	(× 10−6 cm−3)	(× 1046 erg s−1)	(× 1046 erg s−1)
Knot 1	<4	<6	>40	>0.6	<0.5	<3
Knot 2	<6.5	<7	>40	>0.5	<0.2	<1
Knot 3	14 ± 4	7	30	0.2	0.5	3
Knot 4	28 ± 7	7	20	0.1	0.2	1
Knot 5	20 ± 10	7	20	0.2	0.3	1
Knot 6	11 ± 3	6	30	0.3	0.7	4

Notes. Assumed model parameters: γ_min = 50, γ_max = 10⁵, α = 0.8, jet viewing angle = 9° (for de-projection of knot volumes given in Table 2), ratio of proton-to-electron energy densities (_p/_e) ≪ 1. To calculate jet powers, we have assumed Γ = δ.

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The model-dependent results presented in Table 4 suggest that if the results of this model are correct, while the jet Lorentz factor remains approximately constant along the jet, the magnetic field and particle density decrease by a factor of a few between the innermost knots and the outer knots. The results indicate that there is modest, if any, loss of kinetic luminosity along the jet.

The jet of PKS 2101−490 undergoes an apparent bend of approximately 45° at about half way between the core and jet termination, somewhat reminiscent of the jet morphology in PKS 0637−752 (Schwartz et al. 2000). However, unlike PKS 0637−752, the x-ray emission in PKS 2101−490 is detected after the jet bend. This is of interest because a change in the jet viewing angle produces a change in the Doppler factor, which should manifest itself as change in knot brightness and the x-ray to radio flux density ratio. However, there is not a unique relationship between apparent bend angle and change in jet viewing angle. Therefore, the fact that x-ray emission continues beyond the bend in 2101–490 cannot be used to constrain the emission mechanism. It may simply be the case that the bend in 0637–752 is associated with a significant increase in the viewing angle (and hence a significant decrease in the Doppler factor), while the bend in 2101–490 may be associated with a relatively small change, or a decrease in viewing angle (and hence a small change or increase in the Doppler factor). Having said that, on a population basis, the probability distribution of change in the jet viewing angle for a given apparent bend angle may enable a useful constraint on the emission mechanism.

In addition to the morphological similarities to PKS 0637−752, we note a striking similarity between PKS 2101−490 and PKS 0920−397 (Schwartz et al. 2010), particularly in the vicinity of the jet termination. In both these sources, a bright, compact knot approximately 1 arcsec (>10 kpc de-projected) upstream from the jet termination is detected in radio, optical, and x-ray bands. A similar, x-ray bright pre-hotspot jet knot is seen in 1354+195 (Sambruna et al. 2002). One possible interpretation is that these pre-hotspot jet knots may be associated with an oblique shock that is produced as the jet enters the highly turbulent region at the head of the cocoon. Numerical simulations of extragalactic jets show that as the jet approaches the hotspot, it encounters an increasingly violent environment, with strong turbulence that can perturb the jet flow, inducing instabilities or directly causing oblique shocks to form due to density/pressure gradients in the lobes (e.g., Norman 1996). The pre-hotspot jet knots may be the result of the jet entering an increasingly violent environment near the hotspot.

It is possible to estimate the jet power based on observed hotspot properties by applying the conservation of momentum and making a number of reasonable assumptions about the hotspot. In the following section, we carry out this approach and compare the hotspot-derived jet power estimates to those obtained from IC/CMB modeling of Knot 6—the jet feature closest to the hotspot.

Consider a uniform jet of area A, particle energy density , pressure p, mass density ρ, relativistic enthalpy w = + p + ρc², magnetic field perpendicular to the flow direction B_⊥ (as indicated by the polarization), speed cβ, and the corresponding bulk Lorentz factor Γ. The kinetic power (L_jet) and momentum flux (F_M) along the jet are, in c.g.s. units (e.g., Double et al. 2004):

$$\begin{eqnarray} L_{\rm jet} &=& A \Gamma ^2 \beta c \left(w + \frac{B_{\perp }^2}{4 \pi } \right) \qquad \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} F_M &=& A \left[ \Gamma ^2 \beta ^2 \left(w + \frac{B_{\perp }^2}{4 \pi } \right) + p + \frac{B_{\perp }^2}{8 \pi } \right]. \end{eqnarray} \tag{ 3 }$$

Let us first consider the region of the jet upstream of the hotspot. In this region, the IC/CMB model indicates that the jet Lorentz factor, Γ ≫ 1. In such a highly relativistic jet, the kinetic power L_{E, jet} is simply related to the momentum flux F_{M, jet} via

$$\begin{eqnarray} L_{\rm jet} &\approx & c \times F_{\rm M, jet}. \end{eqnarray} \tag{ 4 }$$

We assume a near normal shock at the jet terminus and appeal to conservation of momentum, so that F_{M, jet} = F_{M, hotspot}, and thus L_jet ≈ cF_{M, hotspot}. Assuming that the jet plasma in the hotspot is decelerated to a low velocity so that Γ²β² ≪ 1, the following relation then holds:

$$\begin{eqnarray} L_{\rm jet} &\approx & c A \times \left[ p + \frac{B_\perp ^2}{8 \pi } \right]_{\rm hs}. \end{eqnarray} \tag{ 5 }$$

The above equality holds regardless of assumptions about the jet characteristics such as its composition or the ratio of magnetic to particle energy densities in the jet or hotspot. We estimate the jet kinetic luminosity simply by estimating the hotspot parameters. This technique of jet energy flux estimation will be considered further in an upcoming paper (L. E. H. Godfrey & S. S. Shabala, in preparation).

In order to estimate hot spot parameters, we assume that the lepton population is ultra-relativistic (p = /3) and dominates the particle pressure. Then,

$$\begin{equation} L_{\rm jet} \approx A c \frac{B^2}{8 \pi } \left(1 + \frac{1}{3} \frac{\epsilon _{e^{\pm }}}{\epsilon _B} \right). \end{equation} \tag{ 6 }$$

As a check on this analysis, we apply it to the case of Cygnus A. Synchrotron self-Compton modeling of Cygnus A hotspot A, assuming a power-law electron energy distribution of the form N(γ) = K_eγ^−a, indicates B = 150 μG and B_eq = 280 μG assuming R = 2 kpc and a = 2.1 (Wilson et al. 2000). For hotspot D, B = 150 μG and B_eq = 250 μG assuming R = 2.2 kpc and a = 2.05. Wilson et al. (2000) do not give an estimate of the electron energy density, so we estimate the electron energy density from the SSC and equipartition magnetic field strengths as _e = (B²_eq/8π)(B/B_eq)^{(a + 1)/2}, since _e∝K_e and the synchrotron flux density S_ν∝K_eB^{(a + 1)/2}⇒_e∝B^{−(a + 1)/2}, in the case of a power-law distribution of the form N(γ) = K_eγ^−a. Therefore, _e = 8 × 10⁻⁹ erg cm⁻³ and _e = 5.4 × 10⁻⁹ erg cm⁻³ for hotspots A and D, respectively. Hence, we find L_jet ≈ 1 × 10⁴⁶ erg s⁻¹ for both hotspots A and D. This estimate of the jet power in Cygnus A is in excellent agreement with the value obtained by Wilson et al. (2006), L_jet ≳ 1.2 × 10⁴⁶ erg s⁻¹, based on modeling the cocoon dynamics. An independent estimate of jet power in Cygnus A comes from Lobanov (1998), who show that frequency-dependent shifts of the radio core enable a determination of the jet parameters, which, when applied to the case of Cygnus A, gives a value for the jet power L_jet = (0.55 ± 0.05) × 10⁴⁶ ergs s⁻¹.

We now return to the case of PKS 2101−490. The Chandra x-ray image provides only an upper limit to the hotspot x-ray flux density. Therefore, to calculate the hotspot magnetic field strength and particle energy density, we adopt equipartition estimates. We note that the adoption of an equipartition magnetic field strength does not overproduce hotspot x-ray emission, for which we have obtained an upper limit. We justify the assumption of equipartition by the fact that high-luminosity hotspots typically exhibit x-ray flux density consistent with the equipartition synchrotron self-Compton model predictions (Hardcastle et al. 2004). Moreover, the total energy density, and hence the derived jet power, is only weakly dependent on magnetic field strength for magnetic fields near equipartition conditions, so that moderate departure from equipartition does not alter our conclusions. For example, if the hotspot magnetic field strength were 1/3 the equipartition value, then our equipartition jet power would underestimate the true value by less than a factor of two.

The hotspot synchrotron spectral index is not well constrained, so we apply a single power-law model along with the following assumptions: 10 < γ_min < 1000, γ_max > 10⁴, α = 0.8. This model, combined with the flux densities and volume listed in Table 2, indicates that B_{eq, hs} ∼ 200–450 μG. Due to the prior assumption of equipartition, we have $\epsilon _{e^\pm } = \epsilon _{\rm B}$. Taking A = (6 ± 1) × 10⁴³ cm² as estimated from the 20 GHz ATCA image, the jet energy flux calculated from the hotspot model is

$$\begin{equation} L_{\rm jet} = (0.4 \hbox{--} 2) \times 10^{46} \: {\rm erg\,s^{-1}}. \end{equation} \tag{ 7 }$$

The range of values reflects the range of γ_min assumed in the hotspot model, with the lower jet power corresponding to higher assumed values of γ_min. Minimum Lorentz factors γ_min ≈ 600 have been observed in a number of hotspots (Godfrey et al. 2009 and references therein). This is an order of magnitude greater than the value of γ_min assumed in the jet. As suggested by Godfrey et al. (2009), such an increase in γ_min between the jet and hotspot may be due to the dissipation of jet energy in a relativistic proton/electron jet terminating at a near-normal shock.

A widely used method for estimating jet power in high-luminosity radio sources is based on the model of Willott et al. (1999), in which the jet power is estimated using the 151 MHz radio luminosity as follows:

$$\begin{equation} Q_W \approx f^{3/2} 1.5 \times 10^{38} \left(\frac{S_{151} D_L^2}{10^{28} {\rm \,W\, H{\rm z}^{-1}\, sr}^{-1}} \right)^{6/7} \: {\rm \,W}, \end{equation} \tag{ 8 }$$

where Q_W is the kinetic power per jet, S₁₅₁ is the flux density of the entire source at 151 MHz, D_L is the luminosity distance, and f is a parameter that accounts for uncertainties in various model assumptions, with 1 < f ≲ 20 (see Willott et al. 1999, for details of the model). For high power sources, the value of f has not been constrained empirically, but is typically taken to be f = 10–20 (e.g., Fernandes et al. 2011; Hardcastle et al. 2007). We extrapolate flux density measurements of S_408 MHz = 2.34 Jy (Large et al. 1981), S_843 MHz = 1.48 Jy (Mauch et al. 2003), and S_1.41 GHz = 1.1 Jy (Wright & Otrupcek 1990) to estimate the flux density at 151 MHz—S_151 MHz ≈ 4.25 Jy—and thereby obtain Q_W ∼ 1(3) × 10⁴⁶ erg s⁻¹ assuming f = 10(20).

Here, we estimate the average jet power over the life of the source based on the lobe parameters and an estimate of the source age

$$\begin{equation} L_{\rm jet} \approx \frac{2 U_{\rm lobe}}{t_{\rm age}}, \end{equation} \tag{ 9 }$$

where U_lobe is the total (magnetic plus particle) energy contained in the lobe. The factor of two is an approximate correction accounting for the fraction of jet energy converted to lobe kinetic energy and work done by the expanding lobes (Willott et al. 1999; Bicknell et al. 1997). In Section 7, we estimate the lobe parameters via synchrotron and IC modeling of the radio to x-ray SED, and argue that the source age is likely t_age ∼ 10⁸ yr. Based on the results of Section 7, we estimate the average jet power over the life of the source to be L_jet ≳ 10⁴⁵ erg s⁻¹. This is a factor of 4–20 lower than the other estimates of jet power. We note that we have ignored the possible contribution of protons to the energy density of the lobes, and we have ignored adiabatic cooling in estimating the cooling rate in the above analysis, each of which would increase the jet power estimate. We note that the calculations in Section 7 involve projected volumes. Using de-projected volumes will not increase the total energy, since the product K_e × V remains constant and the particle energy dominates U_lobe in this source.

Three independent order-of-magnitude estimates of jet power indicate L_jet ∼ 10⁴⁵–10⁴⁶ erg s⁻¹. For comparison, the jet power obtained from the one-zone IC/CMB model of jet x-ray emission are given in Table 5, for different values of the minimum Lorentz factor γ_min, and different assumed jet compositions.

It can be seen that, in the case of a purely leptonic composition, there is good agreement between the IC/CMB model and independent estimates of jet power, regardless of the value for γ_min within the stated limits. Under the assumption of a cold proton/electron composition, there is good agreement between the IC/CMB model and independent estimates of jet power, provided γ_min ≳ 50. This assumes one cold proton per radiating electron. If relativistic protons are included, the lower limit on γ_min would increase.

We have shown that in the case of PKS 2101−490, the IC/CMB predicted jet power is ≪10⁴⁸ erg s⁻¹ (the "uncomfortably large" jet power derived for some quasar x-ray jets based on the IC/CMB model; Atoyan & Dermer 2004; Mehta et al. 2009) under a range of reasonable assumptions. Indeed, the IC/CMB derived jet power is comparable to independent estimates of L_j ∼ 10⁴⁵–10⁴⁶ erg s⁻¹ for both leptonic and electron/cold proton compositions, provided γ_min ≳ 50. An important point to note about our jet power estimates is that, for the IC/CMB model, we have used a cylindrical geometry for knot K6 with a length approximately 10 times its width. This highly elongated cylindrical geometry is required by the observation that knot K6 is elongated along the jet axis in the radio maps, and the fact that the jet must be aligned within ≲ 9°of the line of sight if the IC/CMB model is to be applied. Typically, jet knots are assumed to be spherical with the knot diameters equal to the jet widths (e.g., Tavecchio et al. 2000; Mehta et al. 2009). If we were to assume a spherical volume in our analysis, then the IC/CMB derived jet powers would be greater by a factor of ∼4.

The rate of optical jet discovery¹⁷ in objects with confirmed radio and x-ray jets is approximately two out of three. However, few of these sources have resolved optical knots, particularly at high redshift (excepting PKS 0637−752 in which multiple optical knots are resolved; Schwartz et al. 2000; Mehta et al. 2009), and while a systematic study has not yet been performed, it would appear that relatively few have detailed information regarding the spectral slopes in both the radio and optical bands.

In PKS 2101−490, knot 6 is the only jet region that is reliably detected in all three bands (radio, optical, and x-ray). The radio through optical spectrum for knot 6 can be fit using a broken power law with a break of Δα = 0.5 (Figure 7). This suggests that radiative cooling may be responsible for the steeper spectrum at optical frequencies. However, the interpretation of the optical emission from Knot 6 is complicated by the fact that the optical to x-ray spectral index is $\alpha _{6.32 \times 10^{14} {\rm \,Hz}}^{\rm 1 \, keV} \approx 0.8$, consistent with the observed radio and x-ray spectral index of the entire jet. It is therefore plausible that a significant fraction of the observed optical flux density is IC/CMB emission. For the IC/CMB model to produce the observed optical flux at 6.32 × 10¹⁴ Hz, the electron energy distribution must continue with the same slope to Lorentz factors γ ≲ 10, assuming Γ ∼ δ = 6. If the electron distribution cuts off at a Lorentz factor γ ≫ 10, the IC/CMB emission will not make a significant contribution to the observed optical flux density.

With this caveat in mind, in this section, we model the radio through optical spectrum in terms of a broken power-law synchrotron model, and consider whether the jet parameters derived from IC/CMB modeling are consistent with this interpretation of the radio to optical spectrum of knot 6. Figure 8 illustrates the broken power-law fit to the SED of knot 6. The parameters of this fit are α = 0.8, ν_b = 7 × 10¹² Hz and ν_max = 10¹⁶ Hz.

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In modeling the radio to optical spectrum, we consider the effects of synchrotron cooling (Meisenheimer et al. 1989) and adopt the following procedure: we assume that relativistic electrons are injected at a shock with an energy index a (number density per unit Lorentz factor, N(γ) = K_eγ^−a). The electrons cool as a result of synchrotron and IC emission downstream of the shock over a length D, which defines the extent of the emitting region. In this model, we assume that the post-shock magnetic field and velocity are uniform; this is a reasonable assumption for an oblique shock. An electron with Lorentz factor γ cools as a result of synchrotron and IC emission according to

$$\begin{equation} \frac{d\gamma }{dt^\prime } = - \xi \, \gamma ^2 \end{equation} \tag{ 10 }$$

where t' is time in the plasma co-moving frame,

$$\begin{equation} \xi = \frac{4}{3} \frac{\sigma _T}{m_e c} \left(\frac{B^2}{8 \pi } + U_{\rm CMB} \right), \end{equation} \tag{ 11 }$$

in c.g.s. units, and the energy density of the CMB in the plasma co-moving frame is given by

$$\begin{equation} U_{\rm CMB} = U_{{\rm CMB},0} \, (1+z)^4 \, \Gamma ^2, \end{equation} \tag{ 12 }$$

where the current epoch CMB energy density is U_{CMB, 0} ≈ 4.2 × 10⁻¹³ erg cm⁻³ and Γ is the jet Lorentz factor.

The integrated emission from the post-shock region is the superposition of the emission from a number of progressively cooled slices. The volume-averaged number density per unit Lorentz factor is described by

$$\begin{equation} \bar{N}(\gamma) = \left\lbrace \begin{array}{@{}ll}0 & \gamma < \gamma _{{\rm min}} , \quad \gamma > \gamma _{{\rm max}} \\ \frac{K_e \gamma _b}{(a-1)} \gamma ^{-(a+1)} g\left(\frac{\gamma }{\gamma _b} \right) & \gamma _{{\rm min}} < \gamma < \gamma _{{\rm max}} \\ \end{array} \right. \end{equation} \tag{ 13 }$$

where

$$\begin{equation} g \left(\frac{\gamma }{\gamma _b} \right) = \left\lbrace \begin{array}{@{}ll}1 - \left(1 - \frac{\gamma }{\gamma _b} \right)^{a-1} & \gamma < \gamma _b \\ 1 & \gamma > \gamma _b. \\ \end{array} \right. \end{equation} \tag{ 14 }$$

The break Lorentz factor γ_b is given by

$$\begin{equation} \gamma _b = \frac{c}{\xi D} \Gamma _{\rm sh} \beta _{\rm sh}, \end{equation} \tag{ 15 }$$

where β_sh is the velocity of plasma relative to the shock and Γ_sh is the corresponding Lorentz factor. The electron Lorentz factor γ_b is the Lorentz factor to which an electron of initially infinite energy cools following the shock. If the jet shock is stationary, then Γ_sh = Γ.

The electron distribution (13) describes a broken power-law spectrum with the electron spectral index smoothly changing from a to (a + 1) at γ ≈ γ_b. The corresponding synchrotron spectrum is a broken power law with spectral index smoothly changing from α to α + 0.5 at a frequency in the observer's frame,

$$\begin{equation} \nu _b = \frac{\delta }{(1+z)} \frac{3}{4 \pi } \Omega _0 \gamma _b^2, \end{equation} \tag{ 16 }$$

where $\Omega _0(= \frac{q_e B}{m_e c}$ in c.g.s. units) is the non-relativistic gyro-frequency. This equation relates the break frequency to the break Lorentz factor, and it is the break frequency, ν_b, that is actually used in the fit; the other parameter is the injected spectral index, α = (a − 1)/2. The values for the magnetic field B ≈ 30 μG and Doppler factor δ ≈ 6 have previously been determined from the relationship between x-ray and radio emission in Section 3.2.2. In order to minimize the number of parameters, we assume that Γ = δ in the following analysis.

We can compare the fitted break frequency with the value theoretically implied by Equation (15) for γ_b and (16) for ν_b. The theoretical break frequency is

$$\begin{equation} \nu _b = \frac{27}{64 \pi } \, \frac{\delta }{1+z} \, \frac{\Omega _0 m_e^2 c^4}{D^2 \sigma _T^2} \, \frac{\Gamma _{\rm sh}^2 \beta _{\rm sh}^2}{U^2}, \end{equation} \tag{ 17 }$$

where U = U_B + U_CMB is the total magnetic plus radiation energy density. The jet parameters derived from the IC/CMB model for this knot imply that IC losses dominate over synchrotron losses, since the CMB energy density in the jet rest frame is greater than the magnetic energy density. In order to compare the predicted break frequency with that determined by the synchrotron model, we need to estimate the length, D, of the emitting region. The radio image of knot 6 is found to be elongated in the jet direction with de-convolved (intrinsic) length D ≈ 06. The upper limit to the size of the x-ray knot is 06. The maximum allowed viewing angle for δ = 6 is θ_max ≈ 9°, therefore, we take the de-projected knot length to be of order D ≈ 06/sin θ_max ≈ 4'' ≈ 30 kpc ≈ 9 × 10²² cm. If we adopt D ≈ 10²³ cm and insert in Equation (17) the appropriate parameter values from the IC/CMB fit we obtain a predicted break frequency of 1.2 × 10¹³ Hz. This is less than a factor of two above the modeled value of 7 × 10¹² Hz. Hence, the jet parameters derived from IC/CMB modeling of knot 6 are broadly consistent with the break frequency estimated by fitting a broken power law to the radio and optical data points.

We note that the F814W HST image of knot 6 (Figure 3) contains a double-peaked appearance, suggesting that a one-zone model of the emission region, which spans ≳30 kpc in the radio band, may not be applicable.