The Relativistic Jet Orientation and Host Galaxy of the Peculiar Blazar PKS 1413+135

The optical spectrum of the spiral obtained by the Keck I low-resolution imaging spectrometer (LRIS) (Oke et al. 1995) was presented in Figure 3 of Paper 1. It exhibits several narrow emission lines at a redshift of z = 0.24675. We analyzed the ratios of the line equivalent widths to determine the nature of the source of ionizing radiation in the galaxy. The spectrum was obtained with a 1'' slit at a slit position angle of −70° (east of north), aligned with the mean parallactic angle of the observation. Much of the galaxy light was captured in the slit, as the galaxy major axis was only misaligned by ∼30°. The effective spectral resolution of the observation (FWHM), as measured from night-sky emission lines, was a few hundred km s⁻¹ at redshift z = 0.247. An extinction correction of A_V = 0.067 was applied to the spectrum using the extinction curve of Cardelli et al. (1989) with R_V = 3.1. This corresponds to the expected Milky Way foreground extinction (Schlafly & Finkbeiner 2011). No attempt was made to correct for extinction within the spiral galaxy or the host of PKS 1413+135.

We analyzed the spectrum taken with the Keck I/LRIS presented in Paper 1 as follows (see Figure 2). Equivalent widths of the emission lines measured by integrating the line profiles at the known wavelengths are shown in Table 1. The apertures for the integrations were chosen to be 14 Å, which is twice the FWHM of the instrumental resolution. Smaller apertures did not include the full line widths, and larger apertures did not increase the measured equivalent widths. In the case of the [S ii] doublet a 30 Å aperture was used. The Hβ line was found to lie within an absorption trough, which was modeled as part of the continuum. The uncertainties in the equivalent widths were measured by bootstrap sampling from adjacent portions of the continuum. These uncertainties were propagated using standard techniques to the quoted uncertainties in the estimated line ratios discussed below.

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We detected several lines suitable for standard diagnostics of nuclear and star formation activity (Baldwin et al. 1981; Kewley et al. 2006) (see Figures 2 and 3). In particular, we measured the following standard line ratios, all based on the positive line detections shown in Table 1:

• 1.  
  $\mathrm{log}$([O iii]/Hβ) = 0.99 ± 0.05,
• 2.  
  $\mathrm{log}$([N ii]/Hα) = 0.10 ± 0.01,
• 3.  
  $\mathrm{log}$([S ii]/Hα) = −0.01 ± 0.02, and
• 4.  
  $\mathrm{log}$([O i]/Hα) = −1.0 ± 0.1.

The ratios enable us to classify the spiral at z = 0.247 as a Seyfert galaxy. There is no hint of broad emission lines, and all lines are unresolved. The spectrum is therefore well matched to the Seyfert 2 class. We detect the [O ii] line along with the nearby [Ne iii] λ3869 line, which has a substantially higher ionization potential (41 eV as compared to 13.6 eV). The [O ii] line is an excellent tracer of ongoing star formation (Kewley et al. 2004). However, the line ratio $\mathrm{log}$([Ne iii]/[O ii]) is a reasonable diagnostic of AGN activity in metal-rich galaxies, but very few star-forming galaxies have ratios greater than our measured value of −0.56 ± 0.03 regardless of metallicity (Trouille et al. 2011). Finally, we measure a ratio of Hα/Hβ = 3.1 ± 0.4. This is consistent with the Balmer decrement for Case B recombination under densities (∼10²–10⁴ cm⁻³) and temperatures (∼10⁴ K) typical of narrow-line AGNs (Osterbrock 1989). We conclude that a radio-quiet AGN resides in the spiral galaxy.

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However, our observations raise a puzzle. Given the extremely large absorbing column toward the AGN in PKS 1413+135 (Stocke et al. 1992), and the edge-on orientation of the galaxy, how is it that even the narrow-line region is not obscured? A clue is provided by the [O ii] line. Although this line is often associated with AGN activity, the low corresponding ionization potential means that it is only weakly excited in the narrow-line regions of AGNs, and not at all in the broad-line regions (Vanden Berk et al. 2001). Instead, it is often associated with more extended nebulosities bathed in the AGN continuum, known as extended emission-line regions (EELRs) (Spinrad & Djorgovski 1984; Fosbury 1986; Robinson et al. 1987; Husemann et al. 2014). These regions can extend from a few to over 100 kpc from the nucleus, and are sometimes anisotropic, being associated with ionizing cones in AGNs (Osterbrock 1991).

The detection of several narrow emission lines in the optical spectrum of the spiral at z = 0.247 is itself somewhat surprising, because previous observations by Stocke et al. (1992) with a CCD spectrograph detected only the [O ii] λ3727 emission-line doublet. Unfortunately the 1992 spectrum is not available and we cannot determine whether the spectrum may have changed. We inspected a publicly available Sloan Digital Sky Survey (SDSS) Baryon Oscillation Spectroscopic Survey (BOSS) spectrum of the spiral at z = 0.247 obtained on 2012 March 25, and found that several of the lines in the Keck I/LRIS spectrum were also detected in the SDSS spectrum with the same equivalent widths as in our own observations (see Figure 2). The most likely explanation is that the higher excitation lines were simply too weak to detect with the sensitivity of the 1992 observations.

We find no evidence of extended emission in our two-dimensional spectrum of the spiral at z = 0.247 at the position of the [O ii] line beyond the ∼07 seeing smear, which corresponds to a 3 kpc projected size. This is nonetheless consistent with the EELR hypothesis. Additionally, based on our detection of the [O i] λ6300 line, we measure a low value of $\mathrm{log}$([O i]/[O iii]) = −1.5 ± 0.1. This, together with the line ratios discussed above, is entirely consistent with the emission-line diagnostics of EELRs presented in Robinson et al. (1987), but only marginally consistent with a nuclear narrow-line region.

We estimate the mass of supermassive black hole in the spiral galaxy using the empirical relations between black hole mass and bulge luminosity given by McConnell & Ma (2013) and Bentz et al. (2009a). McConnell & Ma (2013) present scaling relations for black hole mass and host galaxy properties for early-type and late-type galaxies. As a sanity check, we also use the empirical relation given in Bentz et al. (2009a) for a sample of AGNs. Bentz et al. (2009a) use black hole mass measurements based on reverberation mapping and bulge luminosities from two-dimensional decompositions of HST images.

For the spiral at z = 0.247, McHardy et al. (1991) showed that optical images in R and I bands clearly exhibit the presence of a small bulge, while the galaxy disk dominates the emission in the B and V bands. Bulge−disk decomposition based on the surface brightness distributions gave the following magnitudes for the disk and bulge: ${m}_{R}^{{\rm{disk}}}=18.9$, ${m}_{R}^{{\rm{bulge}}}=20.6$. At the redshift (z = 0.2467) of the spiral galaxy, R-band magnitudes in the observed frame approximately correspond to V-band magnitudes in the rest frame. We converted the bulge magnitude into a luminosity, which we then used to estimate the mass of the SMBH. Table 2 lists the magnitude, the luminosity, and the SMBH mass estimates for the spiral at z = 0.247.

Table 2.  Bulge and SMBH Mass Estimates

Parameter	Estimated Value	Note
${m}_{V}^{{\rm{disk}}}$	18.9
${m}_{V}^{{\rm{bulge}}}$	20.6
${M}_{V}^{{\rm{bulge}}}$	−19.84
${L}_{V}^{{\rm{bulge}}}$	7.4 × 109L⊙
${\mathrm{log}}_{10}({M}_{\mathrm{SMBH}}/{M}_{\odot })$	7.97 ± 0.25	a
${\mathrm{log}}_{10}({M}_{\mathrm{SMBH}}/{M}_{\odot })$	7.99 ± 0.46	b
${\mathrm{log}}_{10}({M}_{\mathrm{SMBH}}/{M}_{\odot })$	7.88 ± 0.07	c

Notes. Estimates for the spiral galaxy at z = 0.247. The black hole mass (M_SMBH) is estimated using relations between black hole mass and bulge luminosity. Errors shown on the mass estimates of the SMBH are based on the uncertainties in the scaling relations given below and do not take the intrinsic spread in the mass–luminosity relation into account (see text).

^aUsing the relation of McConnell & Ma (2013) for early-type galaxies: $\mathrm{log}({M}_{\mathrm{SMBH}}/{M}_{\odot })$ = $\alpha +\beta {\rm{log}}({L}_{{\rm{V}}}^{{\rm{bulge}}}/{10}^{11}\,{L}_{\odot }$) where α = 9.23 ± 0.10, β = 1.11 ± 0.13. ^bUsing the relation of McConnell & Ma (2013) for late-type galaxies with ${L}_{{\rm{V}}}^{{\rm{bulge}}}\leqslant {10}^{10.8}\,{L}_{\odot }$, the same form as (a) but with α = 9.10 ± 0.23, β = 0.98 ± 0.20. ^cUsing the relation of Bentz et al. (2009b) for AGNs: $\mathrm{log}({M}_{\mathrm{SMBH}}/{M}_{\odot })=\alpha$ + $\beta {\rm{log}}({L}_{{\rm{V}}}^{{\rm{bulge}}}/{10}^{10}\,{L}_{\odot }$); α = 7.98 ± 0.06, β = 0.80 ± 0.09.

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The uncertainties in the derived masses shown in Table 2 are based on the uncertainties in the relations between mean black hole mass and bulge luminosity derived by McConnell & Ma (2013) and do not include the intrinsic scatter in this relationship. From Figure 2 of McConnell & Ma (2013) the intrinsic scatter in the SMBH mass is two orders of magnitude at a given bulge luminosity, and two-thirds of the points lie within a band of width one order of magnitude. Thus the 1σ uncertainty in ${\mathrm{log}}_{10}({M}_{\mathrm{SMBH}}/{M}_{\odot })$ is 0.5, and this is the best estimate of the uncertainty to use since it relies on no fitting and takes systematic errors into account automatically. Thus our estimate for the mass of the SMBH in this early-type spiral galaxy is ${\mathrm{log}}_{10}({M}_{\mathrm{SMBH}}/{M}_{\odot })=7.97\pm 0.5$. We note that a 2σ variation on the low side would place the mass of the SMBH at  ∼10⁷ M_⊙, and a 2σ variation on the high side would place it at  ∼10⁹ M_⊙. Therefore in our discussion of possible gravitational lensing by the SMBH in the spiral galaxy in Section 4.2 we consider the range of masses 10⁷–10⁹ M_⊙ for the SMBH.

As can be seen in the 8.1 GHz image of Figure 5 and in the MOJAVE stacked-epoch 15 GHz image of Figure 4(d), the western jet has a fairly constant position angle of 249° for the first 10 mas. At lower frequencies (see the 1.4 GHz map of Perlman et al. 1996), the jet arcs gradually toward PA = 180°, where it fades out at 90 mas from the core. There is thus direct evidence that the jet ridgeline in PKS 1413+135 is intrinsically bent. This is further supported by the fact that the features in the eastern jet (out to 20 mas in the stacked MOJAVE image, with PA = 59°) are not collinear with the western jet, but are offset by 10°. In contrast, two well-studied two-sided jetted-AGNs in the MOJAVE program, NGC 1052 and Cygnus A, that lie close to the plane of the sky have a high degree of collinearity in the PAs and proper motions of their approaching and receding jets (Lister et al. 2019).

To understand the apparent curvature of the jet in PKS 1413+135, let us first assume that the jet has a small intrinsic bend, ϕ_int, of constant curvature in a single plane as shown in Figure 6. It is shown here for a bend in the sky plane, so here the angle between the jet axis and the line of sight is θ = 90°. The real bend is only a few degrees, but has been exaggerated here for clarity. If we rotate the jet about its midpoint, shown by the dotted line in Figure 6, to angle θ relative to the line of sight, length a is unchanged and the projected length of side b is $b\sin \theta$. The observed bend in the jet is given by ϕ_obs in the relation

$\begin{eqnarray}&&\tan ({\phi }_{\mathrm{obs}}/2)=\tan ({\phi }_{\mathrm{int}}/2)/\sin \theta .\end{eqnarray} \tag{ 1 }$

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In the case of PKS 1413+135, as can be seen from Figure 4(c), the total observed bend along the jet from component N to component G is ϕ_obs ∼ 70°, so Equation (1) yields $\tan ({\phi }_{\mathrm{int}}/2)\approx 0.7\times \sin \theta$. We show in Section 5.1.1 that the angle between the jet axis and the line of sight is certainly ≤7° and is very likely a few degrees. Thus in total we see that the observed bend is ∼70° at angle θ ≈ ϕ_int /1.4. In other words, an intrinsic bend in the jet of only a few degrees will produce the overall projected bend in the jet axis of ∼70° that we see in Figure 4(c) when viewed from within a few degrees of the jet axis. Of course, the observed bend angle tends to 180° as θ → 0.

The simplest possible bending model is a monotonic bend that lies all in a single plane, as assumed above. Given a small viewing angle near the base of the jet, this would normally result in an apparent one-sided jet, due to the strong Doppler boosting of the approaching jet emission and de-boosting of the receding jet emission. The approaching jet would also rapidly fade with distance from the core as the viewing angle increases. However, if the bending plane lies perpendicular to the sky, the jet can cross our line of sight, resulting in apparent emission from the approaching jet on both sides of the core. Statistically, the odds of this alignment appear at first to be low, but in a flux-limited AGN sample, there is a strong preference for those jets that have maximum Doppler boosting factor. The latter occurs when a portion of the jet is pointing directly at us, which is the case for a jet crossing our line of sight. Lister (1999, p99) showed that in fact a simple parabolic bent jet will have a total apparent flux density much larger than a straight jet with identical inner jet viewing angle, provided the bending occurs in a plane nearly perpendicular to our line of sight. One notable case of this azimuthal favoritism is the blazar PKS 1510−089, where Homan et al. (2002) provide strong evidence of a highly superluminal jet that makes a 180° apparent bend in the sky plane, with resulting collinear jet emission on either side of the AGN core. We do therefore have to consider the possibility that all of the radio emission seen in this blazar is associated with the jet, but just happens to appear to straddle the nuclear core due to projection effects.

One clear difference between the radio structures of PKS 1413+135 and PKS 1510−089 is the inner counterjet component C in PKS 1413+135 seen in Figures 4(c)–(f). An even closer component of the inner counterjet, which is connected to the core component, is seen in the 8.1 GHz and 23.8 GHz images of Figure 5. No such inner counterjet, close to the core, is seen in PKS 1510−089, and it is very hard to imagine that these two components close-in to the core, which are so well aligned with the jet position angle projected across the core, are actually components of the approaching curving jet that have been projected across the core to the counterjet side after the approaching jet curves through an apparent bend of ∼70° from components N through G as seen in Figure 4(c). There is also the surface brightness of the components to consider. The components within 10 mas on the northeast side of the core have much higher surface brightness than components E, F, and G so it is extremely unlikely that they are further out along the jet than components E, F, and G. In addition, as Perlman et al. (2002) pointed out, the fact that the apparent counterjet undergoes a brightening at component B of Figure 4(e) just where the jet direction also changes, suggests that this is due to interaction with the interstellar medium.

For these reasons we interpret the bend in the counterjet 20 mas northeast of the core at component B as due to interaction with the surrounding medium and entrainment of material that is decelerating the outer sheath of the jet, giving rise to unbeamed emission from the sheath of the jet. Indeed, this interaction must begin much closer to the core to account for both component C and the component of the counterjet that is attached to the core seen in the 8.1 and 23.8 GHz images of Figure 5, but the interaction must become stronger at the position of component B.

In PKS 1413+135 the observed jet-to-counterjet flux-density ratio, R, is <10, so the counterjet is far brighter relative to the jet than usual for a blazar. We interpret the unusual relative brightness of the counterjet, including components C, B, and A, as due to interaction with the ambient medium and deceleration of the jet. Thus in our view the relatively low jet-to-counterjet flux-density ratio found in PKS 1413+135 cannot be used to place constraints on the jet model. Furthermore this implies that the 60% of the flux density at 1.67 GHz that comes from components A and B is isotropic.

Our main conclusions from this section are that the components southwest of the core are all jet components moving toward us, and the components northeast of the core are all counterjet components moving away from us, and for the remainder of this paper we adopt this interpretation.

A piece of evidence that would appear to argue against PKS 1413+135 being a background radio source is the absence of multiple images of the radio source on arcsecond scales. Multiple images are expected due to gravitational lensing by the mass associated with the nuclear bulge of the spiral galaxy. Lamer et al. (1999) have considered this question in detail and applied the method developed by Narayan & Schneider (1990) to derive a lower limit to the core radius of a lensing galaxy when no multiple images are seen. They show that under the very conservative assumptions of a core radius of 2 kpc and velocity dispersion of 180 km s⁻¹ the redshift of the radio galaxy PKS 1413+135 cannot be greater than z = 0.5. We can therefore be confident that the redshift of the jetted-AGN PKS 1413+135 lies in the range 0.247 < z < 0.5 as indicated in Figure 1.

Given the above range in possible redshifts for the radio source, we calculate the Einstein radius for the case z = 0.5, corresponding to the upper limit, and also for the case z = 0.25, at which the radio source would lie at ∼1% of the distance between z = 0.247 and z = 0.5. For the z = 0.5 case, with the SMBH of ∼10⁸ M_⊙ (see Section 2.2) in the AGN of the spiral at redshift z = 0.247 acting as a gravitational lens, we have L_l = 793 Mpc, L_s = 1.255 Gpc. Hence applying Equation (2) we find that Θ_E = 19.4 mas. For the z = 0.25 case, L_s = 800 Mpc, and the Einstein radius is Θ_E = 3.0 mas.

Since, as discussed in Section 2.2, the scatter in the mass–luminosity relation is two orders of magnitude for early-type galaxies, we also consider the case where the mass of the SMBH is  ∼ 10⁷ M_⊙. In this case the Einstein radius with the radio source at z = 0.5 is Θ_E = 6.1 mas. If the radio source is at z = 0.25, we have Θ_E = 0.95 mas.

The magnitude of the displacement between the radio core, which dominates the 15 GHz emission from the jetted-AGN PKS 1413+135, and the SMBH at the center of the spiral is critical to the question of whether or not we would expect to see multiple gravitational lens images of the radio core. Perlman et al. (2002) showed, on the basis of high-resolution HST near-infrared observations, that the near-infrared core of the blazar PKS 1413+135 is offset by only 13 ± 4 mas from the centroid of the near-infrared isophotes of the spiral galaxy, and in a direction perpendicular to the spiral disk and the dust lane, i.e., roughly in the direction of the jet. With the presence of a dust lane whose normal is slightly inclined to the plane of the sky there is the possibility of shifting the near-infrared spiral isophotes either in the jet direction or in the opposite direction depending on whether the near side of the dust disk lies to the northeast or the southwest. Indeed, Perlman et al. (2002) use this argument to suggest that the near-infrared spiral galaxy isophotes have been distorted by 13 mas and that the real offset is zero, and therefore that the blazar is located in the spiral galaxy. The uncertainty in the offset obtained by Perlman et al. (2002) was ±4 mas, so this would amount to a ∼3σ systematic shift in the near-infrared isophotes.

We are unaware of any direct evidence as to the orientation of the dust disk axis so it is quite possible that the shift is in the opposite direction by 13 mas or more. The evidence that does exist comes from the VLBA H i absorption measurements discussed by Perlman et al. (2002). They find that the absorption seems to occur in front of the northeastern radio emission. This suggests that the H i, and possibly any associated dust, is in the foreground to the northeast of the spiral galaxy. This is in the right place to lead to an underestimate of the magnitude of the separation between the radio core and the spiral nucleus rather than an overestimate, and thus we think it is justified to assume in the following that the displacement might be as much as 26 mas.

We therefore consider a range of projected offsets of the blazar from the SMBH of the spiral galaxy from 0 to 26 mas, and a range of Einstein radii from 1 to 20 mas. We will designate the projected angle between the radio core of PKS 1413+135 and the SMBH in the spiral galaxy by the impact parameter p, and by u when normalized by the Einstein radius, i.e., u = p/Θ_E. We will therefore consider the range u = 0 → 26.

Before we begin this discussion in detail, it is worth pointing out that, given that the range of the offset parameters u = 0 → 26 we do not necessarily expect to see multiple images or other signatures of lensing due to the SMBH in the spiral galaxy in the radio images of the radio source PKS 1413+135. So the absence of multiple radio images of the core does not rule out the background radio source hypothesis.

We now examine the maps in more detail. At first sight the curving structure of the source seen in Figure 4(c), which has a radius of curvature of about 70 mas, might appear to be explicable due to gravitational lensing by an intervening SMBH of mass  ∼ 10⁹ M_⊙ located in the spiral galaxy. Perlman et al. (1996) considered this possibility. They showed that the counterjet features (A, B, and C) have steep negative spectra, whereas the jet features (D and E) have inverted and flat spectra, respectively (see Figure 4(f)). Perlman et al. (1996) were not able to rule out gravitational lensing by a  ∼ 10⁹ M_⊙ SMBH because of the possibility that the steeper spectra of components A and B were due to free–free absorption. However, as we have just seen, Perlman et al. (2002) showed that the nucleus of the blazar is offset by at most 26 mas from the nucleus of the spiral galaxy, which is much too close to the core N (see Figure 4(c)) to account for the curvature. So gravitational lensing can be ruled out as an explanation of the overall curving structure.

We next consider the possibility of lensing on smaller angular scales. The radio source PKS 1413+135 is strongly dominated by the core at frequencies above 5 GHz and the core is very bright down to 1.67 GHz (Perlman et al. 1996).

For illustrative purposes, ignoring any convergence or shear due to the host galaxy, we consider gravitational lensing of a point source by a point mass. This produces two images. The corresponding relationships between the separation of the two images, their magnification, and the deflections of the images are given in Appendix A. Using Equations (A1) and (A2) we have derived the parameters of interest, which are given in Table 3. Here we show the normalized impact parameters, separations of the two images, and deflection of the brighter image from the true source position for magnification ratios of 10, 100, and 1000, as well as for the normalized maximum impact parameter of 26.

The core flux density of PKS 1413+135 in the stacked image of Figure 4(d) is 915 mJy, and the lowest contour level is 0.243 mJy/beam, so the ratio of the core to the lowest contour is ∼3800:1. Given the noise visible on this map at the one-contour level we need a minimum of two to three contours to trust a detection, so the dynamic range of the map is ∼1000.

On the 1 mas scale, as we see from the 43.2 GHz image in Figure 5, there is no hint of multiple images of the flat-spectrum core, only steeper-spectrum components lying linearly along the jet and counterjet. In fact, over the whole range from 1 to 40 mas the morphology of PKS 1413+135 seen in Figures 4(d) and 5 simply does not look like multiple images of a gravitationally lensed object since the jet and counterjet are only gently curved within 20 mas of the core.

The most stringent limits on possible secondary images of the radio core of PKS 1413+135 are provided by the MOJAVE 15 GHz observations. In Figure 7 we show a wide-field image in which the field of view (FOV) is 500 × 500 mas² made from the 2001 epoch observations. The peak flux density is 1.24 Jy/beam and no secondary image of the core is seen down to 1.2 mJy/beam, the level of the second contour. Thus we can definitively rule out a secondary image of the core down to 0.1% of the core brightness out to 116 mas from the core, corresponding to an impact parameter of 5.4456Θ_E (see Table 3). Since the smallest plausible value of the Einstein radius is Θ_E = 1 mas, this means that the impact parameter, i.e., the angle between the jetted-AGN nucleus and the centroid of the near-infrared contours of the spiral galaxy, is at least 5.4456 mas.

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Our conclusion is that, in spite of the presence of a  ∼10⁸ M_⊙ SMBH in the spiral galaxy, we see no evidence of multiple images of the compact radio core.

The absence of a secondary image might be taken as evidence that the radio source is not a background source but is located in the spiral galaxy. But we have seen that Perlman et al. (2002) measured an offset of 13 ± 4 mas, and that the offset could be as much as 26 mas. So the absence of a secondary image is not at all surprising and cannot be used as an argument against the background source hypothesis.

The lack of multiple images down to a magnification ratio of 1:1000 indicates that the displacement of the SMBH in the spiral from the radio core must be at least (5.446 + 0.178) mas = 5.624 mas (see Table 3).

In Paper 1 we reported the discovery of SAV events in the 15 GHz light curve of PKS 1413+135, and we presented a number of arguments supporting the hypothesis that SAV is caused by gravitational milli-lensing by a  ∼10⁴ M_⊙ mass condensate. Vedantham et al. (2017b) have shown that the SAV events are definitely not extreme scattering events (ESEs, Fiedler et al. 1987, 1994). An example of an ESE in the blazar 2023+335 from the 40 m Telescope 15 GHz monitoring program is shown in Figure 8(a) (Pushkarev et al. 2013). This is the highest frequency at which an ESE has been reported. Almost all ESEs detected thus far have not been visible even at 8 GHz (Fiedler et al. 1994). ESEs cannot explain SAV because that would require the angular size of the lensed component to scale as λ² from 15 to 230 GHz, and in addition in PKS 1413+135, given that the SAV events last for one year, aberration would destroy the symmetry of any ESE.

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In Figure 8(b) the original light curve is shown by the black dots, and these have been reflected about the center of the SAV event and are shown by the red diamonds to demonstrate the very high degree of symmetry in this SAV event. As mentioned above, and demonstrated by modeling in Vedantham et al. (2017b), this could not happen in an ESE.

The putative lensed components responsible for SAV are moving at apparent speeds ∼c, which are typical for the apparent speeds of components in the jet of this object, as can be seen on the MOJAVE website (Lister et al. 2016, 2018).²¹ It is interesting to note that an intervening spiral galaxy would provide a natural place to host a milli-lens.

It might be thought that interstellar scattering could blur any multiple images due to gravitational lensing, and hence undermine the arguments of Sections 4.2 and 4.3, but this is evidently not the case since we see from the stacked 15 GHz images shown in Figure 4(d) that components of angular size ∼1 mas are easily seen. Likewise it is very clear from the 43.2 GHz image shown in Figure 5 that components of angular size ∼0.5 mas would easily be seen. As discussed in Paper 1 in regard to milli-lensing, non-detection of interstellar scattering at a wavelength of 18 cm (Perlman et al. 1996) shows that any scattering of the PKS 1413+135 emission by the spiral is low compared to several nuclear sightlines through our own Galaxy at low latitudes.

The radio variability of PKS 1413+135 at 4.8, 8.0, and 14.5 GHz from the University of Michigan Radio Astronomical Observatory (UMRAO) and at 353 GHz from the James Clerk Maxwell Telescope (JCMT) are shown in Figure 9. Examination of these light curves shows that there are some examples of strong quasi-achromatic variability, which in the most extreme cases amounts to a factor of three within a period of a few months—see, for example, the drop in flux density at both 14.5 GHz and 353 GHz in 2001, where the 14.5 GHz variation mimics the 353 GHz variation with a lag of about 70 days, similar to what was seen in Paper 1 in 2015.

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There are also many examples of strong chromatic variability. For example, in the first half of 1997 the 353 GHz flux density dropped by a factor of two while the 14.5 GHz flux density varied little, and then in the second half of 1997 the 353 GHz flux density increased by a factor of two while the 14.5 GHz flux density again varied little. Another example of chromatic variability occurs between early 2002 and early 2003: here the 353 GHz flux density drops by a factor of two while the 14.5 GHz flux density drops by 20%. The chromatic variations in flux density are most likely to be intrinsic source variations and not due to propagation effects. On the other hand the achromatic variations could well be due to gravitational milli-lensing and proper motion of the emission regions, as discussed in detail in Paper 1. In analyzing the radio variability of PKS 1413+135 we have therefore to be cognisant of the fact that some of the variability could be enhanced by gravitational milli-lensing.

5.1.1. The Radio Doppler Variability Factor of PKS 1413+135

Liodakis et al. (2018) carried out a Doppler variability analysis of 1029 sources in the Owens Valley Radio Observatory (OVRO) 40 m Telescope 15 GHz monitoring program (Richards et al. 2011). The results for PKS 1413+135 and two comparison sources, 3C 273 and 3C 279, are shown in Table 4. The blazars 3C 273 and 3C 279 were chosen as comparison objects since in their high variability and luminosity they are similar to PKS 1413+135 and also because their redshifts, z = 0.158 and z = 0.536, respectively, straddle that of PKS 1413+135. Liodakis et al. (2018) assumed z = 0.247 for PKS 1413+135, so we have repeated the calculations for z = 0.5. These results are also shown in Table 4. In the case of PKS 1413+135 there are two SAV events in the OVRO light curve (Paper 1) that we do not think are due to intrinsic fluctuations, and which might affect these values.

Table 4.  Parameters for Blazar Jets Derived from Variability and VLBI

	z	Dvar	γ	θ	θmin	θmax	R
3C 273	0.158	5.2 ± 2.1	9.4 ± 2.8	77 ± 32
		${3.78}_{-0.55}^{+1.1}$	31.31	723	036	1536	1.3 × 107
3C 279	0.536	18.3 ± 1.9	13.3 ± 0.6	19 ± 06
		${11.64}_{-1.11}^{+1.77}$	24.06	422	004	493	1.8 × 108
PKS 1413+135 (OVRO)	0.247	${8.59}_{-1.73}^{+5.58}$	4.49	236	007	668	4.6 × 105
PKS 1413+135 (UMRAO)	0.247	${9.63}_{-2.60}^{+4.53}$	${5.14}_{-1.70}^{+1.74}$	197	003	585
PKS 1413+135 (OVRO)	0.5	${12.19}_{-1.92}^{+2.45}$	${6.44}_{-0.96}^{+1.00}$	20	002	47
PKS 1413+135 (UMRAO)	0.5	${16.43}_{-4.89}^{+6.85}$	${8.58}_{-2.80}^{+2.94}$	109	001	341

Note. The variability Doppler factors, variability γ factors, and variability angles are based on ${\beta }_{\max }$. These are calculated according to Liodakis et al. (2018) using the MOJAVE 15 GHz β_max values, except for the first row on 3C 273 and 3C 279, which are from Jorstad et al. (2017) using their 43 GHz β_max values, with 3C 273 being revised values from S. G. Jorstad (2020, private communication). The UMRAO data cover a period when there are no SAV events (see text). R is the predicted ratio of jet-to-counterjet flux density assuming spectral index α = 0.

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In order to check for the possible effects of SAV on the determination of D_var, γ, and θ, we have analyzed the UMRAO 14.5 GHz light curve of PKS 1413+135 over the period 1982 December 20 to 1992 June 12, when there were no SAV events (see Figure 9). These results are also shown in Table 4 for both z = 0.247 and z = 0.5.

The values from the OVRO light curve including the two SAV events and from the UMRAO light curve with no SAV events are in good agreement, so the SAV events have not significantly altered the result. A key result here, as can be seen in Table 4, is the small angle θ between the jet axis and the line of sight in PKS 1413+135. We return to this point at the end of this section.

5.1.2. The Radio Spectrum of PKS 1413+135

The radio spectra of the three high-luminosity blazars (PKS 1413+135, 3C 273, and 3C 279) are shown in Figure 10. All the points plotted in Figure 10 are total flux densities. The spread of points at a given frequency is an indicator of the variability since 1980 in the total flux density in these objects.

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In this paper we define the spectral index α by S_ν ∝ ν^α. At frequencies above 3 GHz in all three sources shown in Figure 10 the typical spectral index is flatter than α = −0.5 at any given epoch, and the spectrum is sometimes inverted (α > 0). Both the spectrum and the variability of PKS 1413+135 above 5 GHz are very similar to those of 3C 273 and 3C 279, both of which have jet axes aligned close to the line of sight.

The results in Figure 9 show that PKS 1413+135 is a highly variable source from radio to submillimeter wavelengths. However, the first indication of such high variability in PKS 1413+135 came from the infrared observations of Bregman et al. (1981). They reported that PKS 1413+135 is "among the most highly variable extragalactic sources known" and that at 2.2 μm it had shown changes of >20% on timescales of 1 day, and on three occasions the intensity had changed by over a factor of two in one month or less. Impey & Neugebauer (1988) provided further evidence of the high level of variability at mid- to far-infrared wavelengths in PKS 1413+135 from IRAS observations, which showed ∼20% variability at 60 μm on a timescale of 21 days. Since the infrared emission is so highly variable, it must be dominated by synchrotron radiation that is strongly beamed.

A question that naturally arises is whether the infrared radiation comes from the same object as the jetted-AGN radio emission. Given that we see highly variable blazar activity in both the infrared and radio wave bands, it is clear that the infrared and the radio emission both come from the same object. This is also true of the γ-ray emission.

Based on the soft X-ray absorbing column density, Stocke et al. (1992) estimate that A_V ≳ 30 mag of extinction toward the X-ray emission regions in the PKS 1413+135 AGN and they estimate a column density N_H ≥ 2 × 10²² cm^–2 along the line of sight to this source. Because it is so heavily absorbed at soft X-ray energies, Perlman et al. (2002) observed PKS 1413+135 over the 2–10 keV energy range with ASCA and obtained a detectable spectrum. They derived a luminosity of P(2–10 keV) = 2.2 × 10⁴⁴ erg s⁻¹. This is a factor ∼5 lower than in earlier X-ray observations, so it is clear that the jetted-AGN PKS 1413+135 is highly variable at X-ray energies.

The γ-ray and radio emission from blazars has been compared in a number of studies (e.g., Mahony et al. 2010; León-Tavares et al. 2011; Lister et al. 2015; Ramakrishnan et al. 2015; Larionov et al. 2020). In particular it is well known that Fermi-LAT preferentially detects highly Doppler-boosted jets (Kovalev et al. 2009; Abdo et al. 2010; Savolainen et al. 2010; Richards et al. 2014). Lister et al. (2015) show that in blazars there is a strong correlation between Doppler factor and γ-ray flux.

The spectral energy distribution (SED) of PKS 1413+135 peaks at 10^13.4 Hz, then drops precipitously (Planck Collaboration et al. 2011), but this is strongly affected by extinction, so it is likely that the true peak lies at higher frequency. In this regard PKS 1413+135 is typical of γ-ray blazars and of blazars with high Doppler factors in general (Lister et al. 2015).

Light curves at 15 GHz from the OVRO 40 m Telescope and at γ-ray energies from the fourth Fermi-LAT catalog (Abdollahi et al. 2020) for 3C 273, PKS 1413+135, and 3C 279 are shown Figure 11. Note that the ratio of the γ-ray flux to the radio flux density is very similar in all three objects. In addition, all three objects show very similar variability to each other at both radio frequencies and γ-ray energies.

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Recently a burst of γ-ray emission from PKS 1413+135 was detected with Fermi-LAT (Angioni et al. 2019): on 2019 August 28 the daily averaged flux at E > 100 MeV reached (4.8 ± 1.5) × 10⁻⁷ photon cm⁻² s⁻¹, about 38 times the average flux reported in the fourth Fermi-LAT catalog. During the outburst the γ-ray spectrum was significantly harder than the average from this source. This hard-spectrum state was accompanied by the detection of several E > 10 GeV photons, including one with an energy of ∼64 GeV.

We see no other way to explain the observed strong variability of PKS 1413+135 across the electromagnetic spectrum, from radio frequencies to γ-ray energies, than by relativistic beaming, and we find this a compelling demonstration of strong Doppler boosting as expected in a jet close to the line of sight. We return to this in Section 6, where we analyze the subluminal and superluminal motion of PKS 1413+135 and show it to be entirely consistent with our findings based on variability. Our key conclusion of this section is that, as illustrated in Figure 1, the jet axis in PKS 1413+135 is aligned to between 001 and 668 of the line of sight, taking the most conservative values we present in Table 4 from the combination of the OVRO and UMRAO results. This value is refined in Section 6 based on the MOJAVE observations of β_app.

The total flux density of most blazars at 15 GHz is dominated by the core flux density, which is strongly relativistically beamed. The cores of blazars are generally stationary features, which provide a convenient reference point against which the relative motion of components moving out from the core along the jet or counterjet can be measured.

The radio emission from PKS 1413+135 at 8 GHz and higher frequencies is totally dominated by the emission from the core, as can be seen in Figure 5. So in this object it is a simple matter to measure the apparent motions of features in the jet relative to the core at frequencies above 8 GHz. We will focus on the 15 GHz MOJAVE observations.

The apparent transverse speed of a relativistic object is given by Equation (B10) in Appendix B. We note that the highest superluminal motion observed in PKS 1413+135 ((1.72 ± 0.11)c for z = 0.247 and a factor of 1.58 higher for z = 0.5) is much lower than those of 3C 273 and 3C 279 given in the MOJAVE database, which are (20.50 ± 0.82)c and (10.0 ± 0.86)c, respectively.

The apparent velocities, β_app = v_app/c, that have been measured on the MOJAVE program for seven components in PKS 1413+135 are shown in Figure 12 for five components in the jet and two components in the counterjet.

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6.1.1. Implications of β_app for the Counterjet Side

6.1.2. Implications of β_app on the Jet Side

It can be seen in Figure 12 that there is a highly significant increase in v_app on the jet side as components move away from the core. This could be due to (i) a real acceleration in these components as they move away from the core, (ii) a change in angle between the velocity vector of the component and the line of sight, or (iii) a pattern speed not related to the bulk motion of the emitting plasma.

Since we know that the jet is bending, the simplest model is (ii), i.e., a model in which the material is moving at constant speed as it moves away from the core while the angle is changing. We have determined that θ < 668 (Section 5), so θ < 1/γ (see Table 4). In this case, Equation (B10) shows that, in order to explain the increase in apparent speed with component distance from the core, the angle between the jet axis and the line of sight must be smaller at the core than it is out along the jet, i.e., the jet must curve away from the line of sight as we move out along the jet.

Note that it must be the case that the bulk velocity of the plasma emerging from the core on the jet side is highly relativistic, otherwise the high variability of the core could not be explained by relativistic beaming.

In Table 5 we list the angles at which the speeds β = 0.13c and β = 1.72c (the range of speeds seen in the jet in Figure 12 assuming z = 0.247) would be observed for different values of γ. We also show the values for speeds a factor 1.58 greater, corresponding to z = 0.5. We also list the Doppler factors, D₀, for θ = 0°, and the maximum angle θ_×100 at which relativistic beaming would boost the observed flux density by at least a factor 100.

Table 5.  Viewing Angle Constraints from v_app

γ	D0	θ0.13c	θ1.72c	θ0.20c	θ2.71c	θ×100
		z = 0.247	z = 0.247	z = 0.5	z = 0.5
2.2	4.16	09	162	14	⋯	33
2.5	4.79	07	108	105	⋯	100
3	5.83	045	67	07	146	130
4	7.87	025	34	038	5935	141
5	9.90	015	21	0235	349	1386
6	11.92	011	142	0163	233	1335
7	13.93	0075	105	012	1675	1279

Note. The viewing angles, θ, are those at which the apparent speeds are v_app = 0.13c and v_app = 1.72c, corresponding to a redshift of 0.247, and v_app = 0.20c and v_app = 2.71c, corresponding to a redshift of 0.5. D₀ is the on-axis Doppler factor, D(θ = 0). The range of viewing angles over which Doppler boosting in the approaching jet exceeds a factor 100 is from θ = 0° to θ = θ_×100.

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For example, if γ = 5 then β_app ∼ 0.13 requires θ ∼ 015 at a projected distance of 0.3 pc from the core (as shown in Figure 12), and β_app ∼ 1.72 requires θ ∼ 21 at a projected distance of 7 pc from the core (as shown in Figure 12); while for γ = 4 we have that β_app ∼ 0.13 requires θ ∼ 025 at a projected distance of 0.3 pc from the core, and β_app ∼ 1.72 requires θ ∼ 34 at a projected distance of 7 pc from the core.

For illustrative purposes we assume two Lorentz factors for the bulk motion along the jet: γ = 7 and γ = 3. From Equation (B10) we find that the observed maximum apparent speed of 1.72c implies θ ∼ 59° or ∼1° for γ = 7, and θ ∼ 54° or ∼7° for γ = 3. We have shown in Section 5 that the jet axis in PKS 1413+135 is aligned close to the line of sight so we can rule out the larger option, θ ∼ 59° or ∼54°.

The solution γ = 7 is fully consistent with the requirements of beaming, but γ = 3 is more likely since the required alignment of the innermost features is less stringent. Note that for redshifts of the radio source close to z = 0.247, γ = 2.5 is likewise entirely feasible, and that γ = 2.2 also works (just); whereas for redshifts of the radio source close to z = 0.5, γ = 4 is entirely feasible, and γ = 3 also works (just).

Inspection of the angles θ given in Table 5 at which the apparent speeds could be observed shows that we require θ < 1° to account for the very low apparent speeds measured within the first few parsecs of the core shown in Figure 12.

Our conclusions from this section are thus that the PKS 1413+135 jet axis close to the core is inclined at an angle θ < 1° to the line of sight, and bends away from this angle as we move out along the jet, and that the high observed flux density, the variability, and the apparent speeds are easily explained by relativistic beaming in terms of the entirely self-consistent model suggested above. Furthermore, the values of the key parameters in PKS 1413+135 are very similar to those of 3C 273 and 3C 279, shown in Table 4.

If the AGN PKS 1413+135 is located in the spiral galaxy, much of the optical–ultraviolet radiation from AGNs will be absorbed in the galactic disk and re-radiated at infrared wavelengths. We therefore first consider the luminosity of the AGN in the optical–ultraviolet frequency range.

7.1.1. The Luminosity of the Blazar PKS 1413+135

7.1.2. The Mid-infrared Spectrum of PKS 1413+135

The strong variability of PKS 1413+135 at infrared wavelengths discussed in Section 5.2 shows that it is dominated by strongly beamed synchrotron radiation. In Figure 13 we show Spitzer spectra for PKS 1413+135, 3C 273, and 3C 279 (Houck et al. 2004; Werner et al. 2004). We present a re-reduction, using the Cornell Atlas of Spitzer/IRS Sources (CASSIS) (Lebouteiller et al. 2011), of the Spitzer low-resolution PKS 1413+135 spectrum previously published by Willett et al. (2010), together with Spitzer CASSIS spectra of 3C 273 and 3C 279. We used CASSIS version 7, and the AORKEYs as follows: PKS 1413+135: 18322432, 3C 273: 28976640, and 3C 279: 27437312. It can be seen here that 3C 273 shows a mid-infrared bump of emission in the 9.7 μm and 18 μm silicate bands (Hao et al. 2005). This is due to thermal radiation arising from reprocessed optical–ultraviolet emission from the AGN (Armus et al. 2007).

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Millimeter-wavelength observations of PKS 1413+135 show CO absorption of the millimeter continuum (Wiklind & Combes 1994), so the continuum at millimeter wavelengths is being viewed through a molecular screen. The Spitzer IRS emission-frame spectrum, assuming z = 0.247, is shown in Figure 13. This shows silicate absorption at 9.7 μm and 18 μm. The spectrum can be fitted by a power-law continuum absorbed by an intervening screen of cold dust, which is consistent with the CO observations, i.e., there is no evidence of significant heating of the dust. The strength of the rest-frame 9.7 μm silicate absorption corresponds to A_V of 13.5–14 mag for a standard galactic extinction law (Rieke & Lebofsky 1985) and fits the sharp drop in flux density at near-infrared wavelengths. A similar conclusion was reached by Willett et al. (2010) based on an earlier reduction of the Spitzer spectrum.

As shown in Section 7.1.1, the integrated luminosity of the jetted-AGN PKS 1413+135 over the range 10^13.5–10¹⁴ Hz is  ∼10⁴⁵ erg s⁻¹. If PKS 1413+135 were located in the spiral galaxy, the ionizing continuum optical–ultraviolet radiation from the nucleus would be absorbed and reprocessed in the disk of the galaxy by the obscuring dust and we would expect to see a near-infrared dust continuum bump and near-infrared emission lines from dust. However, one does not always see any reprocessed radiation in the form of line emission in such cases. For example, the ultraluminous infrared galaxy 08572+3915 shows no lines at the mid-infrared wavelengths of Spitzer (Armus et al. 2007).

Nevertheless, we would still expect to see evidence of a bump of emission, or unusual relative depths of the silicate absorption features at 9.7 μm and 18 μm in the Spitzer spectrum, if a powerful AGN were embedded in the spiral galaxy. However, the relative depths of the two silicate absorption features are normal for absorption from cold dust. This is strong evidence that a significant source of power, such as a blazar, is not embedded in the obscuring material in the nucleus of the spiral galaxy.

We might expect that any central engine generating a jetted radio source in the nucleus of a spiral galaxy must have derived its angular momentum from the disk of the spiral galaxy, and so, in the absence of a strong interaction via a merger event that torqued the spin axis of the central engine into the galactic disk plane, it would be somewhat aligned with, and not orthogonal to, the spin axis of the spiral host. This would argue that the radio source PKS 1413+135 is not located in the spiral galaxy.

However, we have seen that the spiral is a Seyfert galaxy, and a number of studies have been carried out in Seyfert galaxies of the alignment of the radio jet axis with the spin axis of the spiral host galaxy, and in general Seyfert galaxies show no such alignment of the galaxy spin axis with the jet axis (Ulvestad et al. 1981; Ulvestad & Wilson 1984; Clarke et al. 1998; Gallimore et al. 1999, 2006; Kinney et al. 2000). Indeed in one such case (NGC 2110) the ∼800 kpc jet "appears to propagate directly into the disk of the surrounding host galaxy" (Gallimore et al. 2006). Hopkins et al. (2012) discuss the physical processes, demonstrated with simulations, that give rise to such misalignments.

These radio sources have lower luminosity than PKS 1413+135, so the comparison may be misleading. Indeed the study of Seyfert galaxies by Gallimore et al. (2006) focuses entirely on "radio-quiet," albeit clearly not "radio-silent," Seyfert galaxies. A clue to what may be going on comes from H₂O mega-maser galaxies, which we discuss next.

7.2.1. Alignment of Jet Axes and Accretion Disk Axes in H₂O Mega-maser Galaxies

7.2.2. Alignment of Jet Axes and Accretion Disk Axes in High-luminosity AGNs

There is one argument against this interpretation that should be mentioned here, namely the small angle projected on the sky of 13 ± 4 mas (52 ± 16 pc) between the infrared blazar position and the infrared isophotal center of the spiral galaxy (Perlman et al. 2002).

Following a line of argument similar to that laid out in Paper 1, Appendix D, we calculate the a posteriori probability of this alignment. It is a posteriori because we would not have thought of asking the question a priori, i.e., before noticing that PKS 1413+135 is closely aligned with the spiral galaxy. So we preselected the source before calculating the probability.

We saw in Section 2.2 that the disk and bulge R magnitudes of the spiral are ${m}_{R}^{{\rm{disk}}}=18.9$ and ${m}_{R}^{{\rm{bulge}}}=20.6$ (McHardy et al. 1991), which yield a combined magnitude of ${m}_{R}^{{\rm{disk}}+{\rm{bulge}}}=18.7$. We now calculate the probability of chance alignment of a background source and a foreground galaxy of magnitude m_R = 18.7 to within an angle ζ_off. A compilation of differential number counts of galaxies in the R band measured in 23 separate analyses of different areas of sky is given in Figure 12 of Metcalfe et al. (2001). Integrating the differential counts down to m_R = 18.7, we find there are 631 galaxies per square degree brighter than this limit. Thus the total number of galaxies in the sky brighter than m_R = 18.7 is N_f = 2.60 × 10⁷.

The fraction of sky within a radial distance ζ_off from any of the N_f foreground galaxies is $\pi {\zeta }_{\mathrm{off}}^{2}{N}_{{\rm{f}}}/4\pi$. The expected number of chance alignments is therefore ${N}_{{\rm{f}}}{N}_{{\rm{s}}}{\zeta }_{\mathrm{off}}^{2}/4$, where N_s is the number of sources in the radio sample, here equal to 981 (see Paper 1). Hence the expected number of radio sources in our sample with the required alignment is ${N}_{{\rm{f}}}{N}_{{\rm{s}}}{\zeta }_{\mathrm{off}}^{2}/4=2.53\,\times {10}^{-5}$. As pointed out in Section 4.2, we should allow for the possibility that, due to possible distortion of the infrared isophotes of the galaxy disk, the separation between the infrared blazar nucleus and the centroid of the infrared isophotes of the spiral galaxy could be as much as 26 mas. So the correct probability to consider here is 1.0 × 10⁻⁴. This is a low value, but since this is a posteriori statistics the argument against the background source hypothesis is not compelling.

Of the three arguments presented in this section, as in Paper 1, we here again reject the alignment argument of Section 7.3 against the radio source being a background source because it is a posteriori. The remaining two arguments, based on the mid-infrared spectrum (Section 7.1) and the good alignment of the jet axis in powerful jetted-AGNs in spirals with the spiral spin axis (Section 7.2), combined with the high 1.67 GHz luminosity of PKS 1413+135, lead us to conclude that it is not located in the spiral galaxy but is a background object.

The HST observations of Perlman et al. (2002) show that the jetted-AGN PKS 1413+135 has a very red color, V − H = 6.9 mag. They also deduced from their HST and ASCA observations that ${N}_{{\rm{H}}}={4.6}_{-1.6}^{+2.1}\times {10}^{22}\,{\mathrm{cm}}^{-2}$. All of this is consistent with the high X-ray extinction, which implies A_v ∼ 30 mag, found by Stocke et al. (1992).

As pointed out by Perlman et al. (2002), the high X-ray extinction and much lower near-infrared extinction, which implies A_v ∼ 14, can easily be explained if the medium is patchy. Based on this assumption, they deduce a covering fraction $f={0.12}_{-0.5}^{+0.07}$. Furthermore (Perlman et al. 2002), if this material is in the nuclear regions of the spiral galaxy, and if the jetted-AGN were in the spiral, one would expect to see a bright narrow Fe Kα line, but no such line is seen in their ASCA spectrum, although the sensitivity is only sufficient to place an upper limit of 500 eV on the equivalent width. The non-detection of this line is consistent with the jetted-AGN being a background source.

PKS 1413+135 has low polarization at radio frequencies (Aller et al. 1999). This has long been something of a puzzle given its blazar and BL Lac nature, since blazars and BL Lac objects are well known to be highly polarized (e.g., Cawthorne et al. 1993; Wardle 2013). But PKS 1413+135 is being viewed through an edge-on spiral galaxy, so that Faraday depolarization could explain this. Support for this hypothesis comes from both the infrared observations of Bregman et al. (1981), who measured high infrared polarization in the H band (16% ± 3% in PA 175° ± 5°), and those of Stocke et al. (1992), who measured high infrared polarization in the K band (10.5% ± 1.4% in PA 170° ± 14).

Like most blazars, the SED of PKS 1413+135, shows two peaks: a synchrotron peak in the optical–ultraviolet range and an inverse Compton peak in the GeV energy range. Many studies of AGNs are based on the SED and use a single-zone model in which the emission causing both the synchrotron peak and the inverse Compton peak is assumed to originate from the same particle population.

The first VLBI images of blazars, namely those of 3C 273 and 3C 345, showed that they are one-sided jets with a flat-spectrum, optically thick, core at one end of a steep-spectrum, optically thin, jet (Readhead et al. 1978). This has turned out to be the case in most blazars. The higher-frequency radio emission is dominated by emission regions closer to the center of activity than the lower-energy emission. Thus it is reasonable to expect that the optical synchrotron emission from blazar jets originates closer to the core than the radio emission regions. Kovalev et al. (2017) compared the positions of the radio cores and jets of jetted-AGNs with the optical positions measured with Gaia, and they found differences from less than 1 mas up to 10 mas. They also found that, while in some blazars the centroid of the optical emission lies on the side of the radio core opposite to the jet, and hence closer to the central engine, as expected from the VLBI results discussed above, for a significant fraction of blazars the centroid of the optical positions lies further out along the jet than the nuclear radio jet, which was totally unexpected. It is clear, therefore, that in many blazars a single-zone model cannot explain both the synchrotron radio jet and the synchrotron optical jet.

We have seen in Section 8.1 that the continuum X-ray emission from PKS 1413+135 comes from a region much smaller than the that of near-infrared continuum emission. The single-zone model cannot therefore explain the beamed infrared and X-ray emission in the relativistic jet of PKS 1413+135, and we see from Figure 4 that it likewise cannot explain the relativistically beamed radio emission. Thus a multizone model is needed in all frequency ranges to explain the blazar emission from PKS 1413+135.