On the Diversity of Jet Production Efficiency in Swift/BAT AGNs

Our radio-loudness parameter, as in Gupta et al. (2020), is given as $R={F}_{1.4}/{F}_{{\nu }_{{\rm{W}}3}}$, where F_1.4 and ${F}_{{\nu }_{{\rm{W}}3}}$ are the monochromatic fluxes at 1.4 GHz⁴ and in the Wide-field Infrared Survey Explorer (WISE) at ${\nu }_{{\rm{W}}3}=2.5\times {10}^{13}$ Hz, respectively. This relates to the definition given by Kellermann et al. (1989), i.e., ${R}_{\mathrm{KL}}={F}_{5}/{F}_{{\nu }_{{\rm{B}}}}$, where F₅ and ${F}_{{\nu }_{{\rm{B}}}}$ are the monochromatic fluxes at 5 GHz and in the B band (6.8 × 10¹⁴ Hz), as R ≈ 0.1 × R_KL, assuming the spectral indices of α_1.4–5 = 0.8 and ${\alpha }_{{\nu }_{{\rm{B}}}-{\nu }_{{\rm{W}}3}}=1$ (see Gupta et al. 2018). Based on this, we partitioned the sources into the following radio classes, strictly corresponding to those in Kellermann et al. (1989): RL when R > 10, RI for 1 < R < 10, and RQ when R < 1.

Because we used three radio catalogs differing in radio wavelength, angular resolution, and sensitivity, we decided to adopt the radio flux from the National Radio Astronomy Observatory (NRAO) Very Large Array (VLA) Sky Survey (NVSS), whose data are available for most of the sources and whose sensitivity accounts for all of the extended emission. For objects lacking NVSS data, we took fluxes from the Sydney University Molonglo Sky Survey (SUMSS) and Faint Images of the Radio Sky at Twenty-cm (FIRST).⁵ Such an approach resulted in finding 44 RL, 20 RI, and 250 RQ AGNs in our sample, with their radio-loudness medians of 75.40, 1.69, and 0.15, respectively. The exact radio-loudness distribution is presented in the top panel of Figure 1.

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Among 314 Swift/BAT AGNs, 257 of them have and 57 lack radio data. As one can see in Figure 1, those radio-undetected sources belong entirely to the RQ class. In the group of radio-detected objects, we can specify two subsamples of sources for which we have both: (1) NVSS and SUMSS data (12 sources) and (2) NVSS and FIRST data (76 sources). While the NVSS and SUMSS total fluxes (compared at 1.4 GHz) are almost the same, the ratio of NVSS to FIRST total fluxes is slightly more significant, with a median of 1.2, showing that, indeed, some of the faint extended radio emission might be lost while using only higher angular resolution and better sensitivity radio data.

Within the group of 257 radio-detected AGNs, we can distinguish two main subsamples—those with and without extended radio emission, represented by 52 and 205 objects, respectively. Below we give detailed characteristics of our radio morphological classification.

3.2.1. Compact Sources

3.2.2. Extended Sources

3.2.3. Radio Morphology versus Radio-loudness Classification

3.2.4. FR I/II Classification

The (projected) largest linear size (LLS) was determined for each radio-detected source in our sample. Noting that the true definition of LLS is given as the distance between two hot spots, the most reliable calculations are available for sources with double radio lobes. For complex AGNs with multiple pairs of lobes, the pair with the largest separation was used. The sizes of knotty radio sources correspond either to the distances between the most distant radio matches or to our measurements obtained from their radio maps. The sizes of compact AGNs (exact measurements for resolved and upper limits for unresolved objects) were taken directly from the radio catalogs as the major axis of the fitted Gaussian (a detailed procedure for the size estimation of compact AGNs is given in Appendix B).

The results for the LLS calculations for all the radio-detected sources are shown in the left panel of Figure 2. Knowing that only three out of six morphological groups (namely, complex, triple, and double sources) presented in this paper have direct measurements of the projected size and the sizes of compact (unresolved and resolved) and knotty sources are upper limits (i.e., are possibly shifted to the left on this chart), a clear trend confirming our results from Section 3.2.3 is found, namely, the most extended AGNs are RL whereas the RQ ones do not achieve such big sizes. This distinction is especially visible in the right panel of Figure 2, where only compact sources with radio data from FIRST (which, within the group of 57 compact sources, provides more precise, and is on average 15 times smaller sizes than NVSS; see Appendix B) are included. This figure shows that compact resolved and compact unresolved objects are in fact located in different parts of the diagram.

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From Figure 2 we note that nine sources have projected sizes beyond 700 kpc, which is the definition of giant radio galaxies (GRGs). The fraction of GRGs in our sample compared to the total number of lobed sources (9/40 ≈ 23%) is almost the same as in Bassani et al. (2016), where the authors found that 14 out of their 64 confirmed radio galaxies selected in the soft gamma-ray band and having double-lobe morphologies have giant sizes. Furthermore, Bruni et al. (2019) found that 61% of Bassani et al. giant radio sources have a gigahertz-peaked spectrum (GPS) cores, i.e., young nuclei. We checked that all of our GRGs are in common with those studied by Bruni et al. (2019), with six of them (67%) having GPS cores suggesting the ongoing accretion and reactivation of the jets.⁸ By contrast, four of their AGNs are not found in our sample as they were not observed by Swift/BAT or have blazar-like nuclei, while another two sources (namely 4C +63.22 and PKS 2356–61) listed in Bruni et al. are not recognized as GRGs because their sizes, estimated in this work, are slightly below 700 kpc.

We started our calculations of bolometric luminosity by checking whether or not the method in which MIR W3 fluxes were used to estimate L_bol, as was done in Gupta et al. (2018), can be successfully applied to our sample of AGNs accreting at a moderate accretion rate. Such verification was possible due to the accessibility of the multiband spectra, and in turn, the exact values of bolometric luminosities, which are presented in Gupta et al. (2020), upon which we built our sample of Swift/BAT AGNs.

We decided to analyze a subsample consisting of 131 (20 RL, 111 RQ) sources—all of those being Type 1 and having strict MIR, NIR, optical-UV, and hard X-rays detections (for more details, see Section 4 in Gupta et al. 2020). This choice was dictated by our need of having possibly the most accurate estimations of L_bol in which the obscuration by the dusty torus is minimized.

By representing the given subsample in the $\mathrm{log}({\nu }_{{\rm{W}}3}\,{L}_{{\nu }_{{\rm{W}}3}})$ versus $\mathrm{log}{L}_{\mathrm{bol}}$ plane, where L_bol is taken from Gupta et al. (2020, the specific formulas can be found in Appendix B therein) and for which the distribution is shown in Figure 3, we found that indeed almost all sources exhibit linear correlation (r ≈ 0.90, where r is the correlation coefficient) between monochromatic luminosity in the W3 band and bolometric luminosity in their logarithmic quantities.⁹ Linear regression produces a formula of such a dependence, which is given as

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}\,[\mathrm{erg}\,{{\rm{s}}}^{-1}]=3.25\times {\left({\nu }_{{\rm{W}}3}{L}_{{\nu }_{{\rm{W}}3}}[\mathrm{erg}{{\rm{s}}}^{-1}]\right)}^{1.0086},\end{eqnarray} \tag{ 1 }$$

which for ${\nu }_{{\rm{W}}3}\,{L}_{{\nu }_{{\rm{W}}3}}$ in the range ${10}^{42}\mbox{--}{10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ is well approximated by

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}\,[\mathrm{erg}\,{{\rm{s}}}^{-1}]\simeq 7.77\times {\nu }_{{\rm{W}}3}\,{L}_{{\nu }_{{\rm{W}}3}}\,[\mathrm{erg}\,{{\rm{s}}}^{-1}].\end{eqnarray} \tag{ 2 }$$

The median ratio of bolometric luminosity obtained from the whole spectral energy distribution (SED) to that calculated from Equation (2) for a studied subsample of 131 sources is 0.96. Hence, we conclude that the estimation of bolometric luminosity for AGNs accreting at a moderate accretion rate from the monochromatic MIR luminosity is reliable and can be effectively used for our whole sample.

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In order to obtain BH masses in a uniform way for all AGNs in our sample, we decided to use the relation between BH masses and NIR luminosities of the host galaxies (Marconi & Hunt 2003), with the NIR data being taken from the Two Micron All Sky Survey (Skrutskie et al. 2006). The formula used for our calculations is given as $\mathrm{log}({M}_{\mathrm{BH}}/{M}_{\odot })=-0.37\times {M}_{{\rm{K}}}-0.59$ (Graham 2007), where M_K is the absolute K-band magnitude of the galaxy. A more specific explanation of this strategy is given in Gupta et al. (2020, see Appendix A therein).

Having calculated bolometric luminosities and BH masses, we obtained the Eddington ratio as ${\lambda }_{\mathrm{Edd}}={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}$, where L_Edd is the Eddington luminosity. Its values for the whole sample range from 0.0004 ≤ λ_Edd ≤ 0.1905,¹⁰ with a median value of λ_Edd = 0.011, and its distribution is presented in the top histogram of Figure 4.

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Among various methods of estimating jet powers of AGNs, we decided to use the one based on the calorimetry of radio lobes, originally formulated by Willott et al. (1999) and modified by Shabala & Godfrey (2013), who, by accounting for radiative losses, delivered a more correct relation. Therefore, the formula adopted by us is given as

$$\begin{eqnarray}&&\begin{array}{l}{P}_{j}\,[\mathrm{erg}\,{{\rm{s}}}^{-1}]=1.5\times {10}^{43}\,{\left[{\left(\displaystyle \frac{\nu [\mathrm{MHz}]}{151}\right)}^{\alpha }\displaystyle \frac{{L}_{1.4}[{\rm{W}}{\mathrm{Hz}}^{-1}]}{{10}^{27}}\right]}^{0.8}\\ \,\times (1+z)\,{\left(D[\mathrm{kpc}]\right)}^{0.58},\end{array}\end{eqnarray} \tag{ 3 }$$

where L_1.4 is the monochromatic lobe radio luminosity at ν = 1.4 GHz, α = 0.8, and D is the source size.

The conversion from radio luminosity to jet power derived by Shabala & Godfrey (2013) is defined for FR II AGNs. These objects constitute 11% of our sample and limiting to those most powerful radio galaxies would give us information about only a fraction of jetted AGNs, clearly biasing our understanding of the jet production mechanisms in various classes of AGNs. The only way to avoid this confusion is to obtain jet powers for all radio-detected sources in our sample by establishing their upper limits. Hence, three groups can be identified: (i) FR II–type AGNs, (ii) objects with lobed but not FR II–type radio morphologies, and (iii) sources with radio detections but without visible double lobes.

A substantial difference appears in our calculations of lobed and nonlobed sources, as for the latter ones we use their total instead of lobe radio luminosity which was originally introduced in Equation (3) and such a quantity can be estimated only for lobed AGNs. Regarding the extraction of lobe radio luminosities, we ensured (when possible) that it does not include any excess radio emission, such as the set of wings in X-shaped sources farther from the core, the outer pair of lobes in DDRGs, or any other radio emission regions which, based on their location on radio maps, do not belong to the radio lobes. In general, the accuracy of the jet power estimation decreases in each of the groups mentioned in the previous paragraph as we have less information about the exact radio characteristics (morphology, size, and flux) of a given source. All jet power calculations for objects in our sample, scaled by bolometric luminosities, are shown in Figure 4.

Even though the results presented in Figure 4 rely mostly on upper limit estimations of P_j (in 223 out of 257 objects), the bimodal distribution of P_j/L_bol is evident, noticeably separating objects powered by relativistic jets from sources in which the radio emission may be dominated by star formation, shocks generated by winds from accretion disks and their coronas, or by low-power jets (Panessa et al. 2019 and references therein). No strong dependence between the Eddington ratio and scaled jet power is found in our sample.

Albeit very popular (Wilson & Colbert 1995; Sikora et al. 2007; Fanidakis et al. 2011; Schulze et al. 2017, and references therein), the "spin paradigm" is seriously challenged by the fact that in order to explain the spread of the jet production efficiency by at least three orders of magnitude, the average BH spin in RQ AGNs should be smaller than 0.03, while that estimated using the "Soltan-type argument" is predicted to be ∼0.6 (Soltan 1982; Chokshi & Turner 1992; Small & Blandford 1992; Elvis et al. 2002; Yu & Tremaine 2002; Lacy et al. 2015). Similarly, large BH spins were found by simulations of the cosmological evolution of BHs (Volonteri et al. 2007, 2013). Hence, in order to reconcile the spin paradigm with RQ AGNs having spins as large as ∼0.6, the dependence of the jet production efficiency on the spin for larger values should be much stronger than quadratic. That possibility was recently claimed to be achievable by Ünal & Loeb (2020) in the model, according to which magnetic fields threading the BH are anchored in the accretion disk. However, noting that magnetic tubes generated in the accretion disk carry on average zero magnetic flux, the jet is predicted to be produced in a flaring fashion (see, e.g., Yuan et al. 2019; Mahlmann et al. 2020) and the resulting time-averaged jet powers can be much smaller than deduced from observations.

The above might imply that the large diversity of the jet production efficiency is primarily determined by the amount of magnetic flux collected on the BH. However, in order to convert the electromagnetic outflow generated by the BZ mechanism into narrow, relativistic jets, external confinement is required (Beskin et al. 1998; Chiueh et al. 1998; Lyubarsky & Eichler 2001). Such confinement can be provided by magnetohydrodynamic (MHD) outflows from accretion disks and in the case of powerful jets is presumably associated with magnetically arrested disks (MADs; e.g., Narayan et al. 2003; Tchekhovskoy et al. 2011; McKinney et al. 2012). Because MADs are formed only if the centrally accumulated magnetic flux exceeds the maximum amount that can be confined on the BH by the accretion flow, the division of AGNs into RL and RI/RQ ones is likely to correspond to the division of AGNs with and without MADs, or, equivalently, that the formation of the MAD provides a sort of threshold for launching powerful relativistic jets.

In order to reconcile such an "MAD/non-MAD" bimodality with our calculated distribution of the jet production efficiency tracer P_j/L_bol (see Figure 4), we need to explain what is the cause of the 2 dex span of this ratio for the RL AGNs assuming they all have MADs. Such a large spread can be an artifact resulting from the calculation of jet powers using their statistical correlation with their radio luminosities (Willott et al. 1999). While adequate in a statistical sense, the conversion formula based on such a correlation may give very large errors for individual sources. Those errors can be associated with the possible spread of parameters such as matter content, minimum electron energy (Willott et al. 1999), cooling effects (Shabala & Godfrey 2013), and density of the environment into which the lobes are inflated (Hardcastle & Krause 2013). But an intrinsic spread of the jet powers in MAD AGNs is also expected, contributed to by the spreads in the BH spin and in the efficiency of the jet collimation by the MHD outflows powered by MADs of different sizes.

It is encouraging to see a similar distribution of P_j/L_bol for our RL AGN sample (Figure 4) and for those found by van Velzen & Falcke (2013) and Inoue et al. (2017) for RLQs. They all peak at P_j/L_bol ∼ 0.1 despite the AGNs included in these samples covering very different accretion rates (see also Rusinek et al. 2017, where four various samples of RL AGNs were studied). Such an average value of the jet production efficiency is smaller by a factor ∼100 than the maximum predicted for MAD AGNs by numerical simulations (e.g., Tchekhovskoy et al. 2011). However it should be noted that the fraction of maximal magnetic flux confined on the BH by the accretion flow depends on the geometrical thickness of the accretion flow (see Avara et al. 2016) and that ${P}_{j}\sim \dot{M}{c}^{2}$ (i.e., ${P}_{j}\sim 10{L}_{\mathrm{bol}}$ for ${\epsilon }_{d}\sim 0.1$) is achievable only for geometrically thick accretion flows, which are not representative for our AGNs nor for quasars.

As we can see in Table 1, whereas most of the RL AGNs in our sample are extended (40/44), most RI AGNs are compact (15/20). Hence, there is no doubt that the deficiency of RI AGNs with extended structures is real. Such a deficiency does not mean that there is some threshold for operation of the BZ mechanism, but may simply reflect the very inefficient collimation in AGNs without the help of MHD winds from MADs. Then, the badly collimated BZ outflows would be significantly entrained by winds from stars enclosed within the outflow volume, slowed down and shocked, and then most of their kinetic energy would be expected to be converted to plasma heat, rather than used to accelerate relativistic electrons producing synchrotron radiation. Such radio sources, together with accretion disk coronas, presumably represent the compact radio sources observed in RQ and RI AGNs, with others being associated with star formation rates and accretion disk winds/jets (see review by Panessa et al. 2019 and references therein). Some insights into the nature of compact radio sources of RQ Swift/BAT AGNs are provided by Smith et al. (2016, 2020).

One can envision the following scenarios to form a MAD: by advection of the magnetic flux by accretion flows, by the accumulation of sufficiently large magnetic flux in a galactic core prior to triggering the AGN phase, and by building up the MAD by the so-called "Cosmic Battery" (CB).

Advection of poloidal magnetic fields by accretion disks was predicted to be very efficient by Bisnovatyi-Kogan & Ruzmaikin (1976). Such a possibility was questioned by Lubow et al. (1994), who pointed out that due to the diffusion of magnetic fields in turbulent plasma, such advection cannot proceed in geometrically thin disks. As later studies showed, poloidal magnetic fields can be advected by the surface layers of accretion disks (Bisnovatyi-Kogan & Lovelace 2007; Rothstein & Lovelace 2008; Guilet & Ogilvie 2012, 2013; Zhu & Stone 2018; Cao & Lai 2019). However, the efficiency of such advection can be limited at distances at which models of thin accretion disks predict their fragmentation due to gravitational instabilities (e.g., Hure et al. 1994; Goodman 2003) and ambipolar diffusion in the outer, only partially ionized portions of the accretion flow (e.g., Begelman 1995). Finally, the advected poloidal magnetic field can be multipolar, and therefore, the formation of a unipolar magnetosphere over the BH and in the innermost portions of the accretion disk—the basic attributes of the MAD—may require more time than the typical lifetime of the AGN, t_AGN. Then, only AGNs that are at the age t_AGN > Δt_MAD can produce powerful jets, where Δt_MAD is the time it takes to form the MAD. This possibility seems to be supported by noting that the typical lifetimes of FR II sources are ∼3 × 10⁷ yr (e.g., Bird et al. 2008), while the lifetimes of RQ AGNs are presumably much shorter (e.g., Schawinski et al. 2015; Schmidt et al. 2018; Khrykin et al. 2019). However, it should be noted that the prevalence of MADs in ellipticals over disk galaxies, despite their AGNs having similar lifetimes, can be explained by a larger amount of advected magnetic flux per unit mass in the former.

Regarding the second scenario, Sikora et al. (2013) proposed that the central accumulation of magnetic flux occurs during a hot accretion phase, prior to a cold, higher accretion rate "event" representing the AGN phase. As suggested by Sikora & Begelman (2013), such an event might be triggered by the merger of a giant elliptical galaxy with a disk galaxy.

Finally, MADs can be formed locally via the operation of the CB (Contopoulos & Kazanas 1998; Koutsantoniou & Contopoulos 2014; Contopoulos et al. 2018). The model is based on the Poynting–Robertson radiation drag effect, which generates a nonzero component of the toroidal electric field in the innermost regions of the accretion flow, which in turn gives rise to the growth of poloidal magnetic field loops. Assuming that the outer parts of the loops diffuse outwards, the inner parts must then accumulate onto the BH. The possibility of the formation of a MAD by the CB was numerically confirmed by general-relativistic magnetohydrodynamic (GRMHD) simulations (Contopoulos et al. 2018), though so far only for nonrotating BHs and optically thin accretion flows. Unfortunately, the analytically estimated timescales for building up the MAD for BHs with masses larger than 10⁹ M_⊙ exceeds the Hubble time (Contopoulos et al. 2018). But, noting that the Poynting–Robertson effect can be much stronger in the case of counterrotating disks, one cannot exclude the formations of MADs in such AGNs within their lifetime. Combining this possibility with predictions that counterrotating configurations can only be formed following mergers involving giant ellipticals with gas-rich spirals (Garofalo et al. 2020), one can explain why RL AGNs are extremely rarely hosted by spiral galaxies (see Section 5). Furthermore, because the fraction of counterrotating disks formed in such mergers is expected to be <50%, one can also explain why RQ AGNs can be found in both ellipticals and spirals.¹³

Given that the sources in Gupta et al. (2018) are from the Northern as well as Southern Hemisphere, radio data were collected from two catalogs: NVSS (Condon et al. 1998) and SUMSS (Bock et al. 1999; Mauch et al. 2003). Both are characterized by similar sensitivity (∼2.5 mJy) and resolution (45'' FWHM for NVSS and 45× 45 cosec $| \delta |$ arcsec² for SUMSS). NVSS, a 1.4 GHz continuum survey, covers the northern sky from −40° decl. while SUMSS, a wide-field radio imaging survey conducted at 843 MHz, covers the southern sky from −30° decl., so together they map the whole sky. In addition, we decided to include one more catalog, FIRST (Becker et al. 1995). This 1.4 GHz sky survey is distinguished by its high resolution (54) and its sensitivity down to 1 mJy radio flux. It covers a piece of the sky surveyed by NVSS, and therefore having detections in both of these catalogs helps to determine whether the source is compact or extended, but also to examine the accurate sizes of compact sources (see Appendix B for more details).

For NVSS and SUMSS data, a search within a matching radius of 3' was conducted. For those sources where a single association was found, its exact location was checked—whether the radio match is located within 30'' from the optical center, and if so, this match was assigned to the object. The same procedure was adopted for FIRST data with the only difference being the internal matching radius of 5'' instead of 30''. All the sources having more than one radio association within a matching radius of 3' were checked by eye. This part was done through visual inspection of radio maps with a size of 045 × 045 extracted from NVSS,¹⁴ SUMSS,¹⁵ and FIRST¹⁶ by using the NRAO Astronomical Image Processing System¹⁷ package. In order to avoid false associations of radio matches, we downloaded the images from DSS1 and DSS2, which are digitized versions of several photographic astronomical surveys, in addition to radio maps, which can be found at the ESO archive.¹⁸ Comparison of radio and optical sources on maps from both domains, together with the NASA/IPAC Extragalactic Database,¹⁹ enabled us to distinguish incorrect matches. As some sources turned out to have extremely extended, i.e., beyond 3', radio structures we were gradually increasing the radio search by 1' as long as the association for the whole structure was found. Once the whole radio-emitting region was identified, the radio flux of each of the components was summed up and assigned to the given source. For sources lacking radio detections, we assigned them upper limits corresponding to the value of the sensitivity of the survey that contains the source in its footprint. While the sensitivity of NVSS and SUMSS is the same, for objects located in the area covered by both NVSS and FIRST, we decided on FIRST upper limits, as its sensitivity is lower than that of NVSS.

Because the radio catalogs we used were conducted at two different frequencies, the radio fluxes at 843 MHz were recalibrated to 1.4 GHz using a radio spectral index of α_r = 0.8 (with the convention of F_ν ∝ ν^−α) so that the rest of our calculations are consistent.

The mid-infrared measurements were taken from the AllWISE Data Release (Cutri et al. 2013) which, by combining data from the cryogenic WISE (Wright et al. 2010) and post-cryogenic NEOWISE ("near-Earth object + WISE," Mainzer et al. 2011) survey phases, resulted in a comprehensive view of the mid-infrared sky. Out of four available bands (at 3.4, 4.6, 12, and 22 μm, corresponding to the W1, W2, W3, and W4 bands, respectively), we decided to make use of the W3 band, which was driven by the fact that at this specific wavelength the dusty torus is transparent enough to observe radiation coming directly from the central region of an AGN. At the shorter wavelengths of the W1 and W2 bands, the dusty, circumnuclear tori are optically thick and radiate anisotropically (Hönig et al. 2011; Netzer 2015). At the longer wavelengths of the W4 band, the dusty torus becomes even more transparent, but its measurements are affected by much larger errors than in W3 because (1) the W4 band traces the warm dust continuum at 22 μm and can be contributed to by starbursts (Ichikawa et al. 2019); (2) the sensitivity in the W4 band is much lower than in the W3 band and its signal-to-noise ratio is the lowest of all the WISE channels, resulting in a lower detected fraction for the objects in our sample; and (3) the resolution of W4 images is worse than in other bands (e.g., 12'' in W4 versus 65 in W3), so for some of our sources (point like in optical, with close neighbors), W4 images could be blended.

Searching for counterparts was conducted within a radius of 5'', which allowed for the avoidance of false matches as the angular resolution of WISE in the W3 band is 65.

The conversion from W3 magnitude, m_W3, to the monochromatic flux, ${F}_{{\nu }_{{\rm{W}}3}}$, was done following the formula provided by Wright et al. (2010), given as ${F}_{{\nu }_{{\rm{W}}3}}=30.922\times {10}^{(-{m}_{{\rm{W}}3}/2.5)}$.