AGN Coronae through a Jet Perspective

NuSTAR is a powerful satellite that measures the X-ray spectra between 3 and 79 keV (Harrison et al. 2013). This band is extremely well suited for studying accretion onto black holes, as this is the band where we expect to observe distinct features from the accretion disk. The most prominent features are the relativistically broadened Fe Kα line at 6.4–6.97 keV, and the Compton hump at ∼20 keV (e.g., George & Fabian 1991; Reynolds 1999). In addition, both of these features probe the corona, as a distribution of high-energy particles above the disk is required to shine down and fluoresce the Fe Kα line as well as scatter photons into the Compton hump spectral feature.

Thirty two AGNs are publicly available in the NuSTAR archive as of 2016 August 1 that meet the criterion for this study. This criterion includes Compton-thin AGNs, i.e., ${N}_{H}\lt {10}^{24}$ cm⁻², with at least 20 ks exposure time. The average total exposure time of $\langle {t}_{\exp }\rangle =160\,\mathrm{ks}$, with an average total counts of $\langle {Cts}\rangle =1.8\times {10}^{5}$ photons.

Each of these AGN NuSTAR data were reduced using the standard reduction pipeline nupipeline, during which the screening saamode was set to strict to ensure a low background level. We produced NuSTAR spectra from the cleaned event files using the tool nuproducts. A 60 and 120 arcsec circular extraction region was used for the source and background regions, respectively. Both focal plane module (FPM) A and B spectra were produced. For the targets with multiple observations, the spectra of each focal plane module were summed to create average spectra. The AGNs are expected to be variable in the X-ray band, both in flux and spectral features (e.g., Nandra et al. 2000; Fabian et al. 2012; Parker et al. 2014; Wilkins et al. 2014, 2015). However, as we are comparing the reflection measurements to radio measurements taken years apart from the X-ray observations, variability between the observations likely dominates the systematic errors. In addition, because dynamical timescales scale with mass of the black hole, the X-ray variability will primarily effect our least massive sources. We therefore concentrate on the average X-ray properties of the AGNs and include extra uncertainties in the radio band to account for variability between the measurements (see Section 2.3).

The average FPMA and FPMB spectra were re-binned to include 20 counts per bin for the spectral modeling. The 3–78 keV energy range was used for the spectral modeling. We fit the spectra in ISIS (Houck & Denicola 2000) with a model based on a phenomenological cutoff power-law continuum. We limit the high-energy cutoff of the power law to be larger than 75 keV to avoid confusion with the Compton hump. The xillver model (García et al. 2013) is then used to fit the Fe Kα line and Compton hump. It simply and uniformly quantifies the total reflection fraction, which is measured as $R={\rm{\Omega }}/2\pi$, where Ω is the solid angle that the corona subtends. In most cases, the resolution of NuSTAR does not allow for disentangling the neutral from ionized reflection components, and therefore we only include one xillver component to quantify both. See Section 4.1 for further discussion.

The inclination parameter included in xillver is fixed at the value given in Table 1 because it cannot be precisely constrained from our data. These values were primarily selected from higher-resolution reflection fits which can measure the inclination of the inner disk (e.g., Walton et al. 2014; García et al. 2014; Kara et al. 2014). When this was not available, we utilized inclinations from the jet, assuming the jet axis is aligned perpendicular to the disk (e.g., Giovannini et al. 2001; Jorstad et al. 2005). Finally, we used inclinations of the infrared torus when no other estimates were available (Ruschel-Dutra et al. 2014). The average viewing inclination for our sample is $\langle {\theta }_{\mathrm{inc}}\rangle =41^\circ$, which is to be expected as our sample selects against type II AGNs that are viewed close to edge on, i.e., large inclinations.

To model the Fe Kα line well, the iron abundance is allowed to vary from 0.5 to 3 times solar, which incorporates the typical values measured for most AGNs (Walton et al. 2013). The whole model is modified by the absorption from our own Galaxy described by the model TBnew with the cross section set to vern (Verner et al. 1996) and the abundances set to wilm (Wilms et al. 2000). The absorption column is kept fixed to the average value from the N_H tool for each source. This very basic baseline model provides a good fit (${\chi }_{\mathrm{red}}^{2}\lt 1.5$) to 25 of the 30 sources. For the five sources with reduced ${\chi }^{2}$ larger than 1.5, we included excess absorption and/or a board Fe Kα line (relline, Dauser 2015). These components are denoted in Table 1. The 1σ errors for the reflection fraction are also given in Table 1, and where the reflection fraction is only an upper limit, the 3σ errors are given.

Finally, in several of the radio-loud sources, one might expect an additional jet component in the continuum that would artificially dilute the reflection signal. In our sample, the strongest radio sources are Cyg A, 3C 382, Cen A, 3C 390.3, 3C 273, 4C 74.26, and 3C 120. However, in individual detailed analysis of these sources, we find our reflection fractions are consistent with those published in the literature using the same NuSTAR data sets, albeit sometimes slightly lower (e.g., Madsen et al. 2015; Reynolds et al. 2015; Fürst et al. 2016). As theses studies test for the possibility of an additional jet component but do not find a significant contribution from one, we may be underestimating the reflection fraction in a few of these sources but not by a significant amount.

XMM-Newton is also a powerful X-ray satellite whose higher collecting area and spectral resolution surpasses that of NuSTAR, but with a more limited band pass, 0.3–10 keV (Jansen et al. 2001). With its high collecting area, it is well suited for timing analysis of even the faintest X-ray sources. X-ray reverberation features of the inner accretion disk are expected on dynamical timescales, which for supermassive black holes occur on the order of a few hundred seconds.

In particular, one of the most prominent timing features is the high-frequency lag associated with the Fe Kα region lagging behind the X-ray continuum. The size and shape of the time lags are quite informative, as they encode information about the height and geometry of both the emitting and reflecting regions (Wilkins et al. 2016). We assume the amplitude of the time lag, t_lag, is a good estimate of the extra light path taken by the photons that get reflected off the accretion disc, $H=\tfrac{{t}_{\mathrm{lag}}{c}^{3}}{{GM}}$. However, this does not include the effect of dilution on the lag (see Uttley et al. 2014; Kara et al. 2016, for detailed discussions). In short, the time lag is measured between energy bands that contain both primary and reflected emission, and therefore this dilutes the actual light travel reverberation lag. This dilution effect needs to be modeled individually for each source (e.g., Cackett et al. 2014), and therefore the uncorrected lags presented in Table 1 should be taken as a lower limit on the light travel time. Dilution corrections may increase the reverberation lag, and we account for this in Section 3.2.3.

Our Fe Kα time lag sample comes from Kara et al. (2016). These authors have undertaken a comprehensive X-ray spectral timing analysis of all Seyfert AGNs in the XMM-Newton archive that have at least 40 ks exposures and show substantial variability. Of the 43 AGNs analyzed, 21 sources have revealed X-ray lag signatures. The amplitude of these time lags range from 50 to 1800 s. In addition, eleven of these sources overlap with the NuSTAR reflection sample, although the XMM-Newton observations are not contemporaneous with the NuSTAR observations (Table 1).

As the X-ray band probes the coronal properties, the radio band probes the jet power via synchrotron radiation from the accelerated particles along the magnetic fields in the jet (Allen et al. 2006; Merloni & Heinz 2007). The radio flux densities are primarily taken from the NRAO VLA Sky Survey (NVSS; Condon et al. 1998). This is a 1.4 GHz survey covering the entire sky north of −40° declination. It has a restoring beam of 45 arcsec at full-width half-maximum, and reaches a root-mean-square of 0.45 mJy. Given a $5\sigma$ detection limit, the faintest sources in the catalog are >2.3 mJy. Several sources were either located outside the survey area or undetected. Whenever possible, we included measurements at 1.4 GHz, but a select few sources, PG 1247+267 and NGC 7213, were extrapolated from either 843 MHz or 5 GHz. See Table 1 for the radio luminosities.

The majority of the sources in our sample are point-like sources. However, several radio-loud sources (4C 74.26, 3C309.3, 3C 382, Cen A, Cyg A) do show extended emission even at the NVSS survey resolution. We restrict the measurements to the core component, though some emission from the lobes may have been included due to the large restoring beam.

The radio measurements are not simultaneous with the reflection or reverberation measurements. In fact, most measurements are made over a decade apart. Variability in the radio-quiet and radio-loud is generally on the order of 20% on monthly timescales (Barvainis et al. 2005), and larger variations are expected on yearly timescales (e.g., Chatterjee et al. 2009, 2011). Therefore, to address this variability, we include a factor of 0.3 dex in the systematic uncertainty of the radio luminosities, which dominates over any measurement uncertainties.

The radio luminosities are scaled by their black hole masses, which allows for self-similar comparison between the AGNs. We refer to this radio luminosity as the "radio Eddington luminosity," which we explicitly define as the 1.4 GHz radio luminosity, L_R, divided by the Eddington luminosity, ${L}_{\mathrm{Edd}}=1.38\times {10}^{38}(M)$ erg s⁻¹, where M is the mass of the supermassive black hole. The black hole masses are primarily derived from optical reverberation measurements, assuming a scaling factor $\langle f\rangle =4.3$ (Grier et al. 2013; Bentz & Katz 2015). For those sources where reverberation masses are not available, gas kinematics, virial velocities, optical luminosity, and X-ray scaling relations are all used to estimate the black hole masses. We list the masses in Table 1. The uncertainty for these mass estimates ranges from a few percent to 0.6 dex in the case of the X-ray scaling relations. PDS 456 has the most uncertain mass estimate as Nardini et al. (2015) use the measured X-ray luminosity with an expected Eddington ratio to estimate its mass.

The black hole mass distribution spans four decades in mass, with an average mass of $\langle \mathrm{log}M\rangle =7.6\,{M}_{\odot }$, and a standard deviation of σ = 0.9. The black holes with reflection measurements have an average mass of $\langle \mathrm{log}M\rangle =7.8\,{M}_{\odot }$ and $\sigma =0.9$, while the black holes with X-ray reverberation measurements have a slightly smaller mass distribution at $\langle \mathrm{log}M\rangle =7.1\,{M}_{\odot }$ and $\sigma =0.5$. The black holes are also accreting at relatively high mass-accretion rates ${L}_{{\rm{X}}}/{L}_{\mathrm{Edd}}\gtrsim {10}^{-3}$, indicating the disk should be a thin disk rather than an advection dominated regime(e.g., Narayan & Yi 1995; Narayan 2005). This assumes the bolometric luminosity should be at least an order of magnitude higher than the X-ray luminosity.

Figure 1(a) shows the reflection fraction versus radio Eddington luminosity. There is an apparent inverse correlation between the two parameters, though the sample is dominated by Seyfert galaxies clustered at low radio Eddington fractions. To test the strength of this correlation, we used a Markov chain Monte Carlo (MCMC) model to fit a linear fit to the NuSTAR reflection fraction data in log space with an intrinsic scatter, implemented with python's PyMC v2.3.6 (Fonnesbeck et al. 2015). The detected data points were drawn from a normal distribution determined by their 1σ error bars. The data points with upper limits were drawn from a uniform distribution in linear space between zero and the 3σ upper limit. The fits were only marginally dependent on the upper limit priors. We used 100,000 steps and discarded the initial third of the distribution.

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The resulting best-fit relation is $\mathrm{log}R=-(0.22\pm 0.06)\mathrm{log}({L}_{R}/{L}_{\mathrm{Edd}})-(1.7\pm 0.4)$ and an intrinsic scatter of ${\sigma }_{R}={0.35}_{-0.05}^{+0.06}$ (Table 2). We test this against our null hypothesis of no correlation, i.e., m = 0, and find the relation is inconsistent with this with a probability of p = $1.2\times {10}^{-4}$.

Table 2.  Fit Parameters

Name	Slope	Intercept	$\mathrm{log}(N)$	θ	$\mathrm{log}{L}_{\mathrm{crit}}$	${\sigma }_{R}$	${\sigma }_{H}$
Log-Linear fits
log R versus $\mathrm{log}{L}_{R}/{L}_{\mathrm{Edd}}$	$-(0.22\pm 0.06)$	−(1.7 ± 0.4)				${0.35}_{-0.05}^{+0.06}$
log H versus $\mathrm{log}{L}_{R}/{L}_{\mathrm{Edd}}$	${0.25}_{-0.08}^{+0.07}$	2.5 ± 0.5					${0.26}^{+0.11}_{-0.09}$
Moving Corona			$-(10.1\pm 0.4)$	${13}_{-4}^{+5\circ}$	$-(9.6\pm 0.3)$	${0.46}_{-0.06}^{+0.07}$	${0.17}_{-0.11}^{+0.12}$

Note. This table shows the best-fit values for our log-linear fits as well as with our corona–jet model. The last two columns give the intrinsic scatter in each data set. The models are plotted in Figures 1 and 3, and errors are 1σ.

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Figure 1(b) shows a positive correlation between the corona–disk path length, derived from reverberation time lags, and the radio Eddington luminosity. A linear fit in log space to the XMM-Newton data is given by $\mathrm{log}H={0.25}_{-0.08}^{+0.07}\mathrm{log}({L}_{R}/{L}_{\mathrm{Edd}})+2.5\pm 0.5$ with an intrinsic scatter of ${\sigma }_{H}\,={0.26}_{-0.09}^{+0.11}$ (Table 2). Again, we utilized an MCMC model, drawing the data from a normal distribution using 1σ uncertainties and a uniform distribution between zero and 3σ upper limits in linear space for the non-detections. This fit is also inconsistent with the null hypothesis of no correlation, i.e., m = 0, at $p=3.7\times {10}^{-4}$. In addition, this fit is inconsistent with a slope of unity at $p\lt 1\times {10}^{-5}$. This is important to note as both the path length and radio Eddington luminosity scale inversely with mass, by definition. Therefore, if mass was singularly driving this relation, the slope would be consistent with unity, which we do not find.

Together, these two relations (Figures 1(a) and (b)) indicate that changes in the corona are linked to jet production in AGNs.

We next investigate a physically motivated model that predicts both changes in reflection fraction and disk–corona distance as a function of mass-accretion rate and, consequently, jet power. These changes are assumed to originate in a corona that is moving away from the disk. As the velocity increases, more of the coronal radiation is beamed away from the accretion disk, decreasing the reflection fraction. If the corona is outflowing from the disk, it is reasonable to assume that it is associated with the base of the jet, and that increases in velocity in the corona propagate into the jet. The increased velocity in the jet results in more efficient synchrotron radiation, and thus a more radiatively powerful jet. Finally, an increase in mass-accretion rate is assumed to drive the increase in velocity. The higher mass-accretion rate provides both a larger radiation field to push on the charged particles via Compton scattering, and plausibly a stronger magnetic field, which can dissipate in the corona, also accelerating the charged particles (e.g., Kara et al. 2016). In addition, the local sound speed may also set the corona velocity (Markoff et al. 2005), which increases with mass-accretion rate.

In addition, as the mass-accretion rate increases, a region in the inner disk becomes radiation dominated. The ionization of this region increases and the reflection fraction decreases (García et al. 2013). At high enough mass-accretion rates, the inner region becomes so optically thick that the radiation is advected across the event horizon, again dramatically decreasing the reflection features (Narayan et al. 1998). Therefore, the most prominent "reflection" region moves outward in the disk to the gas-pressure dominated regions, increasing the observed time-lags, and therefore the predicted path length between the corona and emission region in the disk (Figures 2(a) and (b)). In the following subsections each of the particular model components (moving corona, jet, and disk) are described in detail.

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3.2.1. The Moving Corona

Beloborodov (1999) describes a model for computing reflection fractions for a moving corona. He developed this model to fit the small reflection fraction in the stellar-mass black hole Cygnus X-1. As a result, the model not only fit the small reflection fraction in Cygnus X-1 ($R\sim 0.3$) but also the hard spectral index of ${\rm{\Gamma }}=1.6$ with a moving corona with velocity $\beta =v/c=0.3$.

In this model, as the corona moves away from the disk, the photons get Doppler shifted away, resulting in a decrease of flux illuminating the disk. This results in a reflection fraction that is dependent on velocity as:

$$\begin{eqnarray}&&R=\displaystyle \frac{{(1+\beta /2)(1-\beta \mu )}^{3}}{{(1+\beta )}^{2}}\end{eqnarray} \tag{ 1 }$$

where $\mu =\cos {\theta }_{\mu }$, ${\theta }_{\mu }$ is the viewing angle to the source.

3.2.2. Self-similar Jet

In order to compare the predicted reflection fraction to the jet luminosity, we next assume the radio luminosity is well described by the self-similar, optically thick jet model. Synchrotron emission from a self-similar jet that is optically thick, $\alpha =0$, where ${S}_{\nu }\propto {\nu }^{\alpha }$, scales with both black hole mass, M, and mass-accretion rate, $\dot{m}=\dot{M}/{\dot{M}}_{\mathrm{Edd}}$ as

$$\begin{eqnarray}&&{L}_{R}\propto {M}^{187/120}\,{\dot{m}}^{17/15}.\end{eqnarray} \tag{ 2 }$$

where L_R is the radio luminosity, and we assume a standard thin, gas-pressure dominated disk (Heinz & Sunyaev 2003). The magnetic field in a gas-pressure dominated disk also scales with black hole mass and mass-accretion rate as

$$\begin{eqnarray}&&{B}^{2}\propto {\dot{m}}^{4/5}{M}^{-9/10}\end{eqnarray} \tag{ 3 }$$

(Heinz & Sunyaev 2003). Finally, we assume that the gas pressure in the jet is proportional to the magnetic field strength,

$$\begin{eqnarray}&&{B}^{2}\propto {\rho }_{j}(\gamma -1){c}^{2}\end{eqnarray} \tag{ 4 }$$

where $\gamma ={(1-{\beta }^{2})}^{-1/2}$, and ${\rho }_{j}$ is the gas density in the jet. We note that we keep ${\rho }_{j}$ fixed, but this could likely change as a function of mass-accretion rate, mass of the black hole, and even magnetic field strength, depending on how the jets are loaded.

Combing Equations (2), (3), and (4) gives the following radio luminosity dependence on velocity:

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{R}}{{L}_{\mathrm{Edd}}}={NM}{(\gamma -1)}^{17/12}\end{eqnarray} \tag{ 5 }$$

where ${L}_{\mathrm{Edd}}=1.38\times {10}^{38}(M)$ erg s⁻¹, and N is a normalization constant that we fit for in our final MCMC simulations.

3.2.3. Standard Accretion Disk

In addition to predicting the reflection fraction from a moving corona and the luminosity from the optically thick jet, we can use the model to predict the relative path length between the corona and the disk, H. As H is estimated from the X-ray reverberation time-lags, we assume it is the shortest path between the corona and the most efficient reflecting region in the disk, R_refl. R_refl is typically assumed to be the inner radius of a thin disk, as this is the region that has the highest emissivity to produce both the Fe Kα line and Compton hump reflection features (Reynolds 1999). However, high mass-accretion rates can result in a region in the inner disk that is radiation pressure dominated. The inner layers in this region may become overly ionized as it increases in scale height and temperature, thus unable to efficiently produce distinct reflection features (Shakura & Sunyaev 1973; García et al. 2013). The boundary, r_b, between the radiation pressure and the standard, gas-pressure dominated regions scales as:

$$\begin{eqnarray}\left\{\begin{array}{ll}\tfrac{{r}_{b}}{{(1-{r}_{b}^{-1/2})}^{16/21}} & \\ \quad =150{({\alpha }_{D}M)}^{2/21}{\dot{m}}^{16/21}, & \dot{m}\gtrsim \displaystyle \frac{1}{170}{({\alpha }_{D}M)}^{-1/8}\\ {r}_{b}={r}_{\mathrm{isco}}, & \dot{m}\lt \displaystyle \frac{1}{170}{({\alpha }_{D}M)}^{-1/8}\end{array}\right.\end{eqnarray} \tag{ 6 }$$

where ${\alpha }_{D}$ parameterizes the viscosity in the disk, and r_isco is the innermost stable circular orbit (ISCO; Shakura & Sunyaev 1973). Combining Equations (2) and (6) and assuming ${R}_{\mathrm{refl}}={r}_{b}$, we find that the inner disk radius, where the reflection features can easily arise, scales with luminosity as:

$$\begin{eqnarray}\left\{\begin{array}{ll}\tfrac{{R}_{\mathrm{refl}}}{{(1-{R}_{\mathrm{refl}}^{-1/2})}^{16/21}} & \\ \quad ={N}_{2}{\left(\tfrac{{L}_{R}/{L}_{\mathrm{Edd}}}{{M}^{5/12}}\right)}^{80/119}, & \displaystyle \frac{{L}_{R}}{{L}_{\mathrm{Edd}}}\gt {L}_{\mathrm{crit}}{M}^{5/12}\\ {R}_{\mathrm{refl}}={R}_{\mathrm{refl},0}, & \displaystyle \frac{{L}_{R}}{{L}_{\mathrm{Edd}}}\lt {L}_{\mathrm{crit}}{M}^{5/12}\end{array}\right.\end{eqnarray} \tag{ 7 }$$

L_crit is a combination of the factor that scales the mass and mass-accretion rate to an observable 1.4 GHz luminosity (L_R, Equation (2)) and the normalization, $\tfrac{1}{170{\alpha }_{D}^{1/8}}$, in the conditional dependence of Equation (6). We assume ${R}_{\mathrm{refl},0}=1.23697{R}_{G}$, i.e., the smallest inner radius for a maximally spinning black hole, a = 0.998 (Bardeen et al. 1972). Making sure the distribution is continuous, we rewrite N₂ as:

$$\begin{eqnarray}&&{N}_{2}=\displaystyle \frac{{R}_{\mathrm{refl},0}}{{(1-{R}_{\mathrm{refl},0}^{-1/2})}^{16/21}}{L}_{\mathrm{crit}}^{-80/119}.\end{eqnarray} \tag{ 8 }$$

Interestingly, the reflection fraction is only sensitive to changes in the scale height at very small radii, $\lt 10{R}_{G}$ (Dauser et al. 2016). Therefore, for the simplicity of the model, we ignore this effect for the reflection calculations and only include them in the coronal height calculations. We also stress the assumption of ${R}_{\mathrm{refl}}={r}_{b}$ is an oversimplification, and that there likely is still some emission coming from the radiation pressure dominated regions, especially during the initial transitions. However, it is still illustrative in this first-order approximation we are making (ee Section 4.3 for further discussion).

Finally, we take into account the dilution of the time lags due to varying contributions of the reflected spectrum as a function of energy. Dilution is correlated with the percentage of observed reflected emission as compared to the continuum emission, and scales as $D=\tfrac{R+1}{R}$ (see Cackett et al. 2013, 2014; Wilkins & Fabian 2013, for more details). Figure 4(e) shows the effects of including the dilution factor in our model by plotting the corrected path length as a function of reflection fraction. The dilution factor as a function of reflection factor decreases the time lag and therefore inferred path length, H:

$$\begin{eqnarray}&&H=\displaystyle \frac{{R}_{\mathrm{refl}}}{D\sin \theta },\end{eqnarray} \tag{ 9 }$$

where $\sin \theta$ is assumed to be the same for the entire sample.

3.2.4. MCMC Setup

We jointly fit the NuSTAR reflection strength sample and XMM-Newton coronal time-lag sample as functions of radio Eddington luminosity with the moving corona, self-similar jet, and standard thin disk accretion model using Equations (2), (5), and (10) and an MCMC model (see Figures 3 (a) and (b)). The only free parameters in this model are the geometry, θ, the normalization, N, and the critical luminosity, L_crit. Because the dilution factor depends on reflection fraction, the normalization constrains both the NuSTAR and XMM-Newton distributions. This can be seen in Figures 4(a) and (b), where we hold all other parameters constant and change the normalization. Figures 4(c) and (d) show the results of holding all other parameters constant and changing the critical luminosity or viewing angle, respectively. One can see that these later parameters only effect the XMM-Newton time-lag samples.

We assume a log-normal prior on all the detected data, and a uniform prior between 0 and the $3\sigma$ upper limits in linear space for the non-detections. We assumed a log-uniform prior for the critical luminosity, $-13\lt \mathrm{log}{L}_{\mathrm{crit}}\lt -6$, as well as uniform in the normalization, $-15\lt \mathrm{log}N\lt -7$. Finally, we assume a uniform prior on $0\lt \sin \theta \lt 1$. We used a total of 100,000 steps, burning the first third of the distribution for each MCMC run. After checking for convergence using four independent runs utilizing a Gelman–Rubin statistic for each parameter in question (Gelman & Rubin 1992, pp. 625–631), the final results are given in Table 2 and overplotted in Figures 3(a) and (b).

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3.2.5. MCMC Results

The results of the MCMC fits are given in Table 2 and plotted in Figures 3(a) and (b). We find that our model does fit the data well, and has scatters very comparable to the individual linear fits. The small $\theta ={13}_{-4}^{+5\circ}$ indicates that the path length inferred from the time lags between the corona and reflecting regions in the disk is on the order of the corona height above the disk. For emission coming from close to the ISCO of a maximally spinning black hole at mass-accretion rates a few percent of Eddington, this corresponds to a height of $\sim 5\,{R}_{G}$. This is consistent with heights measured in several radio-quiet Seyfert AGNs where the disk extends down to the ISCO (Miniutti & Fabian 2004).

The critical luminosity is measured to be $\mathrm{log}{L}_{\mathrm{crit}}\,=-(9.6\pm 0.3)$. This corresponds to a radio Eddingotn luminosity of $\mathrm{log}({L}_{R}/{L}_{\mathrm{Edd}})\approx -6.5$ using the average mass of the sample of $\langle \mathrm{log}M\rangle =7.6$. In our model, this parameter determines at what radio Eddington luminosity the height of the corona is no longer constant but begins to scale with radio Eddington luminosity and mass of the black hole (Figure 4(b)). Physically, $\mathrm{log}{L}_{\mathrm{crit}}$ determines when the inner disk structure becomes radiation pressure dominated (Shakura & Sunyaev 1973). The shaded regions in Figures 3(a) and (b) mark the region where the inner disk is radiation pressure dominated given the average mass of the distribution, with the darker regions corresponding to larger regions of the inner disk dominated by radiation pressure.

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Finally, we determine the normalization to be $\mathrm{log}N\,=-(10.1\pm 0.4)$. The normalization is responsible for setting the position of the inflection point in Figure 4(a) as well as the amount of dilution which effects H at high radio Eddington fractions in Figure 4(b). The normalization also sets the scale for the velocity of the corona. Taking the average mass of our samples, $\langle \mathrm{log}M\rangle =7.6$, the predicted velocities of the corona span $0.04\lt \beta \lt 0.40$, across our radio Eddington luminosity range.

Our sample indicates an inverse trend between reflection fraction and Eddington scaled radio luminosity. This is well characterized by a linear fit in log space and our physically motivated, moving corona–jet model. In this study, we measure the total reflection fraction with a xillver model. This is likely to be the sum of the relativistically blurred reflection from the inner accretion disk and the distant reflection from the outer accretion disk or torus. Contrary to our interpretation of a continuous distribution of reflection fractions, one could interpret this inverse correlation as a constant distant reflection fraction from a torus with a contribution of blurred reflection in only the low radio Eddington sources. A survey by Nandra et al. (2007) hints that the distant reflection is constant as it shows less scatter compared to the blurred disk reflection. In addition, the authors find the average reflection strength in Seyferts with detected blurred disk lines dominates over the distant reflection at $\langle {R}_{\mathrm{disk}}\rangle =1.00\pm 0.18$ compared to $\langle {R}_{\mathrm{dist}}\rangle =0.46\pm 0.06$. Likewise, many studies of radio AGNs indicate that the reflection quantified by the neutral Fe Kα line is very narrow, possibly originating from the disk (Lee et al. 2002) but likely originating in a torus (Nandra & George 1994).

Assuming the standard AGN unification model (Krolik et al. 1994), where AGNs have the same structure with viewing angle being responsible for the difference in observed spectra, we could expect that all AGNs have a similar distant reflection fraction from the torus. Therefore, we might expect a "floor" in the reflection fraction. Our highest radio Eddington sources strongly argue against this, as they are well below the average reflection fraction for the distant reflectors in Seyfert galaxies given by Nandra et al. (2007). This suggests that either (1) the distant torus is nearly absent in the high radio luminosity sources, contrary to the standard unification model, or (2) that the distant reflecting region is also sensitive to the beaming of the corona.

There is strong evidence that 3C 273, our highest radio Eddington luminosity source, still has a substantial torus as supported by detections of molecular hydrogen in the vicinity of the quasar (Kawara et al. 1989). A follow-up study by these authors indicates that the mass in quasar tori (3C 273 and IRAS 13349+2438) is roughly six times larger than in Seyfert galaxies (Kawara et al. 1990), strongly disfavoring the idea that the torus is diminishing as AGNs move to higher mass-accretion rates. Therefore, we favor the interpretation that both the disk and distant reflection are sensitive to beaming of the corona.

We also examine the hint of larger scatter at low radio Eddington fractions in Figures 1(a) and 3(a). The largest outlier and highest reflection data point is PG 1247+267. At a redshift of z = 2.038, PG 1247+267 is also an outlier in terms of distance as the rest of the sample resides below $z\lt 0.2$. It therefore likely resides in a different environment than the majority of our sample, as the universe was much denser at z = 2.038. This could have effects on both the structure of the disk and jet, and resulting in different measured reflection and jet properties, resulting in PG 1247+267 being an outlier from the sample distribution.

The rest of the scatter at low radio Eddington fractions may be due to varying modes of accretion. At a mass-accretion rate of a few percent of Eddington, the inner disk is predicted to move from a standard, optically thick, geometrically thin accretion disk to an advection dominated flow (Narayan & Yi 1995; Narayan et al. 1998). In such a regime, the inner disk may no longer be as efficient at producing reflection features, and the jet luminosity scales differently with mass and mass-accretion rate as is assumed in Equation (2). Both of these would increase the scatter of our distribution.

The scatter in reflection measurements may also be due to not including relativistic effects as the corona recedes closer to the black hole, as is implied by Figures 1(b) and 3(b). Our model currently saturates at a reflection fraction of unity because we assume that the velocity is always outflowing from the disk as defined by jet structure. If we allowed for velocities to approach the disk, we could of course get $R\gt 1$ (Reynolds & Fabian 1997; Beloborodov 1999), but would not be able to calculate the corresponding jet luminosity in the analytical form presented here. In the future, we aim to include general relativistic effects in our calculations of the reflection fraction with the goal of better characterizing the low radio Eddington fraction data. This regime is where the time-lag analysis data predict the smallest coronal height above the disk (Figure 3(b)), i.e., the region most likely be effected by gravitational light bending and relativistic effects. Such an effect would boost the reflection fraction above unity (Tanaka et al. 1995; Fabian et al. 2000), in agreement with our highest reflection fraction measurements.

In addition to quantifying how the AGN reflection fraction and the corona path length to the reflecting region of the disk scales with radio Eddington fraction, our model predicts the coronal and subsequent jet velocity also scale positively with radio Eddington fraction.

Stochastic motions in the corona have been observed in individual AGNs, including in Mrk 335 (Parker et al. 2014; Wilkins et al. 2015) and 1H 0707-495 (Fabian et al. 2009, 2012; Wilkins et al. 2014). Through the use of detail modeling of the Fe Kα line, the reflection fraction is inversely correlated with flux (e.g., Zdziarski et al. 1999; Parker et al. 2014; Wilkins et al. 2014, 2015). This is consistent with the height of the corona increasing above the disk, reducing the amount of light bending, which ultimately decreases the reflection fraction, and increases the continuum flux. Wilkins (2016) takes this idea one step further and suggests the flaring activity in the low flux states may be reconfigurations of the corona into a base of a jet. Our own analysis suggests more dramatic changes in the corona during even larger changes in accretion states than examined in these works, but qualitatively agree with the findings of a moving corona.

Physically explaining stochastic motions requires dissipation of energy into the corona and subsequently the jet. This motion of the corona has therefore been explained via advection dominated accretion disks creating outflows (e.g., Blandford & Begelman 1999; Quataert & Gruzinov 2000), Compton pressure from the disk and reflected photons on the corona (Beloborodov 1999; Ghisellini & Tavecchio 2010), or magnetically driven corona above an accretion disk (Merloni & Fabian 2002; Hawley & Krolik 2006; Tchekhovskoy & Bromberg 2016). In particular, these last two are most likely applicable to our sample at intermediate to high mass-accretion rates. Interestingly, magnetohydrodynamic models of an outflowing corona above an accretion disk naturally suggest that the velocity increases with height above the accretion disk (Merloni & Fabian 2002) as well as mass-accretion rate (Ghosh & Abramowicz 1997). This is consistent with our models, which assume the velocity scales proportionately to radio luminosity and consequently mass-accretion rate and height of the corona. Conversely, the Compton pressure from the reflected photons as suggested by Beloborodov (1999) is also a viable model, and occurs simultaneously. This effect is most efficient when the plasma is made up of electron–positron pairs. Several works suggest that electron–positron pair production can regulate the temperature of the corona (e.g., Svensson 1984; Zdziarski 1985; Pietrini & Krolik 1995; Stern et al. 1995; Dove et al. 1997; Coppi 1999). Recent work by Fabian et al. (2015) comparing theoretical predictions to NuSTAR observations seems to confirm this, giving viability to this Compton rocket-like scenario.

On much larger scales further down the jet, changes in velocity have been used to explain the differences in jet structure in Fanaroff–Riley (FR) I versus FR II jet morphologies. In our model, we assume the velocity of the corona is proportional to the velocity accelerated further down the jet and therefore is relevant to large-scale jet velocities as well. FR I jets are radio-loud jets with low surface brightness and diffuse structures, while FR II are jets at even higher luminosities and are highly collimated, terminating in large impact lobes. A slower velocity in FR Is compared to FR IIs would would explain the morphology as the jets would be less collimated and more likely to diffuse into the surrounding medium (Gendre et al. 2013). Though it cannot be ruled out that environment also plays a role in these jet morphologies, it is clear that FR Is accrete at a lower rate than FR IIs (Marchesini et al. 2004; Best & Heckman 2012). Thus it would not be surprising if the flow onto the black hole is intimately connect to the large scale jets, as we also suggest here.

In addition, King et al. (2015) examine the velocity of jet knots close to the black hole using Very Long Baseline Array (VLBA) archival data, and find that the velocity of these knots increases with X-ray Eddington luminosity. Whether the X-ray emission is from synchrotron emission from the base of the jet or Comptonized photons from the disk, the correlation indicates the velocity scales with increasing photon energy density close to the black hole, likely a result of increasing mass-accretion rate. These knot velocities are much larger than the predicted for the corona, $0.04\lt \beta \lt 0.40$. This is understood in the context of these VLBA measured knots being measured on order of a ${10}^{4}\,{R}_{G}$ from the black hole, and have already undergone some acceleration along the jet.

Along with mildly relativistic corona velocities, our model predicts that many of the sources are accreting close to their Eddington rates and enter a vertically extended, radiation pressure dominated accretion phase in regions closest to the black hole (shaded region in Figures 3(a) and (b)). Our model assumes that no reflection features are emitted in this radiation pressure dominated regime due to the gas being overly ionized. In reality, the ionization is likely to increase gradually, being dependent on both mass and mass-accretion rate. Thus, the reflection should still originate close to the ISCO in several of the sources above L_crit (lightly shaded region in Figures 3(a) and (b)). This would reconcile many of the analyses of AGNs, like IRAS 13224-3809 (Fabian et al. 2013), that have detailed reflection measurements emitting from the ISCO but where we predict a radiation pressure dominated inner disks.

The exact transition of when the inner disk becomes overly ionized is out of the scope of this paper. Our current model does not allow us to constrain this due to our assumption of "total" ionization as soon as the region becomes radiation pressure dominated. Furthermore, the value of L_crit is ultimately constrained by our assumptions of how the coronal velocity and height scales with mass-accretion rate and mass of the black hole. In the future and with more data, we plan to allow these relations to vary freely to more realistically characterize L_crit.

Though a rough estimate, L_crit is still corroborated by observations of radio-loud AGNs that indicate the reflection region is moving outward. High-resolution observations utilizing XMM-Newton show that, in addition to a narrow line component, 4C 74.26 has a broad Fe Kα line coming from the inner disk but located at a distance $\gt 50\,{R}_{G}$ (Larsson et al. 2008). Such a large distance indicates that not all broad lines are emanating from regions exactly at the ISCO, possibly caused by over-ionization of the inner disk. Therefore, we postulate that there is a continuum of reflecting regions along the disk at the highest mass-accretion rates.

The radiation pressure at the highest mass-accretion rates may also play a large role in jet production. As the scale height of the disk increases as it becomes radiation pressure dominated, the resulting thick inner disk could be necessary for efficiently confining, collimating, and even accelerating the natal corona into highly relativistic jets (Sarazin et al. 1980; Krolik et al. 2007). Simulations of jets generally support this idea, as they require a "thick disk" to produce jets and anchor sufficiently strong magnetic fields (e.g., Meier 2001; McKinney et al. 2012; Foucart et al. 2016). Using the average mass of our sample, $\langle \mathrm{log}M\rangle =7.6$, we put the critical luminosity, $\mathrm{log}{L}_{\mathrm{crit}}\,=-(9.6\pm 0.3)$, in terms of an observable radio luminosity of $\mathrm{log}{L}_{1.4\mathrm{GHz},\mathrm{crit}}\approx 39.3+(17/12\mathrm{log}{M}_{7.6})$ erg s⁻¹. Comparing this value to the divide between "radio-loud" and "radio-quiet" jets, we find the critical luminosity is just below the divide at $\mathrm{log}{L}_{5\mathrm{GHz}}\approx 41$ erg s⁻¹ (Miller et al. 1990). This indicates the inner disk at the divide is well into the radiation pressure dominated regime and may be necessary for launching the strongest jets. We predict the region dominated by radiation pressure at the dichotomy divide extends to $\sim 100\,{R}_{G}$, utilizing Equation (7), which is consistent with the radius measured in the radio-loud AGN 4C 74.26 (Larsson et al. 2008).

Finally, we note that the critical luminosity is also dependent on the mass of the black hole, which is consistent with studies of the radio-loudness dichotomy that demonstrate the importance of including mass as an additional parameter in the characterization of radio-loudness (Broderick & Fender 2011). These authors show that the radio-loudness distribution is not actually a dichotomy but rather a continuum when mass is properly accounted for. This agrees well with our model, as the inner disk gradually transitions to a radiation pressure dominated regime, and thus would gradually be more efficient at producing stronger jets.