AS ABOVE, SO BELOW: EXPLOITING MASS SCALING IN BLACK HOLE ACCRETION TO BREAK DEGENERACIES IN SPECTRAL INTERPRETATION

Accreting black holes (BHs), whether in Galactic X-ray binaries (BHBs) or active galactic nuclei (AGNs), drive a complicated system of inflowing (quasi-)thermalized plasma in an accretion disk, outflowing plasma in the form of winds and/or relativistic jets, and a hot corona that may comprise elements of both phenomena (see, e.g., Markoff et al. 2005). The basic morphological similarities between these systems have led to the proposal that at least some general properties of BH accretion might scale predictably with mass, regardless of outer boundary conditions (i.e., fueling).

Over the past decade, there has been increasing evidence of such a mapping between BHB accretion states (McClintock & Remillard 2006; Belloni 2010) and AGN classifications (e.g., Körding et al. 2006b). The two most compelling examples are the correspondences between variability timescales in BHBs and AGNs (e.g., McHardy et al. 2006, 2007) and the fundamental plane of BH activity (hereafter FP) discovered over a decade ago (Merloni et al. 2003; Falcke et al. 2004) and increasingly refined via several newer studies (e.g., Körding et al. 2006a; McHardy et al. 2006; Gültekin et al. 2009; Plotkin et al. 2012).

The FP is an empirical relation between the radio and X-ray luminosities and masses of accreting BHs in the "hard" BHB state associated with compact, self-absorbed jets (see Fender 2001; McClintock & Remillard 2006) and low-luminosity AGNs with jet cores: i.e., LLAGNs in LINERS and FRI/BL Lacertae objects. Essentially all weakly accreting AGNs with jets seem to adhere to this plane. The planar coefficients can be derived assuming a common reservoir of accretion power linearly dependent on accretion rate $\dot{m}$ (expressed in mass-scaling Eddington units $\dot{m}=\dot{M}/{\dot{M}}_{{\rm{Edd}}},$ where ${\dot{M}}_{{\rm{Edd}}}={L}_{{\rm{Edd}}}/(0.1{c}^{2})$ and ${L}_{{\rm{Edd}}}=4\pi {{GMm}}_{p}c/{\sigma }_{T}$), injected into a region whose size scales linearly with ${M}_{{\rm{BH}}},$ together with conservation laws, optical depth effects, and low radiative efficiencies ($L\propto {\dot{m}}^{q},$ where $q\approx 2;$ Falcke & Biermann 1995; Heinz & Sunyaev 2003; Markoff et al. 2003; Plotkin et al. 2012). The actual physics driving the FP is not yet fully understood, primarily because of persistent degeneracy in the interpretation of the spectral energy distributions (SEDs). Both synchrotron radiation as well as synchrotron self-Compton (SSC) in several flavors of radiatively inefficient accretion flows (RIAFs; Narayan & Yi 1994; Yuan et al. 2003) or outflows (e.g., Markoff et al. 2005; Yuan et al. 2005) have radiative efficiencies consistent with the limits set by the FP (q ≈ 2; though see Plotkin et al. 2012).

The FP predicts that BHs regulate their power output similarly when at similar relative accretion rates (see, e.g., Heinz & Sunyaev 2003; Markoff 2010 for a broader review). In other words, two sources at similar $\dot{m}$ should radiate from regions of similar size (in gravitational radii ${r}_{{\rm{g}}}\equiv {GM}/{c}^{2}$) and with the same physical mechanism (or at least mechanisms with the exact same efficiencies). This Letter explores a new approach to quantitatively test this assumption, with an eye toward breaking the degeneracy between synchrotron and SSC models, via the joint modeling of broadband SEDs from two BHs at extreme ends of the mass scale. In Section 2, we describe the methodology and briefly summarize the model we use for this study. In Section 3, we present our results, and in Section 4, we conclude with an outlook for potential extensions of this approach.

The low-energy spectrum of FP BHs consists of a flat/inverted synchrotron component, associated with self-absorbed emission from stratified regions along a compact jet (e.g., Blandford & Königl 1979). The X-ray bands often show evidence of weak emission from a thermal accretion disk (e.g., Shakura & Sunyaev 1973; Mitsuda et al. 1984) plus a non-thermal component over which debate rages as to the relative contributions of synchrotron and inverse-Compton processes. Real-time radio/X-ray correlations in BHBs clearly demonstrate that the jets and the X-ray source are tightly coupled over orders of magnitude in luminosity. A mass-dependent normalization extends this relation to AGNs, defining the FP.

A straightforward test can isolate the mass-dependent effects: express a given model in terms of mass-scaling units (i.e., all distances expressed in r_g and power in units of ${L}_{{\rm{Edd}}}=1.25\times {10}^{38}\left(M/{M}_{\odot }\right)\;\mathrm{erg}\;{{\rm{s}}}^{-1}$), and see how it fares when applied to data from stellar to supermassive BHs. This type of approach is not new: the standard thin disk paradigm (Shakura & Sunyaev 1973) seems to scale sensibly with mass. The translation of this approach to non-thermal components has not yet been studied. For this Letter, we use the outflow-dominated model of Markoff et al. (2005, hereafter MNW05), with additional modifications as detailed in Maitra et al. (2009). This multi-scale, broadband model has been successfully applied to a variety of BHs at both ends of the mass range individually, but never jointly as we explore here. We emphasize that this test should apply for any model that can address the broadband SEDs of weakly accreting BHs, and thus predict the FP relations.

The details of MNW05 can be found in the above papers, and many applications to both BHBs (see, e.g., Markoff et al. 2005; Gallo et al. 2007; Maitra et al. 2009; Plotkin et al. 2015) and LLAGNs in LINERS (e.g., Markoff et al. 2001, 2008; Maitra et al. 2011; Prieto et al. 2015). Here, we give just a basic summary of the properties of the model and the relevant fitted parameters.

The MNW05 model includes a heuristic, multi-temperature thin disk component (e.g., Shakura & Sunyaev 1973; Mitsuda et al. 1984) whose radius ${R}_{{\rm{in}}}$ and temperature ${T}_{{\rm{in}}}$ are fitted to the data, and whose photons contribute to the photon field for inverse-Compton scattering. Within $r\lt {R}_{{\rm{in}}}$ we assume that radiatively dominant jets are anchored in a RIAF (see, e.g., Yuan et al. 2002), powered by a fraction of $\dot{M}{c}^{2}$ that is divided equally between cold protons and internal pressure (radiating leptons and magnetic fields).

A thermal particle distribution is assumed to enter the jet nozzle, making this region something of an interface with, or proxy for, the inner RIAF/corona. The jet flow solution is based on a self-collimating, freely expanding hydrodynamic wind (see, e.g., Falcke & Biermann 1995; Falcke & Markoff 2000) and thus is decoupled from the internal pressure (see, e.g., Polko et al. 2014 for a relativistic MHD-consistent treatment in development). Thus, once conditions at the launch point are set, the scaling of physical parameters along the jets is fully determined until the location ${z}_{{\rm{acc}}}.$ There, a fixed fraction of particles (60%) is accelerated into a power-law distribution with index p and is assumed to be maintained from that point onward by a distributed process as implied by observations (e.g., Jester et al. 2001). There is also an option to inject particles into the jets already accelerated, in which case ${z}_{{\rm{acc}}}$ is not used and a maximum Lorentz factor ${\gamma }_{{\rm{max}}}$ is instead fit to the data. The fitted parameters are: p, ${z}_{{\rm{acc}}},$ ${R}_{{\rm{in}}}$, and ${T}_{{\rm{in}}}$, the scaled power normalization N_j (in units of ${L}_{{\rm{Edd}}}$) injected into the jets at their base, of radius r₀ and height h₀ (sometimes frozen), with a ratio of magnetic to thermal gas pressure k, the temperature of the initial, mildly relativistic Maxwell–Juttner distribution for the radiating particles T_e (which also sets ${\gamma }_{{\rm{min}}}$ for the injected power-law case), and f_sc, a parameter absorbing uncertainties in the acceleration efficiency when particles are accelerated at ${z}_{{\rm{acc}}}.$

To compare two BHs of different masses requires SEDs of comparable, simultaneous broadband coverage and quality. Currently the only LLAGNs with such extensive coverage are M87 (Prieto et al. 2015), our Galactic center supermassive BH Sgr A*, and M81* from a campaign originally designed to provide a comparison source to Sgr A*. These observations included radio (GMRT, VLA), submillimeter (PdBI, SMA), and X-ray (Chandra-HETG), as described by Markoff et al. (2008), where we also showed that the MNW05 model provides a good description of the M81* SED. The fitted parameter ranges were similar to those found in hard state BHBs; however, we were not able to break the degeneracy between two potential origins for the X-ray emission providing statistically comparable fits: direct synchrotron emission from the inner jets or SSC from the jet base/corona.

To study the potential "self-similarity" in mass and attempt to break the above degeneracy, here we seek to compare the SED from M81* to the BHB V404 Cygni (hereafter V404), with masses $7\times {10}^{7}\;{M}_{\odot }$ (Devereux et al. 2003) and $12\;{M}_{\odot }$ (Shahbaz et al. 1994), respectively. We use the compiled SED of V404 from Hynes et al. (2009), where the X-ray (Chandra-ACIS), UV (Hubble Space Telescope, HST), and radio data (VLA) were simultaneous, while optical/infrared constraints (e.g., from Spitzer and ground-based instruments) were archival. Similarly for M81* we include archival HST (IR/UV) and Spitzer data, as well as ground-based constraints from ISO and MIRLIN (see Markoff et al. 2008 for details). We apply for the first time a multi-zone, multiwavelength model jointly to the data sets from two sources, separated by a huge dynamic range in mass, tying together several model parameters across this mass range. We have developed this new approach within the data analysis software package ISIS (Houck & Denicola 2002). Note that scale-free parameters correspond to different physical values; therefore, features in the model SEDs corresponding to, e.g., optical depth, temperature, and cooling breaks will remain dependent on the actual mass of the object. Importantly, the X-ray luminosities of both sources (${L}_{X}/{L}_{{\rm{Edd}}}\equiv {{\ell }}_{X}\sim {10}^{-6}$) imply similar $\dot{m}$ (see, e.g., Plotkin et al. 2012), a necessary requirement for this exploration.

2.1. Fitting Methods

In Table 1, we list the model parameters for the best fits shown in Figure 1, distinguishing between those free to vary for each source and those tied together for a joint fit to both SEDs.

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Table 1.  Fit Parameters for Synchrotron-dominated and Compton-dominated Fits

Source	${{\rm{N}}}_{H}$	${N}_{{\rm{j}}}$	Tin	Te	h0	fsc	${\gamma }_{{\rm{max}}}$	p	k	rin	r0	zsh	${\chi }_{\nu }^{2}/$dof
	$({10}^{21}\;{\mathrm{cm}}^{-2})$	(10−5)	$({10}^{4}\;{\rm{K}})$	$({10}^{11}\;{\rm{K}})$	(r0)	(10−4)	$({10}^{2})$			$({GM}/{c}^{2})$
${\rm{M}}{81}^{*}$	${0.31}_{-0.03}^{+0.11}$	${0.36}_{-0.04}^{+0.05}$	${1.6}_{-1.3}^{+0.3}$	${16.6}_{-0.7}^{+1.6}$	5a	${47}_{-12}^{+8}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
V404	8.8a	${17}_{-5}^{+20}$	${106}_{-37}^{+5}$	${0.09}_{-0.02}^{+0.02}$	1.5a	${0.8}_{-0.2}^{+0.7}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯
Joint	⋯	⋯	⋯	⋯	⋯	⋯	⋯	${2.74}_{-0.01}^{+0.05}$	${0.73}_{-0.07}^{+0.53}$	${1.1}_{-0.1}^{+8.7}$	${65}_{-3}^{+10}$	${305}_{-21}^{+15}$	1257/582
${\rm{M}}{81}^{*}$	${0.03}_{-0.01}^{+0.02}$	${0.30}_{-0.04}^{+0.06}$	${0.07}_{-0.02}^{+0.49}$	${2.1}_{-0.3}^{+0.3}$	⋯	⋯	${15}_{-7}^{+20}$	${3.22}_{-0.13}^{+0.10}$	${0.45}_{-0.15}^{+0.48}$	⋯	⋯	⋯	⋯
a	${0.65}_{-0.03}^{+0.31}$	${112}_{-11}^{+8}$	${0.35}_{-0.05}^{+0.09}$	⋯	⋯	${0.22}_{-0.08}^{+0.09}$	${2.47}_{-0.36}^{+1.28}$	${0.32}_{-0.02}^{+0.28}$	⋯	⋯	⋯	⋯
Joint	⋯	⋯	⋯	⋯	${10.1}_{-0.1}^{+5.3}$	⋯	⋯	⋯	⋯	${1.1}_{-0.1}^{+3.1}$	${3.7}_{-0.1}^{+1.5}$	⋯	2671/582

Notes. Fit parameters for the synchrotron-dominated (top) and SSC-dominated (bottom) fits. The model components are: blackbody emission from the accretion disk and/or star (magenta/dotted), thermal synchrotron (light green/dashed), post-accelerated non-thermal synchrotron (dark green/solid), inverse-Compton/SSC (orange/dashed–dotted). Note that in the SSC-dominated fit, accelerated particles were injected at the base, thus ${z}_{{\rm{acc}}}$ and ${f}_{{\rm{sc}}}$ are not used, while ${\gamma }_{{\rm{max}}}$ is. These parameters gave the lowest χ² values for all parameter space explored, while error bars are the bounds that encompass 90% of the one-dimensional parameter histograms obtained from MCMC exploration of the model fit (see the text). Other fixed physical parameters: mass (M81: $7\times {10}^{7}\;{M}_{\odot },$ V404: 12 ${M}_{\odot }$), distance (M81: 3.6 Mpc, V404: 2.4 kpc), inclination (M81: 20°, V404: 56°); see Markoff et al. (2008) and Hynes et al. (2009).

^aFrozen parameter.

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The synchrotron-dominated scenario is clearly the most successful, providing a surprisingly good fit to both sources with almost half the parameters tied—including all relevant physical scales. In contrast, no SSC-dominated scenario could fit both sources in a scalable way. While this result does not rule out SSC-dominated scenarios, the idea that these two sources fall on the FP at similar Eddington fractions but via completely different emission mechanisms seems less likely. Even when decoupling some of the tied parameters, we failed to find substantially improved fits. Given that the synchrotron scenario not only had the best χ², but also allowed for the greatest number of tied parameters, we favor the interpretation that synchrotron emission drives the FP correlation for at least the range ${{\ell }}_{X}\sim {10}^{-7}-{10}^{-6}.$

Compared to the best individual fits to M81* (Markoff et al. 2008), several parameters do not coincide within the errors to those found here. Specifically, the joint fitting technique selects a slightly hotter plasma injected within a larger jet base and a slightly steeper injected power law. There are several potential reasons for this difference, including the possibility that the earlier fits were a local rather than a global minimum since they were not obtained with an MCMC approach. It is worth noting that the M81*/V404 observations are close to, but not exactly at, the same $\dot{m}.$ The individual data sets are also not fully simultaneous. Ultimately, one would prefer to repeat this experiment with fully simultaneous data sets at exactly the same ℓ_X. On the other hand, the best-fit parameter values still fall well within the ranges found from earlier modeling of many individual sources. Thus, this new joint fitting approach does not fundamentally change our ideas about the source physics or geometry, but rather serves as a promising method to break the degeneracy between emission scenarios.

The advantage of the MCMC approach is that with the multi-dimensional probability distribution we can a posteriori explore all 120 possible two-parameter correlations. This allows a new level of insight into physical drivers of the FP as well as pinpointing model degeneracy that needs to be addressed in future work. We find that the parameters for the synchrotron model have well-determined means and errors as derived from their one-dimensional histograms. When examining two-dimensional histograms, only a few parameters showed any degree of correlation (see Figure 2). Several of these (not shown) are commonly seen from fits to similar sources, e.g., correlations between fitted neutral column and parameters affecting spectral slope. Likewise for M81*, there is a correlation between disk radius and temperature, indicating that although a soft excess is required by the data, its detailed properties are not well determined. Figure 2 shows the 68%, 90%, and 99% confidence contours from all two-dimensional histograms where we see interesting correlations, indicating either a physical relation or model degeneracy between these parameters. Both sources show a correlation (stronger for V404) between the normalization power N_j and the equipartition parameter k. This correlation indicates degeneracy in how the injected power is divided between the radiating particles and the magnetic field. As k is increased, putting more energy into the magnetic fields, respectively, fewer electrons are required for the same spectral fit, resulting in somewhat lower power. Fewer electrons can provide the same energy density with a higher temperature, thus giving the correlation seen in the middle panel. Taken together, these two figures indicate a degeneracy between N_j, k, and T_e in the model due to the parameterization of the energy partition at the base of the jets. The rightmost panel shows a similar degeneracy between the particle power-law index and ${f}_{{\rm{sc}}}$ on which the power-law cutoff depends. A harder value of p can compensate for a lower cutoff up to a point.

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Our results support an emerging paradigm that the weakly accreting BHs populating the fundamental plane can be treated as self-similar objects, whose physical behavior is determined by accretion properties rather than mass. Specifically, we show that two BHs, separated by seven orders of magnitude in mass but with comparable ℓ_X, can be statistically described as "self-similar" in the physical scale (in units of r_g). For the more successful synchrotron-dominated model, two additional parameters can also be tied: the power-law distribution p, often thought to be universal for a given acceleration process, and k, the partition of energy density between magnetic fields and radiating particles. The fact that k is roughly consistent with unity suggests that this parameter could be eliminated with the assumption of equipartition. The best value for p could imply either weak acceleration efficiency or very efficient accelerations (such as from reconnection; e.g., Sironi & Spitkovsky 2014) in a cooling-dominated regime. The SSC-dominated scenario does not achieve a good description of the data, even with several additional parameters allowed to vary. Interestingly, independent works suggest an interplay exists between synchrotron and SSC as a function of $\dot{m},$ consistent with our results. For example, Russell et al. (2010) empirically show that synchrotron emission dominates the X-ray band around ${L}_{{\rm{bol}}}\sim {10}^{-4}-{10}^{-3},$ while fits to LLAGNs below ${L}_{X}\sim {10}^{-7}{L}_{{\rm{Edd}}}$ seem to prefer SSC radiation (Markoff et al. 2001; Plotkin et al. 2015; Prieto et al. 2015). The FP slope does not seem to change despite this apparent transition (Corbel et al. 2013; Gallo et al. 2014), although the spectral index does show softening below ${{\ell }}_{X}\sim {10}^{-5}$ (Plotkin et al. 2013).

The results of our study suggest that it is possible to exploit mass scaling to break the longstanding degeneracies between the model classes that persist for AGNs (see, e.g., Harris & Krawczynski 2006) as well as BHBs (e.g., Nowak et al. 2011). Compared to individual fitting, the correlations found between parameters pinpoint the interplay between parameter values due to model degeneracies as well as probing meaningful physical relationships and the partition of energy between magnetic, thermal, and kinetic. This new method thus opens the door to several useful applications, such as using BHBs to infer conditions in obscured regions deep in the hearts of galactic nuclei or to study processes that affect galaxy evolution over cosmological timescales.

Using mass scaling for simultaneous joint/multiple fitting also has the potential to constrain the SEDs of BHs with only sparse data coverage, as well as better pegging the contribution of weak accretion activity, particularly in the millimeter/submillimeter band of nearby galaxies. For instance, the discrepancy between the model and data in the submillimeter/OIR regime in Figure 1 is expected due to galactic stellar and dust contributions (e.g., Bendo et al. 2010). We therefore plan to apply this new method to a larger sample of LLAGNs with sub-arcsecond aperture constraints on the galactic component (e.g., Mason et al. 2012; Fernández-Ontiveros et al. 2013) in future work.

Finally, a deeper understanding of why mass-scaling holds will elucidate the respective roles of outer boundary conditions versus intrinsic accretion flow physics, guiding the way toward more reliable prescriptions of BH feedback.

S.M. is grateful to the University of Texas in Austin for its support through a Tinsley Centennial Visiting Professorship.