# Universality of coronal properties in accreting black holes across mass and accretion rate

Sudip Chakraborty, Ajay Ratheesh, Francesco Tombesi, Rodrigo Nemmen, Srimanta Banerjee

► To cite this version:

Sudip Chakraborty, Ajay Ratheesh, Francesco Tombesi, Rodrigo Nemmen, Srimanta Banerjee. Universality of coronal properties in accreting black holes across mass and accretion rate. Astronomy & Astrophysics - A&A, 2023, 676, pp.L13. 10.1051/0004-6361/202347181. hal-04189903

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LETTER TO THE EDITOR

# Universality of coronal properties in accreting black holes across mass and accretion rate

Sudip Chakraborty<fnref n="1" />,*, Ajay Ratheesh<fnref n="2" />,*, Francesco Tombesi<fnref n="3" />,<fnref n="4" />,<fnref n="5" />,<fnref n="6" />,<fnref n="7" />,†,
Rodrigo Nemmen<fnref n="8" />,<fnref n="9" />,‡, and Srimanta Banerjee<fnref n="10" />,||

Received 14 June 2023 / Accepted 2 August 2023

ABSTRACT

**Aims.** Through their radio loudness, lack of thermal UV emission from the accretion disk, and power-law dominated spectra, low-luminosity active galactic nuclei (LLAGNs) display similarity with the hard state of stellar-mass black hole X-ray binaries (BHBs). In this work we perform a systematic hard X-ray spectral study of a carefully selected sample of unobscured LLAGNs using archival NuSTAR data to understand the central engine properties in the lower accretion regime.

**Methods.** We analyzed the NuSTAR spectra of a sample of 16 LLAGNs. We modeled the continuum emission with detailed Comptonization models.

**Results.** We find a strong anti-correlation between the optical depth and the electron temperature of the corona, previously also observed in the brighter Seyferts. This anti-correlation is present irrespective of the shape of the corona, and the slope of this anti-correlation in the log space for LLAGNs (0.68–1.06) closely matches that of the higher accretion rate Seyferts (0.55–1.11) and hard state of BHBs (~0.87). This anti-correlation may indicate a departure from a fixed disk-corona configuration in radiative balance.

**Conclusions.** Our result, therefore, demonstrates a possible universality in Comptonization processes of black hole X-ray sources across multiple orders of magnitude in mass and accretion rate.

**Key words.** accretion, accretion disks – methods: data analysis – galaxies: active – galaxies: Seyfert – black hole physics – radiation mechanisms: non-thermal

# 1. Introduction

Low-luminosity active galactic nuclei (LLAGNs) are intrinsically faint sub-Eddington accreting systems, with average bolometric luminosities $L_{bol} \sim 10^{38-43}$ erg s⁻<fnref n="1" /> and average Eddington ratio $\lambda_{Edd} = L_{bol}/L_{Edd} \sim 10^{-2}-10^{-5}$ (Ho 2009), $L_{Edd}$ being the Eddington luminosity, as opposed to their luminous AGN counterparts ($\lambda_{Edd} \sim 0.1-1$; Kollmeier et al. 2006). It is believed that the bulk of the supermassive black hole (SMBH) population in the local universe reside in the form of underfed LLAGNs (Ho et al. 1995; Ho 2008), making up the major fraction of the lifetime of SMBHs (Martini 2004; Shin et al. 2010), but contributing little to their growth.

Observationally, LLAGNs differ from the luminous AGNs: they lack significant X-ray variability (Pellegrini et al. 2000; Ho 2008), their spectral energy distributions (SEDs) have no quasar-like “big blue bump” in the UV continuum (Quataert et al. 1999; an indicator of the standard optically thick, geometri-

cally thin accretion disk; Ho 1999, 2008; Nemmen et al. 2006), they typically have a weak or nonexistent narrow Fe Kα emission line (Terashima et al. 2002) along with the occasional presence of broad double-peaked Hα lines (Storchi-Bergmann et al. 2003), all consistent with the absence of a thin accretion disk or the presence of a truncated thin accretion disk (truncated at ≥100GM/c<fnref n="2" />, Chen et al. 1989). Furthermore, all of this observational evidence hints at the LLAGN accretion mode likely being advection-dominated accretion flows (ADAFs; Narayan et al. 1998; Nemmen et al. 2014), in which the hot, geometrically thick, optically thin accretion flow has a typically low radiative efficiency ($L \ll 0.1\dot{M}c^2$) and a low accretion rate ($\dot{M} \lesssim 0.01\dot{M}_{Edd}$). Relativistic jets are also thought to play an important role throughout the LLAGN broadband SEDs (Pian et al. 2010; Nemmen et al. 2014; Fernández-Ontiveros et al. 2023). Furthermore, LLAGNs do not follow some of the correlations established in their higher luminosity counterparts; for example, instead of the positive $\Gamma-\lambda_{Edd}$ correlation in brighter

<footnote n="1">¹ Université Paris Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191 Gif-sur-Yvette, France
e-mail: sudip.chakraborty@cea.fr</footnote>

<footnote n="2">² INAF-IAPS, Via del Fosso del Cavaliere 100, 00133 Rome, Italy</footnote>

<footnote n="3">³ Department of Physics, University of Rome 'Tor Vergata', Via della Ricerca Scientifica 1, 00133 Rome, Italy</footnote>

<footnote n="4">⁴ INAF – Astronomical Observatory of Rome, Via Frascati 33, 00078 Monte Porzio Catone, Rome, Italy</footnote>

<footnote n="5">⁵ Department of Astronomy, University of Maryland, College Park, MD 20742, USA</footnote>

<footnote n="6">⁶ NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA</footnote>

<footnote n="7">⁷ INFN – Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy</footnote>

<footnote n="8">⁸ Universidade de São Paulo, Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Departamento de Astronomia, São Paulo, SP 05508-090, Brazil</footnote>

<footnote n="9">⁹ Kavli Institute for Particle Astrophysics and Cosmology (KIPAC), Stanford University, Stanford, CA 94305, USA</footnote>

<footnote n="10">¹⁰ Inter University Centre for Astronomy and Astrophysics, Post bag 4, Ganeshkhind, Pune, India</footnote>
AGNs (Sobolewska & Papadakis 2009), LLAGNs show an anti-correlation between the two parameters (Gu & Cao 2009; Yang et al. 2015; She et al. 2018).

The X-ray spectrum of AGNs is usually dominated by a primary power-law component, attributed to a hot plasma (called the corona; Haardt & Maraschi 1991; Haardt et al. 1997), which Comptonizes the optical-UV photons emitted by the underlying accretion disk into the X-ray band. A high-energy cutoff in the hard X-ray spectrum, indicative of the temperature of the corona, is one of the important signatures of this Comptonization process (Baloković et al. 2015; Brenneman et al. 2014). Additionally, reflection features, in the form of an iron line complex around ~6.4 keV as well as a “Compton hump” at ~20–40 keV, are sometimes present. The simplest description of Comptonizing coronae is obtained by measuring the photon power-law index and the cutoff in the hard X-ray spectrum. The former depends on the interplay between the electron temperature and the optical depth, whereas the latter is directly related to the electron temperature of the corona. While in brighter AGNs, the primary source of X-ray photons is considered as the unsaturated Comptonization of thermal photons from a standard geometrically thin optically thick accretion disk (Shakura & Sunyaev 1973), for LLAGNs the contribution might also originate from the synchrotron self-Compton emission. Therefore, to compare and understand the coronal properties between the brighter AGNs and the LLAGNs, hard X-ray observations are of paramount importance.

While the brightest AGNs have been studied extensively in the past with hard X-ray satellites, such as BeppoSAX (Dadina 2007), INTEGRAL (Molina et al. 2013), and Swift-BAT (Ricci et al. 2017), hard X-ray class studies of LLAGNs is sparse (e.g., Diaz et al. 2023). The Nuclear Spectroscopic Telescope Array (NuSTAR, Harrison et al. 2013), the first focusing X-ray telescope at hard X-rays, allows us to systematically study the hard X-ray signatures of LLAGNs in unprecedented detail thanks to its focusing optics, and broad and high-quality spectral coverage between 3 and 79 keV. Therefore, NuSTAR is suitable for studying the hard X-ray spectra of AGNs with high sensitivity, discriminating between the primary X-ray emission and the reflected component. Alone, or together with simultaneous observations at other X-ray observatories operating below 10 keV, such as XMM-Newton, Suzaku, and Swift-XRT, it has provided strong constraints on the coronal properties of many bright AGNs (Brenneman et al. 2014; Fabian et al. 2015, 2017; Matt et al. 2015; Tortosa et al. 2018).

Recently, Tortosa et al. (2018) studied the coronal properties of a sample of bright Seyferts using NuSTAR spectra. Along with the previously reported correlations, they found clear indications of unexplained anti-correlation between the electron temperature and the optical depth. To explore the validity of these correlations at much lower luminosities, a systematic Comptonization study of LLAGNs with NuSTAR has to be carried out. In our present work, we aim to bridge this gap. In Sect. 2 we discuss our sample selection. In Sect. 3 we explore the selected sample of LLAGNs with different spectral models, and find the correlations between the important parameters. We present the key results in Sect. 4. Finally, in Sect. 5 we discuss the implications of this study.

## 2. Data selection

The sample considered in this work was selected predominantly from the Palomar sample (Saikia et al. 2018; Nagar et al. 2005), and was supplemented with sources from the BASS

DR1¹ (Ricci et al. 2017), and from the following works: Nemmen et al. (2014), Kawamuro et al. (2016), Ho (2009), Hernández-García et al. (2016), González-Martín & Vaughan (2012), Eracleous et al. (2010), Terashima et al. (2002), Ursini et al. (2015). Since the main motive of this work is to study the properties of the central engine in these sources, we identified unobscured Compton-thin LLAGNs by imposing an upper constraint of $10^{24} \text{ cm}^{-2}$ on the hydrogen equivalent column density ($N_H$) and ~$10^{43} \text{ erg s}^{-1}$ on the bolometric luminosity. All the sources and the corresponding NuSTAR data used in this work are outlined in Table D.1. The methods of reduction of the NuSTAR data is given in Appendix A. We also impose a minimum count-rate threshold of $4 \times 10^{-2} \text{ cts s}^{-1}$ in the 3–79 keV energy range in both FPMA and FPMB to reach a satisfactory signal-to-noise level for spectral analysis. This produces the 16 sources that make up our sample (see Table D.1; for details on the uniform reduction of the NuSTAR data, see Appendix A).

## 3. Spectral analysis

The spectral fitting and statistical analysis were carried out using the XSPEC version v-12.12.0 (Arnaud 1996). To jointly fit FPMA and FPMB, a cross-normalization constant (CONSTANT model in XSPEC) was allowed to vary freely for FPMB and was assumed to be unity for FPMA. We also restricted the energy ranges of the individual data sets to 3–25, 3–50, or 3–79 keV, based on the quality of the data. All the models, as described below, included the Galactic absorption through the implementation of the TBABS model. The corresponding abundances were set as per the solar abundances in Wilms et al. (2000). The neutral hydrogen column densities ($N_H$) were fixed to values found in the literature (see Table D.1) for all the described models. All parameter uncertainties were reported at the $1\sigma$ confidence level for one parameter of interest.

To get a detailed understanding of the Comptonization processes and to directly compare with previous studies of other accreting black hole systems, we require reliable estimates of the optical depth ($\tau$) and electron temperatures ($kT_e$) of the coronae. To directly fit the optical depths and electron temperatures in the 3–79 keV NuSTAR data, we used the Comptonization model COMPTT (Titarchuk 1994) in XSPEC. COMPTT models the thermal Comptonization emission of a hot plasma cooled by soft photons with a Wien law distribution and includes special relativistic effects. This model is valid both in the case of optically thick and optically thin plasma. The Comptonized spectrum is determined completely by the plasma temperature and the $\beta$ parameter, which is independent of geometry. We used both the available slab and sphere geometry in this work. For the former we set the geometry switch to 0.5, and for the latter we set it to 2. In both cases, the corresponding $\beta$ values were calculated from optical depth using analytic approximations (Titarchuk 1994). We used the redshift values mentioned in Table D.1 and assumed the seed blackbody temperature to be fixed at 10 eV (Younes et al. 2019). In the cases where excess at Fe-K energies (around 6.4 keV) were observed, we included a Gaussian emission line (GAUSS in XSPEC). Whenever the lines were found to be too narrow to be resolved by NuSTAR, we froze the width of the Gaussian line ($\sigma$) to zero. The details of the best-fit continuum parameters are presented in Table D.2, while emission lines are described in Table D.3. The best-fit parameters and $\chi^2$/d.o.f. values given in Table D.2 are corrected for

<footnote n="1">¹ https://www.bass-survey.com/dr1.html</footnote>
reflection features with physically motivated models as well, as described in the next paragraph. From Table D.2, it can be seen that both the geometries of COMPTT model result in statistically similar fits. Physical constraints over $kT_e$ are found for 12 and 11 cases for the slab and the sphere geometries, respectively. Of these, the values of $kT_e$ for NGC 4258 are found to be unphysical (<10 keV).

Of the ten LLAGNs with prominent iron lines, four were found to have central energies of ≥6.4 keV, and six were found to have line energies ≤6.4 keV. Upon further investigations in the available literature of these ten LLAGNs, the iron lines in the latter six were found to have AGN origin, and the former six were found to have originated from hot diffused gas. Most early-type galaxies emit an extended hot diffuse X-ray component usually fitted with an emission model from an optically thin plasma (Fabbiano 1989). To account for this, we used an Astrophysical Plasma Emission Code (APEC; Smith et al. 2001) component along with the COMPTT continuum in the model instead of the Gaussian. We assumed solar metallicity and fitted for the plasma temperature. For the LLAGN with AGN-origin Fe-K lines, and to account for the occasional reflection hump noted in a few of these sources, we used the PEXMON model (Nandra et al. 2007) for reflection from a neutral medium. The model component is dependent on the inclination angle of the source, which we fixed to 45°, and on elemental abundances, which we assumed to be solar. Compared to other similar models such as PEXRAV, PEXMON has the advantage of self-consistently including reflection due to atomic species such as Fe Kα, Fe Kβ, and Ni Kα (Nandra et al. 2007). The inclusion of APEC and PEXMON improved the fits in all the cases. Finally, in the case of NGC 5506, broad iron lines accompanied by a Compton hump at 30–50 keV were observed. To test the possibility of relativistic broadening, we convolved the PEXMON model with smeared relativistic accretion disk line profiles using RELCONV (Dauser et al. 2010). Hereafter, we use the APEC/PEXMON-corrected $kT_e$ values to study the correlations between the parameters to make them more robust.

## 4. Results

In Fig. 1 we observe an anti-correlation between $\tau$ and $kT_e$ in our sample of LLAGNs, for both the slab and spherical geometry of the corona. This anti-correlation is similar to that found by Tortosa et al. (2018) for the more luminous AGNs.

We find the Spearman's rank correlation coefficient between $\tau$ and $kT_e$ (see Appendix B for details) for the LLAGNs to be $\rho = -0.75 \pm 0.14$ for the spherical corona and $\rho = -0.81 \pm 0.11$ for the slab corona, with corresponding $p$-values of 0.003 and 0.002. The histograms of the $p$-values for the two geometries are displayed in Fig. B.1. Such high negative values of $\rho$ suggest strong anti-correlation between $kT_e$ and $\tau$ for both the coronal geometries, although the anti-correlation is found to be stronger for the slab geometry.

In order to directly perform a regression analysis in the $kT_e-\tau$ plane for our LLAGN sample, we implemented the bi-variate correlated errors and intrinsic scatter (BCES$^2$), following the prescription of Akritas & Bershady (1996). We performed $5 \times 10^4$ trials and adopted the orthogonal least-squares BCES line as our best fit. Fitting the $kT_e$ and $\tau$ in the logarithmic space ($\log(kT_e) = a \log(\tau) + b$), we find the best-fit parameters to be

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Electron temperature ($kT_e$) vs. optical depth ($\tau$) for different geometries of the corona from the COMPTT model. Top panel: slab geometry, bottom panel: spherical geometry. In both panels, the best-fit parameters are shown along with their $1\sigma$ error bars. The data points are color-coded according to their Eddington ratio $\lambda_{Edd}$. The solid lines in both panels indicate the BCES best-fit lines to our LLAGN sample and the shaded gray area contains their confidence regions. For easier comparison, the best-fit line of the Seyferts (Tortosa et al. 2018) has been placed for the corresponding geometry in both panels; the BCES best-fit line is indicated in dashed blue and the corresponding confidence region in light blue (for details of the fit, see Sect. 4).</figcaption></figure>

the following:

For slab geometry $a = -0.77 \pm 0.09$; $b = 1.62 \pm 0.01$, For spherical geometry $a = -0.98 \pm 0.08$; $b = 2.04 \pm 0.03$.

To compare these values directly with those for luminous AGNs, we performed the same BCES-based fit on the $kT_e-\tau$ data from Tortosa et al. (2018). For such bright AGNs, we find the following best-fit values:

For slab geometry $a = -0.97 \pm 0.14$; $b = 1.53 \pm 0.15$, For spherical geometry $a = -0.70 \pm 0.15$; $b = 1.81 \pm 0.05$.

The $kT_e-\tau$ plots along with the BCES best-fit lines and the corresponding confidence regions for the two geometries are displayed in Fig. 1.

## 5. Discussion and conclusions

In this work we probed the coronal properties of a sample of 16 carefully selected unobscured LLAGNs with NuSTAR. We analyzed the NuSTAR spectra using slab and sphere geometries

<footnote n="2">² https://github.com/rsnemmen/BCES</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Electron temperature ($kT_e$) vs. optical depth ($\tau$) for the spherical geometry of the corona across different classes of objects: LLAGNs (green), Seyferts from Tortosa et al. (2018, orange). For a direct comparison with the Galactic BHBs, the hard-state data (with COMPPS model fit) from Banerjee et al. (2020) are also shown (light gray). Furthermore, to indicate the evolution of BHBs along the presented anti-correlation, hard-state data points for MAXI J1820+070 (Buisson et al. 2019) are plotted (red). A discussion on the implication of this universality is presented in Sect. 5.</figcaption></figure>

of COMPTT, a model of thermal Comptonization of the seed disk photons with an assumed seed photon temperature of 10 eV. Assuming a simple parallel with the BHB hard state (see the discussions below), this would imply an inner edge of accretion disks at ~100–500 $R_g$. This is roughly in line with what we find for NGC 5506 (Appendix C). Although the seed photons are assumed to originate from the disk, they could also be of any other origin without considerably affecting the hard X-ray spectra, as any potential changes would primarily be observed on the lower energy side of the spectrum. We also investigated the iron line complex for each LLAGN, and accounted for the emission lines using either a diffused plasma emission via APEC or a reflection from neutral material via PEXMON. Using a uniform analysis over the entire sample, we derive robust values of $kT_e$ and $\tau$ of the coronae. We find an anti-correlation between the optical depth and the electron temperatures for LLAGNs similar to that found in the more luminous AGNs (Tortosa et al. 2018). This anti-correlation, depicted in Fig. 1, is found to hold true for both the slab and spherical geometry of the corona. Overall, the slab geometry is found to have more significant anti-correlation compared to the spherical geometry. The best-fit slope for LLAGNs ($-0.76 \pm 0.09$ for slab and $-0.98 \pm 0.08$ for sphere) closely follows the slope for brighter Seyferts ($-0.98 \pm 0.18$ for slab and $0.70 \pm 0.16$ for sphere).

It has been suggested for LLAGNs in the ADAF regime that the scattering optical depth is related to the thickness and density of the accretion flow. This would imply that, to first order, $\tau$ is related to the mass accretion rate as $\tau = 0.03(\dot{m}/10^{-3})$, assuming a radiatively inefficient accretion flow (Narayan & Yi 1995), where $\dot{m}$ is the accretion rate in units of the Eddington rate ($\dot{M}_{Edd} \equiv L_{Edd}/(0.1c^2)$). Taking the best-fit values found from the observations for the slab geometry, this would suggest that for LLAGNs the electron temperature depends relatively strongly on $\dot{m}$ as $kT_e = 599(\dot{m}/10^{-3})^{-0.77}$ keV or $kT_e = 10^{9.8}(\dot{m}/10^{-3})^{-0.77}$ K. However, no such correlation is evident between $kT_e$ and $\dot{m}$ or $\tau$ and $\dot{m}$ in Fig. 1, although our methodology assumes a standard Comptonizing corona instead of an ADAF in the first place. For example, the Spearman's rank coefficient between $\tau$ and $\lambda_{Edd}$ is only $0.20 \pm 0.16$ and $0.24 \pm 0.14$ for spherical and slab geometry,

respectively, while between $kT_e$ and $\lambda_{Edd}$ it is only $-0.11 \pm 0.16$ and $-0.18 \pm 0.15$ for spherical and slab geometry, respectively. This suggests that in this picture the Comptonizing region in LLAGNs may be completely uncorrelated with the ADAF region and could instead resemble the Comptonizing coronae found in more luminous AGNs.

Remarkably, the same anti-correlation ($\rho = -0.84 \pm 0.01$) between $kT_e$ and $\tau$ has been found for the hard-state spectra of BHBs (Banerjee et al. 2020). A linear regression analysis in logarithmic space yields a strikingly similar slope ($-0.87 \pm 0.02$). As the BHB data set in Banerjee et al. (2020) uses spherical geometry of COMPPS model, this can be compared with the spherical geometries of our LLAGN sample or the bright AGN sample of Tortosa et al. (2018). For the BHB sample, however, the intercept is found to be slightly higher ($2.17 \pm 0.01$). This systematic difference between BHBs and AGNs would indicate that for any given $\tau$ value, the spectra of BHBs are harder (Middei et al. 2019) than their supermassive counterparts. This presence of similar anti-correlation and the remarkable similarity in Comptonization parameters across different classes of accreting black hole sources suggests a universality of the coronal physics in all of them.

The anti-correlation between $kT_e$ and $\tau$ indicates a departure from a fixed disk-corona configuration in radiative balance (Tortosa et al. 2018). The invalidation of a fixed disk-corona configuration can possibly occur due to a change in the coronal geometry. For example, a reduction of coronal height (in the lamppost configuration) would imply a greater Compton cooling from the disk (with an increase of $\tau$), and thereby a smaller $kT_e$. Such a variation in the coronal geometry was earlier proposed to explain the evolution of spectral and timing features of the BHB MAXI J1820+070 in the hard state (Kara et al. 2019; Buisson et al. 2019). As can be seen from the red points in Fig. 2, MAXI J1820+070 also occupies a similar parameter space along the $kT_e-\tau$ anti-correlation line as it evolves from its hard state and its corona contracts. On the other hand, the violation of radiative balance due to a change in the disk fraction (i.e., the fraction of intrinsic disk emission to the total flux) for a fixed disk-corona system can also support this anti-correlation (Tortosa et al. 2018). The disk fraction can alter if the inner edge of the accretion disk evolves as the supply of the soft seed disk photons to the corona depend upon this. Thus, as the disk recedes, the disk fraction falls, and the average number of scattering of thermal seed photons with coronal electrons decreases (i.e., optical depth drops). This leads to a decrease in Compton cooling, and thus to an increase in $kT_e$. Hence for a highly truncated disk, $\tau$ is expected to be lower and $kT_e$ to be higher. In our work, in all three iterations of our blurred reflection model RELCONV⊗PEXMON implementation, we find that the $R_{in}$ for NGC 5506 is significantly large, indicating a highly truncated disk (see Appendix C). While the disk truncation radii for other LLAGNs are not confirmed, there are indications that LLAGNs, as a population, might occupy the hard-state branch (usually associated with large disk truncation; e.g., Done et al. 2007) in the hardness-intensity diagram (Fernández-Ontiveros et al. 2023).

**Acknowledgements.** We thank the anonymous reviewer for the constructive comments in improving this manuscript. This work made use of data from the NuSTAR mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. We thank the NuSTAR Operations, Software and Calibration teams for their support with the execution and analysis of these observations. This research has also made use of the NuSTAR Data Analysis Software (NuSTARDAS), jointly developed by the ASI Science Data Center
(ASDC, Italy) and the California Institute of Technology (USA). This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement n°101004168, the XMM2ATHENA project. The authors thank Prof. A. R. Rao and Prof. Sudip Bhattacharyya for their constructive suggestions and comments for improving this work, and Dr. Tonima Tasnim Ananna for the BASS DR2 estimates. R.N. acknowledges support by FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) under grants 2017/01461-2 and 2022/10460-8.

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# Appendix A: Data reduction

The NuSTAR data used in this work are reduced using v2.1.1 of the NuSTARDAS pipeline using the NuSTAR CALDB v20211020. The *nupipeline* tool was used to generate cleaned level 2 event files. We used DS9 (Joye & Mandel 2003) to select the source and background regions. The source region was selected manually to include the maximum source contribution and the background was selected away from the source to avoid contamination, both using 1' circular regions. The *nuproducts* tool was then used to extract the background-subtracted source spectrum. For non-variable sources with multiple data sets, we performed a combined fit by combining spectra from those observations using the *addspec* (version 1.4.0) tool of the FTOOLS (NASA High Energy Astrophysics Science Archive Research Center (HEASARC) 2014). Before combining the spectra, we checked the source variabilities by probing the consistency between the fluxes and the photon indices for CUTOFFPL model fits of the spectra of different epochs. All the spectra were then binned using *grppha* tool of the FTOOLS to obtain at least 25 counts per bin, in order to use $\chi^2$-statistics.

<figure><img src="image_4.png" /><figcaption>**Fig. B.1.** Histograms of the Spearman's rank correlation coefficients ($\rho$) from the simulated data for the slab (blue) and spherical (red) geometries of the corona in LLAGNs. While the slab geometry shows a more prominent peak at a higher $\rho$ value than the spherical geometry, high negative mean $\rho$ for both the geometries indicates a moderate to significant anti-correlation between $kT_e$ and $\tau$ present in both (for details about the histograms, see section 4).</figcaption></figure>

# Appendix B: Distribution of Spearman's rank correlation coefficient

To check the goodness of our anti-correlation in the presence of error bars in both $\tau$ and $kT_e$, we performed a Monte Carlo calculation by randomly generating simulated data points for each of the sources using the best-fit parameter and the covariance matrix; we repeated the procedure $2 \times 10^4$ times. We then calculated a Spearman's rank correlation coefficient for each of the trials, and found the mean and standard deviation of the resulting distribution. We calculated the $p$-values by determining the number of times $\rho_{sim} \ge \rho_{obs}$ (where $\rho_{sim}$ and $\rho_{obs}$ are the Spearman's rank coefficient derived from the simulated value in consideration and the corresponding value for the actual coefficient calculated from the data points, respectively), and dividing it by the total number of simulations. The distribution of $\rho_{sim}$ for the slab and spherical geometry is portrayed in figure B.1, and the relevant values are stated and discussed in section 4.

# Appendix C: Further details about specific sources: NGC 5506

For NGC 5506, COMPTT+(RELCONV⊗PEXMON) resulted in a much better and more consistent fit than COMPTT+PEXMON. While the simple PEXMON implementation resulted in a $\chi^2$/d.o.f. of 2057/1827, assuming the inclination to be fixed at 45° and freezing the spin parameter *a* to its maximum allowed value of 0.998, the RELCONV⊗PEXMON implementation gives a much better fit with a $\chi^2$/d.o.f. of 1989/1826. The best-fit $kT_e$ is found to be > 36.1 keV and the inner edge of accretion disk $R_{in} > 303 R_g$ (where $R_g$ is the gravitational radius of the central black hole). Changing the inclination to 30° results in an even better fit, with a $\chi^2$/d.o.f. of 1956/1826, the best-fit $kT_e = 78_{-27}^{+131}$ keV, and $R_{in} = 126_{-23}^{+26} R_g$. Finally, letting the inclination vary freely results in the best fit of the three cases, with a $\chi^2$/d.o.f. of 1937/1825. This best-fit result is presented in Table D.2 and is used for the correlation study. The corresponding inclination and $R_{in}$ are found to be $18.3_{-1.3}^{+1.5}$ degrees and $58_{-8}^{+10} R_g$, respectively.
Appendix D: Additional tables

<table><caption>**Table D.1.** Key properties of all sources in our LLAGN sample and details of its corresponding NuSTAR observation used in this work. For the flagged source (*), the different observations were analyzed separately. The spectral types are abbreviated as S: Seyfert, L: LINER. z, $N_H$, and $\lambda_{Edd}$ denote redshift, absorbing column density, and Eddington ratio, respectively.</caption><thead><tr><th>Name</th><th>Obs ID</th><th>Date<br>(yyyy-mm-dd)</th><th>Exposure<br>(ks)</th><th>NuSTAR<br>ref.</th><th>Spectral<br>type</th><th>log(Mass)<br>log(M<sub>☉</sub>)</th><th>z</th><th>N<sub>H</sub><br>(×10<sup>22</sup>cm<sup>-2</sup>)</th><th>log (λ<sub>Edd</sub>)</th></tr></thead><tbody><tr><td>M81 (NGC 3031)</td><td>60101049002</td><td>2015-05-18</td><td>209</td><td>Young et al. (2018)</td><td>S1.5/L1.5</td><td>7.9</td><td>0.001</td><td>1.86</td><td>-6.62<sup>a</sup></td></tr><tr><td>M87 (NGC 4486)</td><td>60201016002<br>60466002002<br>90202052002<br>90202052004</td><td>2017-02-15<br>2018-04-24<br>2017-04-11<br>2017-04-14</td><td>50.0<br>21.1<br>24.4<br>22.5</td><td>Wong et al. (2017)<br>This work<br>Wong et al. (2017)<br>Wong et al. (2017)</td><td>L2</td><td>9.5</td><td>0.004</td><td>0.40</td><td>-5.44<sup>d</sup></td></tr><tr><td>NGC 1052</td><td>60061027002<br>60201056002<br>90701624002</td><td>2013-02-14<br>2017-01-17<br>2021-07-17</td><td>15.5<br>59.7<br>20.5</td><td>Rani et al. (2017)<br>Osorio-Clavijo et al. (2020)<br>This work</td><td>L1.9</td><td>8.7</td><td>0.005</td><td>12.78</td><td>-3.94<sup>b</sup></td></tr><tr><td>NGC 2655</td><td>60160341002<br>60160341004</td><td>2016-11-02<br>2016-11-10</td><td>0.51<br>16.0</td><td>Diaz et al. (2023)<br>This work</td><td>S2</td><td>8.2</td><td>0.005</td><td>4.27</td><td>-3.87<sup>b</sup></td></tr><tr><td>NGC 2992*</td><td>60160371002<br>90501623002</td><td>2015-12-02<br>2019-05-10</td><td>20.8<br>57.4</td><td>Marinucci et al. (2018)<br>Middei et al. (2022)</td><td>S1.9/L1.5</td><td>8.3</td><td>0.00771</td><td>0.78</td><td>-3.30<sup>b</sup></td></tr><tr><td>NGC 3227</td><td>60202002004<br>60202002006<br>60202002008<br>60202002010<br>60202002012<br>60202002014</td><td>2016-11-25<br>2016-11-29<br>2016-12-01<br>2016-12-05<br>2016-12-09<br>2017-01-21</td><td>42.0<br>39.7<br>41.7<br>40.9<br>39.3<br>47.6</td><td>Pal et al. (2022)<br>Pal et al. (2022)<br>Pal et al. (2022)<br>Pal et al. (2022)<br>Pal et al. (2022)<br>Pal et al. (2022)</td><td>L1.9</td><td>6.7</td><td>0.004</td><td>0.09</td><td>-1.70<sup>c</sup></td></tr><tr><td>NGC 3718</td><td>60301031002<br>60301031004<br>60301031006<br>60301031008</td><td>2017-10-24<br>2017-10-27<br>2017-10-30<br>2017-11-03</td><td>24.5<br>90.4<br>57.4<br>57.0</td><td>Diaz et al. (2023)<br>Diaz et al. (2023)<br>Diaz et al. (2023)<br>Diaz et al. (2023)</td><td>L1.9</td><td>8.1</td><td>0.003</td><td>0.14</td><td>-4.49<sup>b</sup></td></tr><tr><td>NGC 3998</td><td>60201050002</td><td>2016-10-25</td><td>104</td><td>Younes et al. (2019)</td><td>L1.9</td><td>8.9</td><td>0.003</td><td>0.03</td><td>-4.47<sup>b</sup></td></tr><tr><td>NGC 4258</td><td>60101046002<br>60101046004</td><td>2015-11-16<br>2016-01-10</td><td>54.8<br>103.6</td><td>Masini et al. (2022)<br>Masini et al. (2022)</td><td>S1.9</td><td>7.6</td><td>0.002</td><td>8.0</td><td>-4.28<sup>b</sup></td></tr><tr><td>NGC 4395</td><td>60061322002</td><td>2013-05-10</td><td>19.2</td><td>Rani et al. (2017)</td><td>S1.8</td><td>5.4</td><td>0.001</td><td>0.04</td><td>-1.90<sup>a</sup></td></tr><tr><td>NGC 4579</td><td>60201051002<br>60201056002</td><td>2016-12-06<br>2017-01-17</td><td>117<br>59.7</td><td>Younes et al. (2019)<br>Osorio-Clavijo et al. (2020)</td><td>S1.9/L1.9</td><td>8.1</td><td>0.004</td><td>0.03</td><td>-3.564<sup>b</sup></td></tr><tr><td>NGC 5033</td><td>60601023002<br>60601023004</td><td>2020-12-08<br>2020-12-12</td><td>103.5<br>53.3</td><td>Diaz et al. (2023)<br>Diaz et al. (2023)</td><td>S1.9</td><td>7.7</td><td>0.002</td><td>0.01</td><td>-4.00<sup>b</sup></td></tr><tr><td>NGC 5273</td><td>60061350002<br>90801618002</td><td>2014-07-14<br>2022-07-03</td><td>21.1<br>18.0</td><td>Rani et al. (2017)<br>This work</td><td>S1.5</td><td>6.7</td><td>0.003</td><td>0.07</td><td>-2.43<sup>a</sup></td></tr><tr><td>NGC 5290</td><td>60160554002</td><td>2021-07-28</td><td>18.9</td><td>Jana et al. (2023)</td><td>S2</td><td>7.8</td><td>0.009</td><td>0.91</td><td>-2.64<sup>a</sup></td></tr><tr><td>NGC 5506</td><td>60061323002<br>60501015002<br>60501015004</td><td>2014-04-01<br>2019-12-28<br>2020-02-09</td><td>57.0<br>61.4<br>47.5</td><td>Matt et al. (2015)<br>This work<br>This work</td><td>S1.9</td><td>6.0</td><td>0.006</td><td>0.31</td><td>-1.22<sup>a</sup></td></tr><tr><td>NGC 7213</td><td>60001031002</td><td>2014-10-05</td><td>109</td><td>Ursini et al. (2015)</td><td>S1.5</td><td>7.1</td><td>0.005</td><td>0.02</td><td>-2.08<sup>a</sup></td></tr></tbody></table>

<footnote><sup>a</sup>:Ananna et al. (2022a,b), <sup>b</sup>: Jana et al. (2023), <sup>c</sup>:Mehdipour et al. (2021), <sup>d</sup>:Prieto et al. (2016)</footnote>
<table><caption>**Table D.2.** Best-fit parameters of COMPTT models for both slab and spherical geometries in our sample of LLAGN. The corresponding error bars are stated at $1\sigma$ levels. Here $kT_e$ is the electron temperature of the corona and $\tau$ is its optical depth. For all the applicable LLAGNs, the best-fit model includes, in addition to COMPTT, suitable APEC and PEXMON models (see Sect. 3 for a detailed discussion).</caption><thead><tr><th rowspan="2">LLAGN name</th><th colspan="3">Sphere</th><th colspan="3">Slab</th></tr><tr><th>kT<sub>e</sub><br>(keV)</th><th>τ</th><th>χ<sup>2</sup>/d.o.f</th><th>kT<sub>e</sub><br>(keV)</th><th>τ</th><th>χ<sup>2</sup>/d.o.f</th><th>3-30 keV Flux<br>(×10<sup>-11</sup> ergs cm<sup>-2</sup> s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>M81</td><td>97.5<sup>+95.7</sup><sub>-51.1</sub></td><td>1.03<sup>+1.35</sup><sub>-0.39</sub></td><td>1277/1232</td><td>122.6<sup>+64.2</sup><sub>-41.6</sub></td><td>0.22<sup>+0.17</sup><sub>-0.11</sub></td><td>1277/1232</td><td>3.40</td></tr><tr><td>M87</td><td>144.3<sup>+3.6</sup><sub>-40.5</sub></td><td>&lt; 0.02</td><td>433.64/292</td><td>83.2<sup>+1.5</sup><sub>-17.4</sub></td><td>&lt; 0.02</td><td>435.78/292</td><td>0.87</td></tr><tr><td>NGC1052</td><td>&gt; 218.9</td><td>0.27<sup>+0.43</sup><sub>-0.01</sub></td><td>980/842</td><td>19.2<sup>+6.1</sup><sub>-2.7</sub></td><td>2.33<sup>+0.27</sup><sub>-0.22</sub></td><td>982/842</td><td>1.55</td></tr><tr><td>NGC 2655</td><td>62.5<sup>+43.9</sup><sub>-24.2</sub></td><td>0.79<sup>+0.68</sup><sub>-0.46</sub></td><td>65.41/50</td><td>49.1<sup>+50.3</sup><sub>-16.5</sub></td><td>0.32<sup>+0.27</sup><sub>-0.24</sub></td><td>65.58/50</td><td>0.65</td></tr><tr><td>NGC 2992</td><td>23.4<sup>+5.7</sup><sub>-3.9</sub></td><td>4.66<sup>+0.56</sup><sub>-0.62</sub></td><td>871.60/940</td><td>22.0<sup>+4.3</sup><sub>-3.2</sub></td><td>2.08<sup>+0.24</sup><sub>-0.26</sub></td><td>872.85/940</td><td>12.07</td></tr><tr><td>NGC 3227</td><td>40.6<sup>+23.1</sup><sub>-12.3</sub></td><td>2.89<sup>+0.93</sup><sub>-0.95</sub></td><td>1901/1674</td><td>37.1<sup>+38.6</sup><sub>-9.7</sub></td><td>1.22<sup>+0.39</sup><sub>-0.72</sub></td><td>1900/1674</td><td>7.56</td></tr><tr><td>NGC 3718</td><td>77.2<sup>+13.8</sup><sub>-11.4</sub></td><td>1.33<sup>+0.3</sup><sub>-0.26</sub></td><td>209.03/199</td><td>65.3<sup>+11.8</sup><sub>-9.6</sub></td><td>0.52<sup>+0.15</sup><sub>-0.12</sub></td><td>209.14/199</td><td>0.22</td></tr><tr><td>NGC 3998</td><td>69.3<sup>+3.9</sup><sub>-3.7</sub></td><td>1.42<sup>+0.1</sup><sub>-0.1</sub></td><td>584.01/613</td><td>87.6<sup>+5.0</sup><sub>-4.7</sub></td><td>0.34<sup>+0.03</sup><sub>-0.03</sub></td><td>584.01/613</td><td>1.20</td></tr><tr><td>NGC 4258</td><td>8.1<sup>+1.4</sup><sub>-0.9</sub></td><td>7.93<sup>+0.58</sup><sub>-0.72</sub></td><td>607/567</td><td>8.0<sup>+1.5</sup><sub>-0.8</sub></td><td>3.63<sup>+0.27</sup><sub>-0.36</sub></td><td>607/567</td><td>0.54</td></tr><tr><td>NGC 4395</td><td>137.4<sup>+198.9</sup><sub>-56.8</sub></td><td>2.38<sup>+1.14</sup><sub>-1.23</sub></td><td>356.63/317</td><td>130.8<sup>+152.9</sup><sub>-52.3</sub></td><td>0.92<sup>+0.54</sup><sub>-0.5</sub></td><td>355.96/317</td><td>2.07</td></tr><tr><td>NGC 4579</td><td>&gt; 84.4</td><td>0.12<sup>+1.08</sup><sub>-0.0</sub></td><td>652/668</td><td>&gt; 105.9</td><td>0.02<sup>+0.26</sup><sub>-0.0</sub></td><td>652/668</td><td>1.20</td></tr><tr><td>NGC 5033</td><td>36.1<sup>+198.7</sup><sub>-19.1</sub></td><td>3.33<sup>+2.16</sup><sub>-2.76</sub></td><td>935.25/842</td><td>26.2<sup>+254.7</sup><sub>-9.3</sub></td><td>1.75<sup>+0.66</sup><sub>-1.39</sub></td><td>935.28/842</td><td>1.44</td></tr><tr><td>NGC 5273</td><td>86.2<sup>+3.5</sup><sub>-3.3</sub></td><td>2.3<sup>+0.13</sup><sub>-0.12</sub></td><td>752.19/721</td><td>79.5<sup>+3.1</sup><sub>-3.0</sub></td><td>0.92<sup>+0.06</sup><sub>-0.06</sub></td><td>752.78/721</td><td>3.77</td></tr><tr><td>NGC 5290</td><td>41.1<sup>+9.4</sup><sub>-7.1</sub></td><td>2.89<sup>+0.67</sup><sub>-0.63</sub></td><td>187/177</td><td>48.8<sup>+11.5</sup><sub>-8.6</sub></td><td>0.93<sup>+0.31</sup><sub>-0.26</sub></td><td>187/177</td><td>1.08</td></tr><tr><td>NGC 5506</td><td>159.7<sup>+123.3</sup><sub>-36.7</sub></td><td>0.55<sup>+0.13</sup><sub>-0.3</sub></td><td>1937.51/1825</td><td>204.8<sup>+111.7</sup><sub>-53.0</sub></td><td>0.09<sup>+0.04</sup><sub>-0.05</sub></td><td>1937.34/1825</td><td>12.37</td></tr><tr><td>NGC 7213</td><td>32.1<sup>+32.9</sup><sub>-13.0</sub></td><td>3.11<sup>+1.44</sup><sub>-1.47</sub></td><td>914/876</td><td>34.1<sup>+40.4</sup><sub>-14.0</sub></td><td>1.15<sup>+0.71</sup><sub>-0.68</sub></td><td>914/869</td><td>2.64</td></tr></tbody></table>
<table><caption>**Table D.3.** Best-fit parameters of modeled reflection features in the subset of our sample of LLAGNs with observed iron emission lines and/or Compton hump (for a detailed discussion, see section 3).</caption><thead><tr><th rowspan="2">LLAGN name</th><th colspan="2">Gauss</th><th>APEC</th><th>PEXMON</th></tr><tr><th>Energy<br>(keV)</th><th>Width<br>(keV)</th><th>Energy<br>(keV)</th><th>Refl frac</th></tr></thead><tbody><tr><td>M81</td><td>6.61<sup>+0.04</sup><sub>-0.04</sub></td><td>0.15<sup>+0.05</sup><sub>-0.05</sub></td><td>1.5<sup>+0.06</sup><sub>-0.05</sub></td><td>...</td></tr><tr><td>M87</td><td>6.58<sup>+0.01</sup><sub>-0.03</sub></td><td>0.0<sup>f</sup></td><td>1.39<sup>+0.03</sup><sub>-0.05</sub></td><td>...</td></tr><tr><td>NGC 1052</td><td>6.3<sup>+0.05</sup><sub>-0.06</sub></td><td>0.16<sup>+0.11</sup><sub>-0.11</sub></td><td>...</td><td>-0.36</td></tr><tr><td>NGC 2992</td><td>6.4<sup>f</sup></td><td>0.38<sup>+0.1</sup><sub>-0.08</sub></td><td>...</td><td>-0.67</td></tr><tr><td></td><td>6.4<sup>f</sup></td><td>0.34<sup>+0.03</sup><sub>-0.03</sub></td><td>...</td><td>-0.7</td></tr><tr><td>NGC 3227</td><td>6.28<sup>+0.02</sup><sub>-0.02</sub></td><td>0.2<sup>+0.04</sup><sub>-0.04</sub></td><td>...</td><td>-0.14</td></tr><tr><td>NGC 4579</td><td>6.49<sup>+0.04</sup><sub>-0.04</sub></td><td>0.35<sup>+0.05</sup><sub>-0.04</sub></td><td>...</td><td>...</td></tr><tr><td>NGC 5033</td><td>6.4<sup>f</sup></td><td>0.0<sup>f</sup></td><td>...</td><td>-0.86</td></tr><tr><td>NGC 5273</td><td>6.4<sup>f</sup></td><td>0.37<sup>+0.08</sup><sub>-0.07</sub></td><td>...</td><td>-0.72</td></tr><tr><td>NGC 5506</td><td>6.3<sup>+0.01</sup><sub>-0.01</sub></td><td>0.23<sup>+0.01</sup><sub>-0.02</sub></td><td>...</td><td>-1.0</td></tr><tr><td>NGC 7213</td><td>6.53<sup>+0.04</sup><sub>-0.04</sub></td><td>0.33<sup>+0.05</sup><sub>-0.04</sub></td><td>7.57</td><td>...</td></tr></tbody></table>