Low-mass Active Galactic Nuclei on the Fundamental Plane of Black Hole Activity

Our parent low-mass AGN sample is the 309 broad-line AGNs with ${M}_{\mathrm{BH}}$ < 2 × 10⁶ ${M}_{\odot }$ compiled by Dong et al. (2012). The ${M}_{\mathrm{BH}}$ was derived from the luminosity and width of the broad Hα emission line, using the virial formalism calibrated by Greene & Ho (2007) based on single-epoch spectra. The statistical uncertainty of the estimated ${M}_{\mathrm{BH}}$ should be around 0.3 dex typically (see, e.g., Wang et al. 2009 for the uncertainty estimation for an AGN sample); yet the uncertainty for some individual sources can be as large as 1 order of magnitude (e.g., Vestergaard & Peterson 2006). We set the uncertainty of ${M}_{\mathrm{BH}}$ (and accordingly the Eddington luminosity ${L}_{\mathrm{Edd}}$) to be 0.6 dex in the subsequent fitting (Section 3.1) and plottings (Section 3.1 and Section 3.2).

Among the sources of Dong et al. (2012), 288 are covered by the VLA FIRST survey. For these sources, we fit the FIRST images,⁷ and measure the fluxes and the corresponding rms (root of mean square) noises (the details of the fitting and noise determination can be found in Section 2.1.1).

The sources with flux greater than three times the rms noise (namely S/N > 3) are deemed to have radio detections. This criterion is a tradeoff between minimizing false detections and maximizing the number of reliable radio sources (or candidates of high probability). We will analyze this criterion at the end of this subsection and check it in Section 3. There are 52 such radio-detected sources. Then we match them to the X-ray archive, NASA's HEASARC,⁸ and find 22 sources that we can obtain their X-ray fluxes (Section 2.1.2).

Among the 22 low-mass AGNs with both radio and X-ray fluxes, there are 3 sources that have already been included in the literature (e.g., included in the low-mass AGN data set used by Gültekin et al. 2014; see also Table 2 of Nyland et al. 2012); they are J0914+0853, J1240−0029 (namely GH10 after Greene & Ho 2004), and J1324+0446. Excluding those 3 sources (they are listed in the literature data set instead; see Table 2), our new data set includes 19 sources.

Here we must evaluate our criterion of radio detection, S/N > 3. We set such a criterion instead of the commonly used flux limit S/N > 5 (or called 5σ if the noise is random and Gaussian), out of the tradeoff between reliability and the purpose to select as many as possible radio-detected sources or candidates. Radio-detected low-mass AGNs are rather rare, and thus even the selection of candidate radio sources has its own merit. Assuming the noise of the FIRST images is Gaussian, the trial penalty, namely the probability of mistaking one or more random fluctuations as radio source(s) with S/N > 3 out of the parent sample of 288 objects covered by the FIRST survey, is $1-{\left({\int }_{-\infty }^{3\sigma }G(x){dx}\right)}^{288}=0.32$. Here G(x) is the Gaussian probability density function, zero-centered and with a standard deviation σ. A chance probability of 0.32 is fairly large. To be worse, there are often correlated errors in radio images, and thus the noise is not purely random Gaussian and the false-detection probability would be greater than the above estimated trail penalty; this is the very reason why the conservative flux limit of 5σ is commonly used. Certainly, on the other hand, our estimation of noise (namely rms; see Section 2.1.1) is not merely the random component, but is able to incorporate other error sources to some degree.

In order to make up the shortcoming of the S/N > 3 criterion, we divide the 19 sources into two groups: 8 sources with S/N > 4.43 (including 4 sources with S/N > 5) are grouped as the reliable radio detections, and the remaining 11 with radio 3 < S/N < 4.43, conservatively speaking, are only candidates. The dividing S/N of 4.43 is set in terms of trial penalty, as follows. For one source, assuming random Gaussian noise, the chance probability of false detection associated with the S/N > 3 criterion is $1-{\int }_{-\infty }^{3\sigma }G(x){dx}$ = 0.0013. We now require the chance probability for our parent sample of 288 objects to be the same level, i.e., $1-{\left({\int }_{-\infty }^{n\sigma }G(x){dx}\right)}^{288}$ ≈ 0.0013, then we get n = 443. In the subsequent analyses (Sections 3.1 and 3.2), we will compare the candidate radio sources with our reliable sources and the literature sources, in the radio/X-ray correlation and in the FP. We find no difference between the two groups of sources.

2.1.1. Radio Flux

We adopt a new method to obtain the (faint) radio flux for as many low-mass AGNs as possible. This method was first used by Liu et al. (2017), to measure the 1.4 GHz flux directly from the VLA FIRST image, when the radio emission is faint and below the flux threshold (1 mJy) set to the official FIRST catalog (Becker et al. 1995; White et al. 1997). Low-mass AGNs are generically radio quiet (${ \mathcal R }\lt 10$), with only a few (<6%) being radio loud (Greene et al. 2006); this is supposedly due to their relatively high accretion rate by selection (as well as small ${M}_{\mathrm{BH}}$ compared with common AGNs with SMBHs; see Section 4). Among the 288 low-mass AGNs covered by the FIRST survey, only 17 are included in the FIRST catalog.

We fit a two-dimensional Gaussian to the FIRST image of every source. The potential radio sources are assumed to be point-like, with the Gaussian FWHM set to be the beam size (54). The center of the Gaussian is fixed to be the optical position of the broad-line nucleus determined by the Sloan Digital Sky Survey (SDSS; York et al. 2000). The only free parameter is the flux of the point source.

In addition, a CLEAN bias always makes an underestimate of the flux, which is typically 0.25 mJy for point sources (see Section 4.3 of White et al. 1997 and Section 7.2 of Becker et al. 1995). We thus correct this bias for the best-fit flux by adding 0.25 mJy, as all versions of the official FIRST catalog released after 1995 October did (White et al. 1997).⁹ As the CLEAN bias arising from that CLEAN algorithm steals flux from discrete sources and spreads it around the image, it also has some influence on the rms noise, which is not yet well understood. We simply measure the rms noise in an empty region of size 9'' × 9'', 20'' away from the center of the Gaussian, and take it as the uncertainty of the 1.4 GHz flux.

Note that our thus-measured rms values are the actual ones directly from the final coadded images, different from those used in the FIRST catalog construction (namely, the so-called 5σ flux threshold) and listed in the catalog (the "rms" column). The latter was based on the weighted combination of noise values derived from the whole-image rms for each grid map that contributes to that image, as displayed in the rms sensitivity map of the FIRST coverage; White et al. (1997) noted that "it [the coverage-map rms value] should not be used to establish a definitive upper limit to the radio flux density from a given location in the sky; rather, the flux density in the relevant coadded image should be measured directly."

We consider the sources with the flux (prior to the correction for the CLEAN bias) higher than 3 times the rms noise (namely S/N > 3) as candidate radio detections. Note that in the S/N calculation the flux is the one prior to the correction for the CLEAN bias. There are 52 such radio-detected sources (including the 17 already in the FIRST catalog), 22 of which have archival X-ray data. We derived their radio flux at 5 GHz from our measured 1.4 GHz flux. Since our sources are relatively bright (namely actively accreting) with X-ray Eddington ratio ${10}^{-3}\lt {L}_{X}/{L}_{\mathrm{Edd}}\lt 1$, we adopt the typical spectral index of bright AGNs, α_R = −0.5 (defined as ${F}_{\nu }\propto {\nu }^{{\alpha }_{R}}$), for the conversion. We simply assume a 20% uncertainty in α_R, which would cause a 13% uncertainty in the 5 GHz flux.

Admittedly, the validity of this new method needs to be tested. For this purpose, we collect radio point sources that have radio flux data and are covered by the FIRST survey, use our new method to measure their radio fluxes, and then make the comparison. We limit our method and the comparison for radio point-like sources only, to minimize the contamination of the radio emission from the host galaxies. The point-like sources are selected in two ways. One part is the 6 unresolved sources among the aforementioned 17 low-mass AGNs included in the FIRST catalog; they are selected to have the beam-corrected major-axis FWHM <15 (the parameter "Deconv. MajAx" in the catalog). The other part is the 6 sources that are in the literature data set (Table 2) and are covered by the FIRST survey (but excluding those with Deconv. MajAx ≥15 in the FIRST catalog). Most of the sources are faint, close to the 1 mJy threshold of the FIRST catalog.¹⁰ The comparison is summarized in Table 3.

Table 3.  Comparison of Radio Flux

Name	f1.4 GHz (our)	f1.4 GHz(cat)
	(mJy)	(mJy)
(1)	(2)	(3)
J121629.13+601823.5	0.94 ± 0.13	1.03 ± 0.11
J162824.49+452811.0	0.75 ± 0.09	0.68 ± 0.09
J074251.09+333403.9	2.75 ± 0.14	3.18 ± 0.16
J074948.33+264734.2	3.55 ± 0.09	4.53 ± 0.13
J132428.24+044629.7	1.79 ± 0.13	2.33 ± 0.14
J132834.37–030744.8	7.10 ± 0.13	8.81 ± 0.14
J082443.28+295923.5	1.67 ± 0.16	1.77 ± 0.11
J101246.49+061604.7	1.04 ± 0.11	1.03 ± 0.15
J121629.13+601823.5	0.94 ± 0.13	1.03 ± 0.11
J132428.24+044629.6	1.79 ± 0.15	2.33 ± 0.14
J155909.62+350147.4	2.72 ± 0.11	3.39 ± 0.12
NGC 4395	1.12 ± 0.12	1.17 ± 0.15

Note. Column (1): Source name. Column (2): Our measured 1.4 GHz flux (corrected for the CLEAN bias) and rms noise. Column (3): The 1.4 GHz flux and the rms retrieved from the FIRST catalog. The upper part lists the 6 unresolved sources among the 17 low-mass AGNs included in the FIRST catalog, while the lower part lists the 6 sources that are in the literature data set (Table 2) and covered by the FIRST survey (but excluding those with Deconv. MajAx ≥15 in the FIRST catalog); see the text in Section 2.1.1 for details.

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We can see that the values by our method (f_our) are in good agreement (all within a factor of ≲1.3) with the fiducial values (f_cat; from the FIRST catalog). The mean and standard deviation of the relative difference (f_our − f_cat)/f_cat are −0.11 and 0.10, respectively. That is, the systematic error and the random uncertainty of our measured flux are both on the level of 10% only. Therefore, it is reliable to apply our method to point-like sources to obtain their radio fluxes. The standard error of the estimated mean (namely, the systematic offset) is only 2.9%, meaning that the systematic offset is fairly stable. Thus we correct this offset of −11% from our measured fluxes in the subsequent fitting (Section 3.1) and plottings (Figures 1 and 2).

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2.1.2. X-Ray Flux

In Table 2, we list all the low-mass AGNs for the FP studies in the literature. These include the 7 sources of Gültekin et al. (2014) that had firm detections in both radio and X-ray (with the 3 upper-limit sources dropped) and had virial ${M}_{\mathrm{BH}}$ from Greene & Ho (2007). In addition, we also include NGC 4395 and NGC 404 (Nyland et al. 2012), and Henize 2–10 (Reines et al. 2016). The ${M}_{\mathrm{BH}}$ of NGC 4395 is obtained by reverberation mapping, and the ${M}_{\mathrm{BH}}$ of NGC 404 is obtained by dynamical measurements. The ${M}_{\mathrm{BH}}$ of Henize 2–10 is estimated from the stellar mass of the host galaxy. See the above references for the details of the radio and X-ray measurements and the ${M}_{\mathrm{BH}}$ estimation.

Before exploring the FP of BH activity, we first examine the radio/X-ray correlation among low-mass AGNs. We consider their radio and X-ray luminosities in terms of Eddington unit, i.e., L_R/${L}_{\mathrm{Edd}}$ and L_X/${L}_{\mathrm{Edd}}$, in order to reduce the impact of ${M}_{\mathrm{BH}}$. Figure 1 shows the L_R/${L}_{\mathrm{Edd}}$–L_X/${L}_{\mathrm{Edd}}$ relationship, where the sources of the new data set are shown as pentagons (the 8 reliable radio sources with S/N > 4.43; see Section 2.1) or triangles (the 11 candidate radio sources with 3 < S/N < 4.43), while the literature data set is shown with open circles. We fit the entire sample with a single power-law model (i.e., linear in the log–log scale), using the LINMIX_ERR program (Kelly 2007) that accounts for measurement errors in both axes. The systematic offset of our measured L_R values with respect to the fiducial ones (−11%; see Section 2.1.1) is corrected in the fitting. The total uncertainty of L_R is the quadrature sum of the following three terms: the rms noise, the error introduced by the assumed α_R, and the random uncertainty with respect to the fiducial (see Section 2.1.1 for the details of the three terms). The total uncertainty of L_X is the quadrature sum of the following two terms: the uncertainty from the archive and the error introduced by the assumed photon index (see Section 2.1.2 for details). The uncertainty of ${L}_{\mathrm{Edd}}$ is simply 0.6 dex (see Section 2.1). The fitting result is as follows (the dotted line in Figure 1),

$$\begin{eqnarray}\mathrm{log}({L}_{R}/{L}_{\mathrm{Edd}}) & = & (0.70\pm 0.05)\mathrm{log}({L}_{X}/{L}_{\mathrm{Edd}})\\ & & -(4.75\pm 0.18),\end{eqnarray} \tag{ 2 }$$

with a reduced χ² = 0.56. We can see from Figure 1 that a data point at log L_X/${L}_{\mathrm{Edd}}$ = −3.75 deviates from the best-fit line by about 3σ; this outlier is NGC 4395. When NGC 4395 is excluded, the best fit becomes

$$\begin{eqnarray}\mathrm{log}({L}_{R}/{L}_{\mathrm{Edd}}) & = & (0.64\pm 0.04)\mathrm{log}({L}_{X}/{L}_{\mathrm{Edd}})\\ & & -(4.77\pm 0.11),\end{eqnarray} \tag{ 3 }$$

with a reduced χ² = 0.25; see the solid line in Figure 1. Such a small reduced χ² indicates that the uncertainties of the data are overestimated to some degree. This best fit is close to the ${L}_{R}\propto {L}_{X}^{0.62}$ relation reported in GX 339-4, a typical BHXB (Corbel et al. 2013), implying that all the low-mass AGNs (probably except NGC 4395) are standard ones (namely, obeying a single power-law ${L}_{R}\propto {L}_{X}^{{\xi }_{X}}$) rather than "outliers" in terms of the radio/X-ray correlation of BH accreting systems (see Section 1).

In order to test the difference between the candidate sources (3 < S/N < 4.43) and the well-measured sources (S/N > 4.43, excluding NGC 4395), we exclude the candidate sources as well as NGC 4395, and perform the fitting again. The best fit is almost the same as Equation (3), being log(L_R/${L}_{\mathrm{Edd}}$) = (0.66 ± 0.04) log(L_X/${L}_{\mathrm{Edd}}$) − (4.69 ± 0.14).

We note in passing that the outlier NGC 4395 deserves further investigation in the future. It has a reliable ${M}_{\mathrm{BH}}$ measurement by the reverberation mapping method (Peterson et al. 2005). From a joint monitoring in radio (VLA) and X-ray (Swift/XRT) in 2011, NGC 4395 seemed to follow a flat radio/X-ray correlation, i.e., ${L}_{R}\propto {L}_{X}^{\sim 0}$ (King et al. 2013). Although that result was not robust due to the very limited dynamic range in both L_R and L_X, it suggested that NGC 4395 might be a source that follows the flat branch of the hybrid radio/X-ray correlation (Xie & Yuan 2016; cf. NGC 7213, Xie et al. 2016).

We then examine the low-mass AGNs in the FP of BH activity. Because our sample is still not large and the data (particularly the radio fluxes) demand to be refined, in this work, we refrain from fitting the data to get a new relation. Instead, we take the same approach as Gültekin et al. (2014), by examining our data with respect to several well-known FP relations in the literature. Three FPs are considered: the Merloni et al. (2003) relation (ξ_X = 0.60, ξ_M = 0.78, and C = 7.33), and the SMBH-only (i.e., fitted with SMBH systems only; ξ_X = 0.50, ξ_M = 2.08, and C = 0.40) and the universal (i.e., fitted with their combined sample of SMBH and stellar-mass BH systems; ξ_X = 0.67 ± 0.12, ξ_M = 0.78 ± 0.27, and C = 4.80 ± 0.24) FP relations of Gültekin et al. (2009); they are illustrated in Figure 2, from left to right, respectively. Note that in the figure the systematic offset of our measured L_R values with respect to the fiducial ones (−11%; see Section 2.1.1) is corrected, and the error bars of our data points are calculated with the error terms listed in the above (Section 3.1) in terms of the standard error propagation formula. The low-mass AGNs match best the universal FP of Gültekin et al. (2009); this confirms the conclusion of Gültekin et al. (2014).

We further test the difference between the candidate sources (our 11 objects with 3 < S/N < 4.43, called Group 1) and the well-measured sources (our 8 new objects with S/N > 4.43 plus the 9 sources from the literature excluding NGC 4395, called Group 2 here), in terms of the FP. We calculate the orthogonal distances of every source to the line of the edge-on viewed universal FP of Gültekin et al. (2009) as depicted in Figure 2 (right panel). The mean and standard deviation of the distances are 0.20 and 0.35, respectively, for Group 1; 0.29 and 0.26, respectively, for Group 2. The standard errors for the two mean values are therefore 0.10 (Group 1) and 0.06 (Group 2). Thus the difference (namely 0.09) between the mean values of the two groups is well within 1σ error. We also perform a Kolmogorov–Smirnov test for the two distributions of the distances. The resultant p-value (chance probability) is 0.70, meaning that we cannot reject the hypothesis that the distributions of the two groups are the same.

It is now generally believed that, in nonblazar AGNs of either low or high accretion rates, the radio emission comes predominantly from the jet, while the high-frequency emission (from the optical through X-ray) comes from the accretion flow and thus is treated as an indicator of accretion rate, namely the Eddington ratio (${\ell }$) in practice (e.g., Heinz & Sunyaev 2003; Saikia et al. 2015). For AGNs, we can simply assume ${\ell }$ ∝ L_X/M_BH. In the literature, the bolometric correction κ_x (defined as ${L}_{\mathrm{bol}}$/L_X) values for AGNs once differed considerably, and depended on ${L}_{\mathrm{bol}}$ and ${\ell }$. This mainly arose from the spectral complexity associated with absorption (see Vasudevan et al. 2010 and references therein). The recent studies, with various improvements in calculating the intrinsic X-ray luminosity and particularly the bolometric luminosity, indicate that κ_x is typically in the range of 10–30 derived from the observational data with an intrinsic scatter of ∼0.2 dex, not as large as previously deemed, and that its dependence on either ${L}_{\mathrm{bol}}$ or ${\ell }$ is mild for the observed ${\ell }$ regime (≈10⁻³ to 1); see, e.g., Vasudevan et al. (2010), Brightman et al. (2017) and references therein. Such a magnitude of κ_x variation does not impact our deduction here. Thus it is easy to understand the dependence of ${ \mathcal R }$ on ${M}_{\mathrm{BH}}$ and ${\ell }$ in terms of the FP (Equation (1)), as follows

$$\begin{eqnarray}&&\mathrm{log}{ \mathcal R }=({\xi }_{X}-1)\mathrm{log}{\ell }+({\xi }_{M}+{\xi }_{X}-1)\mathrm{log}{M}_{\mathrm{BH}}+\mathrm{const}.\end{eqnarray} \tag{ 4 }$$

With the coefficients and their uncertainties of the universal FP of Gültekin et al. (2009), Equation (4) reads:

$$\begin{eqnarray}\mathrm{log}{ \mathcal R } & = & (-0.33\pm 0.12)\mathrm{log}{\ell }\\ & & +(0.45\pm 0.30)\mathrm{log}{M}_{\mathrm{BH}}+\mathrm{const}.\end{eqnarray} \tag{ 5 }$$

This leads us to speculate that the radio loudness of AGNs depends not only on ${\ell }$ but probably also on ${M}_{\mathrm{BH}}$, even possibly to an almost equal degree (tentatively judging from the similar magnitudes of the best-fit power-law indexes). The correlation might be positive with ${M}_{\mathrm{BH}}$ albeit the statistical significance being only 1.5σ (${ \mathcal R }\propto {M}_{\mathrm{BH}}^{0.45\pm 0.30}$), and negative with ${\ell }$ albeit the statistical significance being 3σ (${ \mathcal R }\propto {{\ell }}^{-0.33\pm 0.12}$). If we adopt the fitting result of Merloni et al. (2003), where the best-fit ξ_X, its uncertainty, and the ξ_M value are all similar to the universal relation of Gültekin et al. (2009), but the uncertainty to ξ_M is reduced by a half, then the ${\ell }$ dependence would be of 3.6σ significance, and the ${M}_{\mathrm{BH}}$ dependence would be of 2.4σ significance. The currently large error bars on the indexes of the above FP relations allow a considerable chance probability, 7% (namely single-sided 1.5σ Gaussian deviance), for no correlation or a negative correlation between ${ \mathcal R }$ and ${M}_{\mathrm{BH}}$, thus the above speculation is yet to be verified. On the other hand, this speculation is consistent with—and somehow reinforced by—almost all the significant ${ \mathcal R }$-related correlations in AGNs with either ${M}_{\mathrm{BH}}$ or ${\ell }$ that were discovered mainly by bivariate correlation analysis before (e.g., Laor 2000; Greene et al. 2006). Furthermore, now Equation (5) seems to evoke a panoramic—and probably more insightful (see below)—understanding.

The negative ${ \mathcal R }$–${\ell }$ correlation can be easily understood under the widely accepted coupled accretion-jet models (Yuan & Cui 2005; Heinz & Sunyaev 2003), where the accretion flow responsible for the X-ray is a hot component, either a hot accretion flow (Yuan & Narayan 2014) or a corona located above the cold accretion disk (Shakura & Sunyaev 1973). In this model, the key factor to produce such a negative correlation is the mass accretion rate $\dot{M}$. Hot accretion flow predicts ${L}_{X}\propto {\dot{M}}_{\mathrm{BH}}^{\approx 2-3}$ (Merloni et al. 2003; Yuan & Narayan 2014), while the scale-invariant jet model predicts ${L}_{R}\propto {\dot{M}}_{\mathrm{BH}}^{\approx 1.4}$ (Heinz & Sunyaev 2003).

Regarding the potentially strong and positive correlation between ${ \mathcal R }$ and ${M}_{\mathrm{BH}}$, on the other hand, it is not so easy to understand from a theoretical perspective. Despite the currently large error bars on that index, which could be consistent with no correlation or a negative correlation by a chance probability of 7% as described in the above, we try to give an explanation for a strong, positive correlation as follows. As Heinz & Sunyaev (2003) argued, the dependence on ${M}_{\mathrm{BH}}$ is mainly determined by jet physics itself. A larger ${M}_{\mathrm{BH}}$ leads to a relatively stronger magnetic field near the BH, which would make it easier to launch a jet. Arguably, a stronger magnetic field strength would result in a higher acceleration, and consequently a larger jet velocity. This appears true observationally; i.e., there is likely a positive correlation between the Lorentz factor of AGN jets Γ_jet and ${M}_{\mathrm{BH}}$, as follows. In AGNs with SMBHs, the jets are usually relativistic, with Γ_jet ∼ 10 (Kellermann et al. 2004), whereas in NLS1s (where ${M}_{\mathrm{BH}}$ is not too much higher than the BHs in low-mass AGNs), the jets are only mildly relativistic (Gu et al. 2015). Certainly, it remains unclear whether or not a large-scale magnetic field can be developed around a cold AD, and thus further efforts are required, not the least of which include better constraining any potential ${M}_{\mathrm{BH}}$ dependence of radio loudness.