A Study of the Accretion–Jet Coupling of Black Hole Objects at Different Scales

Abstract

The fundamental plane of black hole activity is a very important tool to study accretion and jets. However, we found that the SEDs of AGNs and XRBs are different in the 2–10 keV energy band, and it seems inappropriate to use 2–10 keV X-ray luminosities to study the fundamental plane. In this work, we use the luminosity near the peak of the blackbody radiation of the active galactic nuclei and black hole binaries to replace the 2–10 keV luminosity. We re-explore the fundamental plane of black hole activity by using the 2500 $\mathring{\mathrm{A}}$ luminosity as the peak luminosity of the blackbody radiation of AGNs and 1 keV luminosity as the peak luminosity of the blackbody radiation of XRBs. We compile samples of black hole binaries and active galactic nuclei with luminosity near the peak luminosity of blackbody radiation and study the fundamental plane between radio luminosity ($L_{\mathrm{R}}$), the peak luminosity of blackbody radiation ($L_{\mathrm{peak}}$), and black hole mass ($M_{\mathrm{BH}}$). We find that the radio–peak luminosity correlations are $L_{5\mathrm{GHz}}/L_{\mathrm{Edd}}\propto {(L_{2500\mathring{\mathrm{A}}}/L_{\mathrm{Edd}})}^{1.55}$ and $L_{5\mathrm{GHz}}/L_{\mathrm{Edd}}\propto {(L_{1\mathrm{keV}}/L_{\mathrm{Edd}})}^{1.53}$ for AGN and XRB, respectively, in the radiatively efficient sample, and $L_{5\mathrm{GHz}}/L_{\mathrm{Edd}}\propto {(L_{2500\mathring{\mathrm{A}}}/L_{\mathrm{Edd}})}^{0.48}$ and $L_{5\mathrm{GHz}}/L_{\mathrm{Edd}}\propto {(L_{1\mathrm{keV}}/L_{\mathrm{Edd}})}^{0.53}$ in the radiatively inefficient sample, respectively. Based on the similarities in radio–peak correlations, we further propose a fundamental plane in radio luminosity, the peak luminosity of blackbody radiation, and black hole mass, which is radiatively efficient: $log{L}_{5\mathrm{GHz}}=1.{57}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}-0.{32}_{-0.16}^{+0.16}log{M}_{\mathrm{BH}}-27.{73}_{-0.34}^{+0.34}$ with a scatter of ${\sigma }_{\mathrm{R}}$ = 0.48 dex, and radiatively inefficient: $log{L}_{5\mathrm{GHz}}=0.{45}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}+0.{91}_{-0.10}^{+0.12}log{M}_{\mathrm{BH}}+12.{58}_{-0.38}^{+0.38}$ with a scatter of ${\sigma }_{\mathrm{R}}$ = 0.63 dex. Our results are similar to those of previous studies on the fundamental plane for radiatively efficient and radiatively inefficient black hole activity. However, our results exhibit a smaller scatter, so when using the same part of blackbody radiation (i.e., the peak luminosity of the blackbody radiation), the fundamental plane becomes a little bit tighter.

1. Introduction

Stellar-mass X-ray binaries (XRBs) and active galactic nuclei (AGNs) capture surrounding material under their own gravity, a process that emits strong radiation, such as radio and X-rays. Radio emission could originate from star formation, disk wind, and jets, and X-ray emission could originate from the corona, accretion disks, and jets [1,2,3]. In addition, the magnetic field is essential for jet formation [4]. It is believed that XRBs and AGNs are scale-invariant across the black hole mass scale and present similar physical properties [5,6,7]. Several studies have pointed out that there may be some sort of connection between the corona and the jets [8,9,10]. However, Ref. [11] posits that there is no connection between the corona and the jets. A tight correlation between the radio and X-ray luminosities was observed in XRBs during their hard states [12,13,14], and this correlation was later extended to AGNs. Considering the mass of the black hole, the authors of Ref. [5] proposed, for the first time, a fundamental plane of black hole activity:

$$log{L}_{5\mathrm{GHz}}=0.{60}_{-0.11}^{+0.11}log{L}_{2-10\mathrm{keV}}+0.{78}_{-0.09}^{+0.11}log{M}_{\mathrm{BH}}+7.{33}_{-4.07}^{+4.05}$$

(1)

The fundamental plane of black-hole-source activity is often believed to be one of the most prominent pieces of evidence of the similarity of XRBs and AGNs, and it provides an important empirical relation to estimate black hole mass through X-ray luminosity and radio luminosity. Subsequently, the authors of Ref. [15] compiled a bright black hole source with $log{L}_{2-10\mathrm{keV}}\le {10}^{-3}{L}_{\mathrm{Edd}}$ and presented a fundamental plane of radiatively efficient BH sources:

$$log{L}_{\mathrm{R}}=1.{59}_{-0.22}^{+0.28}log{L}_{\mathrm{X}}-0.{22}_{-0.20}^{+0.19}log{M}_{\mathrm{BH}}-28.{97}_{-0.45}^{+0.45}$$

(2)

These results support the idea that radio emission is from relativistic jets and X-ray emission is related to the accretion mode and Eddington accretion ratio [16,17]. It is generally accepted that radio emissions come from relativistic jets and X-ray emissions come from the accretion flow or jet base. When the Eddington accretion ratio is relatively high, the most typical model is the disk–corona model, which suggests that X-rays come from the disk–corona system. When the Eddington accretion ratio is relatively small, corresponding to radiatively inefficient accretion modes. The authors of Ref. [18] studied the correlation between the hard X-ray photon indices $\Gamma$ and the Eddington ratio $L_{\mathrm{X}}/L_{\mathrm{Edd}}$ for six black hole X-ray binaries and found that $\Gamma$ and $L_{\mathrm{X}}/L_{\mathrm{Edd}}$ show an anti-correlation when $L_{\mathrm{X}}/L_{\mathrm{Edd}}$ is less than a critical luminosity and a positive correlation between $\Gamma$ and $L_{\mathrm{X}}/L_{\mathrm{Edd}}$ when $L_{\mathrm{X}}/L_{\mathrm{Edd}}$ is higher than a critical luminosity. Similar properties of positive correlation and anti-correlation have been found in AGNs [19,20,21,22]. The results also support the idea that X-ray emissions are related to the accretion mode and Eddington accretion ratio.

However, in these works, the X-ray luminosity of AGNs and XRBs use 2–10 keV luminosity, and by analyzing the spectral energy distributions (SEDs) of AGNs and XRBs, we found that the SEDs of AGNs and XRBs are different in the 2–10 keV energy band, and thus it seems inappropriate to use 2–10 keV X-ray luminosities to study the fundamental plane. For XRBs, the thermal emission from the thin disk peaks in soft X-rays with a peak luminosity roughly around 1 keV. Since the disk temperature in AGNs is cooler compared to that of XRBs, the peak of the AGNs disk appears in the optical/UV band [7,23], with a peak luminosity roughly around 2500 $\mathring{\mathrm{A}}$ luminosity. In AGNs, there is a non-linear relation between X-ray and optical/UV emissions that is usually parameterized as a dependence between the logarithm of the monochromatic luminosity 2500 $\mathring{\mathrm{A}}$ and 2 keV ($log{L}_{2keV}=\beta log{L}_{2500\mathring{\mathrm{A}}}+\gamma$) [24,25]. According to this relation, the X-ray luminosity can be converted to 2500 $\mathring{\mathrm{A}}$ luminosity.

The aim of our work is to use the peak luminosity of blackbody radiation to replace the 2–10 keV luminosity. Although the SED of each AGN and XRB is different, the peak luminosity of blackbody radiation is roughly at 2500 $\mathring{\mathrm{A}}$ for the AGNs and roughly at 1 keV for the XRBs. Therefore, in this work, we re-explore the fundamental plane of black hole activity using 2500 $\mathring{\mathrm{A}}$ luminosity as the peak luminosity of the blackbody radiation of AGNs and 1 keV luminosity as the peak luminosity of the blackbody radiation of XRBs. In terms of the masses of black hole objects, XRBs are homogeneous, with masses of about 3–20 $M_{\odot }$, while AGNs have a broad mass range, with masses of about ${10}^{6-10}$ $M_{\odot }$. The SED of each AGN is different, so different AGN masses have different peak luminosities. Since the disk temperature at 2500 $\mathring{\mathrm{A}}$ is 0.005 keV, considering the $T\propto M_{\mathrm{BH}}^{-1/4}$ scaling of the black hole accretion disk temperature with the black hole mass [26], we found that the peak luminosity of the blackbody radiation of the AGN is 2500 $\mathring{\mathrm{A}}$ luminosity when its black hole mass is $M_{\mathrm{BH}}={10}^9 M_{\odot }$.

This paper is structured as follows: We describe the sample of AGNs and XRBs in Section 2. In Section 3, we present our results and introduce the statistical analysis method. Our discussions are detailed in Section 4. Throughout this paper, we assume the following cosmology for AGNs: $H_0$ = 70 km ${\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$, ${\Omega }_0=0.27$, and ${\Omega }_{\Lambda }=0.73$.

2. Sample

To study the fundamental plane of black hole objects at different scales, we compile a sample with radio luminosity ($L_{\mathrm{R}}$), the peak luminosity of blackbody radiation ($L_{\mathrm{peak}}$), and black hole mass ($M_{\mathrm{BH}}$). Our total sample consists of radiatively efficient and radiatively inefficient samples. Our sample of radiatively efficient AGNs consists of the 64 high-luminosity radio quiet-type I AGNs with $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$≳ 1% compiled by the authors of Ref. [15] (see

Table 1. Data of radiatively efficient AGNs.

Name	${\mathit{M}}_{\mathbf{BH}}$ (${\mathit{M}}_{\odot }$)	${\mathit{L}}_{2-10\mathbf{keV}}$ (erg ${\mathbf{s}}^{-1}$)	${\mathit{L}}_{5\mathbf{GHz}}$ (erg ${\mathbf{s}}^{-1}$)	Refs.	${\mathit{L}}_{2\mathbf{keV}}$ (erg ${\mathbf{s}}^{-1}$)	${\mathit{L}}_{2500\mathring{\mathbf{A}}}$ (erg ${\mathbf{s}}^{-1}$)	$\frac{{\mathit{L}}_{\mathbf{bol}}}{{\mathit{L}}_{\mathbf{Edd}}}$
1H 0419-577	8.58	44.33	39.83	[30,31]	43.54	44.49	−0.60
Ark 120	8.27	43.95	38.56	[30,31]	43.45	44.36	−0.99
Ark 374	7.86	43.46	38.67	[30]	42.93	43.68	-
Ark 564	6.27	43.50	38.79	[30,31]	43.16	43.99	0.07
ESO 323-G77	7.12	42.88	38.50	[30,32,33]	42.29	42.84	-
Fairall 9	7.91	43.97	39.11	[30,31]	43.36	44.25	−1.72
HE 1029-1401	9.08	44.31	39.63	[30,31]	43.77	44.79	−0.88
HE 1143-1810	7.01	43.82	38.61	[30,31]	43.25	44.10	-
IC 4329A	6.77	43.96	38.84	[30,31]	43.38	44.27	-0.83
IRAS 1334+2438	8.62	43.81	40.01	[30,31]	43.32	44.19	−0.58
LZw 1	7.26	43.85	39.09	[30,31]	43.45	44.37	0.12
MC-5-23-16	7.85	43.02	37.68	[30,31]	42.48	43.08	-
MCG-6-30-15	6.19	42.90	36.82	[30,31]	42.37	42.95	−0.81
MR2251-178	9.03	44.30	39.17	[30,34]	43.63	44.60	-
Mrk 110	6.82	43.92	38.16	[30,31]	43.34	44.22	−0.36
Mrk 205	8.68	43.76	38.78	[30]	43.16	43.98	−0.57
Mrk 279	7.62	43.50	38.93	[30,31]	42.94	43.70	−0.81
Mrk 290	7.65	43.25	38.34	[30,31]	42.59	43.24	-
Mrk 335	7.15	43.27	38.36	[30,31]	42.86	43.58	0.05
Mrk 359	6.24	42.50	37.69	[30,31]	41.95	42.39	−0.60
Mrk 493	6.17	43.22	38.05	[30,31]	42.79	43.50	0.21
Mrk 509	7.86	44.68	38.83	[30,31]	44.04	45.15	−1.10
Mrk 586	7.55	44.05	39.81	[30,31]	43.67	44.66	0.53
Mrk 590	7.20	42.86	38.50	[30,31]	42.23	42.76	−0.16
Mrk 766	6.28	43.16	38.05	[30,31]	42.72	43.41	−0.26
Mrk 841	7.88	43.89	38.18	[30,31]	43.36	44.25	−0.36
Mrk 876	7.55	44.36	39.81	[30,35]	43.80	44.82	−1.25
Mrk 1044	6.50	42.55	37.52	[30,31]	42.11	42.60	0.02
Mrk 1383	8.63	44.10	38.89	[30,31]	43.59	44.55	−1.07
Mrk 1513	7.58	43.51	39.00	[30,31]	42.86	43.59	-
NGC 2992	7.72	42.97	38.52	[30,31,36]	42.29	42.84	-
NGC 3516	7.36	42.83	37.72	[27,30,37]	42.25	42.79	−1.89
NGC 3783	6.94	43.03	38.22	[30,31]	42.38	42.95	−1.36
NGC 4051	6.11	41.39	34.95	[5,31,38]	40.89	40.99	−1.50
NGC 4151	7.17	42.87	38.09	[5,39,40]	42.23	42.76	−1.39
NGC 4593	6.91	43.07	37.26	[30,31]	42.45	43.05	−0.79
NGC 5506	7.46	42.83	38.21	[30,31,32]	42.32	42.88	−0.38
NGC 5548	8.03	43.39	38.70	[30,31]	42.77	43.46	−1.63
NGC 7314	6.70	42.28	36.71	[30,31]	41.84	42.24	-
NGC 7469	6.84	43.17	39.26	[30,31]	42.57	43.21	−0.49
PDS 456	8.91	44.77	40.57	[30,31]	44.38	45.59	−0.11
PG 0052+251	8.41	44.61	39.50	[30,31]	44.04	45.14	−0.83
PG 0804+761	8.24	44.46	39.40	[30,31]	43.94	45.01	−1.07
PG 0844+349	8.68	43.74	38.17	[30,31]	43.29	44.15	−0.77
PG 0947+396	8.68	44.37	39.25	[30,31]	43.84	44.88	−0.93
PG 0953+414	8.24	44.73	40.17	[30,31]	44.26	45.44	−0.05
PG 1048+342	8.37	44.00	37.57	[30,35]	43.46	44.38	−0.85
PG 1114+445	8.59	44.11	38.73	[30,41]	43.40	44.29	−0.86
PG 1115+407	7.67	43.93	38.98	[30,31]	43.57	44.52	-
PG 1202+281	8.61	44.43	38.56	[30,31]	43.83	44.87	−0.45
PG 1211+143	7.49	43.70	38.90	[30,31]	43.12	43.93	−0.56
PG 1216+069	9.20	44.72	40.84	[30,31]	44.09	45.21	−1.50
PG 1244+026	6.52	43.15	38.43	[30,31]	42.84	43.56	0.12
PG 1307+085	7.90	44.51	39.10	[5,30]	43.82	44.86	−1.18
PG 1322+659	8.28	44.02	38.88	[30,31]	43.59	44.55	−0.24
PG 1352+183	8.42	44.13	38.96	[30,31]	43.58	44.54	−0.30
PG 1402+261	7.94	44.15	39.56	[30,31]	43.76	44.77	0.26
PG 1415+451	8.01	43.60	38.82	[30,31]	43.10	43.90	−0.77
PG 1416-129	9.05	43.88	39.93	[30,31]	43.20	44.04	−1.45
PG 1427+480	8.09	44.20	38.14	[30,31]	43.67	44.65	−0.41
PG 1440+356	7.47	43.76	38.88	[30,31]	43.42	44.32	0.09
PG 1448+273	6.97	43.29	38.70	[30,31]	42.91	43.65	0.43
PG 1626+554	8.54	44.16	38.66	[30,31]	43.67	44.65	−0.66
RE J1034+396	6.18	42.59	39.29	[30,31,42]	42.27	42.82	0.16

). The $\Gamma$ and $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$ for AGNs show an anti-correlation when $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$ are less than a critical luminosity, and a positive correlation between $\Gamma$ and $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$ for AGNs when the Eddington ratios are higher than a critical luminosity, where the critical luminosity is approx $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$≈1%. To exclude radiatively inefficient sources as much as possible, we therefore only select sources with an Eddington ratio $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$ ≳ 1%. We gather from Ref. [27] a subsample of low-luminosity AGNs with measurements of their black hole mass, nuclear radio (5 GHz), and X-ray (2–10 keV) luminosities. The black hole masses in this subsample are mainly calculated based on the empirical $M_{BH}-{\sigma }_{*}$ relationship ([28,29] review the various ways to estimate black hole masses in detail); in addition, we use other methods to estimate the black hole mass $M_{\mathrm{BH}}$ to obtain more reliable black hole masses, as detailed in the references in

Table 2. Data of radiatively inefficient AGNs.

Name	${\mathit{M}}_{\mathbf{BH}}$ (${\mathit{M}}_{\odot }$)	${\mathit{L}}_{2-10\mathbf{keV}}$ (erg ${\mathbf{s}}^{-1}$)	${\mathit{L}}_{5\mathbf{GHz}}$ (erg ${\mathbf{s}}^{-1}$)	Refs.	${\mathit{L}}_{2\mathbf{keV}}$ (erg ${\mathbf{s}}^{-1}$)	${\mathit{L}}_{2500\mathring{\mathbf{A}}}$ (erg ${\mathbf{s}}^{-1}$)	$\frac{{\mathit{L}}_{2-10\mathbf{keV}}}{{\mathit{L}}_{\mathbf{Edd}}}$
NGC 266	8.37	40.88	37.95	[43]	40.30	40.15	−5.60
NGC 315	9.10	41.60	40.26	[5,36,44]	41.02	41.25	−5.61
NGC 404	5.16	37.02	33.50	[43]	36.44	34.23	−6.25
NGC 474	7.73	38.46	35.55	[45]	37.88	36.44	−7.38
NGC 524	8.94	38.86	37.03	[39,44,45]	38.28	37.05	−8.19
NGC 1167	7.88	39.07	37.59	[38,39,46]	38.49	37.38	−6.91
NGC 1275	8.64	43.32	41.03	[5,47,48]	42.74	43.90	−3.42
NGC 2273	7.30	40.87	37.00	[38,39,46]	40.29	40.14	−4.53
NGC 2768	7.94	39.46	37.39	[43]	38.88	37.97	−6.59
NGC 2787	8.14	38.79	37.01	[43]	38.21	36.95	−7.46
NGC 2841	8.31	38.26	36.00	[43]	37.68	36.13	−8.16
NGC 3031	7.73	39.38	36.03	[43]	38.80	37.85	−6.46
NGC 3079	7.65	39.98	38.17	[5]	39.40	38.77	−5.78
NGC 3147	8.29	41.87	37.91	[43]	41.29	41.67	−4.53
NGC 3169	8.01	41.05	37.19	[43]	40.47	40.41	−5.07
NGC 3226	8.06	40.57	37.01	[39,44,45]	39.99	39.67	−5.60
NGC 3227	7.59	41.78	37.73	[5]	41.20	41.53	−3.92
NGC 3245	8.21	39.29	36.98	[43]	38.71	37.71	−7.03
NGC 3379	8.18	37.53	35.73	[43]	36.95	35.01	−8.76
NGC 3414	8.67	39.92	36.65	[45]	39.34	38.68	−6.86
NGC 3607	8.40	38.63	37.01	[43]	38.05	36.70	−7.88
NGC 3608	8.67	38.20	35.90	[45]	37.62	36.04	−8.58
NGC 3627	7.24	37.60	36.11	[43]	37.02	35.12	−7.75
NGC 3628	7.24	38.24	36.13	[43]	37.66	36.10	−7.11
NGC 3675	7.11	37.86	35.86	[5]	37.28	35.52	−7.36
NGC 3718	8.14	40.62	36.91	[49,50,51]	40.04	39.76	−5.62
NGC 3941	7.37	39.27	35.61	[43]	38.69	37.68	−6.21
NGC 3998	8.93	41.44	37.85	[36,44,49]	40.86	41.01	−5.59
NGC 4138	7.19	40.11	36.13	[43]	39.53	38.97	−5.19
NGC 4143	8.16	39.83	36.98	[39,45,50]	39.26	38.55	−6.43
NGC 4168	8.13	38.89	37.20	[39,45,46]	38.31	37.09	−7.35
NGC 4203	7.79	40.60	37.11	[39,45,50]	40.02	39.72	−5.30
NGC 4216	8.09	38.91	36.58	[43]	38.33	37.13	−7.29
NGC 4258	7.57	40.89	35.78	[43]	40.31	40.17	−4.79
NGC 4261	8.72	41.09	39.22	[36,45,52]	40.51	40.48	−5.73
NGC 4278	8.61	39.25	38.17	[36,44,45]	38.67	37.65	−7.47
NGC 4321	6.80	38.84	36.42	[5]	38.26	37.03	−6.06
NGC 4374	8.96	39.32	38.33	[44,51,53]	38.74	37.75	−7.75
NGC 4395	5.04	39.71	35.77	[5]	39.13	38.36	−3.43
NGC 4450	7.40	40.02	36.78	[43]	39.44	38.83	−5.49
NGC 4457	6.86	39.69	35.42	[5]	39.11	38.33	−5.28
NGC 4459	7.82	38.87	36.09	[43]	38.29	37.07	−7.06
NGC 4472	9.40	39.70	36.43	[39,45,46]	39.12	38.35	−7.80
NGC 4477	7.89	39.60	35.64	[43]	39.02	38.35	−6.40
NGC 4486	9.48	40.67	39.90	[5]	40.09	39.82	−6.92
NGC 4494	7.65	39.45	36.24	[5]	38.87	37.96	−6.30
NGC 4501	7.79	38.89	36.28	[43]	38.31	37.10	−7.01
NGC 4548	7.08	39.74	36.55	[43]	39.16	38.40	−5.45
NGC 4552	8.92	38.24	37.26	[44,45,51]	37.66	36.10	−8.79
NGC 4565	7.41	39.92	36.81	[39,45,46]	39.34	38.68	−5.59
NGC 4579	7.78	41.40	37.78	[38,39,44]	40.82	40.94	−4.49
NGC 4589	8.54	38.90	37.36	[39,44,54]	38.32	37.12	−7.75
NGC 4636	8.14	39.38	36.76	[43]	38.80	37.85	−6.87
NGC 4639	6.77	40.18	35.40	[43]	39.60	39.08	−4.70
NGC 4698	7.57	38.69	35.59	[43]	38.11	36.79	−6.99
NGC 4725	7.49	39.45	36.19	[5]	38.87	37.96	−6.15
NGC 4736	7.05	38.49	35.67	[36,44,45]	37.91	36.48	−6.67
NGC 4762	7.63	38.26	36.58	[43]	37.68	36.13	−7.48
NGC 4772	7.57	39.75	36.38	[39,45,50]	39.17	38.42	−5.92
NGC 5033	7.60	40.70	36.94	[43]	40.12	39.87	−5.01
NGC 5194	6.95	38.95	36.59	[36,38]	38.37	37.19	−6.11
NGC 5363	8.57	39.78	37.68	[39,44,54]	39.20	38.46	−6.90
NGC 5813	8.75	38.79	37.49	[45]	38.21	36.95	−8.07
NGC 5838	9.06	38.97	36.50	[45]	38.39	37.22	−8.20
NGC 5846	8.43	40.83	36.60	[39,44,45]	40.26	40.08	−5.70
NGC 5866	7.81	37.99	36.77	[39,44,45]	37.41	35.72	−7.93
NGC 6500	8.28	40.18	38.97	[5]	39.60	39.07	−6.21
NGC 7626	8.71	40.97	38.48	[43]	40.39	40.29	−5.85
NGC 7743	6.47	39.71	36.99	[5]	39.13	38.35	−4.87

. The nuclear radio luminosities of a subsample of the radiatively inefficient AGNs are mainly from Very Large Arrey (VLA) or Very Long Baseline Array (VLBA) observations, and the X-ray luminosities are mainly from Chandra or XMM-Newton observations. We obtain the redshift of the radiatively inefficient AGNs from the NED, and calculate the luminosities distance of the AGNs based on our assumed cosmology parameters, which convert the radio fluxes and X-ray fluxes to luminosities. As listed in Table 2, our sample of LLAGNs includes 69 sources.

For the radiatively efficient sample, the AGNs photon index $\Gamma$ is taken from Ref. [31]. For the radiatively inefficient sample, we assume that the AGNs photon index $\Gamma$ = 1.8 (a typical value for the X-ray spectrum of AGNs). The 2–10 keV X-rays luminosities are converted to 2 keV X-rays luminosities by the value of AGNs photon index. In AGNs, a non-linear relation between X-ray and optical/UV emission has been found, usually parameterized as a dependence between the logarithm of the monochromatic luminosity at 2500 Å and 2 keV [24]. For the radiatively efficient AGNs, we used the relation:

$$log{L}_{2\mathrm{keV}}=(0.760\pm 0.022)log{L}_{2500\mathring{\mathrm{A}}}+(3.508\pm 0.641)$$

(3)

taken from Ref. [24]. For the radiatively efficient AGNs, we used the relation:

$$log{L}_{2\mathrm{keV}}=(0.652\pm 0.082)log{L}_{2500\mathring{\mathrm{A}}}+(6.269\pm 2.044)$$

(4)

taken from Ref. [25]. Then we plugged the 2 keV into Equations (3) and (4) to obtain the 2500 $\mathring{\mathrm{A}}$ luminosities for the AGNs.

Our radiatively efficient XRBs are the 134 bright hard-state XRBs compiled by the authors of Ref. [15], the main selection criteria of the subsample of XRBs are the positive $\Gamma -F_{\mathrm{X}}$ correlation and the $L_{\mathrm{bol}}/L_{\mathrm{Edd}}$≳ 1%, and follow the “outliers” track [55]. For the radiatively inefficient XRBs, we use the Ref. [43] subsample of black hole X-ray binaries, which includes six black hole X-ray binaries, namely GX 339-4, XTE J1118+480, V404 Cyg, XTE J1752-223, H1743-322, and A0620-00. These sources possess simultaneous or quasi-simultaneous radio and X-ray observations, and only sources with $L_{\mathrm{X}}/L_{\mathrm{Edd}}$≲${10}^{-3}$ can be selected for inclusion. In this paper, the photon index of the radiatively efficient XRBs is assumed to be 2.1. For the radiatively inefficient sample, we assume that the XRBs photon index $\Gamma$ = 1.8. The 2–10 keV X-rays luminosities are converted to 1 keV X-rays luminosities by assuming the value of the photon index.

In this work, our final sample used to make the fundamental plane analysis consists of the radiatively efficient and radiatively inefficient black hole sources that have available estimates for radio luminosity ($L_{\mathrm{R}}$), the peak luminosity of blackbody radiation ($L_{\mathrm{peak}}$), and black hole mass ($M_{\mathrm{BH}}$). We utilized the Eddington ratio ($L_{\mathrm{X}}/L_{\mathrm{Edd}}$) to divide radiatively efficient and radiatively inefficient sources. As shown in Figure 1, we show the distribution of the Eddington ratio ($L_{\mathrm{X}}/L_{\mathrm{Edd}}$) for the radiatively efficient and radiatively inefficient active galactic nuclei and black hole X-ray binaries, respectively. This includes the Eddington ratio for the sample of radiatively efficient AGNs with a median of −2.20, and for radiatively efficient XRBs with a median of −0.90, as well as the Eddington ratio for the sample of radiatively inefficient AGNs with a median of −6.40, and for radiatively inefficient XRBs with a median of −3.35. Our sample of RQ AGNs is limited to those with $L_{\mathrm{X}}/L_{\mathrm{Edd}}\sim {10}^{-1}$. This is because RQ AGNs with $L_{\mathrm{X}}/L_{\mathrm{Edd}}\gtrsim {10}^{-1}$ may have entered a soft state similar to the XRBs, when the radio emission is weak.

3. Results

We first analyze the correlation between the radio luminosities and the peak luminosities of the blackbody radiation of AGNs and XRBs (the peak luminosities of the blackbody radiation are about 2500 $\mathring{\mathrm{A}}$ luminosities for AGNs and 1 keV luminosities for XRBs). We use $L_{\mathrm{R}}/L_{\mathrm{Edd}}$ and $L_{\mathrm{peak}}/L_{\mathrm{Edd}}$ to examine the radio/peak correlation [45]. In this work, following the fundamental plane proposed in Ref. [5], we defined the fundamental plane of black hole activity as follows:

$$log{L}_{\mathrm{R}}={\xi }_{\mathrm{p}}log{L}_{\mathrm{peak}}+{\xi }_{\mathrm{M}}log{M}_{\mathrm{BH}}+b$$

(5)

where $L_{\mathrm{R}}$ is the 5 GHz radio luminosity, $L_{\mathrm{peak}}$ is the peak luminosity of the blackbody radiation, and $M_{\mathrm{BH}}$ is the black hole mass, the luminosity units are erg ${\mathrm{s}}^{-1}$ and the black hole mass units are $M_{\odot }$. We calculate the coefficients of Equation (5) using a method similar to Ref. [5] and minimize the following statistic, i.e.,

$${\chi }^2=\sum _i{\displaystyle \frac{{(log{L}_{5\mathrm{GHz}}-b-{\xi }_{\mathrm{p}}log{L}_{\mathrm{peak}}-{\xi }_{\mathrm{M}}log{M}_{\mathrm{BH}})}^2}{{\sigma }_R^2+{\xi }_p^2{\sigma }_P^2+{\xi }_M^2{\sigma }_M^2}}$$

(6)

where b is a constant, and the AGNs and XRBs adopt typical observational uncertainties, respectively.

3.1. Sample of Radiatively Efficient Black Hole Sources

For radiatively efficient black hole objects, as shown in Figure 2, which shows the $L_{\mathrm{R}}/L_{\mathrm{Edd}}-L_{\mathrm{peak}}/L_{\mathrm{Edd}}$ relationship, we find similar correlations between the radio luminosities and the peak luminosities of the blackbody radiation of AGNs and XRBs, and their best-fit results are as follows:

$$log\left({\displaystyle \frac{L_{5\mathrm{GHz}}}{L_{\mathrm{Edd}}}}\right)=(1.55\pm 0.13)log\left({\displaystyle \frac{L_{2500\mathring{\mathrm{A}}}}{L_{\mathrm{Edd}}}}\right)-(4.23\pm 0.26)$$

(7)

$$log\left({\displaystyle \frac{L_{5\mathrm{GHz}}}{L_{\mathrm{Edd}}}}\right)=(1.53\pm 0.03)log\left({\displaystyle \frac{L_{1\mathrm{keV}}}{L_{\mathrm{Edd}}}}\right)-(5.94\pm 0.04)$$

(8)

Equations (7) and (8) represent the best fit for the subsamples of the radiatively efficient AGNs and XRBs, respectively. The uncertainties in the radio luminosities, peak luminosities, and black hole masses of the AGNs adopt typical observational uncertainties, which are ${\sigma }_{L_R}$ = 0.2 dex [56], ${\sigma }_{L_{\mathrm{peak}}}$ = 0.3 dex [57], and ${\sigma }_{M_{\mathrm{BH}}}$ = 0.4 dex [58]; the typical observational uncertainties for XRBs are ${\sigma }_{L_R}$ = 0.1 dex, ${\sigma }_{L_{\mathrm{peak}}}$ = 0.1 dex [14,59], and ${\sigma }_{M_{\mathrm{BH}}}$ = 0.15 dex [60], where the uncertainties ${\sigma }_{L_{\mathrm{peak}}}$ of XRBs were inferred from the propagation of error method.

Based on the similarity between the radio luminosities and the peak luminosities of blackbody radiation, the best fits for the fundamental plane of AGNs and XRBs is as follows:

$$log{L}_{\mathrm{R}}=1.{57}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}-0.{37}_{-0.16}^{+0.16}log{M}_{\mathrm{BH}}-27.{73}_{-0.34}^{+0.34}$$

(9)

where $L_{\mathrm{peak}}$ is 2500 $\mathring{\mathrm{A}}$ luminosity for AGNs and 1 keV luminosity for XRBs, respectively, and a scatter of ${\sigma }_{\mathrm{R}}$ = 0.48 dex, the results are shown in Figure 3.

To reduce the additional scatter, we use a subsample of AGNs with a narrow range in black hole mass ($M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$), selected from Table 1, to examine the fundamental plane of the AGNs with the black hole mass $M_{\mathrm{BH}}\approx {10}^9$ ${\mathrm{M}}_{\odot }$ and XRBs. The results are shown in Figure 4, for the radiatively efficient sample, the best fits for the fundamental plane of the subsample of AGNs with the black hole mass $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$ and XRBs are as follows:

$$log{L}_{\mathrm{R}}=1.{55}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}-0.{34}_{-0.19}^{+0.18}log{M}_{\mathrm{BH}}-27.{25}_{-0.32}^{+0.32}$$

(10)

with a scatter of ${\sigma }_{\mathrm{R}}$ = 0.37 dex.The best fits for the fundamental plane of the subsample of AGNs with the black hole mass $M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and XRBs are as follows:

$$log{L}_{\mathrm{R}}=1.{55}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}-0.{30}_{-0.30}^{+0.30}log{M}_{\mathrm{BH}}-26.{96}_{-0.30}^{+0.30}$$

(11)

with a scatter of ${\sigma }_{\mathrm{R}}$ = 0.31 dex.

3.2. Sample of Radiatively Inefficient Black Hole Sources

For radiatively inefficient black hole objects, we use the same analysis, as shown in Figure 2, where the radio luminosities and the peak luminosities of the blackbody radiation of radiatively inefficient black hole objects have a similar correlation, with a best fit as follows:

$$log\left({\displaystyle \frac{L_{5\mathrm{GHz}}}{L_{\mathrm{Edd}}}}\right)=(0.48\pm 0.02)log\left({\displaystyle \frac{L_{2500\mathring{\mathrm{A}}}}{L_{\mathrm{Edd}}}}\right)-(5.41\pm 0.16)$$

(12)

$$log\left({\displaystyle \frac{L_{5\mathrm{GHz}}}{L_{\mathrm{Edd}}}}\right)=(0.53\pm 0.01)log\left({\displaystyle \frac{L_{1\mathrm{keV}}}{L_{\mathrm{Edd}}}}\right)-(7.79\pm 0.05)$$

(13)

Equations (12) and (13) represent the best fit for radiatively inefficient AGNs and XRBs subsamples, respectively. The uncertainties in the radio luminosities, peak luminosities, and black hole masses of the AGNs adopt typical observational uncertainties, which are ${\sigma }_{L_R}$ = 0.2 dex [56], ${\sigma }_{L_{\mathrm{peak}}}$ = 0.3 dex [57], and ${\sigma }_{M_{\mathrm{BH}}}$ = 0.4 dex [58]; the typical observational uncertainties for XRBs are ${\sigma }_{L_R}$ = 0.1 dex, ${\sigma }_{L_{\mathrm{peak}}}$ = 0.1 dex [14,59] and ${\sigma }_{M_{\mathrm{BH}}}$ = 0.15 dex [60], where the uncertainties ${\sigma }_{L_{\mathrm{peak}}}$ of XRBs were inferred from the propagation of error method. The best fits for the fundamental plane of radiatively inefficient black hole objects are as follows:

$$log{L}_{\mathrm{R}}=0.{45}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}+0.{94}_{-0.10}^{+0.12}log{M}_{\mathrm{BH}}+12.{58}_{-0.38}^{+0.38}$$

(14)

where $L_{peak}$ is 2500 $\mathring{\mathrm{A}}$ luminosity for AGNs and 1 keV luminosity for XRBs, respectively, and a scatter of ${\sigma }_{\mathrm{R}}$ = 0.63 dex; the results are shown in Figure 3.

Similarly, we use a subsample of AGNs with a narrow range in black hole mass ($M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$), selected from Table 2, to examine the fundamental plane of the AGNs with the black hole mass $M_{\mathrm{BH}}\approx {10}^9$ ${\mathrm{M}}_{\odot }$ and XRBs. The results are shown in Figure 4, for the radiatively inefficient sample, the best fits for the fundamental plane of the subsample of AGNs with the black hole mass $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$ and XRBs are as follows:

$$log{L}_{\mathrm{R}}=0.{48}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}+0.{91}_{-0.11}^{+0.13}log{M}_{\mathrm{BH}}+11.{39}_{-0.33}^{+0.33}$$

(15)

with a scatter of ${\sigma }_{\mathrm{R}}$ = 0.54 dex. The best fits for the fundamental plane of the subsample of AGNs with the black hole mass $M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and XRBs are as follows:

$$log{L}_{\mathrm{R}}=0.{50}_{-0.01}^{+0.01}log{L}_{\mathrm{peak}}+0.{91}_{-0.13}^{+0.15}log{M}_{\mathrm{BH}}+10.{66}_{-0.30}^{+0.30}$$

(16)

with a scatter of ${\sigma }_{\mathrm{R}}$ = 0.49 dex. The peak luminosity of the AGNs with black hole mass $M_{\mathrm{BH}}\approx {10}^9$ ${\mathrm{M}}_{\odot }$ is 2500 $\mathring{\mathrm{A}}$. It can be found that after using the subsamples of AGNs with a narrow range in black hole mass ($M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$), the fundamental plane becomes a little bit tighter, even though the slope of ${\xi }_{\mathrm{p}}$ is roughly unchanged.

4. Discussion

The fundamental plane of black hole activity provides a evidence on disk–jet coupling [5,15], and previous work has mainly investigated the fundamental plane of black hole activity using 5 GHz radio luminosities, 2–10 keV X-ray luminosities, and black hole masses. In this paper, we used the luminosity near the peak of blackbody radiation to replace the 2–10 keV X-ray luminosity; one of the main advantage of using the luminosity near the peak of blackbody radiation to replace the 2–10 keV X-ray luminosity is that 2500 $\mathring{\mathrm{A}}$ and 1 keV are in the same part of the blackbody radiation. We separately compiled radiatively efficient and radiatively inefficient black hole samples that contained the peak luminosity of blackbody radiation. We analyzed the radio–peak luminosity correlations and the fundamental plane for radiatively efficient and radiatively inefficient black hole objects, and we found that the radio–peak luminosity of AGNs and XRBs have a similar correlation, and that black hole objects with different radiative efficiencies have different fundamental planes. The reason for the difference may be that they correspond to different accretion modes. In this paper, we adopt the Eddington ratio to distinguish the radiatively efficient and radiatively inefficient sources, because the Eddington ratio to some extent reflects the accretion efficiency. So far, a growing number of BH sources have been found to follow a steeper radio/X-ray correlation, namely, $L_{\mathrm{R}}$∝$L_{\mathrm{X}}^{\sim 1.4}$, during $L_{\mathrm{X}}/L_{\mathrm{Edd}}\gtrsim {10}^{-3}$. The fundamental plane of black hole activity may depend on Eddington ratio; the authors of Ref. [61] found that Eddington ratio dependence for the fundamental plane, and the samples with different Eddington ratios follow different fundamental planes for both radio-quiet AGNs and radio-loud AGNs.

We further selected two subsamples of AGNs with black hole mass in the narrow range ($M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$) to study the fundamental plane of black hole objects at different scales. As shown in Figure 3 and Figure 4, In the radiatively efficient sample, the fitted slope is ${\xi }_{\mathrm{p}}=1.57$ and the scatter is ${\sigma }_{\mathrm{R}}=0.48$ dex, and the fitted slopes for the fundamental plane of the subsample of AGNs (black hole mass $M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$, respectively) and XRBs are ${\xi }_{\mathrm{p}}=1.55$ and ${\xi }_{\mathrm{p}}=1.55$, respectively, and the scatter is ${\sigma }_{\mathrm{R}}=0.31$ dex and ${\sigma }_{\mathrm{R}}=0.37$ dex. In the radiatively inefficient sample, the fitted slope is ${\xi }_{\mathrm{p}}=0.45$ and the scatter is ${\sigma }_{\mathrm{R}}=0.63$ dex, and the fitted slopes for the fundamental plane of the subsample of AGNs (black hole mass $M_{\mathrm{BH}}={10}^{9\pm 0.4} M_{\odot }$ and $M_{\mathrm{BH}}={10}^{9\pm 1} M_{\odot }$, respectively) and XRBs are ${\xi }_{\mathrm{p}}=0.50$ and ${\xi }_{\mathrm{p}}=0.48$, respectively, and the scatter is ${\sigma }_{\mathrm{R}}=0.49$ dex and ${\sigma }_{\mathrm{R}}=0.54$ dex. We found that after using subsamples of AGNs with a narrow range in black hole mass, the fundamental plane becomes a little bit tighter, even though the slope of ${\xi }_{\mathrm{p}}$ is roughly unchanged.

The authors of Ref. [62] found that LLAGNs and low-/hard-state XRBs follow the fundamental plane most tightly, and that Equation (1) should be most suitable for radiatively inefficient accretion model. The authors of Ref. [15] compiled a sample of radiatively efficient black hole sources, consisting of XRB outliers and bright radio-quiet AGNs, and proposed a new fundamental plane of radiatively efficient BH sources, with ${\xi }_{\mathrm{X}}=1.59$. In this work, we use the luminosity near the peak of the blackbody radiation of the AGNs and XRBs to replace the 2–10 keV luminosity. In the radiatively efficient sample, the slope of ${\xi }_{\mathrm{p}}$ (see Figure 3 and Figure 4) is similar to that of [15], i.e., our work confirms the discovery of [15]. Currently, it is widely recognized that black hole spin may have an impact on the fundamental plane. While we still believe that the accretion disk have a significant effect on the jet power, the spin itself will enter the radio luminosity as a sensitive parameter, introducing scatter in correlation [5]. Therefore, the study of black hole spins may be able to reduce the scatter and obtain a better fit. This will be a focus of our future work.

In the radiatively efficient sample, we find that the radio–peak luminosities of the AGNs and XRBs present a similar correlation slope, and this similar correlation is also present in the radiatively inefficient sample. This is important observational evidence for the coupling of an accretion disk and a jet, and the radiatively efficient and radiatively inefficient samples are found to follow different fundamental planes, which may correspond to different accretion physics for the central engine. The mass accretion rate is an important influence parameter for peak and radio luminosity. Through the mass accretion rate, we may be able to bridge both the radiatively efficient and radiatively inefficient fundamental planes. Therefore, exploring the correlations between peak and radio luminosities and mass accretion rates, and thus constructing a unified fundamental plane, will be a focus of our future work. The radio–peak luminosity correlations and the fundamental plane of AGNs and XRBs further suggest that black hole objects at different scales have similar central engines, and that AGNs and XRBs are likely to have similar accretion and jet mechanisms.