1 Introduction

Active Galactic Nuclei (AGNs) have some special observation properties, such as core-dominated non-thermal continuum, superluminal motions, high and variable luminosity, high and variable polarization, γ-ray emissions and so on^[1-4]. Their properties are explained using a jet model with jet pointing close to the line of sight. As the most powerful subclass of AGNs, blazars can be divided into flat spectrum radio quasars (FSRQs) with strong emission lines and BL Lacertae objects (BL Lacs) with weak emission lines or no emission lines at all. From the Fermi detected blazars, Ghisellini & Tavecchio^[5] proposed that the BL Lacs and FSRQs can be separated in the plot of γ-ray photon index from the γ-ray luminosity^[6-8]. Ghisellini, et al.^[9] also pointed out that BL Lacs and FSRQs can be classified based on the luminosity of the broad line region (L_BLR) measured in Eddington, and the dividing value is about L_BLR/L_Edd~5 × 10^-4, which was confirmed by Sbarrato et al.^[10], see also Yang et al. ^[11-12].

In the theoretical models of jet formation a correlation between jet and accretion process is expected. The power converted into the kinetic power of jet is produced from a spinning black hole which can release its rotational energy^[13], or produced from accretion process and disc^[14]. In order to understand these relationships, a key way is to search for the connection between accretion process and jet, which has been studied in the literatures^{[6, 10, 15-19]}.The accretion disc produces radiation to ionize the surrounding clouds, and further to form the broad emission lines, so the accretion disc luminosity has a certain connection with the broad line region luminosity. In this case, we can use the broad line region luminosity (L_BLR) to measure the accretion disc luminosity (L_d), so L_BLR can be a proxy for the invisible accretion disk luminosity (L_d)^[10].

For a sample of 159 steep-and flat-spectrum quasars, Serjeant et al.^[20] proposed that the radio luminosity L_R is a good agent for the jet. On the other hand, the γ-ray luminosity (L_γ) can represent for bolometric luminosity since that the γ-ray luminosity dominates the bolometric luminosity in Fermi blazars^{[9-10, 21-24]}, so the γ-ray luminosity (L_γ) is an agent for the jet.

Therefore, (L_BLR) is an proxy for accretion disc luminosity while L_R and L_γ are proxies for the jet. In order to explore the connection between jet and accretion radiation, a feasible way is to explore the relationship between radio (or γ-ray) luminosity and broad-line luminosity. In 1999, Cao and Jiang^[17] collected a sample of 198 radio-loud quasars, estimated the total broad-line flux, and obtained a significant correlation between radio and broad-line emission. If the redshift was limited to the range 0.5 < z < 1.5, a correlation with a significant level of 99.93% was obtained between the radio and broad-line fluxes, while a significant level of 99.7% was obtained between the radio and broad-line luminosities. One should note that the effect of the synchrotron self-absorption in blazars can lead to measured radio fluxes lower than the intrinsic radio fluxes, at least at lower frequencies, thus underestimating the radio luminosity. On the other hand, the γ-ray luminosity (L_γ) can represent for bolometric luminosity since that the γ-ray emissions dominate the bolometric luminosity in γ-ray loud blazars^{[9-10, 21]}, so the γ-ray luminosity (L_γ) is a good proxy for the jet.

Therefore, L_BLR is a proxy for accretion disc luminosity while L_γ is a proxy for the jet luminosity. In order to explore the connection between jet and accretion process, a feasible way is to explore the relationship between γ-ray luminosity and broad-line luminosity. Sbarrato et al.^[10] collected 78 blazars (with measured L_BLR, L_γ and black hole mass) detected by the Fermi/LAT and presented in SDSS (Sloan Digital Sky Survey), searched the logarithmic relationship between the L_γ and L_BLR, and found that log L_γ correlates well with log L_BLR. In 2014, Ghisellini et al^[15]. complied a large sample of γ-ray detected sources with measured broad emission lines, and found a correlation between jet power as measured through the γ-ray luminosity, and accretion luminosity as measured by the broad emission lines, and that the jet power dominate over the disk luminosity.

Since there is a correlation between luminosity and redshift, then, even if there is no correlation existing in intrinsic luminosity-luminosity, a correlation still presents itself on observed luminosity-luminosity^[25]. Feigelson and Berg^[26] obtained the correlations on mutual bands, and came to another conclusion that if there is no correlation between the luminosity-luminosity, nor is it in the flux-flux densities. Many works have been done to study the relationship between the γ-ray emission and other monochromatic emission of AGNs^{[2, 22, 27-30]}. Some of the monochromatic emissions in blazars are strongly beamed. When the jet direction is close to the line of sight at rest frame, the luminosity will be enhanced by the beaming effect, which presents as a beaming factor (or a Doppler factor). Therefore the luminosity-luminosity correlations will also be influenced by the correlations between the luminosity and the beaming factor. In this sense, we should consider the effect of the beaming effect when we investigate the luminosity-luminosity correlations.

In the present paper, we compile a large sample of 202 Fermi blazars with available redshift, radio, γ-ray, and broad-line emissions to study the correlation between jet and accretion disk process. The paper is arranged as follows: in section 2, we give the sample and some results; In section 3, some discussions and conclusions are given. The cosmological parameters H₀ = 70 km·s^-1·Mpc^-1, Ω_m = 0.27 and Ω_Λ = 0.73 have been adopted in this work.

2 Samples and Results 2.1 Samples

In order to investigate the connection between the accretion process and jet, we collected the broad-line data for Fermi blazars from the literatures^{[10, 16-18, 31-33]}, and the radio and γ-ray data from Fan et al.^[2]. The total broad-line luminosity can be calculated by the total observed luminosities from all broad lines^[16], for some original data with flux density but no luminosity of broad-line emission, the broad-line flux density (f_BLR) are converted into luminosity (L_BLR): L_BLR=4πνd_L² f_BLR, considering K-correction for broad-line flux density: L_BLR=4πνd_L² f_BLR /(1 + z), where ν is the frequency, d_L is the luminosity distance, z is the redshift.

Therefore, we obtained a sample of 202 blazars with available monochromatic radio, γ-ray emissions and broad-line data, and listed them in Table 1. In Table 1,

col. 1 gives the name of the source;

col. 2 the redshift;

col. 3 the classification,

   F stands for FSRQs,       IB for ISP-BLs,

   B for BL Lacs,       LB for LSP-BLs,

   U for unclassified sources;       HB for HSP-BLs,

   FL stands for LSP-FSRQs,       UI for ISP-unclassified sources,

   FI for ISP-FSRQs,       UL for LSP-unclassified sources;

col. 4 the logarithm of radio luminosity (log L_R) in units of erg·s^-1;

col. 5 the uncertainty for log L_R;

col. 6 the logarithm of the γ-ray luminosity (log L_γ) in units of erg·s^-1;

col. 7 the uncertainty for log L_γ;

col. 8 the references for data in col. 4 - col. 7;

col. 9 the logarithm of the broad line luminosity (log L_BLR) in units of erg·s^-1;

col. 10 references for col. 9,

   C97: CELOTTI et al.(1997)^[16],       X12: XIE et al.(2012)^[32],

   C99: CAO et al.(1999)^[17],       X14: XIONG et al.(2014)^[18],

   S12: SBARRATO et al.(2012)^[10],       X16: XUE et al.(2016)^[33];

col. 11 Doppler factor, δ;

col. 12 references for Doppler factor,

   F09: FAN et al. (2009)^[30],       H09: HOVATTA et al. (2009)^[34],

   LV99: LÄHTEENIMÄKI & VALTAOJA (1999)^[35],

   L17: LIODAKIS et al. (2017)^[36],       S10: SAVOLAINEN et al. (2010)^[37].

For the 202 blazars, 165 are FSRQs, 35 are BL Lacs, and 2 are unclassified blazars. The redshift (z) is in a range from 0.031 to 3.104, the radio luminosity (log L_R) is from 40.18 erg·s^-1 to 44.71 erg·s^-1, γ-ray luminosity (log L_γ) is from 43.39 erg·s^-1 to 48.01 erg·s^-1, and the broad-line luminosity (log L_BLR) is from 41.70 erg·s^-1 to 46.65 erg·s^-1. We also obtained Doppler factors for 66 sources from the papers by FAN et al. (2009)^[30], HOVATTA et al. (2009)^[34], LÄHTEENIMÄKI & VALTAOJA (1999)^[35], LIODAKIS et al. (2017)^[36], and SAVOLAINEN et al. (2010)^[37].

2.2 Results

For the relevant data in Table 1, a linear least regression is applied to the luminosity-luminosity correlation, following results are obtained:

$ \log {L_{\rm{ \mathsf{ γ} }}} = (0.79 \pm 0.05)\log {L_{{\rm{BLR}}}} + (11.21 \pm 2.30) $

(1)

with a correlation coefficient r = 0.73 and a chance probability p = 2.43 × 10^-35, and

$ \log {L_{\rm{R}}} = (0.78 \pm 0.05)\log {L_{{\rm{BLR}}}} + (8.57 \pm 2.10) $

(2)

with r = 0.76 and p = 4.52 × 10^-39 for the 202 blazars.

For the subclasses, we have

$ \log {L_{\rm{ \mathsf{ γ} }}} = (0.72 \pm 0.08)\log {L_{{\rm{BLR}}}} + (14.28 \pm 3.54) $

(3)

with r = 0.58 and p = 4.25 × 10^-16, and

$ \log {L_{\rm{R}}} = (0.79 \pm 0.07)\log {L_{{\rm{BLR}}}} + (8.33 \pm 3.02) $

(4)

with r = 0.67 and p = 4.39 × 10^-23 for the 165 FSRQs; and

$ \log {L_{\rm{ \mathsf{ γ} }}} = (1.07 \pm 0.12)\log {L_{{\rm{BLR}}}} - (1.23 \pm 5.10) $

(5)

with r = 0.84 and p = 9.04 × 10^-11, and

$ \log {L_{\rm{R}}} = (1.06 \pm 0.14)\log {L_{{\rm{BLR}}}} - (3.25 \pm 5.91) $

(6)

with r = 0.79 and p = 4.51 × 10^-9 for the 35 BL Lacs. They are all shown Fig. 1 and Fig. 2.

3 Discussions and Conclusions

In the theoretical models of jet formation, if the squared magnetic field is proportional to the accretion rate, a correlation between jet and accretion process is expected^[15], the power converted into the kinetic power of jet is produced from a spinning black hole which can release its rotational energy^[13], or produced from accretion process and disc^[14].

The broad-line region is photoionized by radiation from the disc, so the broad-line emission can be taken as a proxy of the accretion power of the source^[16], and can be expressed as L_disk=L_BLR/φ, where φ~0.1^[15]. For a simple one-zone leptonic model, the power that the jet spends in producing the non-thermal radiation is ${\mathit{P}_{{\rm{rad}}}} = 2f\frac{{L_{{\rm{jet}}}^{{\rm{bol}}}}}{{{\mathit{\Gamma }^2}}}$, where L_bol jet is bolometric jet luminosity, Γ is the bulk Lorentz factor of the outflowing plasma, the factor "2" accounts for two jets, and f is a factor of order unity. The power in radiation P_rad is believed to be about 10% of the jet power P_jet, namely P_jet = 10P_rad^[15]. So, we can use the bolometric jet luminosity as the proxy of the jet power and the broad emission line luminosity as the proxy of the accretion power.

Based on the detections by EGRET and Fermi/LAT, a series of studies have shown that there is a strong correlation between the γ-ray and radio emission^{[2, 7, 21, 38-43]}. Those correlations suggest that the γ-ray emission has a beaming effect. We also find that the γ-ray luminosity and other monochromatic luminosities are strongly correlated with Doppler factor^[23]. The beaming factor (Doppler factor) is an important parameter for blazars, but it is difficult to determine. Doppler factors can be obtained from a synchrotron self-Compton mechanism^[44], from the radio variability^{[30, 34-37]}, or by model fitting the SED of the sources^[45-46].

People think that the broad-line emission is taken as a good proxy for accretion disc. However, the radio emission is believed from the beamed jet, and it is also found that the γ-ray dominates the bolometric luminosity in Fermi blazars^[15]. In this sense, the radio and γ-ray emissions are good proxies for the jet emission while the broad-line emission is a good proxy for accretion process. The correlation between radio and broad-line emission was studied in some literatures. Many authors think that the correlation between the beamed emission and the broad-line emission is due to the close link between the relativistic jet and accretion disk^{[10, 16-18]}. Cao and Jiang^[17] found a correlation between radio and broad-line flux densities for 198 radio-loud quasars. Sbarrato et al.^[10] obtained a strong correlation between the γ-ray and broad-line luminosity. Fan^[47] also investigated such a relationship using the EGRET data and broad-line emission.

In our present work, we compile a large sample with Fermi/LAT detections. Our results show a close correlation for logL_γ-logL_BLR with a coefficient r = 0.73 and a chance probability p = 2.43 × 10^-35, and also a close correlation for logL_R-logL_BLR with r = 0.76 and p = 4.52 × 10^-39 for the whole sample. Our result is consistent with those literatures^{[6, 10, 15]} for the γ-ray and broad-line emissions. However, the sample from Cao and Jiang^[17] was restricted to the radio flux at 5GHz and the total broad-line flux.

3.1 Redshift Effect

Redshift effect is an important factor influencing the luminosity-luminosity correlation. Considering the cross-correlation in luminosity-luminosity, it is necessary to remove redshift effect. It can be done using a formula: ${r_{ij,z}} = \frac{{{r_{ij}} - {r_{iz}}{r_{jz}}}}{{\sqrt {(1 - r_{iz}^2)\;(1 - r_{jz}^2)} }}$, here, r_ij is the correlation coefficient in luminosity-luminosity, r_iz (or r_jz) is the correlation coefficient between redshift and luminosity and r_{ij, z} is the correlation coefficient in luminosity-luminosity after removing the redshift effect.

From our sample, we have got the correlation coefficients and listed them in Table 2 for the whole and subclasses, the coefficients after removing the redshift effect are: ${r_{{L_{\rm{ \mathsf{ γ} }}}{L_{\rm{B}}},{\rm{z}}}} = \frac{{{r_{{L_{\rm{R}}}{L_{\rm{B}}}}} - {r_L}_{_{{\rm{Rz}}}}{r_L}_{_{{\rm{Bz}}}}}}{{\sqrt {(1 - r_{{L_{{\rm{Rz}}}}}^2)\;(1 - r_{{L_{{\rm{Bz}}}}}^2)} }} = 0.28$ with p = 5.77 × 10^-5 for logL_γ- logL_BLR, and r_{LRLB, z}=0.45 with p = 1.33 × 10^-10 for logL_R-logL_BLR for the whole sample. For FSRQ subclass, we have r_{LγLB, z}=0.25 with p = 1.5 × 10^-3 and r_{LRLB, z}=0.49 with p = 3.36 × 10^-10 for the 165 FSRQs. For BL Lac subclass, we obtain r_{LγLB, z}=0.55 with p = 1.1 × 10^-3 and r_{LRLB, z}=0.47 with p = 5.9 × 10^-3 for the 35 BL Lacs.

So, after removing the redshift effect, we can see that there are still significant correlations in logL_γ-logL_BLR with p = 5.77 × 10^-5 and in logL_R-logL_BLR with p = 1.33 × 10^-10 for the whole sample. The correlations also exist for the 165 FSRQs and the 35 BL Lacs.

3.2 Beaming Effect

Beaming effect is important for blazars, and included in the explanations of their observation properties. The emission in different energy bands is produced by different mechanisms with the radio emissions being from synchrotron emission, while the high energetic γ-rays from a synchrotron-self Compton or external Compton. Therefore, the emission at different band has a different dependence on the Doppler factor. In the correlation analysis of γ-ray and other low energy bands, the γ-ray vs radio relation is found to be very strong in many literatures^{[2, 21, 23, 38-39, 41]}. The strong γ-ray and radio correlation is perhaps from the fact that γ-ray is from an SSC mechanism or that the γ-ray and radio emission are strongly beamed with the same Doppler factor^[30]. So, the Doppler factor estimated from the radio variability is adopted in the discussions of beaming effect in the γ-rays.

In this work, Doppler factors from radio variability are available for 66 sources, and it is interesting to find that there is a correlation between the broad line luminosity and Doppler factor, see Fig. 3. Therefore, it is necessary that the Doppler factor dependence of the monochromatic luminosity and broad-line luminosity should be considered when we investigate the luminosity correlations.

For the 66 sources, we have r = 0.73 and p = 4.60 × 10^-12 for logL_γ-logL_BLR, r = 0.75 and p = 3.89 × 10^-13 for logL_R-logL_BLR. The correlation coefficients are r = 0.59 (p = 2.01 × 10^-7), 0.39 (p = 1.24 × 10^-3) and 0.42 (p = 5.15 × 10^-4) for the correlations between γ-ray luminosity and Doppler factor, between the radio luminosity and Doppler factor, and between broad-line luminosity and Doppler factor. The observed radio and γ-ray luminosities are boosted in the jet, therefore they are correlated with Doppler factor. For the correlation between broad-line emission and Doppler factor, It is possible that some broad-line region enter the jet so that the emissions are boosted or the radiation from broad-line emission is produced by the plasma bubbles in relativistic motion, which show a hint that the broad-line emission clouds have relativistic motion in the direction of sight line, then there is an effect of jet on the broad-line clouds. So, it is not difficult to understand that broad-line vs Doppler factor correlation.

The correlation coefficients between the monoch-romatic luminosity and broad-line luminosity after removing the Doppler factors are: r_{LγLB, δ} = 0.66 with p = 9.86 × 10^-8 for logL_γ- logL_BLR, and r_{LRLB, δ} = 0.70 with p = 1.12 × 10^-8 for logL_R-logL_BLR. We can see clearly that the correlation between the γ-ray luminosity (or radio luminosity) and broad-line luminosity exists even after the Doppler factor effect is removed.

Therefore, there is really a connection between the accretion process and jet even when the redshift and Doppler boosting effects are removed. In this sense, the connection between the accretion process and jet is robust.

4 Conclusions

In this work, we compile a sample of 202 Fermi detected blazars with available radio and emission line data. Out of them, 66 sources have Doppler factors. The correlations between the γ-ray luminosity (radio luminosity) and broad-line luminosity are discussed for the whole sample and the subclasses. We also consider both redshift effect and beaming effect in our discussions. Following conclusions are reached:

(1) Strong correlations between the γ-ray (and radio) luminosity and broad-line luminosity are obtained for the whole and the subclass (FSRQs and BL Lacs) samples. Those correlations exist for the whole sample and FSRQs/BL Lac sub-samples when redshift effect is removed.

(2) The broad-line luminosity, the γ-ray, and the radio luminosities are all correlated with Doppler factor.

(3) The correlation between the γ-ray (and radio) luminosity and broad-line luminosity exists after the beaming effect /(redshift effect) is removed.

(4) There is a real connection between the accretion process and jet.