Is radio jet power linearly proportional to the product of central black hole mass and Eddington ratio in AGN?

A model for the relation between radio jet power and the product of central black hole (BH) mass and Eddington ratio of AGN is proposed, and the model is examined with data from the literature. We find that radio jet power positively correlates but not linearly with the product of BH mass ($m$ in solar mass) and Eddington ratio ($\lambda$), and the power law indices ($\mu$) are significantly less than unity for relatively low accretion ($\lambda<0.1$) AGN, $P_{j}\propto (\lambda m)^{\mu}$, in the radio galaxies and the Seyfert galaxies. This leads to a negative correlation between radio loudness and $\lambda m$ for the low luminosity AGN, i.e. $R\propto (\lambda m)^{\rho}$ with $\rho=(7/6)\mu-1<0$, which may be attributed to a contribution of BH spin to total jet power assuming that the spin induced jet is gradually suppressed as the accretion rate increases. Whereas, for the high-z quasars which often show the slope $\mu\geq1$, a positive correlation between the radio loudness and disc luminosity is predicted. We discuss that the jet powers of the high-z FRII quasars are likely dominated by the accretion disc rather than by the BH spin.

Introduction
============

It is generally accepted that active galactic nuclei (AGN) harbor massive black holes (BHs). The black holes have only three physical parameters: mass, spin and net charges, the net charges are often considered to be zero. The mass (and its time derivative) and spin (and its time derivative) of BHs are crucial for understanding the AGN phenomena; most models and simulations suggest that it is the mass accretion and/or spin of black holes that produce the jets of AGN, e.g. see @bla77 model for the BH spin induced jet, @bla82 model for the disc accretion induced jet, and see recent reviews for the simulations of accretion disc [@fra13] and for hot accretion flows [@yuan14]. The co-evolution of central BH and its host AGN makes it possible for measuring the BH mass via various ways, and the mass accretion rate can be estimated through the measurements of Eddington ratio. While BH spins are still difficult to measure, although there is an explosion of BH spin measurements in recent years, only some 19 AGN may have BH spin measurements [@rey13].

About 10% of quasars are radio loud [@kal12], and many more are in radio weak/quiet state, while a higher radio loud fraction was found in low luminosity AGN [@ho02]. It is not well understood how a radio jet is related to the central BH mass, spin, and accretion rate (or Eddington ratio) in AGN. Observationally, radio jet powers are not well correlated with BH masses [@ho02], whereas there are several findings that radio luminosity correlates with the narrow emission line luminosity or the bolometric luminosity which assumed to be proportional to disc accretion rate [@raw91; @wil99; @cao04; @van13]. In recent development of the @bla77 mechanism, @tch11 and @mck12 demonstrate that the magnetically arrested flows can extract BH spin energy efficiently. A jet-spin correlation has been suggested in X-ray binaries [@ste13], but there are some debates on the correlation [@rus13]. Furthermore, an anti-correlation between the radio loudness and Eddington ratio is found in relatively low luminosity AGN [@ho02; @sik07; @sik13], which is probably due to the BH spin [@sik13; @sikb13]. It is likely that both the disc accretion and BH spin may contribute to the jet power of AGN. In this paper, we propose a simple model between radio jet power and disc accretion, and reanalyze the data from the literature to examine the relation and discuss the result.

Relation of radio jet power to BH mass and Eddington ratio
==========================================================

In the Newtonian approximation – this is suitable to a distance beyond a few Schwarzschild radii of BH [@mei12], where a jet may be formed from the disc accretion of AGN if not to consider a contribution from the BH spin, the binding energy of unit mass in a Keplerian orbit is $GM/2r$, where $G$ is the gravitational constant. For an accreting mass of $\Delta{M}$ in time of $\Delta{t}$, the binding energy $E$ per unit time is

$$E=\Delta{M}/\Delta{t}\times(GM/2r)=G \dot{M} M/2r.$$

Where $\dot{M}$ is the accretion rate defined by accreted mass $\Delta{M}$ per unit time. For the radiative efficiency of $\varepsilon$ in an accretion disc, the disc luminosity:

$$L_{disc}=\varepsilon G \dot{M} M/2r.$$

Similarly assuming radio jet power is also proportional to the binding energy with a jet efficiency $\eta$, we will have the radio jet power $P_{j}$:

$$P_{j}=\eta G \dot{M} M/2r.$$

Considering the ratio of disc luminosity to the Eddington luminosity:

$$\frac{L_{disc}}{L_{Edd}}=\frac{\varepsilon G \dot{M} M/2r}{4\pi G
M m_{p} c/\sigma_{T}}.$$

We have

$$L_{disc}/L_{Edd}=5.28\times 10^{-12} \varepsilon \dot{M}/r.$$

Taking this relation into equation (3), $r$ is cancelled, and we have

$$P_{j}=1.26\times10^{38} (\eta /\varepsilon) [(L_{disc}/L_{Edd})
M/M_{\odot}] (erg/s).$$

This relation shows that the jet power is the disc accretion dominated, which is expected to be linearly proportional to the product of the Eddington ratio ($\lambda=L_{disc}/L_{Edd}$) and the BH mass ($M$) of AGN, with the coefficient of $\eta$/$\varepsilon$. The jet power depends not only on the Eddington ratio but also BH mass in this relation. It is equivalent to the relation (from equations 2 and 3):

$$P_{j}=(\eta /\varepsilon)L_{disc}.$$

Statistical analysis of the jet power, BH mass and Eddington ratio
==================================================================

To test the formula (6), we searched for suitable AGN samples which have information on radio jet luminosity, BH mass and Eddington ratio (or disc luminosity). Firstly, we study from the @sik13 sample, the sample consists of 404 narrow line radio galaxies (NLRGs) consisting of FRIs and FRIIs, double-double radio lobes, X-shaped lobes, and one side lobes (for details, see @sik13), and the sample is limited to redshift $<$0.4. We use only the data of FRIs and FRIIs, they are the majority of the sample. The 1.4 GHz radio luminosity computed from 1.4 GHz fluxes in the NVSS catalog, BH mass and $H_{\alpha}$ (also $[O_{III}]$) line luminosity are available in the Sikora sample, where the black hole masses were estimated from the observed stellar velocity dispersion given in the SDSS using the relation by @tre02 with typical error of log$M$ less than 0.3 dex. The radio jet power $P_{j}$ can be estimated from low frequency (151 MHz) radio luminosity with the minimal energy argument for synchrotron emission [@wil99], assuming that the jet output results in energy stored in radio source lobes together with associated work done on the source environment. We convert the jet power formula of [@wil99] from 151 MHz to 1.4 GHz assuming source spectral indices $S_{\nu}\propto \nu^{-0.8}$, and we have

$$P_{j}=2.32\times10^{20} (f/3)^{3/2} (P_{1.4})^{6/7}[W/Hz] (erg/s)$$

The $f$ (in range 1-20) represents several uncertainties associated with estimating $P_{j}$ from 151 MHz luminosity [@wil99]. We here use the median value $f=10$ [@blu00] and use the equation (8) for that we want to study the power law slope between $P_{j}$ and disc luminosity through $P_{1.4}$. @sik13 used a simplified formula $P_{j} \propto P_{1.4}$, that overestimates a jet power by more than a factor of 10 than that of equation (8). Where the $P_{1.4}$ is the 1.4 GHz luminosity, the radio jet power $P_{j}$ is generally less than the bolometric disc luminosity in our case. The bolometric luminosity $L_{bol}$ estimated with the $H_{\alpha}$ line luminosity, the same as in @sik13, see also @net09:

$$L_{disc}=L_{bol} \simeq 2000 L_{line} (erg/s)$$

The Eddington ratio is computed from $L_{disc}/L_{Edd}$, and the Eddington luminosity $L_{Edd}$ depends only on BH mass.

The radio loudness parameter $R$ is defined by the 1.4 GHz radio luminosity over the $H_{\alpha}$ line luminosity, i.e. $R=P_{1.4}/L_{H_{\alpha}}$.

To test and fit the equations (6)-(7), we rewrite the equation (6)-(7) as a power law form $P_{j}=(\eta^{'}
/\varepsilon^{'})L_{disc}^{\mu}$, i.e.:

$$log(P_{j})=b + \mu \times log(\lambda M/M_{\odot}),$$

where $b=log [1.26\times10^{38} (\eta^{'} /\varepsilon^{'})]$, the ratio $\eta^{'}/\varepsilon^{'}$ is a coefficient for the power law form and it returns to $\eta/\varepsilon$ when $\mu=1$.

For the anti-correlations between the radio loudness and disc luminosity (or Eddington ratio) found by @ho02, @sik07 [@sik13], it is reasonable to assume a power law relation, i.e: $R=\xi L_{disc}^{\rho}$, and rewrite it with the form ($m\equiv M/M_{\odot}$):

$$log(R)=c + \rho \times log(\lambda m)$$

Then, we plot the radio power $P_{j}$ versus the production of BH mass and Eddington ratio for the sample, as well as the radio loudness $R$ versus $\lambda m$, and fit the data with the equation (10),(11) respectively, as shown in Fig. \[fig1\] and Fig. \[fig2\]. The linear regression fitted parameters are listed in Table \[tab1\], as well as the correlation coefficient and the null hypothesis probability in Table \[tab2\].

The result shows that the radio power of the NLRGs (FRIs+FRIIs) positively correlates with the $\lambda m$, with the power law index of 0.52$\pm$0.06 with high correlation coefficient and confidence level in Table \[tab2\], it is not a linear proportionality as one might expect from equations (6)-(7). The radio loudness is anti-correlated with the $\lambda m$ with the power law index of -0.40$\pm$0.08. It shows that the jet powers have no correlation with the BH masses (Fig. \[fig3\]). If we fit separately for FRIIs and FRIs in the sample, the slope $\mu$ is 0.52$\pm$0.08 and 0.49$\pm$0.18 for the FRIIs and FRIs respectively.

![Log\[jet power\] vs. log\[$\lambda m$\] for the narrow line radio galaxies (FRIs+FRIIs), with the best linear fit to the sample, the dash line is fitted with the fixed slope=1. The top axis is the disc luminosity, the right axis is the 1.4 GHz luminosity, and the Eddington ratio is approximately marked in the inside X-axis. <span data-label="fig1"></span>](fig1.eps){width="8cm"}

![Log\[radio loudness\] vs. log\[$\lambda m$\] for the narrow line radio galaxies (FRIs+FRIIs), with the best linear fit to the sample, the dash line is fitted with the fixed slope=0.<span data-label="fig2"></span>](fig2.eps){width="8cm"}

![Log\[jet power\] vs. log\[$m$\] for the narrow line radio galaxies (FRIs+FRIIs), $m$ is black hole mass in solar mass unit.<span data-label="fig3"></span>](fig3.eps){width="8cm"}

We also study the @sik07 sample, that consists of five sub-samples including broad-line radio galaxies (BLRGs), radio-loud quasars (RLQs), FRI radio galaxies, optically selected quasars (PG quasars), and Seyfert galaxies plus LINERS (only 3), all the sources are at redshift z$<$0.5. The 5 GHz radio luminosity, BH mass and Eddington ratio are available in the @sik07 sample. We searched total 1.4 GHz flux densities available from the literature (@con98; and data from the NED and references therein) for the sample, for that the lower frequency radio luminosity would reflect the isotropic properties of emission in the lobes and extended jets. The BH masses in @sik07, were estimated using a broad-line region size-luminosity relation, assuming virial velocities of the gas, which produces broad $H_{\alpha}$ lines [@gre05], or from the appropriate references. The uncertainties of BH mass from @gre05, typically $\sim$20% in the virial mass formula that depends on the $H_{\alpha}$ line alone. The Eddington ratio is $\lambda$=$L_{bol}/L_{Edd}$, and the bolometric luminosity $L_{bol}$ is assumed to be 10 times optical $B$-band nuclear luminosity (at 4400$\AA$), i.e. $L_{bol}=10L_{B}$ (see, e.g., @ric06), where $L_{B}=\nu_{B} L_{\nu_{B}}$, which is from directly measured apparent magnitudes of the nuclear regions or also estimated from the $H_{\alpha}$ line, the errors of $H_{\alpha}$ flux can be $\leq$30% [@gre05], see @sik07 for more details on the estimation of BH mass and nuclear disc luminosity. Here we use the equation (8) to estimate the jet power by using $f=10$ and $P_{1.4}$.

As each subsample of the @sik07 sample is relatively small, to compare with the sample in Fig. \[fig1\] which are narrow line FRI/FRII galaxies, we combine the FRI galaxies and BLRGs into radio galaxies (RG) in the @sik07 sample, also because the FRI and FRII galaxies may have similar accretion mode [@cao04]. The BLRGs and RLQs which divided by absolute magnitude $M_{V}>-23$ and $M_{V}<-23$, are almost all the FRIIs [@sik07]. We plot the jet power $P_{j}$ derived from 1.4 GHz luminosity versus $\lambda m$ in Fig. \[fig4\] as well as the radio loudness versus $\lambda m$ in Fig. \[fig5\]. A few outliers are excluded in our analysis (e.g., the NGC1275 from Seyfert galaxies, which is actually hosted by a giant elliptical galaxy, and 5 PGQs which are in the RLQ area), and those data with only upper/lower limits are excluded. The radio loudness is computed with $R=1\times10^{5}L_{1.4}/L_{B}$ which converted from the formula $R=1.36\times10^{5}L_{5.0}/L_{B}$ at 5 GHz in @sik07, assuming source spectral index $S_{\nu}\propto
\nu^{-0.8}$. Linear regression fittings to the RGs, Seyferts+LINERS, and PG quasars are shown in Fig. \[fig4\] with the fitted parameters in Table \[tab1\], as well as the correlation coefficient and null probability in Table \[tab2\].

The quality of a linear regression can be measured by the coefficient of determination (COD), a value from 0 to 1. If it is close to 0 the relationship between X and Y will be regarded as very poor, the COD theoretically equals to the square of the Pearson coefficient of linear correlation. We try to do linear fitting with slope=1 (dash line in Fig. \[fig1\]), the resulted COD is 0.07, which is much lower than the COD=0.52 of the best linear fit for the FRIs+FRIIs. This is also true for the subsamples in Fig. \[fig4\], except the PG quasars which is close to slope=1 but with larger errors in Table \[tab1\]. The correlation coefficient and confidence level are quite high for the radio galaxies, Seyferts+LINERS in the two samples in Table \[tab2\]. We did not fit for the radio loud quasars (RLQs), because they are clustering in a small area making it difficult to fit them properly.

![Log\[jet power\] vs. log\[$\lambda m$\] for the radio galaxies (RG), Seyfert galaxies and LINERS (Sey+L), and PG quasars (PGQ), with the best linear fit to each subsample, the dash line is fitted with the fixed slope=1. <span data-label="fig4"></span>](fig4.eps){width="8cm"}

![Log\[radio loudness\] vs. log\[$\lambda m$\] for the sub-samples (same as in Fig.4), with the best linear fit to each subsample.<span data-label="fig5"></span>](fig5.eps){width="8cm"}

$$

<span>c|c|c|c|c </span>

Sample & subsample & $\mu$ & $b$ & $\eta^{'}/\varepsilon^{'}$\

Sikora13 & NLRG(0.52) & 0.52$\pm$0.06 & 40.10$\pm$0.34 & 100.0\

Sikora07 & RG(0.51) & 0.35$\pm$0.10 & 41.06$\pm$0.60 & 912.0\
& Sey+L(0.47) & 0.32$\pm$0.12 & 38.32$\pm$0.58 & 1.7\
& PGQ(0.47) & 0.92$\pm$0.44 & 34.08$\pm$3.42 & 9.5E-5\

& & $\rho$ & $c$ &\

Sikora13& NLRG(0.32) & -0.40$\pm$0.08 & 3.46$\pm$0.40 &\

Sikora07& RG(0.69) & -0.59$\pm$0.12 & 7.22$\pm$0.70 &\
& Sey+L(0.73) & -0.63$\pm$0.14 & 4.02$\pm$0.66 &\
& PGQ(0.00) & 0.07$\pm$0.52 & -0.89$\pm$3.96 &\

$$\end{table}

\begin{table}
         \caption[]{The Pearson, Spearman and Kendall linear correlation coefficient between $log[P_{j}]$ and $log[\lambda m]$, and
         that between $log[radio loudness]$ and $log[\lambda m]$ marked `logR' in second line, with
         null hypothesis probability (non correlation probability) in
         brackets.}

         \label{tab2}
\small$$

<span>c|c|c|c</span>

Subsample & Pearson & Spearman & Kendall\
NLRG & 0.72(0) & 0.71(0) & 0.53(0)\
logR & -0.57(0) & -0.52(0) & -0.36(1.6E-18)\

RG & 0.71(5.7E-9) & 0.64(6.4E-7) & 0.45(5.7E-6)\
logR & -0.83(1.1E-13) & -0.81(1.1E-12) & -0.62(2.5E-10)\
Sey+L &0.68(4.2E-6) & 0.68(5.2E-6) & 0.51(1.5E-5)\
logR & -0.85(3.6E-11) & -0.86(2.4E-11) & -0.70(2.2E-9)\
PGQ & 0.69(5.5E-4) & 0.64(0.002) & 0.57(3.9E-4)\
logR & 0.06(0.78) & 0.17(0.45) & 0.17(0.29)\

$$

In the results, we show that the jet pearsonhaley@example.org GHz positively correlate with the $\lambda m$, but not linearly, i.e. the power law indices ($\mu$) are significantly less than unity for the radio galaxies and Seyferts+LINERS. The radio loudness is anti-correlated with the $\lambda m$ for the radio galaxies and Seyferts+LINERS. For PG quasars, there appears to be a linear proportionality between the jet power and disc luminosity but with larger errors, that leads to no correlation between radio loudness and disc luminosity.

To further study the anti-correlation between the radio loudness and the $\lambda m$ we found, considering $P_{j}=\eta^{'}
/\varepsilon^{'} L_{disc}^{\mu}$ and equation (8)-(9), we have radio loudness:

$$R=P_{1.4}/L_{line} \propto P_{j}^{7/6}/L_{disc} \propto
L_{disc}^{(7/6)\mu-1}\propto L_{disc}^{\rho}$$

The ratio $\eta^{'} /\varepsilon^{'}$ can be estimated from the $b$ of the linear fit parameters in Table \[tab1\]. From equation (12) we find a relation $\rho=(7/6)\mu-1$. This relation accords with the fitting results in Table \[tab1\], e.g. for $\mu=0.52$, the $\rho=(7/6)\mu-1$= -0.39 is close to the measured value $\rho=-0.40\pm0.08$ in Table \[tab1\] for the NLRGs. So the anti-correlation is apparently explained, that is due to the $\mu<6/7$, so that $\rho=(7/6)\mu-1<0$.

The physics for $\mu<6/7$ (0.86) needs to be explored further in the $P_{j}= (\eta^{'} /\varepsilon^{'}) L_{d}^{<0.86}$, it implies that the jet power increases less efficiently than the disc luminosity increases. For $\mu<1$, the $\eta$ and $\varepsilon$ may be not constants but vary with the accretion rate. @sik13 used the relation of $\varepsilon\propto\lambda^{2/5}$ for the BH magnetosphere of a truncated disc, in which the jet power is dominated by a BH spin. With $R=P_{1.4}/L_{line}\propto\eta /\varepsilon$ approximately, they explain that the anti-correlation in radio loudness and Eddington ratio could be due to the $\varepsilon\propto\lambda^{2/5}$, i.e. $R\propto\lambda^{-0.4}$.

@van13 find that from 763 FRII quasars with the median redshift of 1.16, a linear correlation between 1.4 GHz luminosity and bolometric disc luminosity with $P_{1.4}\propto L_{bol}$. If we use the equation (8), the jet power will be $P_{j}\propto
P_{1.4}^{6/7} \propto L_{bol}^{0.86}$. They claim that, for this nearly linear proportionality, the power output from the inner part of the accretion disc dominates over the power extracted from the black hole by the Blandford-Znajek mechanism (BZ-jet, hereafter).

@kal12 investigated the $[O_{II}]$ emission line properties of 18508 quasars at z $<$ 1.6 drawn from the Sloan Digital Sky Survey (SDSS) quasar sample. The quasar sample has 1692 radio-loud (RLQs) and 16816 radio-quiet quasars (RQQs), according to the traditional radio-loud/quiet division of the radio-to-optical flux ratio of 10, and the radio loudness computed using the radio flux pearsonhaley@example.org GHz and the optical (7480$\AA$) flux density from the SDSS. They found a strong correlation between 1.4 GHz radio luminosity and narrow emission-line luminosity, with a power law index of $\mu\sim1.6$ and $\sim$1.2 for the RLQs and RQQs respectively. For radio jet power, using the equation (8), the slope reduces to $\mu\sim1.4$ and $\sim$1.0 for the RLQs and RQQs respectively. The optical line luminosity is linearly proportional to the disc luminosity in @kal12.

Discussion
==========

From the results above, in general there is a positive power law correlation between radio jet power and disc luminosity $P_{j}=\eta^{'} /\varepsilon^{'} L_{disc}^{\mu}$. The power law indices are significantly less than unity for the Seyfert galaxies, FRI and FRII galaxies at z $<$ 0.5 and the Eddington ratio $\lambda<0.1$ (or typically $\lambda m<7$ in Fig. \[fig1\] and Fig. \[fig4\]). Second, there is an anti-correlation between radio loudness and disc luminosity for the samples which have the $\mu<$6/7. We find that the negative correlation is caused by $(7/6)\mu <1$ with the relation $\rho=(7/6)\mu-1<0$. On the other hand, in high-z quasars from the literature, we have $\mu\sim0.86$ for the FRII quasars in @van13 and $\mu>1$ for the quasars in @kal12. It will lead to a positive correlation between radio loudness and disc luminosity for $\mu>6/7$ and no correlation for $\mu\sim6/7$ in the high-z samples, and we will investigate these in future.

The radio loudness problem appears to be resolved with our interpretation, the underlying physics is still not clearly identified. Are there different accretion discs or jet forming mechanisms for the power-law correlation index $\mu\ll$1 and $\mu\geq$1? In the BH accretion models, there are cold and hot accretion flows. Cold accretion flow consists of optically thick and geometrically thin gas, e.g. the standard thin disc [@shak73], which occurs at a fraction of the Eddington mass accretion rate, and the slim disc at super-Eddington rates [@abr88]. Hot accretion flows, however, are virially hot and optically thin, they occur at lower mass accretion rates, and are described by models such as the advection-dominated accretion flow (ADAF, @nar94) and luminous hot accretion flow. Observations show that hot accretion flows are often associated with jets, and they are present in low-luminosity AGN (LLAGN) (see @yuan14, for a review on hot accretion flows). @ho01 and @ho02 find that most of the LLAGN in nearby galaxies are radio-loud, and the radio loudness is inversely correlated with the disc luminosity. @sik13 and @sikb13 find that the anti-correlation of radio loudness and disc luminosity could be explained with the $\varepsilon\propto\lambda^{2/5}$ in the BH spin paradigm, implying that the BH spin plays an important role in the jet powers. Therefore the Seyfert galaxies and radio galaxies in our samples are likely powered by both the inner part of accretion disc and the BH spin. If the BZ-jet power is gradually suppressed when the accretion rate increases, the jet power could be less proportional to the disc luminosity and this may lead to an anti-correlation of radio loudness and disc luminosity. This is some similar to the transition from hard/low (with low accretion rate and radio jet) to soft/high (with high accretion rate and no jet) state in the X-ray binaries, in which a contribution of jet power from black hole spin is possible [@fen04]. However, this explanation needs a high fraction of rapid spin BHs in the low-z AGN.

@mar11a [@mar11b] analyzed the ratio of jet power to disc luminosity, they find that at z$<$0.5 the low-excitation galaxies have low accretion rates and bimodal spin distributions, with approximately half of the population having maximal spins, while high exciting galaxies are explained as high-accretion rate but very low spin objects at higher redshifts (z$\sim$1) and only a small population of nearly maximally spinning high accretion rate objects is possible. This may be supported by the findings that much higher fraction of radio loud galaxies at low accretion rates [@ho01; @ho02] than the fraction ($\sim$10%) of radio loud quasars at higher redshifts [@kal12]. And it has been suggested that low redshift FRIs have rapid spinning BHs [@wu11].

In high-z quasars, a steep slope $\mu \sim 1$ or $>$1 between jet power and disc luminosity is often observed, e.g, @van13, @wil99, @fer11, @kal12. The quasars are at high accretion state ($\lambda>0.1$) and hosted by elliptical galaxies. The massive ellipticals are most likely formed via major merger events, and their nuclei may have different nuclear environments than disc galaxies. @fal10 find evidence for the environmental source density to increase with the radio luminosity of AGN at around z=1. The jets of FRII quasars are probably launched through the Bondi accretion of hot interstellar gas [@bon52; @all06; @gas13; @sik13], with efficient jet powers [@wer12]. @all06 find a tight positive correlation between the Bondi accretion rate and the radio power required to inflate cavities observed in the surrounding X-ray emitting gas, suggests that the Bondi formulae provide a reasonable description of the accretion process for powerful jets. Furthermore, the powerful jet may live in a short time for quasars, given that the majority of quasars are radio quiet.

@mei01 proposed a hybrid model combined both the disc accretion and the BH spin/magnetosphere effects, it can produce powerful jets if the BH spin $a_{\ast}>0.9$ ($a_{\ast}$ is dimensionless spin parameter ranging from 0-1, see also @nem07). Therefore, both the hybrid model and the magnetically arrested disc [@tch11; @mck12] needs maximally spinning BHs for producing FRII jets, it is suggested that only a small population of nearly maximally spinning BHs in high accretion rate objects, and spins of BHs are generally low at around z=1 [@mar11a; @mar11b]. As noted by @van13, the powerful jets of FRII quasars are mainly controlled by the inner part of accretion flows rather than by the power extracted from the BH spin (also see @liv99), in contrast to that the BZ-jet may control the negative correlation of radio loudness and disc luminosity in the low luminosity AGN as we discussed above.

@fer11 find from a radio selected sample of 27 radio galaxies at a narrow redshift range z=0.9-1.1 that is unbiased to evolutionary effects, that there is a tight positive correlation between the radio luminosity and the mid-IR (also the $[O_{II}]$) line luminosity with power law indices $\mu>1$. They added optical selected quasars (OSQs) into the analysis, the correlation for the brightest radio sources appears to become an upper envelope. However, @kal12 investigated both radio loud and quiet quasars at z $<$ 1.6 drawn from the SDSS quasar sample. They find a strong correlation between 1.4 GHz radio luminosity and narrow emission-line luminosity, for both RLQs and RQQs, with power law indices $\mu>1$. This sample is much larger than the @fer11 sample, so it could be statistically more significant. Our studies concentrate mainly on radio loud sources, and it is needed to include radio quiet ones in larger and complete samples into analysis in future.

Finally, we note that the different slopes of correlation between radio jet power and disc luminosity might be affected by selection effects, e.g. sample sizes etc., however, we have high confidence for that in high-z quasars the slopes are much steeper than those of low redshift galaxies in our analysis. There are no correlation found between radio luminosities and source sizes in @wil99 and @sik13 for FRIs and FRIIs, however, caution must be taken if there is a wide range of source sizes in a relative small sample [@sha13]. The Doppler boosting effect is not considered here, since our samples are mainly the FRIs, FRIIs and Seyfert galaxies, and the blazars are not included. Furthermore, as noted by @sing14, the radio power and/or the disc luminosity may be also related to the redshift, a redshift – luminosity correlation (Malmquist bias) in a flux-limited sample. We checked our data, find there are weak positive correlations for the radio galaxies while stronger correlation for the PG quasars between the radio jet power (or disc luminosity) and redshift. In fact, the correlation is stronger between the radio power and the disc luminosity than that between the radio (or disc) luminosity and redshift in the radio galaxies of our samples, these type of source are mainly studied in this paper, but the Malmquist bias might still have some effects which we were currently not able to remove from our data. As analyzed by @sing14, a flux limited sample with a wide range of radio luminosities within narrow redshift ranges (and vice versa) will be needed to disentangle the effects in future.

Summary
=======

A model for the relation between radio jet power and the product of central BH mass and Eddington ratio of AGN is proposed. The model is examined with data from the literature.

We find that radio jet power positively correlates but not linearly with the product of BH mass and Eddington ratio, and the power law indices are significantly less than unity for relatively low accretion ($\lambda<0.1$) AGN, $P_{j}\propto (\lambda
m)^{\mu}$, in the radio galaxies and the Seyfert galaxies. This leads to a negative correlation between radio loudness and $\lambda m$ for the low luminosity AGN, i.e. $R\propto (\lambda
m)^{\rho}$ with $\rho=(7/6)\mu-1<0$, which may be attributed to a contribution of BZ-jet to total jet power assuming that the BZ-jet power is gradually suppressed as the accretion rate increases.

On the contrary, for the high-z quasars which often show the slope $\mu\geq1$, a positive correlation between the radio loudness and disc luminosity is predicted. We discuss that the jet powers of the high-z FRII quasars are likely dominated by the accretion disc rather than by the BH spin.

Acknowledgments {#acknowledgments .unnumbered}
===============

We thank the reviewer for useful comments. This work is supported by the National Natural Science Foundation of China (Grant No.11273050) and the 973 Program of China (2009CB824800). This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

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{'source': 'arXiv_out/1409'}