Jet power extracted from ADAFs and the application to Fermi BL Lacertae objects

For the case of BL Lac objects that lack strong broad emission lines, Paliya et al. (2021) obtained the black hole mass ($M_{\rm BH}$) through the following two methods.

First, the black hole mass is calculated by using the stellar velocity dispersion ($\sigma_{*}$). The formula is as follows (Gültekin et al., 2009),

Second, Paliya et al. (2021) used the bulge luminosity to estimate the black hole mass. The formula is as follows (Graham, 2007),

where $M_{\rm R}$ and $M_{\rm K}$ are the absolute magnitudes of the host galaxy bulge in the $R$ and $K$ bands, respectively.

We also note that the black hole mass of our sample is obtained by different methods. The uncertainty of the black hole mass estimated by stellar velocity dispersion is small, $\leq$ 0.25 dex. The uncertainty of the black hole mass estimated by the bulge luminosity is 0.6 dex.

Due to the lack of detection of emission lines, it is impossible to use emission lines to infer the BLR luminosity. Therefore, Paliya et al. (2021) derived a 3$\sigma$ upper limit in the H$\beta$ (or Mg II, depending on the source redshift and wavelength coverage) line luminosity. Paliya et al. (2021) calculated the BLR luminosity by scaling several strong emission lines to the quasar template spectrum of Francis et al. (1991) and Celotti et al. (1997), using Ly$\alpha$ as a reference and giving the total BLR fraction $<L_{\rm BLR}>=555.77$. The BLR luminosity is estimated through the following formula,

where $L_{\rm line}$ is the emission-line luminosity and $L_{\rm ref.frac.}$ is the ratio: 22 and 34 for H$\beta$/Ly$\alpha$ and Mg II/Ly$\alpha$ (Francis et al., 1991; Celotti et al., 1997), respectively. When more than one line luminosity measurements were available, Paliya et al. (2021) took their geometric mean to derive the average $L_{\rm BLR}$. The accretion disk luminosity is estimated by using $L_{\rm disk}=10L_{\rm BLR}$ (e.g., Baldwin & Netzer, 1978), with an average uncertainty of a factor 2 (Calderone et al., 2013; Ghisellini et al., 2014).

We carefully examined the samples of Paliya et al. (2021) and compared them with the classification of the sources of Abdollahi et al. (2020) and Foschini et al. (2022). We only consider that these sources with 1.4 GHz radio flux comes from the The FIRST Survey Catalog (Becker et al., 1995): 14Dec17 Version¹¹1http://sundog.stsci.edu/first/catalogs/readme.html. Finally, we get 276 Fermi BL Lacs (57 LBL, 43 IBL, and 176 HBL). The relevant data is listed in Table 1.

Komossa et al. (2018) used the following formula to estimate the jet kinetic power ($P_{\rm kin}$) in AGN (Bîrzan et al., 2008)

where $P_{\rm kin}$ is in units of $10^{42}$ erg s^-1, and $P_{1.4}$ is the 1.4 GHz radio luminosity in units of $10^{40}$ erg s^-1, $P_{1.4}=4\pi d_{L}^{2}\nu S_{\nu}$. The scatter for this relation is $\sigma=0.85$ dex. We make a K-correction for the observed flux using $S_{\nu}=S_{\nu}^{obs}(1+z)^{\alpha-1}$, where $\alpha$ is the spectral index and $\nu$ is the frequencies, and $\alpha=0$ is adopted (Abdo et al., 2010; Komossa et al., 2018). The $S_{\nu}^{obs}$ is the observed flux. The $d_{L}$ is the distance of luminosity, $d_{L}(z)=\frac{c}{H_{0}}(1+z)\int_{0}^{z}[\Omega_{\rm A}+\Omega_{\rm M}(1+z^{{}^{\prime}})^{3}]^{-1/2}dz^{{}^{\prime}}$ (Venters et al., 2009). Most galaxies have a central supermassive black hole, which may coevolve with the host galaxy, resulting in correlations between bulge luminosity, stellar velocity dispersion, and central black hole mass (e.g., Kormendy & Richstone, 1995; Magorrian et al., 1998). Models show that these correlations are caused by galaxy merging and feedback from AGN (e.g., Silk & Rees, 1998; Kauffmann & Haehnelt, 2000). Around the time of these discoveries, the Chandra X-ray observation found direct evidence of AGN feedback, when observations revealed cavities and shock fronts in X-ray emission gas around many massive galaxies (e.g., Fabian et al., 2000; McNamara et al., 2000). The X-ray cavity provides a direct measurement of the mechanical energy released by the AGN through the work done on the hot, gaseous halo around them (McNamara et al., 2000). This energy is expected to heat the gas and prevent it from cooling and forming stars (Churazov et al., 2001). Studies on X-ray cavities show that AGN feedback provides enough energy to regulate star formation and inhibit the cooling of the hot halos of galaxies and clusters (e.g., Bîrzan et al., 2008; Komossa et al., 2018). Komossa et al. (2018) believes that if the relationship of Cavagnolo et al. (2010) is used to calculate jet kinetic power, the high value of jet kinetic power is predicted up to an order of magnitude. At the same time, all of our samples have a 1.4 GHz radio flux, but there is almost no low-frequency radio flux (for example 150 MHz). We cannot use low-frequency radio flux to calculate jet kinetic power. Therefore, we follow the method of Komossa et al. (2018) and use equation (1) to calculate the jet kinetic power. We note that the total jet power ($P_{\rm jet}$) is given by the sum of two components i.e., the radiation power ($P_{\rm rad}$) and the kinetic power ($P_{\rm kin}$). The jet power of radiation is believed to be about 10% of the jet kinetic power (Nemmen et al., 2012; Ghisellini et al., 2014), namely $P_{\rm kin}\sim 10P_{\rm rad}$. Since the total jet power is mainly dominated by the jet kinetic power, the value of the total jet power can be approximately equal to the jet kinetic power.

The jet power of BZ mechanism ($P_{\rm jet}^{\rm BZ}$) is estimated by the following formula (MacDonald & Thorne, 1982; Thorne et al., 1986; Ghosh & Abramowicz, 1997; Nemmen et al., 2007)

where $\omega_{\rm F}\equiv\Omega_{\rm F}(\Omega_{\rm H}-\Omega_{\rm F})/\Omega_{\rm H}^{2}$ depends on the angular velocity of field lines $\Omega_{F}$ relative to that of the black hole $\Omega_{\rm H}$. We assume $\omega_{\rm F}=1/2$, which implies the output of maximum power (e.g., MacDonald & Thorne, 1982; Thorne et al., 1986; Nemmen et al., 2007). The $B_{\perp}$ is assumed to approximate to the poloidal component of the magnetic field $B_{\rm p}$, $B_{\perp}\approx B_{\rm p}(R_{\rm ms})\approx g(R_{\rm ms})B(R_{\rm ms})$(Livio et al., 1999), $g=\Omega/\Omega^{{}^{\prime}}$. The $R_{\rm ms}$ is the radius of the marginally stable orbit of the accretion disk (see APPENDIX A). The $\Omega^{{}^{\prime}}$ is the angular velocity of the disk. An observer at infinity will see the disk and the magnetic fields near the black hole rotate, in the Boyer–Lindquist coordinate system, not with an angular velocity $\Omega^{{}^{\prime}}$ but $\Omega=\Omega^{{}^{\prime}}+\omega$ (Bardeen et al., 1972), where $\omega$ is the angular velocity of the local metric which is given in Appendix A, equation (A11). The $B$ is the the magnetic field strength near the black hole (see APPENDIX A). The $R_{\rm H}=[1+(1-j^{2})^{1/2}]GM_{\rm BH}/c^{2}$ is the horizon radius (e.g., Ghosh & Abramowicz, 1997), where $G$ is the gravitational constant and $M$ is the mass of the black hole. The $j$ is spin of black hole and $c$ is the speed of light.

The jet power of the hybrid model ($P_{\rm jet}^{\rm Hybrid}$) is estimated by the following formula (Meier, 2001; Nemmen et al., 2007)

where $B_{\phi}$ is the azimuthal component of the magnetic field. Meier (2001) related the amplified, azimuthal component of the magnetic field to the unamplified magnetic field strength derived from the self-similar ADAF solution as $B_{\phi}=gB$, and $\Omega=\Omega^{{}^{\prime}}+\omega$. The $H$ is the vertical half-thickness of the disk. In the case of an ADAF, $H\sim R$ (Nemmen et al., 2007). Following the method of Nemmen et al. (2007), all physical quantities are estimated at $R=R_{\rm ms}$ (see APPENDIX A).

To better evaluate our results and put them into a context, we studied the distributions of the Fermi BL Lacs of our sample in terms of redshift, black hole mass, jet kinetic power, and accretion disk luminosity. The redshift distribution of Fermi LBL extends to larger values than IBL and HBL (Fig. 1). The average values of redshift are $z=0.64$ for LBL, $z=0.26$ for IBL, and $z=0.25$ for HBL. The range of black hole mass is from $10^{6.5}$ to $10^{10}M_{\odot}$. The average values of black hole mass are $\log M_{\rm BH}=8.39$ for LBL, $\log M_{\rm BH}=8.34$ for IBL, and $\log M_{\rm BH}=8.82$ for HBL. Cha et al. (2014) also found that FSRQs and LBL tend to have higher redshift and gamma-ray luminosity than HBL. They suggested that the evolutionary track of Fermi blazars is FSRQs$\rightarrow$LBL$\rightarrow$HBL.

The range of jet kinetic power is mainly from $10^{44.0}$ to $10^{46.0}$ erg s^-1. The LBL tends to have higher jet kinetic power than IBL and HBL (Fig. 1). The average values of jet kinetic power are $\log P_{\rm kin}=45.04$ for LBL, $\log P_{\rm kin}=44.26$ for IBL, and $\log P_{\rm kin}=44.37$ for HBL. The scope of accretion disk luminosity is mianly from $10^{42.0}$ to $10^{46.0}$ erg s^-1. The LBL also tends to have higher accretion disk luminosity than IBL and HBL. The average values of accretion disk luminosity are $\log L_{\rm disk}=44.45$ for LBL, $\log L_{\rm disk}=43.79$ for IBL, and $\log L_{\rm disk}=43.80$ for HBL. Chen et al. (2015a) found that the LBL tends to have higher jet kinetic power and BLR luminosity than HBL and IBL using a smaller sample than the one we consider here (see Figure 7 of Chen et al. (2015a)). Our results are consistent with theirs.

Paliya et al. (2021) suggested that the physical properties of Fermi blazars are likely to be controlled by the accretion rate in Eddington units. Therefore, we study the accretion rate distribution of Fermi BL Lacs. Using the relative contribution of individual lines to the total BLR luminosity (see Section 2.2), we obtain the total line luminosity $L_{\rm lines}$. Wang et al. (2002) defined a "line accretion rate" and its dimensionless form as follows

Assuming that most of the line luminosity ($L_{\rm lines}$ ) is photoionized by the accretion disk (Netzer, 1990), the $L_{\rm lines}$ should be proportional to the total luminosity of the accretion disks. Netzer (1990) defined $\xi\approx 0.1$. The $L_{\rm Edd}$ is Eddington luminosity, $L_{\rm Edd}=1.3\times 10^{38}(M_{\rm BH}/M_{\odot})\rm erg~{}s^{-1}$. Wang et al. (2002) got the relation between $\lambda$ and the dimensionless accretion rate ($\dot{m}$) for an optically thin ADAFs as follows

where $\alpha_{0.3}=\alpha/0.3$, $\xi_{-1}=\xi/0.1$, and $\lambda_{-4}=\lambda/10^{-4}$ (Wang et al., 2003). The viscosity parameters $\alpha$ is 0.3 (Narayan & Yi, 1995). Narayan et al. (1998) suggested that an optical thin ADAFs appears when $\dot{m}\leq\alpha^{2}$. Equation (8) can then be rewritten as

Optically thin ADAFs require $\lambda<\lambda_{1}$. The optically thick, geometrically thin disk (SS) obey (Wang et al., 2003)

and the condition $1>\dot{m}\geq\alpha^{2}$ gives

A standard thin disk satisfies $\lambda\geq\lambda_{2}$ (Wang et al., 2003). When the accretion rate reaches $\dot{m}\geq 1$, we have

A slim disk requires $\lambda\geq\lambda_{3}$ (Wang et al., 2002, 2003), namely Super-Eddington accretion (SEA). It is worth noting that in the transition region between $\lambda_{1}$ and $\lambda_{2}$, the accretion flow may be in a hybrid state, in which the standard disk coexists with ADAFs. Some authors have shown that hybrid states are possible in the accretion disk of AGN (e.g., Quataert et al., 1999; Różańska & Czerny, 2000; Ho et al., 2000). Gu & Lu (2000) suggested that the conversion of a SS disk to an ADAFs is possible through evaporation (Liu et al., 1999). The transition radius depends on accretion rate, black hole mass, and viscosity. However, in such a regime, the disk structure is complex.

Figure 2 shows that the three critical values of $\lambda$ define four regimes in accretion states. The average values of $\lambda$ for LBL, IBL and HBL are $\langle\log\lambda\rangle|_{\rm LBL}=-3.05$, $\langle\log\lambda\rangle|_{\rm IBL}=-3.67$ and $\langle\log\lambda\rangle|_{\rm HBL}=-4.14$, respectively. We find that the LBL has higher accretion rates than IBL and HBL, which could explain why LBL have more luminous disks and more powerful jets than HBL (see Fig.1). Wang et al. (2002) also found that the HBL has lower accretion rates than LBL. We investigated the differences in the distribution of accretion rates using the parameter T-test, nonparametric Kolmogorov Smirnov (K-S) test, and Kruskal Wallis H test. The parameter T-test is mainly used to test whether there is a difference in the mean accretion rate between two independent samples. The nonparametric K-S test and Kruskal Wallis H test are mainly used to test whether there are differences in the distribution of accretion rates between two independent samples. We assume that there are differences in the three tests at the same time, so there is a significant difference in the accretion rate between the two samples. According to the parameter T-test ($P=1.11\times 10^{-15}$, significant probability $P<0.05$), nonparametric K-S test ($P=1.01\times 10^{-15}$, significant probability $P<0.05$), and Kruskal–Wallis H-test ($P=1.99\times 10^{-14}$, significant probability $P<0.05$), we find that the distributions of accretion rates between LBL and HBL are significantly different. The parameter T-test shows that there is a significant difference of the distributions of accretion rates between IBL and LBL ($P=0.0002$). Through a nonparametric K-S test ($P=0.0006$) and a Kruskal–Wallis H-test ($P=9.38\times 10^{-5}$), we find that the distributions of accretion rates between IBL and LBL are significantly different. Through a parameter T-test ($P=0.0008$), nonparametric K-S test ($P=0.006$), and Kruskal–Wallis H-test ($P=0.0008$), we find that the distributions of accretion rates between IBL and HBL are significantly different.

From Figure 2, we find that about 62% HBL and 14% IBL have pure optically thin ADAFs, while about 7% LBL have ADAFs+SS. Cao (2002) found that the accretion flows in all HBL are in the ADAF state. Wang et al. (2003) also found that the low-power HBLs and LBLs have pure optically thin ADAFs. However, the LBLs also may have a hybrid structure consisting of an SS disk plus optically thin ADAFs. Cao (2003) also found that the BL Lacs have different accretion modes. They proposed that the BL Lacs have ADAFs in the inner region of the disk, and it becomes a standard thin disk in the outer region, i.e., ADAFs+SS scenario.

Because our sample has an accretion rate compatible with an ADAFs regime (Fig 2), we considered the jet power of the BZ mechanism and the jet power of the hybrid model in the ADAFs scenario. The relation between the jet power extracted from the ADAFs for the BZ and hybrid models and the black hole mass is shown in Figure 3. In order to estimate the maximum jet power of BZ model and Hybrid model, we use accretion rates $\dot{m}=0.01$, and the disk viscosity parameter $\alpha=0.3$ (Narayan & Yi, 1995). The spin of black hole $j=0.98$ is adopted. The black dashed line is the BZ jet model, and the black solid line is the Hybrid jet model. The red dot is LBL. The black dot is the IBL. The green triangle is HBL. The cyan circle is sources with ADAFs+SS. We find that about 7% LBL, 9% IBL, and 32% HBL are below the maximal BZ jet power expected to be extracted from ADAFs (Fig.3, dashed line). These results show that the jet kinetic power of LBL and IBL can hardly be explained by the BZ jet model. However, about 26% LBLs, 72% IBLs, and 94% HBLs are below the solid line when we consider the hybrid model. These results show that most of the IBL and HBL can be explained by the Hybrid jet model. Most LBLs cannot be explained by the BZ jet model or Hybrid jet model. These sources with ADAFs+SS have high jet power, which makes them unable to be explained by BZ or hybrid models. There are two possible explanations for the jet power of the LBL. One is that these LBLs require other jet models, such as an accretion disk with magnetization-driven outflows (Cao & Spruit, 2013; Li, 2014; Cao, 2016, 2018; Li & Cao, 2022). Cao & Spruit (2013) proposed that if the most angular momentum of the gas in the thin disk is taken away by the magnetically driven outflows, the radial velocity of the disk will increase significantly, so the external field can be significantly enhanced in the inner region of the thin disk with the magnetically driven outflows (Li & Cao, 2019; Cao, 2018). Thereby, the expected jet kinetic power is higher than in the case of the BZ or hybrid model, as we observed in the data. The other is that the LBL has a strong beaming effect. Lister et al. (2011) found that the LBL has generally higher Doppler factors than HBL, which implies that the LBL has a strong beaming effect. Because of the beaming effect, the jet kinetic power could be overestimated. The "real" jet power, in this scenario, may be lower and compatible with models.

It is widely believed that large-scale magnetic fields play a crucial role in the acceleration and collimation of jets and/or outflows (e.g., Pudritz et al., 2007). The origin of the large-scale magnetic field passing through the accretion disk has not been well understood. Some studies suggest that large-scale magnetic fields that accelerate jets or outflows may be formed by weak external field advection (e.g., Bisnovatyi-Kogan & Blinnikov, 1977; Bisnovatyi-Kogan & Ruzmaikin, 1976; Spruit & Uzdensky, 2005). However, Lubow et al. (1994) found that in the geometrically thin accretion disk (H/R $\leq$ 1), the advection in the external field is quite ineffective because of its small radial velocity. This may imply that the field in the inner region of the disk is not much stronger than the external weak field, which cannot accelerate strong jets in radio-loud quasars (Lubow et al., 1994). A few mechanisms were suggested to alleviate the difficulty of field advection in the thin disk (e.g., Spruit & Uzdensky, 2005). Some people believe that the hot corona above the disk can effectively drag the external field inward, that is, the so-called "coronal mechanism" (see Beckwith et al., 2009). The radial velocity of the gas above the disk can be greater than the radial velocity of the mid-plane of the disk, which partly solves the problem of inefficient field advection in the thin disk (Lovelace et al., 2009; Guilet & Ogilvie, 2012, 2013). Recently, Cao (2018) suggested that powerful jets can be accelerated by the coronal magnetic field. Cao (2004) found that the jets are accelerated from the disk coronas for radio-loud quasars. Zhu et al. (2020) found a a close connection between corona-disk-jet for radio-loud quasars. At the same time, some authors found that FSRQs and LBL have similar spectral properties, such as particle acceleration mechanism (e.g., Chen et al., 2021). These results indicate that the jet kinetic power of LBL may also be explained by the coronal magnetic field.

Chen et al. (2023b) found that the jet kinetic power of jetted AGNs are powered by the BZ mechanism based on the relation between jet kinetic power and accretion disk luminosity. Our results are slightly different from those of Chen et al. (2023b). The main reason is that we calculate the maximum jet power of the BZ mechanism and the hybrid model based on the theory, and then compare the maximum jet power of the BZ mechanism and the hybrid model with the observed jet kinetic power. The work of Chen et al. (2023b) is mainly based on whether the slope of the relationship between jet kinetic power and accretion disk luminosity is 1 to judge whether their jet knetic power is dominated by the BZ mechanism. In the work of Chen et al. (2023b), it is assumed that the jet kinetic power of jetted AGN is dominated by Poynting flux, and then it is obtained that the slope of the relationship between the maximum jet power of BZ mechanism and the luminosity of accretion disk is equal to 1 (Ghisellini, 2006). Finally, the slope of the relationship between jet kinetic power and accretion disk luminosity is compared the slope obtained with theory to further judge whether the jet of jetted AGN is dominated by BZ mechanism. However, some studies have found that the jet power of jetted AGN is not dominated by the Poynting flux (e.g., McKinney et al., 2012; Zdziarski et al., 2015; Paliya et al., 2017; Chen et al., 2023a).

In Figure 4, we show the results as in Fig 3, but the dashed lines are $P_{\rm jet}/L_{\rm Edd}$ = 0.01, 0.1, 1, respectively. The jet kinetic power of all Fermi BLLacs is less than 1$L_{\rm Edd}$. We find that about 79% LBL is located in $\sim 0.01-1L_{\rm Edd}$. As a comparison, the jet power expected in case of coronal mechanism is at most 0.05 Eddington (Cao, 2018). Cao & Spruit (2013) suggested that most of the angular momentum of the accretion disk is excluded from magnetization-driven outflows, and the magnetic field will be enhanced compared with the thin disk without outflow. The magnetic field dragged inward by the accretion disc with magnetization-driven outflows may accelerate the jet in source with high jet power. The jet kinetic power of these LBLs may be explained by the magnetization-driven outflows model (Cao, 2018).

We also find that the jet kinetic power of LBL depends on the mass of the black hole, while the jet kinetic power of HBL does not seem to depend on the mass of the black hole. Ghosh & Abramowicz (1997) found that it is possible for the standard disks to find two regimes, which may be radiation pressure dominated (RPD) or gas pressure dominated (GPD). The jet kinetic power depends on the mass of the black hole, indicating that the accretion disk is dominated by radiation pressure. The jet kinetic power depends on the accretion ($L_{\rm bol}/L_{\rm Edd}$), indicating that the accretion disk is dominated by gas pressure (Ghosh & Abramowicz, 1997; Foschini, 2011; Chen et al., 2015a). Our results imply that the accretion disk of LBL may be dominated by radiation pressure, while that of HBL is not dominated by radiation pressure. The accretion disks of HBL may be dominated by gas pressure. In the future, when our source has the bolometric luminosity, we will test these results. Chen et al. (2015a) also found that the accretion disks of FSRQs and LBL are dominated by radiation pressure, while the accretion disks of HBL and IBL seem to be dominated by gas pressure.

There is evidence that there is a positive correlation between jet power and accretion luminosity in jetted AGN (e.g., Rawlings & Saunders, 1991; Wang et al., 2004; Gu et al., 2009; Ghisellini et al., 2010; Sbarrato et al., 2014; Ghisellini et al., 2014; Paliya et al., 2019). The relation between jet kinetic power and accretion disk luminosity is shown in Figure 5. We find a significant correlation between jet kinetic power and accretion disk luminosity for the whole sample ($r=0.74,P=5.10\times 10^{-50}$). The tests of Spearman ($r=0.69,P=1.58\times 10^{-41}$) and Kendall tau ($r=0.52,P=3.81\times 10^{-37}$) also show a significant correlation between jet kinetic power and accretion disk luminosity for the whole sample. The best fitting equation given by least square linear regression is $\log P_{\rm kin}=(0.46\pm 0.03)\log L_{\rm disk}+(24.39\pm 1.09)$. At the same time, we also find that the sources with ADAFs+SS follow the same relation. However, Rajguru & Chatterjee (2022) suggested that the correlations are often driven by the common redshift dependence. They found a weak correlation between jet power and accretion disk luminosity when the redshift was excluded. We test their results. Partial correlation shows a moderately weak correlation between jet kinetic power and accretion disk luminosity when redshift dependence is excluded for the whole sample ($r=0.12,P=0.03$). We use Pearson correlation analysis for other types of AGNs. There is also a significant correlation between jet kinetic power and accretion disk luminosity for the LBL ($r=0.81,P=3.51\times 10^{-14}$). The best fitting equation given by least square linear regression for the LBL is $\log P_{\rm kin}=(0.48\pm 0.05)\log L_{\rm disk}+(23.46\pm 2.13)$. Partial correlation shows a moderately weak correlation between jet kinetic power and accretion disk luminosity when redshift dependence is excluded for the LBL ($r=0.34,P=0.009$). There is a significant correlation between jet kinetic power and accretion disk luminosity for the IBL+HBL ($r=0.60,P=3.09\times 10^{-23}$). The best fitting equation given by least square linear regression for the IBL+HBL is $\log P_{\rm kin}=(0.25\pm 0.02)\log L_{\rm disk}+(33.42\pm 0.98)$. Partial correlation shows a moderately weak correlation between jet kinetic power and accretion disk luminosity when redshift dependence is excluded for the IBL+HBL ($r=0.24,P=0.003$). We find that the slope of the relation between jet kinetic power and accretion disk luminosity for the LBL is slightly different from that of IBL+HBL, and the slope of LBL is greater than that of IBL+HBL. Paliya et al. (2017) found that the slope of the relation bteween jet power and accretion disk luminosity is 0.46 using both 324 $\gamma$-ray detected and 191 $\gamma$-ray undetected blazars. Chen et al. (2023a) also found that the slope of the relation bteween jet kinetic power and accretion disk luminosity is $0.51\pm 0.18$ using 38 gamma-ray-emitting radio galaxies. Our results are similar to the results of Paliya et al. (2017) and Chen et al. (2023a). Ghisellini et al. (2014) found that the slope of the relation bteween jet power and accretion disk luminosity is 0.92 using 217 $\gamma$-ray detected blazars. Chen et al. (2023b) found that the slope of the relation bteween jet kinetic power and accretion disk luminosity is $1.00\pm 0.02$ for the whole sample (FSRQs+BL Lacs+gamma-ray-emitting narrow-line Seyfert 1 galaxies), $0.83\pm 0.04$ for FSRQs, $1.00\pm 0.05$ for BL Lacs, $0.73\pm 0.15$ for gamma-ray-emitting narrow-line Seyfert 1 galaxies. Because we mainly use the jet kinetic power to replace the total jet power. Therefore, we also examined all relationships including 10% of jet kinetic power and found that all relationships were valid in our work.

Figure 6 shows a relation between IC luminosity and synchrotron luminosity for Fermi BL Lacs. We find a significant correlation between IC luminosity and synchrotron luminosity for the whole sample ($r=0.95,P=2.06\times 10^{-143}$). The tests of Spearman ($r=0.92,P=7.66\times 10^{-113}$) and Kendall tau ($r=0.76,P=3.47\times 10^{-80}$) also show a significant correlation between IC luminosity and synchrotron luminosity for the whole sample. The best fitting equation given by least square linear regression is $\log L_{\rm IC}=(1.16\pm 0.02)\log L_{\rm syn}+(-7.58\pm 1.00)$. We find that the slope of the relation between IC luminosity and synchrotron luminosity is close to 1.0. According to Ghisellini (1996)($L_{\rm EC}\sim L_{\rm syn}^{1.5}$, $L_{\rm SSC}\sim L_{\rm syn}^{1.0}$), our results suggest that the IC component of Fermi BL Lacs is dominated by the SSC process. Some authors have confirmed this conclusion (e.g., Celotti & Ghisellini, 2008; Ghisellini et al., 2010; Lister et al., 2011; Ackermann et al., 2012; Wu et al., 2014; Marchesini et al., 2019; La Mura et al., 2022). Xue et al. (2016) found that the slope of the relation between IC luminosity and synchrotron luminosity for 28 Fermi BL Lacs is $k_{\rm BLLacs}=1.12\pm 0.10$. Our work confirms the results of previous studies.