Mapping the distribution of the magnetic field strength along the NGC 315 jet

In this work, we utilize the MHD jet model proposed by Tomimatsu & Takahashi (2003) (hereafter TT03) in the framework of special relativity. In the black hole magnetosphere, due to the balance between the gravitational force of the black hole and magneto-centrifugal force, a stagnation (a.k.a separation) surface is generated that separates the inflow and outflow regions (e.g., Takahashi et al., 1990; McKinney, 2006; Pu et al., 2015; Pu & Takahashi, 2020). Comparing semi-analytical approaches and GRMHD simulation approaches, the semi-analytical approaches have the following advantages. The large spatial extent of the acceleration region has posed a challenge for such calculations by GRMHD simulations and they tend to be eventually limited by computational costs and numerical dissipation (e.g., McKinney, 2006; Komissarov et al., 2007, for details), while semi-analytic approaches are free from these concerns. When discussing properties of axisymmetric and steady MHD flows in general, the magnetic field geometry should be consistent with Grad-Shafranov equation (e.g., Nitta et al., 1991), and the flow should be trans-fast-magnetosonic (Takahashi et al., 1990; Takahashi, 2002). However, it is technically difficult to obtain a solution satisfying both of these conditions (e.g., Beskin, 2010, for review). TT03 model is the only semi-analytic solution to satisfy both of these conditions. Given that a radial profile of the poloidal component of the jet’s four-velocity ($u_{p}$) is sensitive to the magnetic field geometry, we utilize TT03 model in this work. TT03 model is applicable to the region outside the light cylinder ($r_{\rm lc}$). In general, the jet is accelerated outside the light cylinder. ²²2 Flow properties inside the light cylinder have been recently investigated by several authors (e.g., Takahashi et al., 2021; Camilloni et al., 2022; Beskin et al., 2023). Beskin et al. (2023) suggests that a dense cylindrical core appears over long enough distances near the axis with its radius comparable to $r_{\rm lc}$. Camilloni et al. (2022) showed that in the force-free magnetosphere in Kerr geometry, there is a logarithmic term in the angular velocity.

The outer boundary wall condition would be given by a parabolic streamline along the poloidal magnetic field lines. Since multiple normalizations are conducted, it would be helpful to explicitly write down the boundary condition here. The outer boundary wall shape is denoted as ($Z$, $R$) in the cylindrical coordinate and it satisfies the following relation:

where the $\theta_{0}$ is the half-opening angle of the jet at the inlet boundary and the magnetic flux function on the boundary wall satisfies $\Psi(z=Z,r=R)=\Psi_{0}$. Note that the case of $q=1$ corresponds to a conical boundary wall shape. In this work, we will give the value of $q$ with reference to the overall results of the detailed VLBI observations in §4.

On the contrary to the case of varying ${\cal E}$, $\Omega_{F}$ does not alter the profile of $u_{p}$ itself. As already shown, $\Omega_{F}$ is governed by the light cylinder radius $r_{\rm lc}$ and it is given by $\Omega_{F}=c/r_{\rm lc}$. Slower rotation of $\Omega_{F}^{-1}$ leads to more distant starting point of the jet acceleration from the central black hole.

In this work, a location of intersection between the boundary-wall ($\Psi=\Psi_{0}=1$ surface) and the light-cylinder is important. Hereafter, we denote the location as ($R_{0}$, $Z_{0}$). By inserting $R_{0}=r_{\rm lc}$ at Eq. (1), the location of $Z_{0}$, from which the jet acceleration starts (see Figure 1 in Kino et al., 2022), is obtained as follows:

We note that the geometrical factor $\theta_{0}$ also affects the location of $Z_{0}$.

Following Ro et al. (2023), we introduce a spectral index model along the jet. The essence of the model can be explained as follows. A change of a spectral index along the jet reflects a change in the nonthermal electrons’ energy distribution function (eDF hereafter) along the jet. Spectral indices of nonthermal electrons ($p$) and nonthermal synchrotron emission ($\alpha$) are connected as $\alpha=(p+1)/2$ where the sign of the spectral index as $S_{\nu}\propto\nu^{+\alpha}$ is defined and $S_{\nu}$ presents synchrotron flux per unit frequency. Time evolution of eDF is described by the continuity equation including nonthermal electron injection and energy losses. (1) In the case of a constant and continuous injection of non-thermal electrons, the synchrotron spectrum has a break, and the spectral index steepens by $0.5$ before and after the break. (2) For instantaneous injection, the cutoff energy $\gamma_{\rm max}$ decreases with time. Above the cutoff energy, the synchrotron spectrum decreases exponentially. Neither (1) nor (2) alone can explain the properties of the observed spectral index of the M87 jet. To overcome this problem, Ro et al. (2023) proposed a model where the injection-rate function of nonthermal electrons has $z$-dependence written as

where $Q_{0}$ and $p_{\rm inj}$ are a normalization factor and a spectral index of injected nonthermal electrons, respectively. The index $q_{\rm inj}$ describes $z$-dependence of $Q_{e,\rm inj}$ (see details Ro et al., 2023). By matching an observed synchrotron spectral index and a model-predicted one, one can constrain on strengths of magnetic fields along the jet ($B(z)$) and the nonthermal electron injection function. The model generally describes that the stronger magnetic field strength realizes the closer distance for the steeping of the spectral index due to synchrotron cooling in the jet. A smaller $q_{\rm inj}$ leads to a larger injection of the nonthermal electrons, which compensates for the reduction of the number of electrons due to the cooling at a certain distance along the jet. It realizes a suppression of the steepening of the spectral index (i.e., larger $\alpha$).

The jet collimation profile of NGC 315 has been explored in previous works (Park et al., 2021; Boccardi et al., 2021). Transverse intensity profile along the jet axis is obtained by the multi-frequency observations and the jet width at each distance is obtained. The obtained jet width profile defined as $Z\propto R^{q}$ in the previous section shows the parabollic shape characterized by

measured by (Park et al., 2021). In this work, we do not treat the rapidly expanding jet region (so-called geometrical flaring region; Canvin et al. 2005) at $>10^{5}~{}r_{g}$, because it is beyond the application of the TT03 model. Hereafter, we focus on the inner parabolic region of acceleration and collimation zone of the NGC 315 jet with the index $q=1.72$ during the application of TT03 model to NGC 315.

Following the previous work of (Ricci et al., 2022, in Figure 4) where the intrinsic half-opening angle has been measured, hereafter the full-opening angle is set as

This value falls in the typical range of $\theta_{0}$ for various radio galaxies (e.g., Pushkarev et al., 2017). This $\theta_{0}$ value of NGC 315 satisfies guarantees that $\theta_{0}\ll 1~{}{\rm radian}$.

It is generally difficult to constrain on the total power of AGN jets, because of the existence of invisible components such as low-energy electrons/positrons and protons (e.g., Kino et al., 2012; Sikora et al., 2020, and references therein). Following the work of Ricci et al. (2022), we set the allowed range of the total power of the jet ($L_{j}$) as follows:

The lower limit is estimated from the Bondi accretion rate (Nemmen & Tchekhovskoy, 2015), while the upper limit is derived from the empirical relationship between the radio core luminosity and $L_{j}$ using the empirical relationship between core luminosity and jet power (e.g., Heinz et al., 2007).

A mass accretion rate ($\dot{M}$) can be estimated in several ways. Firstly, $\dot{M}$ can be estimated based on a bolometric luminosity ($L_{\rm bol}$) by using the relation of $\dot{M}=L_{\rm bol}/\eta c^{2}$ where $\eta$ is the conversion efficiency of the mass accretion energy to $L_{\rm bol}$. The bolometric luminosity in NGC 315 can be derived from the [O III] 5007 emission line together with the relation $L_{\rm bol}=3500L_{\rm[OIII]}$. The $L_{\rm bol}$ can be obtained also from a direct integration of the nucleus’s SED. Both of them indicate the consistent value of $L_{\rm bol}=2\times 10^{43}~{}{\rm ergs^{-1}}$ (Gu et al., 2007; Ricci et al., 2022) which derives $\dot{M}\sim 10^{-4}~{}M_{\odot}~{}{\rm yr^{-1}}$ when assuming $\eta=0.1$ (Ricci et al., 2022). Secondly, $\dot{M}$ may be estimated when HI broad absorption line is detected (van Gorkom et al., 1989) that can be related to the fueling gas that is falling into the nucleus. Morganti et al. (2009) estimated $\dot{M}$ in NGC 315 and it is suggested as $10^{-4}~{}M_{\odot}~{}{\rm yr^{-1}}\lesssim\dot{M}\lesssim 10^{-3}~{}M_{\odot}~{}{\rm~{}yr^{-1}}$ using the assumed infalling velocity of $50\,{\rm km\,s^{-1}}$. Other independent work of SED model fitting for the central region of NGC 315 also suggest a comparable value of $\dot{M}=4\times 10^{-4}~{}M_{\odot}~{}{\rm yr^{-1}}$ (Kimura & Toma, 2020). Therefore, the allowed range of $\dot{M}$ in this work is set as

in the following discussion.

In addition, Morganti et al. (2009) estimated $\dot{M}$ by using the empirical relationship between $L_{j}$ and Bondi accretion power (e.g., Allen et al., 2006; Balmaverde et al., 2008). Although the estimated maximum mass accretion rate $\dot{M}_{\rm max}\sim 10^{-1}~{}M_{\odot}~{}{\rm yr^{-1}}$ seems extremely large, we will investigate this case as well as an additional one.

Figure 1 presents the best-fit profile of the model-predicted $u_{p}$ overlaid to the VLBI-measured $u_{p}$. As seen here, the jet flow is described as multiple streamlines of the plasma flow along the magnetic surfaces with their curvature is calculated in TT03 model. As emphasized in (Kino et al., 2022), the key feature is the location of the starting point of the jet acceleration. The allowed range of the angular velocity is

Errors bars for the innermost measurement of $u_{p}(z\sim 3\times 10^{3}~{}r_{g})$, introduce uncertainties for the determination of the starting point of the jet acceleration ³³3 It is noteworthy that a recent work of Ricci et al. (2023) shows that if thermal energy in the jet is comparable to or exceeds magnetic energy, thermal acceleration becomes significant at parsec scales. If this is the case, then the discrepancy between the observed velocity field and the model-predicted one would get even larger..

In order to take into account such uncertainties, here we have conducted two cases of the fittings. Case 1 fits the higher value of the data point that realizes a smaller width of $r_{\rm lc}$, while Case 2 fits the lower value of the data point that realizes a larger width of $r_{\rm lc}$. The resultant small difference in the starting point of the jet acceleration is seen in Figure 1. For both cases, we fit the observation data point with the fastest speed at $z\sim 7\times 10^{4}~{}r_{g}$. Since the jet is only logarithmically accelerated, Case 2 requires a larger ${\cal E}$ than Case 1.

The dashed vertical lines show the distance of the jet collimation break. The observation data show the jet deceleration at a distance further away than the jet collimation break (i.e., $z>Z_{\rm brk}$). At the jet deceleration region, TT03 model is not applicable. In addition, it would be worth noting the Bondi radius $R_{\rm Bondi}={2GM_{\bullet}/c_{s}^{2}}=0.031~{}{\rm kpc}(kT_{e}/{\rm keV})^{-1}(M_{\bullet}/10^{9}M_{\odot})$ within which the gravitational influence of the central black hole dominates, where $c_{s}$, and $kT_{e}$ are the sound speed and the temperature of the surrounding hot atmosphere (e.g., Russell et al., 2015). For NGC 315, the temperature of the hot atmosphere within 1 arcsec has $kT_{e}\approx 0.44~{}{\rm keV}$ which leads $R_{\rm Bondi}\approx 1.5\times 10^{6}~{}r_{g}$, which is significantly larger than $Z_{\rm brk}$.

Next, we estimate magnetic field strength on the horizon scale ($B_{H}$) by assuming BZ process is in action at the jet base of NGC 315. The BZ power and the ratio of the angular velocity of the magnetic field line and the event horizon can be given by

and

where $\chi=\int^{\theta_{H}}_{0}\sin^{3}\theta d\theta=\frac{2}{3}-\frac{3}{4}\cos\theta_{H}+\frac{1}{12}\cos 3\theta_{H}$ (e.g., Beskin & Kuznetsova, 2000; Takahashi et al., 2021) and $r_{H}=r_{g}(1+\sqrt{1-a_{*}^{2}})$ is the outer horizon radius of the black hole and $a_{*}\equiv a/M_{\bullet}$ is the dimensionless spin parameter. For simplicity, $r_{H}$ for $a_{*}\approx 1$ is shown here. Here we normalized $\chi$ with a typical value suggested at the jet base $\chi_{-2}=\chi/10^{-2}$ (e.g., Beskin & Kuznetsova, 2000; Tchekhovskoy et al., 2011). From Eqs. (10) and (11), it can be readily seen that $a_{*}$ and $r_{\rm lc}$ are the two parameters that govern the value of $L_{\rm BZ}$.

Note that there is uncertainty in the value of $\chi$ due to $\theta_{H}$. At a glance, it seems natural to substitute $\theta_{H}\approx\theta_{0}$ with the 4-6 degrees measured at the jet downstream. However, this value leads to $\chi\sim O(10^{-4})$, causing a problem of unphysically large BH being required through Eq. (10). This may be probably due to the rapid change in the magnetic field shape near the horizon scale, deviating significantly from the smooth parabolic shape (assumed in TT03). Indeed, GRMHD simulations for MAD state have shown a tendency for the magnetic field to abruptly change to a larger opening angle near the BH, penetrating the horizon (e.g., Tchekhovskoy et al., 2011). Therefore, integrating only within the narrow range of $\theta_{0}=4-6$ degrees on the event horizon significantly underestimate the magnetic flux threading the event horizon. To compromise this problem, we set $\theta_{H}$ as approximately 1 radian in light of GRMHD simulations. In the case of M87, actual 86 GHz VLBI image shows of order of 1 radian half opening angle (Hada et al., 2018; Kim et al., 2018). Hence, the choice of $\theta_{H}$ did not matter (Kino et al., 2022). On the contrary, the jet base of NGC 315 has not yet been spatially resolved to that depth. Hence, we assume $\theta_{H}\approx 1$ radian.

Following the recent GRMHD simulations of highly magnetized jets (e.g., Porth et al., 2019; Ripperda et al., 2019), we set the value of the magnetic field threading angle as $\theta_{H}=1$ radian and it leads to $\chi=0.18$. Here, we assume the allowed range of the black hole spin as $0.5\lesssim a_{*}\lesssim 1$ (e.g., Zamaninasab et al., 2014; Nakamura et al., 2018) since too small spin is not able to explain the required jet power. By combining the estimated $\Omega_{F}$ and assumed $\Omega_{H}$ together with the assumption of BZ-driven jet i.e., $L_{\rm BZ}\approx L_{j}$, we obtain the allowed range of $\Omega_{F}/\Omega_{H}$ and the corresponding $B_{H}$ as follows:

Figure 2 presents the estimated range of $B_{H}$. The lower end of $B_{H}$ is comparable to that estimated in the previous work Ricci et al. (2022). We stress that one of the uniqueness of our analysis is the estimation of $\Omega_{F}$ based on the observed jet velocity profile.

Following the method proposed by Ro et al. (2023), here we will constrain the physical quantities of the NGC 315 jet from the observed spectral index distribution. We use the data obtained on January 05th, 2020 (project code: BP243; Park et al., 2021) to investigate the spectral index distribution of the NGC 315. In particular, we use the higher frequency pair (15 GHz and 22 GHz) in these multi-frequency observations, since this is less effective by the synchrotron self-absorption. The information of data is summarized in Table 1 of Park et al. (2021). To avoid artificial spectral steepening, we used the visibility data with a common $(u,v)$-range of 8 to 445 $M\lambda$, where $\lambda$ is the wavelength of the corresponding radio wave. We convolved the two images into a circular beam with a radius of 0.9 mas. We also correct the core shift between two frequencies. According to Park et al. (2021), the position difference between the 15 GHz core and the 22 GHz core is $0.08\pm 0.06$ mas in right ascension and $0.07\pm 0.05$ mas in declination.

Figure 3 shows a spectral index map of NGC 315 between 15 GHz and 22 GHz. We obtain spectral index values up to $\sim$10 mas from the radio core. It also shows that the core has a relatively flat spectrum while the jet has a steep spectrum, which has already been reported in previous studies (Park et al., 2021; Ricci et al., 2022). Using this spectral index map, we constructed the spectral index distribution of the jet by taking the weighted average of the spectral indices in the direction perpendicular to the jet as it moves down the jet pixel by pixel. When calculating the weighted mean, we used pixel values located within one beam size from the jet axis ($\pm$0.9 mas). The error is estimated by multiplying the formal error of the weighted mean by $\sqrt{n}$, where $n$ is the number of the pixel, to compensate for the biases due to correlations between pixel values.

Figure 4 summarizes the results of the spectral index analysis for the NGC 315 jet. In the left panel, the gray solid line shows the radial distributions of the spectral index between 15 GHz and 22 GHz ($\alpha_{\mathrm{15-22\,GHz}}$) as a function of de-projected distance from the SMBH in units of $r_{g}$. The spectral distribution out to 8 mas (corresponding to 49,000 $r_{g}$ in the de-projected distance) from the central black hole is obtained. A steepening of the spectral index is seen until $\sim$2.5 mas, after which it is constant. This behaviour is qualitatively same to the one observed in the M87 jet (Ro et al., 2023). By applying the spectral index distribution model described in § 2.2, here we constrain $B_{i}$ and $q_{\rm inj}$. Using $B_{i}$, the $z$-profile of the magnetic field strengths $B(z)$ over the region 0.75 – 8 mas (corresponding to 5,200 – 49,000 $r_{g}$ in the de-projected distance) is given by

where the relations of $B(z)\propto 1/R\Gamma$, $\Gamma\propto z^{0.3}$, and $R\propto z^{0.58}$ are used (Zamaninasab et al., 2014; Park et al., 2021). To compare the model and the observations, the allowed region is defined based on the observational spectral index distributions with three parameters: (1) The spectral index $\alpha_{i}$ at $z_{i}$, (2) the location of the end of the steepening region ($z_{f,s}$), and (3) the spectral index range after the steepening region ($\alpha_{f}$). These parameters for the NGC 315 jet are $-0.65<\alpha_{f}<-0.2$. $z_{s,f}=2.5$ mas, $-1.3<\alpha_{f}<-0.65$. We then create many modeled spectral index distributions for different values of the parameters $B_{i}$ and $q_{\rm inj}$. The field strength $B_{i}$ was set from 0.1 G to 1.3 G in 0.1 G increment, and the injection index is set as $1\leq q_{\rm inj}\leq 8$. In all cases, we assumed a constant slope of the nonthermal electron injection function as $p_{\rm inj}=-2.0$ (i.e., $\alpha_{\rm inj}=-0.5$, horizontal dashed line in Figure 4, left). We limited the parameters by excluding models that exist outside the allowed range. Thus, applying the synchrotron-emitting jet model of Ro et al. (2023) to the VLBI observation data, we finally obtain the allowed range as $0.5~{}{\rm G}\leq B_{i}\leq 0.9~{}{\rm G}$ together with $2\leq q_{\rm inj}\leq 5$ (Figure 4, right).

We also applied our model to the spectral index distribution between 22 GHz and 43 GHz in the literature ($\alpha_{\mathrm{22-43\,GHz}}$; Figure 2 of Ricci et al., 2022). From this, we examine the spectrum of the region between 0.22 and 2.2 mas ($\sim$1,600 – $\sim$13,500 $r_{g}$ in de-projected distance) from the black hole. However, $\alpha_{\mathrm{22-43\,GHz}}$ shows a fast steepening at $\lesssim$ 0.4 mas, and the amount of steepening is larger than $\sim 0.5$, suggesting so-called fast-cooling regime for nonthermal electrons (Sari et al., 1998). Therefore, the spectral index between 22 GHz and 43 GHz gives the lower limit of $B(z)$ with $B_{i}=0.8$ G at $z=1.6\times 10^{3}~{}r_{g}$.

In Figure 5, we show the estimated $B_{H}$ and $B(z)$ along the NGC 315 jet. In the subsection 4.1, we have derived $B_{H}$ based on the constraints from $L_{\rm BZ}$ and the estimated $\Omega_{F}$ values and it resides in the range $5\times 10^{3}~{}{\rm G}\lesssim B_{H}\lesssim 2\times 10^{4}~{}{\rm G}$ and this is plotted at $z=r_{g}$. In the subsection 4.2, we have also constrained the field strength at the jet downstream region based on the radial distribution of the spectral index between 15 and 22 GHz, and the black area is the corresponding allowed range of $B(z)$ in the range of $0.75~{}{\rm mas}\lesssim z\lesssim 8~{}{\rm mas}$. The gray-shaded region is the extrapolation of the filled-black region. The lower limit of $B(z)$ (blue-colored) is also appended, which is estimated from the radial distribution of the spectral index between 22 and 43 GHz. In addition, other independent estimations of the magnetic field strengths are also plotted. Kimura & Toma (2020) explored multi-wavelength emission spectrum from NGC 315 and suggested that the observed $\gamma$-ray from NGC 315 is well explained by the hadronic emission from MAD. ⁴⁴4 Igumenshchev et al. (2003) found that, given the right initial conditions in MHD simulations, magnetic fields can become dynamically important in BH accretion flows, to the extent that they impede the inward motion of gas and create a ‘magnetically arrested disc’ and this is named as MAD (Narayan et al., 2003), see also Bisnovatyi-Kogan & Ruzmaikin (1974). Hot accretion flows in the MAD regime can launch powerful jets.

To understand physical properties of the accretion flow and the jet base, we make an order estimation of the magnetic field strengths in a MAD regime as a reference value by using the literature value of $\dot{M}$ for NGC 315. In light of Narayan et al. (2003), the magnetic field supports the plasma against the gravitational pull by the black hole. The strength of the magnetic field $B_{\rm MAD}$ that satisfies the force balance between the gravitational pull and the magnetic support can be given by $B_{\rm MAD}\approx(2M_{\bullet}\dot{M}/3r^{3}v_{r,{\rm disk}}h_{\rm disk}/r_{\rm disk})^{1/2}$ where $v_{r,{\rm disk}}$, $h_{\rm disk}$, and $r_{\rm disk}$ are the radial velocity of the accreting matter, the height of the accretion disk, and the radial distance in the accretion disk, respectively (Ricci et al., 2022, and references therein). The estimated $B_{\rm MAD}$ value (the green-shaded region in Figure 5) indicates the baseline that the SANE region will realize if the estimated value is below that value. Here we assume a scale height of the accretion disk $h_{\rm disk}/r_{\rm disk}\approx 1$ and a medium value of $v_{r,{\rm disk}}=0.1c$ which is between the value assumed in Narayan et al. (2003) (i.e., $v_{r,{\rm disk}}\approx 10^{-2}c$) and the value assumed in Ricci et al. (2022) (i.e, $v_{r,{\rm disk}}\approx c$). Unfortunately, the uncertainty of $\dot{M}$ is so large that the divided region is fairly thick. Therefore, the values of $h_{\rm disk}/r_{\rm disk}$ and $v_{r,{\rm disk}}$ does not affect $B_{\rm MAD}$ significantly. As shown in Fig. 5, we find that the extrapolated region (gray-colored) is consistent with MAD regime. The estimated $B_{H}$ in sub-section 4.1 also suggests that the accretion flow of NGC 315 is in a MAD regime. This claim agrees with the previous study of Ricci et al. (2022). In addition, the magnetic field strengths estimated in Kimura & Toma (2020) also reside in the MAD regime.

Previous works develop an analogy of direct current (DC) circuit analysis for power flow through the black hole magnetosphere, known as the membrane paradigm of black holes (Thorne et al., 1986, and references therein). In this context, we consider a meaning of a small $\Omega_{F}/\Omega_{H}\sim 10^{-2}$ indicated in this work. A plasma-filled (i.e., non-vacuum) black-hole magnetosphere resembles a DC circuit. The magnetized rotating black hole acts as a battery driving the current ($I$). Strong magnetic field lines threading the event horizon function like current-carrying wires, allowing charged particles to slide along them. These magnetic wires connect the event horizon to a distant, weakly-magnetized load region, where the current returns to the black hole battery. This DC circuit analysis between a certain two magnetic-field surfaces can give the relation among, $\Omega_{H}$, $\Omega_{F}$, the resistance across the event horizon ($\Delta R_{\rm H}$),and the load region ($\Delta R_{\rm load}$) as follows:

where $\Delta V_{\rm H}=I\Delta R_{\rm H}$, and $\Delta V_{\rm Load}=I\Delta R_{\rm Load}$, are the voltage drop at the event horizon and that of the load region, respectively (see Section IV in the textbook of Thorne et al., 1986). If assuming the power into the load region is maximized, the analogy of a laboratory DC circuit suggests the impedance matching of $\Delta Z_{\rm H}\approx\Delta Z_{\rm Load}$ (MacDonald & Thorne, 1982), leading to $\Omega_{F}\approx\Omega_{H}/2$. GRMHD simulations have also suggested that the ratio $\Omega_{F}/\Omega_{H}$ fits in the similar range of $0.3-0.5$ (e.g., McKinney & Narayan, 2007; Penna et al., 2013). Therefore, $\Omega_{F}/\Omega_{H}\sim 10^{-2}$ indicated in this work may correspond to an impedance mismatching. In a DC circuit, an impedance mismatching causes energy transmission losses. Therefore, to compensate for this lost energy, a relatively large $B_{H}$ in $L_{\rm BZ}$ is needed in this work to meet the required $L_{j}$ for NGC 315. Although detailed physical processes in the load regions in astropysical AGN jet system are unclear, it seems likely that the electric fields of the load region will accelerate charged particles to high energies (Thorne et al., 1986).

Magnetic flux threading the event horizon ($\Phi_{\rm BH}$) is one of the key parameters that characterizes the jet launching (e.g., Zamaninasab et al., 2014). Although the dimensionless magnetic flux threading one hemisphere of the event horizon ($\phi_{\rm BH}=\Phi_{\rm BH}/\sqrt{\dot{M}r_{g}^{2}c}$ where $\Phi_{\rm BH}\approx B_{H}r_{g}^{2}$ is the magnetic flux threading the black hole) is actively discussed in GRMHD simulations (e.g., Tchekhovskoy et al., 2011; Narayan et al., 2022), the strength of $\phi_{\rm BH}$ is highly uncertain in actual jet objects. Therefore, it is worthwhile to estimate a typical $\phi_{\rm BH}$ in order to compare GRMHD simulations and observations. Using the estimated $B_{H}$, we can make the order estimation of $\phi_{\rm BH}$ (in Gaussian-cgs units) for NGC315 as follows:

or equivalently

Since it is generally difficult to estimate $B_{H}$, Zamaninasab et al. (2014) suggested an alternative way where assuming poloidal magnetic flux threading parsec-scale jets $\Phi_{\rm jet}$ is comparable to $\Phi_{\rm BH}$. Using the relation of $\Phi_{\rm BH}\sim\Phi_{\rm jet}$, they indicated that the jet-launching regions of AGN jets are threaded by dynamically important fields, which obstruct gas infall, compress the accretion disk vertically, slow down the disk rotation by carrying away its angular momentum in an outflow and determine the direction of jets. Our estimated value for NGC 315 shown in Eq. (16) agrees with the result of (Zamaninasab et al., 2014).

In addition, it is also worth checking the value of the efficiency with which the black hole generates the jet power ($\eta_{\rm BZ}$) as the ratio of $L_{\rm BZ}$ to the inflow rate at which rest mass energy flows into the black hole (Tchekhovskoy et al., 2011). Together with the assumption of $L_{\rm BZ}\approx L_{j}$, the value $\eta_{\rm BZ}$ for NGC 315 is estimated as

The estimation of $\eta_{\rm BZ}\gtrsim 100~{}\%$ implies that the NGC 315 jet carries away more energy than the entire rest mass energy of the accreted gas, suggesting the need of extracting the rotational energy of the black hole. The estimated order of $\eta_{\rm BZ}$ in Eq. (17) can be comparable to the maximal value of $\eta_{\rm BZ}\sim 300~{}\%$ seen in GRMHD simulations (e.g., Tchekhovskoy et al., 2011; Narayan et al., 2022).