High-energy Gamma-rays from Magnetically Arrested Disks in Nearby Radio Galaxies

The origins of the GeV gamma-rays from nearby radio galaxies are unknown. Hadronic emission from magnetically arrested disks (MADs) around central black holes (BHs) is proposed as a possible scenario. Particles are accelerated in the MAD by magnetic reconnection and stochastic turbulence acceleration. We pick out the fifteen brightest radio galaxies in the GeV band from the Fermi 4LAC-DR2 catalog and apply the MAD model. We find that we can explain the data in the GeV bands by the MAD model if the accretion rate is lower than 0.1% of the Eddington rate. For a higher accretion rate, GeV gamma-rays are absorbed by two-photon interaction due to copious low-energy photons. If we assume another proposed prescription of the electron heating rate by magnetic reconnection, the MAD model fails to reproduce the GeV data for the majority of our sample. This indicates that the electron heating rate is crucial. We also apply the MAD model to Sgr A* and find that GeV gamma-rays observed at the Galactic center do not come from the MAD of Sgr A*. We estimate the cosmic ray intensity from Sgr A*, but it is too low to explain the high-energy cosmic ray intensity on Earth.


INTRODUCTION
Radio-loud active galactic nuclei (AGN) have powerful relativistic jets that have a strong influence on star formation activities in host galaxies and thermodynamics of gases in galaxy clusters. These AGN also exhibit broadband non-thermal emission signatures from radio to GeV-TeV gamma-rays. However, the production mechanism and physical nature of the jets and the non-thermal emission are still unknown (see e.g., Blandford et al. 2019;Hada 2019, for recent reviews).
Blazars, a subclass of radio-loud AGN seen along the jet axis, provide a dominant contribution to the GeV gamma-ray sky (Ackermann et al. 2015;Abdollahi et al. 2020). Owing to the relativistic beaming effect, emission from the jets dominates over the other emission components. Their rapid variabilities also indicate that the gamma-ray emission site should be as compact as sub-pc scales (e.g., Abdo et al. 2011).
Radio galaxies, off-axis counterparts of blazars, are also detected in GeV-TeV gamma-rays (e.g., Inoue 2011;Stecker et al. 2019;MAGIC Collaboration et al. 2020;H. E. S. S. Collaboration et al. 2020;de Menezes et al. 2020a;Tomar et al. 2021). The gamma-ray production sites for radio galaxies are controversial because relativistic beaming effects should be weaker in these objects. Leptonic compact jet models are actively discussed as a standard scenario (e.g., Abdo et al. 2009;MAGIC Collaboration et al. 2020), but at least in M87, this scenario failed to reproduce the magnetic field strength estimated by core-shift measurements in the radio bands (Kino et al. 2015;Jiang et al. 2021). If we assume the strong magnetic fields given by the radio observations, the resulting gamma-ray spectra are far below the observed flux (Event Horizon Telescope MWL Science Working Group et al. 2021). This motivates ones to investigate another scenarios, such as hadronic jets (Reynoso et al. 2011;MAGIC Collaboration et al. 2020), large-scale jets (Hardcastle & Croston 2011), hybrid jets (Fraija & Marinelli 2016), and blackhole (BH) magnetospheres (Hirotani & Pu 2016;Kisaka et al. 2020). However, all the scenarios have some diffi-culties or conflicts with other observations (see Kimura & Toma 2020, and references therein). Kimura & Toma (2020) proposes hadronic emission in magnetically arrested disks (MADs; Bisnovatyi-Kogan & Ruzmaikin 1974;Narayan et al. 2003) as an alternative scenario. Owing to their strong magnetic fields, MADs can launch powerful relativistic jets via Blandford-Znajek mechanism (Tchekhovskoy et al. 2011;McKinney et al. 2012;Event Horizon Telescope Collaboration et al. 2019). Thus, the presence of jets implies the existence of strong magnetic fields in the vicinity of the BH, which suggests that reconnection-driven particle acceleration (Hoshino & Lyubarsky 2012;Guo et al. 2020) taking place in the MAD is important. The accelerated protons emit GeV gamma-rays via the synchrotron process. This model can reproduce the GeV-TeV gammaray data from M87 and NGC 315, but the majority of the GeV-detected radio galaxies are unexplored yet.
The existence of non-thermal particles in accretion flows is supported in terms of both theories and observations. Recent general relativistic magnetohydrodynamic (GRMHD) simulations revealed that MADs can induce magnetic reconnection in highly magnetized plasmas with the magnetization parameter of σ = B 2 /(4πm p n p c 2 ) 1 (Ball et al. 2018;Ripperda et al. 2020Ripperda et al. , 2022. These reconnection events very efficiently produce non-thermal particles, according to particle-incell simulations (Zenitani & Hoshino 2001;Guo et al. 2016;Zhang et al. 2021). Also, accretion flows are turbulent, under which stochastic acceleration process may produce non-thermal particles efficiently (Kimura et al. 2016;Comisso & Sironi 2018;Zhdankin et al. 2018;Kimura et al. 2019).
The multi-wavelength and multi-messenger observations also provide hints of non-thermal signatures in accretion flows. Aartsen et al. (2020) reported a ∼ 3σ high-energy neutrino signal from NGC 1068, a nearby X-ray bright Seyfert galaxy. This motivates ones to consider non-thermal hadronic emissions in accretion flows (Inoue et al. 2019;Murase et al. 2020;Gutiérrez et al. 2021;Kheirandish et al. 2021;Kimura et al. 2021b). GeV gamma-ray detections are also reported from radio-quiet AGN (Wojaczyński et al. 2015;Abdollahi et al. 2020), indicating non-thermal activity in accretion flows. The flaring activities of Sgr A* in infrared and X-rays are also considered as the non-thermal phenomena triggered by magnetic reconnection (Dexter et al. 2020;Porth et al. 2021;GRAVITY Collaboration et al. 2021).
In this paper, we investigate the characteristics of radio galaxies that can be explained by the MAD model by applying the model to fifteen GeV-loud radio galaxies. We also apply our MAD model to Sgr A* to see whether gamma-rays from the Galactic center can originate from the accretion flow. This paper is organized as follows. In Section 2, we describe the MAD model constructed by Kimura & Toma (2020). In Section 3, we classify the radio galaxies by comparing the calculated photon spectra to the gamma-ray data and discuss the characteristics of radio galaxies. We also examine another prescription of the electron heating rate. In Section 4, we also apply the MAD model to Sgr A* and discuss Sgr A* as a cosmic ray (CR) source. In Section 5, we present our conclusions.

MAD MODEL
We calculate the photon spectra with the MAD model constructed by Kimura & Toma (2020). In this model, particles are accelerated by the magnetic reconnection at the edge of the accretion disk (Ball et al. 2018;Ripperda et al. 2020) and the turbulence in the accretion disk (Yuan et al. 2003;Kimura et al. 2016Kimura et al. , 2019. We consider that plasma is accreted onto a supermassive BH of mass M . The mass accretion rate,Ṁ , and the size of emission region, R, are normalized by the Eddington rate and by the gravitational radius, respectively, i.e., M c 2 =ṁL Edd and R = RR G = RGM/c 2 . We use the notation of Q x = Q/10 X in cgs units, except for the BH mass, M (M 9 = M/[10 9 M ]). This model considers the emission by thermal electrons, non-thermal electrons, non-thermal protons, and secondary electronpositron pairs produced by the Bethe-Heitler process (p + γ → p + e + + e − ) and the two-photon interaction (γ + γ → e + + e − ).
We determine the electron temperature by balancing the electron heating rate with the cooling rate (see Appendix A). The electron heating mechanism in the MAD has not been established yet (Rowan et al. 2017;Kawazura et al. 2019). We consider that magnetic reconnection is the dominant electron heating mechanism and use the formalism of Hoshino (2018) as a fiducial prescription. Then, the electron heating rate is given by where m e and m p are the mass of an electron and a proton, respectively, Q p = NT disṀ c 2 is the proton heating rate, NT is the fraction of the non-thermal particle energy production rate to the dissipation rate, and dis is the fraction of the dissipation rate to the accretion rate. The thermal energy of electrons is lost by radiation cooling or advection to the BH. Forṁ higher thanṁ crit given in Appendix A, the radiation cooling balances the heating rate because the cooling rate is efficient owing to the high density and strong magnetic field. Forṁ lower thanṁ crit , the thermal electrons fall to the central BH before they cool. In this case, the electron temperature is estimated as T e /T p Q e /Q p (see Appendix A) (see also Kimura et al. 2021a).
As the escape process, we consider the infall to the BH and the diffusion. The infall timescale is t fall ≈ R/V R , where V R = αV K /2 is the radial velocity, α is the viscous parameter (Shakura & Sunyaev 1973), and is the Larmor radius, ηr i,L is the effective mean free path, η is the numerical factor, B = 8πρC 2 s /β is the magnetic field, ρ =Ṁ /(4πRHV R ) is the mass density, H ∼ (C s /V K )R is the scale height of an accretion disk, C s ≈ V K /2 is the sound speed, and β is the plasma beta.
We phenomenologically estimate the acceleration timescale as where V A = B/ √ 4πρ is the Alfvén velocity. We consider only the synchrotron cooling as the cooling process of primary electrons and secondary electron-positron pairs because the other processes are negligible. We consider the proton synchrotron, pp collision (p + p → p + p + π), photomeson production (p + γ → p + π), and the Bethe-Heitler process as the protons cooling processes. In the range of our investigation, pp collision and photomeson production are inefficient because of the low number density of thermal protons and high threshold energy for photomeson production than the Bethe-Heitler process.
High-energy protons and photons interact with the low-energy photons. In Kimura & Toma (2020), since they consider the lowṁ radio galaxies, the photons by the thermal electrons are dominant as the target photons for the Bethe-Heitler process and the two-photon interaction. For a highṁ, the number density of photons produced by non-thermal particles is comparable to or higher than that produced by thermal electrons, and thus, we take into account all the photons inside the MAD as the target photons for the two-photon interaction and the Bethe-Heitler process. We iteratively calculate the photon and the electron-positron pairs spectra until converged.
3. RESULTS FOR RADIO GALAXIES

Properties of the MAD Model
We show typical photon spectrum by the MAD model for M = 10 9 M andṁ = 10 −3 in Figure 1. The other parameter values are the same as Kimura & Toma (2020) (see Table 1), by which the spectra for M87  Figure 1. The typical broadband photon spectrum by the MAD model forṁ = 10 −3 and M = 10 9 M . The thick and thin lines are the photon spectra after and before internal attenuation by the two-photon interaction, respectively. The black-solid, red-dashed, green-dot-dashed, yellow-dashed, blue-dotted, and purple-dotted lines are the total luminosity by the MAD model, thermal electrons synchrotron and Comptonization, synchrotron by the secondary electron-positron pairs by the two-photon interaction, primary electrons synchrotron, primary protons synchrotron, and synchrotron by the secondary electron-positron pairs by the Bethe-Heitler process, respectively.  and NGC 315 are explained. The thick and thin lines are the photon spectra after and before internal attenuation by the two-photon interaction, respectively. The non-thermal protons emit GeV gamma-rays (bluedotted line), the secondary electron-positron pairs by the Bethe-Heitler process emit TeV gamma-rays (purpledotted line), and the primary electrons and the secondary electron-positron pairs by the two-photon interaction emit X-rays (yellow-dashed and green-dot-dashed lines, respectively) as seen in Figure 1.
Both of the most efficient energy loss timescales of non-thermal protons at the highest energy range, t syn and t diff , have the same dependence ∝ E −1 p , so that either of the energy loss processes dominates over the other in the entire proton energy range. The equality t syn = t diff gives the critical mass, The synchrotron cooling is dominant if M > M crit for a givenṁ, while the diffusion loss is dominant if M < M crit . For a fixed value of M , the magnetic field is stronger for higherṁ, which leads to higher synchrotron power. The diffusive escape timescale is longer for higheṙ m due to a smaller Larmor radius. For the example shown in Figure 1, the synchrotron cooling is dominant, i.e., M > M crit . We find that it is also the case for vast majority of radio galaxies in our sample (see Figures 2  and 3). The analytical estimates of the peak energy and luminosity of the proton synchrotron spectrum in the synchrotron cooling case are given as follows. Because of the hard spectral index of protons, the proton synchrotron spectrum has a peak at the synchrotron frequency for E p = E p,cut . Balancing the synchrotron cooling and acceleration timescales, we obtain E p,cut as GeV. (4) We obtain the peak frequency of the synchrotron spectrum by the non-thermal protons as Since the synchrotron cooling is the dominant energy loss timescale, we can approximate that all the energies used for non-thermal proton acceleration are converted to the synchrotron photon energy. Then, the photon luminosity for the proton synchrotron process is estimated to be L γ,psyn ≈ 4.0 × 10 42ṁ −3 M 9 NT−0.5 dis−1 erg s −1 . (6) We should note that this estimate provides the integrated photon luminosity. The differential photon luminosity given in Figure 1 is lower than L γ,psyn because of the bolometric correction.

Application to the various Radio Galaxies
We search for bright radio galaxies in the GeV gammaray band from the Fermi 4LAC-DR2 catalog (Ajello et al. 2020). We pick up the fifteen brightest objects after excluding Fornax A, M87, and NGC 315. We exclude the Fornax A because the emission region of gamma-rays is extended and the contribution of the core is lower than 18% (Ackermann et al. 2016). We also omit M87 and NGC 315 since these objects are already explained in Kimura & Toma (2020). For Cen A, we use the gamma-ray data of the core while the extended component is also observed. In particular, the HESS data (E γ > 300 GeV) should be from the extended component (H. E. S. S. Collaboration et al. 2020). Then the 100 GeV data should be the sum of the jet component and disk component. The theoretical model for the HESS data predicts that the extended jet contributes to the 100 GeV data very marginally (H. E. S. S. Collaboration et al. 2020). The MAD contribution to the 100 GeV data is uncertain. Thus, we use the 2-20 GeV data for the fitting procedure and restrict the MAD model not to exceed the data above 100 GeV.
We compare the spectra obtained by the MAD model to the observed ones. To evaluate the goodness of fit, we use χ 2 method. χ 2 is the quantity written as where i represents the observational data points, F data,i is the gamma-ray flux data, F model,i is the calculated gamma-ray flux, and σ i is the observational error. In this calculation, we use only the gamma-ray data and change onlyṁ in the parameters. We consider that the GeV data are explained by the MAD model if Q ≥ 0.01, where Q is the probability that χ 2 exceeds the obtained value by Equation (7). We consider that the emission from the jet predominantly contributes to the lowerenergy data. The photon flux from radio galaxies shows some variability in all the energy bands, and we regard them as the jet contribution. Thus, the contribution by the MAD model should be below the lowest data points in radio to X-ray bands. We classify the results into three; Excellent, Good, and Bad. We classify objects into Excellent if we can explain the gamma-ray data with the parameters in Table 1 and the cataloged value of M . We show the values of M , distance from Earth,ṁ, χ 2 /ν , and Q for the Excellent objects in Appendix B, where ν = N − m is the degree of freedom, N is the number of the data, and m is the number of the changing parameters. Since we only changeṁ, we set m = 1. We also show the photon spectra of these objects in Appendix B. We find that the accretion rates of all the Excellent objects are less than 10 −3 .
For some objects, it is hard to explain the gammaray data with the parameters in Table 1 and the cataloged value of M . This is because the GeV gamma-rays have the cut-off by the two-photon interaction. In order to achieve the high GeV gamma-ray flux, we may use higher values of R and M . The uncertainty of M is about a factor of 3 (e.g., Kormendy & Ho 2013). We classify objects into Good if we can explain the GeV data with R = 30 and M three times higher than the cataloged value. We only changeṁ during the fitting procedure. Thus, we calculate Q with m = 1. We show the resulting quantities and the photon spectra for the Good objects in Appendix B. The accretion rates of the Good objects are around 10 −3 . Owing to the larger emission region, absorption by the two-photon interaction is suppressed, which enables the MAD model to explain GeV data forṁ 10 −3 . The other objects are classified as Bad. We show the photon spectra for the Bad objects and the quantities for these spectra in Appendix B. There are two types of Bad objects. One type has a cut-off due to the twophoton interaction below the GeV energy, which leads to a mismatch in the multi-GeV data. The other type has luminous synchrotron emission by the secondary electron-positron pairs by the two-photon interaction, which overshoots the X-ray data.
To see the features of the radio galaxies, we plot M andṁ for the objects of the three classes in Figure 2, where values of M andṁ for individual objects are tabulated in tables in Appendix B. As can be seen, we can explain the gamma-ray data by the MAD model ifṁ is lower than 10 −3 . The number density of low-energy photons is higher for a higherṁ, and then, the twophoton interaction is more efficient. Thus, the photon spectra by the MAD model have the cut-off below the GeV range, and we cannot explain the gamma-ray data for a higherṁ. For the jet model, GeV gamma-ray absorption is inefficient owing to the large emission region, and thus, we consider that the GeV gamma-rays come from the jet for highṁ radio galaxies.  The electron heating rate by magnetic reconnection has not been established yet. We also examine another prescription of the electron heating rate given by Chael et al. (2018),

Another Formalism of the Electron Heating Rate
where β max = 1/(4σ). We show the photon spectra and the resulting quantities of the objects in Appendix C. We calculate the photon spectra for all the objects with this electron heating rate and classify them as we have done in Section 3.2 by changingṁ with the same parameter set. The classification results are shown in Figure  3, where we see that all the classes (Excellent, Good, Bad) equally scatter in the M -ṁ plane. We find that Q e /Q p ∼ 0.3 if we use Equation (8) with the parameters in Table 1. On the other hand, Q e /Q p ∼ 0.07 by Equation (1). The value of Q e /Q p corresponds to the luminosity of the electrons, and thus, the luminosities in radio and X-ray bands are high if we use Equation (8). This causes the model flux to overshoot the radio and X-ray data if we adjustṁ so that the resulting gammaray spectra match the GeV data. Equation (8) leads to 0.2 < Q e /Q p < 0.4 for 5 ≤ r ≤ 30 and 0.01 ≤ β ≤ 1. Thus, we cannot reconcile the results in Section 3.2 even with a different parameter set. This indicates that the electron heating rate is crucial to explain the gamma-ray data by the MAD model.

SGR A*
Observations in the radio and X-ray bands imply that Sgr A* at the Galactic center has a hot accretion flow (Narayan et al. 1995;Manmoto et al. 1997;Yuan et al. 2003;GRAVITY Collaboration et al. 2021). Sgr A* is thought to have a MAD because the wind accretion by Wolf-Rayet stars can provide sufficiently large-scale magnetic flux (Ressler et al. 2020). A MAD is also expected to be formed in a lowṁ system (Kimura et al. 2021c), and Sgr A* is known to be a very low accretor. According to the observations by Event Horizon Telescope Collaboration, the time variability suggests a weakly magnetized accretion disk, but the other constraints favor a MAD (Akiyama et al. 2022a,b). Here, we apply the MAD model to Sgr A*. We show the parameters in Table 2 and the photon spectrum in Figure  4. For Sgr A*, NT needs to be much lower than that for the other radio galaxies to match the radio and Xray data. We also find thatṁ is too low to explain the GeV-TeV gamma-ray data. For a lowerṁ, the diffusion timescale is much shorter than the synchrotron cooling timescale. Consequently, the radiative efficiency of the non-thermal protons is low. We cannot explain the GeV-TeV data even with NT = 0.5 if we adjustṁ to reproduce radio data and ignore the X-ray data. As long as we use the same value of NT for electrons and protons, it is difficult to reproduce the GeV-TeV data and low-energy (radio to X-ray) data simultaneously. The angular resolution of the GeV-TeV gamma-ray observation is about 0.1 degrees 1 . This corresponds to 200 pc for the length scale at the Galactic center, within which many other GeV-TeV gamma-ray source candidates exist. We consider that other accretion models cannot explain GeV-TeV data because the NT = 0.5 of the MAD model is close to the theoretical upper limit, and thus, we conclude that the sources of GeV-TeV gamma-rays are other objects in the Galactic center region. Recent experiments show the distribution of CRs is anisotropic, and Galactic CRs of higher energies (E p 300 TeV) come from the direction of the Galactic center (Aartsen et al. 2013;Amenomori et al. 2017). We investigate the CR intensity produced at Sgr A*. The luminosity of the CRs injected from the accretion disk of Sgr A* is approximated as L p . This is because the diffusion 1 https://fermi.gsfc.nasa.gov/science/instruments/table1-1.html  timescale is much shorter than the synchrotron cooling timescale for Sgr A*. In Kimura et al. (2018), the CR luminosity escape from the galaxy is given by L esc (E p ) = E p U Ep cM gas /X esc , where U Ep is the differential energy density of the CRs, M gas ∼ 10 10 M is the mass of the gas inside the Galaxy, X esc ∼ = 8.7r −1/3 1 g cm −2 is the grammage, and r 1 = (E p /e)/(10 GV). Assuming the steady state, we estimate the E p U Ep from the balance of the CRs injected to and the escape from the interstellar medium. The cut-off energy for Sgr A* is E p,cut ≈ 2.5 × 10 7 GeV. For E p = E p,cut , we estimate the CR intensity as ≈ 2.1 × 10 −8 GeV s −1 cm −2 sr −1 .
The CR intensity obtained by the CR experiments is 1.2 × 10 −5 GeV s −1 cm −2 sr −1 at E p = 2.8 × 10 7 GeV (Amenomori et al. 2008). Thus, the contribution by Sgr A* is too low with the current activity. Sgr A* is expected to be more active hundreds of years ago (Koyama et al. 1996;Murakami et al. 2000) and may produce a larger amount of CRs that can explain TeV gamma-ray from the Galactic center region (Fujita et al. 2015;HESS Collaboration et al. 2016). The activity of Sgr A* around 10 Myr ago may create the Fermi and eROSITA bubbles (see Su et al. 2010 for the Fermi observation and Predehl et al. 2020 for the eROSITA bubbles; see e.g., Mou et al. 2014;Sarkar et al. 2017;Yang et al. 2022 for theoretical models). If this activity also produces CRs efficiently, Sgr A* can explain the CR intensity of the present-day around the Knee observed on Earth (Fujita et al. 2017). If the past activities are in the MAD state, CRs can be accelerated to higher energies with an enhanced production rate. This may account for the light-mass galactic CRs reported in Buitink et al. (2016)

CONCLUSION
We statistically investigate the features of radio galaxies explained by the MAD model constructed by Kimura & Toma (2020). We apply this model to the fifteen brightest GeV-loud radio galaxies picked out from the Fermi 4LAC-DR2 catalog. We classify these objects into three; Excellent, Good, and Bad, by comparing the spectra by the MAD model to the gamma-ray data. We find that we can explain the gamma-ray data by the MAD model if the accretion rate is lower than 0.1% of the Eddington rate, while it is challenging to reproduce gamma-ray data for highṁ objects (see Figure 2). Foṙ m 10 −3 , the number density of the low-energy photons is so high that GeV gamma-rays cannot escape from the system due to efficient two-photon interactions. In this case, we consider that the GeV gamma-rays come from the jet rather than the disk because GeV gammaray absorption by the two-photon interaction is inefficient owing to the large emission region for the jet model.
For the Bad objects, we cannot reproduce the GeV gamma-rays by the MAD model, but their accretion disks could be in the MAD states. Forṁ 0.1, the accretion disk is radiatively inefficient accretion flow (see e.g., Mahadevan 1997;Xie & Yuan 2012) and could have strong magnetic field owing to the rapid advection. Nevertheless, the thin disk can be formed around 100 − 1000R G for a relatively high accretion rate, saẏ m 0.01 − 0.1, and in this case, the accumulation of the large-scale magnetic field may be so inefficient that the accretion disk around a BH can be weakly magnetized accretion flow (see e,g., Esin et al. 1997;Kimura et al. 2021c). The critical accretion rate above which a MAD is no longer formed is still unclear. Kayanoki & Fukazawa (2022) reported that GeV-loud objects with highṁ tend to have a low X-ray absorption column density, which implies that a viewing angle may be small. On the other hand, GeV-loud objects with loẇ m can have a high column density (see their Figure 5). This implies a large viewing angle, with which emission from the jet should be weaker due to the low Doppler factor. These features may support our conclusions that the lowṁ objects emit gamma-rays by MADs, while highṁ objects emit gamma-rays by jets.
The electron heating rate by magnetic reconnection has not been established yet. We examine another formalism of the electron heating rate given by Chael et al. (2018). The value of the electron heating rate is higher than that of Hoshino (2018). This results in high optical and X-ray fluxes, which easily overshoot the observational data if we adjust theṁ using gamma-ray data. Thus, more than half of our sample are classified as Bad. This feature is independent of the value of M andṁ. Thus, the electron heating rate has a strong influence on whether we can explain the GeV gamma-ray data by the MAD model.
We also apply the MAD model to Sgr A*. Since Sgr A* has a lowṁ, the gamma-ray emission efficiency is very low, and thus, we cannot explain the gamma-ray data by the MAD model. We conclude that the sources of GeV-TeV gamma-rays are other objects in the Galactic Center. We also estimate the CR intensity of Sgr A* and compare the observed one. Because of lowṁ, the contribution by Sgr A* with the current activity is too low. The Sgr A* may have been active in the past, and it may contribute to super-knee cosmic rays observed on Earth.
We thank Masaomi Tanaka for his helpful comments. This work is partly supported by JSPS KAKENHI No. 22K14028 (S.S.K.) and 18H01245 (K.T.). This work is also supported by JST, the establishment of university fellowships towards the creation of science technology innovation, Grant Number JPMJFS2102 (R.K.). S.S.K. acknowledges the support by the Tohoku Initiative for Fostering Global Researchers for Interdisciplinary Sciences (TI-FRIS) of MEXT's Strategic Professional Development Program for Young Researchers.

A. THE CRITICAL MASS ACCRETION RATE FOR THERMAL ELECTRONS
The temperature of the thermal particles is obtained by the balance between the heating and the energy loss rates: Q i = Λ adv,i +Λ rad,i , where i is the particle species, Λ adv,i ≈ n i k B T i /t fall is the advection rate and Λ rad,i ≈ n i k B T i /t rad,i is the radiation cooling rate. The thermal protons do not cool in the range of our interest, and thus, Λ rad,p = 0. Then, the proton temperature is always given by Λ adv,p = Q p . For a very low accretion rate, advection is dominant even for thermal electrons. This leads to Λ adv,e = Q e , and then, we obtain T e /T p = Q e /Q p . For the range of our interest, the thermal synchrotron is the most efficient cooling process, whose cooling rate is given in Kimura et al. (2021b). Equating the thermal synchrotron cooling rate to the heating rate with the elecrtron temperature determined by advection, we obtain the critical mass accretion rate above which the radiative cooling is effective: where x M = ν thml,peak /ν syn , ν thml,peak is the peak frequency of the synchrotron spectrum by the thermal electrons, ν syn = 3θ 2 e eB/(4πm e c) is the synchrotron frequency, and θ e = k B T e /(m e c 2 ).ṁ crit strongly depends on R and x M . For our fiducial parameter set,ṁ crit is very low, and the radiation cooling is dominant for all of our radio-galaxy samples. On the other hand, advection is dominant for the cases with R = 30, i.e., for the Good and Bad objects. Also, we find that advection is dominant for Sgr A*, due to a small value of x M 54.
The radiation timescale is shorter than the advection timescale for the thermal electrons ifṁ >ṁ crit . The radiation cooling leads to a lower electron temperature than that determined by advection. Thus, the electron temperature should be in the range of

B. PHOTON SPECTRA AND RESULTING QUANTITIES FOR VARIOUS RADIO GALAXIES
We show the photon spectra for the various radio galaxies with the electron heating rate of Hoshino (2018). The photon spectra for the Excellent, Good, and Bad objects are shown in Figure 5, Figure 6, and Figure 7, respectively. We tabulate the resulting quantities for the Excellent, Good, and Bad objects in Table 3, Table 4, and Table 5, respectively. For LEDA 58287, the MAD model underpredicts the GeV gamma-ray flux. However, because of their large error bars and the small number of data points, this object statistically results in a Good object.

C. PHOTON SPECTRA AND RESULTING QUANTITIES FOR THE VARIOUS RADIO GALAXIES WITH ANOTHER ELECTRON HEATING PRESCRIPTION
We show the photon spectra of the various radio galaxies with the prescription given by Chael et al. (2018). The photon spectra of the Excellent, Good, and Bad objects are shown in Figure 8, Figure 9, and Figure 10, respectively. We tabulated the resulting quantities in Table 6.       Table 3, but for Bad objects. The BH mass are enhanced by a factor of 3 from the values in the references.    Figure 9. Same as Figure 6, but with the electron heating rate given by Chael et al. (2018). Data points are taken from de Menezes et al. (2020b) for NGC 315.