Interpreting correlated observations of cosmic rays and gamma-rays from Centaurus A with a proton blazar inspired model

The EM SED from central part of Cen A exhibits multiple peaks; the lower energy peak at 0.1 eV (infrared), and the higher energy peak at 170 keV (soft gamma-rays) are generally explained by the leptonic models assuming a compact core emission region Chiaberge01 ; Abdo10a ; Petropoulou14 . Here we consider a moving spherical blob of size $R_{b}^{\prime}$ (primed variables for the jet frame) in the AGN jet is responsible for the non-thermal emission and contains a tangled magnetic field of strength $B^{\prime}$. If $\Gamma_{j}=1/\sqrt{1-\beta_{j}^{2}}$ be the bulk Lorentz factor of the blob then Doppler factor of the moving blob can be written as $\delta_{ob}=\Gamma_{j}^{-1}(1-\beta_{j}\cos\theta)^{-1}$ where $\theta$ is the angle between the line of sight and the jet axis.

Both protons and electrons are accelerated in the spherical blob but due to small radius (sub-persec) of the blob and relatively large magnetic field, the maximum energy of the accelerated cosmic rays will be limited. We assume that the particles are accelerated up to TeV energies in the blob. The shock accelerated accelerated electrons in the blazar jet are assumed to follow a broken power law energy distribution having spectral indices $\alpha_{1}$ and $\alpha_{2}$ before and after the spectral break at Lorentz factor $\gamma_{b}^{\prime}$ to explain the low-energy bump of the EM SED by synchrotron radiation and can be written as Banik19

where $\gamma_{e}^{\prime}=E_{e}^{\prime}/m_{e}c^{2}$ is the Lorentz factor of electrons of energy $E_{e}^{\prime}$. The normalization constant $K_{e}$ can be found using the relation for the kinetic power of accelerated electrons ($L_{e}^{\prime}$) in the blob frame as given in Bottcher13 ; Banik19 . The synchrotron radiation of the primary accelerated electrons describes the low-energy bump of the EM SED emitted from the core of Cen A and can be numerically estimated following Bottcher13 ; Banik19 . The inverse Compton (IC) scattering of primary accelerated electrons with the seed synchrotron photons co-moving with the AGN jet (i.e, the SSC model) describes the lower part of the high energy component (up to few GeV energies) of EM SED of the blazar Cen A following the expressions given in Banik19 ; Blumenthal70 ; Inoue96 .

Due to the large inclination angle of the jet $\theta$, such emission is not supposed to be strongly Doppler boosted. This Doppler factor is considered to be close to unity $\delta_{ob}\sim 1$ in case of Cen A. However, along the jet axis $\theta=0^{\circ}$, the Doppler factor well be large, i.e, $\delta_{j}\approx 2\Gamma_{j}$ with any reasonable values of $\Gamma_{j}$. Here we consider $\Gamma_{j}\sim 3$ for the sub-persec jet of Cen A. Therefore, along the jet axis such emissions should be relativistically boosted which results in a much stronger emission along the jet than measured directly from Cen A. In such a case, the beamed radiation produced in the inner jet can be a dominating radiation field in kpc-scale jet region other thermal emission from an accretion disc, a broad line region (BLR) or a dusty torus (DT) for inverse-Compton scattering with relativistic electrons as discussed in next section Bednarek19 .

In the proton blazar framework, the shock accelerated cosmic rays may interact with the cold matter (protons) of density $n_{H}=(n_{e}^{\prime}-{\small\displaystyle\sum_{Z,A}}Zn_{A}^{\prime})$ in the blob of AGN jet and produce high energy gamma-rays and neutrinos. Here $n_{e}^{\prime}$ and $n_{A}^{\prime}$ are represents total number of electrons and cosmic ray nuclei (atomic number $Z$ and mass number $A$) present in that region of AGN jet respectively as described in next section. However, the contribution of hadronically generated gamma-rays can be restricted far below the inverse-Compton contribution (or Fermi$-$LAT observation) by suitable choice of cosmic ray luminosity. As for example, when cosmic ray luminosity is chosen as $3\times 10^{42}$ ergs the gamma-rays produced in hadronic interaction is found nearly one order less than the inverse-Comption contribution when a power law energy spectrum with slope parameter $-2.5$ is considered and the acceleration efficiency is taken as $10^{-3}$.

EM Emission from the kiloparsec-scale jet of Centaurus A has been studied from the radio, the infrared to the X-ray regime. Hardcastle06 have shown the flux observations by radio VLA (1.4, 4.9 and 8.4 GHz), infrared Spitzer (24 and 5.4 $\mu$m), ultraviolet GALEX (231 nm) and Chandra X-ray (1 keV) for three separate regions of the jet. In order to study the high-energy gamma-ray flux observed by HESS from the large scale jet (up to few kpc in projection) of the source, we consider their observations for the inner region (corresponding to 2.4$-$3.6 kpc in projection).

Similar to the core emission region, here we have considered a second moving spherical blob with a larger radius at 1 kpc away from the core of Cen-A. In our model this is the high-energy gamma-ray emission region responsible for the HESS and partially Fermi$-$LAT observations from Cen A.

The relativistic electrons and cosmic ray nuclei can be assumed to be accelerated (or re-accelerated) in the kpc-scale jet in diffusive shock acceleration by parallel collision-less shock. The accelerated relativistic electrons will obey a broken power-law energy distribution as given in Eq. (1) and their synchrotron emission can describe the radio to X-ray observations of the EM SED from the kpc-scale jet of the source. The synchrotron-self absorption effect is incorporated following Chen17 . The number density and energy density of relativistic (‘hot’) electrons are $n_{e,h}^{\prime}=\int N_{e}^{\prime}(\gamma_{e}^{\prime})d\gamma_{e}^{\prime}$ and $u_{e}^{\prime}=\int m_{e}c^{2}\gamma_{e}^{\prime}N_{e}^{\prime}(\gamma_{e}^{\prime})d\gamma_{e}^{\prime}$ respectively. The synchrotron emission of the electron distribution generally describes the low-energy component of the EM SED of the AGN, which can be numerically estimated following Bottcher13 ; Banik19 . As elaborated in Banik19 ; Banik20 , the acceleration efficiency $\chi_{e}$ of electrons in the AGN jet is likely to be quite low and total number electrons including ‘hot’ and non-relativistic (‘cold’) electrons can be determined as $n_{e}^{\prime}=n_{e,h}^{\prime}/\chi_{e}$. In the present work, we have considered the values of $\chi_{e}\sim 10^{-3}$, which is within the permitted range Eichler05 ; Banik20 and compatible with the hybrid simulation results of diffusive shock acceleration by parallel collision-less shock Gia92 ; Bykov96 .

The Fermi$-$LAT discovery of GeV hardness in the year 2013 and the TeV emission observed during 2004-2008 with H.E.S.S. often require a distinct, separate spectral component to explain. The inverse Compton (IC) scattering of primary accelerated electrons with the soft photon radiation field which is boosted from the core region to the in kpc-scale jet region, has been estimated following the expressions given in Banik19 ; Blumenthal70 ; Inoue96 . The mechanism is found to contribute significantly in the observed EM SED in GeV to TeV energy range of the source. The differential population of soft photons $n_{j}^{\prime}$ coming from the core region at the location of the kpc-scale jet at the distance $r$ along the jet can be estimated as Bednarek19

where $f_{\epsilon}$ (in erg cm^-2 s^-1) represents the observed photon flux from the core of Cen A at earth, $\epsilon_{j}^{\prime}=\delta_{j}\epsilon_{j}$ relates photon energies (in $m_{e}c^{2}$ unit) in the observer and the boosted photon energy in the kpc jet frame respectively with red shift parameter $z$, and $d_{L}$ is the luminosity distance between the AGN and the Earth.

In the framework of proton blazar model, the cosmic ray protons are also supposed to be accelerated in AGN jet and here we consider a mixed composition scenario for cosmic rays accelerated by Cen A. The cosmic ray energy spectrum of element atomic mass number $A$ can be assumed as $N_{A}^{\prime}(\gamma_{A}^{\prime})=f_{A}K{\gamma_{A}^{\prime}}^{-\alpha}\exp({-\frac{\gamma_{A}^{\prime}}{\gamma^{\prime}_{A,max}}})$ where $\alpha$ is the spectral index, $\gamma_{A}^{\prime}=\frac{E_{A}^{\prime}}{Am_{p}c^{2}}$ is the Lorentz factor of injected cosmic rays of energy $E_{A}^{\prime}$ in AGN jet frame and $f_{A}$ (for $\gamma^{\prime}_{A}<\gamma^{\prime}_{p,max}$) represents the relative number abundance of the accelerated cosmic ray nuclei. Also, $\gamma^{\prime}_{A,min}=A\gamma^{\prime}_{p,min}$ and $\gamma^{\prime}_{A,max}=Z\gamma^{\prime}_{p,max}$ are the minimum and maximum Lorentz factor of accelerated cosmic rays with atomic number $Z$ and mass number $A$ respectively. The normalization constant $K$ is related with the kinetic power in relativistic cosmic rays in the AGN jet frame $L_{CR}^{\prime}$ via the relation Banik19

where $n_{A}^{\prime}=\int N_{A}^{\prime}(\gamma_{A}^{\prime})d\gamma_{A}^{\prime}$ denotes the corresponding number density of individual cosmic ray nuclei and $u_{CR}^{\prime}={\small\displaystyle\sum_{Z,A}}\int m_{p}c^{2}\gamma_{A}^{\prime}N_{A}^{\prime}(\gamma_{A}^{\prime})d\gamma_{A}^{\prime}$ represents the corresponding energy density for (mixed compositions) relativistic cosmic rays in the blob of a blazar jet.

The gamma-ray emissivity with photon energies $E_{\gamma}^{\prime}=m_{e}c^{2}\epsilon_{\gamma}^{\prime}$ in the co-moving jet frame produced in cosmic ray interactions with cold proton present in the system under charge neutrality condition, $Q^{\prime}_{\gamma}(\epsilon^{\prime}_{\gamma})={\small\displaystyle\sum_{Z,A}}Q^{\prime}_{\gamma,Ap}(\epsilon^{\prime}_{\gamma})$ can be obtained following Eq. (9) of Banik19 after incorporating the effect of heavier cosmic ray nuclei as given in Banik17 ; Banik17b ; Banik18 . Due to internal photon-photon ($\gamma\gamma$) interactions as given in Aharonian08 ; Banik19 , the produced TeV$-$PeV gamma-rays can be absorbed while propagating through an isotropic low-frequency radiation field. The electrons/ positrons produced in $\gamma\gamma$ pair production and those produced directly due to the decay of $\pi^{\pm}$ mesons created in pp interaction will initiate EM cascades in the AGN blob via the IC scattering and the synchrotron radiation Bottcher13 . We found that this cascade emissions does not have any significant contribution to the overall spectrum in case of Cen A.

Following Banik19 we estimated $Q_{\gamma,esc}^{\prime}(\epsilon_{\gamma}^{\prime})$ which denotes the total gamma-ray emissivity from the blob of AGN jet including all processes stated above i.e, the synchrotron and the IC radiation of accelerated electrons, the gamma-rays produced in $pp$ ($pA$) interaction and also the synchrotron photons of EM cascade electrons. Therefore, the observable differential flux of gamma-rays reaching at the Earth from a blazar can be evaluated from the relation

where the volume of the emission region is $V^{\prime}=\frac{4}{3}\pi R_{b}^{\prime 3}$ and $E_{\gamma}=\delta_{ob}E_{\gamma}^{\prime}/(1+z)$ relates the photon energies in comoving jet of redshift parameter $z$ and the observer frame, respectively. Here we have included the effect of the absorption by the extragalactic background (EBL) light on gamma-ray photons and $\tau_{\gamma\gamma}^{EBL}(\epsilon_{\gamma},z)$ is the corresponding optical depth which can be evaluated using the Franceschini-Rodighiero-Vaccari (FRV) model Franceschini08 ¹¹1http://www.astro.unipd.it/background/.

The correlated muon neutrino flux reaching at the Earth can be evaluated as

where $Q_{\nu}^{\prime}$ denotes the corresponding neutrino emissivity from the AGN jet Banik19 with energies $E_{\nu}^{\prime}=m_{e}c^{2}\epsilon_{\nu}^{\prime}$ in co-moving jet frame, $\xi=1/3$ is a fraction which is included due to neutrino oscillation and $E_{\nu}=\delta_{ob}E_{\nu}^{\prime}/(1+z)$ Atoyan03 provides the relation between the neutrino energies in the observer and co-moving jet frame respectively.

The UHECRs may remain confined over a moderate time period within the AGN jet but thereafter they will slowly escape. Several instabilities grow in the process of confinement of charged particles by magnetic fields and the energetic cosmic rays will eventually, slowly escape along the field lines of the irregular field. The emitted cosmic ray particles may scatter by the extragalactic magnetic field while propagating from the source to us. The strength of the extragalactic magnetic field near the Milky Way is not clearly known. The observation of UHECR events from a sky location within $18^{\circ}$ around Cen-A indicates that the scattering is small (at least for cosmic rays above 55 EeV) and thus the magnetic field is weak. Such weak magnetic fields will only slightly enhance the travel path for Cen-A and one thus may approximate that the cosmic rays of such high energies travel straight from the source to the Earth Gopal10 . The emitted UHECRs while propagating from the source to us can suffer different interaction processes such as pair production, pion production, and photodisintegration Puget76 ; Allard12 . The pair production causes a continuous energy loss which starts to contribute significantly just above its energy threshold ($\sim 10^{18}eV$) while pion production starts to dominate around 70 EeV. The photodisintegration is the dominant energy loss process for UHE cosmic ray nuclei. The main background photon fields in propagation of UHECRs are cosmic microwave background and the radiation field of higher energies viz. the IR/Opt/UV background. The pair and pion production in interaction with the IR/Opt/UV is almost insignificant compare to those produced in interaction with CMB. In the nuclear photodisintegration process the nucleus is supposed to form a compound state due to photoabsorption, followed by a statistical decay process involving the emission of one or more nucleons from the nucleus. In the concerned energy range ($55-85$ EeV) of UHECRs the contribution of photodisintegration by CMB photons dominates over that by IR/Opt/UV backgrounds Allard12 . The expected UHECR flux reaching at the earth from nearby extragalactic sources within 75 Mpc like Cen A can be estimated as

where $Q_{A}^{\prime}(\gamma_{A}^{\prime})=\frac{c}{R_{b}^{\prime}}N_{A}^{\prime}(\gamma_{A}^{\prime})$ represents the effective emissivity of UHECRs in AGN jet frame and $E_{A}=\delta_{ob}E_{A}^{\prime}/(1+z)$ relates the cosmic ray energies in comoving jet and the observer frame, respectively. Here we have considered the possible absorption of UHECRs due to photo-meson and photo-disintegration interactions with cosmic microwave background while traveling towards us from the AGN Cen A and $\tau_{A}$ is the corresponding optical depth which can be evaluated following (Banik17b and references therein).

The corresponding expected UHECR events within the observed energy range between $E_{1}$ and $E_{2}$ in PAO from the Cen A can be evaluated as

where $\Xi=20370$ km²sr yr is the exposure of the detector Abreu10 , $\omega(\delta_{s})=0.64$ is the relative exposure for the declination angle ($\delta_{s}=-47$) and $S_{60}=\pi$ sr is the acceptance solid angle used in PAO corresponding to zenith angle $\leq 60^{\circ}$ Cuoco08 .

To interpret the EM SED of Cen A over the optical to gamma-ray energy range, we have assumed the size of the blob region within the jet to be $R_{b}^{\prime}=2.2\times 10^{15}$ cm with Doppler boosting factor $\delta_{ob}=1.1$ and bulk Lorentz factor of AGN jet $\Gamma_{j}=3.05$. The size of the emitting region is consistent with the size inferred from the variability, namely $R_{b}^{\prime}\lesssim\delta_{ob}ct_{ver}/(1+z)\simeq 3.3\times 10^{15}(\delta_{ob}/1.1)(t_{ver}/10^{5}s)$ cm assuming a typical variability timescale of $t_{ver}\simeq 10^{5}$ s. Adopted choices of Doppler boosting factor and bulk Lorentz factor of the AGN jet in core region is consistent with the upper bounds constrain of those ($\delta_{ob}\sim 1$ and $\Gamma_{j}\lesssim 3-5$) as given in Chiaberge01 .

The synchrotron emission of primary accelerated electron’s distribution obeying a broken power law with spectral indices $\alpha_{1}=1.78$ and $\alpha_{2}=4.5$ respectively before and after the spectral break at the Lorentz factor $\gamma_{b}^{\prime}=1.2\times 10^{3}$ is found to explain well the lower energy bump of the experimental EM SED data of Cen A. The magnetic field of $B^{\prime}=5.6$ G and the kinetic power of primary relativistic electrons of $L_{e}^{\prime}=10^{42}$ erg/s in blazar jet are adopted to establish the observed low energy bump of the EM spectrum. The magnetic field used here is larger than the typical values in the core of BL Lacs which rarely exceed 1 G Tavecchio10 ; Inoue16 . However, several recent analysis Abdo10a ; Tanada19 indicate for a very large magnetic field of a few (4$-$6) Gauss in the core region of Cen-A which is probably because of the close proximity of the core emission region of Cen-A to its nucleus Tanada19 . The primary relativistic electrons may interact with the synchrotron photons co-moving with the AGN jet via IC scattering and consequently found to produce the high-energy bump of the EM spectrum up to few GeV energies. The model fitting parameters to match the observed EM SED from the core region of the source are summarized in Table 1.

In this case, we have chosen the size of the emission region within the jet to be $R_{b}^{\prime}=1.9\times 10^{20}$ cm in the region of kpc-scale jet from the nucleus of the source moving with Doppler boosting factor $\delta_{ob}=1.1$ and bulk Lorentz factor of AGN jet $\Gamma_{j}=1.009$ to describe the high energy EM SED observed apparently from kpc-scale region of Cen A as well the radio to x-ray observations Hardcastle06 from the same part of the jet. Our estimated value of bulk Lorentz factor for kpc-scale jet is consistent with the estimation from the proper motional speeds of the knots monitored by the VLA in kpc-scale jet Goodger10 ; Tanada19 .

The spectral indices $\alpha_{1}$ and $\alpha_{2}$ of the broken power-law electron distribution are taken $2.31$ and $3.85$, respectively, before and after the spectral break at the Lorentz factor $\gamma_{b}^{\prime}=6.4\times 10^{5}$, which is well reproduce the lower energy bump of the experimental EM SED data of Cen A via synchrotron emission. The magnetic field and kinetic power of relativistic electrons in the blazar jet required to explain the EM spectrum are $B^{\prime}=100\times 10^{-6}$ G and $L_{e}^{\prime}=1.83\times 10^{42}$ erg/s, respectively, in blazar jet. The IC scattering of this primary accelerated electrons distribution with soft radiation field which is boosted from the core region as discussed above is found to contribute significantly to the measured EM spectrum by Fermi$-$LAT and HESS in Gev$-$TeV energy range. Unlike the Bednarek19 , the mechanism cannot alone describe observed GeV to TeV energy gamma-rays for this work, as we consider the appropriate accelerated electron distribution required to establish the the radio to x-ray observations from the kpc-scale jet.

Theoretically, the maximum energy achievable in a relativistic source with the comoving size $R_{b}^{\prime}$ and magnetic field strength, $B^{\prime}$, can be obtained as $E_{A,max}^{\prime}\leq ZeB^{\prime}R_{b}^{\prime}\approx Z\times 5.7\times 10^{18}$ eV in comoving jet frame Hillas84 . Here, we consider the mixed composition of primary cosmic ray nuclei with maximum energy $E_{A,max}^{\prime}=Z\times 4.4\times 10^{18}$ eV that consistent with the Hillas criterion as stated above and then we proceed in the following ways:

i) Firstly, we have considered a mixed composition of cosmic rays composed of H¹, He⁴, C¹², O¹⁶, Si²⁸ and Fe⁵⁶ elements and we have chosed abundances ($f_{A}$) of various elements and their production spectrum in the jet frame in such a way to match the average atomic mass of UHECR events observed by PAO. We have estimated the average atomic mass of UHECRs for the mixed composition through the expression

ii) We have taken the same value of $f_{A}$ and cosmic ray production spectral slope $\alpha$ ($-4.3$) Liu12 of UHECRs above 55 EeV as observed by PAO which further restrict the model parameters.

iii) We have chosen injected cosmic ray luminosity $L_{CR}^{\prime}$ in the AGN jet so as to produce observed $9.8$ UHECR events in Auger detector within the energy range between $E_{1}>55$ EeV and $E_{2}=85$ EeV from Cen A.

iv) Finally, we have computed EM (gamma-ray) spectrum given by the proton blazar inspired model using the parameters most of which are already fixed from the above considerations.

The expected UHECR flux of different elements with their relative abundance reaching the Earth from the jet of Cen A along with their probable average atomic mass are shown in Fig. 1. Our findings suggest a relative number abundance of the accelerated cosmic ray nuclei to be $f_{H}:f_{He}:f_{C}:f_{O}:f_{Si}:f_{Fe}\approx 0.54:0.27:0.15:3.75\times 10^{-2}:6.43\times 10^{-3}:2.68\times 10^{-3}$ (for 56 GeV $<\gamma^{\prime}_{A}<\gamma^{\prime}_{p,max}$) and estimated average logarithmic mass of mixed composition is found to fit consistently with the Auger collaboration estimated average logarithmic mass using Epos LHC interaction models Aab14 . The required primary cosmic ray injection luminosity is found out to be $L_{CR}^{\prime}=3\times 10^{45}$ erg/s along with the best fitted spectral slope of $\alpha=-2.5$ which is finally cross-checked by fitting the observed high energy gamma-rays for Cen A simultaneously as discussed below.

In the proton blazar inspired model framework, the EM spectral hardening above few GeV energies observed by Fermi$-$LAT and also HESS data of the source are found to produce well in the interactions of relativistic cosmic rays with the ambient cold protons in the blob as estimated following the equation (4). Under the assumption of a low acceleration efficiency of electrons in AGN jet of $\chi_{e}\approx 10^{-3}$ which is within the permitted range Eichler05 ; Banik20 and compatible with the hybrid simulation results of diffusive shock acceleration by parallel collisionless shock Gia92 ; Bykov96 . Therefore, the cold proton number density in jet found out to be $1.18$ particles cm^-3 under charge neutrality condition. As discussed in the previous section and also in Banik19 ; Banik20 , we found that the contribution to the total EM spectrum of the IC and synchrotron emission of the stationary electron/positron pairs created in EM cascade induced by protons are very small and hence neglected.

The total jet power in the form of the magnetic field, relativistic electron, cosmic rays and cold matter of density $\rho^{\prime}$ (including cold protons and electrons) is evaluated via the relation Banik19

where $u^{\prime}$ (sum of $u_{e}^{\prime}$, $u_{CR}^{\prime}$ and $u_{B}^{\prime}$) represents comoving jet frame energy density and $p^{\prime}=1/3u^{\prime}$ (sum of $p_{e}^{\prime}$, $p_{CR}^{\prime}$ and $p_{B}^{\prime}$) is the total pressure due to relativistic electrons, cosmic rays and magnetic field. The total jet power in the form of magnetic field, relativistic electron, cosmic rays and cold matter in kpc-scale jet is estimated out to be $1.14\times 10^{46}$ erg/s which is consistent with the the Eddington limit. However, there is large uncertainty in the estimated values of the mass of the central black hole of Cen A, spread over almost an order of magnitude. In Fig. 2, we have displayed the estimated differential gamma-ray and neutrino spectrum reaching at the Earth from this FR-I along with the different space and ground based observations. The model fitting parameters to match the EM SED as well as UHECR event are summarized in Table 1. The corresponding expected muon-neutrino events in the IceCube detector from the Cen A in the 32 TeV and 7.5 PeV energy range are found to be about $N_{\nu_{\mu}}=6.4\times 10^{-2}$ events in 10 years of observations. The overall EM SED from the central part of Cen A (including core and kpc-scale jet) has been estimated and displayed in Fig. 3 along with the different experimental observations. We note that our findings can explain not only the observed multi-wavelength EM SED from the central part of Cen A but also the observed association of UHECRs with Cen A as detected by PAO.