Resonant energization of particles by radio AGN

One of the fundamental problems of modern astrophysics is to figure the origin of particle acceleration to extremely high energies. It is very helpful to note that observations in the last decade have established/confirmed a strong correlation of very high energy (VHE) protons with active galactic nuclei (Kim & Kim, 2013). In this paper, we advance a possibly acceleration mechanism operating on particles comprising the magnetospheres in nearby regions of radio AGNs. The mechanism, based on resonant energy transfer from the intense electromagnetic (EM) waves to relativistic quantum particle waves (modeled here by as Klein-Gordon Waves), is an interesting manifestation of the strong affinity between two waves with a similar mathematical structure; the dominant behavior of both the classical electromagnetic wave (the radio waves, for example) and the relativistic Klein-Gordon wave is controlled by essentially the same hyperbolic wave operator.

The literature is full of plausible acceleration mechanisms- the so-called Fermi mechanism (Fermi, 1949) and its several variants (Bell, 1978; Catanese & Weeks, 1999) might account for the origin of extremely high energy cosmic rays; this process, though, has limitations and is efficient when the particles are initially accelerated (Rieger & Mannheim, 2000). One of the papers relying on extreme conditions in the vicinity of compact objects (Blandford & Znajek, 1977) proposes that the poloidal magnetic field close to a black hole induces electrostatic potential, which might efficiently accelerate charged particles; the process of acceleration, however, is limited by a significant screening effect. The so-called turbulent acceleration (Lemoine, 2021) might account for very high energies (Pezzi et al., 2022). In the framework of this mechanism the turbulent plasma is described as a combination of Alfvén waves and magnetosonic modes, where the particle acceleration comes from the phase locking of trajectories of particles with the mentioned waves. As a result, this mechanism might provide energies in the PeV domain (Pezzi et al., 2022). Magneto-centrifugal mechanism of energy pumping has been considered for AGN magnetospheres in a series of papers (Rieger & Mannheim, 2000; Osmanov et al., 2007; Osmanov, 2010) to explain energization of particles up to TeV energies. Taking into account generation of centrifugally driven electrostatic field, similar mechanisms have been invoked for similar class of objects (Osmanov et al., 2014) - the central black hole of Milky Way, for instance (Osmanov et al., 2017).

Harnessing of the strong wave-wave resonant interaction (Mahajan & Asenjo, 2022) for energy transfer is what distinguishes this effort from the aforementioned schemes that also operate in extreme conditions in the vicinity of highly compact objects. In particular, if the phase velocity of the EM wave coincides with the phase velocity of a quantum wave associated with a particle, the latter will be energized to extremely high energies. The resonant enhancement is particularly pronounced when the plasma frequency is much less than the frequency of the EM radiation(Sec.2).

It is, perhaps, important to emphasize that the the manifest wave- wave interaction of this paper is possible only because of the quantum nature of the particle.

The paper is organised as follows: in Sec. 2 we outline a theory of our mechanism, consider radio-load AGN applying the theory, discuss obtained results and in Sec. 3 we summarise them.

In this section we will briefly outline the mechanism of resonant energization developed in (Mahajan & Asenjo, 2022). Enough detail is given to make this paper self contained.

The group velocity of the quantum wave associated with a relativistic particle of energy (momentum) =E (P),

approaches the speed of light ($c=1$) when $P\gg m$, where $P=\gamma m\upsilon$ is the momentum of the particle, $\gamma$ its relativistic factor and $\upsilon$ - the velocity. Thus for extremely high values of the momentum, the particle could, resonantly, exchange energy with an EM wave that propagates in unison. The resonant energization phenomena is well illustrated in a model calculation (Mahajan & Asenjo, 2022) in which a Klein-Gordon (KG) wave is subjected to a circularly polarized EM wave; the latter is described by the EM four potentials $A^{\mu}$ ($A^{0}=A^{z}=0,$),

where $\omega$ and $k$ are, respectively, the frequency and the wavenumber of the EM wave propagating along the $z$ axis. Because, the directions perpendicular to z are ignorable, the KG/EM system obeys the Klein-Gordon equation (see Mahajan & Asenjo (2016) for details),

where $q$ is the particle charge, and $K_{\perp}$ is the wave-vector’s perpendicular component, which on the other hand is the measure of the (conserved) perpendicular momentum and labels (suppressed) the wave function. The explicit $t$ and $z$ dependence in Eq. (3) implies that the energy E and $P_{z}$ will be functions of time. The resonant solutions of Eq. (3) will emerge when we demand $\Psi=\Psi(\omega t-kz)\equiv\Psi(\xi)$ ($\xi\equiv\omega t-kz$); the result is a simple Mathew equation

with $\mu=K_{\perp}^{2}+m^{2}+q^{2}A^{2}$ and $\lambda=2qAK_{\perp}$. Notice that the mathematical operator $\partial^{2}_{t}-\partial^{2}_{z}$ in Eq. (3) translates to $\omega^{2}-k^{2}$ (in Eq. (4) when we impose a solution whose phase factor ($\xi\equiv\omega t-kz$)is exactly that of the EM wave. In some sense this is the obvious mathematics underlying wave resonance. In particular, notice that the signature of resonance is already explicit in Eq. (4); the equation is singular since $\omega^{2}-k^{2}$ tends to zero for EM waves traveling in a tenuous medium. The implication, of course, is that nontrivial solutions must demand derivative of $\Psi$ to become commensurately large. Consequently, the energy and z momentum of the KG state must also become large since both are proportional to the $\xi$ derivative (as we will soon see).

The leading order WKB solution of Eq. (4) is given by (Mahajan & Asenjo, 2022)

where

Following simple rules of quantum mechanics, the expectation value of the energy is

and the acceleration takes place in the range $z=[-l,+l]$. Note that, for this resonant state, the expectation value of the z momentum $\langle K_{z}\rangle=(k/\omega)\langle E\rangle$ .

In order to understand the accessibility of the high energy states, let us calculate the root mean square of the rate of energy gain

From Eqs. (6-8), it is obvious that both the energy $\langle E\rangle$ and the r.m.s rate of energy increase (acceleration) are resonantly enhanced by the factor $\omega/\sqrt{(\omega^{2}-k^{2})}$. If we now invoke the standard relativistic dispersion relation for EM propagation in highly under dense plasmas (see for example Mahajan & Asenjo (2016) )

where $\omega_{p}=\sqrt{4\pi nq^{2}/m}$ denotes the plasma frequency and $n$ is the number density of plasma particles, we may derive an explicit expression for the relativistic factor (associated with the high energy state) $\gamma=E/m\approx S/m$,

the first two factors reflect resonant enhancement. If $\omega>>\omega_{p}$, very high energy states become accessible even for moderate value of $q^{2}A^{2}/m^{2}$. In terms of these parameters, the acceleration rate becomes

which, in physical units, may be approximated as ($\sin(kL)/kL\simeq 1,qA>mc^{2},\hbar K_{\perp}c$)

This completes our review of the theory of resonant energization.

We are now ready to apply the essentials of this theory to the particular case of a plasma in the magnetospheres of a radio AGN that emits copious EM energy in the radio frequency range. Although we will, for this paper, work out the mechanism for the radio AGN, the resonant mechanism of particle energization pertains to any AGNs emitting in the entire electromagnetic spectrum- all the way to gamma rays.

To assess the effectiveness of the resonant energy transfer from the EM waves to the relativistic KG particle-waves, we must consider processes that will impede the acceleration process (summed up in Eq. (12). Perhaps, in the present context, the two most important impeding (cooling) processes will be :1) the inverse Compton (IC) scattering of the charged particles with the ambient photon field, and 2) synchrotron radiation when relatively strong magnetic fields are present. We will deal with them in the following subsections.

We are grateful to an anonymous referee for very interesting and valuable comments.

Data are available in the article and can be accessed via a DOI link.