On the radiation energy density in the jet of high-energy emitting BL Lac objects and its impact on their multi-messenger role

Abstract

We examine the potential multi-messenger role of BL Lac objects emitting at high energy (so-called HBL) focusing on the limits on the energy density of soft radiation in the jet frame, a critical parameter which regulates the proton cooling and the fragmentation of ultra-high energy cosmic ray (UHECR) nuclei possibly accelerated in the jet. We show that (under the assumption that the high-energy emission bump is dominated by inverse Compton emission) the energy density of any external soft radiation field (e.g. produced by a layer surrounding the jet or in the accretion flow) cannot be larger than few times that associated to the observed synchrotron radiation produced in the emission region. Quite interestingly, the constraint that we derive is generally stronger than the limit obtained from the condition that the source is transparent to very-high energy $\gamma$-rays. Using this constraint we can derive a robust upper limit for the efficiency of the photopion reaction leading to the emission of PeV neutrinos, $f_{\pi}\lesssim 10^{-5}$, which makes HBL quite inefficient neutrino sources. For the photodisintegration of nuclei the results are more dependent on the spectral properties of the radiation field. The photodisintegration efficiency is safely below 1 (and nuclei can escape intact) for a `canonical' spectrum of the soft radiation field $\propto \nu^{-0.5}$. For radiation fields characterized by a softer spectrum and extended over a large portion of the jet, the efficiency increases and for an appreciable fraction of the sources nuclei with energies above $10^{19}$ eV might suffer significant photodisintegration.