The origin of ultra-high energy cosmic rays (UHECR) is uncertain, and many possible sources have been proposed. Radio galaxies have long been considered a likely source of UHECR because of their high power, large size, and longevity (e.g. Hardcastle et al. 2009; O’Sullivan, Reville & Taylor 2009; Wykes et al. 2013; Eichmann et al. 2018). In Matthews et al. (2018, 2019), we made the case from observations and theory that UHECR may be accelerated by shocks in the lobes of radio galaxies. Our numerical simulations demonstrated the presence of shocks in concentrated, approximately annular, flows centred on the jet axis, which we refer to as hydromagnetic flux tubes, or more simply as ‘flux tubes’, that emerge from the high pressure hot-spots at the end of relativistic jets. Back-flows in radio lobes have been discussed by many authors (for example Norman et al. 1982; Falle 1991; Scheuer 1995; Saxton et al. 2002, Krause 2005; Keppens et al. 2008; Mignone et al. 2010; Cielo et al. 2014; Tchekhovskoy & Bromberg 2016). The flow expands into the lobes where the pressure is lower. As discussed in Matthews et al. (2019), from application of the Bernoulli equation, the flow becomes supersonic before being slowed down in one or more shocks as it progresses deeper into the lobe as depicted in an idealized model in Fig. 1. Our simulations of radio jets support this picture by demonstrating the presence of a succession of shocks in flux tubes with Mach numbers of a few and flow velocities of the order of c/4. With these velocities, shocks are well suited to UHECR acceleration since fully relativistic shocks pose severe problems for acceleration to the highest energies (Kirk & Reville 2010; Lemoine & Pelletier 2010; Reville & Bell 2014; Bell et al. 2018). Further discussion of our model and the observational context can be found in Matthews et al. (2018, 2019). Simulations show that flux tubes are usually more contorted than the idealized flux tubes in Fig. 1. However, the flux tube need only be long enough to contain the CR diffusion scaleheights upstream and downstream of a shock, and curvature should not invalidate the model since magnetic field lines threading the tube should cause CR to diffuse predominantly parallel to the tube. Cosmic ray (CR) acceleration by diffusive shock acceleration (Krymskii 1977; Axford, Leer & Skadron 1977; Bell 1978a,b, Blandford & Ostriker 1978) has been studied at length, both theoretically and observationally, in the context of supernova remnants (SNR). The maximum energy of CR accelerated by SNR is constrained by various factors. The most general constraint is set by the physical size R of the shock, leading to a maximum CR energy of ZuBR (Hillas 1984), where u is the flow velocity, B is the magnetic field, and the CR has charge Ze. ZuBR is the characteristic maximum CR energy in eV if u, B & R are in SI units. A related constraint is set by the need for the shock to persist for a time longer than the time taken for CR to be accelerated (Lagage & Cesarsky 1983a,b). For a quasi-parallel shock (magnetic field parallel to or at an angle significantly smaller than π/2 to the shock normal) the Lagage & Cesarsky limit is equivalent to the Hillas limit if the shock persists for a flow time R/u. For perpendicular shocks, the acceleration is very rapid and the maximum CR energy is set by the Hillas limit (Jokipii 1982, 1987). In this work, we assume that the hydrodynamic flow through a flux tube is in steady state on the scale of the time taken for CR acceleration, in which case the limit on the maximum CR energy is better understood through the spatial limit of Hillas than the time-scale analysis of Lagage & Cesarsky.
Citation: Krause 2005
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