Particle acceleration at recollimation shocks in sub-relativistic jets A model for jets in Seyfert Galaxies, Microquasars and Protostellar Systems

Abstract

Context: Growing observational evidence suggests that sub-relativistic astrophysical jets may accelerate particles at slowly evolving standing shocks. Recollimation shocks are expected to develop when jets expand in dense environments; their formation may be mediated by the pressure of the cocoon surrounding the jet, while remaining compatible with a quasi-stationary behavior. Despite their high inclination relative to the jet axis, such shocks can be strong and enable efficient particle acceleration. Aims: The aim of this work is to improve the general understanding of particle acceleration via diffusive shock acceleration at recollimation shocks by developing a versatile modeling framework applicable to different classes of astrophysical jets, including Seyfert galaxies, microquasars, and protostellar systems. Methods: We extend an analytic jet hydrodynamics model previously introduced in the literature to the sub-relativistic regime and use it to identify the expected locations of the recollimation shock and the jet head. Within this framework, we formulate a semi-analytic acceleration and transport model for particles injected at the recollimation shock via diffusive shock acceleration. Results: By solving the space-dependent transport equation, we obtain particle distributions and spectra along the jet, as well as robust predictions for the maximum energies achievable as a function of the intrinsic properties of the system and the source class. Conclusions: Our results indicate that recollimation shocks may play a central role in particle acceleration in sub-relativistic jets. In Seyfert galaxies, such shocks may accelerate particles from PeV up to EeV energies, while in microquasars and protostellar jets maximum energies of tens of PeV and up to TeV are expected, respectively. Protons escaping the jets may diffuse through the cocoon, leading to possible hadronic signatures.

Figures (8)

• Figure 1: Sketch illustrating the jet-cocoon system. Particles are accelerated at the recollimation shock and then advected in the jet. The jet material that passes through the reverse shock fills the inner cocoon (yellow area), while the external medium reached by the jet head and, laterally, by the cocoon surface, is swept-up and accumulated in the outer cocoon (green area).
• Figure 2: Sketch illustrating the recollimation shock geometry. The upstream fluid with velocity $u_0$ impacts the shock with angle $\psi$ and is deflected along the jet axis. The component normal to the shock of the upstream velocity is $u_{0, {\rm n}}$. The downstream plasma speed is $u_1$. The shock radius, $r_s$, is the radius of the shock which approximately coincides with the jet radius at $\hat{z}/2$. The recollimation shock forms at $z_*$.
• Figure 3: Seyfert prototype. Spectra of accelerated particles at the recollimation shock. The blue dot-dashed curve represents the Bohm scenario, while the solid black and the dashed red represent the Kraichnan and Kolmogorov cases, respectively.
• Figure 4: Radial behavior of the solution for the case of Seyferts jets. Top panel: particle distribution along $z$ computed at different energies. The vertical dotted line represents the recollimation shock location, while the jet head is located at approximately 20 times $z_{\rm sh}$. Bottom panel: solution computed at different positions in the jet. The solid black line represents the solution at the shock position while dashed lines refer to the upstream (orange for $z= 0.97 z_{\rm sh}$ and red for $z = 0.3 z_{\rm sh}$) and dotted lines to the downstream (cyan and blue for $z \gtrsim 0.98 \, z_{\rm esc}$ and $z \gtrsim 0.99 \, z_{\rm esc}$ respectively).
• Figure 5: Microquasar prototype. Particle spectra at the recollimation shock. The linestyle and colors are identical to Figure \ref{['Fig: SEY']}.