Magnetic reconnection at the termination shock in a striped pulsar wind

Abstract

Context. Most of the rotational luminosity of a pulsar is carried away by a relativistic magnetised wind in which the matter energy flux is negligible compared to the Poynting flux. However, observations of the Crab nebula for instance clearly indicate that most of the Poynting flux is eventually converted into ultra-relativistic particles. The mechanism responsible for transformation of the electro-magnetic energy into the particle energy remains poorly understood. Near the equatorial plane of an obliquely rotating pulsar magnetosphere, the magnetic field reverses polarity with the pulsar period, forming a wind with oppositely directed field lines. This structure is called a striped wind; dissipation of alternating fields in the striped wind is the object of our study. Aims. The aim of this paper is to study the conditions required for magnetic energy release at the termination shock of the striped pulsar wind. Magnetic reconnection is considered via analytical methods and 1D relativistic PIC simulations. Methods. An analytical condition on the upstream parameters for partial and full magnetic reconnection is derived from the conservation laws of energy, momentum and particle number density across the relativistic shock. Furthermore, by using a 1D relativistic PIC code, we study in detail the reconnection process at the termination shock for different upstream Lorentz factors and magnetisations. Results. We found a very simple criterion for dissipation of alternating fields at the termination shock, depending on the upstream parameters of the flow, namely, the magnetisation σ , the Larmor radius r B and the wavelength l of the striped wind. The model depends also on a free parameter $\xi>1$, which is the ratio of the current sheet width to the particle Larmor radius. It is found that for $\sigma \gg l / \xi \, r_{\rm B}$, all the Poynting flux is converted into particle energy whereas for $\sigma \ll (l / \xi \, r_{\rm B} )^{2/3}$, no dissipation occurs. In the latter case, the shock can be accurately described by the ideal MHD shock conditions. Finally, 1D relativistic PIC simulations confirm this prediction and enable us to fix the free parameter ξ in the analytical model. Conclusions. Alternating magnetic fields annihilate easily at relativistic highly magnetised shocks. In plerions, our condition for full magnetic dissipation is satisfied at the termination shock so that the Poynting flux may be converted into ultra-relativistic particles not in the pulsar wind but just at the termination shock. The constraints are more severe for the intra-binary shocks in double pulsar systems. Available models explaining observations require low magnetisation in the downstream flow. The condition that the magnetic field dissipates at the intra-binary shock implies an upper limit on the pair multiplicity in the pulsar wind κ . We found $\kappa \lesssim {\rm few} \times 10^4$ for PSR 1259-63 and PSR 1957+20. In the double pulsar PSR 0737-3039, the radio emission from the pulsar B is modulated with the period of the pulsar A, which implies that the striped structure is not erased completely; this gives a lower limit for $\kappa \gtrsim 310$.