Long wavelength unstable modes in the far upstream of relativistic collisionless shocks

📝 Abstract

The growth rate of long wavelength kinetic instabilities arising due to the interaction of a collimated beam of relativistic particles and a cold unmagnetized plasma are calculated in the ultra relativistic limit. For sufficiently culminated beams, all long wave-length modes are shown to be Weibel-unstable, and a simple analytic expression for their growth rate is derived. For large transverse velocity spreads, these modes become stable. An analytic condition for stability is given. These analytic results, which generalize earlier ones given in the literature, are shown to be in agreement with numerical solutions of the dispersion equation and with the results of novel PIC simulations in which the electro-magnetic fields are restricted to a given k-mode. The results may describe the interaction of energetic cosmic rays, propagating into the far upstream of a relativistic collisionless shock, with a cold unmagnetized upstream. The long wavelength modes considered may be efficient in deflecting particles and could be important for diffusive shock acceleration. It is shown that while these modes grow in relativistic shocks propagating into electron-positron pair plasmas, they are damped in relativistic shocks propagating into electron-proton plasmas with moderate Lorenz factors \Gamma_{sh}\lesssim 100. If these modes dominate the deflection of energetic cosmic rays in electron-positron shocks, it is argued that particle acceleration is suppressed at shock frame energies that are larger than the downstream thermal energy by a factor greater than the shock Lorentz factor.

💡 Analysis

The growth rate of long wavelength kinetic instabilities arising due to the interaction of a collimated beam of relativistic particles and a cold unmagnetized plasma are calculated in the ultra relativistic limit. For sufficiently culminated beams, all long wave-length modes are shown to be Weibel-unstable, and a simple analytic expression for their growth rate is derived. For large transverse velocity spreads, these modes become stable. An analytic condition for stability is given. These analytic results, which generalize earlier ones given in the literature, are shown to be in agreement with numerical solutions of the dispersion equation and with the results of novel PIC simulations in which the electro-magnetic fields are restricted to a given k-mode. The results may describe the interaction of energetic cosmic rays, propagating into the far upstream of a relativistic collisionless shock, with a cold unmagnetized upstream. The long wavelength modes considered may be efficient in deflecting particles and could be important for diffusive shock acceleration. It is shown that while these modes grow in relativistic shocks propagating into electron-positron pair plasmas, they are damped in relativistic shocks propagating into electron-proton plasmas with moderate Lorenz factors \Gamma_{sh}\lesssim 100. If these modes dominate the deflection of energetic cosmic rays in electron-positron shocks, it is argued that particle acceleration is suppressed at shock frame energies that are larger than the downstream thermal energy by a factor greater than the shock Lorentz factor.

📄 Content

1. INTRODUCTION Current understanding of gamma-ray burst (GRB) “afterglows,” the delayed low energy emission following the prompt γ-ray emission, suggests that the radiation observed is the synchrotron emission of energetic non-thermal electrons in the downstream of an ultra-relativistic collisionless shock driven into the surrounding interstellar medium (ISM) or stellar wind (Zhang & Mészáros 2004;Piran 2004).

This model requires a strong magnetic field and a large population of energetic electrons to be present in the downstream. Observations suggest that the fraction of post-shock thermal energy density carried by non-thermal electrons, ǫ e , is large, ǫ e ≈ 0.1 (e.g. Zhang & Mészáros 2004;Frail et al. 2001;Freedman & Waxman 2001;Berger et al. 2003). The fraction of post-shock thermal energy carried by the magnetic field, ǫ B , is less well constrained by observations. However, in cases where ǫ B can be reliably constrained by multi waveband spectra, values close to equipartition, ǫ B ∼ 0.01 to 0.1, are inferred (e.g. Frail et al. 2000).

The non-thermal energetic electron (and proton) population is believed to be produced by the diffusive (Fermi) shock acceleration (DSA) mechanism (for reviews see Drury 1983;Blandford & Eichler 1987;Malkov & O’C Drury 2001).

The required magnetic fields in the shock frame in the downstream (e.g. Frail et al. 2000) and upstream (Li & Waxman 2006) regions are much larger than the ambient field, and thus require substantial amplification. The accelerated particles are likely to have an important itay.rabinak@weizmann.ac.il role in generating and maintaining the inferred magnetic fields.

The main challenge associated with the downstream magnetic field is that the field amplitude must remain close to equipartition deep into the downstream, over distances ∼ 10 10 l sd (Gruzinov & Waxman 1999;Gruzinov 2001a). While near equipartition fields on skin depth scale are likely to be produced in the vicinity of the shock by electromagnetic (e.g. Weibel-like) instabilities (e.g. Blandford & Eichler 1987;Gruzinov & Waxman 1999;Medvedev & Loeb 1999;Wiersma & Achterberg 2004), they are expected to decay within a few skin-depths downstream (Gruzinov 2001a). This suggests that the correlation length of the magnetic field far downstream and possibly upstream must be much larger than the skin depth, L ≫ l sd , perhaps even of the order of the distance from the shock (Gruzinov & Waxman 1999;Gruzinov 2001a;Katz et al. 2007).

The search for a self-consistent theory of collisionless shocks has led to extensive numerical studies using the particle in cell (PIC) based algorithms (e.g. Gruzinov 2001a,b;Medvedev et al. 2005;Silva et al. 2003;Nishikawa et al. 2003;Frederiksen et al. 2004;Jaroschek et al. 2004;Spitkovsky 2005Spitkovsky , 2008a;;Martins et al. 2009).

Such simulations have provided compelling evidence for acceleration of particles and generation of long lasting near-equipartition magnetic fields. However numerically simulating the long term behavior is challenging and is currently restricted to pair (e + e -) plasmas in 2D (e.g. Spitkovsky 2008b; Keshet et al. 2009).

Large scale magnetic fields may possibly be generated in the upstream by the interaction of the beam of CRs propagating ahead of the shock and the upstream plasma (e.g. Katz et al. 2007;Keshet et al. 2009). In particular high energy CRs naturally introduce large scales due to their large Larmor radius, and the large distances to which they propagate into the upstream. Instabilities arising from the interaction of relativistic beams and cold plasmas have long been studied (Akhiezer 1975, and references therein) and are suspected of amplifying the magnetic field in the shock transition layer (Gruzinov & Waxman 1999;Medvedev & Loeb 1999;Wiersma & Achterberg 2004;Bret et al. 2005;Lyubarsky & Eichler 2006;Achterberg et al. 2007;Achterberg & Wiersma 2007;Bret 2009). In (Lemoine & Pelletier 2009) a systematic study of these instabilities for the lowest energy CRs, with energies comparable to the thermal energy of the shocked plasma, and their application for Fermi acceleration is given.

In this paper we analyze long wavelength plasma instabilities resulting from the counter-streaming flow of high energy CRs ,γ ≫ Γ sh , running far ahead of the shock and a non-magnetized upstream plasma. The analysis is restricted to long wavelength modes, k ≪ ω 0 /c, which are expected to deflect particles efficiently. For simplicity it is assumed that the particle distribution is homogenous. The paper is organized as follows. In § 2 we calculate the growth rate of long wavelength modes. We separately discuss highly collimated beams and beams with a significant transverse velocity spread, and derive a condition for the stability of these modes. In § 3 we discuss the possible implications of these results to collisionless shocks. In § 4 we summarize the main results and conclusions. An estimate of the saturation level of the modes is beyond

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