Particle Acceleration at Ultrarelativistic Shocks: An Eigenfunction Method

By John G. Kirk, A. Guthmann, Y. Gallant, A. Achterberg. 2000

299 citations recorded by the source corpus.

We extend the eigenfunction method of computing the power-law spectrum of particles accelerated at a relativistic shock front to apply to shocks of arbitrarily high Lorentz factor. In agreement with the findings of Monte Carlo simulations, we find that the index of the power-law distribution of accelerated particles, which undergo isotropic diffusion in angle at an ultrarelativistic, unmagnetized shock, is s = 4.23 ± 0.01 (where s = -d ln f/d ln p with f the Lorentz invariant phase-space density and p the momentum). This corresponds to a synchrotron index for uncooled electrons of α = 0.62 (taking cooling into account α = 1.12), where α = -d ln Fνα/d ln α, Fν is the radiation flux, and ν is the frequency. We also present an approximate analytic expression for the angular distribution of accelerated particles, which displays the effect of particle trapping by the shock: compared with the nonrelativistic case the angular distribution is weighted more toward the plane of the shock and away from its normal. We investigate the sensitivity of our results to the transport properties of the particles and the presence of a magnetic field. Shocks in which the parameter σ (the ratio of Poynting to kinetic energy flux) upstream is not small are less compressive and lead to larger values of s.

Subfields: Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/1b8fba0bd024224e181373420b6485a8f259ea8f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. L. Sironi
2. D. Giannios
3. A. Spitkovsky
4. U. Keshet
5. M. Lemoine
6. Bing Zhang
7. G. Ghisellini
8. Angela V. Olinto

In Nuclear and High Energy Physics

1. Bing Zhang
2. L. Sironi
3. A. Spitkovsky
4. D. Giannios
5. Pawan Kumar
6. F. Tavecchio
7. G. Ghisellini
8. B. Metzger

Citations in the corpus

• Revised equipartition and minimum energy formula for magnetic field strength estimates from radio synchrotron observations (2005, background): “The limit for ultrarelativistic shocks isγ0 = 2.23 (Kirk et al. 2000).”
• Particle acceleration by ultrarelativistic shocks: theory and simulations (2001, methodology): “The eigenfunction method was recently extended to the case of ultrarelativistic shocks with Gs @ 1 by Kirk et al. (2000).”
• THE AFTERGLOWS OF SWIFT-ERA GAMMA-RAY BURSTS. II. TYPE I GRB VERSUS TYPE II GRB OPTICAL AFTERGLOWS (2008, background): “In the standard fireball model3, if the cooling frequency νc lies blueward of the optical bands, it is β = (p − 1)/2 (e.g., Sari et al. 1998; Zhang & Mészáros 2004; Piran 2005, and references therein), with the canonical value p = 2.2 (Kirk et al. 2000; Achterberg et al. 2001), implying β = 0.6.”
• Particle Acceleration in Relativistic Collisionless Shocks: Fermi Process at Last? (2008, methodology): “Most progress in studying shock acceleration has been made using Monte Carlo test particle simulations (e.g., Ostrowski & Bednarz 1998, Ellison & Double 2004) and semi-analytic kinetic theory methods (e.g., Kirk et al. 2000, Achterberg et al. 2001, Keshet & Waxman 2005).”
• Theories of Gamma-Ray Bursts (2001, background): “Both the internal and the external reverse shocks are mildly relativistic, and are expected to lead to relativistic proton energy spectra of the form dNp/d p ∝ γ−2p (Blandford & Eichler 1987; see also Kirk et al. 2000 and Lloyd-Ronning & Petrosian 2001b).”
• Relativistic Shocks: Particle Acceleration and Magnetization (2015, background): “…matching numerical eigenfunctions of the transport equation between upstream and downstream (Kirk and Schneider 1987; Heavens and Drury 1988; Kirk et al. 2000), by Monte Carlo simulations (Bednarz and Ostrowski 1998; Achterberg et al. 2001; Ellison and Double 2002; Lemoine and Pelletier…”
• Internal shocks in the jets of radio-loud quasars (2001, result): “Following the results on non-relativistic shocks, and the recent results on relativistic ones (Bednarz & Ostrowski 1998; Kirk et al. 2000), we also postulate that the electrons are accelerated (instantaneously) behind the shock as a power law N(γ) ∝ γ−p with γ > γb, and p > 2.”
• UHE nuclei propagation and the interpretation of the ankle in the cosmic-ray spectrum (2005, background): “Instead, both analytical and numerical studies of particle accelerationin relativistic shocks give injection spectra withβ ≃ 2.2–2.3 (Bednarz and Ostrowski, 1998; Kirk et al., 2000; Lemoine and Pelletier, 2003).”
• An Accretion-Jet Model for Black Hole Binaries: Interpreting the Spectral and Timing Features of XTE J1118+480 (2004, methodology): “These shocks accelerate a fraction of the electrons into a power-law energy distribution with index p = 2.24 (e.g., Kirk et al. 2000).”
• INSIGHTS INTO THE HIGH-ENERGY γ-RAY EMISSION OF MARKARIAN 501 FROM EXTENSIVE MULTIFREQUENCY OBSERVATIONS IN THE FERMI ERA (2011, background): “…role for diffusive shock acceleration above the injection energy Ee,min, as this value of the spectral index is often claimed in the literature for particles undergoing first-order Fermi acceleration at relativistic shock (Bednarz & Ostrowski 1998; Kirk et al. 2000; Achterberg et al. 2001).”
• The Binary Neutron Star Event LIGO/Virgo GW170817 160 Days after Merger: Synchrotron Emission across the Electromagnetic Spectrum (2018, background): “…shocks will lie in between the value p = 2 expected at non-relativistic shock speeds (Bell 1978; Blandford & Ostriker 1978; Blandford & Eichler 1987) and p ' 2.22 at ultra-relativistic velocities (Kirk et al. 2000; Achterberg et al. 2001; Keshet & Waxman 2005; Sironi et al. 2013).”
• Signatures of Extragalactic Dust in Pre-Swift GRB Afterglows (2005, result): “This is in general agreement with theoretical predictions for ultrarelativistic shocks (Kirk et al. 2000; Achterberg et al. 2001).”
• GAMMA-RAY ACTIVITY IN THE CRAB NEBULA: THE EXCEPTIONAL FLARE OF 2011 APRIL (2011, background): “Standard diffusive shock acceleration typically results in spectra with p 2 (Gallant et al. 1992; Kirk et al. 2000).”
• Synchrotron Radiation as the Source of Gamma-Ray Burst Spectra (2000, background): “…of internal shocks has not been carefully studied, and it is unclear whether the Fermi mechanism will even work at all (see, e.g. Lyutikov & Blackman, 2000 and Kirk et al., 2000, the latter of whom show that including a magnetic field at the acceleration site will steepen this universal index).”
• PARTICLE ACCELERATION IN RELATIVISTIC MAGNETIZED COLLISIONLESS PAIR SHOCKS: DEPENDENCE OF SHOCK ACCELERATION ON MAGNETIC OBLIQUITY (2009, methodology): “…acceleration in magnetized shocks has been made using semi-analytic kinetic theory methods (e.g., Kirk & Heavens 1989; Ballard & Heavens 1991; Kirk et al. 2000; Achterberg et al. 2001; Keshet & Waxman 2005) or Monte Carlo testparticle simulations (e.g., Bednarz & Ostrowski 1998; Niemiec &…”