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Physical effects and modeling of the radiation reaction force in relativistic astrophysical and laboratory plasmas F. Pegoraro M.D’Angelo, L.Fedeli, A.Macchi, A.Sgattoni Department of Physics “Enrico Fermi”, University of Pisa and CNISM Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy [email protected] SIF, Trieste, 2013 Novel plasma regimes The conceptual framework we generally adopt when describing laboratory and space plasmas is based on coupling Vlasov’s equation to Maxwell’s field equations. The sources of these fields are not the discrete ”particles” that compose the plasma but a continuous distribution of charges and currents. Mean field theory. In this asymptotic description particle correlations (”collisions”) are neglected and the plasma dynamics is determined by a single particle Hamiltonian that depends on the selfconsistent collective fields. Novel plasma regimes The conceptual framework we generally adopt when describing laboratory and space plasmas is based on coupling Vlasov’s equation to Maxwell’s field equations. The sources of these fields are not the discrete ”particles” that compose the plasma but a continuous distribution of charges and currents. Mean field theory. In this asymptotic description particle correlations (”collisions”) are neglected and the plasma dynamics is determined by a single particle Hamiltonian that depends on the selfconsistent collective fields. Radiation reaction comes into the play For an electromagnetic plasma this is in a sense a lucky intermediate energy regime: large enough to make Coulomb collisions negligible and small enough to make dynamical effects due to incoherent radiation unimportant on the time scale of interest. At the extreme laser intensities of next-generation experiments (I ≥ 1023W cm at ∼ 1µm wavelength), electrons experience superstrong accelerations and emit relatively large amounts of high frequency incoherent radiation. Thus, still remaining within the non quantum description, i.e., neglecting (spin and) pair creation, we must address the dynamical effects of the radiation reaction force. This is a tricky problem within classical electrodynamics because it involves the interaction of the fields radiated by a point particle with the particle itself. Radiation reaction comes into the play For an electromagnetic plasma this is in a sense a lucky intermediate energy regime: large enough to make Coulomb collisions negligible and small enough to make dynamical effects due to incoherent radiation unimportant on the time scale of interest. At the extreme laser intensities of next-generation experiments (I ≥ 1023W cm at ∼ 1µm wavelength), electrons experience superstrong accelerations and emit relatively large amounts of high frequency incoherent radiation. Thus, still remaining within the non quantum description, i.e., neglecting (spin and) pair creation, we must address the dynamical effects of the radiation reaction force. This is a tricky problem within classical electrodynamics because it involves the interaction of the fields radiated by a point particle with the particle itself. Radiation reaction comes into the play For an electromagnetic plasma this is in a sense a lucky intermediate energy regime: large enough to make Coulomb collisions negligible and small enough to make dynamical effects due to incoherent radiation unimportant on the time scale of interest. At the extreme laser intensities of next-generation experiments (I ≥ 1023W cm at ∼ 1µm wavelength), electrons experience superstrong accelerations and emit relatively large amounts of high frequency incoherent radiation. Thus, still remaining within the non quantum description, i.e., neglecting (spin and) pair creation, we must address the dynamical effects of the radiation reaction force. This is a tricky problem within classical electrodynamics because it involves the interaction of the fields radiated by a point particle with the particle itself. Radiation reaction in astrophysics The inclusion of the RR term is not only relevant to laser plasma interactions: it is also important for the description of pulsar winds, gamma-ray bursters, and jets of active galactic nuclei (Poynting-flux-dominated plasma outflows) Radiation-Dominated Relativistic Current Sheets1 .. It is important for the interpretation and modeling of the recently discovered phenomenon of the flaring Crab nebula with high-frequency γ-ray emission 2 . 1 C. H. Jaroschek and M. Hoshino PRL 103, 075002 (2009) M., et al., Science. 331, 736 (2011) B Cerruti et al APJ 746 178 (2012) 2 Tavani, Radiation reaction in astrophysics The inclusion of the RR term is not only relevant to laser plasma interactions: it is also important for the description of pulsar winds, gamma-ray bursters, and jets of active galactic nuclei (Poynting-flux-dominated plasma outflows) Radiation-Dominated Relativistic Current Sheets1 .. It is important for the interpretation and modeling of the recently discovered phenomenon of the flaring Crab nebula with high-frequency γ-ray emission 2 . 1 C. H. Jaroschek and M. Hoshino PRL 103, 075002 (2009) M., et al., Science. 331, 736 (2011) B Cerruti et al APJ 746 178 (2012) 2 Tavani, Flares in the Crab nebula Flaring events as detected by Fermi and Agile satellites M.Tavani, 9th AGILE Science Workshop April 2012 Frascati, Italy Flares in the Crab nebula M.Tavani, 9th AGILE Science Workshop April 2012 Frascati, Italy Radiation reaction effects The radiation reaction force in the single particle equation has a number of conceptual difficulties that are already evident in its nonrelativistic form given by F = τ0 d(ma)/dt, where τ0 = (2e2 )/(3mc3 ) It can give rise to runaway solutions where the momentum grows exponentially in the absence of external fields. In a covariant formulation the four-acceleration vector must be ”orthogonal” to the four velocity vector. Covariant Lorentz-Abraham-Dirac equation mc duµ = eF µν uν − eτ0 dτ d 2 uν d 2 uµ + u µ uν dτ 2 dτ 2 ! . (1) Radiation reaction effects The Landau-Lifshitz form is obtained by inserting the unperturbed Lorentz acceleration in LAD equation. This automatically removes the third order derivative (and thus runaway solutions) and brings us back to a more . conventional phase space description In covariant notation the total force mc duµ /dτ = F µ reads mc duµ e e = eF µν uν +eτ0 [uν uα ∂α F µν + F µν Fνα uα + (F νβ uβ Fνα uα )uµ ] dτ mc mc (2) In 3D notation it reads h ∂ i v v ∂ dp =e E + × B +eτ0 γ +v·∇ E+ × +v·∇ B dt c ∂t c ∂t +τ0 v v i 2 v 2 i v e2 h e2 2 h v E×B+ ×B ×B+ · E E − τ0 γ E+ ×B − ·E . mc c c mc c c c (3) Kinetic equation: Numerical Implementation In covariant notation the modified Vlasov equation for the (invariant) particle distribution function f reads ∂ ( f uµ ) ∂ ( f F µ ) + =0 ∂ xµ ∂ pµ (4) Simple numerical scheme to insert the RR force in the PIC code have been implemented assuming that the particle acceleration is dominated by the Lorentz force, with the RR force giving a smaller but non negligible contribution. Pisan references M. Tamburini, F. Pegoraro, A. Di Piazza, C.H. Keitel, A. Macchi, New Journal of Physics 12 123005 (2010). A. Macchi, M. Tamburini, S. Veghini, F. Pegoraro, A. Di Piazza, C.H. Keitel, ICONO-2010, Proceedings of SPIE 7994 LAT(2010). M. Tamburini, F. Pegoraro, A. Di Piazza, C.H. Keitel, T. V. Liseykina, A. Macchi, NIMA Proceedings 653, 181 (2011). A. Macchi, M. Tamburini, F. Pegoraro. T. V. Liseykina, SPIE Optics Optoelectronics conf., Proc. SPIE 8075, 807509 (2011) M. Tamburini, T.V. Liseykina, F. Pegoraro, A. Macchi, Phys. Rev., E 85, 01640 1 (2012) Electrons in circularly polarized laser pulse QED effects and Pair plasmas The above description of RR effects stays within the framework of a Vlasov description based on classical phase space variables. At energies just above those where RR starts to count, QM effects come into play at first through spin effects that may change the particle trajectories in very intense e.m. fields. More interestingly, large projects have been conceived (ELI and Hyper) with the stated objective of reaching even higher e.m. energy densities inside a macroscopic relativistic medium and probe the physics of plasma regimes dominated by QED effects and in particular the generation of electron-positron pair plasmas as the so called Schwinger field ( eES λc ∼ me c2 , λc = }/(me c) is the Compton wavelength) is approached. QED effects and Pair plasmas The above description of RR effects stays within the framework of a Vlasov description based on classical phase space variables. At energies just above those where RR starts to count, QM effects come into play at first through spin effects that may change the particle trajectories in very intense e.m. fields. More interestingly, large projects have been conceived (ELI and Hyper) with the stated objective of reaching even higher e.m. energy densities inside a macroscopic relativistic medium and probe the physics of plasma regimes dominated by QED effects and in particular the generation of electron-positron pair plasmas as the so called Schwinger field ( eES λc ∼ me c2 , λc = }/(me c) is the Compton wavelength) is approached. Pair Plasmas in Space Electron-positron pair plasmas are expected to be an essential feature in a wide range of astrophysical settings, such as Gamma-ray Bursts (GRB), plasma outflows in Pulsar Wind Nebulae (PWN) and relativistic jets from Active Galactic Nuclei (AGN). Characteristically, these processes lead to acceleration of high-energy particles and are thought to be associated to collisionless shock waves or to colliding beams of particles. Counterpropagating particle beams lead to the generation of e.m. fields as is the case the so-called filamentation instability (related to the Weibel instability) that separates currents in the plasma and generates magnetic field3 . 3 Califano, F. Prandi, R. Pegoraro, F. Bulanov, S.V.. Phys.Rev., E 58, 7837 (1998). Pair Plasmas in Space Electron-positron pair plasmas are expected to be an essential feature in a wide range of astrophysical settings, such as Gamma-ray Bursts (GRB), plasma outflows in Pulsar Wind Nebulae (PWN) and relativistic jets from Active Galactic Nuclei (AGN). Characteristically, these processes lead to acceleration of high-energy particles and are thought to be associated to collisionless shock waves or to colliding beams of particles. Counterpropagating particle beams lead to the generation of e.m. fields as is the case the so-called filamentation instability (related to the Weibel instability) that separates currents in the plasma and generates magnetic field3 . 3 Califano, F. Prandi, R. Pegoraro, F. Bulanov, S.V.. Phys.Rev., E 58, 7837 (1998). Pair Plasmas in Space Electron-positron pair plasmas are expected to be an essential feature in a wide range of astrophysical settings, such as Gamma-ray Bursts (GRB), plasma outflows in Pulsar Wind Nebulae (PWN) and relativistic jets from Active Galactic Nuclei (AGN). Characteristically, these processes lead to acceleration of high-energy particles and are thought to be associated to collisionless shock waves or to colliding beams of particles. Counterpropagating particle beams lead to the generation of e.m. fields as is the case the so-called filamentation instability (related to the Weibel instability) that separates currents in the plasma and generates magnetic field3 . 3 Califano, F. Prandi, R. Pegoraro, F. Bulanov, S.V.. Phys.Rev., E 58, 7837 (1998). Pair Plasmas in Space. Relativistic Kinetics The modelling of Electron-positron pair plasmas requires the understanding of such relativistic plasma collective processes using kinetic descriptions and can largely benefit from a comparison with laboratory relativistic plasma results. Fluid-type descriptions certainly fail when modelling the transient γ-ray emission from the Crab Nebula. E.g., observations of synchrotron spectra imply a value of the electric field well beyond the limits imposed by MHD. The inadequacy of fluid description is a central issue in our understanding of Physics at the scale of the Universe. Pair Plasmas in Space. Relativistic Kinetics The modelling of Electron-positron pair plasmas requires the understanding of such relativistic plasma collective processes using kinetic descriptions and can largely benefit from a comparison with laboratory relativistic plasma results. Fluid-type descriptions certainly fail when modelling the transient γ-ray emission from the Crab Nebula. E.g., observations of synchrotron spectra imply a value of the electric field well beyond the limits imposed by MHD. The inadequacy of fluid description is a central issue in our understanding of Physics at the scale of the Universe. Pair Plasmas in Space. Relativistic Kinetics The modelling of Electron-positron pair plasmas requires the understanding of such relativistic plasma collective processes using kinetic descriptions and can largely benefit from a comparison with laboratory relativistic plasma results. Fluid-type descriptions certainly fail when modelling the transient γ-ray emission from the Crab Nebula. E.g., observations of synchrotron spectra imply a value of the electric field well beyond the limits imposed by MHD. The inadequacy of fluid description is a central issue in our understanding of Physics at the scale of the Universe. Pair Plasmas in Space. Counterstreaming beams Geometry of the “neutral” counterstreaming beams Beams propagate along z, modes have a mixed nature: “two stream” type for k = kz and filamentation for k = kx Initial configuration, full current compensation between − + − the “right” (e+ 1 , e2 ) and the “left” ’ (e2 , e1 ) currents. Filamentation instability in a Pair Plasma The filamentation instability arises from the separation of the “right” and of the “‘left” currents producing repelling adjacent current filaments with opposite signs.4 The instability saturates at near equipartition between the beam kinetic energy and the generated magnetic energy. 4 Figures taken from M. D’Angelo Master thesis, Pisa. 2013 Filamentation instability in a Pair Plasma and RR The main effect of RR is on the particle kinetic energy. Filamentation instability in a Pair Plasma and RR The main effect of RR is on the particle kinetic energy. 2D Filamentation instability Two dimensional simulation in the transverse plane5 5 Note “structural analogy” with hydro and MHD results in “Long-time states of inverse cascades in the presence of a maximum length scale”, M. Hossain, W.H. Matthaeus, D. Montgomery, J. Plasma Physics, 30, 479 (1983), 2D Filamentation instability Effect of RR on the particle momentum at different times. THANKS FOR YOUR ATTENTION 2D Filamentation instability Effect of RR on the particle energy at different times.