Download Physical effects and modeling of the radiation reaction force in

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.