ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-34-591-2016Critical pitch angle for electron acceleration in a collisionless shock layerNaritaY.yasuhito.narita@oeaw.ac.atComişelH.MotschmannU.Space Research Institute, Austrian Academy of Sciences, Schmiedlstr. 6, 8042 Graz, AustriaInstitut für Theoretische Physik, Technische Universität Braunschweig, Mendelssohnstr. 3, 38106 Braunschweig, GermanyInstitute for Space Sciences, Atomiştilor 409, P.O. Box MG-23, Bucharest-Măgurele 077125, RomaniaDeutsches Zentrum für Luft- und Raumfahrt, Institut für Planetenforschung, Rutherfordstr. 2, 12489 Berlin, GermanyY. Narita (yasuhito.narita@oeaw.ac.at)12July20163475915931May201614June201629June2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/34/591/2016/angeo-34-591-2016.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/34/591/2016/angeo-34-591-2016.pdf
Collisionless shock waves in space and astrophysical plasmas can accelerate
electrons along the shock layer by an electrostatic potential, and scatter or
reflect electrons back to the upstream region by the amplified magnetic field
or turbulent fluctuations. The notion of the critical pitch angle is
introduced for non-adiabatic electron acceleration by balancing the two timescales under a quasi-perpendicular shock wave geometry in which the upstream
magnetic field is nearly perpendicular to the shock layer normal direction.
An analytic expression of the critical pitch angle is obtained as a function
of the electron velocity parallel to the magnetic field, the ratio of the
electron gyro- to plasma frequency, the cross-shock potential, the width of
the shock transition layer, and the shock angle (which is the angle between
the upstream magnetic field and the shock normal direction). For typical
non-relativistic solar system applications, the critical pitch angle is
predicted to be about 10∘. An efficient acceleration is expected below
the critical pitch angle.
Space plasma physics (charged particle motion and acceleration; shock waves)Introduction
Collisionless shocks in space and astrophysical plasmas are unique
in that electrons are efficiently accelerated there.
Different types are possible for the acceleration mechanisms.
For a quasi-parallel shock at which the upstream magnetic field
is nearly aligned with the normal direction of the shock layer,
the electrons can be efficiently trapped by turbulent fluctuations on the both sides
of the shock. Since the incoming flow speed on the upstream side
is higher than that on the downstream side,
the electron trapping leads to a diffusive shock acceleration .
For a quasi-perpendicular shock, the electrons are accelerated
within the shock transition layer by the electrostatic, cross-shock potential
which is sustained by different bulk motions of the ions to the electrons
.
On the other hand, the electrons can be scattered or reflected away from the shock
layer by a sudden increase of the magnetic field or turbulent fluctuations at the shock ramp.
One may thus formulate that the electrons at the quasi-perpendicular shock
undergo two competing effects: acceleration or scattering.
The purpose of this article is to estimate the critical pitch angle
for the electron acceleration at the quasi-perpendicular shock.
Critical pitch angle
Naively speaking, for a smaller pitch angle, the electrons are nearly
field-aligned (with respect to the magnetic field) and should have
sufficient time to stay in the shock transition layer for an efficient
acceleration by the electrostatic potential. For a larger pitch angle, the
electrons are scattered and eventually kicked away from the shock transition
layer, either in the upstream and downstream directions. We define the
critical pitch angle through the balance between two timescales, the
electron acceleration time τacc and the electron scattering
time τsc, into a formula as τacc=τsc.
One may roughly estimate the acceleration timescale as τacc=ℓv‖, where ℓ stands for the field-aligned length and
v‖ the electron velocity parallel to the magnetic field, respectively.
Likewise, the scattering timescale is estimated as τsc=dv⟂ using d, which is the width of the shock transition layer
and v⟂ the perpendicular velocity. The two length scales, ℓ and
d are related to each other by an angle θnB between the upstream
magnetic field and the shock normal direction as ℓcosθnB=d.
We use the equation of motion (in a non-relativistic sense) for the electron
in the parallel or field-aligned component:
mv˙‖=eE‖,
from which the parallel velocity for a timescale of τacc is
obtained as
v‖=emE‖τacc+v‖0
and the length scale parallel to the field as
ℓ=emE‖τacc2+v‖0τacc.
Here m denotes the electron mass, e the electric charge of the electron,
E‖ the field-aligned component of the electric field,
and v‖0 the initial parallel velocity of the electron.
For the perpendicular component, we have
d=v⟂τsc.
That is, the scattering timescale is of the order of the electron
gyro-period. The pitch angle is defined as tanα=v⟂/v‖. The
electrons are treated as non-adiabatic particles in this theoretical frame by
neglecting the mirror force in Eq. () and the electron drift
in Eq. (). See Eqs. (3.92) and (3.87) in for
the parallel component of equation of motion and the perpendicular component
of electron drift, respectively. The question of whether the electrons are adiabatic
or not at the shock transition needs to be evaluated additionally when
testing the notion of the critical pitch angle against observational or
simulation data.
The acceleration timescale can explicitly be obtained by regarding
Eq. () as a quadratic equation of τacc. The
solution for τacc>0 is
τacc=m2eE‖-v‖0+v‖02+4emE‖ℓ.
The condition for the critical pitch angle is τacc=τsc, from which we obtain an estimate for the critical pitch
angle αcr (after some algebra) as
tanαcr=2eE‖dmv‖02-1+1+4eE‖dmv‖02cosθnB-1=2x-1+1+4xcosθnB,
where the dimensionless quantity x is introduced as
x=eE‖dmv‖02=c2v‖02ΩeωpeE‖Bdλe=c2v‖02ΩeωpeΦBλedλe.
For the particle-in-cell simulation by ,
v‖0c=0.3, Ωeωpe=18, ΦBλe=0.1,
dλe=1, θnB=81∘, the dimensionless
quantity is x=0.14, and the critical angle is αcr=14∘.
Figure shows the profile of the critical pitch angle
αcr as a function of the dimensionless quantity x
(Eq. ). Smaller values of x indicate high-energy,
small-pitch-angle electrons in a thin shock layer. The critical pitch angle
shows an asymptotic behavior at smaller values of x (x<0.1), and
becomes smaller from about 20 to 10∘ and further to
1∘ as the magnetic field angle from the shock normal changes from
70, 80, and 89∘. It is interesting to note that x increases at a larger electrostatic potential and a weaker magnetic field. Therefore,
a stronger electrostatic potential leads to a larger critical pitch angle and
thus enables more particles to be accelerated.
Critical pitch angle as a function of the dimensionless quantity x
for different values of the shock angle between the upstream magnetic field
and the shock normal direction θnB.
Perspective
The critical shock angle is formulated as a function of the electron velocity
parallel to the magnetic field, the ratio of the electron gyro- to plasma
frequencies, the electrostatic potential, the width of the shock transition
layer, and the shock angle (which is the angle between the upstream magnetic
field and the shock normal direction). From the above simple (and algebraic)
estimate, particularly in Eq. (), one draws a lesson that the
electron acceleration is efficient in an increasing sense of the critical
pitch angle (or equivalently, an increasing sense of the dimensionless
quantity x) under the conditions of (1) lower parallel velocities of the
incoming particles, (2) a stronger magnetic field or a lower electron
density, (3) a higher electrostatic potential at the shock transition, or
(4) a thicker transition layer. For typical non-relativistic solar system
applications, the critical pitch angle is predicted to be about 10∘
or even less. Our treatment does not include the effect of turbulent
fluctuations explicitly. Thus it would be interesting to study in detail
whether or how adiabatic electron acceleration and heating are affected by the shape
or time evolution of the shock potential.
A caveat needs to be addressed here.
As we have assumed non-adiabatic motion of electrons,
the spatial gradient of the magnetic or electric field must be
smaller than the electron gyro-radius such that
the magnetic moments of electrons are no longer constant.
The condition of non-adiabatic motion for
the electric field is e|∇E⟂|meΩe2>1.
Here Ωe=eB/me is the electron cyclotron
frequency. Using the Gauss law on the electric field,
∇⋅E=ρeϵ0,
where ρe and ϵ0 denote the charge density and the
permittivity of free space, respectively. One may re-write the non-adiabatic
condition (Eq. ) into a simpler form (using
the relation to the permeability of free space μ0 and the speed of light
c as ϵ0μ0=c-2) as
ErestEmag=12meδnec212μ0B2>1,
where the fluctuating electron number density is estimated through the
electric charge density as δne=ρe/e,
neglecting the ion contribution. In essence, the rest energy of the perturbed
electron fluid must exceed the energy density of the magnetic field.
Understanding the relationship between the cross-shock potential and the
particle dynamics such as trapping, parallel acceleration, and perpendicular
scattering with respect to the mean magnetic field has various applications
to the collisionless shocks in the solar system and in astrophysical systems.
The method of Liouville mapping provides high-time-resolution electron
velocity distribution functions and is a useful tool to evaluate the
cross-shock potential . The
dependence of the critical pitch angle on the shock geometry, the shock
potential, and the electron parallel velocity can be tested not only
numerically using direct numerical simulations such as particle-in-cell
algorithms but also observationally using the novel MMS (Magnetospheric
Multiscale) mission . It is also interesting to note that
the idea of critical pitch angle works for any short-scale gradient with a
parallel electric field such as a steepened electron-scale whistler wave, and
not exclusively at shock waves.
Acknowledgements
H. Comişel thanks Manfred Scholer and Octav Marghitu for stimulating and useful
discussions which led to developing and formulating the notion of the
critical pitch angle for this article. The work by H. Comişel and
U. Motschmann in Braunschweig is supported by an extended program of
Collaborative Research Center 963, “Astrophysical Flow, Instabilities, and
Turbulence” of the German Science Foundation. The work conducted by
H. Comişel in Bucharest is supported by Romanian Ministry for Scientific
Research and Innovation, CNCS – UEFISCDI, project number
PN-II-RU-TE-2014-4-2420. Edited by:
C. Owen Reviewed by: one anonymous referee
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