Download Reflection of electrons in a structured shock front Prof. Michael Gedalin

BSc. Project: Reflection of electrons in a structured shock front
Afik Shachar
Advisor: Prof. Michael Gedalin
Abstract
Collisionless shocks accelerate charged particles. Electron acceleration, which requires reflection off
the shock front, is suppressed in because of the upstream directed electric field inside the ramp. The
objective of this project is to study the effect of the shock substructure on the electron reflection.
We show that in the presence of a sufficiently thin precursor some electrons are reflected.
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Introduction
Shocks are known to be efficient particle accelerators, yet the process of the acceleration is not
completely understood. Particle acceleration requires reflection off the shock front, that is, a particle,
coming from upstream, should be able to return to upstream. For electrons this process is strongly
suppressed because the cross-shock electric field drags the electrons from upstream to downstream.
In a structured shock the direction of the electric field may alternate and force some electrons to
return to the upstream region. In this project we study the electron motion in a quasi-perpendicular
shock transition layer consisting of a main ramp and a large amplitude precursor. When both are
thin they may be considered as a ramp with a substructure. The objective is to establish at least
one set of parameters where electron reflection becomes possible.
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Analytical model
The equations of motion that describe an electron in an electromagnetic field are:
me v̇x = −e(Ex + vy Bz − vz By )
me v̇y = −e(Ey + vz Bx − vx Bz )
me v̇z = −e(Ez + vx By − vy Bx )
we can’t solve these equations for the field of collitionless shck front, therefore we begin by analyticly
solving for a simpler electromagnetic field such that the magnetic field is constant and the electric
field is linear in the x coordinate. we use the following fields:
E = (Ex (x), Ey , 0)
(
Ex (x) =
−βx
β(x − 2x0 )
0
0 ≤ x < x0
x0 ≤ x ≤ 3x0
else
B = const, Ey = const
1
We’ll solve for 0 < x < x0 , the solusion will be the same for x ≥ x0 only with different sign and
starting conditions. we define:
 
x


βe
eB
Bx
vx 
α=±
,
Ω
=
,
cosθ
=
,
~
a
=
 vy 
me Ω2
me
B
vz
and then we can write all our eqations as one vector equation:
~a˙ + M • ~a = ~b
for


0
−1
0
0
−αΩ2
0
Ωsinθ
0 
,
M =
 0
−Ωsinθ
0
Ωcosθ
0
0
−Ωcosθ
0


0
 
~b = − e E0 
me Ey 
0
the homogeneous equation ~a˙ + M • ~a = 0 gives us:
~a =
4
X
ci e−λi t ~ui
i=1
here, λi are the eigenvalues of M, and ~ui are the eigenvectors of M. and from searching for a constant
solution we get a private solution:


E
0
β


0


~ap = 

0


eEy
− me Ωcosθ
therefore a complete solution will be:
~a =
4
X
ci e−λi t ~ui + ~ap
i=1
λ2i
(α − 1) ±
p
(α2 + 2α cos(2θ) + 1)
2

=Ω

λi (λ2i +1)
− αsin(θ)cos(θ)
 cos2 (θ)+λ2 


i

~ui = 
 sin(θ)cos(θ)

λi
 − cos(θ) 
1
When solving this equation we varied the following parameters: the initial velocity, strength of
fields and the direction of the magnetic field in order to see if it may be possible for an electron to
be reflected back in this simple model. Our result are that reflection is possible and the parameters
that allowed for such a trajectory in the simple model were:
π
α = 0.008,
~v0 = (1, 0, 6),
Ey = 1,
θ= ,
Ω=1
3
for:
e = 1,
me = 1
2
we recieve the following trajectory:
Figure 1: the trajectory of an electron
further more, we see that in this simple model the reflection of an electron is dependent of the
velocety of the electron in the z direction when it enters the shock:
(a) vz = 4
(b) vz = 6
(c) vz = 8
Figure 2: trajectory dependence of vz
we notice that there is a range of velosities vz1 < vz < vz2 for which we can get reflection. for
smaller vz (Figure 2a) the particle can’t overcome the driving field, and for large vz (Figure 2c) the
particle is too fast and just passes through the reflecting field without losing enough speed.
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3
Numerical calculation
After having a rough idea of what parameters should give electron reflection, we used a fortran90
script to run a numerical calculation of electron trajectories in collisionless shock front ramp. In
the shock front we have an electric and a magnetic field, both dependent on only one coordinate (x
- directed downstream). The equations of motion are as before:
me v̇x = −e(Ex + vy Bz − vz By )
me v̇y = −e(Ey + vz Bx − vx Bz )
me v̇z = −e(Ez + vx By − vy Bx )
but now with the fields being:
~ = (Ex (x), Ey , 0) ,
E
~ = (Bx , By (x), Bz (x)) ,
B
Ey = vu Bu sin θ,
Bx = Bu cos θ
therefore we get:
me v̇x = −e(Ex + vy Bz − vz By )
me v̇y = −e(vu Bu sin θ + vz Bu cos θ − vx Bz )
me v̇z = −e(vx By − vy Bu cos θ)
To use these equations we first need to simplify by changing to dimensionless form using typical
electron and shock parameters as follows:
X=
eBu x
,
vu me
V =
v
,
vu
by =
By
.
Bu
bz =
and after normalization we get equations of the form:
V̇x = −(e + Vy bz sin θ − Vz by )
V̇y = −(sin θ + Vz cos θ − Vx bz )
V̇z = −(Vx by − Vy Bu cos θ)
Ẋ = Vx
the field approximations commenly used are:
dbz
dX
s mi dbz
e=−
2 me dX
by = a
with magnetic profile:
bz =
R+1 R−1
X
+
tanh
2
2
D
which then gives:
R−1
X
cosh−2
2D
D
s mi R − 1
X
e=−
cosh−2
2 me
2D
D
by = a
4
Bz
,
Bu sin θ
e=
Ex
vu Bu
mi
we denote D = k m
(mi - ion mass, me - electron mass) where k ≤ 0.1. we also denote bm =
e
By
and get:
Bu
max
e=−
s (R − 1)
X
cosh−2
4k
D
X
D
we also add a feature ahead of the shock by using the same relation of by and e to bz while adding:
by = bm cosh−2
bz = b0 e−
(X−X0 )2
2L2
In Figure 3 we show an example of a field with the following parameters: b0 = 0.5, X0 = −3D,
mi
L = 0.5D, k = 0.03, s = 0.5, R = 3, m
= 1836 (proton - electron mass ratio)
e
Figure 3: bz in Black, by in Blue, e in Red
Once we have the shock profile we used a fortran90 script to run electrons through this shock
with different starting parameters and by varying the shock parameters until we see reflection. It
was found that for 0.03 ≤ k ≤ 0.035 and θ = 60◦ about 10% of the electrons were reflected back
upstream. A typical trajectory of a reflected electron in shown in Figure 4b
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(a) vx as a function of x
(b) y as a function of x
Figure 4: e in Green, electron path in Blue, 5 × bz in Red
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Conclusions
Presence of a sufficiently thin precursor may result in the reflection of a fraction of electrons off
the shock front. The reflection point is typically between the downslope of the precursor and the
subsequent rise of the ramp. The analysis suggests that the electrons become demagnetized in
rising side of the precursor and are dragged across the magnetic field to the region where they
are magnetized again and gyrate. A favorable combination of the demagnetization and subsequent
gyration allows some of them to cross the precursor back and escape upstream.
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