Particle transport at CME-driven shocks
Gang Li , Gary P. Zank and W.K.M. Rice†
†
IGPP, University of California, Riverside, CA 92521 USA.
School of Physics and Astronomy, University of St. Andrews, St. Andrews, Fife KY169SS, Scotland.
Abstract. It has been commonly accepted that in large solar energetic particle (SEP) events, particles are often accelerated to
MeV energies (and perhaps up to GeV energies) at shock waves driven by coronal mass ejections (CMEs). As a CME-driven
shock propagates, expands and weakens, particles accelerated diffusively at the shock can escape upstream and downstream
into the interplanetary medium. These escaping energized particles then propagate along the interplanetary magnetic field,
experiencing only weak scattering from fluctuations in the interplanetary magnetic field (IMF). In this paper, we concentrate
on the transport of energetic particles escaping from a CME-driven shock using a Monte-Carlo approach. This work, along
with our previous work on particle acceleration at shocks, allows us to investigate the characteristics (intensity profiles,
spectra, angular distribution, particle anisotropies) of high-energy particles arriving at various distances from the sun and
form an excellent basis with which to interpret observations of high-energy particles made at 1 AU by ACE and WIND.
INTRODUCTION
MODEL DESCRIPTION
It has been well established that gradual Solar Energetic
Particles (SEPs) events result from particle acceleration
at CME-driven coronal and interplanetary shocks (e.g.
[1], [2], [3], and [4]). Here the measured abundances of
ion charge states indicate that gradual events are associated with a temperature of T 2 10 6 K, in good agreement with coronal material (see [5, 6, 7, 8, 9] ).
The commonly accepted mechanism for particle acceleration at a shock is first-order Fermi acceleration, also
known as diffusive shock acceleration (see e.g. [10] ). In
the context of CME-driven shocks, explicit model calculations [11] suggest that particles can be accelerated up
to GeV energies. Subsequent investigations for a shock
with arbitrary strength verifies the conclusion [12]. The
transport of high energy particles accelerated at a shock
front based on the work of [11] and [12] is also underway
[13].
As particles are accelerated at the shock wave, they
are able to escape from shock front and propagate ahead
of the shock wave. The detection of these high energy
particles thus provides a way of predicting subsequent
disruptive events by the shock itself. Upon leaving the
shock, these energetic particles gyrate along interplanetary magnetic field lines, described by the Parker field,
with occasional pitch angle scatterings. In this work, we
consider the transport of these energetic particles using a
Monte-Carlo approach. We are interested particularly in
the intensity, the spectrum and the momentum distribution asymmetry observed at 1 AU.
The Boltzmann-Vlasov equation describes the evolution
of the distribution function f r p t for energetic particles escaping from the shock front into the interplanetary
medium,
∂f
∂t
p
∇ f F ∇p f
m
¬
∂ f ¬¬
∂ t ¬coll
(1)
The right hand side of (1) describes particle collisions. In
our context, this term describes the scattering of charged
particles by fluctuations in the interplanetary magnetic
field (IMF) and is replaced by ∂∂µ Dµ µ ∂∂ µf , where D µ µ
is the corresponding Fokker-Plank coefficient.
Between two consecutive pitch angle scatterings,
charged particles move in the solar wind by gyrating
along a field line. As the magnetic field expands radially, particles will experience focusing effects as a consequence of the adiabatic invariants of the motion. In the
context of solar wind, the B field is given by the usual
Parker spiral,
B B0
R0 2
Ω0 R0 r
1 r
u
R0
12sin2 θ 1 2
(2)
where θ is the colatitude of the solar wind with respect
to the solar rotation axis. Ω 0 is the solar rotation rate, u is
the radial solar wind speed, and B 0 is the interplanetary
magnetic field (IMF) at the co-rotation radius R 0 (typically, R0 10R¬, B0 183 10 6 T , u 400 km/s,
and Ω0 2π 254 days). The components of B along r̂
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
636
and φ̂ , Br and Bφ respectively, satisfy
dr
Br
rsinθ d φ
Bφ
and N t is the function from which we sample the
escape time (and thus location) of our initial particles.
For a given random number ξ , the following equation
(3)
te
On taking θ π 2 (i.e. we consider the equatorial
plane), the path length ds of the particle along the B field
line is then,
ds Ô
dr2 rd φ 2 ts
(4)
Equation (4) describes the “free” motion of charged particles between pitch angle scatterings. The length of this
“free” motion, which is the mean free path λ , is given
in the formalism of the quasi-linear theory [14] by,
λ
λ0
p α r β
1GeV
1AU
pmax
pin j
(5)
ri
vµ t dt
dr
dr
We consider simulations of a weak shock. The CME
driven shock is introduced at 0.1AU by temporarily increasing the number density and solar wind velocity at
0.1AU by a factor of 3 for half an hour. This corresponds
to an energy injection of the order 10 32 erg, a typical
value for a coronal mass ejection. We assume γ 53 to
model the solar wind. The CME driven shock is initiated
at 01 AU where the solar wind is already supersonic.
The weak shock has an initial velocity of 900 km/s and
drops to about 600 km/s at 1 AU, taking 225 days to
reach 1 AU.
Figure 1 plots the relative intensity profile for energetic particles observed at 1 AU from 2 MeV to 50 MeV.
The y-axes show the relative numbers of particles crossing over 1 AU at different times with a total of 100 time
intervals. The three panels in each of the figures represent different choices of α and β as in equation (5). The
curves in each of the panel represent, from top to bottom,
K 25, 55, 11, 21 and 51 MeV respectively. We have
set λ0 in equation (5) to be 08AU.
From Figure 1 we observe that the weak dependence
of particle mean free path on momentum and heliocentric distance has little affect on the simulation. For particles with energy K 11 MeV, the arrival of the shocks
(7)
where ri is the initial position and r f is the final position
of the particle, v is the particle velocity and µ t , the
pitch angle, is a function of time (thus r). If µ t i is less
than zero, then the particle is moving back toward the sun
and it may experience mirroring (when µ becomes 0 and
changes sign). On the other hand, if µ t i is greater than
zero, the particle will experience focusing as it moves
away from the sun.
To facilitate the simulation, we need to know the position of the shock as function of time and the phase space
distribution of charged particle at the shock front. This
is done in our previous work [11, 12]. There, the accelerated particle distribution function f r p t p t is
obtained for a series of times t 1 , t2 , ... , tn . Integrating
f r t p t over p2 d p, we get
N t f r t p t p2 d p
(10)
RESULTS
We can then characterize the motion between two pitch
angle scatterings using l by
rf
f r p ¼ t p ¼ t d p ¼ ξ ¼
where ξ ¼ is another independent random variable. Once
t and p are decided, the corresponding location can then
be inferred from r p t r sh t lesc p, where rsh t represent the location of the shock complex at time t and
lesc p is the escape length for particle with momentum
p.
After the initial condition (r, t, p) of a test particle
is decided, its subsequent motion is simply an interplay
between “free” motion and pitch angle scatterings. The
scatterings themselves are assumed to be isotropic and
Markovian, thus the new pitch angle after a scattering
does not have any memory of the previous pitch angle.
where λ0 , an input parameter, is taken to be 08AU here.
Parameters α and β describe momentum and heliocentric distance dependence, with α between 0 and 13 and
β between 0 and 23. In our simulation, we consider
three cases of α and β , being, 1) α 13, β 0, 2)
α 0, β 0 and 3) α 0, β 23. In the context of a
Monte-Carlo simulation, the motion of charged particles
is followed individually when they escape from shock
front. If we denote l as the distance between two scatterings, then the probability density p l for scattering
satisfies,
l λ p l e
(6)
l
(9)
determines t, which is the time for a test particle leaving
the shock. In the above, t s and te denote the time at which
the shock starts and the time when the shock reaches 1
AU. Once t is determined, the momentum of the particle
can be determined in a similar way through
Õ
1 Bφ Br 2 dr
N t ¼ dt ¼ N t ¼ ξ
(8)
637
α = 1/3, β = 0
3
10
5
α = 1/3, β = 0
10
2
10
3
10
1
Relative Intensity
Relative Intensity
10
0
10
3
α = 0,
10
β=0
2
10
1
10
0
10
3
10
1
10
5
10
α = 0,
β=0
3
10
1
10
5
10
α = 0, β = 2/3
α = 0, β = 2/3
3
2
10
10
1
10
10
0
10
0.5
1
1.5
2
2.5
1
0.1
1
10
100
1000
10000
3
time (days)
Kinetic energy (MeV)
FIGURE 1. Relative intensity for a weak shock, plotted as a
function of time for three possible choices of mean free path as
in equation (5). Each curve represents a distinct particle energy
with the solid curve for K 2 5 MeV, the dashed curve for
K 5 5 MeV, the dotted curve for K 11 MeV, the curve with
square for K 21 MeV and the curve with star for K 51
MeV.
FIGURE 2. The particle spectrum observed at 1 AU for three
possible choices of mean free path as in equation (5). Each
curve represents a distinct time period with the solid curve for
0 15 T, the dashed curve for 15 25 T, the dotted curve
for 25 35 T, the curve with square for 35 45 T and the
curve with star for 45 1 T, where T 2 25 days is the arrival
time for the shock wave.
T, 25 35 T, 15 25 T, 0 15 T respectively. In
Figure 2, the spectra correspond to 0 15 T (the solid
curve) is quite different from those at later times. The
more plateau-like shape for the 0 15 T spectra is due
to the “free-streaming” limit, that particles reaching 1
AU at early times are those freely stream out without
pitch angle scatterings, thus lower energy particles have
smaller speeds and it takes longer for them to reach 1
AU than for high energy particles. The spectra further
softens with time, indicating that fewer very energetic
particles are accelerated by the shock at later times.
The spectra at later times are approximately power laws
which assume a broken form. Comparison of the three
panels reveals again, that the choice of α and β does not
have a pronounced affect on the simulation result.
Finally, we discuss the time evolution of the particle
distribution function f p r 1AU. In particular, we
are interested in the angular distribution of f with B̂ p̂.
Figure 3 plots f p r 1AU at four different times for
the case of λ0 08 AU and α β 0. In the figure, Z x
and Zy are the measures of particle velocity and related
to particle momentum p and the Parker field B through,
at 1 AU corresponds to a peak in the intensity profile.
However, for particles with energy K 20 MeV, no
such peaks exist. The absence of a peak at the shock
in the particle intensity was noted earlier in [11]. The
present work confirms that interplanetary shocks, especially weak shocks, can only accelerate particles up to
high energies during the early stage of their propagation.
As the shock weakens, and the IMF strength becomes
smaller away from the sun, the maximum possible accelerated particle energy decreases. The simple assumption, that longer-lived shocks can accelerate particles to
higher energies in the solar wind, is incorrect. For the
weak shock example of Figure 1, the shock stopped accelerating particles up to energies K 20MeV when it
was still close to the sun. As can also be seen from Figure 1, even though the shock has stopped accelerating
particles above a certain energy by 1 AU, higher energy
particles can remain trapped in the post-shock complex
(K 20 MeV). At even higher energies, particles accelerated earlier have completely escaped and are no longer
trapped in the complex region downstream of the shock
(e.g. K 50 MeV). The intensity profiles of the K 20
MeV particles resemble those that are seen in impulsive
events.
Figure 2 plots particle spectra observed at 1 AU at
different times. Again, the three panels in the figure
represent three choices of α and β . The five curves in
each panel correspond to spectra at 1 AU for different
time periods after the shock was initiated. If we denote
by T the time of shock arrival, which is 225 days for our
simulation, then the curves, from left (curve with stars)
to right (solid curve), correspond to 45 1 T, 35 45
Zx
Zy
cos θB̂p̂ log pMeV 425
sin θB̂p̂ log pMeV 425
(11)
(12)
The Zx axis coincides with the Parker field. The plot
is obtained for a total of eight energy intervals and
forty pitch angle intervals. Note that it is an even function about the Z y axis, reflecting the gyro-symmetry of
pitch angle scattering. From the innermost circle to the
outermost circles, the corresponding particle energies
638
t = 0.20 T
1
Zy
Zy
1
0
-1
0
-1
-1
0
1
2
3
-1
0
Zx
1
2
ACKNOWLEDGMENTS
5000.12
3579.95
1835.14
940.723
482.23
247.199
126.718
64.9578
33.2984
17.0693
8.75
1
Zy
1
0
0
-1
-1
-1
0
3
Zx
t = 0.80 T
t = 0.60 T
Zy
times. Our simulation, together with our earlier theoretical work on particle acceleration at CME-driven shocks,
provides a sophisticated theoretical framework for interpreting corresponding observations.
t = 0.40 T
1
2
-1
3
0
1
2
This work has been supported in part by a NASA grant
NAG5-10932 and an NSF grant ATM-0296113.
REFERENCES
3
Zx
Zx
1.
FIGURE 3. Phase space distribution of the energetic particles observed at 1 AU at 4 different times. The coordinates Zx ,
Zy correspond to v and v (see text for details). Different
colors represent different relative number density. The legends
are the same for all 4 figures ranging from 8 75 to 5000.
2.
3.
are K 488, 812, 1047, 1535, 2106, 3075, 5080,
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∑iNµi
4.
5.
6.
7.
8.
N
9.
(13)
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ative
at µ 1 in the figure shows that particles must have a
component along the B̂ direction to be observed.
11.
CONCLUSION
12.
10.
We have investigated the transport of energetic particles
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weak shocks occur quite frequently. We have also studied
the particle’s angular distribution with respect to particle pitch angle and find a negative asymmetry
at later
13.
14.
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