Download 672_1.pdf

Charge-to-mass Fractionation of Suprathermal Ions
Associated with Interplanetary CMEs
R. Kallenbach , K. Bamert† and R.F. Wimmer-Schweingruber†
†
International Space Science Institute, Hallerstrasse 6, CH-3012 Bern, Switzerland
Physikalisches Institut, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
Abstract. A model for the acceleration and transport of suprathermal ions associated with interplanetary coronal
mass ejections (CMEs) is presented. The combined mechanisms of stochastic acceleration in the turbulence near
interplanetary shocks, first-order Fermi acceleration, stationary spatial diffusion in field irregularities of magnetic
clouds, and time-dependent propagation in the upstream solar wind are described. Ions from the bulk solar wind
are considered as source populations of suprathermal ions. The results of the model are compared to the spectra
of suprathermal ions observed with SOHO/CELIAS/HSTOF during the Bastille Day CME of 14–16 July 2000.
FIGURE 1.
Configuration of a CME [1].
INTRODUCTION
Solar energetic particle (SEP) events are commonly classified as gradual or impulsive. Energetic ions associated with gradual events are believed to be accelerated
near interplanetary shocks driven by CMEs. Very often,
the ejecta of the CME propagate into the solar wind in
form of magnetic clouds with spiral field configurations
(Figure 1). The compression region between the shock
and the cloud contains magnetic turbulence. The Bastille
Day event is a CME with multiple shocks and magnetic
clouds. We evaluate the suprathermal ion distributions
near the main shock, which is preceded by another magnetic cloud upstream (MC1) in addition to the magnetic
cloud (MC2) following the downstream turbulence.
FIGURE 2. Plasma regions associated with the Bastille Day
CME and the dominant processes effecting the spatial and
momentum distribution of suprathermal ions (not to scale).
The processes of stochastic acceleration of solar wind
ions i.e. momentum diffusion in magnetic field irregularities, stationary spatial diffusion in the turbulence region
and inside the magnetic clouds, time-dependent propagation in the upstream solar wind plasma, and first-order
Fermi acceleration are considered (Figure 2). The ion energy range between 35 keV/amu and 2 MeV/amu is studied, the energy range in which the CELIAS/HSTOF instrument is sensitive.
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
672
THE TRANSPORT EQUATION
The evolution of suprathermal ion populations near a parallel shock betweenupstream and downstream plasma
with bulk speeds V
V1 and V
V2 , respectively, is
described by the standard transport equation:
∂f V ∇f
∂t
∇ κ∇ f 1 ∂
p2 ∂p
I S
p2 D pp
p ∂f
∇ V
3 ∂p
∂f
∂p (1)
where p v v0 with v0 the speed of 1 MeV protons.
The left hand side denotes the explicit time dependence
and convection of suprathermal ions, whereas the terms
on the right hand side describe spatial diffusion, sources
(I), sinks (S), adiabatic deceleration, and momentum diffusion. For symmetrically counter-propagating Alfvén
waves and isotropic distribution functions f , spatial and
momentum diffusion [2] are related by
D pp κ 1 2 2 2
V p V 9 A A
B2
D pp µ0 mp np p 2 DT T
(2)
with VA the Alfvén speed, B the ambient magnetic field,
and np the proton density. In the hard-sphere approximation [3], the energy diffusion parameter is
Q 2 α α 1 T
A
B
6 105 2 T m2 s 1
δB
κ0
4 VA2
9 κ0
DT T
2
(3)
where δB is the amplitude of the magnetic field irregularities with power spectral density P P0 f α resonating
with the ion of atomic mass A and charge Q. This yields
D pp
4 VA2
9 κ0
κ
1
κ0
4
Q
A
Q
A
2 α
α 1
p α 2
p3 α
and
into the process of spatial and energy diffusion occurs
at lower energies, the solar wind bulk energies. Finally,
it is assumed that one spatial coordinate x is sufficient
to describe the situation of a parallel shock. We obtain
a partial differential equation solvable by separating the
variables i.e. f x p g x h p :
∂
∂f 1 ∂
∂f
0
κ
p2 D pp
(5)
∂x
∂x p2 ∂p
∂p All Q A- and p-dependent terms are incorporated into
the momentum part of the equation, whereas all xdependent parts are absorbed in the spatial differential
equation. Both parts must equal a constant C because x
and p are independent variables. Therefore,
A 4 2α 4 2α
∂2 h α 1 ∂h
C
p h 0
(6)
∂p2
p ∂p
Q
Neglecting terms with factors of order p 2F 2 and
p 4F 2 , which fall off more rapidly than others for α 3 and for the parameters of our experimental situation
(which needs to be checked if applied to another problem), we find
h p h0 p 2β exp ero p2F Gp2H 3
C A 2 α
β ero 4
2F Q 3 α
9 6α
F G
H F
(7)
2 32F 2 ero The spatial differential equation becomes
9κ0 ∂ κ0 ∂g x Cg x 4VA2 ∂x 4 ∂x With the new dimensionless variable
x 4V
A
ξ : "!
dx#
3κ
0
0
(8)
(9)
the differential equation is transformed to
(4)
Stationary Spatial and Energy Diffusion
If the time scales for acceleration and diffusion of
suprathermal ions in the downstream turbulence near the
main shock are shorter than the dynamical time scales of
the CME structure, the explicite time derivative, the convection term for an observer in the downstream plasma
frame, and the adiabatic deceleration term of the transport equation may be neglected. It is further assumed that
sources and sinks of the populations in the suprathermal
energy range studied here can be neglected, i.e. injection
673
∂2 g ξ 1 ∂VA ∂g ξ Cg ξ $ 0 ∂ξ2
VA ∂ξ ∂ξ
(10)
If, for example, VA VA % 0 exp &' ξ2 ( 2ξ2A *) , then the intensity profile of the suprathermal ion flux, while passing
the CME’s turbulence region, is
ξ2
(11)
g ξ + g0 exp 2ξ20 with ξ0 ξA . The constant C is identified as 1 ξ20 , and,
hence, is related to the scale length L (in meters) of the
turbulence region by
C
1 ξ20
9κ20 16L2VA2
(12)
Stationary Spatial Diffusion
The magnetic clouds of the Bastille Day event also
contain magnetic field irregularities, although with
smaller relative amplitude δB B than in the turbulence region near the shock. Thus, suprathermal ions
are produced locally in the clouds to less extent, but
they diffuse from the turbulence region into the clouds.
This boundary condition prohibits removing the p- and
Q A-dependence in the spatial differential equation
∂f
VMC1 2 ∂x
∂ 1
κ0
∂x 4
Q
A
α 2
p3 α∂f
(13)
∂x where VMC1 2 is the bulk speed of the magnetic cloud
in the plasma frame of the turbulence region, which has,
within about one Alfvén speed, the shock speed.
In steady state, the bulk speeds VMC1 2 are important
parameters in the spatial diffusion profile [4]. With x1 2 as the location of the boundary of the magnetic cloud
towards the shock or turbulence region, we find
f x p
η :
2 α
dx#
(14)
where f x1 2 p is the spectrum evaluated e.g. in the
previous subsection.
The parameter η describes the stationary dispersion in speed and in Q A of the ions, while
diffusing away from the source i.e. the turbulence region.
The speed VMC2 of the downstream magnetic cloud in the
shock frame is probably not much larger than the Alfvén
speed, whereas VMC1 is quite large, on the order of the
upstream plasma speed V1 .
Time-Dependent Solution
The propagation of ions in the upstream solar wind
plasma is explicitely time-dependent as the distance between the CME event and the SOHO spacecraft changes.
The process is described in the SOHO spacecraft frame,
in which the solar wind bulk plasma speed is VSW . Only
the propagation along the direction x of SOHO’s magnetic connection to the CME is considered:
∂f ∂f
VSW
∂t
∂x
∂
∂f κ
∂x ∂x (15)
For a spatially constant diffusion parameter κ in the
upstream solar wind, the solution’s kernel is
K x t + K0
x VSW t 1
exp 4κt
κt
f ζ 2
(16)
674
fMC1 1 erf ζ ζ vx 4κ
(17)
The population f MC1 is the one at xMC1 .
First-order Fermi Acceleration
The injection into first-order Fermi acceleration [4] of
the suprathermal ions that are stochastically accelerated
in the downstream turbulence is given by
fF 0 p f x1 2 p exp η x 4pα 3V
Q
MC1 2 !
κ0
A
x1 2 The suprathermal ions are emitted from the CME as
a moving source. The earliest arrival time for the ions
from the CME spaced by x xMC1 xSOHO , with xMC1
the boundary of the upstream magnetic cloud towards
SOHO, is t x v. We integrate the source term over
time from x v to large t, when the CME was closer to
the Sun (t ∞). We neglect variations in κ and the ∆t of
the arrival times due to diffusive effects. This yields
h0
!
pγ
p
pinj
d p#
exp ero p# 2F Gp# 2H p# 1 2β γ
(18)
This expression yields a power law f F 0 p fF % 0 p γ
with spectral index γ 3V1 V1 V2 , if p pinj . If the
latter is not the case, the expression becomes a more
complicated function of momentum p.
A problem may arise because the ions that participate
in the first-order Fermi process change momentum during one excursion in the turbulence
region downstream.
The momentum diffusion is ∆p
D pp t, whereas the
spatial diffusion is ∆x
κt. The expectation value of
the time for an ion to return
to the shock from the turκ τr V2 , where V2 is
bulence region is τr ∆x V2
the downstream plasma speed in the shock frame. In a
first-order Fermi process, the momentum change ∆p τr by diffusion
should be smaller than the momentum gain
∆P 2 V1 V2 v0 . Therefore,
∆p τr D ppκ
V2
VA
p ∆P
3V2
(19)
The ratio VA V2 is of order 1, and ∆P is of order 0 1,
so that the shock acceleration
is a pure first-order Fermi
process up to energies of 100 keV/amu. This estimate
is only valid for an exactly parallel shock, where perpendicular diffusion and gyro-motion of the ions do not
contribute to the ion return to the shock. Therefore, in
reality the problem should be less severe, i.e. pure firstorder Fermi acceleration can be expected up to higher
energies.
Further parameters are relevant for the processes at the
shock and in the turbulence region, such as the mean free
path λ of the ions, the shock width Lsh , the return time
scale τr , the shock width Lsh , and the injection threshold
pinj (the momentum for which the ion mean free path
equals the shock width). The shock width Lsh typically
has a value between the electron inertial length Le c ωpe and the proton inertial length Lp c ωpp , where
ω2pe p ne p e2 ( 4πε0 me p . For B 10 nT, B2 δB2
100, α 1, L 1010 m, np ne 107 m 3 , and V2
VA 7 104 m s the values for these parameters are:
κ κ0 Q α 2 3 α A 2
τr
p p ;
3 105 s Q
V22
4V22 A 3κ A
λ
4 108 m p;
v0 p
Q
pinj
τL
4cv0 Q
3ωpp κ0 A
Lsh 10 Lp
L v
103 s p
L v0 p
3Q
A
Lsh
;
Lp
(20)
The estimate for the injection threshold certainly is too
low because the model does neither describe the shock
potential nor the ion motion of low-energy particles on
the shock length scale. The momentum for
downstream
escape, however, can be derived from τr τL as
Q 1 3 pesc 0 14
(21)
A
This estimate shows that first-order Fermi accelerated
ions should be observable upstream and downstream
from the shock.
very often
has a Kolmogorov type power spectral in
dex α 5 3, but proton-beam generated waves near the
shock may have a different spectral index. Other theories
for the diffusion coefficients also need to be compared to
the data.
For the case of He, the spectra match the model at
lower energies, if one assumes that the He ions are solar
wind 4 He . However, at higher energies, there appears
to be an additional component, which may be 3 He from impulsive flares [5].
CONCLUSIONS
The presented model and data from the HSTOF sensor
on board SOHO suggest that solar wind ions are preaccelerated by momentum diffusion in the turbulence region downstream from the shock and subsequently injected into first-order Fermi acceleration. The efficiency
for the momentum diffusion scales with the charge-tomass ratio of the ions. The propagation of the suprathermal ions through the magnetic field irregularities shows
the signatures of dispersion in speed and in the chargeto-mass ratios of the ions.
ACKNOWLEDGMENTS
This work was supported by the Swiss National Science
Foundation and by the INTAS grant WP 270.
REFERENCES
SOHO/CELIAS/HSTOF DATA
The suprathermal ions of the Bastille Day event have
been observed at 1 AU with SOHO/CELIAS/HSTOF
[5]. The spectra are fitted by a sum of two distribution
functions. One of them represents the ions stochastically
accelerated in the turbulence downstream of the shock.
They diffuse into the two magnetic clouds upstream and
downstream and farther out. Near the shock, they are
also injected into first-order Fermi acceleration. This
results in a second population which also diffuses into
the magnetic clouds and farther out.
Within the experimental uncertainties of about 20%,
the model is consistent with the spectra of suprathermal
H, CNO, and Fe ions assuming that 1) the ions have
solar wind charge states, 2) the turbulence in the compression region downstream
from the shock is Alfvénic
with spectral index α 1, and 3) the field irregularities
in the magnetic clouds have a spectral index α 1 as
well. The model needs to be verified by an evaluation of
ACE/MAG magnetic field data. Solar wind turbulence
675
1. Lee, M.A., Particle acceleration and transport at CMEdriven shocks, in: N.U. Crooker, J.A. Joselyn, and J.
Feynman (eds.), Coronal Mass Ejections, AGU Press,
p. 227, 1997.
2. Hasselmann, K., and Wibberenz, G., Scattering of charged
particles by random electromagnetic fields, Zeitschrift für
Geophysik, 34, 353, 1968.
3. Möbius, E., Scholer, M., Hovestadt, D., Klecker, B., and
Gloeckler, G., Comparison of helium and heavy ion spectra
in He-3-rich solar flares with Model calculations based
on stochastic Fermi acceleration in Alfvén turbulence,
Astrophys. J., 259, 397, 1982.
4. Forman, M.A., and G.M. Webb, Acceleration of Energetic
Particles, in: R.G. Stone and B.T. Tsurutani (eds.),
Collisionless Shocks in the Heliosphere: A Tutorial Review,
AGU, Washington, D.C., 1985, p. 91.
5. Bamert, K., Wimmer-Schweingruber, R.F., Kallenbach,
R., Hilchenbach, M., and Klecker, B., Charge-tomass fractionation during injection and acceleration
of suprathermal particles associated with the Bastille Day
event: SOHO/CELIAS/HSTOF data, these proceedings.