Acceleration in a Current Sheet and Heavy Ion Abundances
in Impulsive Solar Flares
Yuri E. Litvinenko
Institute for the Study of Earth, Oceans, and Space, University of New Hampshire, Durham, NH 03824-3525,
USA
Abstract. The influence of collisional energy losses on stochastic particle acceleration in impulsive solar flares
is considered in the context of preferential acceleration of heavy ions. It is shown that ion pre-acceleration in a
reconnecting current sheet mitigates the effect of collisional energy losses, thus removing a strong sensitivity
of the resulting anomalous abundances on the initial ion charge states. As an example, the expected Fe/O
enhancement factors are computed and shown to be comparable with those observed with ACE SEPICA in a
series of impulsive flares in 1998. One consequence of the model is that the preferential acceleration of heavy
ions can occur only when the plasma gas pressure is large enough, |3 w mQ/mp, which may explain the observed
correlation between the heavy ion enrichment and selective 3He acceleration in impulsive flares.
INTRODUCTION
Recent observations of impulsive solar flares suggest
relatively low temperatures, T < 3 x 106 K, in the region
of the solar corona where charged particle acceleration
takes place (see [19] for a review). This fact renewed
the interest in two-stage particle acceleration models that
attempt to explain the heavy ion enrichments observed
in impulsive events [21]. Two-stage models postulate
that the selective ion enrichment occurs at the first stage
owing to an acceleration mechanism that depends on
the initial mass-to-charge ratio A/Q of relatively weakly
ionized heavy ions. At the second stage the ions are
ionized, so that they are eventually observed to have
almost identical A/Q ratios.
A specific model for the first-stage ion enrichment
process is based on the effect of collisional energy losses
experienced by the ions as they are being stochastically
accelerated by plasma turbulence in the solar corona.
The losses can play a significant role because of the
above-mentioned low temperature at the flare site. It is
theoretically established that the competition between
stochastic energy gains and Coulomb energy losses leads
to preferential acceleration of heavy ions in space plasma
[6, 5]. Physically for an ion to enter the acceleration
process, the energy gain rate has to exceed the loss rate
due to collisions, giving an advantage ~ Q2/A to heavy
ions.
Several variants of this preferential acceleration model
have been suggested in application to impulsive solar
flares. Particularly detailed calculations [18, 17] demonstrated that the model could indeed reproduce some typical features of heavy ion abundances in impulsive flares.
The fraction of each ionic component accelerated, however, turned out to be very sensitive to the assumed
plasma temperature that determines the ion charge state
distribution in ionization equilibrium. The predicted ion
enhancements were criticized for being much larger than
the observed ones [11] unless additional averaging over
temperature profile in a loop was introduced.
To agree better with observations, the mechanism for
heavy ion enrichment based on Coulomb losses should
depend only weakly on the ionic charge Q and mass A
(defined here in units of the proton mass and electric
charge). The simplest solution is that the ions should be
pre-accelerated as a jet whose speed is independent of
either Q or A [12]. Such high-speed jets are naturally
produced through reconnection of magnetic field lines
in a current sheet. Given that magnetic reconnection in
the corona is the premier candidate for the mechanism
of flare energy release, it appears useful to investigate
the effect of ion acceleration in a flare current sheet on
the formation of anomalous ion abundances in impulsive
solar flares. This is the purpose of this paper.
Section 2 reviews the influence of energy losses on the
composition of the accelerated particles. Application of
the model to a data set from the SEPICA sensor on the
ACE spacecraft confirms the unrealistically extreme sensitivity of the model to the plasma temperature. Section 3
shows that ion pre-acceleration in a reconnecting current
CP598, Solar and Galactic Composition, edited by R. F. Wimmer-Schweingruber
© 2001 American Institute of Physics 0-7354-0042-3/017$ 18.00
311
sheet is indeed important for the following ion enrichment processes in (post)flare loops, decreasing the sensitivity on temperature and improving the agreement with
the SEPICA observations. Another result is a new connection between the conditions for the observed heavy
ion enrichments and those for the selective 3He acceleration. Conclusions are presented in Section 4.
For example O and Ne initially can only have charge
states Q < <2max = 4 whereas Q < Qmax = 7 for Fe.
It is this selection effect that can be responsible for
the observed anomalous ion abundances. Recall that, in
the framework of a two-stage model, the ions are fully
stripped only after their initial acceleration in a lowtemperature (r < 3 x 106 K) plasma.
The predicted ion abundances are computed by summing over the charge states Q < Qmax, making use of the
standard tables of ionization equilibria [1]. In the simplest case the abundances are completely determined by
the temperature. Enhancement factors, which are computed using the data on normal element abundances [19],
can be directly compared with the ion enhancements observed in solar flares. By way of illustration, in this paper the model predictions are compared with the Fe/O
enhancements detected with the SEPICA sensor on ACE
[15, 16] in five well-observed impulsive flares in 1998
(days of year 136,149,249,251,252).
Comparison with the SEPICA data demonstrates that
the collisional loss effect can indeed be responsible for
large heavy ion enhancements in a million-degree plasma
of the solar corona (Fig. 1). The results of the comparison, however, are not quite satisfactory because of the
strong sensitivity of the predicted enrichments on the initial charge state distribution that is determined by the
temperature. A notable characteristic feature of heavy
ion enrichments in impulsive flares is that the relatively
low enhancement factors do not change much from flare
to flare. This is in contrast to the predictions of the simple model based on the action of collisions. One way to
remove the discrepancy is to average the predicted abundances over the flare loop volume [18]. Alternatively, an
additional mechanism may act to remove the strong sensitivity on the plasma parameters. One such mechanism
is proposed below.
ION SELECTION DUE TO
COLLISIONAL ENERGY LOSSES
The starting point of the analysis is the assumption that
ions in the flare loop are being accelerated stochastically
by turbulence generated in the course of the flare:
d<E/dt = ape,
(1)
where p and £ are the momentum and kinetic energy
of an ion. The value of the parameter a is defined by
a particular type of waves and is generally proportional
to the turbulent energy density W. For instance a ~
( v i/ c )^min(Wk/%) in the case of the Alfven turbulence,
where VA is the Alfven speed, UB is the magnetic field
energy density, and km[n is the minimum wave vector
magnitude of the wave spectrum [14].
Besides being accelerated, the ions also lose energy due to collisions with the ambient particles. The
Coulomb loss rate for a particle of speed v, moving in
a plasma with electron temperature TQ and density nQ is
as follows:
(2)
with the function F reaching the maximum Fmax ~ 0.5
for v w l.5(2kTQ/mo)1/2 [2]. For a particle to be accelerated, the energy gain has to exceed the loss. Hence the
condition d^/dt > Pmax determines the fraction of the
ions entering the acceleration process [6, 18].
In order to stress the essential physical points made in
this paper, a constant electron temperature TQ is assumed.
Given a reliable model for the coronal temperature and
density, the results for ion abundances can be integrated
over a coronal loop and properly weighted according to
the differential emission measure. Another simplifying
assumption is that an initially cool flare loop heats up
rapidly until the heating and loss processes are balanced
for the dominant plasma component (protons):
(d*E/dt - /W)protons » 0.
THE EFFECT OF ION ACCELERATION
IN A CURRENT SHEET
(3)
This condition leads to a simple expression [18] for
the ion charge states, which can enter the acceleration
process:
Q<A1'2.
(4)
312
Perhaps the simplest mechanism that can accelerate ions
to suprathermal energies irrespective of their mass is
through the production of fast jets, and the most likely
way for a jet formation in solar flares is through magnetic reconnection in a current sheet (see [12] for a discussion).
One-dimensional magnetic field in such a sheet would
have a zero plane and would not influence the particles
that simply move along the electric field direction. On
the contrary, studies of particle motion in realistic current
sheets with two- and three-dimensional magnetic field
show that the field controls the character of particle orbits
and the escape speed. The escape speed is determined by
1000.01
0)
Cn
100.0 O
cd
10.0 -
O
C
cd
43
ti
5.0
5.5
6.0
6.5
Temperature, log T
7.0
FIGURE 1. Predicted enhancement factors for the Fe/O ratio, compared with the data for five impulsive flares observed with
SEPICA ACE in 1998 (Mobius et al., 1999): 5 = 1.0
both the electric field E in the sheet and the magnetic
field component B± perpendicular to E:
inverse gyro-frequency
(7)
(5)
This result is easily obtained in the reference frame
where the electric field vanishes [22, 9], assuming that
the speed remains nonrelativistic and the influence of
the nonreconnecting magnetic field component directed
along the electric field on the particle motion can be ignored for ions (for a review see [7], where the particle orbits are discussed for various parameter regimes). Thus
the escape speed is independent of either A or Q. Selfconsistent treatment [8] indicates that the speed is determined by the Alfven speed based on the reconnecting
component of the magnetic field:
Vesc = VA =
(6)
which is the same result as in the MHD analysis of
reconnection.
An important aspect of the jet generation through magnetic reconnection is that the process is essentially collisonless [22, 9]. The acceleration time scales with the
313
and is much shorter than the typical collisional energy
loss time:
(8)
for ions with resulting energies of 0.1 — 1 MeV per
nucleon.
Magnetic reconnection is almost universally accepted
as the mechanism of flare energy release. Hence it is
useful to investigate quantitatively how the formation
of anomalous ion abundances by the collisional energy
loss effect is influenced by ion pre-acceleration in a flare
current sheet.
It should be noted that the idea of particle preacceleration through the reconnection jet is different
from the previously suggested model [18] in which the
Coulomb losses modified the abundances in the reconnection inflow region before the ions entered the current
sheet, whereas particle acceleration by the reconnection
electric field in the sheet was treated as the second-stage
process. On the contrary, this paper explores the possibility that pre-acceleration in the sheet is followed by
stochastic turbulent acceleration in the flare loop (cf.,
[12, 13]) to the observed energies.
As before, stochastic ion acceleration is described by
Equation (1), and the collisional loss rate is described
by Equation (2). A significant difference, however, arises
because of the rapid collisionless pre-acceleration of ions
in the current sheet. Since the pre-acceleration timescale is so short, the Alfven speed of the reconnection
jet can be taken as an initial condition for the turbulent
acceleration:
v(t = 0) = vA >0.
The same inequality defines a parameter regime in which
the electromagnetic ion-cyclotron instability has the lowest threshold of any current-driven instability [4]. This
may be more than a coincidence.
Recall that heavy-ion enhancements are well correlated with the spectacular 3He enrichments in impulsive
flares, although the relationship between the two phenomena is rather intricate. It appears that 3 He-rich flares
are also heavy-ion rich but not vice versa, suggesting that
the 3He enrichment process operates at coronal sites that
are enriched in heavy ions due to other reasons, but the
process itself does not preferentially accelerate the heavy
ions [10]. Whereas the heavy ions are likely to be accelerated by turbulent Alfven waves, the selective 3He
acceleration is currently believed to be produced by resonant interaction with the electromagnetic ion-cyclotron
waves generated by electron beams in the corona [23].
The electron beams are produced in the reconnecting current sheet [7].
Equation (13) may provide a new theoretical interpretation for the observed correlation between the heavy-ion
and 3He enrichments, as proposed below. It is in small
impulsive flares that chromospheric evaporation leads
to relatively large values of P. This creates favorable
conditions for both the heavy ion enrichments through
the collisional loss effect and the generation of the ioncyclotron waves by electron beams that are generated
at the site of magnetic reconnection. The waves in turn
interact with the 3He ions, resulting in the 3He enrichments.
Suppose that initially P is small in a loop. As the flare
progresses, MHD Alfven waves are generated in the loop
and start accelerating the heavy ions. If the processes
of plasma heating and evaporation are strong enough,
they eventually result in a larger P thus leading to larger
ion enhancement factors and triggering the generation
of ion-cyclotron waves responsible for the 3He enrichment. A more detailed analysis is required to understand
whether the model can explain the observed heavy-ion
enhancements in the absence of the 3He enrichment in
some flares.
(9)
If the electric field acceleration is strong enough, this
initial speed can greatly exceed 1.5(2kT Q /m Q ) */2, and the
ions will not experience the maximum collisional loss as
they are being accelerated by the waves. For the purposes
of an analytic treatment, it is useful to adopt
711/2
(10)
2x
as a simple limiting case [2].
Assuming again that the heating and loss processes
are balanced for the dominant protons in the loop, which
were not pre-accelerated, Equations (1), (2), and (10) are
combined to give a modified condition for preferential
ion acceleration:
Q<§Al/2
(11)
if 8 > 1 (cf., Equation (4)). Here, ignoring factors of
order unity,
VA
mn
-1/2
(12)
(We/Hie) 1 / 2
and the plasma beta P = ^>TiknQTQ /B2. For example 8 w
1.3 for the typical coronal parameters B w 100 G, T w
10 6 K,andfl«10 9 cnr 3 .
Because the Coulomb loss rate falls off rapidly with
increasing speed, pre-acceleration mitigates the selection
effect of the losses, thus leading to a more modest enhancements in the heavy ion abundances. Comparison
with the ACE SEPICA data, using the same methods and
data set as in the previous section, indeed demonstrates a
much better agreement of the model and the data (Fig. 2).
Further tests of the model should include its application
to the first comprehensive observation of the abundances
of trans-iron elements in solar energetic particle events,
in which the heavy-ion enhancements were measured to
be as high as w 1000 relative to O over their coronal values [20].
Collisional losses can be ignored altogether if 8 > 1.
It follows that the selection effect disappears unless
P > mQ/mp.
DISCUSSION
Collisionless magnetic reconnection in coronal current
sheets is almost unanimously accepted as the flare energy
release mechanism. Highly super-Dreicer electric fields
of the order of a few V/cm are associated with rapid
reconnection for typical parameters in the corona. Hence
efficient particle acceleration in the current sheet is a
signature of rapid magnetic reconnection in flares. This
makes it necessary to introduce ion pre-acceleration into
models for anomalous ion abundances in impulsive solar
(13)
314
1000.0!
0)
fa
100.0 :
o
+J
o
10.0 =
QJ
a<u
o
5.0
5.5
6.0
6.5
7.0
Temperature, log T
FIGURE 2. Predicted enhancement factors for the Fe/O ratio, compared with the data for five impulsive flares observed with
SEPICA ACE in 1998 (Mobius et al., 1999): 5 = 1.3
flares. One such model, investigated in this paper, is
based on an interplay of the processes of ion energy
gain due to acceleration and energy loss due to collisions
[6,5,18,12].
It was demonstrated that pre-acceleration in a reconnecting current sheet mitigates the influence of the losses
on stochastic particle acceleration, thus removing a very
strong sensitivity of the resulting abundances on the initial distribution of ion charge states and leading to a better agreement with observations. This result is a simple
consequence of the fact that the energy loss rate falls off
rapidly with increasing energy. To explain why the eventually observed ions are almost fully stripped, the model
can be developed further by relaxing the assumption of
ionization equilibrium.
It was suggested some time ago [12] that initial energization through plasma jets is necessary to explain why
the abundances of solar energetic particles are not too
different from normal solar abundances. Application of
this idea to the selective acceleration of heavy ions in impulsive flares leads to an interesting quantitative requirement, Equation (13), for the selection to occur. It is very
interesting that the same requirement defines a parameter
315
regime where the electromagnetic ion-cyclotron instability has the lowest threshold of any current-driven instability [4]. Resonant acceleration by the ion-cyclotron
waves is the most likely mechanism for 3He preferential
acceleration in impulsive flares [23]. Thus the approach
of this paper may provide a new theoretical connection between the processes of heavy-ion and 3He enrichments, which is well established observationally [10].
Finally it should be noted that another model for the
heavy ion enrichments invokes cyclotron damping of
cascading turbulence. The damping is more effective for
heavy ions for which the resonance condition is easier
to satisfy than for protons [3]. Recently this approach
has been shown to give good agreement with typical impulsive flare abundances [13]. It appears interesting to
specify under what physical conditions either of the suggested mechanisms could be responsible for the observed
ion abundances in flares.
ACKNOWLEDGMENTS
I am grateful to Dr. Grand Manlier for stimulating discussions and the anonymous referee for useful suggestions. This work was supported by NSF grant ATM9813933 and NASA grant NAG5-7792.
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