Issue |
A&A
Volume 494, Number 2, February I 2009
|
|
---|---|---|
Page(s) | 669 - 675 | |
Section | The Sun | |
DOI | https://doi.org/10.1051/0004-6361:200810099 | |
Published online | 30 October 2008 |
Generation of highly energetic electrons at reconnection outflow shocks during solar flares
G. Mann - A. Warmuth - H. Aurass
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
Received 30 April 2008 / Accepted 17 October 2008
Abstract
Context. During solar flares a large amount of energy is suddenly released and partly transferred into energetic electrons. They are of special interest since a substantial part of the energy released during a flare is deposited into the energetic electrons. RHESSI observations, e.g. of the 2003 October 28 solar event, show that 1036 electrons with energies >20 keV are typically produced per second during large flares. They are related to a power of about 1022 W. It is still an open question in which way so many electrons are accelerated up to high energies during a fraction of a second.
Aims. Within the framework of the magnetic reconnection scenario, jets appear in the outflow region and can establish standing fast-mode shocks if they penetrate the surrounding plasma with super-Alfvénic speed. It is our aim to show that this shock can be the source of the energetic electrons produced during flares.
Methods. The electrons are regarded as energized by shock drift acceleration. The process is necessarily treated in a fully relativistic manner. The resulting distribution function of accelerated electrons is a loss-cone one and it allows to calculate the differential electron flux, which can be compared with RHESSI.
Results. The theoretically obtained fluxes of energetic electrons agree with the observed ones as demonstrated for the 2003 October 28 solar event.
Key words: acceleration of particles - shock waves - Sun: flares - Sun: X-rays, gamma rays - Sun: radio radiation
1 Introduction
During solar flares a large amount of energy is suddenly released and
transferred into a local heating of the coronal plasma, mass motions
(e.g. jets), enhanced emission of
electromagnetic radiation (from the radio up to the -ray range),
and energetic particles (e.g. electrons, protons, and heavy ions).
The electrons play an important role, since they carry a substantial
part of the energy released during a flare (Lin & Hudson 1971, 1976;
Emslie et al. 2004).
RHESSI (Lin et al. 2002) observations show
that 1036 electrons with energies
of >20 keV are typically produced per second during large solar flares
(Warmuth et al. 2007).
They are related to a power of about 1022 W.
It is still an open question as to which way such a huge
number of electrons is accelerated
up to high energies within a fraction of a second during solar flares.
Figure 1 shows the widely accepted flare scenario based on magnetic reconnection in a schematic manner. A prominence is destabilized by its photospheric footpoint motions. The resulting upward motion stretches the underlying magnetic field lines, leading to the formation of a current sheet. If the current within this sheet exceeds a critical value, the resistivity is suddenly increased by plasma wave excitation due to various instabilities (see e.g. Treumann & Baumjohann 1997). Then, magnetic reconnection can take place in the region of enhanced resistivity (called diffusion region). Because of the strong curvature of the magnetic field lines in the vicinity of the diffusion region, the slowly inflowing plasma is shooting away from the reconnection site as oppositely directed hot jets (see Fig. 1). They are embedded between a pair of slow magnetosonic shocks, which heat the plasma. If the speed of this outflow jet is super-Alfvénic, a fast magnetosonic shock, also called termination shock (TS), can be established due to the deceleration of the jet. Such a deceleration can happen because of either an underlying post flare loop or a spread of the outflow region, e.g. by the increasing distance of the slow mode shocks (see Fig. 1). The appearance of TSs is predicted in the numerical simulations by Forbes (1986) and Shibata et al. (1995). Aurass et al. (2002) and Aurass & Mann (2004) report on the signatures of such shocks in the solar radio burst radiation.
![]() |
Figure 1: Sketch of the reconnection process in the solar corona as described in Sect. 1. The termination shock is established by the outflow jet, if it comes from the reconnection site and, subsequently, penetrates into the surrounding plasma of the corona. The grey shaded areas with the black arrows show the reconnection outflow jets. |
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One possible mechanism of electron acceleration within the flare scenario is the direct acceleration at the large scale DC electric field in the diffusion region (e.g. Holman 1985; Benz 1987; Litvinenko 2000). Since the diffusion region is spatially very small, this process can explain the acceleration of a single electron up to high energies, but it is not able to provide a high production rate of energetic electrons (typically 1036 s-1) as required by the observations. Therefore, Tsuneta & Naito (1998) proposed that the TS could be the source of energetic electrons, since the TS has a much greater space than the diffusion region. That is the reason, why the electron acceleration at the TS is studied in a quantitative manner in Sect. 2. In Sect. 3, the theoretically obtained results are compared with RHESSI observations of the solar event on October 28, 2003. In the paper by Warmuth et al. (2009) (henceforth called Paper II) the outcome of this model, i.e. the differential electron fluxes, are compared with 1053 electron spectra derived from the RHESSI photon spectra of 18 flares.
2 Relativistic shock drift acceleration
It is well-known that fast magnetsonic shocks are accompanied by a compression of both the density and the magnetic field (see e.g. Priest 1982). Hence, they represent moving magnetic mirrors at which charged particles can be reflected and accelerated. That is usually called shock drift acceleration (Holman & Pesses 1983; Leroy & Mangeney 1984; Wu 1984). During flares nonthermal hard X-ray fluxes above 100 keV are observed e.g. by RHESSI (Lin et al. 2002) indicating the production of electrons with energies >100 keV. Since it is assumed here that the energetic electrons are generated at the TS, the shock drift acceleration must necessarily be treated in a fully relativistic manner as done by Mann et al. (2006). Here, the relativistic shock drift acceleration is used to determine the resulting fluxes of energetic electrons. Note that these fluxes can directly be compared with RHESSI observations.
Following the treatment of shock drift accleration in the non-relativistic
approach (Ball & Melrose 2001; Mann & Klassen 2003), the vector of the
particle velocity is transformed into the de Hoffmann-Teller frame,
which is defined by removing the motional electric field.
Note that the shock is also at rest in this frame.
Then, the reflection process can be performed under the conservation of
energy and magnetic moment of the particle.
Note that all these transformations must necessarily be Lorentzian ones.
As result, one obtains a relationship between the velocity components
parallel (i.e.
)
and perpendicular
(i.e.
)
to the upstream magnetic field
before (i) and after (r) the reflection
![]() |
(1) |
and
![]() |
(2) |
(Mann et al. 2006), respectively.



Additionally, the reflection conditions
![]() |
(3) |
and
![]() |
(4) |
must be fulfilled by the initial particles in order to be accelerated. The loss-cone angle

![$\alpha_{\rm lc} = \arcsin[(B_{\rm up}/B_{\rm down})^{1/2}]$](/articles/aa/full_html/2009/05/aa10099-08/img19.gif)


The acceleration process actually represents a transformation
in the
-
plane as illustrated in Fig. 2
![]() |
Figure 2:
Illustration of the shock drift acceleration in the
|
Open with DEXTER |



As seen in Fig. 2, the resulting distribution function of
the accelerated electrons is a shifted loss-cone distribution
(see Leroy & Mangeney 1984; and Wu 1984, for the
non-relativisitc case). If f0 is the distribution function
of the initial state in the upstream region, then these particles
with the distribution function
![]() |
(5) |
(






![]() |
(6) |
(


![]() |
(7) |
with the electron mass



![]() |
(8) |
with


![]() |
= | ![]() |
|
![]() |
(9) |
Here, Eqs. (1) and (2) were inverted with respect to


In the magnetized plasma of the corona, the electrons predominantly
move along the magnetic field, since their gyroradius is much
smaller than the typical spatial scale length in the corona. Hence,
the magnetic field aligned flux of electrons is of special interest.
It is defined by
![]() |
(10) |
with N0 as the total electron number density. Then, the differential flux is obtained to be
![]() |
(11) |
Inserting the distribution function (9) into Eq. (10), introducing polar coordinates according to
![]() |
(12) |
![]() |
(13) |
integrating over the pitch angle

![$\beta = [\epsilon (2+\epsilon)]^{1/2}/(1+\epsilon)$](/articles/aa/full_html/2009/05/aa10099-08/img51.gif)

![]() |
![]() |
||
![]() |
|||
![]() |
|||
![]() |
(14) |
with
![]() |
= | ![]() |
|
![]() |
(15) |
and












![]() |
Figure 3:
The differential magnetic field aligned flux (Eq. (14))
normalized to N0 is obtained for electrons
accelerated by shock drift acceleration with a shock
speed of 1500 km s-1 for several angles |
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![]() |
(16) |
It is plotted for the same temperature, i.e.

In-situ spacecraft measurements (Lin et al. 1996) in the quiet solar wind
have revealed that electron distribution functions are usually not
Maxwellian but show an enhanced supra-thermal tail. Such
a distribution can be described in terms of a kappa distribution
defined by
![]() |
(17) |
(see e.g. Maksimovich et al. 1997; Pierrard et al. 1999). In the limit


![]() |
(18) |
The same procedure as previously described for the Maxwellian distribution is done for the kappa distribution for completeness. Inserting it Eq. (17) into Eq. (5), one gets
![]() |
= | ![]() |
|
![]() |
![]() |
||
![]() |
|||
![]() |
(19) |
with
![]() |
(20) |
and
![]() |
(21) |
for the differential flux of accelerated electrons in the case of an initial kappa distribution. The comparison of the theoretically obtained fluxes of accelerated electrons with the observed ones in dependence from the parameter

Figure 3 demonstrates that shock drift acceleration at the TS is able to
produce highly energetic electrons for circumstances typically found
in the flare region. The efficiency of shock drift acceleration
is crucially depending on the angle ,
i.e. the angle between
the shock normal to the upstream magnetic field. In the vicinity of
the reconnection region, the magnetic field lines are flowing away
from the diffusion region toward the TS as sketched in Fig. 4.
Thus, the angle
is varying across the TS
due to the curvature of the magnetic field lines.
![]() |
Figure 4:
Sketch of the magnetic field geometry at the TS.
The magnetic field lines reconnected in the diffusion
region (DR) are propagating away from it through the TS.
The magnetic field line takes a local angle |
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![]() |
(22) |
x gives the distance of the crossing point of the magnetic field line from the center of the TS. In reality, the angle










3 Discussion
In the previous section, it was demonstrated, that shock drift acceleration is able to accelerate electrons up to high energies for circumstances typically found in the flare region. The electrons are accelerated at the shock and reflected back towards the upstream region. They get a substantial gain in the velocity parallel to the magnetic field, whereas the pitch angle is decreasing. Due to the curvature of the magnetic field lines in the upstream region of the TS, the accelerated electrons encounter the TS once again. But now, a substantial part of them do not fulfill the reflection condition (4) because of their low pitch angle. Hence, they transmit into the downstream region and travel along the magnetic field line towards the denser chromosphere, where they emit hard X-ray radiation via bremsstrahlung (Brown 1971).In the present section, the theoretically obtained results from Sect. 2 are compared with observations. The solar event on October 28, 2003 is chosen for that, since it is one of the best observed flares.
In October/November 2003 several flares of X-class importance occured.
Especially, the event on October 28, 2003 showed a strong enhancement of the
hard X- and -ray radiation (see Fig. 5) as observed
by the SPI spectrometer (consisting of the ACS anti-coincidence system
and a Ge detector matrix) aboard
the INTEGRAL spacecraft (Gros et al. 2004; Kiener et al. 2006).
![]() |
Figure 5:
Hard X- and |
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Employing the Nançay radio-heliographic (Kerdraon & Delouis 1996) data
at 327 MHz, the source of the TS is located 250 Mm away
from the hard X-ray sources as seen in Fig. 6.
![]() |
Figure 6: The nonthermal (blue) and thermal (red) hard X-ray sources as observed by RHESSI and of the TS radio source are overplotted on a composite EUV image from SOHO/EIT and TRACE for the event on Ocotber 28, 2003. All outer contours represent the 50% level. The location of the TS is obtained from the Nançay radio heliographic data at 327 MHz. The black line represents the magnetic field lines connecting the hard X-ray footpoints. The white lines illustrate the magnetic field lines connecting the reconnection site, which is thought to be right outside the figure. Note that the magnetic field lines are drawn for illustration and not the result of any magnetic field extrapolation. |
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Figure 7 shows the hard X-ray photon spectrum of the event of discussion
as obserevd by RHESSI.
The spectrum has been corrected for photospheric albedo
(see Kontar et al. 2006) and pulse pile-up (see Smith et al. 2002) using the OSPEX software of
the standard RHESSI analysis tools. Pile-up was treated with the new
PILEUP_MOD function in OSPEX, which should give more accurate results than
the old correction method. Note that the spectrum (see Fig. 7)
has been accumulated from the front segment of detector 1 of RHESSI only,
since PILEUP_MOD works only for single
detectors.
The photon spectrum reveals a thermal and nonthermal component. The nonthermal component
is strongly pronounced above 28 keV. The nonthermal hard X-ray
radiation is assumed to be emitted by thick-target bremsstrahlung
(Brown 1971). Then, the photon spectrum can be converted into
a differential injected nonthermal electron flux, i.e. electrons
per second per keV, by means of a forward-fitting method
(Holman et al. 2003). Performing this procedure, a total flux
of
electrons with energies >28 keV
per second is obtained. It is related to a power of
W (Warmuth et al. 2007).
Both values are to be compared with the theory.
For comparing the theoretically obtained results with the
observations, the basic plasma parameters of the flare region
must be given. Since
the TS appears at 300 MHz for the harmonic emission at the
solar event on October 28, 2003, an electron number density
m-3 has to be used
for this quantity in the upstream region of the TS.
According to the twofold Newkirk (1962) density model
for the corona, such a density level is located 160 Mm above
the photosphere. There, a magnetic field of 4.6 G is expected
according to the model by Dulk & McLean (1978) leading
to an Alfvén speed of 557 km s-1.
In Paper II, the dependence of the theoretically obtained fluxes
of accelerated electrons from the plasma parameters in the
upstream region and the shock (TS) parameters are investigated in detail
and compared with electron spectra derived from X-ray spectra
measured by RHESSI. The study shows that the theoretical fluxes
agree with the observed ones only if high temperatures
of the upstream plasma of about >
K is assumed.
Of course, the coronal plasma inflowing towards the reconnection
site with typical temperatures of 1-2
K is strongly
heated at the slow shocks (see e.g. Cargill and Priest 1982),
so that the outflow jets are hot. Thus, a temperature of
K
seems to be appropriate to assume in the upstream region of the TS
(see the discussion in Paper II).
That results in a thermal electron speed of 24 600 km s-1.
A jump of the density across the TS of about 2
is deduced from the dynamic radio spectrum of the TS (see Fig. 5
bottom). Here, this jump is assumed to be associated with the bandwidth
of the radio signature of the TS in the dynamic radio spectrum
(Mann et al. 1995; Vrsnak et al. 2001).
Then, a jump of the magnetic field
and an Alfvén-Mach number
are found
by means of the Rankine-Hugoniot
relationships (Priest 1982). It corresponds to a real
shock speed
km s-1.
![]() |
Figure 7: The hard X-ray photon spectrum as recorded by RHESSI during the time period 11:06:40-11:07:40 UT on October 28, 2003. The dashed and dotted dashed line represent the thermal and nonthermal component of the X-ray radiation, respectively. |
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The electron number density ,
the temperature T,
and the magnetic field
in the upstream region
as well as the shock speed
,
the loss-cone
angle
,
the shock area
,
and the angle
are the input parameters
for calculating the differential flux according to Eq. (14).
Figure 3 reveals that the magnetic field aligned flux
of accelerated electrons is crucially depending on
the angle
.
As already discussed in Sect. 2,
the value of the angle
is varying across
the TS. Therefore, the resulted flux is finally averaged
over the angles
according to Eq. (22),
where
is varying between 85.3
and 89.6
.
The upper value is found as discussed in Sect. 2
i.e.
.
The lower value is chosen in the following way: the TS
is recognized by its radio radiation. That implies the
presence of supra-thermal electrons for exciting Langmuir
waves (see e.g. Melrose 1985), i.e. their velocity
should exceed the thermal speed
.
Such electrons are produced by shock drift acceleration
if the angle
is greater than
.
In result, the so-obtained flux is drawn in Fig. 8.
![]() |
Figure 8:
The magnetic field aligned differential flux of accelerated
electrons (full line) according to Eq. (14) with the parameters
derived from the solar event on October 28, 2003.
The values of these parameters are given in Sect. 3.
The dashed line represents the corresponding flux (see Eq. (16))
of a pure Maxwellian population for the same temperature of
40 |
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Note, that the dependence of the differential flux
of energetic electrons generated by shock drift acceleration
and the related production rate
and power
on the shock parameter is studied in detail in Paper II.
Since the total flux of the electrons coming towards the TS
is calculated to be
s-1,
4% of the inflowing electrons are accelerated up to energies
>28 keV by the proposed mechanism.
Assuming a barometric height scale
Mm,
which corresponds to a temperature of
K,
the volume
m-3above the TS would completely be evacuated after 35 s,
if the electron number density is
m-3.
But, in the framework of the discussed flare scenario (see Fig. 1)
plasma is continuously supplied by the lateral inflow through
the slow shocks. Since these slow shocks have a great spatial
extension, enough electrons are supplied into the outflow
region, in order to be finally accelerated at the TS.
In the upstream region of the TS,
the inflowing energy consists of the kinetic energy of
the jet, the thermal energy of the electrons and
protons, and the magnetic energy. Adopting the parameters given
above, the following
values
J m-3(mp, proton mass),
J m-3,
and
J m-3 are obtained
for the densities of the kinetic, thermal, and magnetic energy
of the upstream flow, respectively.
Note that the kinetic energy of the reconnection
outflow jet is the dominant one in comparison to the other ones.
Then, the total inflowing power is found to be
W
with
.
In result, 5% of the
total energy inflowing towards the TS is finally transfered into the
energetic electrons with energies >28 keV, if the electrons
are acclerated at the TS.
4 Summary
During solar flares a large amount of electromagnetic radiation
from the radio up to the hard X- and -ray range is emitted
from the corona. In particular, the huge solar event on October 28, 2003
was accompanied by a strongly enhanced emission of hard X- and
-ray
radiation (up to
10 MeV) during the impulsive phase.
These observational facts indicate the generation of highly energetic
electrons during this event. These electrons are not only responsible
for the nonthermal radio and X-ray radiation but also carry
a substantial part of the energy released during flars
(Lin & Hudson 1971; Emslie 2004). That is the reason,
why the acceleration of electrons up to high energies within
a fraction of seconds is one of the open questions in solar physics.
Magnetic reconnection is one possible process for explaining flares
in the solar corona. In the framework of this scenario (see Fig. 1),
a jet of hot plasma streams away from the reconnection site
into the surrounding plasma. If its speed is super-Alfvénic,
a shock wave (TS) can be established in the outflow region.
For the solar event on October 28, 2003 radio data revealed
signatures of such a shock only during the impulsive phase.
The simultaneous appearance of enhanced
photon fluxes up to 10 MeV and the radio signatures
of the TS (see Sect. 2) suggests that the TS is the source of
those electrons, which are needed to generate hard X- and
-ray radiation.
In the present paper, the highly energetic electrons are considered
to be generated by shock drift acceleration at the TS. That requires
a fully relativistic treatment as done by Mann et al. (2006)
and briefly summarized in Sect. 2.
The resulting distribution function of accelerated electrons has
the form of a shifted loss-cone distribution (see. Eq. (9)).
Taking such a distribution function the resulting magnetic field
aligned differential flux can analytically be given by Eq. (14).
Adopting the parameters of the flare plasma as observed
for the event on October 28, 2003, the theoretically obtained
flux (see Eq. (14)) provides a production rate
electrons with energies >28 keV
per second and is related to a power
W.
These values approximately agree with the observations (see Sect. 3).
As discussed above (see Sect. 3), a minor, but not negligible,
part (i.e. about 5%)
of the inflowing energy is tranfered into the energetic electrons
by this mechanism. That was originally presumed by Lin & Hudson
(1971, 1976) and later on confirmed by Emslie et al. (2004).
Note, that the scenario as illustrated in Fig. 1 was found in the observations of the solar event on October 28, 2003, as demonstrated in Fig. 6. In Paper II, the proposed mechanism was applied and compared with 1053 electron spectra derived from the X-ray spectra as measured by RHESSI during 18 individual flares.
In summary, the proposed model is able to explain the observation at least up to an energy of 100 keV. All these results support the presented mechanism, which was originally proposed by Tsuneta & Naito (1998) and quantitatively treated in the presented paper.
Acknowledgements
The authors thank G. Emslie, G. Holman, H. Hudson, R. P. Lin, and L. Vlahos for stimulating discussions. We gratefully acknowledge the use of the SOHO-MDI, RHESSI, and Nançay Multifrequency Radioheliograph data. Furthermore, we thank J. Kiener for the provision of INTEGRAL data. The work was financially supported by the German space agency Deutsches Zentrum für Luft- und Raumfahrt (DLR), under grant No. 50 QL 0001.
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All Figures
![]() |
Figure 1: Sketch of the reconnection process in the solar corona as described in Sect. 1. The termination shock is established by the outflow jet, if it comes from the reconnection site and, subsequently, penetrates into the surrounding plasma of the corona. The grey shaded areas with the black arrows show the reconnection outflow jets. |
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In the text |
![]() |
Figure 2:
Illustration of the shock drift acceleration in the
|
Open with DEXTER | |
In the text |
![]() |
Figure 3:
The differential magnetic field aligned flux (Eq. (14))
normalized to N0 is obtained for electrons
accelerated by shock drift acceleration with a shock
speed of 1500 km s-1 for several angles |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Sketch of the magnetic field geometry at the TS.
The magnetic field lines reconnected in the diffusion
region (DR) are propagating away from it through the TS.
The magnetic field line takes a local angle |
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Hard X- and |
Open with DEXTER | |
In the text |
![]() |
Figure 6: The nonthermal (blue) and thermal (red) hard X-ray sources as observed by RHESSI and of the TS radio source are overplotted on a composite EUV image from SOHO/EIT and TRACE for the event on Ocotber 28, 2003. All outer contours represent the 50% level. The location of the TS is obtained from the Nançay radio heliographic data at 327 MHz. The black line represents the magnetic field lines connecting the hard X-ray footpoints. The white lines illustrate the magnetic field lines connecting the reconnection site, which is thought to be right outside the figure. Note that the magnetic field lines are drawn for illustration and not the result of any magnetic field extrapolation. |
Open with DEXTER | |
In the text |
![]() |
Figure 7: The hard X-ray photon spectrum as recorded by RHESSI during the time period 11:06:40-11:07:40 UT on October 28, 2003. The dashed and dotted dashed line represent the thermal and nonthermal component of the X-ray radiation, respectively. |
Open with DEXTER | |
In the text |
![]() |
Figure 8:
The magnetic field aligned differential flux of accelerated
electrons (full line) according to Eq. (14) with the parameters
derived from the solar event on October 28, 2003.
The values of these parameters are given in Sect. 3.
The dashed line represents the corresponding flux (see Eq. (16))
of a pure Maxwellian population for the same temperature of
40 |
Open with DEXTER | |
In the text |
Copyright ESO 2009
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