A&A 391, 749-756 (2002)
DOI: 10.1051/0004-6361:20020866
G. Mann1 - H. T. Classen1 - E. Keppler 2 - E. C. Roelof3
1 - Astrophysikalisches Institut Potsdam,
An der Sternwarte 16, 14482 Potsdam, Germany
2 -
Max-Planck-Institut für Aeronomie,
Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany
3 -
Applied Physics Laboratory, John Hopkins University,
Laurel, MD 20723, USA
Received 15 March 2002 / Accepted 15 May 2002
Abstract
The interaction of fast and slow speed
solar wind streams leads to the formation of
so-called corotating interaction regions (CIRs) in the heliosphere. These
CIRs are often associated with shock waves, at which
electrons are accelerated as observed by the Ulysses
spacecraft.
A correlation between the ratio of energetic electron fluxes at the
crossing of CIR related shocks to those in the far upstream region
of these shocks and the magnetic field compression of
the associated shocks has been revealed by analysing the data of
the HISCALE instrument aboard Ulysses.
This result can be explained by a model of electron acceleration
at shock waves, where the electrons gain energy due to multiple
reflections at large amplitude magnetic field fluctuations occurring
in the vicinity of the shock transition.
Key words: acceleration of particles - sun: solar wind - shock waves
The Sun is a source of a permanent stream of charged particles (e.g. electrons, protons and heavy ions) penetrating into the interplanetary space. It is the so-called solar wind forming the heliosphere due to its interaction with the interstellar wind (Parker 1958). The solar wind was originally discovered by in-situ measurements of the Mariner 2 spacecraft in 1962 (Neugebauer 1966). It is temporally and spatially structured (see Schwenn 1990 as a review). There are high and slow speed solar wind streams. The coronal holes with open magnetic field structures are the sources of the high speed streams, whereas the slow solar wind is coming from regions with closed magnetic field structures located around the equatorial plane (Schwenn 1990). Due to the rotation of the Sun the fast and slow solar wind streams interact with each other, leading to the formation of a so-called corotating interaction region (CIR) (Pizzo 1978). An interface is located within the CIR. It is a contact discontinuity dividing the fast solar wind plasma from that of the slow solar wind stream. In many cases a pair of forward and reverse shocks forms the boundaries of CIRs. The forward shocks are propagating into the slow solar wind towards the equatorial plane whereas the reverse shocks are travelling pole-ward into the fast solar wind stream (Gosling & Pizzo 1999). The formation of CIRs mainly takes place at distances beyond 1 AU. Since the Ulysses spacecraft was exploring the three-dimensional heliosphere between 1 and 5 AU during the declining period of solar activity (Marsden et al. 1996), the Ulysses mission is highly appropriate to study all phenomena related to CIRs.
The shock waves associated with CIRs are able to generate
energetic electrons, protons and heavy ions, as well-known from
the observations of the Pioneer and Voyager spacecraft (McDonald et al. 1976;
Barnes & Simpson 1976).
The basic features of energetic particles associated with CIRs have been
summarized by Mason & Sanderson (1999). As an example, Fig. 1
offers the
energetic particle and plasma data recorded by the Ulysses spacecraft
during its
passage through the CIR No. 5 occurring during the days 282-286 in 1992.
(The numbering of CIRs observed by Ulysses
was originally introduced by Bame et al. 1993.)
The aim of the present paper is to investigate the efficiency of electron
acceleration at CIR related shocks (see Mann 1999 for a
preliminary study). Note that the generation of energetic
electrons has also been observed at shocks in the solar corona
(Wild & McCready 1950; Cane et al. 1981; Cairns & Robinson 1987),
at travelling interplanetary shocks (Tsurutani & Lin 1985; Lopate 1989), and
at the Earth's bow shock (Anderson et al. 1969; Scarf et al. 1971).
The electron fluxes used in the present paper has been provided by
the HISCALE instrument (Lanzerotti et al. 1992) in the range 30-50 keV
aboard the Ulysses spacecraft. The electron fluxes
at the shock crossing are compared with those j0
measured during quiet solar wind periods before and after the
CIR (see Sect. 2). Furthermore, the ratios
are related to the magnetic field compression
B2/B1
of the associated shock.
Mann & Classen (1995)
proposed a mechanism of generation of energetic electrons
at collisionless shocks. It is well-known that super-critical
shocks are accompanied with large amplitude magnetic field fluctuations
in the vicinity of the shock transition (Kennel et al. 1985). Electrons can be
reflected and subsequently accelerated at these fluctuations. Due to
multiple encounters of electrons with these fluctuations they receive
a considerable acceleration (Mann & Classen 1995). This
special acceleration mechanism will be introduced in a quantitative manner
in Sect. 3, and subsequently adopted to explain the relations
between the flux ratios
and jumps of the magnetic
field
B2/B1 of the associated shocks in Sect. 4.
The HISCALE instrument (Lanzerotti et al. 1992) aboard Ulysses
is able to measure the fluxes of
energetic electrons in four different channels, i.e. DE1: 30-50 keV,
DE2: 50-90 keV, DE3: 90-165 keV, and DE4:
165-300 keV.
The data recorded in the channel DE1 are employed to compare
the electron fluxes
at the shock crossing with those j0,
which are determined during quiet conditions of the solar wind stream
related with the corresponding
shock. It should be recalled, that the forward and reverse shocks are
travelling into the slow and fast speed solar wind (Pizzo 1978), respectively.
The data analysis can be demonstrated for example in Fig. 1. The energetic electron fluxes j0 of quiet solar wind conditions are chosen to be at day 281 and 293 for the forward and reverse shock in this particular case, respectively. These days have been chosen because the particle, plasma and magnetic field data show no strong and rapid changes, i.e. there were really quiet solar wind conditions.
The results of the whole data analysis are summarized in Table 1.
CIRs Nos. 1-18 (Bame et al. 1993) have been employed for this study.
No. |
![]() ![]() |
j0[1/cm
![]() |
![]() |
B2/B1 | ![]() |
T2/T1 |
1R |
![]() |
![]() |
0.208 | 1.73 | 1.9 | 1.4 |
3R1 |
![]() |
![]() |
1.354 | 1.62 | 1.8 | 1.7 |
4R |
![]() |
![]() |
2.162 | 3.73 | 5.3 | 3.8 |
5F |
![]() |
![]() |
0.289 | 1.50 | 1.4 | 1.7 |
5R |
![]() |
![]() |
2.088 | 2.46 | 3.0 | 3.0 |
6F |
![]() |
![]() |
2.327 | 1.88 | 2.4 | 4.0 |
7F |
![]() |
![]() |
0.479 | 1.83 | 3.0 | 5.0 |
7R |
![]() |
![]() |
2.338 | 3.00 | - | 3.3 |
8F |
![]() |
![]() |
0.589 | 1.56 | 1.7 | 2.5 |
8R |
![]() |
![]() |
0.886 | 2.45 | 2.7 | 4.4 |
9F |
![]() |
![]() |
0.447 | 1.57 | 2.5 | 2.0 |
10R |
![]() |
![]() |
0.876 | 2.73 | 3.0 | 4.4 |
11R |
![]() |
![]() |
0.665 | 1.63 | 2.4 | 1.2 |
13R |
![]() |
![]() |
1.163 | 2.07 | 3.1 | 1.5 |
15R |
![]() |
![]() |
1.379 | 2.29 | 2.8 | 1.7 |
upstream magnetic field | B1 = 0.9 nT |
upstream density | N1 = 0.25 cm-3 |
upstream temperature |
![]() |
upstream plasma beta |
![]() |
![]() |
Figure 2:
Correlation between the ratios
![]() |
As already mentioned the CIR related shocks are usually quasi-perpendicular,
i.e.
.
For a plasma-beta
such shocks
are super-critical, if their Alfvén-Mach number is greater than 2.0
(Kennel et al. 1985).
That is mostly the case of the shocks considered in this paper (see Table 1).
Figure 3 shows the behaviour
of the magnitude of the magnetic field during the crossing of the CIR related
shock No. 7F as measured by the magnetometer (Balogh et al. 1992)
aboard the Ulysses spacecraft.
![]() |
Figure 3: Behaviour of the magnitude of the magnetic field during the crossing of the CIR related shock No. 7F as measured by the magnetometer aboard the ULYSSES spacecraft. |
Now, the movement of an electron between two neighbouring mirrors
is considered in detail, following the way described
by Chen (1984) and Mann & Classen (1995).
The computations are lengthy, but straightforward. Thus, only the main line
of thought will be presented. The mirrors M1 and M2 (see Fig. 4)
are accompanied with magnetic field compressions BM1 and BM2
with
BM1 > BM2 and have the velocities V1and V2 with
V1 > V2, respectively.
![]() |
(1) |
![]() |
(2) |
![]() |
(3) |
![]() |
(4) |
Concerning this process it is generally assumed that the electrons are
adiabatically reflected at the magnetic field compressions
acting as magnetic mirrors. This assumption is justified
if the gyroradius
of the electron is essentially smaller than the
characteristic length scale of the magnetic field compressions.
This length scale is typically 10 ion inertial
lengths (Mann et al. 1994).
Since electrons in the energy range 30-50 keV have a typical gyroradius of
1.3 ion inertial lengths under plasma conditions near CIRs (see Sect. 2),
the required condition is well fulfilled in the case considered here.
Since the mirrors in terms of two neighbouring large amplitude magnetic
field compressions cannot penetrate each other, the distance between
them can be reduced only up to a minimum one .
Consequently, the acceleration process is finished either if
or if the electron leaves the region between the mirrors,
since the pitch angle
becomes
.
The movement
of an electron between these two mirrors leads finally to a nonuniform
acceleration as illustrated in Fig. 5. In this special example
the particle has an initial velocity
(
,
Alfvén speed;
,
thermal
electron velocity) parallel to
the ambient magnetic field and an initial pitch angle
.
The distance between the two mirrors is diminished from
L0 = 50diup to
.
The mirrors M1 and M2 are moving with
the velocities
and
,
respectively.
The magnetic field compression at the mirror M2 is assumed to be
BM2/B0 = 2.8 resulting in a final pitch angle
.
Adopting
as typical values
for CIR related shocks,
and
is found.
The result of the numerically evaluated Eqs. (1)-(4) is presented in
Fig. 5.
Since the electrons move much faster than the mirrors, the acceleration process might be considered as a continuous one (see Fig. 5), i.e. the relationship
![]() |
(5) |
is well fulfilled for the cases under consideration.
Then, the acceleration defined by
with
and
may be taken as a differential equation
![]() |
(6) |
with the initial condition
.
Here,
with
has been used in deriving Eq. (6).
Note, that the mirrors are initially separated at a distance L0.
The solution of Eq. (6) is found to be
![]() |
(7) |
with
(Chen 1984; Mann & Classen 1995).
In order to derive a differential equation for the evolution
of the pitch angle Eqs. (1)-(4) will be employed
taking into account the relationship (5).
Then,
can be
calculated to be
![]() |
(8) |
![]() |
(9) |
with the initial condition
.
The solution is given by
![]() |
(10) |
relating the initial pitch angle
and final one
.
Thus, the solutions
of Eqs. (6) and (9) describe the evolution of the particle in
the velocity space
during the acceleration between
t0 = 0 and
(see Fig. 6).
Here,
and
denote the particle velocity parallel
and perpendicular to the ambient magnetic field, respectively.
Now, an ensemble of particles is regarded in the velocity space.
All particles initially located on the path determined by
in the velocity space are accelerated and receive
the final velocity
and final pitch angle
,
i.e.,
all particles with pitch angles
receive an acceleration.
Then, the total number of accelerated particles can be calculated by
![]() |
(11) |
![]() |
(12) |
![]() |
(13) |
![]() |
(14) |
![]() |
(15) |
Now, the mechanism of electron acceleration as introduced in the previous
Section is used to explain the correlation between the ratios
and the jump of the magnetic field
B2/B1as deduced for CIR related shocks from the HISCALE data (see Fig. 2).
The differential flux j(E) is related to the distribution function
in the momentum space by
(Landau & Lifschitz 1975).
Here, the distribution function
is normalized to unity.
If
denotes the distribution function in the energy space,
the conservation of the particle number density in the phase space, i.e.,
d
d
dE, leads to
![]() |
(16) |
Now the flux
of accelerated electrons at the shock crossing is compared
with that j0 in the undisturbed upstream region.
In order to do this a so-called kappa distribution is assumed to
exist for velocity distribution function
in the undisturbed solar wind, i.e. upstream of
CIR related shock waves. A kappa distribution is defined by
![]() |
(17) |
![]() |
(18) |
![]() |
(19) |
![]() |
(20) |
The flux of the accelerated electrons at the shock crossing is determined by
![]() |
(21) |
Now the results theoretically obtained from the presented electron
acceleration mechanism are compared with the observations summarized
in Fig. 2. In order to do this the plasma parameters usually found
upstream of CIR related shocks (see Sect. 2) will be employed.
The jumps of the particle number
density
N2/N1, of the magnetic field
B2/B1, and
of the temperature
T2/T1 are related to the Alfven-Mach number
by the well-known Rankine-Hugoniot relations (Kennel et al. 1985)
as depicted in Fig. 7.
![]() |
Figure 7:
Dependence of the jump of the temperature
T2/T1,
the particle number density
N2/N1, and the magnetic field
B2/B1 across the shock on the Alfven-Mach number ![]() ![]() ![]() |
![]() |
(22) |
The study of the enhancements
of energetic electron fluxes
in the range 30-40 keV during the crossing of CIR related shocks reveals
a relationship between the ratio
and the jump
B2/B1of the magnetic field of the associated shock as shown in Fig. 2.
This relationship
has been explained by an electron acceleration mechanism, which acts
as an interaction of electrons with the large amplitude magnetic field
fluctuations in the vicinity of the shock transition. The enhancements of
the energetic electron fluxes are caused by the heating of the electrons
as a result of the shock crossing and
the subsequent acceleration due to their interaction with the magnetic field
fluctuations in the downstream region. Since the large amplitude magnetic field
fluctuations, which are necessary for the electron acceleration, appear mainly
in the vicinity of the shock transition for quasi-perpendicular
shocks, the proposed acceleration mechanism acts very locally and fast
for electrons at spatial and temporal scales of few ion inertial lengths
and inverse proton cyclotron frequencies, respectively.
In contrast to the shock drift acceleration (Holman & Pesses 1983;
Krauss-Varban & Wu 1989; Kraus-Varban et al. 1989),
where the energy gain is limited
because of a single shock encounter and only efficient at nearly
perpendicular shocks, the mechanism presented is much more efficient
since the electrons accumulate energy due to the multiple reflections
at the large amplitude magnetic field fluctuations. Furthermore, this
mechanism is a deterministic one, unlike diffuse shock acceleration
(Axford et al. 1977), which is a stochastical process acting in the whole
up- and downstream region of the associated shock.
Recently, Classen et al. (1999) reported on
a high correlation between the fluxes of 1 MeV protons and the
low frequency magnetic field turbulence in the downstream region of CIR
related shocks,
especially in the vicinity immediately after the shock transition.
This result implies that wave-particle interactions in the
downstream region even after the shock transition
play an important role for acceleration of particles at CIR-related shock
waves. This observational result confirms additionally the acceleration
mechanism proposed in this paper.