A&A 391, 749-756 (2002)
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
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
energetic particle and plasma data recorded by the Ulysses spacecraft
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.)
|Figure 1: HISCALE data of energetic protons (top panel) and electrons (2nd panel from top) as well as the behaviour of the proton number density N (4th panel), the proton temperature T (5th panel), the solar wind velocity (6th panel) and the magnetic field in polar coordinates (three panels at the bottom) during the passage of the CIR No. 5 by the ULYSSES spacecraft.|
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.
|upstream magnetic field||B1 = 0.9 nT|
|upstream density||N1 = 0.25 cm-3|
|upstream plasma beta||.|
|Figure 2: Correlation between the ratios and the jump of the magnetic field B2/B1 for 15 CIR related shocks (Table 1). The full line represents the theoretical result obtained by means of Eq. (22). The dashed-dotted line shows the line of linear regression.|
As already mentioned the CIR related shocks are usually quasi-perpendicular,
For a plasma-beta
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
BM1 > BM2 and have the velocities V1and V2 with
V1 > V2, respectively.
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
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
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 magnetic field compression at the mirror M2 is assumed to be
BM2/B0 = 2.8 resulting in a final pitch angle
as typical values
for CIR related shocks,
The result of the numerically evaluated Eqs. (1)-(4) is presented in
|Figure 5: Velocity-time diagram of a special example of the mirror acceleration. The chosen parameters are introduced in Sect. 3. The velocity and the time are normalized to the Alfvén speed and the inverse proton cyclotron frequency .|
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
is well fulfilled for the cases under consideration. Then, the acceleration defined by with and may be taken as a differential equation
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
(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).
calculated to be
with the initial condition . The solution is given by
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).
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
all particles with pitch angles
receive an acceleration.
Then, the total number of accelerated particles can be calculated by
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.
denotes the distribution function in the energy space,
the conservation of the particle number density in the phase space, i.e.,
dE, leads to
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
The flux of the accelerated electrons at the shock crossing is determined by
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
N2/N1, of the magnetic field
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 according to the Rankine-Hugoniot relationships (Kennel et al. 1985). An angle between the upstream magnetic field and the shock normal, a plasma beta and an adiabatic index of 5/3 has been used.|
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.