Development of shocks waves in the solar corona and the
interplanetary space
G. Mann , A. Klassen , H. Aurass and H. T. Classen
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany
Abstract. At the Sun shock waves are produced either by flares or by coronal mass ejections and are regarded to be the
source of solar energetic particle events. The propagation of a disturbance away from an active region through the corona into
the interplanetary space is considered by evaluating the radial behaviour of the Alfvén speed. The magnetic field of an active
region is modelled by a magnetic dipole superimposed on that of the quiet Sun. Such a magnetic field structure leads to a
local mimimum of the Alfvén speed in the middle of the corona and a maximum of 740 km/s at a radial distance of 6 solar
radii from the center of the Sun. The occurrence of these extrema of the Alfvén speed has consequences for the formation and
development of shock waves in the corona and interplanetary space.
INTRODUCTION
In the solar corona shock waves can occur as blast waves
due to the flare process [1, 2] or shocks driven by coronal
mass ejections (CMEs) [3]. They can continue as travelling shocks in the interplanetary space. Coronal and interplanetary shocks can be the source of type II radio
radiation [4]. Cane [5] and Gopalswamy et al. [6] suggested the appearance of two different kinds of shocks,
the flare produced blast wave associated with the solar
type II radio bursts in the inner corona and the CME
driven interplanetary shocks, which are regarded to be
the source of solar energetic particle events (SEPs) [7,
8].
Yohkoh and SOHO images of the Sun and complementary spatial radio-heliographic observations revealed that
sources of solar type II radio bursts are mainly propagating non-radially away from active regions [9, 10, 11,
12]. In order to study the formation and development of
shock waves in the solar corona and the interplanetary
space (see Section 3) it is necassary to evaluate the radial behaviour of the Alfvén speed (see Section 2) as the
characteristic speed in a magnetoplasma.
THE ALFVÉN SPEED MODEL
The Alfvén speed is depending on the magnitude B of
the magnetic field and the full particle number density
N. Therefore, a model of the magnetic field of an active
region superimposed on that of the quiet Sun and a
density model is necessary to know, in order to study the
radial behaviour of the Aflvén speed in the solar corona
and the interplanetary space.
Since coronal shock waves are established near but
out of active regions [9, 10, 11, 12], a one-fold Newkirk
model [13]
(1)
Ne N0 104 32RSR
(N0 4 3 104cm 3 ; RS , solar radius; R, radial distance
from the Sun) is assumed as an appropriate model of the
electron number density N e . This model is well fitting
the conditions above quiet equatorial regions as whitelight scattering observations [14] showed. Mann et al.
[15] presented a heliospheric density model as a special
solution of Parker’s [16] wind equation. It agrees well
with the observations from the corona up to the interplanetary space at 5 AU. Since a density model above quiet
equatorial regions is required for this study, the one-fold
Newkirk [13] model is adopted in the region R 1 8R S ,
whereas that by Mann et al. [15] has to be taken at regions R 1 8R S . That results in a radial behaviour of the
electron number density as presented in Figure 1.
In the solar corona the magnetic field B is composed
of that of an active region Bar and of the quiet Sun BqS ,
i. e. B Bar BqS . Here, an active region is modelled
and the length
by a magnetic dipole with the moment M
λ (see Figure 2). It is located in a depth λ 2 under the
photosphere. Therefore, λ 0 1R S is taken because of
the assumed width of the hydrogen convection zone of
the Sun [17]. The field of a magnetic dipole [18] is given
by
3 M
rr M
Bar
(2)
5
r
r3
where r denotes the distance of the point P from the
center of the magnetic dipole (see Figure 2). The axis
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
612
BS 2 2 G as the magnitude of the quiet Sun at the
photospheric level has been deduced from the analysis of
coronal transient (or EIT) waves [19]. Then, the magnetic
field of the quiet Sun is given by
10
10
9
10
8
10
Bqsz R
7
10
BqSρ R
-3
N (cm )
6
10
5
10
BqS cos β
(5)
BqS sin β
(6)
4
10
3
10
2
10
1
10
0
10
1
10
100
R / RS
FIGURE 1. Radial behaviour of electron number density
model from the low corona up to 1 AU
Rs
0.1Rs
Z
1111111111
0000000000
0000000000
1111111111
1111111111
0000000000
υ δ
r
ρ
R α
β
in the same frame of reference (see Figure 2).
The magnetic fields of the dipole (active region) and
the quiet Sun can be superimposed by two different ways.
Along the z-axis both fields can be directed either parallely (”+” sign) or anti-parallely (”-” sign). Henceforth,
these cases are called ”parallel” or ”anti-parallel”. The
composed magnetic field is obtained after vector addition. Its magnitude is finally entering into the calculation
of the Alfvén speed.
Since solar type II radio burst sources are mainly
travelling non-radially away from an active region [9,
10, 11, 12], the behaviour of the Alfvén speed along
a straight line away from the center of the magnetic
dipole should be studied. For a quantitative discussion
this line is chosen to be inclined by an angle δ 45 o
with respect to the z-axis (see Figure 2). Then, each point
P on this line is unambiguously determined either by ρ
or R, which are related by
ρ
RS
P
0 707
R2
R2S
0 50 5
(7)
to each other. Then, the angle ϑ can be calculated by
FIGURE 2.
ϑ
Scheme of reference frame
of the dipole is directed along the z-axis, i. e. M
M ez
with ez as the unit vector along the z-axis, as indroduced
in Figure 2. Then, the magnetic field (see Eq. (2)) is
expressed by
Barz r
Barρ r
B0
λ3
16
RS
r
B0
λ3
16
RS
r
3
3
3 cos ϑ 2
1
(3)
3 cos ϑ sin ϑ (4)
in cylindrical coordinates (see Figure 2). B 0 is defined as
the magnitude of the magnetic field at the z-axis on the
photospheric level, i. e. at r λ 2 and ϑ 0 o . Then, M
can be calculated by M B 0 λ 3 16. Here, B0 0 8 kG is
taken as a typical value of the magnetic field in the center
of an active region [17].
The magnetic field of the quiet Sun is assumed to
be radially directed according to BqS BS RS R2 eR
with eR as the unit vector along the radial direction.
613
arctan
1
1 λ2
RS
R
(8)
leading to the determination of the radial distance r
r
RS
ρ
1
RS sin ϑ
(9)
of the point P from the center of the magnetic dipole
(see Figure 2). Now, Alfvén speed can be calculated at
each point P along the straight line according to v A
B4π µ m pN 12 (m p , proton mass) via the determination of the magnitude of the magnetic field B and the
electron number density N e . The full particle number
density N is related to Ne by N 1 92Ne for a mean
molecular weight µ 0 6 [17]. The radial behaviour of
the so found Alfvén speed is presented for the parallel
and anti-parallel case in Figure 3. As Figure 3 shows, the
local Afvén speed has a local minumum in the middle
of the corona and a local maximum of 740 kms at a distance of 3 8R S from the center of the Sun. The mixture of
the local magnetic field of the active region and the gobal
magnetic field of the quiet Sun causes the appearance
2000
1800
+
vA (km/s)
_
1600
vA (km/s)
vA (km/s)
vA (km/s)
1400
1200
1000
800
600
400
200
0
1
10
R / RS
100
FIGURE 3. Alfvén speed along a straight line with δ 450
as a function of R. The full and dashed lines indicate the parallel
and anti-parallel case, respectively. The dotted line represents
the behaviour of the Alfvén speed only due to the magnetic
field of the quiet Sun.
of the local minimum of the Alfven speed. The influence of the active region on the complete magnetic field
is disappearing in the upper corona beyond a radial distance of 2RS . The occurrence of a local maximum of the
Alfvén speed in the near-Sun interplanetary space was
already deduced by Mann et al. [19]. Gopalswamy et al.
[9] already reported on such local extrema of the Alfvén
speed. They only considered the purely radial behaviour
of the Alfvén speed above active region using the magnetic field model by Dulk and McLean [20] and the density model by Saito et al. [21]. In contrast to Gopalswamy
et al. [9] the presented approach also allows to study the
non-radial propagation of coronal disturbances. That is
much more appropriate because of the observations of a
non-radial movement of type II radio burst sources [9,
10, 11, 12].
dle of the corona, i. e. 1 2 RR S 2 and beyond 6R S .
That fact might be an explanation of the occurrence of
two kinds of shocks as claimed by several authors [5, 6,
22].
The local Alfvén speed has a local minimum of
428kms and 227kms at a radial distance of 1 45R S and
1 32RS in the parallel and anti-parallel case (see Figure 3), respectively. Note that the sound speed has a value
of 180 kms for a temperature of 1 4 10 6 K, which is typical in the corona. Thus, a shock wave is propagating into
region of decreasing Alfvén speed after its formation in
the low corona, i. e. the Alfvén-Mach number is increasing. Especially, it becomes a high Alfvén-Mach number
shock near regions of the Alfvén speed minimum. It is
well-known that high Alfvén-Mach number shocks are
preferably accelerate particles [24]. This fact could be
the root of the time delay between the flare on-set and the
production of highly energetic electrons as recently reported by several authors [25, 26, 27]. These authors [25,
26, 27] claim an additional generator of highly energtic
electrons in the upper corona. According to the presented
approach a shock wave would need about 370s from the
initial energy release (flare) up to reaching the place of
the Alfvén speed mimimum, where the shock waves become efficient particle accelerators because of their high
Alfvén-Mach number.
Furthermore, interplanetary shocks are regarded to be
driven by CMEs. Since CMEs have velocities with a
mean value in the range 400 500 kms [28], they become super-Alfvénic at radial distances beyond 6R S from
the center of the Sun, i. e. in the near-Sun interplanetary
space . They are formed after about 4000 s after the lift
off of the CME and subsequently generate highly energetic protons, which are observed as SEPs [7, 8].
ACKNOWLEDGMENTS
This work was supported by the German Deutsche
Forschungsgemeinschaft, DFG under project number
MA 1376/14-1.
DISCUSSION
The occurrence of such local extrema of the Alfvén speed
has consequences for the formation and development of
shock waves in the solar corona and the interplanetary
space. In order to discuss that, a disturbance is considered to be propagating with a velocity V along a straight
line away from an active region. This straight line should
be inclined with respect to the radial direction. Such a
disturbance has initially been generated by a flare or the
lift off a CME. Shock waves can be established at regions, where the travelling disturbance becomes superAlfvénic, i. e. V v A . The disturbances associated with
solar type II radio bursts have typical radial velocities in
the range 270 1220kms with a mean value at 730kms
[23]. Then, they can become super-Alfvénic in the mid-
614
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