Solar Wind Plasma Experiment on Solar Orbiter: dealing
with the need for a sufficient phase-space resolution
R. D’Amicis , R. Bruno , MB. Cattaneo, B. Bavassano , G. Pallocchia and J.A.
Sauvaud†
Istituto di Fisica dello Spazio Interplanetario (CNR) - Via Fosso del Cavaliere, 100 - 00133 Rome - Italy
†
CESR (CNRS/Université de Toulouse), BP 4346 - 31029, Toulouse Cedex, France
Abstract. Solar Orbiter is proposed as a space mission dedicated to study the solar surface, the corona and the
solar wind by means of remote sensing and in-situ measurements, respectively. At 0.21 AU, closest approach to
the Sun, the full 3-D particle velocity distribution should be sampled as fast as a few tens of msec in order to
study the growth phase of the instabilities. This implies some restrictions on the maximum phase space resolution
given the limited allowed bit-rate for data transmission. In this paper we evaluate consequences of this limitation
for solar wind distributions with different parameters.
INTRODUCTION
NUMERICAL SIMULATIONS
Our present knowledge of the solar corona, the solar
wind and the 3-D heliosphere is based on the results
of missions such as Helios, Ulysses, Yohkoh, Soho and
Trace. Solar Orbiter will provide us with new insights in
the plasma kinetic processes that act in the Sun’s atmosphere and in the extended corona as it will explore the
inner regions of the Solar System (the perihelion will be
at 0.21 AU). The study of the kinetics of the solar wind
is fundamental to fully understand the mechanism acting within the plasma’s micro state. In particular, it helps
us to understand the role of microinstabilities generated
by non-Maxwellian characteristics and anisotropies in
phase-space of the velocity distribution function. In order
to study these microinstabilities it would be extremely
important to study them during their growth phase. To do
so, we should be able to sample the whole 3-D velocity
distribution function fast enough, i.e. with a time resolution of the order of a few tens of msec, which is the time
taken by the s/c to go across a scale length of the order
of the typical Larmor gyroradius at 0.2 AU. Taken into
account the need to follow the instability for about 100
gyroperiods, typical time duration of the growth phase,
and the allowed bit rate of 5 kb/s, it follows that the maximum possible resolution would be 10 energies, 10 azimuths and 10 polar angles (see the Assessment Study
Report, 2000).
In this report we show that this phase-space resolution
is generally not sufficient to fully resolve the ion distribution and to compute correctly its moments.
The solar wind distribution can be represented by a biMaxwellian, defined in the following way:
f
m
n
2π kT
v z
m
2π kT
Vz 2
1 2
exp
m
vx
2kT
m
2kT
Vx v y
Vy 2 2
where m is the proton mass, k is the Boltzmann’s constant and Vxyz are the plasma flow velocity components.
This distribution represents the most general equilibrium
status for a magnetized plasma and it is written in the
magnetic field’s frame, i.e. chosen in such a way that the
x axis is parallel to B. It is characterized by the following
6 variables: the number density, n; the parallel and perpendicular components of the temperature, T and T ,
with respect to the magnetic field’s direction; the particle
velocity components v x ,vy ,vz . We assume that the wind
flow is parallel to the x axis.
Taking into account only the major ions, we have extrapolated the proton parameters to 0.21 AU, closest approach to the Sun for Solar Orbiter, using the radial
trends evaluated from Helios 2 observations. In particular, we chose day 100th (Marsch, 1982) as a reference
for the slow solar wind and day 107th (Marsch, 1982;
Marsch, 1991) for the fast solar wind as shown in Table
1. In Table 2, we show the temperature radial gradients
and their indexes for both slow and fast wind. From the
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
822
radial trends shown in Table 2 and adopting r 2 for the
density radial dependence, we obtained the parameters
of the distribution as shown in Table 3. We chose proton velocity values slightly different from those of Helios
(especially for what concerns the slow wind) because we
intend to study solar wind in quasi-extreme conditions.
The parameters characterizing the alpha particles, as obtained by Helios 2 (Marsch, 1991), are shown in Table
4.
TABLE 4.
0.333
0.291
429
781
71.9
28.3
18.6
48.8
15.7
83.4
TABLE 2. Temperature radial gradients.
Radial
Index Slow
Fast
Gradient
wind wind
T ∝ rβ
β
T ∝ rβ
β
1.0
0.72
0.9
1.12
250
800
181
54.3
29.5
61.7
23.8
120
T
T
Ttot
Protons
α particles
np
0.05 n p
Tp 2 Tp Tp 2 Tp T ptot
2 T ptot
C 2
v cosθ d θ d φ dv
G v4
that becomes:
N
25.7
101
∆θ ∆φ
G
∆v
∑ v2i ∑ ∑ cosθk Ci jk
vi
i
φ j θk
where ∆θ and ∆φ are the amplitudes of the polar and
azimuthal sectors, respectively, while ∆v i is the amplitude of the energy channel. The average over v 2i is an
average over the four adjacent energy sub-channels that
constitute an energy channel while cosθ k is computed using the central angle of the polar sector. Expressions for
the other moments are obtained in a similar way.
We assume a field of view varying from -45 o to +45o
both in azimuthal and polar direction (D’Amicis et al.,
2001). The energy variation law is assumed exponential
and the energy limits vary from 15000 eV to 10 eV in
order to cover the entire energy distribution.
We computed the principal moments as a function of
some possible combinations of the number of azimuths,
polar angles and energy steps, i.e. varying ∆θ , ∆φ and
∆E, starting from the resolution suggested in the Assessment Study Report of 9 o and 10 energy steps, down to
5o and 32 energy steps (Helios resolution). Our model
distribution is characterized by a proton bi-Maxwellian
distribution, whose parameters are shown in Table 3.
Results shown in Table 5 suggest that the lowest resolution causes large uncertainties in the computation of
the principal moments. On the contrary, the other resolutions reproduce quite well the model. It is evident that,
in general, the higher the resolution the closer to the
Computing the moments of the distribution
Given a distribution function f, one can define the
moment of order n of such distribution:
Mn n
N
TABLE 3. Solar Wind Parameters extrapolated at
0.21 AU.
R
vp
np
T p
T p
Ttot
[AU] [km/s] [cm3 ] [eV] [eV] [eV]
0.21
0.21
Parameters
counts
by the instrument, defined as: C
registered
Sv f d 3 v where S is the effective cross-section and
d3 v = v2 cos θ dθ dφ dv, θ is the polar angle and φ is the
azimuthal angle. As the incidence is perpendicular, θ 0, so cos θ 1. Dealing with discrete quantities, one
fi jk S v3 ∆vi ∆φ j ∆θk
can rewrite the preceding as: Ci jk
or in a more compact way as C i jk
fi jk G v4 where
the geometric factor, G = S cosθ ∆φ ∆θ ∆v v has been
introduced. From the preceding equation, the distribution
function can be evaluated: f i jk Ci jk G v4 (for further
reading see Paschmann et al., 1998). As an example, we
will obtain the number density expression. Coming back
to its definition and substituting the preceding expression
for f, one obtains:
TABLE 1. Solar Wind Parameters from Helios 2.
Day
R
vp
np
T p
T p
1976 [AU] [km/s] [cm3 ] [eV] [eV]
100
107
Alpha particles parameters.
f vvn d 3
from which one can evaluate:
• the number density: N
f vd 3 v,
• the number flux density vector: NV
f vvd 3 v
(from which one can compute the velocity dividing
by N),
• the moment flux density tensor: Π m f vvvd 3 v,
m
• the energy flux density vector: Q
f vv2 vd 3 v.
2
Converting the momentum flux tensor and the energy
flux vector to the plasma’s ref. frame, one obtains the
pressure tensor, P Π ρ V V and the heat flux vector,
H Q VP 12 VTrP. Using the definition P NkT ,
one can convert the pressure tensor into a temperature
tensor. These quantities can be written by means of the
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TABLE 5. Moments of the distribution as a function of the number of
energy steps, of polar and azimuthal angles for a fast wind.
Resolution
Moments
no φ
steps
no θ
steps
no E
steps
n
[cm3 ]
Vx
[km/s]
T
[eV]
T 1
[eV]
T2
[eV]
Ttot
[eV]
10
10
10
12
12
18
18
10
10
10
12
12
18
18
10
20
32
20
32
20
32
25.9
52.0
53.4
52.0
53.4
52.0
53.4
731.
803.
800.
803.
800.
803.
801.
48.5
54.0
54.9
54.0
54.9
53.9
54.8
103.
121.
120.
121.
120.
121.
120.
114.
134.
133.
130.
129.
125.
124.
88.0
103.
103.
102.
101.
99.8
99.6
higher resolution, i. e. 12 φ , 12 θ and 32 E and the alpha
peak is now clearly identifiable. We also simulated the
sampling for the maximum resolution (18 18 32),
but we did not obtain a better spatial definition of our
distribution function.
1.0
1.0
0.8
0.8
Normalized counts
Normalized counts
model are the computed moments. Moreover, the number of energy steps plays a very important role. Actually,
the computation of n, Vx and T is more sensitive to this
parameter. The transverse velocity components are very
small, as expected, of the order of unity. As an example, we can compare the 10 10 32 resolution to that
of 18 18 20. The previous parameters are closer to
the model when we use 32 energy steps instead of 20
energy steps, although the total resolution is lower than
in the other case. We also notice that n, Vx and T are
not very sensitive to the resolution in θ and φ provided
that the number of energy steps is kept constant. On the
other side, increasing the angular resolution allows to determine the perpendicular temperature more accurately.
Finally, the highest resolution does not show very different results from the previous cases. An analogous computation for a slow wind (see table 3) showed better results even at the lowest resolution because slow wind is
sampled in energy using narrower energy channels, given
that the wind is peaked at low energies and ∆E E is kept
constant.
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
300
600
900
1200
1500
1800
0
300
600
V [km/s]
200
200
0
B
Vy [km/s]
400
Vy [km/s]
400
-200
1200
1500
1800
0
B
-200
-400
-400
400
900
V [km/s]
600
800
1000
1200
1400
400
600
Vx [km/s]
800
1000
1200
1400
Vx [km/s]
FIGURE 1. Fast wind. In the upper panels: counts vs. velocity. In the lower panels: contour plots integrated over θ . On the
left-hand side panels: 10 φ , 10 θ and 10 E. On the right-hand
side panels: 12 φ , 12 θ and 32 E. The first contours correspond
to 10 % of the maximum phase-space density.
Sampling the distribution function
The next step is to study how the instrument samples
a distribution function, obtained as the sum of two biMaxwellians, one for protons (see Table 3) and the other
one for alpha particles (see Table 4), varying the number
of energy steps, polar and azimuthal angles, both for
slow and fast solar wind. We simulated all the cases
described in Table 5 but we noticed that the alpha peak
emerges when we use 24 energy steps, at least. The left
panels of Fig. 1 which refer to the lowest resolution,
show the counts (top panel) as a function of velocity and
the contour plots (bottom panel) computed integrating
over the polar angle. It is clear that this resolution is not
sufficient to fully resolve the distributions of the major
ions since the alpha particle peak does not show up. In
the right panels the same fast wind is resolved using a
Finally, we simulated a slow solar wind as shown in
Fig. 2 for the highest resolution. We represented three
distributions: protons, alpha particles and protons+alpha
particles. We observe that the proton and alpha particle
distributions can not be separated. We notice that the
proton distribution is dominant and the alpha particles
are entirely masked.
Co-rotating phase
Solar Orbiter will correlate in-situ and remote-sensing
measurements at 45 solar radii from a co-rotational van-
824
CONCLUSIONS
tage point. It is then very important to simulate this phase
of the orbit which will identify the links between the activity on the Sun’s surface and the resulting evolution of
the corona and inner heliosphere. To do this, we have to
add a Vy component, which can be estimated of the order of 100 km/s, to the bulk velocity. The aim of this
study is to verify whether the field of view of the analyzer can cover the whole distribution function for both
slow and fast wind. Fig. 3 shows that this is the case for
a fast wind for both low and high resolution. On the contrary, the slow wind is at the limits of the field of view of
the analyzer. This problem can be simply solved by shifting one angular sector downwards, so that we are able to
observe the entire distribution during the whole orbit.
We studied the moments computation as a function of
the resolution used for energy, polar and azimuthal angles. For a fast wind, the resolution firstly suggested in
the Assessment Study Report seems to be inadequate. It
is important to emphasize how the computation of n, V x
and T mainly depends on the number of energy steps.
Increasing the angular resolution allows only to determine the perpendicular temperature values more accurately. The sampling of the distribution functions shows
that, for a fast wind, the lowest resolution does not show
any evidence for the alpha particles peak. By increasing
the energy steps and leaving almost unchanged the angular resolution, we observe that the alpha peak emerges.
For a slow wind, even with the highest resolution, we are
not able to separate alpha particles from protons. Finally
we find that during the co-rotating phase, a field of view
from 45 o to 45o centered around 0 o seems not to be
appropriate to observe the slow wind distribution. A simple solution can be found by shifting the azimuthal field
of view adequately.
To better resolve the major ions’ distribution, we propose a higher phase-space resolution than the one firstly
suggested in the Assessment Study Report. From our
simulations, it emerges that an angular resolution of 10
φ and 10 θ is sufficient but, a minimum of, at least, 24
energy steps is needed.
Normalized counts
1.0
protons+alpha particles
protons
alpha particles
0.8
0.6
0.4
0.2
0.0
0
200
400
600
800
1000
V [km/s]
FIGURE 2. Slow wind. The picture shows three distributions: protons, alpha particles and protons+alpha particles with
a velocity resolution of 18 φ , 18 θ and 32 E.
200
REFERENCES
400
100
200
Vy [km/s]
Vy [km/s]
0
-100
-200
1. D’Amicis, R., Bruno, R., Bavassano, B., Cattaneo, M. B.,
Baldetti, P., Pallocchia, G., "Numerical Study of a quasi 3D
Top-Hat Analyzer", in Solar Encounter: The First Solar
Orbiter Workshop, ESA SP-493, Tenerife, Spain, 2001, pp.
205–209.
2. Marsch, E., Mühlhäuser, K.-H., Schwenn, R., Rosenbauer,
H., Pilipp, W. and Neubauer, F. M., J. Geophys. Res., 87,
52-72, (1982).
3. Marsch, E., "Kinetics Physics of the Solar Wind Plasma", in
Physics of the Inner Heliosphere II, edited by R. Schwenn,
E. Marsch, Springer-Verlag, Berlin Heidelberg, 1991, Vol.
21, pp. 45–133.
4. Paschmann, G., Fazakerley, A. N. and Schwartz, S. J.,
"Moments of Plasma Velocity Distributions" in Analysis
Methods for Multi-Spacecraft Data, edited by Paschmann
G. and Daly P. W., SR-001, International Space Science
Institute, 1998, pp. 125–158.
5. Solar Orbiter: A High-Resolution Mission to the Sun
and the Inner Heliosphere, Assessment Study Report,
SCI(2000)6, July 2000.
0
-200
-300
-400
0
-400
100 200 300 400 500 600 700
400
Vx [km/s]
200
600
800
1000
1200
1400
1200
1400
Vx [km/s]
400
100
200
Vy [km/s]
Vy [km/s]
0
-100
-200
0
-200
-300
-400
-400
0
100 200 300 400 500 600 700
Vx [km/s]
400
600
800
1000
Vx [km/s]
FIGURE 3. Co-rotating phase. The upper panels show the
distribution with the lowest resolution (10 10 10) for slow
(left) and fast wind (right). The bottom ones show the highest
resolution (18 18 32): slow wind (left) and fast wind(right).
The first contours correspond to 10 % of the maximum phasespace density.
825