Some Basic Aspects of Solar Wind Acceleration
Nicole Meyer-Vernet , Andre Mangeney , Milan Maksimovic , Filippo Pantellini
and Karine Issautier
LESIA, Observatoire de Paris, CNRS UMR8109, 92195 Meudon, France
Abstract. We discuss some effects related to particle coherent orbits in the acceleration region and in the wind. (1) In the
distant wind, when collisions are negligible, the temperature of escaping electrons follows adiabatic anisotropic (CGL)
relations, whereas those reflected by the electrostatic and/or mirror forces behave as an adiabatic isotropic fluid; hence,
contrary to a widespread view, electrons do not follow a single adiabatic law in absence of collisions. (2) In the corona,
if one superimposes a minute hot maxwellian tail to a maxwellian velocity distribution, the relative importance of the tail
increases rapidly with height; this is fundamentally different from a Kappa distribution, whose non maxwellian character
remains constant with height. Suprathermal electrons also produce a large heat flux because many of them escape from the
electric potential. (3) Fluid models, exospheric models, and a numerical simulation including particle orbits and collisions
agree on finding that when an accelerated transonic wind is produced, the potential energy of protons has a maximum, so that
some protons can be reflected; the production of a transonic wind also requires the trapped electron orbits to be populated,
which requires some collisions. Finally, since particles on different kinds of orbits behave very differently, fluid models should
not consider each particle species as a single fluid; each species should be modelled instead as a superposition of several fluids
having different transport properties.
INTRODUCTION
SOME BASIC OBSERVATIONS
Since the solar wind acceleration region is weakly collisional and the particle free paths increase quickly with
energy, suprathermal particles are virtually collisionless,
making the acceleration problem non local, and letting
the velocity distributions develop high energy tails. Thus,
fluid models using local closure relations are not justified. However, kinetic models including a few collisions present major difficulties since with high energy
tails, the Fokker-Planck and other usual approximations
of the Boltzmann-Landau-Balescu formulation are questionable.
As a result, despite the avalanche of data and sophisticated models flowing unremittingly, there is no agreement on how the corona is heated and the solar wind is
accelerated. Since the physics is not fully understood and
the observations are inaccurate, detailed models should
be handled with care, since virtually any mixture put into
the computer kitchen may reproduce observation if the
saucepan contains enough free parameters. A complementary approach is to explore the basic physics with
simple models, which are not expected to reproduce observation, but to suggest instead which physical ingredients should be introduced into the recipe. The present
paper is written in this spirit.
The most recent analyses of observations
in polar coronal
holes yield a proton temperature Tp of 1 3 106 K[3].
The electron temperature Te is about twice as cold: in situ
charge state observations from Ulysses find a maximum
of 1 5 106 K at 1.4r [9], whereas Soho observations
yield values smaller than 106 K[24],[2]. The conflict on
Te might be resolved if the electron velocity distribution
develops a significant suprathermal tail within a short
distance from the coronal base[4].
With these parameters, thermal energy alone with classical conduction cannot drive the wind[1]. Consider the
approximate energy equation between the base r0 of the
wind and large distances:
m pV 2
2
Q0 5 m p M G
k Te0 Tp0 (1)
n0V0 2
r0
With the classical conductive heat flux Q
10 11Te5 2 dTe dr (in S.I. units), Te0 106 K and
β
2
Te ∝ r with β 1, one finds Q0 10 J/m /s at
r0 r . To estimate the flux n 0V0 from the value of
about 2 1012 protons/m2/s observed at 1A.U. in the
high speed wind, we consider the simple case where
the flux tubes expand as r 2 , which may hold at least
in the central part of coronal holes[25]. This yields:
Q0 n0V0 10 16 J/proton. With these parameters, the
heat flux and enthalpy terms cannot even lift the protons
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
263
out of the sun’s gravitational well. The problem is made
worse if the flux tubes expand faster than radially.
This result, however, should not be taken too seriously
for several reasons. First of all, the conductive heat flux
varies as Te7 2 and Te is not well known (as is its gradient). Putting in Eq.(1): Te0 1 5 106 K at r0 1 4r
as measured from
Ulysses[9] would yield instead a wind
speed of 900 β km/s! Secondly, there are not enough
collisions in the corona for the classical heat flux formula
to hold securely because, even though thermal electrons
are somewhat collisional, faster electrons - which contribute most to the heat flux - are not[21]. Thirdly, observations of minor ions suggest heating and acceleration
through dissipation of high-frequency waves[6]; low frequency waves might also play a role. Finally, if the electron velocity distribution in the corona has a suprathermal energy tail, this should increase the heat flux[11] and
contribute to the heating[21] and acceleration[14], as first
suggested by Olbert [17].
The latter effects are related to the coherent motion
of particles. To evaluate their importance, let us estimate
the dynamic time scale. The electric force on electrons,
which acts to ensure charge neutrality, can be estimated
by noting that it approximately balances their pressure
force, i.e.:
eE
1 ∂ nkTe n ∂r
(2)
Since Te varies generally less rapidly than n, we have
eE kTe 1n ∂∂ nr . The last term is the inverse of the
density scale height H, so that the work done by the
electric force along a mean free path l is: eEl kTe l H kTe since l H in the corona and in the wind.
This means that the electric field is roughly equal to the
Dreicer field in these regions (see also [22]).
eEl kTe may be rewritten as
The relation
eE me vthe 1 l vthe , where vthe is the electron
thermal velocity. Hence, the dynamic time scale of the
particles is roughly equal to their collision time, so
that coherent dynamics and collisions are of similar
importance. We discuss below some consequences of
coherent dynamics, and how collisions may change the
picture.
are not reflected by the magnetic mirror force have their
orbits connected to the base r0 ; the other ones are trapped
due to reflection by the mirror force on the sun ward
side and by the electric field on the outward side (Fig.
1, [8],[13]).
FIGURE 1. Sketch of electron orbits and of the corresponding regions in velocity space (the symbols and refer to the
direction of B).
The angular limit for ballistic and escaping particles
(heavy line in Fig.1) is set by energy and magnetic moment conservation between r0 and r. At large distances,
this simplifies to:
v2
v2
The electric field pulls electrons towards the sun, so that
only those whose kinetic energy at distance r satisfies
me v2 2 eφ , where φ is the potential at r, may escape
to infinity in absence of collisions. Electrons of lower
energy are reflected and return inwards. There are two
kinds of such electrons: the ones (labelled “ballistic” in
Fig.1) whose inclination to B is small enough that they
264
θM2
B 1 2e φ0 φ me v2 B0
1 (3)
The diagram is symmetrical with respect to the v 0
axis except that for v 0, the region labelled “escaping” corresponds instead to particles coming to r0 from
infinity and is not populated in absence of collisions (unless there is a reservoir of particles at infinity); the white
region corresponds to particles coming from and returning to infinite distance, and is not populated either. The
population of trapped electrons, which is arbitrary in absence of collisions, is set to be in quasi-equilibrium with
the ballistic population.
The velocity distribution at r is deduced from
the one at r0 from Liouville’s theorem as f v ORBITS OF ELECTRONS
f0 v2 2e φ0 φ me . Let us deduce the radial
temperature profile at large distances[16].
Consider first the non escaping particles. Their total
density nne is the velocity volume integral of f v in the
sphere of radius 2eφ me 1 2 (see Fig.1). Since at large
distances the potential satisfies φ kTe0 e φ0 , f v is
roughly constant (independent of v and r) within the integration volume, so that nne ∝ φ 3 2 . Now, it can be shown
that the non escaping electrons represent the main contri1 2
bution to the total density, even at large distances, so that
nne ∝ r 2 ; thus φ ∝ r 4 3 . The pressure
these par Pne of
ticles is the volume integral of v2 f v
v2 constant]
in the same sphere, i.e., Pne ∝ φ 5 2 . Hence, their temperature varies as Pne nne ∝ φ 5 2 r 2 ∝ r 4 3 . This means
that the temperature of non escaping electrons varies as
for an adiabatic isotropic fluid.
Consider now the escaping electrons. We have seen
that at distance r they populate the domain of velocity
space [ v 2eφ me 1 2 , θ θM ]. To estimate their moments at large distances, the inequality φ
kTe0 e φ0
enables us to make two approximations: (1) f v is independent of φ thus of r; (2) the lower limit of the integral
over v is equal to 0. With these approximations, their density and pressure have the same radial variation, as they
are both proportional to θM2 . Hence the temperature of escaping particles is roughly independent of r at large distances. This is consistent with an adiabatic anisotropic
fluid behaviour of this population since the CGL relations yield at large distances: T T
constant.
Finally, since the densities of escaping and non escaping particles vary similarly (as r 2 ) if B is radial, the
total electron temperature is the sum of a term ∝ r 4 3
and a constant, so that the temperature profile flattens
with distance, as observed[20]. One can show that both
terms have similar orders of magnitude at 1 A.U.[16],
and that the temperature profile agrees with Ulysses
observations[7]. These results can be generalized to a
spiral magnetic field[19].
How do collisions change these results? Because of
the small electron mass, collisions with other particles do
not change the electron energy, and collisions between
electrons do not change their total temperature. Hence,
collisions are not expected to modify locally the total
electron temperature, even though they do affect the
shape of the velocity distribution.
SUPRATHERMAL ELECTRONS
Some observations suggest that the electron velocity distribution in the corona might have a small suprathermal
tail[4]. We make below an order-of-magnitude estimate
to show that adding a minute hot maxwellian tail to a
maxwellian distribution at the coronal base has two consequences: firstly, the relative importance of the tail increases rapidly with altitude[15], which might explain
recent observations[4]; secondly, the heat flux and the
wind speed are significantly enhanced, as was previously
shown with different kinds of distributions presumably
much farther from equilibrium[17], [14].
For illustrative purposes, let us assume that the distribution at the base r0 r is a sum of two maxwellians:
a cold one of temperature Tc 0 8 106 K, and a hot
265
one at Th 5Tc , representing a minute proportion, say,
0.2 % of the total density (so that the total temperature is close to Tc ). For an order-of-magnitude estimate,
we neglect collisions and waves; we approximate the
electric potential close to the sun by the PannekoekRosseland one: eφ m p MG 2r, which is generally an
We also calculate the distribution
underestimate[8],[13].
just above r0 as in a bound atmosphere, i.e. we neglect
the anisotropy due to the escaping particles; this approximation might hold sufficiently close to the base.
With these simplifications, Liouville’s theorem tells us
that the temperatures Tc and Th remain constant with altitude, whereas the density of each maxwellian decreases
with altitude as e eφ kTc and e eφ kTh respectively. Since
eφ kTh eφ kTc , the density of the hot maxwellian decreases more slowly than the cold one, so that the proportion of the hot population increases with altitude. At
the distance where the potential has decreased by ∆φ , the
ratio of hot to cold densities has increased by the factor
exp
e∆φ
k
1
1
Tc Th
(4)
Let us calculate the distribution at, say, r 1 5 r (farther out, our simplifications may be too drastic). We
have e∆φ 0 17 m pMG r , so that the factor in Eq.4
49. Hence the proportion of hot electrons
amounts to
increases from 0.2 % to 10 % over half a solar radius.
This makes the total electron temperature increase from
about Tc to: Tc 1 0 1Th Tc
1 5 Tc.
illustrative,
Although this estimate is merely
it highlights the essence of the physics, that is velocity filtration
by the attractive potential, first calculated by[21] with a
kappa velocity distribution. With a sum of maxwellians which is a well-behaved function, we find a fundamental
difference with the kappa case: the hot tail contributes
more and more to the density as altitude increases, so
that the increase in total temperature stems from the increase in the non thermal character of the distribution.
In contrast, with a kappa distribution, the value of κ ,
which is a measure of the non thermal character, remains constant with altitude, although the total temperature increases[21].
The minute hot maxwellian contributes negligibly to
the density and temperature at r0 . This is not so, however,
for the heat flux, because a significant part of the hot
electron population can escape from the potential well.
Let us make a simple estimate, assuming as previously
a potential of about eφ0 m p M G 2r 0 . The heat flux
is produced by electrons of speed v 2eφ0 me at r0 ,
which are all escaping in absence of collisions. For a
maxwellian of density n and temperature T :
Q0
n
KT me kT 1
eφ kT 2 e 0
eφ0 kT
(5)
Since eφ0 kTh 2 8 , this yields Q0 4 10 10nh0 ,
where nho is the density of the hot population
at r0 . With
a relative concentration of 2 10 3, and a total density of
2 1014m 3 [5], one finds Q0 160 Jm 2 s 1 , which is
flux, and yields a wind
much larger than the conductive
speed of the order of 103 km/s. This is an overestimate,
for at least two reasons. Firstly, the actual electric potential should be different from the Pannekoek-Rosseland
value in order to make the flux of escaping electrons balance the proton one. Secondly, collisions should reduce
the heat flux[11], in particular by populating the orbits of
incoming particles having v 2eφ0 me at r0 .
ORBITS OF IONS
The electron potential energy eφ increases
monotonously with distance; this is not so, however, for
the total potential energy of protons ψ eφ m p M G r,
which is expected to have a maximum. Such a maximum
occurs in simple fluid models because the temperature
should decrease slower than r 1 in order to produce an
accelerated transonic wind[18]. From Eq.2, the electric
force is thus expected to decrease slower than r 2 , so
that this force, which pushes the protons outwards,
should dominate gravity at large distances. If the gravitational attraction dominates at the base of the wind,
the total potential energy of protons has a maximum[8].
The existence of this maximum is of fundamental importance since it governs particle orbits. Indeed, ballistic
and/or trapped ions are present out to the position of
this maximum. In contrast, all ions present farther out
are escaping, thus they all have v 0, so that the
mean parallel velocity is expected to be of the order of
magnitude of the thermal speed. Hence this maximum
is expected not to be very far from the sonic radius[23].
Let us examine this point in more detail.
Consider the simplest fluid model, with equal electron and ion densities, temperatures and speeds, and
assume n ∝ r α , T ∝ r β near the sonic radius
rS as in [18]. At rS , the temperature TS satisfies:
m p M G rS 4kTS 1 β 2 . The slope of the velocity
profile at rS : dV dr rS calculated by Parker, together
with nV r2 constant yields the relation between α and
2
β: α
2
β
β
8 β 2 1 β 4. This
enables us to calculate the slope of the proton potential
energy: d ψ dr eE m p M G r2 at rS . Substituting
E from Eq.2 and the above
relation between TS and
rS yields: d ψ dr rS
4 α β kTS rS . From the
above value of α , elementary arithmetics shows that
d ψ dr rS 0 if and only if: 3 4 β 1.
Since ψ
0 at large distances, the positive slope at
rS means that ψ has a maximum located above rS if and
266
only if the temperature decreases slower than r 1 and
does not increase faster than r 3 4 near rS , which generally
holds. (In the very special case β
3 4 we retrieve
the result found by[23]). If T increases even faster (but
slower than r 2 ), there is still a maximum, but it shifts
below rS . As first noted by [23], the conditions for the
ion potential to have a maximum with this simple model
are similar to those yielding a transonic accelerated wind
(i.e., 2 β 1).
A similar result is obtained with a numerical simulation taking into account particle orbits and model
collisions[12]. With that model, a transonic wind is produced only with enough collisions near the sonic point,
which populate trapped electron orbits and produce a
maximum in proton potential energy located above the
sonic point.
These results are consistent with exospheric models, which assume the orbits of trapped electrons to
be populated in equilibrium with those emerging from
the base, which requires some collisions, and yields a
transonic wind with a maximum in the ion potential
energy[10],[26].
Note, finally, that with a sonic point located below
the maximum in potential energy, the wind acceleration
region contains protons moving on reflected orbits - a
point which is ignored in fluid models.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
Barnes A. et al., Geophys. Res. Lett. 23, 3309 (1995)
David C. et al., A & A 336, L90 (1998)
Esser R. et al., Ap. J. 510, L63-L67 (1999)
Esser R., Edgar R.J. et al., Ap. J. 532, L71 (2000)
Fludra A. et al., J. Geophys. Res. 104, 9709 (1999)
Hollweg J. V., Solar Wind Nine, ed. by S. R. Habbal et al.,
The American Institute of Physics (1999)
Issautier K. et al., Astrophys. Space Sci. 277 (2001)
Jockers K., A & A 6, 219 (1970)
Ko Y.- K. et al., Sol. Phys. 171, 345,(1997)
Lamy et al, Solar Wind 10,(2002)
Landi S., Pantellini F.G., A & A 372, 686 (2001)
Landi S., Pantellini F.G., A & A, in press (2003)
Lemaire J., Scherer M., J. Geophys. Res. 76, 7479 (1971)
Maksimovic M. et al., A & A 324, 725 (1997)
Meyer-Vernet N., Planet. Space Sci. 49, 247 (2001)
Meyer-Vernet N., Issautier K., J. Geophys. Res. 103,
29,705 (1998)
Olbert S., ESA SP161, 135 (1981)
Parker E.N., Ap. J. 139, 72 (1962)
Pierrard V. et al., Geophys. Res. Lett. 28, 223 (2001)
Pilipp W.G. et al., J. Geophys. Res. 95, 6305 (1990)
Scudder J.D., Ap. J. 398, 299-318, 1992
Scudder J.D., J. Geophys. Res. 101, 13,461 (1996)
Scudder J. D., J. Geophys. Res. 101, 11,039 (1996)
Wilhelm K. et al., Ap. J. 500, 1023 (1998)
Woo R., Habbal S.R., J. Geophys. Res. 105, 12,667 (2000)
Zouganelis et al., Solar Wind 10 (2002)