Exospheric models
Exospheric kinetic models assume that there is a sharp level called the exobase,
separating a collision dominated region (where a fluid approximation is valid) from a
region fully collisionless. In fact, there is a transition region where collisions
between particles become less and less numerous when the distance to the Sun increases.
The radial distance of the exobase, r_0, is usually defined as the distance
from the Sun where the Coulomb mean free path becomes equal to the local density scale
height H or, equivalently, when the Knudsen number, K_n, is
of the order of 1 (see Fig. 1) 
Figure 1 : Some basics of the exospheric kinetic models.
In the collisionless region (also called the exosphere), particles move freely
under the influence of the Sun's gravitational field and of the interplanetary electrostatic
and magnetic fields. Their trajectories solely depend on the conservation of the
total energy (sum of their kinetic energy + gravitational potentiel energy +
electrostatic potential energy) and of their magnetic moment.
The correct determination of the radial profile of the electrostatic potential,
V(r), is the keypoint in solar wind exospheric kinetic models. Because
of their lower mass, the electrons tend to escape more easily from the Sun's gravitational
field than the protons (ions). To avoid charge separations and currents on large scales in
the exosphere, the electrostatic potential gives rise to a force which attracts the
electrons towards the Sun and repels the protons. V(r) is induced by a
slight charge separation at the scale of the plasma Debye length. This separation is
produced by a gravitational effect, by magnetic forces and thermoelectric effects.
For the electrons, the gravitational potential is negligible at all radial distances
in the exosphere. Therefore the total potential for an electron is given by the electrostatic
potential and the force acting on an electron is always attractive whatever the radial
distance to the Sun r (see Fig. 2). The electrons moving along a magnetic field line
may then belong to different classes of orbits (see Fig. 1) :

the escaping electrons : these electrons have a
kinetic energy larger than the escape energy and can reach infinity
the ballistic electrons : their kinetic energy is too
low to escape and they fall back into the corona
the trapped particles : these electrons have a
magnetic mirror point and a turning point in the exosphere such that they bounce continuously
up and down along a magnetic field line
the incoming particles : these electrons come from the interplanetary
medium. We assume that their velocity distribution function is empty.

For the protons, gravity cannot be neglected and their total potential
energy is the sum of their gravitational and electrostatic potentials. At large radial
distances, the electrostatic potential dominates the gravitational potential : all the
protons are submitted to a repulsive total force and can escape to infinity. However,
closer to the Sun, the gravitational potential dominates the electrostatic potential : the
total force acting on the protons is therefore attractive.

Figure 2 : An example illustrating that the total potential energy for
the electrons is monotonically increasing with the radial distance (top panel) while
the total potential energy for protons (bottom panel) is first attractive and then
repulsive. The total energy potentials are normalized with kT_0 where k is the
Boltzmann constant and T_0 is the temperature at the exobase (assumed in this example
to be identical for electrons and protons). The exobase r_0 is located at 2.5 solar
radii (Rs) while the r_max radial distance is at 3.68 Rs. The Kappa index determines
the amount of suprathermal electrons assumed to be present at the exobase.

In the coronal holes, the density is lower than in the other regions of the solar
corona resulting in larger mean free paths for the particles. and the exobase is then located
deeper into the solar corona (between 1.1 and ~ 5 solar radii). Therefore, the protons in the
coronal holes are submitted to a nonmonotonic total potential energy which is first attractive
from the exobase r_0 to a radial distance called r_max (where the
two forces balance each other) and then repulsive beyond r_max. Below r_max,
the protons can also be ballistic or trapped, so that the flux of escaping protons is reduced.
The situation is unchanged for electrons since their attractive electrostatic potential is still
much larger than their gravitational potential. Therefore, in order to guarantee equal fluxes
of escaping electrons and protons, a larger electric field is needed to increase the number of
protons with a kinetic energy sufficient to overcome the potential barrier. This simple
mechanism explains how the solar wind emerging from the coronal holes can be accelerated to
large velocities even if the temperatures are lower than in other parts of the solar corona.

Another important ingredient of this exospheric model is the use for the electrons of velocity
distribution functions (VDF) with an excess of high energy particles compared to a Maxwellian
distribution. These suprathermal electrons increase the number of particles which have a
kinetic energy large enough to overcome the potential well and escape to infinity. The
electrostatic field that warrants the quasineutrality of the plasma increases in order to
accelerate the solar wind protons to larger values. Therefore, the solar wind bulk speed
increases as well.

Figure 3 : The suprathermal electrons in the high speed solar wind are well fitted
by Lorentzian (or Kappa) distributions characterized by the value of a kappa index. The lower
the Kappa value, the larger the amount of suprathermal electrons. The Maxwellian distribution
corresponds to a very large value of the Kappa index.
