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Astron. Astrophys. 363, 295-305 (2000)

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2. Solar wind electron hydrodynamics based on truncated Maxwellians

Solar wind electrons are tightly bound to interplanetary magnetic field lines. Thus their magnetic moments essentially behave as invariants of the motion since over electron gyroscales or gyroperiods fluctuating magnetic fields can hardly exert any influence. This also enforces the local electron distribution function to be closely associated to regions of magnetically conjugated footpoints in the corona. Down here of course collison-dominated conditions prevail and quasi-Maxwellian distributions are easily established. Since from such points electrons are emitted to the associated upper space point with a rate defined by the antisunward branch of the coronal Maxwellian, one may find electrons at the upper space point connected by Liouville theorem to the lower corona. (e.g. see papers by Lemaire & Scherer 1971, 1973; Lie-Svendsen et al. 1997; Pierrard & Lemaire 1996; Meyer-Vernet & Issautier 1998; Meyer-Vernet 1999). This distribution at an upper space point, however, without relaxation processes in operation above the coronal exobase, clearly has to differ from a full Maxwellian, because that specific velocity space volume representing trapped particles (i.e. with mirror points above the coronal exobase) in a collisionfree treatment should be unpopulated. The rest of the upward velocity space would, however, be populated according to a Maxwellian at the exobase. In contrast that branch of the distribution function, describing electrons moving downwards towards the lower conjugated footpoint, evidently would be absent in just those velocity space volumes belonging either to trapped or to hyperbolic particles (i.e. particles with energies larger than the effective escape energy). These latter electrons have energies enabling them to overcome the remaining polarisation potential ramp [FORMULA] and escape irreversibly from the heliosphere. Hence such electrons cannot be expected to approach the corona from the top, if not especially generated by other sources further out in the heliosphere.

Collisionfree electrons moving upward from the corona are subject to gravitational forces, electric forces connected with the polarisation potential [FORMULA] and magnetic forces due to the magnetic field divergence and to the electron magnetic moment µ. This allows one to integrate the equation of particle motion yielding two invariants of the motion, namely the total energy [FORMULA] and the invariant magnetic moment [FORMULA]. These invariants lead to the following relations:

[EQUATION]

and:

[EQUATION]

Here [FORMULA] and [FORMULA] are the electron velocity components parallel and perpendicular to the magnetic field [FORMULA], and [FORMULA] is the effective polarisation potential. From these relations one derives that the pitch angle [FORMULA] of an ascending electron (i.e [FORMULA]) varies according to:

[EQUATION]

where "0" characterizes the respective quantities at the coronal exobase (i.e. at [FORMULA]), and where the definitions: [FORMULA] and [FORMULA], have been used. At points [FORMULA] far above the exobase, where [FORMULA], Eq. (4) yields:

[EQUATION]

where [FORMULA] denotes the potential difference between the coronal base and the asymptotic potential at infinity (i.e at large distances [FORMULA]). With relations (2) to (5), astonishingly enough, one already determines the asymptotic solar wind electron flux [FORMULA] which in a quasi-neutral, stationary solar wind also fully determines the asymptotic solar wind mass flow [FORMULA]

Assuming an upward emission of electrons from the exobase according to a Maxwellian one formally arrives at the following expression:

[EQUATION]

where [FORMULA]. Knowledge of the value of [FORMULA] would thus already completely determine the resulting solar wind mass flow [FORMULA].

Some attempts have been made by Meyer-Vernet & Issautier (1998) and Meyer-Vernet (1999) to estimate [FORMULA] purely on the basis of solar coronal plasma properties, simply fulfilling requirements of quasineutrality and vanishing electrical currents, and have led them to the result:

[EQUATION]

dependent on the value taken for [FORMULA] if electrons at the coronal base [FORMULA], are described instead by Maxwellians by so-called Kappa functions like:

[EQUATION]

with:

[EQUATION]

For the Maxwellian case, i.e. [FORMULA], the value [FORMULA] is obtained, whereas for electron distributions with pronounced high-energy tails, i.e. with [FORMULA], a value of [FORMULA] is obtained, leading to asymptotic solar wind velocities of 250 km s-1 ([FORMULA]) and 700 km s-1 ([FORMULA]), respectively. Though this clearly reveals that the solar wind phenomenon is highly sensitive to the escape branch of the electron velocity distribution function in the upward hemisphere of velocity space, it nevertheless does not help to find consistent solar wind solutions.

As pointed out by Jockers (1970), the correct polarisation potential and thus the value of [FORMULA] can only be determined from a consistent knowledge of the solar wind dynamics which is closely connected with a consistent description of trapped electrons unfortunately not emanating from collision-free theories. Thus some relaxation processes of a wave-particle interaction type have to operate above the coronal base to explain observed solar wind quantities. In our view here these processes are due to quasilinear interactions of electrons with preexisting, convected whistler wave turbulences (see Secs. 6 through 8). Up to now trapped particles could not be described adequately by purely kinetic solar wind theories. On the other hand solar wind hydrodynamics can not account for truncated electron distributions. Hence we develop here a semi- hydrodynamic theory parametrizing the electron distribution in a kinetically motivated way permitting the calculation of all relevant velocity moments needed for a hydrodynamic view.

We assume that by wave-induced relaxation operating above the exobase the genuine Liouvillean velocity distribution is converted into one describing a macroscopic drift and the appearance of trapped particles on the basis of quasi-Maxwellians. We assume that the local electron distribution function [FORMULA] can be approximated by a truncated, shifted Maxwellian perhaps with a constant shift of [FORMULA] with respect to the solar rest frame (SF). By some recalibration of the potential this drift can as well, however, also be set equal to [FORMULA], since truncated Maxwellians define an inherent drift. Then with a density normalization [FORMULA] and a local thermal electron velocity spread C(r) the distribution is simply given by:

[EQUATION]

Here polar velocity coordinates [FORMULA] have been used, the polar axis being identical with the magnetic field direction. Hence [FORMULA] then automatically also represents the electron pitch-angle. The step-functions [FORMULA] have to take into account the appropriate local truncation with the effect of suppressing hyperbolic electrons in the sunward magnetic hemisphere of velocity space. The quantity [FORMULA] defines the local escape velocity of electrons, i.e. [FORMULA].

It is evident that, due to this truncation, the functions [FORMULA], [FORMULA] and [FORMULA] in general are not strictly identical to the local density [FORMULA], bulk velocity [FORMULA] and temperature [FORMULA]. They, in contrast, have to be obtained as moments of the above distribution function [FORMULA] by integration over velocity space and thus represent space-variable functions. First, the following relation must be valid:

[EQUATION]

where the potential difference, [FORMULA], has been introduced and where [FORMULA] are step functions (i.e. [FORMULA] for positive arguments X, and [FORMULA] for negative arguments X). Max[FORMULA] is the Maxwellian with a velocity dispersion [FORMULA]. Individual electron velocities are denoted by [FORMULA]. [FORMULA] is the effective electron polarisation potential and [FORMULA] its difference with respect to that of infinity.

Eq. (11) can be evaluated yielding the following relation between [FORMULA] and [FORMULA]:

[EQUATION]

Here the function [FORMULA], for [FORMULA], is defined by the following integral function:

[EQUATION]

The quantity [FORMULA] in Eq. (12) has the following definition:

[EQUATION]

Next we calculate the electron bulk velocity [FORMULA]. The truncated Maxwellian directly determines the radial solar electron flux or the solar wind proton flux in the form

[EQUATION]

where [FORMULA] and [FORMULA] have been introduced as abbreviations with meanings evident by comparison with Eq. (10). The quantity [FORMULA] takes into account the local tilt by an angle [FORMULA] of the Archimedian spiral field with respect to the radial direction. Expression (15) can be evaluated to yield:

[EQUATION]

where [FORMULA] is defined according to Eq. (13) for [FORMULA]. It needs to be mentioned that the electron bulk is not moving in radial direction but into the direction of the local magnetic field with an electron bulk speed of [FORMULA]

Next we calculate the electron pressure and find accordingly:

[EQUATION]

where [FORMULA] is the electron velocity measured in the electron bulk flow frame (EBF) locally moving with the electron bulk velocity [FORMULA]. Thus [FORMULA] is related to the electron velocity [FORMULA] by:

[EQUATION]

The individual electron velocity has a tilt with respect to the radial direction given by: [FORMULA].

Reminding oneself of the symmetry conditions of [FORMULA] the expression (17) evaluates to:

[EQUATION]

Here again [FORMULA] is calculated according to Eq. (13) for [FORMULA].

Of great interest for the thermodynamics and magnetohydrodynamics of the solar wind expansion is the electron heat conduction flow [FORMULA] which on the basis of the parametrized distribution function is represented by:

[EQUATION]

and evidently is oriented purely parallel to the local magnetic field [FORMULA], i.e. [FORMULA]. Again due to symmetry reasons expression (20) simplifies to yield the following modulus of heat conduction flow:

[EQUATION]

The latter expression can be evaluated and finally yields the following form:

[EQUATION]

By the use of truncated Maxwellians one is thus able to represent the heat conduction flow [FORMULA] as a functional of the lowest three velocity moments of this distribution, namely [FORMULA], [FORMULA] and [FORMULA], and thus reach a closed hydrodynamic system of governing differential equations.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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