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

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4. Calculation of the electron heat conduction flow

Now, we test the effect of the newly formulated heat conduction flow as given in Eq. (19) on the distance-dependence of the electron temperature. With the expression (19) we have obtained:

[EQUATION]

With Eqs. (9) and (13) we can now remove either the function [FORMULA] or the function [FORMULA] from the above formula using the following relation:

[EQUATION]

and finally then obtain the heat conduction flow as a function of the solar wind bulk velocity and the argument [FORMULA].

To further evaluate expression (26) and compare results with observational data we first derive an expression for [FORMULA] evaluating the function [FORMULA] or [FORMULA], respectively, in as a consistent form as possible.

For that purpose, we start out from the generally accepted requirement that the hydrodynamical forces acting upon solar wind electrons, due to practical absence of inertial and gravitational forces, should cancel eachother leading in the CGL approximation for anisotropic pressure functions to the following expression (for a general derivation see Fahr et al. 1977):

[EQUATION]

where z is the space coordinate parallel to the field [FORMULA], and where [FORMULA] and [FORMULA] are the electron pressure tensor elements perpendicular and parallel to [FORMULA]. For a purely radial field we can replace the space coordinate z by r and obtain:

[EQUATION]

In this form this relation has also been used by Fichtner & Fahr (1991) or Meyer-Vernet & Issautier (1998). Here, however, we shall evaluate this expression in more detail making use of the parametrized form of the distribution function by a truncated Maxwellian given in Eq. (7). As evident from expression (23) for the scalar pressure [FORMULA] one can easily also derive analogously the following relations:

[EQUATION]

and:

[EQUATION]

With these relations we thus obtain from Eq. (29):

[EQUATION]

Integrating this expression by parts then yields:

[EQUATION]

The outer border of the integration is hereby placed at a distance [FORMULA] where the asymptotic level of the electric potential is achieved. Also the [FORMULA]-drop-off of the electron density at larger distances (i.e. [FORMULA] AU) has been used here.

Assuming, in addition, that the electron temperature drop-off can be represented in a satisfactorily accurate way by [FORMULA] (see e.g. observations by Scime et al. 1994) one finally obtains:

[EQUATION]

To further evaluate Eq. (34) we have to define an adequate point [FORMULA]. Hereby one should pay attention to the following: The above derivation because of the neglect of inertial forces can only be used in the region where solar wind electrons are still subsonic, i.e. inside a region where electron temperatures are larger than a critical value given by:

[EQUATION]

with [FORMULA] being the ratio of electron heat capacities which for electrons bound to the magnetic field (i.e. [FORMULA]) yields [FORMULA]. With [FORMULA] km s-1 one thus finds [FORMULA] K connected with a critical definitions of [FORMULA] by:

[EQUATION]

Using an adequate electron temperature profile taken from observations one now can evaluate the expression (26) for the heat conduction flow and compare it with observational data on [FORMULA].

At larger distances ([FORMULA] AU) [FORMULA] can be taken as constant and thus the solar wind density drops off like [FORMULA]. Solar wind electron temperatures [FORMULA] can for instance be obtained with the help of ULYSSES results published by Scime et al. (1994). These authors give temperatures separately for core ([FORMULA]) and halo ([FORMULA]) electrons in the following form:

[EQUATION]

and:

[EQUATION]

Since the typical abundances of core and halo electrons were found to be (see Feldman et al. 1975):

[EQUATION]

for our purposes here, due to the lack of any better information, one may thus reasonably well represent the effective electron temperature by the following combined expression:

[EQUATION]

Adopting, however, this electron temperature profile and looking for the point where [FORMULA] Kelvin would be achieved, one would get the unreasonable result: [FORMULA] AU.

Here we want to restrict ourselves to regions with measured electron temperatures, i.e. to 0.3 to 5.0 AU. Hence, we decide to finally define the quantity [FORMULA] by:

[EQUATION]

Assuming that the asymptotic level of the electric potential is already reached there.

In Fig. 1 we have displayed the quantity [FORMULA] (i.e. the normalized potential step to the asymptotic plateau level as function of the solar distance r for various values of [FORMULA] = 5, 6, 7 AU. In addition in Fig. 2 we have shown, how the relevant integral functions [FORMULA], [FORMULA], [FORMULA], [FORMULA] needed in expression (26) to calculate the electron heatflow [FORMULA] vary with solar distance r, with [FORMULA] defined by Eq. (41). In this figure the value [FORMULA] AU has been used.

[FIGURE] Fig. 1. Shown is the quantity [FORMULA] (i.e. the electric potential energy difference to the asymptotic point [FORMULA] normalized by the thermal energy of the local electrons) for various values of [FORMULA] (i.e. 5, 6, 7 AU).

[FIGURE] Fig. 2. Shown are the heatflow-relevant integral functions [FORMULA], [FORMULA], [FORMULA] and [FORMULA] as functions of the solar distance r for a boundary value of [FORMULA] AU.

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© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000
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