SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


Astron. Astrophys. 348, 614-620 (1999)

Previous Section Next Section Title Page Table of Contents

2. Derivation of the heliospheric density model

In order to derive the required heliospheric density model the magnetohydrodynamic equations

[EQUATION]

[EQUATION]

[EQUATION]

are employed with the velocity [FORMULA] of flow, the mass density [FORMULA], the thermal pressure p, the magnetic field [FORMULA], and the external force [FORMULA]. They are supplemented by the gravitational force

[EQUATION]

(G, gravitational constant; [FORMULA], mass of the Sun; [FORMULA], unit vector along the radial direction) and the isothermal equation of state

[EQUATION]

(N, electron number density; [FORMULA], Boltzmann's constant, T, temperature). The mass density [FORMULA] and the electron number density N are related by [FORMULA] ([FORMULA], proton mass) with the mean molecular weight [FORMULA] (cf. e. g. Priest (1982)). In the solar corona and the solar wind [FORMULA] has a value of 0.6 (Priest 1982). Here, stationary, spherical symmetric solutions of (2) and (3) are required. Thus, all quantities should only depend on the radial distance r from the center of the Sun. The magnetic field and the flow velocity are assumed to be radially directed, i. e., [FORMULA] and [FORMULA]. Then, (2) can immediately be integrated to

[EQUATION]

with the constant C and, subsequently, (3) transfers into

[EQUATION]

Eliminating [FORMULA] in (8) by using (7), (8) can be integrated to

[EQUATION]

with the critical velocity [FORMULA] and the critical radius [FORMULA] (cf. Priest (1982)). (9) represents the well-known Parker 's wind equation (Parker 1958). The density model [FORMULA] is found by substituting the solution [FORMULA] of (9) into (7). (9) has different solutions. Such a solution is chosen, which is continuously connecting the region of the solar corona, i. e., [FORMULA], with the interplanetary space, i. e., [FORMULA]. Since [FORMULA] is the sound velocity, [FORMULA] represents the distance from the Sun, at which the solar wind becomes supersonic.

In the limit [FORMULA], (7) and (9) can be solved analytically in terms of a barometric height formula

[EQUATION]

with [FORMULA] and [FORMULA]. [FORMULA] denotes the radius of the Sun. Note, that the well-known Newkirk (1961) model given by [FORMULA] with [FORMULA] corresponds to a barometric height formula with the temperature of [FORMULA].

The constant C appearing in (7) is fixed by plasma in-situ measurements at 1 AU. At 1 AU a long duration average of the particle number density and the particle flux was found to be [FORMULA] and [FORMULA] (Schwenn 1990), respectively, by means of the plasma data of the HELIOS 1 and 2 and IMP satellites. This results in a mean solar wind speed of 425 km s-1 at 1 AU. Thus, the constant C is determined to be [FORMULA]. Now, the temperature T is the only parameter appearing in (9). (7) and (9) are numerically evaluated for three values of T, i. e., [FORMULA], [FORMULA], [FORMULA]. The results are summarized in Table 1. The third and fourth column contain the calculated values of the solar wind speed at 1 AU and the particle number density in the low corona, respectively. In the low corona, i. e., slightly above the transition region or 2300 km above the photosphere, a typical electron particle number density of [FORMULA] is given by Vernazza et al. (1981). Inspecting Table 1, the solution of (7) and (9) with a chosen temperature of [FORMULA] is in good agreement with the observed solar wind velocity of 425 km s-1 at 1 AU and the particle number density in the low corona (cf. Vernazza et al. (1981)).


[TABLE]

Table 1. The particle number density [FORMULA] (fourth column) at the bottom of the corona, i. e., 2300 km above the photosphere, the solar wind speed [FORMULA] (third column) as deduced from the numerical solutions of Eqs. (7) and (9) for different values of the temperature T (first column). The correspondig values of the critical radius [FORMULA] are given in the second column.


Thus, the solution of (7) and (9) with [FORMULA], i. e., a special solution of Parker 's (1958) wind equation, is very compatible with the observations in the corona and the interplanetary space and can consequently be regarded as a reasonable global heliospheric density model. Fig. 1 shows the radial dependence of the particle number density N and the solar wind speed v according to (7) and (9) by choosing [FORMULA]. The corresponding radial dependence of the electron plasma frequency [FORMULA] is depicted in Fig. 2.

[FIGURE] Fig. 1. Radial dependence of the particle number density [FORMULA] (full line) and the solar wind velocity [FORMULA] (dashed line) according to the numerical solution of Eqs. (7) and (9) with a temperature of [FORMULA]. The radial distance is given in terms of the solar radius [FORMULA] and astronomical units [FORMULA]

[FIGURE] Fig. 2. Radial dependence of the local electron plasma frequency [FORMULA]

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999
helpdesk.link@springer.de