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

## 2. Derivation of the heliospheric density model

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

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

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

(N, electron number density; , Boltzmann's constant, T, temperature). The mass density and the electron number density N are related by (, proton mass) with the mean molecular weight (cf. e. g. Priest (1982)). In the solar corona and the solar wind 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., and . Then, (2) can immediately be integrated to

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

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

with the critical velocity and the critical radius (cf. Priest (1982)). (9) represents the well-known Parker 's wind equation (Parker 1958). The density model is found by substituting the solution 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., , with the interplanetary space, i. e., . Since is the sound velocity, represents the distance from the Sun, at which the solar wind becomes supersonic.

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

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

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 and (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 . Now, the temperature T is the only parameter appearing in (9). (7) and (9) are numerically evaluated for three values of T, i. e., , , . 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 is given by Vernazza et al. (1981). Inspecting Table 1, the solution of (7) and (9) with a chosen temperature of 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 1. The particle number density (fourth column) at the bottom of the corona, i. e., 2300 km above the photosphere, the solar wind speed (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 are given in the second column.

Thus, the solution of (7) and (9) with , 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 . The corresponding radial dependence of the electron plasma frequency is depicted in Fig. 2.

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

 Fig. 2. Radial dependence of the local electron plasma frequency

© European Southern Observatory (ESO) 1999

Online publication: July 26, 1999
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