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Astron. Astrophys. 344, 317-321 (1999)

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2. Model

Because the mean free path of the interstellar atoms is comparable with the characteristic size of the interface, the kinetic Boltzmann's equation has to be solved (see Eq. 1 in Izmodenov et al. 1997). The equation takes into account the solar gravitation, the direct charge exchange [FORMULA], the reverse charge exchange [FORMULA], the photoionization and the electron impact ionization.

The study of the influence of the electron impact ionization on the interstellar hydrogen atom filtration has been done self-consistently on the basis of the two-shock heliospheric interface model by Baranov & Malama (1996). It is shown in this work that hydrogen filtration due to electron impact ionization is not significant. However, these authors have discovered that electron impact ionization influences the plasma flow in the region between the termination shock (TS) and the heliopause (HP). Indeed, when electron impact ionization is taken into account, there appears a strong density gradient in the whole region of compressed solar wind (see Fig. 3 in Baranov & Malama 1996).

For oxygen the situation is different, because due to its small cosmic abundance neutral oxygen doesn't influence the plasma flow. At the same time the electron impact ionization rate at electron temperatures relevant to the interface is larger for oxygen than for hydrogen (Fig. 1), because these elements have the same first ionization potential [FORMULA], but oxygen has two electrons on the outer orbit and six electrons on the inner orbit (the second ionization potential is also rather s all for oxygen).

[FIGURE] Fig. 1. Electron impact ionization rates for oxygen (solid curve) and hydrogen (dashed curve) as functions of electron temperature.

Lotz (1967) has proposed an empirical formula with three free parameters as a representation of experimental results on the electron impact ionization cross-section. More recently, Arnaud & Rothenflug (1985) have presented a representation of Brook et al. (1978) measurements of the cross section for O atoms. Comparisons between their formula with Lotz (1967) formula show that there is a small difference, which is unimportant to the goals of the present paper. In our calculations we use the more simple formula of Lotz (1967):


Here a,b,c, [FORMULA],[FORMULA], [FORMULA],[FORMULA] are constant numbers. For oxygen (resp. hydrogen) [FORMULA],[FORMULA]. [FORMULA] and [FORMULA] are functions of the electron temperature: [FORMULA], [FORMULA], and [FORMULA].

We assume that the plasma picks up the new oxygen ions immediately after their creation by ionization processes, i.e. ionized atoms acquire immediately the velocity and the temperature of the solar wind. In these conditions, the number density of ions obeys the continuity equation. The ionization balance that probably prevails in the unperturbed medium determines the number density of oxygen ions at the outer boundary.

The boundary conditions for the proton number density, the bulk velocity and the Mach number of the solar wind at the Earth's orbit are taken as [FORMULA].

In the unperturbed LIC we use [FORMULA] = [FORMULA], [FORMULA] K. These values are close to the most recent determinations of interstellar He parameters obtained by Witte et al. (1996) with the GAS instrument on Ulysses. These authors give an interstellar helium velocity [FORMULA] km s-1 and a helium temperature of [FORMULA] K. Our numerical experiments show that the increase of the interstellar temperature up to 6700 K and the interstellar velocity up to 25.6 km s-1, that correspond to the temperature and velocity deduced from ground-based and UV spectra of nearby stars observations (Lallement, 1996), doesn't change the structure of the interface significantly.

The interstellar H atom number density [FORMULA] has been deduced by Gloeckler et al. (1997) using the new SWICS pick-up results and an interstellar HI/HeI ratio of [FORMULA] (the average value of the ratio towards the nearby white dwarfs).

Unfortunately there are no direct ways to measure the circumsolar interstellar electron (or proton) density. There have been measurements of the average electron density in the LIC toward nearby stars. However, resulting densities range from 0.05 (-0.04,+0.14) [FORMULA] up to 0.3 (-0.14, +0.3) [FORMULA] depending on the ions used for the diagnostics or on which line-of-sight is probed (e.g., Lallement & Ferlet, 1997). The most precise, temperature independent value is 0.11 cm -3 toward the star Capella (Wood & Linsky, 1997). In addition, what is measured is always averaged over large distances, while the ionization degree in the local interstellar medium is very likely highly variable and out of ionization equilibrium (e.g., Vallerga, 1998). Therefore there is a need for indirect observations which can bring stringent constraints on the plasma density and on the shape and size of the interface. In our calculations we use for the interstellar proton number density [FORMULA], which is upper limit recently given by Izmodenov et al. (1999). These authors have tried to reconcile the SWICS Ulysses pick-up ion data, [FORMULA] measurements, and low-frequencies radio emissions on the basis of the two-shock heliospheric interface model, and have concluded that the most likely value for the proton density in the LIC is in the range [FORMULA].

The simulations were performed for a number density of oxygen in the unperturbed LIC [FORMULA] equal to [FORMULA]. This corresponds to a ratio of oxygen to hydrogen number densities [FORMULA] of [FORMULA] (Linsky et al., 1995). This is the most recent determination of OI/HI relative abundance measurements in the LIC. Since the influence of the oxygen on the protons and H atoms is negligible, in the case of a different O/H abundance ratio, oxygen atoms and ions density distributions can be obtained from the present results by simply multiplying by the appropriate constant.

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999