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Astron. Astrophys. 317, 193-202 (1997)

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3. Formulation of the problem

We are interested here in the distribution of oxygen of interstellar origin in the computational domain shown in Fig. 1. The flow of atoms must be described kinetically, because, as already said, the free path length of the neutral atoms is of the order of the size of the interaction region. In order to obtain the kinetic distribution function, Boltzman's equation must be solved:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the oxygen atom distribution function, [FORMULA] the local Maxwell distribution function of the protons with the hydrodynamical values [FORMULA], [FORMULA], [FORMULA] from the model described above; [FORMULA] is the hydrogen atom distribution function, [FORMULA] the local Maxwell distribution function of oxygen ions with the hydrodynamic values described later, [FORMULA] [FORMULA] [FORMULA] [FORMULA] the individual velocities of oxygen atoms, protons, oxygen ions and hydrogen atoms, [FORMULA] the charge exchange cross-section of an O-atom with a proton, [FORMULA] the charge exchange cross-section of an hydrogen atom with an oxygen ion, [FORMULA] the photoionization rate, [FORMULA] the mass of the oxygen atom, [FORMULA] the solar gravitational force.

Eq. (1) takes into account the following processes in which oxygen is involved:

a) the solar gravitation: the potential energy is [FORMULA], where [FORMULA], [FORMULA] is the universal gravitational constant, and [FORMULA] - the mass of the Sun.

b) the charge exchange of O-atoms with protons: [FORMULA], with a charge exchange rate [FORMULA], given by formula: [FORMULA], [FORMULA] where [FORMULA] is the relative atom-proton velocity, and [FORMULA] [FORMULA] is the charge exchange cross section. [FORMULA], [FORMULA] are constants, [FORMULA], [FORMULA] (Stebbings & al, 1964, Banks & Kockarts, 1973). The relative velocity is measured in cm/s .

c) the reverse charge exchange of hydrogen atoms with oxygen ions: [FORMULA]. The cross section for this reaction is connected with the cross section of the direct charge exchange by the following relation:

[EQUATION]

d) the photoionization: the rate is given by:

[EQUATION]

(Banks and Kockarts, 1973), where [FORMULA] is the photoionization rate at the Earth's orbit and [FORMULA] - is 1 A.U.

Collisions between oxygen atoms and electrons are not taken into account in the present model. These collisions may be significant in the heliosheath where solar wind electrons are hot, and may ionize substantially the oxygen. Following the arguments of Fahr, Osterbart and Rucinski (1995) and the Lotz (1967) formula, we have estimated this effect for an interstellar plasma density of 0.1 cm-3. An upper limit of 9% extinction between the termination shock and the heliopause can be derived, assuming a 2.10 6 K temperature everywhere in the heliosheath. This upper limit is high enough to deserve further work, and we are considering the possibility of modifying the simulation to include it in the future.

Eq. (1) contains the oxygen ions distribution function [FORMULA]. This distribution function obeys a relation of the type:

[EQUATION]

which is expressed according to the plasma structure.

Actually, we do not calculate explicitly the distribution function [FORMULA], but its first moments, i.e. the number density of oxygen atoms [FORMULA], the bulk velocity [FORMULA], the 'temperature' (average kinetic energy) [FORMULA], and the pick-up ions source term [FORMULA] defined by:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

In the last definition, [FORMULA] is the number of newly created ions per time unit, or equally the number of photoionized or charge- exchanged oxygen atoms per time unit.

Assuming that plasma picks up the new ions instantly, 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:

[EQUATION]

The boundary condition for the distribution function [FORMULA] is the Maxwell distribution:

[EQUATION]

(where [FORMULA] ).

The number density of oxygen ions at the outer boundary is determined by the ionization balance which prevails in the unperturbed medium:

[EQUATION]

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