Astron. Astrophys. 327, 392-403 (1997)

2. Model equations

The dynamics of the system is constrained by the overall mass, momentum and energy equations of the background plasma supplemented by the second moment of the cosmic ray transport equations (called the diffusion equations hereafter). The three steady-state hydrodynamic equations for the gas density , velocity v and pressure P take the form:

where , , , are the GCR and ACR pressures and energy fluxes, and are the gas and magnetized energy fluxes, and the Lorenz force is:

The model takes into account the plasma interaction with the neutral interstellar matter i.e. the photoionisation of the neutral H and He and the charge exchange between the thermal ions and atomic hydrogen. While the process of charge exchange with hydrogen does not affect the flux of mass, the photoionisation introduces a source term in the continuity equation. According to the recent paper by Lee (1995), the source terms arising from this interaction upstream of the shock are:

where is the proton mass, AU, is the charge-exchange cross-section between neutral hydrogen atoms and protons.

In the downwind region, where the flow density increases, photoionization can be neglected and charge-exchange becomes the only important ionization mechanism. It does not contribute to the mass source but, obviously, removes momentum and energy from the flow. Following Holzer (1972) and Lee (1995) we have:

with . The source terms (5)-(7) and (8)-(10) are approximate (see Pauls et al.1995 for the exact forms). The approximation is quite good in the upwind region, but overestimates the momentum and energy transfer to the subsonic solar wind flow in the heliosheath.

We have included the magnetic field into the model. In the presence of the magnetic field the position of the termination shock for a given interstellar field pressure may be closer to the Sun, as described by the Axford-Cranfill effect (Nerney et al., 1991). However, it has been shown recently (Ziemkiewicz 1994, 1995) that the magnetic field has a weak influence on the termination shock position and on the shock structure. On the other hand, in the outer heliosphere near the equatorial plane, as , the magnetic effects become important in the far downwind region, where the velocity value decreases (Nerney et al. 1993).

The cosmic rays are treated as a massless fluid, representing pressure and energy flux (Axford et al. 1982, Webb & McKenzie 1984). The galactic cosmic rays, present everywhere in space, are scattered by turbulent fields travelling in the background flow and diffuse with the diffusion coefficient K. The pressure of the galactic cosmic rays is described by:

The acceleration of charged particles is a mechanism that plays a crucial role in collisionless shock physics (Jones & Ellison, 1993) and it is assumed here that at the solar wind termination shock the anomalous component cosmic rays (ACR) are produced. This population, produced by the shock itself, carries away some part of the background gas energy flux. The ACR diffusion equation has thus the following form:

where the source term Q (Webb et al. 1985), that vanishes everywhere except the shock, describes the injection of the gas energy flux into the ACR energy flux. The energy fluxes of the gas and cosmic rays are defined by:

The specific heats ratio is assumed for the background gas; for the cosmic rays we take and .

In the above equations the terms descibing the magnetic effects can be written (in the spherically symmetric case) as:

for , or:

for . Equation (14) describes the magnetic variables in the ecliptic plane, and (15) in the polar heliosphere.

The conservation laws at the shock (here and in what follows the subscripts 1,2 denote upwind and downwind quantities, respectively and the brackets [.] denote the jump at the shock) are:

where it is assumed that a part of the solar wind energy flux is spent at the shock to produce the ACR, i.e. the term represents the source term Q in Eq.(12). Since more energetic GCRs do not feel the solar wind shock, both their pressure and energy flux are continuous across the shock.

Defining the shock compression ratio as

the conservation laws at the shock yield the following cubic equation for q:

where: , the upwind Mach number , and c, are the sound and the Alfvén speeds.

In the spherically symmetric heliosphere, the set of three hydrodynamic equations (1)-(3) and two diffusion equations (11) and (12) constitute a seventh-order system for , v, P, , , , and . Following Ziemkiewicz (1994) we introduce dimensionless variables:

With the help of formulae (25), Eqs. (1)-(3) and (11)-(12) take the form:

The above equations must be complemented with initial or boundary conditions. In earlier papers attacking a simplified version of the problem (without source terms and for a polytropic gas), an iterative method of solving the boundary value problem has been employed (Ko et al., 1988). In this paper we use a different method, in which Eqs. (26)-(30) are considered as an initial value problem with the values of all unknowns set at 1 AU. In the upwind region we follow the solutions until the assumed position of the shock is reached. The conservation equations together with a prescribed value of yield the initial values in the downwind region, which are then used to integrate the equations to the heliopause. In this way, we obtain a two-parameter () family of solutions. Any solution that: (i) gives the right value of "at infinity", (ii) yields an appropriate pressure value at the heliopause (), and (iii) shows the right (i.e. decreasing and non-negative, see Fig. 1) profile of the anomalous cosmic rays pressure in the downwind region is accepted. There is another reason to solve our system of differential equations as an initial value problem. In some cases the solutions may pass through a critical point or a number of critical points. We have not followed the full critical point analysis that would provide one with the topological nature of the solutions, as it was done for a simpler case by Zank (1989). In our numerical approach Eqs. (26)-(30) either become singular for some models, when one of denominators approaches zero, or provide meaningless solutions starting from the point where a numerator has a (double) root. If it happens in the velocity equation, for example, this variable first decreases and then, after passing through its minimum value, starts to increase. Soon, all of the integrated functions are affected. Such cases can be easily identified and diagnosed in the initial value formulation, but would be more difficult to handle by any of the BVP (boundary value problem) solvers.

Fig. 1. Three types of the ACR pressure profile: (a) corresponds to a source not at the shock, but far in the downwind region (small ), (c) yields negative values downstream of the shock and is, therefore, unphysical (large ), (b) is the sppropriate one (medium ). The dotted line denotes the shock position. The units on both axes are arbitrary.

© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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