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Astron. Astrophys. 358, 759-775 (2000)

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2. MHD models and scaling laws

The plasma flow in a comet can be described by the equations of ideal magnetohydrodynamics (MHD) with source terms (see e.g. Schmidt & Wegmann 1982; Wegmann 1995). One can combine the gas production rate G, the outflow velocity w from the nucleus, the mean mass [FORMULA] of a cometary particle, the ionisation rate [FORMULA] and the solar wind mass flux [FORMULA] to an interaction length scale


which determines the size of the region where the solar wind interacts with the comet.

We adopt a coordinate system [FORMULA] centered at the nucleus with x pointing towards the Sun. The y axis is so that the interplanetary magnetic field (IMF) is parallel to the xy-plane, which we call the IMF plane. The xz-plane is called the 'perpendicular plane'. Mass density [FORMULA], number density n and velocity [FORMULA] are normalized by the corresponding solar wind quantities [FORMULA] and [FORMULA] to


When the spatial coordinate [FORMULA] is measured in units of [FORMULA], and time t is counted in units [FORMULA] then under certain simplifying assumptions which are detailed in the paper by Wegmann et al. (1999), the equations governing the cometary plasma flow, rewritten in the normalized variables of (2), reduce to equations which do not contain any further parameters.

The main consequence of this is a scaling law : All physical quantities scale with the corresponding solar wind quantities, all lengths scale with the interaction scale length [FORMULA], and time intervals scale with [FORMULA]. This scaling can be used to derive from observations information about the comet (in particular the production rate G) and the solar wind (Wegmann et al. 1999 , 2000).

In this paper we are mainly interested in the brightness which is directly proportional to the column density [FORMULA] of cometary ions. We recall (Wegmann 1995) that the density [FORMULA] and the column density [FORMULA] of cometary ions scale like


with functions [FORMULA] and [FORMULA] which are independent of the solar wind and of the cometary parameters. It has been demonstrated by Wegmann et al. (1999) that along a line through the nucleus perpendicular to the sun-comet line,


with the nondimensional coordinate [FORMULA]. This means that at a fixed point on this line the density as well as the brightness


vary like the inverse of the dynamic pressure of the solar wind when all other parameters are kept fixed.

We have calculated models with parameters listed in Table 1. The parameters labeled 'slow1' and 'fast1' are for slow and fast solar wind data taken from Gosling et al. (1978), Fig. 2, eight hours before and after the interface of a corotating interaction region (CIR). The data 'big1' are for a big Halley type comet with solar wind data taken from Gosling et al. (1978), Fig. 1, two days before a CIR. The data 'slow2' and 'fast2' are for the oblique shock model (S4) calculated by Wegmann (1995). We calculate all models with photo-ionisation only, with a constant rate [FORMULA] s-1, an outflow speed [FORMULA] km/s, and a mean mass [FORMULA] amu of a cometary molecule or atom. Only the shock model is calculated with the effect of charge exchange included (Wegmann 1995).

[FIGURE] Fig. 1. Resolution of the discontinuous transition from 'slow1' to 'fast1' flow. Density, pressure and velocity profiles after 1 s.

[FIGURE] Fig. 2. Logarithms of the column densities [cm-2] in a projection onto the IMF plane 0, 1, 2, 3, 4, 5 hours after the HSS entered at the lower left corner. The length scale is 100 000 km.


Table 1. Model parameters

The small models [FORMULA] and [FORMULA] s-1 are calculated on a grid which covers 300 000 km in front of the nucleus and to the sides, and 700 000 km along the tail. The big comet with [FORMULA] s-1 is modeled on a grid five times as large.

We use the same numerical method as in our previous paper (Wegmann 1995). The code is based on a second order Godunov method using an approximate Riemann solver for the MHD equations. Spurious magnetic monopoles generated by numerical errors are removed in each time step (Schmidt-Voigt 1989). The code is able to capture shocks. The flow is calculated on a non equidistant grid with finest resolution of 8000 km near the nucleus for the small models. Advanced codes (e.g. Gombosi et al.1994) for the solar wind - comet interaction use also second order Godunov methods with an approximate Riemann solver. Adaptive grid refinement improves the accuracy of stationary models. These codes have not yet been used for time-dependent calculations.

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© European Southern Observatory (ESO) 2000

Online publication: June 8, 2000