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Astron. Astrophys. 356, 490-500 (2000)

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2. The 3d cataclysmic variable model

Our computer code CVMOD generates N small surface elements (convex quadrangles, some of which are degenerated to triangles), which represent the surfaces of the individual components of the CV (WD, secondary, accretion stream) in three-dimensional space (Fig. 1). Using simple rotation algorithms, the position of each surface element [FORMULA] can be computed for a given orbital phase [FORMULA].

[FIGURE] Fig. 1. Overview of our 3d-Model of a polar. [FORMULA] marks the inner lagrangian point, SR the position of the stagnation or coupling region.

The white dwarf is modelled as an approximated sphere, using surface elements of nearly constant area (G"ansicke et al. 1998). The secondary star is assumed to fill its Roche volume. Here, the surface elements are choosen in such a way that their boundaries align with longitude and latitude circles of the Roche surface, taking the [FORMULA]-point as the origin.

We generate the surface of the accretion stream in two parts, (a) the ballistic part from [FORMULA] to SR, and (b) the dipole-part from SR to the surface of the white dwarf.

(a) For the ballistic part of the stream, we use single-particle trajectories. The equations of motion in the corotating frame are given by


Eq. (3) has been added to Flannery's 1975 set of two-dimensional equations. [FORMULA] is the mass fraction of the white dwarf, [FORMULA] and [FORMULA] are the distances from the point [FORMULA] to the white dwarf and the secondary, respectively, in units of the orbital separation a. The coordinate origin is at the centre of gravity, the x-axis is along the lines connecting the centres of the stars, the system rotates with the angular frequency [FORMULA] around to the z-axis. The velocity [FORMULA] is given in units of [FORMULA], [FORMULA] is the initial velocity in the [FORMULA] point.

If [FORMULA], the trajectories resulting from the numerical integration of Eqs. (1) - (3) are restricted to the orbital plane. However, calculating single-particle trajectories with different initial velocity directions (allowing also [FORMULA]) shows that there is a region approximately one third of the way downstream from [FORMULA] to SR where all trajectories pass within very small separations, corresponding to a striction of the accretion stream.

We define a 3d version of the stream as a tube with a circular cross section with radius [FORMULA] centred on the single-particle trajectory for [FORMULA].

(b) When the matter reaches SR, we switch from a ballistic single-particle trajectory to a magnetically forced dipole geometry. The central trajectory is generated using the dipole formula [FORMULA], where [FORMULA] is the angle between the dipole axis and the position of the particle [FORMULA]. This can be interpreted as the magnetic field line F passing through the stagnation point SR and the hot spots on the WD. Knowing F, we assume a circular cross section with the radius [FORMULA] for the region where the dipole intersects the ballistic stream. This cross section is subject to transformation as [FORMULA] changes. Thus, the cross section of the stream is no longer constant in space but bounded everywhere by the same magnetic field lines.

Our accretion stream model involves several assumptions: (1) The cross section of the stream itself is to some extent arbitrary because we consider it to be - for our current data, see below - essentially a line source. (2) The neglect of the magnetic drag (King 1993; Wynn & King 1995) on the ballistic part of the stream and the neglect of deformation of the dipole field cause the model stream to deviate in space from the true stream trajectory. While, in fact, the location of SR may fluctuate with accretion rate (as does the location of the Earth's magnetopause), the evidence for a sharp soft X-ray absorption dip caused by the stream suggest that SR does not wander about on time scales short compared to the orbital period. (3) The abrupt switch-over from the ballistic to the dipole part of the stream may not describe the physics of SR correctly. This discrepancy, however, does not seriously affect our results, because the eclipse tomography is sensitive primarily to displacements in the times of ingress and egress of SR which are constrained by the absorption dip in the UV continuum (and, in principle, in soft X-rays). The [FORMULA] time bins of our observed light curves correspond to [FORMULA] in space at SR. Hence, our code is insensitive to structure on a smaller scale. In fact, the smallest resolved structures are much larger because of the noise level of our data. While our approach clearly involves several approximations, it is tailored to the desired aim of mapping the accretion stream from the information obtained from an emission line light curve.

In our current code, we restrict the possible brightness distribution so that for each stream segment, which consists of 16 surface elements forming a section of the tube-like stream, the intensity is the same, i.e. there is no intensity variation around the stream. For the current data, this is no serious drawback, since we only use observations covering a small phase interval around the eclipse. Our results refer, therefore, to the stream brightness as seen from the secondary. From the present observations we can not infer how the fraction of the stream illuminated by the X-ray/UV spot on the WD looks like. The required extension of our computer code, allowing for brightness variation around the stream, is straightforward. The present version of the code is, however, adapted to the data set considered here.

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© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000