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Astron. Astrophys. 327, 1023-1038 (1997)

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Appendix A: the eclipse mapping method. Improvement and simulations

Anisotropic emission is clearly evident as orbital humps in the light curve and must be treated in some way because the disk eclipse model predicts constant brightness outside the eclipse. The standard approach (Horne 1985 , Rutten et al. 1992) simply interpolates the out-of-eclipse light curve across the eclipse phases. The resulting uneclipsed light curve is used to divide out the orbital modulations. This simple treatment fails for quiescent dwarf novae, where the sharp white dwarf and bright spot eclipse features clearly show that the bright spot produces the anisotropic radiation while the white dwarf does not. To cope with this problem, light curve decomposition techniques have been employed.

Wood et al. ( 1989) did a light curve decomposition by successively subtracting the white dwarf and the hot spot. This procedure needs a very good signal to noise ratio and a good knowledge of the contact phases of the white dwarf and the hot spot, and therefore can be applied to a few objects only. In the IP Peg system only the egress timings of white dwarf and hot spot are known. Ingress of the white dwarf and hot spot occurs simultaneously and is therefore not clearly separable in UBVRI passbands.

The program package LFIT, developed by Horne and Marsh, (Horne et al. 1994), provides another possibility to decompose the light curve. It makes an automatic decomposition of a light curve by fitting a simple CV model in a multi parameter space, using an amoebafit. This method is still under improvement.

All these methods rely on assumptions about the eclipsed part of the anisotropic lightsource, which can only be obtained by some sort of extrapolation or guess. In order to avoid this problem we developed a method which does not depend on a light curve decomposition. Our method attributes the out-of-eclipse modulation to an azimuth-dependent brightness distribution around the outward-facing rim of the accretion disk.

In order to model anisotropic light curve structures we introduced a "ribbon", wrapped around the disk. Let the disk surface be a cone of fixed opening angle H/R, and add a "ribbon" of height 2H on a cylindrical surface at the outer disk radius [FORMULA]. This results in a concave shape of the disk, which is described by a disk opening angle. Due to the orbital motion and the inclination angle, the additional pixels on the ribbon add a varying contribution to the overall intensity, thereby allowing to map anisotropic light curve structures.

To test whether this method could correctly map anisotropic light curve structures, we used a synthetic map with three hot spots, one located a the rim of the disk, the second one at the same azimuth on the ribbon and the third one half way from the white dwarf on the disk surface. Fig. 10C shows the synthetic map in colour V in the upper left panel of the four subpanels. In Fig. 10A (left panel) the brightness temperature as function of disk radius derived from the synthetic map in colour V is plotted (points). The mean values at constant radii are marked by '+' and connected by lines. Points are related to individual pixels. The brightness temperature of the ribbon as a function of azimuth is shown in the right panel of Fig. 10A.

[FIGURE] Fig. 10. A,B log-T/log-R plots for the radial temperature distributions and log-T/angle plots for the temperature distributions along the rim. A Artificial map with steady state profiles and two Gaussian spots added to it's surface. B Reconstruction of the disk A, using the light curve corresponding to A. C 3-D plots of the intensity distribution in color-V for the artificial map (upper left panel) and reconstructions made with different default images. The most-uniform reconstruction is shown in the lower left panel, the smoothest and the most-nearly-axissymetric reconstruction is plotted in the upper right and lower right panel.

From the synthetic maps an artificial light curve was calculated whereby Gaussian noise was added. The new method was applied to reconstruct the original disk maps. Fig. 10B displays the results of the reconstruction. The log-T/log-R plot (left panel of Fig. 10B) shows that all disk structures are somewhat smeared out and diminished in peak intensity. This is due to the maximum entropy fit which tries to distribute the pixel intensity as close as possible to the default map. The ribbon accounts for all anisotropic features, therefore the spot on the ribbon is reconstructed very well (see Fig. 10B right panel).

In reality there may also be anisotropic light curve variations due to spots, irradiated or heated regions on the surface of the secondary as well as flickering in the disk or anisotropic emission due to shear effects and turbulence (Horne & Marsh 1986 , Horne 1995). This cannot be mapped correctly by the ribbon. It is left to a future work to include the secondary and anisotropic effects in the reconstruction algorithm.

Another source of error is the approximation of a cylindrical outer-disk rim. In reality the disk rim shape may be strongly affected by the impact of the gas stream. We know this to be true in quiescence because the phase of hump maximum in the quiescent lightcurve is different from the azimuth of the bright spot as inferred from eclipse ingress/egress timings, implying that the spot radiates most strongly in a direction that is not radially outward (Wood 1989). We may hope that during decline from an outburst the stream impact is not such a strong perturber of the disk rim as it is in quiescence, so that our cylindrical approximation may be adequate.

Appendix B: the influence of the default map on the reconstruction

In order to investigate the influence of the default map we reconstructed disk maps from the same light curve, using different default maps. Fig. 10C shows the results. The upper left panel of Fig. 10C shows the original map. The panel in the lower left shows the so called most-uniform (MUM) result. Top right and bottom right panel show the smoothest (SM) and the most nearly axissymmetric (MONA) result respectively.

The most uniform reconstruction uses a default map which is totally flat and whose value is the average value of the actual image. The smoothest reconstruction is obtained by replacing the default image after each convergence by a gaussian smeared version of the reconstructed image. The most nearly axissymmetric reconstruction is obtained by replacing the default image after each convergence by an azimutaly smeared version of the reconstructed image.

The data do not completely determine the true location of the eclipsed light at a particular phase, so the eclipse mapping algorithm distributes this light along the ingress and egress arcs of the disk (edges of the secondary's shadow over the disk at a particular phase) to maximize the entropy, whereas the true position of the origin of the light is where these arcs are intersecting. This leads to artificial ghost-like structures in the reconstruction. Light can be accumulated at those positions where two arcs of different light spots are intersecting. This is most prominent in the MUM-map. The SM-image represents the smoothest solution of the problem. The ingress-egress arc problem still occurs, but the ghost-like structures are slightly suppressed and not so prominent. Finally the MONA-picture is the solution which shows the less azimutal structure and a maximum of radial information. By maximizing the entropy along azimutal rings, any non-centric localized spots are smeared out azimutaly resulting in artificial rings.

It is also possible to use a weighted mean of different default maps (Spruit 1994). In order to investigate the influence of such a weighted mean default map, simulations with varying SM/MONA ratios were done. The quality of these reconstructions was measured by a quality number which is described by Baptista & Steiner ( 1993). This quality number measures the similarity between the reconstruction and the original image. Highest entropy was obtained by using a 20 percent SM and 80 percent MONA default image, whereas a ratio of 70 percent SM and 30 percent MONA resulted in the best ribbon reconstruction. Highest quality in disk reconstruction was reached with a 100 percent MONA default image.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998