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3. An example

A number of controlled experiments have been performed to test the quality of the light curve reconstruction routine described above. First a light curve is calculated assuming some set of binary star parameters and a pre-defined brightness distribution across the disk and the star. Next, the brightness distribution over the grid is reconstructed from the light curve, and the result is compared with the original brightness distribution.

From the very large number of grid geometries and brightness structures that can be studied, only one, but fairly realistic example is presented here. A binary star mass ratio of unity, an orbital inclination of 80 degrees (i.e. an eclipsing system), and an accretion disk with an opening angle of 5 degrees (as measured from the orbital plane) have been assumed. The disk extends out to a radius of 80 percent of the distance to the inner Lagrange point. Its outer rim is rounded off, not only to appear more realistic, but also to reduce jumps in the light curve due to the finite size of surface elements. Note that this definition of the disk surface is rather arbitrary, but suitable for this example. For more realistic descriptions of accretion disk surfaces see Warner (1995), and references therein.

The radial brightness distribution of the disk is based on the standard accretion disk model, with tex2html_wrap_inline796 for distances from the center of the disk, R, large compared to the white dwarf radius (see Frank et al. 1992 for the exact formula). On the secondary star's surface the effects of irradiation by the disk are represented by a luminous inner hemisphere and a dark outer hemisphere.

Figure 2 (click here) shows the resulting light curve. Clearly visible are the effects of ellipsoidal variation of the irradiated hemisphere, resulting in large amplitude brightness changes throughout the orbital cycle. Furthermore, a primary eclipse of the disk and a weak secondary eclipse of the star are seen. This light curve is fitted to reconstruct the intensity distribution using the same grid geometry. From the resulting intensity distribution a light curve is constructed and also shown in Fig. 2 (click here); it is nearly indistinguishable from the original light curve. The fitted brightness distribution also correlates closely with the input model, as can be seen (qualitatively) from the grey scale representation in Fig. 3 (click here), which shows the bright accretion disk with the fainter, thick outer rim, and the bright inner hemisphere of the star. The boundary between the inner and outer hemisphere of the star is not as sharply marked in the reconstruction as in the original input model. How well the original model is reproduced depends mainly on the size of errors that are adopted for the light curve.

  figure230
Figure 2: Light curve of an eclipsing CV where the secondary star contributes strongly to the total light, but only from its inner hemisphere. This crudely mimics irradiation of the secondary star by the accretion disk. The thin curve traces the computed light curve, while the bold curve (nearly indistiguishable at most phases) shows the fit to that light curve. The uncertainties on the light curve as assumed in the fit are shown as vertical bars in the lower part of the plot

  figure235
Figure 3: The intensity structure of the accretion disk and the star as derived from the fit to the light curve shown in Fig. 2 (click here). Dark structures in the plot indicate high intensities. Clearly visible is the bright inner hemisphere of the star, and the thick accretion disk with the bright inner disk and the fainter outer rim

As another example of the versatility of the 3D light curve program a light curve is calculated, shown in Fig. 4 (click here), for the case of an opaque but thin accretion disk which is tilted relative to the orbital plane. The top of the disk, as seen from the observer, is tilted pointing away from the secondary star. The tilt angle is taken to be 10 degrees, while the orbital inclination is 75 degrees. How the secondary star and the accretion disk each contribute to the light curve is also shown in Fig. 4 (click here). The uniformly bright secondary star produces elipsoidal variations. The tilted accretion disk produces an orbital variation due to its changing aspect angle, and it is responsible for a fairly pronounced secondary eclipse because a large portion of the star is obscured. At this orbital inclination the bright center of the accretion disk is just being eclipsed at orbital phase zero.

In the same Fig. 4 (click here) the light curve is also plotted for the situation where the disk is assumed optically thin. In that case there are no orbital variations from the tilted disk, and the primary eclipse is much more pronounced. Such comparison offers a way to gauge the optical thickness of accretion disks observationally.

As a final example a model light curve is shown in Fig. 5 (click here) for an eclipsing CV consisting of a thick disk with a bright spot on its outer rim, and a bright white dwarf. These features result in an orbital hump, and in the deep eclipse of the white dwarf and of the bright spot that are so well known from dwarf novae in quiescence.


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