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

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4. Is it possible to detect cooling fronts?

The disk instability model predicts that the high viscosity of the hot outburst disk allows it to reach a roughly quasi stationary temperature profile with [FORMULA]. The disk should reach its hot state via a heating front travelling through the disk. Such a heating front was recovered during a normal outburst of the system OY Car by Rutten et al. ( 1992 ). On decline from an outburst a cooling front starts from the outer edge of the disk when the temperature drops below a critical value and travels inward leaving a cool disk of about 3000-4000 Kelvin behind it.

In order to get an impression of what we should see in the data and reconstructions, we created four artificial disks with cooling fronts at four different radii. We assumed that the width of the cooling front is :

[EQUATION]

as discussed in the paper of Cannizzo et al. ( 1995 ). R is the actual radius of the front and H the corresponding scale-height in the hot state and is calculated by

[EQUATION]

([FORMULA]: mean molecular weight, [FORMULA]: gas constant, T : temperature of the disk, R : Radius, [FORMULA]: mass of primary). If we take about 8000 K for the temperature, [FORMULA] for the primary mass (IP Peg parameters taken from Marsh 1988), then we end up with a value for H/R = [FORMULA] of 0.04 and a total opening angle [FORMULA] for the disk of about [FORMULA]. Therefore we chose for our model a disk opening angle [FORMULA] of [FORMULA] (i.e. [FORMULA] up and down) and also for our reconstructions. An error in temperature would yield a larger scaleheight and therefore an increased disk opening angle. The scaleheight scales with [FORMULA] and an error in temperature of a factor of 2 would change the opening angle by less than a factor of 2. The influence of errors in disk opening angle is discussed in the beginning of Sect. 5 and also in Sect. 6.2.. For this model [FORMULA] is equal to [FORMULA]. We assumed a constant 4000 Kelvin temperature profile "behind" the front and a [FORMULA] shape for the quasi stationary part of the disk, dropping down to 8000 Kelvin where the front has its inner edge. The temperature profile inside the front is assumed to be linear in the log-R/log-T diagram. Fig. 2 presents the synthetic light curves in colours UBVRI. The front radii choosen are 0.2, 0.3, 0.4 and 0.5 [FORMULA] respectively. Fig. 2 shows the log-T/log-R reconstruction in the V-band made with the above mentioned light curves, overplotted with the original profiles. The three inclined solid lines correspond to theoretical effective temperature profiles of stationary disks, depending on mass transfer rates of [FORMULA], [FORMULA] and [FORMULA] per year.

[FIGURE] Fig. 2a-d. UBVRI plot of light curves corresponding to artificial disks with cooling fronts at radius 0.2, 0.3, 0.4 and 0.5 [FORMULA] (from top to bottom at the left side). Gaussian noise was added in order to get a signal to noise ratio of about 50. log-T/log-R plots of reconstructed brightness temperature profiles in colour V for different front radii, corresponding to the light curves are plotted on the right side. The original profiles are overplotted as solid lines. The three inclined straight solid lines correspond to theoretical effective temperature profiles for steady-state disks, depending on mass transfer rates of [FORMULA], [FORMULA] and [FORMULA] per year. Points correspond to individual pixels, crosses connected by lines correspond to the mean value of pixels with same radius.

In the reconstruction plots it can be seen that the shape of the quasi stationary part is reconstructed very well, also the inner edge of the cooling front, where the front matches the hot part of the disk. The position of the outer edge of the front, at the transition to the cool flat part of the disk is not very well reconstructed, because the steepness of the interface between these two regions is too different and the entropy smoothes out the sudden jump. Therefore the boundary region between them can not be reconstructed very well.

Fig. 3 C and D shows that an increase of flickering does not change the result remarkably. The only effect is that the edges of the front are smeared out. Nevertheless quasi-stationary parts in the disk should be recovered by this method! The feature of the front is getting more pronounced as the front radius becomes smaller. Note that the true temperature profile near the outer edge of the disk is not reliably reconstructed for high front radii. This is due to the closeness of the disk edge and the front and the smoothening of the sharp front structure.

[FIGURE] Fig. 3a-d. A,B Influence of different steepness of the disk on the reconstruction and on the light curves. Two UBVRI light curves corresponding to two disks of different steepness and their reconstructed V-band profiles are shown. The steepness of the disks was A [FORMULA] and B [FORMULA] respectively. C,D Influence of flickering on the reconstuction. Two UBVRI light curves with C 10 and D 20 percent flickering and their reconstructed V-band profiles are shown. The original profiles are overplotted as solid lines. The three inclined straight solid lines correspond to theoretical effective temperature profiles of stationary disks, depending on mass transfer rates of [FORMULA], [FORMULA] and [FORMULA] per year.

The influence of different steepness of the disk in the log-T/log-R diagram to the light curves and the reconstructions is shown in Fig. 3 A and B. Note, that in general a flat profile causes V shaped eclipse features in the light curve and a more steeper (more like [FORMULA]) profile makes the eclipse features in the light curve more U-shaped.

The simulations show that the eclipse mapping method is able to detect the predicted cooling fronts and to measure the speed at which they move inward during the decline from outburst. However, structures are more pronounced at smaller radii. Furthermore it is found that a V-shaped eclipse feature in the light curve corresponds to a flat temperature profile of the reconstructed disk. Let's now have a look at the real outburst data and the corresponding reconstructions.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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