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

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4. Tests

To test the quality and the limits of our method, we produce synthetical test light curves with different noise levels ([FORMULA]). We then try to reproduce our initial intensity distribution on the stream from the synthetic data. Two tests with different intensity distributions are performed. For both tests, the geometry of the imaginary system IM Sys is chosen as shown in Table 1. The phase coverage is [FORMULA] with 308 equidistant steps, which is identical to the real HST data of UZ For which we use below for a first application.


[TABLE]

Table 1. System geometry of the imaginary system IM Sys


Additionally, we test our algorithm with a full-orbit light curve with [FORMULA] to demonstrate its capabilities if more than just the ingress information for each stream section is available.

4.1. One bright region near SR

In the first test, we set the stream brightness to 1 on the whole stream except for a small region near the stagnation point SR, where the intensity is set to 10. For this intensity distribution, we show the theoretical light curve in Fig. 3. One clearly sees the fast ingress of the small bright region at phase [FORMULA], whereas the egress occurs beyond [FORMULA] and is not covered as in the real data of UZ For.

The initial intensity map and the reconstructed map with the different noise levels are shown in Figs. 2 and 4. The reconstruction of one bright region near the stagnation point is achieved with no artifacts, not too much smearing and little noise for [FORMULA]. Even with [FORMULA], a reasonable reconstruction can be obtained, but with artifacts: The ballistic stream appears bright near the [FORMULA]-point, and an additional bright region appears on the dipole stream near the northern accretion pole.

[FIGURE] Fig. 2. Maps of the synthetic stream and its reconstructions. From top to bottom: Input data, reconstructions with [FORMULA].

[FIGURE] Fig. 3. Synthetic light curve of an accretion stream which is bright around the stagnation region. Different levels of artifical noise are added: [FORMULA]. The reconstructed light curves are shown as solid lines. The residuals are normalized so that the standard deviation [FORMULA] is 0.1 in the relative flux units.

[FIGURE] Fig. 4. Plot of the reconstruction of the synthetic stream with a bright region around SR. Solid line: Input distribution, dotted line: reconstruction with [FORMULA], dashed line: reconstruction with [FORMULA]. In the left panels (surface elements no. 0 to 104), the intensities of the surface elements on the ballistic stream are shown. In the upper right panel, the northern magnetic stream is to be found (surface elements no. 105 to 182), in the lower right panel the southern magnetic stream (surface elements no. 183 to 227).

4.2. Three bright regions on the ballistic stream

In the second test we assume a rather unphysical intensity distribution with the aim to test the spatial resolution of our mapping method: The ballistic part of the accretion stream between [FORMULA] and SR is divided into 5 sections of equal length. Alternately, the intensity on these sections is set equal to 10 and to 1, producing a `zebra'-like pattern. The intensity on the magnetically funneled stream is set to [FORMULA]. The synthetic light curve for this intensity distribution differs strongly from that in out first test (Fig. 6). Instead of one sharp step in the light curve, there are now - as expected - three steps during the ingress and three during the egress (also not visible in the selected phase interval).

In Figs. 5 and 7 we show the input and the resulting maps for the `zebra'-test. As long as [FORMULA], our algorithm achieves, as in the first test, a good reconstruction of the input map. The bright regions next to SR appear darker than in the original map, since the intensity from that region is spread over the last part of the ballistic stream and the neighbouring parts of the dipole stream, which dissapear behind the limb of the secondary star nearly simultaneously during ingress.

[FIGURE] Fig. 5. Maps of the synthetic stream and its reconstructions. From top to bottom: Input data, reconstructions with [FORMULA].

[FIGURE] Fig. 6. Synthetic light curve of an accretion stream with three bright parts on the ballistic stream. Different levels of artifical noise are added: [FORMULA]. The reconstructed light curves are shown as solid lines. The residuals are normalized so that the standard deviation [FORMULA] is at 0.1 in the relative flux.

[FIGURE] Fig. 7. Plot of the reconstruction of the synthetic stream with three bright regions on the ballistic stream. Solid line: Input distribution, dotted line: reconstruction with [FORMULA], dashed line: reconstruction with [FORMULA]. For an explanation of the plot see Fig. 4

Comparing the two tests for [FORMULA], we find that it is still possible to distinguish between the two different initial intensity distributions (one bright region and `zebra') even with such a noisy signal.

In Table 2, we list the quality parameters of the fits. The increase of [FORMULA] from [FORMULA] to [FORMULA] with the decrease of [FORMULA] shows once more how well the fit algorithm works: For the light curve with no noise, the ideal light curve has - by definition - [FORMULA]. As soon as noise is added, even the input light curve has a [FORMULA], which follows also from the definition of [FORMULA] if [FORMULA] is the real standard deviation of the data set. For the calculation of [FORMULA], one needs a sensibly chosen standard deviation [FORMULA]. Since the light curves of polars show pronounced flickering, we calculate an approximate [FORMULA] for our noisy data sets (synthetic and measured) by comparing the data with a running mean over 10 phase steps. The running mean has, thus, a reduced [FORMULA] of 1. Since the best fit to artificial data with low synthetic noise levels does better than the running mean, we obtain [FORMULA] for these light curves.


[TABLE]

Table 2. Quality of the test calculations


The entropy S of the images is always close to 1. An ideal entropic image would have [FORMULA], the actual values of [FORMULA] come very close to that. One would not expect [FORMULA], since that would imply that there is no variation in the image, i.e., all surface elements have the same brightness. The trend to higher S with lower noise is a result of the weighting of S with respect to [FORMULA] in the one-dimensional quality parameter [FORMULA] (Eq. 7). For the test calculations, we have chosen [FORMULA] constant for all noise levels. This emphasizes the reduction of [FORMULA] for low noise levels and the smoothing for high noise levels.

Our tests show that we are able to detect structures in the brightness distribution on the accretion stream with a size of [FORMULA] from data with phase coverage, phase resolution and noise level similar to that of the HST data set for UZ For (see Sect. 5.2).

4.3. Full-orbit light curve

To test the capabilities of our algorithm for data with wider phase coverage, we fit a synthetic light curve covering the whole binary orbit, computed using the same input map and geometry as in Sect. 4.1. We choose a phase resolution of 0.005 for the simulated data, corresponding to a 30 sec time resolution, and S/N=10. The result of this fit is shown in Fig. 8. Obviously, the additional phase information helps in producing a reliable reconstruction of the initial intensity distribution, which one can see by comparing the map in Fig. 8 with the [FORMULA]-map in Fig. 2. The full-orbit light curve has less phase steps than the ingress-only map, but shows the same quality of the reconstruction. On the other hand, the similarity between the two results allows us to conclude that we can rely on the reconstructions of our algorithm even if only ingress data is available, as will be the case for the HST archive data of UZ For which we use in the following.

[FIGURE] Fig. 8. Top: Full-orbit reconstruction of the `point'-brightness distribution. Compare to Fig. 2, top. Bottom: Full-orbit synthetic light curve and best-fit light curve. For clarity, two full orbits are shown.

The synthetic light curve over the full orbit shows various eclipse and projection features, which are described in detail by Kube et al. (1999).

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© European Southern Observatory (ESO) 2000

Online publication: April 10, 2000
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