## 3. Light curve fittingThe basic idea of our eclipse mapping algorithm is to reconstruct the intensity distribution on the accretion stream by comparing and fitting a synthetic light curve to an observed one. The comparison between these light curves is done with a -minimization, which is modified by means of a maximum entropy method. Sect. 3.1 describes the light curve generation, 3.2 the maximum entropy method, and 3.3 the actual fitting algorithm. ## 3.1. Light curve generationIn order to generate a light curve from the 3d model, it is
neccessary to know which surface elements In general, each of the three components (WD, secondary, accretion
stream) may eclipse (parts of) the other two, and the accretion stream
may partially eclipse itself. This is a typical hidden surface
problem. However, in contrast to the widespread computer graphics
algorithms which work in the image space of the selected output device
(e.g. a screen or a printer), and which provide the information `pixel
Once has been determined, the
angles between the surface normals of
and the line of sight, and the
projected areas of
are computed. Designating the
intensity of the surface element Here, two important assumptions are made: (a) the emission from all surface elements is optically thick, and (b) the emission is isotropic, i.e. there is no limb darkening in addition to the foreshortening of the projected area of the surface elements. The computation of a synthetic light curve is straightforward. It suffices to compute for the desired set of orbital phases. While the above mentioned algorithm can produce light curves for all three components, the WD, the secondary, and the accretion stream, we constrain in the following the computations of light curves to emission from the accretion stream only. Therefore, we treat the white dwarf and the secondary star as dark opaque objects, screening the accretion stream. ## 3.2. Constraining the problem: MEMIn the eclipse mapping analysis, the number of free parameters,
i.e. the intensity of the the default image for the surface element In Eq. (5), and
are the positions of the surface
elements The quality of a intensity map is given as where is chosen in the order of 1. Aim of the fit algorithm is to minimize . ## 3.3. The fitting algorithm: evolution strategyOur model involves approximately 250 parameters, which are the
intensities of the surface elements. This large number is not the
number of the degrees of freedom, which is difficult to define in a
MEM-strategy. A suitable method to find a parameter optimum with a
least and a maximum entropy value is
a simplified imitiation of biological evolution, commonly referred to
as `evolution strategy' (Rechenberg 1994). The intensity information
of the From this parent intensity map, a number of offsprings are created
with randomly changed by a small
amount, the so-called In contrast to the classical maximum entropy optimisation (Skilling
& Bryan 1984), the evolution strategy does not offer a quality
parameter that indicates how close the best-fit solution is to the
global optimum. In order to test the stability of our method, we run
the fit several times starting from randomly distributed maps. All
runs converge to very similar intensity distributions
(see also Figs. 10 and 12).
This type of test is common in evolution strategy or genetic
algorithms (e.g. Hakala 1995). Even though this approach is not a
statistically `clean' test, it leaves us to conclude that we find the
global optimum. Fastest convergence is achieved with 40 to 100
offsprings in each generation. Finding a good fit
() takes only on tenth to one fifth
of the total computation time, the remaining iterations are needed to
improve the smoothness of the intensity map, i.e. to maximize
© European Southern Observatory (ESO) 2000 Online publication: April 10, 2000 |