Issue 
A&A
Volume 622, February 2019



Article Number  A199  
Number of page(s)  7  
Section  Stellar atmospheres  
DOI  https://doi.org/10.1051/00046361/201833482  
Published online  20 February 2019 
Global properties of the light curves of magnetic, chemically peculiar stars as a testbed for the existence of dipolelike symmetry in surface structures^{★}
^{1}
Department of Theoretical Physics and Astrophysics, Masaryk University, Brno,
Kotlářská 2,
611 37 Brno, Czech Republic
email: jagelka@mail.muni.cz
^{2}
American Association of Variable Star Observers (AAVSO),
Cambridge,
USA
^{3}
Bundesdeutsche Arbeitsgemeinschaft für Veränderliche Sterne e.V. (BAV),
Berlin,
Germany
Received:
23
May
2018
Accepted:
8
January
2019
Context. Magnetic, chemically peculiar stars are known for exhibiting surface abundance inhomogeneities (chemical spots) that lead to photometric and spectroscopic variability with the rotation period. It is commonly assumed that the surface structures are causally connected with the global magnetic field that dominates the photospheric and subphotospheric layers of these stars. As a rule, the observed magnetic fields show a simple dipolelike geometry, with the magnetic axis being noncollinear to the rotational one.
Aims. The present study aims at detecting underlying patterns in the distribution of photometric spots in a sample of 650 magnetic, chemically peculiar stars and examines their link to the magnetic field topology.
Methods. Photometric timeseries observations from the ASAS3 archive were employed to inspect the lightcurve morphology of our sample stars and divide them into representative classes described using a principal component analysis. Theoretical light curves were derived from numerous simulations assuming different spot parameters and following the symmetry of a simple dipole magnetic field. These were subsequently compared with the observed light curves.
Results. The results from our simulations are in contradiction with the observations and predict a much higher percentage of doublewave light curves than is actually observed. We thereby conclude that the distribution of the chemical spots does not follow the magnetic field topology, which indicates that the role of the magnetic field in the creation and maintenance of the surface structures may be more subsidiary than what is predicted by theoretical studies.
Key words: stars: variables: general / stars: atmospheres / stars: chemically peculiar
Table A.1 is only available at the CDS via anonymous ftp to cdsarc.ustrasbg.fr (130.79.128.5) or via http://cdsarc.ustrasbg.fr/vizbin/qcat?J/A+A/622/A199
© ESO 2019
1. Introduction
Chemically peculiar (CP) stars are members of the upper part of the main sequence (spectral types B to F). They are generally slow rotators and are characterized by spectral peculiarities that indicate unusual atmospheric abundances of some chemical elements. This is due to the effects of radiative diffusion and gravitational settling (LeBlanc et al. 2009; Alecian & Stift 2010; Alecian 2015), which result in some elements being lifted to the outer parts of the stellar atmosphere while others settle down into the lower parts.
Some groups of CP stars (the socalled Ap/CP2 stars and the Heweak/CP4 stars; cf. Preston 1974) exhibit global magnetic fields with strengths up to several tens of kilogauss (Aurière et al. 2007) and are characterized by surface abundance inhomogeneities (chemical spots)^{1}. Flux is redistributed in these spots (e.g., Peterson 1970; Molnar 1973; Krtička et al. 2013) and, as the star rotates, the changing viewing angle results in periodic spectroscopic and photometric variations. Stars exhibiting this kind of rotational variability are traditionally referred to as α^{2} Canum Venaticorum (ACV) variables.
While a precise understanding of the processes involved in the formation of the surface structures is still missing (Krtička et al. 2013), it is believed that the magnetic field plays a decisive role (LeBlanc et al. 2009; Alecian & Stift 2010; Alecian 2015). The present study aims at detecting underlying patterns in the distribution of photometric spots and examines their link to the magnetic field topology.
2. Analysis of observations
2.1. Data characteristics
The original data for our research were gleaned from the third phase of the All Sky Automated Survey (ASAS) project, which focused on continuous photometric coverage of the southern sky and part of the northern sky with the goal of detecting any kind of photometric variability. During the third phase of the project (ASAS3), observationswere acquired in Johnson V (Pojmanski et al. 2005). In total, 11 509 variable stars brighter than V = 15 mag were cataloged by the ASAS team, of which 7310 remained unclassified. Bernhard et al. (2015) used ASAS3 data to search for new photometric variables among known mCP stars from the catalogue of Renson & Manfroid (2009) and identified 316 stars with photometric characteristics of ACV variables. Using refined search criteria, Hümmerich et al. (2016) subsequently discovered another 334 ACV variables in these data.
As basis for the present investigation, we employed the original ASAS3 data of the 650 ACV variables identified by Bernhard et al. (2015) and Hümmerich et al. (2016). We also used their ephemerides, such as the period and Julian date of photometricmaximum. ASAS3 data contain information about Julian Date, brightness using different photometric apertures, corresponding errors, and quality of measurements. We used measurements from the aperture with the lowest errors for a specific star following the methodology of Pojmanski et al. (2005). For the brightest stars (V < 9 mag), the largest aperture (6 pixels in diameter) was employed; for faint stars (V > 12 mag), we used the smallest aperture (2 pixels in diameter). Furthermore, lower quality measurements (indicated by flags “C” and “D”; for details see Pojmanski et al. 2005) were excluded from the analysis.
2.2. Principal component analysis of the light curves
According to the shape of the light curves of the mCP stars studied by Bernhard et al. (2015) and Hümmerich et al. (2016), we can divide them into simple categories: singlewave, doublewave, and symmetric or asymmetric light curves. To confirm and mathematically describe these classes, we used a principal component analysis (PCA). This method transforms the original data into a system with new coordinates, which are chosen such that they follow the most common patterns in the data. This coordinate system is based on the combination of the eigenvectors with large eigenvalues. Therefore, we can create principal components and use them to describe the light curve data as their linear combinations. For a detailed description of the modifications of PCA, we refer to Mikulášek (2007).
As PCA requires common data points for all light curves (phase points in this case), we divided all measurements for any given star into 25 phase bins. Every phase bin of each star is then represented by its average value of intensity in the V filter. Applying PCA, we acquired full sets of principal components. These components were fitted by harmonic polynomials of second order, and fit parameters and their errors were obtained. We discovered that only the first five components are meaningful, as their fitted parameters have higher values than their errors multiplied by three. The eigenvalues of these components are 1.89, 0.17, 0.16, 0.03, and 0.03. For simplicity we decided to work only with the first three principal components. These are shown in Fig. 1.
The first component represents a typical mCPstar light curve, which is a symmetric single wave with a relatively sharp maximum and very flat minimum. This lightcurve profile can be simulated by assuming one bright photometric spot centered at phase zero. The second principal component is a symmetric doublewave light curve with uneven minima. The third component represents an antisymmetric light curve with the inflection points at phases 0.0 and 0.5, expressing the antisymmetry of light curves. These three components are sufficient to adequately describe all light curves, as indicated by Fig. 2, which illustrates the binned light curves of the first ten stars of our sample overlaid with the smoothed data obtained as a linear combination of the first three principal components.
2.3. Phenomenological model
As has been pointed out, PCA shows that the model function can be satisfactorily described by only the first three principal components, with the first two components being symmetrical and the third one antisymmetrical with regard to phase zero. All of the components can be well approximated by a harmonic polynomial of the second order (see also Mikulášek et al. 2007a; Bernhard et al. 2015). This allows us to express the ASAS3 mCPstar light curves in a simple form with sufficient accuracy (which is quantitatively evaluated in Fig. 3), using only three free parameters describing their shape:
Fig. 1. First three principal components. The first component (top panel) represents a singlewave light curve, the second component (middle panel) a doublewave light curve, and the third component (bottom panel) expresses antisymmetry. The points are real principal components and the solid lines are fits using Eq. (1) with A_{3} = 0 for the first two components and A_{1} = A_{2} = 0 for the thirdone. 

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Here, F represents the apparent magnitude in the V filter and φ is the phase. We used the period values (P) and times of maximum (M_{0}) derived by Bernhard et al. (2015) and Hümmerich et al. (2016). Calculations were complicated by the fact that, employing this model function, we need to work with a nonlinear regression. However, we can assume that the nonlinear parameter φ_{0} has a value near zero, as the light curves have their photometric maxima centered at phase zero, defined by the parameter M_{0}.
In this model, the parameters A_{1}, A_{2}, and A_{3} represent, respectively, the semiamplitudes of the singlewave, the doublewave, and the antisymmetric terms of the light curve. The special form of the antisymmetric part results from two factors: (a) the first derivative in zero phase has to be zero to ensure the position of the maximum; (b) the value of parameter A_{3} is modified by the coefficient , which normalizes it to the other two parameters. The normalization has been derived from integration of partial function squares:
As the parameters A_{1}, A_{2}, and A_{3} are normalized, we can introduce an effective semiamplitude A, where , in agreement with a general definition of the quantity in Mikulášek et al. (2007b).
With the prepared model functions described by Eq. (1), we were able to fit the original data in unbinned form and derive values of the parameters F_{0}, A_{1}, A_{2}, and A_{3} for all analyzed stars, which are given in Table A.1.
Fig. 2. Binned light curves of the first ten stars of our sample. The open circles represent averages of real measurements in a particular bin. The filled circles indicate the fit obtained as a linear combination of the first three principal components. The yaxis describes the relative changes in V flux for eachlight curve. The light curves for different stars have been vertically shifted for clarity. 

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Fig. 3. Standard deviation of the fit (σ) for all sample stars. The full red line represents the fit. Dotted red lines define the 3σ_{exp} interval. 

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Fig. 4. Phased light curve of HD 151610. This illustrates the occasional shifts in the zero level of mean brightness between different ASAS3 data sets (indicated by the use of different colours). 

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2.4. Peculiarities of ASAS3 data
ASAS3 data suffer from occasional inhomogeneities, which cause shifts in the mean brightness level of the observations (see e.g., the case of HD 151610 depicted in Fig. 4). This problem can be solved if different values of the parameter F_{0} in Eq. (1) are assumed for the different data sets (F_{0i}). However, this procedure has to be performed after the elimination of outliers, which we identified using a 3.5 σ interval. In addition, data sets with less than four measurements need to be removed because they contain less points than the number of free parameters. Our analysis showed that, in some extreme and rare cases, the zeropoint shifts between different data sets amount to about 0.05 mag.
The described phenomenological model enables us to fit the ASAS3 data with high accuracy using at least four parameters, which depends on the number of parameters F_{0i}. The standard deviations of the fit increase from 0.01 mag for stars of the seventh magnitude to approximately 0.03 mag forobjects of the eleventh magnitude. These values can be interpreted as the intrinsic accuracy of ASAS3 data and correspond well to the values derived by Pigulski (2014). The relation between the standard deviation of the data fitted by the model described in Eq. (1) and the observed brightness in V was fitted by the simple exponential y = a + bexp(x). Only 1% of the fitted light curves exhibit standard deviations of the fit outside the 3σ_{exp} interval (see Fig. 3), where σ_{exp} is the standard deviation derived from an exponential fit.
Photometric data for different stars from the ASAS survey are identified by a designation consisting of right ascension and declination (e.g., the light curve of HD 10081, the very first star from our sample, is designated ASAS J0136156815.1). This identification, however, is not always sufficiently precise, and data from different stars can be blended, especially in highdensity fields. In fact, this is the case for several stars of our sample, which in consequence exhibit inappropriately high standard deviations. To identify blended data, we separated the measurements from different data sets. Results for a prototypical star are shown in Fig. 5. In this particular case, the source coordinates of the different data sets are slightly different. The scatter in some of these data sets is significantly higher than the intrinsic accuracy of ASAS3 data, which indicates blending issues and, possibly, the variability of another star in the same field of view.
Fig. 5. Phased ASAS3 light curve of HD 88485, exhibiting obvious blending issues. Different colors represent different sets of measurements. While the orange data denote the real light curve of the star, the very different green data are likely influenced by a background object. 

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2.5. Results
The nonlinear regression provides us with values of parameters and corresponding uncertainties. These values are derived under the assumption of a normal distribution of the fit residuals, and thus their determination does not have to be exact in cases of strongly nonlinear model functions. We therefore used the bootstrapping technique to determine corresponding error values and to verify the regression results. The results gained from both methods differ only slightly. Error values δA_{1}, δA_{2}, δA_{3}, and δφ_{0} derived from the bootstrapping procedure reach, on average, 100, 98, 100 and 93% of the values derived from the regression analysis.
As the values of the standard deviations for the corrected source data (i.e., data without blending issues) follow the intrinsic accuracy of the ASAS3 data (Fig. 3), we can assume that the employed phenomenological model fits the data with high accuracy. The key outputs are the values of the model parameters. Obtained values of zero phase φ_{0} are in agreement with the expectations; only in 7% of all cases do the values lie outside the interval [−0.05, 0.05].
Let us assume that the values of the parameters A_{1}, A_{2}, and A_{3} in Eq. (1) are significant only if their absolute values exceed three times their errors δA_{i} as calculated using the bootstrapping technique. As stated above, we employ the term double wave if a light curve fitted with Eq. (1) contains two minima and has a significant value of parameter A_{2}. Similarly, the term single wave is reserved for stars with light curves characterized by a single minimum and a significant value of parameter A_{1}. The number of minima was determined individually for each given star by mathematically inspecting the local minima of the fitted light curve. These definitions of significance and wave character are used in the following to describe the different shapes of the mCPstar light curves.
The most common shape for a light curve among the photometrically variable mCP stars of our sample is the single wave, with 67% of our sample stars showing this characteristic. This is illustrated in the upper panel of Fig. 6 (cf. also Fig. 7). It is furthermore apparent that the parameter A_{1} is, in general, the main source of the effective semiamplitude A. Furthermore, A_{1} is significant in 98% of all cases. The values tend to be negative as the light curve maxima are centered at φ_{0} = 0. This might seem counterintuitive; however, we need to remember that the brightness maximum is defined by minimum magnitude. Also, 49% of light curves show a symmetric single wave, while 18% are characterized by an asymmetric single wave.
The middle panel of Fig. 6 shows the histogram of the normalized parameter A_{2}. Significant values of A_{2} have been found for 46% of our sample stars. Also, 30% of all light curves are characterized by double waves; 15% of our sample stars show symmetric double waves, while 14% show asymmetric double waves; see Fig. 7 (the apparent inconsistency in expected equality of summed rate of symmetric and asymmetric double waves (29%) and all double waves (30%) is caused by rounding). It appears that 1% of stars possess light curves with identical primary and secondary minima, in which case A_{1} and A_{3} are insignificant. However, we caution that the periods were derived from photometric observations only, and some period values may actually represent twice the true value, which is likely to have influenced this result. Without additional (spectroscopic) data, it is sometimes difficult to determine whether a light curve shows a double wave with identical minima or a single wave. Nevertheless, as the following analysis shows, doublewave light curves with identical minima are very rare. Therefore, the above described issue, if at all, applies only to a small portion of our sample stars and will not have influenced our results in a significant way.
The histogram of the values of the normalized parameter A_{3}, which seemsto be significant in 36% of all cases, appears to be symmetrically centered at zero (Fig. 6, bottom panel), which supports the claim that the observed antisymmetry is random. However, the distribution is slightly leftskewed (− 0.11 ± 0.09).
The statistical distribution of the light curve shape according to wave character and symmetry is summarized in Fig. 7, while types and definitions are contained in Table 1. It becomes obvious that among the symmetric light curves, the singlewave character is highly dominant (77% vs. 55% among antisymmetric light curves). It must be noted that the category of light curves with two minima and insignificant A_{2} (3%) suffersfrom observations with high dispersion, and the corresponding values of A_{2} are lower than their estimated errors. This is indicative of lowquality data, and we consequently did not include these light curves in the category of doublewave light curves. They are indicated as lowquality data in Fig. 7.
Fig. 6. Probability histograms of the normalized parameters A_{1}, A_{2}, and A_{3}. 

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Classification of light curves.
Fig. 7. Statistical distribution of the shape of light curves of our sample of 650 mCP stars. 

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3. Simulations of symmetric spot distributions
If we assume that the main factor responsible for the distribution of chemical spots (which, as outlined in Sect. 1, correspond to the photometric spots; cf. Krtička et al. 2007 and Shulyak et al. 2010) is the magnetic field, we can expect some kind of correlation between the distribution of the spots and some magnetic field quantity. The recent study of Alecian (2015) shows that large overabundances are expected near the magnetic equator because this region boasts almost horizontal magnetic field lines, which seems to be an important factor for the creation of surface chemical spots. We here try to confirm the anticipated relation between magnetic field and photometric spots employing the following model simulations.
The basic assumption is that the magnetic field of the model star is central, axially symmetric, and dipolar, with an inclination between rotational and magnetic axis β. We can then assume that the flux distribution can be represented by two circular spots located around the magnetic poles. We tried to simulate this by using circular spots of angular radius α = 0.7 rad (spots of medium size) on a surface grid system assuming random orientation of both rotational and magnetic axes. The surface grid was defined as a system of stellar meridians and parallels with a uniform grid step value of π∕150 rad. For each grid node, the value of flux density was calculated and the total flux was integrated over the whole surface. An exemplary simulation is shown in Fig. 8. Parameter δ expresses the stellar latitude of the centerof the spot and i is the inclination of the rotational axis towards the observer. Random orientation of rotational and magnetic axes was established by choosing δ = arcsin(rand(0, 1)) and i = arccos(rand(0, 1)), where rand(0, 1) picks random numbers between 0 and 1 with uniform distribution. The value of the employed rotational phase step is 0.025. In our calculations, we took into account the limbdarkening formula I(ρ) = I_{0}[1 − q(1 − cosρ)] (Neilson & Lester 2011), where q is a linear limbdarkening coefficient (here we use the value q = 0.5), and ρ is the angular distance from the center of the stellar disk.
At first, we assumed two completely dark spots centered at opposite longitudes and stellar latitude ± δ. The inclination of the stellar rotational axis i is the angle between the rotational axis and the line of sight of the observer. As a result, we obtained a light curve fitted by the model function described in Eq. (1).
This simulation results in light curves that are always symmetric. Therefore, we removed the redundant antisymmetric part from Eq. (1). From lightcurve fits, we obtained values of the parameters A_{1} and A_{2}. In this way, a thousand random light curves were created.
To simplify the process, we assumed dark spots in our simulations, despite the fact that the photometricspots on mCP stars are usually bright in the optical region (Oksala et al. 2015). However, this should not pose a problem as the shape of a light curve containing a bright spot is simply a vertically inverted image of a light curve with a dark spot. This, therefore, does not influence the single or doublewave character of the simulated light curves.
Figure 9 illustrates the dependence of the parameters A_{1}, A_{2}, and A on the inclination of the rotation axis and the declination of the spots. We can see that the highest values of A_{1} are derived for the combination[δ, i] = [45°, 45°] and the highest values of A_{2} for the combination [δ, i] = [0°, 90°]. The main result of this simulation is that doublewave light curves should occur with a probability of 79%, which does not agree with the observed 77% of singlewave light curves among the ASAS3 mCP stars with symmetric light curves. There are at least two main factors which may have influenced these results. First, a double wave with identical minima can be misclassified as a single wave with half the true period. However, the simulation predicts that less than 9% of cases satisfy the relation A_{1} ∕A_{2} < 0.1, which we assume to be a satisfying condition for the identification of light curves with equal minima. Second, the spots are not completely dark, and some fainter spots will not have been detected. If we remove half of the cases with the lowest effective semiamplitudes A, the occurrence rate of doublewave light curves reaches 86%.
We ran similar simulations assuming spots with a radius of 0.3 rad, which resulted in doublewave light curves appearing in 74% of cases up to 83% after removal of half of the stars with the lowest effective amplitudes. However, for such small spots, values of A only reach up to 0.1 mag, even if we assume the spots are completely dark. It is therefore unlikely that we observe spots of this size. Simulations assuming large spots of radius 1.2 rad resulted in 87% of doublewave light curves (up to 95% after removal of the lowestamplitude objects). In addition, we carried out simulations assuming two spots with a radius of 0.7 rad, one being totally dark and the other one of random opacity. Similar results were obtained, with double wave light curves appearing in 74% of cases (up to 88% after removal of half of the stars with the lowest effective amplitudes). All studied cases are summarized in Table 2.
In summary, the results of our simulations suggest that, with the given assumptions, the light curves of mCP stars tend to be doublewaverather than singlewave. However, this is not compatible with the observations, which indicates that the model assuming similar distributions of magnetic field and photometric spots is not adequate. This is in agreement with the results of several studies employing Doppler mapping of magnetic fields in mCP stars and a comparison to the chemical abundance maps of different elements (e.g., Kochukhov et al. 2002; MiMeS Collaboration 2014; Kochukhov et al. 2015).
The symmetry of the simulated light curves is a result of the chosen model. Real light curves are not always symmetric and the presence of nonzero parameter A_{3} can be explained by noncircular spots or shifted spot location as compared to the simulations, which assumed that the spots are located opposite to each other.
Fig. 8. Exemplary simulation of a model spot with parameters α = 0.7 rad, δ = 1.0 rad, i = 0.7 rad. 

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Simulation results for all studied cases.
Fig. 9. Dependence of the parameters A_{1}, A_{2}, and A on the inclination of the rotation axis i and the declination of the spots δ. Values are expressed in magnitudes, as indicated on the right side. The simulations have been based on the assumption of two completely dark spots with radii of 0.7 rad that are located opposite each other on the star. 

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4. Conclusions
Using archival timeseries data from the ASAS3 survey, we analyzed the sample of 650 ACV variables and candidates by Bernhard et al. (2015) and Hümmerich et al. (2016). We focused on the study of the general shapes of their light curves as relating to their symmetry and single or doublewave character. The results show that 67% of the light curves are singlewave, 30% are doublewave, and asymmetry is found in 35% of cases. Among the symmetric light curves, the percentage of singlewave light curves is even higher (77%). We did not observe any kind of systematic antisymmetry.
We assumed a theoretical model of a symmetric dipole magnetic field with two opposite circular spots that follow the symmetry of the magnetic field and are hence located around the magnetic poles. Using this model, we made simulations assuming random positions of spots and random inclination of the rotational axis. Assuming completely dark, mediumsized identical spots with angular radii of 0.7 rad, we find an occurrence rate of 79% for doublewave light curves. If we ignore the simulated light curves with the lowest amplitudes to take into account the lesser probability of their detection, the occurrence rate rises to 86%. In cases of smaller spots with angular radii of 0.3 rad, similar results were obtained: 74% of the light curves are doublewave, rising up to 83% if lowamplitude light curves are ignored. However, the simulations show that the smaller spots frequently result in rotational modulation with amplitudes too low to be detectable in ASAS3 data. Simulations for large spots with angular radii of 1.2 rad result in 87% of light curves being doublewave, and up to 95% after removal of half of the faintest cases.
In summary, the results obtained from our simulations are in contradiction with the observed light curves of mCP stars. While we have assumed quite simple initial conditions for our spotmodelling attempts, we do not see any reason to believe that our results are influenced by significant systematic error. In disagreement with the current understanding, we thereby conclude that the distribution of photometric spots does not follow the magnetic field topology.
Acknowledgements
This research was supported by grant GA ČR 1601116S.
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All Tables
All Figures
Fig. 1. First three principal components. The first component (top panel) represents a singlewave light curve, the second component (middle panel) a doublewave light curve, and the third component (bottom panel) expresses antisymmetry. The points are real principal components and the solid lines are fits using Eq. (1) with A_{3} = 0 for the first two components and A_{1} = A_{2} = 0 for the thirdone. 

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In the text 
Fig. 2. Binned light curves of the first ten stars of our sample. The open circles represent averages of real measurements in a particular bin. The filled circles indicate the fit obtained as a linear combination of the first three principal components. The yaxis describes the relative changes in V flux for eachlight curve. The light curves for different stars have been vertically shifted for clarity. 

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In the text 
Fig. 3. Standard deviation of the fit (σ) for all sample stars. The full red line represents the fit. Dotted red lines define the 3σ_{exp} interval. 

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In the text 
Fig. 4. Phased light curve of HD 151610. This illustrates the occasional shifts in the zero level of mean brightness between different ASAS3 data sets (indicated by the use of different colours). 

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In the text 
Fig. 5. Phased ASAS3 light curve of HD 88485, exhibiting obvious blending issues. Different colors represent different sets of measurements. While the orange data denote the real light curve of the star, the very different green data are likely influenced by a background object. 

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In the text 
Fig. 6. Probability histograms of the normalized parameters A_{1}, A_{2}, and A_{3}. 

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In the text 
Fig. 7. Statistical distribution of the shape of light curves of our sample of 650 mCP stars. 

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In the text 
Fig. 8. Exemplary simulation of a model spot with parameters α = 0.7 rad, δ = 1.0 rad, i = 0.7 rad. 

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In the text 
Fig. 9. Dependence of the parameters A_{1}, A_{2}, and A on the inclination of the rotation axis i and the declination of the spots δ. Values are expressed in magnitudes, as indicated on the right side. The simulations have been based on the assumption of two completely dark spots with radii of 0.7 rad that are located opposite each other on the star. 

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In the text 
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