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Appendix A: The search for periodic variations
The continuously updated archive of the All Sky Automated Survey (ASAS) is a useful source of photometric observations. Observations were obtained from two ASAS observing stations, one in LCO, Chile (since 1997) and the other on Haleakala, Maui (since 2006). Both are equipped with two widefield instruments, observing simultaneously in V and I band. More details and the data archive are on http://www.astrouw.edu.pl/asas/. After removing observations of declared low quality (C, D) and apparent outliers, we obtained a set of 539 Iband and 920 Vband measurements quite homogeneously covering the periods 1997–1999 and 2000–2009, respectively. The robotic telescope “Pi of the Sky” has been designed to monitor a significant fraction of the sky with good time resolution. The final detector consists of two sets of 16 cameras, one camera covering a field of view of 20° × 20°. The set of the HD 101412 measurements covers the time interval of 2006–2009. More details can be found at http://grb.fuw.edu.pl/. CCD photometry was performed without using any filter, so that the results are similar to those for a broadband R color. The star was also observed over eight nights in 2010 April employing the 0.5 m reflector with the classical photometer in SAAO in UBV, using a fairly conventional single channel photometer with a Hamamatsu R943–02 GaAs tube. We obtained 78 triads of measurements with an intrinsic accuracy of 9.4, 6.4, and 6.6 mmag, respectively.
The very good initial estimate of the period allows us to describe the observed periodic variations by a series of phenomenological models described by a minimum number of free parameters, including the period P and the origin of epoch counting M_{0}. The behaviors of the light curves in V and I are nearly the same. For simplicity, we also assumed a similar behavior for light curves in U and B bands, while the light curve R behaves differently. The periodic component in light variations can then be described by means of the periodic functions F(ϑ) and F_{R}(ϑ_{R}) The functions F(ϑ) and F_{R}(ϑ_{R}) are the simplest normalised periodic function that represent the observed photometric variations in detail. The phase of the brightness extreme is defined to be 0.0, and the effective amplitude is defined to be 1.0. The functions, being the sum of three terms, are described by two dimensionless parameters β_{1},β_{2} and β_{3},β_{4}, and ϑ and ϑ_{R} are the phase functions. Assuming linear ephemeris, we respectively obtain (A.3)where Δf_{R} is the phase shift of the basic minimum of the function F_{R} versus zero phase.
Periodic changes in the magnitudes of U,B,V,I (m_{j}(t)) and changes in R (m_{R}(t)) are given by the relations (A.4)where A is the semiamplitude of light changes common to all bands, A_{R} is the amplitude of an additional component of light variability that is nonzero only in R, and and are mean magnitudes in the individual bands.
The periodic variations in the mean value of the longitudinal component of the magnetic field intensity ⟨B_{z}⟩ derived from all lines can be well approximated by the simple cosinusoid (A.5)where is the mean magnetic field intensity, A_{m} is the semiamplitude of the variations, and Δf_{m} is the phase of the magnetic field minimum.
All 18 model parameters were computed simultaneously by a weighted nonlinear least squares method regression applied to the complete observational material representing in total 3134 individual measurements. We found the refined value of the rotational period P = 42ḍ076(17) and the origin of phase counting at the UBVI light maximum M_{0} = 2452797.9(8) to be (A.6)The parameters describing the functions F and F_{R} are β_{1} = 0.29(13), β_{2} = −0.49(10), β_{3} = 0.36(16), and β_{4} = 0.02(18), the function F_{R} primary minimum phase Δf_{R1} = 0.24(4), the phase of the secondary minimum Δf_{R2} = 0.72(12), the semiamplitude of light changes A = −4.7(5) mmag, the semiamplitude A_{R} = 14.1(2.2) mmag, mag, mag, and mag. The phase of the minimum of the mean projected intensity of the magnetic field ⟨B_{z}⟩ is Δf_{m} = −0.091(33), its mean value is G, and its semiamplitude is A_{m} = 465 ± 27 G.
Appendix B: Nonperiodic variations
Relatively precise ASAS V and I data show an apparent evolution of the mean values to the extent of several hundredths of a magnitude over timescales of several years (see Fig. B.1). For this reason, we use detrended magnitudes, i.e., magnitudes from which we have removed the longterm trends, in our search for periodic stellar variability.
Fig. B.1
Longterm changes in ASAS V (open circles) and I (full squares) corrected for variations. 

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An inspection of the light variation in R plotted over the rotational phase in 2006–2009 reveals dramatic seasonal variations in the shape of the light curve. Unfortunately, the low accuracy of the R photometric measurements, the observed longterm instability, and the large stochastic changes do not allow us to analyse this phenomenon more deeply. Using principal component analysis tools, we tested the possibility of seasonal changes in the shape of the V light curve in the first decade of this century. We conclude that the V light curve as defined by ASAS and SAAO measurements was quite stable during this period.
Fig. B.2 Stochastic changes in our SAAO UBV observations. Note that the scatter in individual bands is highly correlated. 

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Herbig Ae/Be stars are known to vary in a complex way. The mechanism causing this variability is not well understood. These more or less stochastic variations amount to up to several hundreds of a magnitude on timescales from several minutes to tens of years (see e.g. Ruczinski et al. 2010). The light changes of HD 101412 in V and I presented in Fig. B.1 can be treated as stochastic variations on scales of tens of years. Nevertheless, there is also evidence of these changes on much shorter timescales.
Our own 78 UBV photoelectric measurements in 2010 April in SAAO were obtained in a total of 12 sets over eight nights. The duration of one measurement set was no longer than 15 min and we assumed that the brightness of the star did not change during the exposure. Then we can estimate relatively well the typical intrinsic uncertainty of one individual measurement as 9.4, 6.4, and 6.6 mmag in U,B, and V, respectively. However, the means in particular sets of U, B, and V data sets, obtained by averaging six individual measurements, show an additional scatter of 14, 10, and 11 mmag (see Fig. B.2). We inferred that this scatter is due to stochastic changes on the scale of hours or tens of hours. It should be noted that the observed stochastic variations in particular colors are highly correlated, whereas their amplitude is largest in the Uband. This behavior is undoubtably related to the physics behind the mechanisms causing these changes.
Because we do not have at our disposal measurements in V and I during one night, we estimate the intrinsic accuracy of these measurements implicitly. The mean scatter of detrended ASAS data in V and I are 16 and 10 mmag, respectively. Assuming that the estimate of stochastic noise in V of 11 mmag is valid or smaller for all ASAS V measurements, we conclude, that the intrinsic uncertainty of one ASAS V observation is smaller than 12 mmag. The ASAS estimates of the uncertainty in the I observations should then be smaller by a factor of 0.6 than for V observations. The intrinsic accuracy should then be about 7 mmag, as well as the measurement of the stochastic scatter. In all cases, the stochastic scatter in I is not allowed to be larger than 10 mmag. This means that the tendency of the stochastic scatter to decrease with increasing wavelength is also supported by ASAS data. A different amount of stochastic variation is expected in R band, where an additional variability mechanism is probably active. The mean scatter in the Rband is about 56 mmag. The intrinsic uncertainty in the “Pi” measurements can be derived from magnitudes obtained in individual nights – it is estimated to be 49 mmag. The stochastic scatter can then be evaluated to be 30 mmag.
© ESO, 2010