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Appendix A: Chromaticism of PIONIER

The chromaticism in our observations is twofold: (1) The transfer function (TF), i.e., the measured but not calibrated V² of a point source given instrumental and atmospheric effects, is wavelength dependent. (2) The atmospheric transmission, the filter function, and the response of the detector are wavelength dependent, and the distribution of the flux from the three spectral channels over the three pixels of the detector depends on the alignment of the instrument. The strength of both effects may change over time and with the internal alignment of the instrument, which is usually redone before starting an observing night. If a science target and the corresponding calibrator have different spectral types, the effect of this will be a shift in the effective wavelength in each spectral channel. The resulting difference between the V² measurement on a science target and the corresponding calibrator may result in systematic calibration errors that have to be characterized.

To investigate the chromaticism, we take advantage of the three spectral channel resolution data obtained during our survey for both science targets and calibrators. This provides us with V² data and photometry at a low spectral resolution. The V² data are corrected for the diameter of the target in order to obtain an estimate of the TF at the time of the observation. The science targets are included in this analysis to investigate the spectral type dependence of the effects studied. This is possible since we only expect a small fraction of our targets to exhibit extended emission beyond the stellar photosphere resulting only in a V² drop on the order of 1%. Thus, the whole sample can still be treated the same way as the calibrators. We fit a parabola to both the spectral shape of the TF and of the apparent flux obtained for each observation. From the data obtained in a seven-channel spectral resolution on β Pic (Defrère et al. 2012a,b), we find this to be a reasonable first-order approximation.

From the parabola fitted to the TF data in each spectral channel i, baseline j, and for each single observation k, we compute a relative change of the TF with wavelength at the center λ_c,i of channel i: (A.1)Here, denotes the TF estimated by measuring the V² of a target and correcting for its diameter. Studying the dependence of α_ijk on different factors reveals that the slope of the

TF depends on the baseline j used (Fig. A.1). In addition, it varies from night to night if PIONIER has been re-aligned in between. It does not change significantly if no realignment was done. Thus, we conclude that it does not significantly change during a night either. Finally, it does not depend significantly on the color of the target (spectral type). As a consequence, we can compute a median slope for each night, baseline, and spectral channel by averaging all observations, respectively.

From the parabola fitted to the spectral distribution of the apparent fluxes, we can compute an effective wavelength λ_eff,ik (the barycenter of the spectral flux distribution) in each channel for each observation: (A.2)where ν is the wave number (i.e., 1 /λ), and ν_0,i and ν_1,i are the upper and lower boundaries of the spectral channel i. The quantity λ_eff,ik depends mostly on spectral type and alignment (night), but it is not expected to significantly depend on baseline or time during a night (Fig. A.2).

Finally, we correct for the chromaticism on a per-observation, per-spectral-channel, and per-baseline basis: (A.3)The corrections found are shown in Fig. A.2. These corrections are applied to both calibrators and science targets, and the corrected V² of the science targets are calibrated with the TF measured on the corrected V² of the calibrators. The introduced correction suffers from idealization. Nonetheless, it gives a good first-order estimate of the magnitude of the chromaticism. We create two sets of calibrated data, only one of which includes the correction for chromaticism. From both data sets, we measure the excess as described in Sect. 3.7 and compare the results. The median difference between the flux ratios measured for single targets on the data with and without applying the correction is found to be 2 × 10^-4. There is the expected trend from K to A type stars, which suggests that the correction works well. Besides a few cases where the correction failed owing to a bad representation of the spectral shape of the apparent flux or the TF by the parabola fitted (usually due to noisy data), the difference in the results is below 5 × 10^-4, clearly negligible compared to our expected accuracy of a few 1 × 10^-3 (1σ).

	Fig. A.1 Spectral slopes of the transfer function of PIONIER for three illustrative nights. Line colors show the different baselines. Data points are averaged over all targets of a night, while error bars illustrate the scatter. Between the nights from July 23 and July 24 PIONIER was realigned. Between July 24 and July 25, no significant realignment was necessary.
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Fig. A.2 Difference between effective and central wavelength for two illustrative nights. Line colors show targets of different spectral type bins (blue: A type; green: F type; orange: G type; red: K type stars). Data are averaged over the targets of one spectral type and over the baselines, while the error bars illustrate the scatter. A clear trend with spectral type is visible for the first spectral channel. For the other spectral channels the trend is there as well, but barely visible given the scale of the figure.

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Fig. A.3 Absolute corrections c(V2) on the single V2 points derived for the chromaticism for the night of July 23rd. We first show the correction for K type stars to illustrate the magnitude of the corrections. Then we show the difference between the corrections for K type stars and stars of other spectral types, which is more illustrative of the actual error made by ignoring the correction but calibrating with K-type stars. Different lines show different baselines. The error bars illustrate the scatter of the correction for the different stars.

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© ESO, 2014