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Appendix A: Reduction and calibration of MIDI data for faint targets

Fig. A.1 The results for the unresolved star HD107485 taken on 2009-05-11. a) Our results for the calibrated correlated flux and its error are shown in solid and dot-dash line, respectively, while the correlated flux from a default reduction with EWS 1.7.1 (see text) is shown in dotted line. The dashed line is the total flux for this star which is obtained by scaling a template spectrum based on the known spectral type and IRAS flux. b) The same as a) but deduced visibilities are shown instead of correlated flux. c) The same as a but raw counts spectra are shown. d) Broad-band correlated flux measurements integrated over 7−13 μm (smoothed over a particular time width w) are shown on the complex plane. e) Histogram of the amplitude of the broad-band correlated flux from the panel d is shown. f) The phase of the correlated flux as a function of frame number, i.e. time. The same track is repeatedly shown with 360° offsets to clearly indicate the track as a function of time. The red points show excluded frames. g) The ratio of the standard deviation σ to mean m of the correlated flux amplitude measurements is shown as a function of smoothing width w in frames (lower axis) and in seconds (upper axis). The solid line with squares is for the target (HD107485) while the dashed line with plus signs is for the calibrator (see Table A.1). The gray squares and gray plus signs (without any connecting lines) indicate the photon noise (see Sect. A.3) estimated for the target and calibrator, respectively. The dotted line with triangles shows the expected fractional dispersion σ/m curve for the target, which is estimated from the σ/m curve for the calibrator (i.e. the dahed line with plus signs) and the photon noise estimations (i.e. gray squares and plus signs with no connecting lines). The dash-dotted line indicates the relative correlated flux counts of the calibrator as a function of w, showing the relative effective system visibility at each w. The green circles show the relative calibrated broad-band correlated flux as a function of w. The green crosses show the relative power-law index of the calibrated correlated flux spectrum also as a function of w. h) The delay function on the plane of delay versus frame number is drawn in gray scale image, showing the group delay track before the iteration described in Sect. A.5. The green and red points indicate non-rejected and rejected fringe peaks, respectively, after the iteration.

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	Fig. A.2 The same as Fig. A.1, but for the data taken on 2009-05-10.
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	Fig. A.3 The same as Fig. A.1, but for the data taken on 2009-05-12. The panel g) shows the σ/m curve after the correction for the time scale difference between the target and calibrator, while the panel i) shows the curves before the correction.
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	Fig. A.4 The same as Fig. A.3 but for another unresolved star HD110392 taken on 2009-05-09.
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We used a part of the software EWS (Jaffe 2004; version 1.7.1) to reduce the MIDI data described in this paper. The mid-IR correlated flux of all the targets here is  ≲ 0.5 Jy, which is much fainter than normally handled with the software. Therefore, we implemented several additional procedures to reduce and calibrate the data as described below, using our own IDL codes. In short, we simply averaged or smoothed over a relatively large number of frames, designated as w here (typically w ~ 20−40) to determine both the group-delay and phase-offset tracks, and applied the same averaging to the calibrator frames to compensate for the side effects of the averaging.

We also observed several calibration stars, listed in Table A.1, which are fainter than  ~0.8 Jy at 12 μm, have known spectral type, and are unresolved with VLTI/MIDI. The deduced correlated flux is directly compared with the total flux, which is obtained by scaling the template spectra of Cohen et al. 1999 (not from less accurate total flux frames obtained with MIDI). This facilitates a reliable evaluation of the accuracy of our reduction and calibration procedures. The scaling factors are derived from both IRAS and AKARI measurements. On the basis of the close match between the two, we estimate that the total flux accuracy is  ~4%.

A.1. Large gsmooth parameter

When the group delay is determined from the delay function (the Fourier transform of each fringe spectrum from the frequency domain to the delay domain), several frames are averaged or frames are smoothed over time direction as specified by gsmooth parameter in EWS to suppress instrumental delay peaks. The default is four, but we used a larger number (typically 10−20) to increase S/N in delay peak determinations, as normally done for faint targets.

A.2. Significant averaging over time for phase-offset determination

Figure A.1(a) compares the calibrated correlated flux of the unresolved star HD107485 with the known total flux spectrum (i.e. the scaled spectrum described above), and Fig. A.1(b) shows the deduced visibility (using the known total flux). The raw count spectrum of the correlated flux is shown in Fig. A.1(c). The star has a peak count rate of  ~800 cts/s in the PRISM-mode spectrum in this particular observation. In these figures, the dotted line shows the spectrum from a default reduction (though with the same large gsmooth parameter), which determines phase offsets with a small averaging width. In this case, the spectrum suffers a red bias, i.e. extra counts at the longer wavelength side.

This seems to be at least partly due to the determination of phase offsets at too low S/N. This can be visualized on the complex plane of the broad-band correlated flux counts integrated over the N-band of  ~7 to 13 μm. The phase offsets are determined from this integrated broad-band flux. When a large enough number of frames are averaged, the distribution of the broad-band correlated flux measurements on the complex plane shows a (partial) circle as shown in Fig. A.1(d), where its radius represents the amplitude of the correlated flux with some phase offset slowly changing over time.

A noisy phase-offset determination (and subsequent rotation using the erroneous phase) would lead to a positive bias in the correlated flux (e.g. it would detect non-zero correlated flux even if the real correlated flux is zero). The red bias is probably caused, since the MIDI count spectra usually have a peak at shorter wavelengths, by the phase-offset determination having a larger weight at shorter wavelengths (or more accurately count peak), thus leading to the longer wavelength side having a stronger positive bias. We note also that, in general, correlated flux spectra would also tend to be redder if larger coherence loss at shorter wavelengths is not correctly calibrated.

Therefore, we chose a large w for the phase offset track determination, and a boxcar smoothing was applied to the group delay track for the same number of frames. By applying the same averaging to the delay and phase tracks of calibrator frames, we estimated the low effective system visibility arising from this averaging process, and calibrated it out from the target frames. As we show below, we chose w to maximize the S/N of the phase-offset determination and the result is insensitive to the exact value of the chosen w.

A.3. Optimum averaging width and evaluation of atmospheric conditions

Here we quantify the effect of this averaging process as a function of w, and describe how we can choose an optimal value of w.

A set of broad-band correlated flux measurements for a given w as shown in the complex plane in Fig. A.1(d) gives a distribution of the correlated flux amplitude measurements. The corresponding histogram is shown in Fig. A.1(e). We can evaluate the distribution as a function of w by calculating a fractional dispersion σ/m (standard deviation σ divided by mean m) for each w. This statistical property would be valid as long as the effective number of correlated flux measurements is not low. Here, the effective number of correlated flux measurements is the number of non-rejected frames (see Sect. A.5 below) divided by w. The reciprocal m/σ corresponds to the S/N of the correlated flux measurements per smoothed frame.

Figure A.1(g) shows the change of σ/m as a function of w (or corresponding averaging time interval, “accumulation time”; see upper axis) for the target and its visibility calibrator. The curve is cut when the effective number of measurements becomes lower than 10. In the small w regime for the faint target, the dispersion naturally decreases with larger w as photon noise decreases. The photon noise here comprises a background Poisson noise and fluctuation due to background subtraction residuals. This can be estimated from the noise spectra (a few hundred frames with large OPD offsets, taken at the start of each fringe track with MIDI) and the fractional fluctuation of the delay function peaks. This is shown in gray squares in Fig. A.1(g), and closely matches the σ/m curve at small w range. The fractional dispersion σ/m starts to increase at some width where the effect of a large time-averaging width to smear and effectively reduce the correlated flux becomes more dominant over the photon noise decrease.

For the bright calibrator, σ/m stays relatively constant at small w where flux fluctuation slowly decreases (indicated by gray plus signs), while the smearing effect of averaging slowly increases. The dash-dotted curve is the relative correlated flux counts of the bright calibrator (normalized at the smallest w), showing a relative effective system visibility at each w. For larger w, σ/m increases quickly (system visibility declines quickly) when the blurring effect of the large-w averaging starts to dominate. This gives an indication of the coherence time in the mid-IR at the time of the observation (~1.5 s in the case of Fig. A.1(g)). In the large w regime, the fractional dispersion σ/m is thus determined predominantly by the atmosphere at the time of the observation, rather than the observed source.

The coherence time and thus the blurring effect is approximately the same between target and calibrator frames when the two observations take place close in time and on the sky. In this case, we can actually calculate the expected σ/m for the target frames as a function of w from the calibrator frames. We subtract the photon noise (gray plus signs) in quadrature from the σ/m of the calibrator (solid plus signs with dashed lines) to obtain the dispersion caused by averaging, and add it in quadrature to the photon noise in target frames. This expected σ/m curve for the target is shown as triangle symbols with a dotted line in Fig. A.1(g), and closely matches the observed σ/m, meaning that atmospheric conditions were very similar. In this case, we simply apply the same averaging width for the calibrator frame to obtain an appropriate estimate of the effective system visibility spectra, or transfer function over wavelengths, for the target frames processed with this particular averaging width.

In Fig. A.1(g), the green circles show the relative calibrated broad-band correlated flux (normalized at w giving minimum σ/m, or maximum S/N) as a function of w, and the green crosses show the relative power-law index of the calibrated correlated flux spectrum. The calibrated correlated flux is systematically higher, and often redder, at small w because of the positive bias from low S/N determination of phase offsets. We have chosen w to yield maximum S/N for the target. If w is chosen to be larger than this value, the result is quite insensitive to the exact value of w, although the effective S/N of correlated flux measurements is lower at larger w owing to the smearing effect of averaging.

	Fig. A.5 Comparison of deduced visibilities for sub-Jy calibrators.
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A.4. Adjustment for the atmospheric difference

Figure A.2 shows a similar case to that shown in Fig. A.1, where the count rate is slightly lower and the coherence time is slightly shorter. The maximum S/N is thus slightly lower. However, the atmospheric conditions between target and calibrator observations seem to be well matched.

In Figs. A.3 and A.4, we show the cases where there seems to be a difference in atmospheric conditions between target and calibrator fringe tracks. In these cases, we try to evaluate the difference by comparing the expected σ/m curve for the target (triangles with dotted line in Figs. A.3(i) and A.4(i)), as derived from the calibrator frames, with the observed curve for the target (squares with solid lines). We find that we can fit these two curves if we simply scale the time length of the calibrator frames (a factor of  ~0.7 in Fig. A.3(g), and  ~2.5 in Fig. A.4(g)). The scaling factor probably corresponds to the difference in the coherence time between target and calibrator frames. Therefore, we infer that we can obtain appropriate calibrations of the target frames if we use this time scaling factor. This correction also stabilizes the calibrated correlated flux over different w.

Figure A.5 shows the calibrated visibility of the 12 low and intermediate S/N observations of unresolved stars as a function of the maximum S/N per smoothed frame. We calculated the visibility in three different wavelength bins, avoiding the range affected by atmospheric ozone features. We confirm that the calibration procedure above correctly calibrates the data, although we see a slight tendency that, toward shorter wavelengths, the correlated flux is underestimated as the maximum S/N decreases. Based on the mean deviation and dispersion of these calibrated visibilities, we assign a systematic error in the correlated flux as 5% of the total flux, when the maximum S/N is at least larger than  ~2.5. Since the cases at lower S/N remain untested, we did not use the target datasets under this level in this paper.

Fig. B.1 All the mid-IR data shown in Figs. 1 and 2 and summarized in Table 5 are plotted here in more conventional ways for a) NGC 4151, b) NGC 3783, and c) ESO323-G77. Top: the total flux and correlated flux spectra taken with MIDI are shown in units of Jy. The filled circles indicate the spectra with a 5-pixel boxcar smoothing (3 pixel for NGC 4151), while the spectra before the smoothing are shown with thin lines. The total flux is shown in black, while the correlated flux is shown in different colors for different baselines as indicated by the legend at top-left where the projected baseline lengths and PAs are written. The error spectra estimated for the smoothed spectra are shown at the bottom with dash-dot lines using the same corresponding colors. The two fluxes from the broad-band VISIR imaging are shown with error bars in thick gray lines. The thin gray spectra mostly at the highest flux levels are from the Spitzer IRS observations. The slightly darker gray spectrum shown for ESO323-G77 is from the VISIR spectroscopy. For NGC 3783, the total flux taken in 2005 scaled to match the flux in 2009 (see Sect. 2.1) is shown in gray color. Bottom: the resulting visibility spectra are shown using the same color scheme, with the error spectra excluding the total flux error contribution. The fractional total flux errors are drawn separately in black dash-dot lines.

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	Fig. B.2 The same as Fig. B.1, but the data shown in solid lines are with the total flux smoothed by 3 pixels while those in filled circles are with total and correlated fluxes smoothed by 7 pixels. The figures are for a) H0557-385, b) IRAS09149-6206, and c) IRAS13349+2438.
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A.5. Refinement of group delay determination

As described in Sect. A.1, we use a large gsmooth parameter for faint targets to smooth over a large number of delay function frames and thus increase the S/N for delay determination and also strongly suppress the instrumental delay peaks. However, this also suppresses the atmospheric delay peaks at the time intervals where the delay is relatively quickly changing. As a result, the delay function image becomes quite “blobby”, and strong delay peaks are left only where the atmospheric delay did not change rapidly.

To partially compensate for this effect, we implemented a simple iteration to determine the delay track. Namely, we first interpolated (over time) between the strong delay peaks to derive an approximate delay track. Then we used this approximate atmospheric delay track (plus instrumental delay) to rotate the original fringe spectra in the complex plane, and re-derived the delay function smoothed with the same gsmooth parameter. This makes the resulting delay peaks line up straighter over time, and thus recovers a larger number of frames with strong delay peaks. We cycled over this process a few times. For calibrator frames, this resulted in almost all the frames having strong peaks (i.e. even with the large gsmooth parameter). For target frames, significant fraction of frames is recovered, resulting in a good representation of the delay track even in a relatively low S/N case as shown in Fig. A.3.

In low S/N cases, we flagged frames that show delay peaks at positions that deviate significantly from the overall track obtained above. This was done by taking the histogram of the difference between the determined group delay track with that smoothed over many frames (typically  ~20−40 frames, set as 2  ×  gmoooth). The distribution can roughly be descibed by a Gaussian, and frames with large deviations (>10σ) were excluded. In this rejection process, we did not select frames based on the strength of the delay peaks in order to avoid possible biasing.

A.6. A note on total flux frames

For the targets observed with UTs that have total flux of less than a few Jy, it is better to implement an additional background subtraction using the sky regions very close to the target position. The residual of the primary background subtraction using chopped frames can still dominate the target flux for these relatively faint targets. An optional way to further reduce the systematic error from the background subtraction is to iteratively reject frames with the sky region counts having large deviations from the average. This was implemented in a small number of cases when the time fluctuation over the photmetry frames was significantly reduced after the frame rejection. With this additional sky subtraction, we averaged many sets of data, even from different nights, to obtain reliable total flux spectra, as confirmed with our VISIR photometry and spectrum.

Appendix B: Mid-IR data plotted in conventional formats

In Figs. 1 and 2 we show our flux and visibility data with normalizations using the inner radius R_in. We show the same data in the mid-IR (after reddening corrections; see Sect. 2.5) using conventional units in Figs. B.1 and B.2, to facilitate direct comparisons with the data in the literature.

Appendix C: Radial structure plot in linear scale

In Fig. 3 we showed observed mid-IR visibilities of all the targets in a single plot to uniformly compare the radial structure of the objects in normalized units with spatial frequency in log scale. Here in Fig. C.1 we show exactly the same figure but with spatial frequency in linear scale, again to facilitate comparisons with the data in the literature.

	Fig. C.1 The same figure as Fig. 3, but with the spatial frequency on a linear scale.
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© ESO, 2011