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Appendix A: Data reduction

In this appendix, we describe the reduction process applied to the PACS data in order to obtain what we call “single” and “combined” maps.

A.1. General PACS photometer data reduction

As explained in the main text (see Sect. 3.1) the data reduction of the maps was performed within HIPE (version 6.0.2044). We modified and adapted the standard HIPE scripts for the needs of our programme. Thus, the PACS photometer data is reduced from Level-0 to Level-2 using a modified version of the standard mini-scanmap pipeline. A major difference of our reduction chain compared with the standard pipeline is the application of frame selection on scan speed rather than on building block identifier (also known as BBID, which is a number that identifies the building block – a consistent part – of an observation). It has been widely tested that the application of the scan speed selection increases the number of useable frames, and finally increases the SNR of the final maps by 10–30% (depending on the PACS observation requests (AOR) setup and on the band used), which is especially important for our faint targets.

Since our targets move slowly (a few ″/h), and the total observation time per scanning direction in one visit is between 10−25 min we do not correct for the proper motion. Performing this kind of correction would smear the background, and would make it impossible to make a correct background subtraction in the later stages (see Sect. A.2).

We use a two-stage high-pass filtering procedure, as in the standard mini scan-map pipeline. In the first stage a “naive” map is created. This map is used in the second stage to mask the neighborhood of the source and the high flux pixels. In the second stage the vicinity of the source has been masked (2 times the FWHM of the beam of the actual band), as well as all pixels with fluxes above 2.5 ×  the standard deviation of the map flux values. We used the standard high-pass filter width parameters of 15, 20 and 35. The final maps are created by the PhotProject() task, using the default dropsize parameter.

These first steps lead to the production of one “single” map per visit, filter, and scan direction (i.e. in total 8 maps per object in the red, and 4 maps in the blue/green created using the MosaicTask() in HIPE). The sampling of the single maps generated with these scripts are 1.1″/pixel, 1.4″/pixel, and 2.1″/pixel for the blue (70 μm), green (100 μm) and red (160 μm) bands respectively.

A.2. Building final maps for photometry

We then use these single maps to generate the final “combined” maps on which the photometry will be performed. Essentially, to generate the combined maps, we determine the background map, subtract it from each single map, and finally co-add all the background-subtracted maps. A detailed description of this process is as follows:

• i)

  We have initially a set of 8 (red) or 4 (blue and green) single maps taken on different dates. Considering the red maps, let us call t1 to t4 (t5 to t8) the dates corresponding to the first (second) visit. t1-t4 and t5-t8 are separated by typically 1–2 days so that the target motion (typically 30–50″ as described in Sect. 2 of the main text) produces a significant change in the observed sky field (see Fig. A.1).

  	Fig. A.1 Left four panels: Typhon red band 1st visit single maps (t1 to t4). Right four panels: Typhon 2nd visit single maps (t5 to t8). t1, t3, t5, and t8 were observed with a scan angle of 110°, and t2, t4, t6, and t7 with a scan angle of 70° with respect to the detector array. The green circle marks the object position.
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• ii)

  A background map is generated using the single maps. To do this we “mask” the target in each of the 8 (4) single maps and co-add the maps in the sky coordinate system. This step produces a background map with high SNR without the target (see the right panel in Fig. A.3 for the Typhon case at 160 micron).

• iii)

  The background map is subtracted from the single maps (Fig. A.1), producing 8 (4) single maps with background removed. We call these images “background-subtracted maps” (Fig. A.2).

  	Fig. A.2 Same as Fig. A.1 for the background-subtracted maps.
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• iv)

  Finally the background-subtracted maps are co-added in the target frame (center panel of Fig. A.3), producing the final combined map on which photometry is performed.

Note that in step ii) before combining the images of the two epochs, we need to calculate the optimal matching offset between the coordinate systems of the two maps since, when we use the original World Coordinate System (WCS) information stored in the file headers, artifacts of bright sources remain in the background subtracted image. The best offset position was found in the following way: we added a small offset – in steps of 0.5″ in both the RA and Dec directions – to the appropriate original WCS parameters in the file headers, and performed the background subtraction. For each of these maps the standard deviation was calculated for pixels with a coverage value of at least 35% of the maximum coverage of the map. The offset values related to the lowest standard deviation correspond to the best matching of the two backgrounds. After this procedure the artifacts of bright sources (e.g. pairs of positive-negative spots) completely disappeared (or, at least, were below detection), and the noise in the background decreased notably. This method was performed on the 160 μm band measurements, where the effect of the background is the largest. However, it has been checked in a series of measurements that the same offsets can be applied to the 70 and 100 μm images as well.

An alternate, simpler method, to obtain final maps is to co-add directly the original single maps (i.e. not background-corrected) in the target frame. We call this the simple coaddition method (left panel of Fig. A.3). This method is obviously less optimal than the previous one in terms of SNR, but provides a useful test to demonstrate that the background subtraction is not introducing any spurious effects.

A more detailed description about the whole PACS data reduction process will be published in Kiss et al. (in prep.-a).

Appendix B: Statistical aspects

B.1. “Rescaled error bar” approach

As indicated in the main text (see Sect. 4.3), error bars on the fitted parameters (diameter, albedo and beaming factor) are determined through a Monte-Carlo approach which essentially consists of randomly building synthetic datasets using uncertainties in the measured fluxes. However, when the measurement errors bars are too small, or similarly when the model is not completely adequate (both cases being indicated by a “poor model fit”), the described Monte-Carlo approach will underestimate the uncertainties on the solution parameters. In this case, we adopted a “rescaled error bar” approach described hereafter.

Quantitatively, the fit quality is determined by the value of the reduced χ²: (B.1)where O, M and σ are the observed, modelled and error bar flux values, and ν is the degree of freedom (ν = N − 1 for fixed-η, and ν = N − 2 for floating-η, where N is the number of thermal wavelengths available). While indicates a good fit, indicates a poor fit. The idea is therefore to empirically “rescale” (uniformly, in the lack of a better choice) all error bars σ by  before running the Monte-Carlo approach. This method leads to much more realistic error bars on the solution parameters. However, the range of for which this approach is warranted depends on the number of data points, as for few observations does not have to be closely equal to 1 to indicate a good fit. Specifically, the variance of the variance estimator of a distribution of N points picked from a Gaussian distribution with dispersion = 1 is itself a Gaussian with a mean equal to unity and a standard deviation equal to . If we had for example N = 1000 observations, 68.2% of the good fits have a reduced χ² between 0.955 and 1.045, i.e. is strongly constrained to 1; for N = 5 (resp. 3) observations in contrast, this range is much broader, 0.368–1.632 (resp. 0.184–1.816). This means for example that for 3 data points (PACS only), a reduced χ² of e.g. 1.6 is acceptable. We considered these N-dependent limits as the thresholds beyond which the fits are statistically bad and the error bars need to be rescaled. Note finally that the rescaling approach is merely an “operative” method to avoid unrealistically low error bars on the fitted diameter, albedo and beaming factor, and that error bars shown in Figs. 1 and 2 of the main text are the original, not rescaled measurement uncertainties.

B.2. Spearman-rank correlation with error bars

As described in the main text the Spearman-ρ correlation test is distribution-free and less sensitive to outliers than most other methods but, like others, treats data-points as “exact” and does not take into account their possible error bars. Any variations in the measured data points within their error bars, may change the correlation coefficient, however. Furthermore, each correlation coefficient has its own confidence interval, which depends on the number of data points and on the magnitude of the correlation. To account for these effects we used the following three procedures:

• 1.

  To estimate the most probable correlation coefficient given the error bars, we generate 1000 samples of data points, building each synthetic dataset from its associated probability function. When errors bars are symmetric this probability function is considered to be Gaussian and the error bar correspond to one standard deviation. When errors bars are asymmetric, we fit a lognormal probability function such that the observed value corresponds to the distribution’s mode, and the interval between the upper and lower limits of the observation corresponds to the shortest interval containing 68.2% of all possible values. The resulting distribution of correlation values is not Gaussian, but its Fisher-transform z = argtanh(ρ) is. So, we can determine the most probable correlation value,  ⟨ ρ ⟩  from our Monte-Carlo simulations, and its upper and lower limits (+ σ_MC/−σ_MC), which are not necessarily symmetric after the reconversion using ρ = tanh(z) (see Peixinho et al. 2004, for more details). It is noticeable (and expected) that observational error bars “degrade” the correlation. The approximate significance p-value, or confidence level (CL), of ρ can be computed from , which follows a t-Student distribution with n − 2 degrees of freedom. However, when n ≲ 15, using t provides a slight overestimation of the significance. We have compared the previous calculation of the significance with exact values tabulated by Ramsey (1989) and computed adjustments required to obtain a more accurate approximation of the ρ’s and p-values in the case of low n.

• 2.

  The confidence level of a given ρ results from testing the hypothesis “the sample is correlated” against the hypothesis “the sample correlation is zero”. Knowing the confidence interval (CI) within which the correlation ρ value of the parent population lies may be more informative. For example, suppose we have ρ = 0.7 with a 3σ confidence level; we would conclude that the sample is correlated. But if the 68% confidence interval of the correlation were say  [0.3,0.9] , we would be unsure if the correlation was very strong or rather weak. To estimate the shortest (+ σ_B/−σ_B) interval containing 68.2% of the population’s ρ (i.e. the equivalent to the Gaussian 1σ interval), we used 1000 bootstrap extractions from the data-sample computing this range as we did for the previous item (e.g. Efron & Tibshirani 1993).

• 3.

  Finally, to be fully correct, one should perform the 1000 bootstraps on each one of our 1000 Monte-Carlo simulations to obtain the true 68% confidence interval of  ⟨ ρ ⟩ , a computationnally heavy task. Fortunately, the combination of the two effects can be obtained by quadratically adding the standard deviations of the Gaussian distributions of the Fisher-transform of the Monte-Carlo simulations and the bootstraps, i.e. , which, after reconversion to ρ, will give our final best estimate for the 68% confidence interval noted as .

© ESO, 2012