Appendix A: Extinction coefficients and colour terms

Extinction coefficients and colour terms were computed using the method outlined by Harris et al. (1991), i.e. via a single-step multi-linear least squares fit to the data provided by the observation of photometric standard star fields (Landolt 1992). This procedure, in fact, allows one to get the photometric solutions using different stars with suitable colour and airmass ranges, without the need for repeating the observations at different airmasses. Due to the FORS1 calibration plan implemented during the time range discussed in this work, this was a mandatory requirement. To get meaningful results, the method needs a fair number of stars well spread in colour and airmass. While the first requirement is almost always fulfilled for the Landolt fields suitable for CCD photometry, the second is more cumbersome to achieve using FORS1 data. In order to overcome this problem we have computed the photometric parameters on a bi-monthly basis, for those time ranges where at least 3 stars were observed at airmass larger than 1.6. Harris et al. (1991) recommend to use data spanning at least 1 airmass, but we had to relax this constraint in order to get a sufficient number of stars.

 

 

Table A.1: Bi-monthly average extinction coefficients (mag airmass^-1) measured at ESO-Paranal with FORS1 during photometric nights. For each entry the total number of data points (n) and the number of data points with airmass larger than 1.6 (n_z) are reported. The last row of the table shows the weighted means and the corresponding rms errors.

Time	U			B			V			R			I
Range	$\kappa$	n	nz	$\kappa$	n	nz	$\kappa$	n	nz	$\kappa$	n	nz	$\kappa$	n	nz
03/04-00	0.41 $\pm$ 0.05	87	3	-	-	-	0.11 $\pm$ 0.02	120	4	-	-	-	0.07 $\pm$ 0.02	88	8
05/06-00	-	-	-	0.25 $\pm$ 0.01	43	3	0.09 $\pm$ 0.01	47	4	0.07 $\pm$ 0.01	46	4	-	-	-
07/08-00	0.47 $\pm$ 0.01	122	12	0.23 $\pm$ 0.01	132	14	0.11 $\pm$ 0.01	162	19	0.07 $\pm$ 0.01	151	15	0.05 $\pm$ 0.01	138	19
11/12-00	0.43 $\pm$ 0.01	78	9	0.21 $\pm$ 0.01	86	16	0.11 $\pm$ 0.01	89	13	0.07 $\pm$ 0.01	83	14	0.03 $\pm$ 0.01	85	10
01/02-01	0.45 $\pm$ 0.02	122	24	0.22 $\pm$ 0.02	154	29	0.10 $\pm$ 0.01	163	33	0.07 $\pm$ 0.01	148	28	0.04 $\pm$ 0.01	145	29
05/06-01	0.40 $\pm$ 0.03	64	4	0.19 $\pm$ 0.02	66	4	0.09 $\pm$ 0.01	60	3	0.06 $\pm$ 0.01	60	3	0.01 $\pm$ 0.01	68	3
07/08-01	0.43 $\pm$ 0.02	117	18	0.22 $\pm$ 0.01	139	17	0.12 $\pm$ 0.01	143	20	0.08 $\pm$ 0.01	124	17	0.04 $\pm$ 0.01	68	17
09/10-01	0.44 $\pm$ 0.02	125	3	0.24 $\pm$ 0.01	143	3	0.11 $\pm$ 0.01	141	4	0.08 $\pm$ 0.01	120	4	0.06 $\pm$ 0.01	130	5
	0.43 $\pm$ 0.02			0.22 $\pm$ 0.02			0.11 $\pm$ 0.01			0.07 $\pm$ 0.01			0.05 $\pm$ 0.02

Some tests have shown that the available photometric data do not allow the computation of the second order colour term and extinction coefficient, which are usually included in the photoelectric photometry solutions (see Harris et al. 1991, their Eqs. (2.9)). In fact, the introduction of such terms in the fitting of our data produces random oscillations in the solutions without any significant decrease in the variance. Both second order coefficients are accompanied by large errors and are always consistent with zero. This clearly means that the accuracy of our measurements is not sufficient to go beyond the first order term. For this reason, we have adopted $M-m=m_0+\gamma~\times~ C \; -\kappa~\times~ z$ as fitting function for the generic passband, where M and m are respectively the catalogue and instrumental magnitudes of the standard stars, m₀ is the zeropoint, $\gamma$ is the colour term with respect to some colour C, $\kappa$ is the extinction coefficient and z is the airmass. To eliminate clearly deviating measurements, we have computed the photometric solutions in two steps. We have first used all data to get a starting guess from which we have computed the global rms deviation $\sigma$. Then we have rejected all data deviating more than $1.5\times \sigma$ and performed again the least squares fit on the selected data.

The results for the extinction coefficients are presented in Table A.1. For each time range and passband we have reported the best fit value of $\kappa$, the estimated rms error (both in mag airmass^-1) and the number of data points used for the least squares fit.

The values of $\kappa$ in the various bands show some minor fluctuations. Of course, we cannot exclude night-to-night variations, which are clearly observed at other sites (see for instance Krisciunas 1990). This would require a dedicated monitoring, which is beyond the scope of this work and not feasible with the available data. Here we can only conclude that, on average, the extinction coefficients do not show any significant evolution or clear seasonal effects during the 20 months covered by the data discussed in this paper. Therefore, given also the purpose of this analysis, we have assumed that the extinction coefficients are constant in time and equal to the average values reported on the last row of Table A.1.

The computed colour terms are shown Table A.2. We note that in those cases where the constraints on the airmass range and the number of data-points were not fulfilled, we have performed the best fit keeping $\kappa$ constant and equal to the average values given in Table A.1. This prevented the best fit from giving results with no physical meaning due to the poor airmass coverage. As in the case of the extinction coefficients, the color terms show fluctuations which are within the errors, stronger oscillations being seen in the U passband, where the accuracy of the photometry is lower.

On the basis of these results we can conclude that there are no strong indications for significant colour term evolution and it is therefore reasonable to assume that they are constant in time. For our purposes we have adopted the weighted mean values shown in the last row of Table A.2.

 

 

Table A.2: Bi-monthly average colour terms for FORS1. The total number of used data points are the same reported in Table A.1, while the estimated rms uncertainty on each entry is 0.01. The last row shows the weighted mean values and their estimated rms errors.

Time Range	$\gamma_{U-B}^U$	$\gamma_{B-V}^B$	$\gamma_{B-V}^V$	$\gamma_{V-R}^R$	$\gamma_{V-I}^I$
03/04-00	0.09	-0.07	0.04	0.02	-0.05
05/06-00	0.08	-0.08	0.04	0.02	-0.04
07/08-00	0.10	-0.08	0.03	0.00	-0.05
09/10-00	0.08	-0.07	0.05	0.04	-0.04
11/12-00	0.04	-0.09	0.04	0.04	-0.04
01/02-01	0.07	-0.08	0.03	0.03	-0.05
03/04-01	0.07	-0.08	0.04	0.03	-0.04
05/06-01	0.03	-0.09	0.05	0.02	-0.04
07/08-01	0.10	-0.08	0.03	0.02	-0.05
09/10-01	0.08	-0.08	0.04	0.04	-0.03
	0.07 $\pm$ 0.02	-0.08 $\pm$ 0.01	0.04 $\pm$ 0.01	0.03 $\pm$ 0.01	-0.04 $\pm$ 0.01