3 Photometric calibration and global errors

Once the sky background $I_{{\rm sky}}$ has been estimated, the flux per square arcsecond and per unit time is given by $f_{{\rm sky}} = I_{{\rm sky}}/(t_{{\rm exp}} \; p^2)$, where p is the detector's scale (arcsec pix^-1) and  $t_{{\rm exp}}$ is the exposure time (in seconds). The instrumental sky surface brightness is then defined as

$$% m_{{\rm sky}} = -1.086\;{\rm ln}(I_{{\rm sky}}) + 2.5\;\log\left(p^2 \; t_{{\rm exp}}\right)$$ (1)

with $m_{{\rm sky}}$ expressed in mag arcsec^-2. Neglecting the errors on p and $t_{{\rm exp}}$, one can compute the error on the sky surface brightness as $\delta_{b_{{\rm sky}}} \simeq \delta_{I_{{\rm sky}}}/I_{{\rm sky}}$. This means, for instance, that an error of 1% on $I_{{\rm sky}}$ produces an uncertainty of 0.01 mag arcsec^-2 on the final instrumental surface brightness estimate. While in previous photoelectric sky brightness surveys the uncertainty on the diaphragm size contributes to the global error in a relevant way (see for example Walker 1988b), in our case the pixel scale is known with an accuracy of better than 0.03% (Szeifert 2002) and the corresponding photometric error can therefore be safely neglected.

The next step one needs to perform to get the final sky surface brightness is to convert the instrumental magnitudes to the standard UBVRI photometric system. Following the prescriptions by Pilachowski et al. (1989), the sky brightness is calibrated without correcting the measured flux by atmospheric extinction, since the effect is actually taking place mostly in the atmosphere itself. This is of course not true for the contribution coming from faint stars, galaxies and the zodiacal light, which however account for a minor fraction of the whole effect, airglow being the prominent source of night sky emission in dark astronomical sites. The reader is referred to Krisciunas (1990) for a more detailed discussion of this point; here we add only that this practically corresponds to set to zero the airmass of the observed sky area in the calibration equation. Therefore, if  $M_{{\rm sky}}$, $m_{{\rm sky}}$ are the calibrated and instrumental sky magnitudes, M_*, m_* are the corresponding values for a photometric standard star observed at airmass z_*, and $\kappa$ is the extinction coefficient, we have that $M_{{\rm sky}}=(m_{{\rm sky}}-m_*) + \kappa\;z_* + M_* +$ $\gamma \; (C_{{\rm sky}}-C_*)$. This relation can be rewritten in a more general way as $M_{{\rm sky}}=m_0+m_{{\rm sky}}+\gamma \; C_{{\rm sky}}$, where m₀ is the photometric zeropoint in a given passband and $\gamma$ is the colour term in that passband for the color $C_{{\rm sky}}$. For example, in the case of B filter, this relation can be written as $B_{{\rm sky}}=B_0+b_{{\rm sky}}+\gamma^B_{B-V} \;(B-V)_{{\rm sky}}$.

Figure 2: UBVRI photometric zeropoints for FORS1 during the time range covered by sky brightness measurements presented in this work (vertical dotted lines). The thick segments plotted on the lower diagram indicate the presence of sky brightness data, while the arrows in the upper part of the figure correspond to some relevant events. A) water condensation on main mirror of UT1-Antu. B) UT1-Antu main mirror re-aluminisation after the water condensation event. C) FORS1 moved from UT1-Antu to UT3-Melipal. Plotted zeropoints have been corrected for extinction and colour terms using average values (see Appendix A).

In the case of FORS1, observations of photometric standard fields (Landolt 1992) are regularly obtained as part of the calibration plan; typically one to three fields are observed during each service mode night. The photometric zeropoints were derived from these observations by means of a semi-automatic procedure, assuming constant extinction coefficients and colour terms. For a more detailed discussion on these parameters, the reader is referred to Appendix A, where we show that this is a reasonable assumption. Figure 2 shows that, with the exception of a few cases, Paranal is photometrically stable, being the rms zeropoint fluctuation $\sigma_{m_0}=$ 0.03 mag in U and  $\sigma_{m_0}=$ 0.02 mag in all other passbands. Three clear jumps are visible in Fig. 2, all basically due to physical changes in the main mirror of the telescope. Besides these sudden variations, we have detected a slow decrease in the efficiency which is clearly visible in the first 10 months and is most probably due to aluminium oxidation and dust deposition. The efficiency loss appears to be linear in time, with a rate steadily decreasing from blue to red passbands, being 0.13 mag yr^-1 in U and 0.05 mag yr^-1 in I. To allow for a proper compensation of these effects, we have divided the whole time range in four different periods, in which we have used a linear least squares fit to the zero points obtained in each band during photometric nights only. This gives a handy description of the overall system efficiency which is easy to implement in an automatic calibration procedure.

 

 

Table 2: Typical night sky broad band colours measured at various observatories.

Site	Year	U-B	B-V	V-R	V-I	Reference
Cerro Tololo	1987-8	-0.7	+0.9	+0.9	+1.9	Walker (1987, 1988a)
La Silla	1978	-	+1.1	+0.9	+2.3	Mattila et al. (1996)
Calar Alto	1990	-0.5	+1.1	+0.9	+2.8	Leinert et al. (1995)
La Palma	1994-6	-0.7	+0.8	+0.9	+1.9	Benn & Ellison (1998)
Paranal	2000-1	-0.4	+1.0	+0.8	+1.9	this work

To derive the colour correction included in the calibration equation one needs to know the sky colours $C_{{\rm sky}}$. In principle $C_{{\rm sky}}$ can be computed from the instrumental magnitudes, provided that the data which correspond to the two passbands used for the given colour are taken closely in time. In fact, the sky brightness is known to have quite a strong time evolution even in moonless nights and far from twilight (Walker 1988b; Pilachowski et al. 1989; Krisciunas 1990; Leinert et al. 1998) and using magnitudes obtained in different conditions would lead to wrong colours. On the other hand, very often FORS1 images are taken in rather long sequences, which make use of the same filter; for this very reason it is quite rare to have close-in-time multi-band observations. Due to this fact and to allow for a general and uniform approach, we have decided to use constant sky colours for the colour correction. In fact, the colour terms are small, and even large errors on the colours produce small variations in the corresponding colour correction.

For this purpose we have used color-uncorrected sky brightness values obtained in dark time, at airmass $z\leq1.3$ and at a time distance from the twilights $\Delta t_{{\rm twi}}\geq2.5$ hours and estimated the typical value as the average. The corresponding colours are shown in Table 2, where they are compared with those obtained at other observatories. As one can see, B-Vand V-R show a small scatter between different observatories, while U-B and V-I are rather dispersed. In particular, V-I spans almost a magnitude, the value reported for Calar Alto being the reddest. This is due to the fact that the Calar Alto sky in Iappears to be definitely brighter than in all other listed sites.

Now, given the color terms reported in Table A.2, the color corrections $\gamma\times C_{{\rm sky}}$ turn out to be -0.02 $\pm$ 0.02, -0.09 $\pm$ 0.02, +0.04 $\pm$ 0.01, +0.02 $\pm$ 0.01 and -0.08 $\pm$ 0.02 in U, B, V, R and I respectively. The uncertainties were estimated from the dispersion on the computed average colours, which is $\sigma_{\rm C}\simeq$ 0.3 for all passbands. We emphasize that this rather large value is not due to measurement errors, but rather to the strong intrinsic variations shown by the sky brightness, which we will discuss in detail later on. We also warn the reader that the colour corrections computed assuming dark sky colours are not necessarily correct under other conditions, when the night sky emission is strongly influenced by other sources, like Sun and moon. At any rate, colour variations of 1 mag would produce a change in the calibrated magnitude of $\sim$0.1 mag in the worst case.

Due to the increased depth of the emitting layers, the sky becomes inherently brighter for growing zenith distances (see for example Garstang 1989; Leinert et al. 1998). In order to compare and/or combine together sky brightness estimates obtained at different airmasses, one needs to take into account this effect. The law we have adopted for the airmass compensation and its ability to reproduce the observed data are discussed in Appendix C (see Eq. (C.3)). After including this correction in the calibration equation and neglecting the error on X, we have computed the global rms error on the estimated sky brightness in the generic passband as follows:

$$% \sigma_{M_{{\rm sky}}}\simeq\sqrt{\delta^2_{m_{{\rm sky}}}+... ...gma^2_{\rm C} +C^2\;\sigma^2_\gamma +(X-1)^2\;\sigma^2_\kappa}$$ (2)

where $\sigma_{m_0}$, $\sigma_C$, $\sigma_\gamma$ and $\sigma_K$ are the uncertainties on the zero point, sky colour, colour term and extinction coefficient respectively. Using the proper numbers one can see that the typical expected global error is 0.03 $\div$ 0.04 mag, with the measures in V and R slightly more accurate than in U, B and I.