Appendix C: Airmass dependency

The results obtained by Garstang (1989) can be used to derive an approximate expression for the sky brightness dependency on the zenith distance, as already pointed out by Krisciunas & Schaefer (1991). If we assume that a fraction f of the total sky brightness is generated by the airglow and the remaining (1-f) fraction is produced outside the atmosphere (hence including zodiacal light, faint stars and galaxies), Garstang's Eq. (29) can be rewritten in a more general way as follows:

$$% b(Z)=b_0\;[ (1-f) + f\;X)]$$ (C.1)

where X, the optical pathlength along a line of sight, is given by

$$% X=\left(1-0.96\;{\rm sin}^2Z\right)^{-1/2}$$ (A.2)

Z being the zenith distance. For the hypothesis on which this result is based, the reader is referred to Garstang's original pubblication and to the analysis presented by Roach & Meinel (1955). Here we just recall that Eqs. (C.1) and (C.2) were obtained assuming that the extra-terrestrial fraction (1-f) comes from infinity, while the airglow emission is generated in a layer (the so called van Rhjin layer) placed at 130 km from the Earth's surface.

While the intensity of the extra-terrestrial component is independent on the zenith distance, this is not true for the airglow, due to the variable depth of the emitting layer along the line of sight. This is taken into account by the term X, which grows towards the horizon.

 

 

Table C.1: Zenith corrections (mag) computed according to Eq. (C.3). The values have been calculated using Paranal average extinction coefficients $\kappa$ (see Table A.1) and f= 0.6. The zenith distance Z is expressed in degrees, while X is in airmasses.

Z	X	$\Delta U$	$\Delta B$	$\Delta V$	$\Delta R$	$\Delta I$
10.0	1.01	0.00	0.01	0.01	0.01	0.01
20.0	1.06	0.01	0.03	0.03	0.03	0.04
30.0	1.15	0.03	0.06	0.08	0.08	0.08
40.0	1.29	0.05	0.11	0.14	0.15	0.16
50.0	1.51	0.07	0.18	0.24	0.26	0.27
60.0	1.89	0.08	0.27	0.37	0.40	0.42

Figure C.1: Upper panel: sky brightness variation (in mag arcsec-2) as a function of zenith distance expected from Eq. (C.3) for f= 0.6. For each passband the mean Paranal extinction coefficients presented in Table A.1 were adopted. Lower panel: expected colour variation as a function of zenith distance. In both plots the upper scale reports the optical pathlength X, expressed in airmasses.

Figure C.2: B,V,R and I sky brightness during dark time as a function of airmass. The solid line traces Eq. (C.3), while the dotted lines are placed at $\pm$0.4 mag from this law.

In his model Garstang (1989) considers three different mechanisms for the background light to reach the observer once it leaves the van Rhjin layer: direct transmission, aerosol scattering and Rayleigh scattering. As a matter of fact, the first channel dominates on the others (see Fig. 5 in Garstang 1989) which, in a first approximation, can be safely neglected. One is therefore left with the propagation of the sky background light across the lower atmosphere, which is described by Garstang's Eq. (30) through the extinction factor EF. Now, if $\kappa$is the extinction coefficient for a given passband (in mag airmass^-1), we can write $EF\simeq 10^{-0.4\;(X-1)}$, so that the expected change in the sky brightness as a function of X is given by:

$$% \Delta m = -2.5\;\log [(1-f)+f\;X] + \kappa \; (X-1)$$ (3)

We notice that using f= 1 in this equation gives Eq. (2) of Krisciunas & Schaefer (1991), which gives slightly larger values for $\Delta m$than Eq. (C.3).

Figure C.3: Sky brightness sequences on six different moonless nights at Paranal in the I passband, corrected for zodiacal light contribution. The dotted line connects chronologically the various data points, while the filled dot indicates the first observation in each time series. Finally, the solid line traces Eq. (C.3) with f= 0.7.

The expected behaviour is plotted in Fig. C.1, where we present the predicted sky brightness (upper panel) and sky colour variations (lower panel) for the different passbands. For f we have assumed a typical value of 0.6 (see Garstang 1989 and Benn & Ellison 1998). As one can see the correction becomes relevant for $Z\geq$ 30$^\circ$ and reaches values as large as $\sim$0.4 mag at Z= 60$^\circ$(see also Table C.1). These predictions are consistent with the measurements obtained by Pilachowski et al. (1989), Mattila et al. (1996), Leinert et al. (1995) and Benn & Ellison (1998). In particular, Pilachowski et al. (1989) have clearly detected the sky inherent reddening (see their Fig. 3), which is in fair agreement with the predictions of Eq. (C.3) for the (B-V) colour (see also Fig. C.1, lower panel).

Our data confirm these findings, as it is shown in Fig. C.2, where we have plotted Paranal dark time measurements for B, V, R and Ipassbands. Even though the scattering due to night to night variations is quite large, the overall trend is consistent with the predictions of Eq. (C.3). The U data, which reach airmass X= 1.36 only, do not show a clear sky brightness increase, and this is again compatible with the expected value ($\sim$0.05 mag).

The airmass dependency is better visible when considering data obtained during single nights, when the overall sky brightness is reasonably stable. This is illustrated in Fig. C.3, where we present six very good series all obtained in the I passband during moonless nights at Paranal. For comparison, we have plotted Eq. (C.3) with f= 0.7, which reproduces fairly well the observed trend from airmass 1.2 to 2.0. As one can see, smooth deviations with peak values of $\sim$0.1 mag arcsec^-1 are detected within each single night, while the range spanned by the zenith extrapolated values is as large as $\sim$0.7 mag. The fact that a comparably large dispersion is seen in the other filters (see Fig. C.1) suggests that night-to-night variations of this amplitude are common to all optical passbands.

We note that on April 7, 2000 and January 1, 2001 the observed slope was higher than predicted by Eq. (C.3). This implies that the contribution by the airglow to the global sky brightness in those nights was probably larger than the value we have used here, i.e. 70%. This hypothesis is in agreement with the fact that on 07-04-2000 and 01-01-2001 the extrapolated zenith value was 0.4-0.5 mag brighter than in all other nights shown in Fig. C.3.