Appendix D: The Walker effect revisited

In this section we discuss and re-examine the findings published by Walker (1988b). In Fig. D.1 we have plotted the B data of his Table 1, with the exception of July 8, 1980 and May 4, 1984 (since they cover a very small time range with 2 and 3 points only) and the two data points marked by Walker with an asterisk in the sequences of May 11, 1983 and Apr 28, 1987. As one can see, there are only two dates in which a clear decreasing trend is visible, i.e. May 11, 1986 and April 25, 1987. Linear squares fitting to the two data sets give slopes of $\sim$0.08 $\pm$ 0.02 mag hour^-1 both in B and V filters, which turn into a sky brightness decrease of 0.48 $\pm$ 0.12 mag during the first 6 hours of the night. Since Walker's data have been collected across a full sunspot cycle, possible systematic variations due to the solar activity have to be removed. This can be achieved shifting all time sequences in order to have the same sky brightness at some reference time, an operation that also has the effect of correcting for night-to-night variations in the overall sky brightness. For this purpose we have performed, for each date, a linear least squares fit to the data and we have interpolated the resulting straight line at a time distance from the evening twilight $\Delta t_{{\rm etwi}}=$ 2 hours. We note that a similar procedure must have been followed by Walker (1988b), since in his Fig. 2 all measurements refer to the magnitude at the end of evening twilight ( $\Delta t_{{\rm etwi}}=$ 0). Even though not explicitly mentioned in the paper, this implies that some extra/interpolation had to be performed, since estimates at $\Delta t_{{\rm etwi}}=$ 0 are practically never available.

Figure D.1: Zenith sky brightness at San Benito Mountain in the B passband from Walker (1988b). For clarity, the mean sky brightness has been subtracted to each time series.

Figure D.2: Variation in zenith sky brightness at San Benito Mountain with time after the end of astronomical twilight. Data have been corrected for differential zodiacal light contribution and all time series were normalised to the interpolated brightness at $\Delta t_{{\rm etwi}}=2$ hours. Dotted and solid lines trace a linear least squares fitting with and without the data of May 11, 1986 and April 25, 1987 respectively (empty circles). Data are from Walker (1988b).

To avoid meaningless extrapolations we have disregarded the data from July 28, 1981, which do not cover a suitable time range. Following Walker, we also have not included the data obtained in 1982, since they were most likely affected by the Chinchonal volcano eruption. Finally, to account for the differential contribution of zodiacal light, we have corrected Walker's measurements using the data by Levasseur-Regourd & Dumont (1980) and assuming that all observations were carried out at the zenith of the observing site. The results one obtains following this procedure are presented in Fig. D.2, which shows a much less convincing evidence for a systematic trend than Walker's Fig. 2. In that case in fact, a decrement of $\sim$0.4 mag is seen during the first four hours, a behaviour which is definitely not visible in our Fig. D.2. As a matter of fact, a linear least squares fitting gives a rate of 0.02 $\pm$ 0.01 and 0.03 $\pm$ 0.01 mag hour^-1 for B and V respectively, which reduce to 0.01 $\pm$ 0.01 and 0.02 $\pm$ 0.01 mag hour^-1 if one excludes the two sequences of May 11, 1986 and April 25, 1987 (empty circles). From these values we can deduce a maximum decrement of 0.1 and 0.2 mag in B and V respectively during the first 6 hours after the end of twilight. Therefore, our conclusion is that even though there is an indication for a systematic trend in Walker's data, the rate is significantly lower than was thought. Moreover, due to the limited number of nights and the small time coverage of several of the time series, the results one gets may depend on the behaviour recorded during a few well sampled nights. For example, all data at $\Delta t_{{\rm etwi}}>$ 5.5 hours were collected on April 28, 1987. This could explain why no other author has found the same strong effect when using larger data bases (see for instance Fig. 9 of Benn & Ellison 1998).