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Appendix C: narrow band filter calibrations

For a given astronomical source, the absolute flux $F_{\lambda}$ measured at the earth in the band-width $\Delta \lambda$, is proportional to the extinction corrected count-rate measured in $\Delta \lambda$, and inversely proportional to the peak of the filter transmission. Let's call the proportionality factor as the ``counts-to-energy'' conversion factor $S_{\lambda}$. Thus,


\begin{displaymath}
F_{\lambda} =
\frac{ S_{\lambda} \cdot C_{\lambda} \cdot
10^{ 0.4 \cdot k_{\lambda} \cdot X } }
{ T_{\lambda} }
\end{displaymath} (9)

where $C_{\lambda}$, $k_{\lambda}$, X and $T_{\lambda}$ are the count-rate, the extinction coefficient, the airmass and the transmission peak in the band-width $\Delta \lambda$.

The ``counts-to-energy'' conversion factor $S_{\lambda}$ is therefore a measure of the efficiency of the whole system (telescope + camera) and can be derived from the measurements of spectrophotometric standard stars. Using n standard stars, one can compute a mean $S_{\lambda}$:

\begin{displaymath}
S_{\lambda} = \frac{ \sum_{i=1}^n S_{\lambda}(i) }{ n}
\end{displaymath} (10)

The $S_{\lambda}(i)$ factor is then determined from the count-rate $C_{\lambda}(i)$ of the ith standard star by convolving the spectral energy distribution, $F_{\lambda}(i)$, of the star with the filter transmission, $T_{\lambda}(i)$, using the following relation:


\begin{displaymath}
S_{\lambda}(i) =
\frac{\int F_{\lambda}(i) \cdot T_{\lambd...
..._{\lambda}(i) \cdot
10^{ 0.4 \cdot k_{\lambda} \cdot X(i) } }
\end{displaymath} (11)

The last step in the above equation is justified by the fact that the filters are narrow and the flux of the standard star can be considered as constant and equal to the value, F0, in the centre of the band. Since the factor $\int T_{\lambda} d\lambda$ gives the equivalent width, $W_{\lambda}$, of the filter, equation (11) can be written as follows:
\begin{displaymath}
S_{\lambda}(i) =
\frac{F_{0}(i) \cdot W_{\lambda} }{ C_{\lambda}(i) \cdot
10^{ 0.4 \cdot k_{\lambda} \cdot X(i) } }
\end{displaymath} (12)

Therefore, the $S_{\lambda}$ factor is determined on the basis of the measured count-rate and the equivalent width of the filter. Adopting the extinction coefficients $k_{\lambda} = k_I$ = 0.08 for all the intermediate bands, and measuring the count-rate $C_{\lambda}(i)$ of each standard by means of aperture photometry, we obtained the $S_{\lambda}$ factors which are reported in Table 2 in units of 10 $^{-16}~erg \cdot s^{-1} \cdot cm^{-2} \cdot cts^{-1}$. Since the second observing run was partially non-photometric, the reported values are those for the first and third runs only.


Table 2: Mean $S_{\Delta \lambda }$ factors
Filter run1 run3  
(nm)      
$\lambda$753 5.57$\pm$0.26 6.36$\pm$0.24  
$\lambda$770 6.20$\pm$0.34 7.14$\pm$0.20  
$\lambda$790 7.70$\pm$0.27 8.94$\pm$0.23  
$\lambda$815 6.60$\pm$0.29 7.69$\pm$0.35  
$\lambda$837 6.67$\pm$0.28 7.28$\pm$0.62  
$\lambda$914 11.67$\pm$0.85 12.77$\pm$0.67  


next up previous
Next: Bibliography Up: The Capodimonte Deep Field Previous: Appendix B: limiting magnitudes
Juan Alcala
2002-02-05