3 Results

3.1 Observations

The results presented here are based in observations taken on 1998 April 6 at the Observatorio del Roque de los Muchachos (ORM) on the island of La Palma, using the INTEGRAL system (Arribas et al. 1998) at the 4.2-m William Herschel Telescope (WHT).

We used the equalized optical-fibre bundle described by Arribas et al. (1998; hereafter AMF98), which at the telescope focal plane consists of a rectangular array of 115 optical fibres of 0.45'' in diameter (see Fig. 1 of AMF98). The transmission of the fibres at the centre of the bundle are reduced in a controlled way in order to allow large dynamic range objects to be studied (i.e. equalized integral-field spectroscopy [EIFS]; see details in AMF98). This bundle was connected to the WYFFOS spectrograph (Bingham et al. 1994), which was equipped with a 316-groove mm^-1 grating, allowing the spectral range 5000-11000 Å to be observed with a mean resolution of 11 Å and a mean dispersion of about 6 Å per pixel. With this configuration we took several exposures of the reference star SAO 17796. The atmospheric transmission was very poor with high clouds. However, the seeing (inferred from the data) was about 1 arcsec during the observations. The object was observed under a mean airmass of 1.28. The individual spectra were reduced following the standard procedure using IRAF, and some private codes.

In Fig. 2 we present the set of spectra after standard reduction obtained for an exposure of the star SAO 17796. Each of these spectra corresponds to the light collected by an aperture (fibre). With the fibre bundle we have sampled the PSF (the image of the star). In absence of DAR all the spectra should be identical (except for their S/N) but due to DAR there are large differences among the spectra taken at different positions (apertures) of the focal plane (see Fig. 2). Note, for instance, spectra 55 and 57. Clearly, spectrum 55 has partially lost the blue light, while the opposite occurs with spectrum 57, which has lost part of the red.

3.2 Differential atmospheric refraction characterization

The first step for the correction of DAR effects is their characterization. For single objects this may be directly determined from the shifts among images generated at different wavelengths. For some extended objects point-like features in the observed field can be used. For instance, in Seyfert 1 galaxies, the location of the BLR emission from different lines (different $\lambda$) can be used to determine the DAR. However, in most cases no point-like features are available for extended objects, for which a displacement of the maximum at different wavelengths should not be directly interpreted as an effect of DAR. In these cases a simple model (e.g. Allen 1976) can be used to determine DAR effects.

In our simple case we have characterized DAR from the location of the maxima obtained from the maps generated at 6 different wavelengths. In Fig. 3 the distribution of the maxima of these maps is shown on sky coordinates. From this plot it is clear that DAR from 10500 Å to 6150 Å amounts to about 0.4 arcsec, the parallactic angle being $\sim$ 20 deg. This is in very agreement with the estimations by Fillipenko (1982) for an altitude of 2 km, taking into account that the object was observed with a mean airmass of 1.28. Figure 3 also indicates the high accuracy when determining the maxima.

  

Figure 2: Observed spectra from each aperture (fibre) of the fibre bundle corresponding to an exposure of the star SAO 17796. Each spectrum is plotted at its relative location in the telescope focal plane. The spectral range represented is 600-900 nm

  

Figure 3: Relative location of the maxima at different wavelengths (indicated in Å). The circle represents the fibre core in adequate scale

3.3 Correction

According to the concepts described in Sect. 2, we have determined the flux corrected from DAR for the aperture ${\it j}$ at the particular wavelength $\lambda_i$,

$$F^{\rm c}_{\lambda_i}(\alpha_j,\delta_j),$$ (1)

following the transformation:

$$F^{\rm c}_{\lambda_i}(\alpha_j,\delta_j)=F_{\lambda_i}(\alpha_j+\Delta\alpha_i,\delta_j+\Delta\delta_i),$$ (2)

where $\Delta\alpha_i$, and $\Delta\delta_i$ correspond to the shift due to DAR at the wavelength $\lambda_i$, taken as reference a given wavelength (10500 Å, in our case). In our case these shifts are measured from the observational data but could be also calculated from the refractive index of the air.

For each aperture ${\it j}$, Eq. (2) is applied for all wavelengths, $\lambda_i$.

Note that $F_{\lambda_i}(\alpha_j+\Delta\alpha_i,\delta_j+\Delta\delta_i)$ is determined by interpolation of the observed values $F_{\lambda_i}(\alpha_j,\delta_j)$. Such an interpolation was done with the help of the NAG routine E01SAF. For each $\lambda$ this routine generates an interpolated two-dimensional surface $F(\alpha,\delta$), which is continuous and has continuous first derivatives. The interpolant is then evaluated at the required position with the routine E01SBF.

In Fig. 4 we present the corrected spectra. We can see now that the large differences between the spectra obtained from the different apertures (Fig. 2) have disappeared.

  

Figure 4: Corrected spectra for each aperture (fibre) of the fibre bundle corresponding to an exposure of the star SAO 17796. Each spectrum is plotted at its relative location in the telescope focal plane. The spectral range represented is 600-900 nm