3. Data analysis and results
Due to lack of observing time the image cubes exposure was relatively short. This forced the subsequent analysis to a restricted part of the coma of about 30 30 103 km. The signal-to-noise ratio in this region is about 40 for the brighter pixels at = 0.7 µm and 15 at = 0.4 µm. Fig. 2 shows the images at 0.7 µm of Hale-Bopp taken on 20/03 (top) and 22/03 (bottom), respectively. Although the images cover roughly the same portion of the coma, a comparison between them is made difficult by the diverse spatial scales and distorsions of the images. Moreover, the comet is not centered in the field due to pointing difficulties.
For comparison purpose, an undistorted CCD image of the comet in the Johnson red filter is shown in Fig.3 . It was taken at the Loiano Observatory (Italy) on 20/03 19.00 U.T..
An arc structure is visible in the sunward direction. In order to study possible colour variations occurring in the coma and connected to the physical properties of the grains, we have processed the two image cubes by means of the principal component analysis (PCA in the following, Davis 1986; Erard et al. 1991; Bell 1992; Chevrel et al. 1994). This is a common technique generally used to study the spectral content of image cubes and to individuate coherent spatial units exhibiting an extreme spectral behaviour. It uses a linear transformation of the data to translate and rotate them in a new coordinate system that maximizes the variance. After this transformation, the new components are statistically independent and the information is contained in few principal component (henceforth PC) images while most of the noise is segregated in the other components. Mathematically, the PCA involves the calculation of the eigenvectors of the variance-covariance matrix of the data and then the transformation of the data into a set of orthogonal axes that are a linear combination of the original data. The first transformed image normally depicts the average brightness of the coma while the other components contain the colour information and are generally pairwise differences between the original spectral channels. In Table 2 the relevant statistic for 3 components is reported. Eigenvalues for bands that contain information are an order of magnitude larger than those that contain only noise. The corresponding images are spatially coherent, while the noise images do not contain any spatial information.
Table 2. Relevant statistics of three principal components. The units are in digital numbers.
Spectral channels from 0.4-0.5 µm were excluded to avoid the C2 emission features. The first principal component images are albedo pictures and are similar to the images shown in Fig.2. In fact, brightness is the first cause of pixel spectral diversity within each image cube. The second PC images are shown in Fig. 4. The regions indicated by arrows (A and B) exhibit different spectra. These two regions are roughly the same in the two images and correspond to the antisunward and sunward directions, respectively.
Here, the spectral difference is linked mainly to the continuum emission, which is the second cause of pixel spectral variability within each image cube. Representative reflectivity spectra (samples from image cube 20/03 18.44 U.T.) of the regions labelled A and B are shown in Fig. 5. The reflectivities are normalized to 5 800 Å. Region B spectra have a rate of increase of the reflectivity greater than the region A spectra. In order to compare the reflectivity gradient to that of other comets, we have followed the method illustrated in Jewitt and Meech (1986b), defining the normalized reflectivity gradient between wavelength and , . Here is computed in the interval to and is the mean reflectivity in the - wavelength range. In the optical, 18% per 1000 Å (B region) and 5% per 1000 Å (A region). These values are within the range defined by other comets (e.g. Levy-Rudenko, P/Halley and P/Shoemaker 1984S, Jewitt & Meech 1986b). Moreover, the spectra of region A exhibit reflectivity values in the 0.4-0.5µm range higher than the respective region B spectra. This fact cannot be attributed to the airmass variation within the image cube. Actually, a systematic increasing of spectral slope with airmass does not exist. Moreover, the previous argumentation excludes also atmospheric refraction of the spectra to be the cause of the observed phenomenon. We do not exclude, however, that minor residual effects could still plague the data. Anyway, the absence of systematic trends of spectral redness in the pixels of regions A and B, rules out the possible instrumental nature of the effect shown in Figs. 4 and 5. The blue excess seen in the region A spectra could be also due to the rather strong C2 gas emission present in this wavelength interval. However, the excess still remains at the continuum wavelengths, such as 4450 Å and 5260 Å.
© European Southern Observatory (ESO) 1998
Online publication: April 15, 1998