Surface Brightness Profiles

From the data taken during Cycle 1, surface brightness profiles have been obtained for 13 globular clusters in M31 (Fusi Pecci et al. 1993), leading also to the detection of a central cusp in a candidate post-core-collapse cluster (Bendinelli et al. 1993). Detailed tests and simulations have been carried out to verify the reliability of the quoted results obtained by applying various deconvolution methods to the observed data.

To further explore the influence on the various procedures of any variation in the size of the adopted PSF compared to the data matrix and in the adopted background, we have carried out here a new set of simulations based on the following assumptions:

a)

Adopted cluster model: King model, = 0.2 arcsec, = 1.75 at the M31 distance (700 kpc), matrix size: 512512. The simulated cluster has then been convolved with the FOC/96 + F430W observed PSF (star GD248, observed on 12/1/92), with no further noise added (i.e., background = 0).

b)

Deconvolutions with: i) Regularized Multi-Gaussians (RMG) (Bendinelli 1991), ii) Richardson-Lucy (R-L) (Richardson 1972, Lucy 1974, 1992), iii) MEM-IDL (Hollis et al. 1992), iv) MEM/MemSys5 (Weir 1991). The deconvolutions have been performed on data matrices of various size, down to 6464, as specified below.

c)

Surface brightness profiles obtained with VISTA.

As shown in Figs. 3 and 4, the following results have been obtained:

1. Both cluster data and PSF matrices have size 512512 and sky = 0 (Fig. 3a): the derived profiles are substantially coincident with the input model independent of the used deconvolution method.

2. PSF 512512, data matrix smaller and smaller (256256, 128128, 6464), sky = 0, RMG (Fig. 3b): the derived profiles remain unaffected a part from the reduction in radial extension.

3. PSF and data matrices smaller and smaller (256256, 128128, 6464), RMG (Fig. 3c): there is an increasing shift, parallel to the model, which could be easily calibrated as a function of the matrix size reduction factor if sky = 0, while an increasing divergence with respect to the model occurs if the ``local'' sky in the data matrix is adopted (Fig. 3d). For ``local'' sky we intend the average background estimated over the 4 corners of each frame.

4. PSF and data matrices smaller and smaller (256256, 128128, 6464), R-L and MEM (Fig. 4a-c): the results are qualitatively similar to those obtained with the RMG (note that the RMG deconvolution operates in the profile domain) but a much stronger edge effect appears.

In Fig. 4d we show the profiles of Fig. 4a differentially with respect to the results of the R-L over 512512. The models indicate that the use of smaller data matrices produces a shift in the overall brightness profile, which is found also when this test is performed on real data (e.g., the cluster Bo373 = G305).

In summary, it is evident that much care must be taken in order to ensure that all the necessary information is amply contained in the frames to be analyzed. The combination of an overestimated background (such as our ``local'' sky) coupled with an insufficiently extended PSF may lead to large discrepancies in the derived profiles, which could be partially reduced by the the use of the RMG deconvolution method, as all the profiles are implicitly extrapolated outside the actual matrix size, albeit with a Gaussian falloff.