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High resolution studies of the cores of elliptical galaxies are needed in order to understand the mechanisms of galaxy formation and evolution. Because ground based imaging is severely impaired by atmopheric seeing, we have undertaken a program of high resolution Hubble Space Telescope (HST) imaging of a magnitude-limited sample of 12 elliptical galaxies in the Virgo cluster. The data has been obtained using the Planetary Camera (PC) and the F555W filter (very close to the Johnson V band). The plate scale of the PC is 0.043 arcsec/pixel = 3.1 pc/pixel at an assumed distance of 14.7 Mpc to the Virgo cluster. In spite of the spherical aberration affecting the HST primary mirror, our modelling shows that deconvolution techniques allow us to recover to surface brightness parameters up to 0.3 arcsec from the centre of the galaxy, an order of magnitude improvement with respect to ground based data. For each galaxy we have determined the ellipticity, position angle of the major axis, deviations of the isophotes from pure ellipses, and the brightness profile inside a 15 arcsec radius, by iteratively fitting the isophotes.\\ All of the galaxies studied show a unique morphology in the inner 10 arcsec. Five galaxies are found to harbour dust, in agreement with previous estimates (e.g. Lauer 1985b, Ebner et al. 1988), seven of the galaxies have isophotes deviating from pure ellipses more than 1\% (cfr Peletier et al. 1990). For three galaxies, NGC4342, NGC4570, and NGC4623, we find the presence of a thin (less than 10 parsecs) elongated ($\approx$150 parsecs) nuclear structure, resembling an edge-on disk, although kinematic data is needed to confirm the presence of a separate disk component. None of the galaxies shows an isothermal core. This may be the due to the presence of a large nuclear mass concentrations, such as a nuclear black hole, the occurence of merging processes, or anisotropies in the velocity distribution.
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