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Physical Conditions in the Broad Line Region of $z\sim3$ Quasars: A Photoionization Method to Derive $r_{BLR}$

We present high S/N UV spectra for eight quasars at $z\sim3$\ obtained with VLT/FORS. The spectra enable us to analyze in detail the strongest emission features in the rest-frame range 1400-2000 Å of each source (\ciii, \siiii, \aliii, \siii, \civ\ and \siiv). Previous work indicates that a component of these lines is emitted in a region with well-defined properties i.e., a high density and low ionization emitting region). Flux ratios \aliii/\siiii, \civ/\aliii, \siiv/\siiii, \civ/\siiv\ and \siii/\siiii\ for this region permit us to strongly constrain electron density, ionization parameter and metallicity through the use of diagnostic maps built from {\sc CLOUDY} simulations. Reliable estimates of the product density times ionization parameter allow us to derive the radius of the broad line region \rb\ from the definition of the ionization parameter. The \rb\ estimate and the assumption of virialized motions in the line emitting gas yields an estimate for black hole mass. We compare our results with estimates obtained from the \rb\ -- luminosity correlation customarily employed to estimate black hole masses of high redshift quasars.