We present the UDF05 project, a HST Large Program of deep ACS (F606W, F775W, F850LP) and NICMOS (F110W, F160W) imaging of three fields, two of which coincide with the NICP1-4 NICMOS parallel observations of the Hubble Ultra Deep Field (HUDF). In this first paper we use the ACS data for the NICP12 field, as well as the original HUDF ACS data, to measure the UV Luminosity Function (LF) of z~5 Lyman Break Galaxies (LBGs) down to very faint levels. Specifically, based on a V-i, i-z selection criterion, we identify a sample of 101 and 133 candidate z~5 galaxies down to z_{850}=28.5 and 29.25 magnitudes in the NICP12 and in the HUDF fields, respectively. Using an extensive set of Monte Carlo simulations we derive corrections for observational biases and selection effects, and construct the rest-frame 1400 A LBG LF over the range M_{1400}=[-21.4, -17.1], i.e. down to ~0.04 L* at z~5, and complement it with data from the Subaru Deep Field (SDF) from Yoshida et al. (2006) to extend it to the brighter end (M_{1400}>-22.2). We show that: (i) Different assumptions regarding the SED distribution of the LBG population, dust properties and intergalactic absorption result in a 25% variation in the number density of LBGs at z~5; (ii) Under consistent assumptions for dust properties and intergalactic absorption, the HUDF is about 30% under-dense in z~5 LBGs relative to the NICP12 field, a variation which is well explained by cosmic variance; (iii) The faint-end slope of the LF is independent of the specific assumptions for the input physical parameters, and has a value of alpha ~ -1.6, similar to the faint-end slope of the LF that has been measured for LBGs at z~3 and z~6.
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