SONYC - Substellar Objects in Nearby Young Clusters - is a survey program to investigate the frequency and properties of substellar objects with masses down to a few times that of Jupiter in nearby star-forming regions. For the ~1Myr old rho Ophiuchi cluster, in our earlier paper we reported deep, wide-field optical and near-infrared imaging using Subaru, combined with 2MASS and Spitzer photometry, as well as follow-up spectroscopy confirming three likely cluster members, including a new brown dwarf with a mass close to the deuterium-burning limit. Here we present the results of extensive new spectroscopy targeting a total of ~100 candidates in rho Oph, with FMOS at the Subaru Telescope and SINFONI at the ESOs Very Large Telescope. We identify 19 objects with effective temperatures at or below 3200 K, 8 of which are newly identified very-low-mass probable members of rho Oph. Among these eight, six objects have Teff <= 3000 K, confirming their likely substellar nature. These six new brown dwarfs comprise one fifth of the known substellar population in rho Oph. We estimate that the number of missing substellar objects in our survey area is ~15, down to 0.003 - 0.03 MSun and for Av = 0 - 15. The upper limit on the low-mass star to brown dwarf ratio in rho Oph is 5.1 +- 1.4, while the disk fractions are ~40% and ~60% for stars and BDs, respectively. Both results are in line with those for other nearby star forming regions.
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