Through photoionization modeling, constraints on the physical conditions of three z ~ 1.7 single-cloud weak Mg II systems (W_r(2796) < 0.3A) are derived. Constraints are provided by high resolution R = 45,000, high signal-to-noise spectra of the three quasars HE0141-3932, HE0429-4091, and HE2243-6031 which we have obtained from the ESO archive of VLT/UVES. Results are as follows: (1) The single-cloud weak Mg II absorption in the three z ~ 1.7 systems is produced by clouds with ionization parameters of -3.8 < logU < -2.0 and sizes of 1-100 pc. (2) In addition to the low-ionization phase Mg II clouds, all systems need an additional 1-3 high-ionization phase C IV clouds within 100 km/s of the Mg II component. The ionization parameters of the C IV phases range from -1.9 < logU < -1.0, with sizes of tens of parsecs to kiloparsecs. (3) Two of the three single-cloud weak Mg II absorbers have near-solar or super-solar metallicities, if we assume a solar abundance pattern. Although such large metallicities have been found for z < 1 weak Mg II absorbers, these are the first high metallicities derived for such systems at higher redshifts. (4) Two of the three weak Mg II systems also need additional low-metallicity, broad Lya absorption lines, offset in velocity from the metal-line absorption, in order to reproduce the full Lya profile. (5) Metallicity in single-cloud weak Mg II systems are more than an order of magnitude larger than those in Damped Lya systems at z ~ 1.7. In fact, there appears to be a gradual decrease in metallicity with increasing N(HI), from these, the most metal-rich Lya forest clouds, to Lyman limit systems, to sub-DLAs, and finally to the DLAs.
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