| Abstract
| - Aims. We aim to understand the nature of the absorbing neutral gas in the galaxies hosting high-redshift long-duration gamma-ray bursts (GRBs) and to determine their physical conditions. Methods. A detailed analysis of high-quality VLT/UVES spectra of the optical afterglow of GRB 050730 and other Swift-era bursts is presented. Results. We report the detection of a significant number of previously unidentified allowed transition lines of Fe +, involving the fine structure of the ground term ( 6D 7/2, 6D 5/2, 6D 3/2, 6D 1/2) and that of other excited levels ( 4F 9/2, 4F 7/2, 4F 5/2, 4F 3/2, 4D 7/2, 4D 5/2), from the $z_{\mathrm{abs}}$ = 3.969, log N(H 0) = 22.10, damped Lyman- α (DLA) system located in the host galaxy of GRB 050730. No molecular hydrogen (H 2) is detected down to a molecular fraction of log f< -8.0. We derive accurate metal abundances for Fe +, S +, N 0, Ni +, and, for the first time in this system, Si + and Ar 0. The absorption lines are best-fit as a single narrow velocity component at $z_{\mathrm{abs}}$ = 3.96857. The time-dependent evolution of the observed Fe + energy-level populations is modelled by assuming the excitation mechanism is fluorescence following excitation by ultraviolet photons emitted by the afterglow of GRB 050730. This UV pumping model successfully reproduces the observations, yielding a total Fe + column density of log N = 15.49 ±0.03, a burst/cloud distance (defined to the near-side of the cloud) of d = 440 ±30 pc, and a linear cloud size of l = $520^{\rm +240}_{-190}$ pc. This application of our photo-excitation code demonstrates that burst/DLA distances can be determined without strong constraints on absorption-line variability provided enough energy levels are detected. From the cloud size, we infer a particle density of nH≈ 5-15 cm -3. Conclusions. We discuss these results in the context of no detections of H 2 and C i lines (with log N(C 0)/ N(S +) < -3) in a sample of seven $z > 1.8$ GRB host galaxies observed with VLT/UVES. We show that the lack of H 2 can be explained by the low metallicities, [X/H] < -1, low depletion factors, and, at most, moderate particle densities of the systems. This points to a picture where GRB-DLAs typically exhibiting very high H 0 column densities are diffuse metal-poor atomic clouds with high kinetic temperatures, $T_{\mathrm{kin}}$$\ga$ 1000 K, and large physical extents, l$\ga$ 100 pc. The properties of GRB-DLAs observed at high spectral resolution towards bright GRB afterglows differ markedly from the high metal and dust contents of GRB-DLAs observed at lower resolution. This difference likely results from the effect of a bias, against systems of high metallicity and/or close to the GRB, due to dust obscuration in the magnitude-limited GRB afterglow samples observed with high-resolution spectrographs.
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