tokens: baryon fraction. Once M g is specified, the gas density is normalized such that, 2.3 Ionization balance In this paper, we consider models that may represent the most metal-poor DLAs currently known. Given the presumably minimal level of recent star formation in these systems, it is reasonable to assume that the local sources of ionizing photons are sub-dominant relative to the extragalactic background. In what follows, we have therefore assumed that the surface of the most metal-poor DLAs is illuminated solely by the Haardt Madau ( 2012 ) ultraviolet/X-ray background radiation from quasars and galaxies with intensity J 0 (ν). We also explore simple power-law models for the ionizing background, which are equivalent to the ‘ table power law slope α ’ command within cloudy , where f (ν) ∝ ν α . We assume that the distributions of gas and dark matter are spherically symmetric. Under this assumption, gas at a distance r from the centre of a dark matter halo is irradiated by background photons from all directions. The radiation field is therefore attenuated by a different amount depending on the density distribution of the gas along a given direction. If J 0 (ν) represents the unattenuated radiation field at a frequency ν, and 2πsin (θ)dθ/4π is the fraction of sky with an optical depth τ(ν, r , θ), then the intensity of the radiation field at a distance r from the centre of the cloud is given by where μ = cos θ, and i corresponds to every element/ion stage that we consider in this work. Unfortunately, our calculations are computationally too demanding to include every ionization stage for all elements with atomic number 30 (unlike cloudy ). We have therefore incorporated only the most abundant metals that are commonly observed in metal-poor DLAs, including H i , D i , He i , He ii , C i –C iv , N i –N iv , O i –O iv , Mg i –Mg iv , and Si i –Si iv . We have adopted the photoionization cross-section, σ i (ν), of each element using the compilation by Verner et al. ( 1996 ). At a distance r from the centre of the halo, the column density of species i in the direction θ can be calculated by integrating the volume density of species i, n i ( R ), from a radial distance r to the virial radius 2 (i.e. integrate along the path defined by the dashed line in Fig. 1 ): where equation ( 12 ) is used for 0 ≤ θ ≤ π/2 and equation ( 13 ) is used for π/2 θ ≤ π. The rate of primary ionizing photons at a given radius from the cloud centre of each species is then where h ν i is the ionization energy 3 of ion i . Each radiation field considered in this work is finely interpolated around the ionization energy of each ion to ensure numerical accuracy in the integrations. Figure 1. The coordinate system used to calculate the angular dependence of the column density of each species at a given radial coordinate (see equations 12 and 13 ). The rate of collisional ionization, Γ ci ( T ), is incorporated in our models using the Dere ( 2007 ) rate calculations.
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