The H i-to-H₂ Transition in a Turbulent Medium

Giant molecular clouds serve as the nurseries for new stars in our Galaxy and in external galaxies (McKee & Ostriker 2007). On global scales, observations of CO and dust show that the star formation rate (SFR) surface density (${{\rm{\Sigma }}}_{\mathrm{SFR}}$) correlates with the H₂ mass surface density (${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$), following an almost linear trend (Bigiel et al. 2008; Genzel et al. 2010; Schruba et al. 2011; Tacconi et al. 2013; Azeez et al. 2016). The presence of H₂ molecules is a basic ingredient for the formation of other heavy molecules, such as CO, OH, and H₂O, that serve as efficient coolants of cold gas (e.g., Herbst & Klemperer 1973; Sternberg & Dalgarno 1995; Tielens 2013; van Dishoeck et al. 2013; Bialy & Sternberg 2015). The study of far-ultraviolet (UV) shielding and the subsequent H i-to-H₂ conversion is of fundamental importance for star and molecule formation in the interstellar medium (ISM).

The H i-to-H₂ transition in the ISM of galaxies has been investigated by numerous authors over the last several decades through analytic and numerical modeling (e.g., Federman et al. 1979; van Dishoeck & Black 1986; Sternberg 1988; Kaufman et al. 1999; Goldsmith et al. 2007; Gnedin & Draine 2014; Liszt 2015), as well as via hydrodynamics simulations (e.g., Robertson & Kravtsov 2008; Gnedin et al. 2009; Glover et al. 2010; Dave et al. 2013; Thompson et al. 2014; Lagos et al. 2015; Hu et al. 2016) and observations (e.g., Savage et al. 1977; Reach et al. 1994; Rachford et al. 2002; Gillmon & Shull 2006; Lee et al. 2012; Balashev et al. 2014; Noterdaeme et al. 2015; Nakanishi & Sofue 2016). Analytic treatments of the H i-to-H₂ transition have been presented by Krumholz et al. (2008), McKee & Krumholz (2010), and Sternberg et al. (2014) using a Strömgren-type analysis for the total steady-state column density of H i that is maintained by an incident photodissociating flux. In particular, S14 derived a scaling relationship for the total H i column density in optically thick, uniformly irradiated slabs as a function of the far-UV flux, gas density, dust absorption cross section, and H₂ formation rate coefficient. Bialy & Sternberg (2016, hereafter BS16) presented an analytic procedure for generating atomic (H i) to molecular (H₂) density profiles for optically thick hydrogen gas clouds in Galactic star-forming regions. These studies have thus far been instrumental in interpreting emission-line observations of H i/H₂ interfaces (Lee et al. 2012; Bialy et al. 2015, 2017; Bihr et al. 2015; Burkhart et al. 2015; Maier et al. 2017), for estimating star formation thresholds in external galaxies (Leroy et al. 2008; Lada et al. 2012; Clark & Glover 2014; BS16; Burkhart & Loeb 2016), and for subgrid components in hydrodynamics simulations (Pelupessy et al. 2006; Thompson et al. 2014; Tomassetti et al. 2015).

Despite the progress toward an analytic theory for the physics of the H i-to-H₂ transition, no current theory includes realistic turbulent density fluctuations. The turbulent nature of molecular clouds is evident from a variety of observations, including nonthermal broadening (Stutzki & Guesten 1990; Dickey et al. 2001; Heiles & Troland 2003; Heyer & Brunt 2004), velocity/density power spectra (Stanimirović & Lazarian 2001; Swift & Welch 2008; Chepurnov & Lazarian 2009; Pingel et al. 2013; Chepurnov et al. 2015), and fractal and hierarchical structures (Elmegreen & Elmegreen 1983; Vazquez-Semadeni 1994; Burkhart et al. 2013a). Simulations and observations have shown that supersonic turbulence creates filaments and regions of high-density contrast (Kowal et al. 2007; Burkhart et al. 2009; Federrath et al. 2010). This behavior suggests that the assumption of uniform density in current analytic models for the H i-to-H₂ transition should be revisited.

The effects of turbulence on the chemical structure of interstellar clouds have been studied from various perspectives. Xie et al. (1995), Willacy et al. (2002), and Bell et al. (2011) studied the effects of the turbulent mixing of chemical species through a diffusion approximation. They found that atomic abundances (e.g., H, C, and O) may be significantly increased in the interiors of molecular clouds if the diffusion coefficient is large. Levrier et al. (2012) studied the chemical structure of turbulent photo-dominated regions (PDRs) using a postprocessing approach and found that the abundances of various molecules (e.g., H₂, CO, CH, and CN) exhibit strong deviations from a homogeneous PDR model (cf. Offner et al. 2013). Glover & Mac Low (2007), Glover et al. (2010), Micic et al. (2011), and Valdivia et al. (2016) performed magnetohydrodynamic (MHD) simulations and followed the time-dependent H₂ formation self-consistently. They focused on the molecular content and showed that strong density compressions created by supersonic turbulent flows rapidly produce H₂ on timescales of a few Myr.

In this paper, we study the effects of turbulent density perturbations on the H i/H₂ structure in PDRs, focusing on the atomic gas produced by photodissociation at the cloud boundaries. As we show, this gas is particularly useful for constraining the nature of turbulence, using 21 cm observations. We consider a twofold approach via (i) numerical MHD simulations supplemented by H₂/H i chemical balance calculation and (ii) analytic modeling, as well as introducing a novel method for constraining the Mach number and turbulence driving scale.

The paper is organized as follows. In Section 2, we provide a basic overview of H i-to-H₂ theory for uniform-density gas, as presented by S14 and BS16. In Section 3, we discuss the effect of density fluctuations and the validity of our chemical steady-state assumption when considering turbulence. In Section 4, we present the results of our MHD simulations. In Section 5, we present computations of H i/H₂ profiles and H i column density PDFs for turbulent media. In Section 6, we develop an analytic model for the H i column density distribution. In Section 7, we demonstrate how our analytic model may be used to constrain turbulent parameters from 21 cm observations. We discuss and summarize our results in Section 8.

In this section, we briefly review the theory of the H i-to-H₂ transition in steady-state, uniform-density gas. For a thorough discussion, we refer the reader to S14 and BS16.

At any cloud depth and for unidirectional radiation normal to the cloud surface, the H₂ formation-destruction equilibrium is given by

$$\begin{eqnarray}&&{Rn}\ {n}_{{\rm{H}}{\rm{i}}}=\displaystyle \frac{1}{2}{D}_{0}\ {f}_{\mathrm{shield}}({N}_{{{\rm{H}}}_{2}}){e}^{-{\sigma }_{g}N}\ {n}_{{{\rm{H}}}_{2}},\end{eqnarray} \tag{ 1 }$$

where R (cm³ s⁻¹) is the H₂ formation rate coefficient, $n={n}_{{\rm{H}}{\rm{i}}}+2{n}_{{{\rm{H}}}_{2}}$ is the total (atomic plus molecular) hydrogen volume density, and D₀ (s⁻¹) is the free-space H₂ photodissociation rate. In this expression, ${f}_{\mathrm{shield}}$ is the H₂ self-shielding function that depends on the H₂ column density N_H2 and the absorption-line Doppler broadening parameter b_D (km s⁻¹; Draine & Bertoldi 1996; S14). The factor ${e}^{-{\sigma }_{g}N}$ is the dust absorption attenuation term, where ${\sigma }_{g}$ (cm²) is the dust-grain absorption cross section per hydrogen nucleus integrated over the Lyman–Werner dissociation band (11.2–13.6 eV; hereafter LW band) and $N={N}_{{\rm{H}}{\rm{i}}}+2{N}_{{{\rm{H}}}_{2}}$ is the total (atomic plus molecular) column density. The factor of 1/2 accounts for absorption of half the radiation by the optically thick slab.

Assuming that all of the photodissociating radiation is absorbed in the cloud, the total H i column density converges to a finite value, N_{H i}. As shown by S14, assuming slab geometry and for constant density,

$$\begin{eqnarray}{N}_{{\rm{H}}{\rm{i}}}\ & = & \displaystyle \frac{1}{{\sigma }_{g}}\ \mathrm{ln}\left[\displaystyle \frac{\alpha G}{2}\ +\ 1\right]\\ & = & 5.3\,\times \,{10}^{20}\displaystyle \frac{1}{{\phi }_{g}Z^{\prime} }\ \mathrm{ln}\left[\displaystyle \frac{\alpha G}{2}\ +\ 1\right]\ {\mathrm{cm}}^{-2},\end{eqnarray} \tag{ 2 }$$

where, in the second equality,

$$\begin{eqnarray}&&{\sigma }_{g}=1.9\times {10}^{-21}{\phi }_{g}Z^{\prime} \ {\mathrm{cm}}^{2},\end{eqnarray} \tag{ 3 }$$

where $Z^{\prime}$ is the dust-to-gas ratio relative to the Galactic value, and ${\phi }_{g}$ is a factor of order unity that characterizes the dust absorption properties. In Equation (2), $\alpha \equiv {D}_{0}/({Rn})$ is the (dimensionless) ratio of free-space H₂ photodissociation and H₂ formation rates. The dimensionless factor $G\equiv {\sigma }_{g}\int {f}_{\mathrm{shield}}({N}_{{{\rm{H}}}_{2}}){e}^{-2{\sigma }_{g}{N}_{{{\rm{H}}}_{2}}}{{dN}}_{{{\rm{H}}}_{2}}$ is the "effective shielding factor" and may be expressed as $G=3.0\,\times {10}^{-5}{\phi }_{g}Z^{\prime} {(9.9/[1+8.9{\phi }_{g}Z^{\prime} ])}^{0.37}$ (BS16). The combination

$$\begin{eqnarray}&&\alpha G=\displaystyle \frac{{D}_{0}G}{{Rn}}=2.0\ {I}_{\mathrm{UV}}\left(\displaystyle \frac{30\ {\mathrm{cm}}^{-3}}{n}\right)\end{eqnarray} \tag{ 4 }$$

has the physical meaning of an effective dissociation parameter taking into account H₂-shielding and the competition with dust absorption. The numerical value in Equation (4) is for $R=3\times {10}^{-17}$ cm³ s⁻¹, ${\phi }_{g}Z^{\prime} =1$, and ${D}_{0}=5.8\,\times {10}^{-11}\ {I}_{\mathrm{UV}}$ s⁻¹ (S14), where ${I}_{\mathrm{UV}}$ is the radiation strength relative to the Draine (1978) field. The product $\alpha G$ may be small or large for realistic astronomical environments. For example, for the star-forming region W43 and the Perseus molecular cloud, Bialy et al. (2017) and Bialy et al. (2015) derived $\alpha G\sim 20$ and $\alpha G\sim 10$, respectively, whereas for a sample of dwarf irregular galaxies in the LITTLE THINGS survey (Hunter et al. 2012), Maier et al. (2017) deduced $\alpha G\lt 1$. For the cold neutral medium (CNM), which is in pressure equilibrium with the warm neutral medium (WNM), the $n/{I}_{\mathrm{UV}}$ ratio is restricted to the range $\approx 8\mbox{--}70$ cm⁻³ (Wolfire et al. 2003), giving ${(\alpha G)}_{\mathrm{CNM}}\approx 1\mbox{--}8$.

The left panel of Figure 1 shows the H i and H₂ fractional abundance profiles, ${x}_{{\rm{H}}{\rm{i}}}\ \equiv {n}_{{\rm{H}}{\rm{i}}}/n$ and $2{x}_{{{\rm{H}}}_{2}}\equiv 2{n}_{{{\rm{H}}}_{2}}/n$, as functions of cloud depth, as parameterized by the gas column density N. The various curves are for ${\phi }_{g}Z^{\prime} =1$, $\alpha G=2$, and a Doppler parameter ranging from ${b}_{D}=0.5$ to 8 km s⁻¹. With increasing depth, the radiation is absorbed by H₂ photodissociation events and dust absorption, and the gas makes the transition from atomic to molecular form. For larger b_D, the Doppler cores of the H₂ lines are broader, and the onset of self-shielding occurs at larger cloud depths. However, the H i-to-H₂ transition point is insensitive to b_D because it occurs deeper in the cloud, where the LW flux is absorbed in the H₂ damping wings (BS16).

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The right panel of Figure 1 shows the total H i column density, N_{H i}, as a function of $\alpha G$. For large $\alpha G$ (the "strong field limit"), dust absorption determines the H i column density, and N_{H i} is weakly dependent on $\alpha G$. For small $\alpha G$ (the "weak field limit"), H₂ self-shielding dominates and ${N}_{{\rm{H}}{\rm{i}}}\ \propto \alpha G$. As for the H i-to-H₂ transition points, N_{H i} is insensitive to b_D (for the same reason). The insensitivity of N_{H i} to the Doppler parameter is important in our analysis of turbulent media and the line broadening induced by turbulent motions.

As discussed above, two basic assumptions for the H i-to-H₂ density profiles and the H i column density (as shown in Figure 1 and given by Equation (2)) are a constant gas density n and a chemical steady state. In this paper, we relax the assumption of constant density by considering turbulent density fluctuations while retaining the assumption of a chemical steady state. Density fluctuations are naturally produced in the cold ISM by supersonic turbulence. Since the H₂ formation–to–removal rate ratio is proportional to density, a local increase in density increases the local H₂ fraction. Such a perturbation also affects deeper locations in the cloud through enhanced H₂ self-shielding, which depends on the H₂ column density.

For a given density n and a local (attenuated) dissociation rate D, the chemical time is

$$\begin{eqnarray}&&{t}_{\mathrm{chem}}=\displaystyle \frac{1}{2{Rn}+D}.\end{eqnarray} \tag{ 5 }$$

In the outer atomic layers, $D\gg 2{Rn}$, and ${t}_{\mathrm{chem}}\simeq 1/D$ is the photodissociation time, which is typically very short. For example, for unshielded gas, $D={D}_{0}/2=2.9\times {10}^{-11}$ s⁻¹ and ${t}_{\mathrm{chem}}=1.1\times {10}^{3}\,\mathrm{yr}$. Beyond the atomic-to-molecular transition, $D\lt 2{Rn}$, and ${t}_{\mathrm{chem}}\simeq 1/(2{Rn})$ is the H₂ formation time, which can become long. The strongest effect of the density perturbations will occur near the H i-to-H₂ transition points, where Rn and D are comparable. At the transition point, $D=2{Rn}$, and

$$\begin{eqnarray}&&{t}_{\mathrm{chem}}=\ \displaystyle \frac{1}{4{Rn}}\ \approx 2.7\displaystyle \frac{1}{{\phi }_{R}Z^{\prime} {T}_{2}^{1/2}{n}_{2}}\ \mathrm{Myr},\end{eqnarray} \tag{ 6 }$$

where the rate coefficient

$$\begin{eqnarray}&&R=3\times {10}^{-17}\ {T}_{2}^{1/2}\ {\phi }_{R}\ Z^{\prime} \ {\mathrm{cm}}^{3}\ {{\rm{s}}}^{-1}\end{eqnarray} \tag{ 7 }$$

Here, ${T}_{2}\equiv T/{10}^{2}$ K, ${n}_{2}\equiv n/{10}^{2}$ cm⁻³, and ${\phi }_{R}$ is a factor of order unity. Typically, ${\phi }_{R}\approx 1$; however, in some environments, the H₂ formation rate may be enhanced. For example, Habart et al. (2003, 2004) found that moderately illuminated PDRs, such as Oph W, S140, and IC 63, may have ${\phi }_{R}\approx 5$, considerably reducing the chemical time.

Irrespective of any density fluctuations, the chemical time must be short compared to the cloud lifetime, ${t}_{\mathrm{cloud}}$. This requires

$$\begin{eqnarray}&&n\geqslant \displaystyle \frac{1}{4{{Rt}}_{\mathrm{cloud}}}\approx 27\displaystyle \frac{1}{{\phi }_{R}Z^{\prime} {T}_{2}^{1/2}}\left(\displaystyle \frac{10\ \mathrm{Myr}}{{t}_{\mathrm{cloud}}}\right)\ {\mathrm{cm}}^{-3},\end{eqnarray} \tag{ 8 }$$

where we have normalized ${t}_{\mathrm{cloud}}$ to the characteristic lifetime of 10 Myr.

In a turbulent medium, we also require ${t}_{\mathrm{chem}}\leqslant {t}_{\mathrm{turb}}$, where ${t}_{\mathrm{turb}}$ is the characteristic time over which turbulent density fluctuations are formed and destroyed. The turbulent time is

$$\begin{eqnarray}&&{t}_{\mathrm{turb}}\simeq \displaystyle \frac{{L}_{{\rm{H}}{\rm{i}}}}{\delta {\nu }_{1d}({L}_{{\rm{H}}{\rm{i}}})},\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray}&&{L}_{{\rm{H}}{\rm{i}}}=\displaystyle \frac{1}{{\sigma }_{g}n}=1.7\ \displaystyle \frac{1}{{\phi }_{g}Z^{\prime} {n}_{2}}\ \mathrm{pc}\end{eqnarray} \tag{ 10 }$$

is the characteristic length scale of the H i layer and $\delta {\nu }_{1d}({L}_{{\rm{H}}{\rm{i}}})$ is the 1D velocity dispersion over L_{H i}. Following the line width–size relation (Larson 1981; McKee & Ostriker 2007), the 3D velocity dispersion over a length scale ℓ is

$$\begin{eqnarray}&&\delta \nu ({\ell })=\delta \nu ({L}_{\mathrm{drive}}){\left(\displaystyle \frac{{\ell }}{{L}_{\mathrm{drive}}}\right)}^{1/2}\ \ \ ({L}_{s}\leqslant {\ell }\leqslant {L}_{\mathrm{drive}}),\end{eqnarray} \tag{ 11 }$$

where ${L}_{\mathrm{drive}}$ is the outer driving scale and L_s is the sonic length for which $\delta \nu ({L}_{s})\equiv {c}_{s}$, where c_s is the sound speed. Defining the Mach number ${{ \mathcal M }}_{s}\equiv \delta \nu ({L}_{\mathrm{drive}})/{c}_{s}$ and assuming an isotropic velocity field, we get

$$\begin{eqnarray}{t}_{\mathrm{turb}} & = & \displaystyle \frac{\sqrt{3}}{{c}_{s}{{ \mathcal M }}_{s}}\ \displaystyle \frac{1}{{\sigma }_{g}n}{\left(\displaystyle \frac{{L}_{{\rm{H}}{\rm{i}}}}{{L}_{\mathrm{drive}}}\right)}^{-1/2}\\ & \approx & 4.0\displaystyle \frac{1}{{{ \mathcal M }}_{s}{T}_{2}^{1/2}{n}_{2}{\phi }_{g}Z^{\prime} }{\left(\displaystyle \frac{{L}_{{\rm{H}}{\rm{i}}}}{{L}_{\mathrm{drive}}}\right)}^{-1/2}\ \mathrm{Myr},\end{eqnarray} \tag{ 12 }$$

where, in the second equality, we use ${c}_{s}=0.72{T}_{2}^{1/2}$ km s⁻¹, assuming a mean particle mass of 1.6 times the proton mass, at the transition point where ${x}_{{\rm{H}}{\rm{i}}}=2{x}_{{{\rm{H}}}_{2}}$ and including helium.

The ratio of the chemical and turbulent times is then

$$\begin{eqnarray}&&\displaystyle \frac{{t}_{\mathrm{chem}}}{{t}_{\mathrm{turb}}}\approx 0.66{{ \mathcal M }}_{s}\displaystyle \frac{{\phi }_{g}}{{\phi }_{R}}{\left(\displaystyle \frac{{L}_{{\rm{H}}{\rm{i}}}}{{L}_{\mathrm{drive}}}\right)}^{1/2}.\end{eqnarray} \tag{ 13 }$$

Equations (12) and (13) are for ${L}_{s}\leqslant {L}_{{\rm{H}}{\rm{i}}}\leqslant {L}_{\mathrm{drive}}$. For ${L}_{{\rm{H}}{\rm{i}}}\gt {L}_{\mathrm{drive}}$, $\delta \nu \to \delta \nu ({L}_{\mathrm{drive}})$, and ${L}_{{\rm{H}}{\rm{i}}}/{L}_{\mathrm{drive}}$ should be replaced with unity. For ${L}_{{\rm{H}}{\rm{i}}}\lt {L}_{s}$, density perturbations are negligible, as they are smoothed out by pressure waves. At ${L}_{{\rm{H}}{\rm{i}}}={L}_{s}$, ${L}_{{\rm{H}}{\rm{i}}}/{L}_{\mathrm{drive}}=1/{{ \mathcal M }}_{s}^{2}$. Thus, for moderate Mach numbers, ${t}_{\mathrm{chem}}/{t}_{\mathrm{turb}}$ remains close to unity, possibly ranging from $0.66({\phi }_{g}/{\phi }_{R})$ to $0.66({\phi }_{g}/{\phi }_{R}){{ \mathcal M }}_{s}$. Throughout this paper, we assume chemical equilibrium. We will consider the more complicated time-dependent problem elsewhere.

Turbulent motions may also affect the H i/H₂ structure by shifting the frequencies of the H₂ absorption lines and reducing the efficiency of H₂ self-shielding. This affects the H i/H₂ profiles at intermediate depths, 10¹⁴ cm⁻² $\leqslant \,{N}_{{{\rm{H}}}_{2}}\ \leqslant {10}^{18}$ cm⁻², where absorption is dominated by the Doppler cores (Gnedin & Draine 2014). However, since most of the H i gas is accumulated at greater cloud depths, where the radiation is absorbed in the H₂ line-damping wings, the velocity shifts do not affect N_{H i} (see the discussion in Section 2). The Doppler broadening is important for the optically thin medium (to the LW radiation) but not for optically thick clouds that have fully converted H i to H₂. Since our focus is on optically thick gas, we assume a constant b_D = 2 km s⁻¹ throughout our calculations and do not include any corrections to the Doppler parameter or the H₂ self-shielding function (e.g., Gnedin & Draine 2014).

In this section, we use MHD simulations to obtain realistic density profiles for sub- and supersonic turbulent gas. Our simulations are isothermal and non–self-gravitating (cf. Jappsen et al. 2005). This allows a natural extension of the S14 H i-to-H₂ transition model, which is inherently isothermal, into the turbulent regime. Furthermore, an isothermal equation of state (EOS) allows a simple estimate of the density dispersion (Equation (17) below). In reality, the H i-to-H₂ transition takes place in a nonisothermal medium with active heating and cooling processes, e.g., depth-dependent photoelectric heating versus [C ii] emission-line cooling (Tielens & Hollenbach 1985; Sternberg & Dalgarno 1989). However, for moderate sonic Mach numbers (${{ \mathcal M }}_{s}\lesssim 5$), the density and column density PDFs are similar in simulations of isothermal and nonisothermal EOSs (e.g., Glover & Mac Low 2007; Federrath & Banerjee 2015).

We use a third-order-accurate hybrid, essentially nonoscillatory scheme (Cho & Lazarian 2002) to solve the ideal MHD equations,

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+{\rm{\nabla }}\cdot (\rho {\boldsymbol{\nu }})=0\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho {\boldsymbol{\nu }}}{\partial t}+{\rm{\nabla }}\cdot \left[\rho {\boldsymbol{\nu }}{\boldsymbol{\nu }}+\left(p+\displaystyle \frac{{B}^{2}}{8\pi }\right){\boldsymbol{I}}-\displaystyle \frac{1}{4\pi }{\boldsymbol{BB}}\right]={\boldsymbol{f}},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}-{\rm{\nabla }}\times ({\boldsymbol{\nu }}\times {\boldsymbol{B}})=0,\end{eqnarray} \tag{ 16 }$$

where ρ is the density, ${\boldsymbol{B}}$ is the magnetic field, p is the gas pressure, ${\boldsymbol{I}}$ is the identity matrix, and ${\boldsymbol{f}}$ is the specific force. We assume the zero-divergence condition ${\rm{\nabla }}\cdot {\boldsymbol{B}}=0$, periodic boundary conditions, and an isothermal EOS $p={c}_{s}^{2}\rho$. For the source term ${\boldsymbol{f}}$, we assume random large-scale solenoidal driving at a wave number $k\approx 2.5$ (i.e., 1/2.5 the box size). The simulations have 512³ resolution elements and have been employed in many previous works (Cho & Lazarian 2003; Kowal et al. 2007, 2009, 2011; Burkhart et al. 2009, 2010).

Each simulation is defined by the sonic Mach number ${{ \mathcal M }}_{s}\equiv | {\boldsymbol{\nu }}| /{c}_{s}$ and the Alfvénic Mach number ${{ \mathcal M }}_{A}\equiv | {\boldsymbol{\nu }}| /\langle {\nu }_{A}\rangle$, where ${\boldsymbol{\nu }}$ is the velocity, c_s and ${\nu }_{A}$ are the isothermal sound speed and Alfvén speed, and $\langle \cdot \rangle$ denotes averages over the entire simulation box. We show results for the ${{ \mathcal M }}_{s}=0.5$, 2, and 4.5 simulations, i.e., subsonic, transonic, and supersonic gas. As we show below, the value of the sonic Mach number strongly affects the variance of the density field. The simulations are sub-Alfvénic with ${{ \mathcal M }}_{A}=0.7$ (i.e., a strong magnetic field). We have also considered super-Alfvénic (${{ \mathcal M }}_{A}=2.0$) simulations and found that the results are weakly sensitive to the value of ${{ \mathcal M }}_{A}$. Because the simulations are non–self-gravitating, they are scale-free, and we may assign any desired physical scale for the box length and density (see Hill et al. 2008, Appendix). In this section, we keep the results general and do not apply any physical scaling to the simulations. We scale the simulations to physical units in Section 5.

4.1. The 3D Density Distribution

Figure 2 shows three random density cuts (upper panels) through the ${{ \mathcal M }}_{s}=0.5,2$, and 4.5 simulations. The color axis corresponds to $\mathrm{ln}x$, where $x\equiv n/\langle n\rangle$. The density is nearly uniform for the subsonic simulation, but once the Mach number exceeds unity, strong density fluctuations are generated. The lower panels show the PDFs of $\mathrm{ln}x$ for these simulations (shaded areas). The $\mathrm{ln}x$ distributions are nearly Gaussian with a standard deviation that increases with Mach number. This is in agreement with previous studies (e.g., Padoan et al. 1997; Passot & Vázquez-Semadeni 1998; Federrath et al. 2008; Price et al. 2011; Burkhart & Lazarian 2012; Molina et al. 2012) that found that x is lognormally distributed (and $\mathrm{ln}x$ is Gaussian), with

$$\begin{eqnarray}&&{\sigma }_{x}\simeq b{{ \mathcal M }}_{s},\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ln}x}^{2}\simeq \mathrm{ln}[1+{(b{{ \mathcal M }}_{s})}^{2}].\end{eqnarray} \tag{ 18 }$$

In these expressions, ${\sigma }_{x}$ and ${\sigma }_{\mathrm{ln}x}$ are the standard deviations of the x and $\mathrm{ln}x$ distributions, where x has a unit mean (by definition) and the mean of $\mathrm{ln}x$ is ${\mu }_{\mathrm{ln}x}=-(1/2){\sigma }_{\mathrm{ln}x}^{2}$. The proportionality constant b depends on the nature of the turbulent driving and ranges from 1/3 to 1 for pure solenoidal or compressive driving, respectively (Nordlund & Padoan 1999; Federrath et al. 2008, 2010). The blue curves in Figure 2 are Gaussians with ${\sigma }_{\mathrm{ln}x}$ as given by Equation (18) with $b=1/3$, appropriate for our solenoidally driven simulations. The agreement is not perfect due to small deviations from the phenomenological $\sigma \mbox{--}{{ \mathcal M }}_{s}$ relation, given by Equation (18).

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4.2. Line-of-sight–averaged Densities

We now discuss an important distribution that will play a crucial role in determining the H i column density PDF. For a column of length ℓ, we define the average density along a line of sight (LOS),

$$\begin{eqnarray}&&{x}_{{\ell }}\equiv \displaystyle \frac{{\int }_{0}^{{\ell }}x\ d{\ell }^{\prime} }{{\ell }}\ \ ,\end{eqnarray} \tag{ 19 }$$

where $0\leqslant {\ell }\leqslant {L}_{\mathrm{box}}$ and ${L}_{\mathrm{box}}$ is the simulation box length. For any given ℓ, different sight lines have different density profiles, and thus the set x_ℓ forms a random variable. We refer to the distribution of x_ℓ as the "LOS-averaged density distribution."

For a turbulent cascade, the density is correlated over all scales up to the driving scale. However, the correlation decreases with increasing spatial separation (Vazquez-Semadeni & Garcia 2001, hereafter VG01). To obtain an analytic description for the x_ℓ distribution, we assume that the correlation may be described with a single parameter, ${L}_{\mathrm{dec}}$, hereafter the "decorrelation scale," such that, for ${\ell }\lt {L}_{\mathrm{dec}}$, the density is effectively constant (i.e., maximally correlated), and, for ${\ell }\geqslant {L}_{\mathrm{dec}}$, the density cells are uncorrelated. The number of independent density cells along a LOS of length ℓ is then

$$\begin{eqnarray}&&{ \mathcal N }({\ell })=1+\displaystyle \frac{{\ell }}{{L}_{\mathrm{dec}}}\ \ ,\end{eqnarray} \tag{ 20 }$$

and the x_ℓ distribution may be viewed as the sampling distribution of the mean, for which

$$\begin{eqnarray}&&{\sigma }_{{x}_{{\ell }}}\simeq \displaystyle \frac{{\sigma }_{x}}{\sqrt{{ \mathcal N }({\ell })}}\ \ .\end{eqnarray} \tag{ 21 }$$

This distribution is often encountered in the calculation of errors in repeated measurements (Barlow 1989). For ${\ell }\ll {L}_{\mathrm{dec}}$, the LOS contains a single fluctuation ${ \mathcal N }\approx 1$, and ${\sigma }_{{x}_{{\ell }}}={\sigma }_{x}$. For ${\ell }\gg {L}_{\mathrm{dec}}$, ${ \mathcal N }\gg 1$, the LOS contains many turbulent fluctuations, and ${\sigma }_{{x}_{{\ell }}}\ll {\sigma }_{x}$ as the fluctuations are averaged out.

Figure 3 shows the ratio of the standard deviations, ${\sigma }_{{x}_{{\ell }}}/{\sigma }_{x}$, as a function of ${\ell }/{L}_{\mathrm{box}}$, as calculated for our ${{ \mathcal M }}_{s}=2$ and 4.5 simulations. We consider sight lines along the X (solid), Y (dashed), and Z (dotted) directions. The blue curve is a fit for the predicted relation

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{{x}_{l}}}{{\sigma }_{x}}={\left(1+\displaystyle \frac{{\ell }}{{L}_{\mathrm{dec}}}\right)}^{-1/2}={\left(1+\displaystyle \frac{{L}_{\mathrm{box}}}{{L}_{\mathrm{dec}}}\displaystyle \frac{{\ell }}{{L}_{\mathrm{box}}}\right)}^{-1/2},\end{eqnarray} \tag{ 22 }$$

with the best-fitted parameter

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{dec}}}{{L}_{\mathrm{box}}}=7.8\times {10}^{-2}\end{eqnarray} \tag{ 23 }$$

(equivalent to 40 out of 512 cells). Evidently, our simplified treatment for the density correlation gives a reasonable estimate for the x_l dispersion.

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In Figure 4, we show PDFs of $\mathrm{ln}{x}_{{\ell }}$ (shaded areas) for ${{ \mathcal M }}_{s}=2$ and 4.5 and ${\ell }/{L}_{\mathrm{box}}=0.25$, 0.5, and 1. The PDFs have distorted Gaussian shapes, becoming narrower with increasing ${\ell }$, as expected from Equation (22). The blue curves are Gaussians with standard deviations according to Equations (17) and (20)–(23). We conclude that x_ℓ is indeed well described by a lognormal with ${\sigma }_{{x}_{{\ell }}}=b{{ \mathcal M }}_{s}/\sqrt{{ \mathcal N }}$. We note that this relationship derived for ${\sigma }_{{x}_{{\ell }}}$ is complimentary to the column density variance–${{ \mathcal M }}_{s}$ relationship derived in Burkhart & Lazarian (2012; their Equation (4) with A = 0.11) in the limit that ${\ell }={L}_{\mathrm{box}}$. However, the relationship present here is more general and provides a method to determine the driving and decorrelation scales via measurement of the column density variance.

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It is instructive to write ${L}_{\mathrm{dec}}$ in terms of the driving scale. For all our simulations, ${L}_{\mathrm{drive}}={L}_{\mathrm{box}}/2.5$, and, with Equation (23), we get

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{dec}}}{{L}_{\mathrm{drive}}}=0.20.\end{eqnarray} \tag{ 24 }$$

The decorrelation scale is smaller than, but of order of, the driving scale. This is because the driving process introduces density (and velocity) correlations, which cascade down to smaller scales. Equation (24) is a general relation for ${L}_{\mathrm{dec}}$ and ${L}_{\mathrm{drive}}$, although the prefactor may depend on the driving details (e.g., compressional versus solenoidal). VG01 and Fischera & Dopita (2004) also studied the ${L}_{\mathrm{dec}}-{L}_{\mathrm{drive}}$ relation (using alternative methods) and obtained ${L}_{\mathrm{dec}}=0.33{L}_{\mathrm{drive}}$ and ${L}_{\mathrm{dec}}\,=0.13{L}_{\mathrm{drive}}$, respectively. Our value for the ${L}_{\mathrm{dec}}-{L}_{\mathrm{drive}}$ relation is also in good agreement with that of Kowal et al. (2007).

While ${L}_{\mathrm{dec}}\lt {L}_{\mathrm{drive}}$, it is typically larger than the sonic scale. For example, Equations (11) and (24) suggest that ${L}_{\mathrm{dec}}\geqslant {L}_{s}$ as long as ${{ \mathcal M }}_{s}\geqslant 2.2$. This is important for our model, because ${L}_{\mathrm{dec}}$ represents the scale below which the density becomes effectively uniform. But if ${L}_{\mathrm{dec}}\lt {L}_{s}$, then ${L}_{\mathrm{dec}}$ should be replaced everywhere with L_s.

In this section, we present atomic and molecular density profiles and integrated H i column density distributions for nonhomogeneous turbulent gas. We use the density field obtained from our MHD simulations and assume a unidirectional UV flux incident of the box from one side. As we discuss in the Appendix, our results depend weakly on the geometry of the radiation field (e.g., beamed versus isotropic). We solve Equation (1) to obtain the atomic and molecular density profiles along each LOS. We then integrate the H i densities to obtain the H i column density PDF.

5.1. Basic Parameters

For homogeneous gas, two parameters fully characterize the H i/H₂ equilibrium problem: (i) the $\alpha G$ parameter, which is proportional to ${I}_{\mathrm{UV}}/n$, and (ii) the dust absorption cross section ${\sigma }_{g}$, or, equivalently, ${\phi }_{g}Z^{\prime}$. Since n is no longer a constant when turbulent fluctuations are present, we define

$$\begin{eqnarray}&&\overline{\alpha G}\ \equiv \ \displaystyle \frac{{D}_{0}G}{R\langle n\rangle }=2.0\ {I}_{\mathrm{UV}}\left(\displaystyle \frac{30\ {\mathrm{cm}}^{-3}}{\langle n\rangle }\right),\end{eqnarray} \tag{ 25 }$$

where we have replaced n with the volume average $\langle n\rangle$ in Equation (4). We consider a wide range of $\overline{\alpha G}$ values from the weak ($\overline{\alpha G}\ll 2$) to the strong ($\overline{\alpha G}\gg 2$) field limits. For ${\sigma }_{g}$, we assume the standard value ${\sigma }_{g}=1.9\times {10}^{-21}\ {\mathrm{cm}}^{2}$, corresponding to ${\phi }_{g}Z^{\prime} =1$.

The H i column is accumulated over a typical length of

$$\begin{eqnarray}&&{L}_{{\rm{H}}{\rm{i}}}\equiv \displaystyle \frac{1}{{\sigma }_{g}\langle n\rangle }=5.7\left(\displaystyle \frac{30\ {\mathrm{cm}}^{-3}}{\langle n\rangle }\right)\displaystyle \frac{1}{{\phi }_{g}Z^{\prime} }\ \mathrm{pc}.\end{eqnarray} \tag{ 26 }$$

For a turbulent medium, the density fluctuations have typical lengths of the decorrelation scale ${L}_{\mathrm{dec}}$ (Section 4.2). Thus, for a turbulent medium, the ratio

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{dec}}}{{L}_{{\rm{H}}{\rm{i}}}}={\sigma }_{g}\langle n\rangle {L}_{\mathrm{dec}}\equiv {\tau }_{\mathrm{dec}}\end{eqnarray} \tag{ 27 }$$

enters as an additional parameter. The ${L}_{\mathrm{dec}}$-to-L_{H i} ratio has the physical meaning of a mean dust opacity over the decorrelation width, denoted by ${\tau }_{\mathrm{dec}}$. The ratio ${L}_{\mathrm{dec}}/{L}_{{\rm{H}}{\rm{i}}}={\tau }_{\mathrm{dec}}$ further determines the characteristic number of fluctuations along the H i length, through

$$\begin{eqnarray}&&{ \mathcal N }({L}_{{\rm{H}}{\rm{i}}})=1+\displaystyle \frac{{L}_{{\rm{H}}{\rm{i}}}}{{L}_{\mathrm{dec}}}=1+{\tau }_{\mathrm{dec}}^{-1}\end{eqnarray} \tag{ 28 }$$

(i.e., Equation (20) with ${\ell }={L}_{{\rm{H}}{\rm{i}}}$), which then controls the H i/H₂ structure. Following Equations (24) and (27), the driving scale is related to ${\tau }_{\mathrm{dec}}$ through

$$\begin{eqnarray}&&{L}_{\mathrm{drive}}=5.0{\tau }_{\mathrm{dec}}{L}_{{\rm{H}}{\rm{i}}}.\end{eqnarray} \tag{ 29 }$$

5.2. Profiles

We scale our simulations such that the average optical depth over the box is ${\tau }_{\mathrm{box}}\equiv {\sigma }_{g}\langle n\rangle {L}_{\mathrm{box}}=5.7$, ensuring H i-to-H₂ conversion for all sight lines (since ${\tau }_{\mathrm{box}}\gg 1$). Following Equations (23) and (28), ${\tau }_{\mathrm{box}}$ sets ${\tau }_{\mathrm{dec}}$ and ${ \mathcal N }({L}_{{\rm{H}}{\rm{i}}})$, giving ${\tau }_{\mathrm{dec}}=0.45$ and ${ \mathcal N }({L}_{{\rm{H}}{\rm{i}}})=3.2$. We use the above scaling for our results in this section and in Section 5.3. Following Equations (13) and (24), ${\tau }_{\mathrm{dec}}=0.45$ implies ${t}_{\mathrm{chem}}/{t}_{\mathrm{turb}}\,=0.44({\phi }_{g}/{\phi }_{R}){{ \mathcal M }}_{s}$. Thus, for the highest Mach number we consider (${{ \mathcal M }}_{s}=4.5$), the chemical time may exceed the turbulent time, unless the H₂ formation efficiency is enhanced (${\phi }_{R}\gt 2$) or the dust absorption efficiency is reduced (${\phi }_{g}\lt 1/2$).

In the upper panels of Figure 5, we show the density profiles, $x\equiv n/\langle n\rangle$, for three arbitrary sight lines for ${{ \mathcal M }}_{s}=0.5$, 2, and 4.5. The cloud depth in the abscissa is represented by the mean opacity $\langle \tau ({\ell })\rangle \equiv {\sigma }_{g}\langle n\rangle {\ell }$, ranging from 0 to ${\tau }_{\mathrm{box}}=5.7$. The horizontal bars in each panel represent the decorrelation opacity width ${\tau }_{\mathrm{dec}}=0.45$, which is comparable to the typical scales of density fluctuations. The middle and lower panels show the calculated H i profiles, ${x}_{{\rm{H}}{\rm{i}}}\ \equiv {n}_{{\rm{H}}{\rm{i}}}/n$, and the integrated H i column densities, ${N}_{{\rm{H}}{\rm{i}}}({\ell })\equiv {\int }_{0}^{l}{n}_{{\rm{H}}{\rm{i}}}\ d{\ell }^{\prime}$, for the corresponding LOSs, assuming $\overline{\alpha G}=2.0$ and ${\phi }_{g}Z^{\prime} =1$. For comparison, the dashed curves show the exponential decay of x_{H i} and the gradual buildup of ${N}_{{\rm{H}}{\rm{i}}}({\ell })$ as obtained by the uniform-density solution.³

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For the subsonic simulation (${{ \mathcal M }}_{s}=0.5$), the density remains nearly homogeneous, and the H i/H₂ profiles and integrated H i column densities remain close to the homogeneous density solution. As the Mach number increases and exceeds unity, the density fluctuations become substantial, and the H i (and H₂) density profiles become highly distorted. For the highly supersonic case (${{ \mathcal M }}_{s}=4.5$), the H i profiles exhibit an extreme scatter, differing by orders of magnitude in some locations. For example, at cloud depth $\langle \tau ({\ell })\rangle =3$, ${x}_{{\rm{H}}{\rm{i}}}\approx {10}^{-5}$, $2\times {10}^{-2}$, and 0.8 for the red, yellow, and blue LOSs, respectively. This increasing scatter with ${{ \mathcal M }}_{s}$ is further reflected in ${N}_{{\rm{H}}{\rm{i}}}({\ell })$ and the total (asymptotic) H i column density, N_{H i}. However, because N_{H i} is an integrated quantity, the perturbations are (partially) averaged out, and the scatter is much smaller than for x_{H i}. For example, for the three LOSs of ${{ \mathcal M }}_{s}=4.5$, the scatter in N_{H i} is less than 0.4 dex. In Section 5.3, we show the calculated PDFs for the total H i columns.

5.3. The H i Column Density Distribution

We integrate the H i profiles along all of the LOSs for the ${{ \mathcal M }}_{s}=0.5$, 2, and 4.5 simulations and obtain the H i column density distributions. In the upper panels of Figure 6, we show PDFs of ${\mathrm{log}}_{10}{N}_{{\rm{H}}{\rm{i}}}$ for $\overline{\alpha G}=0.2$, 2, and 20. The lower panels show the median (solid curves), mean (dotted curves, which almost converge with the median), and 68.3, 95.5, and 99.7 percentiles (about the median) as functions of $\overline{\alpha G}$. Evidently, the distributions become wide with increasing ${{ \mathcal M }}_{s}$ and decreasing $\overline{\alpha G}$. For example, for the ${{ \mathcal M }}_{s}=4.5$ simulation, the width of the 68.3 percentile is 0.2 and 0.5 dex for $\overline{\alpha G}=10$ and 0.1, respectively. For the ${{ \mathcal M }}_{s}=0.5$ simulation, the width of the 68.3 percentile ranges from 0.03 to 0.1 dex for $\overline{\alpha G}=10$ to 0.1. The widening of the H i PDF with increasing ${{ \mathcal M }}_{s}$ reflects the increasing spread of the density PDF, as illustrated in Figure 2 and the $\sigma -{{ \mathcal M }}_{s}$ relation (Equation (17)). The H i PDFs become narrow at large $\alpha G$ because, in the strong field limit, the H i-to-H₂ is very sharp as the radiation is absorbed by dust (exponential attenuation), resulting in a weak dependence of the H i column on gas density (Equation (2)).

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The median and mean are above the homogeneous solution for small $\overline{\alpha G}$ and below it for large $\overline{\alpha G}$. Importantly, for all $\overline{\alpha G}$, the deviations from the homogeneous solution remain small. For example, for $\overline{\alpha G}$ ranging from 0.1 to 10, the deviation is 0.08–0.24 dex for the ${{ \mathcal M }}_{s}=4.5$ case and 0.02–0.13 dex for ${{ \mathcal M }}_{s}=2$. Thus, the mean H i column is well approximated by the S14 formula for homogeneous gas, as given by Equation (2), with $n=\langle n\rangle$.

Interestingly, the shapes of the H i PDFs deviate from a lognormal. For all supersonic simulations, the N_{H i} distributions are strongly truncated at the highest ends and have extended tails at the lower ends of the distributions. This is due to the interaction of the propagating radiation with the nonuniform gas and the effect on the H₂ self-shielding. At any point inside the cloud, the H₂ and H i fractions depend on the local volume density of the gas. However, the H₂ self-shielding introduces a dependence on the accumulated H₂ column from the edge to the point of interest. This introduces a nonlinear dependence of the H i/H₂ fractions on the volume density profile of the gas. Any positive density perturbation along the column results in a disproportional increase of the H₂ fraction and reduced H i fraction from that point onward. This effect introduces a bias toward lower values of N_{H i}.

5.4.  ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ versus $\overline{\alpha G}$, ${{ \mathcal M }}_{s}$, and ${\tau }_{\mathrm{dec}}$

The width of the $\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}$ distribution depends on three parameters: $\overline{\alpha G}$, the Mach number ${{ \mathcal M }}_{s}$, and the decorrelation opacity ${\tau }_{\mathrm{dec}}$. The first encapsulates H₂ formation versus destruction, the second determines the width of the density distribution, and the last determines the frequency of density fluctuations along an H i column. We vary ${\tau }_{\mathrm{dec}}$ by modifying the simulation scaling ${\tau }_{\mathrm{box}}$ (see Equation (23)) and consider ${\tau }_{\mathrm{dec}}$ from 0.01 to 8.

In Figure 7, we plot ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ (natural logarithm) in the $\overline{\alpha G}-{\tau }_{\mathrm{dec}}$ plane for the ${{ \mathcal M }}_{s}=0.5$, 2, and 4.5 simulations. The dashed line marks the ${\tau }_{\mathrm{dec}}=0.45$ value used for Figures 5–6. As discussed in Section 5.3, ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ increases with increasing ${{ \mathcal M }}_{s}$ or decreasing $\overline{\alpha G}$. However, Figure 7 shows that ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ also has a strong dependence on ${\tau }_{\mathrm{dec}}$. For small ${\tau }_{\mathrm{dec}}$, the number of density fluctuations along a LOS, ${ \mathcal N }$, is large (${ \mathcal N }=1+{\tau }_{\mathrm{dec}}^{-1}$). Each LOS then samples a large portion of the (same) parent density PDF, and the different sight lines become more alike. As ${\tau }_{\mathrm{dec}}$ increases, ${ \mathcal N }$ decreases until, finally, for ${\tau }_{\mathrm{dec}}\gg 1$, ${ \mathcal N }\to 1$. Each LOS is then correlated over the entire L_{H i} scale, and its density is approximately uniform, drawn from the parent density PDF. The width of the H i column density is then maximal and reflects the width of the parent density PDF.

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In the following section, we derive the analytic formula for ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ as a function of $\overline{\alpha G}$, ${{ \mathcal M }}_{s}$, and ${\tau }_{\mathrm{dec}}$.

As shown in Figure 5, each LOS has a unique density profile with a complicated H i density structure. To obtain a simple analytic representation for the distribution of the H i columns, we approximate each LOS as containing a uniform density equal to the average along the H i length; i.e., for each LOS, we set the density equal to $\langle n\rangle {x}_{{L}_{{\rm{H}}{\rm{i}}}}$ (i.e., Equation (19) with ${\ell }={L}_{{\rm{H}}{\rm{i}}}$). Hereafter, we use a shortened notation and omit the subscript H i in ${x}_{{L}_{{\rm{H}}{\rm{i}}}}$. For each LOS, the H i column density is then given by

$$\begin{eqnarray}&&{N}_{{\rm{H}}{\rm{i}}}=\displaystyle \frac{1}{{\sigma }_{g}}\ \mathrm{ln}\left[\displaystyle \frac{\overline{\alpha G}}{2}\displaystyle \frac{1}{{x}_{L}}\ +\ 1\right].\end{eqnarray} \tag{ 30 }$$

The PDF of $\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}$ is thus

$$\begin{eqnarray}&&\displaystyle \frac{{df}}{d\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}=\left(1+\displaystyle \frac{2{x}_{L}}{\overline{\alpha G}}\right)\mathrm{ln}\left(\displaystyle \frac{\overline{\alpha G}}{2{x}_{L}}+1\right)\ \displaystyle \frac{{df}}{d\mathrm{ln}{x}_{L}}\ ,\end{eqnarray} \tag{ 31 }$$

where ${df}/d\mathrm{ln}{x}_{L}$ is the PDF of $\mathrm{ln}{x}_{L}$. Following the discussion in Section 4.2, x_L and $\mathrm{ln}{x}_{L}$ are approximately Gaussian and lognormal with standard deviations

$$\begin{eqnarray}&&{\sigma }_{{x}_{L}}=\displaystyle \frac{{\sigma }_{x}}{\sqrt{{ \mathcal N }}}\simeq \displaystyle \frac{b{{ \mathcal M }}_{s}}{\sqrt{1+{\tau }_{\mathrm{dec}}^{-1}}}\end{eqnarray} \tag{ 32 }$$

 and

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ln}{x}_{L}}={\mathrm{ln}}^{1/2}(1+{\sigma }_{{x}_{L}}^{2})\simeq {\mathrm{ln}}^{1/2}\left(1+\displaystyle \frac{{[b{{ \mathcal M }}_{s}]}^{2}}{1+{\tau }_{\mathrm{dec}}^{-1}}\right),\end{eqnarray} \tag{ 33 }$$

respectively, where the second equality follows from Equations (17) and (28).

Figure 8 shows the $\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}$ PDFs as obtained from the simulations (also shown in Figure 8), along with the analytic PDFs as given by Equation (31). The locations and widths of the analytic PDFs roughly follow the PDFs from the simulations, with some differences. First, the shapes of the analytic PDFs are symmetric, whereas the simulated PDFs are truncated at the high end and have left tails. This difference is expected because the analytic approximation does not account for the interaction of the radiation with the density perturbations along each LOS, and, as discussed in Section 5.3, this interaction introduces a preference for small H i columns.

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The lower panels of Figure 8 show the median (solid curves), mean (dotted curves), and 68.3, 95.5, and 99.7 percentiles about the median (shaded regions) as functions of $\overline{\alpha G}$. The dashed curves are the homogeneous solutions for comparison. The trend of an increasing dispersion with increasing ${{ \mathcal M }}_{s}$ or decreasing $\overline{\alpha G}$ is in agreement with the numerical results shown in Figure 6. The median and mean H i columns remain close to the homogeneous solution, also in agreement with the numerical results. An analytic expression for the median N_{H i} is obtained by substituting the median,

$$\begin{eqnarray}&&{x}_{L,\mathrm{med}}={e}^{-\tfrac{1}{2}{\sigma }_{\mathrm{ln}{x}_{L}}}\simeq {\left(1+\displaystyle \frac{{[b{{ \mathcal M }}_{s}]}^{2}}{1+{\tau }_{\mathrm{dec}}^{-1}}\right)}^{-1/2},\end{eqnarray} \tag{ 34 }$$

into Equation (30), giving

$$\begin{eqnarray}&&{N}_{{\rm{H}}{\rm{i}},\mathrm{med}}=\displaystyle \frac{1}{{\sigma }_{g}}\mathrm{ln}\left[\displaystyle \frac{\overline{\alpha G}}{2}{\left(1+\displaystyle \frac{{[b{{ \mathcal M }}_{s}]}^{2}}{1+{\tau }_{\mathrm{dec}}^{-1}}\right)}^{1/2}+1\right].\end{eqnarray} \tag{ 35 }$$

For moderate $b{{ \mathcal M }}_{s}$ (or if ${\tau }_{\mathrm{dec}}\ll 1$), ${x}_{L,\mathrm{med}}\approx 1$ and the median N_{H i} remain close to the homogeneous solution (see also Figure 8). Then, to a good approximation,

$$\begin{eqnarray}&&\langle {N}_{{\rm{H}}{\rm{i}}}\rangle \simeq {N}_{{\rm{H}}{\rm{i}},\mathrm{med}}\simeq \displaystyle \frac{1}{{\sigma }_{g}}\mathrm{ln}\left[\displaystyle \frac{\overline{\alpha G}}{2}+1\right],\end{eqnarray} \tag{ 36 }$$

similar to Equation (2) for homogeneous gas but with $\alpha G$ replaced by $\overline{\alpha G}$. This result is also confirmed by our numerical computations shown in Figure 6.

The standard deviation of $\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}$ may be approximated by

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}\simeq {\sigma }_{\mathrm{ln}{x}_{L}}{\left.\displaystyle \frac{d\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}{d\mathrm{ln}{x}_{L}}\right|}_{\mathrm{ln}{x}_{L}=0},\end{eqnarray} \tag{ 37 }$$

where the approximation becomes increasingly accurate for small ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ values. Plugging in Equations (30) and (33), we get

$$\begin{eqnarray}&&{\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}\simeq \displaystyle \frac{{\mathrm{ln}}^{1/2}\left(1+\tfrac{{[b{{ \mathcal M }}_{s}]}^{2}}{1+{\tau }_{\mathrm{dec}}^{-1}}\right)}{\left(1+\tfrac{2}{\overline{\alpha G}}\right)\mathrm{ln}\left(\tfrac{\overline{\alpha G}}{2}+1\right)}.\end{eqnarray} \tag{ 38 }$$

In this expression, the nominator is ${\sigma }_{\mathrm{ln}{x}_{L}}$ (Equation (33)), introducing the dependence on the turbulence parameters b, ${{ \mathcal M }}_{s}$, and ${\tau }_{\mathrm{dec}}$. As expected, ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ increases with increasing $b{{ \mathcal M }}_{s}$. It also increases with ${\tau }_{\mathrm{dec}}$ and becomes independent of τ_dec once ${\tau }_{\mathrm{dec}}\gg 1$. For ${\tau }_{\mathrm{dec}}\to 0$, ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}\propto {\tau }_{\mathrm{dec}}\to 0$. The dependence on the radiation intensity enters through the $\overline{\alpha G}$ parameter (Equation (25)). For $\overline{\alpha G}\ll 1$ (the weak field limit), ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}\simeq {\sigma }_{\mathrm{ln}{x}_{L}}$, and the dispersion is maximal and independent of $\overline{\alpha G}$. For $\overline{\alpha G}\gg 1$ (the strong field limit), ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}\,\simeq {\sigma }_{\mathrm{ln}{x}_{L}}/\mathrm{ln}(\overline{\alpha G}/2)$, and the distribution becomes narrow with increasing $\overline{\alpha G}$.

Figure 9 shows ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ as a function of ${\tau }_{\mathrm{dec}}$ and $\overline{\alpha G}$, as given by Equation (38). Like the numerical results (shown in Figure 7), the standard deviation increases as (a) ${{ \mathcal M }}_{s}$ increases, (b) ${\tau }_{\mathrm{dec}}$ increases, and (c) $\overline{\alpha G}$ decreases. Deviations from the numerical results exist and are expected, given that the analytic model introduces simplifying assumptions: (a) the density correlations are described by a single decorrelation scale, ${\tau }_{\mathrm{dec}}$; (b) the density distribution is lognormal and follows the ${\sigma }_{x}-{{ \mathcal M }}_{s}$ relation (Equation (17)); and (c) the ansatz that for each LOS the H i column is given by Equation (30) is adopted. The advantage of the analytic approximation is that it provides a smooth solution for ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ as a function of ${{ \mathcal M }}_{s}$ and b.

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In this section, we present a brief example demonstrating how our results for the width and mean of the H i column PDF may be used to analyze 21 cm observations toward molecular clouds, setting constraints on the Mach number and turbulence driving scale.

Based on 21 cm emission lines from the GALFA-H i Survey (Peek et al. 2011), Lee et al. (2012) obtained an H i map for the Perseus molecular cloud. Burkhart et al. (2015) derived the H i PDF and found it to be very narrow, with ${\sigma }_{{N}_{{\rm{H}}{\rm{i}}}}/\langle {N}_{{\rm{H}}{\rm{i}}}\rangle \,=0.13$. Based on absorption-line data from Stanimirović et al. (2014), Burkhart et al. (2015) obtained the Mach number distribution for the CNM around Perseus. They found that ${{ \mathcal M }}_{s}$ ranges from ${{ \mathcal M }}_{s}=1$ to 11, with a median ${{ \mathcal M }}_{s}=4$.

Unlike the Mach number, the H i PDF, being observed in emission, contains contributions from both the WNM and CNM phases. The former are typically subsonic and the latter supersonic (Heiles & Troland 2003; Wolfire et al. 2003). We decompose the observed ${\sigma }_{{N}_{{\rm{H}}{\rm{i}}}/\langle {N}_{{\rm{H}}{\rm{i}}}\rangle }=0.13$ into CNM and WNM components,

$$\begin{eqnarray}&&{\sigma }_{{N}_{{\rm{H}}{\rm{i}}}/\langle {N}_{{\rm{H}}{\rm{i}}}\rangle }={({\phi }_{C}{\sigma }_{{N}_{{\rm{H}}{\rm{i}},C/\langle {N}_{{\rm{H}}{\rm{i}},C}\rangle }}^{2}+{\phi }_{W}{\sigma }_{{N}_{{\rm{H}}{\rm{i}},W/\langle {N}_{{\rm{H}}{\rm{i}},W}\rangle }}^{2})}^{1/2},\end{eqnarray} \tag{ 39 }$$

where the subscripts C and W refer to CNM and WNM and ${\phi }_{C}$ and ${\phi }_{W}$ are the gas mass fractions in these phases. Stanimirović et al. (2014) found that, around Perseus, ${\phi }_{C}$ ranges between 0.1 and 0.5, with a median ${\phi }_{C}=0.35$. Assuming the median ${\phi }_{C}=0.35$ and that the WNM is subsonic and thus has a negligible H i dispersion (see Figures 6–9 above), we obtain ${\sigma }_{{N}_{{\rm{H}}{\rm{i}},C/\langle {N}_{{\rm{H}}{\rm{i}},C}\rangle }}=0.22$, or, equivalently,⁴  ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}},C}}=0.22$, for the CNM in Perseus.

Inspecting the ${{ \mathcal M }}_{s}=4.5$ panel in Figure 7, we see that, for $\overline{\alpha G}$ within the CNM range ${(\overline{\alpha G})}_{\mathrm{CNM}}=1\mbox{--}8$ (see Section 2), the 0.22 contour is obtained for ${\tau }_{\mathrm{dec}}={L}_{\mathrm{dec}}/{L}_{{\rm{H}}{\rm{i}}}=0.06$–0.3. These values of ${\tau }_{\mathrm{dec}}$ correspond to ${L}_{\mathrm{drive}}/{L}_{{\rm{H}}{\rm{i}}}=0.3\mbox{--}1.5$ (Equation (29)), i.e., the driving scale is of order of the H i scale length. Assuming a typical CNM density, $\langle n\rangle \approx 30$ cm⁻³, and for ${\phi }_{g}Z^{\prime} =1-2$, as suggested by Lee et al. (2012) for Perseus, we obtain (with Equation (26)) ${L}_{\mathrm{drive}}\sim 1\mbox{--}8$ pc.

If we do not neglect the WNM contribution in Equation (39), ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}},C}}$ would be smaller, further reducing ${\tau }_{\mathrm{dec}}$ and ${L}_{\mathrm{drive}}$. Our numerical computations in Figure 7 are based on the MHD simulations that assume solenoidal driving ($b=1/3$). For compressional or mixed driving, the contours in Figure 7 will shift downward (since increasing b is similar to increasing ${{ \mathcal M }}_{s};$ Equation (17)), and ${\tau }_{\mathrm{dec}}$ and ${L}_{\mathrm{drive}}$ will again decrease. Thus, our derived driving scale is an upper limit.

A similar conclusion may be drawn from the analytic model. Equation (38) predicts the standard deviation of $\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}$ as a function of b, ${{ \mathcal M }}_{s}$, ${\tau }_{\mathrm{dec}}$, and $\overline{\alpha G}$. Plugging in ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}},C}}\,=0.22$ and $\overline{\alpha G}=1\mbox{--}8$ and inverting Equation (38), we get

$$\begin{eqnarray}&&b{{ \mathcal M }}_{s}=C\sqrt{1+{\tau }_{\mathrm{dec}}^{-1}},\end{eqnarray} \tag{ 40 }$$

where $C=0.28\mbox{--}0.47$ for $\overline{\alpha G}=1\mbox{--}8$. This relation between the Mach number, the driving forcing parameter b, and the driving scale (${\tau }_{\mathrm{dec}}=0.2{L}_{\mathrm{drive}}/{L}_{{\rm{H}}{\rm{i}}};$ Equation (29)) is shown in Figure 10 for $b=1/3$, 1/2, and 1, corresponding to solenoidal, mixed, and compressional drivings. The widths of the strips correspond to the width of the $\overline{\alpha G}=1\mbox{--}8$ CNM range. For the median ${{ \mathcal M }}_{s}=4$ (dashed line), ${L}_{\mathrm{drive}}/{L}_{{\rm{H}}{\rm{i}}}$ ranges within 0.2–0.7, 0.1–0.3, and 0.02–0.07 for $1/3$, $1/2$, and 1, respectively.

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Interestingly, narrow H i PDFs were recently reported for more Galactic clouds—Ophiuchus, Orion A, Orion B, California, MonR2, and Rosette (Imara & Burkhart 2016)—suggesting that the presence of a small driving scale, of order of or smaller than the H i length, might be a general feature in molecular clouds and/or a considerable WNM component to the H i column density.

However, there are caveats to this analysis. First, the decomposition of the H i PDF from emission-line measurements into CNM and WNM components, which requires a CNM/WNM fraction from absorption measurements, is uncertain. Thus, it would be valuable to use absorption-line measurements directly to infer the H i PDF of the CNM. Second, accurate measurements of the Mach number are difficult due to LOS blending of features in position-position–velocity space. Third, the width of the H i PDF may be affected by (a) variations in ${I}_{\mathrm{UV}}$ across the observed region; (b) the chosen cutoff criteria of the observed H i cube, velocity range, and spatial extent; (c) the finite angular resolution, which may smooth out density fluctuations; and (d) optically thick H i that may produce artificially narrow H i PDFs (Burkhart et al. 2013b). While optical thickness is probably not a major issue for Perseus (Lee et al. 2015 obtained ∼10% corrections for the optically thick gas), it might be important in other giant molecular clouds (GMCs). A thorough observational analysis that addresses these uncertainties will be provided elsewhere. Model limitations are discussed in Section 8.

In this paper, we have studied the H i-to-H₂ transition and H i column densities maintained by far-UV radiation in turbulent media. We have used a suite of MHD simulations to produce realistic turbulent density distributions (Section 4). The density field is nearly lognormal, and the dispersion follows the PDF variance-sonic Mach number ($\sigma -{{ \mathcal M }}_{s}$) relation (Section 4). We find that, for supersonic gas, the density decorrelation length is related to the driving scale through ${L}_{\mathrm{dec}}=0.2{L}_{\mathrm{drive}}$. The decorrelation length determines the number of density fluctuations along a LOS, through ${ \mathcal N }=1+{\ell }/{L}_{\mathrm{dec}}$, where ℓ is the length scale along the LOS. This simple description allows us to analytically model the scale-dependent LOS-averaged density distribution (Section 4.2) and the H i column density PDF. We note that, while pure MHD simulations are scale-free, when the depth-dependent H i-to-H₂ transition (or any other chemical species) is of interest, the adopted simulation box length plays an important role. The adopted box scale determines ${L}_{\mathrm{dec}}$, which in turn defines the number of density fluctuations along an H i column (${ \mathcal N }=1+{L}_{{\rm{H}}{\rm{i}}}/{L}_{\mathrm{dec}}$), thus affecting the H i/H₂ structure (see Sections 5.1 and 5.4).

As we show in Section 5.2, once the turbulence becomes supersonic, strong density fluctuations are developed, and the atomic-to-molecular density profiles are significantly distorted relative to those for homogeneous uniform-density media. As a result, different LOSs differ in their total accumulated H i column density, N_{H i}. We calculate the PDF for N_{H i} (Sections 5.3–5.4) as a function of the governing physical parameters: (a) the sonic Mach number ${{ \mathcal M }}_{s}$, (b) the effective dissociation parameter $\overline{\alpha G}$, and (c) the decorrelation opacity ${\tau }_{\mathrm{dec}}$, which is related to the turbulence driving scale (${\tau }_{\mathrm{dec}}\equiv {L}_{\mathrm{dec}}/{L}_{{\rm{H}}{\rm{i}}}=0.2{L}_{\mathrm{drive}}/{L}_{{\rm{H}}{\rm{i}}}$).

We find that the mean and median N_{H i} are affected by turbulence; however, as long as $b{{ \mathcal M }}_{s}\sim 1$, or if ${\tau }_{\mathrm{dec}}\ll 1$, N_{H i} may be well approximated by the S14 and BS16 formula for uniform-density gas,

$$\begin{eqnarray*}&&{N}_{{\rm{H}}{\rm{i}}}=\displaystyle \frac{1}{{\sigma }_{g}}\mathrm{ln}\left(\displaystyle \frac{\overline{\alpha G}}{2}+1\right)\ \ ,\end{eqnarray*}$$

where

$$\begin{eqnarray*}&&\overline{\alpha G}\equiv 2.0\ {I}_{\mathrm{UV}}\left(\displaystyle \frac{30\ {\mathrm{cm}}^{-3}}{\langle n\rangle }\right)\ \end{eqnarray*}$$

is the effective dimensionless dissociation parameter, where ${I}_{\mathrm{UV}}$ is the free-space intensity of the far-UV field and $\langle n\rangle$ is the volume density. Here, ${\sigma }_{g}$ (cm²) is the dust-grain absorption cross section per hydrogen nucleus averaged over the LW band.

The major effect of turbulent density fluctuations is in producing a spread in the H i column distribution. For subsonic gas, the density is nearly uniform, the H i PDF is very narrow, and the solutions converge to the S14 and BS16 formula for uniform-density gas. As the Mach number increases, density fluctuations become substantial, and the H i PDF widens. As discussed in Section 5, the H i PDF also depends on $\overline{\alpha G}$ and ${\tau }_{\mathrm{dec}}$, becoming wider for small $\overline{\alpha G}$ or ${\tau }_{\mathrm{dec}}$. In Section 6, we present an analytic formula, Equation (38), for the standard deviation of the H i PDF as a function of ${{ \mathcal M }}_{s}$, $\overline{\alpha G}$, and ${\tau }_{\mathrm{dec}}$.

We demonstrate how our model may be combined with 21 cm observations toward GMCs to constrain turbulent parameters (Section 7). For Perseus (and in other Galactic clouds), the very narrow observed H i PDF may suggest small-scale driving, potentially pointing to the importance of multiscale turbulent driving in the CNM (Haverkorn et al. 2008; Yoo & Cho 2014). Alternatively, the narrow PDF may be caused by small-scale decorrelation lengths induced by the abrupt change in the chemical and thermal properties at the H i-to-H₂ transition. Observational caveats are discussed in Section 7.

Our results are for single-phased gas irradiated by far-UV radiation. In Section 7, we describe how our results may be applied to a mixed CNM/WNM medium, given estimates of the CNM and WNM fractions. This marks the importance of observations of both emission and absorption 21 cm lines, which provide constraints on the CNM fraction and its Mach number.

The values of observed H i column densities may depend on geometry (i.e., slabs versus spheres, or slab inclination) and the number of H i-to-H₂ transition layers along the observed sight lines. However, importantly, the turbulent parameters ${{ \mathcal M }}_{s}$ and ${\tau }_{\mathrm{dec}}$ are obtained from the standard deviation of the logarithmic H i column, ${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}$ (Equation (38)). Thus, our analysis is robust against any multiplication of the H i column, including number of clouds, geometry factors, and inclination corrections.

A basic assumption made in our analysis is that H i and H₂ are in a chemical steady state (see Section 3 for a discussion of the timescales). The coupling of time-dependent chemistry and turbulence may affect the H i PDF in various ways. For example, rapidly changing density fluctuations may alter the dispersion in the H i distribution, since H i and H₂ in different LOSs do not have time to react to the fast density changes. Turbulent mixing may increase the mean H i column by transferring molecular gas from inner shielded to outer unshielded regions, where it can rapidly dissociate and form H i. We plan to investigate time-dependent effects in a future work.

We conclude that the atomic-to-molecular density profiles and H i column density are affected by the turbulent nature of the CNM. For moderate Mach numbers, the homogeneous solution still provides a good estimate for the mean H i column density. The standard deviation of the H i PDF contains useful information regarding the turbulence properties. Our model, combined with 21 cm observations, may be used to constrain the sonic Mach number and turbulence driving scale of cold atomic gas.

We thank Enrique Vazquez-Semadeni, Sahar Shahaf, and Ewine van Dishoeck for fruitful discussions. We thank the referee for constructive comments that improved our paper. This work was supported in part by the PBC Israel Science Foundation I-CORE Program grant 1829/12, DFG/DIP grant STE1869/2-1 GE625/17-1, and the Raymond and Beverly Sackler Tel Aviv University—Harvard/ITC Astronomy Program. B.B. acknowledges support from the NASA Einstein Postdoctoral Fellowship and the Raymond and Beverly Sackler TAU-ITC Visiting Researcher fund.

In our radiative transfer analysis, we have assumed unidirectional beamed irradiation. Beamed irradiation is expected in the proximity of strong far-UV sources. For clouds immersed in the diffuse Galactic interstellar radiation field, isotropic irradiation may be a better approximation. S14 showed that, for uniform-density clouds exposed to isotropic fields, the H i column density is given by

$$\begin{eqnarray}&&{N}_{{\rm{H}}{\rm{i}}}=\displaystyle \frac{\langle \mu \rangle }{{\sigma }_{g}}\mathrm{ln}\left[\displaystyle \frac{\alpha G}{4\langle \mu \rangle }+1\right],\end{eqnarray} \tag{ 41 }$$

where $\langle \mu \rangle \equiv 0.8$ is a geometrical factor. For a given $\alpha G$, the flux (normal to the slab) for isotropic radiation is half the beamed flux; hence, the H i column density is smaller (see Section 2.3 in S14 for a full discussion and derivation of Equation (41)). Comparing Equation (41) with the beamed solution (Equation 2), we see that

$$\begin{eqnarray}&&{N}_{{\rm{H}}\,{\rm{i}}}^{\mathrm{iso}}(\alpha G)=0.8\ {N}_{{\rm{H}}\,{\rm{i}}}^{\mathrm{bm}}(\alpha G/1.6),\end{eqnarray} \tag{ 42 }$$

where ${N}_{{\rm{H}}\,{\rm{i}}}^{\mathrm{iso}}$ and ${N}_{{\rm{H}}\,{\rm{i}}}^{\mathrm{bm}}$ are the H i columns produced by isotropic and beamed fields, respectively. Equation (42) provides a simple transformation between the beamed and isotropic solutions for uniform-density gas. Such a transformation is particularly useful given that an isotropic field calculation is much more time-consuming.

However, Equation (42) may be less accurate for turbulent gas, because the density fluctuations will have different effects on the H i/H₂ structures for the different field geometries. As an illustrative example, consider a diffuse medium and a very dense clump. Let the clump be a thin disk of radius R, located near the slab surface at $z=r=0$ (cylindrical coordinates), where z is the axis normal to the slab along which we integrate N_{H i}. For beamed radiation, all rays originating at $r\leqslant R$ will pass through the clump, and N_{H i} will be very small for these sight lines, whereas for LOSs with $r\gt R$, the gas is diffuse, and N_{H i} will be very large. For isotropic radiation, inclined rays (originating at $r\gt R$) will penetrate behind the clump, and thus ${N}_{{\rm{H}}{\rm{i}}}(r\leqslant R)$ will increase compared to the beamed case. By contrast, for the isotropic field, gas at $r\gt R$ may still be partially shielded by the clump, leading to a decrease in ${N}_{{\rm{H}}{\rm{i}}}(r\gt R)$ compared to the beamed case. These opposite effects will tend to smooth out the contrast in the N_{H i} map and reduce the width of the H i PDF.

To test the effects of isotropic versus beamed irradiation, we have carried out a calculation of the H i/H₂ structure for isotropic radiation impinging the ${{ \mathcal M }}_{s}=4.5$ simulation box with $\overline{\alpha G}=2.6$ (the mean CNM value). The resulting H i column density map is shown in Figure 11 (left panel). The corresponding PDF is shown in the right panel (red shaded area). For comparison, the middle panel and the gray shaded area in the right panel show $0.8\times {N}_{{\rm{H}}{\rm{i}}}$ for beamed irradiation with $\overline{\alpha G}=2.6/1.6=1.63$ (as suggested by the scaling of Equation (42)). We see that the two maps and PDFs are not identical, as would be the case for a uniform-density medium. For the isotropic case, the H i map is smoother, and the low-N_{H i} features are more extended, as expected. The width of the isotropic PDF is somewhat narrower (${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}=0.15$) and has a lower mean ($\langle {N}_{{\rm{H}}{\rm{i}}}\ \rangle =1.5\times {10}^{20}$ cm⁻²) compared to the beamed field (${\sigma }_{\mathrm{ln}{N}_{{\rm{H}}{\rm{i}}}}=0.18$ and ${N}_{{\rm{H}}{\rm{i}}}=2.1\times {10}^{20}$ cm⁻²).

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Importantly, while there are differences between the beamed and isotropic cases, especially when considering individual LOSs, the relative differences in the mean and standard deviations are small (28% and 18%, respectively). We conclude that, to a good approximation, Equation (42) may be used as a transformation from our (efficiently computed) beamed field results to isotropic radiation fields.