© ESO, 2017

1. Introduction

Several observations highlight the presence of tiny, unresolved structure in atomic gas across a wide range of astrophysical environments. For instance, the wide, smooth emission lines in quasar spectra suggest the atomic gas close to the black hole has both a suprathermal velocity dispersion but low volume filling factor (e.g., Rees 1987; Arav et al. 1997). Moreover, studies of the diffuse gas in the halos of massive galaxies at redshifts z ~ 2−3 routinely find that these galaxies are filled with tiny clouds of neutral gas, again with a high covering factor, but with a low overall volume filling factor (e.g., Rauch et al. 1999; Cantalupo et al. 2014; Hennawi et al. 2015; Cai et al. 2017). Similar evidence for tiny-scale structure in neutral gas may be found in galactic winds and in high-velocity clouds in the Milky Way (see, e.g., McCourt et al. 2016, for a summary).

The physical origin of these clumps has been investigated recently by McCourt et al. (2016), who have found that cooling gas clouds are prone to rapid fragmentation akin to the Jeans instability. These authors have suggested this fragmentation rapidly “shatters” cold gas into tiny cloudlets of a characteristic size l ~ 0.1 pc(n/ cm^-3)^-1, or equivalently a column density N_cloudlet ~ 10¹⁷ cm^-2. McCourt et al. (2016) have argued that this length scale is consistent with a number of observational upper limits, but unfortunately such small scales are extremely difficult to probe directly in distant objects. In this paper, we show that radiative transfer of the resonant Lyα line can indeed probe sub-parsec scales, even in distant galaxies.

There are several reasons why the Lyα emission line hydrogen is an ideal probe for tiny-scale structures. As the most prominent transition line of the most abundant element, Lyα is a sensitive probe of neutral gas enabling us to study even the most distant objects in the Universe. Recently, instruments such as MUSE (Bacon et al. 2010) reveal the ubiquity of Lyα emission throughout the observable space. In particular, Lyα is used to study our galactic neighborhood (Hayes 2015), galaxies at the peak of cosmic star formation (Barnes et al. 2014), and the later stages of reionization (Dijkstra 2014).

Apart from this practical reason, the resonant nature of the Lyα transition gives Lyα observations a potentially great constraining power in studying otherwise unresolvable structure. This is due to the strong frequency dependence of the neutral hydrogen scattering cross section, which leads to many orders of magnitude of variation in the photon mean free path. For instance, in a medium with one neutral hydrogen atom per cm^-3, a Lyα photon travels on average only ~ 1 AU if it is in the core of the line; however, this distance grows by nearly five orders of magnitude to ~ 0.5 pc if the frequency is shifted merely five Doppler widths (~ 60 km s^-1) away from line center. The mean free path is therefore also sensitive to gas motions on the scale of ~ (1−100) km s^-1, providing powerful constraints on the kinematic properties of galaxies and their surrounding environments.

In this paper, we revisit Lyα radiative transfer through a simplified clumpy medium consisting of spherical “clouds” of neutral hydrogen embedded in an ionized surrounding medium. While this setup has been considered many times before (e.g., Neufeld 1991; Hansen & Oh 2006; Laursen et al. 2013; Duval et al. 2014; Gronke & Dijkstra 2016), all of these previous studies have focused on a part of the parameter space with a relatively low (≲ 10) number of clumps per sightline. In light of the observations and shattering scenario discussed above, we now consider the limit with many more clouds per sightline, exploring the full range from ~ 1 to ~ 1000 s. We show that this has tremendous influence on the propagation of Lyα and we provide simple scaling relations that enable simple, order-of-magnitude calculations for Lyα radiative transfer through clumpy media.

Our paper is structured as follows. In Sect. 2, we discuss the problem analytically and estimate the expected results. In Sect. 3 we describe briefly our Lyα radiative transfer calculations and introduce our model. We present the simulation results in Sect. 4 with particular focus on the spectral shape and the Lyα escape fraction as well as their connections to a corresponding homogeneous medium. We then discuss the results in Sect. 5 before we conclude in Sect. 6.

2. Analytic results

We find several distinct regimes for Lyα radiative transfer in multiphase media, which we summarize in Fig. 1. In this section, we describe the physics relevant to each regime and provide analytic estimates for the boundaries separating them. Since it proves essential for our analysis, we first review some general results about Lyα escape from a homogeneous slab (Sect. 2.2) before describing radiative transfer in clumpy medium (Sect. 2.3). We confirm these analytic results numerically in Sect. 4 using Monte Carlo radiative transfer simulations.

2.1. Definitions and notation conventions

The basics of Lyα radiative transfer have been described in the literature (e.g., recently in a review by Dijkstra 2014) and are not repeated in detail here. Instead, we summarize the most relevant quantities for our present applications.

We express the photon frequency ν in terms of its Doppler parameter $\mathit{x}\mathrm{=}\frac{\mathit{\nu }\mathrm{-}{\mathit{\nu }}_{\mathrm{0}}}{\mathrm{\Delta }\mathit{\nu }}\mathit{,}$(1)where ν₀ ≈ 2.47 × 10¹⁵ s^-1 is the frequency at line center, and $\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{D}}\mathrm{=}{\mathit{v}}_{\mathrm{th}}{\mathit{\nu }}_{\mathrm{0}}\mathit{/}\mathit{c}\mathrm{=}\sqrt{\mathrm{2}{\mathit{k}}_{\mathrm{B}}\mathit{T}\mathit{/}{\mathit{m}}_{\mathrm{H}}}{\mathit{\nu }}_{\mathrm{0}}\mathit{/}\mathit{c}$ is the line width due to thermal motions of the atoms.

Temperature dependence is expressed through the Voigt parameter ${\mathit{a}}_{\mathit{v}}\mathrm{=}\frac{\mathrm{\Delta }{\mathit{\nu }}_{\mathrm{L}}}{\mathrm{2}{\mathrm{\Delta }}_{\mathrm{D}}}\mathrm{\approx }\mathrm{4.7}\mathrm{\times }{\mathrm{10}}^{-4}{\left(\frac{\mathit{T}}{{\mathrm{10}}^{\mathrm{4}}\hspace{0.17em}\mathrm{K}}\right)}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{·}$(2)Here, Δν_L = 9.939 × 10⁷ s^-1 is the natural (i.e., quantum mechanical) line broadening due to the finite lifetime of the transition.

The Lyα cross section of neutral hydrogen is $\begin{array}{ccc}{\mathit{\sigma }}_{\mathrm{HI}}\mathrm{\left(}\mathit{x,T}\mathrm{\right)}& \mathrm{=}& {\mathit{\sigma }}_{\mathrm{0}}\mathit{H}\mathrm{\left(}{\mathit{a}}_{\mathit{v}}\mathit{,x}\mathrm{\right)}\\ & \mathrm{=}& \frac{{\mathit{\sigma }}_{\mathrm{0}}{\mathit{a}}_{\mathit{v}}}{\mathit{\pi }}\underset{\mathrm{-}\mathrm{\infty }}{\overset{\mathrm{\infty }}{\mathrm{\int }}}\mathrm{d}\mathit{y}\frac{{\mathrm{e}}^{\mathrm{-}{\mathit{y}}^{\mathrm{2}}}}{\mathrm{(}\mathit{y}\mathrm{-}\mathit{x}{\mathrm{)}}^{\mathrm{2}}\mathrm{+}{\mathit{a}}_{\mathit{v}}^{\mathrm{2}}}\mathit{,}\end{array}$(3)where σ₀ ≈ 5.895 × 10^-14 (T/ 10⁴ K)^{− 1/2} cm² denotes its value at line center and H(a_v,x) is the Voigt function, which can be approximated as H(a_v,x) ~ e^{− x2} in the core of the line and as $\mathrm{~}{\mathit{a}}_{\mathit{v}}\mathit{/}\mathrm{\left(}\sqrt{\mathit{\pi }}{\mathit{x}}^{\mathrm{2}}\mathrm{\right)}$ in the wing of the line. The transition occurs at a frequency x_∗ ≈ 3.26 for T = 10⁴K. The normalized Voigt distribution $\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathit{H}\mathrm{\left(}{\mathit{a}}_{\mathit{v}}\mathit{,x}\mathrm{\right)}\mathit{/}\sqrt{\mathit{\pi }}$ represents the probability of a photon in the frequency interval [x ± dx/ 2] to interact with an atom.

The optical depth per length d is, hence, $\mathit{\tau }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\underset{\mathrm{0}}{\overset{\mathit{d}}{\mathrm{\int }}}\mathrm{d}\mathit{s}\hspace{0.17em}{\mathit{\sigma }}_{\mathrm{HI}}\mathrm{\left(}\mathit{x}\mathrm{\right)}{\mathit{n}}_{\mathrm{HI}}\mathrm{\left(}\mathit{s}\mathrm{\right)}\mathit{,}$(4)where n_HI denotes the number density of neutral hydrogen atoms. We did not include the contribution of dust in the above expression as its impact is modeled in the post processing (see Sect. 4.5 for details).

2.2. Radiative transfer in a homogeneous slab

Since it is crucial for our analytic work below, we briefly review some classical solutions for Lyα radiative transfer in a semi-infinite (that is, only one dimension is finite) slab with half-height B and optical depth τ₀.

Lyα escape can be seen as a random walk in both real space and frequency space, as every scattering event (that is, absorption and quick re-emission from a neutral hydrogen atom) alters the frequency and direction of the Lyα photon. However, because of the large value of σ₀, the mean free path of a photon close to line center is very small (λ_mfp ~ 5.5 × 10^-6(n/ cm^-3)-1pc for T = 10⁴K), and most scatterings are spatially close to each other. Thus, the vast majority of scatterings do not contribute significantly toward the escape of the Lyα photon (at least in optically thick media¹). Instead, Adams (1972) found that Lyα photons escape in several consecutive wing scatterings where the mean free path is significantly enhanced (for instance, λ_mfp ~ 0.48 pc at x = 5 for the above setup). The random walk in frequency space is therefore crucial to the escape of Lyα. These series of wing scatterings are referred to as excursion and this is thought of as the common way Lyα photons propagate in an astrophysical context.

To estimate the average displacement per “excursion”, one has to take into account its random walk in frequency. Specifically, a Lyα photon in the wing of the line at frequency x has a slight tendency to return to the core of the line with mean frequency shift per scattering event of −1 /x (Osterbrock 1962). This means it scatters ~ x² times before returning to the core with a mean free path of l = Bσ₀/ (σ_HI(x)τ₀) = B/ (H(x)τ₀) between each scatter. Since an excursion itself can be seen as a random walk, Adams (1972) obtained ${\mathit{d}}_{\mathrm{exc}}\mathrm{=}\sqrt{{\mathit{N}}_{\mathrm{sct}\mathit{,}\mathrm{exc}}}\mathit{l}\mathrm{=}\mathit{xB}\mathit{/}\mathrm{\left(}\mathit{H}\mathrm{\right(}\mathit{x}\mathrm{\left)}{\mathit{\tau }}_{\mathrm{0}}\mathrm{\right)}$ as mean distance per excursion. Furthermore, by using the wing approximation $\mathit{H}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{~}{\mathit{a}}_{\mathit{v}}\mathit{/}\mathrm{\left(}\sqrt{\mathit{\pi }}{\mathit{x}}^{\mathrm{2}}\mathrm{\right)}$ described above, and setting ${\mathit{d}}_{\mathrm{exc}}\mathrm{=}\sqrt{\mathrm{3}}\mathit{B}$ (where the factor $\sqrt{\mathrm{3}}$ arises due to geometrical considerations; Adams 1975), he obtained ${\mathit{x}}_{\mathrm{esc}}\mathrm{=}{\left({\mathit{\tau }}_{\mathrm{0}}{\mathit{a}}_{\mathit{v}}\sqrt{\mathrm{3}\mathit{/}\mathit{\pi }}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}\mathrm{\approx }\mathrm{6.5}{\left(\frac{{\mathit{N}}_{\mathrm{HI}}}{{\mathrm{10}}^{\mathrm{19}}\hspace{0.17em}{\mathrm{cm}}^{-2}}\frac{{\mathrm{10}}^{\mathrm{4}}\hspace{0.17em}\mathrm{K}}{\mathit{T}}\right)}^{\mathrm{1}\mathit{/}\mathrm{3}}$(5)as an expression for the most likely escape frequency.

Adams (1972) continued to calculate the number of scatterings it takes for a photon to reach a frequency | x | ≥ x_esc, which allows for escape. In an optically thick medium, photons undergo many scatterings and the frequency distribution J(x) is roughly constant². Thus, the probability to find an arbitrary photon with frequency in the interval [x ± dx/ 2] is ~ φ(x)dx (complete redistribution approximation³). However, a given photon scatters ~ x² times at the frequency ~ x. Consequently, ~ x² scattering events are not into a frequency interval, which allows for escape, and thus the probability of scattering into [x ± dx/ 2] is ~ φ(x) /x²dx. This implies a cumulative escape probability ${\mathit{P}}_{\mathrm{esc}}\mathrm{=}\mathrm{2}\underset{{\mathit{x}}_{\mathrm{esc}}}{\overset{\mathrm{\infty }}{\mathrm{\int }}}\mathrm{d}\mathit{x}\frac{\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}}{{\mathit{x}}^{\mathrm{2}}}\mathrm{=}\frac{\mathrm{2}{\mathit{a}}_{\mathit{v}}}{\mathrm{3}\mathit{\pi }{\mathit{x}}_{\mathrm{esc}}^{\mathrm{3}}}\mathit{,}$(6)where in the last equality we used the wing approximation for φ(x). The number of scatterings required to escape is 1 /P_esc and plugging in x_esc from Eq. (5)one obtains ${\mathit{N}}_{\mathrm{sct}}^{\mathrm{esc}}\mathrm{\approx }\mathrm{4.6}{\mathit{\tau }}_{\mathrm{0}}\mathrm{\approx }\mathrm{2.7}\mathrm{\times }{\mathrm{10}}^{\mathrm{6}}\left(\frac{{\mathit{N}}_{\mathrm{HI}}}{{\mathrm{10}}^{\mathrm{19}}\hspace{0.17em}{\mathrm{cm}}^{-2}}\right){\left(\frac{\mathit{T}}{{\mathrm{10}}^{\mathrm{4}}\hspace{0.17em}\mathrm{K}}\right)}^{\mathrm{-}\mathrm{1}\mathit{/}\mathrm{2}}\mathrm{·}$(7)This relation differs only by a factor of a few with the exact solution of Harrington (1973) which has been backed up by numerical results (e.g., Bonilha et al. 1979; Dijkstra et al. 2006).

In summary, Adams (1972) found that a typical Lyα photon leaving an optically thick slab scatters a large number of times essentially in place (Eq. (7)), until it reaches the frequency x_esc (Eq. (5)), after which it escapes undergoing ${\mathit{N}}_{\mathrm{sct}}^{\mathrm{exc}}\mathrm{~}{\mathit{x}}_{\mathrm{esc}}^{\mathrm{2}}$ scattering interactions in the wing of the line.

2.3. Radiative transfer in clumpy medium

Resonant line transfer in a clumpy medium has fundamentally different behavior than in a homogeneous medium because much of the distance can be traversed in the optically thin medium between the clumps. As we discussed in the previous section, owing to its highly variable interaction cross section, Lyα escapes through “excursion” in regimes for which the mean free path at the initial frequency is short. In a multiphase medium, however, a significant fraction of the volume may have no neutral hydrogen at all. The gas opacity thus varies strongly with position, even at line center. This opens up an alternate escape route in which Lyα photons “solve the maze” by scattering into the optically thin medium between clouds. This possibility is essential to consider, since astrophysical systems such as the ISM and CGM are thought to have a multiphase nature (e.g., McKee & Ostriker 1977).

2.3.1. Model parameters

In this section, we describe the expected propagation of Lyα photon in a clumpy medium, which we model using spherical clumps of radius r_cl and HI number density n_HI,cl placed in an otherwise empty, semi-infinite slab of height 2B. In what follows, we use the clump column density N_HI,cl = r_cln_HI,cl and optical depth τ_cl(x,T) = N_HI,clσ_HI(x,T) as convenient notation. The most important parameter of a clumpy medium is the covering factor f_c, which describes the average number of clumps per orthogonal sightline between the midplane and the surface of the slab. These sightlines intercept a column density of ${\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{total}}\mathrm{=}\frac{\mathrm{4}}{\mathrm{3}}{\mathit{f}}_{\mathrm{c}}{\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{cl}}$ where the factor 4/3 is due to the spherical geometry of the clumps⁴.

2.3.2. Escape regimes

In a static, clumpy medium several regimes are possible for the escape of a Lyα photon. We introduced some of these regimes in Gronke et al. (2016), but further describe each regime below. Furthermore, we sketch (i) under which circumstances each regime is active; (ii) on average, how many clumps a photon encounters N_cl; and (iii) which emergent spectrum can be expected. Additionally, Fig. 1 provides a visual overview of the regimes, and a similar overview for a non-static setup is given in Appendix A.

Porous regime. If a substantial number of sightlines do notintercept any clumps, many photons will not scatter and simplyescape at their intrinsic frequency. The fraction of sightlineswithout any clumps can be estimated assuming the clumps arePoisson distributed with mean f_c yielding exp(−f_c) (cf. Gronke & Dijkstra 2016; Dijkstra et al. 2016). This area of the parameter space has been explored in previous work (Hansen & Oh 2006; Laursen et al. 2013; Gronke & Dijkstra 2016) and is of interest as the empty sightlines allow for ionizing photon escape (Verhamme et al. 2015; Dijkstra et al. 2016) and might allow for directionally dependent photon escape (Gronke & Dijkstra 2014). This is the regime suggested by cosmological simulations of the CGM (e.g., Faucher-Giguère et al. 2015; Liang et al. 2016), although that may be a consequence of their limited resolution, which strongly limits the number of clumps to be no more than ~a few.

Random walk regime. If the clumps are optically thick to the photons, i.e., τ_cl ≳ 1, the photons are expected to scatter at every clump encounter. When τ_cl ≫ 1 the photon scatters close to the surface of each clump and effectively random walks between the clumps, rather than through the clumps. This regime was studied by Neufeld (1991) and by Hansen & Oh (2006). In this “random walk regime” the number of clumps a photon intercepts scales as ∝ f_c². This is because after N_cl interactions a photon has traveled on average a distance $\sqrt{{\mathit{N}}_{\mathrm{cl}}}\mathit{B}\mathit{/}{\mathit{f}}_{\mathrm{c}}$ away from the midplane. Thus, to escape this distance has to be ~ B, which yields N_cl = f_c². Hansen & Oh (2006) found the scaling to be ${\mathit{N}}_{\mathrm{cl}}\mathrm{~}{{\mathit{f}}_{\mathrm{c}}}^{\mathrm{2}}\mathrm{+}{\mathit{f}}_{\mathrm{c}}\mathit{,}$(8)with pre-factors of order unity that depend somewhat on the geometry (see Hansen & Oh 2006, for details).

Optically thin regime. If the clumps are, on the other hand, optically thin to the intrinsic radiation (τ_0,cl ≲ 1), not every cloud interception necessarily causes the photon to scatter. In particular, the probability of a scattering event to happen in this case is 1−exp(−τ_0,cl) ≈ τ_0,cl. This implies that to describe the expected scaling in this regime, we can replace in the above considerations f_c by τ_0,clf_c. As in this regime τ_0,cl ≲ 1, this corresponds to decrease effectively the pre-factors in Eq. (8). Specifically, if all clumps in a given sightline are optically thin, the intrinsic photons at line center, that is, τ_0,total ≡ 4/3f_cτ_0,cl ≲ 1, most Lyα photons do not interact before escaping. This means they simply stream through all the clumps (leading to N_cl ∝ f_c) keeping their intrinsic frequency (i.e., a peak frequency of x_p ~ 0). We call this an optically thin regime.

Homogeneous regime. Since τ_cl depends strongly on the frequency of the photon, which changes throughout the scattering process, Lyα might also escape from clumpy media in a frequency excursion as discussed in the homogeneous slab. In particular, during the course of the ~ f_c² scatterings needed to random walk through the clumpy medium, the photon may scatter far enough into the wing of the line to escape the medium in a single excursion as described in Sect. 2.2. If this happens, most clumps become optically thin for the photon and one can generalize the argument made above when describing the optically thin regime by replacing τ_0,cl by τ_cl(x). Since this possibility becomes increasingly likely with every scattering event, we anticipate that above some critical value of f_c, clumpy media behave like homogeneous slabs in the sense that photos escape via frequency excursion.

Conclusively, the four regimes are different in the photon’s preferential escape route, which depends in the static case on the covering factor f_c and the optical depth of individual clumps. Each escape route implies that the photon experiences a clumpy medium differently, which leaves a clear imprint on the emergent spectrum. One way to characterize these regimes is the optically thin and porous regimes represent escape without significant interaction, while the random walk and homogeneous regimes represent escape primarily via spatial or frequency diffusion, respectively. Table 1 provides a brief summary of the different regimes.

2.3.3. Division between the regimes

In the last section we introduced the four different routes for a Lyα photon to escape from a clumpy medium. We also briefly discussed the physical conditions under which each escape route is favored. In this section, we quantify these boundaries more precisely.

We denote the boundary between the homogeneous and other regimes with f_{c, crit}, and specifically to the random walk regime with ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}^{\mathrm{exc}}$. Physically, this value characterizes the critical number of clumps per sightline when a excursion-like escape becomes faster than a random-walk diffusion. In order to find this boundary we follow this argument and compute the criteria for when it is possible for the photons to stream through the clumps⁵. As stated above, the characteristic escape frequency is given by Eq. (5), and in a clumpy medium the total line center optical depth is τ₀ = 4/3f_cN_HI,clσ₀. The transition occurs when photons can stream through individual clumps, that is when 4/3τ_cl(x_esc) = 1. Using the wing approximation for the Lyα cross section, this yields ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}^{\mathrm{exc}}\mathrm{=}\frac{\mathrm{2}\sqrt{{\mathit{a}}_{\mathit{v}}{\mathit{\tau }}_{\mathrm{0}\mathit{,}\mathrm{cl}}}}{\mathrm{3}{\mathit{\pi }}^{\mathrm{1}\mathit{/}\mathrm{4}}}\mathrm{\approx }{\left(\frac{{\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{cl}}}{{\mathrm{10}}^{\mathrm{17}}\hspace{0.17em}{\mathrm{cm}}^{-2}}\right)}^{\mathrm{1}\mathit{/}\mathrm{2}}{\left(\frac{\mathit{T}}{{\mathrm{10}}^{\mathrm{4}}\hspace{0.17em}\mathrm{K}}\right)}^{-1}\mathrm{·}$(9)For optically thinner medium, an escape through excursion is not possible as a frequency shift into the wings of the lines leads to immediate escape. Specifically, this happens if the wings become optically thin, i.e., if $\sqrt{\mathrm{3}}\mathit{\tau }\mathrm{\left(}{\mathit{x}}_{\mathrm{\ast }}\mathrm{\right)}\mathit{<}\mathrm{1}\mathit{,}$ which translates to $\sqrt{\mathrm{3}}{\mathit{\tau }}_{\mathrm{0}}{\mathit{a}}_{\mathit{v}}\mathit{<}\sqrt{\mathit{\pi }}{\mathit{x}}_{\mathrm{\ast }}^{\mathrm{2}}\mathrm{\approx }\mathrm{18.78}\mathit{,}$ where we included factors of $\sqrt{\mathrm{3}}$ due to the rectangular geometry. This transition happens for an homogeneous medium as well as a clumpy medium and sets a lower limit to f_{c, crit}. However, if the individual clumps possess an optical depth at line center of τ_0,cl ≲ 1, not every clump encounter leads necessarily to a scattering, and thus introduces the additional factor of 1−e^{− τ_0,cl} (as described for the optically thin regime in Sect. 2.3.3).

In conclusion, we expect the transition to the homogeneous regime to occur if ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\mathrm{=}\{\begin{array}{c}\\ & \mathrm{for}\sqrt{\mathrm{3}}{\mathit{a}}_{\mathit{v}}{\mathit{\tau }}_{\mathrm{0}}\mathit{>}\sqrt{\mathit{\pi }}{\mathit{x}}_{\mathrm{\ast }}^{\mathrm{2}}\\ & \mathrm{otherwise,}\end{array}$(10)where ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}^{\mathrm{sf}}\mathrm{=}\mathrm{2}{\mathit{x}}_{\mathrm{\ast }}\mathit{/}{\mathrm{3}}^{\mathrm{5}\mathit{/}\mathrm{4}}\mathrm{\approx }\mathrm{1.65}$ is due to continuity of f_{c, crit} at $\sqrt{\mathrm{3}}{\mathit{a}}_{\mathit{v}}{\mathit{\tau }}_{\mathrm{0}}\mathrm{=}\sqrt{\mathit{\pi }}{\mathit{x}}_{\mathrm{\ast }}^{\mathrm{2}}$, that is, at the transition from excursion to single-flight escape.

2.3.4. Non-static case

Since the Lyα cross section depends sensitively on the frequency x, clump motions can dramatically influence radiative transfer. If a clump moves with a velocity ≳x_∗v_th ~ 50 km s^-1, a single clump interaction puts the photon far enough into the wing of the line to allow the photon to escape directly in a single excursion. This possibility is important to consider because random velocities v ~ 100 km s^-1 may be typical for the CGM in galaxies and in galactic winds, and velocities v ~ 1000 km s^-1 may be typical in the regions around black holes. This means that for large f_c the medium behaves as a slab with an increased temperature of ${\mathit{T}}_{\mathrm{eff}}\mathrm{=}\mathit{T}\mathrm{+}\frac{{\mathit{\sigma }}_{\mathrm{cl}}^{\mathrm{2}}{\mathit{m}}_{\mathrm{H}}}{\mathrm{2}{\mathit{k}}_{\mathrm{B}}}\mathit{,}$(11)where σ_cl is the 1D velocity dispersion of the clumps.

For a lower number of clumps the overall velocity distribution is not well sampled, which leads to sightlines with no clumps in the core of the line. In this case the photons escape without any clump interaction. We can estimate this to happen if the mean separation of two clumps in velocity space becomes larger than the velocity range over which a clump can provide τ_cl ≳ 1. For a Gaussian velocity distribution with variance of ${\mathit{\sigma }}_{\mathrm{cl}}^{\mathrm{2}}$, the average separation is approximately given by σ_cl/ (αf_c) where α is the fraction of clumps within core of the velocity distribution, i.e., in our case α ≈ 0.68. Consequently, the transition to the homogeneous regime for randomly moving clumps occurs at 4/3τ_cl(σ_cl/ (αf_cv_th)) = 1, which – using the core approximation and including geometrical factors – can be written as a critical covering factor for the randomly moving case ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\mathrm{=}\{\begin{array}{c}\\ & \mathrm{if}{\mathit{f}}_{\mathrm{c}}\mathit{>}\mathrm{1}\mathit{/}\mathit{\alpha }\\ & \mathrm{otherwise}\mathrm{.}\end{array}$(12)Here, the lower boundary of 1 /α results simply from the requirement that at least one clump within the core of the velocity distribution function is necessary to sample the core of the velocity distribution. We expect, for larger covering factors, the system to behave as a homogeneous slab of temperature T_eff. See also Appendix A for more details about the expected behavior in the case of uncorrelated clump motion.

In the case of clumps with a systematic velocity structure (for instance, outflowing clumps), the above requirement of a well-sampled velocity field is fulfilled if the adjacent clump is optically thick to the Lyα photon, i.e., if 4/3τ_cl(x_next) ≳ 1, where x_next depends on the exact velocity profile. For a linearly scaled (Hubble-like) outflow from 0 at midplane to | v_max | at the boundaries of the slab, we have x_next = v_max/ (f_cv_th). In addition, a photon might be artificially forced into the wing of the line if x_next>x_∗ owing to the sampling of the velocity field. This does not occur in a homogeneous medium, and thus, for a Hubble-like outflow the criterion to be fulfilled for the homogeneous regime is ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\mathrm{=}\{\begin{array}{c}\\ & \mathrm{if}{\mathit{v}}_{\max }\mathit{>}\mathit{v̂}\max \\ & \mathrm{otherwise,}\end{array}$(13)where $\mathit{v̂}{}_{\max }\mathrm{=}{\mathit{v}}_{\mathrm{th}}{{}^{\mathrm{(}}\mathit{a}_{\mathit{v}}{\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{total}}{\mathit{x}}_{\mathrm{\ast }}{\mathit{\sigma }}_{\mathrm{0}}\mathit{/}{\sqrt{\mathit{\pi }}}^{\mathrm{)}}}^{\mathrm{1}\mathit{/}\mathrm{3}}$.

3. Numerical method

3.1. Lyα radiative transfer

Because of the complexity of the resonant line transfer, Monte Carlo radiative transfer codes are commonly in use (e.g., Auer 1968; Ahn et al. 2002; Zheng & Miralda-Escudé 2002). This algorithm works by following individual photon packages in a stochastic manner through real and frequency space until their escape. In this work, we used the code tlac, which has been used and described previously, for example, in Gronke & Dijkstra (2014). In particular, we made use of tlac’s features (i) to handle embedded spherical grids within a Cartesian grid; and (ii) employ a dynamical core-skipping scheme (as described in Smith et al. 2015; Gronke & Dijkstra 2016). We also turned off the dynamical core skipping for a few models and checked that the emergent spectra are identical. We ran most setups using ~ 10⁴ photon packages but occasionally used more to obtain a higher resolution spectrum.

3.2. Model parameters

Analogous to Sect. 2.3, our setup consisted of a slab with half-height B in which we distributed non-overlapping spherical clumps with radius r_cl randomly in the box until a fraction of the total volume F_V was filled⁶. This means the number density of clumps is ${\mathit{n}}_{\mathrm{cl}}\mathrm{=}{\mathit{F}}_{\mathit{V}}\mathit{/}\mathrm{(}\mathrm{4}\mathit{/}\mathrm{3}\mathit{\pi }{\mathit{r}}_{\mathrm{cl}}^{\mathrm{3}}\mathrm{)}\mathit{,}$ where r_cl is the clump radius. The connection between the volume filling factor F_V and the previously introduced covering factor f_c, which describes the average number of clumps a line orthogonal to the slab intercepts between the midplane and boundary of the box, is given by the integration along the finite axis of the slab, that is, ${\mathit{f}}_{\mathrm{c}}\mathrm{=}\underset{\mathrm{0}}{\overset{\mathit{B}}{\mathrm{\int }}}\mathrm{d}\mathit{r\pi }{\mathit{n}}_{\mathrm{cl}}{\mathit{r}}_{\mathrm{cl}}^{\mathrm{2}}\mathrm{=}\frac{\mathrm{3}{\mathit{F}}_{\mathit{V}}\mathit{B}}{\mathrm{4}{\mathit{r}}_{\mathrm{cl}}}\mathrm{·}$(14)The clumps are filled with neutral hydrogen with a number density of n_HI,cl and temperature T, leading to a column density between the center of the clumps to their outskirts of N_HI,cl = r_cln_HI,cl. As described in Sect. 2.3, this means on average the shortest path between the midplane and the boundary of the box intercepts a column density of ${\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{total}}\mathrm{=}\frac{\mathrm{4}}{\mathrm{3}}{\mathit{f}}_{\mathrm{c}}{\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{cl}}\mathrm{=}{\mathit{F}}_{\mathit{V}}\mathit{B}{\mathit{n}}_{\mathrm{HI}\mathit{,}\mathrm{cl}}\mathit{.}$(15)In general, we considered three cases: the static case with no motion, the randomly moving case, and an outflowing case. Astrophysical sources commonly show signs of turbulent motion (probed, e.g., through Hα line profiles; Herenz et al. 2016). For instance, turbulence in the ISM is thought to be driven by supernova explosions. Furthermore, outflows such as galactic winds are driven by star formation or AGN mediated feedback (probed, for instance, via absorption lines; Steidel et al. 2010; Rivera-Thorsen et al. 2015). The static case, on the other hand, allowed us to study the pure radiative transfer without the additional complications of Doppler shifts due to bulk motions.

In the randomly moving case, we assigned each clump a random velocity by drawing each component from a Gaussian with standard deviation σ_cl. This represents a “white noise” spectrum with velocity differences that are statistically equally probable on all spatial scales. For the outflow, we chose a simple linear velocity scaling from 0 km s^-1 to v_max at the midplane and boundary of the slab, respectively. We will investigate models with correlated turbulence and different velocity profiles in future work.

Furthermore, we studied two different emission sites for the Lyα photons: first, simply the midplane of the box and, secondly, randomly chosen emission within the clumps. While the former is useful to study merely the radiative transfer processes through the clumpy medium from an external source such as a star-forming region, the latter case represents a physically motivated scenario in which Lyα are produced via recombination events within the clumps. Both scenarios might be responsible, for example, for the Lyα halos found around galaxies (e.g., Dijkstra & Kramer 2012; Mas-Ribas & Dijkstra 2016).

4. Numerical results

In this section we present the results from our numerical radiative transfer simulations. In particular, we focus on three quantities, namely the number of clumps encountered by the photons N_cl, and the emergent Lyα spectra. The value N_cl is a useful diagnostic, since we expect N_cl ~ f_c² for escape via random walk in position space, and N_cl ~ f_c for escape via excursion and single flight as described in the previous section with the transition occurring at f_c ~ f_{c, crit}.

The section is approximately ordered by ascending complexity. In Sect. 4.1, we discuss the static case, in Sects. 4.2 and 4.3 we introduce random clump motions and outflows, respectively. Moreover, in Sect. 4.4 we change the emission site of the photons to be inside the clumps, which resembles a case of fluorescent emission. Finally, we study the effect of dust inside the clumps on the Lyα escape in a clumpy medium (Sect. 4.5), which we quantify through the Lyα escape fraction.

4.1. Static case

Figure 2 shows the number of clumps a Lyα photon passed through before escaping the box versus the covering factor f_c, which we vary over ~ 3 orders of magnitude. Each symbol and color represents different values of N_HI,cl and, thus, different clump optical depths at line center τ_cl,0 which we vary from ~ 0.06 (optically thin) to ~ 6 × 10⁸ (optically thick). We also ran each combination of (N_HI,cl,f_c) with two different cloud radii r_cl = { 10^-2, 10^-3 } pc to confirm that this parameter is not important (Hansen & Oh 2006).

The dashed lines in the corresponding color show curves following ${\mathit{N}}_{\mathrm{cl}}\mathrm{=}\{\begin{array}{c}\\ & \mathrm{for}{\mathit{f}}_{\mathrm{c}}\mathit{<}{\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\\ & \mathrm{for}{\mathit{f}}_{\mathrm{c}}\mathrm{\ge }{\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\end{array}$(16)with the resulting best-fit parameters for (f_{c, crit},a₂) shown in the legend in each figure. Prior we fit the data points for N_HI,cl = 10²² cm^-2 for f_c ≤ 100 to determine (a₁,b₁) = (3/2, 2); the best-fit values are (1.51, 1.90) which – given the uncertainty – we rounded to the nearest convenient fraction for simplicity. These coefficients represent geometrical factors in the surface scattering regime, where clouds are optically thick, and thus independent of N_HI,cl. We directly verified this numerically. These values are then fixed for all N_HI,cl in order to fit each N_HI,cl-curve for (f_{c, crit},a₂), while b₂ is fixed by requiring continuity at f_{c, crit}. Figure 2 shows the resulting fits and the obtained values for (f_{c, crit},a₂). The break in the scaling relation at f_{c, crit} is clearly visible; for visual aid, Fig. 2 also shows the f_c = a₁f_c² + b₁f_c curve from which the scaling departures for f_c>f_{c, crit}.

We found that the obtained f_{c, crit} for high column densities (N_HI,cl ≳ 10²⁰ cm^-2) matches the prediction from Sect. 2.3.3 reasonably well; this breaks down for lower optical depths. Also, for N_HI,cl = 10¹² cm^-2, i.e., when the clumps are always optically thin for Lyα photons, we obtained ${\mathit{a}}_{\mathrm{2}}\mathrm{\approx }{\mathit{\tau }}_{\mathrm{0}\mathit{,cl}}^{\mathrm{2}}\mathrm{\approx }\mathrm{0.010}$ as discussed in Sect. 2. With increasing N_HI,cl we found a decreasing a₂ to match the data. Thus, we can identify the escape regimes (characterized by the number of clumps encountered) described in the analytic model in Sect. 2. In summary, Fig. 2 shows that for f_c<f_{c, crit}, N_cl ∝ f_c² (as expected for a random walk), while for f_c>f_{c, crit}, N_cl ∝ f_c (as expected for escape through excursion).

Figure 3 shows the corresponding spectra for N_HI,cl = 10¹⁷ cm^-2. For this column density, we found f_{c, crit} ~ 2, which corresponds roughly to the boundary between single and double-peaked spectra. In particular, we recovered the spectral shape of Hansen & Oh (2006) for f_c ≪ f_{c, crit} while obtaining wide, double-peaked spectra with zero flux at line center for f_c ≫ f_{c, crit}. This means that the escape regimes not only impact the paths of the photons but also modify the escape frequencies and, hence, leave a clear observational signature on the emergent spectra.

4.2. Random motion

Figure 4 shows the N_cl−f_c scaling relation in the case of random clump motion for a fixed clump’s column density of N_HI,cl = 10¹⁷ cm^-2. As conjectured in Sect. 2.3.4, compared to the static case the photons spend less time until escape and thus the number of clumps passed is smaller. This is because in the case of fast moving clumps, the photons escape either through “holes” in velocity space (where, N_cl ∝ f_c) or via single flight (in which case also N_cl ∝ f_c). Departures from that are either due to convergence to the static case (for σ_cl → 0) or when escape via single flight involves multiple surface scatterings (for f_c ≪ f_{c, crit}) when the interaction with another clump is non-negligible.

In Fig. 5 we show the emergent spectra from this setup. In particular, we focus on the case with f_c ≈ 1000 and N_HI,cl = 10¹⁶ cm^-2 and four different values of clump velocity dispersion σ_cl. Also in Fig. 5 we overlay spectra from homogeneous slabs with an effective temperature T_eff (see Eq. (11)) corresponding to the respective value of σ_cl. Clearly, the spectra match very well and, specifically, the peak separation. However, with increasing T_eff the matches become worse, which makes sense since the wider velocity space is more poorly sampled.

4.3. Outflows

Figure 6 shows the N_cl−f_c relation in the presence of linearly scaled outflows with maximum velocity v_max (as described in Sect. 3.2). We can see that a flattening of the curve still exists, which we interpret again as the transition between the “random walk” and “homogeneous regime”. As expected with increasing outflow speed this threshold decreases.

Figure 7 illustrates the change in spectral shape when introducing outflows. In each subpanel, the solid lines show the emergent spectrum from the clumpy model and the dashed lines in matching color show those from a homogeneously filled slab with the same total column density and velocity structure. We focus on the case with constant clump column density N_HI,cl = 10¹⁷ cm^-2 and show three cases, f_c = { 1, 100, 1000 } , which match the slab case increasingly well for all four values of v_max. This implies that the spectra become more asymmetric as they converge toward the homogeneous limit. The asymmetry develops because the outflow shifts the scattering cross section in the observers reference frame toward the blue. Thus, the optical depth for photons with frequency redward of line center (e.g., the back-scattered photons off the far side of the system) is lowered allowing for easier escape. We discuss the result of higher asymmetry with increased number of clumps further in Sect. 5.

Figure 8 shows this increase in asymmetry with greater f_c for fixed outflow velocities of v_max = { 100, 1000 } km s^-1 and total column densities of N_HI,total = 4/3 × { 10¹⁸, 10¹⁹, 10²⁰ } cm^-2. We characterize the spectral asymmetry by the integrated flux ratio of the blue over the red part of the spectra (minus one), i.e., a value of −1 means that all photons escape redward of line center (x ≤ 0) and if this quantity is zero the spectrum is symmetric (around x = 0). While the transition from symmetric to dominantly red spectra for v_max = 100 km s^-1 is nearly independent of the column density at f_c ~ 10, this is not the case for v_max = 1000 km s^-1 where a larger total column density implies a shift at lower f_c. This is because of the dependence of f_{c, crit} on v_max and N_HI,total that is described in Sect. 2.3.4. In particular, as seen in Eq. (13), f_{c, crit} does not depend on the column density if v_max is small.

4.4. Emission within the clumps

In this section we study the effect of the emission originating from inside the clumps. This case resembles Lyα production due to cooling in the inner parts of the clumps or to recombination events in the outer layer of (self-shielding) clumps caused by an external ionizing source. The latter is sometimes referred to as fluorescence (e.g., Hogan & Weymann 1987; Mas-Ribas & Dijkstra 2016). In both cases, Lyα are produced in the reference frame of the clumps and experience an initial optical depth before entering the inter-clump medium; both effects shape the “intrinsic” spectrum.

Figure 9 compares some spectra with starting position inside the clumps to those previously presented, i.e., with starting position at midplane. The clumps in this case possess a column density of N_HI,cl = 10¹⁷ cm^-2, which means they are optically thick to Lyα radiation. The effect of this can be seen best in the spectrum with f_c = 1 (red curve in Fig. 9). This contrasts with a double-peaked profile due to the escape from the clump to the single-peaked profile from the random walk process between the clumps. For greater values of f_c, however, this “initial feature” gets washed out from the scatterings off subsequent clumps and the spectra are independent of the emission site.

In Fig. 10 we show a similar plot for moving clumps. In this case them frequencies of the photons are rescaled according to the value of σ_cl for presentation purposes. This means that in the spectra for σ_cl = 10⁴ km s^-1 (shown in purple) are within a full width at half maximum (FWHM) of Δx ~ 5000 the widest of the presented spectra. As previously stated, for the large f_c ≫ 1 cases, the spectra with the emission sites within the clumps resemble closely those with emission sites in the midplane. The only difference is that the latter are slightly wider and possess a smaller flux at line center, which is simply because a number of clumps are located at the boundary of the slab. This is encouraging as it shows that our results are more general, that is, not dependent on the exact emission site. However, a small caveat is that for spherical geometries most clumps are located at large radii, which might make this setup more sensitive to in-clump emission; on the other hand, the outermost clumps might emit less Lyα photons as some are “shadowed” by clumps closer to the ionizing source.

4.5. Dusty clumps

When placing absorbing dust in the clumps, which we characterize by the all-absorbing dust optical depth τ_cl,d, Lyα can be destroyed leading to an escape fraction f_esc ≤ 1. Interestingly, in clumpy medium, the Lyα escape fraction might be larger than the continuum fraction as predicted by Neufeld (1991). This Neufeld effect occurs because Lyα photons may “surface scatter” off the neutral clumps, thus, effectively shielding the dust from these clumps. Therefore, one expects the observed Lyα equivalent widths to be potentially much larger than the intrinsic Lyα equivalent widths. Hansen & Oh (2006) characterized this effect more systematically using Lyα radiative transfer simulations for a wide range of parameters. Building upon their work, Laursen et al. (2013) found, however, that the boosting vanishes in an area of the parameter space that they tried to constrain by observations. Specifically, out of their 4 × 10³ models only a few percent showed an equivalent width boost (see also Duval et al. 2014, for a study of the Neufeld effect in clumpy shells). Laursen et al. (2013) thus concluded, “consider the Neufeld model to be an extremely unlikely reason for the observed high equivalent widths (EWs)”. All these studies focused on values of f_c ~ 1 and we want to revisit Lyα escape in clumpy medium with several orders of magnitude greater covering factors. Thus, it is not entirely clear from the literature whether radiative transfer effects from clumpy media can explain the extreme equivalent width measurements observed in some galaxies. However, Laursen et al. (2013) identified some criteria that have to be fulfilled such as relatively slowly moving clumps with high-dust optical depths.

Instead of re-running the radiative transfer simulations for various dust contents, we use the information of the hydrogen column density “seen” by each photon package to compute the Lyα escape fraction as in Gronke et al. (2015), which yields an escape fraction for each photon package that is ${\mathit{f}}_{\mathrm{esc}\mathit{,}\mathit{i}}\mathrm{=}\exp \left[\mathrm{-}\frac{\mathit{N̂}\mathrm{HI}\mathit{,i}}{{\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{cl}}}{\mathit{\tau }}_{\mathrm{d}\mathit{,}\mathrm{cl}}\right]\mathit{.}$(17)Here, is the column density experienced by photon package i. Given f_esc,i for a certain setup one can now obtain (a) the overall Lyα escape fraction as the average of f_esc,i; and (b) the spectral shape altered through dust by simply assigning each photon package the weight f_esc,i when assembling the spectrum.

In Fig. 11 we plot the Lyα escape fraction versus f_c for a constant total dust and hydrogen number content. A similar trend is visible for all three values of τ_d,total shown: with increasing f_c, first an approximately linear fall off in escape fraction before a flattening occurs, that is, f_esc ~ const. for f_c ≳ 40. Interestingly, the position of this threshold is independent of τ_d,total, which hints at an origin in the nature of the radiative transfer. The flattening occurs at the boundary between the “free streaming” and “homogeneous” regime because in the former the probability of absorption is proportional to the number of clump interactions (and, thus, f_c), whereas in the latter the escape fraction is set by the total dust content only and does not grow further with f_c. We discuss this phenomenon in more detail in Sect. 5.2.

An implication of the respective escape fractions of the two regimes is visible in Fig. 12. Here we show several values of N_HI,cl for the static setup using τ_d,cl = 10^-4 (empty symbols) and τ_d,cl = 1 (filled symbols), which correspond to metallicities of $\mathit{Z}\mathit{/}{\mathit{Z}}_{\mathrm{\odot }}\mathrm{=}\mathrm{0.63}{}^{\mathrm{(}}\mathit{\tau }_{\mathrm{d}}\mathit{/}{\mathrm{10}}^{-4}{}^{\mathrm{)}}{}^{\mathrm{(}}{\mathrm{10}}^{\mathrm{17}}\hspace{0.17em}{\mathrm{cm}}^{-2}\mathit{/}{\mathit{N}}_{\mathrm{HI}\mathit{,}\mathrm{cl}}{}^{\mathrm{)}}$ (Pei 1992; Laursen et al. 2009); this reaches clearly unrealistic values. However, as in this paper we are interested in the fundamental impact of the individual parameters, we also study these extreme values. Also shown in Fig. 12 (with a black [gray] solid line for the low [high] dust content) is the proposed analytic solution for f_esc by Hansen & Oh (2006)${\mathit{f}}_{\mathrm{esc}}^{\mathrm{HO}\mathrm{06}}\mathrm{=}\mathrm{1}\mathit{/}\cosh \mathrm{\left(}\sqrt{\mathrm{2}{\mathit{N}}_{\mathrm{cl}}\mathit{ϵ}}\mathrm{\right)}\mathit{,}$(18)where for N_cl we used Eq. (8)(with (a₁,b₁) = (3/2, 2) as found in Sect. 4.1) and for the clump albedo (i.e., the fraction of incoming photons that are reflected) ϵ, we adopted a value of c₁(1−e^{− τ_d,cl}) with c₁ = 1.6 [c₁ = 0.06] to match the N_HI,cl = 10²² cm^-2 data points for τ_d,cl = 10^-4 [τ_d,cl = 1]. The behavior for the low- and high-dust contents is quite different. On the one hand, the escape fractions versus N_HI,cl scales for τ_d,cl = 1 (filled symbols in Fig. 12) as predicted by Hansen & Oh (2006) in their “surface scatter” approximation, that is, a larger clump hydrogen column density “shields” the dust better from the Lyα photons and thus prevents their destruction more efficiently. On the other hand, however, this is not the case for the low-dust scenario presented in Fig. 12 (with unfilled symbols) where a larger value of N_HI,cl implies a lower f_esc. This is because here the dust optical depth through all the clumps (shown in the black dotted line in Fig. 12) is lower than the accumulated dust optical depth through the subsequent random-walk clump encounters (black solid line), i.e., $\exp \mathrm{(}\mathrm{-}\mathrm{4}\mathit{/}\mathrm{3}{\mathit{f}}_{\mathrm{c}}{\mathit{\tau }}_{\mathrm{d}\mathit{,}\mathrm{cl}}\mathrm{)}\lesssim {\mathit{f}}_{\mathrm{esc}}^{\mathrm{HO}\mathrm{06}}$. Consequently, configurations in the “free-streaming” regime can possess enhanced Lyα escape fractions compared to the “random walk” regime (see Sect. 5.2 for a more detailed discussion). Still, both cases possess (much) larger escape fractions than a homogeneous slab, which is shown in Fig. 12 with a black dashed line. Here, we use the derived escape fraction by Neufeld (1990) with N_HI = 4/3 × f_c10²² cm^-2 and τ_d = 4/3f_cτ_d,cl, i.e., with equal column densities as in the N_HI,cl = 10²² cm^-2 case.

The same quantity, i.e., f_esc versus f_c, for the case of randomly moving clumps is plotted in Fig. 13. As previously, the escape fraction departures from the curve given by Eq. (18)for f_c ≳ f_{c, crit}. The lower number of clump encounters in this regime leads to a significantly higher escape fraction, for example, f_esc ~ 10^-1 for σ_cl = 500 km s^-1.

5. Discussion

In this section, we discuss our results in light of the various escape regimes discussed in Sect. 2 (Sect. 5.1). Furthermore, we analyze what implications our results have for “Lyα equivalent width boosting” (in Sect. 5.2), and make the connection to observational results (of “shell-model” fitting; Sect. 5.3) and to radiative transfer results through hydrodynamical simulations (Sect. 5.4).

5.1. Regimes of the clumpy model

Figure 14 summarizes our findings for the static case. Here, color shows the flux at line center expressed in units of flux at the peak of the spectra F(x = 0) /F_peak. This measure is ~ 1 for a single-peaked spectra and is less for double-peaked spectra; a value of ~ 0 corresponds to an optically thick, slab-like spectrum. We highlighted the dividing value of F(x = 0) /F_peak = 1/2 specifically.

Also visible in Fig. 14 are the three regimes described Sect. 2, along with our analytic estimates. They can be summarized as follows:

Optically thin regime. For an overall optical depth,τ_0,total = 4/3f_cτ_0,cl ≲ 1, the N_cl−f_c scaling is shallow, and the emergent spectra are single peaked. The dotted line in Fig. 14 indicates this boundary.

Homogeneous regime. If not in the “optically thin regime”, for f_c ≳ f_c,crit we also found a shallower N_cl−f_c scaling than Hansen & Oh (2006). This is because of the preferential escape in an optically thick medium through single excursion, which causes broad, double-peaked spectra. The dashed line in Fig. 14 indicates f_c,crit as a function of N_HI,cl. Above this line we find F(x = 0) /F_peak → 0 denoting double-peaked spectra as predicted. Similarly, below this line the numerical results show single-peaked spectra.

Random-walk regime. For optically thick clumps and f_c ≲ f_{c, crit} we recovered the results of Hansen & Oh (2006), i.e., N_cl ∝ f_c² and single-peaked spectra owing to a surface-scattering escape of the photons.

As noted in Sect. 2.3.3 these regimes break down for f_c ≲ 3, which is an area of the parameter space that was previously by studied by Hansen & Oh (2006), Laursen et al. (2013), Gronke & Dijkstra (2016), where the probability of not finding a clump in a certain sightline is non-negligible (this allows for nonzero ionizing photon escape fraction; see Dijkstra et al. 2016).

Figure 15 shows the transition from double- to single-peaked spectra for randomly moving clumps. The color coding shows in this case the peak position of the spectrum, where white is x_peak ~ 3, that is, when the peak position moves outside the core of the line⁷. For faster clumps, this boundary moves to greater values of f_c, making it more likely to obtain a single-peaked spectrum (at line center). The black dashed lines in Fig. 15 denote f_{c, crit} from Eq. (12)⁸; in other words below this line the velocity space is not well sampled and allows photons at line center to escape.

The same line is also indicated in Fig. 16, where we focus on the clump column density N_HI,cl = 10¹⁷ cm^-2 as predicted by “shattering” (McCourt et al. 2016). Here the peak position (in log scale) is color coded as a function of covering factor and clump velocity dispersion. For large values of σ_cl, the transition to double-peaked spectra occurs at a larger covering fraction, since more clumps are required to sample the broader velocity distribution. Below this threshold, we see a single-peaked spectrum from photons that escape through holes in velocity space.

Figure 17 shows this increase in asymmetry with greater f_c for fixed outflow velocity and total column density of N_HI,total = 4/3 × 10¹⁹ cm^-2. Here, the color corresponds to the asymmetry of the spectra, which we define as in Sect. 4.3 to be the ratio of the integrated blue over the red flux minus one. In Fig. 17 we also indicate graphically the conditions for homogeneous escape discussed in Sect. 2.3.4, that is, that the adjacent clump is optically thick (4/3τ_cl(x_next) ≳ 1 with x_next ≡ v_max/ (f_cv_th)), and that the initial scatterings occur in the core of the line (x_next<x_∗). If both conditions are fulfilled (and sufficient outflows are present, i.e., v_max ≳ 50 km s^-1), the emergent spectrum is asymmetric toward the red side (as visible from the red region in Fig. 17).

5.2. Escape fractions and equivalent width boosting

The different regimes for Lyα radiative transfer through a multiphase gas have different implications for the Lyα escape fraction when dust is present within the clumps.

Inside the “optically thin regime”(τ_0,total ≲ 1), the Lyα escape fraction is equal to the continuum escape fraction as Lyα photons stream through all the clumps and are affected by the dust content within these clumps. Hence, f_esc ≈ exp(−τ_d,total). This can be seen for the N_HI,cl ≲ 10¹⁴ cm^-2, f_c ≲ f_{c, crit} data points in Fig. 12.

In the “random-walk regime”, we confirm the escape fraction given by Hansen & Oh (2006, see Eq. (18), apart from geometrical pre-factors). In this regime the governing quantity for the escape fraction is ϵ, i.e., the absorption probability per clump interaction and the number of clumps encountered N_cl. In this regime, the latter is merely a function of f_c (see Eq. (8)), but ϵ depends non-trivially on N_HI,cl and the clump movement as a result of variations in how deep the photons penetrate into the clumps. This is why there is some scatter in f_esc in this regime; this is visible, for instance, in Fig. 12. This is the only regime in which Lyα photons are shielded from dust, thus, allowing for “equivalent-width-boosting” (Hansen & Oh 2006). That is, the ratio between the Lyα and UV escape fraction might be greater than unity.

Finally, in the “homogeneous regime”, the behavior is a combination of the above two behaviors. Initially, the photons (on average) interacts with ~ N_cl(f_{c, crit}) clumps before diffusing to the line wings and escaping through free-streaming, which leads to other ~ f_c clump encounters (cf. Fig. 2). Consequently, in this regime the escape fraction is approximately given by ${\mathit{f}}_{\mathrm{esc}}\mathrm{~}{\mathit{f}}_{\mathrm{esc}}^{\mathrm{HO}\mathrm{06}}\mathrm{\left(}{\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\mathrm{\right)}{\mathrm{e}}^{\mathrm{-}{\mathit{\tau }}_{\mathrm{d}\mathit{,}\mathrm{total}}}$. This causes the flattening of f_esc versus f_c in Fig. 11 as in this case τ_d,total is kept constant.

From the above considerations, one can see that the escape fraction depends on several parameters and is therefore non-trivial to predict. As a consequence, in Sect. 4.5 we demonstrated that the Lyα escape fraction may either increase or decrease with increasing metallicity, which is Z ∝ τ_d/n_HI (Pei 1992; Laursen et al. 2009), depending on the dust optical depth through an individual clump τ_d,cl (see the trends in the filled and unfilled symbols in Figs. 12 and 13). The controlling parameter is essentially the ratio of absorption probability per surface scatter to the absorption probability per clump passing. Moreover, we have shown that Lyα escape fractions can be large, even for large values of f_c. Thus, we find that homogeneous, “slab-like” spectra can be observable even in models with significant dust content (as is realistic; see Sects. 5.3 and 5.4).

Regarding the equivalent width boosting we found, one necessary requirement for the “Neufeld effect” to be active is that Lyα photons escape via surface scatterings off the clumps, i.e., in the “random walk” regime. This implies that the emergent spectrum is narrow and single peaked at line center (as already noted by Laursen et al. 2013), which is a clear observational signature for equivalent width boosting to be active⁹.

5.3. Connection to a homogeneous medium

Observed Lyα spectra can often be successfully modeled using a simple model called the shell model (see, for instance, Hashimoto et al. 2015; Karman et al. 2017). This shell model consists of a central Lyα (and continuum) emitting region that is surrounded by a moving shell of hydrogen and dust (Ahn et al. 2003; Verhamme et al. 2006). It is somewhat surprising that this simple, six-parameter model can account for the likely radiative transfer effects happening in the complex, multiphase medium of a variety of galaxies and their environments. Since the shell model is clearly very idealized, it is unclear what the extracted shell-model parameters mean physically. In Gronke & Dijkstra (2016) we found that a simple one-to-one mapping between the shell-model parameters and those from a clumpy medium is not possible; for the most part, the shell model cannot reproduce the spectra emergent from a multiphase medium. This failure mostly results from the high fluxes at line center from the multiphase simulations, which are hard to obtain through radiative transfer through a uniform gas distribution (such as a shell).

However, in Gronke & Dijkstra (2016) we restricted our analysis to covering factors of (and σ_cl ≲ 100 km s^-1), i.e., the “random-walk” and “optically thin” regime. As we showed here, for a (much) greater number of clouds the system approaches a slab-like state; this state leads to, for example, much lower fluxes at line center for the resulting Lyα spectrum. Hence, these multiphase spectra might be closer to observed spectra. Whether or not the shell-model parameters correspond to physical parameters of a clumpy medium with large f_c will be analyzed for a future work. However, our results show that for f_c ≳ f_{c, crit} the spectra are similar to a slab with the same column density. Furthermore, we fitted shell-models to three spectra originating from a clumpy medium with N_HI,total = 4/3 × 10¹⁹ cm^-2, v_max = 50 km s^-1 and various covering factors. The column density of our system and the outflow velocity were chosen to be well within the range of shell-model parameters recovered from observations (e.g., Yang et al. 2017). Prior to fitting, we smoothed the spectra using a Gaussian kernel with FWHM W ~ 24 km s^-1. Figure 18 shows the three spectra and the best-fit shell model spectra. The resulting shell-model parameters are also shown in the figure. While the fits for f_c = 3 and f_c = 10 are rather poor and the recovered shell-model column densities are more than an order of magnitude off, the spectra for f_c = 50 can be remarkably well recovered. Interestingly, here the shell column density is very close to the input value, and the recovered shell of v_exp ≈ 25 km s^-1 outflow velocity corresponds to the mass weighted mean of the used Hubble-like outflow. Also, the dust content and, to some extent, the temperature of the gas are recovered. On the other hand, as the photons are injected at line center, the recovered widths of the intrinsic spectra (σ_i) are too large. This may be to compensate for the narrow coverage of the shell in velocity space. A similar discrepancy in the intrinsic profile width is also found in the literature (e.g., Yang et al. 2016, by comparing σ_i with the width of the Hα line), in which this discrepancy might also originate from radiative transfer effects.

All these points suggest that at least some of the shell-model parameters might have a true physical meaning. In this work, we provide an equally simple but physically meaningful model that serves as a theoretical justification for the shell model. The full mapping from the shell model to the parameters of a multiphase medium with large f_c will be part of future work. However, from our single example it is already apparent that if an observed Lyα spectra can be modeled using a simple, homogeneous shell, one can think instead about a fog of droplets (with f_c ≫ f_{c, crit}), which is more realistic given our knowledge about gas properties.

5.4. Implications for ab initio Lyα radiative transfer simulations

Our findings suggest a possible cause for the mismatch between observed Lyα spectra and those computed with snapshots of hydrodynamical simulations as input; these are sometimes referred to ab initio Lyα radiative transfer simulations.

Observed Lyα spectra from z ~ 0 to higher redshifts show several common features:

A significant shift redward of the main emitting peak. Forinstance, at z ~ 2–3 galaxies selected owing to their strong Lyα emission and dropout-selected galaxies (Lyα emitters and Lyman-break galaxies or LAEs and LBGs, respectively) show shifts of several hundred  km s${-1}^{}$ (e.g., Steidel et al. 2010; Kulas et al. 2012; Erb et al. 2014; Song et al. 2014; Trainor et al. 2015; Hashimoto et al. 2015).

Asymmetric profiles with mostly stronger red than blue components. For instance, Erb et al. (2014) measured the median equivalent width ratio W_blue/W_red in their sample of 36 LAEs at z ~ 2–3 to be ~ 0.4. This is consistent with the findings at z ~ 0.2, which also show a dominant red side (by a factor of a few, Henry et al. 2015; Yang et al. 2016, 2017).

There are roughly as many single as double-peaked spectra. For instance, in the Lyα selected galaxy sample presented by Trainor et al. (2015) of 318 LAEs at z ~ 2.5–3, 41% show a double-peaked spectra. This fraction agrees well with the double-peaked fraction of 45% they found in the KBSS-MOSFIRE LBG sample (Steidel et al. 2014) and those of other studies (e.g., Kulas et al. 2012; Yamada et al. 2012, found ratios of ~ 1/3 and 1/2, respectively). Only a small part of the double-peaked profiles show a dominant blue peak, which agrees with flux ratio discussed above. Trainor et al. (2015) have quantified these to be ~ 10% of the double-peaked spectra.

For double-peaked spectra, the flux in the “valley” between the peaks is small. Because of smoothing and resolution effects due to the observational aperture, measuring this quantity is challenging, in particular for higher redshifts. However, at lower redshifts Yang et al. (2016) have found in their sample of 12 galaxies at z ~ 0.2 the flux ratio between the valley and the red peak to be $\mathrm{0.0}{\mathrm{3}}_{-0.02}^{\mathrm{+}\mathrm{0.08}}$ and never greater than 0.27. Also the 14 galaxies of the “Lyman-α reference sample” (LARS; Östlin et al. 2014) at 0.02 <z< 0.2 have a flux ratio between the maximum and minimum of <0.1, mostly even consistent with zero (Rivera-Thorsen et al. 2015).

These findings seem to be in stark contrast to the Lyα radiative transfer simulations that use a snapshot of a (high-resolution) hydro-dynamical simulation of a galaxy as input geometry (e.g., Tasitsiomi 2006; Laursen & Sommer-Larsen 2007; Zheng et al. 2010; Barnes et al. 2011; Verhamme et al. 2012; Behrens & Braun 2014; Smith et al. 2015; Trebitsch et al. 2016). Owing to the computational cost and probable directional dependence of the emergent spectrum (Verhamme et al. 2012; Behrens & Braun 2014), no statistical compilation of simulation-based spectra has yet been assembled. However, existing predicted spectra are generally too symmetric and/or possess a flux that is too high at line center. This is commonly attributed to (i) CGM in combination with instrumental effects (as discussed in Gronke & Dijkstra 2016); (ii) radiative transfer effects in close proximity to the origin of the photon; and/or (iii) IGM absorption (Dijkstra et al. 2007; Laursen et al. 2011). All these arguments move the problem to different spatial location (in case of (ii) even to a subgrid scale). However, the last solution cannot be universally invoked, especially at lower redshifts. For instance, Laursen et al. (2011) found that for z ≲ 3.5 only ≲ 30% of the sightlines show a full absorption feature, which would lead to a low flux in the “valley”. Furthermore, while the IGM opacity increases with redshift, we see that the Lyα escape fraction from star-forming galaxies also increases with redshift (Hayes et al. 2011; Blanc et al. 2011; Dijkstra & Jeeson-Daniel 2013). Both arguments strongly suggest that IGM absorption cannot be the dominant mechanism regulating the visibility of Lyα emission.

We have shown here that this discrepancy between observations and simulations can be understood easily. Simulations with Lagrangian-type techniques such as adaptive-mesh-refinement or smooth-particle-hydrodynamics (AMR and SPH, respectively) reach their highest resolution in the densest regions such as the midplane of the galaxy disk. While future simulations likely reach peak resolutions approaching the ~ 0.1 pc scale we expect, this still does not capture clump formation and evolution at large distances in the CGM, where the density and resolution remains low. This means the clumps are unresolved and, thus, the covering factor per resolution element is less than unity compared to potentially hundreds as suggested theoretically by McCourt et al. (2016). This lower f_c (while keeping the column density and global structure unchanged) leads to a higher flux at line center (as shown in Fig. 3), less asymmetric spectra (Fig. 17), and in general more “unrealistic spectra” (cf. Gronke & Dijkstra 2016). Therefore, small-scale structure in the CGM is crucial for modeling radiative transfer through the galaxy. We expect that direct simulation of the multiphase CGM is essentially impossible precisely because it requires very high spatial resolution, even in parts of the galaxy that are typically empty: for example, a spatial resolution of ~ 0.1 pc in the outskirts of a galaxy corresponds to a mass resolution of ~ 10^-10–10^-9 solar masses. Instead, we propose to study both Lyα radiative transfer and hydrodynamical behavior on the smallest scales and then to use this knowledge as a subgrid recipe.

6. Conclusions

Motivated by several observations and a recent theoretical work by McCourt et al. (2016), we studied Lyα radiative transfer in an extremely clumpy medium, i.e., with large number of clumps per sightline (up to f_c ~ 1000). Our main findings on Lyα radiative transfer through clumpy media are

The behavior of a multiphase medium depends strongly on the“clumpiness” of the system even when keeping the otherparameters, such as the total column density, constant.

In particular, we identify a threshold above which Lyα photons escape preferentially via frequency excursion, i.e., above which multiphase media affect Lyα as if they were homogeneous. This transition depends on clump column density and can be estimated analytically. We found the threshold for the static case to be ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\mathrm{\approx }\{\begin{array}{c}\\ & \mathrm{for}\sqrt{\mathrm{3}}{\mathit{a}}_{\mathit{v}}{\mathit{\tau }}_{\mathrm{0}}\gtrsim \mathrm{19}\\ & \mathrm{otherwise}\mathrm{.}\end{array}$(19)The value of this threshold between a clumpy and homogeneous nature further depends on the clump kinematics in a way that can also be estimated analytically. If the clump motion is uncorrelated and Maxwellian, we find that the threshold is given by ${\mathit{f}}_{\mathrm{c}\mathit{,}\hspace{0.17em}\mathrm{crit}}\mathrm{\approx }\max \left(\frac{\mathrm{3}{\mathit{\sigma }}_{\mathrm{cl}}}{\mathrm{2}{\mathit{v}}_{\mathrm{th}}\sqrt{\ln \mathrm{(}\mathrm{4}\mathit{/}\mathrm{3}{\mathit{\tau }}_{\mathrm{0}\mathit{,}\mathrm{cl}}\mathrm{)}}}\mathit{,}\mathrm{1.5}\right)\mathit{.}$(20)This is valid for sufficiently large clump motion, i.e., σ_cl ≳ v_th. For smaller values, the system approaches the static case above. Furthermore, we expect large-scale correlations in velocity the transition between these extreme cases. We will investigate this in a future study.

A similar threshold was found for outflowing clumps (Sect. 2.3.4). We also showed that for outflowing clumps, increasing f_c naturally leads to more asymmetric line profiles, which is in much better agreement with what has been observed in observations of galaxies.

These results suggest important implications for the interpretation of observed Lyα spectra, as a multiphase medium is physically more motivated than simplified homogeneous geometries such as the “shell model”. Nevertheless since shell models successfully reproduce observed spectra, they are frequently used to model observations. Because a medium with sufficiently large covering factor behaves as a homogeneous medium, the success of shell models may indicate large covering factors are typical in galaxies as predicted by McCourt et al. (2016). Specifically, we found typical values of f_{c, crit} ~ 10–50, which are much smaller than f_c ≳ 1000 predicted in their work. In this picture, it is easy to understand the convergence to the shell model.

Motivated by these results, we fitted shell models to spectra emerging from extremely clumpy outflows undergoing Hubble flow. We found that the column density from the shell closely matches that of the collection of clumps as a whole and the shell expansion velocity appears to be the mass weighted average velocity. This result is very promising as it suggests that the shell model provides us with a fast method of extracting some physical properties of the interstellar and circumgalactic medium from the Lyα spectral line shape. In addition, the value of other shell parameters (e.g., intrinsic Lyα line width prior to scattering) should not necessarily be interpreted literally as physical. We will explore this systematically in future work.

Another implication concerns the mismatch between observed Lyα spectra and those predicted by theoretical studies of Lyα radiative transfer utilizing hydrodynamical simulations for their input geometry. Our work suggests that this mismatch can be due to the existence of tiny clumps in the observed systems that cannot form even in the most modern hydrodynamical simulations of galaxies because of their limited resolution. Thus, setups of these simulations might yield effective covering factors that are too low, causing the spectra to possess, e.g., a flux at line center that is too large. We will use our results for radiative transfer on small scales to develop an effective theory that can be implemented as a subgrid model in global simulations of galaxies.

Movie

Movie of Fig. B.2 Access here

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Acknowledgments

This research made use of NASA’s Astrophysics Data System, and a number of open source software such as the IPython package (Pérez & Granger 2007); matplotlib (Hunter 2007); and SciPy (Jones et al. 2001). M.D. and M.G. thank the astronomy group at UCSB for their hospitality, and the organizers and participants of “SnowCLAW 2017” for an inspiring conference. M.M. was supported by NASA grant HST-HF2-51376.001-A, under NASA contract NAS5-26555. S.P.O. acknowledges NASA grant NNX15AK81G.

References

Appendix A: Regimes of a medium with uncorrelated clump motion

For randomly moving, optically thick clumps the photons either escape through holes in velocity space (if f_c ≲ f_{c, crit}, Eq. (12)) or escape in single flight (for f_c ≳ f_{c, crit}), as described in Sect. 2.3.4. In the former case, the emergent spectrum is similar to the intrinsic spectrum, that is, narrow and single peaked. If, however, f_c>f_{c, crit} the radiative transfer process is similar to a slab with temperature T_eff (Eq. (11)), which means the photons escape in a single flight after interaction with one (fast-moving) clump, and so will the emergent double-peaked spectrum, i.e., a peak position of x_p ~ x_∗ or in (observed) velocity units ${\mathit{v}}_{\mathrm{p}}\mathrm{\approx }{\mathit{x}}_{\mathrm{\ast }}{\mathit{v}}_{\mathrm{th}}\mathrm{\left(}{\mathit{T}}_{\mathrm{eff}}\mathrm{\right)}\mathrm{\approx }\mathrm{3.8}{\mathit{\sigma }}_{\mathrm{cl}}\mathit{.}$(A.1)Figure A.1 shows a visual overview of these regimes. In this figure, we also indicated that a smaller clump motion than the internal thermal motion of the atoms (for the parameters used in this work of σ_cl ≲ 13 km s^-1) leads to a convergence back to the static case.

Another interesting part of the parameter space is between the two regimes, for f_c ~ f_{c, crit}. Here, the velocity space is sufficiently sampled so that hardly any photons can escape without clump interaction. However, after an interaction with a (slowly moving) clump the probability of interacting with another clump is small – even if the photon is still in the core of the line. This is because the velocity distribution is not that well sampled to provide τ₀ ≫ 1. As a result, the emergent spectrum directly represents the clumps’ velocity dispersion of the clumps, which means a single-peaked spectrum a line center of width ~ σ_cl.

In summary, with increasing covering factor, a medium with uncorrelated clump motion can lead to a narrow or wide single-peaked spectrum (of widths of the intrinsic spectrum or clump velocity dispersion, respectively) at line center or a wide double-peaked spectrum (if f_c ≳ f_{c, crit}).

Appendix B: Additional numerical results

Appendix B.1: Transmission through a clumpy slab

Figure B.1 shows the fraction of photons that passed through a clumpy medium when emitted at the boundary of the box for a fixed total column density but various number of clumps per sightline. The transmitted fraction of photons decreases with increasing covering factor and approaches the limit of a homogeneous slab. We attribute this dependence on f_c and the differences compared to a homogeneous medium to surface effects, i.e., because of the roughness of the boundary it is easier for photons to get trapped in the slab.

Appendix B.2: Examples of photon trajectories

Figure B.2 shows examples for the three different escape mechanisms discussed in this work. The left panel shows a random walk in a static medium, the central panel shows escape through excursion, and the right panel shows the escape through single flight. An animated version of Fig. B.2 is available online¹⁰.

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