The Lyα Reference Sample. VIII. Characterizing Lyα Scattering in Nearby Galaxies

The LARS galaxies were imaged by the HST using a synthetic narrowband centered on Lyα. The technique used to measure the Lyα emission line is described in detail in Östlin et al. (2014) and Hayes et al. (2009). Briefly, a synthetic narrowband image was created by subtracting the F140LP and F150LP or the F125LP and F140LP long-pass filters (depending on the galaxy redshift) on the Solar Blind Channel to observe the Lyα line. Either the Wide Field Camera 3 (WFC3) or the Advanced Camera for Surveys (ACS) were then used to obtain Hα and Hβ emission maps. These three emission maps serve as the basis for our following analysis.

The images were reduced using the Drizzlepac¹² package in the standard HST reduction algorithm. We used custom-built software called Lyα Extraction Software (LaXs; Hayes et al. 2009; Östlin et al. 2014) to subtract the continuum from the Lyα images. To find the underlying continuum flux in Lyα (and to lesser extent Hα and Hβ), we performed spectral energy distribution fitting using observations from five continuum filters on HST (two in the far-UV (FUV) and three in the optical). To briefly summarize: we fit each pixel with a three-component model, comprised of a young, star-forming population, a stellar continuum from older stars, and line+continuum produced by photoionization. These were fit to the data with a ${\chi }^{2}$ fitting method and population spectra from Starburst99 templates (Leitherer et al. 1999). The free parameters in this procedure are the equivalent width of the Lyα line, the age (of the young population), E(B–V)_s, and the masses of the young and old populations. The fitting routine also accounts for flux from the nebular continuum by utilizing the Hα and Hβ observations. For more details see Hayes et al. (2009). The images' point spread functions were also matched using custom-made software (Hayes et al. 2016; J. Melinder et al. 2017, in preparation).

Using the appropriate aperture for each galaxy is important, as we need to determine a size that encompasses as much of the Lyα emission as possible without including too much background. The Petrosian (1976) radius determined from the UV continuum does not encompass the outer regions of the Lyα halo for every galaxy. We therefore created an aperture by convolving the UV continuum image with a $\sigma =50$ pixel Gaussian kernel, creating a larger aperture. Using this convolved image, we calculated an isophotal radius at which the local intensity is 20% of the average flux contained within the radius, and used this radius to define the aperture for each galaxy. However, in cases where the Lyα halo morphology is particularly extended or more complex, even this radius is not large enough to encompass all of the Lyα emission. For LARS01, LARS02, LARS05, LARS07, and LARS08, we therefore used 1.5 times this radius to define the aperture to include all of the Lyα emission from the galaxy. Using an isophotal measurement of the radius, rather than a fixed circular aperture, maximizes the signal while decreasing the background. The aperture is large enough to contain the extended Lyα emission while excluding excess background. An average of 75% of the Lyα emission is captured within the calculated apertures.

In Figure 1, we show the integrated line ratios for the 14 LARS galaxies, with the fluxes given in Table 1. Following Scarlata et al. (2009), we show several models describing dust geometry. A simple dust screen model with R_V = 3.07 (Cardelli et al. 1989) is denoted by the black dashed line. The black dashed-dotted line demonstrates a smooth internal dust model (Mathis 1972), where the stars, gas, and dust are evenly mixed. This model is described as ${I}_{\lambda }/{I}_{\lambda }^{0}=(1-{e}^{-{\tau }_{\lambda }})/{\tau }_{\lambda }$, where I_λ is the observed intensity of the source, ${I}_{\lambda }^{0}$ is the assumed intrinsic intensity, and ${\tau }_{\lambda }$ is the optical depth of the medium. Finally, we show models for various clumpy media following the formalism of Natta & Panagia (1984),

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{\lambda }}{{I}_{\lambda }^{0}}={\rm{\exp }}(-N(1-{e}^{-{\tau }_{c,\lambda }})),\end{eqnarray} \tag{ 1 }$$

where I_λ and ${I}_{\lambda }^{0}$ are defined as before, N is the mean number of clumps along the line of sight, and ${\tau }_{c,\lambda }$ is the optical depth of each clump. The number of clumps per sightline follow a Poisson distribution and individually obey the Cardelli model, with each clump having the same optical depth. These models do not include Lyα radiative transfer effects. Scarlata et al. (2009) found that a majority of $z\sim 0.3$ Lyα emitting galaxies were best described by the clumpy dust model. Meanwhile, for the galaxies that lie below the uniform dust screen model, they noted that a mechanism that would preferentially dampen the Lyα photons but leave the Balmer photons unaffected would be required, such as a neutral medium through which the Lyα photons would pass without scattering.

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Table 1.  LARS Integrated Fluxes

LARS ID	z	Lyα (erg s−1 cm−2)	Hα (erg s−1 cm−2)	Hβ (erg s−1 cm−2)
01	0.028	$(5.25\pm 0.03)\times {10}^{-13}$	$2.82\times {10}^{-13}$	$8.44\times {10}^{-14}$
02	0.030	$(2.56\pm 0.01)\times {10}^{-13}$	$7.36\times {10}^{-14}$	$2.69\times {10}^{-14}$
03	0.031	$(9.87\pm 0.07)\times {10}^{-14}$	$2.72\times {10}^{-13}$	$3.75\times {10}^{-14}$
04	0.033	$(-1.12\pm 0.01)\times {10}^{-13}$	$1.91\times {10}^{-13}$	$5.67\times {10}^{-14}$
05	0.034	$(2.75\pm 0.02)\times {10}^{-13}$	$1.60\times {10}^{-13}$	$5.76\times {10}^{-14}$
06	0.034	$(-2.30\pm 0.45)\times {10}^{-15}$	$2.48\times {10}^{-14}$	$9.11\times {10}^{-15}$
07	0.038	$(2.30\pm 0.02)\times {10}^{-13}$	$1.17\times {10}^{-13}$	$3.00\times {10}^{-14}$
08	0.038	$(1.01\pm 0.01)\times {10}^{-13}$	$3.44\times {10}^{-13}$	$5.31\times {10}^{-14}$
09	0.047	$(4.58\pm 0.16)\times {10}^{-14}$	$4.33\times {10}^{-13}$	$1.60\times {10}^{-13}$
10	0.057	$(1.56\pm 0.05)\times {10}^{-14}$	$3.37\times {10}^{-14}$	$1.40\times {10}^{-14}$
11	0.084	$(1.47\pm 0.00)\times {10}^{-13}$	$9.06\times {10}^{-14}$	$1.56\times {10}^{-14}$
12	0.102	$(1.71\pm 0.01)\times {10}^{-13}$	$7.28\times {10}^{-14}$	$1.65\times {10}^{-14}$
13	0.147	$(6.78\pm 2.68)\times {10}^{-14}$	$4.11\times {10}^{-14}$	$1.15\times {10}^{-14}$
14	0.181	$(1.63\pm 0.00)\times {10}^{-13}$	$2.75\times {10}^{-14}$	$1.03\times {10}^{-14}$

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We find that while several of the LARS galaxies lie in the region of Figure 1 described by a clumpy dust model, others show Lyα to be significantly lower than that inferred from the Balmer decrement. Of particular note are the galaxies for which the Hα/Hβ ratios are consistent with the intrinsic value, but have Lyα/Hα values anywhere from ∼75% (LARS14) to nearly 100% (LARS09) less than than the intrinsic value of 8.7. This is in agreement with previous studies that have found that integrated Lyα emission is often much weaker than exp-ected from recombination theory (e.g., Terlevich et al. 1993; Giavalisco et al. 1996; Atek et al. 2008; Scarlata et al. 2009). Overall, the LARS galaxies show Lyα/Hα ratios that span the whole range from no additional destruction of photons beyond that done by dust to the formation of damped Lyα absorption profiles.

The internal dust model is difficult to differentiate from a clumpy dust model at low optical depth and a small number of clumps. However, given the global values shown in Figure 1, it is unlikely that the internal dust best explains the ISM in the LARS galaxies.

Using the resolved line maps of the LARS galaxies, we are able to probe the dust geometry down to the minimum phy-sical scales allowed by the HST point spread function (see J. Melinder et al. 2017, in preparation), which for these galaxies is approximately 40 to 130 parsecs, depending on redshift. By using the emission line ratios and knowing where exactly in the galaxies different line ratios occur, we can trace what regions in the galaxies are affecting the integrated line ratios. In this section, we will compare two galaxies that are each representative of differing dust models, LARS01 and LARS12. The HST Lyα and Hα emission maps are shown for these two galaxies in top panels of Figures 2 and 3, respectively. Because the Hβ images have low signal-to-noise in the outer regions of the galaxies, we have implemented a cut in the Hβ line maps of ${f}_{{\rm{H}}\beta }=1\times {10}^{-18}$ erg/s/cm². We then use only the pixels from all three emission maps that correspond to Hβ values greater than this cut.

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LARS01, which is displayed in Figure 1 as a purple square, is an example of a galaxy that is not well described by a clumpy dust model. We show the Lyα/Hα versus Hα/Hβ values for the galaxy in Figure 4. As in Figure 1, we have plotted the various dust models. While some pixels fall in the clumpy model region, most lie below the plotted dust models. Scarlata et al. (2009) noted that the dust models shift slightly downwards if the scattering of Lyα photons within the clumps is included. This effectively increases the optical depth of these photons. Further dampening of Lyα may be due to these photons traveling through the neutral hydrogen in the ISM. Additionally, a number of pixels correspond to regions where the Lyα is the same or brighter than predicted by recombination physics. In the lower panels of Figure 2, the Lyα/Hα and Hα/Hβ ratios are shown for the galaxy.

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In addition to the bright Lyα, there are also a significant number of pixels that have Lyα/${\rm{H}}\alpha \lt 0$. Some of these may be due to noise, producing artificially negative Lyα values. Nevertheless, our cut in Hβ minimizes the number of pixels affected by noise. Therefore, the negative Lyα/Hα values are more likely to be where Lyα is seen in absorption. This is due to continuum light being either absorbed or scattered by the n = 2 to n = 1 resonance. The Lyα emission maps shown in the top right panels of Figures 2 and 3 display the negative Lyα pixels in dark purple. The fact that the negative pixels are spatially correlated, particularly in the more central regions of the galaxies, indicates that these are real absorption features.

Unlike LARS01, LARS12, denoted in Figure 1 by a yellow star, falls directly in the region of the clumpy dust models. We show the Lyα/Hα versus Hα/Hβ values for the pixels in LARS12, along with the same dust models, in Figure 5. As would be expected from the global Lyα/Hα and Hα/Hβ ratios, a majority of the pixels fall in the regions of the plot well described by the clumpy dust models with varying N (numbers of clumps along the line of sight) and τ (optical depth). Overall, LARS12 has much higher Hα/Hβ values than LARS01, indicating that the galaxy is significantly dustier. We show the Lyα/Hα and Hα/Hβ ratios of LARS12 in the bottom panels of Figure 3.

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To quantify the comparison of Lyα and Hα in a galaxy, we examine their respective surface brightnesses (Östlin et al. 2009). We plot the surface photometry for LARS01 in the left panel of Figure 6. Shown in black are the logarithmic Lyα surface brightnesses plotted against the logarithmic Hα surface brightnesses. The solid red line indicates the intrinsic Lyα/${\rm{H}}\alpha \sim \,8.7$ expected under the assumption of Case B recombination. Lower Lyα/Hα ratios indicate either dust absorption or scattering of the Lyα photons out of the line of sight, while Lyα/Hα values greater than ∼8.7 show where Lyα has likely been scattered into the line of sight.

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Overplotted in blue in the left panel of Figure 6 are the individual pixels of LARS01 that fall in the region corresponding to the clumpy dust models of Natta & Panagia (1984; defined as any pixel lying above the dust screen curve in Figure 4). Most of the blue pixels show Lyα/Hα ratios well below the intrinsic relationship, indicating that the Lyα photons are either absorbed by dust or scattered out of the line of sight by neutral hydrogen (Hayes 2015). Some pixels indicate that Lyα photons have also been scattered into the line of sight, as they are overluminous for the corresponding Hα surface brightnesses. Both of these scenarios can be explained by the presence of a clumpy medium that can both dampen as well as scatter the Lyα. The brightest Lyα pixels (in the upper right region of the left panel of Figure 6) are not affected by the clumpy medium.

In the center panel of Figure 6, we trace the pixels in the clumpy dust region of Figure 4 back to the LARS01 Lyα image. What is immediately obvious is that the Lyα photons that are escaping from the center of the galaxy are not doing so through a clumpy dust medium. Not only is the central ∼0.65 kpc diameter region the source of the brightest Lyα emission, but its Hα/Hβ ratio is very close to the intrinsic value of 2.86. Whatever minimal extinction exists in the region is consistent with that expected from a dust screen model. This suggests that the region has been cleared of static neutral hydrogen, either by the winds from a preponderance of supernovae or by a bubble blown out by the region's active star formation. The resulting outflows then allow Lyα photons in the wings of the line to escape (Mas-Hesse et al. 2003; Rivera-Thorsen et al. 2015; Herenz et al. 2016). Moreover, the concentration of ionized hydrogen in the galaxy's central region (see Figure 2) indicates that most of the medium has been ionized, and is therefore transparent to Lyα photons. Laursen et al. (2009) showed that the dust model used in Lyα radiative transfer calculations is relatively unimportant, as the photons either travel through fairly dust-free regions, or they travel through dusty regions and are absorbed. This is in line with the general lack of Lyα attenuation in the center of LARS01.

Most of the pixels that lie above the Cardelli et al. (1989) dust screen model in Figure 4 come from a concentrated annular region around the center of the galaxy. This indicates that an annulus of clumpy dust exists outside the central star-forming region.

Rivera-Thorsen et al. (2015) observed the LARS galaxies using the Cosmic Origin Spectrograph (COS) on the HST. They found that, in general, gas outflows with velocities greater than 50 km s⁻¹ occurred in all of the galaxies where Lyα is observed. The model used to explain this behavior (Tenorio-Tagle et al. 1999; Mas-Hesse et al. 2003) posits that feedback from the actively star-forming regions in the center of the galaxy creates Rayleigh–Taylor instabilities at the border between the hot bubble and the neutral phase of the ISM. This results in a clumpy medium through which Lyα photons can either escape or be absorbed. This model could explain the regions we observe in LARS01 and the other LARS galaxies.

In contrast to LARS01, the integrated ratios of LARS12 place it solidly in the region of Figure 1 that is described by the clumpy dust models. LARS12 is consistent with the clumpy dust model almost all the way to the nucleus (see Figure 7). While Lyα escapes freely from the inner ∼0.65 kpc of LARS01, the same cavity in LARS12 is only ∼0.4 kpc in diameter. This region is denoted by several pixels in the center of the galaxy, shown in the center panel of Figure 7. This difference can also be seen in the curve of growth analysis performed by Hayes et al. (2014; Figure 4): in LARS01, the integrated Lyα flux peaks close to the galactic center, while in LARS12, this peak is 5 kpc from the galaxy's center. In other words, in the center of LARS12, dust attenuation is suppressing Lyα emission in the center of LARS12 compared to what would be expected from a clumpy dust medium.

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We next isolate the pixels of the LARS01 image that fall above the intrinsic Lyα/Hα value. The regions of the galaxy in which these pixels are located are of particular interest as they provide insight into the mechanisms that can transport Lyα to large radii. Previous studies have found that in some galaxies, as much as 70% of the Lyα emission comes from the diffuse regions that are outside the more central areas of star formation (Hayes et al. 2005; Atek et al. 2008). These halos often reach galactocentric radii of 10 kpc. Several processes can theoretically modify the Lyα/Hα ratio, such as extreme high or low densities, shocks, and other non-equilibrium conditions, but the more likely explanation of an upward deviation from Lyα/${\rm{H}}\alpha \sim 8.7$ is the physics of Lyα radiative transfer (e.g., Hayes 2015). The fact that Lyα/${\rm{H}}\alpha \gg 10$ is frequently observed tells us that scattering into the line of sight not only happens, but that it can dominate the integrated luminosity of a galaxy.

In the right panel of Figure 6, we show only the regions of LARS01 where Lyα photons are scattered into the line of sight. The pixels that contain overly bright Lyα make up the Lyα halo of LARS01. Figure 7 shows the same quantity but for LARS12. This galaxy has the same characteristics as LARS01, with the highest Lyα/Hα pixels being the most significant contributor to the emission from the outer halo. For Lyα photons to be this bright relative to Hα, the photons have to be scattered into the line of sight. It has been shown that the size of extended Lyα halos around low-redshift Lyα emitters is inversely correlated with dust content, indicating that low dust abundance is necessary for Lyα photons to resonantly scatter to large radii (Hayes et al. 2013). The fact that the high Lyα/Hα pixels correspond directly to the Lyα halo indicates that the halo is generated by a scattering process where little dust exists, rather than escaping directly through holes blown through the ISM by galactic winds.

We present figures similar to Figures 6 and 7 for the other LARS galaxies in the Appendix. As might be expected from the results for LARS12, all of the galaxies that lie on or above the Cardelli model in Figure 1 (LARS03, 08, 11, and 13) show little to no cavity in the center of the galaxies. Rather than forming an annulus, the pixels corresponding to the clumpy dust model exist all the way to the center. In contrast, most of the galaxies of this study possess a scattering halo where Lyα is overly bright compared to Hα.

Our scattering model is based on the observed properties of the dust absorption and photon scattering discussed in Section 3.2. LARS01 shows that the pixels corresponding to a clumpy dust model congregate in an annulus around a central region within which the Lyα emission is brightest and only slightly extinguished. Meanwhile, the pixels that are overbright in Lyα compared to Hα, which is an indication of scattering back into the line of sight, form the Lyα halo. Following this, we posit a three-component galaxy model: a central region in which the Lyα photons are attenuated by only a uniform dust screen, an annular region that contains a clumpy dust geometry, and finally an outer halo region with no dust where the Lyα photons can scatter out to large distances. Of particular interest are how and how much the Lyα photons scatter in the halo region, as one of the big questions about Lyα escape is how the dust affects the escape of Lyα photons.

We begin multiplying the 2D Hα emission line map by the intrinsic ratio of 8.7 to obtain an estimate of the intrinsic Lyα distribution that is unaffected by dust or scattering. We then create three separate images: one derived using a dust screen model, another using a clumpy dust model, and a third where the scattering of Lyα photons is the only physical process. After each image is created, we piece them together, using the dust screen model for the central region, the clumpy dust model in an annulus around the center, and the scattering model for the outer regions. Below, we describe the details of each segment, and how the sizes of each region are determined.

We modulate the central region Lyα emission with a simple dust screen model with

$$\begin{eqnarray}&&E(B-V)=\displaystyle \frac{2.5}{{k}_{{\rm{H}}\beta }-{k}_{{\rm{H}}\alpha }}{{\rm{log}}}_{10}\left(\displaystyle \frac{{\rm{H}}\alpha /{\rm{H}}\beta }{2.86}\right),\end{eqnarray} \tag{ 2 }$$

where ${k}_{{\rm{H}}\alpha }$ = 2.455 and ${k}_{{\rm{H}}\beta }$ = 3.520, as described by Cardelli et al. (1989). Here, the Hα/Hβ values used are drawn randomly from the Balmer decrement distribution for the galaxy being modeled and applied to each pixel in the central region. The reason we draw randomly from a distribution of Balmer decrements rather than the measured value for each pixel is because the Hβ is not always well measured, making a reliable value at each pixel impossible to determine. This extinction is then applied as

$$\begin{eqnarray}&&{L}_{{\rm{Ly}}\alpha }=8.7\times {L}_{{\rm{H}}\alpha }\times {10}^{-0.4E(B-V){k}_{{\rm{Ly}}\alpha }}.\end{eqnarray} \tag{ 3 }$$

The size of the central region is a free parameter that is fit for in the model.

The annulus between the central region and the Lyα halo experiences a more complex dust component than the central region as it is assumed to contain clumpy media. This changes the effect that the dust has on the Lyα photons, resulting in a different absorption cross-section. We model this effect with Equation (1). Using the intrinsic Hα/Hβ ratio of 2.86, we calculate the number of clumps along the line of sight for each pixel in the annular region. From this set of clump values we then draw randomly to create a distribution of clumps along the line of sight in the annulus and apply the random values to all of the pixels in the region. As with the central region, the size of the annulus is determined during the model fitting. Additionally, we assume a normal distribution of τ centered around 1 with a standard deviation of 0.25. This has the effect of magnifying the absorption slightly as compared to the dust screen model applied to the central region. Other optical depth values would slightly increase or decrease the absorption in the annular clumpy dust region. The choice of this optical depth distribution was largely empirical, chosen because higher optical depths could not reproduce the surface brightness distributions from the annular region. Using a global distribution of τ centered at 1 is a simplifying assumption, but its use does not change qualitatively the overall conclusions drawn from the model outputs.

To model the Lyα halo at large radii, we "scatter" the photons by convolving the simulated Lyα (where Lyα = 8.7 × Hα) emission with a symmetric two-dimensional Gaussian kernel. The convolution operation empirically mimics the effect of scattering the Lyα from the centralized emission that is seen in the Hα maps toward larger angular distances seen in the Lyα halo. That a Gaussian two-dimensional kernel is appropriate can be understood from the radiative transfer point of view: once in the wings of the line, a Lyα photon performs a random walk (Adams 1972). An ensemble of random "walkers" in an inho-mogenous medium will result in a Gaussian distribution after sufficient steps (Gronke et al. 2016, 2017). Other kernels were tested, such as a Lorentzian kernel, with similar results.

The resulting model has three input parameters: the average scattering distance of Lyα photons in the halo region, represented by the Gaussian kernel width, the isophotal size of the central region, and the isophotal size of the annulus. We parameterize the sizes of the two regions as a fraction of the aperture calculated in Section 2.

To find the best-fit model for each galaxy, we use a Markov Chain Monte Carlo (MCMC) implementation called emcee (Foreman-Mackey et al. 2013). This technique ensures that the parameter spaces for each input are fully explored without being excessively computationally expensive. For each iteration of input parameters, we compare the simulated Lyα versus Hα surface brightness distribution to the observed surface brightness distribution to determine which set of parameters best fits the data. We do this by first creating a 2D histogram of the original Lyα versus Hα surface brightness distribution. After each iteration in the calculation, a new surface brightness distribution is created and a 2D histogram of that distribution is made. To compare the observed and modeled surface brightnesses, we test the fit by minimizing the quantity

$$\begin{eqnarray}&&\displaystyle \sum _{i,j=0}^{n}\displaystyle \frac{({\rm{Observation}}[i,j]-{\rm{Model}}[i,j])}{{\rm{Observation}}[i,j]},\end{eqnarray} \tag{ 4 }$$

where n is the number of bins, and (i, j) represents the bins of the 2D histogram. The output of emcee is the kernel width and region sizes that best match the observed surface brightness distributions.

In Figures 8 and 9, we compare the observed Lyα versus Hα surface brightnesses to our simulated values. For LARS01, the resulting scattering kernel is $\sigma ={0.72}_{-0.27}^{+0.38}$ kpc, where the errors give the 16th and 84th percentiles of the walkers from the MCMC algorithm. This kernel encodes the average scattering distance that each Lyα photon undergoes in any given galaxy. In contrast, the halo surrounding LARS12 has a Lyα photon scattering distance of $\sigma ={1.03}_{-0.69}^{+1.74}$ kpc. The best-fit scattering model for all of the LARS galaxies is given in Table 2.

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Table 2.  LARS Scattering Properties

	Halo Scattering	Central Region	Annular Region
LARS ID	Distance (kpc)	Radius (kpc)	Radius (kpc)
01	${0.72}_{-0.27}^{+0.38}$	${0.36}_{-0.26}^{+1.35}$	${2.28}_{-0.26}^{+0.16}$
02	${0.36}_{-0.034}^{+0.04}$	${1.52}_{-1.3}^{+0.59}$	${3.50}_{-1.30}^{+0.49}$
03	${0.35}_{-0.05}^{+0.05}$	${3.08}_{-0.24}^{+0.31}$	${3.71}_{-0.24}^{+0.36}$
04	${0.70}_{-0.42}^{+0.54}$	${2.17}_{-0.54}^{+0.14}$	${2.92}_{-0.54}^{+2.63}$
05	${0.40}_{-0.09}^{+0.08}$	${1.59}_{-0.32}^{+0.46}$	${2.67}_{-0.32}^{+0.58}$
06	${0.41}_{-0.09}^{+0.06}$	${1.94}_{-1.07}^{+0.73}$	${4.39}_{-1.07}^{+0.58}$
07	${0.37}_{-0.26}^{+0.14}$	${1.43}_{-0.33}^{+0.32}$	${2.22}_{-0.33}^{+0.47}$
08	${0.44}_{-0.08}^{+0.07}$	${2.71}_{-1.46}^{+1.09}$	${5.58}_{-1.46}^{+1.75}$
09	${0.56}_{-0.11}^{+0.11}$	${0.40}_{-0.68}^{+0.13}$	${7.18}_{-0.68}^{+1.55}$
10	${0.52}_{-0.06}^{+0.08}$	${1.35}_{-0.11}^{+0.23}$	${2.30}_{-0.11}^{+0.27}$
11	${1.56}_{-0.49}^{+0.03}$	${6.41}_{-1.51}^{+0.50}$	${6.92}_{-1.51}^{+0.21}$
12	${1.03}_{-0.69}^{+1.74}$	${1.23}_{-1.06}^{+2.46}$	${4.77}_{-1.06}^{+1.04}$
13	${2.24}_{-0.29}^{+0.18}$	${3.29}_{-0.33}^{+0.52}$	${6.62}_{-0.33}^{+0.31}$
14	${1.68}_{-1.13}^{+2.83}$	${1.96}_{-1.68}^{+3.91}$	${7.58}_{-1.68}^{+1.65}$

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The right panels of Figures 8 and 9 show the models for the Lyα versus Hα surface brightnesses. Note that the pixels in the brightest Lyα region lie directly on the Case B recombination line or just below it—this is because in the central regions, only a dust screen model is applied, meaning that there is no way for the photons to scatter to brighter Lyα values. The clumpy dust annulus corresponds to the region further down and to the left in the surface brightness distribution, where there is more reddening due to the clumpy nature of the dust, so Hα is dampened compared to the Lyα. The region of the figure with the overbright Lyα compared to Hα is from the Lyα halo. Note that in our model, the borders between the three zones are sharp, whereas in the data, the transitions are more gradual. For example, both Figures 8 and 9 show a distinct corner in the simulated surface brightness distributions where the annular clumpy dust butts up against the brighter Lyα pixels that are above the intrinsic Lyα/Hα line. The natural conclusion is that there is a continuum to the region edges as they bleed into each other, and/or that some radiative transfer physics is not being captured by the model. This difference is apparent in the models for both LARS01 and LARS12. For the model comparisons for all LARS galaxies, see the Appendix.

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In Figures 10 and 11, we show the fractional differences between the model and the observed Lyα and Hα surface brightnesses. While the model itself is performed on individual pixels, in order to determine how well the model matches the observed distribution, we binned both the observed and modeled distributions, and compared how many pixels exist in each bin. We chose a large number of bins (60 × 60) to be able to compare on a fairly fine scale, while still maintaining enough points in each bin. The differences were then calculated by finding the observed minus predicted differences in each bin. A positive difference (purple) indicates that there are more points in the observed bin than the model bin. Negative differences (yellow) occur where there are more points in the model bin than the observed. From the figures, it is apparent that our model matches the surface brightness distribution for LARS01 very well. For LARS12, the differences are more stark, with the model distributing points that should be in the Lyα halo (shown in yellow in Figure 11) to the area of the surface brightness distribution that corresponds to the clumpy annulus (shown in purple in the same figure.)

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The characteristic scattering distances for the LARS galaxies range from a fraction of a kiloparsec to several kiloparsecs, and are shown in Figure 12. We discuss the implications of these results in the following section.

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