The Effects of Biconical Outflows on Lyα Escape from Green Peas

The SALT model approximates the galaxy as a spherical source (radius R = R_SF) of isotropically emitted radiation. The source is located at the center of a radially extended envelope (wind) of material extending from the radius R_SF to the terminal radius, R_W. A schematic representation of a cross-sectional cut of a biconical outflow is presented in Figure 1. The ξ-axis runs perpendicular to the line of sight and is measured using normalized units, r/R_SF. The s-axis runs parallel with the line of sight and is measured using the same normalized units.

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The wind is characterized by a velocity field, v, and a density field, n. The velocity field is described by the following power law.

$$\begin{eqnarray}\begin{array}{rcl}v & = & {v}_{0}{\left(\displaystyle \frac{r}{{R}_{\mathrm{SF}}}\right)}^{\gamma }\quad \mathrm{for}\ r\lt {R}_{{\rm{W}}}\\ v & = & {v}_{\infty }\qquad \quad \mathrm{for}\ r\geqslant {R}_{{\rm{W}}},\end{array}\end{eqnarray} \tag{ 1 }$$

where v₀ is the wind velocity at the surface of the source (i.e., at R_SF), and v_∞ is the terminal velocity of the wind at R_W. The corresponding density field (we assume a constant outflow rate) is

$$\begin{eqnarray}&&n(r)={n}_{0}{\left(\displaystyle \frac{{R}_{\mathrm{SF}}}{r}\right)}^{\gamma +2},\end{eqnarray} \tag{ 2 }$$

where n₀ is the gas density at R_SF.

We adopt the Sobolev approximation (Ambartsumian 1958; Sobolev 1960) and consider the outflow as an ensemble of thin spherical shells of a given radius, r, velocity, v, and optical depth, τ(r, ϕ). Here, ϕ is the angle between the velocity and the trajectory of the photon. In this context, the optical depth can be written (see Carr et al. 2018) as

$$\begin{eqnarray}&&\tau (r,\phi )=\displaystyle \frac{{\tau }_{0}}{1+(\gamma -1){\cos }^{2}\phi }{\left(\displaystyle \frac{{R}_{\mathrm{SF}}}{r}\right)}^{2\gamma +1},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{\tau }_{0}=\displaystyle \frac{\pi {e}^{2}}{{mc}}{f}_{{lu}}{\lambda }_{{lu}}{n}_{0}\displaystyle \frac{{R}_{\mathrm{SF}}}{{v}_{0}},\end{eqnarray} \tag{ 4 }$$

and f_ul and λ_ul are the oscillator strength and wavelength, respectively, for the ul transition. All other quantities take their usual definitions.

The modeled spectrum is computed in terms of the observed velocities that refer to the projections of the velocity field onto the line of sight. The SALT model constructs the observed spectrum one shell at a time, whether by removing energy from the spectrum via absorption, or by adding energy via reemission. The distribution of energy in observed velocity space will depend on the geometry of the outflow, the presence of holes or clumps in the wind, and the observational limits imposed by the observing aperture. The complete SALT model for a normalized line profile is

$$\begin{eqnarray}\begin{array}{rcl}I(x) & = & 1-{\displaystyle \int }_{\max (x,1)}^{{y}_{1}}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}(1-{e}^{-\tau (y)})}{y-{y}_{\min }}{dy}\\ & & +\,{\displaystyle \int }_{{y}_{1}}^{{y}_{\infty }}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}(1-{e}^{-\tau (y)})}{2y}{dy}\\ & & +\,{\displaystyle \int }_{\max (x,1)}^{{y}_{\infty }}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}(1-{e}^{-\tau (y)})}{2y}{dy},\end{array}\end{eqnarray} \tag{ 5 }$$

where x and y are the observed velocity and intrinsic velocity of a shell, respectively, both normalized by v₀. The first integral on the right computes the absorption profile and depends only on the material located along the line of sight. The integral includes all shells with intrinsic velocities ranging from x to y₁, where y₁ > x is the highest velocity a shell can have and still contribute to the absorption at x. A single shell of intrinsic velocity y can only contribute to the absorption spectrum at velocities in the range [y, y_min]. This range corresponds to the projected velocities attributed to the portion of the shell blocking the observer's view of the source. The remaining integrals account for the emission profile and depend on the entire outflow. They include shells extending from the source out to the terminal shell with intrinsic velocity, y_∞. The scale factors f_g, f_c, and Θ_AP are called the geometric scale factor, the covering fraction, and the aperture factor, respectively. Factor f_g is a function of the opening angle, α, and the orientation angle, ψ, and controls how the geometry of the outflow influences the line profile. Factor f_c can take any value between zero and one and is assumed to be constant with radius. In this way, f_c acts to scale the entire spectrum uniformly and is meant to account for radiation escaping the outflow through small holes. Factor Θ_AP acts to remove any contributions to the would-be spectrum from material lying outside the observational limits imposed by a circular observing aperture with projected radius R_AP. We define here an important quantity to be used later on, v_ap, which is defined as the value of the velocity field at radius R_AP. See Carr et al. (2018) for more details regarding the construction of the profiles.

Here, we introduce a new component to our model that accounts for dust in the CGM. We will discuss the model in depth with further applications in a future paper and only provide a brief introduction here. We assume the outflow is immersed within a dusty spherical medium (radius R = R_W) extending from the star formation region to the edge of the outflow. Furthermore, we assume the dust is moving at low velocity relative to the galactic system. Under these conditions we must treat the full equation of radiation transfer (i.e., the Sobolev approximation is no longer valid). Following Prochaska et al. (2011), we make the following assumptions:

• (1)  
  dust opacity scales with the density of the gas (i.e., a fixed dust to gas ratio);
• (2)  
  dust opacity is independent of wavelength;
• (3)  
  photons absorbed by dust are reemitted in the infrared;
• (4)  
  dust absorbs but does not scatter photons.

By invoking these assumptions, the dust optical depth problem essentially reduces to that of a point source behind a foreground dust screen. Moreover, within the confines of the SALT model, any dust extinction that occurs prior to the absorption and reemission of photons by ions in the outflow will be erased from the normalized spectrum. Hence, dust extinction will act only to decrease the emission line profile. In this context, the solution to the dust optical depth problem becomes

$$\begin{eqnarray}&&\displaystyle \frac{I}{{I}_{0}}={e}^{-{\tau }_{\mathrm{dust}}}\end{eqnarray} \tag{ 6 }$$

where

$$\begin{eqnarray}&&{\tau }_{\mathrm{dust}}=\int {{kn}}_{\mathrm{dust}}(s){ds}\end{eqnarray} \tag{ 7 }$$

is the dust optical depth. Here, n_dust is the density of the dust, k is the opacity of the dust, and s is the distance from the edge of the CGM to the location of reemission by the relevant ion as one travels along the line of sight. To compute Equation (7) to a point in the outflow, we will find it convenient to use the parameter $\kappa := {{kn}}_{{n}_{\mathrm{dust}},0}{R}_{\mathrm{SF}}$ where n_dust,0 is the density of the dust at R_SF. We will refer to this parameter as the dust opacity from now on. The complete SALT model, including a dusty CGM, becomes

$$\begin{eqnarray}\begin{array}{rcl}I{(x)}_{\mathrm{MS},\mathrm{dust}} & = & 1-{\displaystyle \int }_{\max (x,1)}^{{y}_{1}}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}(1-{e}^{-\tau (y)})}{y-{y}_{\min }}{dy}\\ & & +\,{\displaystyle \int }_{\max (x,1)}^{{y}_{\infty }}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}{e}^{-{\tau }_{\mathrm{dust}}}(1-{e}^{-\tau (y)})}{2y}{dy}\\ & & +\,{\displaystyle \int }_{{y}_{1}}^{{y}_{\infty }}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}{e}^{-{\tau }_{\mathrm{dust}}}(1-{e}^{-\tau (y)})}{2y}{dy}.\end{array}\end{eqnarray} \tag{ 8 }$$

The influence of dust on the emission profile of a spherical SALT model is shown in Figure 2. The dark red line shows the emission profile with a κ = 10 dust component. The light red line shows a normal emission profile without dust (i.e., κ = 0). Overall, the effect of the dust is to shift the emission profile blueward of the transition line since photons redward of the source must travel farther through the CGM, and hence, suffer more extinction. These results are in agreement with Prochaska et al. (2011), who also accounted for the effects of dust on line profiles in the context of galactic winds using Monte Carlo simulations.

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In their paper, Scarlata & Panagia (2015) accounted for the effect of fluorescent reemission—where the ionic species decays into an excited ground level⁶ —on the emission line profile. They described both a single and MS case.⁷ Both cases can be easily adapted to include the effects of a dusty CGM. We accommodate the MS scenario here and leave single scattering as a special case.

Scarlata & Panagia (2015) illustrate the MS process occurring in each shell by describing probability trees whose branches represent the fractions of photons exiting the outflow via the resonant and the fluorescent channels after each scatter. They show that, for each shell, the fraction of absorbed photons converted into fluorescent photons, F_F, is given by:

$$\begin{eqnarray}&&{F}_{{\rm{F}}}(\tau ,x,y)={p}_{{\rm{F}}}\displaystyle \sum _{n=0}^{\infty }{[{p}_{{\rm{R}}}(1-\beta )]}^{n}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&=\,{p}_{{\rm{F}}}/[1-{p}_{{\rm{R}}}(1-\beta )],\end{eqnarray} \tag{ 10 }$$

and the fraction of absorbed photons escaping as resonant photons, F_R, is given by:

$$\begin{eqnarray}&&{F}_{{\rm{R}}}(\tau ,x,y)={p}_{{\rm{R}}}\beta \displaystyle \sum _{n=0}^{\infty }{[{p}_{{\rm{R}}}(1-\beta )]}^{n}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&=\,\beta {p}_{{\rm{R}}}/[1-{p}_{{\rm{R}}}(1-\beta )],\end{eqnarray} \tag{ 12 }$$

where β = (1 − e^−τ)/τ is the probability of escape of a photon from a given shell (Mathis 1972).⁸ p_R is the fraction of photons absorbed by the shell and reemitted via the resonant channel, and p_F is the fraction of photons absorbed by the shell and remitted via the fluorescent channel. The sum represents the limit of infinitely many scatters—that is, each term represents a fraction of photons escaping the outflow. Now, for a specific location in the outflow, we can assume the above probabilities hold. Thus, after each scatter, the fraction of photons leaving that location and traveling to the observer through a dusty CGM will be reduced by the factor ${e}^{-{\tau }_{\mathrm{dust}}}(x,y)$. The above equations become

$$\begin{eqnarray}&&{F}_{{\rm{F}}}{(\tau ,x,y)}_{\mathrm{dust}}={e}^{-{\tau }_{\mathrm{dust}}(x,y)}{p}_{{\rm{F}}}/[1-{p}_{{\rm{R}}}(1-\beta )],\end{eqnarray} \tag{ 13 }$$

and

$$\begin{eqnarray}&&{F}_{{\rm{R}}}{(\tau ,x,y)}_{\mathrm{dust}}={e}^{-{\tau }_{\mathrm{dust}}(x,y)}\beta {p}_{{\rm{R}}}/[1-{p}_{{\rm{R}}}(1-\beta )].\end{eqnarray} \tag{ 14 }$$

Therefore, the normalized profiles including the effects of dust in the case of MS become

$$\begin{eqnarray}\begin{array}{rcl}I{(x)}_{\mathrm{MS},\mathrm{dust}} & = & 1-{\displaystyle \int }_{\max (x,1)}^{{y}_{1}}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}(1-{e}^{-\tau (y)})}{y-{y}_{\min }}{dy}\\ & & +\,{\displaystyle \int }_{\max (x,1)}^{{y}_{\infty }}{{F}_{i}}_{,\mathrm{dust}}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}(1-{e}^{-\tau (y)})}{2y}{dy}\\ & & +\,{\displaystyle \int }_{{y}_{1}}^{{y}_{\infty }}{{F}_{i}}_{,\mathrm{dust}}\displaystyle \frac{{{\rm{\Theta }}}_{\mathrm{AP}}{f}_{c}{f}_{g}(1-{e}^{-\tau (y)})}{2y}{dy},\end{array}\end{eqnarray} \tag{ 15 }$$

where F_i,dust = F_F,dust(F_R,dust) for the fluorescent(resonant) channel, respectively.

The complete SALT model is described by nine parameters: the opening angle, α, the orientation angle, ψ, the power-law index of the velocity field, γ, the optical depth, τ,⁹ the initial velocity, v₀, the terminal velocity, v_∞, the velocity at the aperture radius, v_ap, the dust opacity, κ, and the covering fraction, f_c. We can reduce the number of parameters to eight by assuming a value for R_SF and writing v_ap as a function of v₀ and γ. We take R_SF to be equal to the Petrosian radius, R_P—a reasonable assumption given the compact morphology of the Green Peas. See Section 2 for a description of the Petrosian radius. By taking the ratio of the COS aperture radius, R_AP, (i.e., the distance spanned by 125 at the angular diameter distance appropriate to each galaxy) with R_P as the base of the power law defined by the velocity field, we get

$$\begin{eqnarray}&&{v}_{\mathrm{ap}}={v}_{0}{\left(\displaystyle \frac{{R}_{\mathrm{AP}}}{{R}_{{\rm{P}}}}\right)}^{\gamma }.\end{eqnarray} \tag{ 16 }$$

To derive the best-fit parameters for each model, we use the emcee package in Python (Foreman-Mackey et al. 2013), which relies on the Python implementation of Goodman's and Weare's Affine Invariant Markov Chain Monte Carlo (MCMC) Ensemble sampler (Goodman & Weare 2010). The MCMC sampler draws from the following parameter ranges: 0° ≤ α ≤ 90°, 0° ≤ ψ ≤ 90°, 0.5 ≤ γ ≤ 2, 0.01 ≤ τ₀ ≤ 200, 2 ≤ v₀ ≤ 80 km s⁻¹, 200 ≤ v_∞ ≤ 800 km s⁻¹, 0.01 ≤ κ ≤ 200, and 0 ≤ f_c ≤ 1. We fit the SALT model to spectra after convolving them to the appropriate resolution using SciPy's Gaussian filter. We limit the power-law index of the velocity field to values γ > 0, i.e., we exclude velocity gradients that decay with increasing radius, as the evidence for these outflows is sparse in the literature (Burchett et al. 2020). Adapting the SALT model to include γ < 0 is nontrivial, as in this case, photons of a given wavelength can interact with multiple positions in the outflow. We will address this aspect in a future paper. We do find, however, that we are able to achieve reasonable fits using the given parameter range.

The emcee analysis returns posterior distributions (PDFs) for each parameter dependent upon the likelihood of the SALT model fits given the prior distributions (i.e., the parameter ranges specified above). We assume a Gaussian likelihood function. The PDFs for a typical fit have been provided in the Appendix along with a discussion of potential model degeneracies. More details on the degeneracies of the various parameters can be found in Carr et al. (2018). Because many of the PDFs are asymmetric, the best-fit parameters are chosen to represent the most likely value (i.e., the mode) for a given parameter's PDF. Similarly, we choose the median of the absolute deviation around the mode (MAD) to describe the width for each distribution (and use this value as an estimate of the error associated with each parameter). Since the MCMC process samples the prior distributions by moving in the direction that maximizes the quality of fit, these errors represent our ability to constrain the SALT model within the prior distributions.

In the wavelength range covered by the COS spectra, we can probe Si⁺ using the 1190.42 Å, 1193.28 Å Si ii doublet, as well as the 1260.42 Å transition. We model the doublet independently from the 1260 Å transition and use the scatter in the derived parameters to discuss the uncertainty in the models.

We fit the SALT model to the 1190.42 Å, 1193.28 Å Si ii doublet (1185–1201 Å) for all of the Green Peas except galaxies 1133+6514 and 1219+1526, whose spectra suffer contamination from Milky Way absorption. The best fit for each galaxy is provided in Figure 3 and the associated parameters have been recorded in Table 2. For galaxies 1133+6514, 1219+1526, and 1054+5238, best-fit parameters were extrapolated from the (1260.42 Å) Si ii transitions, and are highlighted in green in Figure 3. In the case of galaxy 1054+5238, the 1190.42 Å, 1193.28 Å Si ii doublet shows absorption redward of the transition lines—a feature typically associated with an ISM component, turbulence, or galactic inflows, and which cannot be modeled with the current version of the SALT model—however, this feature is absent from the 1260.42 Å Si ii transition line. Therefore, we have chosen to use the latter fit to describe the cool outflows and assume the red absorption present in the 1190.42 Å, 1193.28 Å Si ii doublet may be due to noise or low resolution.

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Table 2.  Parameters Obtained by the SALT Model from Fits to Transition Lines of Si ii, Si iii, and Si iv

Galaxy	Transition	α	ψ	γ	τ	v0	vw	vap	fc	κ	NSi*	RW	NH i
		(deg)	(deg)			(km s−1)	(km s−1)	(km s−1)			(1016cm−2)	(kpc)	(1020cm−2)
0303–0759	(1190.42 Å, 1193.28 Å) Si ii	${46}_{-8}^{+21}$	${2}_{-15}^{+1}$	${1.98}_{-0.6}^{+0.008}$	${0.2}_{-0.1}^{+2}$	${23}_{-8}^{+19}$	${1133}_{-312}^{+591}$	${123}_{-38}^{+203}$	${0.5}_{-0.08}^{+0.1}$	${1.1}_{-1}^{+18}$	${0.033}_{-0.006}^{+0.3}$	${4.6}_{-0.9}^{+8}$	${2.2}_{-0.6}^{+1.9}$
	(1260.42 Å) Si ii	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
	(1206.5 Å) Si iii	${50}_{-12}^{+14}$	${3}_{-1}^{+18}$	${0.61}_{-0.05}^{+0.48}$	${70}_{-33}^{+57}$	${7}_{-1.3}^{+5.5}$	${525}_{-44}^{+149}$	${29}_{-3.1}^{+17}$	${0.8}_{-0.05}^{+0.06}$	${1.5}_{-1.4}^{+30}$	${8.2}_{-3.3}^{+8.4}$	${11}_{-5.4}^{+76}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${46}_{-9}^{+20}$	${3}_{-1}^{+17}$	${1.1}_{-0.3}^{+0.4}$	${35}_{-14}^{+81}$	${23}_{-7}^{+18}$	${445}_{-15}^{+28}$	${71}_{-13}^{+129}$	${0.6}_{-0.06}^{+0.07}$	${0.80}_{-0.8}^{+15}$	${9.3}_{-3}^{+23}$	${3.8}_{-0.8}^{+6}$	⋯
0911+1831	(1190.42 Å, 1193.28 Å) Si ii	${88}_{-7}^{+0.8}$	${65}_{-24}^{+11}$	${1.4}_{-0.3}^{+0.2}$	${1.3}_{-0.4}^{+5}$	${25}_{-8}^{+12}$	${542}_{-23}^{+389}$	${172}_{-31}^{+43}$	${0.6}_{-0.04}^{+0.05}$	${0.026}_{-0.01}^{+0.2}$	${1.1}_{-0.3}^{+0.9}$	${12}_{-2}^{+10}$	${1.7}_{-0.4}^{+0.7}$
	(1260.42 Å) Si ii	${89.6}_{-8}^{+0.2}$	${64}_{-22}^{+10}$	${1.8}_{-0.4}^{+0.1}$	${0.1}_{-0.04}^{+0.8}$	${31}_{-10}^{+18}$	${736}_{-80}^{+554}$	${241}_{-74}^{+201}$	${0.5}_{-0.04}^{+0.13}$	${0.021}_{-0.006}^{+0.13}$	${0.079}_{-0.02}^{+0.2}$	${10}_{-2}^{+6}$	⋯
	(1206.5 Å) Si iii	${44}_{-8}^{+20}$	${3}_{-2}^{+17}$	${1.6}_{-0.5}^{+0.2}$	${59}_{-25}^{+61}$	${7}_{-1}^{+5}$	${1045}_{-74}^{+133}$	${45}_{-4}^{+23}$	${0.6}_{-0.04}^{+0.06}$	${0.7}_{-0.7}^{+18}$	${4.6}_{-1}^{+6}$	${20}_{-3}^{+48}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${62}_{-11}^{+12}$	${2.8}_{-2}^{+14}$	${1.96}_{-0.4}^{+0.02}$	${162}_{-57}^{+16}$	${8}_{-0.9}^{+3}$	${921}_{-139}^{+810}$	${71}_{-10}^{+44}$	${0.8}_{-0.07}^{+0.08}$	${1.4}_{-1.1}^{+39}$	${12}_{-4}^{+4}$	${19}_{-5}^{+14}$	⋯
0926+4427	(1190.42 Å, 1193.28 Å) Si ii	${79}_{-5}^{+3}$	${78}_{-5}^{+2}$	${1.7}_{-0.18}^{+0.15}$	${22}_{-9}^{+24}$	${16}_{-2}^{+5}$	${430}_{-11}^{+27}$	${142}_{-18}^{+42}$	${0.4}_{-0.04}^{+0.1}$	${0.015}_{-0.003}^{+0.06}$	${1.4}_{-0.4}^{+2}$	${6.7}_{-1}^{+1}$	${1.2}_{-0.2}^{+0.5}$
	(1260.42 Å) Si ii	${76}_{-3}^{+2}$	${65}_{-3}^{+5}$	${1.2}_{-0.3}^{+0.3}$	${0.2}_{-0.1}^{+2}$	${42}_{-9}^{+39}$	${524}_{-19}^{+139}$	${264}_{-43}^{+81}$	${0.3}_{-0.03}^{+0.1}$	${0.016}_{-0.003}^{+0.06}$	${0.068}_{-0.01}^{+0.3}$	${7.5}_{-2}^{+4}$	⋯
	(1206.5 Å) Si iii	${53}_{-7}^{+16}$	${0.8}_{-0.4}^{+11}$	${1.98}_{-0.3}^{+0.008}$	${46}_{-18}^{+55}$	${6}_{-0.6}^{+2}$	${681}_{-74}^{+16}$	${36}_{-4}^{+46}$	${0.6}_{-0.03}^{+0.04}$	${9}_{-4}^{+17}$	${3.4}_{-0.9}^{+6}$	${9.5}_{-3}^{+8}$	⋯
	(1393.76 Å,1402.77 Å) Si iv	${59}_{-3}^{+8}$	${0.3}_{-0.2}^{+6}$	${1.99}_{-0.08}^{+0.002}$	${190}_{-23}^{+5}$	${8}_{-0.5}^{+0.6}$	${622}_{-9}^{+23}$	${98}_{-9}^{+11}$	${0.7}_{-0.03}^{+0.02}$	${20}_{-6.7}^{+25}$	${11}_{-0.9}^{+1}$	${9.2}_{-0.5}^{+1}$	⋯
1054+5238	(1190.42 Å,1193.28 Å) Si ii	${87}_{-24}^{+1}$	${13}_{-25}^{+7}$	${0.7}_{-0.06}^{+0.5}$	${0.6}_{-0.3}^{+6}$	${18}_{-5}^{+18}$	${313}_{-28}^{+876}$	${75}_{-17}^{+84}$	${0.4}_{-0.04}^{+0.09}$	${0.18}_{-0.12}^{+2}$	${0.26}_{-0.1}^{+1}$	${4.7}_{-1}^{+30}$	⋯
	(1260.42 Å) Si ii	${88}_{-19}^{+1.2}$	${42}_{-19}^{+15}$	${1.89}_{-0.4}^{+0.05}$	${0.1}_{-0.07}^{+1}$	${65}_{-22}^{+25}$	${481}_{-58}^{+235}$	${247}_{-87}^{+353}$	${0.5}_{-0.08}^{+0.2}$	${0.08}_{-0.05}^{+1}$	${0.032}_{-0.01}^{+0.1}$	${3.5}_{-0.4}^{+3}$	${0.57}_{-0.2}^{+0.9}$
	(1206.5 Å) Si iii	${69}_{-19}^{+10}$	${0.7}_{-0.3}^{+15}$	${1.3}_{-0.3}^{+0.3}$	${1}_{-0.5}^{+5}$	${12}_{-2}^{+7}$	${627}_{-33}^{+506}$	${48}_{-8.7}^{+54}$	${0.9}_{-0.08}^{+0.03}$	${12}_{-5}^{+36}$	$.{28}_{-0.09}^{+0.9}$	${16}_{-5}^{+34}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${56}_{-14}^{+16}$	${0.8}_{-0.4}^{+24}$	${0.65}_{-0.07}^{+0.6}$	${9}_{-4}^{+79}$	${7}_{-2}^{+10}$	${628}_{-140}^{+773}$	${31}_{-5}^{+42}$	${0.68}_{-0.13}^{+0.12}$	${1.0}_{-0.86}^{+20}$	${5.6}_{-2}^{+10}$	${9.1}_{-3}^{+73}$	⋯
1133+6514	(1190.42 Å, 1193.28 Å) Si ii	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
	(1260.42 Å) Si ii	${60}_{-13}^{+9}$	${79}_{-14}^{+5}$	${1.98}_{-0.4}^{+0.007}$	${0.4}_{-0.3}^{+5}$	${67}_{-22}^{+33}$	${444}_{-97}^{+779}$	${241}_{-82}^{+183}$	${0.99}_{-0.3}^{+0.007}$	${0.027}_{-0.01}^{+0.2}$	${0.23}_{-0.2}^{+3}$	${7.0}_{-1}^{+7}$	${1.1}_{-0.5}^{+6}$
	(1206.5 Å) Si iii	${81}_{-20}^{+5}$	${1}_{-0.4}^{+21}$	${1.2}_{-0.3}^{+0.3}$	${8}_{-3}^{+71}$	${9}_{-2}^{+12}$	${491}_{-21}^{+295}$	${26}_{-5}^{+39}$	${0.5}_{-0.07}^{+0.05}$	${0.5}_{0.4}^{+11}$	${2.5}_{-0.9}^{+12}$	${11}_{-3}^{+83}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${64}_{-24}^{+11}$	${39}_{-20}^{+22}$	${1.6}_{-0.4}^{+0.2}$	${156}_{-74}^{+21}$	${6.4}_{-2}^{+51}$	${325}_{-49}^{+494}$	${22}_{-7}^{+143}$	${0.25}_{-0.09}^{+0.3}$	${0.5}_{-0.4}^{+10}$	${2.1}_{-1}^{+20}$	${6.7}_{-2}^{+18}$	⋯
1137+3524	(1190.42 Å, 1193.28 Å) Si ii	${89}_{-15}^{+0.3}$	${13}_{-6}^{+18}$	${1.5}_{-0.3}^{+0.2}$	${26}_{-7}^{+28}$	${7}_{-1}^{+1}$	${1571}_{-425}^{+414}$	${34}_{-4.4}^{+10}$	${0.9}_{-0.04}^{+0.06}$	${0.12}_{-0.08}^{+1.5}$	${2.7}_{-0.5}^{+2}$	${34}_{-7}^{+42}$	${1.8}_{-0.4}^{+0.7}$
	(1260.42 Å) Si ii	${89}_{-12}^{+0.3}$	${0.94}_{-0.5}^{+20}$	${1.9}_{-0.3}^{+0.07}$	${2}_{-0.7}^{+3}$	${13}_{-4}^{+2}$	${2063}_{-776}^{+198}$	${57}_{-12}^{+37}$	${0.9}_{-0.04}^{+0.03}$	${0.07}_{-0.04}^{+0.9}$	${0.4}_{-0.1}^{+0.4}$	${18}_{-4}^{+14}$	⋯
	(1206.5 Å) Si iii	${43}_{-11}^{+23}$	${8}_{-4}^{+16}$	${0.7}_{-0.07}^{+0.3}$	${44}_{-22}^{+76}$	${7}_{-2}^{+3}$	${1171}_{-368}^{+593}$	${16}_{-2}^{+7}$	${0.9}_{-0.07}^{+0.06}$	${0.9}_{-0.8}^{+16}$	${3.6}_{-1}^{+7}$	${62}_{-27}^{+578}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${88}_{-20}^{+1}$	${2}_{-1}^{+23}$	${0.94}_{-0.2}^{+0.5}$	${197}_{-90}^{+1.4}$	${5}_{-1}^{+4}$	${328}_{-30}^{+981}$	${20}_{-2}^{+9}$	${0.98}_{-0.06}^{+0.01}$	${1.3}_{-1}^{+49}$	${8.2}_{-3}^{+5}$	${18}_{-7}^{+108}$	⋯
1219+1526	(1190.42 Å, 1193.28 Å) Si ii	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
	(1260.42 Å) Si ii	${71}_{-19}^{+7}$	${77}_{-25}^{+6}$	${1.3}_{-0.3}^{+0.3}$	${0.15}_{-0.1}^{+2}$	${48}_{-18}^{+36}$	${239}_{-22}^{+1111}$	${303}_{-138}^{+485}$	${0.8}_{-0.3}^{+0.1}$	${0.16}_{-0.1}^{+2}$	${0.038}_{-0.02}^{+0.5}$	${3.7}_{-1}^{+8}$	${2.6}_{-1}^{+6}$
	(1206.5 Å) Si iii	${69}_{-21}^{+11}$	${4}_{-2}^{+21}$	${0.6}_{-0.03}^{+0.5}$	${30}_{-13}^{+55}$	${16}_{-3}^{+34}$	${733}_{-56}^{+444}$	${67}_{-8}^{+153}$	${0.8}_{-0.07}^{+0.07}$	${0.8}_{-0.7}^{+15}$	${12}_{-4}^{+40}$	${3.8}_{-1}^{+29}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${69}_{-17}^{+10}$	${7}_{-4}^{+20}$	${1.1}_{-0.3}^{+0.3}$	${168}_{-75}^{+16}$	${29}_{-8}^{+33}$	${893}_{-27}^{+71}$	${200}_{-62}^{+184}$	${0.6}_{-0.05}^{+0.1}$	${0.43}_{-0.4}^{+10}$	${19}_{-8}^{+50}$	${4.4}_{-1}^{+14}$	⋯
1244+0216	(1190.42 Å, 1193.28 Å) Si ii	${89.7}_{-2}^{+0.2}$	${1}_{-0.5}^{+25}$	${1.99}_{-0.12}^{+0.003}$	${35}_{-10}^{+30}$	${17}_{-3}^{+1}$	${271}_{-12}^{+13}$	${214}_{-28}^{+55}$	${0.992}_{-0.05}^{+0.007}$	${0.023}_{-0.008}^{+0.2}$	${7.9}_{-3}^{+5}$	${5.1}_{-0.5}^{+0.5}$	${1.9}_{-0.4}^{+1.4}$
	(1260.42 Å) Si ii	${89}_{-8}^{+0.5}$	${12}_{-6}^{+19}$	${1.96}_{-0.3}^{+0.02}$	${2}_{-0.5}^{+4}$	${11}_{-3}^{+3}$	${519}_{-51}^{+152}$	${115}_{-33}^{+55}$	${0.999}_{-0.03}^{+0.0006}$	${0.033}_{-0.015}^{+0.3}$	${0.66}_{-0.02}^{+0.5}$	${9.4}_{-2}^{+6}$	⋯
	(1206.5 Å) Si iii	${74}_{-25}^{+8}$	${1}_{-0.5}^{+20}$	${0.8}_{-0.1}^{+0.4}$	${197}_{-82}^{+1}$	${4}_{-0.6}^{+2}$	${329}_{-26}^{+267}$	${15}_{-1}^{+7}$	${0.98}_{-0.06}^{+0.007}$	${1.4}_{-1.3}^{+32}$	${6.2}_{-2}^{+6}$	${19}_{-10}^{+175}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${53}_{-13}^{+15}$	${4}_{-2}^{+19}$	${1.6}_{-0.5}^{+0.2}$	${174}_{-83}^{+12}$	${6}_{-1}^{+8}$	${339}_{-40}^{+643}$	${40}_{-5}^{+53}$	${0.7}_{-0.08}^{+0.1}$	${1.0}_{-0.9}^{+18}$	${6.0}_{-3}^{+8}$	${3.0}_{-0.4}^{+18}$	⋯
1249+1234	(1190.42 Å, 1193.28 Å) Si ii	${85}_{-9}^{+2}$	${49}_{-23}^{+12}$	${0.6}_{-0.03}^{+0.3}$	${9}_{-3}^{+13}$	${28}_{-7}^{+26}$	${971}_{-139}^{+512}$	${70}_{-13}^{+11}$	${0.6}_{-0.03}^{+0.02}$	${0.03}_{-0.01}^{+0.25}$	${3.3}_{-0.8}^{+3.9}$	${72}_{-26}^{+185}$	${2.4}_{-0.5}^{+3.5}$
	(1260.42 Å) Si ii	${86}_{-14}^{+2}$	${6}_{-3}^{+35}$	${1.4}_{-0.4}^{+0.3}$	${0.25}_{-0.17}^{+3}$	${17}_{-5}^{+31}$	${350}_{-34}^{+169}$	${87}_{-28}^{+95}$	${0.6}_{-0.06}^{+0.1}$	${0.047}_{-0.03}^{+9}$	${0.068}_{-0.03}^{+0.6}$	${4.4}_{-0.5}^{+8}$	⋯
	(1206.5 Å) Si iii	${79}_{-23}^{+5}$	${3}_{-1}^{+22}$	${1.97}_{-0.7}^{+0.015}$	${70}_{-33}^{+62}$	${6}_{-1}^{+22}$	${447}_{-24}^{+35}$	${19}_{-2}^{+57}$	${0.7}_{-0.03}^{+0.06}$	${1}_{-0.9}^{+27}$	${1.9}_{-0.8}^{+30}$	${6}_{-0.7}^{+19}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	${89}_{-23}^{+0.4}$	${5}_{-2}^{+23}$	${0.6}_{-0.04}^{+0.4}$	${49}_{-20}^{+57}$	${9}_{-1}^{+11}$	${455}_{-44}^{+353}$	${26}_{-3}^{+23}$	${0.7}_{-0.07}^{+0.09}$	${0.63}_{-0.6}^{+13}$	${9.3}_{-3}^{+10}$	${13}_{-6}^{+119}$	⋯
1424+4217	(1190.42 Å, 1193.28 Å) Si ii	${87}_{-21}^{+1}$	${52}_{-16}^{+12}$	${0.7}_{-0.08}^{+0.4}$	${0.6}_{-0.4}^{+6}$	${19}_{-4}^{+25}$	${572}_{-135}^{+599}$	${93}_{-21}^{+75}$	${0.4}_{-0.05}^{+0.2}$	${0.22}_{-0.2}^{+3}$	${0.12}_{-0.05}^{+1}$	${13}_{-6}^{+43}$	${1.0}_{-0.4}^{+6}$
	(1260.42 Å) Si ii	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
	(1206.5 Å) Si iii	${89}_{-27}^{+0.5}$	${35}_{-17}^{+10}$	${1.6}_{-0.6}^{+0.1}$	${124}_{-56}^{+38}$	${6}_{-0.5}^{+54}$	${449}_{-17}^{+26}$	${60}_{-13}^{+222}$	${0.8}_{-0.07}^{+0.1}$	${1}_{-0.9}^{+18}$	${4.6}_{-1}^{+70}$	${2.5}_{-0.2}^{+6}$	⋯
	(1393.76 Å, 1402.77 Å) Si iv	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯

Note. Column densities, N_Si*, and N_{H i}, along with the wind radii, R_W, are also shown here.

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Similarly, we fit the 1260.42 Å Si ii resonant transition in all galaxies except for 0303–0759 and 1424+4217. The spectrum of galaxy 0303–0759 suffers contamination from Milky Way absorption, and galaxy 1424+4217 does not have data at this wavelength range due to a failed observation. The best fit for each galaxy is shown in Figure 4 and the associated parameters are recorded in Table 2. We note that for 0926+4427 there appears to be a double absorption feature at the 1260 Å transition, and we suspect that a more sophisticated model may be required to better interpret this object.

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In Figure 5, we compare the best-fit parameters obtained from the 1260.42 Å Si ii transitions (1254–1269 Å) with those obtained from the 1190.42 Å, 1193.28 Å Si ii doublet. Parameter values describing the geometry and the spatial extent of the Si ii gas agree well. Impressively, the model fits give values of α that all agree within 3°! The values obtained for the orientation angle ψ from the different transitions do not agree as well; however, this can be expected in part, for models with a spherical outflow geometry (i.e., α = 90°), ψ is no longer well defined and changing its value has no affect on the resulting line profile. The estimates of τ differ significantly between the different transitions, with lower estimates always attained from the 1260.42 Å Si ii transition. We suspect the values derived from the 1190.42 Å, 1193.28 Å Si ii doublet to be more accurate because these measurements rely on two resonant transitions, which is known to break the degeneracies between optical depth and geometric effects that are present in similar models when analyzing single resonant lines (e.g., Chisholm et al. 2018a). As expected, the errors associated with the best fits to the Si ii doublet are smaller than those obtained from the 1260.42 Å Si ii transition lines, reflecting the larger number of observational constraints. Finally, we mention that all modeled transitions originating in the Si⁺ gas have associated fluorescent channels with different probabilities. Therefore, the amount of blue emission infilling is different for all transitions, which helps to remove degeneracies among different models that otherwise fit the absorption profile (e.g., Carr et al. 2018).

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The best-fit parameters from either the Si ii doublet or the 1260 Å transition (or both, when available) show that six out of the 10 galaxies in the sample have close to spherical (i.e., α > 85°) outflow geometries as traced by the Si⁺ gas, while the remaining objects have biconical outflows. We find that the outflows in 1133+6514, and 1219+526 are oriented almost perpendicularly with respect to the line of sight. Galaxies with this geometry tend to have weak or shallow absorption dips while still maintaining prominent fluorescent emission components in their spectra (Carr et al. 2018). Galaxy 0303–0759 also shows a biconical geometry, but the outflow in this case is parallel to the line of sight, thus showing strong blueshifted absorption. Finally, 0926+4427 has an almost spherical outflow but oriented edge on with respect to the observer (i.e., α ∼ ψ).

For all galaxies, we find small values for the dust opacity associated with the Si⁺ gas, which suggests that the CGM of these Green Peas is not very dusty. This is in agreement with the separate observations of Henry et al. (2015), who reached the same conclusion through more traditional techniques. The terminal velocities of the outflows range from ${271}_{-12}^{+13}$ km s⁻¹ for galaxy 1244+0216 to ${1570}_{-425}^{+414}$ km s⁻¹ for galaxy 1137+3524. We note that the MCMC sampler was unable to converge in terminal velocity for galaxy 1137+3524 and its large value likely represents this fact. (The MCMC sampler was also unable to converge in terminal velocity for this galaxy using the 1260.42 Å Si ii transition lines, where we obtained a value of 2000 km s⁻¹.) Aside from galaxy 1137+3524, galaxy 1249+1234 shows the terminal velocity at ${971}_{-139}^{+512}$ km s⁻¹.

We were able to fit the Si iii 1206 Å absorption line profiles (1200–1212 Å) for all galaxies (see Figure 6). For 0926+4427, the double absorption feature seen in the 1260 Å transition is visible also in the 1206 Å line. Six of the 10 galaxies in the sample are well described by a biconical outflow geometry oriented parallel to the line of sight (i.e., ψ ∼ 0), while the remaining galaxies have close to spherical outflows (i.e., α ≥ 85°). Biconical geometries oriented parallel to the line of sight tend to imprint strong absorptions in the spectra, while having a weak emission component, as the scattered resonant and fluorescent photons can preferentially escape perpendicularly with respect to the line of sight (Carr et al. 2018). In general, the Si²⁺ gas displays a higher terminal velocity than Si⁺. This kinematic difference likely reflects the mechanism by which the gas enters the CGM. The hotter Si iii gas appears to be more energetic and collimated (i.e., smaller α) than in comparison to the cool Si ii gas.

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We find best-fit values of κ that are typically higher than the values obtained from the fit to the 1190.42 Å, 1193.28 Å Si ii doublet. Since the opening angle of the Si²⁺ gas is typically smaller than the opening angle of the Si⁺ gas, the values of κ obtained from the Si iii lines probe a smaller volume of the CGM. This may result in an overall larger value of κ if the majority of dust traces the warmer gas. However, we should caution against speculating too much about these values. Since the Si iii profiles have relatively weak emission components due to their geometry, recovering κ is a much more difficult process: indeed, its obtained value depends entirely on the emission component of the line profile. This difficulty is reflected by the large errors associated with the values shown in Table 2.

We were able to fit the absorption profile of the 1393.76 Å, 1402.77 Å Si iv doublet (1386–1405 Å) to the spectrum of every Green Pea except for galaxy 1424+4217, whose spectrum does not cover this wavelength range due to a failed observation (see Figure 7). Unlike in the case of the 1190.42 Å, 1193.28 Å doublet, we do not have to account for fluorescent transitions. Similar to the results obtained for the Si²⁺ gas, the best-fit parameters to the Si iv doublet suggest that the Si³⁺ gas is characterized by a biconical geometry, with the cone oriented parallel to the line of sight to the observer. The line-of-sight orientations likely reflect a selection bias: these Green Peas were selected for being UV bright, which suggests we are looking down the more ionized portion of the outflow (Tenorio-Tagle et al. 1999). The opening angles and terminal velocities of the Si²⁺ and the Si³⁺ gases are comparable, but suggest these outflows are not co-spatial in general.

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The obtained values of κ are comparable to the values obtained from the Si iii gas and suggest the dustier portion of the outflow lies along the line of sight; however, these values are still subject to large uncertainties given the small emission component observed with these geometries.

We begin the analysis by examining ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ as a function of the Si⁺ outflow properties (derived either from the 1190 Å doublet or the 1260 Å Si ii transition). Figure 8 shows how ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ varies as a function of v_∞, v₀, τ, f_c, γ, v_ap/v_∞, α, and f_c ∗ V_ff. The new quantity, V_ff, is defined as the fraction of the volume of the CGM between the source and the observer filled with outflowing material and is referred to as the line-of-sight volume filling factor. By further multiplying by f_c, we are able to account for any holes in the outflow through which photons might escape. For example, a spherical outflow has V_ff = 1.0, but f_c ∗ V_ff may be less than one if there are holes present in the outflow. Since all galaxies presented little-to-no dust in the outflow, we find no correlations with κ, and exclude this parameter from Figure 8.

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Considering all galaxies, regardless of the shape of the outflow, we find no significant correlation between ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ and most of the parameters describing the Si⁺ outflows (the only exceptions are γ and f_c × V_ff). Some of these results are surprising. As an example, let us consider the lack of correlation between τ and ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$. Given its ionization potential of 16.3 eV, Si⁺ is expected to trace partially ionized gas and to act as a good tracer of H i.¹⁰ Since the Si⁺ gas density at R_SF is proportional to τ, to first approximation, a higher value of τ corresponds to a higher density of neutral hydrogen, and a larger amount of scattering that Lyα photons undergo in the outflow. Even though we find that the dust content is small, a negative correlation would have been expected. Similarly, we would have expected that a low covering fraction, as in 1244+0216, would correspond to a high ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$. We find no correlation between ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ and v₀ (ρ = 0.359) and v_∞ (ρ = −0.304). This result is in agreement with Henry et al. (2015).

We detect a weak negative correlation (ρ = −0.627, P = 0.053) between γ, the power-law index of the velocity field, and ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$. For larger values of γ, or steeper velocity fields, we expect Lyα photons to scatter and clear out of the outflow more easily (Lamers & Cassinelli 1999), so this result is rather surprising. Recalling the power law of the density field, ${n}_{0}{\left(\tfrac{{R}_{\mathrm{SF}}}{r}\right)}^{\gamma +2.0}$, we see that the density field depends on both γ and n₀. Therefore, γ may be a better indicator of the overall amount of Lyα scattering. Still, the density equation suggests larger values of γ correspond to lower optical depths in favor of Lyα escape (Verhamme et al. 2015).

In addition to the parameters directly fit in the SALT model, we also explore how ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ depends on the effective covering fraction of the outflow, that includes both f_c and the partial geometrical covering of the source that we quantify using the line-of-sight volume filling factor, V_ff. We find a weak negative correlation (ρ = −0.559, P = 0.093) between ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ and f_c ∗ V_ff. This correlation, which is driven by V_ff, as ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ does not correlate with f_c, suggests that when less H i gas is blocking the source along the line of sight, Lyα can more easily escape to the observer.

The Lyα line profiles of Green Pea galaxies are known to show a characteristic double peak shape. All galaxies in our sample have this profile type, with the exception of 1249+1234.

In order to describe the shape of the profile, we use the parameter, Δ_peak, the velocity difference between the two peaks. The term Δ_peak is expected to correlate with the column density of H i along the line of sight, as well as with the velocity of the bulk of the outflow (Verhamme et al. 2015). In what follows, we investigate how the properties of the Si⁺ outflows affect Δ_peak, by performing a similar analysis to that of the previous section (see Figure 9).

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Considering the sample of nine Green Peas with double Lyα emission peaks, we find no correlations between Δ_peak and the parameters f_c and v_ap/v_∞, and a weak correlation with τ. These results are surprising, as the peak separation is expected to increase as the column density of the scattering gas increases (Verhamme et al. 2015). Among the kinematic parameters, we find no correlation between Δ_peak and v_∞, or γ, while we detect a strong negative correlation between Δ_peak and v₀ (ρ = −0.720 with P = 0.03). The last correlation suggests that both low-velocity and high-density material is required to promote further scattering. By fitting the Doppler broadening of UV lines in Lyman break galaxies using models of spherical outflows with constant velocity fields, Verhamme et al. (2008) found that the galaxies with double-peaked Lyα profiles in their sample were best fit by outflows with low (10–25 km s⁻¹) velocity fields which appears to be in agreement with our findings. Indeed, Lyα profiles are thought to be sensitive to outflow velocity (e.g., Verhamme et al. 2008; Gronke et al. 2015), but more work, both observationally and theoretically, is necessary to understand how outflow kinematics influence the shape of the Lyα profile. In particular, exploring the ramifications of more complicated velocity fields (i.e., dV/dr ≠ 0) in radiative transport studies of Lyα radiation is necessary.

Finally, when we focus on the geometrical parameters describing the outflows, we find no correlations between between Δ_peak and α, and a strong positive correlation with f_c ∗ V_ff (ρ = 0.770 with P = 0.02), suggesting that outflow geometry plays a strong role in determining Δ_peak.

From the analysis of the previous section, it is clear that determining which characteristics of galactic outflows support Lyα escape in general is complicated. Outflow kinematics—including density, velocity, and the velocity gradient—likely play a role, and surely the outflow geometry and orientation have an impact as well. In an attempt to untangle these two effects, we have identified galaxies according to their outflow geometry and orientation with respect to the observer. We separate galaxies into two groups: those in which the Si⁺ gas is consistent with a spherical distribution (i.e., where α > 85°), and those in which the Si⁺ gas is biconical. The latter group includes objects with biconical outflows oriented perpendicular to the line of sight (α < 85°, and ψ > α), edge-on outflows (α < 85° and α = ψ), and biconical outflows oriented parallel to the line of sight (α < 85° and α > ψ).

A picture showing the cross-sectional view of these outflow geometries, as seen by an observer from the right, is provided at the bottom right in Figures 8 and 9. Galaxies 0911+1831, 1054+5238, 1137+3524, 1244+0216, 1249+1234, and 1424+4217 have spherical outflows in Si ii, while galaxies 1133+6514 and 1219+1526 have perpendicular biconical outflows. Galaxy 0926+4427 is our only galaxy with a biconical outflow observed edge on. (Even though α > ψ in galaxy 0926+4427, we have chosen to include 0926+4427 in this group since α ∼ ψ within our uncertainties.) Lastly, galaxy 0303–0759 is our only galaxy with a biconical outflow oriented parallel to the line of sight. Members of these groups are circled in blue, red, violet, and green, respectively, in Figures 8 and 9.

Because Lyα photons scatter multiple times in the H i medium, the geometry of the gas plays a crucial role. Specifically, Lyα photons will escape toward directions with the minimum H i column density. In the case of biconical geometry, photons will encounter the lowest H i column density in the direction perpendicular to the outflow. The orientation of the outflow with respect to the line of sight, then, will determine the visibility of Lyα photons by an observer. This seems to be confirmed with the data: galaxies 1133+6514 and 1219+1526, with outflows oriented perpendicular to the line of sight, have the first and third largest values of the escape fraction out of our 10 Green Pea galaxies, while galaxy 0303–0759, our only galaxy with a biconical outflow oriented along the line of sight, has the lowest ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ with respect to all galaxies.

For a similar reason, we also expect galaxies with outflows observed edge on with respect to the observer to still show some Lyα. However, since in this geometry roughly half the source is blocked, we expect this group to have lower values of ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ in comparison to the galaxies with perpendicular biconical outflows. This ordering appears to agree with 0926+4427's value of f_esc = 0.2, which places it near the median of the ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ values from our Green Peas.

Finally, when the geometry of the scattering material is spherical, we expect that the conditions within the outflow will control the Lyα escape, as opposed to the geometry. To test this hypothesis, we have isolated the galaxies with spherical Si⁺ outflows in Figures 8 and 9, and performed a separate Spearman correlation test using only their values. As visible in Figure 8, limiting the sample to the galaxies with spherical outflows strengthened the correlation between ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ and γ (ρ = −0.986 and P < 0.001). In addition, a very weak negative trend (ρ = −0.382 with P = 0.45) emerged between f_c and ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$, in better agreement with our expectations. Lastly, a weak trend between v_∞ and ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ (ρ = 0.638 and P = 0.173) also emerged. Overall, though, γ appears to be the dominant outflow parameter controlling Lyα escape in spherical outflows. In terms of kinematics, this suggests that outflows with shallow velocity gradients favor Lyα escape; however, as noted in Section 5.1, this relationship likely reflects the dependence of the density field on γ.

The peak separation in the Lyα profile will also depend on the geometry of the outflow as seen in Figure 9. For example, biconical outflows oriented perpendicular to the line of sight are expected to have the smallest, if any, Δ_peak, as in this case there is almost no gas in front of the source to scatter Lyα photons at the largest projected velocity. Spherical outflows will show the largest Δ_peak, while biconical outflows oriented along the line of sight should show values covering a broad range in between. Indeed, galaxies 1133+6514 and 1219+1526 agree with our expectations, having the two smallest peak separations while having biconical outflows oriented perpendicular to the line of sight, and galaxies 1137+3524 and 1244+0216 have the two largest peak separations while having spherical outflows. The remaining galaxies are distributed in the middle of these two extremes.

The positive correlation between R_W and ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ in galaxies with spherical outflows suggests that a large contribution to the observed Lyα flux may be coming from the galactic wind itself, in addition to the Lyα photons produced by recombination in the H ii regions and scattering their way through the outflow. In situ Lyα production has been studied before in simulations of protogalaxies and Lyα blobs, (Dijkstra et al. 2006; Dijkstra & Loeb 2009; Faucher-Giguère et al. 2010; Goerdt et al. 2010; Rosdahl & Blaizot 2012; Mitchell et al. 2020), but these studies tend to focus on infalling gas or galactic inflows, not outflows. Our results suggest in situ production of Lyα photons may be a key aspect of the supernovae-driven outflows notably left out of many state-of-the-art Lyα radiation transfer simulations (Rosdahl & Blaizot 2012; Mitchell et al. 2020). Further support for in situ Lyα production comes from the strong correlation (ρ = 0.829 and P = 0.042) observed between R_W and the Lyα luminosity, L_Lyα, shown in the middle panel of Figure 11. If the observed Lyα flux were entirely due to Lyα photons produced in the central H ii regions, one would expect ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ to anti-correlate with R_W, as the more extended the outflow of neutral gas (i.e., the larger R_W) is, the greater the chance that Lyα photons will be absorbed by dust and disappear.

If our interpretation is correct, then the quantity ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ does not represent the fraction of Lyα photons that escape the ISM into the CGM and IGM, but rather includes a component that is produced directly in the outflow of the galaxies. This component may have already been observed in the extended Lyα haloes observed around star-forming galaxies both locally and at high redshift (Hayes et al. 2013; Wisotzki et al. 2016).

In order to explain the correlation between R_W and ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$, we compute the expected range of Lyα luminosities that could be expected due to collisional excitation of H  i by free electrons. We estimate the Lyα emissivity, j_Lyα (energy radiated per unit time, volume, and solid angle) as $4\pi {j}_{\mathrm{Ly}\alpha }=h{\nu }_{\mathrm{Ly}\alpha }{n}_{e}{n}_{H{\rm\small{I}}}{q}_{\mathrm{Ly}\alpha }^{\mathrm{eff}}$, where n_e and n_{H i} are the number density of electrons and neutral hydrogen, respectively, and ${q}_{\mathrm{Ly}\alpha }^{\mathrm{eff}}$ is the effective collisional excitation coefficient defined by Cantalupo et al. (2008), and includes excitation processes up to the level n = 3.

We aim to compute the number density of neutral hydrogen, n_{H i}, from the number density of ionized silicon, ${n}_{{\mathrm{Si}}^{+}}$. First, we use ${n}_{{\mathrm{Si}}^{+}}$ to estimate the total hydrogen number density (i.e., ${n}_{{\rm{H}}}=\tfrac{{n}_{{\mathrm{Si}}^{+}}}{A{\chi }_{{\mathrm{Si}}^{+}}}$, where ${\chi }_{{\mathrm{Si}}^{+}}$ is the fraction of all silicon in Si⁺). We assume that the metallicity of the outflowing material probed by the Si  ii absorption is $\sim \tfrac{1}{3}{Z}_{\odot }$, i.e., the same as that of the ionized gas probed by the Hα emission.¹¹ With this assumption, and assuming the solar photospheric abundance of Si relative to H given in Asplund et al. (2009, n_Si/n_H = 3.2 × 10⁻⁵), we derive A = 1.1 × 10⁻⁵.

To constrain the ionization state of Si (${\chi }_{{\mathrm{Si}}^{+}}$), we divide the outflows in two regions: (1) the region in which silicon is detected in multiple ionization states; and (2) the volume in which we predict that most of the silicon is in the form of Si⁺ (see Figure 12). In the latter region, given the relative ionization potentials of Si (8.16 eV) and Si⁺ (16.3 eV) compared to that of H, the fact that ${n}_{{\mathrm{Si}}^{2+}}/{n}_{{\mathrm{Si}}^{+}}\gg 1$ implies that the observed absorption is associated with hydrogen in neutral form, and therefore the Si⁺ ions trace the neutral phase. For the former, we use the actual measured densities of silicon in the various ionization states, and make the assumption (confirmed by, e.g., Chisholm et al. 2018a; McKinney et al. 2019, for similar galaxies), that the majority of silicon lies within the first three ionization states.

Finally, for simplicity we assume that n_e = n_p = χn_H = n_H − n_{H I}, and a hydrogen ionization parameter χ = 0.99, as observed in absorption line studies of the CGM of starburst galaxies (e.g., Werk et al. 2014). The compact morphology and high star formation rates of Green Peas, however, would suggest that the ionization fraction may be greater for our galaxies (see Jaskot & Oey 2013). We consider χ a lower limit. The Lyα emissivity due to collisional excitation depends strongly on temperature. In the calculation of j_Lyα, therefore, we consider a range of reasonable temperatures expected in the outflow, 1 × 10⁴ K ≤ T ≤ 2 × 10⁴ K, corresponding to values of q_Lya^eff between 1.5 × 10⁻¹³ and 5.0 × 10⁻¹¹, respectively (Katz et al. 1996).

The results of this calculation are provided in Figure 13, where we compare the predicted Lyα luminosities from the outflow with the observed total Lyα luminosities. The predicted Lyα luminosities were calculated over the full extent of the outflows using the densities assigned to the appropriate regions detailed in Figure 12—we neglected any aperture effects. The vertical bars associated with the predictions account for our uncertainties in the measured parameters as well as the considered temperature range. We find a strong correlation (ρ = 0.829 with P = 0.042) between our predicted Lyα luminosities and the observed Lyα luminosities of the Green Peas with spherical outflows. This suggests that the contribution to the observed Lyα flux from the Lyα radiation produced in the winds is not only significant, but the dominate source of Lyα radiation in these detections. We explore this issue further by testing for a correlation between the column density of neutral hydrogen, N_{H i}, which we estimate along the line of sight as

$$\begin{eqnarray}&&{N}_{{\rm{H}}{\rm\small{I}}}={\int }_{{R}_{\mathrm{SF}}}^{{R}_{W,\mathrm{Si}{\rm\small{II}}}}{f}_{c,\mathrm{Si}{\rm\small{II}}}{n}_{0,{\rm{H}}{\rm\small{I}}}{\left(\displaystyle \frac{{R}_{\mathrm{SF}}}{r}\right)}^{2+{\gamma }_{\mathrm{Si}{\rm\small{II}}}}{dr},\end{eqnarray} \tag{ 18 }$$

and the observed Lyα luminosity. We find no correlation in the right panel of Figure 11. This result likely reflects the fact that both Lyα photons produced within central H ii regions and within the galactic winds of the galaxies are competing to control the correlation—the latter of which becoming more dominant when we include the full extent of the wind.

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These calculations show that the Lyα produced in the outflows could account for a significant portion of the observed Lyα luminosity, between 1% and 100%, depending on the uncertain value of the gas temperature in the outflow material. Hydrodynamical simulations suggest that galaxy winds are multiphase, with temperatures reaching up to ~10⁵ K (e.g., Schneider et al. 2020). Resolved direct measurements of the electron temperature would help to constrain the wind contribution to the total Lyα luminosity. Additionally, in this simple order of magnitude calculation we neglected radiative transfer effects, such as scattering and dust attenuation, that could lower the observed Lyα luminosity. The role of Lyα production from the wind should be investigated further both theoretically and observationally.

A significant contribution to the observed Lyα flux from the wind itself would have noticeable consequences. For example, the Hα/Lyα ratio in the wind would be much lower than expected from recombination theory (see Fardal et al. 2001). Additionally, if our scenario is correct, there may exist objects with ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ larger than 100%. There does appear to be some support of this claim coming from the literature, see, e.g., Yang et al. (2017), who reported a Green Pea galaxy with ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }=1.18$.

As another example, one would not expect the Lyα flux produced via cooling over an extended region to correlate with the strength of central UV sources (Matsuda et al. 2004). We found a weak trend (ρ = 0.638, P = 0.173) between L_Lyα and the FUV absolute magnitude, M_UV, among the Green Peas with Si⁺ spherical outflows. This weak trend suggests that dimmer Green Peas in the FUV are brighter in Lyα—the opposite of what is expected from Lyα produced in central H ii regions. These results could have major implications for the relationship between Lyα and LyC radiation, as they imply that Lyα photons do not trace the ionizing radiation directly. This could be one of the reasons why ${f}_{\mathrm{esc}}^{\mathrm{Ly}\alpha }$ and ${f}_{\mathrm{esc}}^{\mathrm{LyC}}$ do not perfectly correlate in empirical studies (e.g., Verhamme et al. 2017).

Finally, we mention that the Lyα photons produced in the wind would contribute only to the emission component of the Lyα profile. If their contribution is large enough, these photons may be able to determine the overall shape of the Lyα profile. This might explain why it is difficult for the majority of Lyα profiles observed in Green Peas to be reproduced by the classical picture of a central source of Lyα radiation combined with an expanding shell of neutral gas responsible for the scattering of the Lyα photons (Orlitová et al. 2018).

PDFs returned by the MCMC analysis for galaxy 1244+0216 are provided in Figure 14. The parameter values associated with the best fit are taken from the mode of the marginalized PDFs and are shown in red in Figure 14. The errors attributed to each parameter were taken to be the median absolute deviation from above and below the mode and are shown in blue in Figure 14. The errors represent our ability to constrain the SALT model given the prior distributions. Therefore, any model degeneracies given the prior distributions are included in these uncertainties.

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We finish this section by discussing the possibility of a degenerate parameter space. Carr et al. (2018) first studied the ability of the SALT model to recover parameters from mock data generated to encompass a variety of galactic systems with biconical outflows spanning a range of different orientations and kinematic properties. They found that parameters are best recovered when both strong emission and absorption features are present in the spectrum of the relevant transition line. This reflects the fact that while some parameters, for example, may influence the absorption profile in a similar way, this degeneracy can be broken when including how the energy is distributed in the emission profile. In this regard, our parameter predictions inferred from the Si ii lines should be the most reliable—typical spectra contain both absorption and emission features—while our parameter predictions from the Si iii and Si iv lines are comparatively less reliable—typical spectral lines appear primarily in absorption.

Carr et al. (2018) concluded that parameters are overall well recovered by the SALT model from mock data—that is, there is little, if any, degenerate parameter space—however, they did not include the effects of an observing aperture, dusty CGM, or the covering factor, f_c, in their analysis. We now discuss how these parameters can influence the line profile in light of our results from Section 4.

A restrictive observing aperture, a strong dust component to the CGM, and the outflow geometry associated with a narrow bi-cone oriented along the line of sight can all act to diminish the emission component of a line profile while maintaining a strong absorption component. Since these parameters do not affect the line profile in the exact same way—that is, there are subtle differences among all three scenarios (see the discussions of Equations (5) and (8) in Section 3), when working with actual data these differences may get unresolved, rendering the three scenarios degenerate. One can eliminate the possibility of an aperture effect from this scenario by finding other lines with an emission component (an aperture affects each line profile in the same way since it depends on how much of the galactic system is resolved). For example, since the majority of Si ii line profiles taken from our Green Peas have visible emission components, this implies the lack of emission observed in the Si iii and Si iv line profiles must be the result of either a narrow biconical outflow geometry oriented parallel to the line of sight (the geometry favored in general from these line profiles by the MCMC analysis) or a strong dust presence in the CGM. This may explain why the SALT model generally predicts higher values of κ from the Si iii and Si iv transition line profiles.

In regard to the covering factor, f_c, a galaxy with a spherical outflow filled with holes (i.e., f_c < 1) will have equal absorption and emission equivalent widths (neglecting other effects). It was shown by Carr et al. (2018) that galaxies with various biconical outflow geometries (controlled by the geometric factor, f_g, as opposed to f_c in Equation (5)) can also produce line profiles with roughly equal absorption and emission equivalent widths. While photons reemitted from biconical outflows with different orientations and opening angles have different energy distributions, if the resolution is not high enough, these two scenarios can also become degenerate. In this instance, it can become difficult to distinguish a biconical outflow from a spherical outflow with a global covering fraction. This is important because this is where the SALT model and other models that interpret UV absorption features with a covering fraction (e.g., Martin et al. 2012, 2013; Rubin et al. 2014; Chisholm et al. 2018b) become degenerate. For galaxies 1133+6514 and 1219+1526, the geometries recovered from the transition line of Si ii favor emission equivalent widths to be greater than the absorption equivalent widths and, therefore, cannot be the result of a spherical outflow with a global covering fraction.

The 1190.42 Å, 1193.28 Å Si ii doublet represents the wavelengths associated with two resonant transitions from the ground state of Si ii to the hyperfine energy levels of the 2P orbital. For each of these resonant transitions, the atomic structure of Si ii admits a probable fluorescent transition—that is, there is a nonnegligible probability that electrons in the ²P_J energy levels transition into an excited hyperfine energy level of the ground state. The associated fluorescent transitions have wavelengths 1194.5* Å, 1197.39* Å, respectively. A diagram representing the relevant energy levels to the 1190.42 Å, 1193.28 Å Si ii doublet is provided in the upper left corner of Figure 15.

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To test the consistency of the SALT model, we have independently fit to the 1260.42 Å resonant transition of Si ii, which is associated with an electron transition from the ground state to the 2D orbital. The atomic structure of Si ii admits a fluorescent transition from the 2D orbital to an excited hyperfine energy level of the ground state at a wavelength of 1265.02* Å. The 1260.42 Å resonant transition is typically associated with two emission lines, 1264.73* Å and 1265.02* Å, where the 1264.73* Å corresponds to the transition between excited hyperfine energy levels. We do not see any absorption at the 1264.73 Å wavelength (cf., Jaskot et al. 2019), and therefore we have chosen not to include this transition in our modeling.¹² The relevant atomic energy levels for this transition are shown in the upper right corner of Figure 15.

The 1206 Å Si iii resonant transition is associated with the transfer of an electron from the ground state to the first P orbital of Si iii. There are no associated fluorescent transitions. The relevant energy levels for Si iii have been provided in Figure 15.

The 1393.76 Å, 1402.77 Å Si iv doublet represents the wavelengths of two resonant transitions from the ground state to the 2P orbital of Si iv. Like the 1206 Å Si iii resonant transition, there are no associated fluorescent transitions. The relevant energy levels for Si iv have been provided in Figure 15. All relevant atomic information for Si ii, Si iii, and Si iv has been provided in Table 3.