THE LYMAN CONTINUUM ESCAPE FRACTION OF THE COSMIC HORSESHOE: A TEST OF INDIRECT ESTIMATES^{* †}

For our analysis, we have used two images from the HST WFC3 UV channel (UVIS). First, we used a one-orbit (2412 s) image in the F606W filter from the Hubble Program ID 11602 to measure the non-ionizing UV flux density at ${\lambda }_{\mathrm{rest}}\sim 1800\,\mathring{\rm A}$.

Because the rest-UV SED is nearly flat in ${f}_{\nu }$ ($g-i=0.04$ AB; Belokurov et al. 2007), we assume that the flux density at 1500 Å (typically used for escape fraction measurements) is the same as the flux density at 1800 Å.

We obtained a 10-orbit image in the F275W filter to sample the rest-frame LyC (transmission-weighted wavelength of 2704 Å, or 800 Å in the rest frame of the Horseshoe). Figure 2 shows these filter transmission curves with a typical Lyman break galaxy (LBG) spectrum at z = 2.38 and the smoothed spectrum of Quider et al. (2009).

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Below 4000 Å, read noise is the main source of noise in our UVIS imaging, although, for more recent UVIS observations, the dark current has become more important. Hence, to minimize the number of readouts, the F275W exposure times were half an orbit in duration (1404 s), with a total of 20 exposures and a total exposure time of 28,080 s.

The UVIS CCDs suffer from significant charge transfer inefficiency (CTI) when both the background and the targets are faint (MacKenty & Smith 2012). As there is effectively no sky background in F275W (∼0.35 ${e}^{-}$ per pixel per half orbit exposure), the dark current (∼1.5 ${e}^{-}$ per pixel per half orbit exposure) dominates the total background but is very low. In addition, the expected LyC signal is very faint. Therefore, CTI is of particular concern for these observations, and we have to understand its effects on our measurements. In order to minimize the number of pixels traversed by the electrons upon readout and therefore reduce the CTI, we intentionally placed the Horseshoe closer to the readout edge of the detector, with its center $\sim 40 \%$ of the CCD width from the readout edge. In F606W, both the high signal and high background minimize CTI, so it will not significantly affect our measurements in that filter.

Part of the WFC3/UVIS calibration process is the subtraction of a dark reference file to correct dark current structure and flag hot pixels that can cause significant artifacts in the images. The current standard processing of the dark calibration is insufficient for the WFC3/UVIS data of this program due to the radiation damage causing poor charge transfer efficiency (CTE) and the low background levels in the F275W filter. The STScI dark frames have the following inadequacies: First, the STScI process uses an outdated definition of a hot pixel that has not been updated to account for CTE degradation, resulting in unmasked warm-to-hot pixels remaining. Second, the standard processing uses the median value of the average darks as the value of all pixels in the dark frame. This median dark file is not suitable for the low background of the near-UV, because it leaves a low-level gradient and a blotchy pattern in the dark that is not subtracted. Lastly, the STScI darks are not corrected for CTE, furthering the improper hot pixel masking, and contributing to incorrect median levels in the STScI darks.

To solve these issues, custom CTE-corrected superdarks are created in a two-step process as detailed in Rafelski et al. (2015). First, all darks from a 4-day window at the same cadence as the STScI darks are used to create a superdark, where the background is modeled with a third-order polynomial to remove the background gradient temporarily to find the hot pixels with a uniform updated threshold level. This step is necessary due to the large number of new hot pixels per day and the drastic change in hot pixels after each anneal, where the CCD is warmed up for several hours to reduce the number of hot pixels caused by radiation damage. Then, all darks from a single anneal cycle are averaged together, avoiding the hot pixels from each 4-day window, to determine the actual dark level for each good pixel. This averaged background is then used in conjunction with the hot pixel map from the first step to create a new superdark, which is used for calibrating the science exposures. These new darks properly flag the hot pixels, remove the background gradient, and significantly reduce the blotchy pattern in the science exposures.

We first CTE-corrected and dark-subtracted all of the flat-fielded images. We then used the AstroDrizzle package (Gonzaga et al. 2012), provided by the Space Telescope Science Institute, to combine the dithered images into a final image with a 004 pixel scale. AstroDrizzle outputs inverse variance maps, which were used to determine the expected Poisson uncertainties in each pixel. The alignment of individual exposures was accurate to $\lt 4$ mas in both filters. After combining all images, we subtracted a slight, residual background gradient in the readout direction by fitting a third-order polynomial to both CCDs in order to guarantee a uniform background around the Horseshoe.

The reduced images in both filters are shown in Figure 3. To determine the image depths, we corrected the pixel rms for the adjacent pixel correlations based on Casertano et al. (2000) (in our case, multiplied by 1.5) to estimate sigma per pixel. The derived $5\sigma$ image depths in an aperture with a 04 radius are 27.34 and 26.79 AB mag for the F606W and F275W images, respectively. We note that this depth does not reflect the CTI that will be discussed later and affects the calculated upper limit on f_esc.

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We searched for F275W (LyC) flux in a number of apertures defined by isophotes of varying F606W (non-ionizing UV) surface brightness, because it is possible that LyC is only escaping from certain regions (like those of high star formation surface density, for example). We used three apertures with boundaries defined by low (L), medium (M), and high (H) surface brightness isophotes, corresponding to $1.3\sigma ,3\sigma$, and $5\sigma$ above the background in F606W. These apertures constitute a range from a near-total flux aperture up to a high surface brightness aperture and are displayed in Figure 4. No signal is detected through the F275W filter in any of the apertures.The F275W signal-to-noise ratio (S/N) in these apertures varies between −1.2 and +1.1 and is therefore not a significant detection.

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We can use the depths of the F275W image to place upper limits on the LyC escape fraction. However, first, we must assess the effect (if any) of CTI as it may move some of the expected flux out of our aperture, erasing some of the signal. Teplitz et al. (2013) have shown cases where the signal can be lost completely in smaller apertures. Unfortunately, at the time at which the data were taken the "post-flash" option (Biretta & Baggett 2013) to add background to the image was not available to mitigate CTI, and pixel-based empirical corrections of CTI are impossible in reconstructing images if no signal has remained. Therefore, our images in the F275W filter will be affected by CTI.

We first make an "ideal" image in the F275W filter, add the expected background, simulate the effect of CTE, and finally add the read noise to produce an individual F275W exposure. We then add them to produce a simulated full-depth F275W image. Below we discuss details of each of these steps.

4.1.1. Assumed Image

4.1.2. Background

4.1.3. Charge Transfer Phase and Read Noise

4.1.4. Stacking the Images

We simulated 20 of these F275W images and stacked them to obtain a full-depth stacked simulated image. Now we are able to investigate whether, for an assumed LyC flux, the Horseshoe would be detected with $3\sigma$ confidence in our three different apertures.

Of course, our "detection level" derived above is noisy, because it relies only on a single simulated stacked image, and the signal in this image will, by definition, vary by the input standard deviation multiplied by the square root of the number of pixels. Therefore, to more accurately assess the expected detection threshold, we produce five instances of these simulated images at the same LyC flux level and use the average of their detection levels. We had to using a higher-S/N background, because the Poisson variations in the background are large compared to the background and such variations can affect the level of CTI in each pixel.

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We generated simulated images with eight different input magnitudes, spanning a range of 2 mag with steps of 0.25 mag (close to where we expect to have $3\sigma$ detection within our apertures). For each of the stacked simulated images, after removing the background, we measured the level of signal and noise within each aperture. Figure 6 shows the measured S/N of each of the images in the three apertures, as a function of the input magnitude within that aperture. For each of the input magnitudes, the average S/N is then plotted as filled circles. To find the input magnitude that results in a $3\sigma$ detection, we fit a curve to the filled circles, assuming that input flux is linearly proportional to the detection level, and determine at what magnitude the significance of the detection would be greater than ${\rm{S}}/{\rm{N}}\gt 3$. These threshold magnitudes and the corresponding F606W magnitudes are listed in Table 1. All magnitudes have been corrected for Galactic extinction at the location of the Horseshoe based on Schlafly & Finkbeiner (2011) (${A}_{{\rm{F}}275{\rm{W}}}=0.258$ mag and ${A}_{{\rm{F}}606{\rm{W}}}=0.117$ mag).

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Table 1.  Characteristics and Measured Quantities in Each Aperture

Apertures	L	M	H
σ above background in F606Wa	1.3	3.0	5.0
Number of pixels	14302	7514	3278
Magnitude in F606W	19.83	20.12	20.62
Detection level in F275W (total S/N)	−1.23	−0.99	1.08
3σ limiting magnitude in F275W imageb	25.82	26.17	26.62
${f}_{\mathrm{esc},\mathrm{rel}}$ for limiting magnitude	0.079	0.075	0.078
3σ threshhold magnitude in F275Wc	25.82	26.11	26.45
${f}_{\mathrm{esc},\mathrm{rel}}$ for threshold magnitude	0.079	0.079	0.092

Notes.

^aThe selection criteria to choose the apertures. ^bBased on the depth of the image. ^cBased on Figure 6.

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For comparison, we have also listed in Table 1 the escape fraction limits based solely on the background noise and not considering signal loss due to CTI. Interestingly, the limits are very similar, meaning that the CTI losses are not large for our measurements. The CTI results in 0%, 5%, and 17% increase in the limiting flux for L, M, and H apertures, respectively. Not surprisingly, the CTI effect on photometry is not significant in the largest apertures, as the typical trapped e⁻ is released $\lt 10$ pixels behind the original pixel (Rafelski et al. 2015). Therefore, CTI losses in the photometry only become a concern when the aperture width (in the readout direction) approaches this length scale.

With the F275W $3\sigma$ threshold magnitudes and the F606W magnitudes in the corresponding apertures, we can calculate the limits on the non-ionizing-to-ionizing flux ratio and, ultimately, the relative escape fraction, using Equation (2). In Table 1, we list the $3\sigma$ limits on the relative escape fraction in all three apertures.

After including the effects of CTI, the upper limits on the relative escape fraction in each aperture are similar: 0.079, 0.079, and 0.092 in the L, M, and H apertures, respectively, as the significant reduction in F275W noise in smaller apertures is offset by reduced F606W flux. Using a Calzetti extinction curve (Calzetti 1997) and ${E}_{(B-V)}=0.15$ (Quider et al. 2009), we convert ${f}_{\mathrm{esc},\mathrm{rel}}\lt 0.08$ to ${f}_{\mathrm{esc},\mathrm{abs}}\lt 0.02$, well below the average value needed to maintain ionization at $z\sim 7$ (${f}_{\mathrm{esc},\mathrm{abs}}\sim 0.2$) (e.g., Bouwens et al. 2015a; Robertson et al. 2015).

As mentioned in Section 1, the high-resolution, rest-frame UV spectrum of the Horseshoe has multiple, resolved interstellar absorption lines of low ions with depths of ∼60% of the continuum flux density (Quider et al. 2009). Some of these lines are shown in Figure 7. These lines have very different oscillator strengths but similar absorption depths, especially for the case of Si ii transitions, which all originate from the same ion, and the relative depths are therefore independent of the metallicity or ionization. Quider et al. (2009) therefore conclude that the depth of the lines is not due to the column density of a foreground screen. Rather, the depth is dictated by the fraction of the UV-bright disk that is covered by clouds that are opaque in these lines. Hence, for the Horseshoe, we expect a ∼0.6 "covering fraction" of low-ionization gas where neutral hydrogen would reside. Based on the picket fence model and the discussion in the Appendix, this in turn implies ${f}_{\mathrm{esc},\mathrm{rel}}\sim 0.4$.

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The reported $3\sigma$ limits in our apertures of ${f}_{\mathrm{esc},\mathrm{rel}}\lt 0.08$ are roughly five times lower than the expected escape fraction inferred from the depth of the low-ion absorption lines in the spectrum of the Horseshoe. One reason for this discrepancy could be the existence of a rather opaque IGM along the LOS to the Horseshoe. To obtain our measured ${f}_{\mathrm{esc},\mathrm{rel}}$, we adopt the average transmission of the IGM through the F275W filter (${e}^{-{\tau }_{{H}_{I,\mathrm{IGM}}}}=0.402$). But, as is evident in Figure 1, this value is not representative of the nearly bimodal distribution of the IGM transmission. Specifically, there is an almost 20% chance that the Horseshoe lies along an LOS that is at least $5\times$ more opaque than the average transmission assumed here (less than 0.08), which could thus explain our nondetection, even if ${f}_{\mathrm{esc},\mathrm{rel}}$ is 0.4 as suggested by the depth of the low-ionization interstellar absorption lines. On the other hand, the IGM transmission could also be significantly higher than the average value, in which case our ${f}_{\mathrm{esc},\mathrm{rel}}$ limit would be even lower.

There are still a number of additional possible causes for a discrepancy between the transmission in the low ions and the relative escape fraction. Many of these reasons have been mentioned by Jones et al. (2013b), but we discuss them here in the context of the Horseshoe:

• 1.  
  The absorption depths of the low ions give only a covering fraction of the non-ionizing UV disk (from ∼1250 to 1700 Å). However, because the LyC-emitting regions are short-lived, they likely compose only a subset of the area that is emitting non-ionizing UV light. Therefore, a clear LOS toward a non-ionizing UV-emitting region does not necessarily imply that LyC photons will escape. Furthermore, it is likely that the LyC-emitting regions have higher columns of dust and gas toward them, as they are younger and more embedded in their birth clouds. Indeed, this is the reason that extinction of H ii regions is typically higher than extinction of other regions at the same wavelengths (Calzetti 1997). Therefore, the transmitted emission in the low-ion absorption lines should be considered an upper limit to the possible LyC escape fraction.
• 2.  
  The LyC absorption by hydrogen acts at all wavelengths below the Lyman limit, whereas the absorption lines absorb at one specific wavelength. Therefore, one must be careful in interpreting the absorption lines. The velocity structure of the absorbing gas is critical when converting the depth of the absorption lines to covering fractions. For example, if half of the galaxy is covered by gas outflowing at 0–100 km s⁻¹ and the other half is covered by gas outflowing at 100–200 km s⁻¹, then the depth of the absorption line (in a high-resolution spectrum) will never be more than 50% of the continuum. However, the LyC in such a scenario would be 100% absorbed because it is insensitive to the velocity of the outflow. Thus, once again, the LyC escape fraction should generally be lower than expected given the absorption-line depth.
• 3.  
  If there are foreground neutral clumps with low velocity dispersion that are not resolved in wavelength, the profile of the absorption line will be smoothed, and the measured transmission will reflect only an upper limit on ${f}_{\mathrm{esc},\mathrm{rel}}$. In our case the Keck/Echellette Spectrograph and Imager (ESI) spectra of the Horseshoe have a resolution of ∼75 km s⁻¹. Though the velocity width of the absorption profiles is much larger than this value, it is still possible that unresolved narrow components exist.
• 4.  
  These absorption lines are due to resonance transitions from the ground state. Therefore, the photon can be scattered and refill the absorption line if the slit encompasses a large fraction of the scattering gas cloud (Prochaska et al. 2011; Rubin et al. 2011; Scarlata & Panagia 2015). Thus, the covering fraction may be larger than implied by the depth of the absorption lines, and the relative escape fraction will be lower.
• 5.  
  If the LyC absorbing gas is very low metallicity (Fumagalli et al. 2011), perhaps because it is inflowing from the IGM, then the metal-line absorption will not be strong, but the hydrogen opacity will still be large and can absorb the LyC. However, given the significant amount of enriched outlawing material and the large accumulated stellar mass, it seems unlikely that a significant fraction of the absorbing gas would be very low metallicity. However, in such a scenario, the absorption-line transmission should be treated as an upper limit.

Because of the reasons outlined above, we believe that a nonuniform coverage of low-ionization metals is a necessary, but not sufficient, condition for significant escape of LyC. Hence, any estimates of ${f}_{\mathrm{esc},\mathrm{rel}}$ based on these absorption lines should be interpreted as upper limits.

Although ${f}_{\mathrm{esc},\mathrm{rel}}$ may be significantly lower than predicted by the transmission in low-ion absorption lines, it is worth noting that the two values will be better correlated in more compact galaxies because many of the issues raised above that can cause the two to differ are mitigated significantly if the galaxy is extremely compact ($\lt 100$ pc). First, the ionizing and non-ionizing UV continua are likely emitted from the same regions (a single star-forming region). In contrast, in a large galaxy, much of the non-ionizing flux is likely emitted from regions with no current star formation (and thus no LyC production). Second, the smaller the galaxy, the likelier it is that the absorption of light from different parts of the galaxy is caused by the same absorbers, especially if the galaxy size approaches the typical size of an absorbing neutral clump in the ISM or CGM. Thus, one has to worry less about clumps of different velocities covering different parts of the disk.

Indeed, Heckman et al. (2011), Borthakur et al. (2014), and Alexandroff et al. (2015) have been investigating the relative escape fraction, both directly and indirectly from luminous and extremely compact galaxies, which they refer to as dominant central objects (DCOs, ionizing UV sizes of $\lt 100$ pc). Thus, it may be true that the absorption-line transmission reasonably predicts the relative escape fraction in these galaxies, but it may not be the case in all galaxies.

The intrinsic size of the Cosmic Horseshoe galaxy has been measured in both the non-ionizing UV and Hα by Jones et al. (2013a). In both cases, the emission is coming from an elongated region that is ∼0.2 kpc $\times \,0.4\,\mathrm{kpc}$, significantly smaller than the average UV size of galaxies of similar luminosity that typically have diameters of $\sim 3\mbox{--}4\,\mathrm{kpc}$ (Law et al. 2007). With such a small area, we might expect that some of the issues above would be mitigated. However, this area is still large enough that it likely consists of many distinct star-forming regions, and may still be covered by a range of clump distributions along the LOS. To understand the efficacy of these indirect estimates of ${f}_{\mathrm{esc},\mathrm{rel}}$, we must directly image the LyC of a large sample of galaxies with high transmission in the low-ion absorption lines.

For the dust-free case, the LyC flux density and the flux density in the absorption lines get completely absorbed by the clouds, while the 1500 Å continuum flux density remains unaffected by the gas clouds:

$$\begin{eqnarray}&&{F}_{\mathrm{LyC},\mathrm{out}}={F}_{\mathrm{LyC},\mathrm{stel}}\times (1-{C}_{F});{F}_{1500,\mathrm{out}}={F}_{1500,\mathrm{stel}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&1-{C}_{F^{\prime} }=\displaystyle \frac{{F}_{\mathrm{line},\mathrm{obs}}}{{F}_{\mathrm{cont},\mathrm{obs}}}=\displaystyle \frac{{F}_{1500,\mathrm{stel}}\times (1-{C}_{F})}{{F}_{1500,\mathrm{stel}}}=1-{C}_{F}.\end{eqnarray} \tag{ 4 }$$

So here ${C}_{F}={C}_{F^{\prime} }$, which results in

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{rel}}=\displaystyle \frac{{({F}_{\mathrm{LyC}}/{F}_{1500})}_{\mathrm{out}}}{{({F}_{\mathrm{LyC}}/{F}_{1500})}_{\mathrm{stel}}}=1-{C}_{F}=1-{C}_{F^{\prime} }\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{abs}}=\displaystyle \frac{{F}_{\mathrm{LyC},\mathrm{out}}}{{F}_{\mathrm{LyC},\mathrm{stel}}}=1-{C}_{F}=1-{C}_{F^{\prime} }.\end{eqnarray} \tag{ 6 }$$

In this case, ${f}_{\mathrm{esc},\mathrm{rel}}$ and ${f}_{\mathrm{esc},\mathrm{abs}}$ are the same. In the literature, this is sometimes referred to as the dust-free escape fraction.

In this case we add a layer of uniform dust to the previous geometry. Now all of the flux densities are the dust-attenuated flux densities in case (a):

$$\begin{eqnarray}&&{F}_{\mathrm{LyC},\mathrm{out}}={F}_{\mathrm{LyC},\mathrm{stel}}\times (1-{C}_{F})\times {e}^{-{\tau }_{\mathrm{dust},\mathrm{LyC}}};\\ &&\quad {F}_{1500,\mathrm{out}}={F}_{1500,\mathrm{stel}}\times {e}^{-{\tau }_{\mathrm{dust},\mathrm{LyC}}}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&1-{C}_{F^{\prime} }=\displaystyle \frac{{F}_{\mathrm{line},\mathrm{obs}}}{{F}_{\mathrm{cont},\mathrm{obs}}}=\displaystyle \frac{{F}_{1500,\mathrm{stel}}\times (1-{C}_{F})}{{F}_{1500,\mathrm{stel}}}=1-{C}_{F}.\end{eqnarray} \tag{ 8 }$$

Again, in this geometry we have ${C}_{F}={C}_{F^{\prime} }$, which results in

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{rel}}=\displaystyle \frac{{({F}_{\mathrm{LyC}}/{F}_{1500})}_{\mathrm{out}}}{{({F}_{\mathrm{LyC}}/{F}_{1500})}_{\mathrm{stel}}}=1-{C}_{F^{\prime} }\times {e}^{-({\tau }_{\mathrm{dust},\mathrm{LyC}}-{\tau }_{\mathrm{dust},1500})}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{abs}}=\displaystyle \frac{{F}_{\mathrm{LyC},\mathrm{out}}}{{F}_{\mathrm{LyC},\mathrm{stel}}}=1-{C}_{F^{\prime} }\times {e}^{-{\tau }_{\mathrm{dust},\mathrm{LyC}}}.\end{eqnarray} \tag{ 10 }$$

Here neither ${f}_{\mathrm{esc},\mathrm{rel}}$ nor ${f}_{\mathrm{esc},\mathrm{abs}}$ is equivalent to the $1-{C}_{F^{\prime} }$. We note that this is the geometry adopted by Borthakur et al. (2014).

In this case the dust closely traces the dense gas. We believe that this is the model that most closely resembles reality, as any significant dust will be accompanied by opaque columns of gas (in both the LyC and the interstellar absorption lines; Gnedin et al. 2008). Though the dust has practically no effect on the line or LyC flux (they get absorbed predominantly by the gas), it attenuates the observed continuum flux. So we have

$$\begin{eqnarray}{F}_{\mathrm{LyC},\mathrm{out}} & = & {F}_{\mathrm{LyC},\mathrm{stel}}\times (1-{C}_{F});{F}_{1500,\mathrm{out}}\\ & = & {F}_{1500,\mathrm{stel}}\times (1-{C}_{F}+{C}_{F}\times {e}^{-{\tau }_{\mathrm{dust},1500}})\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}1-{C}_{F^{\prime} } & = & \displaystyle \frac{{F}_{\mathrm{line},\mathrm{obs}}}{{F}_{\mathrm{cont},\mathrm{obs}}}\\ & = & \displaystyle \frac{{F}_{1500,\mathrm{stel}}\times (1-{C}_{F})}{{F}_{1500,\mathrm{stel}}\times (1-{C}_{F}+{C}_{F}\times {e}^{-{\tau }_{\mathrm{dust},1500}})}\\ & \ne & 1-{C}_{F}.\end{eqnarray} \tag{ 12 }$$

Interestingly, in such a scenario the covering fraction based on the depth of the low-ionization absorption lines is actually different from the physical covering fraction, i.e., ${C}_{F}\ne {C}_{F^{\prime} }$. As such, we end up with the following relations for the escape fractions:

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{rel}}=\displaystyle \frac{{({F}_{\mathrm{LyC}}/{F}_{1500})}_{\mathrm{out}}}{{({F}_{\mathrm{LyC}}/{F}_{1500})}_{\mathrm{stel}}}=1-{C}_{F^{\prime} }\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{f}_{\mathrm{esc},\mathrm{abs}}=\displaystyle \frac{{F}_{\mathrm{LyC},\mathrm{out}}}{{F}_{\mathrm{LyC},\mathrm{stel}}}\\ &&\quad =1-{C}_{F}=1-\displaystyle \frac{{C}_{F^{\prime} }}{{C}_{F^{\prime} }+(1-{C}_{F^{\prime} })\times {e}^{-{\tau }_{\mathrm{dust},1500}}}.\end{eqnarray} \tag{ 14 }$$

Therefore, ${f}_{\mathrm{esc},\mathrm{rel}}$ is, in fact, equal to the ratio of the observed flux density in the line to the observed continuum flux density, ${f}_{\mathrm{esc},\mathrm{rel}}=1-{C}_{F^{\prime} }$. Since $1-{C}_{F^{\prime} }$ is a common observable, it is best to compare this measurement to that of ${f}_{\mathrm{esc},\mathrm{rel}}$, as we have done in this work.