Sp1149. II. Spectroscopy of H ii Regions near the Critical Curve of MACS J1149 and Cluster Lens Models

Galaxy-cluster gravitational lenses enable the study of faint galaxies even at large lookback times, and, recently, time-delay constraints on the Hubble constant. There have been few tests, however, of lens model predictions adjacent to the critical curve (≲8″) where the magnification is greatest. In a companion paper, we use the GLAFIC lens model to constrain the Balmer L–σ relation for H ii regions in a galaxy at redshift z = 1.49 strongly lensed by the MACS J1149 galaxy cluster. Here we perform a detailed comparison between the predictions of 10 cluster lens models that employ multiple modeling assumptions with our measurements of 11 magnified, giant H ii regions. We find that that the models predict magnifications an average factor of 6.2 smaller, a ∼2σ tension, than that inferred from the H ii regions under the assumption that they follow the low-redshift L–σ relation. To evaluate the possibility that the lens model magnifications are strongly biased, we next consider the flux ratios among knots in three images of Sp1149, and find that these are consistent with model predictions. Moreover, while the mass-sheet degeneracy could in principle account for a factor of ∼6 discrepancy in magnification, the value of H 0 inferred from SN Refsdal’s time delay would become implausibly small. We conclude that the lens models are not likely to be highly biased, and that instead the H ii regions in Sp1149 are substantially more luminous than the low-redshift Balmer L–σ relation predicts.


Introduction
Gravitational lensing by galaxy clusters provides a uniquely powerful tool for both finding and studying intrinsically faint galaxies and stars that existed when the Universe was a fraction of its present age.The magnifying power of galaxy-cluster lenses amplifies the flux and increases the angular sizes of background galaxies, enabling measurements that would otherwise not be possible (e.g., Swinbank et al. 2009;Wuyts et al. 2014;Wang et al. 2017;Curti et al. 2020;Williams et al. 2023a).Refsdal (1964) showed that, in principle, the time delays between multiple images of the same strongly lensed supernova (SN) could be used to measure the Hubble constant (H 0 ).This method of measuring H 0 using time delays provides an independent means to address the "Hubble tension" between measurements of H 0 from Type Ia supernovae (SNe Ia) in the local Universe (Riess et al. 2022) and from early-Universe observations of the cosmic microwave background (CMB; Planck Collaboration et al. 2020).If the Hubble tension represents a true difference, reconciling the inferred values may require revision to the standard Λ-cold-dark-matter (ΛCDM) cosmological model.
Uncertainties associated with cluster lens models limit our ability to study magnified populations of galaxies, as well as measure H 0 from time delays.Models use systems of multiple images of lensed background sources as constraints, yet also require assumptions about the connection between the distribution of luminous matter and dark matter.For wellconstrained galaxy clusters such as the Hubble Frontier Fields (HFF; Lotz et al. 2017), the latest modeling techniques predict magnifications that are consistent among different models up to a magnification factor of μ ≈ 40-50 (Bouwens et al. 2022).These modeling techniques have been tested on simulated images of galaxy clusters, and most models can reliably predict the magnification due to lensing within an accuracy of 30% for magnifications up to μ ≈ 10 ( Meneghetti et al. 2017).Uncertainties become more pronounced in regions near the critical curve where μ  60, where different lens modeling techniques can predict magnifications ranging from ∼40 to ∼100 at the same positions (Bouwens et al. 2022).Identifying the most accurate lens modeling assumptions would improve our ability to use gravitational lenses as tools to study the galaxies that existed in earlier epochs of the Universe as well as measure H 0 from time-delay cosmography.
SN Refsdal, the first known multiply imaged SN, was discovered in 2014 (Kelly et al. 2015;Treu et al. 2016) in an Einstein-cross configuration around an elliptical galaxy in the HFF (Lotz et al. 2017) galaxy cluster MACS J1149.5+2223(redshift z = 0.544, hereinafter referred to as MACS J1149), and reappeared ∼8″ away in 2015 (Kelly et al. 2016).SN Refsdal provided the first opportunity to use a cluster-scale gravitational lens to infer the value of H 0 .The systematic uncertainties associated with cluster-scale lens models differ from the galaxy-scale models that have been used in previous measurements of H 0 from multiply imaged quasars (Grillo et al. 2020).The time delay between the 2014 and 2015 appearances of SN Refsdal was measured within 1.5% (Kelly et al. 2023).Given a perfect lens model, this would yield an equally precise constraint on H 0 .In the case of SN Refsdal, the greatest contribution to the uncertainty in H 0 is the uncertainty associated with the MACS J1149 cluster mass models (Grillo et al. 2020;Kelly et al. 2023) km s −1 Mpc −1 using the full set of pre-reappearance lens models, and 66.6 3.3 4.1 -+ km s −1 Mpc −1 using the two models that best reproduce the H 0 -independent observables (Kelly et al. 2023).
After the discovery of SN Refsdal, an individual, highly magnified (μ ≈ 600) blue supergiant star was discovered in the same host galaxy as SN Refsdal in the MACS J1149 field (Kelly et al. 2018).Known as "Icarus," the star was the first individual star discovered at a cosmological distance.An image of Icarus is always detectable in sufficiently deep images.The light curve of Icarus can place constraints on the abundance of primordial black holes (Oguri et al. 2018) and the initial mass function of stars responsible for intracluster light, but inferences depend on the ability of lens models of the MACS J1149 cluster to predict the magnification of Icarus.
The MACS J1149 cluster lens has been modeled using several different methods.So-called simply parameterized models have been constructed by Keeton (2010), Sharon & Johnson (2015), Grillo et al. (2016), the GLAFIC team (Oguri 2010;Kawamata et al. 2016), and the Clusters As Telescopes (CATS) team (Jauzac et al. 2016).The simply parameterized method assigns dark-matter halos to individual cluster galaxies and to the cluster, and uses halos with simple, physically motivated profiles each described by a small number of parameters.These models use the positions of multiply imaged galaxies, and assign the mass to each galaxy halo using a proxy such as the stellar mass.Williams & Liesenborgs (2019) instead make no assumptions about the connection between luminous and dark matter, and use only the positions of multiply imaged galaxies as constraints.They apply a "freeform" approach that involves a large number of components that are not associated with any cluster galaxies but instead only uses the strong lensing image positions as input.Bradač et al. (2009) created a free-form model that uses both strong and weak lensing image positions.A hybrid model was created by Diego et al. (2015), which uses a free-form approach to model the overall cluster halo and a parametric approach for the individual cluster members.Zitrin et al. (2013) constructed a "light-traces-mass" (LTM) model, which reconstructs the mass distribution of the cluster by smoothing and rescaling the surface brightnesses of the individual cluster members.Zitrin (2021) also constructed a parametric model using a Navarro-Frenk-White (NFW; Navarro et al. 1996) density profile.We designate the two Zitrin models as "Zitrin-LTM" and "Zitrin-NFW." The strongly lensed host galaxy of SN Refsdal and Icarus presents an opportunity to constrain the cluster lens' magnification at positions near the critical curve of the MACS J1149 cluster, which has not been possible for lens models of clusters.Known as Sp1149, the host galaxy is a triply imaged and highly magnified face-on spiral galaxy at z = 1.49(Smith et al. 2009;Di Teodoro et al. 2018).The three images of this galaxy are some of the largest images ever observed of a spiral at z > 1 (see Figure 1).The high magnification and relative lack of image distortion make it possible to study the spatial structure of the galaxy in detail.In 2011, Yuan et al. (2011) acquired integral field unit (IFU) spectroscopy of the largest image of Sp1149 with the OH-Suppressing Infra-Red Spectrograph (OSIRIS; Larkin et al. 2006) on the Keck II 10 m telescope.The Hα map measured from these observations revealed more than 10 resolved H II regions located less than 10″ from the critical curve (see Figure 2).
The H II regions in Sp1149 should, in principle, allow a direct measurement of the magnification due to the MACS J1149 cluster by utilizing the empirical relationship between Balmer luminosities and velocity dispersions of H II galaxies and giant H II regions (the L-σ relation; Melnick et al. 1988).The L-σ relation for H II galaxies has been shown to be consistent with luminosity distances expected for standard cosmological parameters to z ≈ 4, and observations of H II galaxies have been used to measure H 0 and constrain the dark energy equation of state (e.g., Chávez et al. 2016;Fernández Arenas et al. 2018;Tsiapi et al. 2021).Terlevich et al. 2016 used the L-σ relation to estimate the intrinsic Hβ luminosity of a single compact H II galaxy at z = 3.12 that is gravitationally lensed by the HFF cluster Abell S0163, and inferred a magnification of 23 ± 11, in agreement with the value of ∼17 predicted by a simply parameterized model presented by Caminha et al. (2016).Due to their compact size and intrinsic faintness, there are few constraints on the L-σ relation for giant H II regions at redshifts beyond z ≈ 1.If the L-σ relation measured at low redshift is an accurate description of the giant H II regions in Sp1149 at z ≈ 1.5, the Balmer luminosities of the H II regions can be used as standardizable candles and the magnification due to lensing at their positions can be constrained.Previously, direct measurements of a galaxy cluster's magnification have only been made using SNe Ia at offsets of more than 20″ from the critical curves where the magnifications are 2 (Nordin et al. 2014;Patel et al. 2014;Rodney et al. 2015;Rubin et al. 2018).
In Paper I of this series, Williams et al. (2023b), we used a combination of archival OSIRIS IFU data and newly acquired spectroscopy from the Multi-Object Spectrometer For Infra-Red Exploration (MOSFIRE) to measure the Hα luminosities and intrinsic velocity dispersions of 11 H II regions in Sp1149.After correcting for magnification using the GLAFIC mass model (v3) of MACS J1149 (Oguri 2010;Kawamata et al. 2016), we found that the H II regions in Sp1149 were 6.4 2.0 2.9 -+ times more luminous than expected from the locally calibrated L-σ relation.However, if we instead assume that the L-σ relation for giant H II regions calibrated using low-redshift galaxies accurately describes those in Sp1149, then this result would suggest that the GLAFIC model underpredicts the magnification at the positions of the H II regions by a factor of ∼6.
In this work, we make the assumption that the L-σ relation in Sp1149 is identical to that in low-redshift galaxies, and infer the magnification due to lensing at the positions of 11 H II regions in Sp1149.The H II regions are adjacent to the critical curve of the MACS J1149 cluster (∼2″-8″), where magnifications are expected to reach up to ∼20.We compare our magnification measurements with the predicted magnifications from 10 different lens models of MACS J1149.Magnification depends on the second derivative of the gravitational potential, so these measurements test a different aspect of the cluster models than the relative time delays from SN Refsdal, which depend on the difference in the potential between the multiple appearances.
In Section 2 we describe the OSIRIS and MOSFIRE observations and data reduction.Section 3 details our method for inferring the magnification due to lensing at the position of each H II region.We compare our measurements to the magnifications predicted by the models in Section 4 and discuss the implications of our results in Section 5.

Observations and Data Reduction
Rest-frame optical IFU spectroscopy of the largest image of Sp1149 was obtained by Yuan et al. (2011) with the OSIRIS instrument on the Keck II 10 m telescope (Larkin et al. 2006).The observations were taken using the Hn3 filter (15940-16760 Å; resolution R ≡ λΔλ ≈ 3400), capturing the Hα emission line at the redshift of Sp1149.Adaptive optics provided a corrected spatial resolution of 0 1, corresponding to ∼300 pc in the source plane for a typical magnification of μ = 8.The total exposure time was 4.75 hr.Data-reduction details are described by Yuan et al. (2011).We extracted the one-dimensional spectra of 11 H II regions in Sp1149, inside circular apertures with radius r = 500 pc in the source plane, given the GLAFIC model magnification predictions.Figure 2 shows the Hα emission-line intensity map of Sp1149 and the H II region extraction apertures.
To measure the Balmer decrement and infer the extinction due to dust at the positions of the H II regions, we acquired multislit spectroscopy using MOSFIRE on the Keck II 10 m telescope (McLean et al. 2012) over two half nights, 2020 February 25 and 26 UTC.We observed each H II region in the J and H bands to detect the Hβ and Hα emission lines, respectively, and total exposure times ranged from 8 to 36 minutes on six different slit masks.The spectra were reduced with the MOSFIRE Data Reduction Pipeline (Konidaris et al. 2019).We used observations of telluric standard stars to correct for telluric absorption and acquired spectra of field stars on each slit mask to measure the absolute flux calibration for each mask.See Paper I for a detailed description of the MOSFIRE observations and data reduction.

Magnification Measurements
To infer magnification values using the Balmer L-σ relation, we require a calibration of the L-σ relation that uses aperture sizes and spectral resolution comparable to those of our OSIRIS measurements of the Hα luminosities and velocity dispersions of the H II regions in Sp1149.Using archival IFU spectroscopy of nine nearby spiral galaxies taken with the Multi Unit Spectroscopic Explorer (MUSE; Bacon et al. 2010) on the Very Large Telescope, we extract the spectra of 347 H II regions at z ≈ 0 using the same physical aperture sizes, given the GLAFIC model predictions, that we used to extract the H II regions in Sp1149 from the OSIRIS data.
We employ Markov Chain Monte Carlo (MCMC) sampling with the pymc3 package (Salvatier et al. 2016) to measure the intrinsic velocity dispersion and Hα luminosity of each H II region in the local sample from the Hα and Hβ emission lines.From the posteriors for all of the H II regions, we apply a hierarchical linear mixture model implemented by the linmix where L Hα is in units of erg s −1 and σ is in units of km s −1 .The rms intrinsic scatter is 0.34 dex.Additional descriptions of our calibration of the relation at low redshift may be found in Paper I.
In Paper I, we presented measurements of the intrinsic Hα luminosity and velocity dispersion of 11 H II regions in Sp1149.We used our MOSFIRE observations of the Hβ and Hα emission lines for each H II region to constrain the extinction due to dust, and the OSIRIS observations of the Hα emission lines to infer their intrinsic velocity dispersions and, in combination with our constraints on extinction, intrinsic Hα luminosities.
To measure the magnification at the position of each H II region, we use the posteriors from Paper I to compute the observed (magnified) Hα luminosity, L obs (Hα).We compute the expected Hα luminosity, L H exp ( ) a of each H II region using the posteriors for the velocity dispersion from Paper I and applying our calibration for the local L-σ relation (Equation ( 1)).The posteriors on the slope and intercept of the low-redshift relation are used to propagate the uncertainties Figure 4.The statistical tension between our measurements of the magnification compared with each models prediction for the magnification at the position of each H II region.A positive value for the tension indicates that our measured magnification is greater than the model's prediction.The black lines indicate the median tension for each model.We find that all 10 models underpredict the magnification compared to the measurements by a median value among the H II regions of 1.8σ-2.5σ.
Figure 5.The factor by which each model underpredicts the magnification of the H II regions in Sp1149.The average underprediction factor for all the models is 6.2.associated with the calibration.The observed magnification, μ obs , at the position of each H II region is given by We list our measurements of the magnifications at the positions of 11 H II regions in Sp1149 in Table 1, and show these as a function of distance from the critical curve of MACS J1149 at z = 1.49 in Figure 3.

Comparison with the Models
We next compare our observed magnifications of the 11 H II regions in Sp1149 with the magnifications predicted by 10 different models of the MACS J1149 cluster.Each lens modeling team provided ∼100 model magnification maps corresponding to different MCMC realizations. 7We use the median value at the position of each H II region as the predicted magnification, adopting the 16th and 84th percentile values to compute the 1σ uncertainties.The model magnification values are listed in Table 1, and the model predictions together with our constraints are plotted in Figure 3.
To evaluate the level of agreement between our measurements and the model predictions, we calculate the tension between each measurement and prediction.We take the difference between the measurement and the prediction, and divide by the 1σ uncertainty in the difference.A positive tension indicates that the value we measure is greater than the model's prediction.
In Figure 4, we plot the tension for each model and H II region, as well as the median tension for each model.Almost all of our magnification measurements are ∼1σ-3σ greater than the models' predictions.The median statistical tension among the set of 11 H II regions for each model is 1.8σ-2.6σ.
We next compute the average factor by which each model underpredicts the magnification compared to our measurements by computing μ obs /μ model for each H II region and assigning a weight to each measurement based on the combined 1σ uncertainties of the observed magnification and the model prediction.As Figure 5 shows, the 10 available models of MACS J1149 underpredict the magnification by a factor of ∼5-8.The average underprediction factor among the models is 6.2, and the median factor among the models is 6.6.

Discussion
Using the Balmer L-σ relation, we measured the magnification due to gravitational lensing by the MACS J1149 cluster of 11 giant H II regions in the spiral galaxy Sp1149 (z = 1.49).We have compared our measurements to the magnifications predicted by 10 different cluster mass models and found that all of the models predict magnifications that are smaller than our inferred values at the positions of the H II regions by factors of ∼5-8.The tension between our measurements and the model predictions is 1.8σ-2.6σ.
The models that are in the least tension with our measurements (∼1.8σ) are the simply parameterized Sharon model and the Zitrin-LTM model, which underpredict the magnification in comparison to our constraints by an average factor of 4.1 ± 1.4 and 4.8 ± 1.7, respectively.The Williams free-form model underpredicts the magnification by the largest average factor among the models, with 〈μ obs /μ model 〉 = 8.3 ± 2.9.
Here we have calculated the magnification under the assumption that the L-σ relation for giant H II regions that we have calibrated at low redshift in matching apertures and spectral resolution applies to the H II regions in SN Refsdal's host galaxy at z = 1.49.Previous studies have shown that the L-σ relation for H II galaxies does not evolve strongly with redshift at least z ≈ 4, and it has been used to calculate H 0 and the dark energy equation of state (e.g., Chávez et al. 2016;Fernández Arenas et al. 2018;Tsiapi et al. 2021).If we instead assume that the magnifications predicted by the models are accurate, then our results indicate that H II regions are substantially more luminous in Sp1149 than predicted by the low-redshift L-σ relation (3σ tension), and would imply a physical difference between the two populations of H II regions (see Paper I).
If our results correspond to a systematic bias of the lens models of the MACS J1149 cluster, the magnification Figure 6.The flux ratios of five bright knots in Image 1.1 and Image 1.3 of Sp1149, measured from the F606W HST imaging of the MACS J1149 cluster field.We compare these flux ratios to the predicted magnification ratios at their positions and find that the measured ratios agree with the predictions from the models within the 1σ uncertainties.corrections applied to background galaxies that are lensed by MACS J1149 would cause us to overestimate their intrinsic luminosities and emission-line fluxes.Physical properties of these galaxies, such as star formation rate and stellar mass, would also be overestimated.The value of H 0 inferred from time-delay measurements is sensitive to the details of cluster models, so a systematic bias in the lens models used to infer H 0 from SN Refsdal could affect the interpretation of that measurement.For instance, increasing the magnification by a factor of 6 would require a mass sheet of κ = 0.6 under the mass-sheet degeneracy (μ ∝ (1 − κ) −2 ; Oguri & Kawano 2003), which would reduce the derived value of H 0 by a factor of 1 − κ = 0.4.The value of H 0 inferred from the time delays of SN Refsdal is 64.8 4.3   4.4 -+ km s −1 Mpc −1 , so this interpretation would imply an implausible H 0 ≈ 26 km s −1 Mpc −1 .
To test the likelihood of the magnification of Image 1.1 of Sp1149 being a factor of ∼3 times higher than the models predict, we compare the observed flux ratios of bright knots in Image 1.1 and Image 1.3 to the model-predicted magnification ratios at their positions.As shown in Figure 1, Image 1.3 is substantially farther from the critical curve than mirrored Images 1.1 and 1.2.We identify five bright knots in Sp1149 and measure their flux densities in both Image 1.1 and Image 1.3 from the Hubble Space Telescope (HST) F606W image of MACS J1149.Background subtraction is performed by measuring the background in five positions around each image.We find that, for all five knots, the observed flux ratios agree with the predicted magnification ratios from a majority of the models (see Figure 6).The reported uncertainties in the knots' photometry include the uncertainty in the relative aperture sizes between Image 1.1 and Image 1.3 as determined by the relative model-predicted magnifications.The photometric uncertainty is background-dominated. Figure 7 shows the bright knots in HST F606W close-up images of Sp1149.
In other galaxy-cluster fields, the magnifications of SNe Ia have been measured at similar offsets from the critical curve of ∼20″, and obtained approximate agreement (Nordin et al. 2014;Patel et al. 2014;Rodney et al. 2015;Rubin et al. 2018).Additionally, a factor of 6 bias in the predicted magnifications in Image 1.1 would imply an improbably small value of H 0 ≈ 26 km s −1 Mpc −1 .Consequently, we conclude that the discrepancy we identify between the predicted and measured Hα luminosities of the giant H II regions in Sp1149 is most likely not due to a systematic problem in the lens models.Instead, the Balmer L-σ relation of the H II regions in Sp1149 is likely offset to higher luminosities by a factor of ∼6.
As an additional test of excess luminosity of Sp1149, we compare the galaxy's rotation velocity and inferred stellar mass km s −1 (see also Chen et al. 2018).We compare these measurements to the local TFR as calibrated by 729 star-forming galaxies in the SydneyAAO Multi-object Integral field spectrograph (SAMI) galaxy survey (Bloom et al. 2017).We find that the stellar mass of Sp1149 is 0.93 ± 0.31 dex (a factor of 8.5) higher than predicted by the local TFR for the rotation velocity derived by Livermore et al. (2015).Using the rotation velocity from Di Teodoro et al. (2018), the stellar mass of Sp1149 is consistent with the prediction from the local TFR within the 1σ uncertainties (see Figure 8).
If the rotation velocity derived by Livermore et al. (2015) is correct, the factor of 8.5 positive offset from the TFR would be similar to the factor of 6.2 positive offset from the L-σ relation.This may suggest that a bias in the magnification predictions is responsible for both offsets.However, we note that the stellar mass of Sp1149 was derived from photometry of Image 1.3, while the H II region luminosities were measured from Image 1.1.Image 1.3 is substantially farther from the critical curve than Image 1.1, so it is unlikely that both images would be The Balmer L-σ relation is a well-established empirical correlation, but the physical origin of this relation is not yet understood.While the L-σ relation is constant with redshift for H II galaxies out to at least z ≈ 4, our results suggest that it may not apply for individual H II regions at z  1. Observations of the luminosities and velocity dispersions of H II regions in other magnified galaxies are needed to confirm this conclusion.

Figure 1 .
Figure 1.False-color image of a portion of the MACS J1149 lensing cluster from HST imaging of the field.The three images of the face-on spiral galaxy Sp1149 are labeled, and the critical curve of the cluster (GLAFIC model) is shown as a white line.The appearances of SN Refsdal in an Einstein-cross configuration are labeled S1-S4.Images 1.1 and 1.3 are shown in the right panels, with the H II regions identified in white.

Figure 2 .
Figure 2. Emission-line intensity map of Hα in Image 1.1 of Sp1149, with the 11 H II regions we use to infer the magnification labeled in white.Labels are the same as Figure 1.This map was created from archival OSIRIS IFU observations of Sp1149 (PI: Kewley).

Figure 3 .
Figure 3. Magnification measurements for each of the observed H II regions (large black points) overlaid on each of the models' predictions for the magnification at their positions (colored points).Error bars correspond to 1σ uncertainties.
to the local Tully-Fisher relation (TFR; Tully & Fisher 1977).Using HST photometry of Image 1.3 of Sp1149 and correcting for magnification using the GLAFIC model, Wang et al. (2017) estimated a stellar mass of M disagreement in the literature regarding the rotation velocity of Sp1149.Using the same OSIRIS data as this work, Livermore et al. (2015) found a rotation velocity of V rot = 59 ± 3 km s −1 (see also the rotation curve published by Wang et al. 2017).Alternatively, Di Teodoro et al. (2018) used MUSE observations of Image 1.3 of Sp1149 and found a substantially higher rotation velocity,

Figure 7 .
Figure 7. HST F606W close-up images of Image 1.1 and Image 1.3 of Sp1149, with five bright knots identified in each image.We measure the flux density of each knot in both images and compare their flux ratios to the magnification ratios predicted by each model.The white regions show the apertures used to measure the flux of each knot.

Figure 8 .
Figure 8.The rotation velocity and inferred stellar mass of Sp1149 compared to the Tully-Fisher relation for local galaxies.The green square shows the rotation velocity of Sp1149 derived by Livermore et al. (2015) from OSIRIS data, and the blue square indicates the rotation velocity of Sp1149 derived by Di Teodoro et al. (2018) from MUSE data.Purple points represent local galaxies from the SAMI survey (Bloom et al. 2017), and the black dashed line shows the best-fit TFR for the local galaxies.The shaded yellow region indicates the 1σ scatter of the local TFR.
. The value of H 0 derived from the time delays of SN Refsdal is H

Table 1
Magnification Measurements a Model-predicted magnifications at the position of each H II region, and our measured magnifications.The ratio μ obs / mod m is the weighted average of the ratios of the measured to model-predicted magnifications for the 11 H II regions.Reported uncertainties are 1σ.These measurements are shown in Figure3. a