The Cosmic Telescope That Lenses the Sunburst Arc, PSZ1 G311.65–18.48: Strong Gravitational Lensing Model and Source Plane Analysis^*

PSZ1 G311.65−18.48 was the target of several HST programs in Cycle 25. GO-15101 (PI: Dahle) obtained multiband imaging and grism spectroscopy of PSZ1 G311.65−18.48 with the goals of enabling a robust lensing analysis and investigating the physical conditions of the lensed galaxy. That program used five orbits of broadband imaging in the F555W, F814W, F105W, and F140W filters; four orbits of F410M medium-band imaging; and five orbits of WFC3 G141 grism spectroscopy. A second program, GO-15377 (PI: Bayliss), complemented these data with four orbits of HST imaging in F606W, F098M, F125W, and F160W as part of a Chandra Cycle-19 program, to determine whether the source galaxy hosts an active galactic nucleus (AGN) and to observe the diffuse X-ray gas of the lensing cluster. Third, the Cycle 25 midcycle program GO-15418 (PI: Dahle) used three HST orbits to probe the LyC of the lensed source in F275W. Some of the visits failed owing to gyroscope problems and were repeated. Table 1 tabulates the successful and failed HST Cycle 25 observations from the programs listed above.

The target was also observed by two programs in HST Cycle 27. A two-orbit integration in F390W was executed in 2020 July 13 by GO-15949 (PI: Gladders) and is used in this work; the remaining data from Cycle 27 will be presented and analyzed in forthcoming publications.

The HST imaging data were reduced following standard procedures, after inspecting each image for quality assurance, given the higher-than-usual failure rate due to the gyroscope problems in Cycles 25–26. We used the Drizzlepac ¹⁰ software package to reduce the data and align the frames to a common reference grid, as follows. First, exposures in each filter that were taken within a single visit were drizzled using the astrodrizzle routine using a Gaussian kernel with a drop size (final_pixfrac) of 0.8. Next, for each filter in which observations were executed over multiple visits, the drizzled images from each visit were aligned to a common world coordinate system (WCS) using the tweakreg routine. These WCS solutions were propagated back to the individual exposures using tweakback before all exposures in a single filter were drizzled together using astrodrizzle, with the same parameters listed above. Finally, the drizzled images were aligned in WCS space, again using tweakreg, and drizzled using astrodrizzle with the same kernel and drop size onto a common reference grid with north up and a pixel scale of 003 pixel^–1.

WFC3 IR G141 grism observations from GO-15101 were executed with two telescope roll angles, ORIENT=2737 and 35537. The data were reduced using the reduction package Grizli, ¹¹ using standard reduction procedures. We used these data to search for or attempt to confirm candidate lensed galaxies in the grism spectra.

The field containing the Sunburst Arc was observed with the Chandra X-ray Observatory under observation ID 20442. The purpose of this observation was to constrain any bright X-ray emission from the lensed galaxy, while also producing a robust detection of the foreground cluster lens. The observation was executed as a single 39.53 ks exposure with the aim point located near the center of the I3 chip in the ACIS-I array. To minimize background, the observation was performed in VFAINT telemetry mode. We reduced the Chandra data using Chandra Interactive Analysis of Observations (CIAO v4.13) with CALDB v4.9.6 to apply routine processing. The data were filtered for flares using the lc_sigma_clip function in the lightcurves Python package that is included in CIAO, resulting in a usable integration time of 38.53 ks. We apply a 0.5–7 keV energy filter to the reduced event file and correct for a small (∼1'') astrometric offset between the Chandra and HST data by comparing the coordinates of a galaxy that appears both in the HST field of view and as an X-ray point source in the Chandra data. We then use the 0.5–7 keV Chandra image to measure the basic X-ray properties of the foreground cluster lens. The X-ray peak (brightest pixel) is at R.A. 15:50:07.053, decl. −78:11:29.165, and the X-ray centroid is at R.A. 15:50:06.82, decl. −78:11:29.921, both within 1'' of the BCG in projection.

The total observed 0.5 − 7 keV X-ray flux is 2.54 × 10⁻¹² erg cm⁻² s⁻¹. Excising the central r ∼ 150 kpc to estimate a core-excised flux results in f_0.5−7keV = 1.06 × 10⁻¹² erg cm⁻² s⁻¹. The ratio of those two values (core-excised to total fluxes) makes it clear that the cluster has an extremely strong cool core. For example, a comparison to other Planck clusters analyzed by Mantz et al. (2018) places this cluster on the extreme end in terms of f_ce/f_total, meaning that its X-ray emission is among the most core dominated. This is consistent with what is observed in the optical and IR; the BCG appears dust obscured, with evident filamentary star formation activity in both imaging and spectroscopy, similar to what is seen in other extremely strong cool cores (e.g., the Phoenix Cluster; McDonald et al. 2012, 2019).

PSZ1 G311.65−18.48 was observed with MUSE on the VLT on 2016 May 13 (ESO program 297.A-2012(A); PI Aghanim). The data were retrieved from the ESO archive; the data reduction process is detailed in Weilbacher et al. (2020) and Urrutia et al. (2019). The field was observed for 2966 s under good conditions, with 07 seeing, spanning 4750−9300 Å with R ≃ 2100. The MUSE cube results in a spectrum for each resolution unit, which is a powerful way to derive redshifts for numerous sources in the field of view, since there is no need to decide in advance which sources to target. We use the MUSE data to confirm counterimages in lensed systems and measure their redshifts.

The field was observed with Magellan Echellette on the Magellan-I 6.5 m Baade telescope, covering multiple distinct image plane regions in the north and northwest arcs. The details of the observation and data reduction are given in Rivera-Thorsen et al. (2017, 2019) and Lopez et al. (2020), and most comprehensively in J. Rigby et al. (2022, in preparation). These data were used to confirm the identifications of some of the clumps in the sunburst arc.

The most prominent lensing feature in the field of PSZ1 G311.65−18.48 (Figure 1) is the highly magnified galaxy discovered by Dahle et al. (2016). In the discovery paper, based on ground-based R- and z-band imaging data from the ESO New Technology Telescope (NTT), Dahle et al. (2016) reported that the source is lensed into three giant arcs, which appear north, northwest, and west of the BCG (S1, S2, S3 in their notation), with a possible fourth image to the southeast. They measure a spectroscopic redshift of z = 2.3686 ± 0.0006 from nebular emission lines and z = 2.3708 ± 0.0004 from the Lyα emission line. Rivera-Thorsen et al. (2017) determined the systemic redshift of the system as z = 2.37094 ± 0.00001 from nebular line emission in the FIRE spectra. A systemic redshift of z = 2.37034 ± 0.00024 was obtained from an average of the velocities of the narrow components of the [O iii] λλ4959, 5007 line in several FIRE pointings (Mainali et al. 2022). Differences of this order are within expectation owing to velocity differences between the galaxy components, e.g., winds and relative motions, and insignificant for the purpose of lens modeling. In this work we adopt z = 2.3703 as the systemic redshift.

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High-resolution HST imaging data are invaluable in understanding the lensing configuration. Studying the first F814W and F275W observations, obtained in 2018 February and April, respectively, Rivera-Thorsen et al. (2019) found that the north and northwest arcs are each composed of multiple images, and in particular, the bright emission clump in the source is lensed into a dozen images: six in the north arc, four in the northwest arc, and one each in the west and southeast arcs. With the multiband HST imaging available to date, we are able to identify numerous emission knots in each image and match these emission knots between images. Table 2 and Figure 2 list the positions of the emission knots and label their mapping between the lensed images. The identified emission knots within the Sunburst Arc are labeled with prefixes 1 through 19. ID 1.xx is assigned to the bright LyC-emitting knots identified by Rivera-Thorsen et al. (2019). The suffix xx denotes the ID of the lensed image within the multiple-image family. For consistency, we follow the multiplicity ID numbers 1−12 that were introduced by Rivera-Thorsen et al. (2019).

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In Figure 2 we show the four images of the Sunburst Arcs: north, northwest, west, and southeast, in color rendition from HST F140W, F606W, and F410M. These bands are selected for this color rendition in order to display the clump-to-clump color variations. For a source at z = 2.3703, the broadband filters F140W and F606W sample emission redward of the 4000 Å break and rest-frame UV, respectively, while the medium-band filter F410M is centered on the Lyα emission. The scale and stretch in this figure are tuned to bring out the clumpiness of the arc and visually resolve its numerous emission knots. We label and color-code the emission knots in the panel adjacent to each color rendition. Knot 1.x (i.e., the one emitting LyC radiation, hereafter the LyC emitter (LCE) clump) is labeled with a red square in all the arcs. We label the next three brightest knots with 2.x, 3.x, 4.x, and the remaining knots are labeled and color coded to guide the eye to their projected location and arc-to-arc symmetry rather than by brightness.

The north arc (arc 1) consists of six partial images of the source (labeled x.1 through x.6 in Table 2 and Figure 2). In two of the images, x.5 and x.6, only the bright LyC-emitting knot is seen. The other four include several more emission knots in addition to the bright knot: knots 2, 5, 6, 7, and 8 in images x.1 and x.2; and knots 2–12 in images x.3 and x.4. The unusually high multiplicity within the north arc is due to contributions to the lensing potential from three cluster-member galaxies, which complicate the shape of the critical curve in this region.

The west arc (arc 3) and the southeast arc (arc 4) are both full images of the source. As such, these are the only instances where the properties of the galaxy as a whole can be measured (J. K. Kim et al. 2022, in preparation). Only in these two images do we observe a likely companion galaxy, labeled 50 in Table 3 (see also the next section). The knots associated with it, 51.x and 52.x, as well as knots 18.x and 19.x, are not observed in the north and northwest arcs owing to the lensing geometry. Because these images have a lower magnification, some of the knots that are identified in other arcs are either not resolved or too faint to be uniquely matched in these arcs. We map knots 16.x and 17.x between these two arcs, but note that they could be matched to knots 11.x and/or 12.x in the north arc (see also Section 5.4).

Table 3. List of Lensing Constraints from Secondary Lensed Galaxies

ID	R.A. (deg)	Decl. (deg)	zspec	zmodel	Notes
	J2000	J2000
21.1	237.507879	−78.195669	⋯	${2.00}_{-0.03}^{+0.03}$
21.2	237.563285	−78.184445
22.1	237.507377	−78.195596
22.2	237.563645	−78.184605
31.1	237.501071	−78.194245	2.4600	⋯
31.2	237.572190	−78.186764
32.1	237.501173	−78.194403
32.2	237.570735	−78.186045
41.1	237.508731	−78.189852	1.1860	⋯
41.2	237.558868	−78.193976
51.11	237.495214	−78.192641	2.3709	⋯	Companion galaxy to the Sunburst Arc
51.12	237.567111	−78.194878
52.11	237.495142	−78.192566
52.12	237.566914	−78.194972
53.1	237.570080	−78.193511
53.2	237.572192	−78.191494
53.3	237.572516	−78.191086
c61.1	237.538392	−78.190088	⋯	⋯	Radial arc, candidate
c61.2	237.545362	−78.188855
c61.3	237.493848	−78.194733
c61.3	237.485901	−78.196788
71.1	237.544744	−78.190197	⋯	${2.20}_{-0.06}^{+0.04}$	Radial arc
71.2	237.541128	−78.190545
72.1	237.545902	−78.189977
72.2	237.539452	−78.190676
72.3	237.480527	−78.192377
81.1	237.571984	−78.194627	⋯	${2.14}_{-0.01}^{+0.05}$
81.2	237.501340	−78.190593
82.1	237.571210	−78.195048
82.2	237.501364	−78.190318
83.1	237.571123	−78.195091
83.2	237.501366	−78.190278
91.1	237.496776	−78.193334	3.5100	⋯
91.2	237.578283	−78.188071
92.1	237.496803	−78.193415
92.2	237.578121	−78.187978
93.1	237.496902	−78.193517
93.2	237.577906	−78.187816
94.1	237.497320	−78.193754
94.2	237.577150	−78.187272
95.1	237.498103	−78.194186
95.2	237.573889	−78.185414
96.1	237.497953	−78.194128
96.2	237.574173	−78.185593
97.1	237.498285	−78.194236
97.2	237.573730	−78.185329
131.1	237.553434	−78.191877	⋯	${1.52}_{-0.02}^{+0.02}$
131.2	237.493765	−78.187681
132.2	237.492902	−78.187827
132.1	237.552558	−78.191747
141.1	237.537716	−78.196427	⋯	${1.76}_{-0.04}^{+0.02}$
141.2	237.510804	−78.182610
151.1	237.533169	−78.197672	⋯	${3.27}_{-0.14}^{+0.14}$
151.2	237.518525	−78.179783
161.1	237.531076	−78.197533	⋯	${2.48}_{-0.10}^{+0.03}$
161.2	237.517617	−78.180774
162.1	237.530599	−78.197467
162.2	237.518346	−78.180675
c170.1	237.533765	−78.182837	⋯	⋯	Candidate
c170.2	237.553368	−78.198716
c170.3	237.523858	−78.200091
181.1	237.556363	−78.180047	⋯	${3.15}_{-0.08}^{+0.19}$	Low confidence zspec 2.5820
181.2	237.518409	−78.196005
191.1	237.547077	−78.180167	⋯	${2.08}_{-0.01}^{+0.12}$
191.2	237.524546	−78.194008

Note. The IDs, positions, and redshifts of lensed multiply imaged galaxies other than the Sunburst Arc that were used as constraints in this work. Where possible, individual emission knots in each image are identified and used as lensing constraints. The IDs of images of lensed galaxies are labeled as AB.X or AAB.X, where A or AA is a number indicating the source ID (or system name), B is a number indicating the ID of the emission knot within the system, and X is a number indicating the ID of the lensed image within the multiple-image family. A prefix c identifies candidates. The reference for spectroscopic redshifts is this work; see Section 3. The redshifts of sources 40 and 90 are consistent with the independent measurements of Pignataro et al. (2021), using the same data, of 1.186 and 3.505, respectively.

Download table as:  ASCIITypeset images: 1 2

The northwest arc (arc 2) is the least understood from the perspective of lensing geometry. It has at least two and up to four partial images of the source. Rivera-Thorsen et al. (2019) compared the MagE spectra of several spatially distinct regions along the arc, including images 1.1, 1.4+1.5, 1.8, and 1.10 (images 1, 4+5, 8, and 10 in their notation, respectively). They found that the spectral features of these images are indistinguishable, arguing in favor of these being images of the same source, which implies that the LCE knot is observed with a multiplicity of 4× in this arc. While in the north arc the 6× multiplicity of the LCE knot can be readily explained by the complexity of the lensing potential owing to nearby cluster-member galaxies, no such perturbers are observed near the west end of the northwest arc.

Indeed, near images 1.9 and 1.10, the lensing geometry is not trivial. One explanation for the apparent multiplicity in this region, as suggested by Diego et al. (2022), is that the critical curve serpentines around image 1.9; this requires a lensing potential that is more complex than can be explained from the observed galaxy distribution, e.g., due to a very low surface brightness or a yet-unobserved mass component. They forced a high multiplicity in this region by constraining the lens model with critical points between images 1.8, 1.9, and 1.10 (h, i, and l in their notation). The model presented in this paper does not reproduce the multiplicity of images 1.9 and 1.10 because it lacks the flexibility on small scales that is needed in order to wind the critical curve around image 1.9 using only lens components with an observed counterpart. Our early attempts to force the critical curve through those positions (by using critical curve constraints, similar in principle to what was recently done by Diego et al. 2022) resulted in unrealistic magnification ratios between the lensed images. We ruled out those models based on the unrealistic magnifications and abandoned this direction. We will explore the possibility that a low surface brightness or a "dark" clump is responsible for the lensing complexity in this region in future work. Despite its limitation in the southern part of the northwest arc, our lens model correctly recovers the numerous lensing constraints from multiple source planes (Section 3) elsewhere; the global lens properties and measurements are not affected.

An alternative explanation of the observed images 1.9 and 1.10 is that the LCE clump itself is composed of multiple smaller clumps, unresolved in the other instances but resolved in this image owing to its high tangential distortion. Additional data, such as the approved JWST NIRSpec Integral Field Unit (IFU) spectroscopy of this part of the arc (JWST-GO program 02555; PI: Rivera-Thorsen), will provide more clues. The IFU spectroscopy of images 1.9 and 1.10 could reaffirm that they are indeed identical to the other images of the LCE, thus confirming its 4× multiplicity within this arc and requiring a more complex lens model with sufficient flexibility to wind a critical curve around image 1.9, as was attempted by Diego et al. (2022). Deep imaging may reveal a faint interloper that complicates the lensing potential, or put limits on the surface brightness of such a component if undetected. The first public JWST data already showcase its power to detect low surface brightness structures (e.g., the intracluster light; Mahler et al. 2022; Montes & Trujillo 2022). Alternatively, the IFU data may reveal that one or two of these clumps differ from the main LCE source, indicating that the LCE clump is actually composed of several distinct LyC leaking regions.

A further anomaly observed in the northwest arc is a bright, unresolved emission clump that displays significantly different colors and spectral features, compared to other clumps of similar brightness. This emission clump can be seen in Figure 2, as the bright clump in the left part of the 4.8 ellipse. Because a counterimage with similar brightness does not appear in any of the other giant arcs, in particular those that display full images of the source, Vanzella et al. (2020b) argued that this unresolved emission clump (labeled Tr in Vanzella et al. 2020a) is a transient. They further identified Bowen fluorescence lines (Bowen 1934) in its spectrum. Diego et al. (2022) examined the possibility that the observed brightness and morphology of this candidate (which they nicknamed "Godzilla" in their paper) is due to an unobserved lensing perturber in close proximity. They found that such a perturber requires mass of order ∼10⁸ M_⊙. In Section 5.3.1, we discuss the discrepant emission clump in more detail and demonstrate that it is unlikely to be a transient.

From the multiband HST imaging and archival MUSE spectroscopy, we identify multiple images of several other galaxies that are also strongly lensed by PSZ1 G311.65−18.48. Their positions and, where available, spectroscopic redshifts are used as constraints in the lensing analysis. To avoid confusion with the IDs of clumps within the Sunburst Arc, we assign these secondary lensed galaxies ID numbers starting at 20. The tens digit denotes the source ID, and the ones digit denotes the label of each emission knot. The digit to the right of the decimal point labels each image within the multiple-image "family" of the same source. For example, images 21.1 and 21.2 are two lensed images of clump number 1 in source number 20. Candidate features have an additional prefix c. Figure 1 shows the locations of the images of the secondary systems. The positions of individual emission knots and the available redshifts are tabulated in Table 3. In Figure 3 we present a close-up view of the identified multiple images of each system. We note that for each of the tangential systems the model predicts a demagnified (0 < ∣μ∣ < 1) counterimage behind the BCG. These highly demagnified images are predicted to be several magnitudes fainter than the observed arcs, and as such they are not expected to be detectable in the current data. As noted in Sections 1 and 2.2, the BCG area is active with star formation, which is expected for a cool-core cluster such as PSZ1 G311.65−18.48. Since star formation chains near the BCG can mimic the appearance of radial arcs (e.g., Sharon et al. 2014; Tremblay et al. 2014; McDonald et al. 2019), we treat the central region with extra care in order not to misidentify star formation as lensed features.

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Source 101.—Likely associated with the Sunburst Arc, source 101 (Figure 2) appears as a single emission knot projected less than half an arcsecond from the main arc, but with only five multiple images: two by the north arc, one by the northwest arc, one by the west arc, and one by the southeast arc. A sixth image is predicted by the northern tip of the northwest arc, and we tentatively identify it blended in the light of a foreground galaxy. The sixth image candidate is not used as a constraint. Due to its proximity to the main arc and the lack of independent redshift measurement, we cannot spectroscopically rule out its association with the main arc. When leaving its redshift as a free parameter, the model-predicted redshift for this source converges to that of the main arc. Furthermore, narrowband HST imaging reveals strong line emission from all the images of Source 101 at a wavelength consistent with [O iii] λ4959 at the redshift of the Sunburst Arc (J. Rigby et al. 2022, in preparation). We therefore proceed by assigning this clump the same redshift as the Sunburst Arc.

Source 20.—This source appears with two images, one southwest and one northeast of the BCG. Although we do not have a spectroscopic redshift of this galaxy, the morphology and color variations along the images provide confidence in their identification as multiple images of the same source.

Source 30.—We detect two images of source 30, with similar morphology and color. A single emission line appears in both locations in the MUSE data, placing this source at z_spec = 2.460, assuming that this line is from C iii]. The secure lensing identification and high-confidence redshifts of other sources rule out other possible emission lines, which would result in too high or too low redshifts (e.g., Lyα, [O ii], [O iii]). The east image of this source (30.2) and image 2 of source 90 appear in a region with higher tangential shear than the west images of the same sources (Figure 1).

Source 40.—Source 40 is lensed into two images, projected 17'' and 22'' from the cluster center. Archival MUSE spectroscopy indicates that it is a star-forming galaxy at z_spec = 1.1484, based on several emission lines that are observed in both of its images, 40.1 and 40.2.

Source 50.—We identify two images of a galaxy, one extending the west Sunburst Arc to the south, and the other extending the southeast Sunburst Arc to the east. Images of this source do not appear near any of the other Sunburst Arc images because of the location of the source with respect to the caustics in the source plane; this is consistent with expectation from the lensing geometry. Source 50 is smoother and redder than the main arc, with a distinctive color gradient. We identify two clumps within the images to be used as constraints.

MUSE spectroscopy of images 50.11 and 50.12 places this source at the same redshift as the Sunburst Arc, z_spec = 2.3709, based on weak C iii] emission lines and Si ii and C iv in absorption in both images of the source. We note that the spectrum of 50.11 contains features from an interloping galaxy at z_spec = 0.7373, based on a bright double emission feature at 6477.0 and 6482.1 Å from the [O ii] λ3737 doublet, corroborated by other lines including Hβ emission and [O iii] λ5008 emission. We identify the interloper as a faint galaxy in close projected proximity, 094, northwest of 51.11. The identified interloper has the same spectroscopic features that contaminate the MUSE spectrum of 50.11.

Our redshift analysis of source 50 disagrees with the redshift reported for this galaxy by Pignataro et al. (2021), who measured z = 2.393 (Sys-3 in their paper). While their paper does not provide information regarding the spectroscopic features that were used to establish this redshift, it could possibly be based on a misidentification of the [O ii] line from the z = 0.7373 interloper, as a C iii] line at z = 2.393. Source 50 is close in projection to the Sunburst Arc (∼6 kpc away in projected distance in the source plane according to our lensing analysis; Section 5.4), which could explain the absorption lines had source 50 been in the background. Nevertheless, the identification of the C iii] emission line adds confidence to our z_spec = 2.3709 interpretation.

A close inspection of the imaging data reveals a thin elongated arc that extends 50.12 toward the northeast. This emission is likely the magnified image of just the edge of source 50, which is lensed to this location by contribution to the lensing potential from a nearby cluster-member galaxy. The three images that make this extension are labeled 53.1, 53.2, and 53.3.

Source 60 (candidate).—We identify this image as a candidate radial arc. At this time, we have no spectroscopic redshift for it. There are several possible counterimages at the expected locations that match its surface brightness, colors, and expected lensing shear, but none can be spectroscopically confirmed as counterimages. We therefore do not include this system as a constraint in the lens model.

Source 70.—This source appears as a radial arc, observed east of the cluster core. We identify a candidate counterimage of this arc, west of the west Sunburst Arc. The candidate has similar colors to the radial arc. The direction in which the arcs are sheared is consistent with expectation from lensing geometry. The radial arc appears in the MUSE data with low significance, and a redshift could not be secured from these data. The location of the counterimage falls outside of the footprint of the archival MUSE data, and we are therefore unable to confirm it or rule it out with the MUSE data. Similarly, the arc is too faint in the Cycle 25 grism data available to this study. It is possible that the HST-WFC3/G280 data (GO-15966) are deep enough for spectroscopic redshift confirmation. The redshift of this source is left as a free parameter in our modeling.

Source 80.—The two images of source 80 present distinctive color and morphology, which secure their identification despite not having a spectroscopic redshift confirmation.

Source 90.—Source 90 has two images; one appears south of the west Sunburst Arc, and the second one appears northeast of the cluster core. We measure the redshift of both images, z_spec = 3.5053, based on MUSE detection of extended Lyα emission coincident with this galaxy. We identify five unique emission knots in the HST imaging of this source, while the Lyα emission appears much more extended than the optical/IR emission. The eastern image of source 90 spans 106, about three times more extended than its west counterpart, at 34, indicating that there is significantly more lensing shear in the east region. We observe similar behavior in the nearby system 30.

Source 130.—System 130 has two images, both resolved, with a red center and two distinct blue emission knots. We use the blue emission knots as constraints, as their location can be more precisely determined than that of the extended red center. We have no spectroscopic redshift for this system, but its unique morphology and colors provide confidence in its identification.

Sources 140, 150, 160.—Each of these systems has two images, with no spectroscopic redshift. We identify them based on their colors and morphology.

Source 170 (candidate).—We identify three images of this candidate system, with similar surface brightness and morphology, which are predicted by the lens model to be counterimages of each other. Image 170.1 appears in the north part of the field, close to image 1.1. The lens model predicts its counterimages in the south of the field. We identify arcs with similar morphology that match the lensing parity and shear direction that are predicted by the model. We note that image 170.1 appears slightly greener than 170.2 in the rendition shown in Figure 3, likely due to contamination from bright nearby cluster galaxies. The colors of 170.3 are contaminated by a nearby star. To be conservative, these candidate images were not used as constraints.

Source 180.—We identify two images of this source, one in the northeast and one southwest of the BCG. The MUSE spectra of both images suggest a low confidence line, which could be C iii] at z = 2.582. Due to its low confidence, the suggestive redshift is not used as a constraint. The morphology of the two images is similar and consistent with the expected lensing parity and shear direction.

Source 190.—We identify two faint candidate images of this source: a radial arc west of the BCG, and a counterimage northwest.

We use the lens modeling software Lenstool (Jullo et al. 2007), which is a parametric lens modeling algorithm, i.e., it assumes that the lens plane can be constructed from a linear combination of individual parametric mass halos. In this work, we model each mass component with a pseudo-isothermal ellipsoidal mass distribution (PIEMD, also known as dPIE; Elíasdóttir et al. 2007), with seven parameters: position (x, y, measured in arcseconds from a reference point), ellipticity e, position angle θ, core radius r_c , truncation radius r_cut, and a normalization σ.

Following procedures that have become standard in the field, the lens plane is modeled iteratively. The most obvious lensing evidence (i.e., the most prominent features in the Sunburst Arc) is used to constrain an initial lens model. The model is then used to identify additional lensing constraints and predict the locations of their counterimages. We use the archival MUSE data to spectroscopically confirm candidates. Following the recommendations of Johnson & Sharon (2016), we reinitiate the modeling process when including new spectroscopically confirmed constraints, in order not to bias the lens model by its own predictions. As new constraints are identified, we increase the lens model complexity as needed by either freeing the parameters of galaxies in close proximity to the giant arc or adding dark matter halos that represent correlated substructure (e.g., Mahler et al. 2018).

This lens model is a significant improvement on the model published by Rivera-Thorsen et al. (2019), which only used the Sunburst Arc images as constraints and consequently had a limited flexibility with which to model the lens plane. Two important data sets enabled this improvement: multiband HST imaging to identify lensed sources and resolve substructure within lensed images, and MUSE spectroscopy to derive reliable redshifts. The large number of constraints coming from the clumpy, highly magnified Sunburst Arc and the other lensed sources discussed in Section 3 enables construction of a lens model with high flexibility and complexity, allowing a large number of free parameters. The availability of constraints from secondary systems, i.e., multiply imaged galaxies at different source planes, provides important leverage for constraining the global properties of the cluster mass distribution and lowers the uncertainties (Johnson & Sharon 2016).

The components of the lens model are described in the next section. The model uses positional constraints from 146 images of 48 identified clumps within 15 multiply imaged lensed galaxies (19 of the clumps are in the Sunburst Arc), with multiplicities ranging from 2 to 12, and has a total of 57 free parameters, including 9 free redshifts. The best-fit model rms scatter between predicted and observed images is 085.

The lens plane is represented by one cluster-scale and two group-scale dark matter halos, supplemented by galaxy-scale halos. We position the cluster-scale halo near the BCG. We free all the parameters of this halo with the exception of the cut radius, which is fixed at 1500 kpc, as for cluster-scale halos this radius is far beyond the observed lensing evidence. This mass component accounts for the dark matter halo of the cluster, as well as the hot X-ray gas, which we find to be centered within 1'' of the BCG. We place one group-size halo in the general direction of a small group of bright cluster galaxies approximately 25'' northeast of the BCG and one group-size halo 41'' south of the BCG fixed to the position of a luminous cluster member. All the other parameters of both halos are left free, with broad priors. The cluster halos are supplemented with 250 galaxy-scale halos, positioned on the observed locations of cluster-member galaxies. The selection procedure of these galaxies is explained in the next section. The galaxies are modeled as PIEMDs, with positional parameters fixed at their observed properties (R.A., decl., ellipticity, and position angle) and the other parameters scaled to their luminosity following a parametric scaling relation given in Limousin et al. (2005, Equation (28)), with pivot parameter M(L*) = 19.45 mag. Several galaxies were modeled as individual halos, allowing Lenstool to solve for their best-fit parameters. These are either galaxies that lie in close projected proximity to the arcs or galaxies that are not expected to follow the same scaling relations as the red sequence cluster galaxies, e.g., the BCG, and spectroscopically confirmed star-forming galaxies at the cluster redshift.

At the core of the cluster, the UV/optical light of the central galaxy appears to be obscured by dust. We therefore measure its position in the reddest band, F160W, which should be the least affected by dust obscuration. We leave all the parameters of the BCG free, including its position, with a ±05 prior around the F160W centroid.

We include in the lens model a lensing galaxy that is positioned close to the northern part of the northwest arc, behind the cluster, at z_spec = 0.733 (galaxy G1 in Lopez et al. 2020). We approximate this interloper as contributing to the lensing potential at the same plane by allowing high flexibility in modeling its mass. The degeneracy between the normalization and the distance term means that the mass of this galaxy cannot be computed reliably with this approximation. Similarly, a faint interloper galaxy near image 50.11 (z_spec = 0.737; see Section 3) is also included in the lens model at the cluster redshift. We refer the reader to Raney et al. (2020) for a thorough examination of the implications of the modeling approach of projecting the interloper onto the same lens plane of the main lensing structure.

The mass associated with cluster-member galaxies contributes to the lensing potential of the cluster. We use Source Extractor (Bertin & Arnouts 1996) in dual-image mode to generate a photometric catalog of the field, with the F814W image used as the detection image, and MAG_AUTO measured in both the F814W and F606W images within the same aperture. We use a detection threshold of 5σ and a deblending contrast of 0.001. Stars were flagged by their locus in the MU_MAX versus MAG_AUTO plane and excluded from the catalog. We select cluster-member galaxies based on their F814W–F606W color in a color–magnitude diagram, following Gladders & Yee (2000). These particular bands were selected because they span the characteristic 4000 Å break of passive galaxies at the cluster redshift. The ACS data provide the widest field of view around the cluster core. Finally, we visually inspect the galaxy catalog and remove objects that were erroneously picked as cluster members, such as diffraction spikes, faint stars, and parts of over-deblended galaxies.

Figure 4 shows the projected mass density profile of PSZ1 G311.65−18.48, measured as a function of distance from the BCG. We find that the projected mass density of the core of PSZ1 G311.65−18.48, enclosed within R = 250 kpc, is M(<250 kpc) = ${2.93}_{-0.02}^{+0.01}\times {10}^{14}$ M_⊙. For comparison with previous mass estimates, we report the mass within 40'' = 228.3 kpc, $M(\lt 40^{\prime\prime} )={2.629}_{-0.015}^{+0.005}\times {10}^{14}$ M_⊙, and the mass within 200 kpc, $M(\lt 200\,\mathrm{kpc})={2.238}_{-0.007}^{+0.003}\times {10}^{14}$ M_⊙, which is similar to what was estimated by Pignataro et al. (2021) and Diego et al. (2022). It is unclear whether there is a discrepancy since no uncertainties were provided in those publications. Interestingly, we find that the mass within the radius of the giant arc, $M(\lt 169\,\mathrm{kpc})={1.809}_{-0.002}^{+0.005}\times {10}^{14}$ M_⊙, is in perfect agreement with what was estimated by Dahle et al. (2016), M_E = (1.8 ± 0.6) × 10¹⁴ M_⊙ for r_E = 169 ± 25 kpc. Their mass estimate is based on the relationship between the Einstein radius and the projected mass enclosed within it, $M(\lt {r}_{E})=\pi {r}_{E}^{2}{{\rm{\Sigma }}}_{\mathrm{crit}}$, for a circularly symmetric lens. Such a close agreement between the detailed lens model measurement and the mass enclosed by the Einstein radius method indicates that the circular symmetry assumption is valid for this cluster (Remolina González et al. 2021).

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The main cluster halo coincides with the BCG and the X-ray gas centroid to within 1'', which is consistent with PSZ1 G311.65−18.48 being a relaxed cluster.

Figure 5 maps the absolute value of the lensing magnification and its statistical uncertainty for a source at z = 2.3703, with a zoom-in on the north arc shown in Figure 6. Consistent with the visual interpretation of the lensing evidence, we find that the north and northwest images of the Sunburst Arc form in regions of highest magnification; the west and southeast arcs have the lowest magnification.

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Estimated over the entire arc, the average magnifications of the west and southeast arcs are $\langle {\mu }_{{\rm{W}}}\rangle ={13.5}_{-1.0}^{+2.4}$ and $\langle {\mu }_{\mathrm{SE}}\rangle ={13.1}_{-0.4}^{+1.0}$, respectively. Since these are the only two complete images of the source galaxy, a magnification measurement of the entire galaxy is only possible for these two arcs. The average magnifications of the west and southeast arcs are computed by defining an aperture in the image plane, ray-tracing the aperture to the source plane, and dividing the image plane area by the source plane area of the ray-traced aperture. For small enough regions that are far from the critical curves, the average magnification is not different from the value given by the magnification map at the center of the feature of interest. Table 2 lists the magnification for each of the knots in the Sunburst Arc, measured at the center of each feature. To estimate the statistical modeling uncertainties, we generate ∼100 lens models, each with a set of parameters from the Markov Chain Monte Carlo (MCMC) chain, which sample a 1σ confidence interval in the parameter space. We then run our calculation on the ∼100 magnification and deflection outputs.

As a test of the lens model, we compare the model-predicted magnifications to the observed flux of the LCE clump images. The absolute magnification prediction can be tested only when the absolute luminosity of the source can be estimated (e.g., for a standard candle; Rodney et al. 2015). Since the intrinsic fluxes of the sources in this field are unknown, we evaluate predicted relative magnifications against relative fluxes. We choose to run this analysis on the images of the LCE clump, which is the brightest lensed feature with the highest multiplicity in the field. Fluxes are measured within a 012 aperture in the F555W image, after matching it to the point-spread function (PSF) of the F160W filter (a full description of the aperture photometry measurement will be presented in J. K. Kim et al. 2022, in preparation). We divide the flux of each image of the LCE by that of image 1.8. Since the magnification uncertainties of the different images are likely correlated, we calculate the magnification relative to that of image 1.8 in each of the ∼100 models sampled from the MCMC and combine these measurements to estimate the uncertainty on the relative magnification. The formal uncertainty of the flux measurement is negligible compared to the lensing uncertainty, even in cases where the light from the source is somewhat obscured by foreground galaxies. The comparison is shown in Figure 7. Most of the images have a relative magnification within a factor of 2 from the relative fluxes, within errors. The relative magnifications of images 4, 5, 6, and 7 are lower than expected. This could be due to their proximity to foreground galaxies whose local contribution to the lensing potential is not adequately reproduced by the modeling process. The results of this exercise emphasize that there are systematic uncertainties that are unaccounted for in the lensing analysis, as was repeatedly highlighted in the literature (e.g., Rodney et al. 2015; Zitrin et al. 2015; Johnson & Sharon 2016; Priewe et al. 2017; Meneghetti et al. 2017; Kelly et al. 2018; Mahler et al. 2018; Remolina González et al. 2018). Based on this evaluation, we conclude that our statistical magnification uncertainties likely underestimate the full uncertainty by a factor of 4–5. It also indicates that the strong-lens model can be improved on by tapping into relative magnification constraining power, which can provide leverage over the second derivative of the lensing potential. For example, flux anomalies in lensed quasars were found to be indicative of substructure in the lens plane (Bradač et al. 2002).

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The arrival time surface, also known in the literature as the Fermat potential, is described by the following equation (e.g., Schneider 1985):

$$\begin{eqnarray}&&\tau (\vec{\theta },\vec{\beta })=\displaystyle \frac{1+{z}_{l}}{c}\displaystyle \frac{{D}_{l}{D}_{s}}{{D}_{{ls}}}\left[\displaystyle \frac{1}{2}{\left(\vec{\theta }-\vec{\beta }\right)}^{2}-\psi (\vec{\theta })\right],\end{eqnarray} \tag{ 1 }$$

where $\vec{\beta }$ is the position of the source in the source plane; $\vec{\theta }$ is a coordinate in the image plane; z_l is the lens redshift; D_l and D_s are the angular diameter distances from the observer to the lens and to the source, respectively; D_ls is the angular diameter distance from the lens to the source; and ψ is the lensing potential. Multiple images of a strongly lensed source form in stationary points in this surface, i.e., minima, maxima, and saddle points. The time delay between any two images can be calculated as the difference in arrival time, ${\rm{\Delta }}\tau =\tau ({\vec{\theta }}_{1},\vec{\beta })-\tau ({\vec{\theta }}_{2},\vec{\beta })$. The source location $\vec{\beta }$ is formally the same for all images of the same source; however, since lens models have finite accuracy, the calculated source positions of the different images are expected to have a small scatter in the source plane. It is therefore common to use the average or magnification-weighted average of the N model-derived source positions of the N multiple images as $\vec{\beta }$.

Figure 8 shows the Fermat potential result from the best-fit lens model. The relative arrival time is calculated with respect to image 1.1 of the LCE. We find that a packet of light emerging from the galaxy in the source plane arrives at the southeast arc first (image 12), preceding the other arc positions by 16–17 yr. Next to appear are the images of the north arc (1–6), followed by the northwest arc (7–10) and the west arc. Finally, the light from the source plane is predicted to arrive at the location of the (unobserved) demagnified central image some 26 yr after the north arc, due to gravitational time delay by the deep potential well of the cluster. Interestingly, we find that images 1–11 of the source occur within the span of several months to a year, a result that is qualitatively quite robust to the details of the lens model. This relatively short time delay implies that counterimages of transient events in the Sunburst Arc, such as supernova explosions, could potentially be observed, and their time delays measured, within the expected lifetime of current observational facilities. We discuss this point further in the next section.

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5.3.1. The Discrepant Clump: Variability or Something Else?

As noted in Section 3, a bright point source appears in the northwest arc, with similar brightness but different spectral energy distribution (SED) from the LCE images. A counterimage of this point source is not apparent in any of the other arcs. Vanzella et al. (2020b) interpreted the occurrence of this source to be due to time variability and postulate that this source is transient in nature. They estimate its magnification at 20 ≤ μ ≤ 100, based on lensing symmetry arguments (Vanzella et al. 2020a). Multiband HST imaging of PSZ1 G311.65−18.48 spans more than 2 yr, from 2018 February 21 to 2020 December 30, and while not designed as a cadenced survey, the field was imaged with a wide-throughput filter between two and four times each year, with the longest gap of approximately 1 yr between 2019 June 30 and 2020 July 14. During this time, the discrepant clump does not show significant variability. A quantitative epoch-to-epoch comparison is challenging, since different filters were used in different epochs, with the exception of F140W, which was repeated four times (2019 May and 2020 August, November, and December). Nevertheless, the brightness of the discrepant clump remains qualitatively similar to that of the LCE for over 2 yr and does not fade away. Moreover, Vanzella et al. (2020b) note that the distinct spectral features of this source are observed in MUSE data as early as 2016. If indeed this discrepant emission is transient in nature, its light curve appears to be flat on a several-year observed timescale and an intrinsic timescale of more than a year (cosmological expansion stretches the rest-frame time by a factor of 1 + z = 3.37), significantly longer than most supernovae.

Our lensing analysis points to a short time delay between the north and northwest arcs, meaning that a transient event in any of these arcs should appear in the other arcs within a few months; time variability alone cannot explain why multiple images of the discrepant source are not detected. Unless this image experiences extreme magnification at the position of the discrepant clump, its counterimages should be detectable in close proximity to images 4.3 and 4.4 in the north arc. In Figure 9 we show the location of the discrepant arc (top row) and the location where a counterimage should appear (bottom row) in the reddest available HST broadband filter in each of eight epochs. The discrepant source is clearly detected in each epoch in the northeast arc, with observed brightness comparable to that of the LCE clump. A dashed box marks a broad region around the expected location of the counterimage, near clump 4. If the transient hypothesis were correct, a counterimage of the discrepant clump should be visible in these regions with brightness similar to that of the LCE. However, throughout the observing window, no such sources appear in or disappear from these regions. The lensing analysis indicates that images 7 and 11 of the source, the northern tip of the northwest arc and the west arc, respectively, lag less than a year behind image 8 and the discrepant clump. Although these images are less magnified than image 8, a clump with comparable brightness to the LCE clump would have been detectable. We observe no counterimage of the discrepant clump in either location.

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To summarize, the combined model-predicted time delays and the observational data tell us that the fact that the discrepant clump appears only in the northwest arc must be lensing related and not related to the variability of the clump. Had it been only due to variability, we should have observed its counterimage.

The only explanation that remains for why the discrepant clump appears in the northwest arc but is absent from all other counterimages of the source galaxy is an extreme magnification at this image plane location. Such extreme magnification can occur if the source of emission is intrinsically small and located extremely close to the source plane caustic. For a point-like source, source plane caustics formally map to loci of infinite magnification. Such a lensing geometry was invoked in order to explain candidate extremely magnified single stars owing to caustic crossing (Kelly et al. 2018) or proximity (Welch et al. 2022). The average tangential magnification of image 8 is higher than the other arcs, evident by its overall higher distortion; for example, compare the image plane distance between the three brightest clumps, 1, 2, and 3, in each of the images. In image 8, a grazing critical curve could be resolving clump 4 into two emission knots plus the discrepant clump, the latter being further magnified by its proximity to the critical curve.

An interesting solution was recently proposed by Diego et al. (2022); they explore the possibility that the critical curve near image 8 is perturbed by a small mass, ∼10⁸ M_⊙, with no observed counterpart. They show that such a perturber could place the source of the discrepant clump right on top of a caustic, thus magnifying it extremely, and cause image 4.8 to split into the two observed clumps. A similar approach was also employed by Mahler et al. (2023), where supermassive black holes were shown to be a plausible reason for lensing anomalies. Upcoming JWST imaging might be able to detect a faint perturber in this location.

The overall effect of gravitational lensing magnification is in amplifying the solid angle of a source while conserving its surface brightness, thus increasing the amount of light that reaches the observer from the source by a factor of μ. The angular amplification is not spatially uniform; it is a combination of an isotropic magnification (convergence, κ) and distortion (shear, γ). κ and γ are combinations of second partial derivative components of the lensing potential and thus are a function of the image plane position (Narayan & Bartelmann 1996).

The extreme tangential shear that makes the images of the Sunburst Arc acts to resolve the source galaxy in multiple directions; in each arc, the shear acts most strongly in a preferential primary orientation. We illustrate this behavior in Figure 10, showing a mock source and its resulting lensed images in four regions of the image plane: image 2 in the north arc, image 8 in the northwest arc, image 11 in the west arc, and image 12 in the southeast arc. While the source does appear to be magnified in the radial direction, most of its magnification is in the tangential direction. The mock source is designed to highlight the three primary source plane angles in which the shear acts in this line of sight, with parallel lines plotted at −15°, 90°, and 130° (measured counterclockwise from north), perpendicular to the primary shear directions, 75°, 0°, and 40°, respectively. By lensing the mock source through the lens model, we find that the shear near the west arc (image 11) acts to resolve the source in the north–south direction, separating the red lines in the mock source, which are angled at 90°. Near the southeast arc (image 12), the shear resolves the source in the northeast−southwest direction, increasing the separation between the color-sequence lines that are plotted at a 130° angle. It acts in a similar direction near image 8. Finally, near image 2 in the north arc, the resolution is almost perpendicular to that of image 11, acting to separate the blue parallel lines that are plotted at −15°.

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The multidirectional gravitational lensing distortion can therefore enable a tomographic reconstruction of the source plane, facilitating a high-resolution analysis, by carefully combining information from differently distorted images. Structures that may appear unresolved in one image of the source may be resolved in a different image. Understanding the directional distortion also enabled an informed selection of our upcoming JWST NIRSpec IFU pointings, to resolve the same source component in different directions (JWST-GO program 02555, PI: Rivera-Thorsen).

A comprehensive computed tomography (CT) reconstruction and forward-modeling of the source plane of the Sunburst Arc is reserved for future work. In this section, we use the lens model and our understanding of the multidirectional lensing distortion of the source to qualitatively construct the galaxy in the source plane by hand. Figure 11 shows an "artist's impression" of the source, as an approximate visualization of the Sunburst Arc galaxy in the source plane. The conceptual reconstruction was produced by ray-tracing the position of each of the emission clumps from the image plane to the source plane through the lens model. The approximate location of the features was determined by taking into account the ray-tracing of the different images of the Sunburst Arc, taking advantage of the high magnification of some of the images from the northern arcs, as well as the less magnified but more complete images in the west and southeast. We then populate these approximate locations with circular artificial sources with colors and surface brightness approximately matching those of the composite color view in Figure 2. The source image of the extended galaxy (source 50) and connecting "bridge" to the north is a direct ray-tracing of the image plane color rendition.

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We identify 19 discrete clumps in the main source galaxy. Clump 1 (the LCE) is positioned in the source plane such that it is multiply imaged 12 times (a 13th undetected image is predicted by the lens model to be a central demagnified image). The clumps occupy a region sized ∼1 × 2 kpc in the source plane, with the LCE clump positioned at one edge of the galaxy.

The closest clump to the LCE, clump 5, is resolved from the LCE mainly in the north arc; other arcs lack the resolving power in the direction connecting clumps 1–5. As Figure 10 showcases, the eastern parts of the source are not imaged in the north arc. Clumps 1, 2, and 5–8 appear in all the images of the source except images 5 and 6. Images 3 and 4 contain more of the source galaxy, including clumps 1–12, but lack the clumps at the east edge, which are seen in images 7 and 8. Images 5 and 6 include only clump 1: image 5 because it is only isolating clump 1 within its caustics (red caustics in Figure 12); image 6 would have included any emission west of clump 1 (i.e., west of the orange caustic in Figure 12), if there were any. It does show an image of clump 101, which may be related to the same galaxy. From color considerations and the source plane analysis, we identify that the nonleaking clump in images 9 and 10 is clump 4. A lensing caustic can separate clumps 1 and 4 from the rest of the galaxy, thus creating their multiple images at these locations, as can be seen in Figure 12. We identify an apparent double image of clump 4 in image 8, which, along with the discrepant clump (see Section 5.3.1), can be a result of a more complex caustic or a more complex source that is extremely magnified and tangentially resolved into several components only in this location. Finally, all the clumps appear in images 11 and 12, which are complete images of the galaxy. However, due to the lower magnification, only the brightest clumps are readily identified. From their source plane positions, it is possible that the clumps we label 16 and 17 are the same as clump 11 or clump 12 but are less resolved, or that they are outside of the caustic of images 3 and 4.

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For studies of the source properties of the Sunburst Arc galaxy, and in particular the physical properties of its LyC leaking versus nonleaking regions, it is informative to know the source plane separation between each clump and the LCE. In Table 4 we summarize this information. For each clump, we measure the source plane distance between the center of the clump and the LCE and note in which of the 12 Sunburst Arc images it appears. The lower and upper limits on the distance to the LCE take into account the statistical lens modeling uncertainties from the MCMC, as well as the variations of source plane mapping from different images. The quoted distance is the average of the distances calculated from the n different images of the same clump, i.e., $\langle d{\rangle }_{j}=(1/n){\sum }_{i=1}^{n}{d}_{j,1}(i)$, where d_j,1(i) is the distance between clump j and clump 1 in the source plane projection of image i. The projections from image 7 were excluded from this calculation owing to the effect of modeling the foreground galaxy in this region on the lensing magnification. Including the source predictions from image 7 would increase the upper uncertainty by a factor of ∼2.

Table 4. Source Plane Distances between Clumps and the LCE

ID	Distance from	Images	Notes
	Clump 1 (kpc)
1	⋯	1,2,3,4,5,6,7,8,9,10,11,12	LCE
2	${0.60}_{-0.32}^{+0.87}$	1,2,3,4,7,8,11,12
3	${0.91}_{-0.39}^{+1.12}$	3,4,7,8,11,12
4	${1.05}_{-0.16}^{+0.72}$	3,4,7,8,9,10,11,12
5	${0.11}_{-0.06}^{+0.25}$	1,2,3,4
6	${1.08}_{-0.16}^{+0.21}$	1,2
7	${1.77}_{-0.42}^{+0.82}$	1,2,3,4,7,8,11,12
8	${0.83}_{-0.31}^{+0.83}$	1,2,3,4,7,8,11,12
9	${0.91}_{-0.36}^{+0.81}$	3,4,8
10	${1.21}_{-0.42}^{+1.00}$	3,4,7,8
11	${1.76}_{-0.42}^{+0.97}$	3,4
12	${1.70}_{-0.44}^{+0.98}$	3,4
13	${1.40}_{-0.09}^{+0.24}$	7,8
14	${1.45}_{-0.09}^{+0.25}$	7,8
15	${1.44}_{-0.09}^{+0.25}$	7,8
16	${1.20}_{-0.32}^{+0.18}$	11,12
17	${1.74}_{-0.44}^{+0.24}$	11,12
18	${3.70}_{-0.74}^{+0.41}$	11,12	"Bridge"
19	${3.73}_{-0.74}^{+0.38}$	11,12	"Bridge"
51	${5.86}_{-1.21}^{+0.84}$	11,12	Companion
52	${5.61}_{-1.18}^{+0.82}$	11,12	Companion
53	${6.37}_{-1.12}^{+1.66}$	North of 12	Companion
101	${2.16}_{-0.32}^{+0.46}$	1,6,8,11,12

Note. Average projected distance in kiloparsecs between each of the non-LyC leaking clumps and the LCE (clump 1) in the source plane of the Sunburst Arc galaxy. The quoted distance is measured as the average of the source plane distance predictions from different multiple images. Uncertainties take into account both the image-to-image variation and the strong-lens modeling uncertainty.

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5.4.1. Clump Sizes

The extreme shear and anisotropic magnification, combined with the telescope PSF, complicates a traditional ray-tracing-based source reconstruction of barely resolved star-forming regions. Simply ray-tracing an unresolved lensed image to the source plane can only place upper limits on the clump size, as it will return the telescope PSF corrected for the linear magnification. A properly unmagnified, undistorted source plane can be recovered by using forward-modeling techniques (e.g., Johnson et al. 2017), where a parameterized model of the source is constructed and ray-traced through the lens equation to the observer, and the raw lensed image is convolved with the telescope PSF and perturbed by noise and other instrument effects. The resulting image is then compared to the observed data iteratively to solve for the set of source plane model parameters that best minimize the scatter between the model and the data. These and other techniques have been demonstrated to resolve structures down to tens of parsecs, especially when the magnification is particularly high (e.g., Zitrin et al. 2011; Johnson et al. 2017; Vanzella et al. 2017; Welch et al. 2022). A full forward-modeling of the Sunburst Arc is saved for a future publication. We can, however, deduce an upper limit for clump sizes from their apparent circular morphology and size in the image plane.

The different directions in which the source is distorted in the different images of the same part of the Sunburst Arc source galaxy allow not only for a tomographic analysis of the galaxy morphology but also for a tomographic measurement of clump sizes, i.e., to constrain the spatial extent of individual star-forming regions in multiple directions. We show this for each of the 19 clumps we identify, in Figure 13. For each of the images of each clump, we place a circle with R = 008 in the image plane, representing the HST/ACS PSF. We ray-trace the PSF circle to the z = 2.3703 source plane using the lens equation and the lens model outputs. For clarity, we only plot the results from the best-fit model. With the exception of image 1.5 of the LCE, plotting the statistical uncertainties would have resulted in only slight broadening of the shown ellipse lines. We find that typically the HST PSF circles delens into high-eccentricity ellipses in the source plane, with semiminor axes of order <50 pc and semimajor axes of order 500 pc. This means that to be unresolved or barely resolved in the image plane, a clump must be smaller than 50 pc in the source plane. Regions with larger spatial extent in the source plane should appear extended and elongated in the image plane, as they span several source plane resolution elements. We note that the distinct star-forming clumps observed in the Sunburst Arc are largely barely resolved.

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Our upper limits on star-forming clump sizes are significantly smaller than those reported for unlensed galaxies at cosmic noon, which is consistent with expectations from numerical simulations. For example, Meng & Gnedin (2020) conclude that clump sizes measured in field galaxies are overestimated by a factor of 2−3 owing to a combination of the insufficient resolution of the telescope (usually HST) and projection effects. They show that without the benefit of high lensing magnification, the observed clumps are likely an aggregate of several smaller clumps.

Since the LCE clump is of high interest in the literature, we plot the delensed PSF for it separately, in Figure 14, and include lines from the 100 models that sample the parameter space. The projection from each image of the clump is shown in a different shade of gray; image 1.5 is not shown because the high magnification uncertainty due to a foreground galaxy prevents it from providing useful constraints. The clump, which is unresolved in our HST imaging data, is constrained to r ≲ 20 pc in its short axis and r ≲ 50 pc in its long axis, leading to an effective radius of ${r}_{\mathrm{eff}}\lesssim \sqrt{20\times 50}=32$ pc. The estimate of Vanzella et al. (2020a) of r_e ≲ 20 in the tangentially resolved direction, which they obtained by measuring the image plane size and assuming μ > 25, is consistent with our lens-model-based upper limit.

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5.4.2. Evidence for Galaxy Interaction in the Plane of the Sunburst Arc