arXiv subjects

Patrick L. Kelly

Publications and source records attributed to Patrick L. Kelly.

At least 19 recordsLinked to original sources

arXiv2026

First Statistical Study of Over 100 Magnified Stellar Events at Redshift $z \approx 0.725$ with JWST

Highly magnified stars at cosmological distances ($z \gtrsim 0.7$) become detectable thanks to microlensing by intracluster stars near the critical curves of galaxy clusters. Multi-epoch photometric campaigns targeting caustic crossing galaxies magnified by massive galaxy clusters enable the detection of these objects as transient events. Such stars provide unique opportunities to study stellar populations at early cosmic times, probe the nature of dark matter, reveal small-scale structure in the cluster, and improve lens models. To date, only a few dozen high-redshift stars have been reported, with a single lensed galaxy, the Dragon, holding the current record of 44 detections. These numbers, however, remain insufficient to exploit their full potential. In this paper, owing to the inclusion of new observations, we report the identification of more than 100 magnified stellar events in the Dragon, behind the massive galaxy cluster Abell 370. The relatively low redshift of the Dragon ($z\approx0.725$) facilitates the detection of its most massive stars. Using imaging data from three different cycles (2022--2024) with the James Webb Space Telescope, we apply a time-domain technique to identify flux variations associated with caustic-crossing events. From the spatial distribution of stellar events we constrain the high-end slope of the stellar luminosity function, finding $\beta=2.18^{+0.20}_{-0.30}$. Alternatively, assuming a fixed slope, we constrain the microlens surface mass density. In addition, we examine the parity asymmetry of the detected caustic-crossing events, a proposed probe of wave dark matter, and find that it remains present. We also use the events to trace the regions of highest magnification, offering an alternative way to map the system critical curves.

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arXiv2026

Other red dots: A possible GLIMPSE of normal AGB stars at Cosmic Noon through extreme lensing

We report the discovery of four extremely faint ($m_{\mathrm{F444W}}\gtrsim29$) red point sources in recent ultra-deep JWST/NIRCam images of the strong lensing galaxy cluster Abell S1063. All four sources sit in lensed arcs, on the symmetry points very close to the critical curves for their host-galaxies' redshifts ($z\sim1-4$). Remarkably, these point sources appear in most arcs that are sufficiently faint close to the critical curve's position ($<21\,\mathrm{nJy}\,\mathrm{arcsec}^{-2}$ in F115W). This suggests that -- unlike previous caustic-crossing events or lensed stars -- thanks to the unprecedented depth of the GLIMPSE observations paired with the extreme lensing magnification (up to $\mu\sim10^4$) we might be resolving the lower-mass ($M\sim1-11\,\mathrm{M}_{\odot}$) red stellar population. Concretely, we detect three likely extremely magnified asymptotic giant branch (AGB) stars ($T_{\mathrm{eff}}\sim3200-3750$ K), and one yellow super-giant star ($T_{\mathrm{eff}}\sim6750$ K) -- possibly a yellow hyper-giant or a Cepheid. In addition to offering the first glimpse at low-mass extremely magnified stars, these detections open a possible window into stellar populations, evolution, and chemical enrichment at high redshifts, and could pave the way for using lensed stars such as these as standard candles to populate the distance ladder at cosmological redshifts.

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arXiv2026

SN 2022riv in RX J2129: Discovery, Spectroscopic Classification, and Microlensing of a Strongly Lensed Type Ia Supernova from JWST and HST Observations

The multiply imaged SN 2022riv was discovered through a search of galaxy cluster fields as part of a Hubble Space Telescope (HST) SNAP program to find highly magnified stars. The supernova (SN) was detected in the image corresponding to the longest time delay of a galaxy at redshift $z=1.522$ strongly lensed by the foreground galaxy cluster RX J2129.7+0005. Follow up James Webb Space Telescope (JWST) NIRSpec G140M and PRISM spectroscopy yields a Type Ia SN classification. Using the SALT3-NIR light-curve fitter, we obtain a cosmology-independent measurement of the magnification of $5.35\pm1.01$ for the last-to-arrive image of the SN, with multiple SALT SN spectral time-series models yielding consistent constraints. The last-to-arrive image of SN 2022riv we detect appeared adjacent to the brightest cluster galaxy (BCG) at a location with an exceptionally high stellar mass density ($\sim 1-2$ dex higher than that of SN Refsdal), where microlensing is expected to introduce a 20-50% modulation of the magnification. Analyzing six independent lens models of the cluster, we find that four predict the magnification with much greater precision ($p < 0.05$) than would be expected by random chance, given the large effect anticipated from microlensing. Five models yield magnifications of roughly $4-7$ (within $1\sigma$) prior to accounting for microlensing, whereas HoliGRALE favors a significantly higher value of $15.39 \pm 0.85$. After incorporating nominal microlensing, the HoliGRALE prediction is within $1\sigma$ tension with our measurement. A companion paper (Dalrymple et al.) will present constraints on the relative time delay of the image that arrived earlier.

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arXiv2026

Caught in Swallowtails: Discovery of Two Swallowtail Image Formations in MS 0451.6-0305

We report the discovery of two swallowtail image formations at $z=2.91$ and $z=6.70$ behind the galaxy cluster MS 0451.6-0305 in JWST-NIRCam imaging. We find that in both of the above lensed systems, the complex image morphology cannot be reproduced by simple fold/cusp caustics, and detailed lens modeling reveals higher-order swallowtail caustic configurations. In the $z=2.91$ lens system, a small part of the source galaxy (which itself is part of a galaxy group) containing atleast two compact knots sits inside the swallowtail caustic, producing a quadruply imaged arc. At two of the image positions of these knots, we infer point source magnifications of $\gtrsim 300$, implying lensing-corrected effective radii of $\lesssim 0.8-1.5$ pc. The $z=6.70$ system exhibits even more complex image formation. We therefore only use the most confidently identified counter-images of knots in this system as constraints in our lens modeling. The resulting model predicts magnifications $\sim20-200$ and lensing-corrected effective radii of $\lesssim 0.8-18.5$ pc for various knots. Together, these two systems represent the first example of observations of multiple swallowtail image formations in a single galaxy cluster and demonstrate the ability of swallowtail caustics to magnify individual substructures at sub-parsec scales, from intermediate redshifts to the first billion years of the Universe.

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arXiv2026

A spectroscopically confirmed, strongly lensed, metal-poor Type II supernova at z = 5.13

Observing supernovae (SNe) in the early Universe (z > 3) provides a window into how both galaxies and individual stars have evolved over cosmic time, yet a detailed study of high-redshift stars and SNe has remained difficult due to their extreme distances and cosmological redshifting. To overcome the former, searches for gravitationally lensed sources allow for the discovery of magnified SNe that appear as multiple images - further providing the opportunity for efficient follow-up. Here we present the discovery of "SN Eos": a strongly lensed, multiply-imaged, SN II at a spectroscopic redshift of z = 5.133 +/- 0.001. SN Eos exploded in a Lyman-{\alpha} emitting galaxy when the Universe was only ~1 billion years old, shortly after it reionized and became transparent to ultraviolet radiation. A year prior to our discovery in JWST data, archival HST imaging of SN Eos reveals rest-frame far ultraviolet (~1,300{\AA}) emission, indicative of shock breakout or interaction with circumstellar material in the first few (rest-frame) days after explosion. The JWST spectroscopy of SN Eos, now the farthest spectroscopically confirmed SN ever discovered, shows that SN Eos's progenitor star likely formed in a metal-poor environment (<= 0.1 Z_{\odot}), providing the first direct evidence of massive star formation in the metal-poor, early Universe. SN Eos would not have been detectable without the extreme lensing magnification of the system, highlighting the potential of such discoveries to eventually place constraints on the faint end of the cosmic star-formation rate density in the very early Universe.

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arXiv2025

Lensed stars in galaxy-galaxy strong lensing -- a JWST prediction for the Cosmic Horseshoe

We explore for the first time the possibility of detecting lensed star transients in galaxy-galaxy strong lensing systems upon repeated, deep imaging using the {\it James-Webb Space Telescope} ({\it JWST}). Our calculation predicts that the extremely high recent star formation rate of $\sim 140\,M_{\odot}\textrm{yr}^{-1}$ over the last 50 Myr (not accounting for image multiplicity) in the ``Cosmic Horseshoe'' lensed system ($z = 2.381$) generates many young, bright stars, of which their large abundance is expected to lead to a detection rate of $\sim 60$ transients per pointing in {\it JWST} observations with a $5\sigma$ limiting magnitude of $\sim 29\,m_{AB}$. With the high expected detection rate and little room for uncertainty for the lens model compared with cluster lenses, our result suggests that the Cosmic Horseshoe could be an excellent tool to test the nature of dark matter based on the spatial distribution of transients, and can be used to constrain axion mass if dark matter is constituted of ultra-light axions. We also argue that the large distance modulus of $\sim46.5\,$mag at $z \approx 2.4$ can act as a filter to screen out less massive stars as transients and allow one to better constrain the high-mass end of the stellar initial mass function based on the transient detection rate. Follow-up {\it JWST} observations of the Cosmic Horseshoe would allow one to better probe the nature of dark matter and the star formation properties, such as the initial mass function at the cosmic noon, via lensed star transients.

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arXiv2025

JWST's PEARLS: Temperatures of Nine Highly Magnified Stars in a Galaxy at Redshift 0.94 and Simulated Stellar Population Dependence on Stellar Metallicity and the Initial Mass Function

We present stellar atmosphere modeling of JWST NIRCam photometry of nine highly magnified individual stars in a single galaxy at redshift z=0.94 known as the Warhol arc, which is strongly lensed by the galaxy cluster MACSJ0416. Seven of these transients were identified by Yan et al. (2023). The nine sources are all likely red supergiants with temperatures of T~4000K. We present new longslit spectroscopy of the Warhol arc acquired with Keck-I and the Large Binocular Telescope, and use these data to constrain the arc's oxygen abundance to be 12+log(O/H)=8.45+-0.08. We perform a microlensing simulation on synthetic stellar populations using a range of stellar metallicities and initial mass function slopes. The temperature distribution of the simulated detectable stars is sensitive to the choice of stellar metallicity, and setting the stellar metallicity equal to the arc's nebular metallicity (log(Z*/Zsun)=-0.24) produces a simulated temperature distribution that is consistent with the observations, while lower stellar metallicities (log(Z*/Zsun)<-0.75) produce simulated temperatures that are inconsistent with the observations. The expected detection rate is strongly anticorrelated with the IMF slope for alpha>1.2. For the canonical IMF slope alpha=2.35, the simulation yields expected transient detection rates that agree with the observed detection rates in the HST Flashlights filters, but over predicts the detection rate by a factor of ~3-12 (<2sigma tension) in the JWST filters. The simulated detection rate is sensitive to the choice of stellar metallicity, with lower metallicities (log(Z*/Zsun)<-0.75) yielding a significantly lower simulated detection rate that further reduces the modest tension with the observations.

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arXiv2025

JWST's PEARLS: A Candidate Massive Binary Star System in a Lensed Galaxy at Redshift 0.94

Massive stars at cosmological distances can be individually detected during transient microlensing events, when gravitational lensing magnifications may exceed ~1000. Nine such sources were identified in JWST NIRCam imaging of a single galaxy at redshift z=0.94 known as the "Warhol arc,'' which is mirror-imaged by the galaxy cluster MACSJ0416.1-2403. Here we present the discovery of two coincident and well-characterized microlensing events at the same location followed by a third event observed in a single filter approximately 18 months later. The events can be explained by microlensing of a binary star system consisting of a red supergiant (T ~ 4000 K) and a B-type (T ~ 13,000 K) companion star. The timescale of the coincident microlensing events constrains the estimated projected source-plane size to tens of AU. The most likely binary configurations consistent with the observational constraints on the temperature and luminosity of each star are stars with initial masses M1=23.6+5.3-4.3 Msol and an initial mass ratio between the two stars close to unity. A kinematic model that reproduces the observed light curve in all filters gives a relatively small transverse velocity of 50 km/s. This requires the dominant velocity component of several hundreds of km/s to be roughly parallel to the microcaustic. An alternative possibility would be that the three microlensing events correspond to unrelated stars crossing distinct microcaustics, but this would imply a highly elevated rate of events at their common position, even though no underlying knot is present at the location.

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arXiv2025

Transient star B/R ratio and star formation in $z\gtrsim 1$ lensed galaxies

The extreme magnification from galaxy clusters and microlenses therein allows the detection of individual, luminous stars in lensed galaxies as transient events, and hence provides a valuable window into the high mass stellar population in $z>1$ galaxies. As these bright stars can only be formed at specific ages, the relative abundance of transient events at blue (B) and red (R) optical wavelengths ($B/R$ ratio) can provide insights into the recent star formation history of galaxies that are not well constrained by their spectral energy distributions (SEDs). Here, we forward model the transient detection rates in an idealized mock scenario to find that the $B/R$ ratio of strongly lensed $z>1$ galaxies decreases quickly with increasing age. This ratio has moderate sensitivity to metallicity and comparatively low sensitivity to dust attenuation, with no significant dependency on the stellar initial mass function. Fitting model stellar populations to either the SED or $B/R$ ratio alone of ``Warhol'' arc ($z = 0.94$), we find that neither a simple single starburst nor a more complex star formation can simultaneously reproduce both constraints. We then demonstrate that a best-fit model constrained by both the B/R ratio and SED requires a star-formation rate that has varied quite dramatically over the past $\sim$50 Myr, for which the total stellar mass formed over this time is a factor of 10 (with $2-3\sigma$ significance) different from the best-fit models to the SED alone. Our work shows that the transient $B/R$ ratio can be used as an additional powerful constraint on the recent star formation history of higher-redshift galaxies in future works that are strongly lensed by galaxy clusters.

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arXiv2025

Constraining the z $\sim$ 1 Initial Mass Function with {\it HST} and {\it JWST} Lensed Stars in MACS J0416.1-2403

Our understanding of galaxy properties and evolution is contingent on knowing the initial mass function (IMF), and yet to date, the IMF is constrained only to local galaxies. Individual stars are now becoming routinely detected at cosmological distances, where luminous stars such as supergiants in background galaxies strongly lensed by galaxy clusters are temporarily further magnified by huge factors (up to $10^{4}$) by intracluster stars, thus being detected as transients. The detection rate of these events depends on the abundance of luminous stars in the background galaxy and is thus sensitive to the IMF and the star-formation history (SFH), especially for the blue supergiants detected as transients in the rest-frame ultraviolet/optical filters. As a proof of concept, we use simple SFH and IMF models constrained by spectral energy distributions (SEDs) to see how well we can predict the {\it HST} and {\it JWST} transient detection rate in a lensed arc dubbed ``Spock'' ($z = 1.0054$). We find that demanding a simultaneous fit of the SED and the transient detection rate places constraints on the IMF, independent of the assumed simple SFH model. We conclude our likelihood analysis indicates that the data definitively prefers the ``Spock'' galaxy to have a Salpeter IMF ($\alpha = 2.35$) rather than a Top-heavy IMF ($\alpha = 1$) -- which is thought to be the case in the early universe -- with no clear excess of supergiants above the standard IMF.

astro-ph.CO↗

arXiv2025

Microlensing at Cosmological Distances: Event Rate Predictions in the Warhol Arc of MACS 0416

Highly magnified stars ($\mu$ $>$ 100) are now outinely identified as transient events at cosmological distances thanks to microlensing by intra-cluster stars near the critical curves of galaxy clusters. Using the {\it James Webb} Space Telescope (JWST) in combination with the {\it Hubble} Space Telescope (HST), we outline here an analytical framework that is applied to the Warhol arc (at $z=0.94$) in the MACS 0416 galaxy cluster (at $z=0.396)$ where over a dozen microlensed stars have been detected to date. This method is general and can be applied to other lensed arcs. Within this lensed galaxy we fit the spatially resolved SED spanned by eight JWST-NIRCam filters combined with three ACS filters, for accurate lensed star predictions in 2D. With this tool we can generate 2D maps of microlensed stars for well resolved arcs in general, including dependence on wavelength and limiting apparent magnitude, for comparison with with planned cadenced campaigns for JWST and Hubble, for constraining directly the IMF and the level of dark matter substructure.

astro-ph.CO↗

arXiv2025

Flashlights: Prospects for constraining the Initial Mass Function around cosmic noon with caustic-crossing events

The Flashlights program with the Hubble Space Telescope imaged the six Hubble Frontier Fields galaxy clusters in two epochs and detected twenty transients. These are primarily expected to be caustic-crossing events (CCEs) where bright stars in distant lensed galaxies, typically at redshift $z\approx1$--3, get temporarily magnified close to cluster caustics. Since CCEs are generally biased toward more massive and luminous stars, they offer a unique route for probing the high end of the stellar mass function. We take advantage of the Flashlights event statistics to place preliminary constraints on the stellar initial mass function (IMF) around cosmic noon. The photometry (along with spectral information) of lensed arcs is used to infer their various stellar properties, and stellar synthesis models are used to evolve a recent stellar population in them. We estimate the microlens surface density near each arc and, together with existing lens models and simple formalism for CCEs, calculate the expected rate for a given IMF. We find that, on average, a Salpeter-like IMF ($\alpha=2.35$) underpredicts the number of observed CCEs by a factor of ${\sim}0.7$, and a top-heavy IMF ($\alpha=1.00$) overpredicts by a factor of ${\sim}1.7$, suggesting that the average IMF slope may lie somewhere in between. However, given the large uncertainties associated with estimating the stellar populations, these results are strongly model-dependent. Nevertheless, we introduce a useful framework for constraining the IMF using CCEs. Observations with JWST are already yielding many more CCEs and will soon enable more stringent constraints on the IMF at a range of redshifts.

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arXiv2024

BUFFALO Wild Wings: A High Precision Free-Form Lens Model of MACSJ0416 with Constraints on Dark Matter from Substructure and Highly Magnified Arcs

We present new free-form and hybrid mass reconstructions of the galaxy cluster lens MACS J0416.1$-$2403 at $z=0.396$ using the lens inversion method GRALE. The reconstructions use 237 spectroscopically confirmed multiple images from Bergamini et. al. 2023 as the main input. Our primary model reconstructs images to a positional accuracy of 0.191", thus representing one of the most precise reconstructions of this lens to date. Our models find broad agreement with previous reconstructions, and identify two $\sim 10^{12} M_{\odot}$ light-unaffiliated substructures. We focus on two highly magnified arcs: Spock and Mothra. Our model features a unique critical curve structure around the Spock arc with 2 crossings. This structure enables sufficient magnification across this arc to potentially explain the large number of transients as microlensing events of supergiant stars. Additionally, we develop a model of the millilens substructure expected to be magnifying Mothra, which may be a binary pair of supergiants with $\mu \sim 6000$. This model accounts for flexibility in the millilens position while preserving the observed flux and minimizing image position displacements along the Mothra arc. We constrain the millilens mass and core radius to $\lesssim 10^6 M_{\odot}$ and $\lesssim 17$ pc, respectively, which would render it one of the smallest and most compact substructures constrained by lensing. If the millilens is dominated by wave dark matter, the axion mass is constrained to be $\lesssim 3.0 \times 10^{-21}$ eV. Further monitoring of this lens with JWST will uncover more transients, permitting tighter constraints on the structure surrounding these two arcs.

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arXiv2024

A high-resolution view of the source-plane magnification near cluster caustics in wave dark matter models

We present the highest resolution images to date of caustics formed by wave dark matter ($\psi$DM) fluctuations near the critical curves of cluster gravitational lenses. We describe the basic magnification features of $\psi$DM in the source plane at high macromodel magnification and discuss specific differences between the $\psi$DM and standard cold dark matter (CDM) models. The unique generation of demagnified counterimages formed outside the Einstein radius for $\psi$DM is highlighted. Substructure in CDM cannot generate such demagnified images of positive parity, thus providing a definitive way to distinguish $\psi$DM from CDM. Highly magnified background sources with sizes $r\approx 1pc$, or approximately a factor of ten smaller than the expected de Broglie wavelength of $\psi$DM, offer the best possibility of discriminating between $\psi$DM and CDM. These include objects such as very compact stellar clusters at high redshift that JWST is finding in abundance.

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arXiv2024

Dark Matter distinguished by skewed microlensing in the "Dragon Arc"

Microlensed stars recently discovered by JWST & HST follow closely the winding critical curve of A370 along all sections of the ``Dragon Arc" traversed by the critical curve. These transients are fainter than $m_{AB}>26.5$, corresponding to the Asymptotic Giant Branch (AGB) and microlensed by diffuse cluster stars observed with $\simeq 18M_\odot/pc^2$, or about $\simeq 1$\% of the projected dark matter density. Most microlensed stars appear along the inner edge of the critical curve, following an asymmetric band of width $\simeq 4$kpc that is skewed by $-0.7\pm0.2$kpc. Some skewness is expected as the most magnified images should form along the inner edge of the critical curve with negative parity, but the predicted shift is small $\simeq -0.04$kpc and the band of predicted detections is narrow, $\simeq 1.4$kpc. Adding CDM-like dark halos of $10^{6-8}M_\odot$ broadens the band as desired but favours detections along the outer edge of the critical curve, in the wrong direction, where sub-halos generate local Einstein rings. Instead, the interference inherent to ``Wave Dark Matter" as a Bose-Einstein condensate ($ψ$DM) forms a symmetric band of critical curves that favours negative parity detections. A de Broglie wavelength of $\simeq 10$pc matches well the observed $4$kpc band of microlenses and predicts negative skewness $\simeq -0.6$kpc, similar to the data. The implied corresponding boson mass is $\simeq 10^{-22}$eV, in good agreement with estimates from dwarf galaxy cores when scaled by momentum. Further JWST imaging may reveal the pattern of critical curves by simply ``joining the dots" between microlensed stars, allowing wave corrugations of $ψ$DM to be distinguished from CDM sub-halos

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arXiv2024

Flashlights: Transients among Gravitationally-Lensed Star Clusters in the Dragon Arc. I. Stellar Microlensing vs Stellar Outbursts

We report the discovery of transients among star clusters in a distant galaxy that is gravitationally lensed by a foreground galaxy cluster, and explore whether these transients correspond to: (i) intrinsic variations associated with stellar outbursts; or (ii) extrinsic variations imposed through microlensing by intraclusters stars along, perhaps, with primordial black holes. From images at two epochs separated by nearly a year, we discovered ten such transients -- displaying brightness variations of $\sim$10\%--20\% -- among 55 persistent knots identified as young star clusters in the Dragon arc. Two of these transients are associated with a triply-lensed star cluster, permitting a test of intrinsic variability by checking whether their light variations are repeated among the different lensed counterparts with a suitable time delay given their different light arrival times at the observer. Despite considerable care in constructing a lens model for Abell 370 that is optimized at the Dragon arc, we found that the predicted lensing magnifications are not sufficiently accurate to provide a definitive test of intrinsic variability based only on two images -- although such a test will become feasible as more observations are made. On the other hand, we perform simulations demonstrating that the observed level of brightness variations, as well as the observed transient event rate, can be explained entirely by stellar microlensing: as stars in the background star cluster move across the sky relative to intracluster stars, changes in their individual brightnesses can result in an overall change in the brightness of their parent star cluster.

astro-ph.CO↗

arXiv2024

Identification of $>$40 gravitationally magnified stars in a galaxy at redshift of 0.725

Strong gravitational magnifications enable to detect faint background sources, resolve their internal structures, and even identify individual stars in distant galaxies. Highly magnified individual stars allow various applications, including studies of stellar populations in distant galaxies and constraining dark matter structures in the lensing plane. However, these applications have been hampered by the small number of individual stars observed, as typically one or a few stars are identified from each distant galaxy. Here, we report the discovery of more than 40 microlensed stars in a single galaxy behind Abell 370 at redshift of 0.725 when the Universe was half of its current age (dubbed ``the Dragon arc''), using James Webb Space Telescope (JWST) observations with the time-domain technique. These events are found near the expected lensing critical curves, suggesting that these are magnified stars that appear as transients from intracluster stellar microlenses. Through multi-wavelength photometry, we constrain stellar types and find that many of them are consistent with red giants/supergiants magnified by factors of hundreds. This finding reveals an unprecedented high occurrence of microlensing events in the Dragon arc, and proves that {\it JWST}'s time-domain observations open up the possibility of conducting statistical studies of high-redshift stars.

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arXiv2024

Imaging dark matter at the smallest scales with $z\approx1$ lensed stars

Observations of caustic-crossing galaxies at redshift $0.7 2$ and the number density of events is greater around substructureand the number density of events is greater around substructures, and (ii) negative imaging regime where $\beta<2$. We study the particular case of seven microlensing events found by HST in the Dragon arc (at z=0.725). We find that a population of supergiant stars with a steep LF with $\beta=2.55$ fits the distribution of these events. We identify a small region of high density of microlensing events, and interpret it as evidence of a possible invisible substructure, for which we derive a mass of $\sim 1.3 \times 10^8\,\Msun$ (within its Einstein radius).

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