# JWST Discovery of $40+$ Microlensed Stars in a Magnified Galaxy, the
“Dragon” behind Abell 370

Yoshinobu Fudamoto Center for Frontier Science, Chiba University, 1-33 Yayoi-
cho, Inage-ku, Chiba 263-8522, Japan  Fengwu Sun Steward Observatory,
University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA  Jose M.
Diego Instituto de Física de Cantabria (CSIC-UC). Avda. Los Castros 39005
Santander, Spain  Liang Dai Department of Physics, University of California,
366 Physics North MC 7300, Berkeley, CA. 94720, USA  Masamune Oguri Center for
Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522,
Japan  Department of Physics, Graduate School of Science, Chiba University,
Chiba 263-8522, Japan  Adi Zitrin Department of Physics, Ben-Gurion University
of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel  Erik Zackrisson
Observational Astrophysics, Department of Physics and Astronomy, Uppsala
University, Box 516, SE-751 20 Uppsala, Sweden  Swedish Collegium for Advanced
Study, Linneanum, Thunbergsvägen 2, SE-752 38 Uppsala, Sweden  Mathilde Jauzac
Centre for Extragalactic Astronomy, Durham University, South Road, Durham DH1
3LE, UK  Institute for Computational Cosmology, Durham University, South Road,
Durham DH1 3LE, UK  Astrophysics Research Centre, University of KwaZulu-Natal,
Westville Campus, Durban 4041, South Africa  School of Mathematics, Statistics
& Computer Science, University of KwaZulu-Natal, Westville Campus, Durban
4041, South Africa  David J. Lagattuta Centre for Extragalactic Astronomy,
Durham University, South Road, Durham DH1 3LE, UK  Institute for Computational
Cosmology, Durham University, South Road, Durham DH1 3LE, UK  Eiichi Egami
Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ
85721, USA  Edoardo Iani Kapteyn Astronomical Institute, University of
Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands  Rogier A.
Windhorst School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-6004, USA  Katsuya T. Abe Center for Frontier Science, Chiba
University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan  Franz Erik Bauer
Instituto de Astrofísica and Centro de Astroingeniería, Facultad de Física,
Pontificia Universidad Católica de Chile, Campus San Joaquín, Av. Vicuña
Mackenna 4860, Macul Santiago, Chile, 7820436  Millennium Institute of
Astrophysics, Nuncio Monseñor Sótero Sanz 100, Of 104, Providencia, Santiago,
Chile  Space Science Institute, 4750 Walnut Street, Suite 205, Boulder,
Colorado 80301  Fuyan Bian European Southern Observatory, Alonso de Córdova
3107, Casilla 19001, Vitacura, Santiago 19, Chile  Rachana Bhatawdekar
European Space Agency (ESA), European Space Astronomy Centre (ESAC), Camino
Bajo del Castillo s/n, 28692 Villanueva de la Cañada, Madrid, Spain  Thomas J.
Broadhurst Dept. of Physics, University of the Basque Country, Bilbao, Spain
DIPC, Donostia International Physics Centre, San Sebastian, Spain  Ikerbasque,
Basque Foundation for Science, Bilbao, E-48011, Spain  Zheng Cai Department of
Astronomy, Tsinghua University, Beijing 100084, China  Chian-Chou Chen
Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No. 1, Sec.
4, Roosevelt Road, Taipei 10617, Taiwan  Wenlei Chen Department of Physics,
Oklahoma State University, 145 Physical Sciences Bldg, Stillwater, OK 74078,
USA  Seth H. Cohen School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287-1404, USA  Christopher J. Conselice Jodrell Bank
Centre for Astrophysics, University of Manchester, Oxford Road, Manchester M13
9PL, UK  Daniel Espada Departamento de Física Teórica y del Cosmos, Campus de
Fuentenueva, Edificio Mecenas, Universidad de Granada, E-18071, Granada, Spain
Instituto Carlos I de Física Teórica y Computacional, Facultad de Ciencias,
E-18071, Granada, Spain  Nicholas Foo School of Earth and Space Exploration,
Arizona State University, Tempe, AZ 85287-1404, USA  Brenda L. Frye Department
of Astronomy/Steward Observatory, University of Arizona, 933 N. Cherry Avenue,
Tucson, AZ 85721, USA  Seiji Fujimoto Hubble Fellow  Department of Astronomy,
The University of Texas at Austin, Austin, TX 78712, USA  Lukas J. Furtak
Department of Physics, Ben-Gurion University of the Negev, P.O. Box 653,
Be’er-Sheva 84105, Israel  Miriam Golubchik Department of Physics, Ben-Gurion
University of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel  Tiger Yu-
Yang Hsiao Center for Astrophysical Sciences, Department of Physics and
Astronomy, The Johns Hopkins University, 3400 N Charles St. Baltimore, MD
21218, USA  Jean-Baptiste Jolly Max-Planck-Institut für Extraterrestrische
Physik (MPE), Giessenbachstraße 1, D-85748 Garching, Germany  Hiroki Kawai
Department of Physics, The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
Center for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba
263-8522, Japan  Patrick L. Kelly Minnesota Institute for Astrophysics,
University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA
Anton M. Koekemoer Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  Kotaro Kohno Institute of Astronomy, Graduate School
of Science, The University of Tokyo, 2-21-1 Osawa, Mitaka, Tokyo 181-0015,
Japan  Research Center for the Early Universe, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan  Vasily
Kokorev Kapteyn Astronomical Institute, University of Groningen, 9700 AV
Groningen, The Netherlands  Mingyu Li Department of Astronomy, Tsinghua
University, Beijing 100084, China  Zihao Li Department of Astronomy, Tsinghua
University, Beijing 100084, China  Xiaojing Lin Department of Astronomy,
Tsinghua University, Beijing 100084, China  Steward Observatory, University of
Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA  Georgios E. Magdis Cosmic
Dawn Center (DAWN), Denmark  DTU-Space, Technical University of Denmark,
Elektrovej 327, 2800, Kgs. Lyngby, Denmark  Niels Bohr Institute, University
of Copenhagen, Jagtvej 128, 2200, Copenhagen N, Denmark  Ashish K. Meena
Department of Physics, Ben-Gurion University of the Negev, P.O. Box 653,
Be’er-Sheva 84105, Israel  Armin Nabizadeh Observational Astrophysics,
Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20
Uppsala, Sweden  Johan Richard Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS,
Centre de Recherche Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval,
France  Charles L. Steinhardt Dark Cosmology Centre  Niels Bohr Institute,
University of Copenhagen, Jagtvej 120, DK-2100 Copenhagen Ø  Yunjing Wu
Department of Astronomy, Tsinghua University, Beijing 100084, China  Yongda
Zhu Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tucson, AZ
85721, USA  Siwei Zou Department of Astronomy, Tsinghua University, Beijing
100084, China

###### Abstract

Strong gravitational magnification by massive galaxy clusters enable us to
detect faint background sources, resolve their detailed internal structures,
and in the most extreme cases identify and study individual stars in distant
galaxies. Highly magnified individual stars allow for a wide range of
applications, including studies of stellar populations in distant galaxies and
constraining small-scale dark matter structures. However, these applications
have been hampered by the small number of events observed, as typically one or
a few stars are identified from each distant galaxy. Here, we report the
discovery of 46 significant microlensed stars in a single strongly-lensed
high-redshift galaxy behind the Abell 370 cluster at redshift of 0.725 when
the Universe was half of its current age (dubbed the “Dragon arc”), based on
two observations separated by one year with the James Webb Space Telescope
(JWST). These events are mostly found near the expected lensing critical
curves, suggesting that these are magnified individual stars that appear as
transients from intracluster stellar microlenses. Through multi-wavelength
photometry and colors, we constrain stellar types and find that many of them
are consistent with red giants/supergiants magnified by factors of thousands.
This finding reveals an unprecedented high occurrence of microlensing events
in the Dragon arc, and proves that JWST’s time-domain observations open up the
possibility of conducting statistical studies of high-redshift stars and
subgalactic scale perturbations in the lensing dark matter field.

††facilities: JWST(NIRCam)††software: astropy (Astropy Collaboration et al.,
2013, 2018), DAOFIND (Stetson, 1987), Photutils (Bradley et al., 2023), Pypher
(Boucaud et al., 2016), SciPy (Virtanen et al., 2020), SNCosmo (Barbary et
al., 2024), Source Extractor (Bertin & Arnouts, 1996)

##  1 Introduction

Galaxy clusters are the most massive gravitationally bound objects in the
Universe and produce strong gravitational lens events. The high magnification
afforded by such massive clusters (Bartelmann, 2010; Natarajan et al., 2024)
allows us to identify faint sources to the highest redshifts (Atek et al.,
2023; Fujimoto et al., 2023; Wang et al., 2023), study their detailed internal
structure (Adamo et al., 2024; Fudamoto et al., 2024; Fujimoto et al., 2024),
and in most extreme cases allows us to identify and study even individual
stars in distant galaxies. In particular, Miralda-Escude (1991) suggested that
highly magnified individual stars may be observed within strongly lensed
galaxies near critical curves produced by massive foreground clusters. Later,
Kelly et al. (2018) indeed reported transients near the critical curve of the
cluster MACS J1149.5+2223, which were interpreted as background stars
magnified by the foreground cluster by a factor of thousands. Such phenomena
are explained by microlensing events. Namely, the gravitational potential of
galaxy clusters sets macrolensing magnifications of order $\sim 100$ or more
near critical curves, and on top of that, masses in the foreground
intracluster medium act as microlenses resulting in total magnifications of
$\gtrsim 1000$. Subsequent microlens observations and analyses using the
Hubble Space Telescope (HST) and the James Webb Space Telescope (JWST) show
that such lensed stars are ubiquitous and provide us with a window to study
stellar physics at cosmological distances (e.g., Rodney et al., 2018; Kaurov
et al., 2019; Chen et al., 2019; Welch et al., 2022; Kelly et al., 2022; Meena
et al., 2023; Diego et al., 2023a, b; Furtak et al., 2024).

Such highly magnified individual stars allow for a wide range of applications,
including constraints on compact dark matter (Venumadhav et al., 2017; Oguri
et al., 2018; Diego et al., 2018; Vall Müller & Miralda-Escudé, 2024) or self-
gravitating micro-structures (Dai & Miralda-Escudé, 2020), the measurement of
the abundance of dark matter subhalos (Dai et al., 2018, 2020; Williams et
al., 2024), direct constraints on the stellar initial mass function (Kelly et
al., 2018), direct observations of Population III stars (Windhorst et al.,
2018; Schauer et al., 2022; Zackrisson et al., 2023) and stellar properties of
individual magnified stars (Han & Dai, 2024), and direct constraints on the
Hubble constant ($H_{0}$) when supernovae (SNe) are multiply imaged (Kelly et
al., 2023; Frye et al., 2024; Pascale et al., 2024). However, some of these
applications have been hampered by the small number of events observed so far,
as typically one or a few stars are identified from each galaxy. With its
superb light-collecting power and excellent spatial resolution, JWST has been
expected to significantly increase the number of detected events and open up
these exciting applications.

Here, we report JWST identifications of 46 microlensing events in a single
$z=0.725$ galaxy behind Abell 370 ($z=0.375$), which has been magnified into
an arc known as the “Dragon”. Kelly et al. (2022) reported the discovery of
several high-significance microlensing events in the Dragon with deep HST
observations, which already indicated that the microlensing rate in the Dragon
is particularly high due to its relatively low redshift and the high surface
brightness. Our finding with JWST confirms a very high occurrence of
microlensing events in the Dragon and opens up the possibility of conducting
statistical studies of highly magnified stars with a large number of events.
Multi-wavelength high-resolution observations with JWST’s NIRCam up to
$\lambda_{\rm obs}\sim 5\,{\rm\mu m}$ allow us to constrain stellar types of
some of the brightest microlensed stars.

This paper is organized as follows: in §2 we describe JWST observations used
in this study. In §3, we present data analyses and measurements. In §4, we
present the results of the study and discuss our findings. Finally, we
conclude with the summary in §5. Throughout this paper, we assume a cosmology
with $(\Omega_{m},\Omega_{\Lambda},h)=(0.3,0.7,0.7)$.

##  2 Observations

We used JWST NIRCam imaging (Rieke et al., 2023b) observations from two
programs: Cycle-1 GTO-1208 (the CAnadian NIRISS Unbiased Cluster Survey,
CANUCS, PI: C. Willot; Willott et al. 2022) and Cycle-2 GO-3538 (PI: E. Iani)
which are publicly available. These two programs targeted the Abell 370
cluster at $z=0.375$ using multiple NIRCam filters. Observations were executed
in December 2022 and December 2023, respectively, i.e., separated in a time
period of $\sim 1\,{\rm year}$. The Abell 370 field includes a giant arc of a
star-forming galaxy at $z=0.725$ also known as the “Dragon” (Soucail et al.,
1987a, b, 1988; Richard et al., 2010; Dai et al., 2018; Patrício et al., 2018;
Kelly et al., 2022), which is the target of this study (Upper panel of Figure
1).

GTO-1208 CANUCS obtained images in filters of F090W, F115W, F150W, F200W,
F277W, F356W, F410M, F444W, each with integration times of $\sim 6400\,{\rm
s}$. GO-3538 performed NIRCam wide-field slitless spectroscopy and direct
imaging in filters of F300M, F335M, F410M, and F460M, accompanied with short-
wavelength imaging using F182M ($\sim 19400\,{\rm s}$) and F210M ($\sim
19090\,{\rm s}$) filters. Direct imaging for long wavelength filters were also
taken with integration time of 2770–3092 ${\rm s}$.

All observations are calibrated using the standard JWST
pipeline111https://github.com/spacetelescope/jwst (Bushouse et al., 2023)
version v1.11.2, with reference file jwst_1188.pmap. We also included
customized routines to remove well-known artifacts, including the subtraction
of 1/f noise stripes in both row and column directions, template-based wisp
subtraction in NIRCam short-wavelength detectors, hot-pixel masking in long-
wavelength detectors, and manual masking of the persistence from bright
objects (e.g., Rigby et al., 2023; Rieke et al., 2023b, a). The astrometry of
each image was carefully corrected using a combined catalog of HST sources
detected with HFF/BUFFALO images (Steinhardt et al. 2020; primary) and DESI
Legacy Imaging Survey catalog (Dey et al. 2019; secondary), both registered to
Gaia (Gaia Collaboration et al., 2018). As a result, each image is well
aligned, with an absolute astrometric error of $<0\farcs 03$, and an internal
RMS astrometric error of $\approx 0\farcs 004$ for all $>5\sigma$ sources. The
final mosaicked images were drizzled with pix_frac=1.0 and a pixel size of
0$\farcs$03.

Figure 1:  Upper panel: A false-color image of the entire “Dragon” arc behind
the Abell 370 cluster (Soucail et al., 1987a), using JWST filters F090W,
F150W, and F200W. North is up, East to the left, and a reference angular scale
of $1^{\prime\prime}$ is shown by the solid horizontal bar at the bottom right
corner The dashed white rectangle shows the region of interest further
analyzed in Figure 2. Lower panels: F200W zoom in on a part of the Dragon in
the 2022 image (left panel) and in the effective F200W in 2023 (right panel).
The effective F200W image in 2023 was made using F182M and F210M images (see
§3.1). Examples of the apparently bright microlensing events are indicated,
where dashed half-crosses show bright sources seen only in 2022 data and solid
half-crosses show sources only in 2023. Many additional microlensing events
exist, but are only visible in the differential image in Figure 2. Horizontal
bars in the lower right corners show $1^{\prime\prime}$ scales.  Figure 2: Top
Panel: The observed $\sim 2\,{\rm\mu m}$ differential image between the 2022
and 2023 epochs. Positive signals (red) show objects that appear only in 2023,
while negative signals (blue) show sources only seen in 2022. Contours show
critical curves for $z=0.725$ estimated from several programs: the solid black
critical curve is derived with the WSLAP+ code (Diego et al., 2007), the
dashed green lines are from the lenstool software (Jullo & Kneib, 2009)
analyzed in Niemiec et al. (2023), and the dash-dotted purple lines are from
the Light-Traces-Mass (LTM) method described by Zitrin et al. (2009). Star
symbols indicate locations of previous HST detections of microlensing events
(Kelly et al., 2022). Lower Panels: Zoom-ins of crowded transient regions
indicated with A, B, C, and D letters in the upper panel. Circles ($r=0\farcs
12$) show detected microlensed events. Solid circles show lensed stars
significantly detected in 2023 epoch, while dashed circles indicate stars
detected events in 2022 epoch. In total, 46 microlensed stars are
significantly detected.

##  3 Analysis

###  3.1 Identifications of microlensing events

We first investigated the “Dragon” by directly comparing the F200W (2022) and
F182M (2023) images. where we find that several bright sources are only seen
in one of the epochs, representing potential transient events across the
Dragon arc. Then, to more precisely identify fainter transient events, we
subtracted a combined F182M + F210M image (2023) from the F200W image (2022).
We use $\lambda_{\rm obs}=2\,{\rm\mu m}$ images to identify transients instead
of the repeated F410M filter because the $2\,{\rm\mu m}$ data are deeper by
$\sim 1$ mag and higher spatial resolution. To make the 2023 epoch F182M +
F210M image (i.e., ’effective’ F200W image), we first performed point spread
function (PSF) homogenization for the F182M and F200W images to match their
PSFs to that of F210M. The PSF for each image was made using WebbPSF (Perrin
et al., 2014) applying the wavefront measurements for the closest date of each
observation. Convolution kernels were produced using pypher (Boucaud et al.,
2016) by applying a regularization factor of $0.0001$. The kernels were
convolved using the convolve2d function of the scipy.signal submodule
(Virtanen et al., 2020). Using the PSF-homogenized F182M and F210M images, we
created an effective F200W image by linearly interpolating fluxes at the pivot
wavelength of each filter. The differential image was then created by
subtracting the effective F200W image (2023) from the PSF-homogenized F200W
image (2022).

Finally, we subtracted the global background of the differential image. This
is to remove any large-scale background in the differential image, mostly
because of the slight difference in background (sky and intracluster light)
subtraction during the production of mosaicked images. As a result, a clean
differential image was created with the pixel distribution of $\sim 0.0\pm
0.8\,{\rm nJy}$. The differential image clearly shows significant positive and
negative peaks across the Dragon arc, indicating a large number of transient
events observed over the two epochs (Figure 2).

###  3.2 Transient Detection

We used DAOFIND (Stetson, 1987) incorporated in the photutils.detection
submodule and SExtractor (Bertin & Arnouts, 1996) to detect microlensing
transients from the differential image. These two source-finding algorithms
are complementary in a way that DAOFIND efficiently detects circular Gaussian
sources in crowded regions while Sextractor can detect less circular sources
that can also be found in the crowded regions in the differential image.

For the Sextractor run, to detect compact sources, we applied DETECT_MINAREA
$=5$ and DETECT_MINTHRESH $=2.68$ (i.e., $>6\,\sigma$ detection). To deblend
crowded sources, we applied a relatively low contrast parameter of
DEBLEND_MINCONT $=0.0005$. With these settings, we run Sextractor for positive
and negative instances of the differential image, corresponding to transient
events in 2023 and 2022, respectively. For DAOFIND run, we used a $4\sigma$
threshold of $0.03\,{\rm MJy/sr}$ ($0.6$ nJy pix-1) and a kernel full width at
half maximum of $2.5\,{\rm pixels}$. Similarly, DAOFIND was used for positive
and negative instances of the differential image.

For both of the source detections, we masked regions where bright sources
(e.g., bright cluster member galaxies and bulges of the Dragon arc) exist in
the original F200W image. This is to avoid the contamination from their
relatively strong residuals caused by minor PSF differences between 2022 and
2023.

To assess the impact of noise signal on detection, we ran the same source-
finding routine over the same F200W differential image of several types of
galaxies (e.g., clumpy or smooth morphology with different luminosities).
Except for genuine transients such as supernovae that can be also detected in
other bands, we do not detect any significant bad pixels that only appear in
the single band with similar significance. Therefore, we conclude that the
vast majority of transients detected on the Dragon arc are not data-quality
artifacts but real transient events.

We note that strong emission lines from H ii regions still have some
probability to contaminate as transients due to the imperfect match between
F182M$+$F210M and F200W filter transmission curves. At $z=0.725$, He i
$\lambda$1.083 $\mu$m and Paschen $\gamma$ can elevate the flux density from
F182M photometry, and Paschen $\beta$ may boost the F200W photometry if they
are strong enough. In the “head” of the Dragon arc (the least magnified
multiple image of the Dragon arc seen in the most eastern part of Figure 1),
three bright clumps can be tentatively seen as faint F200W (2022) excess
sources with peak brightness $\sim 1$ nJy pix-1 in the difference image.
However, counter images of these clumps are not selected in our sample, as
they are much fainter than our selected transients. Additionally, there are no
underlying compact star-forming clumps seen in continuum emission. For these
reasons, we conclude that the emission line contamination in our transient
sample is negligible.

Using the PSFphotometry routine of the photutils.psf module, we assessed the
sizes of the individually detected transient events by subtracting scaled PSFs
at the positions of detected transients. The residuals appear consistent with
background noise, leading us to conclude that the transients are consistent
with point sources.

###  3.3 Photometry

To measure the F200W fluxes of detected transients, we performed forced
aperture photometry on the difference images at the detected positions. We
used a fixed aperture photometry with an aperture radius of $0.06\,{\rm
arcsec}$. Photometric errors are estimated by placing random apertures with
the same size on the transient-free regions of differential images. To take
into account realistic Poisson noise from the host galaxy, we only place
random apertures where the surface brightness in F200W image is 0.4–50$\,{\rm
nJy}\,{\rm pixel}^{-1}$, i.e., similar to the surface brightness of the
background Dragon arc around the transient positions. We performed aperture
correction based on the PSF models obtained using WebbPSF (Perrin et al.,
2014). The $5\sigma$ detection limit is therefore determined as 28.75 AB mag
in the difference image.

We also performed photometry of transients using F410M images because F410M is
the only repeated filter used in both epochs. During this long wavelength
photometry, we limited our sample only to F410M bright sources, namely F410M
detection of $>4.5\,\sigma$. Because of the larger PSF sizes and higher sky
background, longer wavelength images only have limited detection capability.
Especially, within the bright region of the Dragon arc, large correlated noise
buries weak transient signals. Therefore, we only focused on F410M photometry
and analysis for the brightest microlensing events. For these bright sources,
we similarly performed forced aperture photometry with a fixed aperture radius
of $0.1\,{\rm arcsec}$. The F410M detection limit of transient is $\sim 27.7$
AB mag.

Table 1: F200W-based detections of microlensing events in 2022 and 2023 ID | RA | Dec | year | magF200W |  $\mu$ glafica |  $\mu$ WSLAP+b |  $\mu$ Zitrinc |  $\mu$ BUFFALOd |  $\mu$ HFFe
---|---|---|---|---|---|---|---|---|---
2201 | 39.969029 | -1.584647 | 2022 | $28.72\pm 0.2$ | 49 | 106 | 47 | 86 | 44
2202 | 39.969409 | -1.584811 | 2022 | $28.33\pm 0.14$ | 58 | 65 | 88 | 52 | 3439
2203 | 39.969423 | -1.584756 | 2022 | $28.56\pm 0.17$ | 38 | 39 | 82 | 31 | 926
2204 | 39.96945 | -1.58484 | 2022 | $28.31\pm 0.14$ | 112 | 333 | 245 | 97 | 38
2205 | 39.969477 | -1.584746 | 2022 | $27.54\pm 0.07$ | 102 | 50 | 92 | 59 | 1480
2206 | 39.969667 | -1.584871 | 2022 | $28.55\pm 0.17$ | 26 | 40 | 875 | 30 | 30
2207 | 39.96995 | -1.584894 | 2022 | $28.26\pm 0.13$ | 34 | 102 | 154 | 49 | 55
2208 | 39.970076 | -1.584973 | 2022 | $28.55\pm 0.17$ | 34 | 118 | 180 | 50 | 54
2209 | 39.970102 | -1.585041 | 2022 | $28.36\pm 0.14$ | 30 | 81 | 319 | 43 | 44
2210 | 39.970144 | -1.584943 | 2022 | $27.98\pm 0.1$ | 44 | 4639 | 108 | 75 | 84
2211 | 39.970337 | -1.585007 | 2022 | $28.53\pm 0.17$ | 59 | 114 | 88 | 312 | 235
2212 | 39.970382 | -1.585057 | 2022 | $28.65\pm 0.19$ | 56 | 141 | 96 | 741 | 162
2213 | 39.970386 | -1.585017 | 2022 | $27.9\pm 0.09$ | 69 | 88 | 84 | 2020 | 914
2214 | 39.970408 | -1.585102 | 2022 | $27.5\pm 0.06$ | 52 | 189 | 118 | 470 | 121
2215 | 39.970427 | -1.585014 | 2022 | $28.12\pm 0.11$ | 84 | 65 | 74 | 5875 | 1368
2216 | 39.970428 | -1.585096 | 2022 | $27.96\pm 0.1$ | 55 | 122 | 102 | 1453 | 167
2217 | 39.970453 | -1.584975 | 2022 | $28.08\pm 0.11$ | 136 | 48 | 64 | 239 | 2477
2218 | 39.970476 | -1.585282 | 2022 | $28.36\pm 0.14$ | 35 | 161 | 2061 | 95 | 54
2219 | 39.970735 | -1.585283 | 2022 | $28.4\pm 0.15$ | 64 | 98 | 167 | 113 | 668
2220 | 39.971087 | -1.584895 | 2022 | $28.38\pm 0.14$ | 35 | 16 | 37 | 11 | 16
2221 | 39.971776 | -1.584971 | 2022 | $28.45\pm 0.15$ | 160 | 120 | 199 | 4925 | 67
2222 | 39.971912 | -1.584815 | 2022 | $28.43\pm 0.15$ | 82 | 119 | 146 | 132 | 50
2223 | 39.97196 | -1.584844 | 2022 | $27.27\pm 0.05$ | 73 | 116 | 166 | 49 | 67
2224 | 39.972053 | -1.584797 | 2022 | $27.44\pm 0.06$ | 57 | 158 | 161 | 25 | 122
2225 | 39.972098 | -1.584749 | 2022 | $28.24\pm 0.13$ | 57 | 568 | 168 | 23 | 171
2226 | 39.972171 | -1.584843 | 2022 | $27.44\pm 0.06$ | 28 | 38 | 109 | 8 | 32
2227 | 39.972256 | -1.584715 | 2022 | $27.19\pm 0.05$ | 19 | 17 | 90 | 5 | 19
2228∗ | 39.972791 | -1.584742 | 2022 | $27.36\pm 0.06$ | 46 | 363 | 456 | 11 | 25
2301 | 39.969129 | -1.584608 | 2023 | $28.57\pm 0.17$ | 19 | 18 | 166 | 16 | 128
2302† | 39.96926 | -1.584196 | 2023 | $28.0\pm 0.1$ | 13 | 11 | 189 | 8 | 16
2303 | 39.969484 | -1.584777 | 2023 | $28.5\pm 0.16$ | 484 | 108 | 122 | 272 | 104
2304 | 39.970082 | -1.584929 | 2023 | $27.19\pm 0.05$ | 38 | 286 | 116 | 63 | 71
2305 | 39.970199 | -1.58508 | 2023 | $28.45\pm 0.16$ | 32 | 114 | 221 | 52 | 50
2306 | 39.97037 | -1.585106 | 2023 | $28.1\pm 0.11$ | 45 | 382 | 132 | 137 | 89
2307 | 39.970444 | -1.585255 | 2023 | $28.09\pm 0.11$ | 35 | 145 | 649 | 90 | 54
2308 | 39.970454 | -1.585061 | 2023 | $27.96\pm 0.1$ | 75 | 71 | 82 | 2142 | 856
2309 | 39.970586 | -1.585195 | 2023 | $28.59\pm 0.18$ | 64 | 84 | 109 | 237 | 398
2310 | 39.970627 | -1.585159 | 2023 | $27.59\pm 0.07$ | 94 | 54 | 83 | 84 | 3038
2311 | 39.971445 | -1.584845 | 2023 | $28.39\pm 0.15$ | 93 | 48 | 66 | 36 | 108
2312 | 39.971454 | -1.58492 | 2023 | $28.13\pm 0.12$ | 117 | 54 | 80 | 52 | 316
2313 | 39.971979 | -1.584869 | 2023 | $28.56\pm 0.17$ | 71 | 108 | 183 | 36 | 328
2314 | 39.97202 | -1.584807 | 2023 | $27.44\pm 0.06$ | 63 | 131 | 164 | 32 | 84
2315 | 39.972023 | -1.584868 | 2023 | $28.48\pm 0.16$ | 61 | 95 | 182 | 25 | 200
2316 | 39.972027 | -1.584802 | 2023 | $27.43\pm 0.06$ | 62 | 131 | 164 | 31 | 87
2317 | 39.972177 | -1.584793 | 2023 | $28.16\pm 0.12$ | 29 | 53 | 134 | 8 | 39
2318∗∗ | 39.972776 | -1.584769 | 2023 | $26.75\pm 0.03$ | 40 | 225 | 162 | 9 | 18

##  4 Results and Discussion

###  4.1 Summary of transient events in 2022 and 2023

From two independent source detections, we found 44 and 33 sources from
Sextractor and DAOFIND, respectively. 31 sources are detected with both
software. We merged sources identified in Sextractor and DAOFIND to make the
final transient catalog. In total, we detected 46 microlensing events from the
Dragon; 28 in the 2022 observation and 18 in the 2023 observation (Table 1).
Compared with previous identifications of microlensing events and magnified
single star observations (e.g., 8 in Dragon arc identified with HST
Flashlights program down to $3.3\sigma$ significance; Kelly et al., 2022), our
finding of 46 microlensing events in a single galaxy far surpasses previous
records, which shows the extremely high occurrence rate of microlensing events
in the Dragon arc captured by JWST. This is because the dragon arc has a lower
redshift than other galaxies that host microlensing events (e.g., Kelly et
al., 2018; Welch et al., 2022; Diego et al., 2023a), and therefore more stars
are visible to the JWST when highly magnified.

We also note that in Figure 2, many point-like residuals in the dragon arc are
currently not selected as transients because of our conservative $5\sigma$
detection limit. These surface brightness fluctuations are also likely caused
by fainter microlensed events, or events off from the peak phase. As the
magnitudes get fainter, we expect it increasingly probable that multiple,
independently micro-lensed stars blended within the PSF collectively
contribute to the observed flux variability. Such signals have been discussed
in the contexts of pixel variability (Tuntsov et al., 2004) and collective
variability of unresolved stellar associations (Dai, 2021). Future deep multi-
epoch JWST/NIRCam imaging observations will be able to examine this
interpretation.

The detection of microlensing events requires extremely high magnification
($\mu\gtrsim 10^{3}$). Therefore, they occur frequently around the critical
curves of the large-scale gravitational lens powered by major dark matter
halos of the galaxy cluster (macrolens). Our detections of microlensing events
are generally consistent with macrolens critical curves for $z=0.725$ objects
previously estimated in different studies and with different software and
methods (see Figure 2). The estimated magnifications from macrolens based on
existing cluster mass models are typically at $\mu\gtrsim 10^{2}$ despite
large uncertainties as indicated by the model-to-model scattering in Table 1.

Although the event positions are generally consistent with the macrolens
critical curves, we note that existing lens models of the Abell 370 cluster do
not simultaneously explain all the microlensing events. In particular, several
transient events have large angular offsets from the expected positions of
macrolens critical curves (e.g., region C in Figure 2). These large offsets
between the macrolens critical curve and microlensing events may indicate the
complex structure of the dark matter sub-halo structure of the lensing cluster
(e.g., Meneghetti et al., 2017; Dai et al., 2018, 2020). In-depth analyses of
detailed mass models with complex sub-halo distributions will be presented in
future works.

Furthermore, we identified an event which has a large angular offset ($\sim
1.8^{\prime\prime}$) from the Dragon arc. We call this event as event $\alpha$
(see Figure 2), and detailed analysis and discussion is presented in Section
4.3.1.

Figure 3: Color-magnitude diagram of F410M-detected transients. Square symbols
show events in 2022 and circular symbols show events in 2023. Filled space
shows the F410M detection limit. Except for event $\alpha$, $\beta$,
$0.4\,{\rm mag}$ of dust attenuation corrections are applied for all F200W -
F410M colors estimated from measurements of Patrício et al. (2018). Solid line
in the upper panel shows typical F200W - F410M color of different temperature
giant stars redshifted at $z=0.725$ (Lejeune et al., 1997). Right panel shows
a histogram of detected F200W magnitude and absolute magnitude at the rest-
frame J-band. The observed J-band absolute magnitude of these sources show
that these stars are strongly magnified by $\mu\gtrsim 1000-10000$ assuming
they are red giants or asymptotic giant stars (e.g., Madore et al., 2022).

###  4.2 Color magnitude diagram

The color magnitude diagram of the detected sources shows that most of the
lensed sources have red colors in F200W – F410M (roughly corresponding to
$J-K$ in the rest frame; Figure 3). Previous Herschel observations at 100–500
$\mu$m have shown that the Dragon arc is a dusty star-forming galaxy (Rawle et
al., 2016), and it exhibits significant interstellar dust attenuation of up to
${\rm E(B-V)}=0.4\,{\rm mag}$ (Patrício et al., 2018). This implies a
necessity of color corrections between F200W and F410M of up to $0.4\,{\rm
mag}$ to derive intrinsic F200W – F410M colors of each source, assuming a
uniform dust screen. We simply applied a color correction of $0.4\,{\rm mag}$
to demonstrate possible stellar type classification for detected sources,
except for the off-arc event alpha and beta, but we caution the potential
variation of dust attenuation of observed stars, which cannot be corrected
with the data taken so far.

Sources detected in microlensing events have F200W – F410M colors of
$-0.5\sim+0.5\,{\rm mag}$ (Figure 3). By comparing with theoretical stellar
atmosphere spectra of giant stars (lowest surface gravity available at each
effective temperature in the set of Lejeune et al. 1997, under the assumption
of $[\mathrm{M}/\mathrm{H}]=0$), these colors are consistent with typical
ranges of stellar spectra with temperatures of $T\lesssim 5000\,{\rm K}$. Due
to their low temperature and high luminosity, most of these sources are likely
to be red giants/supergiants (and potentially binaries) magnified by factors
of $\mu>1000$, and thus reaching absolute J-band magnitude of $-14\sim-16$ AB
mag (see also §4.3 below). These color distributions are particularly biased
toward red sources because we performed source detection in F200W filters and
because of the shallowness of F410M observations. Detecting higher temperature
stars at this redshift (e.g., $\gtrsim 8000\,{\rm K}$) requires multi-epoch
observations at $\lambda_{\rm obs}<1\,{\rm\mu m}$ (e.g., Kelly et al., 2022).

Figure 4: JWST single-epoch SEDs of event $\alpha$ (left) and event $\beta$,
$\beta^{\prime}$ (right). For event $\alpha$, we consider three possible
scenarios, including (i) microlensed red supergiant star in the halo of Dragon
arc (model from Lejeune et al. 1997; top pannel); (ii) nebular-phase SN Ia in
the foreground Abell 370 cluster (spectrum from Kwok et al. 2023; middle
panel); (iii) lensed SN Ia at $z=1.32$ with no obvious host detection (129-day
post-peak, Hsiao et al. 2007; bottom panel). All three templates fit the
observed SED with similar goodness. For event $\beta$ / $\beta^{\prime}$,
although they are found at almost identical positions with similar F200W
fluxes, their SEDs are largely different, showing that event $\beta$ and event
$\beta^{\prime}$ are different objects experiencing different microlensing
events.

###  4.3 SEDs analysis of bright microlensed sources

For three particularly bright events, event $\alpha$, event $\beta$ and event
$\beta^{\prime}$, we study their SEDs using multi-wavelength photometry. As
these events occurred in regions that are away from the bright arc, relatively
accurate background subtraction is possible using single-epoch data. We
conducted photometry of these events using background subtracted images of
observations in 2022 and 2023. Aperture photometry was performed with aperture
radii of $0.06\,{\rm arcsec}$ and $0.10\,{\rm arcsec}$ for short wavelength
filter images and long wavelength filter images, respectively. Annuli with a
width of $0.10\,{\rm arcsec}$ were used to measure local backgrounds, and flux
density uncertainties were measured from random aperture experiments.

####  4.3.1 Event $\alpha$

From our analysis of the difference images between the 2022 and 2023
observation, Event $\alpha$ has the largest angular separation from the arc
($\sim 1.8$″; Figure 2). The SED of event $\alpha$ is shown in the top-left
panel of Figure 4, and we find that a $T=2200\,{\rm K}$ template among the
examples in the stellar spectral library of Lejeune et al. (1997) provides the
best match to the observed SED. Therefore, if event $\alpha$ is a microlensing
event of a single star, it should be a cool star in the galaxy halo of the
Dragon arc. We also examine the F182M and F210M light curve of event $\alpha$
over epoch-2 (3.4-day span in the observer frame). Despite the data being
noisy per each telescope visit, we detect no evidence of significant
brightness change ($<0.7$ mag) within this time frame.

However, most of the existing cluster mass models do not explain the required
high magnification of event $\alpha$ needed to be observed as a microlensing
event (typical macrolens $\mu\sim$10–100 in Table 1). Assuming a macro$+$micro
lensing magnification of $\mu=1000$ at $z_{s}=0.725$ and no dust extinction,
event $\alpha$ should have absolute $J$-band magnitude of $-7.2$ AB mag in the
rest frame. Given the expected low number density of very luminous, short-
living stars in galaxy halos, the likelihood of event $\alpha$ being a
microlensing supergiant star in Dragon arc is very low. Although one can
assume a significantly larger lensing magnification to accommodate a lower
intrinsic luminosity of the source, this would require a much smaller impact
parameter (and thus a rarer chance) for the alignment of the microlens with
the background star. In addition, the separation of $\alpha$ from the host
galaxy is too large for an evolved star to be found at this distance, making
the interpretation of $\alpha$ as a microlensing event even less likely.
Therefore, we also consider the following physical explanations of event
$\alpha$ and explore their likelihood:

– Type Ia Supernova in the Abell 370: Possible. Given its proximity to one
cluster member galaxy of Abell 370, event $\alpha$ is likely a SN at the
cluster redshift. Because most of galaxies in Abell 370 cluster are quiescent
and although some cluster galaxies are known to show core-collapse supernovae
(e.g., Graham et al., 2012; Golubchik et al., 2023), we consider event
$\alpha$ a likely SN Ia. At the cluster redshift $z=0.375$, the observed F200W
magnitude implies an absolute magnitude of $-13.2$ AB mag at 1.5 $\mu$m. With
this brightness, it can only be a nebular phase SN Ia that is $\sim 200$ days
post-peak in the rest frame (e.g., Graur et al., 2020). We note that the two
epochs of JWST observations have a separation of 260 days in the $z=0.375$
rest frame, placing a stringent upper limit of the age of event $\alpha$ if it
is a cluster SN. The middle-left panel of Figure 4 compares the SED of event
$\alpha$ with the JWST spectrum of the nearby Type Ia supernova SN 2021aefx at
$+$255 days post peak (i.e., nebular phase; Kwok et al., 2023). We find that
the observed SED of event $\alpha$ can be explained by this spectral template
with similar $\chi^{2}$ ($\sim 8$) to that of the aforementioned cool star
template, and the residual is mostly dominated by the F460M non detection.
Therefore, we conclude that event $\alpha$ could be a SN Ia at the redshift of
the foreground cluster.

– Kilonova in the Abell 370: Unlikely. At $z=0.375$, the observed F200W
magnitude could match that of a kilonova like AT 2017gfo (the electromagnetic
counterpart of GW170817; Abbott et al. 2017) at $\sim 10$ days post merger
(Villar et al., 2017). At this epoch, AT 2017gfo has already shown a red rest-
frame $H-K$ color of 1 AB mag. However, this is not observed for event
$\alpha$ in F210M – F300M. Therefore, we conclude that event $\alpha$ is
unlikely a kilonova at the cluster redshift. However, our knowledge on
kilonovae’s light curves and SEDs could still be very limited, which could
show a large diversity because of different ejecta properties (Kawaguchi et
al., 2020).

– Luminous red nova in Abell 370 or Dragon arc: Unlikely. Massive star merger
can trigger luminous red novae reaching absolute V-band magnitude of $\sim-15$
mag (e.g., Smith et al., 2016). They can also show similar rest-frame near-
infrared color as that of event $\alpha$. However, we argue that the
likelihood of having a massive stellar progenitor in either the halo of Dragon
or Abell 370 quiescent member galaxies is considerably low.

– “Hostless” SN at $z\simeq 1-2$: Possible. Event $\alpha$ could also be a
lensed SN hosted by a low-mass galaxy that was not detected even by JWST. This
is also known as “hostless” supernova. If event $\alpha$ is a SN at $z\gtrsim
2$, based on glafic lens model (Oguri, 2010; Kawamata et al., 2018), we expect
the detection of a counter image of the SN with a short time delay to event
$\alpha$ ($\lesssim 20$ days) but potentially larger magnification. If event
$\alpha$ is a SN at $z\lesssim 0.8$, then with the large magnification
($\mu\sim 10$) and limited evolution time ($<1$ yr in observed frame), event
$\alpha$ should have been as bright as $\lesssim 27$ AB mag at 2 $\mu$m, i.e.,
1 mag brighter than the actual observed value. Therefore, if event $\alpha$ is
a hostless SN, it can only be at $z\simeq 1-2$. With the SN Ia spectral
templates from Hsiao et al. (2007) and software SNCosmo (Barbary et al.,
2024), we find that the SED of event $\alpha$ can be explained by SN Ia at
$z\sim 1.32$ with $\chi^{2}\sim 8$, for which the age is $t\sim+129$ days
post-peak ($+300$ days in the observer frame; Fig 4, bottom-left panel). At
this redshift, the multiple images are expected as nondetections in obtained
JWST data, either because of large difference of arrival time or proximity to
bright cluster member galaxies. Therefore, we do not rule out the possibility
of event $\alpha$ being a hostless SN at $z\sim 1.3$.

As a final remark, naturally there are other transient sources (e.g., tidal
disruption events) that could be considered to interpret event $\alpha$.
However, we believe that the above discussions cover most of the known
possibilities, and we conclude that event $\alpha$ could be also interpreted
as a nebular-phase SN Ia at the Abell 370 redshift or a “hostless” SN at
$z\sim 1.3$.

####  4.3.2 Event $\beta$ and $\beta^{\prime}$

The event $\beta$ and event $\beta^{\prime}$ are found in the 2022 and 2023
observations, respectively, at the most south-east position of the Dragon arc
(Figure 2). Both event $\beta$ and event $\beta^{\prime}$ are some of the
brightest events among the current microlensing events found in the 2022 and
2023 images. As the locations of these events are in the outer disk region of
the Dragon arc, we expect low dust attenuation for these two sources. Thus, we
assume no dust attenuation for sources observed in both event $\beta$ and
event $\beta^{\prime}$.

Right panel of Figure 4 shows the observed fluxes of event $\beta$ and event
$\beta^{\prime}$. Using the spectra of stellar atmosphere models of Lejeune et
al. (1997), we performed $\chi^{2}$ minimization. We find that low temperature
stars with surface temperature of $T=3000\,{\rm K}$ to $T=3300\,{\rm K}$ fit
well and a $T=3200\,{\rm K}$ star gives the minimum $\chi^{2}$ for the SED of
event $\beta$. Similarly, the observed SED of event $\beta^{\prime}$ is well
represented with a star with surface temperature of $T=3500-3700\,{\rm K}$
while a $T=3500\,{\rm K}$ star gives the minimum $\chi^{2}$. With these fits,
we find the apparent luminosity of the event $\beta$ and event
$\beta^{\prime}$ to be $\mu\,L_{\rm bol}\sim 1.1\times 10^{8}\,L_{\odot}$ and
$\mu\,L_{\rm bol}\sim 7.8\times 10^{7}\,L_{\odot}$, respectively.

Due to their low temperatures represented by their red colors, these events
would be microlensed red giant/supergiant stars that are strongly magnified
with $\mu\gtrsim 1000$. The different spectral shapes of event $\beta$ and
event $\beta^{\prime}$ suggest that they would be different stars, rather than
multiple images of a single star. The existence of such massive stars suggests
that there could be a star-forming region under the critical curve of this
position where we can expect more microlensing events in future observations.
At the same time, as low temperature stars at high-redshift are only bright in
$\lambda_{\rm obs}\gtrsim 2\,{\rm\mu m}$, these sources demonstrate the
importance of such near-IR observations to build complete stellar samples at
high-redshift.

An alternative interpretation would be that these two stars are indeed the
same star multiply imaged as a result of astrometric perturbation induced by
dark matter subhalos. Such events are predicted by Dai et al. (2018) and will
occur specifically around the critical crossing region such as the location
near event $\beta$ and $\it event{\beta^{\prime}}$ (see also Kelly et al.,
2018). In this case, the difference of the observed SEDs are interpreted by
intrinsic variability of the magnified star. In fact, such variability of
stars accompanying changes of surface temperature and SEDs are observed in
nearby stars such as Cepheid. Although a definitive conclusion is difficult to
obtain with existing photometry using single epochs, confirmations of such
magnified close-paired star in large numbers allow to investigate the subhalo
structure of dark matter as well as to constrain dark matter physics. Future
JWST’s time-domain observations using matched filters allow us to obtain
accurate multi-wavelength photometry of individually magnified stars and allow
us to conduct such studies.

##  5 Conclusion

In this study, we report the discovery of multiple microlensing events in a
strongly lensed $z=0.725$ galaxy Abell 370 “Dragon” arc. Using JWST
observations taken in 2022 and 2023, the microlensing events are found as
transients occurring in each epoch. In total, 46 microlensing events are
detected with $>5\,\sigma$ across the Dragon arc, showing the high rate of
microlensing events that can be captured by JWST observations.

Although the locations of the majority of microlensing events are consistent
with the macrolensing critical curves of existing mass models, a few of them
are significantly away from the expected critical curves. Events farther away
from the critical curve can be obtained in two ways; i) adding substructure on
small scales that can boost the magnification (and hence the probability of
microlensing) around them, such as millilenses, or ii) a shallow luminosity
function for the lensed stars increases the number of lensed stars farther
away from the critical curve (see Diego et al., 2024, for a discussion of
these two competing effects). A detailed study of these two scenarios based on
the events presented in this work is left for a future work.

Using multi-wavelength photometry, we compare rest-frame colors of the
microlensing events. Many of the detected sources have relatively red colors
due to dust attenuation from host galaxy’s interstellar medium as well as the
low atmospheric temperatures of the detected microlensed stars.

We then study multi-wavelength SEDs of a few transient events that are
relatively free from the background emission of the Dragon arc. In particular,
we study the SED of the event $\alpha$ that has the largest angular offset
from the Dragon arc ($1.8^{\prime\prime}$), and conclude that it is unlikely a
microlensed star because of the low surface density of massive stars in the
halo of Dragon arc. Instead, it may be a nebular-phase SN Ia at the redshift
of Abell 370 cluster ($z=0.375$), or a lensed SN Ia at $z\sim 1.32$, for which
the host galaxy is undetected. We also investigate SEDs of two events (event
$\beta/\beta^{\prime}$). We find their SEDs are consistent with stars with
atmospheric temperatures of $T=3200-3500\,{\rm K}$. Their proximity to the
critical curve and the similarity of their SEDs suggest that they could be a
multiply-imaged single variable star, while it is also possible that they are
two different stars individually microlensed.

Our findings demonstrate the unique power of time-domain observations in near-
IR wavelength using JWST. Further JWST multiple-epoch observations of the
Dragon arc, including the F182M and F210M imaging observations of GO-2883 (PI:
Sun), as well as other strongly lensed galaxies at high redshift will provide
us with essential information about stellar populations in high-redshift
galaxies. Further detailed analysis of such microlensing events will also
provide essential information about the lensing properties of dark matter sub-
halo distribution in galaxy clusters.

## Acknowledgments

We thank Lindsey Kwok for kindly sharing the JWST spectrum of SN 2021aefx, and
Jordi Miralda-Escude for insightful comments. This work is based on
observations made with the NASA/ESA/CSA James Webb Space Telescope. The data
were obtained from the Mikulski Archive for Space Telescopes at the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127
for JWST. These observations are associated with program #1208, 3538. The
authors acknowledge the JWST GO-3538 team led by PI E. Iani for developing
their observing program with a zero-exclusive-access period. All the BUFFALO
HST data used in this paper can be found in MAST: 10.17909/t9-w6tj-wp63
(catalog 10.17909/t9-w6tj-wp63). This work was supported by JSPS KAKENHI Grant
Number JP22K21349, JP23K13149, JP20H05856, JP22H01260, JP22J21. F.S. and E.E.
acknowledge JWST/NIRCam contract to the University of Arizona NAS5-02015. F.S.
acknowledges support for program #2883 provided by NASA through a grant from
the Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127.
J.M.D. acknowledges the support of project PID2022-138896NB-C51
(MCIU/AEI/MINECO/FEDER, UE) Ministerio de Ciencia, Investigación y
Universidades. L.D. acknowledges research grant support from the Alfred P.
Sloan Foundation (Award Number FG-2021-16495), and support of Frank and Karen
Dabby STEM Fund in the Society of Hellman Fellows. A.Z. acknowledges support
by Grant No. 2020750 from the United States-Israel Binational Science
Foundation (BSF) and Grant No. 2109066 from the United States National Science
Foundation (NSF); by the Ministry of Science & Technology, Israel; and by the
Israel Science Foundation Grant No. 864/23. E.Z. acknowledges project grant
2022-03804 from the Swedish Research Council (Vetenskapsrådet) and has also
benefited from a sabbatical at the Swedish Collegium for Advanced Study. M.J.
is supported by the United Kingdom Research and Innovation (UKRI) Future
Leaders Fellowship ‘Using Cosmic Beasts to uncover the Nature of Dark Matter’
(grants number MR/S017216/1 & MR/X006069/1). D.J.L. acknowledges support from
the UKRI FLF grants number MR/S017216/1 & MR/X006069/1. D.J.L. is also
supported by Science and Technology Facilities Council (STFC) grants
ST/T000244/1 and ST/W002612/1. E.I. acknowledges funding from the Netherlands
Research School for Astronomy (NOVA). R.A.W. and S.H.C. acknowledge support
from NASA JWST Interdisciplinary Scientist grants NAG5-12460, NNX14AN10G and
80NSSC18K0200 from GSFC. C.-C.C. acknowledges support from the National
Science and Technology Council of Taiwan (111-2112M-001-045-MY3), as well as
Academia Sinica through the Career Development Award (AS-CDA-112-M02). D.E.
acknowledges support from a Beatriz Galindo senior fellowship (BG20/00224)
from the Spanish Ministry of Science and Innovation, projects
PID2020-114414GB-100 and PID2020-113689GB-I00 financed by
MCIN/AEI/10.13039/501100011033, project P20-00334 financed by the Junta de
Andalucía, and project A-FQM-510-UGR20 of the FEDER/Junta de Andalucía-
Consejería de Transformación Económica, Industria, Consejerá, Universidades.
K.K. acknowledges the support by JSPS KAKENHI Grant Number JP17H06130,
JP22H04939, and JP23K20035. G.E.M. acknowledges financial support from the
Villum Young Investigator grants 37440 and 13160 and the Cosmic Dawn Center
(DAWN), funded by the Danish National Research Foundation (DNRF) under grant
No. 140. A.N. acknowledges funding from Olle Engkvists Stiftelse.

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# The last chapter of the Disquisitiones of Gauss

Laura Anderson, Jasbir S. Chahal, Jaap Top

###### Abstract

This exposition reviews what exactly Gauss asserted and what did he prove in
the last chapter of Disquisitiones Arithmeticae about dividing the circle into
a given number of equal parts. In other words, what did Gauss claim and
actually prove concerning the roots of unity and the construction of a regular
polygon with a given number of sides. Some history of Gauss’s solution is
briefly recalled, and in particular many relevant classical references are
provided which we believe deserve to be better known.

##  1 Introduction

A remark by J-P. Serre made towards one of us concerning what the textbook [2]
asserts about Gauss’ results on the construction of regular $n$-gons, provided
the initial motivation to write this note. A second source of inspiration came
from D. Surya Ramana’s expository paper [17] on Carl Friedrich Gauss
(1777–1855). Various useful additions as well as interesting historical notes
are presented in a slightly later text by B. Sury [21]. In 1801 Gauss
published his Disquisitiones Arithmeticae [5]. The purpose of the present
article is to elaborate on the remark of Serre and the comments by Ramana and
Sury concerning the last (seventh) chapter of this celebrated textbook. Its
last chapter is devoted to the study of roots of unity, i.e., the (complex)
solutions to an equation $x^{n}=1$. Gauss applied the results he obtained on
roots of unity to the problem of constructing a regular polygon of $n$ sides
or, what amounts to the same, divide the circle into $n$ equal parts. Here
(and everywhere in the present paper) ‘constructing’ means constructing by
using a straightedge and compass alone.

Ramana writes (p. 64 of [17])

> $\ldots$ Gauss proved that a regular polygon with $n$ sides, where $n$ is an
> odd integer, can be constructed using a ruler and compass alone if and only
> if $n$ can be expressed as a product of distinct Fermat primes.

Gauss indeed discovered and formulated a general result, namely a necessary
and sufficient condition on positive integer $n$ for a regular polygon with
$n$ sides to be constructible. He states it in the very last Article 366 of
Chapter VII of the Disquisitiones as follows (whenever we quote from Gauss’s
text, the 1986 revised English translation [5] is used) :

“In general therefore in order to be able to divide the circle geometrically
into $N$ parts, $N$ must be 2 or a higher power of 2, _or_ a prime number of
the form $2^{m}+1$, _or_ the product of several prime numbers of this form,
_or_ the product of one or several such primes into 2 or a higher power of 2;”

However, Gauss also writes (Article 365) that “the limits of the present work
exclude $\ldots$ demonstration here.” and (Article 335) “$\ldots$ although we
could discuss them (he refers to roots of unity in ${\mathbb{C}}$ here) in all
their generality, we reduce them to the simplest case $\ldots$, both for the
sake of brevity and in order that the new principles of this theory may be
more easily understood.” With this last sentence he apparently restricts
himself to $p$-th roots of unity, for (odd) prime numbers $p$.

The precise modern formulation of the statement by Gauss quoted above is

###### Theorem 1

A regular $N$-gon is constructible if and only if $N>1$ is of the form

$N=2^{r}p_{1}p_{2}\ldots p_{s}$

with $r,s\in{\mathbb{Z}}_{\geq 0}$ and $p_{j}$ distinct Fermat primes, i.e.,
primes of the form $2^{m}+1$.

Article 365 of the Disquisitiones contains a proof of the fact that if
$2^{m}+1$ is prime, then $m=2^{\nu}$, as well as the observation that Fermat
studied these numbers and that Euler discovered $2^{32}+1$ to be divisible by
$641$. Although Fermat believed all integers $2^{2^{\nu}}+1$ to be prime, it
is an old conjecture (for example, see [15, p. 315] that they are prime for
only finitely many integers $\nu$. Likely, $\nu=4$ is the largest of these.

Gauss finishes the Disquisitiones by listing all integers below $300$
satisfying the criterion of Theorem 1:

Many sources assert that Gauss proved Theorem 1, for example Chapter 5 of the
history textbook [2] written by the second author of the current paper. After
receiving a copy of these history notes, Jean-Pierre Serre pointed out to him
(as already alluded to in the first lines of this paper) that the
Disquisitiones does not contain a complete proof of the result. In the
remainder of the present note we review some of the famous history of the
problem of constructing regular $n$-gons, and its current mathematical
formulation. Note, incidentally, that Serre’s remark was observed by others
long before him: see, for example, the title of Pierpont’s 1895 text [16], a
paper discussing among other things a proof of Theorem 1. Also Sury’s text
[21, p. 48] points out that this theorem is stated but not proven in the
Disquisitiones.

###  1 Constructions with straightedge and compass

Constructions rely on two axioms already presented in Euclid’s Elements (Book
1):

  1. 1)

Any two distinct points may be joined by one and only one (straight) line.

  2. 2)

In the plane, a circle may be drawn with any point as its center and any
finite length segment as its radius.

Clearly, they are equivalent to allowing the use of two basic tools:

  1. 1)

A straightedge to join two distinct points, and

  2. 2)

a compass to draw circles with already determined centers and radii.

A convenient approach to this identifies the plane with the complex plane,
i.e., with the field of complex numbers ${\mathbb{C}}$. This is explained in
the first chapter of Klein’s 1895 textbook [10], and in terms of
${\mathbb{R}}^{2}$ in §37 of Hilbert’s textbook [9] of 1899, and using field
theory and ${\mathbb{C}}$ in §65 of Van der Waerden’s 1930 classic [22]. It
leads to the general criteria below.

###### Theorem 2

A complex number $z$ is constructible starting from $0$ and $1$ if and only if
one of the following equivalent statements holds:

  * (i)

Field extensions ${\mathbb{Q}}=K_{0}\subseteq K_{1}\subseteq\ldots\subseteq
K_{e}\subset{\mathbb{C}}$ exist such that every $K_{j}\subseteq K_{j+1}$ is
quadratic, and $z\in K_{e}$.

  * (ii)

The splitting field of ${\mathbb{Q}}(z)$ over $\mathbb{Q}$ has degree a power
of $2$.

The Greeks were interested in knowing what kind of figures are possible to
construct. In Book 1 (Proposition 1) of his Elements, Euclid shows how to
construct an equilateral triangle on a given line segment $AB$. In Book 1
(Proposition 46) it is shown how to construct a square. Finally, Book 4
(Proposition 11) describes a method to construct a regular pentagon.

A regular $n$-gon, or simply $n$-gon, is an $n$ sided convex figure in the
Euclidean plane with all sides equal, or equivalently with all the interior
angles equal to each other. For example, an equilateral triangle is a 3-gon
and a regular pentagon is a 5-gon. Since it is possible to bisect any given
angle, one can obtain a $2n$-gon from an $n$-gon.

More generally, given two positive integers $n,m$, constructing the regular
$n$-gon and the regular $m$-gon is equivalent to constructing the regular
$\mbox{lcm}(n,m)$-gon. Namely, put $k:=\mbox{lcm}(n,m)$. Constructing a
regular $k$-gon is the same as constructing an angle of size $2\pi/k$. Having
this, evidently all integral multiples of this angle can be constructed, so in
particular those of size $2\pi/n$ and $2\pi/m$. Vice versa, from these two
angles one can also construct angles of size $2\pi x/n+2\pi y/m$ for arbitrary
integers $x,y$. Taking $x,y$ such that $xm+yn=\mbox{gcd}(n,m)$ (such integers
$x,y$ exist by the extended Euclidean algorithm), one obtains

$x\frac{2\pi}{n}+y\frac{2\pi}{m}=\frac{2\pi}{nm}(xm+yn)=\frac{2\pi}{\mbox{lcm}(n,m)}.$

This is classical, and briefly recalled in Article 336 of the Disquisitiones.

The construction of an $n$-gon is equivalent to the construction of the
numbers $e^{2\pi im/n}=\cos(2\pi m/n)+i\sin(2\pi m/n)$ in the complex plane,
given the numbers $0$ and $1$. In other words, one tries to construct from $0$
and $1$ the complex solutions of $x^{n}=1$.

Ramana’s exposition [17] amply discusses the breakthrough by Gauss concerning
the construction of $n$-gons. In the words of Gauss himself (see [1]),

> “During the winter of 1796 (my first semester in Göttingen), I had already
> discovered everything related to the separation of the roots of the equation
>
> $\frac{x^{p}-1}{x-1}=0$
>
> into _two_ groups, on which the beautiful theorem…depends…. After intensive
> consideration of the relation of all the roots to one another on
> arithmetical grounds, I succeeded during a holiday in Braunschweig, on the
> morning of the day alluded to [March 29, 1796] (before I had got out of
> bed), in viewing this relation in the clearest way, so that I could
> immediately make special application to the 17-side and to the numerical
> verification…. I announced this discovery…in May or June 1796.”

In Article 365 of the _Disquisitiones_ Gauss observes,

> “It is certainly astonishing that although the geometric divisibility of the
> circle into three and five parts was already known in Euclid’s time, nothing
> was added to this discovery for 2000 years. And all geometers had asserted
> that, except for those sections and the ones that derive directly from them
> (that is, division into 15, $3\cdot 2^{\mu}$, $5\cdot 2^{\mu}$, and
> $2^{\mu}$ parts), there are no others that can be effected by geometric
> constructions.”

According to the German mathematician Heinrich Weber (1842–1913) famous for
the Kronecker-Weber theorem, Gauss requested that the $17$-gon be engraved on
his tombstone (footnote in § 97 of the first edition of [24] :

Translation: 1) Disq. arithmeticae, Chapter VII. As is recorded that
Archimedes ascertained that his epitaph show a ball with a cylinder, Gauss
expressed his wish that his epitaph immortalizes the figure of the $17$-gon.
This little tale shows the importance Gauss himself put on his discovery. On
the epitaph this wish was not fulfilled, yet on the monument erected for Gauss
in his native town Braunschweig, the statue is placed on a (barely visible for
the visitor) $17$-gon.

Both Ramana [17] and Sury [21] present a detailed derivation of Gauss’ formula
for $\cos(2\pi/17)$; whereas Ramana’s paper does not contain the resulting
formula itself, Sury’s one does. Note that the formula given in Article 365 of
the Disquisitiones actually contains a typo: it reads

and the rightmost minus sign on the first line should be a $+$, as confirmed
by (e.g.) Sury’s short and clear derivation [21, bottom of p. 46] (using his
notation, $\alpha_{3}/2=\cos(2\pi/17)$) and the resulting correct formula on
[21, p. 47]. From the modern perspective as given in Theorem 2 above, Gauss
found the explicit tower of quadratic field extensions

${\mathbb{Q}}\subseteq{\mathbb{Q}}(\sqrt{17})\subseteq{\mathbb{Q}}\left(\sqrt{34-2\sqrt{17}}\right)\subseteq{\mathbb{Q}}(\cos(2\pi/17))\subseteq{\mathbb{Q}}(e^{2\pi
i/17}).$

Although both the formula and the series of quadratic extensions lead to a
construction method of the $17$-gon, an actual construction is not discussed
in the Disquisitiones. According to R.C. Archibald’s historical survey [1],
J.F. Pfaff in 1802 wrote a letter to Gauss in which he quotes the construction
from a letter he received from C.F. Pfleiderer. Archibald reproduces the part
of Pfaff’s letter containing the construction:

(taken from R.C. Archibald [1].)

The next Fermat prime after $2^{4}+1=17$ is $2^{8}+1=257$. In 1831, Friedrich
Julius Richelot in his PhD thesis [18] (supervised by Carl Jacobi) presents a
construction of the $257$-gon. This was also done in 1887 by E. Pascal [14]
and in 1893 in a short book by Hugo Schwendenwein [19]. For the final known
Fermat prime $2^{16}+1=65537$, Johann Gustav Hermes in 1894 published a
construction of the 65537-gon [7]. The practical difficulty in constructing a
precise 65537-gon is obvious, for even 257 is so large that on a page of a
book, the 257-gon will look like a circle.

##  2 Proving Gauss’s claim

A contemporary proof of Theorem 1 goes like this; compare [16], [4, § 19.3]
and many similar texts.

For any integer $n\geq 1$, put

$\omega=e^{2\pi i/n}\in{\mathbb{C}}.$

The complex numbers

$1,\omega,\omega^{2},\ldots,\omega^{n-1}$

are the zeros of the polynomial

$x^{n}-1$

of degree $n$.

As remarked earlier, constructing a regular $n$-gon is equivalent to
constructing regular $p^{m}$-gons, for every prime power $p^{m}$ (with $m>0$)
exactly dividing $n$. Writing $\zeta=e^{2\pi ip^{-m}}$, by Theorem 2 the
latter is equivalent to the existence of field extensions

${\mathbb{Q}}=K_{0}\subset K_{1}\subset\ldots\subset
K_{e}={\mathbb{Q}}(\zeta)$

in which every $K_{j}\subset K_{j+1}$ is quadratic.

The extension ${\mathbb{Q}}(\zeta)/{\mathbb{Q}}$ has degree
$\phi(p^{m})=p^{m-1}(p-1)$ where $\phi$ is Euler’s $\phi$-function. By
assumption the extension $K_{e}/K_{0}$ has degree $2^{e}$. Hence if the
$p^{m}$-gon is constructible then either $p=2$ (and $m>0$ is arbitrary), or
$m=1$ and $p$ is a Fermat prime. Vice versa, if $\phi(p^{m})=2^{e}$, then the
observation that ${\mathbb{Q}}(\zeta)/{\mathbb{Q}}$ is a Galois extension of
degree $2^{e}$ with abelian Galois group, implies by the fundamental theorem
of Galois theory that $\zeta$ satisfies the condition for constructibility
given in Theorem 2. This finishes the proof of Theorem 1.

A special instance of Theorem 1 is that an $18$-gon, and therefore an angle of
$20^{\circ}$, cannot be constructed whereas a $6$-gon, so an angle of
$60^{\circ}$, can. This shows:

###### Corollary 3

Given only a compass and straightedge, it is impossible to trisect all given
angles.

In 1837, Pierre Laurent Wantzel [23] published a (different) proof of
Corollary 3, although it is an immediate consequence of Theorem 1. An
interesting discourse why Gauss might have emphasized constructibility while
somewhat neglecting the impossibility of constructions was recently given by
Lützen [12], [13]. These papers moreover discuss earlier proofs of Wantzel’s
theorem by R. Descartes (1637), J.É. Montucla (1754), and N.J.A.N. de
Condorcet (1775), all preceding the publication of the Disquisitiones.

In what follows, we quote what Gauss himself says of the proof of Theorem 1 he
left out.

Concerning $p^{\lambda}$-th roots of unity with $\lambda>1$ and $p$ an odd
prime, Article 336 of the Disquisitiones asserts

> “to determine a polygon of $p^{\lambda}$ sides we necessarily require the
> solution of $\lambda-1$ equations of degree $p$. Even though the following
> theory could be extended to this case also, nevertheless we could not avoid
> so many equations of degree $p$, and there is no way of reducing their
> degree if $p$ is prime.”

In today’s terminology, this seems to say that if $\zeta=e^{2\pi i/p}$ and
$\omega=e^{2\pi ip^{-\lambda}}$, then the extension $K(\omega)/K(\zeta)$ has
degree $p^{\lambda-1}$ and in fact a chain of intermediate fields

${\mathbb{Q}}(\zeta)=L_{0}\subset K_{1}\subset\ldots\subset
K_{\lambda-1}={\mathbb{Q}}(\omega)$

exists in which each $K_{j+1}$ has degree $p$ over $K_{j}$.

In his discussion of polynomials such that $\zeta=e^{2\pi i/p}$ can be
expressed in terms of their zeros (with $p$ an odd prime number not of the
form $1+2^{m}$), he writes (_Disquisitiones_ , Article 365), the
capitalization is by Gauss himself:

> “we can show with all rigor that these higher-degree equations cannot be
> avoided in any way nor can they be reduced to lower-degree equations. The
> limits of the present work exclude this demonstration here, but we issue
> this warning lest anyone attempt to achieve geometric constructions for
> sections other than the ones suggested by our theory (e.g. sections into 7,
> 11, 13, 19, etc. parts) and so spend his time uselessly.”

In modern terms, this is interpreted as the claim that the degree of
${\mathbb{Q}}(\zeta)$ over ${\mathbb{Q}}$, which equals $p-1$, is divisible by
an odd number $>1$.

For $n$ a Fermat prime, Gauss states (Disquisitiones, Article 365; note that
he uses $P$ for $2\pi$),

> “Whenever$\ldots$ the value of $n$ is 3, 5, 17, 257, 65537, etc. the
> sectioning of the circle is reduced to quadratic equations only, and the
> trigonometric functions of the angles $P/n$, $2P/n$, etc. can be expressed
> by square roots which are more or less complicated (according to the size of
> $n$). Thus in these cases the division of the circle into $n$ parts or the
> inscription of a regular polygon of $n$ sides can be accomplished by
> geometric constructions.”

In todays language, this claims that the field ${\mathbb{Q}}(\zeta)$ with
$\zeta=e^{2\pi i/p}$ and $p$ a Fermat prime, admits a chain of subfields

${\mathbb{Q}}=K_{0}\subset K_{1}\subset\ldots\subset
K_{e}={\mathbb{Q}}(\zeta),$

with each $K_{j+1}$ quadratic over $K_{j}$.

Write $\Phi_{n}$ for the minimal monic polynomial with rational coefficients
having $\omega=e^{2\pi i/n}$ as a zero. The degree of $\Phi_{n}$ equals the
degree of ${\mathbb{Q}}(e^{2\pi i/n})$ over ${\mathbb{Q}}$. Gauss determined
in Article 341 of the Disquisitiones that for $n=p$ a prime number,
$\Phi_{p}=x^{p-1}+x^{p-2}+\ldots+x+1$. This may be regarded as a proof of the
assertion in the second quote by Gauss mentioned in this section. Note that
Entry #136 in Gauss’s Diary [6] written in 1808 essentially says that he was
able to prove that for every $n$ the degree of $\Phi_{n}$ equals $\phi(n)$.
Section IV of Wantzel’s 1837 paper [23] contains the assertion that a slight
adaptation of Gauss’s argument for determining $\Phi_{p}$ gives $\Phi_{p^{r}}$
for all $r>0$. Although he refrained from explicitly observing it, Klein’s
determination (1895) of $\Phi_{p^{2}}$ sketched below in fact extends to the
case $\Phi_{p^{r}}=(x^{p^{r}}-1)/(x^{p^{r-1}}-1)$, confirming Wantzel’s
assertion. Earlier in 1850 Serret [20] in a less elementary way computed
$\Phi_{n}$ when $n$ is a power of a prime number. Kronecker [11] in 1854
determined $\Phi_{n}$ for all $n$, confirming what Gauss asserted in his diary
46 years earlier. A historical survey containing many interesting classical
references may be found in [15, Kapitel IV § 12]. In Section III.8 of [10]
Klein determines $\Phi_{n}$ for $n=p^{2}$ and $p$ prime, namely,
$\Phi_{p^{2}}=(x^{p^{2}}-1)/(x^{p}-1)$. In fact, replacing $x$ by $x+1$ allows
one to apply the well known Eisenstein criterion for irreducibility. Klein
continues by observing that the degree of this polynomial is a multiple of
$p$. Hence for $p$ odd and $n$ any multiple of $p^{2}$, the $n$-gon cannot be
constructed.

##  3 Conclusion

Gauss in the Disquisitiones stated a correct and complete criterion regarding
constructibility of a regular $n$-gon (Theorem 1 above).

Moreover, by determining (stated in modern terms) for any odd prime $p$ the
minimal polynomial $\Phi_{p}$ of $e^{2\pi i/p}$, he shows that indeed only the
prime $2$ and Fermat primes can occur as prime factors of $n$, in case the
$n$-gon is constructible.

Gauss correctly states the reason why no odd prime $p$ can have the property
$p^{2}|n$ in case the $n$-gon is constructible. However, he did not provide a
proof in this case. Based on a diary entry he wrote $7$ years after the
publication of the Disquisitiones, and on a very simple argument published
later by Klein and probably already known to Wantzel, it seems reasonable to
assume that indeed Gauss did have a complete proof of this, although maybe not
yet in 1801.

Finally by classical arguments, to show that indeed for integers $n$ of the
kind described by Gauss the $n$-gon is constructible, it suffices to show that
for any Fermat prime $F_{\nu}=2^{2^{\nu}}+1$ the $F_{\nu}$-gon is
constructible. Although we have not discussed it here, arguably Gauss knew how
to achieve this (see Articles 343-364 and the summary he presents in the first
lines of Article 365). The example $\nu=2$, so $F_{\nu}=17$ presented by him
illustrates this method.

## Acknowledgement

It is our pleasure to thank Sudhir Ghorpade for helpful advise concerning
publication of the present note. We are grateful for the comments of a
referee, in particular informing us of the paper [21] and of the typo in the
original publication of the Disquisitiones which we had not noticed.

## References

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Laura Anderson, 1065 West 10210 South, South Jordan, UT 84095, USA,
<EMAIL_ADDRESS>
Jasbir S. Chahal, Department of Mathematics, Brigham Young University, Provo,
UT 84602, USA<EMAIL_ADDRESS>
Jaap Top, Bernoulli Institute, University of Groningen, P. O. Box 407, 9700 AK
Groningen, The Netherlands<EMAIL_ADDRESS>