11institutetext: Department of Experimental Physics, Institute of Physics,
University of Szeged, Dóm tér 9, 6720 Szeged, Hungary
11email<EMAIL_ADDRESS>22institutetext: HUN-REN–SZTE Stellar
Astrophysics Research Group, Szegedi út, Kt. 766, 6500 Baja, Hungary
33institutetext: MTA-ELTE Lendület ”Momentum” Milky Way Research Group, Szent
Imre H. st. 112, 9700 Szombathely, Hungary 44institutetext: HUN-REN Research
Centre for Astronomy and Earth Sciences, Konkoly Observatory, Konkoly Th. M.
út 15-17., 1121 Budapest, Hungary 55institutetext: CSFK, MTA Centre of
Excellence, Konkoly Thege Miklós út 15-17, 1121 Budapest, Hungary
66institutetext: ELTE Eötvös Loránd University, Gothard Astrophysical
Observatory, 9400 Szombathely, Hungary 77institutetext: Center for
Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA
02138-1516, USA 88institutetext: The NSF AI Institute for Artificial
Intelligence and Fundamental Interactions 99institutetext: The School of
Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
1010institutetext: CIFAR Azrieli Global Scholars Program, CIFAR, Toronto,
Canada 1111institutetext: Department of Astronomy, University of Washington,
3910 15th Avenue NE, Seattle, WA 98195-0002, USA 1212institutetext: Las
Cumbres Observatory, 6740 Cortona Drive, Suite 102, Goleta, CA 93117-5575, USA
1313institutetext: Department of Physics, University of California, Santa
Barbara, CA 93106-9530, USA 1414institutetext: Center for Interdisciplinary
Exploration and Research in Astrophysics and Department of Physics and
Astronomy, Northwestern University, 1800 Sherman Ave., 8th Floor, Evanston, IL
60201, USA 1515institutetext: Space Telescope Science Institute, 3700 San
Martin Dr, Baltimore, MD 21218, USA 1616institutetext: Baja Astronomical
Observatory of University of Szeged, Szegedi út, Kt. 766, 6500 Baja, Hungary
1717institutetext: SETI Institute, 339 N. Bernardo Ave., Ste. 200, Mountain
View, CA 94043, USA 1818institutetext: Department of Physics and Astronomy,
Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, 08826, South Korea
1919institutetext: ELTE Eötvös Loránd University, Institute of Physics and
Astronomy, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary

# The story of SN 2021aatd – a peculiar 1987A-like supernova with an early-
phase luminosity excess

T. Szalai 112233 R. Könyves-Tóth 4455 A. P. Nagy 1122 D. Hiramatsu 7788 I.
Arcavi 991010 A. Bostroem 1111 D. A. Howell 12121313 J. Farah 12121313 C.
McCully 12121313 M. Newsome 12121313 E. Padilla Gonzalez 12121313 C.
Pellegrino 12121313 G. Terreran 12121313 E. Berger 7788 P. Blanchard 1414 S.
Gomez 1515 P. Székely 1122 D. Bánhidi 1616 I. B. Bíró 221616 I. Csányi 1616 A.
Pál 4455 J. Rho 17171818 J. Vinkó 4455111919

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Context. While reviewing the current classification scheme of supernovae (SNe)
is quite challenging on its own, there is a growing number of peculiar events
that cannot be assigned to any of the main classes. SN 1987A and a handful of
similar objects, thought to be explosive outcomes of blue supergiant stars,
belong to them: while their spectra closely resemble those of H-rich (IIP)
SNe, their light-curve (LC) evolution is very different (reaching their
primary LC peak in almost 3 months after a short and rapid rise, without any
plateaus). Other exciting phenomena are SNe with long-lasting LCs; some of
them, like the famous iPTF14hls, have shown multiple LC peaks and an extremely
slow spectral evolution.

Aims. Here we present the detailed photometric and spectroscopic analysis of
SN 2021aatd, a peculiar Type II explosion: while its early-time evolution
resembles that of the slowly evolving, double-peaked SN 2020faa (however, at a
lower luminosity scale), after $\sim$40 days, its LC shape becomes similar to
that of SN 1987A-like explosions.

Methods. Beyond comparing LCs, color curves, and spectra of SN 2021aatd to
that of SNe 2020faa, 1987A, and of other objects, we compare the observed
spectra with our own SYN++ models and with the outputs of published radiative
transfer models. We also carry out a detailed modeling of the pseudo-
bolometric LCs of SNe 2021aatd and 1987A with a self-developed semi-analytical
code, assuming a two-component (‘core’ + ‘shell’) ejecta and involving the
rotational energy of a newborn magnetar in addition to radioactive decay.

Results. We find that both the photometric and spectroscopic evolution of SN
2021aatd can be well described with the explosion of a $\sim$15 $M_{\odot}$
blue supergiant star. Nevertheless, SN 2021aatd shows higher temperatures and
weaker Na i D and Ba ii 6142 Å lines than SN 1987A, which is reminiscent of
rather to IIP-like atmospheres. With the applied two-component ejecta model
(counting with both decay and magnetar energy), we can successfully describe
the bolometric LC of SN 2021aatd, including the first $\sim$40-day long phase
showing an excess compared to 87A-like SNe but being strikingly similar to
that of the long-lived SN 2020faa. Nevertheless, finding a unified model that
also explains the LCs of more luminous events (like SN 2020faa) is still a
matter of concern.

###### Key Words.:

supernovae: general – supernovae: individual: SN 2021aatd, SN 1987A, SN
2020faa – stars: massive

##  1 Introduction

Type II supernovae (SNe), the cataclysmic endings of H-rich massive stars,
constitute a significant fraction of of all detected SN explosions (reaching
$\sim$35-40% in galaxies closer than $z\sim 0.05$, see e.g. Perley et al.
2020). Based on their spectral and light-curve (LC) properties, which seem to
strongly connect to the rate of lost H envelopes prior to explosion, Type II
SNe are divided into various subgroups.

Progenitors of Type IIP events, showing a plateau in their LC, are thought to
be red supergiant (RSG) stars with most of their H envelope preserved; while
Type IIL SNe show a linear decline from peak magnitude, probably due to the
lower amount of H that could be ionized and recombined after explosion (see
e.g. Patat et al. 1994; Hiramatsu et al. 2021). However, despite some obvious
photometric and spectral differences, there are doubts on whether Type IIL and
IIP SNe truly originate from different types of progenitors (see, e.g.,
Anderson et al. 2014; Valenti et al. 2016).

A further kind of such explosions, Type IIb ones, constitute an intermediate
subgroup between Type II and the H-poor, stripped envelope Type Ib/c SNe; they
can be recognized due to the prominent He features that appear in their
spectra after a few days, and the double-peaked shape of their LC (note,
however that the first peak is not always detectable). Moreover, we also
define Type IIn SN explosions showing narrow lines in their spectra,
indicating interaction between the SN ejecta and a dense circumstellar medium
(CSM), see e.g. Schlegel (1990); Filippenko (1997). More recently, extremely
bright events, called superluminous (SL)SNe, have also been found; those
showing a H-rich spectrum are called SLSNe-II (Gal-Yam 2012; Branch & Wheeler
2017). While their origin is still unclear, it has been already revealed that
some of them also show signs of strong CSM interaction (similarly to Type IIn
ones), while in other cases, alternatives are necessary to explain the
extremely high luminosity of these events.

While it is challenging to handle this classification scheme itself, there is
a growing number of peculiar events that cannot be assigned to any of the
groups described above. SN 1987A, the closest and best-studied extragalactic
SN appeared in the Large Magellanic Cloud (LMC), is a typical example: while
its spectra closely resemble to those of SNe IIP, its LC evolution is very
different (reaching its primary LC peak in almost 3 months after a short and
rapid rise, without any plateaus, see e.g. Hamuy et al. 1988; Whitelock et al.
1988; Catchpole et al. 1988). Shortly after the first studies, it was revealed
that the progenitor of SN 1987A was a compact blue supergiant (BSG),
contradicting the stellar evolutionary models developed until that time (e.g.
Woosley et al. 1988; Podsiadlowski et al. 1992). Since the discovery of SN
1987A, extended efforts have been made to find the reason of exploding BSG
stars including low metallicity, strong mixing of atmospheric layers, rapid
rotation, binary interaction, or stellar mergers (see a recent study on
modeling strategies in Dessart & Hillier 2019). Up to now, only a handful of
similar events have been found; however, their number is also growing thanks
to the effective analysis of data obtained during transient sky surveys (see
e.g. Arcavi et al. 2012; González-Gaitán et al. 2015; Taddia et al. 2016; Sit
et al. 2023).

Nowadays, also thanks to the expanding capacities for long-term follow-up of
more and more transient events, researchers have the chance to focus on a
similarly exciting phenomenon: the existence of SNe with long-lasting,
sometimes multipeaked LCs. A strong motivation to look for such kind of events
was the discovery of the luminous, extraordinarily long-lived Type II
iPTF14hls that has showed at least five LC peaks and an extremely slow
spectral evolution (reaching the nebular phase after more than 600 days,
Arcavi et al. 2017; Sollerman et al. 2019). Among others, enhanced CSM
interaction, magnetar formation, fall-back accretion, common-envelope jets,
and pulsational pair-instability SNe have been invoked to explain this unique
phenomenon (see Woosley 2018, for a review). During the extended analysis of
Zwicky Transient Facility (ZTF) data, Soraisam et al. (2022) identified almost
40 SNe of various types showing multipeaked LCs, including SN 2020xkx, a true
photometric analog of iPTF14hls (unfortunately, without a well-sampled
spectroscopic follow-up). Another ZTF object was SN 2020faa, which seemed to
be also quite similar to iPTF14hls (at least, during the first $\sim$200 days
of its evolution) and has been analyzed in detail by Yang et al. (2021) and
Salmaso et al. (2023).

In this work, we present the detailed photometric and spectroscopic analysis
of SN 2021aatd, another peculiar Type II explosion. While its early-time
evolution resembles to that of SN 2020faa and iPTF14hls (however, at a lower
luminosity scale), after $\sim$40 days, its LC shape becomes definitely
similar to that of SN 1987A-like explosions. This paper is organised as
follows. First, we present our ground-based photometric and spectroscopic
observations in Sect. 2. Then, we describe the steps of comparative LC and
spectral modeling analysis in Section 3. Finally, we discuss the results and
provide our concluding remarks in Section 4.

##  2 Observations

Figure 1: $BVgri$ photometry of SN 2021aatd. Filled and empty circles denote
measurements are from LCO sites and from Baja Observatory, respectively.
Dotted vertical lines mark the epochs of spectroscopic observations shown in
Table LABEL:tab:spec.

SN 2021aatd (ATLAS21bjwk) was discovered by the Asteroid Terrestrial-impact
Last Alert System (ATLAS, Tonry et al. 2018; Smith et al. 2020) on 2021
October 7.38 UT (MJD 59494.38), at an apparent brightness of mo =
18.35$\pm$0.10 mag (Chen et al. 2021), with an epoch of last non-detection on
2021 October 5.51 UT (MJD 59492.51). The SN is offset by 2.1 arcsecs south,
5.7 arcsecs west from its likely host galaxy GALEXASCJ005904.40-001210.0. The
first spectroscopic classification was reported by Wyatt et al. (2021) who
found the object being consistent with a young, peculiar Type II SN (showing
the best match with the spectrum of SN 1987A).

The redshift of the host galaxy is given as z = 0.01524$\pm$0.00004 in
NASA/IPAC Extragalactic Database (NED111https://ned.ipac.caltech.edu/),
corresponding to a Hubble-flow distance of d = 62.6$\pm$4.4 Mpc (assuming H0 =
67.8 $\pm$ 4.7 km s-1 Mpc-1). The Galactic extinction of $E(B-V)=0.025$ was
also adopted from NED (Schlafly & Finkbeiner 2011). Because of the lack of
early-time high-resolution spectra, we do not have any information on the
extinction in the host galaxy; however, based on the position of the SN, host
reddening is probably not significant.

###  2.1 Photometry

Figure 2: Optical spectra of SN 2021aatd obtained from LCO sites (black), and
with Magellan (blue).

We performed optical Johnson $BV$ and Sloan $gri$ (hereafter $BVgri$)
photometry of SN 2021aatd with follow-up observations from Las Cumbres
Observatory (LCO) utilizing a world-wide network of telescopes under the
Global Supernova Project (GSP, Howell 2019), and with the 0.8m Ritchey-
Chrétien telescope at the Baja Observatory, Hungary (Barna et al. 2023). The
median combination and the aperture photometry of LCO data was carried out
applying self-written pipelines using the fitsh software (Pál 2012). Data
obtained at Baja Observatory were processed with standard IRAF222IRAF is
distributed by the National Optical Astronomy Observatories, which are
operated by the Association of Universities for Research in Astronomy, Inc.,
under cooperative agreement with the National Science Foundation. routines,
including basic corrections. Then we co-added three images per filter per
night aligned with the wcsxymatch, geomap and geotran tasks. We obtained PSF
photometry on the co-added frames using the daophot package in IRAF, and image
subtraction photometry based on other IRAF tasks like psfmatch and linmatch,
respectively. For the image subtraction we applied a template image taken at a
sufficiently late phase, when transient was no longer detectable on our
frames.

The photometric calibration was carried out using stars from Data Release 1 of
Pan-STARRS1 (PS1 DR1) 333https://catalogs.mast.stsci.edu/panstarrs/. In order
to get reference magnitudes for our $B$\- and $V$-band frames, the PS1
magnitudes were transformed into the Johnson $BVRI$ system based on equations
and coefficients found in Tonry et al. (2012). Finally, the instrumental
magnitudes were transformed into standard $BVgri$ magnitudes by applying a
linear color term (using $g-i$) and wavelength-dependent zero points. Since
the reference stars fell within a few arc-minutes around the target, no
atmospheric extinction correction was necessary. S-corrections were not
applied. $BVgri$ photometry of SN 2021aatd obtained from LCO sites and from
Baja Observatory are collected in Tables LABEL:tab:phot_data_LCO and
LABEL:tab:phot_data_Baja, respectively; light curves (LCs) are plotted in Fig.
1.

###  2.2 Spectroscopy

Figure 3: Comparison of the early- and late-time $r$-band photometric
evolution of SN 2021aatd to that of iPTF14hls ($r+R$ filters, Arcavi et al.
2017), SNe 1987A ($R$ filter, Hamuy et al. 1988; Whitelock et al. 1988;
Catchpole et al. 1988, 1989), 2005cs (Pastorello et al. 2009), 2006au (Taddia
et al. 2012), 2020faa (Yang et al. 2021; Salmaso et al. 2023), and 2020xkx
(Soraisam et al. 2022, downloaded from the ALeRCE ZTF Explorer). Note that in
the case of SN 2006au and iPTF14hls, there is a large uncertainty regarding
the date of explosion; thus, their LCs are shifted to match with the LC maxima
of SNe 1987A and 2020faa, respectively (following the method of Taddia et al.
2012; Yang et al. 2021). Figure 4: Comparison of early-phase absolute
magnitudes of SNe 2020faa and 2021aatd. Latter values are shifted by $-$1.8
mag.

Figure 5: Comparison of the light- and color curves of SN 2021aatd to that of
SNe 1987A and 2020faa.

We obtained 11 sets of optical spectra at the LCO with the FLOYDS
spectrographs mounted on the 2m Faulkes Telescope North (FTN) at Haleakala
(USA) and the identical 2m Faulkes Telescope South (FTS) at Siding Spring
(Australia), through the GSP, between 2021 Oct 9 and 2022 January 13. A
2″-wide slit was placed on the target at the parallactic angle (Filippenko
1982). We extracted, reduced, and calibrated 1D spectra following standard
procedures using the FLOYDS pipeline (Valenti et al. 2014).

One late-time spectrum was obtained on 2022 July 26 with the Low Dispersion
Survey Spectrograph 3 (LDSS-3) on the 6.5m Magellan Clay telescope at Las
Campanas Observatory in Chile. Standard reductions were carried out using
IRAF, including bias subtraction, flat-fielding, cosmic-ray rejection, local
sky subtraction, and extraction of one-dimensional spectra. The slit was
aligned along the parallactic angle to minimize differential light losses, and
flux calibration was done using a spectrophotometric standard taken that night
at similar airmass.

The log and a summarizing plot of our spectroscopic observations on SN
2021aatd are shown in Table LABEL:tab:spec and in Fig. 2, respectively.

Table 1: Peak times and magnitudes in the LCs of SN 2021aatd |  $T_{\textrm{peak,1}}$ (MJD) | Peak mag1 | Peak abs. mag1 |  $T_{\textrm{peak,2}}$ (MJD) | Peak mag2 | Peak abs. mag2
---|---|---|---|---|---|---
$B$ | 59497.8 $\pm$ 2.2 | 17.94 $\pm$ 0.08 |  $-$16.13 $\pm$ 0.17 | 59581.1 $\pm$ 0.8 | 18.25 $\pm$ 0.04 |  $-$15.83 $\pm$ 0.16
$V$ | 59503.9 $\pm$ 2.3 | 17.83 $\pm$ 0.04 |  $-$16.22 $\pm$ 0.16 | 59581.4 $\pm$ 1.0 | 17.45 $\pm$ 0.05 |  $-$16.60 $\pm$ 0.16
$g^{\prime}$ | 59499.9 $\pm$ 0.6 | 17.76 $\pm$ 0.04 |  $-$16.31 $\pm$ 0.16 | 59581.6 $\pm$ 0.7 | 17.67 $\pm$ 0.04 |  $-$16.40 $\pm$ 0.16
$r^{\prime}$ | 59506.4 $\pm$ 1.5 | 17.66 $\pm$ 0.08 |  $-$16.38 $\pm$ 0.17 | 59581.6 $\pm$ 0.5 | 17.28 $\pm$ 0.03 |  $-$16.76 $\pm$ 0.16
$i^{\prime}$ | 59508.2 $\pm$ 3.6 | 17.80 $\pm$ 0.06 |  $-$16.23 $\pm$ 0.16 | 59583.0 $\pm$ 1.0 | 17.26 $\pm$ 0.06 |  $-$16.77 $\pm$ 0.16
Table 2: Basic data of Type II SNe used for comparison

Name | Date of discovery | $z$ | $D$ | E(B$-$V) | Source
---|---|---|---|---|---
| (MJD) |  | (Mpc) | (mag) |
SN 2021aatd | 59494.4 | 0.015a | 62.6$\pm$4.4a | 0.025a | This work
SN 1987A | 46849.8 | 0.0009 | 0.051 | 0.180 | Hamuy et al. (1988)
SN 2005cs | 53549.0 | 0.0018 | 7.1$\pm$1.2 | 0.050 | Pastorello et al. (2009)
SN 2006au | 53801.2 | 0.0098 | 46.2$\pm$3.2 | 0.300 | Taddia et al. (2012)
iPTF14hls | 56922.5 | 0.035 | 156 | 0.014 | Arcavi et al. (2017)
SN 2020faa | 58933.1 | 0.041 | 187 | 0.022 | Yang et al. (2021)
SN 2020xkx | 59140.0 | 0.042 | 190 | 0.070 | Soraisam et al. (2022)

Notes. aAdopted from NASA/IPAC Extragalactic Database (NED,
https://ned.ipac.caltech.edu).

##  3 Analysis and Results

###  3.1 Light-curve & color evolution

SN 2021aatd shows a double-peak LC in all filters. we fit single Gaussian
functions to the brightness values around the peaks to get proper estimations
on the peak magnitudes and epochs (see Table 1). Blue-band LCs peak
systematically earlier than redder bands (note that $B$-peak is very close to
the epoch of our first observation). After the initial peak, the SN shows a
declining phase up to $\sim$40 days when it starts to brighten again. In all
filters, except $B$-band, the SN reaches a second peak magnitude higher, than
the previous one. The second peak of the LC is monitored with a higher cadence
– while, considering the mean values of the second peak times, there is a
2-days shift from $B$ to $i$-band maximum, their uncertainty ranges actually
overlap. After the second peak, the LCs of SN 2021aatd show a rapid decline of
about 0.6$-$1.1 mag (from the longest to the shortest wavelength,
respectively) in $\sim$20 days before the SN disappeared behind the Sun.
Observations continued from $\sim$240 d up to $\sim$400 d after discovery; in
this phase, the SN showed a slow linear declining tail.

The late-time LC evolution of SN 2021aatd – peak brightness reached $\sim$3
months after explosion, followed by a rapid decline and, finally, by a linear
declining tail – resembles quite well to the behaviour of 1987A-like SNe.
Nevertheless, neither SN 1987A, nor the few known similar events assumed to be
explosions of blue supergiant stars, show the first peak in their LCs (see a
recent overview in Xiang et al. 2023). This suggests that, during the first
several weeks of the evolution, a different LC powering mechanism may exist in
the case of SN 2021aatd. We searched objects with a similar early-time
behaviour in recently published studies focusing on optically rebrightening
SNe. As we show in detail below, the early-phase LC evolution of SN 2021aatd
is very similar to that of special Type II SN 2020faa (Yang et al. 2021;
Salmaso et al. 2023) – however, SN 2020faa was a more luminous event.

In Fig. 3, we present the comparison of the early- and late-time $r$-band
photometric evolution of SN 2021aatd to those of SN 2020faa, SN 1987A ($R$
filter, Hamuy et al. 1988; Whitelock et al. 1988; Catchpole et al. 1988, 1989)
the 87A-like event SN 2006au (Taddia et al. 2012), the low-luminosity Type IIP
SN 2005cs (Pastorello et al. 2009), as well as the rebrightening objects
iPTF14hls ($r+R$ filters, Arcavi et al. 2017) and SN 2020xkx (Soraisam et al.
2022, downloaded from the ALeRCE ZTF Explorer444https://alerce.online/). Basic
data of the SNe used for generating absolute magnitudes are shown in Table 2.
Note that, in the case of SN 2006au and iPTF14hls, there is a large
uncertainty regarding the date of explosion; thus, their LCs are shifted to
match with the LC maxima of SNe 1987A and 2020faa, respectively (following the
method of Taddia et al. 2012; Yang et al. 2021).

As it can be seen in Fig. 3, 87A-like and rebrightetning SNe (iPTF14hls,
2020faa, 2020xkx) form two distinct luminosity groups. Note, however, that LC
of SN 2020faa starts to quickly decline $\sim$200 days after explosion, on the
contrary to the other two objects (see Salmaso et al. 2023). SN 2021aatd,
apparently, belongs to the group of 87A-like events. At the same time, the
first $\sim$40 days of its LC evolution closely resemble to that of bright,
slowly evolving SNe, especially of 2020faa. Early-phase LC similarities of SNe
2021aatd and 2020faa can be seen even better in Fig. 4, which shows the
comparison of early-phase absolute LCs not only in $r$, but also in $g$ and
$i$ filters. To get an almost perfect match, one should shift the LCs of SN
2021aatd by $\sim$1.8 mag.

SN 2006au seems to also show some early-phase flux excess compared to the LC
shape of SN 1987A (note that, in general , SN 2006au was a bit more luminous
event). Unfortunately, the earliest part of the LC of SN 2006au is probably
missing (the discovery of the object occured more than one year since the
previous non-detection, see Taddia et al. 2012); thus, it is difficult to use
this object for a detailed comparative analysis. We also omitted iPTF14hls and
SN 2020xkx from the further LC analysis, since their late-time evolution is
completely different from that of SN 2021aatd. Therefore, we only show a
detailed comparison of light and color curves of SN 2021aatd to that of SNe
1987A and 2020faa (based on $B$, $V$, $R/r$, and $g$, $r$, $i$ data,
respectively), see Fig. 5.

SN 2021aatd, as has already been presented in Fig. 4, shows a very similar
early-time LC evolution to that of SN 2020faa, but at lower luminosities,
which can be also well seen in both $g-r$ and $r-i$ curves (Fig. 5, bottom
right panel). We also discussed above the general differences between the
late-time LCs of the two events. Moreover, taking another look at the right
panels of of Fig. 5, the slope of the LCs of SNe 2020faa and 2021aatd seem to
be quite similar in $i$ band, again, while this is not the case in the $g$ and
$r$ bands (note however, that very late-time $g$\- and $r$-band data of SN
2020faa are close to the detections limit, see Salmaso et al. 2023). After
comparing the light and color curves of SN 2021aatd with that of SN 1987A, we
see similar evolution after the first $\sim$50 days. However, SN 2021aatd
seems to be a bit bluer in general – but, note again that we have no
information on the host-galaxy extinction in the direction of SN 2021aatd.

###  3.2 Spectral analysis

Figure 6: Left: The comparison of the +6d spectrum of SN 2021aatd to that of
SNe 2020faa (Yang et al. 2021), 2017eaw (IIP, Szalai et al. 2019) and 1987A
(Pun et al. 1995) obtained at +6, +8, and +7 days phase, respectively. All the
spectra are corrected for redshift and extinction. Spectra of SNe 1987A and
2020faa have been downloaded from WISeREP database. Right: The same comparison
of spectra of the four SNe obtained at +83, +88, +83, and +84 days,
respectively.

After the detailed comparative LC analysis, in Fig. 6, we present a set of
spectra of SNe 2021aatd, 2020faa, and 1987A obtained at both a very early
epoch (6-8 days after explosion) and around the second LC peak of SN 2021aatd
(+83-88 days). For a further comparison, we also plot the same-age spectra of
SN 2017eaw, one of the best-studied Type IIP explosion.

As can be seen, in accordance with the color evolution shown in Fig. 5, SNe
2021aatd and 2020faa have basically the same temperature evolution up to the
peak brightness, while SN 1987A has a redder spectrum at both epochs. SN
2017eaw shows the bluest spectrum at early time, but, at +83/88 days, its
temperature becomes definitely lower than that of SNe 2021aatd and 2020faa.
The slow spectral evolution is a well-known characteristic of multiple LC-
peaked SNe like iPTF14hls or SN 2020faa (Arcavi et al. 2017; Sollerman et al.
2019; Yang et al. 2021; Salmaso et al. 2023); up to $\sim$100 days after
explosion (when its last ”photospheric” spectrum was obtained), 2021aatd seems
to also belong to this group. At both early and later epochs, spectra of SN
2021aatd show strong H Balmer lines, similarly to that of SNe IIP and 1987A.
At the same time, one can see differences in the cases of Na i D and Ba ii
6142 Å two other spectral features definitely worth analyzing during the
comparison of Type II SNe. In the case of SNe 1987A and 2017eaw, Na i D has a
deep P Cygni profile at +83/88 days, while SNe 2021aatd and 2020faa shows only
weak Na i D lines. In the spectra of SN 1987A, Ba ii 6142 Å is also a dominant
line, while this is not the case in spectra of other three objects. It has
been described in several works that there is a diversity in the strength of
Ba ii 6142 Å line among 87A-like SNe, and weakness of this line is rather a
characteristic of IIP-like SN atmospheres (see e.g. Takáts et al. 2016;
Dessart & Hillier 2019; Xiang et al. 2023).

Using our optical spectra – following the general method also used in the
cases of SN 2020faa (Yang et al. 2021; Salmaso et al. 2023) and of 87A-like
SNe (e.g. Xiang et al. 2023) –, we determined the H$\alpha$ and Fe ii 5169Å
line velocities for SN 2021aatd. Taking advantage of the adequate signal-to-
noise ratio of our spectra, we fit single Gaussian profiles to the regions of
the absorption minima of the two lines for calculating the line velocities
($v_{\text{H}\alpha}$ and $v_{\text{FeII}}$). Velocity curves of SN 2021aatd,
together with that of SNe 1987A, 2020faa, and Type IIP 2012aw (also used for
comparison by Salmaso et al. 2023), are shown in Fig. 8. We note that, in the
lack of spectrum for SN 2012aw around +80-90 days, we used the data of its
spectral twin, SN 2017eaw (Szalai et al. 2019) for the spectral comparison
shown in Fig. 6.

In order to monitor the chemical composition, as well as the evolution of the
(photospheric) temperatures and velocities in more detail, we carried out
SYN++ modeling for the spectra of SNe 2021aatd and 2020faa; moreover, we also
compared our observational data to the outcomes of models published by Dessart
& Hillier (2019).

####  3.2.1 SYN++ modeling of the photospheric spectra

We applied the parameterized resonance scattering code
SYN++555https://c3.lbl.gov/es/ (Thomas et al. 2011) for the analysis of
2$-$98d spectra of SN 2021aatd. Note that, basically, the code can be used for
analyzing photospheric spectra; however, because of its slow spectral
evolution, SN 2021aatd can be still considered in the photospheric phase at
+98 days. The code allows to make estimations on some global parameters, such
as the photospheric temperature ($T_{\rm ph}$), and the velocity at the
photosphere ($v_{\rm ph}$). The contribution of the single ions can be taken
into account by setting some local parameters – the optical depth ($\tau$),
the minimum and the maximum velocity of the line forming region ($v_{\rm min}$
and $v_{\rm max}$), the scale height of the optical depth above the
photosphere (aux), and the excitation temperature ($T_{\rm exc}$) – for each
identified ion. Since we did not find any similar analyses for SN 2020faa in
the literature, we also involved its photospheric spectra in our modeling
process. Moreover, as a consistency check, we also fit SYN++ models on the
+84d spectrum of SN 1987A.

Figure 7: Ion contributions from SYN++ models calculated for +83, +88, and +84
days spectra of SNe 2021aatd, 2020faa, and 1987A, respectively.

Figure 8: Measured H$\alpha$ (top left) and Fe II 5169 Å (top right) line
velocities of SN 2021aatd, compared to that of SNe 1987A (Xiang et al. 2023),
2020faa (Yang et al. 2021), and Type IIP 2012aw (Bose et al. 2013). Calculated
line velocities from a3, a4, and a5 explosion models of Dessart & Hillier
(2019) (DH19), as well as photospheric velocities of SNe 2021aatd and 2020faa,
determined from our SYN++ models, are also plotted. See text for further
details. Bottom left: Calculated photospheric temperatures ($T_{\rm phot}$)
from DH19 a3, a4, and a5 models, compared to $T_{\rm phot}$ values of SNe
2021aatd and 2020faa determined from our SYN++ models. Bottom right:
Calculated V-band absolute magnitudes ($M_{\rm V}$) from DH19 a3, a4, and a5
models, compared to the measured values of SN 2021aatd.

Fig. 17 shows the results of SYN++ modeling carried out on all the 11 spectra
of SN 2021aatd obtained during the first $\sim$3 months after explosion. The
observed spectra were corrected for extinction and redshift before plotting.
The values of the best-fit global and local parameters are collected in Table
LABEL:tab:local_21aatd. At the earliest epochs, H i and He ii lines dominate
the spectrum of SN 2021aatd, which can be best described with the global
parameters of $v_{\rm ph}\sim$18 000 km s-1 and $T_{\rm ph}\sim$6500 K. At
later phases (after $\sim$20 days), lines of Na i, Mg i, Mg ii, Ca ii, Ti ii,
Sr ii, and Ba ii emerge as well and give a significant contribution to the
spectrum formation. By +98 days (the last observed epoch of the photospheric
phase), $v_{\rm ph}$ decreases to 5000 km s-1, while $T_{\rm ph}$ shows a
flat, nearly constant evolution around 5000 K. We note that there is a known
inconsistency during the parallel fitting of both the absorption and emission
components of the strong H$\alpha$ line profile – this is probably due to the
basic assumption of the SYN++ code of using local thermodynamic equilibrium
(LTE) for calculating level populations (see e.g. Szalai et al. 2016).

For comparison, we also modeled the spectra of SN 2020faa (obtained at 13
different epochs from +6 to +165 days, published by Yang et al. 2021) with
SYN++. Plots and parameters are shown in Fig. 18 and in Table
LABEL:tab:local_20faa, respectively, in the same way as in the case of SN
2021aatd. Both the list of identified ions and the overall chemical evolution
of SN 2020faa (at least, during the first 3 months) shows a great resemblance
with that of SN 2021aatd – even though SN 2020faa was a more luminous event.

In Fig. 7, we highlight the complete SYN++ models and elemental contributions
calculated for +83/88 days spectra of SNe 2021aatd and 2020faa (also used for
comparison in Fig. 6), respectively. We also generated a SYN++ model for the
+84d spectrum of SN 1987A. The identified ions are broadly similar in all
three cases; however, modeling the blue part of the spectrum of SN 2020faa is
complicated because of its lower quality. In the spectrum of SN 1987A, as
mentioned above, Na i D and Ba ii 6142 Å have significantly deeper profiles
than in the cases of SNe 2021aatd and 2020faa. Additionally, we added two
further ions – N ii and O i – to the model of SN 1987A to describe its +84d
spectrum.

In Fig. 8, beyond H$\alpha$ and Fe ii 5169Å line velocities, photospheric
velocity ($v_{\rm phot}$) evolution of both SNe 2021aatd and 2020faa,
determined from SYN++ models, is also shown. In the case of SN 2020faa, the
first $\sim$2 months are poorly covered by spectra; however, it can be clearly
seen at later phases that both of its photospheric and line velocities are
much higher than that of the other two objects. SN 2021aatd has, at the same
time, also higher line velocities than that of SN 1987A (especially in the
case of the Fe ii 5169Å line). It can be seen also that $v_{\rm phot}$ values
are definitely larger than that of $v_{\text{FeII}}$ in both SNe 2021aatd and
2020faa. In SNe IIP, $v_{\text{FeII}}$ is thought to be a good indicator of
the photospheric velocity ($v_{\text{phot}}$) after $\sim$20 days, since the
minimum of the Fe ii 5169Å absorption profile tends to form near the
photosphere (see Branch et al. 2003). However, the detailed investigation of
Takáts & Vinkó (2012) showed that the true $v_{\text{phot}}$ may significantly
differ from single line velocities. In the cases of SNe 2021aatd and 2020faa,
as our SYN++ models suggest, Fe ii lines can be blended e.g. with Mg i.

####  3.2.2 Comparison of spectral data with Dessart & Hillier (2019) models

Figure 9: Comparison of +83 and +84d spectra of SNe 2021aatd and 1987A to the a3 (top left), a4 (top right), and a5 (bottom) model spectra published by Dessart & Hillier (2019); see details in text. All spectra are normalized to the flux values at 6000 Å. Table 3: Ejecta masses, kinetic energies, and initial 56Ni masses in the adopted Dessart & Hillier (2019) explosion models. Model | $M_{\rm ej}$ | $E_{\rm kin}$ | $M_{\rm Ni}$
---|---|---|---
| ($M_{\odot}$) | (1051 erg) | ($M_{\odot}$)
a3 | 13.53 | 0.87 | 0.047
a4 | 13.22 | 1.26 | 0.084
a5 | 13.10 | 2.46 | 0.157

Figure 10: Comparison of nebular spectra of SNe 2021aatd, 2020faa and 1987A to
the a3 (top left), a4 (top right), and a5 (bottom) model spectra published by
Dessart & Hillier (2019); see details in text. All spectra are normalized to
the flux values at 6000 Å.

We also compare our spectral dataset and the determined velocities and
temperatures to the outputs of radiative transfer (cmfgen) models published by
Dessart & Hillier (2019, hereafter DH19). DH19 presented various sets of model
spectra (covering the first $\sim$250-300 days after explosion), representing
the final outcomes of the explosion of a single $M_{\rm ZAMS}$ = 15
$M_{\odot}$ low-metallicity progenitor that dies as a BSG, in order to study
in detail the origin of 87A-like SNe. As the authors state, their model set
was not designed to properly reproduce the observations; however, model
outputs can be well used for giving constraints on certain physical
parameters.

We selected three of the DH19 models (a3, a4, a5), which main parameters –
ejecta masses, kinetic energies, and initial 56Ni masses – are shown in Table
3. Comparison of these DH19 model spectra to the observed +83/84d spectra of
SNe 2021aatd and 1987A are shown in Fig. 9. As DH19 has already shown, spectra
of SN 1987A can be best described with the a4 model. At the same time, a3 and
a5 models reproduce the main features and shape of the spectra; note however
that a5 model seems to be definitely too energetic.

Unfortunately, available photospheric spectra of SN 1987A have a cut-off at
$\sim$7200Å ; nevertheless, the selected DH19 models seem to describe the red
part of the spectrum of SN 2021aatd quite well. At the same time, SN 2021aatd
shows a clear blue-part excess (i.e. larger temperature) that was also
discussed in Sect. 3.1 regarding the comparison of both color curves and
observed spectra (Figs. 5 and 6, respectively). Indeed, all DH19 models give
much lower photospheric temperatures ($\sim$4300$-$4700 K) at $\sim$83 days
than value of $T_{\rm phot}$=6300K we determined from the observed data using
SYN++, see Fig. 8. Model velocity curves published by DH19 (also shown in Fig.
8), however, strengthen the picture that both a3 and a4 models can be valid
descriptions of the event of SN 2021aatd. Note that, at the same time, a5
velocity curves are in a quite good agreement with that of SN 2020faa.

Furthermore, as DH19 have already described in detail, LCs calculated from
their spectral models do not give back the proper shape of 87A-like events –
the model LCs are overall too broad and, specifically for SN1987A, a4 model LC
shows a slower rise to a fainter maximum. Comparing the value of the observed
(second) $V$ peak magnitude of SN 2021aatd ($V_{\rm max}=-16.60\pm 0.16$ mag,
see Table 1) with the calculated $V$ peak magnitudes in DH19, there is an even
larger difference: this value is much larger than that of a3 and a4 model LCs
($V_{\rm max}\approx-$15.0 and $-$15.6 mag, respectively), and even slightly
larger than that of a5 model ($V_{\rm max}\approx-$16.4 mag), see bottom right
panel of Fig. 8. Note that the V-band LC calculated from the 56Ni-enhanced
a4ni model of DH19 actually reaches the observed peak brightness of SN
2021aatd; however, its shape is even broader than that of a4 and a5 model LCs.

DH19 list several possible explanations for resolving the LC problem at the
given 15 $M_{\odot}$ progenitor configuration – clumping of ejecta matter,
weaker 56Ni mixing, or a greater He-to-H abundance ratio –, and also note that
a smaller ejecta mass would result in a faster rise to maximum and a brighter
peak. In the cases of both SNe 2021aatd and 2020faa, the remarkable early-
phase flux excess needs a further explanation. We discuss the potential
explanations of the LC evolution in Sec. 3.3.

####  3.2.3 Progenitor mass from the nebular spectrum

Figure 11: Comparison of the +293d nebular spectrum of SN 2021aatd to the
model spectra published by Jerkstrand et al. (2012, 2014). Model spectra with
the closest epochs to that of the observed one (+306 days for the 12, 15, and
25 $M_{\odot}$ models, and +332 days for the 19 $M_{\odot}$ model,
respectively) have been selected; see details in text. All spectra are
normalized to the flux values at 6000 Å. The [O I] line doublet
($\lambda\lambda$6300, 6364) for the model and the observed spectra is shown
in the inset; the observed line profile resembles mostly to the 15 $M_{\odot}$
model suggesting that the initial mass of the progenitor is close to this
value.

Beyond the comparison with the DH19 models, we can also use our single nebular
(+293d) spectrum of SN 2021aatd for the estimation of the initial progenitor
mass adopting the spectral models published by Jerkstrand et al. (2012, 2014).
Based on their works, a monotonic relation exists between the intensity of the
[OI] doublet ($\rm{\lambda\lambda 6300,6364}$) and the mass of the progenitor
star. Here we compare the observed [O i] emission line using the model spectra
presented in Jerkstrand et al. (2012, 2014). The adopted synthesized spectra
were produced for $\rm{M_{ZAMS}=12,15,19,}$ and 25 $\rm{M_{\odot}}$ at epoch
of 306 days for the 12, 15, and 25 $\rm{M_{\odot}}$ models and at 332 days for
the 19 $\rm{M_{\odot}}$ model, respectively. These models were originally
generated for the Type IIP SN 2004et using a nickel mass of
$\rm{M_{{}^{56}Ni,mod}}$ = 0.062 $\rm{M_{\odot}}$ and $\rm{d_{mod}}$ = 5.5
Mpc. To apply these models to SN 2021aatd, the synthetic spectra are scaled to
the inferred distance (62.6 Mpc) and nickel mass (0.056 $\rm{M_{\odot}}$, see
Table 4) via the following relation (described e.g. in Szalai et al. 2019):

$F_{\rm obs}=F_{\rm mod}\times\left(\frac{d_{\rm mod}}{d_{\rm
obs}}\right)^{2}\left(\frac{M_{\rm{}^{56}Ni,obs}}{M_{\rm{}^{56}Ni,mod}}\right)e^{\frac{t_{\rm
mod}-t_{\rm obs}}{111.4}},$ (1)

where $\rm{F_{obs}}$ and $\rm{F_{mod}}$ are the observed and model fluxes,
$\rm{d_{obs}}$ and $\rm{d_{mod}}$ are the observed and model distances, and
$\rm{M_{{}^{56}Ni,obs}}$ and $\rm{M_{{}^{56}Ni,mod}}$ are the observed and
model nickel masses synthesized during the explosion.

The result of this comparison can be seen in Fig. 11. We find that both the
shape and the intensity level the observed line profile resemble mostly to
those of the 15 $M_{\odot}$ model, suggesting that the initial mass of the
progenitor is close to this value. This supports our previous findings
(Section 3.2.2) showing that spectral observables of SN 2021aatd can be well
described with the explosion of a $M_{\rm ZAMS}\sim 15M_{\odot}$ star. Note
that, however – instead of cases of Type IIP SNe 2004et (Jerkstrand et al.
2012) and 2017eaw (Szalai et al. 2019) assumed to also arise from a $M_{\rm
ZAMS}\sim 15M_{\odot}$ star –, the general match between model and observed
spectra is not perfect. A potential reason behind that could be that
Jerkstrand et al. developed these models not for studying exploding BSG but
RSG stars.

###  3.3 Bolometric LC modeling

####  3.3.1 Constructing the bolometric LC of SN 2021aatd

Figure 12: BVgri (pseudo)bolometric LCs of SN 2021aatd calculated with the
SuperBol code. Orange and red filled circles denote ’pure’ pseudo-bolometric
LC (i.e. integration of BVgri fluxes) and the one corrected after integration
of a blackbody (BB) function fitting of the available photometry,
respectively. Black circles show the final bolometric LC of SN 2021aatd
calculated from a suppressed BB estimation (see details in the text).
Bolometric LCs of SNe 1987A and 2020faa (adopted from Nagy & Vinkó 2016 and
Salmaso et al. 2023, respectively) are also shown.

In principle, generating the bolometric luminosities requires sampling the
spectral energy distribution along the whole electromagnetic spectrum.
However, proper observational coverage in all photometric bands is not the
case for SN 2021aatd.

The bolometric LC of SN 2021aatd was constructed using SuperBol code (Nicholl
2018). As a first step, we generated a pseudobolometric LC due to the simple
integration of the observed fluxes measured in the BVgri bands. To do that, we
used only that epochs where we have data measured in all the filters (29 out
of 35 nights in total), corrected the magnitudes for extinction, and converted
them to flux densities. For the integration of the flux densities, the code
uses a simple trapezoidal rule and assumes zero flux below and above the limit
defined by the filter equivalent width of the bluest and reddest filter,
respectively. Finally, the measured BVgri flux was converted into luminosity
using the host distance given in Table 2.

Next, we made efforts on calculating the flux contribution from the unobserved
ultraviolet (UV) and infrared (IR) regimes. A basic step is to fit a blackbody
(BB) function to the observed flux densities to estimate the UV/IR part of the
spectral energy distributions (SEDs). Generally, as presented e.g. in Salmaso
et al. (2023), IR contribution can be well estimated with the integration of
the Rayleigh-Jeans tail of the fitted BB function.

This is not the case in the UV. As also noted by Salmaso et al. (2023) (and
described in detail previously by Dessart & Hillier 2005), strong depletion
caused by metal lines leads depressed UV fluxes with respect to the BB
function. In the case of SN 2021aatd, no observational UV data are available
to directly estimate the degree of this suppression. While, atmospheric models
of 87A-like explosions calculated by Dessart & Hillier (2019) do cover the UV
regime, too, these probably cannot be reliably used for estimating the UV
contribution in the case of SN 2021aatd: as we show in Figs. 5 and 6, early-
time ($\lesssim$40 d) spectral properties and color evolution of SN 2021aatd
differ quite much from those of 1987A, especially in the blue region. At the
same time, in this earliest phase, SN 2021aatd seems to be very similar to the
special Type II SN 2020faa regarding both their color curves and spectra, just
at a lower luminosity scale (as we also described in detail above).

Thus, we followed a method somewhat similar to that of Salmaso et al. (2023)
in the case of SN 2020faa. Based on their Fig. 2, we adopted the UV photometry
of Type II-P SN 2012aw obtained at an epoch of +28 days, shows a definite
suppression regarding the BB function ($T_{BB}$=6500K) fitted to its optical-
near-IR flux densities. Additionally, we also adopted the +30 days UV spectrum
of SN 1987A666The spectrum was downloaded from WISeREP
(https://www.wiserep.org/). We used the option in SuperBol taking into account
line blanketing at UV wavelengths applying the formula
$L_{UV}(\lambda<\lambda_{\textrm{cutoff}})=L_{BB}(\lambda)*(\lambda/\lambda_{\textrm{cutoff}})^{x}$,
where users can choose the values of the cutoff wavelength
($\lambda_{\textrm{cutoff}}$) and of the suppression index ($x$). We found
that applying values of $\lambda_{\textrm{cutoff}}$=4000 Å and $x$=4.0 results
in a reliable shape for the suppressed BB curve comparing to the measured UV
SED/spectrum of SNe 2012aw and 1987A.

Thus, the ’final’ bolometric LC of SN 2021aatd was constructed using this
approximation; note however, that this final luminosity curve only slightly
differs from the one calculated via full BB correction after the first $\sim$2
weeks (when the largest part of the SED of the SN shifts to the optical
regime). ’Final’ bolometric LC of SN 2021aatd, together with the results of
other approximations used in SuperBol, are shown in Fig. 12. Comparison of the
final bolometric LC of SN 2021aatd to that of SN1987A and 2020faa are in
agreement with the trends seen in the filtered LCs of these objects (see Fig.
5).

####  3.3.2 Two-component semi-analytic modeling of the bolometric LCs of SNe
1987A and 2021aatd

Figure 13: Bolometric LC of SN 1987A (circles) with the best-fit two-component
Arnett-Fu models – assuming either only radioactive energy input (blue line,
Nagy & Vinkó 2016), or Ni-Co decay with an additional magnetar power source
(black line, Model I from this work), respectively.

Figure 14: Bolometric LC of SN 2021aatd (black circles) with the best-fit two-
component Model I curves (red line); left and right panels show the results of
our model fitting assuming a recombination temperature of $T_{\rm rec}$= 6500
and 7000 K, respectively. The contributions of the ’core’ and the ’shell’
parts of the ejecta are plotted with blue and green lines, respectively. The
light blue and magenta lines represent the contribution of the radioactive
decay and magnetar energy input to the luminosity evolution of the ’core’
component, respectively. Details and parameter values can be found in the text
and in Table 4.

Figure 15: Left: the same as in Fig. 14, but showing the fit of Model II to
the bolometric LC of SN 2021aatd. Right: best-fit Model II curves fitted to
the average values and to the upper and lower limits of the bolometric
luminosities, respectively. Details and parameter values can be found in the
text and in Table 4.

For estimating the physical properties of SN explosions, we use a semi-
analytical diffusion model originally introduced by Arnett & Fu (1989), which
assumes a homologous expanding and spherically symmetric SN ejecta with a
uniform density profile and also takes into account the effect of
recombination. However, the LC of SN 2021aatd shows a double peak, which
cannot be fitted by the Arnett model. To deal with this cooling phase of the
shock breakout, we assume a two-component structure for the SN ejecta
(applying a model developed by some of us, see Nagy et al. 2014; Nagy & Vinkó
2016). Here, the first peak of the LC is dominated by the adiabatic cooling of
the shock-heated, H-rich envelope (referred to as the ’shell’), and the second
peak is powered by radioactive decay deposited in the denser inner region
(’core’). Thus, the main difference between the two components is that the
outer, low-density ’shell’ is assumed to be powered only by shock heating (and
not by radioactive decay).

As a starting point, we adopted the model parameters we used for modeling the
bolometric LC of SN 1987A in Nagy & Vinkó (2016), hereafter NV16, assuming a
moderate value of ejecta mass ($M_{\rm ej}\sim 8M_{\odot}$) and a
recombination temperature of Trec = 5500K. Nevertheless, we note that our
original model presented in NV16 is unable to fit the second (’main’) LC peak
of SN 1987A properly. For modeling the observed LCs as closely as possible, we
apply here an improved version of our previously published LC fitting method:
we involve an alternative energy source in addition to the radioactive decay,
which considers the energy deposition of the rotational energy of a newborn
magnetar (just as applied by Salmaso et al. 2023, in the case of SN 2020faa).
The magnetar model has the advantage of reducing the required nickel mass as
well as altering the slope of the late-time LC. Fig. 13 shows the comparison
of the original (NV16) and the improved model without and with the magnetar
contribution, respectively. It can be seen that the late decreasing part of
the LC of SN 1987A can be well explained with pure Ni-Co decay, but involving
the magnetar energy source i) results in a successful modeling of the main LC
peak at roughly +90 days, ii) does not affect the early-time LC fit (’shell’
component), and iii) allows the fine-tuning of the fit of the late-time LC.

Thus, we decided to continue the LC modeling process applying this improved
method (called hereafter Model I). The main model parameters – beyond
recombination temperature (Trec) and ejecta mass ($M_{\rm ej}$) mentioned
above – are: the progenitor radius $R_{0}$ (in 1011 cm units), the initial
mass of the radioactive 56Ni ($M_{\rm Ni}$, in $M_{\odot}$), the total energy
($E_{\rm tot}$, in 1051 erg), the ratio of the kinetic and thermal energies of
the ejecta ($E_{\rm kin}/E_{\rm Th}$), the average opacity of the ejecta
($\kappa$, in cm2/g), and the spin-down energy of the magnetar ($E_{\rm p}$,
in 1051 erg) and its characteristic time-scale ($t_{\rm p}$, in days). As can
be seen in Table 4, ’Model I’ LC parameter set of SN 1987A show some
differences regarding that of previously published in NV16. The most
significant changes are in the values of Trec (7000 vs. 5500 K), and in ’core’
parameters $R_{\rm 0,core}$ (2.8 vs. 29.0 $\times 10^{11}$ cm) and $E_{\rm
kin,core}/E_{\rm Th,core}$ (3.2 vs. 11.7). At the same time, it was not
necessary to change any ’shell’ parameters.

As a next step, we fit Model I to the final bolometric LC of SN 2021aatd
calculated with SuperBol (see above). In Fig. 14, we present two slightly
different solutions assuming recombination temperatures of $T_{\rm rec}$ =
6500 K and 7000 K (left and right panels, respectively). The reason of that is
the presence of a modest re-brightening on the bolometric LC around +300 days,
which may be a hint for CSM interaction – but the late part of the LC is not
well-covered enough to consider it as a strong evidence. Nevertheless, late-
time LC evolution of SN 2021aatd is not linear and cannot be perfectly
described with one model. Thus, we decided to take into account both cases:
the left panel of Fig. 14 ($T_{\rm rec}$ = 6500 K) shows the best fit for data
up to $\sim$300 days, while the right panel ($T_{\rm rec}$ = 7000 K) is the
best solution for fitting including the latest data points, too. Note,
however, that both parameter sets represent a hydrogen-depleted ejecta and
that they are quite similar; there are only small differences in some ’core’
parameters but no changes in ’shell’ parameters at all (see Table 4). In both
panels of Fig. 14, we also present the contributions of both the radioactive
decay and magnetar energy input to the ’core’ luminosity component,
respectively.

Table 4: Parameters of the best-fit two-component bolometric LC models for SNe
2021aatd and 1987A. See text for a detailed description.

| SN 2021aatd | SN 1987A
---|---|---
| Model I | Model II | NV16 | Model I | Model II
Trec (K) | 6500 K | 7000 K | 7000 $\pm$ 500 K | 5500 K | 7000K | 7000 K
| ’Core’
$R_{0}$ (1011 cm) | 1.8 | 1.3 | 0.64 $\begin{subarray}{c}+0.1\\\ -0.2\end{subarray}$ | 29.0 | 2.8 | 1.3
$M_{\rm ej}$ ($M_{\odot}$) | 7.4 | 7.7 | 14.35 $\begin{subarray}{c}+0.88\\\ -1.12\end{subarray}$ | 8.6 | 7.9 | 14.35
$M_{\rm Ni}$ ($M_{\odot}$) | 0.13 | 0.13 | 0.13 $\begin{subarray}{c}+0.03\\\ -0.05\end{subarray}$ | 0.069 | 0.058 | 0.058
$E_{\rm tot}$ (1051 erg) | 1.75 | 1.7 | 6.4 $\begin{subarray}{c}+0.7\\\ -1.5\end{subarray}$ | 1.5 | 1.7 | 3.61
$E_{\rm kin}$/ETh | 2.5 | 3.25 | 1.2 $\begin{subarray}{c}+0.03\\\ -0.11\end{subarray}$ | 11.7 | 3.2 | 1.91
$\kappa$ (cm2/g) | 0.3 | 0.3 | 0.24 | 0.19 | 0.20 | 0.21
$E_{\rm p}$ (1051 erg) | 0.012 | 0.013 | 0.016 $\begin{subarray}{c}+0.006\\\ -0.002\end{subarray}$ | – | 0.014 | 0.014
$t_{\rm p}$ (days) | 6.0 | 7.0 | 7.0 $\pm$ 1.0 | – | 6.0 | 6.0
| ’Shell’
$R_{0}$ (1013 cm) | 1.5 $\pm$ 0.3 | 1.0 | 1.5
$M_{\rm ej}$ ($M_{\odot}$) | 0.3 | 0.1 | 0.05
$M_{\rm Ni}$ ($M_{\odot}$) | – | – | –
$E_{\rm tot}$ (1051 erg) | 0.303 $\begin{subarray}{c}+0.002\\\ -0.013\end{subarray}$ | 0.4 | 0.095
$E_{\rm kin}$/ETh | 2.26 $\begin{subarray}{c}+0.01\\\ -0.05\end{subarray}$ | 20.0 | 18.0
$\kappa$ (cm2/g) | 0.34 | 0.34 | 0.34
777Parameters are: the recombination temperature (Trec), the progenitor radius
$R_{0}$, the ejecta mass ($M_{\rm ej}$), the initial mass of the radioactive
56Ni ($M_{\rm Ni}$), the total energy ($E_{\rm tot}$), the ratio of the
kinetic and thermal energies of the ejecta ($E_{\rm kin}/E_{\rm Th}$), the
average opacity of the ejecta ($\kappa$), and the spin-down energy of the
magnetar ($E_{\rm p}$) and its characteristic time-scale ($t_{\rm p}$).

Best-fit parameters sets of ’Model I’ for the data of both SNe 1987A and
2021aatd give an ejecta mass of $M_{\rm ej}\sim 8M_{\odot}$, similarly to the
original conclusion of NV16. However, taking into account both our findings on
comparing observed and model spectra (Sections 3.2.2 and 3.2.3) and the
results of previous works on stellar evolution and explosion processes of BSG
stars (e.g. Blinnikov et al. 2000; Utrobin et al. 2015; Dessart & Hillier
2019; Xiang et al. 2023), it seems to be plausible to assume a larger ejecta
mass. This problem was already discussed in NV16; from the point of view of
the applied LC modeling method, the key is the degeneration of ejecta mass
($M_{\rm ej}$) and average opacity ($\kappa$). As described in NV16 and
references therein in detail, for H-rich CCSNe, the potential range of the
average opacity in the core is $\kappa_{\rm core}\approx 0.2\pm 0.1$ cm2/g; if
one let $\kappa_{\rm core}$ vary within the whole interval, value of $M_{\rm
ej,core}$ (and, thus, mass of the progenitor) may constrained within a factor
of $\sim$2\. There is the same relation between $\kappa_{\rm shell}\approx
0.3-0.4$ and $M_{\rm ej,shell}$; however, since $M_{\rm ej,shell}$ has a much
lower value, it contributes less to the mass of the progenitor star.

Thus, as a next step, we defined ’Model II’, fixing $T_{\rm rec}$ = 7000 K and
$M_{\rm ej,core}$ = 14.35 $M_{\odot}$ for both SNe 2021aatd and 1987A; these,
together with $M_{\rm ej,shell}\approx 0.1-0.2M_{\odot}$, are in a very good
agreement with the assumptions of the atmospheric models a3 and a4 of Dessart
& Hillier (2019). As ’Model II’ seems to be the more plausible solution, we
estimated the uncertainty of the model parameters for SN 2021aatd. To do so,
we create models and fit them to not only the average values but also to the
upper and lower limits of the bolometric luminosities. At the same time, it
was necessary to change the ’shell’ parameters for both SNe 2021aatd and 1987A
to get similarly good fits as in the case of ’Model I’; see Fig. 15 and Table
4 for the results. During the fitting of ’Model II’, we used a power-law
density profile (using an index of n=1.8) for SN 2021aatd, while we got a
better fit by assuming a constant density profile for SN 1987A. While core Ni-
mass and magnetar parameters remain unchanged, we get smaller values for the
radius and larger values for the total energy of the core component;
$\kappa_{\rm core}$, being 0.24 and 0.21 cm2/g for SN 2021aatd and 1987A,
respectively, is still in the desired range.

Figure 16: Comparison of the early-phase ’Model II’ bolometric LC fits of SNe
2021aatd and 1987A.

Overall, ’core’ parameters of SNe 2021aatd and 1987A seem to be similar using
either ’Model I’ or ’Model II’ configurations – this is exactly what one
expects from the similar late-time LC evolution of the two objects. However,
SN 2021aatd was a more luminous event than SN 1987A, which is reflected in the
values of Ni-mass (0.13 vs 0.06 $M_{\odot}$) and, in ’Model II’, also of the
total energy (6.4 vs. 3.6 $\times 10^{51}$ erg). At the same time, the ’core’
radius of SN 2021aatd seems to be significantly smaller than that of SN 1987A.

Taking a look at the final ’shell’ parameters, differences between the two
events are more remarkable, just as suggested by the differences in their
early-time luminosity and spectral properties (shapes of the shell LC
components of both SNe can be seen better in Fig. 16). This is especially
true, if we compare the ’Model II’ ’shell’ parameters of the objects – based
on that parameter sets, we definitely see a more massive and more energetic
shell in the case of SN 2021aatd. A further evidence on the reliability of
’Model II’ is that we get back a larger kinetic energy (and, thus, larger
ejecta velocities) in the shell for SN 2021aatd than for SN 1987A, in
agreement with the results we received from the analysis of the spectral
datasets (see Sec. 3.2) – while this is not the case if we apply ’Model I’.

##  4 Discussion & Conclusions

Based on our detailed spectral and LC analysis, the LC evolution of SN
2021aatd after the first $\sim$40 days is very similar to that of SN 1987A. We
found that this phase can be well described with the explosion of a $M_{\rm
ZAMS}\sim$15 $M_{\odot}$ BSG star, as it was previously assumed and
investigated in detail for 87A-like explosions by Dessart & Hillier (2019).
However, as we noted above, photospheric temperatures are considerably higher
for SN 2021aatd than for SN 1987A. Moreover, Na i D and Ba ii 6142 Å lines,
which appear as dominant lines in the spectra of SN 1987A, are very weak in
the case of SN 2021aatd. These properties of SN 2021aatd, just as the
relatively high $v_{\rm H\alpha}$ and $v_{\rm FeII}$ velocity curves shown in
Fig. 8, resemble rather to SN IIP-like atmospheres. Nevertheless, such
transitions between 1987A- and IIP-like SNe have been already described in
several works (e.g. Takáts et al. 2016; Dessart & Hillier 2019; Xiang et al.
2023).

As described in Sect. 3.2.2, the proper explanation of the LC evolution of
87A-like SNe based on the outcomes of radiative transfer spectral models is
challenging. Extended efforts have been made for modeling bolometric LC of
1987A and of similar events using hydrodynamic codes, taking into account
various aspects like rotation, mixing of 56Ni, clumping of ejecta matter, or
the presence of a binary companion (see a review in Dessart & Hillier 2019,
and even more recent results in e.g. Dessart & Audit 2019, Menon et al. 2019,
or Utrobin et al. 2021).

In this work (see Sect. 3.3), we present a more simple, semi-analytical model
assuming a two-component ejecta – developed by some of us (Nagy et al. 2014;
Nagy & Vinkó 2016), based on the method originally introduced by Arnett & Fu
(1989) –, which seems to be also appropriate for describing the bolometric LC
of 87A-like SNe. While this kind of LC modeling of SN 1987A has been already
published in Nagy & Vinkó (2016), for achieving an even more proper fit, we
apply here an improved version of the LC model: involving the rotational
energy of a newborn magnetar in addition to radioactive decay allows us to
fine-tune the fitting at both around the peak and at later phases.

In the case of the primary object of this study, SN 2021aatd, we can also
successfully apply this improved bolometric LC model. While, as mentioned, its
late-time LC evolution is very similar to that of SN 1987A (which leads to
very similar model parameters regarding the ’core’ ejecta component), the
early-time peculiar LC of SN 2021aatd can be also well described with changing
the outer (’shell’) parameters towards a larger ejecta mass and total energy
than in the case of SN 1987A, choosing our ’Model II’ configuration (which has
a fixed recombination temperature of $T_{\rm rec}$ = 7000 K and a fixed core
ejecta mass of $M_{\rm ej,core}$ = 14.35 $M_{\odot}$ for both SNe 2021aatd and
1987A). Using this parameter set, we find that summed (’core’ + ’shell’)
values of key parameters ($M_{\rm ej}\approx 14.6M_{\odot}$, $M_{\rm
Ni}\approx$0.13 $M_{\odot}$, and $E_{\rm kin}=3.7\times$ 1051 erg) harmonize
well with those used in the adopted DH19 spectral models for the case of SN
2021aatd (see Table 3).

We also note that the parameter set belongs to ’Model I’ – assuming a lower
total ejecta mass of $\sim 8M_{\odot}$, but a somewhat larger core radius but
smaller energies, without any change in ’shell’ parameters – also gives a
proper LC solution for SN 2021aatd (and also for SN 1987A, but with modified
’shell’ parameters, see Table 4). This is probably due to the known
degeneration of our model between the mass and the average opacity of the
ejecta (see Nagy & Vinkó 2016). Furthermore, as noted by Dessart & Hillier
(2019), assuming a low ejecta mass (and, thus, a progenitor mass of $<$15
$M_{\odot}$) could solve the LC fitting problems of their radiative transfer
models. However, we note that the ’shell’ parameters resulted by fitting
’Model I’ do not reflect the true energy/velocity relations between SNe
2021aatd and 1987A (while ’Model II’ parameters do). Moreover, providing
another argument in favor of a $\sim$15 $M_{\odot}$ progenitor, we refer here
to the results of the comparison of the nebular spectrum of SN 2021aatd with
the models of Jerkstrand et al. (2012, 2014) (see Sect. 3.2.3).

We also showed that both spectral and early-time LC evolution of SN 2021aatd
(i.e. before $\sim$90 days) resembles remarkably that of the long-lived SN
2020faa (see Sect. 3.1). Taking a look again at Figs. 3$-$5, the striking
similarity of light and color curves of these two events during the first
$\sim$3 months calls for a similar physical background; even that the
luminosity scales of the two SNe differ by a factor of $\sim$5-10. For
describing the pseudo-bolometric (g’+r’+i’) LC of SN 2020faa, Salmaso et al.
(2023) also apply the Nagy & Vinkó (2016) model with the built-in magnetar
energy component, just as we do in the case of SNe 2021aatd and 1987A. They
successfully model the early LC of SN 2020faa, however, assuming a more
massive ($M_{\rm ej}\approx 23M_{\odot}$) and larger ($R_{\rm 0,shell}\approx
10^{14}$ cm) progenitor. At the same time, this model does not provide a
satisfactory power explanation for the peak without an unphysically large
value of Ni-mass, or, assuming magnetar spin-down parameters that are in
contrast with the rapid drop of LC after the peak (which, as can be seen in
Figs. 3 and 5, is quite different from the post-peak evolution of both SNe
2021aatd and 1987A). Thus, in order to describe the peak luminosity, Salmaso
et al. (2023) involved further potential explanations (hidden interaction with
an inner disc, or, effects of a delayed, choked jet); however, proving these
alternatives requires further investigations.

Nevertheless, our analysis on 2021aatd, as well as on SNe 1987A and 2020faa,
seems to strengthen that we do see similar kind of explosions, which all can
be described with a two-component ejecta (assuming extra luminosity source(s)
in the case of SN 2020faa). Up to now, only a handful of 1987A-like SNe and
even less long-lived, multipeaked SNe are known; thus, their connection is not
clear now. However, ongoing and near-future large-scale transient surveys are
expected to find more and more similar objects to that of SN 2021aatd, helping
to give a clearer view of the reason behind the similarities and diversity of
exploding H-rich massive stars.

###### Acknowledgements.

We used WISeREP (https://www.wiserep.org/) for downloading previously
published spectra. We gratefully thank Irene Salmaso for sharing the nebular
spectra of SN 2020faa with us. This work makes use of the Las Cumbres
Observatory global telescope network. The LCO group is supported by NSF grants
AST-1911151 and AST-1911225. This project has been supported by the
GINOP-2-3-2-15-2016-00033 project of the National Research, Development and
Innovation (NRDI) Office of Hungary funded by the European Union, as well as
by NKFIH OTKA FK-134432, PD-134434 and K-142534 grants, and from the HUN-REN
Hungarian Research Network. T.S. is supported by the János Bolyai Research
Scholarship of the Hungarian Academy of Sciences. R.K.T., T.S., and D.B. are
supported by the ÚNKP 22-4 and ÚNKP 22-5 New National Excellence Programs of
the Ministry for Culture and Innovation from the source of the NRDI Fund of
Hungary, respectively.

## References

  * Anderson et al. (2014) Anderson, J. P., González-Gaitán, S., Hamuy, M., et al. 2014, ApJ, 786, 67
  * Arcavi et al. (2012) Arcavi, I., Gal-Yam, A., Cenko, S. B., et al. 2012, ApJ, 756, L30
  * Arcavi et al. (2017) Arcavi, I., Howell, D. A., Kasen, D., et al. 2017, Nature, 551, 210
  * Arnett & Fu (1989) Arnett, W. D. & Fu, A. 1989, ApJ, 340, 396
  * Barna et al. (2023) Barna, B., Nagy, A. P., Bora, Z., et al. 2023, A&A, 677, A183
  * Blinnikov et al. (2000) Blinnikov, S., Lundqvist, P., Bartunov, O., Nomoto, K., & Iwamoto, K. 2000, ApJ, 532, 1132
  * Bose et al. (2013) Bose, S., Kumar, B., Sutaria, F., et al. 2013, MNRAS, 433, 1871
  * Branch et al. (2003) Branch, D., Baron, E. A., & Jeffery, D. J. 2003, in Supernovae and Gamma-Ray Bursters, ed. K. Weiler, Vol. 598 (Lecture Notes in Physics), 47–75
  * Branch & Wheeler (2017) Branch, D. & Wheeler, J. C. 2017, Supernova Explosions (Springer)
  * Catchpole et al. (1988) Catchpole, R. M., Whitelock, P. A., Feast, M. W., et al. 1988, MNRAS, 231, 75P
  * Catchpole et al. (1989) Catchpole, R. M., Whitelock, P. A., Menzies, J. W., et al. 1989, MNRAS, 237, 55P
  * Chen et al. (2021) Chen, T. W., Cai, L., Smith, K. W., et al. 2021, Transient Name Server AstroNote, 262, 1
  * Dessart & Audit (2019) Dessart, L. & Audit, E. 2019, A&A, 629, A17
  * Dessart & Hillier (2005) Dessart, L. & Hillier, D. J. 2005, A&A, 437, 667
  * Dessart & Hillier (2019) Dessart, L. & Hillier, D. J. 2019, A&A, 622, A70
  * Filippenko (1982) Filippenko, A. V. 1982, PASP, 94, 715
  * Filippenko (1997) Filippenko, A. V. 1997, ARA&A, 35, 309
  * Gal-Yam (2012) Gal-Yam, A. 2012, Science, 337, 927
  * González-Gaitán et al. (2015) González-Gaitán, S., Tominaga, N., Molina, J., et al. 2015, MNRAS, 451, 2212
  * Hamuy et al. (1988) Hamuy, M., Suntzeff, N. B., Gonzalez, R., & Martin, G. 1988, AJ, 95, 63
  * Hiramatsu et al. (2021) Hiramatsu, D., Howell, D. A., Moriya, T. J., et al. 2021, ApJ, 913, 55
  * Howell (2019) Howell, D. 2019, in American Astronomical Society Meeting Abstracts, Vol. 233, American Astronomical Society Meeting Abstracts #233, 258.16
  * Jerkstrand et al. (2012) Jerkstrand, A., Fransson, C., Maguire, K., et al. 2012, A&A, 546, A28
  * Jerkstrand et al. (2014) Jerkstrand, A., Smartt, S. J., Fraser, M., et al. 2014, MNRAS, 439, 3694
  * Menon et al. (2019) Menon, A., Utrobin, V., & Heger, A. 2019, MNRAS, 482, 438
  * Nagy et al. (2014) Nagy, A. P., Ordasi, A., Vinkó, J., & Wheeler, J. C. 2014, A&A, 571, A77
  * Nagy & Vinkó (2016) Nagy, A. P. & Vinkó, J. 2016, A&A, 589, A53
  * Nicholl (2018) Nicholl, M. 2018, Research Notes of the American Astronomical Society, 2, 230
  * Pál (2012) Pál, A. 2012, MNRAS, 421, 1825
  * Pastorello et al. (2009) Pastorello, A., Valenti, S., Zampieri, L., et al. 2009, MNRAS, 394, 2266
  * Patat et al. (1994) Patat, F., Barbon, R., Cappellaro, E., & Turatto, M. 1994, A&A, 282, 731
  * Perley et al. (2020) Perley, D. A., Fremling, C., Sollerman, J., et al. 2020, ApJ, 904, 35
  * Podsiadlowski et al. (1992) Podsiadlowski, P., Joss, P. C., & Hsu, J. J. L. 1992, ApJ, 391, 246
  * Pun et al. (1995) Pun, C. S. J., Kirshner, R. P., Sonneborn, G., et al. 1995, ApJS, 99, 223
  * Salmaso et al. (2023) Salmaso, I., Cappellaro, E., Tartaglia, L., et al. 2023, A&A, 673, A127
  * Schlafly & Finkbeiner (2011) Schlafly, E. F. & Finkbeiner, D. P. 2011, ApJ, 737, 103
  * Schlegel (1990) Schlegel, E. M. 1990, MNRAS, 244, 269
  * Sit et al. (2023) Sit, T., Kasliwal, M. M., Tzanidakis, A., et al. 2023, arXiv e-prints, arXiv:2306.01109
  * Smith et al. (2020) Smith, K. W., Smartt, S. J., Young, D. R., et al. 2020, PASP, 132, 085002
  * Sollerman et al. (2019) Sollerman, J., Taddia, F., Arcavi, I., et al. 2019, A&A, 621, A30
  * Soraisam et al. (2022) Soraisam, M., Matheson, T., Lee, C.-H., et al. 2022, ApJ, 926, L11
  * Szalai et al. (2019) Szalai, T., Vinkó, J., Könyves-Tóth, R., et al. 2019, ApJ, 876, 19
  * Szalai et al. (2016) Szalai, T., Vinkó, J., Nagy, A. P., et al. 2016, MNRAS, 460, 1500
  * Taddia et al. (2016) Taddia, F., Sollerman, J., Fremling, C., et al. 2016, A&A, 588, A5
  * Taddia et al. (2012) Taddia, F., Stritzinger, M. D., Sollerman, J., et al. 2012, A&A, 537, A140
  * Takáts et al. (2016) Takáts, K., Pignata, G., Bersten, M., et al. 2016, MNRAS, 460, 3447
  * Takáts & Vinkó (2012) Takáts, K. & Vinkó, J. 2012, MNRAS, 419, 2783
  * Thomas et al. (2011) Thomas, R. C., Nugent, P. E., & Meza, J. C. 2011, PASP, 123, 237
  * Tonry et al. (2018) Tonry, J. L., Denneau, L., Heinze, A. N., et al. 2018, PASP, 130, 064505
  * Tonry et al. (2012) Tonry, J. L., Stubbs, C. W., Kilic, M., et al. 2012, ApJ, 745, 42
  * Utrobin et al. (2015) Utrobin, V. P., Wongwathanarat, A., Janka, H. T., & Müller, E. 2015, A&A, 581, A40
  * Utrobin et al. (2021) Utrobin, V. P., Wongwathanarat, A., Janka, H. T., et al. 2021, ApJ, 914, 4
  * Valenti et al. (2016) Valenti, S., Howell, D. A., Stritzinger, M. D., et al. 2016, MNRAS, 459, 3939
  * Valenti et al. (2014) Valenti, S., Sand, D., Pastorello, A., et al. 2014, MNRAS, 438, L101
  * Whitelock et al. (1988) Whitelock, P. A., Catchpole, R. M., Menzies, J. W., et al. 1988, MNRAS, 234, 5P
  * Woosley (2018) Woosley, S. E. 2018, ApJ, 863, 105
  * Woosley et al. (1988) Woosley, S. E., Pinto, P. A., & Ensman, L. 1988, ApJ, 324, 466
  * Wyatt et al. (2021) Wyatt, S., Sand, D., Andrews, J., Daly, P., & Hosseinzedah, G. 2021, Transient Name Server Classification Report, 2021-3458, 1
  * Xiang et al. (2023) Xiang, D., Wang, X., Zhang, X., et al. 2023, MNRAS, 520, 2965
  * Yang et al. (2021) Yang, S., Sollerman, J., Chen, T. W., et al. 2021, A&A, 646, A22

##  Appendix A Observational data

Table 5: $BVgri$ photometry of SN 2021aatd obtained from LCO sites MJD | B (mag) | V (mag) | g (mag) | r (mag) | i (mag)
---|---|---|---|---|---
59496.0 | 17.93 (0.07) | 17.97 (0.06) | 17.81 (0.03) | 18.03 (0.03) | 18.26 (0.04)
59498.1 | 17.93 (0.06) | 17.86 (0.03) | 17.73 (0.03) | 17.87 (0.03) | 18.07 (0.03)
59499.0 | 17.95 (0.08) | 17.88 (0.04) | 17.75 (0.04) | 17.84 (0.02) | 18.05 (0.03)
59501.0 | 18.02 (0.07) | 17.89 (0.03) | 17.78 (0.04) | 17.78 (0.02) | 17.98 (0.03)
59501.9 | 18.01 (0.06) | 17.88 (0.03) | 17.78 (0.04) | 17.76 (0.02) | 17.95 (0.02)
59504.9 | 18.09 (0.06) | 17.83 (0.03) | 17.86 (0.04) | 17.69 (0.03) | 17.86 (0.03)
59509.1 | 18.35 (0.09) | 17.91 (0.05) | 17.98 (0.06) | 17.70 (0.04) | 17.80 (0.05)
59511.6 | 18.56 (0.08) | 18.03 (0.03) | 18.08 (0.03) | 17.77 (0.04) | 17.92 (0.03)
59515.1 | 18.77 (0.07) | 18.12 (0.03) | 18.27 (0.03) | 17.87 (0.02) | 18.00 (0.03)
59518.0 | 18.91 (0.06) | 18.16 (0.03) | 18.37 (0.03) | 17.90 (0.03) | 18.03 (0.04)
59521.9 | 18.98 (0.06) | 18.21 (0.04) | 18.43 (0.04) | 17.92 (0.03) | 17.98 (0.03)
59527.2 | 19.20 (0.09) | 18.24 (0.03) | 18.56 (0.03) | 17.96 (0.03) | 18.03 (0.04)
59532.0 | 19.26 (0.09) | 18.24 (0.03) | 18.58 (0.04) | 18.02 (0.03) | 18.02 (0.04)
59549.9 | 18.77 (0.08) | 17.94 (0.04) | 18.17 (0.03) | 17.76 (0.03) | 17.77 (0.04)
59562.8 | 18.43 (0.11) | 17.73 (0.08) | 17.91 (0.08) | 17.49 (0.07) | …
59569.1 | 18.32 (0.06) | 17.57 (0.03) | 17.79 (0.04) | 17.37 (0.03) | 17.42 (0.03)
59576.8 | 18.23 (0.07) | 17.47 (0.03) | 17.68 (0.03) | 17.28 (0.03) | 17.30 (0.05)
59583.1 | 18.28 (0.11) | 17.50 (0.03) | 17.68 (0.03) | 17.28 (0.03) | 17.29 (0.04)
59590.9 | 18.25 (0.16) | 17.47 (0.07) | 17.70 (0.07) | 17.32 (0.09) | 17.29 (0.06)
59605.1 | 19.20 (0.15) | 17.93 (0.08) | 18.34 (0.06) | … | 17.72 (0.09)
59609.1 | 19.39 (0.10) | 18.26 (0.05) | 18.63 (0.05) | 17.86 (0.04) | 17.89 (0.05)
59731.2 | 20.75 (0.10) | 19.92 (0.06) | 20.13 (0.05) | … | 19.57 (0.09)
59741.1 | 20.78 (0.13) | 19.91 (0.07) | 20.21 (0.06) | 19.18 (0.04) | 19.61 (0.07)
59749.4 | 20.86 (0.14) | 20.07 (0.11) | 20.32 (0.09) | 19.18 (0.06) | 19.63 (0.09)
59765.3 | 20.99 (0.11) | 20.12 (0.05) | 20.48 (0.04) | 19.34 (0.03) | 19.65 (0.05)
59785.2 | 21.02 (0.10) | 20.43 (0.07) | 20.67 (0.06) | 19.56 (0.04) | 19.97 (0.15)
59792.7 | 21.15 (0.09) | 20.47 (0.14) | 20.76 (0.06) | 19.56 (0.03) | 20.02 (0.06)
59800.7 | 21.24 (0.16) | 20.37 (0.07) | 20.71 (0.08) | 19.59 (0.04) | 20.15 (0.07)
59813.3 | 21.15 (0.08) | 20.46 (0.06) | 20.82 (0.06) | 19.61 (0.03) | 20.10 (0.05)
59846.7 | 21.47 (0.09) | 20.85 (0.06) | 21.00 (0.05) | 20.02 (0.03) | 20.16 (0.06)
59850.4 | 21.49 (0.10) | 20.82 (0.06) | 20.99 (0.05) | 20.06 (0.04) | 20.26 (0.06)
59865.3 | 21.57 (0.21) | … | 21.13 (0.15) | 20.21 (0.08) | 20.50 (0.16)
59895.2 | 21.68 (0.20) | … | … | 20.44 (0.08) | 20.68 (0.31)

Table 6: $BVgri$ photometry of SN 2021aatd obtained from Baja Observatory MJD | B (mag) | V (mag) | g (mag) | r (mag) | i (mag)
---|---|---|---|---|---
59501.9 | … | 17.94 (0.04) | 17.83 (0.09) | 17.78 (0.03) | 17.91 (0.04)
59510.9 | 18.59 (0.23) | 18.01 (0.06) | … | 17.77 (0.03) | 17.78 (0.14)
59511.9 | 18.57 (0.24) | 18.00 (0.08) | 18.17 (0.07) | 17.78 (0.04) | 17.85 (0.05)
59512.8 | 18.69 (0.21) | 18.05 (0.05) | 18.17 (0.07) | 17.77 (0.02) | …
59514.9 | 18.78 (0.21) | 17.99 (0.05) | 18.20 (0.04) | 17.87 (0.02) | 17.93 (0.04)
59515.9 | 18.73 (0.18) | 18.14 (0.06) | 18.30 (0.04) | 17.96 (0.02) | 18.03 (0.03)
59516.9 | 18.89 (0.21) | 18.09 (0.08) | 18.36 (0.04) | 17.92 (0.02) | 18.00 (0.04)
59517.9 | 18.95 (0.21) | 18.21 (0.06) | 18.34 (0.05) | 17.97 (0.03) | 17.96 (0.03)
59518.9 | 19.03 (0.18) | 18.22 (0.06) | 18.33 (0.06) | 17.92 (0.02) | 18.08 (0.03)
59538.8 | … | 18.18 (0.09) | 18.47 (0.05) | 17.94 (0.03) | 17.85 (0.03)
59539.8 | … | 18.22 (0.05) | 18.48 (0.09) | 17.97 (0.05) | 17.89 (0.05)
59555.8 | 18.62 (0.23) | 17.85 (0.03) | 18.04 (0.05) | 17.61 (0.02) | 17.68 (0.04)
59565.8 | … | … | 17.82 (0.24) | 17.39 (0.13) | 17.46 (0.07)
59585.8 | 18.28 (0.19) | 17.48 (0.06) | 17.73 (0.05) | 17.32 (0.02) | 17.29 (0.02)
59586.7 | 18.25 (0.21) | 17.47 (0.03) | 17.69 (0.04) | 17.29 (0.02) | 17.24 (0.04)
59591.7 | 18.26 (0.26) | 17.66 (0.08) | 17.73 (0.07) | 17.31 (0.05) | 17.30 (0.04)
59593.7 | 18.38 (0.23) | 17.60 (0.10) | 17.75 (0.08) | 17.37 (0.05) | 17.32 (0.05)
59594.7 | 18.37 (0.21) | 17.67 (0.07) | 17.88 (0.05) | 17.42 (0.03) | 17.40 (0.05)
59597.7 | … | 17.71 (0.16) | 17.99 (0.05) | 17.45 (0.05) | 17.48 (0.05)
59598.8 | … | 17.77 (0.06) | 18.09 (0.07) | 17.43 (0.06) | 17.45 (0.03)
59603.7 | 19.00 (0.17) | 17.92 (0.07) | 18.21 (0.04) | 17.67 (0.07) | 17.62 (0.04)
Table 7: Log of spectroscopic observations UT Date | Phase | Instrument | Range | R
---|---|---|---|---
| (days) |  | (Å) | ($\lambda$/$\Delta\lambda$)
2021-10-09 | +2 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-10-11 | +4 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-10-13 | +6 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-10-21 | +14 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-10-26 | +19 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-10-31 | +24 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-11-09 | +33 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-11-20 | +44 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-12-16 | +70 | LCO FLOYDS | 3250$-$10 000 | 400-700
2021-12-29 | +83 | LCO FLOYDS | 3250$-$10 000 | 400-700
2022-01-13 | +98 | LCO FLOYDS | 3250$-$10 000 | 400-700
2022-07-26 | +293 | Magellan LDSS-3 | 3700$-$10 650 | ?

##  Appendix B Results of SYN++ modeling

###  B.1 SN 2021aatd

Figure 17: The modeling of the available spectra of SN 2021aatd using the SYN++ code. The observed spectra (black) are corrected for redshift and interstellar reddening before plotting. Their best-fit models are marked with red color, while the vertically shifted orange lines represent the single-ion contributions to the spectra.  Table 8: Best-fit SYN++ parameter values of the modeled spectra of SN 2021aatd. The global parameters are: $T_{\rm ph}$ (1000K) and $v_{\rm ph}$ ($10^{3}$ km s-1). The local parameters are: $\log\tau$ (–), $v_{\rm min}$ ($10^{3}$ km s-1), $v_{\rm max}$ ($10^{3}$ km s-1), aux ($10^{3}$ km s-1) and $T_{\rm exc}$ (1000 K). Ions | H I | He I | Na I | Mg I | Mg II | Ca II | Ti II | Fe II | Sr II | Ba ii
---|---|---|---|---|---|---|---|---|---|---
SN 2021aatd +2 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $9; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $16)
$\log\tau$ | 1.3 | 0.0 |  |  |  |  |  |  |  |
$v_{\rm min}$ | 16.0 | 16.0 |  |  |  |  |  |  |  |
$v_{\rm max}$ | 30.0 | 30.0 |  |  |  |  |  |  |  |
aux | 1.0 | 2.0 |  |  |  |  |  |  |  |
$T_{\rm exc}$ | 8.0 | 25.5 |  |  |  |  |  |  |  |
SN 2021aatd +4 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $11; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $15)
$\log\tau$ | 1.2 | -0.2 |  |  |  |  |  |  |  |
$v_{\rm min}$ | 15.0 | 15.0 |  |  |  |  |  |  |  |
$v_{\rm max}$ | 30.0 | 30.0 |  |  |  |  |  |  |  |
aux | 1.0 | 2.0 |  |  |  |  |  |  |  |
$T_{\rm exc}$ | 7.5 | 7.5 |  |  |  |  |  |  |  |
SN 2021aatd +6 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7.5; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $14)
$\log\tau$ | 1.0 |  |  |  |  |  |  |  |  |
$v_{\rm min}$ | 14.0 |  |  |  |  |  |  |  |  |
$v_{\rm max}$ | 30.0 |  |  |  |  |  |  |  |  |
aux | 2.0 |  |  |  |  |  |  |  |  |
$T_{\rm exc}$ | 2.6 |  |  |  |  |  |  |  |  |
SN 2021aatd +14 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6.5; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $10)
$\log\tau$ | 1.9 | 0.0 |  |  |  |  |  | 0.5 |  |
$v_{\rm min}$ | 10.0 | 20.0 |  |  |  |  |  | 10.0 |  |
$v_{\rm max}$ | 30.0 | 30.0 |  |  |  |  |  | 30.0 |  |
aux | 2.0 | 10.0 |  |  |  |  |  | 1.0 |  |
$T_{\rm exc}$ | 6.5 | 5.5 |  |  |  |  |  | 5.5 |  |
SN 2021aatd +19 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $9)
$\log\tau$ | 1.7 |  |  | 0.0 | 0.0 | 2.5 |  |  |  |
$v_{\rm min}$ | 9.0 |  |  | 9.0 | 9.0 | 9.0 |  |  |  |
$v_{\rm max}$ | 30.0 |  |  | 30.0 | 30.0 | 30.0 |  |  |  |
aux | 2.0 |  |  | 2.0 | 2.0 | 2.0 |  |  |  |
$T_{\rm exc}$ | 5.0 |  |  | 5.0 | 5.0 | 5.5 |  |  |  |
SN 2021aatd +24 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $8)
$\log\tau$ | 1.7 |  |  | -0.3 |  | 2.3 |  | 0.0 |  | 0.5
$v_{\rm min}$ | 8.0 |  |  | 8.0 |  | 8.0 |  | 8.0 |  | 8.0
$v_{\rm max}$ | 30.0 |  |  | 30.0 |  | 30.0 |  | 30.0 |  | 30.0
aux | 2.0 |  |  | 2.0 |  | 2.0 |  | 1.0 |  | 1.0
$T_{\rm exc}$ | 5.0 |  |  | 5.0 |  | 5.5 |  | 5.0 |  | 5.0
SN 2021aatd +33 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7)
$\log\tau$ | 1.7 |  |  | -0.3 |  | 2.3 | -0.5 | 0.6 | 2.2 | 0.9
$v_{\rm min}$ | 7.0 |  |  | 7.0 |  | 7.0 | 7.0 | 7.0 | 7.0 | 7.0
$v_{\rm max}$ | 30.0 |  |  | 30.0 |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  |  | 2.0 |  | 2.0 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  | 5.0 |  | 5.5 | 5.0 | 5.0 | 5.0 | 5.0
SN 2021aatd +44 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6.5; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6)
$\log\tau$ | 1.7 |  | -0.3 | 0.0 |  | 2.6 | 0.5 | 0.5 | 2.2 | 0.9
$v_{\rm min}$ | 6.0 |  | 6.0 | 6.0 |  | 6.0 | 6.0 | 6.0 | 6.0 | 6.0
$v_{\rm max}$ | 30.0 |  | 30.0 | 30.0 |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  | 2.0 | 2.0 |  | 2.0 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  | 5.0 | 5.0 |  | 5.5 | 5.0 | 5.0 | 5.0 | 5.0
SN 2021aatd +70 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6.5; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $5)
$\log\tau$ | 1.3 |  | -0.3 | 0.0 |  | 2.9 | 0.1 | 0.5 | 0.5 | 0.5
$v_{\rm min}$ | 5.0 |  | 5.0 | 5.0 |  | 6.0 | 5.0 | 6.0 | 5.0 | 5.0
$v_{\rm max}$ | 30.0 |  | 30.0 | 30.0 |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  | 2.0 | 2.0 |  | 2.0 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  | 5.0 | 5.0 |  | 5.5 | 5.0 | 5.0 | 5.0 | 5.0
SN 2021aatd +83 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6.3; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $5)
$\log\tau$ | 1.3 |  | -0.3 | 0.0 |  | 2.9 | -0.3 | 0.3 | 0.5 | 0.5
$v_{\rm min}$ | 5.0 |  | 5.0 | 5.0 |  | 6.0 | 5.0 | 6.0 | 5.0 | 5.0
$v_{\rm max}$ | 30.0 |  | 30.0 | 30.0 |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  | 2.0 | 2.0 |  | 2.0 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  | 5.0 | 5.0 |  | 5.5 | 5.0 | 5.0 | 5.0 | 5.0
SN 2021aatd +98 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $5)
$\log\tau$ | 1.3 |  | -0.3 | 0.0 |  | 2.9 | 0.3 | 0.5 | 1.0 | 0.5
$v_{\rm min}$ | 5.0 |  | 5.0 | 5.0 |  | 6.0 | 5.0 | 6.0 | 5.0 | 5.0
$v_{\rm max}$ | 30.0 |  | 30.0 | 30.0 |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  | 2.0 | 2.0 |  | 2.0 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  | 5.0 | 5.0 |  | 5.5 | 5.0 | 5.0 | 5.0 | 5.0

###  B.2 SN 2020faa

Figure 18: Spectrum modeling of SN 2020faa, the comparison object of SN 2021aatd. The redshift- and reddening corrected spectra (black) are plotted together with the best-fit model obtained in SYN++ (green), and the single-ion contributions (orange) similarly to Fig. 17. Table 9: Best-fit parameters of the SYN++ modeling of SN 2020faa in the same way as in Table LABEL:tab:local_21aatd.  Ions | H I | He I | C II | O I | Na I | Mg I | Ca II | Ti II | Fe II | Sr II | Ba ii
---|---|---|---|---|---|---|---|---|---|---|---
SN 2020faa +6 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $10; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $20)
$\log\tau$ | 0.3 |  |  |  |  |  |  |  |  |  |
$v_{\rm min}$ | 20.0 |  |  |  |  |  |  |  |  |  |
$v_{\rm max}$ | 30.0 |  |  |  |  |  |  |  |  |  |
aux | 2.0 |  |  |  |  |  |  |  |  |  |
$T_{\rm exc}$ | 6.0 |  |  |  |  |  |  |  |  |  |
SN 2020faa +11 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $8; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $20)
$\log\tau$ | 0.3 | 0.0 |  |  |  |  |  |  |  |  |
$v_{\rm min}$ | 20.0 | 20.0 |  |  |  |  |  |  |  |  |
$v_{\rm max}$ | 30.0 | 30.0 |  |  |  |  |  |  |  |  |
aux | 2.0 | 2.0 |  |  |  |  |  |  |  |  |
$T_{\rm exc}$ | 6.0 | 6.3 |  |  |  |  |  |  |  |  |
SN 2020faa +12 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7.8; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $20)
$\log\tau$ | 0.3 | 0.0 |  |  |  |  |  |  |  |  |
$v_{\rm min}$ | 20.0 | 20.0 |  |  |  |  |  |  |  |  |
$v_{\rm max}$ | 30.0 | 30.0 |  |  |  |  |  |  |  |  |
aux | 2.0 | 2.2 |  |  |  |  |  |  |  |  |
$T_{\rm exc}$ | 6.0 | 6.3 |  |  |  |  |  |  |  |  |
SN 2020faa +68 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7.3; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $10)
$\log\tau$ | 0.5 |  |  |  | -0.1 |  | 2.7 |  |  |  | 0.5
$v_{\rm min}$ | 10.0 |  |  |  | 10.0 |  | 10.0 |  |  |  | 10.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 |  | 30.0 |  |  |  | 30.0
aux | 2.0 |  |  |  | 1.0 |  | 0.7 |  |  |  | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 20.0 |  | 5.5 |  |  |  | 5.0
SN 2020faa +88 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7.3; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $10)
$\log\tau$ | 0.0 |  |  |  | -0.4 | 0.0 | 2.5 |  |  |  | 0.6
$v_{\rm min}$ | 10.0 |  |  |  | 10.0 | 10.0 | 10.0 |  |  |  | 10.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 | 30.0 | 30.0 |  |  |  | 30.0
aux | 2.0 |  |  |  | 2.0 | 1.0 | 2.0 |  |  |  | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 20.0 | 5.0 | 5.5 |  |  |  | 5.0
SN 2020faa +100 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6.8; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $8)
$\log\tau$ | 0.6 |  | -0.3 | -0.2 |  | 0.0 | 3.5 | -0.7 |  |  | 0.6
$v_{\rm min}$ | 8.0 |  | 8.0 | 8.0 |  | 8.0 | 8.0 | 8.0 |  |  | 8.0
$v_{\rm max}$ | 30.0 |  | 30.0 | 30.0 |  | 30.0 | 30.0 | 30.0 |  |  | 30.0
aux | 2.0 |  | 2.0 | 2.0 |  | 1.0 | 1.0 | 1.0 |  |  | 2.0
$T_{\rm exc}$ | 5.0 |  | 5.0 | 5.0 |  | 5.0 | 5.5 | 5.0 |  |  | 5.0
SN 2020faa +118 and +120 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6.6; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7)
$\log\tau$ | 0.4 |  |  | -0.5 | -0.1 | -0.5 | 2.3 |  |  | 1.0 | 0.7
$v_{\rm min}$ | 7.0 |  |  | 7.0 | 7.0 | 7.0 | 7.0 |  |  | 7.0 | 7.0
$v_{\rm max}$ | 30.0 |  |  | 30.0 | 30.0 | 30.0 | 30.0 |  |  | 30.0 | 30.0
aux | 2.0 |  |  | 1.0 | 2.0 | 2.0 | 1.0 |  |  | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  | 5.0 | 5.0 | 5.0 | 5.5 |  |  | 5.0 | 5.0
SN 2020faa +129 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7.5; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $9)
$\log\tau$ | 0.2 |  |  |  | -0.1 | -0.5 | 3.0 |  |  | 1.0 | 0.8
$v_{\rm min}$ | 9.0 |  |  |  | 9.0 | 9.0 | 9.0 |  |  | 9.0 | 9.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 | 30.0 | 30.0 |  |  | 30.0 | 30.0
aux | 2.0 |  |  |  | 2.0 | 2.0 | 0.6 |  |  | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 5.0 | 5.0 | 5.5 |  |  | 5.0 | 5.0
SN 2020faa +139 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $9)
$\log\tau$ | 0.2 |  |  |  | -0.1 | -0.5 | 3.0 | 0.0 |  | 1.0 | 0.9
$v_{\rm min}$ | 9.0 |  |  |  | 9.0 | 9.0 | 9.0 | 9.0 |  | 9.0 | 9.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 | 30.0 | 30.0 | 30.0 |  | 30.0 | 30.0
aux | 2.0 |  |  |  | 2.0 | 2.0 | 0.6 | 2.0 |  | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 5.0 | 5.0 | 5.5 | 5.0 |  | 5.0 | 5.0
SN 2020faa +143 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7.2; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $9)
$\log\tau$ | 0.4 |  |  |  | -0.1 | -0.5 | 3.0 | 0.0 | 0.6 | 1.0 | 0.7
$v_{\rm min}$ | 9.0 |  |  |  | 9.0 | 9.0 | 9.0 | 9.0 | 9.0 | 9.0 | 9.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  |  |  | 2.0 | 2.0 | 0.6 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 5.0 | 5.0 | 5.5 | 5.0 | 5.0 | 5.0 | 5.0
SN 2020faa +149 and +152 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $8.0; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $8)
$\log\tau$ | 0.0 |  |  |  | -0.1 | -0.2 | 2.2 | 0.0 |  | 1.0 | 0.7
$v_{\rm min}$ | 8.0 |  |  |  | 8.0 | 8.0 | 8.0 | 8.0 |  | 8.0 | 8.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 | 30.0 | 30.0 | 30.0 |  | 30.0 | 30.0
aux | 2.0 |  |  |  | 2.0 | 2.0 | 0.6 | 2.0 |  | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 5.0 | 5.0 | 5.5 | 5.0 |  | 5.0 | 5.0
SN 2020faa +158 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $6.0; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $8)
$\log\tau$ | 0.5 |  |  |  | -0.3 | -0.4 | 2.9 | 0.0 | 0.8 | 1.0 | 0.7
$v_{\rm min}$ | 8.0 |  |  |  | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 | 8.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  |  |  | 2.0 | 2.0 | 0.6 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 5.0 | 5.0 | 5.5 | 5.0 | 5.0 | 5.0 | 5.0
SN 2020faa +165 days phase ($T_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $7.5; $v_{\rm ph}\leavevmode\nobreak\
=\leavevmode\nobreak\ $8)
$\log\tau$ | 0.0 |  |  |  | -0.1 | -1.3 | 1.7 | 0.0 | 0.5 | 1.0 | 0.7
$v_{\rm min}$ | 8.0 |  |  |  | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 | 8.0 | 8.0
$v_{\rm max}$ | 30.0 |  |  |  | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 2.0 |  |  |  | 2.0 | 2.0 | 0.6 | 1.0 | 1.0 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 |  |  |  | 5.0 | 5.0 | 5.5 | 5.0 | 5.0 | 5.0 | 5.0

###  B.3 SN 1987A

Table 10: Best-fit parameters of the SYN++ modeling of SN 1987A in the same way as in Table LABEL:tab:local_21aatd.  Ions | H I | N II | O I | Na I | Mg II | Ca II | Ti II | Fe II | Sr II | Ba ii
---|---|---|---|---|---|---|---|---|---|---
SN 1987A +84 days phase ($T_{\rm ph}\leavevmode\nobreak\ =\leavevmode\nobreak\
$4.5; $v_{\rm ph}\leavevmode\nobreak\ =\leavevmode\nobreak\ $3)
$\log\tau$ | 1.7 | 0.0 | 1.0 | 1.3 | -0.3 | 2.3 | -0.5 | 1.8 | 2.2 | 1.2
$v_{\rm min}$ | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0
$v_{\rm max}$ | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0 | 30.0
aux | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 2.0 | 1.0 | 0.5 | 1.0 | 1.0
$T_{\rm exc}$ | 5.0 | 14.5 | 5.0 | 5.0 | 5.0 | 5.5 | 5.0 | 5.0 | 5.0 | 4.5