11institutetext: Instituto de Astrofísica de La Plata (IALP), CCT-CONICET-
UNLP. Paseo del Bosque S/N, B1900FWA, La Plata, Argentina
11email<EMAIL_ADDRESS>22institutetext: Facultad de
Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo
del Bosque S/N, B1900FWA, La Plata, Argentina 33institutetext: Universidad
Nacional de Río Negro. Sede Andina, Mitre 630 (8400) Bariloche, Argentina
44institutetext: European Southern Observatory, Alonso de Córdova 3107,
Casilla 19, Santiago, Chile 55institutetext: Kavli Institute for the Physics
and Mathematics of the Universe (WPI), The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa, Chiba 277-8583, Japan 66institutetext: Vice President
and Head of Mission of AURA-O in Chile, Avda. Presidente Riesco 5335 Suite
507, Santiago, Chile 77institutetext: Hagler Institute for Advanced Studies,
Texas A&M University, College Station, TX 77843, USA 88institutetext: CENTRA-
Centro de Astrofísica e Gravitaçäo and Departamento de Física, Instituto
Superio Técnico, Universidade de Lisboa, Avenida Rovisco Pais, 1049-001
Lisboa, Portugal 99institutetext: Consejo Nacional de Investigaciones
Científicas y Técnicas (CONICET), Argentina. 1010institutetext: Department of
Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,
Denmark 1111institutetext: Carnegie Observatories, Las Campanas Observatory,
Casilla 601, La Serena, Chile 1212institutetext: Finnish Centre for Astronomy
with ESO (FINCA), FI-20014 University of Turku, Finland 1313institutetext:
Tuorla Observatory, Department of Physics and Astronomy, FI-20014 University
of Turku, Finland 1414institutetext: Observatories of the Carnegie Institution
for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA 1515institutetext:
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu,
HI 96822, USA 1616institutetext: Department of Astronomy, University of
California, 501 Campbell Hall, Berkeley, CA 94720-3411, USA 1717institutetext:
Data and Artificial Intelligence Initiative, Faculty of Physical and
Mathematical Sciences, University of Chile, Santiago, Chile 1818institutetext:
Centre for Mathematical Modelling, Faculty of Physical and Mathematical
Sciences, University of Chile, Santiago, Chile 1919institutetext: Millennium
Institute of Astrophysics, Santiago, Chile. 2020institutetext: Department of
Astronomy, Faculty of Physical and Mathematical Sciences, University of Chile,
Santiago, Chile 2121institutetext: Institute of Space Sciences (ICE, CSIC),
Campus UAB, Carrer de Can Magrans, s/n, E-08193 Barcelona, Spain
2222institutetext: Department of Physics, Florida State University, 77
Chieftan Way, Tallahassee, FL 32306, USA 2323institutetext: George P. and
Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy,
Department of Physics and Astronomy, Texas A&M University, College Station, TX
77843

# Type II supernovae from the Carnegie Supernova Project-I

III. Understanding SN II diversity through correlations between physical and
observed properties

L. Martinez 112233 J. P. Anderson 44 M. C. Bersten 112255 M. Hamuy 6677 S.
González-Gaitán 88 M. Orellana 3399
M. Stritzinger 1010 M. M. Phillips 1111 C. P. Gutiérrez 12121313 C. Burns 1414
T. de Jaeger 15151616 K. Ertini 1122 G. Folatelli 112255
F. Förster 1717181819192020 L. Galbany 2121 P. Hoeflich 2222 E. Y. Hsiao 2222
N. Morrell 1111 P. J. Pessi 2244
N. B. Suntzeff 2323

(Received XXX; accepted XXX)

Type II supernovae (SNe II) show great photometric and spectroscopic diversity
which is attributed to the varied physical characteristics of their progenitor
and explosion properties. In this study, the third of a series of papers where
we analyse a large sample of SNe II observed by the Carnegie Supernova
Project-I, we present correlations between their observed and physical
properties. Our analysis shows that explosion energy is the physical property
that correlates with the highest number of parameters. We recover previously
suggested relationships between the hydrogen-rich envelope mass and the
plateau duration, and find that more luminous SNe II with higher expansion
velocities, faster declining light curves, and higher 56Ni masses are
consistent with higher energy explosions. In addition, faster declining SNe II
(usually called SNe IIL) are also compatible with more concentrated 56Ni in
the inner regions of the ejecta. Positive trends are found between the initial
mass, explosion energy, and 56Ni mass. While the explosion energy spans the
full range explored with our models, the initial mass generally arises from a
relatively narrow range. Observable properties were measured from our grid of
bolometric LC and photospheric velocity models to determine the effect of each
physical parameter on the observed SN II diversity. We argue that explosion
energy is the physical parameter causing the greatest impact on SN II
diversity, that is, assuming the non-rotating solar-metallicity single-star
evolution as in the models used in this study. The inclusion of pre-SN models
assuming higher mass loss produces a significant increase in the strength of
some correlations, particularly those between the progenitor hydrogen-rich
envelope mass and the plateau and optically thick phase durations. These
differences clearly show the impact of having different treatments of stellar
evolution, implying that changes in the assumption of standard single-star
evolution are necessary for a complete understanding of SN II diversity.

###### Key Words.:

supernovae: general — stars: evolution — stars: massive

##  1 Introduction

Type II supernovae (SNe II111Throughout this paper we use ‘SNe II’ to refer to
all hydrogen-rich core-collapse SNe that show slow- and fast-declining light
curves (historically referred to as SNe IIP and IIL, respectively), excluding
type IIb, IIn, and SN 1987A-like events.) result from the terminal explosion
of massive stars (¿8$-$10 $M_{\odot}$) that have retained a significant
fraction of their hydrogen-rich envelope at the time of core collapse. SNe II
are classified by the presence of prominent hydrogen lines in their spectra
(Minkowski 1941) and are the most common type of core-collapse SN in nature
(Li et al. 2011; Shivvers et al. 2017). The direct detection of progenitors of
SNe II in pre-explosion images provides increasing evidence for red supergiant
(RSG) progenitors (e.g. Van Dyk et al. 2003; Smartt 2009) with initial masses
in the range of $\sim$8$-$18 $M_{\odot}$ (Smartt 2015).

Statistical studies of SNe II have revealed the great diversity in their
photometric and spectral properties. In addition, a continuum of light-curve
(LC) parameters (e.g. absolute magnitudes, declination rates, time durations
of different phases), colours, expansion velocities, and equivalent widths of
a number of spectral lines are observed (e.g. Hamuy 2003; Bersten 2013;
Anderson et al. 2014b; Sanders et al. 2015; Valenti et al. 2016; Gutiérrez et
al. 2017a, b; de Jaeger et al. 2018). The diversity of spectroscopic
properties is also observed in the near-infrared (NIR), although in this
regime there may be spectroscopic differences between slow- and fast-declining
SNe II (Davis et al. 2019). Early-time LCs ($\lesssim$ 30 days from explosion)
also contribute to SN II diversity (e.g. González-Gaitán et al. 2015; Gall et
al. 2015; Ganot et al. 2020). Early LCs are mostly sensitive to the
characteristics of the outer envelope. The measured rise times, that is, the
time from the explosion epoch to the date of maximum light, are much shorter
than model predictions indicating either small and dense pre-SN envelopes, or
a delayed and prolonged shock breakout because of interaction with an extended
atmosphere or a shell of dense circumstellar material (CSM) (González-Gaitán
et al. 2015; Förster et al. 2018; Morozova et al. 2018).

The observed diversity of SNe II may be attributed to differences in stellar
evolution processes that produce progenitors with different pre-SN properties
(hydrogen-rich envelope mass, progenitor size, chemical abundances, CSM
properties) and/or differences in the properties of the explosion (e.g. energy
deposited by the shock and 56Ni mass). While the underlying physical processes
involved in SNe II are generally well understood, significant gaps remain in
our understanding of how massive stars evolve to produce these type of
hydrogen-rich events, particularly in regard to the different mass-loss
processes involved (stellar winds, eruptions, and mass transfer in binary
systems). Therefore, it is important to constrain the predominant physical
properties that cause the observed diversity, and to determine the full range
of parameters that produce SN II events.

Theoretical works have studied the diversity of SNe II in terms of physical
properties and found that changes in the hydrogen-rich envelope mass,
progenitor radius, and explosion energy produce large differences in the
luminosity, duration of the optically thick phase, and expansion velocities
(e.g. Litvinova & Nadezhin 1985; Young 2004; Utrobin 2007; Kasen & Woosley
2009; Bersten et al. 2011; Goldberg et al. 2019). Kasen & Woosley (2009)
showed that 56Ni mass extends the plateau duration, although with a smaller
contribution than the aforementioned physical parameters (see also Bersten
2013), while the mixing of 56Ni within the ejecta tends to modify the shape of
the LC (Bersten et al. 2011; Kozyreva et al. 2019). In addition, it has been
shown that the progenitor hydrogen-rich envelope mass affects the plateau
declination rate, where smaller masses produce SNe II with faster declining
LCs (e.g. Bartunov & Blinnikov 1992; Blinnikov & Bartunov 1993). Dessart et
al. (2013) presented synthetic multi-band LCs and spectra varying progenitor
and explosion properties —such as the hydrogen-rich envelope mass, explosion
energy, radius, and metallicity— that support previous findings with respect
to the changes in LCs (see also Hillier & Dessart 2019). Additionally, Dessart
et al. (2013) showed that the progenitor radius strongly influences early
colours. The hydrogen-rich envelope mass and the explosion energy seem to be
the physical parameters that most influence SN II LC diversity.

In this, the third paper of a series where we analyse a statistically
significant sample of SNe II, we focus on understanding their observed
diversity in terms of progenitor and explosion properties. We used a much
larger set of observations than previous studies, which were directly modelled
using hydrodynamical simulations that explore the most important physical
parameters.

The first paper of this series (Martinez et al. 2021b, hereafter Paper I)
focuses on the calculation of bolometric LCs for the SNe II observed by the
Carnegie Supernova Project-I (CSP-I, Hamuy et al. 2006). We found that NIR
observations are crucial for accurate bolometric luminosity calculations, and
that the absence of these data produces significantly different bolometric
LCs. These differences are then transferred to the physical parameters derived
from LC and expansion velocity modelling. Paper I provides relations to
address the absence of NIR data, in addition to new prescriptions for
bolometric corrections as a function of optical colours to be used by the
community.

The bolometric LCs presented in Paper I, together with the Fe ii $\lambda$5169
line velocities measured by Gutiérrez et al. (2017b), were then used for
deriving progenitor and explosion properties via hydrodynamical modelling in
Martinez et al. (2021a, hereafter Paper II), where a large set of explosion
models were used that were previously presented in Martinez et al. (2020,
hereafter M20). In Paper II we also analysed the distributions of the inferred
physical parameters. The modelling of the initial-mass ($M_{\rm ZAMS}$)
distribution gives an upper mass of $<$ 21.5 $M_{\odot}$, consistent with the
existence of the RSG problem (Smartt et al. 2009), especially when the power-
law slope of the $M_{\rm ZAMS}$ distribution is constrained to be that of a
Salpeter massive-star initial-mass function (IMF). However, a much steeper
distribution than that for a Salpeter IMF is favoured by our modelling. We
named this result ‘the IMF incompatibility’. This incompatibility is due to
the large number of progenitors compatible with low-ejecta-mass models,
possibly implying that massive stars lose more mass during their lives than
predicted by standard single-star evolution, although a more thorough analysis
of all stellar evolution assumptions is required to test this hypothesis.

As part of the studies presented in Paper I and Paper II, we built the most
homogeneous and largest sample of SN II bolometric LCs to date for which the
physical properties of the objects are determined via hydrodynamical
modelling. Here, in Paper III, we present an analysis of correlations between
observed and physical properties in order to further our understanding of SN
II diversity in terms of the physics of the explosions and their progenitors.
The paper is organised as follows. Section 2 briefly describes the
observational and modelling samples. Section 3 presents the analysis of
correlations when physical properties are derived using pre-SN models assuming
non-rotating single-star evolution, while Sect. 4 shows the results when non-
standard stellar evolution is considered. In Sect. 5 we discuss the most
interesting of the correlations in detail, and present our conclusions in
Sect. 6. Figures not included in the main body of the manuscript are presented
in the Appendices.

##  2 Observational and theoretical sample

###  2.1 Supernova sample

The sample of SNe II used in this work is the same as that analysed in Paper I
and Paper II. The dataset was observed by the CSP-I and contains 74 SNe II.
The CSP-I was a follow-up programme of SNe based at the Las Campanas
Observatory that observed between 2004 and 2009 (Hamuy et al. 2006). The CSP-I
sample consists of optical ($uBgVri$) and NIR ($YJH$) LCs and optical spectra
for most objects, resulting from high-cadence and high-quality observations.
CSP-I $V$-band photometry was published by Anderson et al. (2014b, A14
hereafter) while the optical spectra were published by Gutiérrez et al.
(2017a, b). The final photometry for the CSP-I SN II sample is presented in
Anderson et al. (in prep.). Details of these SNe II are available in the
above-mentioned studies and Paper I.

###  2.2 Observed properties

Previous studies used the SNe II in the CSP-I sample to analyse their spectral
and photometric diversity. A14 analysed the $V$-band LC properties through the
measurement of magnitudes at different epochs, declination rates, and
durations of different phases. Anderson et al. (2014a) and Gutiérrez et al.
(2014) presented studies of the Hα profiles, specifically of the blueshifted
emission-peak offset, velocity, and ratio of absorption to emission. A
posterior analysis was presented by Gutiérrez et al. (2017b) and Gutiérrez et
al. (2017a, hereafter G17), where the authors studied expansion velocities and
pseudo-equivalent widths (pEWs) of numerous spectral lines, together with
additional $V$-band LC and spectral properties. CSP-I SN II colour curves were
studied by de Jaeger et al. (2018, hereafter dJ18).

The number of photometric and spectroscopic parameters studied in the
literature for the CSP-I SN II sample is extremely large, and a complete
analysis of all previously mentioned parameters would be too long for one
publication. Therefore, in this section, we summarise the observed parameters
used in the present study. These are chosen to elucidate previous questions
posed in the literature with respect to how observed parameters are determined
from progenitor and/or explosion properties.

Progenitor and explosions properties for our SN II sample were derived via
hydrodynamical modelling of their bolometric LCs and photospheric velocities
(Paper II). Therefore, we used bolometric LC parameters for the analysis of
correlations, instead of the $V$-band LC parameters measured by A14 and G17.
SN II bolometric LC parameters for the CSP-I sample were measured and analysed
in Paper I, to which we refer the reader for details. We briefly outline the
parameters used in the present work (see also Fig. 8 from Paper I, ): (1)
$M_{\rm bol,end}$ is the bolometric magnitude measured 30 days before the mid-
point of the transition from plateau to the radioactive tail ($t_{\rm PT}$);
(2) $M_{\rm bol,tail}$ is the bolometric magnitude measured 30 days after
$t_{\rm PT}$; (3) $s_{\rm 1}$ is the declination rate in magnitudes per 100
days of the cooling phase; (4) $s_{\rm 2}$ is the declination rate in
magnitudes per 100 days of the plateau phase (which is not necessarily a phase
of constant magnitude); (5) $s_{\rm 3}$ is the declination rate in magnitudes
per 100 days of the slope in the radioactive tail phase; (6) optd is the
duration of the optically thick phase and it is equal to $t_{\rm PT}$; (7) pd
is the duration of the plateau phase and it is equal to $t_{\rm PT}$ $-$
$t_{\rm trans}$222$t_{\rm trans}$ is the epoch of transition between $s_{\rm
1}$ and $s_{\rm 2}$.; and (8) Cd is the duration of the cooling phase, defined
between the explosion epoch and $t_{\rm trans}$. In addition, we used ten
spectral parameters from G17, all measured at 50 days post-explosion: (1)
expansion velocity ($v$) for the absorption component of Hα, Hβ, and Fe ii
$\lambda$5169 lines; (2) pEW of Hα (absorption and emission components), Hβ,
Fe ii $\lambda$4924, Fe ii $\lambda$5018, and Fe ii $\lambda$5169; and (3)
flux ratio of the absorption to emission component of Hα P-Cygni profile
($a/e$). Colour information was also included. We used ($g-r$) colours
measured at 15 and 70 days post-explosion (dJ18).

###  2.3 Progenitor and explosion models

Table 1: Physical parameters for the grid of hydrodynamical models.

Parameter | Range | Increment | Extra values
---|---|---|---
$M_{\rm ZAMS}$ | 9$-$25 $M_{\odot}$ | 1 $M_{\odot}$ | —
_E_ | 0.1$-$1.5 foeaa$a$1 foe $\equiv$ 1051 erg.,bb$b$Models for the largest masses and lowest energies were not computed due to numerical difficulties (see M20, for details). | 0.1 foe | —
$M_{\rm Ni}$ | 0.01$-$0.08 $M_{\odot}$ | 0.01 $M_{\odot}$ | 10-4, 5$\times$10-3 $M_{\odot}$
${}^{56}\rm Ni$ mixing | 0.2$-$0.8cc$c$Given in fraction of the pre-SN mass. | 0.3 | —

333

The LCs and expansion velocities of SNe II are sensitive to the physical
properties of the progenitor star and the explosion, such as the ejecta mass
($M_{\rm ej}$: pre-SN mass minus the mass of the forming compact remnant),
hydrogen-rich envelope mass ($M_{\rm H,env}$), progenitor radius ($R$),
explosion energy (_E_), 56Ni mass ($M_{\rm Ni}$), and its distribution within
the ejecta (${}^{56}\rm Ni$ mixing). In Paper II, we used a large grid of
bolometric LC and photospheric velocity models applied to stellar evolution
progenitors presented in M20 for deriving the physical properties of the CSP-I
SN II sample. Here, a summary of the models is presented, but we refer the
reader to M20 for additional information.

Non-rotating solar-metallicity pre-SN RSG models were calculated for $M_{\rm
ZAMS}$ between 9 and 25 $M_{\odot}$ with increments of 1 $M_{\odot}$ using the
public stellar evolution code MESA444http://mesa.sourceforge.net/ version
10398 (Paxton et al. 2011, 2013, 2015, 2018, 2019). The stellar models were
evolved from the pre-main sequence until core collapse, except for the 9, 10,
and 11 $M_{\odot}$ progenitor models that were calculated up to core carbon
depletion, because more advanced burning phases are computationally expensive.
Stellar evolution was followed using the standard prescriptions for mass loss
and convection. For convection, the Ledoux criterion was adopted with a
mixing-length parameter of 2.0. The wind mass loss was calculated from the
‘Dutch’ prescription defined in the MESA code with an efficiency of $\eta$ = 1
(de Jager et al. 1988; Vink et al. 2001; Glebbeek et al. 2009). Hydrogen-rich
envelope masses for this set of progenitor models cover the range of
7.1$-$10.4 $M_{\odot}$, while progenitor radii are found in the range of
445$-$1085 $R_{\odot}$. These pre-SN stellar models were used as input to the
1D Lagrangian hydrodynamical code presented in Bersten et al. (2011) to
compute bolometric LCs and photospheric velocities of SNe II. The grid of
explosion models covers a wide range of explosion parameters (_E_ , $M_{\rm
Ni}$, and ${}^{56}\rm Ni$ mixing), which is given in Table 1. We refer to
these models as ‘standard models’555The grid of bolometric LCs and
photospheric velocity models can be downloaded from
https://doi.org/10.5281/zenodo.6228795.. The explosion models were calculated
without including any CSM shell surrounding the progenitor star.

In Paper II, the determination of the physical properties of the SNe II in the
CSP-I sample is based on describing the bolometric LC and photospheric
velocities simultaneously by means of comparing hydrodynamical models with
observations. The bolometric LCs for the CSP-I sample were presented in Paper
I. Fe ii line velocities measured by Gutiérrez et al. (2017b) were used,
assuming this line as a proxy for the photospheric velocity.

The posterior probability distributions of the physical parameters ($M_{\rm
ZAMS}$, _E_ , $M_{\rm Ni}$, and ${}^{56}\rm Ni$ mixing) for the SNe II in the
CSP-I sample were found by employing Markov chain Monte Carlo (MCMC) methods
via the emcee package (Foreman-Mackey et al. 2013) following Förster et al.
(2018). For this, the interpolation method presented in Förster et al. (2018)
was used to quickly interpolate between the set of hydrodynamical models
described above. The MCMC sampler assumes flat distributions as priors for the
four physical parameters of our model ($M_{\rm ZAMS}$, _E_ , $M_{\rm Ni}$, and
${}^{56}\rm Ni$ mixing) and also for the explosion epoch ($t_{\rm exp}$). The
sampler was allowed to run within the ranges of the physical parameters in our
set of hydrodynamical models (Table 1) and within the observational error of
$t_{\rm exp}$ (see Paper I, ). An additional parameter was defined, the
‘scale’, which multiplies the bolometric luminosity by a constant
dimensionless factor to allow for errors in the bolometric LC introduced by
the uncertainties in the distance and host-galaxy extinction. A Gaussian prior
was used for the scale (see Paper II, for further details). Thus, in the
current work, the physical parameters determined in Paper II were used:
$M_{\rm ZAMS}$, _E_ , $M_{\rm Ni}$, and ${}^{56}\rm Ni$ mixing. Additionally,
given that we used stellar evolution calculations as pre-SN models, each
$M_{\rm ZAMS}$ relates to a pre-SN structure, for which $M_{\rm ej}$, $R$, and
$M_{\rm H,env}$ have been determined. Therefore, we also used the last three
parameters to analyse possible correlations. The errors on our estimated
physical parameters are statistical in nature, and do not take into account
systematic errors such as the uncertainties in stellar evolution modelling. As
a consequence, the errors on the progenitor parameters are likely to be
underestimated. The size of the errors indicates the high quality of the
observations and the robustness of our fitting technique.

Together with the observed parameters mentioned in Sect. 2.2, we also measured
observables from our extensive grid of hydrodynamical LC and photospheric
velocity models for determining the effect of each physical parameter with
respect to the observed SN II diversity. We used the same grid of explosion
models previously mentioned, that is, our standard grid of models. While this
grid was built covering a wide range of progenitor and explosion parameters,
we used the interpolation technique presented in Förster et al. (2018) to
obtain an even larger number of measurements. The generated grid of models
covers the following parameter space: $M_{\rm ZAMS}$ between 9 and 25
$M_{\odot}$ in steps of 1 $M_{\odot}$ and explosion energies between 0.1 and
1.5 foe (1 foe $\equiv$ 1051 erg) in intervals of 0.05 foe, with the exception
of the largest masses and lower explosion energies, because these models could
not be calculated for numerical problems (see M20, for details). $M_{\rm Ni}$
ranges between 0.01 and 0.08 $M_{\odot}$ in steps of 0.005 $M_{\odot}$ with a
degree of mixing of between 0.2 and 0.8 (given as a fraction of the pre-SN
mass) in intervals of 0.1. For each model, we measured optd, $M_{\rm
bol,end}$, $M_{\rm bol,tail}$, $s_{\rm 2}$, and $s_{\rm 3}$ using the same
definitions as in Sect. 2.2. Additionally, we measured the photospheric
velocity at 50 days from explosion ($v_{\rm ph,50}$).

Models were constructed covering regular ranges of physical parameters, but a
subset of SN II models present bolometric LC parameters that have not (yet)
been observed in nature, although they are theoretically possible. Some of the
bolometric LC models for low _E_ and relatively high $M_{\rm H,env}$ yield
optd values that are larger than any SNe II observed to date. For this reason,
we only analysed models with optd shorter than 160 days. This criterion is
somewhat arbitrary but it is $\sim$15 days longer than the longest SN II
plateau observed to date666This choice does not affect the conclusions of this
study, but we prefer to remove these models to understand the SNe II in our
sample.: SN 2009ib (Takáts et al. 2015). In total, 38757 measurements of each
observational parameter are available.

We did not measure pd in our models for the following reason: according to the
parameter definitions in Sect. 2.2, pd needs previous measurements of two
parameters: $t_{\rm PT}$ (optd) and $t_{\rm trans}$. The latter parameter is
defined as the epoch of transition between $s_{\rm 1}$ and $s_{\rm 2}$,
although it can also be understood as the epoch of transition between the
cooling and plateau phases (based on the definitions from Sect. 2.2). However,
it is usually found that the observed cooling phase is longer than model
predictions (and rise times to maximum light of optical LCs are shorter, e.g.
González-Gaitán et al. 2015) for which the presence of additional material
confined close to the progenitor star has been suggested as an explanation.
Our explosion models were calculated without the presence of any possible
circumstellar material. Therefore, the analysis of model Cd and pd would be
biased to smaller and larger values, respectively.

##  3 Results

Table 2: Relative importance of the physical parameters for each observable measured in our LC and photospheric velocity models. | $M_{\rm ZAMS}$ | _E_ | $M_{\rm Ni}$ |  56Ni mixing
---|---|---|---|---
$optd$ | 0.28 | 0.66 | 0.06 | 0.00
$M_{\rm bol,end}$ | 0.22 | 0.69 | 0.03 | 0.06
$M_{\rm bol,tail}$ | 0.01 | 0.22 | 0.74 | 0.03
$s_{\rm 2}$ | 0.21 | 0.50 | 0.15 | 0.14
$s_{\rm 3}$ | 0.58 | 0.03 | 0.20 | 0.19
vph,50 | 0.33 | 0.67 | 0.00 | 0.00

In this section, we search for correlations between the observed and physical
parameters described in Sects. 2.2 and 2.3. Physical parameters for the CSP-I
SN II sample were inferred in Paper II, where the results were classified into
two groups: the ‘gold’ and ‘full’ samples. SNe II with extensive data coverage
that are well reproduced by our models were classified as ‘gold events’ (24
objects), but in total we inferred physical parameters for 53 objects (see
Paper II, for details). In the subsequent figures, we label these distinct
samples with different symbols and colours.

Throughout the rest of the paper, we used the Pearson test on the full sample
of SNe II to determine the existence and strength of correlations by employing
10000 Monte Carlo bootstrapping simulations. For each simulation, $N$ random
values were drawn allowing multiple events to be taken (where $N$ is the
number of events for each correlation), and the Pearson correlation
coefficient was calculated. The distribution of the correlation coefficient is
symmetric. Therefore, the mean correlation coefficient ($\rho$) of these 10000
simulations and the standard deviation ($\sigma$) were used to characterise
the correlations. These values are presented in each figure. In addition, an
upper limit to the probability of finding such a correlation strength by
chance ($P$) is presented. The gold sample was also analysed, showing similar
correlation coefficients to those of the full sample. In most cases, the
correlation coefficients estimated for the gold sample are within the error
bars of the coefficients for the full sample. Only a small number of trends
show notably different correlations. These are the trends involving $(g-r)$
colours at 15 days post-explosion, because of the low number of events in the
gold sample for which this parameter has been measured. For these reasons, we
only present our analysis of correlations for the full sample. We used the
following descriptions to characterise the strength of correlations:
correlation coefficients between 0 and 0.19 show zero correlation, 0.20$-$0.39
weak, 0.40$-$0.59 moderate, 0.60$-$0.79 strong, and 0.8$-$1.0 very strong
(Evans 1996).

The observables measured from the grid of explosion models allow us to perform
a statistical analysis of the influence of each individual physical parameter
on the observed diversity of SNe II. We performed a key driver analysis (KDA)
using the python library Kruskals777https://github.com/Rambatino/Kruskals to
determine the effect of physical properties on the observed parameters
(Kruskal 1987). KDA is a technique used to identify which of a set of
independent variables causes the largest impact on a given dependent variable.
Table 2 reports the relative importance of the physical parameters ($M_{\rm
ZAMS}$, _E_ , $M_{\rm Ni}$, and ${}^{56}\rm Ni$ mixing) for each parameter
measured from our hydrodynamical models. We used $M_{\rm ZAMS}$ given that it
is the independent variable related to a unique pre-SN structure (in the
context of standard single-star evolution) emphasising that $M_{\rm ZAMS}$
represents the effect of $M_{\rm ej}$ and $R$ simultaneously, and that they
cannot be separated given their dependency on $M_{\rm ZAMS}$. We note that our
results are relevant for the standard stellar evolution adopted in this study
and the inclusion of additional pre-SN models could modify the relative effect
of each physical parameter on the observables (see Sect. 4.2). A description
of the results from Table 2 is found in the following section together with
the analysis of correlations using the CSP-I SN II sample. In addition,
Appendix A includes figures that show observables measured from the models
against the physical parameter yielding the highest relative importance, and a
supplementary analysis of the relations found.

We separate the analysis of correlations in the following subsections.
Correlations between observed and physical parameters for the CSP-I SN II
sample are presented in Sect. 3.1. In Sect. 3.2, we then look for correlations
between physical parameters only. In addition, an Appendix is included where
the main trends between different observed spectral, colour, and bolometric LC
parameters are presented888We present those in the Appendix given that most of
that work duplicates previous efforts using this same sample. The only main
difference is that here we use bolometric LC parameters in place of the $V$
band. (Appendix B). The reader is referred to those pages for a complete
analysis of the correlations. In Sect. 4 we add pre-SN models evolved with an
enhanced mass-loss rate to our grid of explosion simulations to fit some SNe
II that are not well reproduced by standard single-star models. Additionally,
these models are included in the fitting procedure to the full sample and
correlations are re-analysed. Some of the correlations increase in strength
when including models with enhanced mass loss.

###  3.1 Correlations between physical and observed parameters using 53 SNe II
from the CSP-I sample

Figure 1: Correlation matrix between observed and physical parameters of SNe
II. For each pair, the Pearson correlation coefficient is given and is colour-
coded. The observed parameters shown are: pd, optd, Cd, $s_{\rm 1}$, $s_{\rm
2}$, $s_{\rm 3}$, $M_{\rm bol,end}$, $M_{\rm bol,tail}$, $(g-r)$15, $(g-r)$70,
velocities of Hα, Hβ, and Fe ii $\lambda$5169, pEW(Hα) of absorption
component, pEW(Hα) of emission component, pEW(Hβ), pEW(Fe ii $\lambda$4924),
pEW(Fe ii $\lambda$5018), pEW(Fe ii $\lambda$5169), and $a/e$. Figure 2:
Correlations between pd and three physical parameters: $M_{\rm H,env}$ (left
panel), _E_ (middle panel), and $M_{\rm Ni}$ (right panel). The yellow and red
markers refer to the results obtained from the CSP-I SN II sample (yellow
markers indicate gold events). Each subplot contains the number of events
($N$), the Pearson correlation coefficient ($\rho$), and the probability of
detecting a correlation by chance ($P$) using the full sample. Figure 3:
Correlations between optd and physical parameters: $M_{\rm H,env}$ (top left),
_E_ (top right), $M_{\rm Ni}$ (bottom left), and ${}^{56}\rm Ni$ mixing
(bottom right). The yellow and red markers refer to the results obtained from
the CSP-I SN II sample (yellow markers indicate gold events). Each subplot
contains the number of events ($N$), the Pearson correlation coefficient
($\rho$), and the probability of detecting a correlation by chance ($P$) using
the full sample. Results from the models are colour-coded based on different
physical parameters. The physical parameters not being varied —if they do not
appear in the plot— are fixed at $M_{\rm ZAMS}$ = 10 $M_{\odot}$, $M_{\rm Ni}$
= 0.01 $M_{\odot}$, and ${}^{56}\rm Ni$ mixing = 0.5. Some observations fall
outside the range of the model parameters because of the fixed physical
parameters. Changes in the fixed values represent different ranges of model
parameters. Figure 4: Correlations between physical parameters and magnitude
at different epochs. The top panels show correlations between $M_{\rm
bol,end}$ and two physical parameters: _E_ (top left) and $M_{\rm Ni}$ (top
right). The bottom panels show correlations between $M_{\rm bol,tail}$ and two
physical parameters: _E_ (bottom left) and $M_{\rm Ni}$ (bottom right). The
yellow and red markers refer to the results obtained from the CSP-I SN II
sample (yellow markers indicate gold events). Each subplot contains the number
of events ($N$), the Pearson correlation coefficient ($\rho$), and the
probability of detecting a correlation by chance ($P$) for the full sample.
Results from the models are colour-coded based on different physical
parameters. The physical parameters not being varied —if they do not appear in
the plot— are fixed at $M_{\rm ZAMS}$ = 10 $M_{\odot}$, $M_{\rm Ni}$ = 0.03
$M_{\odot}$, and ${}^{56}\rm Ni$ mixing = 0.5. Some observations fall outside
the range of the model parameters because of the fixed physical parameters.
Changes in the fixed values represent different ranges of model parameters.
Figure 5: Correlations between $s_{\rm 2}$ and _E_ (left panel), $s_{\rm 2}$
and ${}^{56}\rm Ni$ mixing (middle panel), and Fe ii $\lambda$5169 line
velocities at 50 days post-explosion and _E_ (right panel). The mixing of 56Ni
within the ejecta is given as a fraction of the pre-SN mass. The yellow and
red markers refer to the results obtained from the CSP-I SN II sample (yellow
markers indicate gold events). Each subplot contains the number of events
($N$), the Pearson correlation coefficient ($\rho$), and the probability of
detecting a correlation by chance ($P$) using the full sample. Results from
the models are colour-coded based on different physical parameters. The
physical parameters not being varied —if they do not appear in the plot— are
fixed at $M_{\rm ZAMS}$ = 15 $M_{\odot}$, $M_{\rm Ni}$ = 0.04 $M_{\odot}$, and
${}^{56}\rm Ni$ mixing = 0.5. Some observations fall outside the range of the
model parameters because of the fixed physical parameters. Changes in the
fixed values represent different ranges of model parameters. Figure 6:
Correlations between $s_{\rm 3}$ and three physical parameters: $M_{\rm ej}$
(left panel), $M_{\rm Ni}$ (middle panel), and ${}^{56}\rm Ni$ mixing (right
panel). The dashed line indicates the expected declination rate for full
trapping of 56Co decay. The yellow and red markers refer to the results
obtained from the CSP-I SN II sample (yellow markers indicate gold events).
Each subplot contains the number of events ($N$), the Pearson correlation
coefficient ($\rho$), and the probability of detecting a correlation by chance
($P$) using the full sample. Results from the models are colour-coded based on
different physical parameters. The physical parameters not being varied —if
they do not appear in the plot— are fixed at _E_ = 1.2 foe, $M_{\rm Ni}$ =
0.04 $M_{\odot}$, and ${}^{56}\rm Ni$ mixing = 0.5. Some observations fall
outside the range of the model parameters because of the fixed physical
parameters. Changes in the fixed values represent different ranges of model
parameters.

Figure 1 shows the correlation matrix for the progenitor properties and
observables that we consider in this study. For each pair, the Pearson
coefficient is given and colour-coded: darkest colours represent the highest
correlations, while white colour indicates no correlation. In Sect. 3.2, we
show the strong correlations between progenitor parameters ($M_{\rm ZAMS}$,
$M_{\rm ej}$, $M_{\rm H,env}$, and $R$) which are inherent to the progenitor
models calculated assuming standard stellar evolution (i.e. they are all
determined by $M_{\rm ZAMS}$). Despite this, we searched for correlations
between all progenitor parameters and observables for a more appropriate
comparison with previous works.

The figures presented in this section show correlations between physical and
observed properties for the CSP-I SN II sample. Each figure also shows results
from the models (if the observed parameter has been measured). These are
colour-coded based on the physical parameter that produces the largest impact
on the observable being analysed (adopting the results from Table 2) unless
the physical parameter is already in the plot. In that case, the parameter
that produces the second-largest impact is used. The other physical parameters
are fixed. Some observations fall outside the range of the model parameters,
which is due to the fixed physical parameters. Changes in these values will
produce different ranges of model observables.

Figure 2 shows relations between pd and three physical parameters: $M_{\rm
H,env}$, _E_ , and $M_{\rm Ni}$. A positive trend is found between pd and
$M_{\rm H,env}$ (left panel), with a correlation coefficient of $\rho$ = 0.43
$\pm$ 0.15 ($N$ = 20). At the same time, pd shows an evident trend with the
explosion energy ($\rho$ = $-$0.54 $\pm$ 0.16, $N$ = 20). It is interesting to
note that while the two SNe II in our sample with the longest pd are
consistent with the lowest-energy explosions, the SN II with the shortest pd
is the most energetic. A trend is also found between pd and $M_{\rm Ni}$,
although our sample size is too small (only six points) for us to make any
strong conclusions.

Figure 3 shows correlations between optd and physical parameters: $M_{\rm
H,env}$ (top-left panel), _E_ (top-right panel), $M_{\rm Ni}$ (bottom-left
panel), and ${}^{56}\rm Ni$ mixing (bottom-right panel). optd displays a
strong correlation with $M_{\rm H,env}$ ($\rho$ = 0.60 $\pm$ 0.11 $N$ = 31).
Given that this correlation coefficient is larger than that for the
pd$-$$M_{\rm H,env}$ relation, in addition to the higher significance, these
might suggest that optd is a better indicator of the hydrogen-rich envelope
mass than pd. This is the opposite of what is claimed by G17. We note that
different definitions are used for the ‘plateau duration’ and ‘optically thick
phase duration’ in the two studies (G17 named these parameters as Pd and OPTd,
respectively; see Sect. 2.2 and Paper I). Therefore, we also estimated the
correlation coefficients between $M_{\rm H,env}$ and both Pd and OPTd using
the observed values measured by G17, finding again that the optically thick
phase duration shows a stronger correlation with $M_{\rm H,env}$.

In opposition to pd, we find that optd exhibits a higher degree of correlation
with $M_{\rm H,env}$ than with _E_ ($\rho$ = $-$0.40 $\pm$ 0.17, $N$ = 31). A
trend is found with $M_{\rm Ni}$, in the sense that longer optd are consistent
with higher 56Ni values, although with a large dispersion. We observe a strong
correlation between optd and the mixing of 56Ni within the ejecta ($\rho$ =
0.61 $\pm$ 0.11, $N$ = 31). Figure 3 also displays the results measured
directly from the models. In line with previous theoretical predictions, it is
seen that larger optd values are found for higher $M_{\rm H,env}$ and $M_{\rm
Ni}$, and lower _E_. However, different ${}^{56}\rm Ni$ mixing in the ejecta
does not alter the duration of the optically thick phase, in contrast to the
strong correlation found using the CSP-I SN II sample (see discussion in Sect.
5).

The analysis presented in Table 2 shows that the explosion energy is the
physical parameter that produces the largest impact on optd, while $M_{\rm
ZAMS}$ (directly related to $M_{\rm H,env}$, see Fig. 7) produces the second-
largest impact, similar to what is found from the correlations. As previously
mentioned, our analysis is based on standard single-star evolution and models
with different input physics could modify our statistical analysis. As
expected, $M_{\rm Ni}$ influences optd, but with a lower relative importance
than _E_ and $M_{\rm ZAMS}$.

Figure 4 presents correlations involving magnitude measurements and physical
properties. Very strong correlations are found between $M_{\rm bol,end}$ and
_E_ ($\rho$ = $-$0.80 $\pm$ 0.05, $N$ = 51), and $M_{\rm bol,tail}$ and _E_
($\rho$ = $-$0.85 $\pm$ 0.11, $N$ = 12), with more energetic explosions
producing brighter SNe II during the plateau and radioactive tail. Table 2
argues that $M_{\rm bol,end}$ is mostly affected by _E_ , while $M_{\rm ZAMS}$
is the physical parameter that produces the largest deviation in the _E_
$-$$M_{\rm bol,end}$ relation. A strong correlation is found between $M_{\rm
bol,end}$ and $M_{\rm Ni}$ ($\rho$ = $-$0.63 $\pm$ 0.17, $N$ = 16). The
additional heating of the ejecta at late times by the 56Ni decay chain not
only extends the duration of the plateau (e.g. Kasen & Woosley 2009), but it
also increases the luminosity in the late-plateau phase (Bersten 2013;
Kozyreva et al. 2019). Our models show that $M_{\rm Ni}$ has its major effect
on $M_{\rm bol,end}$ in the low-_E_ regime (_E_ $\lesssim$ 0.7 foe) when SNe
II are fainter (Fig. 12, middle panel). Because of the low luminosity, 56Ni
plays a more important role producing higher luminosities at the end of the
plateau. Therefore, $M_{\rm Ni}$ also affects $M_{\rm bol,end}$, although
other physical properties such as _E_ and $M_{\rm ZAMS}$ have larger effects
(Table 2). Additionally, Fig. 4 (bottom-right panel) shows a very strong
correlation between $M_{\rm Ni}$ and $M_{\rm bol,tail}$ ($\rho$ = $-$0.88
$\pm$ 0.05, $N$=14). This is to be expected given that the tail luminosity is
predominantly related to the amount of 56Ni in the ejecta. The explosion
energy and ${}^{56}\rm Ni$ mixing deviate the tight correlation between
$M_{\rm Ni}$ and $M_{\rm bol,tail}$ only on small scales (see Table 2 and Fig.
13).

The left and middle panels of Fig. 5 explore the correlations found between
the $s_{\rm 2}$ declination rate and physical parameters. We observe strong
correlations with the explosion energy ($\rho$ = 0.73 $\pm$ 0.07, $N$ = 51),
which suggests that more steeply declining SNe II during the plateau phase are
produced by more energetic explosions. In addition, it is found that $s_{\rm
2}$ and ${}^{56}\rm Ni$ mixing show a strong anti-correlation with fast-
declining SNe II, which is compatible with more concentrated 56Ni in the inner
regions of the ejecta (Bersten et al. 2011). According to Table 2, the
explosion energy causes the largest effect on the $s_{\rm 2}$ diversity, while
${}^{56}\rm Ni$ mixing displays smaller changes. The KDA finds that $M_{\rm
Ni}$ also influences variations in $s_{\rm 2}$. As stated above, higher
$M_{\rm Ni}$ can boost the late-plateau luminosity producing slowly declining
events (see also Fig. 14, left panel). However, here we find zero correlation
between $M_{\rm Ni}$ and $s_{\rm 2}$ when the CSP-I SN II sample is used (Fig.
1). Previous studies suggested that progenitors with smaller hydrogen-rich
envelope masses produce faster declining SNe II (i.e. larger $s_{\rm 2}$; e.g.
Blinnikov & Bartunov 1993). However, we find zero correlation between $M_{\rm
H,env}$ and $s_{\rm 2}$ in the present work (see Fig. 17). The study of
correlations shows that the $s_{\rm 2}$ diversity is mostly related to changes
in _E_ and ${}^{56}\rm Ni$ mixing, although the analysis of the KDA technique
indicates that _E_ is the physical parameter that most influences variations
of $s_{\rm 2}$. However, the analysis we are carrying out in this section only
involves standard pre-SN models, where none of the progenitors were evolved
with significant mass loss. Therefore, the effect of $M_{\rm H,env}$ is
probably underestimated in this analysis.

As expected, the explosion energy shows a strong correlation with Fe ii
$\lambda$5169 velocity (Fig. 5, right panel), although there are outliers from
this relation that display high Fe ii velocities but lower energies than
expected. These SNe II are SN 2004er and SN 2007sq. In Paper II, the
bolometric LCs of these two SNe II are generally well reproduced, but for both
SNe our models underestimate their photospheric velocities. SN 2004er shows a
moderately luminous and considerably long optd. While a more energetic
explosion would give much better agreement with the observed velocities, it
would also lead to shorter optd. This is similar to what happens for SN 2007sq
(see Paper II for discussion). This shows that higher explosion energies and
different pre-SN structures (e.g. the core to envelope mass) are required in
both cases for a more appropriate modelling. Unfortunately, for both SNe II,
these proposed models fall outside our grid.

The _E_ and $M_{\rm ZAMS}$ are the only two physical parameters that produce
different photospheric velocities during the optically thick phase, with _E_
showing the highest relative importance (Table 2). However, a moderate
correlation is found between ${}^{56}\rm Ni$ mixing and the Fe II
$\lambda$5169 expansion velocities (Fig. 1). This relation is possibly driven
by other parameters that simultaneously correlate with the mixing of 56Ni and
expansion velocities. Previously, we noted that faster declining SNe II are
well reproduced by low degrees of ${}^{56}\rm Ni$ mixing. At the same time,
faster declining SNe II present higher expansion velocities (e.g. G17, ).
Thus, this combination of correlations may produce the trend between
${}^{56}\rm Ni$ mixing and Fe II $\lambda$5169 velocities.

The declination rate of the radioactive tail phase ($s_{\rm 3}$) takes the
theoretical value of 0.98 mag per 100 days assuming full trapping of the
gamma-ray emission from 56Co decay (dashed lines in Fig. 6). The full trapping
may be possible by a large ejecta mass, which results in a long diffusion
time. Observations show higher $s_{\rm 3}$ for a considerable number of SNe II
(; ; ). Less trapping of gamma rays as a result of low ejecta mass and/or low
density can explain the observed higher $s_{\rm 3}$ values. In Fig. 6, the
$s_{\rm 3}$ declination rate is plotted against three physical parameters. The
$s_{\rm 3}$ shows moderate trends with $M_{\rm ej}$ (left panel) and
${}^{56}\rm Ni$ mixing (right panel). In addition, a weak trend is detected
with $M_{\rm Ni}$ (middle panel). However, in all of these cases, the number
of events is low and the scatter is large, which prevents us from making
definitive conclusions. Some of these relations are reanalysed in Sect. 4.2.
For a small number of SNe II, the luminosity decays slower than what is
assumed for full trapping (i.e. lower $s_{\rm 3}$ values). This may be caused
by the presence of residual thermal energy of the explosion in the ejecta
(Utrobin 2007), interaction with CSM (Fraser et al. 2015; Pastorello et al.
2018), late-time accretion onto a compact remnant (e.g. Moriya et al. 2018;
Gutiérrez et al. 2020), or by a more quickly receding photosphere (e.g.
increasing helium abundances, Chieffi et al. 2003), in addition to the
contribution of 56Co decay.

The analysis of the model parameters shows that $s_{\rm 3}$ is mostly affected
by $M_{\rm ZAMS}$ (Table 2, see also Fig. 15). A more massive star implies
larger $M_{\rm ej}$ and $R$, at least within the range of initial masses of
our standard pre-SN models. Therefore, the high $s_{\rm 3}$ values are
consistent with the lowest ejecta masses in our grid of models. Our models
show that the degree of 56Ni mixing within the ejecta also contributes to the
differences found in $s_{\rm 3}$ (Table 2). Extensively distributed 56Ni in
the envelope allows gamma-ray photons emitted from the decay of the 56Co in
the outermost layers of the ejecta to easily escape before being thermalised
due to the low mass beyond the location where they are emitted. This is seen
in the right panel of Fig. 6 for the observables measured from the models,
where more extended ${}^{56}\rm Ni$ mixing produces SNe II declining more
rapidly during the radioactive tail. However, this is in tension with the
relation found using the $s_{\rm 3}$ declination rate from the observations.
Extended mixing of 56Ni is necessary to fit some aspects of the observations,
although this does not explain the large $s_{\rm 3}$ values measured. This
might suggest that the ${}^{56}\rm Ni$ mixing is not the main driver of the
observed diversity of $s_{\rm 3}$ declination rates.

We do not find correlations between physical properties and the early-time LC
parameters Cd and $s_{\rm 1}$ (with the exception of $s_{\rm 1}$ and _E_ , see
Fig. 1). This is not surprising because a large fraction of SNe II may
experience interaction between the SN ejecta and a dense CSM shell surrounding
the progenitor star (González-Gaitán et al. 2015; Förster et al. 2018; Bruch
et al. 2021). Thus, early phases of SN evolution would be sensitive to the CSM
characteristics, which are not studied in the present work.

The $(g-r)$ colour at 15 days from explosion shows a trend with $M_{\rm Ni}$,
although only five points are available to compute the correlation strength.
At 70 days from explosion, the $(g-r)$ colour moderately correlates with the
explosion energy, in the sense that more energetic SNe II are redder at late
times. We further analyse this correlation in Sect. 5.

Finally, correlations between _E_ and the pEWs of some lines are mentioned.
Figure 1 shows an evident trend with $a/e$ ($\rho$ = $-$0.69 $\pm$ 0.07) in
the sense that more energetic explosions display lower $a/e$ values. We also
find that _E_ anti-correlates with the pEWs of Fe ii $\lambda$4924 and Fe ii
$\lambda$5018 ($\rho$ = $-$0.70 $\pm$ 0.08 and $\rho$ = $-$0.49 $\pm$ 0.14,
respectively), although positive correlations are found with the pEWs of Fe ii
$\lambda$5169, the emission component of Hα, and the absorption component of
Hβ. None of the spectral and colour parameters show any significant
correlation with those physical parameters dependent on $M_{\rm ZAMS}$.

###  3.2 Correlations between physical parameters

Figure 7: Correlation matrix of the progenitor and explosion parameters. The
Pearson correlation coefficients are presented in the upper triangle, while in
the lower triangle the correlation coefficients are colour-coded. Figure 8:
Correlations between three physical parameters: $M_{\rm ZAMS}$, _E_ , and
$M_{\rm Ni}$. Each subplot contains the number of events ($N$), the Pearson
correlation coefficient ($\rho$), and the probability of detecting a
correlation by chance ($P$) using the full sample. Yellow markers indicate
gold events.

Figure 7 shows the correlation matrix of the physical parameters. Given that
the progenitor models used were calculated assuming single-star evolution with
a standard wind efficiency, all the progenitor parameters ($M_{\rm ZAMS}$,
$M_{\rm ej}$, $M_{\rm H,env}$, and $R$) are strongly correlated. In this
context, $M_{\rm ZAMS}$ is the only independent variable related to a unique
pre-SN structure. This structure is characterised by $M_{\rm ej}$, $M_{\rm
H,env}$, and $R$ which increase almost monotonically with $M_{\rm ZAMS}$
within the range of initial masses of our pre-SN models (see M20, their Fig.
2).

In Fig. 8 (left panel), $M_{\rm ZAMS}$ is plotted against _E_. It is found
that these physical parameters show a weak correlation ($\rho$ = 0.34 $\pm$
0.10, $N$ = 51) in the sense that higher $M_{\rm ZAMS}$ progenitors display
higher energy explosions. Most studies compare _E_ with $M_{\rm ej}$ instead
of $M_{\rm ZAMS}$. We also find a weak correlation between $M_{\rm ej}$ and
_E_ ($\rho$ = 0.36 $\pm$ 0.11) which was expected because of the tight
relation between $M_{\rm ZAMS}$ and $M_{\rm ej}$.

Similar trends were also inferred by previous studies that obtained physical
parameters from observations via hydrodynamical modelling or analytic scaling
relations (e.g. Hamuy 2003; Morozova et al. 2018; Utrobin & Chugai 2019),
although most of these studies find stronger correlations. This difference
could be associated to the type of pre-SN structure adopted to initialise the
explosion model. Particularly, works assuming polytropic progenitor models
find tight relations between $M_{\rm ej}$ and _E_ , while weaker trends are
found using pre-SN models from standard single-star evolution calculations
(see Fig. 6 by Burrows & Vartanyan 2021, for a comparison of different
studies). This may be due to the contribution of two factors. On the one hand,
polytropic models allow the construction of progenitor structures over a wide
range of pre-SN masses, hydrogen-rich envelope masses, and progenitor radii.
On the other hand, parameter degeneracy —where different physical parameters
sometimes produce similar LCs and spectral properties during the recombination
phase— sometimes admits solutions for larger masses and higher explosion
energies simultaneously, given that the effect of both parameters on LCs and
photospheric velocities is cancelled. However, the small range of masses
predicted by standard single-star evolution, in addition to the wide range of
inferred explosion energies, leads to scatter in the mass–energy relation in
the current work. While the dispersion of this relation is large, our results
show a lack of high-mass progenitors. This may imply an important constraint
to the explosive end of more massive stars. Most of the inferred progenitor
masses range between 9 and 13 $M_{\odot}$ and are consistent with models of
varied energy covering the range of 0.15 to 1.40 foe. That is, while the
inferred explosion energies span the full range explored in our models, the
constrained masses generally only sample the low-mass end.

The middle panel of Fig. 8 indicates a positive trend between $M_{\rm ZAMS}$
and $M_{\rm Ni}$ similar to previous findings in the literature (e.g. Hamuy
2003; Utrobin & Chugai 2019). We find a correlation coefficient of $\rho$ =
0.61 $\pm$ 0.23; however, given the low number of events ($N$ = 15), this
trend may be driven by the three objects with high $M_{\rm ZAMS}$ and $M_{\rm
Ni}$. We also note that low-$M_{\rm ZAMS}$ progenitors cover a large range of
$M_{\rm Ni}$ values between $\sim$0.005 and 0.05 $M_{\odot}$. These objects
represent 12 of the 15 SNe II with derived $M_{\rm Ni}$. The other three
progenitors are compatible with stars more massive than 15 $M_{\odot}$, and
all have $M_{\rm Ni}$ values above 0.06 $M_{\odot}$. That is, modelling of our
SN II observations shows that progenitors with $M_{\rm ZAMS}$ larger than 15
$M_{\odot}$ exclusively show high 56Ni masses.

In the right panel of Fig. 8, the explosion energy is plotted against $M_{\rm
Ni}$. It is found that these parameters show a positive correlation ($\rho$ =
0.63 $\pm$ 0.17, $N$ = 15) with higher energy explosions showing more 56Ni.
56Ni originates from explosive nucleosynthesis during a SN explosion. After
this explosive burning phase, the innermost layers of the ejecta can become
bound and fall back onto the compact remnant carrying a significant fraction
of 56Ni. Thus, in principle, the amount of 56Ni powering the radioactive tail
phase is not that produced during explosive nucleosynthesis, which may produce
a bias in the _E_ $-$$M_{\rm Ni}$ correlations. However, fallback is expected
to be less important at higher explosion energies, and in our case, only a
small number of SNe II are compatible with low explosion energies. Therefore,
we do not expect a large bias in our trend. In Sect. 5.1 we discuss the
physical origin of this correlation and its connection with the observed
diversity of SNe II.

###  3.3 Unveiling the SN II physical parameter space of observed explosions

Throughout the previous subsections we analyse several correlations between
physical and observed properties of SNe II. Here we focus on examining the
parts of the parameter space that are not reached with our observations
because this may afford significant constraints to the evolution and explosion
of massive stars. The large number of SNe II in the CSP-I sample allows such
an analysis.

Observed optd ranges from 42 and 146 days (Paper I). While the optd values in
our sample longer than 100 days are well reproduced by models for a variety of
$M_{\rm H,env}$ (from the shortest to the largest $M_{\rm H,env}$ within our
grid of models; see Fig. 3), the SNe II in the full sample displaying the
shortest optd ($\sim$80 days) are always found for low $M_{\rm H,env}$,
specifically $M_{\rm H,env}$ $\lesssim$ 8 $M_{\odot}$. Therefore, such a short
optd is not found for $M_{\rm H,env}$ $\gtrsim$ 8 $M_{\odot}$. Short optd for
high $M_{\rm H,env}$ could be obtained for sufficiently energetic explosions,
but this does not seem to be the case here. As discussed previously, we also
note that there are models with optd values much larger than any of those
found for observed SN II to date.

$M_{\rm bol,end}$ and _E_ show a tight correlation with a low dispersion, and
therefore less luminous SNe II are not found for higher explosion energies and
vice versa (Fig. 4). Additionally, SNe II with $M_{\rm bol,end}$ dimmer than
around $-$16 mag are not consistent with 56Ni masses in the ejecta higher than
$\sim$0.05 $M_{\odot}$.

We already mentioned the strong dependency of $M_{\rm bol,tail}$ on $M_{\rm
Ni}$; therefore, more (less) luminous SNe II during the radioactive tail phase
are not compatible with low (high) 56Ni masses. Moreover, some regions of the
$M_{\rm bol,tail}$-_E_ parameter space are not covered, specifically those for
bright (dim) radioactive tails and low (high) explosion energies. However,
this absence is related to the physical connection between _E_ and $M_{\rm
Ni}$ (Sect. 3.2). Therefore, higher energy explosions with low 56Ni masses
(and vice versa) are also missing. Furthermore, our results show a lack of
high-mass progenitors particularly with low explosion energies, similar to
previous studies (see Morozova et al. 2018, and references therein).

Lastly, we analysed the parameter space that is not covered by the observed
$s_{\rm 2}$ declination rate. Although some dispersion is found, it is seen
that the largest (smallest) values of $s_{\rm 2}$ are not compatible with low
(high) explosion energies (see Sect. 5 for explanation). At the same time,
while only one slowly declining SN II with an inner distribution of 56Ni
within the ejecta is found, more steeply declining SNe II are not consistent
with 56Ni extended to the outer regions of the ejecta.

##  4 Results from non-standard pre-SN evolution

The grid of explosion models used in Paper II employs pre-SN structures from
stellar evolution calculations to initialise the explosion, which were
calculated using the standard prescription for the wind mass loss defined in
the MESA code (see Sect. 2.3). While the CSP-I SN II sample consists of 74
objects, in Paper II we derived physical and explosion properties for 53
events. Some SNe II were excluded for the various reasons outlined in Paper
II, and these are not repeated in the current work. Here we only mention the
two relevant (for the current analysis) SNe II that were excluded: SN 2006Y
and SN 2008bu. These SNe present atypically short optd values of 64 $\pm$ 4
days and 52 $\pm$ 7 days for SN 2006Y and SN 2008bu, respectively (Paper I).
Given that none of our LC models present such short optd values, SN 2006Y and
SN 2008bu could not be well fitted.

Table 2 shows that the explosion energy is the parameter that produces the
largest impact on optd. A higher explosion energy would lead to a shorter
optd, that is, in the direction needed to model the previous two short-plateau
SNe II. However, the more energetic explosions also produce more luminous SNe
II that display higher velocities, differing even further from observations.
Therefore, here we present new pre-SN models evolved with a higher mass-loss
rate, which reduces the extent of the hydrogen-rich envelope at the time of
core collapse, and therefore the optd. With this assumption, we are able to
model these two short-plateau SNe II (see also Hiramatsu et al. 2021).
Furthermore, in order to analyse the extent of variation in the correlations
when the pre-SN models change, we modelled all the well-observed SNe II in the
CSP-I sample (previously modelled in Paper II using standard stellar models)
employing the explosion models with additional mass stripping.

###  4.1 New progenitor and explosion models

Table 3: Progenitor properties for the pre-SN models evolved with enhanced
mass loss and the standard pre-SN models.

$M_{\rm ZAMS}$ | $\eta$ | $M_{\rm presn}$ | $R$ | $M_{\rm H,env}$ | $M_{\rm H}$ | $M_{\rm He}$
---|---|---|---|---|---|---
($M_{\odot}$) |  | ($M_{\odot}$) | ($R_{\odot}$) | ($M_{\odot}$) | ($M_{\odot}$) | ($M_{\odot}$)
10 | 1.0 | 9.53 | 462 | 7.06 | 4.79 | 2.47
10 | 3.0 | 8.56 | 476 | 6.22 | 4.21 | 2.34
10 | 6.0 | 7.32 | 472 | 5.08 | 3.40 | 2.24
10 | 9.0 | 6.28 | 455 | 4.28 | 2.74 | 2.00
12 | 1.0 | 11.08 | 594 | 8.03 | 5.42 | 3.05
12 | 3.0 | 9.24 | 627 | 6.21 | 4.16 | 3.03
12 | 5.0 | 7.78 | 618 | 4.86 | 3.24 | 2.92
12 | 7.0 | 6.38 | 635 | 3.53 | 2.33 | 2.85
14 | 1.0 | 13.19 | 742 | 9.35 | 6.12 | 3.84
14 | 2.5 | 10.01 | 817 | 6.15 | 4.12 | 3.86
14 | 3.5 | 8.49 | 841 | 4.69 | 3.11 | 3.80
14 | 4.0 | 7.74 | 851 | 3.96 | 2.62 | 3.78
999The table includes the initial mass ($M_{\rm ZAMS}$), the wind scaling
factor ($\eta$), the final mass at core collapse ($M_{\rm presn}$), the
progenitor radius ($R$), the hydrogen-rich envelope mass ($M_{\rm H,env}$),
the total mass of hydrogen ($M_{\rm H}$), and the helium-core mass ($M_{\rm
He}$).

In order to assess the physical parameters for the short-plateau SNe II in the
CSP-I sample, we constructed new non-rotating solar-metallicity pre-SN RSG
models using the stellar evolution code MESA version 10398 (Paxton et al.
2011, 2013, 2015, 2018, 2019). Given that we want to reproduce the short-
plateau phase of SN 2006Y and SN 2008bu, higher wind efficiencies during the
evolution of the progenitor star were assumed to reduce the final mass of the
hydrogen-rich envelope101010However, it is important to note that this stellar
wind efficiency was used to mimic any mechanism that may produce additional
envelope stripping (including for instance binary interactions and eruptive
mass loss).. Stars were evolved for three $M_{\rm ZAMS}$ values (10, 12, and
14 $M_{\odot}$) to obtain models with different pre-SN radii. The final radii
range from 455 to 851 $R_{\odot}$, which is a good starting point for our
following analysis. We used the ‘Dutch’ recipe for mass loss defined in MESA.
The wind scaling factor ($\eta$) linearly modifies the mass-loss rate and is
equal to unity in the standard pre-SN models. Here, $\eta$ is arbitrarily
chosen to produce $M_{\rm H,env}$ in the range of $\sim$3.5$-$7.1 $M_{\odot}$,
the latter value being the lowest $M_{\rm H,env}$ in the standard pre-SN
models. The other stellar evolution parameters took the same values as
summarised in Sect. 2.3. Table 3 lists the physical characteristics of the
pre-SN models with enhanced mass loss, together with the standard pre-SN
models. We then computed a grid of synthetic bolometric LCs and photospheric
velocities using a 1D hydrodynamical code (Bersten et al. 2011) in the same
manner as we constructed our standard set of models used in Paper II. We
varied the explosion energy between 0.2 and 1.5 foe and $M_{\rm Ni}$ between
0.005 and 0.08 $M_{\odot}$ for each progenitor model. In addition, three
values of ${}^{56}\rm Ni$ mixing for each explosion model were considered to
account for the effect of the spatial distribution of 56Ni within the ejecta:
out to the 20%, 50%, and 80% of the pre-SN structure in mass coordinates.

We used the same fitting procedure as in Paper II, which is based on MCMC
methods and allows one to find the posterior probability of the model
parameters given the observations (see Sect. 2.3). In Paper II, we used six
parameters to model the observables: the explosion epoch ($t_{\rm exp}$),
scale, $M_{\rm ZAMS}$, _E_ , $M_{\rm Ni}$, and ${}^{56}\rm Ni$ mixing. The
pre-SN models used in Paper II were calculated assuming the standard
prescriptions for single-star evolution; consequently, $M_{\rm ZAMS}$ is the
only independent variable related to a unique pre-SN structure. Here, we
obtained different pre-SN structures for the same $M_{\rm ZAMS}$ given that we
also varied the efficiency of wind mass loss. Therefore, we included $\eta$ as
an additional parameter to break the previous degeneracy between $M_{\rm
ZAMS}$ and the pre-SN structures. We used uniform priors for six parameters:
$t_{\rm exp}$, $M_{\rm ZAMS}$, $\eta$, _E_ , $M_{\rm Ni}$, and ${}^{56}\rm Ni$
mixing. The sampler is allowed to run within the observational uncertainty of
the $t_{\rm exp}$ (see Table 1 from Paper I) and within the parameter space
for the other parameters. For the scale parameter, we used a Gaussian prior
centred at one, with a standard deviation controlled by the uncertainty in the
distance and host-galaxy extinction (see Paper II for details). We employed
400 walkers and 10000 steps per sampler, with a burn-in period of 5000 steps.
The walkers were randomly initialised, covering the entire parameter space.

###  4.2 Correlations between physical and observed parameters using non-
standard pre-SN models

Table 4: Comparison of the strength of correlations ($\rho$) with the largest
differences between standard (std) and non-standard (non-std) results.

Correlation |  $N$ (std) |  $\rho$ (std) |  $P$ (std)
---|---|---|---
|  $N$ (non-std) |  $\rho$ (non-std) |  $P$ (non-std)
$M_{\rm H,env}-pd$ | 20 | 0.43 $\pm$ 0.15 |  $\leq$0.23
| 22 | 0.64 $\pm$ 0.10 |  $\leq$9.2 $\times$ 10-3
$M_{\rm H,env}-optd$ | 31 | 0.60 $\pm$ 0.11 |  $\leq$5 $\times$ 10-3
| 33 | 0.71 $\pm$ 0.09 |  $\leq$1.1 $\times$ 10-4
$M_{\rm ej}-s_{3}$ | 11 |  $-$0.47 $\pm$ 0.32 |  $\leq$0.66
| 12 |  $-$0.59 $\pm$ 0.24 |  $\leq$0.26
$M_{\rm Ni}-pd$ | 6 | 0.70 $\pm$ 0.36 |  $\leq$0.50
| 7 | 0.26 $\pm$ 0.48 |  $\leq$0.63
$M_{\rm Ni}-optd$ | 15 | 0.49 $\pm$ 0.21 |  $\leq$0.31
| 16 | 0.22 $\pm$ 0.31 |  $\leq$0.74

111111Columns: (1) correlation; (2) number of SNe II within that correlation;
(3) correlation coefficient; (4) upper limit of the significance of
correlations.

Figure 9: Correlations between $M_{\rm H,env}$ and pd (left panel), $M_{\rm
H,env}$ and optd (middle panel), and $M_{\rm ej}$ and $s_{\rm 3}$ (right
panel) using the results from non-standard pre-SN models. In the right panel,
the dashed line indicates the expected declination rate for full trapping of
56Co decay. Each subplot contains the number of events ($N$), the Pearson
correlation coefficient ($\rho$), and the probability of detecting a
correlation by chance ($P$). Blue triangles show the results of using non-
standard pre-SN models, while green squares indicate results from standard
evolution. SNe labelled as ‘non-standard’ have Bayes factors greater than
101/2. Figure 10: Correlations between physical and observed parameters using
the results from non-standard pre-SN models. Left panel: $M_{\rm Ni}$ versus
pd. Right panel: $M_{\rm Ni}$ versus optd. Each subplot contains the number of
events ($N$), the Pearson correlation coefficient ($\rho$), and the
probability of detecting a correlation by chance ($P$). Blue triangles show
the results obtained using non-standard pre-SN models, while green squares
indicate results from standard evolution. SNe labelled as ‘non-standard’ have
Bayes factors greater than 101/2.

The bolometric LC and expansion velocity models presented in the previous
section were designed to determine the physical properties of SN 2006Y and SN
2008bu, that is, the two shortest-plateau SNe II in the CSP-I sample. The
posterior distributions of the parameters were constructed using the MCMC
fitting procedure described in Sect. 4.1. In Appendix C, models drawn from the
posterior distribution in comparison with the observations are presented for
these two SNe II.

The short-plateau LC and expansion velocities of SN 2006Y are well reproduced
by models with $M_{\rm ej}$ $\simeq$ 5.5 $M_{\odot}$, $M_{\rm H,env}$ $\simeq$
4.4 $M_{\odot}$, $R$ $\simeq$ 460 $R_{\odot}$, and _E_ $\simeq$ 1.2 foe. The
progenitor structure is the result of the evolution of a star with an initial
mass of 10.3 $M_{\odot}$ and a high wind scaling factor of $\eta$ $\sim$ 8\.
The LC of SN 2006Y was recently modelled by Hiramatsu et al. (2021). We find a
progenitor radius consistent with the results of these latter authors,
although they found a less massive hydrogen-rich envelope mass of 1.7
$M_{\odot}$ and a lower explosion energy of 0.8 foe. These differences may be
attributed to the wider range of progenitor properties studied by Hiramatsu et
al. (2021) —particularly $M_{\rm ZAMS}$ and the wind scaling factor— or to the
fact that they determine progenitor and explosion parameters without using any
spectral information in their modelling (M20). The observations of SN 2008bu
are consistent with $M_{\rm ej}$ $\simeq$ 5.2 $M_{\odot}$, $M_{\rm H,env}$
$\simeq$ 3.8 $M_{\odot}$, $R$ $\simeq$ 630 $R_{\odot}$, and _E_ $\simeq$ 0.5
foe. The pre-SN model used to reproduce the observations of SN 2008bu is the
consequence of the evolution of a star with $M_{\rm ZAMS}$ $\simeq$ 11.5
$M_{\odot}$, with a wind scaling factor of around 6. Therefore, as expected,
SN 2006Y and SN 2008bu are reasonably well reproduced with models calculated
with significant envelope stripping. These results are reported in Appendix C.
They are characterised by the median of the marginal distributions, while we
adopt the 16th and 84th percentiles as our lower and upper uncertainties,
respectively. We also include estimates of $M_{\rm ej}$, $M_{\rm H,env}$,
$M_{\rm H}$, and $R$. These values are not model parameters, and are therefore
not fitted. These values are interpolated from the $M_{\rm ZAMS}$ and $\eta$
derived in the fitting.

Initially, the pre-SN models with additional mass stripping together with
their corresponding explosion models (Sect. 4.1) were calculated to reproduce
the observations of SN 2006Y and SN 2008bu because of their short optically
thick phases. However, we used these synthetic bolometric LCs and expansion
velocities to model several other SNe II in the CSP-I sample that were already
modelled in Paper II in order to determine possible variations in the
correlations found in Sect. 3.1. Given the degeneracies in inferring physical
properties from LC modelling, we were able to obtain more than one set of
physical parameters for the same SN.

We explored new solutions for those SNe II in the CSP-I sample that fulfil the
following criteria: (a) enough photometric observations to cover the
photospheric phase and at least the beginning of the transition to the
radioactive tail; and (b) at least one measurement of the Fe ii velocity. This
information is crucial to determining the hydrogen-rich envelope mass (among
other physical properties). We find 31 SNe II (without considering SN 2006Y
and SN 2008bu) that fulfilled our criteria. For these 31 SNe II, we used the
MCMC procedure described in Sect. 4.1 to determine a set of physical
parameters consistent with partially stripped progenitors. For each SN, we
then compared their observations with models drawn from the posterior
distribution. For some SNe, we find large differences between models and
observations, specifically in the duration of the optically thick phase.
However, for some others, the models with additional mass stripping are able
to reproduce the observations. We quantified the significance for this
solution over that from Paper II using the Bayes factor ($B$). According to
Jeffreys (1998), $B$ ¿ 101/2 implies that the support for the solution with
reduced hydrogen-rich envelope is substantial, strong if $B$ is between 10 and
102, and decisive if $B$ ¿ 102. Following the above statistical analysis, we
find that the solution with additional mass lost is favoured against the
standard solution ($B$ ¿ 101/2) for nine SNe II. Together with SN 2006Y and SN
2008bu, we therefore reach a sample of 11 SNe II that are better fit with
these higher mass-loss models. Their physical parameters, in addition to
comparisons between models and observations, are presented in Appendix C.

These results were then used to reanalyse the correlations between observed
and physical parameters. The results for the 11 SNe II that favour non-
standard pre-SN models were used. At the same time, the results from standard
single-star evolution (i.e. from Paper II) were employed for the remaining SNe
II. In general, similar trends to those presented in Sect. 3.1 were found. The
largest differences are described below.

Figure 9 presents $M_{\rm H,env}$ against pd (left panel), $M_{\rm H,env}$
against optd (middle panel), and $M_{\rm ej}$ against $s_{\rm 3}$ (right
panel). $M_{\rm H,env}$ shows strong correlations with pd and optd. The
correlation coefficients are $\rho$ = 0.64 $\pm$ 0.10 ($N$ = 22) and $\rho$ =
0.71 $\pm$ 0.09 ($N$ = 33), respectively. In addition, a moderate trend is
found between $M_{\rm ej}$ and $s_{\rm 3}$ with $\rho$ = $-$0.59 $\pm$ 0.24
($N$ = 12). In Table 4, the strengths of correlations are compared between
standard and non-standard results for those correlations that show the largest
differences. We find that for these three cases, the strengths of the
correlations are now significantly greater than in Sect. 3.1, where we
analysed the results from standard evolutionary models. Figure 10 shows
$M_{\rm Ni}$ against pd and optd. In Sect. 3, we show a trend between these
parameters in the sense that SNe II with higher 56Ni masses develop a longer
plateau and optically thick phases. However, we now find weak trends (see
Table 4). We note at least one obvious outlier in these plots (SN 2006Y), with
the correlations being stronger when this event is removed. SN 2006Y presents
the shortest pd and optd with the largest $M_{\rm Ni}$ estimate ($M_{\rm Ni}$
= 0.075). Therefore, this shows that a large amount of 56Ni present in the SN
ejecta does not necessarily imply long pd and optd.

The rest of the correlations show little or no variation between standard and
non-standard evolution models and they are not discussed further. The
differences found in the correlations show the impact of the parameter
degeneracy on the results, given that this may alter the strength of some
correlations. However, in the cases where the parameter degeneracy admitted
different results, only the strength of correlations changed (Table 4). The
parameter degeneracy did not produce new trends, nor did it cause the
disappearance of previous correlations. We note that the trends between
$M_{\rm Ni}$ and pd and $M_{\rm Ni}$ and optd disappear because of one outlier
(SN 2006Y; see Fig. 10) and not because of parameter degeneracies, given that
SN 2006Y was modelled only once (SN 2006Y could not be modelled with our grid
of standard models). The correlation matrix between observed and physical
parameters using non-standard pre-SN models is presented in Appendix C.

We determined the effect of each physical parameter on SN II observables using
the models constructed with the moderately stripped progenitors presented in
Sect. 4.1. We measured observables from the non-standard LC and photospheric
velocity models as described in Sect. 2.3. We then performed a KDA in a
similar manner to in Sect. 3. For the KDA, we used the standard models and
those with additional mass stripping together. The inclusion of models from
non-standard pre-SN calculations raises a problem, given that there is more
than one model for several sets of physical parameters. This issue is solved
by including the wind efficiency $\eta$ as an additional physical parameter
for the KDA. Values of $\eta$ equal unity for the standard models, and its
corresponding value (see Table 3) for each model with additional mass
stripping. $M_{\rm ZAMS}$ and $\eta$ are highly related to the ejecta mass. As
a consequence, the combination of $M_{\rm ZAMS}$ and $\eta$ represent, at
first order, the effect of the ejecta mass. Two major changes are found with
respect to the results presented in Table 2. We find a lower influence of the
explosion energy on optd, while both $M_{\rm ZAMS}$ and $\eta$ together
present a higher effect than that corresponding to $M_{\rm ZAMS}$ in Table 2.
Moreover, the impact of $M_{\rm ZAMS}$ and $\eta$ on $s_{\rm 3}$ increases
considerably, reaching a relative importance of 0.73 against the estimate of
0.58 found for $M_{\rm ZAMS}$ in Table 2.

##  5 Discussion

###  5.1 The effect of explosion energy

Following the analysis in Sect. 3, it seems that the majority of the diversity
in SN II LCs can be confidently ascribed to differences in explosion energy
when standard single-star evolution is assumed. A higher explosion energy
leads to more luminous SNe II with higher expansion velocities, which cool and
recombine the ejecta more rapidly, producing a shorter plateau and optically
thick phases. Moreover, faster expansion of the SN ejecta induces lower
densities at earlier epochs. At sufficiently low ejecta densities, the cooling
wave —which is responsible for the plateau— cannot be formed, resulting in
steeper plateau phases (Grassberg et al. 1971; Blinnikov & Bartunov 1993). A
faster expansion also produces a more rapid cooling of the SN ejecta, which is
consistent with the picture of faster declining SNe II having redder colours
at late times (de Jaeger et al. 2018). Additionally, higher energy explosions
produce more 56Ni, which cause increased variation in the late-plateau and
tail brightness (see discussion below). We now compare our results with those
found in previous studies.

The declination rate during the plateau ($s_{\rm 2}$) is one of the parameters
that show great diversity and a continuum of values (A14; Sanders et al. 2015;
Galbany et al. 2016; Valenti et al. 2016; Rubin & Gal-Yam 2016; de Jaeger et
al. 2019). Theoretical studies suggest that fast declining SNe II are produced
by reducing the progenitor hydrogen-rich envelope mass (e.g. Blinnikov &
Bartunov 1993; Hillier & Dessart 2019; Pessi et al. 2019). Observational
studies support this scenario based on correlations between $s_{\rm 2}$ and
observed parameters that are mostly related to the progenitor envelope mass,
such as the plateau and optically thick phase durations (; ). While low
hydrogen-rich envelope masses satisfactorily produce the low ejecta densities
necessary to avoid the cooling wave, these low densities can also be achieved
by sufficiently high explosion energies. Our results show a lack of
correlation between $M_{\rm H,env}$ and $s_{\rm 2}$ (see Fig. 17), but this
analysis does not contradict the prediction that small hydrogen-rich envelope
masses produce fast-declining LCs; indeed some of the fastest decliners are
consistent with the lowest $M_{\rm H,env}$ values of our models. Rather, this
analysis may imply that $M_{\rm H,env}$ is not the main driver of the
diversity observed in plateau declination rates. However, our grid of pre-SN
models was calculated assuming standard prescriptions for the wind-mass loss,
which does not account for highly stripped progenitors. While this may bias
our conclusions, the results using the explosion models from moderately
stripped progenitors (non-standard models) also suggest a lack of correlation
between $M_{\rm H,env}$ and $s_{\rm 2}$. We note that only a narrow range of
$M_{\rm H,env}$ was studied here and a more exhaustive analysis is needed.

It has also been suggested that 56Ni mass can explain the $s_{\rm 2}$
diversity. Figure 12 (middle panel) indicates that the late-plateau luminosity
increases if 56Ni is present in the SN ejecta, yielding a shallower slope. In
addition, the left panel of Fig. 14 clearly shows the 56Ni effects on $s_{\rm
2}$, which can vary by up to $\sim$1 mag per 100 days for the models presented
in that plot. This was also proposed in theoretical studies by Bersten (2013)
and Kozyreva et al. (2019). From an observational point of view, Nakar et al.
(2016) analysed a sample of 24 SNe II and concluded that 56Ni can flatten the
plateau phase. While low-$M_{\rm Ni}$ events can produce fast-declining SNe
II, we find no clear relation between $M_{\rm Ni}$ and $s_{\rm 2}$ for the
CSP-I SN II sample, in accordance with the results from G17.

An alternative scenario for the diversity of $s_{\rm 2}$ includes the
interaction of the SN ejecta with a dense CSM surrounding the progenitor star.
The presence of CSM can boost the early-time luminosity, thus resulting in
rapidly declining LCs. Morozova et al. (2017) show that the multi-band LCs of
slow- and fast-declining SNe II are well reproduced by RSG explosions that
collide with a dense CSM of different properties (see also Hillier & Dessart
2019). However, for low to moderate CSM masses, only the early LC is affected.
Without considering the effects of CSM interaction, we find that our models
reproduce most well-sampled SNe II after 30 days post-explosion, including
fast declining SNe II. Therefore, we do not expect the effects of CSM
interaction to dominate the late-time declination rate.

Kozyreva et al. (2019) suggest that more energetic events evolve more rapidly
and have faster declination rates if the contribution of 56Ni is ignored. This
is somewhat similar to our findings. In Sect. 3.1 we present our findings of
evident correlations between _E_ and $s_{\rm 2}$, which imply that fast-
declining SNe II in the CSP-I sample are consistent with high-energy
explosions. Additionally, from the analysis of the model parameters through
the KDA (Table 2), we find that the explosion energy is the physical parameter
that most influences the $s_{\rm 2}$ declination rate. For these reasons, we
conclude that the explosion energy is the main driver of $s_{\rm 2}$
diversity, that is, under the assumption of standard single-star evolution.

Previous theoretical studies found that the explosion energy is strongly
related to pd and optd, because more energetic explosions lead to higher
expansion velocities, and therefore more rapid cooling and hydrogen
recombination (e.g. Bersten 2013; Dessart et al. 2013). This is supported by
our results given that we find moderate anti-correlations between _E_ and pd
and also between _E_ and optd. In addition, previous observational studies
found an anti-correlation between pd and expansion velocities, which are
closely related to the explosion energy (Faran et al. 2014, G17). The pd is
also affected by the mass of the hydrogen envelope as is further discussed in
Sect. 5.2.

Very strong correlations between the explosion energy and the SN II brightness
at the photospheric phase are found, in line with theoretical predictions
(Bersten 2013). This is also seen in observational studies, where it is found
that more luminous SNe II develop higher expansion velocities, and must
therefore arise from more energetic explosions (e.g. Hamuy & Pinto 2002, G17).
There are exceptions to this trend, with SNe II displaying low velocities and
high luminosities (Rodríguez et al. 2020), although these observational
features are probably due to the interaction of the SN ejecta with a massive
CSM. In the current paper, we also find a very strong correlation between _E_
and $M_{\rm bol,tail}$ (see also G17, ). However, $M_{\rm bol,tail}$ strongly
depends on the amount of 56Ni present in the ejecta (Woosley et al. 1989) and
not —directly— on the explosion energy. Moreover, we find that $M_{\rm
bol,tail}$ is not greatly impacted by explosion energy (Fig. 13). Therefore,
we should not expect a correlation between $M_{\rm bol,tail}$ and _E_. The
caveat here is that we treated _E_ and $M_{\rm Ni}$ as purely independent
variables while it appears that they may be closely connected.

56Ni production takes place during a SN explosion. After core collapse, a
shock wave forms accelerating and heating the stellar material above the
nascent proto-neutron star. The total amount of 56Ni synthesised depends on
the mass of the ejecta exposed to temperatures above 5 $\times$ 109 K. Above
this temperature, explosive Si burning occurs where 56Ni dominates the
production of nuclear species (Woosley & Weaver 1995; Thielemann et al. 1996;
Umeda & Nomoto 2002). For a low explosion energy, the temperature needed for
explosive Si burning is only reached in the innermost layers of the SN ejecta,
while a higher explosion energy increases the mass exposed to high
temperatures (Sukhbold et al. 2016). Therefore, a positive correlation between
_E_ and $M_{\rm Ni}$ is expected. Our findings show a strong correlation
between these two physical parameters (see Fig. 8). In addition, a correlation
between $M_{\rm Ni}$ and Fe ii $\lambda$5169 velocities is found ($\rho$ =
0.58 $\pm$ 0.21), in accordance with previous observational studies (e.g.
Hamuy 2003; Spiro et al. 2014, G17). The correlations between _E_ , $M_{\rm
Ni}$, $M_{\rm bol,end}$, and expansion velocities support the hypothesis that
the explosion energy is the driver of these relations: more energetic
explosions produce brighter SNe II with faster expansion velocities and more
56Ni, which increments the tail luminosity.

G17 found an inverse correlation between expansion velocities and the strength
of metal lines and concluded that higher energy explosions produce higher
temperatures for a longer time, leading to lower metal-line pEWs. In the
current analysis, we recover those trends using the pEWs of Fe ii
$\lambda$4924 and Fe ii $\lambda$5018\. The emission component of Hα and the
absorption component of Hβ show the opposite behaviour, namely moderate
correlations with _E_. This was also inferred by G17.

Our study suggests that the explosion energy is the key parameter underlying
SN II diversity, which raises the question of what determines the energy of
the explosion. Massive stars develop an iron core in the centre during the
last stage of nuclear burning. At the end of massive-star evolution, the iron
core becomes gravitationally unstable and starts to collapse. The collapse of
the central parts is halted when nuclear densities are reached, at which point
this collapse rebounds, producing a shock wave. However, this shock wave is
not sufficiently energetic and fails to trigger the explosion (see Janka 2012,
for a review). The mechanism of the energy deposition into the envelope to
reverse the stalled shock into explosion is still debated, but the so-called
‘delayed neutrino-driven mechanism’ is the most favoured scenario (Bethe &
Wilson 1985; Janka 2012). The gravitational binding energy of the neutron star
($\gtrsim$1053 erg) is released as neutrinos when the core of a massive star
collapses to a neutron star. The interaction of a fraction of the neutrino
flux with the material of the mantle powers the explosion (Colgate & White
1966; Janka et al. 2016, for a recent review). The rapid rotation of the
collapsing core and magnetic fields may also play a role (LeBlanc & Wilson
1970; Bisnovatyi-Kogan 1971).

Many observational and theoretical works have analysed a possible connection
between _E_ and $M_{\rm ZAMS}$ (or $M_{\rm ej}$). From the observational point
of view, Hamuy (2003) and Pejcha & Prieto (2015) used the analytic scaling
relations of Litvinova & Nadezhin (1985) and found a positive correlation
between _E_ and $M_{\rm ej}$. Hydrodynamical modelling of several SNe II
reveals a general trend with more massive progenitors undergoing more
energetic explosions, although with large scatter (e.g. Pumo et al. 2017;
Morozova et al. 2018; Eldridge et al. 2019; Utrobin & Chugai 2019; Ricks &
Dwarkadas 2019; Martinez & Bersten 2019). The same is found in the current
analysis (Sect. 3.2). However, explosion models based on the neutrino
mechanism have found little evidence of a correlation between _E_ and the
progenitor mass (e.g. Ugliano et al. 2012; Pejcha & Thompson 2015; Ertl et al.
2016; Sukhbold et al. 2016). None of these studies recover a monotonic
increase in _E_ with $M_{\rm ZAMS}$. Recently, 3D core-collapse simulations
found that lower mass progenitors produce lower explosion energies, while
higher mass progenitors experience higher energy explosions (Burrows et al.
2020). Burrows & Vartanyan (2021) show that the general trend of the
mass–energy relation inferred by observational studies is reproduced by
theoretical models, although some scatter is present both in observations and
models. In addition, Burrows & Vartanyan (2021) argue that a diversity of
explosion energies, among other physical properties, could be obtained for the
same initial pre-SN structure.

###  5.2 The effect of hydrogen-rich envelope mass

During the plateau phase, hydrogen recombination occurs at different layers of
the ejecta as a recombination wave recedes (in mass coordinate) through the
expanding envelope (Grassberg et al. 1971; Bersten et al. 2011). Because the
opacity is dominated by electron scattering, it decreases outwards from the
recombination front allowing the radiation to escape. Therefore, the duration
of the plateau phase is connected to the hydrogen-rich envelope mass at the
time of core collapse, among other physical properties. Short plateau SNe II
are usually associated with low hydrogen envelope masses (see Sect. 4.2),
while the opposite is found for long-plateau SNe II (e.g. Litvinova & Nadezhin
1983). Our results are consistent with this picture given that we find
moderate trends between $M_{\rm H,env}$ and both pd and optd.

A14 and G17 introduced an additional parameter as a tracer of the hydrogen-
rich envelope mass. Both studies found that the declination rate during the
radioactive tail phase ($s_{\rm 3}$) is usually higher than that expected from
the decay rate of 56Co if full trapping of gamma-ray photons is assumed
(Woosley et al. 1989). In addition, A14 and G17 found that $s_{\rm 3}$ is
strongly related to $OPTd$ and $Pd$ (the uppercase acronyms refer to the
definitions from A14 and G17 which are slightly different from ours), in the
sense that shorter plateau SNe II display higher declination rates in the
radioactive phase. This may suggest that, in these cases, $M_{\rm ej}$ is too
small for full trapping of the gamma-rays, which causes higher $s_{\rm 3}$
values. While in principle we recover a weak trend between $M_{\rm H,env}$ and
$s_{\rm 3}$, the significance of the correlation is low (see Fig. 6, right
panel).

The previously mentioned correlations are much stronger when using the results
from moderately stripped progenitors (Sect. 4.2), which leads to the following
conclusions: (1) $s_{\rm 3}$ is affected by the ejecta mass, which is
consistent with the analysis presented in Table 2 and Sect. 4.2, where we find
that the mass is the physical parameter causing the greatest effect on $s_{\rm
3}$, and with previous studies in the literature; (2) additional mass
stripping than predicted by standard single-star evolution is necessary to
reproduce the great dependency of the hydrogen-rich envelope mass with pd and
optd (see Fig. 9). Moreover, in Paper II, we found a clear inconsistency
between our $M_{\rm ZAMS}$ distribution and a Salpeter IMF, where we invoked
additional mass loss as one possible explanation (see Paper II for details).
Therefore, our findings seem to suggest that higher mass-loss rates are
required to reproduce SN II observations. However, recent studies suggest that
prescriptions for wind mass-loss rates used in stellar evolution already
overestimate the mass loss by winds (Puls et al. 2008; Smith 2014; Beasor et
al. 2020). The increment of the wind mass-loss rate in our models was used to
mimic any mechanism producing additional envelope stripping. Alternatively,
stellar eruptions, rotation, and/or mass transfer by Roche-Lobe overflow in a
binary system should be studied more thoroughly.

The hydrogen-rich envelope mass is an important parameter in our study, and
therefore we compare the $M_{\rm H,env}$ distribution of our progenitor models
with that predicted for binary evolution using the Binary Population and
Spectral Synthesis (BPASS) version 2.1 (Eldridge et al. 2017). At the same
time, we included pre-SN single-star models calculated with a different code.
Sukhbold et al. (2016) present single-star pre-SN models for a large $M_{\rm
ZAMS}$ range calculated with the KEPLER code (Weaver et al. 1978; Woosley et
al. 2002), but here, the initial masses were limited to the 9$-$25 $M_{\odot}$
range for consistency with the stellar models used in our work. A similar
range of values between our distribution and that from Sukhbold et al. (2016)
is found. As before, the initial masses of primary stars (i.e. the initially
more massive star of the system) of the BPASS models were restricted to the
9$-$25 $M_{\odot}$ range. We also required that the star finish as a RSG.
Stars that lose mass through Roche-lobe overflow finish their evolution with
smaller $M_{\rm H,env}$ than if they had evolved as single stars, producing a
wider distribution that reach lower $M_{\rm H,env}$ values than our models. If
the effects of binary evolution are taken into account, significant diversity
in hydrogen-rich envelope masses is found, from a minimal value of 0.1
$M_{\odot}$ to a maximum value of 29 $M_{\odot}$121212However, some of the
stars in the BPASS distribution will not produce the SNe II studied here.
Stars with low $M_{\rm H,env}$ might produce SN IIb-like events. In addition,
some stars can experience a merger and finish their evolution with $M_{\rm
H,env}$ $\gtrsim$ 10 $M_{\odot}$, which might produce SNe II with optd longer
than $\sim$150 days (called ‘long-SNe IIP’; Eldridge et al. 2018)..
Differences in the $M_{\rm H,env}$ distribution between single- and binary-
star models do exist, with standard single-star evolution giving a narrower
range of $M_{\rm H,env}$ that may affect our results. However, the goal of
this study is to test the predictions of standard single-star evolution
against correlations between physical and observed properties of SNe II. A
study including the effects on binary evolution is warranted in the future.

###  5.3 56Ni mixing

Our analyses of the relative importance of each physical parameter for $s_{\rm
2}$ suggest that the explosion energy is the parameter that is driving
variation to the greatest degree, with ${}^{56}\rm Ni$ mixing showing only a
minor influence (see Table 2). However, in Sect. 3.1 we find a strong anti-
correlation between ${}^{56}\rm Ni$ mixing and $s_{\rm 2}$ in the sense that
fast-declining SNe II are consistent with a more concentrated 56Ni.

The effect of ${}^{56}\rm Ni$ mixing in shaping LCs of SNe II was previously
studied in the literature. The degree of ${}^{56}\rm Ni$ mixing within the
envelope determines the time at which 56Ni starts to affect the LC (e.g.
Bersten et al. 2011; Kozyreva et al. 2019). While an extensively mixed 56Ni
influences the LC from the early plateau phase, a more concentrated 56Ni
distribution starts affecting the LC during the late recombination phase (see
Bersten et al. 2011, their Fig. 12). Before the photosphere recedes to the
shells containing 56Ni, the behaviour of the LC is similar to that in the
absence of 56Ni, which naturally causes a decline during the plateau phase.
Therefore, 56Ni mostly concentrated in the inner ejecta is necessary to
produce rapid decline in the plateau phase (higher $s_{\rm 2}$ values). In
summary, we find that fast declining SNe II are consistent with higher energy
explosions and small-scale ${}^{56}\rm Ni$ mixing. We now analyse possible
connections between both physical parameters.

The mixing of heavy elements through the ejecta is caused by Rayleigh$-$Taylor
instabilities that appear once the shock wave passes through the composition
interfaces between the carbon-oxygen core and the helium core, and this latter
and the hydrogen-rich envelope. Numerous numerical simulations have been
carried out using blue supergiant progenitors to reproduce the large-scale
mixing of 56Ni presented in SN 1987A, but only a small number of explosion
models with RSGs have been calculated. Recently, Wongwathanarat et al. (2015)
computed 3D simulations and found that the degree of ${}^{56}\rm Ni$ mixing
depends on the progenitor structure (e.g. the size of the carbon-oxygen core,
the density structure of the helium core, and the density gradient at the
interface between the helium core and the hydrogen-rich envelope), the
explosion energy, and the asymmetries created by the neutrino-driven
mechanism. More energetic explosions produce a faster propagation of 56Ni
within the ejecta, but also result in a faster shock wave (Wongwathanarat et
al. 2015). Therefore, in principle, the explosion energy is not expected to
correlate with the mixing of 56Ni, implying that the correlation found between
${}^{56}\rm Ni$ mixing and the plateau declination rate is authentic.

The analysis of the effect of each physical parameter on the observed
diversity of SNe II shows that the mixing of 56Ni within the ejecta produces
minor changes in optd, although at the same time, a strong correlation is
found between both parameters based on the results achieved with the CSP-I
sample (see Sect. 3.1). Thus, this correlation may be a consequence of some
additional correlations, such as: SNe II declining faster have shorter pd
(e.g. Pskovskii 1967, A14; Fig. 18), and are consistent with more concentrated
mixing of 56Ni (Fig. 5).

###  5.4 Observational biases

The observed SN II sample used in this work is of high quality in terms of LC
cadence, wavelength coverage, and photometric accuracy. However, it still
contains observational biases that may affect our results and conclusions. The
CSP-I SN II sample is magnitude-limited —SNe were only chosen for followup if
they were brighter than around 18th magnitude (at optical wavelengths) at
maximum light. Thus, our sample is biased towards intrinsically brighter SNe
II, meaning we lack lower luminosity events (as compared to their absolute
rate of explosion). Figure 4 shows that low-luminosity events are almost
exclusively constrained to arise from low-energy explosions, and therefore our
sample is probably less populated with such explosions compared to the number
that exist in nature. However, it is not clear that such a bias will affect
our results on correlations between observed and physical parameters, or our
conclusions on the dominant physical parameters driving SN II diversity.

The magnitude-limited nature of our sample also means that we will be biased
against heavily extinguished events. However, there is no evidence (to our
knowledge) that the amount of host-galaxy extinction suffered by a SN II
correlates with any of its intrinsic physical properties, and therefore we do
not believe that such a bias will affect our results. Finally, the CSP-I SN II
sample was obtained at a time when most surveys discovering nearby SNe II
(that would be candidates for followup by the CSP-I) were targeted in nature,
with relatively small fields of view. Such surveys were biased against
discovering SNe II in low-luminosity hosts. Galaxy luminosity correlates with
galaxy metallicity, which probably means that our sample lacks SNe II arising
from low-metallicity progenitors. Gutiérrez et al. (2018) analysed SNe II
exploding in such dim galaxies, concluding that in general they had more
slowly declining LCs, but otherwise had observational parameters similar to
the rest of the SN II population. As presented in Figure 5, we find more
slowly declining SNe II (lower $s_{\rm 2}$) to generally arise from lower
energy explosions. Therefore, similarly to the above, the lack of SNe II in
low-luminosity hosts in our sample may have led to a lack of low-energy
explosions. This may somewhat increase the uncertainties when making more
general statements, but this should not significantly affect our results or
conclusions for the sample as it is.

##  6 Conclusions

In this study, we present an analysis of correlations between the observed and
physical properties of a statistically significant sample of SNe II from the
CSP-I in order to understand their diversity in terms of progenitor and
explosion properties. We used the physical parameters determined in Paper II
through hydrodynamical modelling of SN II bolometric LCs and expansion
velocities, such as $M_{\rm ZAMS}$ (which relates to $M_{\rm ej}$, $M_{\rm
H,env}$, and $R$), _E_ , $M_{\rm Ni}$, and the distribution of 56Ni within the
ejecta, together with many photometric and spectroscopic parameters.

This study shows that the explosion energy is the physical parameter that
correlates with the highest number of observed parameters, including pd, optd,
$M_{\rm bol,end}$, $M_{\rm bol,tail}$, $s_{\rm 2}$, expansion velocities, the
pEWs of some metal lines, and the flux ratio of the absorption to emission
component of the Hα P-Cygni profile. In addition, we find that higher energy
explosions manifest more 56Ni in the ejecta. Faster declining SNe II are
consistent with higher energy explosions and more concentrated 56Ni in the
inner regions of the ejecta. In contrast to previous results, we find zero
correlation between the declination rate during the plateau and the hydrogen-
rich envelope mass. However, we note the caveat that only a narrow range of
hydrogen-rich envelope masses was studied. We measured parameters from our
grid of bolometric LC and photospheric velocity models to determine the effect
of each physical parameter on the observables through a statistical study.
According to our findings, the explosion energy is the parameter causing the
greatest impact on SN II diversity.

New progenitor pre-SN structures evolved with enhanced mass loss were
considered to model the observations of two short-plateau SNe II in the CSP-I
sample: SN 2006Y and SN 2008bu. These models were also used to reproduce
several other SNe II in the CSP-I sample successfully modelled in Paper II,
finding that nine additional objects are better reproduced with the non-
standard models. Furthermore, using these new results from non-standard
stellar evolution, we find a significant increase in the strength of
correlations between $M_{\rm H,env}$ and pd, $M_{\rm H,env}$ and optd, and
$M_{\rm ej}$ and $s_{\rm 3}$. This implies that more mass loss is needed to
reproduce short-plateau SNe II than is predicted by standard single-star
evolution (as noted by previous studies), but also that non-standard models
are necessary for a complete understanding of SN II diversity. The differences
in the strength of correlations when different pre-SN models are used clearly
shows the impact of having different treatments of stellar evolution.

A significant number of SNe II decline faster during the radioactive tail
phase (11 out of 16, see Paper I) than the expected rate whilst assuming full
trapping of gamma-ray photons. The moderate correlation found between $s_{\rm
3}$ and $M_{\rm ej}$ —when non-standard models are included— implies that
$s_{\rm 3}$ is directly affected by the SN ejecta mass in accordance with
previous suggestions (; ).

This paper concludes a three-part series using the CSP-I SN II sample to
understand the SN II phenomenon. Constraints are presented on progenitor and
explosion properties through derivation of physical parameters from comparison
between observed and modelled SNe. The current work further constrains the
importance of different SN II physical parameters in explaining SN II
diversity. However, these investigations also highlight limitations in our
understanding of the lives and explosive deaths of massive stars, thus
encouraging further modelling and additional high-quality observations.

###### Acknowledgements.

We thank the referee for the useful comments that improved the manuscript. The
work of the Carnegie Supernova Project was supported by the National Science
Foundation under grants AST-0306969, AST-0607438, AST-1008343, AST-1613426,
AST-1613472, and AST-1613455. L.M. acknowledges support from a CONICET
fellowship. L.M. and M.O. acknowledge support from UNRN PI2018 40B885 grant.
M.H. acknowledges support from the Hagler Institute of Advanced Study at Texas
A&M University. S.G.G. acknowledges support by FCT under Project CRISP
PTDC/FIS-AST-31546/2017 and Project No. UIDB/00099/2020. M.S. is supported by
grants from the VILLUM FONDEN (grant number 28021) and the Independent
Research Fund Denmark (IRFD; 8021-00170B). F.F. acknowledges support from the
National Agency for Research and Development (ANID) grants: BASAL Center of
Mathematical Modelling AFB-170001, Ministry of Economy, Development, and
Tourism’s Millennium Science Initiative through grant IC12009, awarded to the
Millennium Institute of Astrophysics, and FONDECYT Regular #1200710. L.G.
acknowledges financial support from the Spanish Ministry of Science,
Innovation and Universities (MICIU) under the 2019 Ramón y Cajal program
RYC2019-027683 and from the Spanish MICIU project PID2020-115253GA-I00. P.H.
acknowledges the support by National Science Foundation (NSF) grant
AST-1715133. This work made use of v2.2.1 of the Binary Population and
Spectral Synthesis (BPASS) models as described in Eldridge et al. (2017) and
Stanway & Eldridge (2018).
Software: emcee (Foreman-Mackey et al. 2013), NumPy (Oliphant 2006; Van Der
Walt et al. 2011), matplotlib (Hunter 2007), MESA (Paxton et al. 2011, 2013,
2015, 2018, 2019), SciPy (Virtanen et al. 2020), Kruskals, Pandas (McKinney
2010), ipython/jupyter (Perez & Granger 2007).

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##  Appendix A SN II diversity and its physical origin

In Sect. 3.1 we present an analysis of correlation between physical and
observed parameters for the CSP-I SN II sample. In addition, with the aim of
determining the effect of physical parameters on SN II diversity, we measured
some parameters directly from our grid of bolometric LC and photospheric
velocity models (Sect. 2.3). Here, in Figs. 11 to 16, we show observables
measured from our grid of explosion models against the physical parameter
yielding the highest relative importance, following the results from Table 2.
Each panel presents the effect of a distinct physical parameter denoted by
different colours given in the colour bar. The physical parameters not being
varied are presented in each panel together with their fixed values.

Figure 11 shows that the large range of optd values seen in the previous
sections are mainly produced by $M_{\rm ZAMS}$ and _E_ , in line with previous
theoretical studies. The left panel clearly shows the great dependency of
these two physical parameters on the optd, while middle and right panels show
the effect of $M_{\rm Ni}$ and its mixing in the _E_ $-$optd relation,
respectively. As expected, higher $M_{\rm Ni}$ extends the optically thick
phase, although 56Ni distribution in the ejecta does not seem to produce
significant changes.

The brightness at the end of the plateau ($M_{\rm bol,end}$) is highly
affected by the explosion energy, with higher energy explosions delivering
more luminous SNe II (Fig. 12). $M_{\rm ZAMS}$ is the physical parameter that
produces the largest deviation in the _E_ $-$$M_{\rm bol,end}$ relation (Fig.
12, left panel), while $M_{\rm Ni}$ and its degree of mixing within the ejecta
alter this relation on smaller scales. However, $M_{\rm Ni}$ has its major
effect on $M_{\rm bol,end}$ in the low-_E_ regime where SNe II are fainter and
the effects of 56Co decay are more significant during the late photospheric
phase, causing higher luminosities at late-plateau phases (Fig. 12, middle
panel). Figure 13 indicates that $M_{\rm bol,tail}$ is mostly affected by the
amount of 56Ni in the ejecta. The explosion energy and the mixing of 56Ni do
not significantly contribute to $M_{\rm bol,tail}$.

The explosion energy produces the largest effect on the $s_{\rm 2}$
declination rate, followed by $M_{\rm Ni}$, at least with the models used in
this series of papers (Fig. 14). $M_{\rm ZAMS}$ also shows variations on
$s_{\rm 2}$, but with a lower effect than the explosion energy. However, this
analysis only includes standard pre-SN models, where none of the progenitors
were evolved with significant mass loss. The relation between _E_ and $s_{\rm
2}$ has an additional contribution to the dispersion produced by the mixing of
56Ni, with faster declining SNe II consistent with low degrees of ${}^{56}\rm
Ni$ mixing.

The declination rate of the radioactive tail phase ($s_{\rm 3}$) takes the
theoretical value of 0.98 mag per 100 days assuming full trapping of the
gamma-ray emission from 56Co decay (dashed lines in Fig. 15). However, a
considerable number of SNe II are found with higher $s_{\rm 3}$ (; ; ). Here,
we find that $M_{\rm ZAMS}$ is the main driver affecting the $s_{\rm 3}$
declination rate (Table 2, Fig. 15). At least within the range of initial
masses of our standard pre-SN models, high-$M_{\rm ZAMS}$ stars imply higher
$M_{\rm ej}$. Therefore, the high $s_{\rm 3}$ values are consistent with lower
ejecta masses. Low ejecta masses are less efficient in thermalising the gamma-
ray emission from 56Co decay, leading to a more rapid decline in the
luminosity during the radioactive tail phase. The degree of 56Ni mixing within
the ejecta also contributes to the differences found in $s_{\rm 3}$. This is
seen in the middle panel of Fig. 15 where more extended ${}^{56}\rm Ni$ mixing
produces a more rapid declination of SNe II during the radioactive tail.
Figure 16 shows that _E_ and $M_{\rm ZAMS}$ are the only two physical
parameters that produce different photospheric velocities at 50 days from
explosion.

Figure 11: Optically thick phase duration (optd) measured from the synthetic
bolometric LCs as a function of the explosion energy. The physical parameters
not being varied are presented in each subplot together with their fixed
values. Each subplot shows the influence of the other physical parameters:
$M_{\rm ZAMS}$ (left panel), $M_{\rm Ni}$ (middle panel), and ${}^{56}\rm Ni$
mixing (right panel). Only models with optd values smaller than 160 days are
analysed. Black dots represent the observations from the CSP-I SN II sample.
Some observations fall outside the range of the model parameters due to the
fixed physical parameters. Changes in the fixed values represent different
ranges of model parameters. Figure 12: $M_{\rm bol,end}$ measured from the
synthetic bolometric LCs as a function of the explosion energy. The physical
parameters not being varied are presented in each subplot together with their
fixed values. Each subplot shows the influence of the other physical
parameters: $M_{\rm ZAMS}$ (left panel), $M_{\rm Ni}$ (middle panel), and
${}^{56}\rm Ni$ mixing (right panel). Only models with optd values smaller
than 160 days are analysed. Black dots represent the observations from the
CSP-I SN II sample. Some observations fall outside the range of the model
parameters due to the fixed physical parameters. Changes in the fixed values
represent different ranges of model parameters. Figure 13: $M_{\rm bol,tail}$
measured from the synthetic bolometric LCs as a function of $M_{\rm Ni}$. The
physical parameters not being varied are presented in each subplot together
with their fixed values. Each subplot shows the influence of the other
physical parameters: _E_ (left panel), $M_{\rm ZAMS}$ (middle panel), and
${}^{56}\rm Ni$ mixing (right panel). Only models with optd values smaller
than 160 days are analysed. Black dots represent the observations from the
CSP-I SN II sample. Some observations fall outside the range of the model
parameters due to the fixed physical parameters. Changes in the fixed values
represent different ranges of model parameters. Figure 14: Declination rate
during the plateau ($s_{\rm 2}$) measured from the synthetic bolometric LCs as
a function of explosion energy. The physical parameters not being varied are
presented in each subplot together with their fixed values. Each subplot shows
the influence of the other physical parameters: $M_{\rm Ni}$ (left panel),
$M_{\rm ZAMS}$ (middle panel), and ${}^{56}\rm Ni$ mixing (right panel). Only
models with optd values smaller than 160 days are analysed. Black dots
represent the observations from the CSP-I SN II sample. Some observations fall
outside the range of the model parameters due to the fixed physical
parameters. Changes in the fixed values represent different ranges of model
parameters. Figure 15: Declination rate of the radioactive tail ($s_{\rm 3}$)
measured from the synthetic bolometric LCs as a function of $M_{\rm ZAMS}$.
The physical parameters not being varied are presented in each subplot
together with their fixed values. Each subplot shows the influence of the
other physical parameters: $M_{\rm Ni}$ (left panel), ${}^{56}\rm Ni$ mixing
(middle panel), and _E_ (right panel). The horizontal dashed lines indicate
the expected declination rate for full trapping of emission from 56Co decay.
Only models with optd values smaller than 160 days are analysed. Black dots
represent the observations from the CSP-I SN II sample. Some observations fall
outside the range of the model parameters due to the fixed physical
parameters. Changes in the fixed values represent different ranges of model
parameters. Figure 16: Photospheric velocity at 50 days measured from the
models ($v_{\rm ph,50}$) as a function of explosion energy. The physical
parameters not being varied are presented in each subplot together with their
fixed values. Each subplot shows the influence of the other physical
parameters: $M_{\rm ZAMS}$ (left panel), $M_{\rm Ni}$ (middle panel), and
${}^{56}\rm Ni$ mixing (right panel). Only models with optd values smaller
than 160 days are analysed. Black dots represent the observations from the
CSP-I SN II sample. Some observations fall outside the range of the models
(see text). Figure 17: Same as in Fig. 5 but for $M_{\rm H,env}$ versus
$s_{\rm 2}$.

##  Appendix B Photometric and spectroscopic correlations for the CSP-I SN II
sample

In the main text of this paper, we present our analysis of correlations
between physical and observed parameters, without discussing the relations
between observed SN II parameters. For completeness, in this Appendix we
present correlations between bolometric LC parameters, colours at different
epochs, and expansion velocities and pEWs measured at 50 days post-explosion.
Similar analyses have been carried out in the literature (; ; ). The only
difference is that in those works the authors used $V$-band LC properties in
their analyses, while here we used the bolometric LC parameters measured in
Paper I.

Figure 18 shows the correlation matrix of the parameters. Most of the trends
found in previous studies are recovered with similar or higher degree of
correlation. Similar to G17, we find that SNe II with shorter pd are brighter,
have faster declining LCs at the three measured epochs, lower pEW(Hα) of the
absorption component and $a/e$, and higher expansion velocities. In addition,
similar to dJ18, we find that fast-declining SNe II are bluer at early epochs
but redder at later epochs. Given that similar results are obtained, we do not
go into the details of the correlations and their possible explanations. The
reader is referred to A14, G17, and dJ18 for further details.

While we recover the trend between pd and $s_{\rm 3}$ ($\rho$ = $-$0.70 $\pm$
0.25, $N$ = 6), we find zero correlation between optd and $s_{\rm 3}$. The
lack of correlation between $s_{\rm 3}$ and optd may be possible because, by
definition, optd includes the plateau phase, mostly related to the hydrogen-
rich envelope mass, and other phases that are powered by different mechanisms
(the release of shock deposited energy, ejecta-CSM interaction) that are
related to different progenitor properties. However, G17 found a moderate
correlation between optd and $s_{\rm 3}$.

Figure 18: Correlation matrix of the bolometric LC, spectral, and colour
parameters used in the current study. The Pearson correlation coefficients are
presented in the upper triangle, while in the lower triangle the correlation