# Inferencing Progenitor and Explosion Properties of Evolving Core-collapse
Supernovae from
Zwicky Transient Facility Light Curves

Bhagya M. Subrayan Purdue University, Department of Physics and Astronomy, 525
Northwestern Ave, West Lafayette, IN 47907  Dan Milisavljevic Purdue
University, Department of Physics and Astronomy, 525 Northwestern Ave, West
Lafayette, IN 47907  Integrative Data Science Initiative, Purdue University,
West Lafayette, IN 47907, USA  Takashi J. Moriya National Astronomical
Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  School of Physics and Astronomy, Faculty of
Science, Monash University, Clayton, Victoria 3800, Australia  Kathryn E. Weil
Purdue University, Department of Physics and Astronomy, 525 Northwestern Ave,
West Lafayette, IN 47907  Geoffery Lentner Purdue University, Department of
Physics and Astronomy, 525 Northwestern Ave, West Lafayette, IN 47907  Mark
Linvill  Purdue University, Department of Physics and Astronomy, 525
Northwestern Ave, West Lafayette, IN 47907  John Banovetz Purdue University,
Department of Physics and Astronomy, 525 Northwestern Ave, West Lafayette, IN
47907  Braden Garretson Purdue University, Department of Physics and
Astronomy, 525 Northwestern Ave, West Lafayette, IN 47907  Jack Reynolds
Purdue University, Department of Physics and Astronomy, 525 Northwestern Ave,
West Lafayette, IN 47907  Niharika Sravan  California Institute of Technology,
Pasadena, USA  Ryan Chornock Department of Astronomy, University of
California, Berkeley, CA 94720-3411, USA  Raffaella Margutti Department of
Astronomy, University of California, Berkeley, CA 94720-3411, USA

###### Abstract

We analyze a sample of 45 Type II supernovae from the Zwicky Transient
Facility (ZTF) public survey using a grid of hydrodynamical models in order to
assess whether theoretically-driven forecasts can intelligently guide follow
up observations supporting all-sky survey alert streams. We estimate several
progenitor properties and explosion physics parameters including zero-age-
main-sequence (ZAMS) mass, mass-loss rate, kinetic energy, 56Ni mass
synthesized, host extinction, and the time of explosion. Using complete light
curves we obtain confident characterizations for 34 events in our sample, with
the inferences of the remaining 11 events limited either by poorly
constraining data or the boundaries of our model grid. We also simulate real-
time characterization of alert stream data by comparing our model grid to
various stages of incomplete light curves ($\Delta t<$ 25 days, $\Delta t<50$
days, all data), and find that some parameters are more reliable indicators of
true values at early epochs than others. Specifically, ZAMS mass, time of
explosion, steepness parameter $\beta$, and host extinction are reasonably
constrained with incomplete light curve data, whereas mass-loss rate, kinetic
energy and 56Ni mass estimates generally require complete light curves
spanning $>100$ days. We conclude that real-time modeling of transients,
supported by multi-band synthetic light curves tailored to survey passbands,
can be used as a powerful tool to identify critical epochs of follow up
observations. Our findings are relevant to identify, prioritize, and
coordinate efficient follow up of transients discovered by Vera C. Rubin
Observatory.

Supernovae, Type II supernovae, Surveys, Hydrodynamical simulations

††software:  KEPLER (Weaver et al., 1978), STELLA (Blinnikov et al., 1998,
2000, 2006; Moriya et al., 2017, 2018; Ricks & Dwarkadas, 2019),
astropy(Astropy Collaboration et al., 2013, 2018), dynesty (Skilling, 2004)

##  1 Introduction

The upcoming Legacy Survey of Space and Time (LSST) to be conducted by the
Vera C. Rubin Observatory is highly anticipated to revolutionize time domain
astronomy (LSST Science Collaboration et al., 2009). Its sensitivity ($\sim$
24 mag), six broadband filters ($u$-$g$-$r$-$i$-$z$-$y$), regular southern-sky
patrolling (cadences anticipated between hourly and every few days; LSST
Science Collaboration et al. 2017), and prompt reporting of transient activity
(latency of $\approx 60$ seconds from exposure read-out to alert distribution)
will provide opportunities to discover and investigate millions of supernovae
(SNe) over its planned ten year lifetime (Ivezić et al., 2019).

However, managing the massive data sets associated with LSST will be
demanding. It will produce $\sim$ 20 TB of raw images every single night,
which will be processed rapidly via template subtraction to send out real-time
alerts of residual source variability (approximately ten million alerts
nightly). Moreover, because LSST photometry alone will generally be
insufficient to adequately investigate the transients it will discover (Alves
et al., 2022), the survey’s success will be partially dependent on other
telescopes for supporting observations (see, e.g., Najita et al. 2016), with
electromagnetic and multi-messenger facilities (Huerta et al., 2019).

The LSST Corporation (LSSTC) architects along with emerging alert stream data
brokers are developing the processes, cyberinfrastructure, and software needed
to confront this challenge and help manage upcoming LSST discoveries (Borne,
2008; Narayan et al., 2018). Data brokers utilize transient classification
methods that often employ machine learning (Möller & de Boissière, 2020;
Förster et al., 2021; Sooknunan et al., 2021; García-Jara et al., 2022).
Current data brokers include ALeRCE111Automatic Learning for the Rapid
Classification of Events, http://alerce.science/ (Sánchez-Sáez et al., 2021),
ANTARES222Arizona–NOIRLab Temporal Analysis and Response to Events System,
https://antares.noirlab.edu/ (Matheson et al., 2021),
Lasair333https://lasair.roe.ac.uk/ (Smith, 2019), MARS444Make Alerts Really
Simple, https://mars.lco.global/, and FINK555https://fink-broker.org/ (Möller
et al., 2021), Babamul and PITT-Google. These data brokers are taking on
different responsibilities to promptly process, value-add, cross-reference,
and classify survey alert streams, which in turn permits users to filter and
prioritize targets.

Working downstream of these data brokers are additional services to coordinate
follow up observations, including Target and Observation Managers (TOMs) that
permit observers to sort through broker alert streams to plan and trigger
follow up (Street et al., 2018). TOMs have some level of automation, but
generally rely largely on humans to make decisions about target prioritization
and coordination. The Recommender Engine For Intelligent Transient Tracking
(REFITT; Sravan et al. 2020) is an attempt to completely automate transient
follow up as an Object Recommender for Augmentation and Coordinating Liaison
Engine (ORACLE). REFITT uses data ingested from surveys to predict the light
curve evolution of transients, prioritize events based on confidence in its
prediction, and finally makes recommendations to observers on targets that
need follow up, specific to their observing facility and coordinated among all
observing agents.

To date, decisions about which transients to prioritize for follow up
observations with supporting facilities are generally data-driven; i.e., based
on comparisons to data sets of previously observed events. In this paper, we
explore the feasibility of guiding follow-up of core–collapse supernovae
(CCSNe) using parameters of theoretically-driven forecasts. Our expectation is
that prioritizing the underlying physics of transients will make it possible
to 1) rapidly recognize transients of desired physical parameter spaces, and
2) identify information-rich epochs in transient evolution for efficient
follow-up with limited facilities.

To this end, in this paper we characterize a sample of 45 light curves of Type
II supernovae using publicly available data from the Zwicky Transient Facility
(ZTF) (Bellm et al., 2019) survey with a grid of theoretical hydrodynamical
models spanning various progenitor properties (zero-age-main-sequence (ZAMS)
mass, mass-loss history, etc.) and explosion physics (e.g., kinetic energy,
56Ni synthesized). We compare results between model fits with both complete
and incomplete light curves in order to assess whether theoretically-driven
forecasts can intelligently guide follow up observations supporting all-sky
survey alert streams. ZTF data is used because, among currently operating all-
sky surveys that include ATLAS666Asteroid Terrestrial-impact Last Alert System
(Tonry et al., 2018) and ASAS–SN777All-Sky Automated Survey for Supernovae
(Shappee et al., 2014), ZTF’s functioning alert stream best mimics LSST data
flow, but at a more manageable scale (Masci et al., 2019). Given that ZTF’s
cadence, depth, and filters differ from those of LSST, it still serves as an
excellent testing ground for developing the infrastructure and software that
LSST will require as it starts operating.

The paper is organized as follows: Section 2 describes the treatment of ZTF
data in multiple passbands with forced photometry and extinction. In Section
3, we describe our hydrodynamical models with circumstellar material (CSM)
structure constructed using the stellar evolutionary code KEPLER and the
radiative transfer code STELLA. Section 4 outlines the results from our
fitting method and trends in the parameters derived from the fits, and Section
5 describes our real-time fitting analysis for ZTF events and an assessment of
how model parameters evolve as a function of time. The implications and
utility of this work in the backdrop of current and upcoming all–sky surveys
including LSST are discussed in Section 6.

##  2 Survey Data from ZTF

We used data from the public ZTF alert stream, available as photometry in the
ztf–$g$ and ztf–$r$ passbands. We selected 45 ZTF events from Garretson et al.
(2021) that are spectroscopically classified as Type II or IIP (Table 1). For
the events that were classified as Type II, we confirmed that the light curves
clearly demonstrated some portion of the plateau phase through visual
inspection, differentiating them from Type IIL supernovae. The events span
across the first four years of ZTF survey starting from 2018–2021. All events
in this sample have a minimum of 5 detections, as defined below, in each
ztf–$g$ and ztf–$r$ passbands. We treat this as a representative test sample
as the light curves represent a variety of phases in evolution, reasonably
span expected redshifts of the survey ($0.010<z<0.055$), and span peak
apparent magnitudes approximately between 18–20 mag.

Table 1: ZTF events in this paper

ZTF ID | TNS | TNS | RA | Dec | Redshift | No. $g$-band | No. $r$-band
---|---|---|---|---|---|---|---
| Classification | Name | (J2000) | (J2000) | ($z$) | detections | detections
ZTF18abcpmwh$*$$*$The reported number of measurements in ztf–$g$ and ztf–$r$ obtained within 150 days of first detection, after running forced photometry for each ZTF event. Additional detections beyond 150 days not included in the analysis are quoted in parenthesis for each band. | IIP | SN 2018cur | 12:59:09.12 | +37:19:00.19 | 0.015 | 8(12) | 8(40)
ZTF18acbvhit | II | SN 2018hle | 3:39:28.11 |  $-$13:07:02.50 | 0.014 | 16 | 15
ZTF18acbwasc$\dagger$$\dagger$footnotemark:  | IIP | SN 2018hfc | 11:01:58.61 | +45:13:39.26 | 0.020 | 51 | 65
ZTF18acrtvmm$*$$*$The reported number of measurements in ztf–$g$ and ztf–$r$ obtained within 150 days of first detection, after running forced photometry for each ZTF event. Additional detections beyond 150 days not included in the analysis are quoted in parenthesis for each band. | II | SN 2018jfp | 3:17:56.27 |  $-$0:10:10.82 | 0.023 | 17 | 20(2)
ZTF18acuqskr | II | SN 2018jrb | 8:09:33.69 | +15:31:10.55 | 0.045 | 7 | 10
ZTF19aakiyed | II | SN 2019awk | 15:07:02.58 | +61:13:42.37 | 0.044 | 17 | 23
ZTF19aaqdkrm | II | SN 2019dod | 13:25:49.97 | +34:29:43.58 | 0.034 | 27 | 30
ZTF19aauqwna | IIP | SN 2019fem | 19:44:46.13 | +44:42:49.13 | 0.041 | 20 | 47
ZTF19aavbkly | IIP | SN 2019fmv | 12:29:33.80 | +35:46:12.15 | 0.041 | 31 | 29
ZTF19aavhblr | II | SN 2019fuo | 15:31:43.38 | +16:42:49.30 | 0.050 | 19 | 27
ZTF19aavkptg | IIP | SN 2019gqs | 11:36:15.78 | +49:09:12.55 | 0.038 | 30 | 22
ZTF19abguqsi | II | SN 2019lsh | 22:52:58.52 | +0:26:50.53 | 0.052 | 14 | 20
ZTF19abhduuo$\dagger$$\dagger$footnotemark:  | II | SN 2019lre | 1:58:50.79 |  $-$9:35:05.65 | 0.018 | 14 | 16
ZTF19abiahko | IIP | SN 2019lsj | 19:36:59.13 |  $-$11:57:13.63 | 0.023 | 8 | 11
ZTF19abqyouo$\dagger$$\dagger$footnotemark:  | IIP | SN 2019pbk | 7:46:23.89 | +64:13:23.79 | 0.045 | 11 | 13
ZTF19abvbrve | IIP | SN 2019puv | 19:12:36.67 |  $-$19:25:01.67 | 0.020 | 14 | 42
ZTF19acbvisk$*$$*$The reported number of measurements in ztf–$g$ and ztf–$r$ obtained within 150 days of first detection, after running forced photometry for each ZTF event. Additional detections beyond 150 days not included in the analysis are quoted in parenthesis for each band. | II | SN 2019rms | 9:00:45.39 | +19:44:42.32 | 0.037 | 19 | 37(1)
ZTF19ackjvtl$\dagger$$\dagger$footnotemark:  | IIP | SN 2019uwd | 13:16:20.75 | +30:40:48.72 | 0.019 | 53 | 99
ZTF19acmwfli$\dagger$$\dagger$footnotemark:  | II | SN 2019tza | 13:14:04.68 | +59:15:04.91 | 0.028 | 39 | 59
ZTF19acszmgx | II | SN 2019vew | 5:27:49.43 |  $-$5:21:39.88 | 0.042 | 22 | 22
ZTF20aahqbun | II | SN 2020alg | 14:27:10.55 | +35:55:20.22 | 0.028 | 26 | 41
ZTF20aamlmec | II | SN 2020chv | 7:46:27.78 | +1:57:34.73 | 0.034 | 8 | 9
ZTF20aamxuwl | II | SN 2020ckv | 11:03:26.48 |  $-$1:32:27.04 | 0.037 | 13 | 12
ZTF20aatqgeo | II | SN 2020fcx | 13:40:10.01 | +23:20:29.56 | 0.032 | 25 | 16
ZTF20aatqidk | II | SN 2020fbj | 12:47:47.03 | +22:17:10.44 | 0.034 | 26 | 29
ZTF20aaullwz | II | SN 2020fch | 11:33:24.03 |  $-$9:20:55.94 | 0.027 | 12 | 10
ZTF20aausahr | II | SN 2020hgm | 8:31:22.09 | +49:13:35.43 | 0.043 | 9 | 10
ZTF20aazcnrv | II | SN 2020jjj | 14:31:19.63 |  $-$25:39:31.04 | 0.023 | 12 | 13
ZTF20aazpphd | II | SN 2020jww | 16:10:51.58 | +27:09:42.02 | 0.046 | 36 | 43
ZTF20abekbzp | II | SN 2020meu | 15:34:44.68 | +6:38:53.35 | 0.041 | 9 | 11
ZTF20abuqali | II | SN 2020rht | 2:30:17.30 | +28:36:02.64 | 0.040 | 13 | 12
ZTF20abwdaeo$\dagger$$\dagger$footnotemark:  | II | SN 2020rvn | 21:06:36.34 | +17:59:34.86 | 0.021 | 28 | 29
ZTF20abyosmd$\dagger$$\dagger$footnotemark:  | II | SN 2020toc | 8:28:30.14 | +17:28:08.52 | 0.021 | 20 | 26
ZTF20acjqksf$\dagger$$\dagger$footnotemark:  | IIP | SN 2020tfb | 6:08:52.14 |  $-$26:24:46.02 | 0.048 | 17 | 18
ZTF20acnvtxy$\dagger$$\dagger$footnotemark:  | IIP | SN 2020zkx | 11:18:31.84 | +6:44:28.84 | 0.030 | 10 | 13
ZTF20acptgfl | IIP | SN 2020zjk | 5:22:00.02 |  $-$7:11:20.81 | 0.037 | 22 | 27
ZTF21aabygea$*$$*$The reported number of measurements in ztf–$g$ and ztf–$r$ obtained within 150 days of first detection, after running forced photometry for each ZTF event. Additional detections beyond 150 days not included in the analysis are quoted in parenthesis for each band. | II | SN 2021os | 12:02:54.08 | +5:36:53.15 | 0.019 | 31 | 49(2)
ZTF21aaevrjl | II | SN 2021arg | 4:31:18.78 |  $-$10:23:46.95 | 0.031 | 14 | 15
ZTF21aafkktu | II | SN 2021avg | 11:39:59.00 | +14:31:40.65 | 0.031 | 31 | 32
ZTF21aafkwtk | II | SN 2021apg | 13:41:19.24 | +24:29:43.88 | 0.027 | 26 | 37
ZTF21aagtqpn$\dagger$$\dagger$footnotemark: $*$$*$The reported number of measurements in ztf–$g$ and ztf–$r$ obtained within 150 days of first detection, after running forced photometry for each ZTF event. Additional detections beyond 150 days not included in the analysis are quoted in parenthesis for each band. | II | SN 2021bkq | 18:20:34.83 | +40:56:36.28 | 0.036 | 33(4) | 44(6)
ZTF21aaigdly$\dagger$$\dagger$footnotemark:  | II | SN 2021cdw | 14:05:31.80 |  $-$25:21:54.51 | 0.040 | 17 | 28
ZTF21aaluqkp | II | SN 2021dhx | 11:05:10.38 |  $-$15:21:10.13 | 0.025 | 34 | 25
ZTF21aamzuxi | II | SN 2021dvl | 7:49:56.10 | +71:15:42.11 | 0.034 | 27 | 29
ZTF21acchbmn | II | SN 2021zaa | 23:46:41.12 | +26:44:45.11 | 0.032 | 22 | 19

††footnotetext: No upper–limit constraints before first detection are
available for informed priors on explosion date.

We utilize point spread function (PSF)–fit photometry measurements using
difference imaging as provided by the ZTF forced-photometry service(IRSA,
2022). Complete light curves were constructed from the differential flux
measurements (forcediffimflux and forcediffimfluxunc) that the service returns
in each band along with upper limits. A signal-to-noise threshold (SNT) $=3$
and a signal-to-noise ratio (SNU) $=5$ were used to declare the measurements
as a detection versus an upper-limit (Masci et al., 2019). The photometric
zero-points in each passband were used to calculate the differential
magnitudes.

We use the redshift reported by the Transient Name Server (TNS) (Gal-Yam,
2021) to calculate the distance modulus using astropy (Astropy Collaboration
et al., 2013, 2018) for every ZTF event assuming standard flat $\Lambda$CDM
cosmology model with $H_{0}=70\>\text{km}\>\>\text{Mpc}^{-1}\>\text{s}^{-1}$
and $\Omega_{0}=0.3$. The measurements are corrected for Milky Way extinction
using dust maps as prescribed by Schlegel et al. 1998 for each passband,
assuming a $R_{V}=3.1$. In our analysis, for each ZTF event, we only use
measurements up to 150 days in the rest-frame from the first detection. We do
not account for cosmological K-corrections in this work since we use a sample
of low-redshift events. However, because LSST is expected to discover events
at higher redshifts, K-corrections for a similar analysis of LSST events would
be non-negligible.

##  3 Model Fitting

###  3.1 Model grid using STELLA

The hydrodynamic models used in this work are specific to Type IIP,
constructed using the multi-group radiation hydrodynamics code STELLA
(Blinnikov et al., 1998, 2000, 2006; Moriya et al., 2017, 2018; Ricks &
Dwarkadas, 2019). In this work, our models have the following parameters: ZAMS
mass, kinetic energy of explosion ($E_{k}$), mass-loss rate ($\dot{M}$),
steepness of velocity law ($\beta$) associated with the stellar wind, and 56Ni
mass synthesized. The model parameters along with their corresponding values
in the grid are described in Table 2. Red supergiant (RSG) pre-supernova
progenitors from Sukhbold et al. (2016) were used, which were calculated using
the KEPLER code (Weaver et al., 1978), with physics previously discussed
(e.g., Woosley et al. 2002). A neutron star remnant mass of 1.4
$\rm{M}_{\odot}$ is assumed and the SN explosions are triggered by putting
thermal energy above the mass cut. 56Ni is assumed to be uniformly mixed up to
half of the hydrogen-rich envelope in the mass coordinate.

Table 2: Prior Distribution on physical parameters used in our sampling method
along with values of parameters in our hydrodynamical model grid.

Parameter | Hydrodynamical Model Values | In steps | Prior Distribution | Units
---|---|---|---|---
$t_{\text{exp}}$ | – | – | $U(0,t_{\text{upper-limit}})$ | day
ZAMS | 12 – 16 | 2 | $N(14,3)\in(12,16)$ | $M_{\odot}$
$E_{k}$ | 0.5 – 5.0 | 0.5 | $N(1,1)\in(0.5,5)$ |  $10^{51}$ erg
56Ni | 0.01 – 0.1 | 0.01 (0.001,0.2,0.3) | $N(0.05,0.01)\in(0.001,0.3)$ | $M_{\odot}$
$\beta$ | 1 – 5 | 1 | $N(3,2)\in(1,5)$ | –
$-\text{log}_{10}\dot{M}$ | 1 – 5 | 0.5 | $U(4,2)\in(5,1)$ |  $M_{\odot}$ $\text{yr}^{-1}$
$A_{V}$ | – | – | $N(\text{ln}(0.05),2)\in(10^{-4},2)$ | mag

Note. — The “In steps” values in parentheses for 56Ni are the additional
values for the parameters present in the model grid.

Our hydrodynamical model grid also incorporates an associated circumstellar
material (CSM) density structure attached to these RSG progenitors. The CSM
density is given by

$\rho_{CSM}(r)=\frac{\dot{M}}{4\pi v_{wind}r^{2}},$ (1)

where $\dot{M}$ is the mass-loss rate and $v_{wind}$ is the velocity structure
associated with the stellar wind. The radial dependency of $v_{wind}$ is given
by the velocity law

$v_{wind}(r)=v_{0}+\left(v_{\infty}-v_{0}\right)\left(1-\frac{R_{0}}{r}\right)^{\beta},$
(2)

where $v_{0}$ is the initial wind velocity with a value $<0.01$ km s-1,
$v_{\infty}$ is the terminal wind velocity $=10$ km s-1, $R_{0}$ is the wind
launching radius set to the photosphere of the star, and $\beta$ is the
steepness parameter that gives a measure of wind acceleration. RSG progenitors
typically have $\beta>2$, owing to slower acceleration of the stellar winds
(Mauron & Josselin, 2011). A fixed CSM radius of $10^{15}$ cm is assumed in
our models.

The RSG progenitors with associated CSM density structures were then exploded
as thermal bombs with energies ranging from $0.5$ – $5.0\times 10^{51}$ erg
using STELLA. The code calculates the resulting light curve of the explosion
following the evolution of spectral energy distributions (SEDs) with time at
every epoch. The light curves in various passbands are obtained by convolving
the ZTF filter transmission functions to the numerical SEDs. Our final grid of
varying progenitor, explosion, and CSM properties is made up of 4206 unique
models. A similar model grid used in this work can be found in Förster et al.
(2018) for further reference.  The full details of the numerical model grid
will be presented in a separate paper (T.J. Moriya, in prep.).

###  3.2 Fast Interpolation Method

Using Bayesian Inference methods requires the models to be finely sampled
within the parameter space. However, because our model grid is neither
complete nor uniform, a scale–independent fast interpolation process that can
use a non–uniform grid of models was incorporated into our Monte Carlo
sampling method following Förster et al. (2018) and Martinez et al. (2020).
This allowed us to quickly interpolate between the models in the grid to
sample any combination of values in the parameter space. For a given parameter
vector $\vec{\theta}$, the method finds the closest models
$\vec{\theta}_{close}$ and weighs them appropriately using

$m(t,\vec{\theta})=\sum_{\vec{\theta}_{\it{i}}\in\vec{\theta}_{\textit{close}}}\hat{w}(\vec{\theta},\vec{\theta}_{\it{i}})m(t,\vec{\theta}_{\it{i}}),$
(3)

where $m(t,t_{\text{exp}},\vec{\theta})$ is the magnitude for a given
$\vec{\theta}$ at time $t$ and the normalized weights are given by

$\hat{w}(\vec{\theta},\vec{\theta}_{\it{i}})=\frac{w(\vec{\theta},\vec{\theta}_{\it{i}})}{\sum_{\vec{\theta}_{\it{j}}\in\vec{\theta}_{\text{close}}}w(\vec{\theta},\vec{\theta}_{\it{j}})},$
(4)

where

$w(\vec{\theta},\vec{\theta}_{\it{i}})=\Big{(}\prod_{\it{j}}\big{|}\theta^{\it{j}}-\theta_{\it{i}}^{\it{j}}\big{|}+\delta^{\it{j}}\Big{)}^{-1}.$
(5)

Equation 5 uses a very small vector $\vec{\delta}$ with the same units as
$\vec{\theta}$, which ensures that the weights do not diverge when a given
parameter combination exactly matches a model in the grid. This fast
interpolation method can be used to calculate light curves for any combination
of parameter vector $\theta$ bound by the limits of our model grid.

###  3.3 Explosion date and host extinction

Along with fitting for the parameters of our models, we also fit for the
explosion date and host extinction. The time of explosion is calculated as the
number of days before the first photometric measurement in each passband. We
defined an informed prior distribution for the time of explosion leveraging
the constraints given by the upper limits in each passband for each event. The
priors are more constraining if upper limits preceding the first detection are
well defined. To calculate the approximate bounds of time of explosion priors,
the deepest upper limit available before the first detection is identified. If
no upper-limits are available for an event, we allow a wider distribution for
the prior.

Observed Type II SN light curves can be significantly affected by host
extinction (Kasen & Woosley, 2009; Mattila et al., 2012; Kochanek et al.,
2012). We fit for total extinction in the visual band assuming a prior
distribution listed in Table 2. We do this by simultaneously fitting ztf–$g$
and ztf–$r$ passbands to infer host extinction. The derived extinction is used
to calculate $E(B-V)$ for the host of each event assuming $R_{V}$ = 3.1.

###  3.4 Nested Sampling Methods for Parameter Inference

We derive posterior distributions of the parameters involved using the python
based MIT-licensed Dynamic Nested Sampling package dynesty (Skilling, 2004)
that estimates the Bayesian target distributions. We assigned a combination of
uniform and Gaussian distributions as our priors for different parameters. The
prior distributions considered in this work are listed in Table 2. The
likelihood function incorporates fast interpolation and evaluates how close
the observations are with a sample model drawn. Using the above framework,
multi-band ZTF observations in ztf–$g$ and ztf–$r$ passbands in their absolute
magnitudes are fit to the hydrodynamical models.

Nested sampling methods require all the samples to be identically and
independently distributed (i.i.d) random variables drawn from the prior
distribution. We use uniform sampling method widely used for dimensions $<10$
in the dynesty package (Skilling, 2006). In the Bayesian framework all of the
inference is contained in the final multi-dimensional posterior, which can be
marginalised over each parameter to obtain constraints. The Bayesian evidence
is represented by the overall normalisation of this posterior. The nested
sampling algorithm converges until the evidence has been estimated to a
desired accuracy after accepting or rejecting the samples that are drawn from
the prior distribution.

Table 3:  Median values from the posterior distribution with 1–$\sigma$ uncertainty when fitted with all data from the events.  ZTF ID | ZAMS | $E_{k}$ | -log${}_{10}\dot{M}$ | MNi56 | $t_{exp}$ | $\beta$ | $A_{V}$
---|---|---|---|---|---|---|---
| ($M_{\odot}$) | ($10^{51}$ ergs) | ($M_{\odot}\,\text{yr}^{-1}$) | ($M_{\odot}$) | (days) |  | (mag)
ZTF18abcpmwh | $15.77_{-1.39}^{+0.20}$ | $1.67_{-0.29}^{+0.35}$ | $4.09_{-0.40}^{+0.38}$ | $0.07_{-0.02}^{+0.02}$ | $14.81_{-1.58}^{+1.88}$ | $2.56_{-0.27}^{+1.20}$ | $0.23_{-0.10}^{+0.11}$
ZTF18acbvhit$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $13.35_{-0.91}^{+1.68}$ | $0.51_{-0.01}^{+0.02}$ | $4.00_{-0.09}^{+0.09}$ | $0.04_{-0.01}^{+0.01}$ | $25.54_{-4.04}^{+3.93}$ | $3.98_{-0.10}^{+0.09}$ | $0.31_{-0.09}^{+0.09}$
ZTF18acbwasc$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $12.47_{-0.31}^{+0.40}$ | $0.52_{-0.01}^{+0.01}$ | $4.22_{-0.36}^{+0.35}$ | $0.10_{-0.02}^{+0.005}$ | $48.48_{-1.79}^{+1.06}$ | $3.06_{-0.20}^{+0.17}$ | $0.02_{-0.01}^{+0.02}$
ZTF18acrtvmm | $12.86_{-0.50}^{+0.80}$ | $0.99_{-0.05}^{+0.04}$ | $2.43_{-0.25}^{+0.15}$ | $0.10_{-0.03}^{+0.04}$ | $8.81_{-1.00}^{+0.75}$ | $3.64_{-1.22}^{+0.14}$ | $0.03_{-0.02}^{+0.03}$
ZTF18acuqskr | $13.89_{-0.98}^{+1.02}$ | $2.90_{-0.39}^{+0.30}$ | $3.91_{-0.45}^{+0.47}$ | $0.04_{-0.02}^{+0.02}$ | $16.72_{-2.38}^{+2.40}$ | $3.01_{-0.45}^{+0.46}$ | $0.06_{-0.04}^{+0.07}$
ZTF19aakiyed | $14.85_{-1.74}^{+1.04}$ | $1.02_{-0.20}^{+0.21}$ | $3.75_{-0.54}^{+0.52}$ | $0.06_{-0.02}^{+0.02}$ | $17.29_{-3.09}^{+3.22}$ | $2.93_{-0.53}^{+0.82}$ | $0.10_{-0.07}^{+0.10}$
ZTF19aaqdkrm | $13.14_{-0.81}^{+2.52}$ | $1.08_{-0.28}^{+0.25}$ | $1.67_{-0.21}^{+0.25}$ | $0.05_{-0.01}^{+0.02}$ | $16.68_{-2.88}^{+1.95}$ | $3.06_{-1.20}^{+0.89}$ | $0.12_{-0.08}^{+0.11}$
ZTF19aauqwna | $13.71_{-1.01}^{+1.17}$ | $1.19_{-0.31}^{+0.30}$ | $1.32_{-0.20}^{+0.26}$ | $0.08_{-0.02}^{+0.02}$ | $16.29_{-3.80}^{+6.28}$ | $3.03_{-0.50}^{+0.55}$ | $0.05_{-0.03}^{+0.07}$
ZTF19aavbkly | $13.07_{-0.75}^{+2.09}$ | $1.06_{-0.21}^{+0.43}$ | $2.43_{-0.25}^{+0.18}$ | $0.02_{-0.01}^{+0.01}$ | $7.27_{-0.82}^{+0.51}$ | $3.00_{-0.10}^{+0.10}$ | $0.18_{-0.10}^{+0.25}$
ZTF19aavhblr | $13.46_{-0.97}^{+1.36}$ | $0.86_{-0.19}^{+0.28}$ | $1.09_{-0.04}^{+0.41}$ | $0.07_{-0.01}^{+0.02}$ | $24.13_{-10.53}^{+4.25}$ | $2.98_{-0.47}^{+0.49}$ | $0.05_{-0.04}^{+0.09}$
ZTF19aavkptg | $13.14_{-0.80}^{+1.97}$ | $1.06_{-0.26}^{+0.45}$ | $2.22_{-0.20}^{+0.30}$ | $0.03_{-0.02}^{+0.02}$ | $8.21_{-1.40}^{+1.09}$ | $2.95_{-0.50}^{+0.71}$ | $0.38_{-0.17}^{+0.25}$
ZTF19abguqsi | $12.57_{-0.40}^{+1.02}$ | $1.24_{-0.45}^{+0.73}$ | $1.51_{-0.04}^{+0.06}$ | $0.01_{-0.01}^{+0.01}$ | $13.71_{-1.34}^{+0.88}$ | $3.00_{-0.44}^{+0.45}$ | $0.17_{-0.10}^{+0.19}$
ZTF19abhduuo$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $12.95_{-0.59}^{+0.85}$ | $0.52_{-0.02}^{+0.05}$ | $4.00_{-0.08}^{+0.08}$ | $0.03_{-0.01}^{+0.01}$ | $51.02_{-6.87}^{+5.20}$ | $3.97_{-0.36}^{+0.47}$ | $0.79_{-0.11}^{+0.17}$
ZTF19abiahko | $13.11_{-0.71}^{+0.92}$ | $0.75_{-0.21}^{+0.32}$ | $3.99_{-0.10}^{+0.11}$ | $0.02_{-0.01}^{+0.01}$ | $25.15_{-5.27}^{+7.76}$ | $3.99_{-0.10}^{+0.10}$ | $0.54_{-0.36}^{+0.23}$
ZTF19abqyouo$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $12.98_{-0.70}^{+1.36}$ | $1.39_{-0.23}^{+0.15}$ | $3.95_{-0.70}^{+0.58}$ | $0.06_{-0.02}^{+0.02}$ | $31.37_{-3.42}^{+3.16}$ | $3.00_{-0.84}^{+0.85}$ | $0.09_{-0.06}^{+0.09}$
ZTF19abvbrve${\ddagger}$${\ddagger}$Fits to explosion energies are very close to the model grid parameter boundaries. | $12.85_{-0.59}^{+1.59}$ | $0.51_{-0.01}^{+0.02}$ | $3.56_{-0.25}^{+0.57}$ | $0.06_{-0.02}^{+0.02}$ | $8.27_{-1.16}^{1.47}$ | $2.82_{-0.43}^{+0.89}$ | $0.13_{-0.07}^{+0.09}$
ZTF19acbvisk | $13.82_{-0.86}^{+0.77}$ | $0.54_{-0.03}^{+0.07}$ | $1.63_{-0.41}^{+0.41}$ | $0.10_{-0.02}^{+0.02}$ | $20.02_{-6.41}^{+9.31}$ | $3.11_{-0.49}^{+0.48}$ | $0.02_{-0.02}^{+0.03}$
ZTF19ackjvtl$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $14.43_{-1.26}^{+1.05}$ | $0.51_{-0.01}^{+0.02}$ | $3.88_{-0.80}^{+0.71}$ | $0.06_{-0.00}^{+0.01}$ | $57.60_{-1.99}^{+2.22}$ | $2.47_{-0.83}^{+1.37}$ | $0.40_{-0.07}^{+0.07}$
ZTF19acmwfli | $13.34_{-0.97}^{+2.19}$ | $1.01_{-0.04}^{+0.06}$ | $3.82_{-0.79}^{+0.71}$ | $0.04_{-0.01}^{+0.01}$ | $24.75_{-4.51}^{+2.12}$ | $2.69_{-0.21}^{+1.07}$ | $0.21_{-0.08}^{+0.10}$
ZTF19acszmgx | $13.21_{-0.86}^{+1.40}$ | $1.39_{-0.40}^{+0.42}$ | $1.03_{-0.03}^{+0.07}$ | $0.08_{-0.02}^{+0.02}$ | $19.37_{-0.88}^{+0.45}$ | $2.98_{-0.47}^{+0.50}$ | $0.43_{-0.36}^{+0.25}$
ZTF20aahqbun | $15.39_{-2.27}^{+0.51}$ | $0.58_{-0.04}^{+0.05}$ | $3.01_{-0.48}^{+1.11}$ | $0.06_{-0.01}^{+0.01}$ | $27.73_{-2.11}^{+1.51}$ | $2.55_{-0.18}^{+1.20}$ | $0.17_{-0.07}^{+0.07}$
ZTF20aamlmec | $13.53_{-1.01}^{+1.48}$ | $1.00_{-0.26}^{+0.38}$ | $2.37_{-0.40}^{+0.72}$ | $0.03_{-0.02}^{+0.02}$ | $22.35_{-3.56}^{+3.76}$ | $2.99_{-0.49}^{+0.53}$ | $0.25_{-0.16}^{+0.20}$
ZTF20aamxuwl | $13.48_{-1.00}^{+1.49}$ | $1.03_{-0.28}^{+0.37}$ | $1.78_{-0.39}^{+0.46}$ | $0.06_{-0.01}^{+0.02}$ | $14.41_{-3.18}^{+4.21}$ | $3.05_{-0.54}^{+0.57}$ | $0.08_{-0.06}^{+0.11}$
ZTF20aatqgeo | $12.60_{-0.43}^{+1.67}$ | $0.92_{-0.25}^{+0.15}$ | $2.02_{-0.06}^{+0.10}$ | $0.06_{-0.02}^{+0.02}$ | $18.60_{-1.54}^{+0.97}$ | $2.76_{-0.40}^{+0.94}$ | $0.29_{-0.13}^{+0.09}$
ZTF20aatqidk | $13.81_{-1.04}^{+1.21}$ | $0.61_{-0.04}^{+0.05}$ | $2.67_{-0.26}^{+0.37}$ | $0.10_{-0.04}^{+0.04}$ | $16.88_{-2.26}^{+2.47}$ | $3.52_{-1.11}^{+0.26}$ | $0.03_{-0.02}^{+0.04}$
ZTF20aaullwz | $14.01_{-0.69}^{+0.73}$ | $1.03_{-0.08}^{+0.38}$ | $2.73_{-0.28}^{+0.33}$ | $0.08_{-0.02}^{+0.01}$ | $10.65_{-1.94}^{+1.76}$ | $3.02_{-0.35}^{+0.37}$ | $0.04_{-0.03}^{+0.10}$
ZTF20aausahr | $13.74_{-1.02}^{+1.18}$ | $2.38_{-0.34}^{+0.46}$ | $3.69_{-1.01}^{+0.74}$ | $0.03_{-0.02}^{+0.02}$ | $27.38_{-3.29}^{+3.29}$ | $3.00_{-0.47}^{+0.48}$ | $0.12_{-0.08}^{+0.12}$
ZTF20aazcnrv | $14.80_{-1.61}^{+1.02}$ | $1.26_{-0.39}^{+0.51}$ | $3.78_{-0.52}^{+0.51}$ | $0.02_{-0.01}^{+0.01}$ | $19.69_{-2.43}^{+2.65}$ | $3.94_{-0.27}^{+0.57}$ | $0.35_{-0.20}^{+0.21}$
ZTF20aazpphd | $13.73_{-0.90}^{+0.82}$ | $1.00_{-0.02}^{+0.03}$ | $1.86_{-0.15}^{+0.33}$ | $0.10_{-0.02}^{+0.03}$ | $13.94_{-1.60}^{+1.67}$ | $3.08_{-0.55}^{+0.45}$ | $0.02_{-0.01}^{+0.03}$
ZTF20abekbzp | $13.62_{-1.05}^{+1.38}$ | $1.54_{-0.46}^{+0.75}$ | $3.19_{-0.78}^{+1.08}$ | $0.02_{-0.02}^{+0.02}$ | $15.27_{-3.34}^{+2.84}$ | $2.99_{-0.49}^{+0.51}$ | $0.28_{-0.18}^{+0.27}$
ZTF20abuqali | $13.81_{-1.22}^{+1.62}$ | $1.01_{-0.15}^{+0.18}$ | $2.41_{-0.33}^{+0.31}$ | $0.08_{-0.02}^{+0.02}$ | $17.38_{-2.96}^{+3.00}$ | $3.00_{-0.56}^{+0.75}$ | $0.07_{-0.05}^{+0.08}$
ZTF20abwdaeo | $15.22_{-1.95}^{+0.71}$ | $0.57_{-0.03}^{+0.03}$ | $2.55_{-0.27}^{+0.26}$ | $0.09_{-0.02}^{+0.01}$ | $20.09_{-2.49}^{+2.75}$ | $3.50_{-1.08}^{+0.28}$ | $0.03_{-0.02}^{+0.03}$
ZTF20abyosmd$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $12.72_{-0.51}^{+1.61}$ | $0.97_{-0.25}^{+0.08}$ | $4.00_{-0.44}^{+0.44}$ | $0.06_{-0.01}^{+0.01}$ | $47.15_{-3.01}^{+1.89}$ | $2.97_{-0.49}^{+0.55}$ | $1.03_{-0.11}^{+0.09}$
ZTF20acjqksf$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $12.83_{-0.59}^{+1.62}$ | $1.59_{-0.45}^{+0.63}$ | $4.05_{-0.63}^{+0.58}$ | $0.03_{-0.02}^{+0.02}$ | $28.34_{-1.99}^{+1.16}$ | $2.98_{-0.47}^{+0.49}$ | $0.40_{-0.22}^{+0.24}$
ZTF20acnvtxy$\S$$\S$Events with relatively less confident inferences due to poor data quality (including missing phases of light curve along with no constraints on upper limits) | $13.32_{-0.89}^{+1.49}$ | $1.38_{-0.49}^{+0.59}$ | $3.89_{-0.76}^{+0.65}$ | $0.03_{-0.02}^{+0.02}$ | $54.43_{-5.08}^{+3.58}$ | $3.00_{-0.48}^{+0.44}$ | $0.98_{-0.26}^{+0.28}$
ZTF20acptgfl${\ddagger}$${\ddagger}$Fits to explosion energies are very close to the model grid parameter boundaries. | $12.94_{-0.41}^{+0.42}$ | $0.52_{-0.01}^{+0.02}$ | $3.97_{-0.11}^{+0.15}$ | $0.05_{-0.01}^{+0.01}$ | $19.41_{-2.55}^{+2.76}$ | $3.00_{-0.10}^{+0.10}$ | $0.08_{-0.05}^{+0.07}$
ZTF21aabygea | $15.96_{-0.19}^{+0.03}$ | $0.57_{-0.02}^{+0.02}$ | $3.76_{-0.14}^{+0.18}$ | $0.05_{-0.02}^{+0.02}$ | $9.84_{-0.98}^{+0.87}$ | $3.75_{-0.05}^{+0.05}$ | $0.02_{-0.01}^{+0.02}$
ZTF21aaevrjl | $13.72_{-1.05}^{+1.19}$ | $1.42_{-0.60}^{+0.76}$ | $4.00_{-0.10}^{+0.10}$ | $0.02_{-0.01}^{+0.02}$ | $14.82_{-2.54}^{+3.13}$ | $3.00_{-0.09}^{+0.10}$ | $0.81_{-0.32}^{+0.30}$
ZTF21aafkktu | $14.35_{-1.29}^{+1.21}$ | $0.56_{-0.05}^{+0.09}$ | $2.11_{-0.40}^{+0.47}$ | $0.09_{-0.01}^{+0.01}$ | $8.56_{-2.31}^{+3.54}$ | $2.66_{-0.33}^{+1.09}$ | $0.04_{-0.03}^{+0.05}$
ZTF21aafkwtk${\ddagger}$${\ddagger}$Fits to explosion energies are very close to the model grid parameter boundaries. | $14.36_{-0.85}^{+1.12}$ | $0.51_{-0.00}^{+0.01}$ | $2.75_{-0.20}^{+0.35}$ | $0.04_{-0.01}^{+0.01}$ | $13.52_{-2.11}^{+1.84}$ | $2.78_{-0.40}^{+0.74}$ | $0.05_{-0.03}^{+0.04}$
ZTF21aagtqpn | $12.11_{-0.08}^{+0.17}$ | $1.44_{-0.10}^{+0.10}$ | $4.04_{-0.45}^{+0.45}$ | $0.10_{-0.002}^{+0.002}$ | $19.70_{-1.40}^{+1.21}$ | $2.50_{-0.30}^{+1.23}$ | $0.04_{-0.03}^{+0.04}$
ZTF21aaigdly | $14.64_{-1.76}^{+1.24}$ | $0.67_{-0.07}^{+0.13}$ | $2.79_{-0.26}^{+0.43}$ | $0.09_{-0.06}^{+0.07}$ | $10.60_{-2.14}^{+2.38}$ | $3.71_{-1.24}^{+0.08}$ | $0.05_{-0.04}^{+0.06}$
ZTF21aaluqkp | $15.14_{-2.06}^{+0.77}$ | $0.54_{-0.02}^{+0.04}$ | $3.70_{-0.42}^{+0.49}$ | $0.09_{-0.03}^{+0.01}$ | $16.11_{-2.75}^{+2.74}$ | $2.60_{-0.19}^{+1.16}$ | $0.07_{-0.04}^{+0.06}$
ZTF21aamzuxi | $14.48_{-1.36}^{+1.25}$ | $1.49_{-0.09}^{+0.08}$ | $3.72_{-0.51}^{+0.49}$ | $0.09_{-0.06}^{+0.05}$ | $7.85_{-1.31}^{+1.41}$ | $2.56_{-0.23}^{+1.20}$ | $0.05_{-0.04}^{+0.05}$
ZTF21acchbmn | $14.14_{-1.35}^{+1.62}$ | $1.51_{-0.08}^{+0.13}$ | $2.17_{-0.24}^{+0.32}$ | $0.09_{-0.02}^{+0.01}$ | $8.96_{-0.89}^{+0.66}$ | $3.32_{-0.92}^{+0.46}$ | $0.05_{-0.03}^{+0.05}$
Figure 1: The observed ZTF light curves and our model fits plotted with
respect to derived time of explosion . The family of light curves in each
panel represents 150 models randomly sampled from the derived posterior
probability distribution in individual bands respectively. The dashed curve
represents the best fit model for each ZTF event. The upper limits before
first detection that are used to constrain explosion date are plotted. The
median values and their $1-\sigma$ uncertainity for the parameters are listed
in Table 3. ZTF events with no upper limits and poorly constraining rise data
are flagged with $\S$. ZTF events whose fits approached model boundaries are
flagged with ${\ddagger}$. Additional plots for the remaining ZTF events
listed in Table 1 are provided in the Appendix.

##  4 Parameter Inference and Trends of Complete Light Curves

Using our Bayesian inference procedure to fit all ZTF events with our model
grid, we derived the posterior probability distribution for all parameters of
each Type II supernova. Our models were fit confidently to 34 events. The fits
for the remaining 11 events were limited by poor upper limits prior to first
detection (flagged with $\S$) and the hydrodynamical model grid boundaries
(flagged with ${\ddagger}$). Figure 1 shows the samples of the posterior
probability distributions for the first 15 ZTF events. The light curve fits
for the remaining events are plotted in an extended version of Figure 1 in
Appendix. The observed data points are shifted with respect to the inferred
explosion date and corrected for host extinction derived from the fits. The
uncertainity in host extinction and explosion date is not represented in
Figure 1. The dashed line in each passband represents the best fit model.
Table 3 lists the inferred median values for the seven parameters with their
16th and 84th percentile confidence regions. We note that these $1-\sigma$
uncertainities of the parameters only reflect how well the model fits the
observed data and do not take into account any uncertainties related to the
assumptions used to create the model grid itself. Representative corner plots
of posterior probability distributions are shown in the Appendix.

Figure 2: Top panel: ZAMS mass plotted against 56Ni mass, mass–loss rate, and
kinetic energy. The blue and grey circles are values that were found for SNe
II in Förster et al. 2018 and Martinez et al. (2020). Bottom panel: Kinetic
energy plotted against time of explosion with respect to first detection, 56Ni
mass, and host extinction ($A_{V}$). The grey dotted lines represents the
parameter bounds of our model grid. The purple circle in the bottom left panel
demarcates events that required a narrower prior distribution for the
explosion date constrained either from upper limits or rise-time data. The
green circles represent flagged ZTF events with low confidence model fits.

ZTF21aabygea (SN 2021os) and ZTF18abcpmwh (SN 2018cur) yielded the highest
ZAMS masses ($15.96^{+0.03}_{-0.19}$ M⊙ and $15.77^{+0.20}_{-1.39}$ M⊙,
respectively), while ZTF21aagtqpn (SN 2021bkq) has the lowest
($12.11^{+0.17}_{-0.08}$ M⊙). The highest explosion energies are seen for
ZTF18acuqskr (SN 2018jrb) and ZTF20aausahr (SN 2020hgm)
($2.90^{+0.30}_{-0.39}\times 10^{51}$ erg and $2.38^{+0.46}_{-0.38}\times
10^{51}$ erg , respectively). Shown in Figure 1, these two events correspond
to steeper and early declines in the light curves as compared to low energetic
events ($\sim 0.5\times 10^{51}$ erg ) including ZTF19ackjvtl (SN 2019uwd) and
ZTF19acbvisk (SN 2019rms). These observed short-lived plateaus and fast
declines are in agreement with previous results for other high energy CCSNe
(Valenti et al., 2016; Rubin & Gal-Yam, 2017; de Jaeger et al., 2019; Barker
et al., 2021). The more energetic events in our sample also show increased
peak luminosity in both ztf–$g$ and ztf–$r$ bands (Galbany et al., 2016;
Valenti et al., 2016; Sanders et al., 2015).

For seven events, our estimates of kinetic energy favor the minimum parameter
value of our grid (0.5 $\times 10^{51}$ erg). We flag these events as ones for
which our model fits are less confident, since the actual kinetic energy may
be significantly lower.

Most events (37 out of 45) in our sample tend to favor mass–loss rates between
$10^{-4.0}-10^{-2.0}\text{M}_{\odot}\text{yr}^{-1}$. However, a non-negligible
number of events (8 out of 45) yield higher mass-loss rate estimates $\leq
10^{-2.0}\text{M}_{\odot}\text{yr}^{-1}$. This finding is consistent with
Förster et al. (2018), who reported higher mass loss rates to be correlated
with early and steep rises in the light curves of many Type II SNe from the
High cadence Transient Survey (HiTS) (Förster et al., 2016), possibly due to
shock-breakout in dense CSM (Moriya et al., 2011; Morozova et al., 2015;
Moriya et al., 2017; Morozova et al., 2018; Moriya et al., 2018; Bruch et al.,
2021; Haynie & Piro, 2021). We find that the fits for the steepness parameter
$\beta$ are most likely prior dominated and favour values closer to $\beta\sim
3$, consistent with slowly accelerating winds found in red supergiants (Baade
et al., 1996).

ZTF19abguqsi (SN 2019lsh) produced the least 56Ni mass of $0.01\pm 0.01$
$\text{M}_{\odot}$. Other ZTF events in the sample have estimates of 56Ni mass
ranging from 0.02–0.1 $\text{M}_{\odot}$. ZTF19aabvkly (SN 2019fmv),
ZTF19abiahko (SN 2019lsj) and ZTF20aazcnrv (SN 2020jjj) have relatively short-
lived plateau regions in their light curves and are associated with lower
estimates of synthesized 56Ni mass (see Figure 1). Events with higher
estimates of 56Ni mass have long–lived plateaus as compared to the the events
with lower 56Ni mass estimates and faster declines in their light curves.
These results are in agreement with previous analyses of SNe Type II that
consider how 56Ni mass affects light curve evolution (Eastman et al., 1994;
Bersten, 2013; Faran et al., 2014; Kozyreva et al., 2019).

ZTF20abyosmd (SN 2020toc) has the highest host extinction
($A_{V}=1.03^{+0.19}_{-0.11}$ mag), followed by ZTF21aaevrjl (SN 2021arg)
($A_{V}=0.81^{+0.30}_{-0.32}$ mag). ZTF18acbwasc (SN 2018hfc) and ZTF21aabygea
(SN 2021os) have negligible host extinction values (both
$A_{V}=0.02^{+0.02}_{-0.01}$ mag). All ZTF events in our sample show host
extinction values ranging from $A_{V}=0.01-1.1$ mag, typical for CCSNe hosts
(Pastorello et al., 2006; Maguire et al., 2010; Faran et al., 2014).

###  4.1 Correlations between physical parameters

We plot our inferred values with comparison to other similar works in
literature (Förster et al., 2018; Martinez et al., 2020) in Figure 2. Förster
et al. (2018) in their work uses hydrodynamical models on HiTS Type II
supernovae that include estimates of circumstellar environment, while Martinez
et al. (2020) do not include CSM structure in their models. Figure 2 shows the
bounds of our model grid with grey dotted lines for each parameter in the
plot. The ZTF events whose energy value fits approached the bounds of the
model grid are marked in Table 3. We represent events that had relatively
narrower prior distribution for explosion date inside the purple circle in
Figure 2. These events had either constraining upper limits before first
detection or enough rise time data to make an educated guess on the upper
bound of prior distribution for explosion date.

Generally, our parameter estimates are less confident for ZTF events with poor
data quality, such as those missing phases of light curves combined with
unavailable upper limit constraints and whose parameter values approached the
parameter boundaries of the model grid (see Table 3). These events are
represented by green circles in Figure 2 as flagged events. Among 7 out of 11,
ZTF18acbwasc (SN 2018hfc), ZTF19abhduuo (SN 2019lre), ZTF19abqyouo (SN
2019pbk), ZTF19ackjvtl (SN 2019uwd), ZTF20abyosmd (SN 2020toc), ZTF20acjqksf
(SN 2020tfb) and ZTF20acnvtxy (SN 2020zkx) can be grouped together as events
that have higher values for time of explosion ($>$ 25 days). This could be
attributed to the fact that the prior distribution was too broad as there were
no upper limits for these events as provided by the ZTF survey. The remaining
4 ZTF events whose fits approached model boundaries are ZTF18acbvhit (SN
2018hle), ZTF19abvbrve (SN 2019puv), ZTF20acptgfl (SN 2020zjk), and
ZTF21aafkwtk (SN 2021apg).

Moreover, we performed a Pearson correlation analysis to check if we can find
any correlations between the physical parameters. However, we were unable to
find any significant correlations (r $>+0.5$ or r $<-0.5$) within the sample
studied. The correlation matrix is shown in Figure 3. The modest mass range in
our model grid (12 - 16 M⊙) limits our ability to make strong claims on any
potential ZAMS dependencies. For example, in Figure 2, the correlations in
Martinez et al. (2020) becomes only clear when the ZAMS range is extended to
10 M⊙. Our hydrodynamical grid explores new parameter spaces that include
mass-loss properties and this can possibly introduce additional degeneracies.

Figure 3: Pearson correlation matrix showing correlation coefficients that are
color-coded for different physical parameters in the analysis. No significant
correlations between physical parameters were found within the sample of Type
II SNe used in this study.

##  5 Real-time Parameter Evolution

We analyzed how each of the parameter fits evolved as a function of time,
termed real-time characterization. We were motivated to characterize how well
or poorly the model parameter fits and their uncertainties at fractional light
curve stages anticipated the values found from complete light curves. We
compared our model grid to three regimes of incomplete light curves with
respect to the first detection: 1) $\Delta t\leq 25$ days; 2) $\Delta t\leq
50$ days; 3) all available data.

Figure 4: Multi-epoch real-time characterization for ZTF19aaqdkrm (SN
2019dod). The left panel shows model fits to only data within first 25 days of
detection in each individual band. The middle panel shows model fits using
data within the first 50 days of detection. The right panel uses all data. The
light curves are plotted with respect to the derived time of explosion and
host extinction from model fits at each epoch. The table below shows parameter
estimates derived from fits at each epoch.

Figure 4 shows a detailed evolving characterization for the event ZTF19aaqdkrm
(SN 2019dod). As expected, the fits and estimates of the parameters change
with time as the supernova evolves and additional measurements are
incorporated into the fitting process. Within $\Delta t\leq 25$ days and
$\Delta t\leq 50$ days, the fits favor lower 56Ni masses and higher energies.
As data in both bands accumulate, the fits favor higher 56Ni masses and lower
energies. At $\Delta t\geq 50$ days, the portion of the light curve powered by
hydrogen recombination of the event starts to fall off, giving better
estimates on ZAMS and 56Ni masses.

We performed this same analysis on all 45 ZTF events, where similar trends are
seen. Figure 5 shows the difference in parameter values of our fits with
respect to the final epoch with all the data as the event unfolds. Our
analysis shows that the explosion energies and mass–loss rates for the events
are initially overestimated and tend to favor lower final values when all data
is included in the fit. This is opposite to what we see in case of 56Ni mass,
as it is underestimated with only few measurements and consistently favors
higher values for many events during later stages of light curve evolution as
the recombination drop–off starts to unfold.

The most confident estimate of 56Ni mass is inferred when the hydrogen
recombination phase ends and the radioactive decay phase starts. This results
in epochs with all the data yielding higher estimates of 56Ni mass. With
kinetic energy, the peak luminosity and decline rate of the light curve plays
a significant role in estimating the energetics of the event. As a result, as
more data become available at later epochs, more reliable estimates of kinetic
energy are seen. The inferred values of ZAMS, host extinction, explosion date
and the $\beta$ remain nearly constant as the light curve evolves.

Figure 5: Real-time parameter evolution for all the listed ZTF events. The
average change in parameter values (averaged over all 45 ZTF events) at each
epoch with respect to the final epoch with all data are represented with the
blue squares from top left to bottom right. The order of the parameters are as
follows: ZAMS mass, kinetic energy of explosion ($E_{k}$), mass-loss rate
($\dot{M}$), steepness of velocity law ($\beta$) associated with the stellar
wind, and 56Ni mass synthesized, explosion date with respect to first
detection, and host extinction ($\text{A}_{V}$).

##  6 Discussion

###  6.1 Parameter Space Degeneracy

A major challenge encountered when modelling only two passbands provided by
public ZTF light curves was model degeneracy. Specifically, the probability
that different combinations of explosion and progenitor parameters can
potentially lead to the same light curve posed difficulties in converging to a
unique solution in our fitting method. As noted in previous works, the
hydrodynamical modelling approach has led to larger estimates of progenitor
properties, especially ZAMS estimates, when compared to other approaches like
pre-SN explosion imaging (Utrobin & Chugai, 2008, 2009; Maguire et al., 2010;
Sanders et al., 2015). In the Appendix, posterior distribution of three ZTF
events at various epochs are examined using kernel density estimation (KDE)
analysis, along with a calculation of the number of modes. In a similar
analysis for all 45 events, we find that all the ZTF events have multi-modal
posteriors and temporal evolution of modes cannot be generalized.

Goldberg et al. (2019) recognize the challenges involved in breaking the
degeneracies between ejecta mass, explosion energy and progenitor radius, and
argue that in order to do so requires an independent measurement of one of the
parameters. The scaling relationships used in their work yield families of
explosions with varied parameters that can reproduce similar light curves.
Hillier & Dessart (2019) also highlight calculations of similar photospheric
phases for well-sampled Type II supernovae in their multi-band and
spectroscopic modelling.

Martinez et al. (2020) attempt to partially lift this degeneracy issue by
fitting photospheric velocity information from their models to velocity
measurements obtained of SNe during the plateau phase. However, Goldberg et
al. (2019) argue that only the ejecta velocities measured during the initial
shock cooling phase can be useful to break these degeneracies seen in the
parameters. Our analysis focused on photometry only, and future work can
investigate whether use of kinematic information from spectroscopy can better
constrain parameter selection.

###  6.2 Bolometric vs Pre-computed Multi-band Inference

Our analysis adopts the approach of fitting observations to synthetic multi-
band photometry derived from theoretical hydrodynamical models. Similar works
like Nicholl et al. (2017) and Guillochon et al. (2018) use semi-analytical,
black body spectral energy distribution (SED) models to fit multi-band
photometry for transients. These procedures contrast with typical methods that
first construct bolometric light curves from multi-filter and/or
multi–wavelength observations, that are in turn compared to model bolometric
light curves. Each method has associated uncertainties. In the case of
synthetic model photometry, uncertainties arise from assumptions in opacity
treatment at different frequencies in STELLA. In the case of creating
bolometric light curves from interpolated observed data sets, the
uncertainties stem from potential gaps in photometry cadence, limited
passbands, and potentially few data points overall to fit against.

For real time characterization of events, we found that a proper Bayesian
inferencing of explosion parameters for large numbers of supernovae is most
efficiently conducted with pre-computed grids of models. Computing models in
real-time per event will lead to duplicative efforts and incur computational
time costs. With our method, the fitting is more rapid, is more flexible, and
the associated uncertainties are less significant.

###  6.3 Intelligent Augmentation

Our analysis only uses ZTF public data in ztf–$g$ and ztf–$r$ passbands to
make inferences about Type II explosion parameters. Our parameter fits could
be further constrained with observations at other passbands, and it is
worthwhile to consider which passbands at which epochs are most constraining.
To this end, inferencing transients in real time in order to make on-the-fly
decisions about optimal follow-up in complementary passbands is needed
(Carbone & Corsi, 2020; Sravan et al., 2021). Generally, most constraining for
inferring Type II properties are observations at early and late phases of
light curve evolution. At early phases, UV observations best sample shock
breakout and circumstellar interaction (Gezari et al., 2015; Ganot et al.,
2016; Soumagnac et al., 2020; Haynie & Piro, 2021; Jacobson-Galán et al.,
2022). At late phases, near- and mid-infrared passbands provide diagnostics
that best follow ejecta cooling and dust formation (Szalai & Vinkó, 2013;
Bianco et al., 2014; Tinyanont et al., 2016). Optimally augmenting all-sky
survey photometry in real-time in this way can enhance opportunities to
generate large samples of core-collapse supernovae sufficiently observed to
perform population and host-environment studies (D’Andrea et al., 2010;
Anderson et al., 2014; Sanders et al., 2015; Schulze et al., 2021).

Figure 6: Spectroscopically classified Type II supernovae with anomalous light
curves identified in this work. Only the upper limits prior to the first
detection used for deriving the fits are plotted.

###  6.4 Anomaly Detection

Our work uncovered two examples of anomalous Type II supernovae, which are not
included in our sample of 45 events. We present the unusual light curves of
ZTF18acgvgiq (SN 2018fru) and ZTF20acwxrgp (SN 2020acjg) in Figure 6. The fits
for the entire light curves converged to a model solution with very poor
likelihood scores resulting in inaccurate inferences. Consequently, we were
unable to characterize these events with our current model grid. The anomalous
nature of these events could be identified via poor model fits as early as
$\Delta t<25$ days. The cumulative log-evidence (logZ) inferred for these two
events were above -200 indicating very poor likelihood estimates when
comparing the models with the data. For the events with good fits, the logZ
estimates were under -20. These high logZ values indicate that the models were
unable to converge to a target distribution of the parameters for the two
anomalous events.

This experience shows that real-time inferencing can be used as a way to
identify targets that deviate from normal theoretical predictions. Such real-
time analyses for detecting anomalies (Pruzhinskaya et al., 2019; Soraisam et
al., 2020; Villar et al., 2020; Ishida et al., 2021; Villar et al., 2021;
Martínez-Galarza et al., 2021) can be automated into an ORACLE such as REFITT
to motivate rapid spectroscopic follow–up of non-traditional CCSNe.

##  7 Conclusions

In this paper we have characterized 45 Type II supernovae using only products
from the public ZTF survey (i.e., in ztf-$g$ and ztf-$r$ passbands) using a
grid of theoretical hydrodynamical models. Our grid parameters span multiple
supernova progenitor and explosion properties, as well as the time of
explosion with respect to the first detection and host extinction. We compare
results between complete and fractional light curves to determine which
parameters are most robust to incomplete photometric data sets. This effort is
to assess whether opportunities exist for theoretically-driven forecasts to
inform when follow up observations are needed to support all-sky survey alert
streams. The following conclusions are made:

  * •

We obtain confident characterizations for 34 SNe II in our sample. Inferences
of the remaining 11 events are limited either by poorly constraining data or
the boundaries of our model grid. The properties of these well-fitted events
broadly follow those reported in previous analyses of SNe II.

  * •

In cases where fitted parameters derived from complete vs. incomplete data
sets are compared, some parameters are more reliably determined at early
epochs than others. The explosion energy, host extinction and mass-loss rate
parameters are overestimated during initial phases of evolution, while the
56Ni mass is underestimated. The ZAMS mass and $\beta$ estimates do not change
significantly at different phases.

  * •

The explosion date is a very sensitive parameter that requires
well–constrained pre-explosion upper–limits from the survey for confident
inferences. Generally, we found parameter estimates to be less reliable for
ZTF events with poor data quality, such as missing phases of the light curve
along with poor upper limit constraints.

  * •

Real-time Bayesian inferencing of progenitor and explosion parameters for
large numbers of CCSNe from all-sky surveys demands a pre-computed grid of
models. Creating synthetic model light curves in respective all-sky survey
passbands catalyzes real-time characterization of evolving transients by
avoiding challenges associated with constructing bolometric light curves with
sparse and incomplete photometry.

Our work has demonstrated that hydrodynamical model grids for CCSNe along with
statistical analyses can provide opportunities to enhance scientific return
from all-sky surveys that provide live alert streams. Theoretically-driven
predictions can be leveraged to efficiently coordinate worldwide observing
facilities to conduct follow up observations that augment survey light curves
to optimally achieve scientific objectives (Bianco et al., 2014; Modjaz et
al., 2019; Kennamer et al., 2020; Sravan et al., 2020; Anand et al., 2021).

For example, real-time characterization can identify and prioritize transients
that fall within certain parameter spaces of interest, including the extreme
high and low ends of kinetic energy or 56Ni mass. Likewise, theoretical
forecasts can identify and prioritize follow up photometry at critical phases
of transient evolution, including monitoring the plateau drop-off of SNe II
light curves that provides information needed to improve estimates of kinetic
energy and ZAMS and 56Ni masses. Ideally, predicting transient evolution using
the underlying physics of transients can be incorporated into a TOM or ORACLE
that can efficiently recommend targets for follow-up at information-rich
epochs (Djorgovski et al., 2016; Street et al., 2018; Kasliwal et al., 2019;
Sravan et al., 2020; Agayeva et al., 2021).

Our future work relies on an expanded grid of hydrodynamical models exploring
larger parameter ranges, including varying degrees of 56Ni mixing within the
inner layers of the progenitors, and information on photospheric velocity that
can be used to potentially break degeneracies between parameters. It will also
expand synthetic photometry to all six passbands of LSST. Although our work
focuses on Type II CCSNe, our methods can be easily applied to identify,
prioritize and coordinate follow-up of other transients discovered by Vera C.
Rubin Observatory.

## Acknowledgements

The authors would like to thank the anonymous referee for helpful comments
that have significantly improved this paper. We also acknowledge helpful
discussions with Thomas Matheson, Mariana Orellana, and Melina Bersten. The
ZTF forced-photometry service was funded under the Heising-Simons Foundation
grant #12540303 (PI: Graham). Numerical computations were in part carried out
on PC cluster at Center for Computational Astrophysics (CfCA), National
Astronomical Observatory of Japan. D. M. acknowledges NSF support from grants
PHY-1914448, PHY- 2209451, AST-2037297, and AST-2206532.

Figure 7: Continued from Figure 1.  Figure 8: Continued from Figure 1.  Figure
9: Continued from Figure 4 for event ZTF20abwdaeo (SN 2020rvn). Figure 10:
Continued from Figure 4 for event ZTF21aabygea (SN 2021os). Figure 11: Corner
plot showing the posterior probability distribution of various parameters for
the event ZTF20acptgfl (SN 2020zjk).

Figure 12: Corner plot showing the posterior probability distribution of
various parameters for the event ZTF20aausahr (SN 2020hgm).

Figure 13: Corner plot showing the posterior probability distribution of
various parameters for the event ZTF19abqyouo (SN 2019pbk).

##  Appendix A Multi-epoch Evolution of Posterior Distribution of Parameters

We performed a kernel density estimation (KDE) analysis in order to find the
modality of the posterior distributions at various epochs. The samples in the
posterior distribution were collected and smoothed using Silverman’s bandwidth
with a gaussian kernel. The KDE approximated distribution was then used to
calculate the number of modes at every epoch. The modes were found by
identifying inflection points in the distribution i.e positions where the
first derivative changes the sign.

We found that all the objects have multi-modal posteriors for at least one
parameter in our analysis. From this analysis, we conclude that the change in
the posteriors for physical parameters over different epochs cannot be
generalized for all the events. Figure 14 shows examples of multi-modal
posteriors for ZAMS, Kinetic Energy and 56Ni for three ZTF events The red
circles represent different modes found in the distribution using inflection
point analysis. We note that the modes of the distribution for kinetic energy
shift from higher to lower values as time proceeds in Figure 14 for each event
as discussed in the paper. The trend in 56Ni with time is reflected in the
modes with earlier epochs favouring lower values as compared with final
epochs. The degeneracies in parameter space as discussed in 6.1 are clearly
reflected in these distributions through multi-modality.

Figure 14: Kernel Density Estimates (KDE) of physical parameters along with
modes represented by red circles obtained at three epochs for three ZTF
events. The order of physical parameters from top to bottom row are as
follows: (a) ZAMS mass, (b) Kinetic Energy and (c) 56Ni mass.

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