# Embracing Uncertainty Flexibility: Harnessing a Supervised Tree Kernel to
Empower Ensemble Modelling for 2D Echocardiography-Based Prediction of Right
Ventricular Volume

Tuan A. Bohoran 1,*,${}^{\textrm{\Letter}}$ Polydoros N. Kampaktsis2  Laura
McLaughlin2  Jay Leb2  Serafeim Moustakidis3  Gerry P. McCann4  Archontis
Giannakidis1 1School of Science and Technology  Nottingham Trent University
Nottingham  UK.;
;
2Division of Cardiology<EMAIL_ADDRESS>Columbia University Irving
Medical Center  New York City  NY  USA.;
3AiDEAS  Tallinn  Estonia.;
4Department of Cardiovascular Sciences  University of Leicester and the NIHR
Leicester Biomedical Research Centre  Glenfield Hospital  Leicester  UK.;
* Authors contributed equally

###### Abstract

The right ventricular (RV) function deterioration strongly predicts clinical
outcomes in numerous circumstances. To boost the clinical deployment of
ensemble regression methods that quantify RV volumes using tabular data from
the widely available two-dimensional echocardiography (2DE), we propose to
complement the volume predictions with uncertainty scores. To this end, we
employ an instance-based method which uses the learned tree structure to
identify the nearest training samples to a target instance and then uses a
number of distribution types to more flexibly model the output. The
probabilistic and point-prediction performances of the proposed framework are
evaluated on a relatively small-scale dataset, comprising 100 end-diastolic
and end-systolic RV volumes. The reference values for point performance were
obtained from MRI. The results demonstrate that our flexible approach yields
improved probabilistic and point performances over other state-of-the art
methods. The appropriateness of the proposed framework is showcased by
providing exemplar cases. The estimated uncertainty embodies both aleatoric
and epistemic types. This work aligns with trustworthy artificial intelligence
since it can be used to enhance the decision-making process and reduce risks.
The feature importance scores of our framework can be exploited to reduce the
number of required 2DE views which could enhance the proposed pipeline’s
clinical application.

###### keywords:

uncertainty estimation, echocardiography, regression, machine learning, right
ventricle, instance-based learning, ensemble models.

##  1 Introduction

Right ventricular systolic (RV) dysfunction is a powerful and independent
mortality predictor [1] which may occur from a large variety of cardiovascular
disorders that result in inability to pump enough blood for oxygenation.
Machine learning methods have recently shown [2] great potential in
quantifying RV volumes using tabular data (such as area measurements, age,
gender and cardiac phase information) obtained from the widely available and
highly portable two-dimensional echocardiography (2DE). However, for clinical
deployment where patient safety is at stake, it is crucial to complement these
RV volume predictions with uncertainty scores that reflect the degree of trust
in these predictions.

The goal of this paper is to present an uncertainty quantification framework
when predicting RV volumes through the use of ensemble models, in particular
Gradient-Boosted Regression Trees (GBRTs), on 2DE-derived tabular data. GBRTs
are regarded [3] the method of choice for tabular data. To get an estimate of
the prediction uncertainty, we propose to make use of a $k$-nearest neighbour
method [4] that relies on a supervised tree kernel [5, 6]. Unlike other state-
of-the-art (SOTA) gradient-boosted algorithms [7, 8, 9] that provide
probabilistic predictions, this method performs well on both probabilistic and
point performances, and can also use a number of distribution types to more
flexibly model the output. It can also be applied to any GBRT model, adding
further flexibility.

The probabilistic and point-prediction performances of our framework are
evaluated on a relatively small-scale dataset, comprising 100 end-diastolic
(ED) and end-systolic (ES) RV volumes. The reference values for point
performance were obtained from cardiovascular MRI (CMR). Our pipeline is also
compared to other SOTA methods. Lastly, we provide conditional output
distributions and the respective confidence intervals for a couple of high and
low accuracy (test set) predictions.

##  2 Materials & Methods

###  2.1 Dataset

The study population was a retrospective cohort of 50 adult patients for which
2DE and CMR were acquired. Data acquisition and annotation were as described
in [2]. In brief, for each patient the RV endocardial-myocardial interface was
manually traced in end-systole and end-diastole (making a total of 100 data
points) for the following eight standardised echocardiographic views:
parasternal long axis (PLAX), right ventricular inflow (RV Inflow),
parasternal short axis at the level of the aortic valve (PSAX AV), basal (PSAX
Base), mid (PSAX mid) and apical left ventricular segments (PSAX Distal),
four-chamber (Four C) and subcostal (Sub C) views. The eight area measurements
along with the patient age were the numerical input variables of our model,
whereas the gender and cardiac phase information were the categorical ones.
The short-axis cine CMR-derived ED and ES RV volumes were recorded in a semi-
automated way and served as the reference values. The study was approved by
the Columbia University Irving Medical Center Institutional Review Board and
the Nottingham Trent University Ethics Committee.

###  2.2 Gradient-Boosted Regression Trees

Assume $D\coloneqq\\{(x_{i},y_{i})\\}_{i=1}^{n}$ is the training set where
$x_{i}=(x_{i}^{j})_{j=1}^{p}\in X\subseteq\mathbb{R}^{p}$ and $y_{i}\in
Y\subset\mathbb{R}$. Gradient-boosting [10] is a powerful machine learning
algorithm that constructs a model $f:X\rightarrow Y$ by relying on stage-wise
additive modelling and minimising the expected value of some empirical loss
function $L$. The model is obtained through the recursive relationship:
$f_{0}(x)=\gamma$, $\ldots$, $f_{t}(x)=f_{t-1}(x)+\eta\cdot m_{t}(x)$. In this
formulation, $f_{0}$ is the base learner, $\gamma$ is an initial
approximation, $f_{t}$ is the model at iteration $t$, $m_{t}$ represents the
weak learner added during iteration $t$ to boost the model, and $\eta$ is the
learning rate.

In the case of GBRTs, the most frequently employed $L$ is the mean squared
error (MSE), $\gamma$ is chosen as the average outcome of the training
instances ($\frac{1}{n}\sum_{i=1}^{n}y_{i}$), and regression trees represent
the weak learners. The decision tree at iteration $t$ is chosen to approximate
the residual (or else the negative derivative of the loss function with
respect to $\hat{y_{i}}=f_{t-1}(x_{i})$),
$m_{t}=\arg\min_{m}\frac{1}{n}\sum_{i=1}^{n}(-g_{i,t}-m(x_{i}))^{2}$, where
$g_{i,t}=\frac{\partial L(y_{i},\hat{y_{i}})}{\partial\hat{y_{i}}}$ is the
functional gradient of the $i$-th training instance at iteration $t$. The
decision tree at iteration $t$ recursively creates $M_{t}$ disjoint regions
$\\{r_{j}^{t}\\}_{j=1}^{M_{t}}$ through partitioning the instance space. Each
of these regions is termed a leaf. The parameter value $\theta_{t}^{j}$ for
leaf $j$ at tree $t$ is commonly determined (given a fixed structure) through
a one-step Newton method: $\theta_{t}^{j}=-\frac{\sum_{i\in
I_{t}^{j}}g_{t}^{i}}{(\sum_{i\in I_{t}^{j}}h_{t}^{i}+\lambda)}$, where
$I_{t}^{j}=\\{(x_{i},y_{i})\>|\>x_{i}\in r_{j}^{t}\\}_{i=1}^{n}$ is the
instance set of leaf $j$ for tree $t$, $h_{i}^{t}$ is the second derivative of
the $i$-th training instance with respect to $\hat{y_{i}}$, and $\lambda$ acts
as a regularization parameter. Hence, $m_{t}$ can be denoted as:
$m_{t}(x)=\sum_{j=1}^{M_{t}}\theta_{t}^{j}\mathds{1}[x\in r_{t}^{j}]$ where
$\mathds{1}$ is the indicator function. Lastly, to generate a prediction for a
target sample $x_{te}$, the final GBRT model sums up the values of the leaves
traversed by $x_{te}$ over all $T$ iterations:
$\hat{y}_{te}=\sum_{t=1}^{T}m_{t}(x_{te})$.

###  2.3 Instance-based Uncertainty Quantification

Figure 1: IBUG flow chart. For a target instance, IBUG collects the training
instances at each leaf it traverses, keeps the $k$ most frequent samples, and
then uses those instances to model the output distribution.

The goal is to estimate the conditional probability distribution $P(y|x)$ for
some target variable $y$ given some input vector $x$. To allow probabilistic
predictions for any GBRT point predictor, we propose to make use of a method
that adopts ideas from instance-based learning [11] and a supervised tree
kernel, namely the Instance-Based Uncertainty quantification for GBRTs (IBUG)
method [4]. To start with, the method capitalises on the fact that GBRTs yield
accurate point predictions and uses this scalar output to model the
conditional mean in a probabilistic forecast. Next, to further model the
conditional output distribution, IBUG uses a supervised tree kernel [12] to
more effectively identify the $k$ training examples with the largest affinity
to the target example. In particular, the affinity of the $i$-th training
example $x_{i}$ to a target example $x_{te}$ is given by

$A(x_{i},x_{te})=\sum_{i=1}^{T}\mathds{1}[R_{t}(x_{i})=R_{t}(x_{te})]$ (1)

where $R_{t}(x_{i})$ is the leaf (of the tree $t$) to which $x_{i}$ is
assigned. Such a metric uses the structure of the learned trees in the
ensemble. Lastly, the method employs the set of those $k$ affinity scores,
$A^{(k)}$, to produce a probabilistic prediction. The overall IBUG workflow is
illustrated in Fig. 1.

Unlike other SOTA methods, IBUG offers numerous choices for modelling the
conditional output distribution; the simplest way is, of course, through a
normal distribution. In this case, the scalar output of the GBRT model is used
to approximate the conditional mean ($\mu_{\hat{y}_{te}}=f(x_{te})$), and then
the set $A^{(k)}$ is manipulated to compute the variance
$\sigma^{2}_{\hat{y}_{te}}$. To further optimise the calculation of the
prediction variance, the following calibration

$\sigma^{2}_{\hat{y}_{te}}\leftarrow\gamma\sigma^{2}_{\hat{y}_{te}}+\delta$
(2)

is commonly applied, where $\gamma$ and $\delta$ are tuned on the validation
set after the choice of $k$ has been made. The method acts as a wrapper around
any GBRT model allowing one to try various GBRT point predictors and then
select the model with the best performance. To more flexibly model the output
distribution using any parametric or non-parametric distribution, IBUG can use
the set $A^{(k)}$ to directly fit (using maximum likelihood estimation) any
continuous distribution $D$, including those with high-order moments:

$\hat{D}_{te}=D\left(A^{(k)}\mid\mu_{\hat{y}_{te}},\sigma^{2}_{\hat{y}_{te}}\right).$
(3)

Choosing an appropriate value for $k$ is critical for producing accurate
probabilistic predictions. In this study, the tuning of $k$ was performed in a
held-out validation dataset $D_{\text{val}}\subset D$ using the negative log
likelihood (NLL) probabilistic scoring metric. To accelerate the tuning
process, through avoiding the repetition of the computationally expensive
affinity calculations, the procedure described in Algorithm 3 of [4] was
adopted, where parameter $\rho$ was used to model instances of abnormally low
variance.

###  2.4 Implementation

All implementations were in Cython. The experiments were conducted utilising
an Intel Core i9 CPU 10900K Comet Lake, 10 Cores, 20 Threads @ 5.3GHz system
equipped with 128GB of DDR4 RAM operating @ 2.6GHz. IBUG was applied to
XGBoost [13], LightGBM [14], and CatBoost [15]. We tuned $k$, using values:
[3, 5, 7, 9, 11, 15, 31, 61, 91, 121, 151, 201, 301, 401, 501, 601, 701]. The
parameters $\gamma$ and $\delta$ were tuned using values ranging from $1\times
10^{-8}$ to $1\times 10^{3}$ with additional multipliers $[1.0,2.5,5.0]$. The
number of trees, $T$, was tuned using values
$[10,25,50,100,250,500,1000,2000]$ (early stopping [7] was used for NGBoost).
The learning rate was tuned using values $[0.01,0.1]$. We also optimised: the
maximum number of leaves $h$ using values $[15,31,61,91]$, the minimum number
of leaves using values $[1,20]$, and the maximum depth $d$ using values
$[2,3,5,7,-1]$ (indicating no limit). The $p$ parameter was adjusted based on
the minimum variance obtained from the validation set predictions. We employed
5-fold cross-validation to generate 5 different 80/20 train/test folds. For
each fold, the 80% training set was randomly divided into a 60/20
train/validation set for hyperparameter tuning. Upon tuning the
hyperparameters, the model was retrained using the complete 80% training set.
To test IBUG’s flexibility in posterior modelling, we modelled each
probabilistic prediction using the following distributions: normal,
skewnormal, lognormal, Laplace, student t, logistic, Gumbel, Weibull, and KDE.

###  2.5 Evaluation Metrics

To evaluate probabilistic performance, the continuous ranking probability
score (CRPS), NLL, check score, and interval score were used [16]. For all
metrics, the lower the better. To gauge point performance, the root mean
squired error (RMSE), the mean absolute error (MAE), the mean absolute
percentage error (MAPE), the R2 measure, and correlation were used. IBUG was
compared to three recent gradient boosting algorithms that provide
probabilistic predictions, namely NGBoost [7], PGBM [8], and CatBoost with
uncertainty (CBU) [9].

###  2.6 Exemplars

To facilitate a comprehensive understanding of our results, we provide
conditional output distributions and the respective confidence intervals (CIs)
for two high and two low accuracy (test set) predictions.

###  2.7 Explainability

A benefit of using GBRTs is that it is straight-forward to retrieve feature
importance scores that indicate how valuable each attribute was in the
construction of the model. In this study, feature importance scores were
calculated for all models using the ’Gain’ metric which quantifies the
relative contributions.

##  3 Results

The final set of hyperparameters for each method and the corresponding tuning
and training times are listed in Table 1. Table 2 compares the probabilistic
performance of the IBUG model against the three SOTA probabilistic prediction
methods. IBUG model with CatBoost as the base learner provided the lowest
average scores in all CRPS, NLL, Check Score and Interval Score indices. In
Table 3, the point performance of all methods is provided. In overall, IBUG
method with CatBoost base learner displayed the best performance once again.
Table 4 shows the importance of variance calibration in the probabilistic
performance of IBUG. Table 5 demonstrates that the logistic (parametric)
distribution better fits the underlying data than assuming normality. In Fig.
3 and Fig. 3, we illustrate the conditional output distributions for four
representative test cases (two that were predicted with high accuracy and two
that were predicted with low accuracy), when normal and logistic probabilistic
density functions were used for modelling, respectively. Table 6 lists the 95%
and 99% confidence intervals for the above cases. These results showcase the
appropriateness of the proposed framework for providing uncertainty scores for
RV volume predictions. Lastly, in Fig. 4, we illustrate the ”Gain” feature
importance score for all eleven features in the best IBUG model. The
parasternal long axis (PLAX), four chambers (Four C) and parasternal short
axis at base level (PSAX Base) standard views were the top three contributors
to the model predictions.

Table 1: The final set of hyperparameters used for each method. Also shown are the tuning and training times. IBUG was applied to CatBoost, XGBoost, and LightGBM. Parameter | CatBoost | XGBoost | LightGBM | NGBoost | PGBM | CBU
---|---|---|---|---|---|---
K | 5 | 15 | 3 | - | - | -
$\delta$ | 1 | 0.5 | 0.5 | 5 | 10 | 1
Operation | add | mult | mult | add | add | add
min scale | 6.164 | 13.826 | 2.055 | - | - | -
n estimators (trees) | 100 | 25 | 25 | 244 | 250 | 250
maximum depth | 5 | 2 | -1 | - | - | -
learning rate | 0.1 | 0.1 | 0.1 | 0.01 | 0.01 | 0.1
minimum data in_leafv | 1 | - | - | - | 20 | 1
minimum child weight | - | 20 | 20 | - | - | -
number of leaves | - | - | 15 | - | 15 | 15
max bin | 255 | 255 | 255 | 255 | 255 | 255
tune+train time (s) | 67.369 | 19.932 | 14.260 | 5.370 | 872.663 | 81.929
Table 2: Probabilistic performance comparison on the test set (five folds). IBUG results are for the case when CatBoost was the base learner. IBUG results have been averaged over all nine posterior output distributions. Method | NLL | CRPS | Check Score | Interval Score
---|---|---|---|---
IBUG | 4.747 | 15.398 | 7.775 | 73.380
NGBoost | 7.571 | 22.174 | 11.177 | 141.618
PGBM | 6.136 | 20.796 | 10.492 | 122.401
CBU | 5.780 | 19.524 | 9.853 | 110.140
Table 3: Point performance comparison on the test set (five folds). IBUG results are for the case when CatBoost was the base learner. Method | MAE | RMSE | MAPE | R2 | Correlation
---|---|---|---|---|---
IBUG | 22.75 | 26.292 | 20.22 | 0.666 | 0.824
NGBoost | 28.114 | 32.269 | 24.406 | 0.496 | 0.736
PGBM | 27.479 | 31.27 | 25.147 | 0.527 | 0.768
CBU | 26.378 | 30.127 | 22.974 | 0.561 | 0.772
Table 4: Probabilistic performance comparison of IBUG method with and without variance calibration. IBUG results are for the case when CatBoost was the base learner. Operation | NLL | CRPS | Check Score | Interval Score
---|---|---|---|---
With Calibration | 4.747 | 15.398 | 7.775 | 73.38
Without Calibration | 4.781 | 15.457 | 7.805 | 74.044
Table 5: Probabilistic performance comparison when assuming normal and logistic distributions for modelling the underlying data. Distribution | NLL
---|---
Normal | 5.10466
Logistic | 5.00837

(a)

(b)

(c)

(d)

Figure 2: The conditional output normal distributions for test instances that
were predicted with high [(a) and (b)] and low [(c) and (d)] accuracy.

(e)

(f)

(g)

(h)

Figure 3: The conditional output logistic distributions for test instances
that were predicted with high [(e) and (f)] and low [(g) and (h)] accuracy.

Table 6: The 95% and 99% Confidence Intervals for both Normal and Logistic distributions for four representative test set cases, two that were predicted with high accuracy (low APE) and two that were predicted with low accuracy (high APE). Prediction |  APE (%) |  Point | Normal Distribution | Logistic Distribution
---|---|---|---|---
Accuracy |  |  Prediction |  95% CI |  99% CI |  95% CI |  99% CI
High |  3.090 |  103.090 |  35.979 |  47.286 |  30.196 |  42.697
|  0.508 |  87.553 |  46.739 |  61.428 |  39.227 |  55.466
Low |  10.169 |  163.492 |  91.932 |  120.825 |  77.157 |  109.099
|  10.119 |  61.119 |  83.650 |  109.939 |  70.206 |  99.270
Figure 4: Feature importance plot for the IBUG model with CatBoost as the base
learner.

##  4 Discussion and Conclusions

In this paper, we presented an uncertainty quantification framework when
predicting RV volumes through the use of GBRTs on 2DE tabular data (such as
area measurements, age, gender and cardiac phase information). To get an
estimate of the prediction uncertainty, we employed the IBUG method which uses
the learned tree structure to identify the $k$ nearest training samples to a
target instance. The results on a small-scale dataset demonstrate that this
simple wrapper yields improved probabilistic and point performances over other
SOTA methods. The appropriateness of the proposed framework for providing
uncertainty scores for RV volume predictions was showcased by providing
conditional output distributions and confidence intervals for four exemplar
cases. Additional research is required, involving a larger sample size of
patients and encompassing a broader range of RV volumes, to substantiate these
findings. The estimated uncertainty embodies both aleatoric and epistemic
types of uncertainty since IBUG is an instance-based approach and also
predictions on the training set were used to tune $k$, $\gamma$, and $\delta$.
Overfitting was observed in IBUG’s point performance which is a typical
finding when the size of the dataset is small. This work aligns with
trustworthy artificial intelligence [17] since it can be used to enhance the
decision-making process and reduce risks. It could help overcome mistrust
which is a major barrier to the deployment of machine learning systems in the
clinical setting. The calculated feature importance scores can be used for
reducing the number of required 2DE views, which in turn could enhance the
proposed pipeline’s clinical application.

###### Acknowledgements.

## Funding

Tuan Aqeel Bohoran is funded by the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
801604.

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## AUTHORS’ BACKGROUND

Name |  Title | Research Field | Personal website
---|---|---|---
Tuan Aqeel Bohoran |  Researcher |  Data Science, Computer Science, Machine Learning, Deep Learning, Medical Imaging, Mathematics, Computer Vision, Cardiovascular MRI |  https://github.com/tuanaqeelbohoran
Polydoros N. Kampaktsis |  Assistant Professor |  Valvular Disease, Coronary Artery Disease, transcatheter therapies, artificial intelligence, cardiovascular imaging |  https://www.columbiacardiology.org/profile/polydoros-kampaktsis-md
Laura McLaughlin |  Medical Student |  |  https://www.linkedin.com/in/laura-mclaughlin-4112ab82
Jay Leb |  Assistant Professor |  Cardiovascular Disease, CT, MRI |  https://www.columbiaradiology.org/profile/jay-s-leb-md
Serafeim Moustakidis |  Co-founder/partner |  Healthcare, Energy, Agriculture, Environment, Manufacturing, Security and Safety, Construction and building technology, Infrastructure and Transportation |  https://www.aideas.eu/serafeimmoustakidis
Gerry P. McCann |  Full Professor |  Cardiac Imaging, Aortic Stenosis, Diabetes and Heart Failure, CAD, Clinical Imaging |  https://le.ac.uk/people/gerry-mccann
Archontis Giannakidis |  Senior Lecturer |  Data Science, Computer Science, Machine Learning, Deep Learning, Medical Imaging, Mathematics, Computer Vision, Cardiovascular MRI |  https://www.ntu.ac.uk/staff-profiles/science-technology/archontis-giannakidis


11institutetext: Department of Experimental Physics, Institute of Physics,
University of Szeged, Dóm tér 9, 6720 Szeged, Hungary
11email<EMAIL_ADDRESS>22institutetext: Eötvös Loránd
University, Department of Astronomy, Pázmány Péter sétány 1/A, 1117 Budapest,
Hungary 33institutetext: Konkoly Observatory, Research Centre for Astronomy
and Earth Sciences, Konkoly Th. M. út 15-17., 1121 Budapest, Hungary
44institutetext: ELKH-SZTE Stellar Astrophysics Research Group, Szegedi út,
Kt. 766, 6500 Baja, Hungary 55institutetext: Baja Astronomical Observatory of
University of Szeged, Szegedi út, Kt. 766, 6500 Baja, Hungary 66institutetext:
CSFK, MTA Centre of Excellence, Konkoly Thege Miklós út 15-17, 1121 Budapest,
Hungary 77institutetext: ELTE Eötvös Loránd University, Gothard Astrophysical
Observatory, 9400 Szombathely, Hungary 88institutetext: Eötvös Loránd
University, Institute of Physics, Pázmány Péter sétány 1/A, 1117 Budapest,
Hungary 99institutetext: Max-Planck-Institut für Radioastronomie, Auf dem
Hügel 69, 53121 Bonn, Germany 1010institutetext: Scottish Universities Physics
Alliance (SUPA), School of Physics and Astronomy, University of St Andrews,
North Haugh, St Andrews, KY16 9SS, UK

# Three is the magic number - distance measurement of NGC 3147 using SN
2021hpr and its siblings

B. Barna 11 A. P. Nagy 11 Zs. Bora 2233 D. R. Czavalinga 4455 R. Könyves-Tóth
116677 T. Szalai 1144 P. Székely 11 Sz. Zsíros 11 D. Bánhidi 55 I. B. Bíró 44
5 5 I. Csányi 55 L. Kriskovics 22 6 6 A. Pál 22 6 6 Zs. M. Szabó 33 6 6 9 9 10
10 R. Szakáts 22 6 6 K. Vida 22 6 6 Zs. Bodola 11 J. Vinkó 22 6 6 1 1 8 8

(Received December 12 2022)

###### Abstract

Context. The nearby spiral galaxy NGC 3147 hosted three Type Ia supernovae
(SNe Ia) in the past decades, which have been subjects of intense follow-up
observations. Simultaneous analysis of their data provides a unique
opportunity for testing the different light curve fitting methods and distance
estimations.

Aims. The detailed optical follow-up of SN 2021hpr allows us to revise the
previous distance estimations to NGC 3147, and compare the widely used light
curve fitting algorithms to each other. After the combination of the available
and newly published data of SN 2021hpr, its physical properties can be also
estimated with higher accuracy.

Methods. We present and analyse new $BVgriz$ and Swift photometry of SN
2021hpr to constrain its general physical properties. Together with its
siblings, SNe 1997bq and 2008fv, we cross-compare the individual distance
estimates of these three SNe given by the SALT code, and also check their
consistency with the results from the MLCS2k2 method. The early spectral
series of SN 2021hpr are also fit with the radiative spectral code TARDIS in
order to verify the explosion properties and constrain the chemical
distribution of the outer ejecta.

Results. After combining the distance estimates for the three SNe, the mean
distance to their host galaxy, NGC 3127, is $42.5\pm 1.0$ Mpc, which matches
with the distance inferred by the most up-to-date LC fitters, SALT3 and
BayeSN. We confirm that SN 2021hpr is a Branch-normal Type Ia SN that ejected
$\sim 1.12\pm 0.28$ M⊙ from its progenitor white dwarf, and synthesized $\sim
0.44\pm 0.14$ M⊙ of radioactive 56Ni.

###### Key Words.:

supernovae – distance measurement

##  1 Introduction

The Type Ia supernovae (SNe Ia), being high-luminosity standardizable candles,
have essential importance in the cosmic distance measurements providing
extension of the distance ladder toward higher redshifts. Since at present
there is a significant tension between the cosmological parameters, like
$H_{0}$, inferred locally and from the Cosmic Microwave Background (CMB), it
is important to further reduce the potential biases in the measured distances,
which may help in revealing the cause of the discrepancy. Local galaxies that
hosted SNe Ia and have observable Cepheid populations are especially important
in this respect (Riess et al., 2022).

The basis of the standardization process of Type Ia light curves (LCs) is the
Phillips relation (Phillips, 1993), i.e. the empirical correlation between the
peak absolute brightness (typically in the $B$-band) and the $\Delta
m_{\mathrm{15}}$ decline rate measured during the first 15 days after the
moment of maximum light ($t_{\mathrm{0}}$). Later, multiple studies tried to
link the shape of the LC to the peak luminosity. The most widely used SN Ia LC
synthesis and distance estimator codes are the newest versions of SALT (Guy et
al., 2005) and MLCS (Riess et al., 1998; Jha et al., 2006), but other
approaches, like BayeSN (Thorp et al., 2021; Mandel et al., 2022) and SNooPy
(Burns et al., 2011) have also been published (see Section 3.2).

However, LC fitters and distance estimations still suffer from intrinsic
scattering due to spectrophotometric calibration issues and maybe some sort of
unknown systematic effects. One way to reduce the sources of uncertainties is
using SN siblings, i.e. SNe discovered in the same galaxy. These SNe share the
same distance, redshift, and other physical properties of their common host
galaxy. Thus, the expected dispersion in their individually estimated
distances should be significantly lower than the intrinsic scatter in the
Hubble-diagram at the same redshift. Thus, SN siblings may also allow us to
test distance measurement methods and may support their further improvements
(see e.g. Burns et al., 2020; Gallego-Cano et al., 2022; Scolnic et al., 2020;
Hoogendam et al., 2022; Ward et al., 2022).

Due to the decade-long observations by recent transient discovery programs,
the number of galaxies hosting multiple SNe is gradually increasing. The
absolute record holder of modern times is NGC 6946, also called the Firework
Galaxy, which hosted ten SNe in a century. However, the datasets of early SNe
usually suffer either from high uncertainties, data gaps or lack of wavelength
coverage, thus, not all of these SNe can be used for high-precision distance
estimations.

We searched for additional supernova siblings in the Open Supernova Catalog
(OSC, Guillochon et al., 2017). By narrowing our search to the simple Ia
category in the OSC, we found 67 galaxies that hosted two or more “normal”
Type Ia supernovae. The actual number is higher as the OSC has a sub-
classification of Ia supernovae that we omitted from our search, and the
catalog has not been updated since April 2021. Recent studies by Burns et al.
(2020) and Gallego-Cano et al. (2022) have increased the sample further. In
the most recent study by Kelsey (2023), 113 galaxies were found to host 236
thermonuclear SNe (including both “normal” SNe Ia and other subclasses) in the
OSC.

There are only 6 galaxies in which at least three Type Ia supernova siblings
have been discovered. Both M84 (SNe 1957B, 1980I, and 1991bg) and NGC 1316
(SNe 1980N, 1981D, 2006dd, and 2006mr) include two Type Ia supernovae before
the CCD era, which increases the importance of NGC 5468 (SNe 1999cp, 2002cr,
and 2005P), NGC 5018 (SNe 2017isq, 2002dj, 2021fxy), NGC 3367 (SNe 2018kp,
2003aa, 1986A), and NGC 3147 (SNe 1972A, 1997bq, 2008fv, and 2021hpr). The
last one hosted the recently discovered SN 2021hpr, which was the subject of
intense follow-up observations by several observatories. Therefore, it became
a key object in the distance estimations using Type Ia supernova siblings.

In this paper we present new optical photometric observations of SN 2021hpr,
and combine them with published LCs of SNe 1997bq and 2008fv to determine an
improved distance to their common host galaxy, NGC 3147. Based on the improved
distance, we infer and discuss the physical parameters for SN 2021hpr by
building models for its spectra and the bolometric LC. The paper is structured
as follows: in Section 2, we introduce the datasets of the three SNe Ia hosted
by galaxy NGC 3147, with a special interest in the newly obtained LC of SN
2021hpr published in this paper first. Methods for the spectral synthesis and
LC analysis, as well as the fitting algorithms used for distance estimations,
are described in Section 3. The results are presented and discussed in Section
4. Finally, we summarize our conclusion in Section 5.

##  2 Three supernovae of NGC 3147

NGC 3147 is a barred spiral galaxy in the Draco constellation, at
$\alpha$(2000.0) = $10^{\mathrm{h}}16^{\mathrm{m}}50^{\mathrm{s}}$,
$\delta$(2000.0) = +73∘ 24’, being in the focus of interest due to its low-
luminosity Type II Seyfert active galactic nucleus (Panessa & Bassani, 2002).
Its heliocentric redshift is $z=0.00934$ (Epinat et al., 2008). The historical
distance estimations show a wide range between 27.7 Mpc (from Tully-Fisher
relation, Bottinelli et al., 1984) and 55.2 Mpc (based on the Type Ia SN
1972A, Parodi et al., 2000), but the latest pre-SN 2021hpr results narrowed
down to 39.3 Mpc (SN 1997bq, Tully et al., 2013) and 43.7 Mpc (SN 2008fv,
Biscardi et al., 2012). The most recent Cepheid-based distance is derived by
the comprehensive analysis of Riess et al. (2022) where the authors estimated
40.1 Mpc $\pm 3.3$ Mpc including the analysis of 27 Cepheid variables of NGC
3147.

The proximity and face-on orientation of NGC 3147 makes it a prominent host
for discovering transient events. In the past half-century, this galaxy hosted
six SNe, four of which were classified as Type Ia (the other two, SN 2006gi
and SN 2021do, were Type Ib/c events). For SN 1972H, only photographic
photometry was published (Barbon et al., 1973). Thus, only SNe 1997bq, 2008fv,
and the recently discovered 2021hpr can be used in a modern LC analysis.

The galactic component of the interstellar reddening in the direction of NGC
3147 is $E(B-V)=0.021$ mag (Schlafly & Finkbeiner, 2011). The explosions took
place at distant sites within NGC 3147, sampling different regions of the
galaxy. The host galaxy component of the reddening is taken from previous
studies for each object, as listed below.

SN 1997bq was discovered on 50546.0 MJD by Laurie & Challis (1997). The SN was
located outside the observable spiral arms of NGC 3147 with 60” offset (R
$\approx$ 16.0 kpc) southeast from the bulge. $UBVRI$ LCs were obtained at the
Fred Lawrence Whipple Observatory of the Harvard-Smithsonian Center for
Astrophysics and published by Jha et al. (2006). Due to the outskirt location
of SN 1997bq, no significant host galaxy reddening is expected, but the
individual LC fit performed with the code BayeSN indicated a total extinction
of $A_{\mathrm{V}}=0.45$ mag (Ward et al., 2022).

SN 2008fv was first detected on 54736.0 MJD by K. Itagaki to 37” north-east
from the bulge of NGC 3147 (Nakano et al., 2008). Biscardi et al. (2012)
reported a significant host-galaxy reddening of $E(B-V)_{\mathrm{host}}=0.22$
with selective extinction coefficients of $R_{\mathrm{V}}=2.9$ based on
Cardelli et al. (1989). The SN peaked at 14.55 mag on $\sim$54749.0 MJD in the
$B$-band. Optical and near-infrared (NIR) photometry was obtained by Biscardi
et al. (2012) and Tsvetkov & Elenin (2010). However, there is an enormous
difference between the two datasets in the $I$-band. The choice of the adopted
magnitudes for the LC fitting is explained in Sec. 4.1.2.

The discovery of SN 2021hpr was reported by Itagaki (2021) based on the first
observation on 2021-04-02 at 10:46 UTC (59306.4 MJD). Later a pre-discovery
detection (59304.92 MJD) was reported by Tsvetkov et al. (2021) by 0.3 day
before the observation of the Zwicky Transient Factory. The first, yet
ambiguous classification (Tomasella et al., 2021) claimed that the new
transient is probably a Type Ia SN due to its Si II $\lambda$6355 feature with
an expansion velocity of 21,000 km s-1. The strong high-velocity feature (HVF)
and the rapid brightening of the object suggested that SN 2021hpr was
discovered at its very early phase.

SN 2021hpr was the subject of two previous studies, which published and
analyzed their independent follow-up observations: Zhang et al. (2022)
presented $BVRI$ LCs and optical spectra between -14 and +64 days to $B$-band
maximum, while Ward et al. (2022) provided $grizy$ photometry between -10 and
+40 days and one optical spectrum at -4.2 days. In this study, we publish a
new set of optical photometry obtained at two Hungarian observatories.
Furthermore, we include the analysis of UV-photometry taken by the Neil
Gehrels Swift Observatory Ultraviolet and Optical Telescope (UVOT). Hereafter,
we use our new $BVgri$ photometry, supplemented by Swift $UBV$ data, for the
LC analysis of SN 2021hpr. The previously published LCs mentioned above, are
also used for comparison, but not for re-analysis. We also model the spectra
of SN 2021hpr available from the literature (see Tab.
LABEL:tab:spectroscopic_data).

The host extinction was assumed to be negligible by Zhang et al. (2022), but
the LC analysis of Ward et al. (2022) suggested $A_{V,host}\simeq 0.20$ mag.
Without any more established estimation, we use their mean value,
$A_{V,host}=0.1$ mag, for the rest of the paper.

The main properties of NGC 3147, as well as of the three SNe, are listed in
Tab. 1.

###  2.1 Observations

In this paper we present a SN photometric dataset that was obtained with two
recently installed 0.8m telescopes in Hungary: one at the Piszkéstető mountain
station of Konkoly Observatory, and one at Baja Observatory. The twin
instruments are two 0.8m Ritchy-Chrétien telescopes (hereafter KRC80 and
BRC80, respectively), manufactured and deployed by the company AstroSysteme
Austria (ASA). The focal length of 5700 mm provides an f/7 light-gathering
power. The telescope is equipped with Johnson $BV$ and Sloan $ugriz$ filters
and a 2048x2048 back-illuminated FLI PL230 CCD chip with a pixel scale of
0.55”. Due to the similarities between the two telescopes, the combined LCs of
SN 2021hpr can be considered as a homogeneous dataset.

We carried out standard Johnson–Cousins BV and Sloan griz CCD observations on
50 nights between April and September 2021 (Fig. 1). The achieved photometric
accuracy varied between 0.01–0.05 mag depending on the weather conditions. The
exposure times were 180 seconds except for the B filters where we used 300
sec.

All data were processed with standard IRAF111IRAF is distributed by the
National Optical Astronomy Observatories, which are operated by the
Association of Universities for Research in Astronomy, Inc., under cooperative
agreement with the National Science Foundation. routines, including bias, dark
and flat-field corrections. Then we co-added three images per filter per night
aligned with the wcsxymatch, geomap and geotran tasks. We obtained PSF
photometry on the co-added frames using the daophot package in IRAF, and image
subtraction photometry based on other IRAF tasks like psfmatch and linmatch,
respectively. For the image subtraction we applied a template image taken at a
sufficiently late phase, when the transient was no longer detectable on our
frames.

For PSF photometry we built an automated pipeline using self-developed C-codes
and bash shell scripts, and the necessary IRAF tasks are called as system
binaries outside the IRAF environment. The IRAF executables are collected into
a single parallel processing script using gnu-parallel (Tange, 2011). This
“all inclusive” method enabled us to reduce the processing time for the $\sim
1$ GB of data per night to a few minutes on a normal PC with 16 CPU cores.

The photometric calibration was carried out using stars from Data Release 1 of
Pan-STARRS1 (PS1 DR1) 222https://catalogs.mast.stsci.edu/panstarrs/. The
selection of the photometry reference stars and the calibration procedures are
as follows. First, sources within a 5 arcmin radius around the SN with
$r$-band brightness between 15 and 17 mag (to avoid saturation, (Magnier et
al., 2013)) were downloaded from the PS1 catalog. Next, non-stellar sources
were filtered out based on the criterion
$i_{\mathrm{PSFmag}}-i_{\mathrm{Kronmag}}$ ¡ 0.05 for stars
333https://outerspace.stsci.edu/display/PANSTARRS/. In order to get reference
magnitudes for our Johnson $B$\- and $V$-band frames, the PS1 magnitudes were
transformed into the Johnson BVRI system based on equations and coefficients
found in Tonry et al. (2012). Finally, the instrumental magnitudes were
transformed into standard $BVgriz$ magnitudes by applying a linear color term
(using $g-i$) and wavelength-dependent zero points. Since the reference stars
fell within a few arcminutes around the target, no atmospheric extinction
correction was necessary. S-corrections were not applied.

The obtained $BVgriz$ LCs are plotted in Fig. 1. Direct comparison with $B$\-
and $V$-band LCs of Zhang et al. (2022) further confirm that our data are free
of systematic errors.

In the case of SN 2021hpr the ground-based optical observations were
supplemented by the available archival data of the Neil Gehrels Swift
Observatory (Swift, Gehrels et al., 2004; Burrows et al., 2005) taken with the
Ultraviolet-Optical Telescope (UVOT, Roming et al., 2005) in April 2021 (see
in Fig. 2). Data were collected in six filters from optical to ultraviolet
wavelengths (u, b, v, uvw1, uvm2, uvw2). The SN was detectable as a point
source on the images, although it was located in a complex galactic
environment. In order to model its background flux, we applied five different
background regions distributed around the SN, and determined the background as
the average of the flux values taken from each region. The Swift/UVOT data
were processed using the HEAsoft software package. We summed the individual
frames using the uvotimsum task and carried out aperture photometry on the
summed images using the uvotsource task.

Two spectra used in this study (see Tab. LABEL:tab:spectroscopic_data) were
published by Zhang et al. (2022), and another was obtained at Smolecin
Observatory (L25). All spectra are available at WISeREP online supernova
database (Yaron & Gal-Yam, 2012).

Figure 1: Comparison of $BVgriz$ photometry of SN 2021hpr from different data
sources. Figure 2: Swift photometry of SN 2021hpr.

##  3 Methods

In this section, we briefly introduce the modeling codes adopted for the
photometric and spectroscopic analyses.

Table 1: Main parameters of the three SNe Ia in NGC 3147. NGC 3147 | Ref.
---|---
heliocentric redshift | 0.00934 | 1
$E(B-V)_{\mathrm{MW}}$ | 0.021 mag | 2
SN 1997bq
RA | 10h 17m 04s | 3
DEC | +73∘ 23’ 03” | 3
T${}_{\mathrm{max}}(B)$ | 50558.0 MJD | 4
Bmax | 14.57 mag | 4
$\Delta m_{\mathrm{15}}(B)$ | 1.01 mag | 4
$E(B-V)_{\mathrm{host}}$ | 0.11 mag | 5
SN 2008fv
RA | 10h 16m 57s.28 | 6
DEC | +73∘ 24’ 36” | 6
T${}_{\mathrm{max}}(B)$ | 54749.3 MJD | 7
Bmax | 14.55 mag | 7
$\Delta m_{\mathrm{15}}(B)$ | 0.94 mag | 7
$E(B-V)_{\mathrm{host}}$ | 0.22 mag | 7
SN 2021hpr
RA | 10h 16m 38s.68 | 8
DEC | +73∘ 24’01 00”.80 | 8
T${}_{\mathrm{max}}(B)$ | 59321.9 MJD | 9
Bmax | 14.017 mag | 9
$\Delta m_{\mathrm{15}}(B)$ | 0.949 mag | 9
$E(B-V)_{\mathrm{host}}$ | 0.00/0.07 mag | 9,5
1 - Epinat et al. (2008); 2 - Schlafly & Finkbeiner (2011);
3 Laurie & Challis (1997); 4 - Jha et al. (2006);
5 - Ward et al. (2022); 6 Nakano et al. (2008);
7 - Biscardi et al. (2012); 8 - Itagaki (2021);
9 - Zhang et al. (2022)

###  3.1 The radiative transfer code TARDIS

Our approach for the fitting of the spectral series of SN 2021hpr is performed
with the one-dimensional radiative transfer code TARDIS (Kerzendorf & Sim,
2014). TARDIS calculates synthetic spectra on a wide wavelength range in
exchange for a low computational cost, providing an ideal tool for fitting
both the continuum and spectral features of homologously expanding ejecta.

The main assumption of the code is a sharp photosphere emitting blackbody
radiation modeled via indivisible energy-packets representing bundles of
photons (for further description see Abbott & Lucy, 1985; Lucy & Abbott, 1993;
Lucy, 1999, 2002, 2003). The model atmosphere is divided into radial layers
with densities and chemical abundances defined by the user. The algorithm
follows the propagation of the packets, while calculating the wavelength- and
direction-changing light-matter interactions based on a Monte Carlo scheme in
every shell. The final output spectrum is built up by summarizing the photon
packets escaping from the model atmosphere.

The approach of TARDIS offers several improvements from the simple LTE
assumptions. We followed the same settings as in most of the studies that used
TARDIS for spectral fitting (see e.g. Magee et al., 2016; Boyle et al., 2017;
Barna et al., 2018). As an example, the ionization state of the material used
here are estimated following an approximate non-LTE (NLTE) mode, the so-called
nebular approximation, which significantly deviates from the LTE method by
accounting for a fraction of recombinations returning directly to the ground
state (Mazzali & Lucy, 1993). The excitation state is calculated according to
the dilute-LTE approximation which is also not purely thermal. The summary of
TARDIS numerical parameters and modes adopted in this study are listed in Tab.
2. The limitation of the simulation background, as well as the detailed
description of the NLTE methods, are presented in the original TARDIS paper
(Kerzendorf & Sim, 2014).

Table 2: TARDIS fitting parameters Setting | Approximation
---|---
Radiation mode | dilute-blackbody
Ionization mode | nebular
Excitation mode | dilute-LTE
Line interaction mode | macroatom
Parameter | Best-fit value
Time of explosion | 59304.0 MJD
Core density ($\rho_{0}$) | 4.7 $g\,cm^{-3}$
Density slope ($v_{\mathrm{0}}$) | 2750 $km\,s^{-1}$

###  3.2 Light curve fitter codes for SNe Ia

We applied the recently released version of Spectral Adaptive Lightcurve
Template (SALT3, Kenworthy et al., 2021) for estimating the distance to SN
2021hpr and its two siblings. We also utilized the earlier version, SALT2.4
(Betoule et al., 2014), as well as an independent LC fitter, MLCS2k2 (Jha et
al., 2007) for checking the consistency of the inferred distance moduli with
those from earlier calibrations.

MLCS2k2 is the improved version of the original Multi-Color Light Curve Shape
(MLCS) code introduced by the High-z Supernova Search Team (Riess et al.,
1996). The LCs obtained in Johnson-Cousins $UBVRI$ filters are fitted with two
tabulated functions ($P_{\lambda}$ and $Q_{\lambda}$) trained on a carefully
selected sample of 133 SNe (Jha et al., 2007) and with the $\Delta$ parameter
that linked the shape of the LC with the peak absolute brightness in the
$V$-band. The code fits the observed LC $m_{\lambda}(t)$ with the following
function

$m_{\lambda}(t)=M_{\lambda}(t)+\mu_{0}+\zeta_{\lambda}\big{(}\alpha_{\lambda}+\beta_{\lambda}/R_{V}\big{)}\cdot
A_{V}+P_{\lambda}(t)\cdot\Delta+Q_{\lambda}(t)\cdot\Delta^{2},$ (1)

where $t$ is the rest-frame time (corrected for time dilation) elapsed from
the moment of $B$-band maximum, $M_{\lambda}(t)$ is the LC of the fiducial SN
Ia in absolute magnitudes, and $\mu_{0}$ is the true (extinction-free)
distance modulus. MLCS2k2 also takes into account the effect of interstellar
reddening with $A_{V}$ extinction in the $V$-band. $\alpha$, $\beta$ describe
the interstellar reddening law as a function of wavelength, while $\zeta$
takes into account the temporal variation of the reddening correction due to
the spectral evolution of the SN.

SALT2 was introduced by the SuperNova Legacy Survey Team (Guy et al., 2007).
Unlike MLCS2k2, this code models the entire spectral energy distribution (SED)
as a function of time. To do this, a combination of multiple colour-dependent
vectors is trained on a large sample of thoroughly chosen SNe. The components
of the model function are

$F(\lambda,t)=x_{\mathrm{0}}\cdot[M_{\mathrm{0}}(\lambda,t)+x_{\mathrm{1}}\cdot
M_{\mathrm{1}}(\lambda,t)]\cdot\exp(c\cdot CL(\lambda)),$ (2)

where $M_{0}$, $M_{1}$ and CL are the trained vectors of SALT, while $x_{0}$,
$x_{1}$ and $c$ are fitting parameters, representing the normalization, the
stretch and the colour of the SED, respectively.

SALT3 is a recent improvement to SALT2.4 developed by Kenworthy et al. (2021).
SALT3 was trained on more than one thousand SN Ia spectra, an order of
magnitude larger sample than the previous version of the code, providing lower
uncertainties and fewer systematics compared to the previous versions.

The SALT model does not contain the distance as a direct fitting parameter.
Instead, it is inferred from the fitting parameters of the SALT3 code by the
following formula (Tripp, 1998):

$\mu=-2.5\log_{10}(x_{0})+\alpha x_{1}-\beta c-M_{0}$ (3)

We adopt the recent calibration of Pierel et al. (2022) for $\alpha$ and
$\beta$ parameters as $\alpha=0.133\pm 0.003$ and $\beta=2.846\pm 0.017$,
respectively.

##  4 Results

###  4.1 Distance measurements

Recently, a comprehensive analysis of the distance of a multiple-SNe host
galaxy was performed by Ward et al. (2022). The authors presented a new
version of the BayeSN model (Thorp et al., 2021; Mandel et al., 2022) fitting
the optical-NIR regime (in $grizy$ bands) of SN 2021hpr, but also included the
photometry of SNe 1997bq and 2008fv, both individually and simultaneously.
They inferred the mean value of the distance moduli as $\mu=33.14$ mag with a
standard deviation of 0.01, smaller than the intrinsic scattering of SNe Ia.
The common $\mu$-fit to the data of the three SNe resulted in $\mu=33.13\pm
0.08$ mag.

Ward et al. (2022) also compared BayeSN to the widely used LC-fitter code
SNooPy. The latter code provided a higher standard deviation of $\sigma\sim
0.1$ mag, but the mean value of the individual three distance moduli
($\mu=33.18$ mag) was close to that of the BayeSN code.

In the present paper, we apply SALT3 for a similar analysis as Ward et al.
(2022) and use SALT2.4 and MLCS2k2 for cross-comparing the inferred distance
moduli. All LC fittings are computed separately on the individual datasets of
SNe 1997, 2008fv, and 2021hpr. While SALT3 can be fed by a collection of
various filters and magnitude systems, MLCS2k2 was trained for Johnson-Cousins
$UBVRI$ filters and Vega magnitudes. In the case of SN 2021hpr, where the
object was not observed in Johnson-Cousins $R$ and $I$ filters, only $B$ and
$V$ data were used during the fitting with MLCS2k2. SALT3 fits are shown in
Figs. 3 \- 5, while MLCS2k2 and SALT2 fits can be seen in Figs. 11 \- 16.

Due to the low ($z<0.01$) redshift of the host galaxy, K-correction for
transforming observed LCs to rest-frame bands is estimated in the order of
0.01 mag, which is lower than the random observational uncertainties of the
individual data, thus, K-corrections were neglected. To avoid any discrepancy
due to the different H0,ref values that were assumed to tie LC fitting codes
to the distance scale, all distances are transformed to a common value of
H${}_{0}=73$ km s-1 (Riess et al., 2022):

$\mu(H_{0})=\mu_{\mathrm{fit}}-5\,\log(H_{0}/H_{\mathrm{0,ref}})$ (4)

where $\mu_{\mathrm{fit}}$ is the distance modulus inferred directly by the LC
fitter. Despite that all three LC fitter codes have been trained for the
U-band wavelengths, we refrain from using the U-band (both Johnson and Swift)
magnitudes. The near-UV diversity of SNe Ia is not fully covered, because of
the limited sample of observations in the U-band. Thus, the training of the LC
fitters cannot be sufficient for this spectral regime and increase the
uncertainty of the distance estimation.

The individual distance estimations based on the three SNe and carried out
with the three LC fitters are summarized in Tab. 3.

####  4.1.1 SN 1997bq

The $BVRI$ LCs of SN 1997bq are deficient around the peak, which makes
constraining the date of maximum light more difficult. The polynomial fit of
$B$-band LC provided $T_{max}(B)=50557.5$ MJD which is consistent with the
result of SALT when the time of maximum is a fitting parameter
($T_{max}(B)=50558.0$ MJD). Adopting the latter date for the MLCS2k2, the
inferred host galaxy extinction is $A_{\mathrm{host}}=0.60$ mag, and the
constrained distance modulus, $\mu=33.00$ mag, is close to the Cepheid-
distance ($\mu_{Ceph}=33.014\pm 0.165$ mag, Riess et al., 2022).

The SALT distances perfectly coincide with each other and also support our
final conclusion about the distance of NGC 3145 (see below). Moreover, it is
in between the previous estimates published by Tully et al. (2013) and Jha et
al. (2007) (32.97 and 33.16 mag, respectively, assuming $H_{0}=73$ km s-1, see
Eq. 4) based on the same photometric data of SN 1997bq.

####  4.1.2 SN 2008fv

For SN 2008fv the publicly available $I$-band observations of Biscardi et al.
(2012) are significantly brighter than expected, and they exceed the published
$I$-band magnitudes of Tsvetkov & Elenin (2010) by $\sim$0.8 mag. Ward et al.
(2022) investigated the colours of SN 2008fv and concluded that the $I$-band
data of Biscardi et al. (2012) are probably inaccurate, thus, they were
discarded from the fitting process. The Biscardi et al. (2012) photometry in
other filters was also discredited by Ward et al. (2022), but the estimated
discrepancy is in the order of 0.1 mag. In this study, we aimed to use most of
the available data, thus, we adopted the complete $BVR$ photometry of Biscardi
et al. (2012) together with post-maximum observations of Tsvetkov & Elenin
(2010) in $BVRI$.

The MLCS2k2 code provides $\mu_{\mathrm{08fv}}=32.97$ mag, which is in good
agreement with the Cepheid-based distance modulus of the host galaxy (Riess et
al., 2022). The total $A_{\mathrm{V}}$ is constrained as 0.96 mag,
significantly higher than that estimated by Ward et al. (2022) with the BayeSN
code. At the same time, the nearly identical models of SALT2.4 and SALT3 fail
to fit the $I$-band fluxes and infer significantly higher distance moduli of
$\mu\sim 33.33\pm 0.09$ and $\mu\sim 33.35\pm 0.05$ mag, respectively.

####  4.1.3 SN 2021hpr

SN 2021hpr has the most densely covered light curves among the three SNe
extending from $-15$ to $+100$ days in $BVRI$ and $griz$ bands. However, the
$z$-band LC suffers from higher uncertainties due to inferior sky conditions
during the observations, thus, it was omitted from the fitting.

The MLCS2k2 fit is made using only the $BV$ LCs from the (B)RC80 observations
to keep the homogeneity of the dataset and avoid any systematic errors.
However, the result ($\mu=32.85$ mag) greatly differs from any other distance
modulus of this study, which underlines the importance of having a photometric
dataset covering wide spectral range. At the same time, the SALT codes provide
good fits to all of the LCs in all bands around the maximum light, resulting
in similar distance moduli within a $1\sigma$ agreement. As a further
validation, we also include the $BVRI$ photometry of Zhang et al. (2022) to
the homogeneous $BVgriz$ dataset for an additional modeling with SALT3. The
extra LCs and data points barely change the parameters of the fits (Fig. 17),
including the inferred distance modulus of $\mu=33.15$ mag.

As a conclusion, we propose the distance modulus $\mu=33.14\pm 0.05$ mag of SN
2021hpr estimated by SALT3 for the distance of NGC 3147. This distance shows a
good match with the mean value of (all) the distance moduli (including SNe
1997bq and 2008fv) estimated in this study ($\mu=33.12\pm 0.10$ mag), and also
with that of Ward et al. (2022) ($\mu=33.14\pm 0.12$ mag). Moreover, the
inferred distance is consistent with the result of the Cepheid-based distance
($\mu=33.01\pm 0.165$ mag Riess et al., 2022). The summary of the distance
estimation of NGC 3147 published in the literature and inferred in this study
can be found in Fig. 6 and in Tab. 4.

Table 3: Comparison of the distance moduli estimated with different LC fitter codes for the three studied SNe of NGC 3147. Code | SN 1997bq | SN 2008fv | SN 2021hpr
---|---|---|---
SALT3 | 33.11 $\pm$0.05 | 33.35 $\pm$ 0.05 | 33.14 $\pm$ 0.05
MLCS2k2 | 33.00 $\pm$0.10 | 32.97 $\pm 0.20$ | 32.85 $\pm$ 0.05
SALT2 | 33.11 $\pm$0.13 | 33.33 $\pm$ 0.09 | 33.21 $\pm$ 0.07
Figure 3: SALT3 LC model fitting of the photometry of SN 1997bq Figure 4: SALT3 LC model fitting of the photometry of SN 2008fv. Figure 5: SALT3 LC model fitting of the photometry of SN 2021hpr. Table 4: Summary of the most recent distances estimated with approaches. All distance moduli are scaled according to $H_{0}=73$ km s-1 Mpc-1.  Method | $\mu_{\mathrm{NGC3147}}$ | Reference
---|---|---
Cepheid-variables | 33.01 $\pm$ 0.165 | Riess et al. (2022)
Redshift | 33.15 $\pm$ 0.15 | Mould et al. (2000)
Tully-Fisher relation | 33.12 $\pm$ 0.80 | Tully & Fisher (1988)
SN 1997bq (MLCS2k2) | 33.16 $\pm$ 0.11 | Jha et al. (2006)
SN 2008fv (Phillips-relation) | 33.20 $\pm$ 0.10 | Biscardi et al. (2012)
SN 2021hpr (Phillips-relation) | 33.46 $\pm 0.21$ | Zhang et al. (2022)
SN 2021hpr (BayeSN) | 33.14 $\pm$ 0.12 | Ward et al. (2022)
SN 2021hpr (SALT3) | 33.14 $\pm$ 0.05 | This study
Common $\mu$-fit of SNe (BayeSN) | 33.13 $\pm$ 0.08 | Ward et al. (2022)
Average of LC fits of SNe (MLCS2k2, SALT2, SALT3) | 33.12 $\pm$ 0.16 | This study
Figure 6: Summary of the individual distance measurements involving the three
SNe of NGC 3147. The MLCS2k2, the SALT2 and the SALT3 modeling is published in
this study; the distance estimates conducted with BayeSN and SNooPy were
published by Ward et al. (2022), while the ’Others’ refers to the work of Jha
et al. (2006), Biscardi et al. (2012) and Zhang et al. (2022). The most recent
distance estimations based on Cepheid-variables and cosmological redshift are
also shown with horizontal lines.

###  4.2 The physical properties of SN 2021hpr

####  4.2.1 Rise time

To constrain the moment of the explosion of SN 2021hpr, we adopted the
assumption of the expanding fireball model. According to this, the emerging
pre-maximum flux increases as a power-law function of time (Arnett, 1982;
Nugent et al., 2011):

$F=a\cdot\left(T-T_{\mathrm{first}}\right)^{n}$ (5)

Here, $T_{\mathrm{first}}$ is the time of first light, which is not equivalent
to the time of the explosion ($T_{\mathrm{exp}}$), as it refers to the moment
when the first photons emerge, while the latter refers to the actual moment
when the explosion starts. The intermediate ‘dark phase’ may last for a few
hours to days (Piro & Nakar, 2013; Firth et al., 2015).

In theory, the power-law function with $n=2$ is valid only for the bolometric
flux. However, studies pointed out that it is still more-or-less valid for
quasi-monochromatic fluxes in the optical bands, but in those cases the value
of the exponent may differ from the textbook example, varying between $n=2.2$
(Ganeshalingam et al., 2011) and $n=2.44$ (Firth et al., 2015) for normal SNe
Ia.

We fit the early magnitudes of the most densely sampled LCs of SN 2021hpr,
i.e. the B and V bands, simultaneously with the function Eq. 5, but using the
same $T_{\mathrm{first}}$ for both light curves. At first, we fix the exponent
as $n=2$, which corresponds to the classical fireball model. The resulting
moment of first light is $59305.4\pm 1.5$ MJD, which is slightly late in the
aspect of the first detection a day later.

Next, we allow $n$ to vary between 2.0 and 2.5 for each LCs (Fig. 7). The
inferred date of $T_{\mathrm{first}}=59304.6\pm 1.6$ MJD is a more realistic
time for the explosion, considering the early discovery 1.9 days later. The LC
in the $B$-band peaks at 59323.0 MJD as it is constrained by polynomial fit.
As a key parameter, this date is used as input for the MLCS2k2 and SALT2 fits
in the following. The estimated 18.4 days rise time is in good agreement with
the average of normal SNe Ia (18.98 $\pm$ 0.54 days, Firth et al., 2015).

####  4.2.2 Abundance tomography

We carried out modeling the spectroscopic evolution of SN 2021hpr with the
radiative transfer code TARDIS (Kerzendorf & Sim, 2014). To synthesize the
spectral luminosities, we fixed the distance of SN 2021hpr to 42.5 Mpc
corresponding the distance modulus carried out with SALT3 (see in Sec. 4.1.3).
For de-reddening, a total of $A_{\mathrm{V}}=0.17$ mag was adopted as the
average value of the total extinction assumed by Zhang et al. (2022) and Ward
et al. (2022).

Three of the earliest spectra of SN 2021hpr have been subject to fit (see in
Tab. LABEL:tab:spectroscopic_data). The best-fit synthetic spectra are
presented in Fig. 8. The key parameters of the spectral synthesis are the
total bolometric luminosity ($L$), the photospheric velocity
($v_{\mathrm{phot}}$), and the time since the explosion ($t_{\mathrm{exp}}$,
derived from $T_{\mathrm{exp}}$). We fixed the density function of our input
to the well-known W7 model (Nomoto et al., 1984) to reduce the number of free
parameters. The latter simplification results in a discrepancy, if we assume
the constrained $T_{\mathrm{first}}$ (see in Sec. 4.2.1) directly as
$T_{\mathrm{exp}}$, because the diluted density structure causes a too-dense
and too-hot model ejecta, especially for the first epoch (59307.5 MJD). To
compensate for this, we choose to fit $T_{\mathrm{exp}}$ in our abundance
tomography within the range of one day before $T_{\mathrm{first}}$, taking
into account an approximate dark phase. The best value is characterized as
$T_{\mathrm{exp}}=59304.0$ MJD.

The spectral tomography taken at the earliest epoch samples the outermost
layers, which is located above 18 000 km s-1 based on the $v_{\mathrm{phot}}$
at $t_{\mathrm{exp}}=3.5$ days. Two other spectra were taken within half a day
after the first epoch, but these datasets do not carry additional information,
thus, we do not include them in our analysis.

The next spectrum was taken at $t_{\mathrm{exp}}=13.6$ days, when the
photosphere receded to 11 000 km s-1. This agrees with the conclusion of Zhang
et al. (2022), where the authors classified SN 2021hpr as a high-velocity
gradient (HVG, Benetti et al., 2005) SN Ia based on the $\sim 800$ km s-1
day-1 decrease of $v_{\mathrm{phot}}$.

Figure 7: The fit of $BV$ LCs with the function 5, allowing the exponent to
vary between 2.0 ¡ $n$ ¡ 2.5. A shared time of first light
($T_{\mathrm{first}}$) is assumed for both LCs.

Because of the exponentially decreasing density function toward higher
velocities, the dominant light-matter interaction occurs in the few thousand
km s-1 wide region above the photosphere (except for a few elements like Fe).
Thus, the majority of the velocity domain over 10,800 km s-1 is poorly
sampled.

Finally, a third epoch at $t_{\mathrm{exp}}=19.9$ days is chosen for spectral
synthesis ($v_{\mathrm{phot}}$ = 9,800 km s-1). Assuming a linear decrease of
$v_{\mathrm{phot}}$ between the second and third epoch, we characterize
$v_{\mathrm{phot}}$ = 10,000 km s-1 at the moment of maximum light. After the
maximum, the assumption of the blackbody emitting photosphere becomes weak in
the case of the normal Type Ia SNe, which prevents the computation of
realistic spectral fits with TARDIS.

To reduce the number of the fitting parameters, we set the densities fixed to
the exponential fit of the W7 profile (see the upper panel of Fig. 9),
adopting $\rho_{0}=4.7$ g cm-3 as the central density at the reference time
$t_{0}=100$ s after the explosion, and $v_{\mathrm{0}}=2750$ km s-1 as the
exponential decrease in the function:

$\rho(t_{\mathrm{exp}},v)=\rho_{0}\cdot\left(\frac{t_{\mathrm{exp}}}{t_{0}}\right)^{-3}\cdot\exp\left({-\frac{v}{v_{\mathrm{0}}}}\right)$
(6)

The summary of the input physical parameters producing the well-fitting
synthetic spectra of Fig. 8 can be found in Table 5.

Despite the low time resolution, stratification of the chemical mass fractions
can be mapped (Fig. 9), but the steep changes in the abundance functions
cannot be tracked due to limited constraints. The first epoch samples the
outermost region ($v>18\,200$ km s-1), which was designed with the initial
assumption of a pure C/O layer. The abundance of C is fit with a monotonously
decreasing trend inwards the ejecta to reproduce the C II $\lambda 6580$ line.
The model of the outermost region also includes increased intermediate-mass
element (IME) abundances in order to reproduce the HVFs observed at Ca II H&K,
Si II, Ca II NIR lines frequently reported in the literature. Moreover, Mg is
present in the model ejecta with a mass fraction of $X(Mg)>0.1$, but here the
only constraint is the tentative need for the Mg II $\lambda$4481, which, if
real, is severely overlapped with features of ionized iron. The Fe II and Fe
III features are very sensitive to the abundance at this early epoch, and we
characterize it as $X(Fe)=0.01$ above 18 200 km s-1. All these Fe and IME mass
fractions are introduced on the conto of C/O abundances. Note that O is not a
well-constrained element is our fitting process, as the O I $\lambda 7774$
feature is relatively insensitive to the mass fraction of the element. Thus,
we use $X(O)$ as a filler in our chemical composition.

The majority of the model ejecta is designed according to the fit of the
second spectral epoch ($v_{\mathrm{phot}}=10\,900$ km s-1), however, some
compromise has to be implemented to achieve a better agreement for the third
epoch with only a slightly lower photospheric velocity
($v_{\mathrm{phot}}=9\,800$ km s-1). The changes in the abundances of elements
can be tracked by the absorption profile of the prominent lines. The red wing
of the Ca II H&K profile caught by the second epoch, constrains the inner
abundances of the element with an upper limit of $X(Ca)<0.0005$ below 16 000
km s-1. The Si and S abundances peak around $\sim$13 000 km s-1 according to
the fit of Si II $\lambda 6355$ and S II W feature. The red wings of these
absorptions also indicate reduced mass fractions toward the lower velocities.

The IGE elements (except the high-velocity Fe) are limited below $\sim$14 000
km s-1, otherwise, the complex Fe feature around 5000 Å would be excessive at
both epochs, especially towards the shorter wavelengths. Due to the
limitations of IME abundances (see the paragraph above), Fe and Ni become the
dominant elements here.

Note again that the derived abundance structure should be handled with caution
due to the limited fitting constraints provided by the small spectral sample
and the high number of fitting parameters. Between 9,800 and 20,000 km s-1,
where we can the following regions can be distinguished in the model ejecta by
the general trend of the most prominent elements:

  * •

C/O outer region: the chemical profile is dominated by O with an inwards
decreasing C contribution, while there is only a moderate IME and almost no
IGE abundance.

  * •

IME region: the mass fraction of Si higher than 0.5 in a narrow, 1000-2000 km
s-1 wide region; the abundance of C/O drops, while the Ni abundance rises with
decreasing velocity.

  * •

IGE inner region: the 56Ni produced in the explosion and its daughter isotopes
(56Co, 56Fe) dominate that ejecta

This kind of stratification is not specific to only one explosion scenario,
instead, most of the deflagration-to-detonation transition and pure detonation
models show similar regions with varying locations and relative strengths. The
exact velocities, where these regions are separated from each other, just like
the chemical abundances, are sensitive to the initial conditions of the
hydrodynamic simulations. Thus, a direct quantitative comparison with
predictions of any possible explosion scenario is not feasible from the
present dataset.

Figure 8: The TARDIS fits (red lines) of the three spectral epochs (grey), time since explosion ($t_{\mathrm{exp}}$) is indicated in upper right corner. Figure 9: The inferred density (upper panel) and chemical abundances from the abundance tomography carried out via TARDIS fits of the three spectral epochs. Table 5: Summary of input parameters of the TARDIS model ejecta for each epochs of SN 2021hpr. The abundances of groups of elements show the average mass fractions within the 2000 km s-1 range of the photosphere, where the greater part of optical depths emerges. Note that the fitting paramter $L$ indicated here overestimates the bolometric luminosity (see in Sec. 4.2.3) due to the blackbody assumption of TARDIS. $t_{\mathrm{exp}}$ | L | $v_{\mathrm{phot}}$ | $\rho_{\mathrm{phot}}$ | X(IGE) | X(IME) | X(C/O)
---|---|---|---|---|---|---
(d) | (1043 erg s-1) | (km s-1) | (g cm-3) | Averages within 2000 km s-1 above $v_{\mathrm{phot}}$
3.5 | 0.073 | 18.200 | 0.006 | 0.00 | 0.27 | 0.73
13.6 | 1.30 | 10.900 | 0.087 | 0.30 | 0.45 | 0.25
19.9 | 1.05 | 9.800 | 0.130 | 0.48 | 0.27 | 0.25

####  4.2.3 Analysis of the bolometric light curve

By using the distance modulus $\mu=33.14$ mag constrained in Sec. 4.1.3,
physical properties like the initial radioactive nickel mass produced in the
explosion (MNi) can be constrained. To do so, we construct the pseudo-
bolometric light curve of SN 2021hpr using two different methods. In both
cases, we apply the available fluxes from Swift UVM2 filter to SDSS z-band,
but exclude the Swift UVW1 and UVW2 filters from the process because of their
significant red leak (Brown et al., 2016). As a first approximation (hereafter
referred to as BOL1), we use the
SuperBol444https://zenodo.org/badge/latestdoi/73849147 code for computing
polynomial fits and extrapolations on the individual light curves. Then,
before the integration, we extrapolate the observed SEDs with blackbody fits
in the bluest and reddest bands for each epoch. By integrating these
individual light curves we generate the pseudo-bolometric LC with BB-
correction.

As a second method (BOL2), the flux contribution from the unobserved UV- and
IR regimes is estimated in another way. For integrating the UV contribution,
the trapezoidal rule is applied with the assumption that the flux reaches zero
at 1000 Å (Marion et al., 2014; Bora et al., 2022). The infrared contribution
is taken into account by the exact integration of a Rayleigh-Jeans tail
attached to the observed flux at the longest observed wavelength (i.e. $I$\-
or $i$-band in the present case). Hence, in the following sessions, we refer
to this light curve as the pseudo-bolometric LC with Rayleigh-Jeans tail.

There are multiple ways to estimate the initial 56Ni mass from the bolometric
LC, e.g. the t15 method (Sukhbold, 2019) or the tail luminosity method
(Afsariardchi et al., 2021). Here we fit the estimated pseudo-bolometric LC
(supplemented with blackbody corrections) with a semi-analytic code based on
the model of Arnett (1982) assuming radiative diffusion in a homologously
expanding SN ejecta heated by the radioactive decay of 56Ni and 56Co. This
model has been developed further by Valenti et al. (2008) and Chatzopoulos et
al. (2012).

The fitting parameters of the Arnett-model are the mean diffusion timescale,
$t_{\mathrm{d}}$ (also called the light curve timescale, which is practically
the geometric mean of the diffusion and the expansion timescales, see Arnett
1982), the gamma-ray leakage timescale, $T_{\gamma}$. and the initial mass of
the radioactive 56Ni, $M_{\mathrm{Ni}}$. These parameters are directly related
to the global physical parameters of the ejecta, namely the total ejecta mass
$M_{\mathrm{ej}}$, the characteristic expansion velocity $v_{\mathrm{exp}}$,
the effective optical opacity $\kappa$, and the opacity for gamma-rays
$\kappa_{\gamma}$ (see e.g. Könyves-Tóth et al., 2020).

Since the Arnett-model provides only two timescales ($t_{\mathrm{d}}$ and
$T_{\gamma}$) for three physical parameters ($M_{\mathrm{ej}}$,
$v_{\mathrm{exp}}$ and $\kappa$), the physical parameters cannot be
constrained independently from only photometry. To overcome this difficulty,
Könyves-Tóth et al. (2020) used an approximate, iterative method by estimating
a lower and an upper limit for the optical opacity first, then using the
average of them to constrain the ejecta mass and the expansion velocity.

Recently the diffusion model was challenged by Khatami & Kasen (2019), who
suggested an alternative formalism to estimate the initial nickel masses for
various types of SN explosions. However, Bora et al. (2022) demonstrated that
for Type Ia SNe in particular, the nickel masses inferred from the two methods
are consistent within their uncertainties.

As a first attempt, we fit the pseudo-bolometric light curve (see in Fig. 10)
provided by SuperBol (BOL1). The best-fit model parameters estimated by the
Minim code (Chatzopoulos et al., 2013) are shown in Table 6 together with the
inferred physical parameters. These data suggest a nickel mass of
$M_{\mathrm{Ni}}\sim 0.48$ M⊙, which is consistent with the absolute peak
magnitude of $M_{\mathrm{max}}(B)\sim 19$ mag. However, the BOL1 LC suffers
from higher uncertainties mainly due to the imperfect BB-extensions to the
near-UV and near-IR regime fit by the SuperBol code. The strong bump between
+20 and +30 days is partially non-physical as the BB-fits of the code
overestimate the flux contribution from the not-observed spectral regimes.
Since these uncertainties lead to an inferior fitting, we disfavour the
results from the BOL1 LC.

The BOL2 light curve is also fit (Fig. 10) following the same methodology, and
the inferred physical quantities from the best-fit parameters (see Table 6
indicate a realistic M${}_{\mathrm{Ni}}\sim 0.44$ M⊙, and
$v_{\mathrm{exp}}\sim 11200$ km s-1. The mean optical opacity is constrained
as $\kappa\sim 0.144$ according to the iterative method by Könyves-Tóth et al.
(2020), which corresponds to an ejecta mass of $\sim 1.12$ M⊙.

As an alternative solution, the same set of best-fit parameters can be used
with a fixed expansion velocity adopted from the spectral analysis (Section
4.2.2). The results are shown in the 4th column of Table 6. By assuming
$v_{\mathrm{phot}}$ = 10 000 km s-1 as vexp, we get a lower ejecta mass of
$M_{\mathrm{ej}}\sim 0.89$ M⊙. Note, however, that even though
$v_{\mathrm{phot}}$ is generally used as an approximation for
$v_{\mathrm{exp}}$, these two velocities are not the same quantity by
definition (see Arnett, 1982). Thus, this estimate can be considered only as a
lower limit for the ejecta mass.

For further validation, we can also constrain the value of the effective
optical opacity ($\kappa$) directly from fitting the light curve with the
Arnett-model-based LC2 code (Nagy & Vinkó, 2016) coupled with Minim. In LC2
the ejecta mass, the radius of the progenitor, the nickel mass, the opacity
and the kinetic energy of the ejecta can be chosen as fitting parameters.
Since our data span only the first few months after the explosion, we ignore
the leaking of positrons from the ejecta, and assume only the usual gamma-ray
leaking with an effective gamma-ray opacity of $\kappa_{\gamma}\sim 0.03$
cm2g-1. The advantage of this approach is that the highly uncertain expansion
velocity is not a fitting (and neither an input) parameter, however, the
results are sensitive to the value of $\kappa$ included in the fitting. Note,
that due to significant parameter correlations in the Arnett-model (e.g. Nagy
et al., 2014; Nagy & Vinkó, 2016) all the previously described methods are
tainted with significant systematic uncertainties. We take into account these
correlations while estimating the average errors of the inferred physical
properties of SN 2021hpr (see Table 6).

The main parameters inferred from the fitted $\kappa$ method
($M_{\mathrm{ej}}\sim 1.28$ M⊙) are consistent with those estimated by the
previous approaches and are rather close to the results of the mean $\kappa$
method. Thus, we accept $M_{\mathrm{Ni}}=0.44\pm 0.14$ M⊙ and the other
corresponding parameters (see the 3rd column with boldface fonts in Table 6)
as the final result of the bolometric LC analysis.

Figure 10:  Pseudo-bolometric light curves estimated from the integration of observed magnitudes with trapezoidal approximation for the UV and Rayleigh-Jeans tail for the NIR contribution (red), and the LC estimated by blackbody fits to the UV/NIR regions (blue), and their best-fit Arnett-models (see Sec. 4.2.3.) Table 6: Comparison of the physical parameters of SN 2021hpr from fitting the pseudo-bolometric light curves, taking into account gamma-ray leakage. The three methods are a) assuming the average of the upper and lower limit of $\kappa$ as effective opacity (mean $\kappa$, Könyves-Tóth et al., 2020); b) assuming a fixed expansion velocity from the spectroscopic analysis (adopted $v_{\mathrm{exp}}$, Könyves-Tóth et al., 2020); c) direct fit of the effective opacity (fitted $\kappa$, Nagy et al., 2014). The approximate uncertainties are computed from error propagation in the corresponding method. The only exception is $M_{\mathrm{Ni}}$, where the Minim algorithm provided too low uncertainties from the fits; therefore, the uncertainty of the nickel masses is estimated as in Nagy & Vinkó (2016).  Bolometric light curve | BOL1 | BOL2
---|---|---
Parameter | mean $\kappa$ | mean $\kappa$ | adopted $v_{\mathrm{exp}}$ | fitted $\kappa$
$M_{\mathrm{Ni}}$ [M⊙] | 0.48 (0.16) | 0.44 (0.14) | 0.44 (0.14) | 0.46 (0.15)
$t_{\mathrm{d}}$ [day] | 13.72 (0.30) | 13.68 (0.227) | 13.68 (0.227) | 16.3 (1.22)
$T_{\mathrm{\gamma}}$ [day] | 44.83 (0.84) | 41.47 (0.58) | 41.47 (0.58) | 50.1 (4.6)
$\kappa$ [cm2 g-1] | 0.139 (0.010) | 0.144 (0.017) | 0.161 (0.018) | 0.16 (0.01)
$v_{\mathrm{exp}}$ [$10^{3}$ km s-1] | 10.8 (1.0) | 11.2 (1.2) | 10.0 (1.0) | 9.9 (0.7)
Mej [M⊙] | 1.22 (0.24) | 1.12 (0.28) | 0.89 (0.21) | 1.28 (0.15)
Ekin [$10^{51}$ erg] | 0.85 (0.38) | 0.84 (0.29) | 0.54 (0.28) | 0.75 (0.10)

##  5 Conclusion

We analyzed the optical/UV light curve of the normal Type Ia SN 2021hpr
together with its two well-observed siblings, SNe 1997bq and 2008fv, that
appeared in the same host galaxy, NGC 3147. The three SNe provide a unique
opportunity to revise the distance of their host, and allow the testing of
systematic effects influencing the various distance estimation methods.

We took new photometric data on SN 2021hpr with the twin robotic telescopes,
RC80 and BRC80, from two Hungarian observatories (Konkoly and Baja). The light
curves, including $BVgriz$ filters, start from $t=\,-14.9$ days, adopting
59323.0 MJD as the moment of maximum light in the $B$-band. Based on fitting
the early data points, we constrained $T_{first}\sim 59304.4$ MJD as the date
of first light, which gives $t_{\mathrm{rise}}=18.4$ days for the rise time.
Besides the optical follow-up, we also downloaded and analyzed the available
Swift UVOT photometry taken from $-16.1$ to $+12.3$ days relative to the
$B$-band maximum.

We estimated the distance to SN 2021hpr by fitting our new $BVgri$ data with
the LC-fitter codes MLCS2k2, SALT2 and SALT3. The same three codes were
applied for the two sibling SNe, and the inferred individual distances were
compared to each other. We found that the results scatter between $\mu=32.97$
and $33.35$ mag with a mean value of $\mu=33.12\pm 0.16$ mag. This estimation
for the distance of NGC 3147 is in good agreement with previous SN-based
studies (see Table 4 and Figure 6), and also marginally consistent with the
most recent Cepheid-based distance ($\mu_{\mathrm{Ceph}}=33.01\pm 0.165$ mag,
Riess et al., 2022).

We adopted the distance inferred from the SALT3 fitting to SN 2021hpr, where
the most complete self-consistent optical dataset was applied and the models
provided good fits to all bands. The SALT3-based distance modulus
$\mu=33.14\pm 0.05$ mag is also consistent with that of Ward et al. (2022)
estimated by the BayeSN code from fitting to an independent LC of SN 2021hpr.

In order to study the physical parameters of SN 2021hpr based on our new,
improved distance, abundance tomography was performed by fitting three pre-
maximum spectra with the radiative transfer code TARDIS. The constrained
chemical structure is consistent with multiple explosion models, including
deflagration-to-detonation transition and pure detonation scenarios, with an
additional overabundance of Mg, Si and Ca to reproduce the observed blue-
shifts and high-velocity features of the corresponding absorption lines.

We calculated the bolometric LC of SN 2021hpr from its optical data
supplemented by near-UV data from Swift, and fit it with the radiative
diffusion Arnett-model. The peak luminosity turned out to be $1.02\times
10^{43}$ erg/s, while the total mass of 56Ni produced in the explosion is
estimated as $0.44\pm 0.14M_{\odot}$, while the ejecta mass and expansion
velocity were calculated as $1.12\pm 0.28$ M⊙ and $11\,200\pm 1200$ km s-1,
respectively. Considering all the estimated physical properties and
spectroscopic characteristics of SN 2021hpr, it belongs to the Branch-normal
and high velocity gradient classes of Type Ia SNe.

###### Acknowledgements.

This research made use of tardis, a community-developed software package for
spectral synthesis in supernovae (Kerzendorf & Sim, 2014; Kerzendorf et al.,
2023). The development of tardis received support from GitHub, the Google
Summer of Code initiative, and from ESA’s Summer of Code in Space program.
tardis is a fiscally sponsored project of NumFOCUS. tardis makes extensive use
of Astropy and Pyne. The authors acknowledge the Hungarian National Research,
Development and Innovation Office grants OTKA K-131508, K-138962, K-142534,
FK-134432, KKP-143986 (Élvonal), and 2019-2.1.11-TéT-2019-00056. LK
acknowledges the Hungarian National Research, Development and Innovation
Office grant OTKA PD-134784. BB, RKT and ZsB is supported by the ÚNKP-22-2 New
National Excellence Program of the Ministry for Culture and Innovation from
the source of the National Research, Development and Innovation Fund. APN is
supported by NKFIH/OTKA PD-134434 grant, which is founded by the Hungarian
National Development and Innovation Fund. LK, KV, and TS are Bolyai János
Research Fellows of the Hungarian Academy of Sciences. KV and TS are supported
by the Bolyai+ grants ÚNKP-22-5-ELTE-1093 and ÚNKP-22-5-SZTE-591,
respectively. SZ is supported by the National Talent Programme under NTP-
NFTÖ-22-B-0166 Grant. ZMS acknowledges funding from a St Leonards scholarship
from the University of St Andrews. ZMS is a member of the International Max
Planck Research School (IMPRS) for Astronomy and Astrophysics at the
Universities of Bonn and Cologne.

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##  Appendix A Observational data

Table 7: Log of SN 2021hpr photometry from Observatory of Baja.  MJD | B | V | g | r | i | z
---|---|---|---|---|---|---
59312.1 | 15.312 (0.08) | 15.237 (0.06) | 15.201 (0.01) | 15.201 (0.01) | 15.509 (0.01) | 15.528 (0.02)
59312.9 | 15.050 (0.16) | 15.022 (0.08) | 15.004 (0.01) | 15.012 (0.01) | 15.324 (0.01) | 15.398 (0.02)
59314.8 | 14.684 (0.06) | 14.684 (0.04) | 14.671 (0.01) | 14.733 (0.02) | 14.998 (0.01) | 15.070 (0.02)
59329.1 | 14.418 (0.10) | 14.266 (0.05) | 14.327 (0.01) | 14.321 (0.01) | 15.045 (0.01) | 15.085 (0.04)
59339.0 | 15.481 (0.22) | 14.762 (0.05) | 14.948 (0.03) | 14.893 (0.02) | 15.484 (0.03) | 15.192 (0.10)
59403.8 | 17.678 (0.34) | 17.114 (0.08) | 17.331 (0.05) | 17.119 (0.03) | 17.610 (0.04) | 18.012 (0.14)
59407.9 | 17.700 (0.07) | 17.174 (0.03) | 17.350 (0.02) | 17.242 (0.02) | 17.600 (0.03) | 17.871 (0.11)
59426.0 | 18.023 (0.37) | 17.675 (0.09) | 17.708 (0.07) | 17.916 (0.05) | 18.222 (0.06) | 18.631 (0.22)
59432.8 | 18.269 (0.95) | 17.672 (0.10) | 17.789 (0.04) | 17.978 (0.98) | 18.403 (0.06) | 18.720 (0.16)
59446.8 | 18.362 (0.29) | 18.062 (0.08) | 18.109 (0.07) | 18.398 (0.06) | 18.748 (0.16) | 18.209 (0.29)
59459.7 | 18.356 (0.78) | 18.313 (0.13) | 18.112 (0.05) | 18.793 (0.06) | 18.989 (0.10) | 19.095 (0.30)
59467.8 | 18.854 (0.15) | 18.433 (0.11) | 18.349 (0.05) | 19.224 (0.08) | 19.347 (0.12) | 18.937 (0.27)
Table 8: Log of SN 2021hpr photometry from Observatory of Piszkesteto.  MJD | B | V | g | r | i | z
---|---|---|---|---|---|---
59308.01 | 17.72 (0.10) | 16.78 (0.06) | 17.06 (0.05) | 16.81 (0.04) | 17.27 (0.07) | 16.90 (0.13)
59308.91 | 17.00 (0.10) | 16.31 (0.04) | 16.49 (0.03) | 16.36 (0.03) | 16.74 (0.03) | 16.50 (0.06)
59310.93 | 15.83 (0.08) | 15.45 (0.05) | 15.52 (0.02) | 15.53 (0.02) | 15.87 (0.02) | 15.81 (0.04)
59311.90 | 15.45 (0.09) | 15.15 (0.05) | 15.27 (0.04) | 15.23 (0.03) | 15.57 (0.02) | 15.61 (0.04)
59312.92 | 15.19 (0.07) | 14.93 (0.04) | 14.98 (0.03) | 15.00 (0.02) | 15.31 (0.02) | 15.35 (0.04)
59314.90 | 14.84 (0.08) | 14.62 (0.04) | 14.63 (0.02) | 14.66 (0.02) | 15.00 (0.02) | 15.12 (0.04)
59319.97 | 14.45 (0.09) | 14.20 (0.04) | 14.24 (0.03) | 14.27 (0.03) | 14.83 (0.02) | 14.89 (0.04)
59320.87 | 14.42 (0.08) | 14.18 (0.04) | 14.22 (0.03) | 14.24 (0.03) | 14.83 (0.02) | 14.89 (0.04)
59324.79 | 14.46 (0.08) | 14.13 (0.04) | 13.95 (0.09) | 14.17 (0.04) | 14.90 (0.03) | 14.91 (0.05)
59325.88 | 14.49 (0.20) | 14.21 (0.05) | 14.29 (0.07) | 14.27 (0.02) | 14.92 (0.03) | 14.97 (0.05)
59328.89 | 14.61 (0.16) | 14.30 (0.05) | 14.34 (0.09) | 14.30 (0.05) | 15.06 (0.04) | 15.10 (0.05)
59331.88 | 14.83 (0.13) | 14.35 (0.05) | 14.48 (0.05) | 14.49 (0.03) | 15.22 (0.04) | 15.17 (0.05)
59334.84 | 15.11 (0.08) | 14.57 (0.04) | 14.68 (0.04) | 14.71 (0.03) | 15.45 (0.03) | 15.16 (0.09)
59335.99 | 15.28 (0.15) | 14.65 (0.06) | 14.77 (0.04) | 14.80 (0.04) | 15.47 (0.03) | –
59336.84 | 15.25 (0.10) | 14.69 (0.04) | 14.83 (0.04) | 14.81 (0.03) | 15.47 (0.03) | 15.14 (0.08)
59338.85 | 15.56 (0.09) | 14.79 (0.04) | 14.98 (0.03) | 14.87 (0.03) | 15.49 (0.02) | 15.23 (0.21)
59341.94 | 15.90 (0.12) | 14.97 (0.04) | 15.22 (0.03) | 14.93 (0.03) | 15.43 (0.02) | 15.08 (0.04)
59342.85 | 16.00 (0.09) | 15.01 (0.05) | 15.31 (0.03) | 14.94 (0.03) | 15.41 (0.02) | 15.08 (0.05)
59344.92 | 16.21 (0.08) | 15.11 (0.03) | 15.48 (0.03) | 14.96 (0.03) | 15.36 (0.02) | 15.05 (0.05)
59345.86 | 16.28 (0.09) | 15.17 (0.03) | 15.55 (0.02) | 14.98 (0.02) | 15.34 (0.02) | 15.07 (0.05)
59350.87 | 16.69 (0.08) | 15.44 (0.04) | 15.95 (0.03) | 15.13 (0.03) | 15.32 (0.02) | 15.07 (0.05)
59352.83 | 16.86 (0.13) | 15.57 (0.05) | 16.09 (0.03) | 15.26 (0.02) | 15.42 (0.02) | 15.11 (0.04)
59354.92 | 16.98 (0.08) | 15.73 (0.04) | 16.24 (0.04) | 15.41 (0.03) | 15.54 (0.03) | 15.25 (0.04)
59359.94 | 17.34 (0.15) | 16.04 (0.06) | 16.50 (0.07) | 15.72 (0.03) | 15.90 (0.02) | 15.62 (0.05)
59360.88 | 17.42 (0.16) | 16.04 (0.06) | 16.49 (0.09) | 15.73 (0.04) | 15.94 (0.02) | 15.67 (0.06)
59365.91 | 17.46 (0.08) | 16.19 (0.06) | 16.66 (0.04) | 15.99 (0.04) | 16.27 (0.04) | 15.99 (0.08)
59368.91 | 17.34 (0.08) | 16.28 (0.04) | 16.71 (0.04) | 16.09 (0.03) | 16.36 (0.02) | 16.17 (0.06)
59369.84 | 17.40 (0.16) | 16.37 (0.06) | 16.78 (0.05) | 16.14 (0.05) | 16.39 (0.03) | 16.17 (0.08)
59370.95 | 17.51 (0.10) | 16.38 (0.05) | 16.78 (0.04) | 16.18 (0.03) | 16.47 (0.03) | 16.28 (0.08)
59381.86 | 17.55 (0.13) | 16.67 (0.05) | 16.98 (0.05) | 16.54 (0.03) | 16.92 (0.05) | 16.64 (0.07)
59387.90 | 17.82 (0.14) | 16.82 (0.05) | 17.23 (0.07) | 16.80 (0.04) | 17.13 (0.05) | 17.14 (0.11)
59387.90 | 17.82 (0.14) | 16.82 (0.05) | 17.23 (0.07) | 16.80 (0.04) | 17.13 (0.05) | 17.14 (0.11)
59393.88 | 17.74 (0.10) | 16.97 (0.04) | 17.20 (0.03) | 16.93 (0.03) | 17.37 (0.03) | 17.29 (0.11)
59402.86 | 17.82 (0.11) | 17.17 (0.04) | 17.38 (0.04) | 17.16 (0.02) | 17.61 (0.04) | 17.85 (0.17)
59407.88 | 17.95 (0.09) | 17.27 (0.04) | 17.43 (0.03) | 17.36 (0.03) | 17.80 (0.03) | 17.91 (0.09)
59419.01 | 18.25 (0.23) | 17.58 (0.11) | 17.66 (0.12) | 17.75 (0.07) | 18.06 (0.11) | 18.45 (0.35)
59423.97 | 18.13 (0.09) | 17.59 (0.04) | 17.75 (0.04) | 17.82 (0.04) | 18.21 (0.08) | 18.58 (0.24)
59429.04 | 17.95 (0.25) | 17.76 (0.10) | 18.32 (0.32) | 18.08 (0.09) | 18.20 (0.32) | –
59434.02 | 18.36 (0.08) | 17.84 (0.05) | 17.81 (0.04) | 18.11 (0.03) | 18.60 (0.11) | 19.28 (0.62)
Table 9: Log of Swift UVOT photometry of SN 2021hpr.  MJD | UW2 | UM2 | UW1 | U | B | V
---|---|---|---|---|---|---
59335.240 | 18.005 (0.102) | 18.741 (0.159) | 16.788 (0.059) | 15.379 (0.033) | 15.062 (0.020) | 14.553 (0.026)
59332.190 | 17.723 (0.084) | 18.390 (0.125) | 16.354 (0.045) | 15.030 (0.027) | 14.770 (0.017) | 14.401 (0.024)
59331.060 | 17.875 (0.191) | 18.785 (0.358) | 16.249 (0.087) | 14.944 (0.053) | 14.544 (0.033) | 14.292 (0.047)
59326.290 | 17.178 (0.079) | 18.192 (0.150) | 15.848 (0.047) | 14.284 (0.025) | 14.293 (0.020) | 14.085 (0.028)
59321.590 | – | – | 15.552 (0.026) | – | – | –
59321.570 | 17.084 (0.062) | – | – | – | – | –
59318.730 | 17.014 (0.059) | 18.414 (0.154) | 15.445 (0.032) | 13.882 (0.018) | 14.220 (0.016) | 14.255 (0.026)
59316.660 | 17.206 (0.060) | 18.509 (0.146) | 15.633 (0.030) | 14.031 (0.017) | 14.326 (0.015) | 14.415 (0.025)
59311.890 | 18.208 (0.260) | 19.094 (0.564) | 16.925 (0.137) | 15.358 (0.066) | 15.251 (0.045) | 15.184 (0.079)
59309.830 | – | – | 18.046 (0.161) | 16.727 (0.063) | 16.115 (0.031) | 15.754 (0.067)
59308.460 | – | – | 18.651 (0.266) | 17.704 (0.128) | 17.010 (0.054) | 16.438 (0.091)
59307.240 | – | – | 18.983 (0.353) | 18.217 (0.195) | 17.753 (0.096) | 17.503 (0.244)
59306.800 | – | 18.759 (0.346) | 18.180 (0.240) | 17.320 (0.161) | 18.078 (0.277) | –
Table 10: Log of the spectra; $t_{\mathrm{exp}}$ shows the time since the date of explosion constrained in the abundance tomography (MJD $59\,304.0$); while the phases are given relative to the maximum in $B$-band (MJD $59\,321.9$).  MJD | texp [d] | Phase [d] | Telescope/Instrument | Wavelength range [Å]
---|---|---|---|---
59307.5 | 3.5 | -14.4 | XLT/BFOSC | 3700 - 8800
59317.6 | 13.6 | -4.3 | XLT/BFOSC | 3700 - 8800
59323.9 | 19.9 | +2.0 | Smolecin Observatory | 3900 - 7100

##  Appendix B Light curve fits

In the followings, the plots of MLCS2k2, SALT2.4 and SALT3 fits for SNe
1997bq, 2008fv and 2021hpr are listed. The distance moduli estimated from
these fits are discussed in Sec. 4.1, while the corresponding fitting
parameters are listed in Tab. B.

Table 11: Log of the light curve fits.  | SALT3 | SALT2.4 | MLCS2k2
---|---|---|---
| x0 | x1 | c | x0 | x1 | c | Ahost | $\Delta$
SN 1997bq | 0.03142 | -1.025 | 0.0696 | 0.0305 | -0.6073 | 0.1094 | 0.60 | 0.00
SN 2008fv | 0.0260 | 0.9040 | 0.1452 | 0.02688 | 0.7690 | 0.1433 | 0.90 | -0.29
SN 2021hpr | 0.04091 | -0.044 | 0.0051 | 0.03739 | 0.4483 | 0.0423 | 0.45 | 0.03

Figure 11: MLCS2k2 LC model fitting of the photometry of SN 1997bq Figure 12:
SALT2.4 LC model fitting of the photometry of SN 1997bq Figure 13: MLCS2k2 LC
model fitting of the photometry of SN 2008fv. Figure 14: SALT2.4 LC model
fitting of the photometry of SN 2008fv. Figure 15: MLCS2k2 LC model fitting of
the photometry of SN 2021hpr. Figure 16: SALT2.4 LC model fitting of the
photometry of SN 2021hpr. Figure 17: SALT3 LC model fitting of the extended
photometric dataset of SN 2021hpr.