11institutetext:  Max-Planck-Institute für Astrophysik, Karl-Schwarzschild-
Str. 1, 85748 Garching, Germany 22institutetext: Physics Department,
Technische Universität München, James-Franck-Str. 1, 85748 Garching, Germany
33institutetext: Exzellenzcluster ORIGINS, Boltzmannstr. 2, 85748 Garching,
Germany 44institutetext: GSI Helmholtzzentrum für Schwerionenforschung,
Planckstraße 1, 64291 Darmstadt, Germany 55institutetext: Aix Marseille Univ,
CNRS, CNES, LAM, Marseille, France 66institutetext: European Southern
Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany
77institutetext: School of Natural Sciences, Technische Universität München,
James-Franck-Str. 1., 85748 Garching, Germany 88institutetext: Zentrum für
Astronomie der Universität Heidelberg, Institut für Theoretische Astrophysik,
Philosophenweg 12, 69120 Heidelberg, Germany 99institutetext: Heidelberg
Institute for Theoretical Studies, Schloss-Wolfsbrunnenweg 35, 69118
Heidelberg, Germany

# Family dispute: do Type IIP supernova siblings agree on their distance?

G. Csörnyei 1122 C. Vogl 1133 S. Taubenberger 1122 A. Flörs 44 S. Blondin 55
M. G. Cudmani 116677 A. Holas 8899 S. Kressierer 6677 B. Leibundgut 2266 W.
Hillebrandt 112233

(Received XXX)

###### Abstract

Context. Type II supernovae provide a direct way to estimate distances through
the expanding photosphere method, which is independent of the cosmic distance
ladder. A recently introduced Gaussian process-based method allows for a fast
and precise modelling of spectral time series, which puts accurate and
computationally cheap Type II-based absolute distance determinations within
reach.

Aims. The goal of the paper is to assess the internal consistency of this new
modelling technique coupled with the distance estimation empirically, using
the spectral time series of supernova siblings, i.e. supernovae that exploded
in the same host galaxy.

Methods. We use a recently developed spectral emulator code, which is trained
on Tardis radiative transfer models and is capable of a fast maximum
likelihood parameter estimation and spectral fitting. After calculating the
relevant physical parameters of supernovae we apply the expanding photosphere
method to estimate their distances. Finally, we test the consistency of the
obtained values by applying the formalism of Bayes factors.

Results. The distances to four different host galaxies were estimated based on
two supernovae in each. The distance estimates are not only consistent within
the errors for each of the supernova sibling pairs, but in the case of two
hosts they are precise to better than 5%. The analysis also showed that the
main limiting factor of this estimation is the number and quality of spectra
available for the individual objects, rather than the physical differences of
the siblings.

Conclusions. Even though the literature data we used was not tailored for the
requirements of our analysis, the agreement of the final estimates shows that
the method is robust and is capable of inferring both precise and consistent
distances. By using high-quality spectral time series, this method can provide
precise distance estimates independent of the distance ladder, which are of
high value for cosmology.

###### Key Words.:

Radiative transfer – Stars: distances – supernovae: general – supernovae:
individual (2002gw, 2003hl, 2003iq, 2004et, 2008ho, 2008in, 2017eaw, 2020jfo)

##  1 Introduction

Sibling supernovae are transients that exploded in the same host galaxy. Given
that these supernovae occurred at essentially the same distance from us, they
allow us to test distance estimation methods and investigate their systematics
empirically. Such consistency checks have been performed in the literature for
pairs of Type Ia supernovae (see e.g. Burns et al. 2020; Scolnic et al. 2020)
and for a pair of a Type Ia and a Type IIP supernova (Graham et al., 2022)
recently, yielding good matches. However, no similar tests were conducted on
Type II supernovae specifically yet.

Previously, Type II supernova siblings were analysed with different goals. For
example, Poznanski et al. (2009) used such siblings to check the colour term
in their standardizable candle model under the assumption that they share the
same distance. More recently, Vinkó et al. (2012) and Szalai et al. (2019)
used a Type IIP-IIb and a IIP-IIP pair respectively to constrain the distances
to their hosts better than with a single transient. In contrast, here we
perform the first dedicated distance consistency check for Type IIP supernova
siblings.

Type II supernovae (SNe II) correspond to the final gravitational collapse of
massive ($\geq 8M_{\odot}$), red or blue supergiant stars, which is supported
by pre-explosion images (Smartt, 2009) and theoretical models (Tinyanont et
al., 2022). These supernovae are historically subdivided into two main
classes, Type IIP (plateau) and Type IIL (linear), based on their light curves
(e.g. Patat et al. 1994). However, there are several indications that these
objects can be rather explained as a continuous distribution, as opposed to
distinct classes (Anderson et al., 2014; Sanders et al., 2015; Galbany et al.,
2016; Morozova et al., 2017; Pessi et al., 2019).

Due to their high intrinsic luminosity and the fact that these supernovae are
the most frequent stellar explosions in the Universe (Li et al., 2011), Type
II SNe make for excellent distance indicators. To date, mainly two methods
have been used to estimate the distances to Type II SNe: the expanding
photosphere method (EPM, Kirshner & Kwan 1974), which is a geometric technique
relating the photospheric radius of the SN to its angular size, and the
standardized candle method (SCM, Hamuy & Pinto 2002), which is based on an
empirical relation between the photospheric expansion velocity and the plateau
luminosity of the supernovae. However, only the EPM provides a distance
estimation that does not require any external calibration, as opposed to the
SCM and other distance ladder formalism methods. It is independent of any
other distance measurements and hence of the cosmic distance ladder. Owing to
these advantages, it has already been applied to various sets of SNe to derive
the Hubble constant (see e.g. Schmidt et al. 1992). The classical EPM analysis
is, however, prone to several uncertainties as, for example, shown by Jones et
al. (2009): the results can be subject to systematic differences depending on
which atmospheric model and thus dilution factors are assumed (e.g. Eastman et
al. 1996 or Dessart & Hillier 2005a) or which photometric passband set is used
for the photometry. To avoid such issues, as pointed out by Dessart & Hillier
(2005a), one is required to carry out the EPM based on the radiative transfer-
based modelling of the spectra and estimate the input parameters through these
models. We call this augmented version the tailored-EPM analysis, which bears
more similarities to the Spectral-fitting Expanding Atmosphere Method (SEAM)
introduced by Baron et al. (2004). This step not only allows for a more
precise estimation of physical parameters but also avoids the detour of
choosing the dilution factors for the EPM.

Although the number of available spectroscopic observations has grown
significantly over the past years, the spectral modelling remains a time-
consuming and laborious process. To change this situation, Vogl et al. (2020)
developed an emulator based on spectra calculated with the Tardis radiative
transfer code (Kerzendorf & Sim, 2014), which allows for a maximum likelihood
parameter estimation and modelling of the spectral time series several orders
of magnitude faster than the conventional methods. To showcase the code, Vogl
et al. (2020) performed the tailored-EPM analysis for SNe 1999em and 2005cs,
showing that a few per cent precision in the derived distance can be achieved.

Here, we attempt to further test the method and investigate its internal
consistency empirically by applying it to sibling supernovae. In their case,
the maximum possible separation between the siblings is set by the line-of-
sight extension of the thick disk of their host galaxies. For face-on galaxies
(such as all the hosts in our sample), this is at most about 10 kpc (Gilmore &
Reid, 1983). At a distance of $\sim$6 Mpc, for the closest host galaxy in our
sample, this corresponds only to a maximum relative error of 0.2%. Hence, the
EPM should yield the same distance for the sibling pairs within the
uncertainties. As a result, siblings provide a simple empirical way to assess
the consistency and robustness of the algorithm: they allow us to test whether
we find the same distances for the pairs even though the underlying conditions
(such as the metallicity, the mass of the progenitor, the amount of ejecta-CSM
interaction, the reddening), as well as the overall data quality and the level
of calibration are different, and whether the inferred uncertainties are
reasonable.

The paper is structured as follows. In Sect. 2 we give a brief overview of the
data collected for this study. Sect. 3 describes the calibration steps we
applied to the data to achieve a similar level of calibration for all objects,
provides the background of the emulator-based modelling, and finally outline
the distance estimation. Sect. 4 shows the results of the modelling for the
individual host galaxies, while Sect. 5 discusses these results and details
the consistency check of the method. In Sect. 6 we summarize and conclude the
paper.

##  2 Data

To obtain a set of objects compatible with the goals of this study, we
filtered the catalogue of known supernovae through the Open Supernova
Catalog111https://sne.space/ (OSC, Guillochon et al. 2017). Apart from looking
for supernovae that exploded in the same host, we posed further constraints on
the individual objects to make sure that our method could be applied to them:
the objects should have at least one observation in their early photospheric
phase (more precisely, in the epoch range of 10 to 35 days with respect to the
time of explosion, see Sect. 3.2) to be compatible with our radiative-transfer
modelling, a well-covered light curve during these epochs (optimally, in
multiple bands for calibration purposes, see Sect. 3.1) and a well-constrained
time of explosion from either non-detections or from fitting of the rise of
the light curve.

By filtering the catalogue, we found that SN IIP pairs in four host galaxies
met all the conditions that we described above: NGC 772, NGC 922, NGC 4303 (M
61), and NGC 6946 (Fig. 10). To retrieve the data for the supernovae in these
hosts, we made use of the OSC and WISeREP222https://www.wiserep.org/ (Yaron &
Gal-Yam, 2012). The properties of the final dataset are summarized in Table 1.

Host | $z_{\textrm{helio}}$ | $E(B-V)_{\textrm{MW}}$ | SN | $N_{\textrm{spec}}$ |  $N_{\textrm{phot}}$ (bands) | References
---|---|---|---|---|---|---
M61 | 0.00522 | 0.0193 | 2008in | 6 | 22 (BVRI) | Nakano et al. (2008); de Jaeger et al. (2019)
2020jfo | 7 | 21 ($gri$) | Schulze et al. (2020); Sollerman et al. (2021)
NGC 772 | 0.00825 | 0.0623 | 2003hl | 1 | 36 (BVRI) | Moore et al. (2003); Faran et al. (2014)
2003iq | 4 | 44 (BVRI) | Llapasset et al. (2003); Faran et al. (2014)
NGC 922 | 0.01028 | 0.0163 | 2002gw | 3 | 34 (BVI) | Monard (2002); Galbany et al. (2016)
2008ho | 2 | 11 (BV) |  Pignata et al. (2008); Anderson et al. submitted
NGC 6946 | 0.00013 | 0.2967 | 2004et | 6 | 40 (BVRI) | Zwitter et al. (2004); Faran et al. (2014)
2017eaw | 8 | 96 (BVRI) | Wiggins (2017); Szalai et al. (2019)
Table 1: Summary table of the compiled SN sample, along with important
properties of the host, namely its heliocentric redshift (as adopted from the
OSC) and reddening caused by the dust in the Milky Way (Schlafly & Finkbeiner,
2011). References are provided for the discovery and photometry of the
objects; for details on the spectroscopy, see Sect. 4.

##  3 Methods

###  3.1 EPM and spectral modelling inputs

Before we could fit the spectral time series and perform the EPM analysis, we
needed to obtain the necessary input data: an estimate of the time of
explosion $t_{0}$, photometry interpolated to the spectral epochs, and flux-
calibrated spectra. We describe how we inferred $t_{0}$ from a parametric fit
in Sect. 3.1.1. Section 3.1.2 explains the Gaussian process (GP) interpolation
of the light curves and Sect. 3.1.3 details how we used the interpolated
magnitudes to recalibrate the spectra.

####  3.1.1 Time of explosion

The time of explosion is a crucial parameter in the EPM. It sets the size of
the photosphere and thus the model luminosity. Any error in $t_{0}$ causes an
error in the distance. In case the distance to a supernova is estimated using
a single epoch, the uncertainty in the time of explosion $t_{0}$ translates
directly into the distance uncertainty:

$\frac{\Delta D}{D}\approx\sqrt{\left(\frac{\Delta
t_{0}}{t-t_{0}}\right)^{2}+\left(\frac{\Delta(\Theta/v)}{\Theta/v}\right)^{2}}$
(1)

based on linear relative error propagation, in accordance with the equations
shown in Sec. 3.3. The parameters with the $\Delta$ denote the uncertainty of
the given values. Assuming a 10% error on the $\Theta/v$ measurement for a
single spectrum at the epoch of 20 days, a $t_{0}$ uncertainty of $\pm 2$ days
would yield $\sim 14$% error on the final distance, which is too large for our
purposes.

Analyzing multiple spectral epochs helps limiting the final uncertainties of
the fit parameters, partially owing to the additional constraining effect
exerted on $t_{0}$ by the EPM regression. For example, having four high-
quality observations in the epoch range of 10 to 25 days along with the above
$t_{0}$ uncertainty of 2 days would yield an EPM distance error of about $\sim
11$%. While the improvement in precision is significant compared to the above
case, bringing it to the required levels would require a factor of a few more
spectral epochs, which is rarely available. This demonstrates that even with a
sufficiently large number of observations having a well-constrained prior
estimate on $t_{0}$ is crucial for the distance determination.

We can use the constraints on $t_{0}$ from the early light curve to
significantly reduce these uncertainties. Also, if the time of explosion is
known to a high precision (to an uncertainty of less than a day ideally),
independently of the EPM analysis, even a single spectral epoch is enough to
obtain a meaningful distance. Observationally, the common approach for
estimating the time of explosion has been to take the midpoint between the
first detection and last non-detection (e.g. Gutiérrez et al. 2017). This is
not accurate enough for our analysis. The method does not take into account
the depth of the non-detections in comparison to the first detection; this can
bias the estimated time of explosion and thus the distance. Also, the approach
neglects the information from data on the rise of the light curve.

To minimize these possible biases and improve the precision, we estimate the
time of explosion for each SN through the fitting of their early light curves
(the initial plateau, i.e. the first $10-40$ days, depending on the individual
SNe). Following Ofek et al. (2014) and Rubin et al. (2016), we fit the flux
$f$ in band $W$ with a model

$f_{W}(t)=f_{m,W}\left[1-\exp{\left(-\frac{t-t_{0}}{t_{e,W}}\right)}\right],$
(2)

where $t$ is the time, $t_{0}$ is the time of explosion, $f_{m,w}$ is the peak
flux, and $t_{e,W}$ is the characteristic rise time in the particular band.
Whenever the data allowed, we carried out this fitting for multiple
photometric bands simultaneously to increase the accuracy of the method. In
this joint fit, each of the different bands had its own $f_{m,W}$ and
$t_{e,W}$, but the fits were connected through the global $t_{0}$ value, which
was the same for all bands. To improve the fitting, we introduced two
additional constraints: (i) we took the depth of the non-detections into
account by placing the constraint that the model flux should not exceed the
limiting flux (but the light curves were allowed to extend to times before the
non-detection), and (ii) we required that the characteristic time scale
$t_{e}$ for the light curve rise increases with wavelength for a given
supernova, as it was found previously (see, e.g., González-Gaitán et al.,
2015). This way, even the $t_{e}$ values, which otherwise correspond to their
own band, were constrained globally in the joint fit.

To fit the model to the set of light curves in different bands simultaneously,
we applied the UltraNest333https://johannesbuchner.github.io/UltraNest/
package (Buchner, 2021), which allows for Bayesian inference on complex,
arbitrarily defined likelihoods and which derives the necessary posterior
probability distributions based on the nested sampling Monte Carlo algorithm
MLFriends (Buchner, 2016, 2019). We assumed a flat prior for each of the input
parameters and used a Gaussian likelihood for the parameter inference. To
account for a possible underestimation of the photometric errors, which can
influence the inferred parameters, we extended the likelihood with an
additional term corresponding to the error inflation as described in Hogg et
al. (2010). Using this procedure, we obtained a $t_{0}$ posterior distribution
for each supernova. We then used the maximum of this distribution as our
single ”best estimate” value for $t_{0}$ to set the phase of the spectral
observations for the fitting. The full distribution is used as a prior for the
EPM analysis, as described in Sect. 3.3. This unique approach for obtaining
$t_{0}$ in parallel to the EPM significantly enhances the precision of the
distance measurements, as will be shown in Sect. 4.

####  3.1.2 Interpolated photometry

The epochs of spectral observations are not necessarily covered by individual
photometric datapoints. To obtain photometry at all phases, we fit the light
curves using GPs by applying the
george444https://george.readthedocs.io/en/latest/ python package (Ambikasaran
et al., 2015). GPs present a useful way to interpolate the light curves as
they provide a non-parametric way of fitting, while taking into account the
uncertainties of the datapoints. Following several works that already used GPs
for the modelling of supernova light curves (Inserra et al., 2018; Yao et al.,
2020; Kangas et al., 2022), we adopted covariance functions from the Matérn
family for our calculations (Rasmussen & Williams, 2006). We chose a
smoothness parameter of $3/2$ for our analysis. We attempted to keep the
length scale of the best-fit GP curve high enough to retain its robustness
against the scatter present in the datapoints. To achieve this, we omitted the
rise of the light curve and the drop from the plateau from the fits. Figure 20
shows the interpolated light curves and Table 6 lists the magnitudes at the
spectral epochs. The interpolated magnitudes are direct inputs for the EPM, as
described in Sect. 3.3.

Figure 1: Images of the sample host galaxies. Top left: SDSS
DR9888https://www.sdss3.org/index.phpimage of M61, along with the positions of
SN 2008in and 2020jfo. Top right: SDSS DR9 image of NGC 772, along with the
positions of 2003iq and 2003hl. Bottom left: PanSTARRS
DR1999https://archive.eso.org/dss/dssimage of NGC 922, along with the
positions of 2002gw and 2008ho. Bottom right:
DSS101010https://outerspace.stsci.edu/display/PANSTARRS/image of NGC 6946,
along with the positions of 2004et and 2017eaw.

####  3.1.3 Flux calibrated spectra

A reliable flux calibration is crucial for the spectral modelling and the
determination of the host extinction. We therefore apply a linear flux
correction to avoid possible biases in the analysis. This correction was based
on the photometry: for every epoch, we calculated a set of synthetic
magnitudes using the transmission curves from Bessell & Murphy (2012) (for
$BVRI$ magnitudes) or Dekany et al. (2020) (in case of ZTF photometry), and
then compared these values to the corresponding interpolated magnitudes. The
correction curve was then calculated by fitting the linear trend present in
the ratios of synthetic and interpolated fluxes against the effective
wavelengths of the passbands. Finally, the spectra were multiplied with the
obtained correction trends. In some cases, additional more complex flux
calibration steps were required, which are described for the individual
supernovae separately.

###  3.2 Spectral modelling

To fit the individual spectra, we applied the methodology introduced by Vogl
et al. (2020). The fitting method is based on a spectral emulator that
predicts synthetic spectra for a set of supernova parameters. The training set
of synthetic spectra was calculated with a modified version of the Monte Carlo
radiative transfer code Tardis (Kerzendorf & Sim, 2014; Vogl et al., 2019),
and is described in detail in Vogl et al. (2020). To allow for a fast and
reliable interpolation of model spectra for a given set of physical
parameters, along with correct absolute magnitudes, an emulator was built on
this training set with the following procedure: first, the dimensionality of
the spectra was greatly reduced by the use of principal component analysis
(PCA), then GPs were trained on the physical input parameters (photospheric
temperature and velocity, $T_{\textrm{ph}}$ and $v_{\textrm{ph}}$, metallicity
$Z$, time since explosion $t_{\textrm{exp}}$, and the exponent of the density
profile $n=-\textrm{d}\ln\rho/\textrm{d}\ln r$) to interpolate spectra and to
predict absolute magnitudes. As described in Vogl et al. (2020), the emulator
predicts spectra that in almost all cases match the simulations with a
precision of better than 99%, as measured by the mean fractional error. The
original version of the emulator covered the early photospheric phase, from
6.5 to 22.5 days after explosion (Vogl et al., 2020). This was extended
towards later epochs up until 38 days, and additional high-temperature models
with an NLTE (non-local thermal equilibrium) treatment of He were also added
(Vogl, 2020; Vasylyev et al., 2022; Vogl, 2022). The 38-day upper end is set
by the modelling limitations, since characteristically after this epoch time-
dependent effects in the excitation and ionization-balance become
important.111111The choice of 38 days is arbitrary to a certain degree and
arguments can be made for lower and higher values for the cutoff: time-
dependent effects already play a minor role at earlier epochs, while single
snapshot models are probably still accurate enough for slightly later epochs.
Modelling such late epochs is more complex and time intensive, henceforth the
emulator is limited to earlier times, where models can be calculated in single
snapshot simulations. The physical range covered by the emulator is summarized
in Table 2.

|  $v_{\mathrm{ph}}$ [km$\,\mathrm{s}^{-1}$] |  $T_{\mathrm{ph}}\,$[K] | $Z\,[Z_{\odot}]$ |  $t_{\mathrm{exp}}\,$[days] | n | NLTE
---|---|---|---|---|---|---
$\mathbf{t_{\mathrm{exp}}\,}$ ¡ 16 days | H | He
Min | 4500 | 7200 | 0.1 | 6.0 | 9 | ✓ | ✓
Max | 12000 | 16000 | 3.0 | 16.0 | 26
$\mathbf{t_{\mathrm{exp}}\,}$ ¿ 16 days | H | He
Min | 3600 | 5800 | 0.1 | 6.5 | 6 | ✓ | ✗
Max | 10700 | 10000 | 3.0 | 40.0 | 16

Table 2: Parameter range covered by the extended spectral emulator. Depending
on the spectral epoch, one of these two simulation sets was chosen for
building the emulator.

To infer physical parameters, we used a Gaussian likelihood:

$\ln
p(f_{\lambda}^{\textrm{obs}}|\theta_{\textrm{SN}},E(B-V))=-\frac{1}{2}(R^{\textrm{T}}C^{-1}R+\ln\textrm{det
}C+N\ln 2\pi),$ (3)

where,

$R=f_{\lambda}^{\textrm{obs}}-f_{\lambda}(\theta_{\textrm{SN}},E(B-V)),$ (4)

where
$\theta_{\textrm{SN}}=(v_{\textrm{ph}},T_{\textrm{ph}},Z,t_{\textrm{exp}},n)$
denotes the set of physical parameters, $f_{\lambda}^{\textrm{obs}}$ and
$f_{\lambda}$ denote the observed and the reddened emulated spectra,
respectively, $N$ is the number of spectral bins and $C$ is the bin-to-bin
covariance matrix (see, e.g., Czekala et al. 2015). The matrix $C$ is
important for the inference. It should account for uncertainties in the data
and the model, and capture the correlations of these uncertainties across
wavelength; otherwise, we would significantly underestimate the parameter
uncertainties (see, Czekala et al., 2015). Constructing a matrix with these
properties is a challenging, unsolved problem.

We thus used a simple diagonal matrix with constant values as in Vogl et al.
(2020). Since we cannot infer reasonable uncertainties under these
circumstances, we performed only maximum likelihood estimation for the
parameters. Throughout the fitting, the $t$ epoch of each spectrum was fixed.
We treated the reddening towards the supernova separately from the other
physical parameters: instead of directly including it in the likelihood
function, we set up a grid of reddened spectra corresponding to various
$E(B-V)$ values, and then evaluated the likelihood for each separately. Apart
from reducing computational time, this allowed us to quantify the
uncertainties caused by the reddening using the treatment described below. We
applied the reddening correction according to the Cardelli et al. (1989) law
with $R_{V}=3.1$. For the lower limit of the $E(B-V)$ grid, we always assumed
the Galactic colour excess towards the supernova, which was determined based
on the Schlafly & Finkbeiner (2011) dust maps. The best-fit $E(B-V)$ was
chosen as the average of the $E(B-V)$ values that resulted in the lowest
$\chi^{2}$ for the individual spectra.

Apart from calculating the best-fit physical parameters, we also evaluated the
angular diameter values for every $E(B-V)$ gridpoint, to allow us to quantify
the distance uncertainties resulting from the unknown amount of host reddening
as described in the next section.

Figure 2: The distribution of best-fitting reddening estimates for all the
epochs of SN 2017eaw (blue bars), and the constructed KDE distribution
(background shape). The dashed grey line at $E(B-V)=0.3$ mag denotes the
Galactic reddening towards NGC 6946 based on the Schlafly & Finkbeiner (2011)
dust map, which sets the lower limit for the KDE. The green and grey curves
show the Gaussian kernels for a single observation obtained by our setting and
the Silverman’s Rule respectively. A normalization was applied on the KDE
histogram for a better comparison. For more details on this supernova, see
Sect. 4.4.

###  3.3 Distance determination

To infer the distances $D$ for the supernovae, we used a variant of the
tailored-EPM method (Dessart & Hillier, 2006; Dessart et al., 2008; Vogl et
al., 2020). As a first step, the photospheric angular diameter of the
supernova ($\Theta=R_{\textrm{ph}}/D$, where $R_{\textrm{ph}}$ denotes the
radius of the photosphere) had to be inferred for each of the spectral epochs.
The predicted apparent magnitude for a passband $S$ depends on $\Theta$ as
follows:

$\begin{split}&m_{S}=M_{S}^{\textrm{ph}}(\Sigma^{*})-5\log(\Theta)+A_{S}\\\
&M_{S}^{\textrm{ph}}=M_{S}+5\log\frac{R_{\textrm{ph}}}{10\textrm{
pc}}\end{split}$ (5)

Here, $\Sigma^{*}$ denotes the set of physical parameters corresponding to the
best fit, $M_{S}^{\textrm{ph}}$ is the absolute magnitude predicted by the
radiative transfer model at the position of the photosphere and $A_{S}$
denotes the broadband dust extinction in the bandpass. $M_{S}$ denote the
absolute magnitude as defined by the distance modulus formula. The absolute
magnitudes are transformed with above formula so that the variations in the
size will have no effect. With this definition, we determined the best-fitting
angular diameter $\Theta^{*}$ by minimizing the square difference between the
observed $m_{S}^{\textrm{obs}}$ and model magnitudes:

$\Theta^{*}=\arg\min_{\Theta}\sum_{\textrm{S}}\left(m_{\textrm{S}}-m_{\textrm{S}}^{\textrm{obs}}\right)^{2}$
(6)

Figure 3: Exponential fit on the ROTSE light curve of SN 2008in (left), and
the ZTF early light curves of SN 2020jfo (right). The blue shaded areas shows
the obtained $t_{0}$ posteriors. The black curves show the mean fit, i.e. the
fit that results by taking the mean of the posterior distributions. In case of
SN 2020jfo, the $g$-band light curve was further rescaled after the
normalization to improve clarity on the figure. The shaded regions denote the
68$\%$ and 95$\%$ confidence intervals. Figure 4: Spectral time series of SNe
2008in (Roy et al., 2011; Hicken et al., 2017) and 2020jfo (Sollerman et al.,
2021), along with our fits. The grey shaded areas denote telluric regions.
Figure 5: Evolution of $\Theta/v_{\textrm{ph}}$ for SNe 2008in (top panel) and
2020jfo (bottom panel). The derived distance $D$ and the final time of
explosion estimate (with respect to the initial light curve fit based value)
$\Delta t_{0}$ are displayed in the top left corner of each panel. The shaded
region shows the 68$\%$ confidence interval of the fit. The displayed
uncertainties denote the $1\sigma$ errors on $\Theta/v_{\textrm{ph}}$.

Finally, assuming homologous expansion $R_{\textrm{ph}}=v_{\textrm{ph}}t$
(which is well motivated by observations and models for normal SNe IIP, see
e.g. Woosley 1988; Dessart & Hillier 2005b) we determined the distance to the
supernova and its time of explosion through a Bayesian linear fit to the
ratios of the angular diameters and the photospheric velocities
($\Theta/v_{\textrm{ph}}$) against time $t$. In the fit, we assumed Gaussian
uncertainties for $\Theta/v_{\textrm{ph}}$ of 10% of the measured values for a
given colour excess as in Dessart & Hillier (2006); Dessart et al. (2008);
Vogl et al. (2020). We used a flat prior for the distance, whereas for the
time of explosion we used a normalized histogram of the $t_{0}$ posterior from
the early light curve fit as the prior. However, instead of applying the
standard $\chi^{2}$-based likelihood, we used a different approach to account
for the correlated errors caused by the reddening.

This is important since the reddening can affect the final EPM distance, as
$E(B-V)$ influences the $\Theta$ measurements through the de-reddening of the
observed magnitudes and changes in the best-fit parameters such as the
photospheric temperature. Hence, the uncertainty of the reddening introduces a
correlated error to the final $\Theta/v_{\textrm{ph}}$ measurements, which
transitions over to the distance obtained from the EPM fit. Depending on the
exact model, higher extinction usually leads to shorter inferred distances. To
take this into account, we extended the likelihood with the uncertainty of the
reddening by applying kernel density estimation.

Kernel density estimation (KDE) is an effective tool for approximating the
underlying probability density distribution of an observable using only a
number of realizations. We use the scipy KDE implementation assuming Gaussian
kernels (Virtanen et al., 2020) to estimate an underlying distribution for the
reddening values obtained from the individual spectral fits. The distributions
calculated by KDE depend on the bandwidth of the individual Gaussian kernels.
We set the bandwidth to a constant value of 0.025. This choice was based on
empirical comparison with Silverman’s Rule (Silverman, 1986), which is
regularly used for KDE: for multiple epochs, the bandwidth estimate obtained
using Silverman’s Rule is in agreement with our preset value, hence the
resulting distributions match in the two cases. However, Silverman’s Rule
cannot be used for a single epoch, or when a sole value of $E(B-V)$ is
favoured by all fits; in these cases, our bandwidth choice still provided a
realistic KDE. Finally, we set the lower limit of the estimated distribution
to the Galactic reddening based on the Schlafly & Finkbeiner (2011) map, to
exclude non-physical cases. One of the obtained KDE distributions can be seen
in Fig. 2.

To incorporate the correlated uncertainty in the fit, we first drew a large
number of reddening samples using the obtained KDE. Then, a respective sample
of $\Theta/v_{\textrm{ph}}$ values was generated based on the
$\Theta/v_{\textrm{ph}}-E(B-V)$ linear interpolation, and by adding a random
offset to each value assuming 10% uncertainty. This sample not only contained
the $\Theta/v_{\textrm{ph}}$ values for the relevant reddening values only,
but it also carried information about the correlated errors present in them.
To represent these distributions and set up the final EPM regression, we
applied a multi-dimensional Gaussian KDE on the set of
$\Theta/v_{\textrm{ph}}$ values, and then used the corresponding probability
density function as the likelihood for the fitting. We then evaluated this
likelihood using UltraNest: at each step, the sampler drew an assumed distance
and time of explosion value, for which the model
$(\Theta/v_{\textrm{ph}})^{*}$ were calculated, and then the log probability
density function was evaluated. As a result, we obtained a posterior
distribution for the distance that includes the correlated error introduced by
the uncertainty in the reddening.

##  4 Results

In this section, we present the distances obtained for the individual host
galaxies based on their two sibling supernovae.

Figure 6: Early light curve fits of SN 2003hl (left) and SN 2003iq (right),
and the determined time of explosion posteriors (blue shaded regions). Figure
7: Spectral time series of SNe 2003hl and 2003iq, along with their fits. The
grey shaded areas denote telluric regions. Figure 8: Evolution of
$\Theta/v_{\textrm{ph}}$ for SNe 2003hl (top panel) and 2003iq (bottom panel).

###  4.1 M61

The spiral galaxy M61 (NGC 4303, Fig. 10) hosted two Type II supernovae, SN
2008in and 2020jfo. This pair of transients has not been investigated together
in the literature yet, since the second supernova occurred only recently.

The first supernova, SN 2008in was discovered on JD 2454827.29 by Nakano et
al. (2008). As M61 was monitored by the Robotic Optical Transient Search
Experiment (ROTSE) starting from the week before the discovery (Roy et al.,
2011), the fitting of the early light curve allowed for the estimation of the
time of explosion independent of the EPM, from which we obtained
$t_{0}=\textrm{JD }2454824.51^{+0.19}_{-0.14}$ (Fig. 3). This supernova was
found to belong to the peculiar, subluminous group of SNe IIP (Roy et al.,
2011). The spectral sequence for our analysis consisted of spectra obtained by
Roy et al. (2011) complemented with more recently published spectra from
Hicken et al. (2017).

SN 2020jfo was discovered on JD 2458975.70 by the Zwicky Transient Facility
(ZTF, Schulze et al. 2020). Owing to the relatively short cadence of
observations of ZTF (Bellm et al., 2019) the time of explosion $t_{0}$ is not
only constrained by a non-detection 4 days pre-discovery, but the supernova
was discovered on the rise, hence it was possible to estimate $t_{0}$
precisely by the fitting of the early light curve (Fig. 3). We obtained a time
of explosion of $t_{0}=\textrm{JD }2458975.37^{+0.10}_{-0.10}$. The spectral
time series of this object was presented by Sollerman et al. (2021) and Teja
et al. (2022). For our study, we used seven early time spectra ($t<20$ days).

The spectral time series and the emulator fits are shown in Fig. 4. For the
calculation of the best-fitting models the telluric regions (as marked on the
figure) were masked. After fitting, we performed the EPM analysis on both
supernovae (Fig. 5). The light curve fits used for the flux calibration of the
spectra are displayed in Appendix C, while the best-fit parameters are listed
in Table 5. The estimated distances to the supernovae are $D=15.06\pm 0.71$
Mpc and $D=14.95\pm 0.78$ Mpc for SN 2008in and SN 2020jfo respectively. The
classical EPM analysis of SN 2008in was conducted previously by Bose & Kumar
(2014), yielding a distance of $D=14.51\pm 1.38$ Mpc, which is consistent with
our current estimate. No distance measurements of SN 2020jfo were carried out
previously. Although the calibration quality varied significantly from epoch
to epoch (since multiple instruments were used for both spectral sequences),
the obtained distances are consistent within the uncertainties and agree
within 1% percent.

Figure 9: Early light curve fits of SN 2002gw (left) and SN 2008ho (right),
and the determined time of explosion posteriors (blue shaded regions). Figure
10: Spectral time series of SNe 2002gw and 2008ho, along with their fits. The
grey-shaded areas denote telluric regions. Figure 11: Evolution of
$\Theta/v_{\textrm{ph}}$ for SNe 2002gw (top panel) and 2008ho (bottom panel).
Figure 12: The EPM regression performed for SN 2002gw on the
$\Theta/v_{\textrm{ph}}$ values calculated using the $BV$ (green) and $BVI$
(blue) bandpass sets.

###  4.2 NGC 772

The spiral galaxy NGC~772 (Fig. 10) is unique, as it not only had been host to
two Type II supernovae, but both objects were also observable simultaneously,
as they exploded one and a half months after one another. Both supernovae were
followed up by the Carnegie Type II Supernova Program (CATS), and their
spectral sequences were previously analysed by Jones et al. (2009).

The first supernova, SN 2003hl, was discovered on JD 2452872.0 by Moore et al.
(2003). By applying the method described in Sect. 3.1 on the early light curve
along with the unfiltered pre-discovery KAIT (Katzman Automatic Imaging
Telescope, Filippenko et al. 2001) non-detection on JD 2452863.0 we estimated
the time of explosion to be JD $2452864.62^{+1.18}_{-1.15}$ (Fig. 6). We
assigned the KAIT non-detection to the $V$-band, given that the colour of the
supernova is close to zero owing to the combined effect of the very blue
spectral energy distribution in such early phases and the reddening, which
places the effective wavelength close to the $V$-band even for a red-sensitive
CCD. Only one spectrum was obtained for this supernova in the temporal range
covered by our emulator. Nevertheless, using the time of explosion, we could
still derive an approximate EPM distance.

The second supernova, SN 2003iq was discovered by Llapasset et al. (2003) on
JD 2452921.5 during the photometric follow-up observations of SN 2003hl. Owing
to the monitoring of the host, a pre-discovery image was taken on JD 2452918.5
by Llapasset et al. (2003) shortly before the first detection, which already
constrained the explosion date of SN 2003iq to a range of 3 days. By fitting
the early light curve, we estimated its time of explosion to be JD
$2452919.71^{+0.59}_{-0.65}$ (Fig. 6). Apart from the more precisely known
$t_{0}$, this supernova has a more thorough spectral record during the
photospheric phase than its sibling, as four spectra were acquired during its
early evolution.

The fitted spectral time series of the supernovae are displayed in Fig. 7. The
EPM regression derived from the obtained physical parameters is shown in Fig.
8. We obtain $D=33.95\pm 4.49$ Mpc and $26.02\pm 1.85$ Mpc for the distance of
NGC 772 from SN 2003hl and SN 2003iq respectively. Although we could use only
one spectrum for SN 2003hl, and hence the resulting distance estimate is
fairly uncertain, both the final result and the $\Theta/v_{\textrm{ph}}$ value
around 15 days show broad consistency with those of SN 2003iq.

By comparing our results with the classical EPM carried out by Jones et al.
(2009), we find significant differences for SN 2003hl and SN 2003iq as well,
both for the Eastman et al. (1996) and Dessart & Hillier (2005a) correction
factors: in the case of 2003hl, our distance is larger than the two Jones et
al. (2009) estimates ($D=17.7\pm 2.1$ Mpc and $D=30.3\pm 6.3$ Mpc for the
different correction factors respectively), while for SN 2003iq our distance
is shorter than the previous estimates ($D=36.0\pm 5.6$ Mpc and $D=53.3\pm
17.1$ Mpc, Jones et al. 2009). However, the differences between the estimates
can be explained by the significantly higher colour excess we obtained for SN
2003iq (with our best estimate being $E(B-V)=0.14$ mag), and by the fact that
we could not make use of more than one spectrum for SN 2003hl, due to the
limitations of our modelling approach. On the other hand, by comparing our
estimates with previous SCM distances, we find our solution for SN 2003iq is
consistent with the previous result of Poznanski et al. (2009) ($D=26.6\pm
1.25$ Mpc), while our distance for SN 2003hl is not in tension with the SCM
estimate of Olivares E. et al. (2010) ($D=25.6\pm 3.30$ Mpc).

Figure 13: Exponential fit on the early light curves of SN 2004et (left) and
SN 2017eaw (right), and the determined time of explosion posteriors (blue
shaded regions). Figure 14: Spectral time series of SNe 2004et and 2017eaw,
along with their fits. Figure 15: Evolution of $\Theta/v_{\textrm{ph}}$ for
SNe 2004et (top panel) and 2017eaw (bottom panel).

###  4.3 NGC 922

The peculiar SBcd type galaxy NGC~922 (Fig. 10) hosted two Type II supernovae
six years apart: SN 2002gw and SN 2008ho. SN 2002gw was discovered on JD
2452560.8 (Monard, 2002). Although the latest non-detection occurred too far
from the discovery to be useful for constraining the explosion date, the
fitting of the early light curve (including the unfiltered CCD observations
obtained by Itagaki et al. 2002 and Monard 2002) gave an estimate of JD
$2452556.58^{+0.97}_{-1.40}$ (Fig. 9), which is consistent with the value
obtained through performing the EPM regression in Jones et al. (2009). In
total, three optical spectra were acquired for this supernova by the CATS
program (Hamuy et al., 2006) in the epoch range that could be used for our EPM
analysis.

The second supernova, SN 2008ho, was discovered on JD 2454796.61 by Pignata et
al. (2008). The last pre-discovery non-detection occurred on JD 2454787.77,
which provides a weak constraint on the time of explosion: we found
$t_{0}=\textrm{JD }2454789.63^{+3.63}_{-2.54}$ from the light curve fit (Fig.
9). In total, two spectra were obtained for this supernova through the
Carnegie Supernova Project (Hamuy et al., 2006). The spectra were recalibrated
using the $BV$ observations obtained through the CSP project (Anderson et al.
submitted, private communication). For the EPM, we made use of the $BV$
photometry from the same campaign.

The fitted spectral sequences of SN 2002gw and SN 2008ho are shown in Fig. 10.
From the EPM regression we obtained a distance for SN 2008ho of $D=$ $40.02\pm
5.07$ Mpc (Fig. 11). On the other hand, we found $D=43.46\pm 3.77$ Mpc for
$BV$ bands only and $D=43.85\pm 3.78$ Mpc for the full $BVI$ set for SN 2002gw
(see Fig. 12 for the comparison). Our distance estimates for SN 2002gw fall
between the values derived by Jones et al. (2009) for the different dilution
factors ($D=37.4\pm 4.9$ Mpc and $D=63.9\pm 17.0$ Mpc), and agree well with
the previous SCM ($D=48.1\pm 6.2$ Mpc, Olivares E. et al. 2010) and
photospheric magnitude method estimates ($D=45.10\pm 3.11$ Mpc, Rodríguez et
al. 2014). Although the difference is only $\sim 1$%, we use the $BV$ instead
of the $BVI$ distance for SN 2002gw for the plots and the distance consistency
check (Sect. 5) to make the analysis of the siblings as similar as possible.
We suspect that the majority of the offset between the distances of the two
SNe can be attributed to the relatively large uncertainties on the times of
explosion. Nevertheless, the two distances are fully consistent within
$1\sigma$.

###  4.4 NGC 6946

NGC~6946 (Fig. 10) is a bright face-on SABcd type galaxy, which produced
several Type II supernovae: SNe 1948B, 1980K, 2002hh, 2004et and 2017eaw. Due
to its proximity, the distance of this galaxy could be estimated through the
tip of the red giant branch (TRGB) method (Anand et al., 2018) and the
planetary nebulae luminosity function (PNLF) relation (Herrmann et al., 2008).
Three of the aforementioned supernovae (SNe 2002hh, 2004et and 2017eaw) were
observed spectroscopically during the early photospheric phase, which makes it
possible to measure their distances through the tailored-EPM. However, only
SNe 2004et and 2017eaw were optimal for our purposes: even though the spectral
time series and the photometric coverage would have allowed the analysis of SN
2002hh, it was found that this supernova exhibited very strong, two-component
reddening (one component arising from the joint effect of the interstellar
dust in the host and the Milky Way, the other from large quantities of local
dust, see Barlow et al. 2005, Pozzo et al. 2006 and Welch et al. 2007). With
the currently available data and methodology it is not possible to obtain an
accurate extinction correction. Considering the very high reddening, any error
in our extinction correction would significantly impact the inferred distance.
Hence we chose to exclude SN 2002hh from the analysis.

SN 2004et was discovered on JD 2453273 by Zwitter et al. (2004), and
extensively followed up by the 2-m Himalayan Chandra Telescope (HCT) of the
Indian Astronomical Observatory (IAO) and the 3-m Shane telescope at the Lick
Observatory. The resulting spectral times series has been previously studied
by Sahu et al. (2006). However, the spectra taken with the HCT were subject to
calibration issues, which could not be corrected with our standard linear flux
re-calibration. Since these issues can influence the fitting significantly, we
attempted to empirically correct them, using spectra taken with the Keck and
Lick telescopes. This procedure is detailed in Appendix A. We completed this
spectral time series with three additional early-time spectra obtained at the
David Dunlap Observatory (Takáts & Vinkó, 2012, courtesy of József Vinkó).
However, since these spectra covered only a very narrow wavelength range, we
could not apply the re-calibration procedure from Sect. 3.1.3. The pre-
discovery non-detection of the supernova and the subsequent early photometry
from Li et al. (2004) allowed for an accurate determination of the time of
explosion through the fitting of the early light curve (Fig. 13), which
yielded $t_{0}=\textrm{JD }2453271.19^{+0.16}_{-0.17}$. Along with the tight
constraints on the time of explosion, the six spectra obtained in the first
month after explosion also allowed for a precise EPM analysis.

SN 2017eaw, was discovered by Wiggins (2017) on JD 2457885.78, and was
extensively followed up with spectroscopic observations at the Las Cumbres
Observatory 1 m telescope and at the McDonald Observatory with the Low
Resolution Spectrograph 2 mounted on the 10 m Hobby–Eberly Telescope. The
observations are described in Szalai et al. (2019). The supernova was
discovered early, on the rise, and a non-detection is also available from a
pre-discovery observation of the host, which yielded a time of explosion of
$t_{0}=\textrm{JD }2457886.01^{+0.25}_{-0.20}$. In terms of spectroscopy, this
supernova had the best-sampled and most homogenous spectral sequence in our
sample, resulting in eight well-calibrated spectra.

The calibrated and fitted spectral sequences are shown in Fig. 14. For SN
2004et, the EPM regression yielded a distance of $D=5.99\pm 0.36$ Mpc, while
for SN 2017eaw the resulting value was $D=6.44\pm 0.29$ Mpc (Fig. 15). The
distance for SN 2004et is 20% higher than the classical EPM value inferred by
Takáts & Vinkó (2012) and Szalai et al. (2019), while for SN 2017eaw the value
we obtained is about 5-20% lower than the value from Szalai et al. (2019)
(depending on the assumed reddening and the chosen epoch range of spectra of
the reference work, see Szalai et al. 2019 for details). We note that our
distance estimates are in much better agreement with one another, which
highlights the advantage of the tailored-EPM over the classical EPM approach.
The distances are also consistent with the various independent distances
obtained for NGC 6946: apart from the previous EPM-based values, our estimates
are in agreement with SCM distances of Poznanski et al. (2009) and de Jaeger
et al. (2017) (both of which yielded $D=6.69\pm 0.30$ Mpc for SN 2004et) and
the PNLF distance (Herrmann et al. 2008, $D=6.1\pm 0.6$ Mpc), but is slightly
lower than the latest TRGB estimate ($D=6.95\pm 0.20$ Mpc, Anand et al. 2021,
from the Extragalactic Distance Database121212https://edd.ifa.hawaii.edu/).

Host | SN |  $D$ [Mpc] |  $t_{0}$ (MJD) |  $E(B-V)$ [mag]
---|---|---|---|---
M61 | 2008in | $15.06(0.71)$ | $54824.15(0.15)$ | 0.06(0.03)
| 2020jfo | $14.95(0.78)$ | $58975.08(0.34)$ | 0.07(0.04)
NGC 722 | 2003hl | $33.95(4.49)$ | $52864.26(1.20)$ | 0.31*
| 2003iq | $26.02(1.85)$ | $52919.20(0.54)$ | 0.15(0.05)
NGC 922 | 2002gw | $43.46(3.67)$ | $52555.84(1.15)$ | 0.12(0.03)
| 2008ho | $40.02(5.07)$ | $54789.17(2.12)$ | 0.21*
NGC 6946 | 2004et | $5.99(0.36)$ | $53270.62(0.16)$ | 0.38(0.10)
| 2017eaw | $6.44(0.29)$ | $57885.52(0.22)$ | 0.35(0.04)

Table 3: Summary table of the EPM results. $t_{0}$ denotes the time of explosion as obtained by the EPM regression. The $E(B-V)$ values listed here are those determined by the spectral fitting. The values in brackets denote the uncertainties of the given quantities. In the starred (*) cases we either had only one spectrum to fit, or both spectra yielded the same reddening value, which did not allow us to give a reasonable empirical uncertainty. Galaxy | SN 1 | D${}_{\textrm{SN\,1}}$ | SN 2 | D${}_{\textrm{SN\,2}}$ | Bayes factor | Prior
---|---|---|---|---|---|---
NGC 772 | 2003hl |  $33.95\pm 4.49$ Mpc | 2003iq |  $26.02\pm 1.85$ Mpc | 0.890 | 13.44 – 53.77 Mpc
NGC 922 | 2002gw |  $43.46\pm 3.67$ Mpc | 2008ho |  $40.02\pm 5.07$ Mpc | 3.448 | 21.29 – 85.16 Mpc
NGC 4303 | 2008in |  $15.06\pm 0.71$ Mpc | 2020jfo |  $14.95\pm 0.78$ Mpc | 7.944 | 7.31 – 29.25 Mpc
NGC 6946 | 2004et |  $5.99\pm 0.36$ Mpc | 2017eaw |  $6.44\pm 0.29$ Mpc | 4.272 | 2.77 – 11.09 Mpc
Table 4: Bayes factors for the comparisons of the distance posteriors of the
various SN pairs.

##  5 Discussion

As shown in the previous sections, the fitting procedure not only yields
distances with a claimed uncertainty of $\sim$10% or better (Table 3), it does
so by requiring only a limited amount of modelling choices. Due to various
uncertainties in the observations and the modelling, these distances can be
slightly different for the supernova siblings. The main question, in this
case, is whether the estimated distances of the siblings are consistent with
one another. Assuming an average galaxy, a reliable upper limit for the
thickness of the disc is 10 kpc, while the width of the disc is on the order
of 100 kpc (Gilmore & Reid, 1983; Zanisi et al., 2020); consequently, in an
ideal case, assuming face-on galaxies, the distances inferred for the
supernova siblings should match to an uncertainty of $\pm 0.01$ Mpc, which
corresponds to only $\sim 0.2\%$ even for the most nearby pair. For more
inclined galaxies the uncertainties can be one order of magnitude larger;
however, since the closest hosts we investigated are very likely low
inclination galaxies (M 61 and NGC 6946), and the farther away ones are not
viewed edge-on either (NGC 722 and NGC 922), the offset between the siblings
should remain sub-1% relative to the distance of the galaxy. Hence, the
distance estimation to siblings provides an empirical test to assess the
effect of these uncertainties and systematics (e.g. the CSM interaction, or
data calibration issues, among others). It allows us to test, whether our
results are not only _precise_ , but _accurate_.

To test the consistency of the obtained distances, we performed Bayesian model
comparison. We adopted the methodology used by Wong et al. (2020) for
assessing whether pairs of strongly lensed quasars favour a single global set
of cosmological parameters or individual cosmologies for each lens that are
inconsistent with one another. We applied this method to the distance
estimates of the siblings: we thus asked the question whether the measured
distance posteriors of a pair have more likely been generated from a single
underlying distance $D^{\textrm{gal}}$ or from two distinct distances
$D^{\textrm{ind,1}}$ and $D^{\textrm{ind,2}}$ (and are thus inconsistent). The
probability ratio of the two scenarios is called the Bayes factor and can be
calculated as

$F_{ij}=\frac{P(\mathbf{d_{i}},\mathbf{d_{j}}|D^{\textrm{gal}})}{P(\mathbf{d_{i}}|D^{\textrm{ind,i}})P(\mathbf{d_{j}}|D^{\textrm{ind,j}})}=\frac{\int^{D_{2}}_{D_{1}}\mathbf{d_{i}}\mathbf{d_{j}}P(D)dD}{\int^{D_{2}}_{D_{1}}\mathbf{d_{i}}P(D)dD\int^{D_{2}}_{D_{1}}\mathbf{d_{j}}P(D)dD},$
(7)

where $\mathbf{d_{i}}$ and $\mathbf{d_{j}}$ denote the distance posteriors of
the individual supernovae in each pair. We chose a uniform prior $P(D)$ around
the average distance of the host (based on previous measurements, as quoted on
NASA/IPAC Extragalactic Database, NED131313https://ned.ipac.caltech.edu/),
ranging from half to twice that distance. The quoted Bayes factors can thus be
understood as lower limits since they scale linearly with the width of the
distance prior. If the obtained Bayes factor exceeds unity, this can be
interpreted as a sign of consistency between the two posterior distributions
in accordance with the table of Kass & Raftery (1995). The higher the $F$
value, the stronger the consistency between the siblings.

Figure 16: Distance posteriors obtained from the EPM analysis of the
supernovae for the various host galaxies. The calculated Bayes factors are
displayed in the top-left corners of the panels.

The distance posteriors from the EPM analysis are shown in Fig. 16, while the
obtained Bayes factors are listed in Table 4. The consistency of the distance
posteriors for the various supernovae is good, with Bayes factors being close
to or over 5 for two cases, showing remarkable agreement. In the case of NGC
772 and NGC 922 the values of the Bayes factors are lower, which might be
attributed to the lack of a well-sampled spectral sequence for SN 2003hl, and
the lack of well-constrained explosion date for SNe 2002gw and 2008ho.
However, the remaining cases show that the fitting method and the analysis
result in distances that are not only consistent for supernovae in the same
host but are also precise to a degree of a few per cent. The analysis showed
as well that some of the greatest limiting factors of the method are an
insufficient amount of spectra, poor quality and calibration-level of the
obtained spectral times series and weak constraints on the time of explosion.
If instead all the conditions for the above are met, the EPM analysis can
yield highly precise and consistent values for the supernova pairs, as in the
case of SNe 2004et and 2017eaw or in the case of SNe 2008in and 2020jfo.

We offer the caveat that some of our results may be influenced by the choice
of $R_{V}=3.1$. If the true $R_{V}$ is different, this could in principle
cause notable offsets between the sibling distances, provided the differential
reddening between the objects is non-negligible and the measurement
uncertainties are low enough. In our sample, however, the pairs with the most
significant differential reddening had also the largest uncertainty on their
distance and vice versa. Hence it is currently not possible to estimate the
scale of this effect on the basis of our four sibling pairs.

##  6 Summary

In this work, we investigated the consistency of the EPM distances of sibling
Type II SNe, calculated based on the spectral fitting method introduced by
Vogl et al. (2020). According to our analysis, the method not only yields
precise results with an estimated individual distance uncertainty that is
better than $\sim$10%, despite using literature data from a wide range of
sources, but the resulting distances are consistent for SNe that are within
the same host galaxy. The degree of consistency depends on how well the
supernovae were observed, with the worst results occurring when the quality of
the data barely allows for the EPM analysis, while the best consistency is
achieved when both supernovae are similarly well observed and have a good
constraint on the time of explosion. In cases of M61 and NGC 6946 highly
consistent distances are derived with mismatches of less than 5%.

The high level of consistency between siblings also shows that if there are
any other systematics present between the various SNe, they are likely
subdominant compared to the effect of observation quality. Such systematics
may include different reddening, CSM interaction, explosion energetics and
metallicities. Checking for the effect and scale of such systematics will
require a larger set of siblings with data quality matching that of the
siblings of M 61 or NGC 6946.

Apart from checking for the internal consistency of the method, we obtained
precise distances for the four investigated host galaxies, even though we used
literature data, with a highly varying level of calibration. Furthermore, the
tailored-EPM provides absolute distances, is physics-based, is affected by
different systematics than the other existing distance estimation techniques
and can be utilized completely independently of them. These properties and the
presented analysis in this paper demonstrate the potential of the tailored-EPM
to provide not only precise distances but a precise Hubble parameter as well.
Such an analysis is currently being conducted on high-quality spectral time
series obtained for SNe II in the Hubble Flow by the adH0cc collaboration
(accurate determination of H0 with core-collapse
supernovae141414https://adh0cc.github.io/), which will provide important clues
to the Hubble tension.

##  7 Contributor roles1515footnotemark: 15

1616footnotetext: Using the CRT standard (see https://credit.niso.org/)

  1. 1.

Conceptualization: B. Leibundgut, W. Hillebrandt, C. Vogl

  2. 2.

Data curation: G. Csörnyei

  3. 3.

Formal analysis: G. Csörnyei

  4. 4.

Funding acquisition: W. Hillebrandt

  5. 5.

Investigation: G. Csörnyei, C. Vogl, S. Taubenberger, A. Flörs

  6. 6.

Methodology: B. Leibundgut, W. Hillebrandt, C. Vogl

  7. 7.

Project administration: B. Leibundgut, W. Hillebrandt, C. Vogl

  8. 8.

Resources: W. Hillebrandt

  9. 9.

Software: G. Csörnyei, C. Vogl

  10. 10.

Supervision: B. Leibundgut, W. Hillebrandt, C. Vogl

  11. 11.

Validation: G. Csörnyei

  12. 12.

Visualization: G. Csörnyei

  13. 13.

Writing – original draft: G. Csörnyei, C. Vogl

  14. 14.

Writing – review and editing: B. Leibundgut, W. Hillebrandt, S. Taubenberger,
A. Flörs, S. Blondin, M. G. Cudmani, A. Holas, S. Kressierer

## Acknowledgements

We specially thank J. Anderson, M. Stritzinger and the CSP collaboration for
providing us pre-published data, which helped us to improve our analysis. The
research was completed with the extensive use of Python, along with the numpy
(Harris et al., 2020), scipy (Virtanen et al., 2020) and astropy (Astropy
Collaboration et al., 2018) modules. This research made use of Tardis, a
community-developed software package for spectral synthesis in supernovae
(Kerzendorf & Sim, 2014; Kerzendorf et al., 2022). The development of Tardis
received support from the Google Summer of Code initiative and from ESA’s
Summer of Code in Space program. Tardis makes extensive use of Astropy and
PyNE. Based on observations obtained with the Samuel Oschin 48-inch Telescope
at the Palomar Observatory as part of the Zwicky Transient Facility project.
ZTF is supported by the National Science Foundation under Grant No.
AST-1440341 and a collaboration including Caltech, IPAC, the Weizmann
Institute for Science, the Oskar Klein Center at Stockholm University, the
University of Maryland, the University of Washington, Deutsches Elektronen-
Synchrotron and Humboldt University, Los Alamos National Laboratories, the
TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, and
Lawrence Berkeley National Laboratories. Operations are conducted by COO,
IPAC, and UW. This project utilizes data obtained by the Robotic Optical
Transient Search Experiment. ROTSE is a collaboration of Lawrence Livermore
National Lab, Los Alamos National Lab, and the University of Michigan
(www.umich.edu/~rotse). CV and WH were supported for part of this work by the
Excellence Cluster ORIGINS, which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy-EXC-2094-390783311. The work of AH was supported by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
138713538 – SFB 881 (”The Milky Way System”, subproject A10). AH acknowledges
support by the Klaus Tschira Foundation. AH is a Fellow of the International
Max Planck Research School for Astronomy and Cosmic Physics at the University
of Heidelberg (IMPRS-HD). This work was supported by the ’Programme National
de Physique Stellaire’ (PNPS) of CNRS/INSU co-funded by CEA and CNES. SB
acknowledges support from the ESO Scientific Visitor Programme in Garching. AF
acknowledges support by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation program (ERC Advanced Grant
KILONOVA No. 885281). ST has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program (LENSNOVA: grant agreement No 771776).

## References

  * Ambikasaran et al. (2015) Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W., & O’Neil, M. 2015, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 252
  * Anand et al. (2018) Anand, G. S., Rizzi, L., & Tully, R. B. 2018, AJ, 156, 105
  * Anand et al. (2021) Anand, G. S., Tully, R. B., Rizzi, L., Riess, A. G., & Yuan, W. 2021, arXiv e-prints, arXiv:2108.00007
  * Anderson (2022) Anderson, J. 2022, submitted
  * Anderson et al. (2014) Anderson, J. P., González-Gaitán, S., Hamuy, M., et al. 2014, ApJ, 786, 67
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123
  * Barlow et al. (2005) Barlow, M. J., Sugerman, B. E. K., Fabbri, J., et al. 2005, ApJ, 627, L113
  * Baron et al. (2004) Baron, E., Nugent, P. E., Branch, D., & Hauschildt, P. H. 2004, ApJ, 616, L91
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002
  * Bessell & Murphy (2012) Bessell, M. & Murphy, S. 2012, PASP, 124, 140
  * Bose & Kumar (2014) Bose, S. & Kumar, B. 2014, ApJ, 782, 98
  * Buchner (2016) Buchner, J. 2016, Statistics and Computing, 26, 383
  * Buchner (2019) Buchner, J. 2019, PASP, 131, 108005
  * Buchner (2021) Buchner, J. 2021, The Journal of Open Source Software, 6, 3001
  * Burns et al. (2020) Burns, C. R., Ashall, C., Contreras, C., et al. 2020, ApJ, 895, 118
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245
  * Czekala et al. (2015) Czekala, I., Andrews, S. M., Mandel, K. S., Hogg, D. W., & Green, G. M. 2015, ApJ, 812, 128
  * de Jaeger et al. (2017) de Jaeger, T., González-Gaitán, S., Hamuy, M., et al. 2017, ApJ, 835, 166
  * de Jaeger et al. (2019) de Jaeger, T., Zheng, W., Stahl, B. E., et al. 2019, MNRAS, 490, 2799
  * Dekany et al. (2020) Dekany, R., Smith, R. M., Riddle, R., et al. 2020, PASP, 132, 038001
  * Dessart et al. (2008) Dessart, L., Blondin, S., Brown, P. J., et al. 2008, ApJ, 675, 644
  * Dessart & Hillier (2005a) Dessart, L. & Hillier, D. J. 2005a, A&A, 439, 671
  * Dessart & Hillier (2005b) Dessart, L. & Hillier, D. J. 2005b, A&A, 437, 667
  * Dessart & Hillier (2006) Dessart, L. & Hillier, D. J. 2006, A&A, 447, 691
  * Eastman et al. (1996) Eastman, R. G., Schmidt, B. P., & Kirshner, R. 1996, ApJ, 466, 911
  * Faran et al. (2014) Faran, T., Poznanski, D., Filippenko, A. V., et al. 2014, MNRAS, 442, 844
  * Filippenko et al. (2001) Filippenko, A. V., Li, W. D., Treffers, R. R., & Modjaz, M. 2001, in Astronomical Society of the Pacific Conference Series, Vol. 246, IAU Colloq. 183: Small Telescope Astronomy on Global Scales, ed. B. Paczynski, W.-P. Chen, & C. Lemme, 121
  * Galbany et al. (2016) Galbany, L., Hamuy, M., Phillips, M. M., et al. 2016, AJ, 151, 33
  * Gilmore & Reid (1983) Gilmore, G. & Reid, N. 1983, MNRAS, 202, 1025
  * González-Gaitán et al. (2015) González-Gaitán, S., Tominaga, N., Molina, J., et al. 2015, MNRAS, 451, 2212
  * Graham et al. (2022) Graham, M. L., Fremling, C., Perley, D. A., et al. 2022, MNRAS, 511, 241
  * Guillochon et al. (2017) Guillochon, J., Parrent, J., Kelley, L. Z., & Margutti, R. 2017, ApJ, 835, 64
  * Gutiérrez et al. (2017) Gutiérrez, C. P., Anderson, J. P., Hamuy, M., et al. 2017, ApJ, 850, 90
  * Hamuy et al. (2006) Hamuy, M., Folatelli, G., Morrell, N. I., et al. 2006, PASP, 118, 2
  * Hamuy & Pinto (2002) Hamuy, M. & Pinto, P. A. 2002, ApJ, 566, L63
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357–362
  * Hastie & Tibshirani (1990) Hastie, T. & Tibshirani, R. 1990, Generalized Additive Models (Routledge)
  * Herrmann et al. (2008) Herrmann, K. A., Ciardullo, R., Feldmeier, J. J., & Vinciguerra, M. 2008, ApJ, 683, 630
  * Hicken et al. (2017) Hicken, M., Friedman, A. S., Blondin, S., et al. 2017, ApJS, 233, 6
  * Hogg et al. (2010) Hogg, D. W., Bovy, J., & Lang, D. 2010, arXiv e-prints, arXiv:1008.4686
  * Inserra et al. (2018) Inserra, C., Smartt, S. J., Gall, E. E. E., et al. 2018, MNRAS, 475, 1046
  * Itagaki et al. (2002) Itagaki, K., Nakano, S., Monard, L. A. G., & Africa, S. 2002, IAU Circ., 7996, 4
  * Jones et al. (2009) Jones, M. I., Hamuy, M., Lira, P., et al. 2009, ApJ, 696, 1176
  * Kangas et al. (2022) Kangas, T., Yan, L., Schulze, S., et al. 2022, MNRAS, 516, 1193
  * Kass & Raftery (1995) Kass, R. E. & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773
  * Kerzendorf et al. (2022) Kerzendorf, W., Sim, S., Vogl, C., et al. 2022, tardis-sn/tardis: TARDIS v2022.2.27, Zenodo
  * Kerzendorf & Sim (2014) Kerzendorf, W. E. & Sim, S. A. 2014, MNRAS, 440, 387
  * Kirshner & Kwan (1974) Kirshner, R. P. & Kwan, J. 1974, ApJ, 193, 27
  * Li et al. (2004) Li, W., Filippenko, A. V., van Dyk, S. D., & Cuillandre, J. C. 2004, IAU Circ., 8414, 2
  * Li et al. (2011) Li, W., Leaman, J., Chornock, R., et al. 2011, MNRAS, 412, 1441
  * Llapasset et al. (2003) Llapasset, J. M., Yamaoka, H., & Ayani, K. 2003, Central Bureau Electronic Telegrams, 48, 1
  * Monard (2002) Monard, L. A. G. 2002, IAU Circ., 7995, 2
  * Moore et al. (2003) Moore, M., Li, W., & Boles, T. 2003, IAU Circ., 8184, 2
  * Morozova et al. (2017) Morozova, V., Piro, A. L., & Valenti, S. 2017, ApJ, 838, 28
  * Nakano et al. (2008) Nakano, S., Kadota, K., & Wells, W. 2008, Central Bureau Electronic Telegrams, 1636, 1
  * Ofek et al. (2014) Ofek, E. O., Arcavi, I., Tal, D., et al. 2014, ApJ, 788, 154
  * Olivares E. et al. (2010) Olivares E., F., Hamuy, M., Pignata, G., et al. 2010, ApJ, 715, 833
  * Patat et al. (1994) Patat, F., Barbon, R., Cappellaro, E., & Turatto, M. 1994, A&A, 282, 731
  * Pessi et al. (2019) Pessi, P. J., Folatelli, G., Anderson, J. P., et al. 2019, MNRAS, 488, 4239
  * Pignata et al. (2008) Pignata, G., Maza, J., Hamuy, M., et al. 2008, Central Bureau Electronic Telegrams, 1587, 1
  * Poznanski et al. (2009) Poznanski, D., Butler, N., Filippenko, A. V., et al. 2009, ApJ, 694, 1067
  * Pozzo et al. (2006) Pozzo, M., Meikle, W. P. S., Rayner, J. T., et al. 2006, MNRAS, 368, 1169
  * Rasmussen & Williams (2006) Rasmussen, C. E. & Williams, C. K. I. 2006, Gaussian Processes for Machine Learning
  * Rodríguez et al. (2014) Rodríguez, Ó., Clocchiatti, A., & Hamuy, M. 2014, AJ, 148, 107
  * Roy et al. (2011) Roy, R., Kumar, B., Benetti, S., et al. 2011, ApJ, 736, 76
  * Rubin et al. (2016) Rubin, A., Gal-Yam, A., De Cia, A., et al. 2016, ApJ, 820, 33
  * Sahu et al. (2006) Sahu, D. K., Anupama, G. C., Srividya, S., & Muneer, S. 2006, MNRAS, 372, 1315
  * Sanders et al. (2015) Sanders, N. E., Soderberg, A. M., Gezari, S., et al. 2015, ApJ, 799, 208
  * Schlafly & Finkbeiner (2011) Schlafly, E. F. & Finkbeiner, D. P. 2011, ApJ, 737, 103
  * Schmidt et al. (1992) Schmidt, B. P., Kirshner, R. P., & Eastman, R. G. 1992, ApJ, 395, 366
  * Schulze et al. (2020) Schulze, S., Sollermann, J., Perley, D. A., et al. 2020, Transient Name Server AstroNote, 99, 1
  * Scolnic et al. (2020) Scolnic, D., Smith, M., Massiah, A., et al. 2020, ApJ, 896, L13
  * Servén et al. (2018) Servén, D., Brummitt, C., Abedi, H., & Hlink. 2018, Dswah/Pygam: V0.8.0, Zenodo
  * Silverman (1986) Silverman, B. W. 1986, Density estimation for statistics and data analysis
  * Smartt (2009) Smartt, S. J. 2009, ARA&A, 47, 63
  * Sollerman et al. (2021) Sollerman, J., Yang, S., Schulze, S., et al. 2021, arXiv e-prints, arXiv:2107.14503
  * Szalai et al. (2019) Szalai, T., Vinkó, J., Könyves-Tóth, R., et al. 2019, ApJ, 876, 19
  * Takáts & Vinkó (2012) Takáts, K. & Vinkó, J. 2012, MNRAS, 419, 2783
  * Teja et al. (2022) Teja, R. S., Singh, A., Sahu, D. K., et al. 2022, ApJ, 930, 34
  * Tinyanont et al. (2022) Tinyanont, S., Ridden-Harper, R., Foley, R. J., et al. 2022, MNRAS, 512, 2777
  * Vasylyev et al. (2022) Vasylyev, S. S., Filippenko, A. V., Vogl, C., et al. 2022, arXiv e-prints, arXiv:2203.08001
  * Vinkó et al. (2012) Vinkó, J., Takáts, K., Szalai, T., et al. 2012, A&A, 540, A93
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * Vogl (2020) Vogl, C. 2020, PhD thesis, Technical University of Munich, Germany
  * Vogl (2022) Vogl, C. 2022, In prep.
  * Vogl et al. (2020) Vogl, C., Kerzendorf, W. E., Sim, S. A., et al. 2020, A&A, 633, A88
  * Vogl et al. (2019) Vogl, C., Sim, S. A., Noebauer, U. M., Kerzendorf, W. E., & Hillebrandt, W. 2019, A&A, 621, A29
  * Welch et al. (2007) Welch, D. L., Clayton, G. C., Campbell, A., et al. 2007, ApJ, 669, 525
  * Wiggins (2017) Wiggins, P. 2017, Central Bureau Electronic Telegrams, 4390, 1
  * Wong et al. (2020) Wong, K. C., Suyu, S. H., Chen, G. C. F., et al. 2020, MNRAS, 498, 1420
  * Woosley (1988) Woosley, S. E. 1988, ApJ, 330, 218
  * Yao et al. (2020) Yao, Y., De, K., Kasliwal, M. M., et al. 2020, ApJ, 900, 46
  * Yaron & Gal-Yam (2012) Yaron, O. & Gal-Yam, A. 2012, PASP, 124, 668
  * Zanisi et al. (2020) Zanisi, L., Shankar, F., Lapi, A., et al. 2020, MNRAS, 492, 1671
  * Zwitter et al. (2004) Zwitter, T., Munari, U., & Moretti, S. 2004, Central Bureau Electronic Telegrams, 95, 1

##  Appendix A Correction of the SN 2004et HCT spectra

By comparing the spectra of SN 2004et taken at the HCT and Lick Observatory at
similar epochs, we found that the former show a significant flux deficit in
the wavelength range covered mainly by the Bessell $V$ and $R$ bands as shown
in the Fig. 17. This trend could not be corrected by a linear flux correction
based on the photometry.

Figure 17: Comparison of the spectra of SN 2004et taken at the HCT and the
Lick observatory. The discrepancy between the two spectral sets below the 7000
$\AA$ is clear.

To fix this issue, we took the ratios of the uncorrected HCT and the re-
calibrated Keck and Lick spectra taken at close epochs (which otherwise are
too old for the emulator-based fitting, see top panel of Fig. 18), then
averaged the obtained trend, assuming that it is the same for every HCT
spectrum, and fitted it using Generalized Additive Models (GAMs, Hastie &
Tibshirani 1990, by applying the pyGAM package from Python, Servén et al.
2018), as shown in the bottom panel of Fig. 18. We then applied the fitted
trend on all the HCT spectra by dividing them with this calibration curve
(assuming this calibration trend remains the same regardless of epoch). With
this empirical correction we obtained the HCT spectral sequence shown in Fig.
19, which we passed to the regular relative flux calibration routines
described in Sect. 3.1.

Figure 18: Top panel: the ratios of selected HCT and Keck/Lick spectra that
are close in time. Bottom panel: the average curve calculated from the four
individual ones on the top panel, along with its GAM fit. Figure 19: The
spectral sequence of SN 2004et obtained by HCT. The grey curves show the
uncorrected spectra, while the coloured ones show those for which we applied
the non-linear correction derived from the Keck/Lick spectra.

##  Appendix B Model parameters

In Table 5 we show the physical parameters obtained for the various supernovae
using the emulator.

Epoch [d] |  $T_{\textrm{ph}}$ [K] |  $v_{\textrm{ph}}$ [km/s] | $n$ |  $\Theta$ [$10^{8}$ km / Mpc] |  $\Theta/v_{\textrm{ph}}$ [d / Mpc]
---|---|---|---|---|---
M61 – NGC 4303
2008in
7.59 | 11955 | 8019 | 11.30 | 3.617 | 0.517
8.53 | 11081 | 7656 | 11.70 | 3.402 | 0.515
9.50 | 10542 | 7547 | 10.93 | 3.834 | 0.586
13.99 | 6948 | 6094 | 11.62 | 5.484 | 1.042
16.30 | 6570 | 6324 | 10.92 | 5.924 | 1.084
29.32 | 5962 | 3988 | 7.70 | 6.853 | 1.989
2020jfo
7.08 | 10665 | 10518.75 | 9.76 | 4.675 | 0.537
9.44 | 10176 | 8898.72 | 9.68 | 5.136 | 0.685
11.39 | 8984 | 8606.20 | 9.08 | 5.936 | 0.799
12.00 | 9552 | 8071.05 | 9.02 | 5.638 | 0.815
13.98 | 8215 | 8450 | 9.86 | 6.420 | 0.884
15.00 | 8156 | 7786 | 9.37 | 6.448 | 0.965
19.42 | 6563 | 6519 | 10.50 | 7.984 | 1.417
NGC 772
2003hl
15.28 | 10425 | 9049 | 10.15 | 3.610 | 0.462
2003iq
8.98 | 10196 | 10557 | 11.58 | 3.429 | 0.376
16.19 | 8111 | 8935 | 10.38 | 4.561 | 0.591
20.79 | 6571 | 7629 | 11.01 | 5.530 | 0.839
29.04 | 5950 | 5904 | 8.76 | 6.048 | 1.186
NGC 922
2002gw
16.52 | 6861 | 7396 | 9.47 | 2.501 | 0.391
20.12 | 6229 | 6464 | 8.21 | 2.723 | 0.484
21.12 | 6221 | 6304 | 9.13 | 2.677 | 0.491
2008ho
20.48 | 6242 | 5053 | 9.53 | 2.404 | 0.551
25.43 | 5946 | 4423 | 8.00 | 2.607 | 0.682
NGC 6946
2004et
10.44 | 10377 | 12401 | 11.06 | 19.818 | 1.866
12.31 | 10308 | 11562 | 10.36 | 20.626 | 2.047
14.30 | 10181 | 10604 | 9.68 | 21.557 | 2.334
20.19 | 8505 | 8185 | 12.95 | 22.185 | 3.312
24.00 | 7386 | 7708 | 13.25 | 26.994 | 4.064
29.98 | 6294 | 6436 | 14.90 | 30.373 | 5.315
2017eaw
11.78 | 9764 | 10388 | 14.69 | 16.971 | 1.891
13.77 | 9450 | 10170 | 12.14 | 18.339 | 2.087
15.74 | 8373 | 9975 | 11.18 | 20.771 | 2.410
19.73 | 7199 | 8987 | 11.91 | 23.629 | 3.043
21.73 | 6764 | 8569 | 11.88 | 25.082 | 3.388
22.72 | 6550 | 8451 | 11.81 | 25.686 | 3.518
24.72 | 6318 | 7866 | 11.46 | 26.551 | 3.907
32.83 | 5946 | 6247 | 8.16 | 28.740 | 5.325
Table 5: Table of the inferred physical parameters for the various supernovae.
$T_{\textrm{ph}}$ and $v_{\textrm{ph}}$ denote the photospheric temperature
and velocity respectively, $n$ the density exponent, and $\Theta$ the
estimated angular diameter of the supernovae.

##  Appendix C Light curve fits

In Table 6 we list the interpolated magnitudes required for the angular
diameter calculations, while Fig. 20 shows the individual light curve fits.

Figure 20: The interpolated light curves of the individual supernovae. The vertical grey lines show the epochs of spectral observations. Epoch [d] | $B$ | $V$ | $R$ | $I$ | $g$ | $r$
---|---|---|---|---|---|---
M61 – NGC 4303
2008in
7.59 | 15.17(.02) | 15.11(.01) | 15.01(.01) | 14.87(.02) | — | —
8.53 | 15.21(.02) | 15.12(.01) | 15.02(.01) | 14.87(.02) | — | —
9.50 | 15.22(.02) | 15.12(.01) | 15.02(.01) | 14.87(.02) | — | —
13.99 | 15.27(.02) | 15.14(.01) | 15.03(.01) | 14.87(.02) | — | —
16.30 | 15.30(.02) | 15.15(.01) | 15.03(.01) | 14.87(.02) | — | —
29.32 | 15.44(.02) | 15.19(.01) | 15.05(.01) | 14.86(.02) | — | —
2020jfo
7.08 | — | — | — | — | 14.54(.02) | 14.48(.02)
9.44 | — | — | — | — | 14.59(.01) | 14.50(.02)
11.39 | — | — | — | — | 14.61(.01) | 14.50(.02)
12.00 | — | — | — | — | 14.67(.01) | 14.51(.01)
13.98 | — | — | — | — | 14.69(.01) | 14.52(.01)
15.00 | — | — | — | — | 14.82(.01) | 14.54(.01)
19.42 | — | — | — | — | 14.82(.01) | 14.54(.01)
NGC 772
2003hl
15.28 | 16.93(.02) | 16.46(.01) | 16.04(.02) | 15.87(.01) | — | —
2003iq
8.98 | 15.76(.02) | 15.75(.01) | 15.42(.01) | 15.33(.02) | — | —
16.19 | 15.97(.02) | 15.72(.01) | 15.42(.01) | 15.33(.01) | — | —
20.79 | 16.17(.02) | 15.74(.01) | 15.38(.01) | 15.30(.01) | — | —
29.04 | 16.57(.02) | 15.82(.01) | 15.43(.01) | 15.33(.01) | — | —
NGC 922
2002gw
16.52 | 17.66(.02) | 17.32(.01) | — | 16.87(.01) | — | —
20.12 | 17.80(.01) | 17.35(.01) | — | 16.84(.01) | — | —
21.12 | 17.83(.02) | 17.36(.01) | — | 16.83(.01) | — | —
2008ho
20.48 | 18.51(.01) | 18.00(.01) | — | — | — | —
25.43 | 18.68(.01) | 18.02(.01) | — | — | — | —
NGC 6946
2004et
10.44 | 12.91(.02) | 12.63(.01) | 12.29(.01) | 12.03(.02) | — | —
12.31 | 12.96(.02) | 12.65(.01) | 12.29(.01) | 12.03(.02) | — | —
14.30 | 13.00(.02) | 12.66(.01) | 12.30(.01) | 12.02(.01) | — | —
20.19 | 13.13(.02) | 12.70(.01) | 12.31(.01) | 12.02(.01) | — | —
24.00 | 13.27(.02) | 12.73(.01) | 12.32(.01) | 12.01(.01) | — | —
29.98 | 13.58(.02) | 12.78(.01) | 12.34(.01) | 12.00(.01) | — | —
2017eaw
11.78 | 13.24(.01) | 12.87(.01) | 12.45(.01) | 12.21(.01) | — | —
13.77 | 13.30(.01) | 12.90(.01) | 12.46(.01) | 12.23(.01) | — | —
15.74 | 13.37(.01) | 12.92(.01) | 12.48(.01) | 12.25(.01) | — | —
19.73 | 13.51(.01) | 12.95(.01) | 12.49(.01) | 12.26(.01) | — | —
21.73 | 13.60(.01) | 12.97(.01) | 12.49(.01) | 12.26(.01) | — | —
22.72 | 13.64(.01) | 12.98(.01) | 12.49(.01) | 12.26(.01) | — | —
24.72 | 13.73(.01) | 12.99(.01) | 12.50(.01) | 12.27(.01) | — | —
32.83 | 14.08(.01) | 13.10(.01) | 12.57(.01) | 12.28(.01) | — | —
Table 6: Table of the interpolated magnitudes for the various supernovae. The
values in the brackets denote the uncertainties of the estimates.