the $g$-band Type Ibc template. We thus obtain a stretch correction factor
$S^{r}_{Ibc}/S^{g}_{Ibc}\approx 1.22$. Multiplying the stretch values measured
in the $g$ band by this factor corrects them to the $r$-band template stretch-
space, which is what we plot for the $g$ band in the right panels of Fig. 2.

Looking at the figure, the distribution of stretch values for Type Ibc SNe
seem to indicate the presence of two populations. The majority of SE SNe
cluster around a stretch value of 1.0, with a secondary distribution located
at high broadness values ($S^{r}_{Ibc}\gtrsim 1.5$ in stretch value).

It can be seen in the bar plot of $g$-band stretch values plotted in Fig. 13
that the broad objects seem to be have statistically significant higher
stretch values161616This figure includes Type IIb SNe (Appendix B)..
Interestingly, there is also an object with a significantly lower stretch
value, iPTF14gqr, which was found to be an ultra-stripped SE SN (De et al.
2018) as discussed in Sect. 8. We tested the apparent bimodality in lightcurve
broadness with two approaches: Gaussian mixture modeling and K-means
clustering. We will use the results of these statistical tests to derive our
final broad sample.

Figure 15: Histogram of the distribution of stretch values plotted after
running a k-means clustering algorithm on the one-dimensional stretch
parameter dataset. The SNe put in to a second cluster by the algorithm have
been plotted in blue, and labeled as broad. The ordinary (non-broad) SNe in
each band have been plotted with the respective color of that band ($r$ band
left and $g$ band to the right). The histogram was not normalized so the
counts represent the number of SNe in each bin. Top: Type Ibc SNe, Bottom:
Type IIb SNe.

####  A.4.1 Gaussian mixture modeling

To test whether the stretch values prefer one or more distributions, we fit
them using multi-component Bayesian Gaussian Mixture Models (GMMs) using the
SKLearn library in Python 3. These models allow infinitely large component
Gaussian fits, but attach very low weights when further components are not
helpful in reproducing the observed distribution. As shown in the upper and
lower panels of Fig. 14, the Type Ibc stretch values prefer two Gaussian
components to explain the distribution, with one component to explain the
ordinary broadness objects and another to explain the broad ones. Further
components are not found to be necessary, as the likelihood has already
reached a maximum with two Gaussian components.

As an additional test, we made 500 realizations of the stretch distribution by
varying all values randomly within the standard errors. We fit each of these
realizations using ordinary multi-component GMMs and select the preferred
model (i.e. the preferred number of Gaussian components) based on the lowest
value of the BIC goodness-of-fit indicator. Based purely on the lowest BIC, we
find that 100% of these fits prefer $\geq 2$ components. In the $r$ band,
$99.6\%$ prefer exactly two, while this is $63.6\%$ in the $g$ band, with the
remaining primarily preferring three components. If adjusted for the “rule of
thumb” that models within $\Delta BIC\leq 2.0$ are not worth distinguishing,
the percentage of $g$ band two-component favored fits increases to $80.4\%$.
Single component fits are still never favored, neither in the $g$ nor the $r$
band.

That a certain number of Gaussians is preferred means that this combination
best represents the data compared to any other combination of Gaussians. The
resulting GMM fit is then a good representation of the data under the
assumption that the underlying distribution of the data is a combination of
Gaussians. Based on these results, we choose to split the stretch distribution
into two components and identify the broad Type Ibc SNe using the component
centered around higher stretch values.

####  A.4.2 K-means clustering

In order to obtain the broad sample, we also ran a k-means clustering
algorithm171717Using the SKLearn package in Python 3. with two initial seeds
that will form two clusters in the data. These seeds are placed pseudo-
randomly and grow by acquiring their nearest neighbors while trying to
minimize the sum of the distances within a cluster. The process is repeated to
find the best overall fit. In this manner, the algorithm will yield the two
most tightly bound clusters possible in this distribution. The one dimensional
case used here is trivially solved, so while the k-means clustering algorithm
is not a great statistical arbiter, it does show the natural clustering
present in our dataset. A benefit of this method is that we do not have to
assume any underlying distribution, as is the case for GMM based tests.

As shown in Fig. 15, and mirroring the result from the GMM fits, we find that
the data readily clusters into two distributions, one centered around a
stretch value of 1.0 and the other located at stretch values $>1.5$, which we
label “ordinary” and “broad”, respectively. The result indicates that a hard
cut at a stretch value between 1.5–1.8 is optimal for splitting the Type Ibc
sample into two distributions, and we pick the mid-point value of $1.65$.
Several SNe have both $g$ and $r$-band lightcurves. All of our SNe with
multiband stretch values show up in the “ordinary” or “broad” categories
regardless of the band. Even when taking into account 1-sigma fitting errors,
the SNe cannot move categories when using $1.65$ as the cut-off value.

##  Appendix B Type IIb SNe

In this paper, we developed a method of finding SE SNe with broad lightcurves,
and applied it to Type Ibc SNe. Studying the resulting broad SE SNe in detail,
we found them to be likely associated with massive stars, at least when
applying the Arnett formalism (however, see Sect. 8). Then, we estimated the
fraction of broad Type Ibc SNe to be ${\sim 13}\%$, which corresponds to
${\sim 6}\%$ after correcting for observational biases181818Using the $r$ band
to account for the follow-up bias as discussed in Sect. 7.. In this section,
we show that Type IIb SNe in (i)PTF also hint at having a similar fraction of
broad (and likely high-mass) SNe. However, a detailed investigation of Type
IIb SNe, especially broad Type IIb SNe, is beyond the scope of the paper.

###  B.1 Template fitting for Type IIb SNe

Following exactly the same steps as with Type Ibc SNe, we fit template
lightcurves to the $g$ and $r-$band lightcurves of the (i)PTF Type IIb sample.
For Type IIb SNe, we use the well-observed and well-behaved SN 2011dh (Ergon
et al. 2014, 2015) as a template. We are confident in using a single SN as a
template since Fremling et al. (2016) showed a remarkable similarity between
the lightcurves of two ordinary Type Ib and IIb SNe that exploded in the same
galaxy at the same distance close in time. The template fitting is done near
peak only, so the variable early cooling phase seen in some Type IIb SNe does
not have an effect on the fits.

We use the same templatability criteria as for Type Ibc SNe, with the
exception that we use the Type IIb template. As reported in Table 1, there are
58 Type IIb SNe, which reduce to 39 after all cuts on templatability. These 39
Type IIb SNe are fit using a template to obtain the stretch parameter
$s^{g/r}_{IIb}$ for each SN. Just like for the Type Ibc SNe, we verify the
fits visually. We noticed that it was not possible to find a good fit to
PTF10pzp, although it technically barely passes the templatability criteria.
This object has a stretch value of $1.6\pm 0.1$ in the $r$ band. Although we
nominally find it to be in the broad category (see below), we are not
confident of the stretch value. Therefore, the broad fractions we report in
this section can be 20% lower if this object is excluded. Out of all 107 Type
Ibc and Type IIb SNe, this object is the only one with an obvious visually-
poor fit.

In order to be able to compare the resulting distributions, we also fit the
Type IIb $g$ and $r$-band template lightcurves to the Type Ibc equivalent and
obtain correction factors of $1.202$ for the $r$ band and $1.147$ for the $g$
band. The resulting distribution is plotted in the middle panel of Fig. 2.
These distributions have all been corrected to the space of $r$ band Type Ibc
stretch values.

band | observed | duration | Malmquist
---|---|---|---
$r$ | $0.139^{+0.06}_{-0.06}$ | $0.132^{+0.06}_{-0.06}$ | $0.061^{+0.04}_{-0.03}$
$g$ | $0.222^{+0.07}_{-0.08}$ | $0.208^{+0.08}_{-0.09}$ | $0.081^{+0.06}_{-0.05}$
combined | $0.154^{+0.06}_{-0.06}$ | $0.146^{+0.06}_{-0.06}$ | $0.068^{+0.04}_{-0.03}$
Table 6: The fraction of broad Type IIb SE SNe after correcting for
observational biases, reported with 90% Poissonian uncertainties.

We find that Type IIb SNe also show tentative evidence of two clusters: one
located around a stretch value of ${\sim}1.2$ in the $r$ band and ${\sim}1.0$
in the $g$ band (before correcting to the $r$-band stretch), with the other at
a higher stretch value. Interestingly, applying the corrections that were
simply derived on the templates, not on the SNe, leads us to seeing a possible
gap in stretch values located exactly around $s=1.65$, the same value as for
Type Ibc SNe. However, the gap is much less clear for the Type IIb SNe. This
can also be seen in the result of the K-means test plotted in the lower panel
of Fig. 15. In Fig. 14, the GMM test for multiple distributions in Type IIb
SNe favors one distribution for the $r$ band and two distributions for the $g$
band. Thus, while the stretch distribution in Type IIb SNe also seems to
indicate a population of broad events, it is more difficult to determine an
exact cut-off using our methodology from Type Ibc SNe to select a broad
sample. The lower number of Type IIb SNe, and the smaller range of stretch
values inhabited, exacerbates the difficulty of selecting a broad sample
especially in the $r$ band.

iPTF13aby (SN 2013bb), the most massive Type IIb SNe discovered to date
(Prentice et al. 2019), has a stretch value of ${\sim}1.85$ in the corrected
$g$ and $r$-band stretch values, making it one of the broadest Type IIb SNe in
our sample. Interestingly, there are several other Type IIb SNe with similarly
broad lightcurves, which suggests that many more broad and likely massive Type
IIb SNe might exist in the (i)PTF database. Therefore, the broad sample of
(i)PTF Type IIb SNe should include iPTF13aby (and others like it).

First, we consider simply using the same cut-off as with the Type Ibc SNe,
since we see an indication that there is an apparent gap at this value, and
since SE SN lightcurves behave rather similarly (Fremling et al. 2016). This
would also correctly label iPTF13aby as a broad SN. However, the GMM and
K-means based methodologies we employed for Type Ibc SNe (Sect. A.4.2) are not
fully consistent in this simple approach. GMM fits to the Type IIb $r$-band
stretch distribution prefer a single Gaussian to explain the observed
distribution. When we fit multi-component GMMs to 500 realizations of the Type
IIb $r$-band stretch distribution obtained by randomly varying the stretch
values within the errors, a two-component fit was favored over a single
component fit in only ${\sim}3\%$ of those realizations, as determined by the
lowest Bayesian information criteria (BIC) of the fit. In those $3\%$, a
stretch cut of $1.65$ would be sufficient to separate the broad SNe from the
ordinary, matching the value found for Type Ibc SNe in either band as well as
for the $g$-band Type IIb SNe. In the remaining $97\%$ of realizations of the
GMM fits, a single Gaussian component fit is preferred.

This fact can also explain the unclear result from our K-means test. We are
searching for two clusters when only one seems to be present. Therefore, we
consider the stretch cut implied by our K-means test on the Type IIb $r$ band
to be a lower limit (see Fig. 15). The most conservative upper limit is
established by considering all $r$-band Type IIb SNe as ordinary (including
those that would in the broad category in $g$ band), leaving only iPTF15cna as
broad, which has a $g$-band stretch of $2.1$. While iPTF15cna is only the
second broadest after PTF11po, iPTF15cna lacks an $r$-band peak that might
have disqualified it from being labeled broad. A more sensible upper limit
might instead be derived using the well-determined stretch value of iPTF13aby
from Prentice et al. (2019), which is a known broad and massive object, and
then considering those that have a stretch value greater than or equal to that
of iPTF13aby within 1-sigma errors. Interestingly, this yields the same result
as using a stretch cut of $1.65$.

Therefore, in our limited investigation, we use a stretch cut of $1.65$ also
for Type IIb SNe, even though the presence of a well-separated second
population of high stretch events is less certain, especially in the $r$ band.
Nevertheless, we do know of several broad events that become correctly labeled
using this division, including iPTF13aby.

###  B.2 Corrected fraction of broad Type IIb SNe

Based on a stretch cut of $1.65$, we are left with 6 Type IIb SNe out of 36
being broad, meaning an observed fraction of $15^{+6}_{-6}\%$. The numbers for
each band are reported in Table 1, while the broad fraction is reported in
Table 6. After correcting for observational biases in the same way as for Type
Ibc SNe, we calculate the broad fraction in the $r$ band as $6^{+4}_{-3}\%$
and in the $g$ band as $8^{+6}_{-5}\%$, reported with $90\%$ confidence
intervals.

Since we correct the stretch distribution to that of the Type Ibc $r$-band
template, we are able to use the same calculation for the lightcurve duration
bias. For the Malmquist bias, we follow the steps outlined in Appendix C using
values from the Type IIb sample. However, we have kept the same $z_{max}$ as
with Type Ibc SNe for consistency, even though the most distant Type IIb SN is
located at redshift $0.099$. Using $z_{max}=0.1$ would instead yield a
corrected fraction of $10\%$ in the $r$ band and $13\%$ in the $g$ band.
Similar to Type Ibc SNe, there also seems to be a follow-up bias towards broad
SE SNe in the $g$ band.

###  B.3 Broad Type IIb SNe

PTF10qrl, PTF10pzp191919We remind that PTF10pzp is the only lightcurve in our
sample where visual inspection shows a poor fit. Thus, its identification as a
broad SN could be incorrect., PTF11po, iPTF13aby, iPTF14bas, iPTF15cna are
found to be broad Type IIb SNe, for which we briefly discuss some early
indications. iPTF13aby (SN 2013bb) was studied by Prentice et al. (2019) and
found to be the most high-mass Type IIb SN discovered to date. iPTF14bas was
identified as an unusually long rising Type IIb SN by Rubin & Gal-Yam (2016).

iPTF15cna has late-time Keck spectra which can be used to estimate the
metallicity as in Sect. 6. It also seems to be located in a dwarf host galaxy
with high SFR (Schulze et al. 2021). Using the late time Keck spectrum, we
calculate the metallicity at the location of iPTF15cna using the O3N2 method
and find log(O/H)+12=${\approx}8.2$ dex in both the PP04 and M13 methods. It
seems that at least some broad Type IIb SNe, like iPTF15cna, are located in
low-metallicity environments similar to the broad Type Ibc SNe.

Using the nebular Keck spectrum, we also estimate the
$L_{[{Ca\textsc{ii}}]}/L_{[{O\textsc{i}}]}$ for iPTF15cna and obtain
${\sim}0.5$. This hints that iPTF15cna might be similar to iPTF13aby in having
a high-mass origin. Thus, in addition to the fraction of broad Type Ibc SNe
(Table 4), we also report the fraction of all broad SE SNe including Type IIb
SNe (see Table 5).

##  Appendix C The Malmquist Bias

Correcting for the Malmquist bias is not a straightforward endeavor.
Typically, there are three conceptually distinct ways to correct for it. The
first and easiest method is to use the distribution only out to a distance
where the entire brightness range is not truncated, thereby converting a
magnitude-limited survey into an effectively volume limited survey. A second
way is to use an empirical $1/\text{V}_{\text{Max}}$ correction based on the
distance (Schmidt 1968). Finally, a more complicated correction that uses the
distribution observed in the lower distance (non-biased) part, or knowledge
about the underlying distribution, in order to correct the higher distance
(Malmquist-biased) part can be employed (see e.g., Richardson et al. 2014).
The last of these methods makes the most use of the data, but cannot be used
in our case because we are interested in the distribution of a correlated
third parameter, stretch, whose inherent distribution we have no knowledge of.
Instead, in this paper, we employed a novel method to correct for the peculiar
case of the Malmquist bias on two parameters, which are correlated with a
third parameter.

Figure 16: Top: A plot of peak absolute magnitude in $r$ band versus distance
modulus (DM) for ordinary (red circles) and broad (purple circles) SE SNe,
which display the Malmquist bias associated with the (i)PTF detection limit
thresholds. The detection thresholds at $r=20.5$ mag and $r=21.0$ mag are
plotted as grey dashed and black dot-dashed lines, respectively. They are
approximate estimates of the detection threshold for (i)PTF during bright and
dark times, respectively. This plot shows that the broad SNe are
preferentially brighter than the ordinary SE SNe, meaning that they are
detected at a higher fraction at higher redshifts. Bottom: Histogram of peak
magnitude with the gaussian fits overlaid for the broad and ordinary groups in
$g$ and $r$ bands. This type of distribution is often referred to as a
truncated distribution (Feigelson & Babu 2012), and correcting for this
truncation is a complicated but necessary step for all magnitude limited
surveys.

Since the luminosity function of our SNe is not known, especially for the
broad class, we attempt to estimate and correct for this before calculating
the ratio of broad versus ordinary SE SNe. The distribution resulting from the
Malmquist bias is a truncated distribution (Feigelson & Babu 2012), more
specifically right or left truncated depending on if the peak brightness is
reported in magnitudes or flux.

The method works as follows. To perform the fit we assumed that the peak
brightness for each class (ordinary and broad) was normally distributed with
some unknown mean and variance. Using a constant mean peak brightness is
justified if we do not expect cosmic evolution of the brightness with respect
to distance, which we do not for the low redshifts of our SNe. Thus, we fit a
truncated linear regression model202020Using the truncreg function in Stata
version 16.0, with a left truncation on the absolute value of the peak
absolute magnitudes. to the peak magnitude for ordinary and broad SE SNe
separately, to estimate the mean of the peak brightness of each sample. We did
this separately for each band and obtained the values reported in Table 7
where $\mu_{0}$ and $\mu_{1}$ are the mean values of the “ordinary” and
“broad” samples, respectively, while $\sigma$ is the standard deviation
(assumed to be the same for simplicity). The resulting gaussian fits are
plotted in the lower panel of Fig. 16 for the $g$ and $r$ bands of the Type
Ibc stretch distribution.

Band | Parameter | Estimate | Standard Error
---|---|---|---
$r$ | $\mu_{0}$ | 17.33 | 0.107
| $\mu_{1}$ | 18.44 | 0.256
| $\sigma$ | 0.717 | 0.069
$g$ | $\mu_{0}$ | 17.44 | 0.172
| $\mu_{1}$ | 17.77 | 0.358
| $\sigma$ | 0.697 | 0.111
Table 7: Results of the truncated regression model fit for Type Ibc SNe.
$\mu_{0}$ and $\mu_{1}$ are the means of the “ordinary” and “broad” samples,
respectively, while $\sigma$ is the standard deviation (assumed to be the same
for simplicity).

Next we correct the fraction of broad to ordinary SE SNe by the statistical
exposure, which in this case represents the volume size of each SN. The
Malmquist bias means that brighter SNe had a larger volume size and thus were
measured with a longer exposure. The formula for this correction is simply:

$\displaystyle\text{Corrected
ratio}=\frac{n_{1}/E_{1}}{n_{0}/E_{0}+n_{1}/E_{1}},$ (2)

where $n_{x}$ is the number of SNe in group $x$, 0 for ordinary, 1 for broad,
and $E_{x}$ is the exposure for each group. According to the Malmquist bias
the intrinsically brighter group will have a larger exposure. Next we estimate
the $E_{x}$ for each group from our truncated linear regression model results.

To estimate the $E_{x}$ we need to know the observable fraction for each
group. For this we assume that the SNe brighter than apparent magnitude 21,
the nominal (i)PTF dark time brightness limit for the standard one minute
exposure, are detected. Then we calculate using our normal distributions of
peak brightness the observable fraction for each group with the following
formula:

$\displaystyle\text{p}_{x}=1-\phi(\frac{dm-21.0-\mu_{x}}{\sigma}).$ (3)

Here $\text{p}_{x}$ is the observable fraction of group $x$ and $\phi$ is the
cumulative normal distribution function. In essence we subtract from unity the
normalized sum of the normal distribution that passes the survey brightness
threshold. Meaning that we get an estimate of the fraction of SNe in that
group which were actually observed. Finally $E_{x}$ is calculated following

$\displaystyle E_{x}=\int^{V_{C,max}}_{V_{C,min}}\text{p}_{x}dV_{C},$ (4)

where $V_{C}$ is the co-moving volume. The differential co-moving volume is
integrated from $z=0$ to $z=z_{max}$, where $z_{max}$ is the maximum
distance/redshift allowed in to the survey. Ideally, $z_{max}$ should be set
ahead of time. At the least, it is crucial to not bias the results by
selecting $z_{max}$ to maximize or minimize the desired outcome. In our case,
we set $z_{max}$ to be the redshift of the furthest SN in our sample at
$z=0.138$. If using a lower cut, the fractions and regression should be
updated accordingly before applying the bias correction. We convert $z$ to
distance modulus and calculate the differential co-moving distance using the
same cosmology as in the rest of the paper based on parameters as measured by
WMAP5. Finally, we use the exposures $E_{x}$ and Eq. 2 to calculate the
corrected fraction of broad/ordinary in each band and obtain the values
reported in Table 4. Poissonian uncertainties are calculated by normalizing
the exposures to keep the number of detected broad objects the same
($E_{1}^{\text{norm}}=E_{1}/E_{1}\equiv 1$,
$E_{0}^{\text{norm}}=E_{0}/E_{1}$).

In general, our method can be represented algorithmically as follows:

  1. 1.

Determine $z_{max}$ of the sample.

  2. 2.

Fit a Gaussian to the peak brightness using truncated regression, with an
appropriate left or right truncation, to each group in the sample. (In our
case, the groups were ordinary and broad.)

  3. 3.

Using the resulting Gaussian mean(s) and standard deviation(s), calculate
$\text{p}_{x}$ as a function of $z$.

  4. 4.

$E_{x}$ at a given $z$ is the integral from $z=0$ to $z$ of
$\text{p}_{x}(z)\times dV_{C}$. It is important to not neglect the solid
angle.

  5. 5.

Integrate $E_{x}$ up to $z_{max}$ to obtain the statistical exposure of each
group.

  6. 6.

The Malmquist bias corrected fraction for any combination of parameters
calculated between the groups is simply obtained by dividing the contribution
of each group by its associated exposure.

Malmquist-bias corrected ratios can be obtained for groups created using any
measured or derived parameter (e.g., host metallicity or model parameters such
as ejecta mass or nickel mass). As a concrete example, if the groups instead
represent high- and low-velocity sub-groups within a sample of SNe, the above
method allows calculating the fraction of high and low velocity objects.

It is further possible to generalize this method to use other well-behaved
distributions where $\text{p}_{x}$ can be calculated, instead of having to
assume a normal distribution. For example, it would be possible to obtain the
Malmquist-bias corrected fraction of Type Ia SNe subtypes versus normal SNe
Ia, whose luminosity function usually follow a Schechter (1976) function
(Brown et al. 2019).

##  Appendix D Supernova classifications

Schulze et al. (2021), henceforth S21, comprehensively classified all (i)PTF
CC SNe from the (i)PTF database. Since we independently constructed and
verified our SESN sample from the (i)PTF database, there are some differences
with the S21 sample, as can be seen in Table 1. Generally, the differences
were either due to missed classifications from external sources by S21, or due
to S21 explicitly excluding or reclassifying the SN different than us,
sometimes based on the work of Modjaz et al. (2020), (M20). However, these
minor differences do not impact our results, since most of the differing
classifications discussed belong to SNe which do not pass our selection
criteria for template fitting anyway.

  * •

PTF09dha: missing from S21; this is classified as a Type Ib SN in the (i)PTF
Marshal and by us.

  * •

PTF10wg: Excluded by S21 due to uncertain classification; classified as Type
Ib/c in (i)PTF Marshal and by us. Later cut in a further step in Table 1 due
to lacking a peak but included in our total number.

  * •

PTF11bky: Missing from S21; this is a SN Ib/c classified by Central Bureau for
Astronomical Telegrams (CBET) 2677 (Prieto et al. 2011).

  * •

PTF11gcj: Excluded by S21 as a Ic-BL based on M20 reclassifying it from a Ic;
Type Ic SN re-classified again by Barbarino et al. (2021) based on good
spectral template matches to several Type Ic SNe.

  * •

PTF12mps: Missing from S21; this is a Type Ib/c externally classified by
(Hadjiyska et al. 2012).

  * •

iPTF13doq: Excluded by S21 due to host contamination of the spectrum; we
classify it as a Type Ib with an $r$-band stretch value around 0.8.

  * •

iPTF13ou: Missing from S21; this is SN 2012fh classified as a Type Ic SN by
Nakano et al. (2012). The (i)PTF lightcurve is only significantly after peak.

  * •

iPTF14jhf: Excluded from S21 due to low data quality; Type Ib/c with an
$r$-band stretch value around 0.9, also classified by Fremling et al. (2018);
Barbarino et al. (2021) .

  * •

iPTF15cla: Missing from S21; this is a Type Ib/c classified by Terreran et al.
(2015).

  * •

iPTF16ewz: Missing from S21; we tentatively classified it as a Type Ic-BL SN
as in the (i)PTF marshal, although it was noted as a possible Type Ia or other
uncertain classification as well, and hence was excluded from (Taddia et al.
2019a). Later, it was re-classified as a Type Ic by (Sollerman et al. 2021).
Due to the uncertain classification, we exclude it from the 183 Type Ibc/IIb
SNe in our sample, but it is included in our topline number of 220 SESNe in
Table 1.

Excluded from our sample:

  * •

PTF10ysd: Included in S21 as a Type Ic as reclassified by M20; we classify as
a Type Ic-BL as in Taddia et al. (2019a); Barbarino et al. (2021).

  * •

PTF10aauo: Included in S21 as a Type IIb; excluded by us as Type II with a
blue continuum, noted as Type II Blue in the (i)PTF Marshal.

  * •

PTF10xfh: Included in S21 as a Type Ib; we exclude it as a peculiar gap
transient as in the (i)PTF Marshal.

  * •

PTF13dzy: Included in S21 as Type IIb; we exclude it as a Type II as in the
(i)PTF Marshal.

By including the 9 SE SNe excluded from the CC SN sample of S21, we update the
number of CC SNe in (i)PTF to 897 (from 888 in S21), which includes all
peculiar SNe such as SLSN and Type Ibn SNe. Excluding them, the total CC SNe
would instead become 833 (from 824 in S21).

##  Appendix E Supporting figures and tables

In this section, we provide supporting figures and tables for each SN in our
broad Type Ibc sample. We include plots with apparent magnitude lightcurves,
K-corrections, Arnett model fits to the bolometric lightcurves, spectral
sequences, and line-velocity evolution. We also include tables with all
spectral observations and host metallicity measurements.

Figure 17: The apparent magnitudes of our broad SE SNe in the observer frame.
No extinction or redshift correction were applied. For clarity, data after 150
days and all upper limits have been omitted. The photometry are available on
Wiserep. Figure 18: K-corrections in the $r$ band compared to template
K-corrections. Open diamonds mark K-corrections calculated from template Type
Ibc spectra, see Sect. 7.4. At the epochs of the existing spectra, colored
symbols are K-corrections calculated from the spectra of our broad sample, and
the dashed lines represent the actual K-corrections used, which use a second
degree polynomial fit for interpolation or a flat extrapolation. Figure 19:
Bolometric lightcurves of the broad SE SN sample overplotted with weighted
(orange) and unweighted (red) Arnett model fits. The fit to the main peak of
PTF11mnb is shown with a blue line (see Sect. 5). Figure 20: Absorption
velocity of Fe ii $\lambda\lambda 4924,5018,5169$, He i $\lambda 5876$, Na i
$\lambda\lambda 5890,5896$, O i $\lambda\lambda 7772,7774,7775$, and Si ii
$\lambda 6355$ in each SN.

Figure 21: Gaussian smoothed spectral sequence of PTF09dfk and PTF10inj,
labeled with restframe phase.

Figure 22: Gaussian smoothed spectral sequence of PTF11bov and PTF11mnb,
labeled with restframe phase.

Figure 23: Gaussian smoothed spectral sequence of PTF11rka and iPTF15dtg,
labeled with restframe phase.

Figure 24: Gaussian smoothed spectral sequence of iPTF6flq and iPTF16hgp, labeled with restframe phase. SN | JD | Telescope | Instrument | Restframe Phase
---|---|---|---|---
|  |  |  | (days)
PTF09dfk | 2455068.95 | Keck I 10m | LRIS |  $-$27.0
| 2455095.88 | P200 | DBSP |  $-$0.5
| 2455115.50 | WHT 4.2m | ISIS | 18.8
| 2455126.92 | Keck I 10m | LRIS | 30.1
| 2455501.50 | Keck I 10m | LRIS | 398.7
PTF10inj | 2455355.50 | Keck I 10m | LRIS |  $-$21.3
| 2455391.50 | P200 | DBSP | 12.5
| 2455410.50 | WHT 4.2m | ISIS | 30.3
| 2455447.50 | KPNO 4m | RC Spec | 65.1
| 2455472.50 | Keck I 10m | LRIS | 88.6
| 2457982.73 | Keck I 10m | LRIS | 2447.2
PTF11bov | 2455662.50 | KPNO 4m | RC Spec |  $-$20.0
| 2455692.50 | KPNO 4m | RC Spec | 9.3
| 2455719.79 | KPNO 4m | RC Spec | 36.0
| 2455770.50 | WHT 4.2m | ISIS | 85.6
| 2455913.74 | TNG | DOLoReS | 225.8
| 2455915.50 | WHT 4.2m | ISIS | 227.5
| 2455945.01 | P200 | DBSP | 256.4
PTF11mnb | 2455842.01 | UH88 | SNIFS |  $-$17.1
| 2455864.78 | P200 | DBSP | 4.3
| 2455891.91 | Keck I 10m | LRIS | 29.9
| 2455916.68 | P200 | DBSP | 53.3
| 2455953.59 | KPNO 4m | RC Spec | 88.1
PTF11rka | 2455922.15 | Keck I 10m | LRIS |  $-$10.0
| 2455953.92 | KPNO 4m | RC Spec | 19.6
| 2455977.99 | Keck I 10m | LRIS | 42.0
| 2456044.88 | Keck I 10m | LRIS | 104.3
| 2456069.77 | Keck I 10m | LRIS | 127.4
| 2456364.50 | VLT | FORS2 | 401.7
| 2457363.10 | Keck I 10m | LRIS | 1331.2
iPTF15dtg | 2457336.60 | TNG | DOLoReS |  $-$30.4
| 2457337.98 | Keck I 10m | LRIS |  $-$29.1
| 2457350.46 | TNG | DOLoReS |  $-$17.3
| 2457358.85 | DCT | Deveny+LMI |  $-$9.3
| 2457362.88 | Keck I 10m | LRIS |  $-$5.5
| 2457367.43 | NOT | ALFOSC |  $-$1.1
| 2457398.32 | TNG | DOLoReS | 28.2
| 2457416.37 | NOT | ALFOSC | 45.4
| 2457430.42 | TNG | DOLoReS | 58.7
| 2457456.73 | Gemini North | GMOS | 83.7
| 2457693.02 | Keck I 10m | LRIS | 308.2
| 2457730.53 | NOT | ALFOSC | 343.9
| 2457982.97 | Keck I 10m | LRIS | 583.7
iPTF16flq | 2457628.84 | Palomar 1.5m | SEDM |  $-$17.8
| 2457646.56 | NOT | ALFOSC |  $-$1.1
| 2457652.62 | NOT | ALFOSC | 4.7
| 2457661.48 | NOT | ALFOSC | 13.0
| 2457672.60 | NOT | ALFOSC | 23.5
| 2457692.95 | Keck I 10m | LRIS | 42.7
| 2457716.40 | NOT | ALFOSC | 64.8
iPTF16hgp | 2457683.80 | DCT | Deveny+LMI |  $-$25.2
| 2457694.81 | Keck I 10m | LRIS |  $-$15.0
| 2457720.90 | Keck I 10m | LRIS | 9.2
| 2457729.43 | NOT | ALFOSC | 17.1
| 2457755.78 | Keck I 10m | LRIS | 41.4
Table 8: Log of spectral observations from (i)PTF of the eight broad SE SNe. Acronyms in addition to those previously introduced: William Herschel Telescope (WHT), Low Resolution Imaging Spectrometer (LRIS), Gemini Multi-Object Spectrograph (GMOS), Large Monolithic Imager (LMI), The Double Spectrograph (DBSP), SuperNova Integrated Field Spectrograph (SNIFS), Intermediate-dispersion Spectrograph and Imaging System (ISIS), SEDMachine (SEDM) SN | $M13$ | $PP04$
---|---|---
| $12+log(O/H)$ | $12+log(O/H)$
09dfk | $8.17^{8.2}_{8.14}$ | $8.19^{8.22}_{8.14}$
10inj | $8.18^{8.21}_{8.14}$ | $8.2^{8.23}_{8.14}$
11bov | $8.21^{8.24}_{8.18}$ | $8.25^{8.28}_{8.18}$
11mnb | $8.18^{8.21}_{8.14}$ | $8.2^{8.23}_{8.14}$
11rka | $8.0^{8.04}_{7.95}$ | $7.93^{7.97}_{7.95}$
15dtg | $8.06^{8.1}_{8.02}$ | $8.03^{8.07}_{8.02}$
16flq | $8.37^{8.39}_{8.34}$ | $8.48^{8.5}_{8.34}$
16hgp | $8.13^{8.16}_{8.1}$ | $8.13^{8.16}_{8.1}$
Table 9: Metallicity of the host galaxies of the broad sample as measured using the PP04 and M13 line flux ratio methods, with the confidence interval given as the superscript upper and subscript lower bound. SN | $M_{ej}$ | $M_{Ni}$ | $E_{k}$ | $E_{E}xp$ | $V_{ph}$
---|---|---|---|---|---
| ($M_{\sun}$) | ($M_{\sun}$) | ($10^{51}erg$) | ($restframedays$) | ($kms^{-1}$)
09dfk | $22.6\pm 1.8$ | $0.58\pm 0.01$ | $7.3\pm 2.3$ | $2.2\pm 0.9$ | $7300\pm 1100$
10inj | $26.4\pm 1.0$ | $0.39\pm 0.01$ | $16.4\pm 1.7$ | $0.0\pm 0.3$ | $10200\pm 500$
11bov | $13.0\pm 1.0$ | $0.42\pm 0.02$ | $4.6\pm 0.6$ | $1.5\pm 0.3$ | $7700\pm 400$
11mnb | $32.4\pm 5.0$ | $0.48\pm 0.05$ | $5.6\pm 0.9$ | $-9.3\pm 0.5$ | $5400\pm 200$
11mnb_peak1 | $13.3\pm 2.5$ | $0.55\pm 0.03$ | $2.3\pm 0.4$ | $-4.3\pm 5.3$ | $5400\pm 200$
11rka | $20.3\pm 4.0$ | $0.54\pm 0.03$ | $18.2\pm 3.6$ | $7.7\pm 10.2$ | $12200\pm 200$
15dtg | $11.6\pm 0.4$ | $0.36\pm 0.01$ | $4.0\pm 0.9$ | $4.6\pm 0.2$ | $7600\pm 900$
16flq | $10.1\pm 2.6$ | $0.23\pm 0.01$ | $9.2\pm 2.6$ | $11.7\pm 2.3$ | $12400\pm 700$
16hgp | $9.7\pm 0.4$ | $0.48\pm 0.01$ | $3.2\pm 0.5$ | $6.8\pm 0.3$ | $7400\pm 600$
Table 10: Arnett model parameters with their statistical fitting uncertainties
and measured characteristic ejecta velocities. _11mnb_peak1_ refers to the fit
to only the first peak of 11mnb. The explosion epochs were fit as described in
the text and their shift and uncertainty is listed under the $E_{E}xp$ column.
There is a significant correlation between errors on $E_{E}xp$ and $M_{ej}$.
$M_{Ni}$ could be driven up by host extinction correction, which we do not
apply. The distance uncertainty (e.g. random from peculiar velocities or
systematic from $H_{0}$) is not included in the error terms. Table 11: Line
velocities of Fe ii $\lambda\lambda 4924,5018,5169$, He i $\lambda 5876$, Na
iD $\lambda\lambda 5890,5986$ (as 5890), Si ii $\lambda 6355$, and O i
$\lambda 7774$. Phases are in restframe.

SN | Date | Phase | $\lambda 4924$ | $\lambda 5018$ | $\lambda 5169$ | $\lambda 5876$ | $\lambda 5890$ | $\lambda 6355$ | $\lambda 7774$
---|---|---|---|---|---|---|---|---|---
|  | (days) | ($kms^{-1}$) | ($kms^{-1}$) | ($kms^{-1}$) | ($kms^{-1}$) | ($kms^{-1}$) | ($kms^{-1}$) | ($kms^{-1}$)
09dfk | 20090825 | $-27.0$ | $8620\pm 670$ | $9270\pm 660$ | $7200\pm 670$ | $10190\pm 60$ | – | – | –
09dfk | 20090921 | $-0.5$ | $7770\pm 560$ | $8280\pm 500$ | $6110\pm 750$ | $9130\pm 50$ | – | – | $9650\pm 80$
09dfk | 20091011 | $18.8$ | $7310\pm 750$ | $7970\pm 340$ | $5490\pm 490$ | $8620\pm 70$ | – | – | $7130\pm 110$
09dfk | 20091022 | $30.1$ | $70.0\pm 740$ | $7670\pm 380$ | – | $8230\pm 80$ | – | – | $6680\pm 80$
09dfk | 20101101 | $398.7$ | – | – | – | – | – | – | –
10inj | 20100608 | $-21.3$ | – | – | $160.0\pm 480$ | $170.0\pm 340$ | – | – | $13340\pm 570$
10inj | 20100714 | $12.5$ | – | – | $6370\pm 420$ | $9210\pm 280$ | – | – | $7950\pm 370$
10inj | 20100802 | $30.3$ | – | – | – | $9350\pm 370$ | – | – | $7330\pm 290$
10inj | 20100908 | $65.1$ | – | – | – | – | – | – | –
10inj | 20101003 | $88.6$ | – | – | $5260\pm 560$ | $7410\pm 270$ | – | – | $7470\pm 210$
10inj | 20170817 | $2447.2$ | – | – | – | – | – | – | –
11bov | 20110411 | $-20.0$ | $9150\pm 690$ | $9810\pm 660$ | $9420\pm 610$ | – | $9960\pm 200$ | $8630\pm 270$ | $9510\pm 110$
11bov | 20110511 | $9.3$ | $7540\pm 470$ | $7540\pm 530$ | $6430\pm 460$ | – | $80.0\pm 180$ | – | $80.0\pm 110$
11bov | 20110607 | $36.0$ | $7310\pm 580$ | $6510\pm 610$ | $5260\pm 630$ | – | $6350\pm 70$ | – | $7140\pm 80$
11bov | 20110728 | $85.6$ | $6690\pm 10.0$ | $5440\pm 260$ | $4310\pm 790$ | – | $50.0\pm 160$ | – | $6140\pm 170$
11bov | 20111218 | $225.8$ | – | – | – | – | – | – | –
11bov | 20111220 | $227.5$ | – | – | – | – | – | – | –
11bov | 20120118 | $255.9$ | – | – | – | – | – | – | –
11mnb | 20111007 | $-17.1$ | – | – | – | – | $5470\pm 290$ | – | $5850\pm 580$
11mnb | 20111030 | $4.1$ | $5750\pm 690$ | $50.0\pm 440$ | $4730\pm 570$ | – | $4930\pm 160$ | – | $5280\pm 90$
11mnb | 20111126 | $29.9$ | $4980\pm 880$ | $4740\pm 690$ | $4570\pm 350$ | – | $4900\pm 130$ | – | $5100\pm 100$
11mnb | 20111221 | $53.3$ | $4160\pm 580$ | $4480\pm 340$ | $4540\pm 1390$ | – | $4800\pm 150$ | – | $50.0\pm 140$
11mnb | 20120127 | $88.1$ | – | – | – | – | – | – | –
11rka | 20111226 | $-10.0$ | $11910\pm 600$ | $12110\pm 450$ | $13110\pm 620$ | – | $6330\pm 310$ | – | $9200\pm 110$
11rka | 20120220 | $42.0$ | $10420\pm 380$ | $10670\pm 470$ | $10530\pm 10.0$ | – | $5710\pm 260$ | – | $7120\pm 70$
11rka | 20120127 | $88.1$ | $10240\pm 630$ | $10670\pm 690$ | – | – | $6390\pm 270$ | – | –
11rka | 20120427 | $104.3$ | $5450\pm 550$ | $60.0\pm 740$ | $50.0\pm 800$ | – | $5220\pm 120$ | – | $6530\pm 170$
11rka | 20120522 | $127.4$ | $5550\pm 2680$ | $5480\pm 820$ | $50.0\pm 400$ | – | $4750\pm 280$ | – | $6360\pm 120$
11rka | 20130313 | $401.7$ | – | – | – | – | – | – | –
11rka | 20151206 | $1331.2$ | – | – | – | – | – | – | –
15dtg | 20151110 | $-30.4$ | $14380\pm 690$ | $12570\pm 350$ | $13320\pm 930$ | – | $5730\pm 550$ | $10820\pm 230$ | $12120\pm 140$
15dtg | 20151111 | $-29.1$ | – | – | – | – | $5480\pm 710$ | $10940\pm 60$ | $11870\pm 810$
15dtg | 20151123 | $-17.3$ | $70.0\pm 740$ | $7270\pm 1340$ | $60.0\pm 700$ | – | $5320\pm 630$ | – | $11990\pm 160$
15dtg | 20151202 | $-9.3$ | $70.0\pm 1590$ | $5960\pm 10.0$ | – | – | – | $7980\pm 150$ | –
15dtg | 20151210 | $-1.1$ | – | – | $6560\pm 720$ | – | – | $6890\pm 130$ | $9780\pm 100$
15dtg | 20160110 | $28.2$ | $6510\pm 590$ | $6390\pm 1430$ | $4150\pm 1690$ | – | $5470\pm 500$ | – | $7940\pm 100$
15dtg | 20160128 | $45.4$ | $60.0\pm 640$ | $6320\pm 270$ | $4140\pm 300$ | – | $5550\pm 360$ | $2210\pm 300$ | $7780\pm 270$
15dtg | 20160211 | $58.7$ | $5920\pm 650$ | $6550\pm 530$ | $3710\pm 720$ | – | $5110\pm 470$ | – | $7330\pm 140$
15dtg | 20160309 | $83.7$ | $5230\pm 1330$ | $5530\pm 1290$ | $4540\pm 570$ | – | – | – | $6630\pm 130$
15dtg | 20161031 | $308.2$ | – | – | – | – | – | – | –
15dtg | 20161208 | $343.9$ | – | – | – | – | – | – | –
15dtg | 20151206 | $1331.2$ | – | – | $7430\pm 900$ | – | – | $6890\pm 100$ | $10310\pm 150$
15dtg | 20170817 | $2447.2$ | – | – | – | – | – | – | –
16flq | 20160828 | $-17.8$ | – | – | – | – | $15140\pm 830$ | – | $9160\pm 430$
16flq | 20160915 | $-1.1$ | $11150\pm 650$ | $120.0\pm 560$ | $11230\pm 900$ | – | $10480\pm 740$ | – | $8200\pm 690$
16flq | 20160921 | $4.7$ | – | $11300\pm 620$ | $10160\pm 600$ | – | $9740\pm 750$ | – | $80.0\pm 380$
16flq | 20160929 | $13.0$ | – | – | $10870\pm 770$ | – | $9520\pm 260$ | – | $6560\pm 370$
16flq | 20161011 | $23.5$ | $8610\pm 340$ | – | $6180\pm 470$ | – | $9190\pm 460$ | – | $60.0\pm 220$
16flq | 20161123 | $64.8$ | $5940\pm 710$ | – | $4880\pm 690$ | – | $7610\pm 330$ | – | $6260\pm 200$
16flq | 20161031 | $308.2$ | $7790\pm 10.0$ | – | $3960\pm 600$ | – | $7610\pm 290$ | – | $5880\pm 310$
16hgp | 20161022 | $-25.2$ | – | – | – | – | $12700\pm 860$ | – | –
16hgp | 20161102 | $-15.0$ | – | – | – | – | $100\pm 420$ | $18530\pm 100$ | $9890\pm 130$
16hgp | 20161128 | $9.2$ | $6640\pm 720$ | $6810\pm 290$ | $6310\pm 670$ | – | $6810\pm 840$ | $14800\pm 190$ | $7500\pm 210$
16hgp | 20161206 | $17.1$ | $6520\pm 440$ | $5950\pm 720$ | $5480\pm 360$ | – | $6870\pm 230$ | $12180\pm 180$ | $7390\pm 90$
16hgp | 20170102 | $41.4$ | $6450\pm 530$ | $5430\pm 360$ | $4150\pm 10.0$ | – | $50.0\pm 250$ | $12410\pm 60$ | $6560\pm 80$

Cut-off | $s_{0}$ | $\tau$ | $c$ | Normalization
---|---|---|---|---
(mag) | (stretch) |  |  |
20.5 | 0.457 | 0.387 | 1.827 | 2.073
21.0 | 0.377 | 0.380 | 1.456 | 1.650
21.5 | 0.311 | 0.358 | 1.167 | 1.313
Table 12: Lightcurve duration bias exponential fits with their normalizations
$W_{s}(1.0)$ as described by Equation 1. The weight $W_{s}$ obtained can be
used to correct the number of (i)PTF SNe by their stretch to obtain the
lightcurve duration bias corrected occurrence. Cut-off values are nominal
limiting magnitudes for (i)PTF roughly corresponding to bright, grey, and dark
time.