# SN2017jgh - A high-cadence complete shock cooling lightcurve of a SN IIb
with the Kepler telescope

P. Armstrong,1 B. E. Tucker,1,2,3 A. Rest,4,5 R. Ridden-Harper,4,5 Y. Zenati,5
A. L. Piro,6 Email<EMAIL_ADDRESS>S. Hinton,7 C. Lidman,1,56 S.
Margheim,9 G. Narayan,10,11 E. Shaya,12 P. Garnavich,24 D. Kasen,31,32,33 V.
Villar,47 A. Zenteno,50 I. Arcavi,13,14 M. Drout,6,15 R. J. Foley,16 J.
Wheeler,17 J. Anais,18 A. Campillay,19 D. Coulter,16 G. Dimitriadis,16 D.
Jones,16 C. D. Kilpatrick,21,22 N. Muñoz-Elgueta,20 C. Rojas-Bravo,16 J.
Vargas-González,23 J. Bulger,25 K. Chambers,25 M. Huber,25 T. Lowe,25 E.
Magnier,25 B. J. Shappee,25 S. Smartt,30 K. W. Smith,30 T. Barclay,26,27 G.
Barentsen,28 J. Dotson,29 M. Gully-Santiago,17 C. Hedges,28,29 S. Howell,29 A.
Cody,29 K. Auchettl,3,16,57 A. Bódi,34,35 Zs. Bognár,34,35 J. Brimacombe,36 P.
Brown,51 B. Cseh,34 L. Galbany,37 D. Hiramatsu,38,39 T. W.-S. Holoien,6 D. A.
Howell,38,39 S. W. Jha,55 R. Könyves-Tóth,34,40 L. Kriskovics,34 C.
McCully,38,39 P. Milne,45 J. Muñoz,41,42 Y. Pan,43 A. Pál,34 H. Sai,44 K.
Sárneczky,34 N. Smith,45 Á. Sódor,34,35 R. Szabó,34,35,46 R. Szakáts,34 S.
Valenti,53 J. Vinkó,34,40,46,17 X. Wang,44,48 K. Zhang,49 G. Zsidi,34,46,54
Affiliations are given at Section 7.2

(Accepted 2021. Received 2021; in original form 2021)

###### Abstract

SN 2017jgh is a type IIb supernova discovered by Pan-STARRS during the C16/C17
campaigns of the Kepler/K2 mission. Here we present the Kepler/K2 and ground
based observations of SN 2017jgh, which captured the shock cooling of the
progenitor shock breakout with an unprecedented cadence. This event presents a
unique opportunity to investigate the progenitors of stripped envelope
supernovae. By fitting analytical models to the SN 2017jgh lightcurve, we find
that the progenitor of SN 2017jgh was likely a yellow supergiant with an
envelope radius of $\sim 50-290\leavevmode\nobreak\ R_{\odot}$, and an
envelope mass of $\sim 0-1.7\leavevmode\nobreak\ M_{\odot}$. SN 2017jgh likely
had a shock velocity of $\sim 7500-10300$ km s-1. Additionally, we use the
lightcurve of SN 2017jgh to investigate how early observations of the rise
contribute to constraints on progenitor models. Fitting just the ground based
observations, we find an envelope radius of $\sim 50-330\leavevmode\nobreak\
R_{\odot}$, an envelope mass of $\sim 0.3-1.7\leavevmode\nobreak\ M_{\odot}$
and a shock velocity of $\sim 9,000-15,000$ km s-1. Without the rise, the
explosion time can not be well constrained which leads to a systematic offset
in the velocity parameter and larger uncertainties in the mass and radius.
Therefore, it is likely that progenitor property estimates through these
models may have larger systematic uncertainties than previously calculated.

###### keywords:

supernovae: general – supernovae: individual (SN 2017jgh)

††pubyear: 2021††pagerange: SN2017jgh - A high-cadence complete shock cooling
lightcurve of a SN IIb with the Kepler telescope–References

##  1 Introduction

Massive stars end their lives as a core-collapse supernovae (CCSNe), leaving
behind a black hole or neutron star and a supernova remnant. CCSNe come in a
variety of observed sub-types depending on the properties of the progenitor
star prior to explosion, the explosion mechanism itself, and its circumstellar
environment. Type II supernovae display strong hydrogen lines in their early
spectra, with this population consisting of Type II-P and II-L SNe (plateauing
and linear lightcurve decay, respectively), Type IIn SNe (narrow line
spectra), and Type IIb SNe (He in spectrum with disappearing H; Filippenko,
1997). Type I supernovae lack strong H lines, such as Type Ib SNe (He in
spectrum), and Type Ic SNe (no He in spectrum; Janka, 2012; Gal-Yam, 2017).
SNe IIb along with Ib and Ic are known as stripped envelope supernova as they
lose their hydrogen over time.

Weeks after the explosion of Type IIb Sne, hydrogen lines begin to disappear
and helium lines begin to dominate with the spectrum more closely resembling a
type Ib supernova (Filippenko, 1997), suggesting a progenitor which is mostly
stripped of its hydrogen envelope. The mechanism which strips the hydrogen is
still unknown, with possibilities including stellar winds, stellar rotation,
binary interaction, and nuclear burning instabilities (e.g. Podsiadlowski,
1992; Woosley et al., 1993; Bersten et al., 2012; Liu et al., 2016; Yoon et
al., 2017; Bersten et al., 2018; Naiman et al., 2019; Sravan et al., 2019).

Type IIb SNe are of particular interest as many IIb SNe progenitors have been
identified. Several have evidence of binary interaction (e.g. SN 1993J; Maund
et al. 2004, SN 2011dh; Bersten et al. 2012). A well studied example of a type
IIb supernova is SN 1993J (Richmond et al., 1994; Nomoto et al., 1993). The
progenitor of this supernova was identified from direct imaging as a late G or
early K supergiant with an effective temperature between 4000-4500 K and an
initial mass prior to explosion of 17 M⊙. There is evidence that the
progenitor of SN 1993J had a binary companion which may be responsible for
stripping the progenitor of its hydrogen (Maund et al., 2004).

Stripped supernovae can show two prominent peaks in their optical lightcurve.
The first peak is a burst of emission after the explosion known as the shock
cooling lightcurve (SCL; Gal-Yam, 2017; Arcavi, 2017; Sravan et al., 2020),
and the second peak is nuclear powered emission fueled by the decay of 56Ni.

The SCL provides an opportunity to probe physical properties of the
progenitor. This emission is produced when photons trapped behind optically
dense material within the progenitor finally escape (Waxman & Katz, 2017). The
associated optical emission lasts on the order of days, and the shape of this
curve depends on both the behaviour of the shock wave and the physical
properties of the progenitor star (Nakar & Sari 2010; Waxman & Katz 2017 both
provide a more complete review of shock breakout physics). As such,
investigating the SCL provides a unique opportunity to probe both the
properties of the shock wave and the progenitor star.

There have been many supernovae observed with a SCL - SN 1993J (Richmond et
al., 1994), SN 2011dh (Arcavi et al., 2011), SN 2011fu (Kumar et al., 2013),
SN 2011hs (Bufano et al., 2014), SN 2013df (Morales-Garoffolo et al., 2014),
and SN 2016gkg (Arcavi et al., 2017; Kilpatrick et al., 2017). Hydrodynamical
modelling of SN 2011dh suggests that a star with a compact core and low mass
envelope is most likely responsible for the observed SCL (Bersten et al.,
2012). This was later confirmed by Nakar & Piro (2014) who employed semi-
analytical models to show that a progenitor with an extended envelope was
required. The Nakar & Piro (2014) model includes two relationships: one
between the bolometric luminosity at the peak of the SCL and the radius of the
extended envelope, and the other between the time of the peak of the SCL and
the mass concentrated at the envelope’s radius. Piro (2015) (hereafter P15)
produced an analytical expression for the complete SCL. P15 enforced
conditions necessary to produce a SCL which require that the material is
massive enough for the shock to propagate and extended enough for the peak to
be bright in the optical band. No explicit assumptions are made as to the
density profile of the supernova progenitor. An alternative to the P15 model,
Rabinak & Waxman (2011), explicitly assume a polytropic density profile to
provide a more physically realistic model of the progenitor. This was expanded
upon by Sapir & Waxman (2017) (hereafter SW17) by scaling their model to
better agree with hydrodynamical simulations, allowing the model to extend to
later times. Piro et al. (2020) (hereafter P20), like the P15 model, makes no
assumptions about the density profile of the progenitor star, instead the P20
model makes use of a two-component velocity profile to attain better fits to
the SCL than P15. All attempts at modelling the SCL thus far rely on the
assumption of hydrostatic equilibrium within the progenitor (Chevalier, 1993;
Nakar & Sari, 2010).

Capturing the shock cooling lightcurve of a supernova is difficult owing to
the short lifetime of these events. Most examples of SCLs have only included
the decline of the SCL, sometimes managing to capture the peak as well. The
rise of the SCL was yet to be captured at high cadence, which presents a
problem for the development and improvement of SCL models and, thus, our
ability to probe the progenitor of these supernovae. As mentioned in P15, the
shape of the rise is highly dependant on the density structure of the
progenitor which makes the rise difficult to accurately model. Without data of
the SCL rise, one cannot test how effective models are at this critical early
stage (Piro, 2015; Sapir & Waxman, 2017; Piro et al., 2020).

In order to effectively capture the complete SCL, continuous, high cadence
observations are needed. A cadence of < 1 day allows one to capture the
earliest emission from the supernova, and high cadence observations are
required in order to observe the rapidly evolving SCL. Telescopes like the
Kepler Space Telescope (Kepler/K2; Howell et al., 2014) and the Transiting
Exoplanet Survey Satellite (TESS; Ricker et al., 2014) allow for this type of
observing strategy. As such, we can expect more shock cooling lightcurves with
higher cadence data to be observed, presenting an exciting opportunity to gain
a much better understanding of the progenitors of type IIb supernovae (Vallely
et al., 2021; Fausnaugh et al., 2021). In order to effectively use this new
data, we need to determine which class of analytical SCL models fits the data
the best and how much of the SCL must be observed to constrain progenitor
models.

In this paper we present SN 2017jgh, a type IIb supernova which was observed
by Kepler/K2 in Campaign 16. Kepler/K2 was able to capture the full lightcurve
of SN 2017jgh, including the complete evolution of the SCL. This is the first
time a high cadence, complete shock cooling lightcurve has been observed. This
represents a unique opportunity to evaluate the effectiveness of the
analytical models and to investigate how important the rise is for accurate
modelling of progenitor properties.

##  2 Observations

SN 2017jgh111EPIC: 211427218 was discovered by Pan-STARRS1 (PS1; Chambers et
al., 2016) on 2017 Dec 23 at 11:09:36 (MJD 58110.965) with $g=20.21$ mag
(Chambers et al., 2017). SN 2017jgh occurred at
$(\alpha,\delta)=(09^{\text{h}}02^{\text{m}}56^{\text{s}}.736,+12\degree{}03^{\prime}04^{\prime\prime}27)$,
at a separation of 0.157 arcseconds from the centre of its host galaxy, 2MASX
J09025612+1202596 with redshift 0.079. We estimated the Milky Way reddening
towards SN 2017jgh as $E(B-V)=0.02$ mag (Schlegel et al., 1998). We do not
take into account host galaxy extinction. This extinction is likely to redden
the lightcurve, however the shape of the SCL is more important than the colour
when fitting so we believe the effects of this extinction to be negligible. We
measure the radioactive maximum in the Kepler/K2 band to be $t_{max}=58127\pm
1$ MJD. Throughout the paper, epochs are presented relative to this maximum,
as $t-t_{max}$.

Figure 1: The combined Kepler/K2 and ground-based lightcurve of SN 2017jgh.
The ground-based data is composed of Swope (Squares; Folatelli et al., 2010)
and Pan-STARRS (Diamonds; Chambers et al., 2016) data. The black circles are
the 30 minute cadence Kepler/K2 data. In order to better see the Kepler/K2
lightcurve, we include 6hr bins of this lightcurve (red circles). The solid
vertical line indicates where the SCL ends and the nuclear powered portion
begins. The dashed vertical line shows when the spectrum of SN 2017jgh was
taken.  Figure 2: A comparison between the Kepler/K2 lightcurve of SN 2017jgh,
the r’-band photometry of SN 2016gkg (Arcavi et al., 2017), and the R-band
photometry of SN 1993J (Richmond et al., 1994). Clear filter observations are
also included for SN 2016gkg (Otero & Buso, 2016). The SCL does not have the
sharp rise and decline of SN 2016gkg and appears similar to SN 1993J.

###  2.1 Ground-Based photometry

Ground-based photometry was obtained from both PS1 (Chambers et al., 2016;
Dotson et al., 2018) and the Swope Supernova Survey (SSS; Folatelli et al.,
2010).

PS1 is a 1.8m telescope located at Haleakala on Maui, Hawaii. The telescope
contains a 1.4 gigapixel camera, GPC1, mounted at the Cassegrain f/4.4 focus.
GPC1 has sixty orthogonal transfer array devices, each with a
4846$\times{}$4846 pixel detector area. Each pixels measures 10 $\mu$m in
size, giving a focal plane of 418.88 mm in diameter (3.3°). This gives a field
of view 7.06 square degrees. PS1’s filter system ($grizy$) is similar to SDSS
(Abazajian et al., 2009), with the addition of a composite $gri$ ’wide’ filter
$w$ (Chambers et al., 2016; Tonry et al., 2012).

Reduction of PS1 images (described in detail in Magnier et al. (2020)) is
performed by the PS1 Image Processing Pipeline (IPP; Ryan, 2006; Magnier et
al., 2008) which includes standard reductions, astrometric solution, stacking
of nightly images, source detection, and photometry. The stacks are then sent
to the Harvard FAS Cannon Research Computing cluster where the photpipe
pipeline (Rest et al., 2005) performs difference imaging and transient
identification.

SSS’s optical photometry of SN 2017jgh was obtained in $gri$ with the Swope
1-meter telescope at Las Campanas, Chile, from 17 Dec 2017 to Feb 7 2018.
Following the description in Kilpatrick et al. (2018), all image processing
and optical photometry on the Swope data was performed using photpipe (Rest et
al., 2005). The photometry were calibrated using standard sources from the
Pan-STARRS DR1 catalog (Flewelling et al., 2020) in the same field as SN
2017jgh and transformed following the Supercal method (Scolnic et al., 2015)
into the Swope natural system (Krisciunas et al., 2017). Deep $gri$ template
images of the SN 2017jgh field were obtained on Jan 8 2019 and Jan 14 2019.
Final image subtraction was performed using hotpants (Becker, 2015). Forced
photometry was performed on the subtracted images.

###  2.2 Kepler/K2 Photometry

The K2 mission consists of a series of campaigns which observe different
fields within the ecliptic plane. Kepler/K2 uses a 0.95 m aperture Schmidt
telescope, orbiting the Earth in a heliocentric orbit. The telescope has one
broad filter spanning 4000 to 9000Å, with peak transmission roughly coinciding
with the Pan-STARRS $r$-band (Bryson et al., 2010).

The primary advantage of Kepler/K2 is the 30 minute cadence. This rapid
cadence, combined with the roughly 80 day campaigns, allows for early and
detailed lightcurves of transient events. Throughout campaign 16, the Kepler
spacecraft had been running with the loss of 2 reaction wheels, so was
operating on 2 remaining wheels and thrusters. This introduced a number of
systematic effects into the lightcurve. These included reaction wheel jitter,
which introduces additional short term noise, and solar pressure induced
drifting, which introduced long term systematics with a characteristic
"sawtooth" pattern.

As described in Shaya et al. (2015), the data reduction pipeline begins by
correcting the CCD images for bias level, dark current, smear, nonlinear gain,
undershooting pixels, and flat fields (Quintana et al., 2010). After initial
calibration, the Presearch Data Conditioning (PDC) (Stumpe et al., 2012)
applies corrections for both instrumental and spacecraft anomalies, and
removes contamination from nearby stars. During this process, a set of 14
cotrending basis vectors (CBVs) are generated by singular value decomposition
to represent correlated instrumental artifacts such as flexing of the
telescope structure, thermal transients, and drifting which occurs due to
solar pressure. The PDC lightcurve then has a superposition of these CBVs
removed such that the root-mean-squared (rms) deviations are minimised.
However, for SNe, this process would remove most of the physical variations in
the light curve. We therefore carefully create a set of CBVs using only quiet
galaxies that are on the same CCD channel as the SN itself. Then we solve for
the coefficients of a superposition of CBVs that minimise the rms deviations
before the onset of the SN event.

In order to calculate the Kepler/K2 zero point we follow the Kepler Instrument
Handbook (Van Cleve & Caldwell, 2016) which gives the following equation for
the Kepler/K2 magnitude ($K_{p}$) in terms of the $(g-r)$ colour (where $g$
and $r$ are the SDSS filters):

$\displaystyle K_{p}=\begin{cases}r+0.2(g-r)&\text{if $(g-r)\leq 0.8$ mag}\\\
r+0.1(g-r)&\text{if $(g-r)>0.8$ mag}\end{cases}$ (1)

The $(g-r)$ colour is calculated from the ground-based photometry. Once the
psuedo-Kepler/K2 magnitude is calculated, it is compared to the raw Kepler/K2
flux in order to calculate the Kepler/K2 zeropoint. We calculate the Kepler/K2
zeropoint to be $25.3\pm 0.1$. We validate this zeropoint by calculating
synthetic Kepler photometry using the Kepler bandpass listed on SVO filter
profile service (Rodrigo et al., 2012; Rodrigo & Solano, 2020), normalised to
AB magnitudes. Since the GMO-S spectrum discussed in Section 2.3 did not cover
the full wavelength range, we use the SN 2016gkg spectra taken 2 days before
radioactive maximum by (Jha et al., 2016), normalised to the ground-based
photometry for SN 2017jgh at the same phase. Comparing to the Kepler counts,
we find a zeropoint of $23.33\pm 0.15$, which is within 10% of the zeropoint
derived from ground-based photometry. Since there are unknown differences
between the SN 2017jgh and SN 2016gkg spectra we use the ground-based
photometric zeropoint of $25.3\pm 0.1$.

The combined Kepler/K2 and ground-based lightcurve is presented in Figure 1.
The double-peaked profile is evident in the Kepler/K2 photometry and is
mirrored in the ground-based data, although the ground-based data is
significantly sparser and does not cover the rise. A comparison between SN
2017jgh, SN 1993J (Richmond et al., 1994) and SN 2016gkg (Arcavi et al., 2017)
is presented in Figure 2, and the evolving colours of SN 2017jgh, SN 2016gkg
and SN 1993J are shown in Figure 3222The lightcurves and spectra of SN 2016gkg
and SN 1993J were gathered from https://sne.space/ (Guillochon et al., 2017).

The lightcurve of SN 2017jgh appears most similar to SN 1993J, which has a
very similar SCL decline. The SCL of SN 2016gkg has a much sharper decline,
which may indicate it had less extended material. Overall, this suggests that
the progenitor of SN 2017jgh was closer to the yellow supergiant progenitor of
SN 1993J than the blue supergiant progenitor of SN 2016gkg. Additionally SN
1993J’s colour evolution closely matches the g-r colour evolution of SN
2017jgh, which again suggests that the progenitor of SN 1993J and SN 2017jgh
were very similar.

Figure 3: Colour evolution of SN 2017jgh (black), SN 2016gkg (blue), and SN
1993J (orange). The colour evolution of SN 2017jgh appears most similar with
the SN 1993J evolution.

An excerpt of the lightcurve data is shown in Table 1, with the full dataset
available online.

Table 1: Excerpt of lightcurve data for SN 2017jgh. The full lightcurve is available as supplementary material. Time (MJD) | Flux (erg s${}^{-}1$ cm${}^{-}2$ | Flux Error (erg s${}^{-}1$ cm${}^{-}2$) | Band | Instrument
---|---|---|---|---
58117.33 | 1.50e-28 | 9.16e-30 | r | Swope
58117.34 | 1.54e-28 | 1.41e-29 | i | Swope
58117.35 | 1.17e-28 | 8.38e-30 | g | Swope
58118.21 | 1.21e-28 | 1.89e-29 | r | Swope
58118.22 | 1.15e-28 | 1.95e-29 | i | Swope
58118.22 | 9.51e-29 | 1.85e-29 | g | Swope
58119.36 | 1.72e-28 | 3.54e-29 | r | Swope
58119.36 | 1.47e-28 | 3.61e-29 | i | Swope
58125.37 | 1.58e-28 | 2.82e-29 | r | Swope
58126.36 | 1.66e-28 | 1.46e-29 | r | Swope

###  2.3 Spectroscopy

Figure 4: The spectrum of SN 2017jgh 2 days before radioactive maximum,
compared to SN 1993J 1 day before radioactive maximum (Jeffery et al., 1994)
and SN 2016gkg 2 days before radioactive maximum (Jha et al., 2016). All of
the spectra have been normalised and then shifted in order to easily
differentiate between them.

We obtained optical spectroscopy on Jan 7th 2018, two days before radioactive
maximum (14 day after discovery) using the Gemini Multi-Object Spectrograph on
the Gemini South telescope (GMOS-S; Hook et al., 2004). GMOS-S was configured
using the R400 grating and a 1 arcsec longslit, with the detectors in 2x2
binning mode, covering a spectral range from 475 to 925 nm at a resolution of
$\sim{}$900\. Data reduction was completed using the Gemini IRAF package. The
data was first corrected for bias and flat-fielded using standard techniques.
Cosmic rays were removed (van Dokkum, 2001), as well as any detector bad
columns. Note that the width of a cosmic ray detection or bad column is small
compared to the size of the resolved features in our spectra. The sky
background was removed by first subtracting nodded pairs of images, followed
by removing any residual by linear fitting the sky background near the target.
One-dimensional spectra were then produced using variance weighted extraction,
utilizing a 1 arcsec spatial aperture. A similarly processed standard star,
EG-131, was used to correct each extracted spectra for instrument response,
prior to producing the final co-added spectrum. This Gemini baseline
correction does not provide an absolute flux-calibrated spectrum, as the
standard is not contemporaneous with the science observation, nor are slit-
losses or second-order contamination a part of the standard correction. These
calibration issues will likely compound at shorter wavelengths, which may be
responsible for the increased flux around 5000 Åin the SN 2017jgh spectrum
compared to similar SNe.

The reduced Gemini spectrum is shown in Figure 4. The spectrum of SN 2017jgh
is compared to both SN 1993J at 1 day before radioactive maximum (Jeffery et
al., 1994; note that they state the spectra is ‘not photometrically
accurate’), and SN 2016gkg at 2 days before radioactive maximum (Jha et al.,
2016). The red ($>6000$Å) spectrum of SN 2017jgh is similar to both the SN
1993J spectrum, and the SN 2016gkg spectrum. Below 6000Å the spectrum of SN
2017jgh diverges from the other supernovae, likely due to the calibration
issues we have described earlier.

We used the Supernova Identification (SNID; Blondin & Tonry, 2007) program
which classified SN 2017jgh as a Type IIb supernova. However it is worth
noting that with only one spectrum we do not have the robust IIb
classification criteria of the observation of strong He features and the
disappearance of H features.

##  3 Shock Cooling Curve Models

Here we summarise the formalism of the P15, P20, and SW17 analytical models.

###  3.1 P15 Model

The P15 model makes no assumptions about the density profile of the progenitor
and assumes a simple expanding photosphere. P15 used these assumptions along
with simple thermodynamic equations to derive the bolometric luminosity of a
SCL. We follow Arcavi et al. (2017), who recast original equation for
luminosity in P15 into the following equation which depends on the radius of
the extended envelope ($R_{e}$), the mass of the envelope ($M_{e}$), and the
velocity of the envelope ($v_{e}$):

$\frac{L(t)}{\text{erg
s${}^{-1}$}}=\frac{8.27\times{}10^{42}}{\text{cm}}\kappa_{0.34}^{-1}v_{9}^{2}R_{13}\left(\frac{M_{c}}{M_{\odot}}\right)^{0.01}\times\\\
\exp\left[\frac{-4.135\times{}10^{-11}}{\text{cm g${}^{-1}$
s${}^{-1}$}}t(tv_{9}+2\times{}10^{4}R_{13})\times\\\
\kappa_{0.34}^{-1}\left(\frac{M_{c}}{M_{\odot}}\right)^{0.01}\left(\frac{M_{e}}{0.01M_{\odot}}\right)^{-1}\right]$
(2)

where $\kappa_{0.34}$ is the opacity in $0.34\leavevmode\nobreak\
\rm{}cm^{2}\,g^{-1}$, $v_{9}$ is the envelope velocity in
$10^{9}\rm{}cm\,s^{-1}$, $R_{13}$ is the envelope radius in $10^{13}\rm{}cm$,
and $t$ is the time since explosion in seconds. Both $M_{e}$ and $M_{c}$ (the
core mass) are in solar masses.
Assuming the emission is a blackbody with expanding radius
$R(t)=R_{e}+v_{e}t$, we can estimate the temperature as:

$\frac{T(t)}{\text{K}}=\left[\frac{L(t)}{4\pi{}R^{2}(t)\sigma}\right]^{1/4}$
(3)

where $\sigma$ is the Stefan-Boltzmann constant. With both the bolometric
luminosity and the temperature defined, we are able to calculate the observed
flux in arbitrary bands.

###  3.2 P20 Model

P20 provide an improvement upon the P15 model by considering a two component
model. This model consists of outer material with a steep velocity gradient,
and inner material with a shallow velocity gradient. The transition between
these layers occurs at the transition velocity $v_{t}$, related to the total
energy of the extended material $E_{e}$ by:

$\displaystyle
v_{t}=\sqrt{\frac{(n-5)(5-\delta)}{(n-3)(3-\delta)}}\sqrt{\frac{2E_{e}}{M_{e}}}$
(4)

Where $t$ is the time in days, $n$ and $\delta$ are numerical factors which
control the radial dependence of the outer and inner density structure,
respectively. P20 show that the solution is not sensitive to the values of $n$
and $\delta$ as long as $n\gg 1$ and $\delta\gtrsim 1$. They suggest using
typical value of $n\approx 10$ and $\delta\approx 1.1$. The P20 model has a
two component luminosity – the luminosity of the inner region and the
luminosity of the outer region. Within the inner region, the luminosity is
defined as:

$\displaystyle\frac{L(t\leq{}t_{d})}{\text{erg s${}^{-1}$}}\approx 1.157\times
10^{-5}\frac{E_{th}(v_{t},t_{d})}{t_{d}}\left(\frac{t_{d}}{t}\right)^{4/(n-2)}$
(5)

where $t_{d}$ is the time when the SCL reaches the transition between the
inner and outer regions, and is equal to:

$\displaystyle\frac{t_{d}}{\text{days}}=5.16\times
10^{6}\sqrt{\frac{3\kappa{}KM_{e}}{(n-1)v_{t}c}}$ (6)

where $c$ is the speed of light in km s-1, and the numerical factor $K$ is
equal to $\frac{(n-3)(3-\delta)}{4\pi(n-\delta)}$; for our values of $n$ and
$\delta$, $K=0.119$. $E_{th}(v_{t},t_{d})$ is the thermal energy at a given
velocity and time:

$\displaystyle\frac{E_{th}(v_{t},t_{d})}{\text{erg}}=5.147\times
10^{30}t_{d}\frac{\pi(n-1)}{3(n-1)}\frac{cR_{e}v_{t}^{2}}{\kappa}$ (7)

The luminosity of the outer region is:

$\displaystyle\frac{L(t\geq{}t_{d})}{\text{erg s${}^{-1}$}}=1.157\times
10^{-5}\frac{E_{th}(v_{t},t_{d})}{t_{d}}\exp\left[-\frac{1}{2}\left(\frac{t^{2}}{t_{d}^{2}}-1\right)\right]$
(8)

In order to calculate the luminosity in any band we once again assume the SCL
radiates as a black body, with a temperature:

$\displaystyle\frac{T}{\text{K}}=\left(\frac{L}{4\pi{}r_{ph}^{2}\sigma}\right)^{1/4}$
(9)

Here $r_{ph}$ is the photospheric radius (the point where the optical depth
$\tau$ is equal to $2/3$). The photospheric radius is also a two-component
function dependent on $t_{ph}$, the time when the photosphere reaches the
transition between the inner and outer regions.

$\displaystyle\frac{t_{ph}}{\text{days}}$ $\displaystyle=5.16\times
10^{6}\sqrt{\frac{3\kappa{K}M_{e}}{2(n-1)v_{t}^{2}}}=\sqrt{\frac{c}{2v_{t}}}t_{d}$
(10) $\displaystyle\frac{r_{ph}(t\leq{}t_{ph})}{\text{cm}}$
$\displaystyle=8.64\times 10^{9}\left(\frac{t_{ph}}{t}\right)^{2/(n-1)}v_{t}t$
(11) $\displaystyle\frac{r_{ph}(t\geq{}t_{ph})}{\text{cm}}$
$\displaystyle=8.64\times
10^{9}\left[\frac{\delta-1}{n-1}\left(\frac{t^{2}}{t_{ph}^{2}}-1\right)+1\right]^{-1/(\delta-1)}v_{t}t$
(12)

The free parameters of the P20 model are $M_{e}$ in solar masses, $R_{e}$ in
solar radii and $v_{t}$ in km s-1 (or $E_{e}$ as they are related by equation
4).

###  3.3 SW17 Model

Rabinak & Waxman (2011) assume a polytropic progenitor density profile to
derive an analytical form of the early lightcurve, characterised by the
polytropic index $n$. This allows us to differentiate between progenitors with
a convective envelope and a polytropic index of $n=3/2$, such as red
supergiants (RSG), and progenitors with a radiative envelope and a polytropic
index of $n=3$, such as blue supergiants (BSG). Sapir & Waxman (2017) improve
upon the Rabinak & Waxman (2011) model by introducing an additional term that
suppresses the luminosity at later times. This accounts for the shock phase
days after the initial explosion, when the shock cooling emission begins to
emerge from the inner layers. The SW17 bolometric luminosity (after the
recasting done by Arcavi et al. (2017)333Note that Arcavi et al. (2017) had a
typo in their equation 6 as the factor of 19.5 was included in the square root
term within the exponent. This has been corrected here.) is:

$\displaystyle\frac{L_{\text{n=3/2}}(t)}{\text{erg s${}^{-1}$}}$
$\displaystyle=\frac{1.88\times 10^{42}}{\text{cm}}\times$ (13)
$\displaystyle\left(\frac{v_{s,8.5}t^{2}}{f_{p}M\kappa_{0.34}}\right)^{-0.086}\frac{v_{s,8.5}^{2}R_{13}}{\kappa_{0.34}}\times$
$\displaystyle\exp{\left(-\left\\{\frac{1.67t}{\frac{19.5}{\text{days
cm${}^{-0.5}$
s${}^{-0.5}$}}(\kappa_{0.34}M_{e}v^{-1}_{s,8.5})^{0.5}}\right\\}^{0.8}\right)}$
$\displaystyle\frac{L_{\text{n=3}}(t)}{\text{erg s${}^{-1}$}}$
$\displaystyle=\frac{1.66\times 10^{42}}{\text{cm}}\times$ (14)
$\displaystyle\left(\frac{v_{s,8.5}t^{2}}{f_{p}M\kappa_{0.34}}\right)^{-0.175}\frac{v_{s,8.5}^{2}R_{13}}{\kappa_{0.34}}\times$
$\displaystyle\exp{\left(-\left\\{\frac{4.57t}{\frac{19.5}{\text{days
cm${}^{-0.5}$
s${}^{-0.5}$}}(\kappa_{0.34}M_{e}v^{-1}_{s,8.5})^{0.5}}\right\\}^{0.73}\right)}$

Here $v_{s,8.5}$ is the shock velocity in $10^{8.5}\leavevmode\nobreak\
\rm{}cm\,s^{-1}$, $M$ is equal to $M_{e}+M_{c}$, and $t$ is the time since
explosion in days. The factor $f_{p}$ is equal to $\sqrt{\frac{M_{e}}{M_{c}}}$
for n=3/2, and $0.08(M_{e}/M_{c})$ for n=3.
The temperature is given as:

$\displaystyle\frac{T_{n=3/2}(t)}{\text{K}}$
$\displaystyle=2.05\times{}10^{4}\times$ (15)
$\displaystyle\left(\frac{v^{2}_{s,8.5}t^{2}}{f_{p}M\kappa_{0.34}}\right)^{0.027}\left(\frac{R_{13}}{\kappa_{0.34}}\right)^{0.25}t^{-0.5}$
$\displaystyle\frac{T_{n=3}(t)}{\text{K}}$
$\displaystyle=1.96\times{}10^{4}\times$ (16)
$\displaystyle\left(\frac{v^{2}_{s,8.5}t^{2}}{f_{p}M\kappa_{0.34}}\right)^{0.016}\left(\frac{R_{13}}{\kappa_{0.34}}\right)^{0.25}t^{-0.5}$

Under the assumption that the emission is characterised by a black body, this
once again allows us to calculate the luminosity in any band.

We make a number of assumptions for all models, which follow the assumptions
made by Arcavi et al. (2017). We assume $\kappa_{0.34}=1$, typical for solar
composition materials to exhibit electron scattering. We also set
$M_{c}=1M_{\odot}$. As stated in Arcavi et al. (2017), the early lightcurve is
weakly dependant on this factor and our results are insensitive to it.

##  4 Fitting the Shock Cooling Lightcurve

For each model, we fit the $gri$ bands and the binned Kepler/K2 data
simultaneously. We fit from the first observation up to -9.5 days before
radioactive maximum. This was chosen to ensure our fits are not contaminated
by emission from the main radioactive peak. For the P15, SW17, and P20 models,
we fit the parameters $M_{e}$, $R_{e}$, $v_{e}$ or $v_{s}$ or $v_{t}$,
respectively, and the offset time, $t_{\text{off}}$, between the earliest
observation and the start of the SCL.

We make use of the Python emcee package (Foreman-Mackey et al., 2019), which
provides an implementation of an affine-invariant ensemble Monte Carlo Markov
Chain (MCMC) sampler. This algorithm attempts to produce an approximation of
the posterior given a model, data, and a likelihood function which states how
well the model fits the data. A number of initial positions are randomly
chosen and evaluated with the likelihood function. From these initial
positions, walkers traverse the parameter space, at each step deciding to
either move to a random new position or stay at their current position based
on how well the new parameter position fits the data. After each step, the
walkers record their position in a chain. After a large number of steps these
chains will approximate the posterior. We use the reduced chi-squared as our
likelihood function. We note that a chi-squared loss function naturally arises
from a Gaussian log-likelihood, which assumed that each observations has white
(uncorrelated) noise. The reduced chi-squared is inversely proportional to the
degrees of freedom, which can be thought of as a regularization term to ensure
that we are not biasing our fits to favor the higher cadence of the Kepler/K2
data.

For all fits, we use 500 walkers with a burn-in phase of 100 steps followed by
1000 additional steps. These were chosen after manually investigating the
posterior and the walker chains to ensure they converged. The uniform priors
used are provided in Table 2. These were initially chosen manually to fit the
data while producing physically reasonable masses, radii, and velocities.

Table 2: Uniform prior used for our MCMC fits. $R_{e}$ (R⊙) |  $M_{e}$ (M⊙) |  $v$ (km s-1) |  $t_{off}$ (Days)
---|---|---|---
$0\to 500$ | $0\to 5$ | $0\to 4\times 10^{4}$ | $0\to 15$
Figure 5: A Gaussian fit to the continuum subtracted H$\alpha$ emission of SN
2017jgh. The solid black line is our Gaussian fit and the dashed line the full
width at tenth maximum, used to estimate the velocity.

As an additional method of evaluating these fits, we compare their estimates
of the velocity with that estimated from our spectrum. Following Jha et al.
(2016), we approximate the expansion velocity of the supernova from the full
width at tenth max of the H$\alpha$ line. Figure 5 shows a Gaussian fit to the
H$\alpha$ emission of SN 2017jgh in velocity space. The expansion velocity is
measured to be $9100\pm 470kms^{-1}$.

Figure 6: Model fits to the lightcurve of SN 2017jgh. The coloured lines are
1000 randomly drawn samples from the MCMC chain which give a visual
understanding of the shape of the posterior. The black line is the median
value of the posterior. Note that the median is taken from each parameter
posterior independently so will differ from the randomly drawn samples.
Additionally the P15 and P20 models contain non-Gaussian posteriors so the
median model is not an accurate reflection of the best fitting model.

Figure 7: Corner plot & results of our fits to the lightcurve of SN 2017jgh. The dark blue regions show the 1 sigma posterior. Created using the chainconsumer API. The best fitting value was calculated using the cumulative method of chainconsumer, which takes the 50th percentile of the parameter posterior as the best fitting value. The 16th and 84th percentiles are then the lower and upper bounds, respectively. The shaded region of each posterior, and the joint posteriors, show the lower and upper bounds (the 16th and 84th percentile). Note that $t_{off}$ is given in terms of days relative to radioactive maximum. Model | $R_{e}\leavevmode\nobreak\ \left(R_{\odot}\right)$ | $M_{e}\leavevmode\nobreak\ \left(M_{\odot}\right)$ | $v\leavevmode\nobreak\ \left(10^{3}\leavevmode\nobreak\ \mathrm{km}\mathrm{s}^{-1}\right)$ | $t_{off}\leavevmode\nobreak\ \left(t-t_{max}\right)$ | Reduced $\chi^{2}$ | DOF
---|---|---|---|---|---|---
P15 | $42.41^{+135.66}_{-22.32}$ | $0.46^{+0.28}_{-0.23}$ | $14.25^{+4.14}_{-4.26}$ | $-22.50^{+0.38}_{-0.55}$ | 17.79 | 4
P20 | $45.82^{+36.48}_{-11.04}$ | $3.07^{+1.37}_{-2.12}$ | $17.79^{+1.53}_{-3.16}$ | $-21.02^{+0.20}_{-0.31}$ | 12.82 | 4
SW17 n=3/2 | $127.82^{+160.35}_{-76.11}$ | $0.56^{+1.17}_{-0.56}$ | $8.81^{+1.50}_{-1.26}$ | $-21.05^{+0.25}_{-0.27}$ | 6.57 | 4
SW17 n=3 | $310.62^{+119.80}_{-97.59}$ | $3.09^{+1.09}_{-0.75}$ | $8.17^{+0.83}_{-0.76}$ | $-20.96^{+0.23}_{-0.21}$ | 6.39 | 4
Table 3: Best fitting parameters for each model, found by taking the 16th,
50th, and 84th percentile of each parameter as the lower bound, best value,
and upper bound, respectively. Note that this does not take into account the
dependencies between parameters. The reduced chi squared of the best fitting
value is also included. The SW17 models fit the lightcurve best, and have very
similar velocity and offset times, however differ greatly in both the envelope
radius and mass parameters. Since SW17 n=3/2 has a best fitting velocity which
is less than one standard deviation of the spectral velocity (9100$\pm$470km
s-1), we choose this as the preferred model for SN 2017jgh.

The best fit to the lightcurve of SN 2017jgh is shown in Figure 6 for each of
our models of interest, the corner plots of each model are shown in Figure 7,
and the best fitting values are given in Table 3. The best fitting value is
taken as the 50th percentile of the parameter posterior, with the 16th and
84th percentile as the lower and upper bounds, respectively. Note that this
does not take into account the dependencies between parameters.

As can be seen in Figure 6, P15 appears to perform the worst of the four
models, with a small divergence at the start of the SCL rise and a larger
deviation at about -15 days before radioactive maximum. Additionally, the
randomly drawn MCMC samples appear to diverge from the best fit. The offset
time found by P15 (-22.5 days) is different to the offset time found by every
other model (about -21 days) which accounts for the smaller deviation. The
larger deviation and the divergence from the randomly drawn MCMC samples can
be explained by the non-Gaussian posterior of P15’s $R_{e}$ parameter. This
non-Gaussian profile means that the median of the posterior does not lie at
the maximum of the posterior. P20 suffers a similar issue of non-Gaussianity,
with the posterior of $M_{e}$ being multimodal (two peaks), although it does
perform significantly better than P15 despite making similar assumptions.
Unlike the SW17 models which have smaller residuals, neither the P15 nor P20
model make any assumption about the density of the progenitor which could
account for their non-Gaussian posteriors.

The SW17 n=3/2 and SW17 n=3 models have a reduced chi-squared of 6.57 and
6.39, respectively; these are betters fits to the ground-based data than P15
and P20 which have a reduced chi squared of 17.79 and 12.82, respectively. All
models have relatively large reduced chi squared (with a value of 1 being
desirable). This could be due to the inherent systematics in the Kepler/K2
data, or could be indicative of some physics not paramaterised by these
models.

When comparing the best fitting velocities to the velocity measured from the
spectrum, 10,200 km s-1, we see that the SW17 n=3/2 once again best matches
the data with a best fitting velocity of $\sim$8,800 km s-1, as opposed to
$\sim$14,000 km s-1, $\sim$18,000 km s-1, and $\sim$8,100 km s-1 for P15, P20,
and SW17 n=3, respectively. As such we conclude that SW17 n=3/2 is the best
model for SN 2017jgh.

Overall it seems that the SW17 models are more physically accurate than P15
and P20 owing to their density assumption with SW17 n=3/2 being the best
fitting model for this supernova. This suggests that the progenitor of SN
2017jgh had an envelope radius of $\sim$130$R_{\odot}$ and an envelope mass of
$\sim$0.50$M_{\odot}$. This radius is similar to the progenitor radius of SN
1993J (Maund et al., 2004), and another well studied IIb supernova SN 2011dh
($\sim 200_{\odot}$ Bersten et al. (2012)), both of which are believed to have
yellow supergiant progenitors. We conclude that the progenitor of SN 2017jgh
was likely a yellow supergiant. This is reinforced by the similarities SN
2017jgh had with SN 1993J, in both its lightcurve and colour evolution.

##  5 The Importance of the Rise

Figure 8: Fits to the ground-based lightcurve, ignoring all Kepler/K2 data. This emulates the conditions of most previous SCL observations. The coloured lines are 1000 randomly drawn samples from the MCMC chain which give a visual for the shape of the posterior. The black line is the median value of the posterior. Note that the median is taken from each parameter posterior independently so will differ from the randomly drawn samples. It is obvious that these fits do not constrain the rise time, and even though they seem reasonably when compared to the ground-based data, when applied to the Kepler/K2 data we see that they are quite inaccurate. Figure 9: A comparison of the posteriors of the ground-based fits (blue) and the fits to the full lightcurve (green) for the SW17 n=3/2 model. The ground-based fits do not constrain the rise time and as such have a systematic offset in the $v_{s}$ parameter. However the ground-based fits do appear to constrain the $R_{e}$ and $M_{e}$ parameters at least as well as the full lightcurve fits. Model | $R_{e}\leavevmode\nobreak\ \left(R_{\odot}\right)$ | $M_{e}\leavevmode\nobreak\ \left(M_{\odot}\right)$ | $v\leavevmode\nobreak\ \left(\mathrm{km}/\mathrm{s}\right)$ | $t_{off}\leavevmode\nobreak\ \left(t-t_{max}\right)\leavevmode\nobreak\ (days)$ | $\chi^{2}$
---|---|---|---|---|---
SW17 n=3/2 | $150^{+180}_{-100}$ | $0.55^{+1.16}_{-0.23}$ | $\left(11.5^{+3.7}_{-2.7}\right)\times 10^{3}$ | $-22.9^{+3.0}_{-3.1}$ | 8.66
Table 4: Best fitting parameters for fits to the ground-based lightcurve with
the SW17 n=3/2 model, found by taking the 16th, 50th, and 84th percentile as
the lower bound, best value, and upper bound, respectively. The reduced chi
squared of the best fitting value is also included.

As this is the first complete example of a high cadence shock cooling
lightcurve, it provides an opportunity to evaluate how important the
information contained in the rise is to getting an accurate fit. We perform
fits to the ground-based lightcurve of SN 2017jgh without any Kepler/K2 data.
The ground-based lightcurve contains no rise information and is quite similar
in coverage and cadence to the lightcurves of SN 2016gkg and SN 1993J. Figure
8 shows the results of these fits for the SW17 n=3/2 model, the model which
best fit the full lightcurve. The best fitting values are given in Table 4,
and a comparison between the posteriors of the ground-based fits and the fits
to the full lightcurve are shown in Figure 9.

Although the fits to the ground-based portion of the lightcurve appear
reasonable, the extrapolated fits to the rise portion of the SCL are very
poorly constrained. Additionally, the best fit does not match the Kepler/K2
lightcurve. Overall, this suggests that the decline of the SCL is not enough
to constrain the rise of the SCL.

The best fitting radius and mass of the ground-based fits ($\sim
150\leavevmode\nobreak\ R_{\odot}$ and $\sim 0.55\leavevmode\nobreak\
M_{\odot}$) are consistent with the radius and mass of the full lightcurve
fits ($\sim 130$ and $\sim 0.56$), however both the velocity and offset time
are significantly different at $\sim 11,000$ km s-1 and $\sim-23$ days
compared to the full lightcurve fits of $\sim 8,800$ km s-1 and $\sim-21$
days. This suggests that the rise is required to constrain the velocity and
explosion time, however the envelope radius and envelope mass can be
constrained with just the decline.

In order to estimate the numerical impact of the SCL rise on the quality of
the fits, we calculate the percentage difference between the fits with and
without the rise. We make use of bootstrap resampling to approximate the
population mean and uncertainty. For both the full lightcurve fit and the
decline fit we resample 10000 chains, each of which contain 500000 samples of
the posterior. For each of these resampled chains we find the median of the
parameters, and then calculate the mean and standard deviation of these
medians. The mean, and percentage difference between the full lightcurve fit
and the decline fit is show in table 5. The envelope mass is the least
affected, with only a 1% difference between the fits. The envelope radius,
offset time, and velocity are more greatly affected, at 15%, 20%, and 25%,
respectively. These values provide a rough approximation for the systematic
uncertainty of not having the rise of the SCL.

| $R_{e}\leavevmode\nobreak\ (R_{\odot})$ | $M_{e}\leavevmode\nobreak\ (M_{\odot})$ |  $v\leavevmode\nobreak\ (10^{3})$ km s-1 | $t_{off}\leavevmode\nobreak\ (t-t_{max})\leavevmode\nobreak\ (days)$
---|---|---|---|---
Full lightcurve fit | $127.82\pm 0.21$ | $0.56\pm 0.0008$ | $8.81\pm 0.0023$ | $9.5\pm 0.00024$
Decline fit | $149.11\pm 0.25$ | $0.55\pm 0.00055$ | $11.52\pm 0.0047$ | $7.62\pm 0.0062$
Percentage difference |  $15.38\pm 0.67$% |  $1.13\pm 0.49$% |  $26.67\pm 0.14$% |  $22\pm 0.23$%
Table 5: Average parameter median over 10000 bootstrap resamples for both the
full lightcurve fit and decline fit, as well as the percentage difference
between them.

##  6 Conclusions

The high cadence Kepler/K2 lightcurve of SN 2017jgh provides a unique
opportunity to investigate the complete shock cooling lightcurve of a Type IIb
SNe. We fit the lightcurve with four models, the linearly expanding P15 model,
the two-component P20 model, and the polytropic SW17 models with $n=3/2$ and
$n=3$, modelling a red supergiant, and blue supergiant, respectively.

After fitting the P15, P20, SW17 $n=\nicefrac{{3}}{{2}}$ and SW17 $n=3$ models
to SN 2017jgh, we found that the SW17 $n=\nicefrac{{3}}{{2}}$ model provides
the best fit (with a reduced $\chi^{2}$ of 6.57). This fit suggests that the
progenitor of SN 2017jgh was a yellow supergiant with an envelope radius of
$\sim 50-290\leavevmode\nobreak\ R_{\odot}$, an envelope mass of $\sim
0-1.7\leavevmode\nobreak\ M_{\odot}$, a shock velocity of
$\left(7.5-10.3\right)\times 10^{3}$ km s-1, and an offset time of $\sim-21$
days until radioactive maximum. SW17 $n=\nicefrac{{3}}{{2}}$ is also best able
to reproduce the measured shock velocity of 9100$\pm$470 km s-1.

Furthmore, we showed that the P15 and P20 models had difficulty reproducing
the decline of the SCL, overestimating and underestimating it, respectively.
This is possibly due to the lack of density information in these models. By
comparison, both the SW17 $n=3$ and SW17 $n=\nicefrac{{3}}{{2}}$ models follow
the shape of the lightcurve better with very little deviation in the residual.
The polytropic density model appears to be a better approximation for the true
density of the shock cooling lightcurve.

In addition to determining the parameters of the progenitor of SN 2017jgh, we
use the complete SCL as an opportunity to investigate how important the rise
of the shock cooling lightcurve is to the quality of fits. To do so, we fit
the ground-based lightcurve of SN 2017jgh ignoring any Kepler/K2 data. These
fits were unable to constrain the explosion time and could not recover the
Kepler/K2 lightcurve. This lead to a systematic offset in $v_{s}$, but both
$R_{e}$ and $M_{e}$ seem mostly unaffected. In all models velocity and
explosion time are degenerate, and radius and mass are degenerate. It appears
that the rise constrains the offset time and velocity whilst the decline
constrains the radius and mass. The rise is more sensitive to temperature and
density (Piro, 2015; Sapir & Waxman, 2017; Piro et al., 2020), so the shock
velocity will also be sensitive to the shape of the rise.

Overall this suggests that the cadence and rise information contained within
the Kepler/K2 lightcurve is invaluable to getting a good fit, specifically to
constraining the offset time and shock velocity. This is less important for
constraining the envelope radius and mass. Long term, high cadence surveys
akin to Kepler/K2 will be essential for improving these analytical SCL models.

##  7 Acknowledgments

This is a pre-copyedited, author-produced PDF of an article accepted for
publication in the Monthly Notice of the Royal Astronomical Society following
peer review. The version of record (Armstrong et al., 2021) is available
online at: https://doi.org/10.1093/mnras/stab2138.

P.A. and B.E.T. acknowledge parts of this research was carried out on the
traditional lands of the Ngunnawal people. We pay our respects to their elders
past, present, and emerging. This paper includes data collected by the K2
mission. Funding for the K2 mission is provided by the NASA Science Mission
directorate. KEGS is supported in part by NASA K2 cycle 4, 5, and 6 grants
NNX17AI64G and 80NSSC18K0302, and 80NSSC19K0112, respectively. Pan-STARRS
(PS1) is supported in part by the National Aeronautics and Space
Administration under Grants NNX12AT65G and NNX14AM74G. The PanSTARRS1 Surveys
(PS1) and the PS1 public science archive have been made possible through
contributions by the Institute for Astronomy, the University of Hawaii, the
Pan-STARRS Project Office, the Max-Planck Society and its participating
institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max
Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins
University, Durham University, the University of Edinburgh, the Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las
Cumbres Observatory Global Telescope Network Incorporated, the National
Central University of Taiwan, the Space Telescope Science Institute, the
National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation Grant No. AST–1238877, the
University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos
National Laboratory, and the Gordon and Betty Moore Foundation Based on
observations at Cerro Tololo Inter-American Observatory, National Optical
Astronomy Observatory (NOAO 2017B-0279; PI: A Rest, NOAO 2017B-0285; PI: A
Rest), which is operated by the Association of Universities for Research in
Astronomy (AURA) under a cooperative agreement with the National Science
Foundation Based on observations obtained at the international Gemini
Observatory, which is managed by the Association of Universities for Research
in Astronomy (AURA) under a cooperative agreement with the National Science
Foundation on behalf of the Gemini Observatory partnership: the National
Science Foundation (United States), National Research Council (Canada),
Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia,
Tecnología e Innovación (Argentina), Ministério da Ci$\hat{\text{e}}$ncia,
Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space
Science Institute (Republic of Korea). Observations in this program were
obtained through program ID GS-2017B-LP-13. P. A. was supported by an
Australian Government Research Training Program (RTP) Scholarship. B. E. T.
and his group were supported by the Australian Research Council Centre of
Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through
project number CE170100013. The UCSC transient team is supported in part by
NASA/K2 grants 80NSSC18K0303 and 80NSSC19K0113, the Gordon & Betty Moore
Foundation, the Heising-Simons Foundation, and by a fellowship from the David
and Lucile Packard Foundation to R.J.F. D. O. J. acknowledges support provided
by NASA Hubble Fellowship grant HST-HF2-51462.001, which is awarded by the
Space Telescope Science Institute, operated by the Association of Universities
for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. I. A. is
a CIFAR Azrieli Global Scholar in the Gravity and the Extreme Universe Program
and acknowledges support from that program, from the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation program
(grant agreement number 852097), from the Israel Science Foundation (grant
number 2752/19), from the United States - Israel Binational Science Foundation
(BSF), and from the Israeli Council for Higher Education Alon Fellowship. M.
R. D. acknowledges support from the NSERC through grant RGPIN-2019-06186, the
Canada Research Chairs Program, the Canadian Institute for Advanced Research
(CIFAR), and the Dunlap Institute at the University of Toronto. D. A. C.
acknowledges support from the National Science Foundation Graduate Research
Fellowship under Grant DGE1339067. This project has been supported by the
LP2018-7 Lendület grant of the Hungarian Academy of Sciences. L. G.
acknowledges financial support from the Spanish Ministry of Science,
Innovation and Universities (MICIU) under the 2019 Ramón y Cajal program
RYC2019-027683 and from the Spanish MICIU project PID2020-115253GA-I00. B. J.
S. is supported by NASA grant 80NSSC19K1717 and NSF grants AST-1920392 and
AST-1911074. Support for T. W. S. H was provided by NASA through the NASA
Hubble Fellowship grant HST-HF2-51458.001-A awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, Inc., for NASA, under contract NAS5-26555. Research by
S.V. is supported by NSF grants AST-1813176 and AST-2008108 L. K. acknowledges
the financial support of the Hungarian National Research, Development and
Innovation Office grant NKFIH PD-134784. L. K. and Zs. B. are supported by the
Jànos Bolyai Research Scholarship of the Hungarian Academy of Sciences. The
Konkoly team has been supported by the project "Transient Astrophysical
Objects" GINOP 2.3.2-15-2016-00033 of the National Research, Development and
Innovation Office (NKFIH), Hungary, funded by the European Union. S. W. J.
acknowledges support from US National Science Foundation award AST-1615455.
This research has made use of the SVO Filter Profile Service
(http://svo2.cab.inta-csic.es/theory/fps/) supported from the Spanish MINECO
through grant AYA2017-84089. The LCO team is supported by NASA grant
80NSSC19K0119 and NSF grants AST-1911225 and AST-1911151.

###  7.1 Data availability

The data underlying this article are available in the article and in its
online supplementary material.

###  7.2 Affiliations

1Mt Stromlo Observatory, The Research School of Astronomy and Astrophysics,
Australian National University, ACT 2601, Australia
2National Centre for the Public Awareness of Science, Australian National
University, Canberra, ACT 2611, Australia
3The ARC Centre of Excellence for All-Sky Astrophysics in 3 Dimensions (ASTRO
3D)
4Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218,
USA
5Department of Physics and Astronomy, The Johns Hopkins University, 3400 North
Charles Street, Baltimore, MD 21218, USA.
6The Observatories of the Carnegie Institution for Science, 813 Santa Barbara
St., Pasadena, CA 91101, USA
7School of Mathematics and Physics, University of Queensland, Brisbane, QLD
4072, Australia
8Australian Astronomical Observatory, North Ryde, NSW 2113, Australia
9Gemini Observatory/NSF’s NOIRLab, Casilla 603, La Serena, Chile
10University of Illinois at Urbana-Champaign, 1003 W. Green St., IL 61801, USA
11Centre for Astrophysical Surveys, National Centre for Supercomputing
Applications, Urbana, IL 61801, USA
12Astronomy Department, University of Maryland, College Park, MD 20742-2421,
USA
13The School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978,
Israel
14CIFAR Azrieli Global Scholars program, CIFAR, Toronto, Canada
15David A. Dunlap Department of Astronomy and Astrophysics, University of
Toronto, 50 St. George Street, Toronto, Ontario, M5S 3H4 Canada
16Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA 95064, USA
17Department of Astronomy, University of Texas at Austin, Austin, TX, 78712,
USA
18Centro de Astronomía (CITEVA), Universidad de Antofagasta, Avenida U. de
Antofagasta 02800, Antofagasta, Chile
19Las Campanas Observatory, Carnegie Observatories, Casilla 601, La Serena,
Chile
20Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str 1, D-85748
Garching bei München, Germany
21Center for Interdisciplinary Exploration and Research in Astrophysics
(CIERA)
22Department of Physics and Astronomy, Northwestern University, Evanston, IL
60208, USA
23Centre for Astrophysics Research, School of Physics, Astronomy and
Mathematics, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK
24University of Notre Dame, Notre Dame, IN 46556, USA
25Institute for Astronomy, University of Hawai`i at Manoa, 2680 Woodlawn
Drive, Honolulu, HI 96822, USA
26Exoplanets and Stellar Astrophysics Laboratory, Code 667,NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA
27University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD
21250, USA
28Bay Area Environmental Research Institute, P.O. Box 25, Moffett Field, CA
94035, USA
29NASA Ames Research Center, Moffett Field, CA 94035, USA
30Astrophysics Research Centre, School of Mathematics and Physics,Queen’s
University Belfast, BT7 1NN, UK
31Department of Astronomy and Theoretical Astrophysics Center, University of
California, Berkeley, CA 94720, USA
32Department of Physics, University of California, Berkeley, CA 94720, USA
33Lawrence Berkeley National Laboratory, Berkeley, CA, USA
34Konkoly Observatory, Research Centre for Astronomy and Earth Sciences,
Eötvös Lorànd Research Network (ELKH), H-1121 Konkoly Thege Miklòs ùt 15-17,
Budapest, Hungary
35MTA CSFK Lendület Near-Field Cosmology Research Group
36Coral Towers Observatory, Cairns, QLD 4870, Australia
37Institute of Space Sciences (ICS, CSIC), Campus UAB, Carrer de Can Magrans,
s/n, E-08193 Barcelona, Spain
38Department of Physics, University of California, Santa Barbara, CA
93106-9530, USA
39Las Cumbres Observatory, 6740 Cortona Dr, Suite 102, Goleta, CA 93117-5575,
USA
40Institute of Physics, University of Szeged, Dom ter 9, Szeged, 6720 Hungary
41Departamento de Astronomía y Astrofísica, Universidad de Valencia, E-46100
Burjassot, Valencia, Spain
42Observatorio Astronómico, Universidad de Valencia, E-46980 Paterna,
Valencia, Spain
43Graduate Institute of Astronomy, National Central University, 300 Jhongda
Road, 32001 Jhongli, Taiwan
44Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua
University, Beijing, 100084, China
45Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson,
AZ 85721-0065, USA
46ELTE Eötvös Loránd University, Institute of Physics, Budapest, Hungary
47Department of Astronomy, Columbia University, New York, NY 10027-6601, USA
48Beijing Planetarium, Beijing Academy of Science and Technology,Beijing,
100044
49Physics Department, Tsinghua University, Beijing, 100084
50Cerro Tololo Inter-American Observatory, NSF’s National Optical-Infrared
Astronomy Research Laboratory, Casilla 603, La Serena, Chile
51Department of Physics and Astronomy, Texas A&M University, 4242 TAMU,
College Station, TX 77843, USA
52Department of Mathematics, University of York, Heslington, York, YO10 5DD,
United Kingdom
53Department of Physics and Astronomy, University of California, 1 Shields
Avenue, Davis, CA 95616-5270, USA
54European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching
bei M ünchen, German
55Department of Physics and Astronomy, Rutgers the State University of New
Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA
56Centre for Gravitational Astrophysics, College of Science, The Australian
National University, ACT 2601, Australia
57School of Physics, The University of Melbourne, Parkville, VIC 3010,
Australia

###  7.3 Software

NumPy (Harris et al., 2020), MatPlotLib (Hunter, 2007), SciPy (Virtanen et
al., 2020), AstroPy (The Astropy Collaboration et al., 2018), emcee (Foreman-
Mackey et al., 2019), scikit-learn (Pedregosa et al., 2011), chainconsumer
(Hinton, 2016)

###  7.4 Facilities

Kepler (Howell et al., 2014), Gemini (Hook et al., 2004), Swope (Folatelli et
al., 2010), PS1 (Chambers et al., 2016)

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