# A search for transits among the $\delta$ Scuti variables in Kepler

Daniel R. Hey School of Physics, Sydney Institute for Astronomy (SIfA)
The University of Sydney, NSW 2006, Australia  Stellar Astrophysics Centre,
Department of Physics and Astronomy
Aarhus University, DK-8000 Aarhus C, Denmark  Benjamin T. Montet School of
Physics, University of New South Wales, Sydney, NSW 2052, Australia  UNSW Data
Science Hub, University of New South Wales, Sydney, NSW 2052, Australia
Benjamin J.S. Pope School of Mathematics and Physics, The University of
Queensland, St Lucia, QLD 4072, Australia  Centre for Astrophysics, University
of Southern Queensland, West Street, Toowoomba, QLD 4350, Australia  Simon J.
Murphy School of Physics, Sydney Institute for Astronomy (SIfA)
The University of Sydney, NSW 2006, Australia  Stellar Astrophysics Centre,
Department of Physics and Astronomy
Aarhus University, DK-8000 Aarhus C, Denmark  Timothy R. Bedding School of
Physics, Sydney Institute for Astronomy (SIfA)
The University of Sydney, NSW 2006, Australia  Stellar Astrophysics Centre,
Department of Physics and Astronomy
Aarhus University, DK-8000 Aarhus C, Denmark

###### Abstract

We search for transits around all known pulsating $\delta$ Sct variables (6500
K < Teff < 10 000 K) in the long-cadence _Kepler_ data after subtracting the
pulsation signal through an automated routine. To achieve this, we devise a
simple and computationally inexpensive method for distinguishing between low-
frequency pulsations and transits in light curves. We find 3 new candidate
transit events that were previously hidden behind the pulsations, but caution
that they are likely to be false positive events. We also examined the
_Kepler_ Objects of Interest catalog and identify 13 additional host stars
which show $\delta$ Sct pulsations. For each star in our sample, we use the
non-detection of pulsation timing variations for a planet that is known to be
transiting a $\delta$ Sct variable to obtain both an upper limit on the mass
of the planet and the expected radial velocity semi-amplitude of the host
star. Simple injection tests of our pipeline imply 100% recovery for planets
of 0.5 RJup or greater. Extrapolating our number of _Kepler_ $\delta$ Sct
stars, we expect 12 detectable planets above 0.5 RJup in TESS. Our sample
contains some of the hottest known transiting planets around evolved stars,
and is the first complete sample of transits around $\delta$ Sct variables. We
make available our code and pulsation-subtracted light curves to facilitate
further analysis.

††journal: AJ††software: astropy (Astropy Collaboration et al., 2013; Price-
Whelan et al., 2018), lightkurve (Vinícius et al., 2018; Barentsen et al.,
2019), PyMC3 (Salvatier et al., 2016), TLS (Hippke & Heller, 2019), matplotlib
(Hunter, 2007), numpy (Oliphant, 2015).

##  1 Introduction

Planetary systems around the hot A/F stars are of considerable interest from
the perspective of planetary formation, atmospheres, and habitability. The
discovery of the ‘radius gap’ in the distribution of _Kepler_ planetary radii
(Fulton et al., 2017; Van Eylen et al., 2018) strongly supports the theory of
photoevaporation of primary planetary atmospheres under the harsh light of
their hot young stellar hosts (Owen & Wu, 2013, 2017). Extremely highly
irradiated planets around hot stars such as the A0 star KELT-9 (Gaudi et al.,
2017) have therefore been heavily studied as laboratories for understanding
atmospheric loss generally, with evidence for significant exospheric envelopes
of lost atmospheric gas (Yan & Henning, 2018; Hoeijmakers et al., 2018).
Further examples of planets transiting hot stars are likely to attract similar
interest.

Furthermore, while they are challenging for RV and photometry, many A/F stars
are popular targets for direct imaging and astrometric planet searches. These
include the A6 host $\beta$ Pictoris (Lagrange et al., 2010, 2020), with its
edge-on debris disk, the F0 star HR 8799 (Brandt et al., 2021) and its four
well-studied giant planets (Marois et al., 2008, 2010); or the directly-imaged
debris disks around the A0 $\delta$ Sct Vega (Matrà et al., 2020) and the A4
star Fomalhaut (Kalas et al., 2005). Direct imaging is most-sensitive to
planets and disk structures far from their hosts, and the inner architecture
of hot star planetary systems is less-well understood.

The A/F stars’ far-out habitable zone is easily accessible to astrometric
planet detection (Gould et al., 2003). While the habitability of such planets
is expected to suffer from a harsh radiation environment (Rugheimer et al.,
2015) and short main-sequence lifespan (Cuntz & Guinan, 2016), it is not yet
clear whether this is an insurmountable barrier to life (Sato et al., 2014,
2017) and any such planets will be valuable to study. Planets around hot
stars, whether in the habitable zone or otherwise, are valuable in exploring
the architecture and habitability of these poorly-understood systems.

One class of A/F stars are the $\delta$ Sct variables – a class of
intermediate-mass stars that pulsate in pressure modes with frequencies
between 5 and 100 d-1 (Guzik, 2021). They exist along the classical
instability strip across the HR diagram, typically between temperatures of
6,500 and 10,000 K (Murphy et al., 2019; Cunha et al., 2019). The $\delta$ Sct
variables lack thick convective outer layers and thus retain much of their
primordial angular momentum (Cantiello & Braithwaite, 2019). As a result, they
typically experience rotational velocities greater than 100 km/s (Royer, 2009;
Zorec & Royer, 2012) which heavily broadens their spectral lines, making it
challenging to obtain precise RV measurements (Ahlers et al., 2019). For this
reason, they are usually excluded from radial velocity exoplanet surveys which
favor cooler, more slowly rotating host stars (Howard et al., 2010; Grandjean
et al., 2020; Rosenthal et al., 2021), although some searches directly target
these stars (see, for example, Desort et al. (2007); Borgniet et al. (2017)).
Because of this difficulty in obtaining planetary precision RV measurements,
few exoplanets are known, or even suspected, to exist around $\delta$ Sct
stars. Those known are typically rather massive hot Jupiters (Galland et al.,
2006a; Herrero et al., 2011; Borgniet et al., 2014; Murphy et al., 2016;
Temple et al., 2017; Martínez et al., 2020; Wong et al., 2020).

While commonly seen as a nuisance, the pulsations in variable stars can be
used to better constrain the evolutionary state of the system. For example, in
solar-like oscillators, global asteroseismic parameters have been used to
better characterize the system under a joint asteroseismic and transit
analysis (see, for example, Gilliland et al. 2010; Huber 2018; Chontos et al.
2020). For $\delta$ Sct variables, the splitting of pulsation modes can reveal
the stellar rotation rate and the plane of obliquity (Kurtz et al., 2014; Chen
& Li, 2017). Unfortunately, the rapid rotation of $\delta$ Sct variables not
only spoils their absorption line features, but also causes large and uneven
rotational splittings in the excited modes of oscillation, such that
multiplets of adjacent modes can overlap (Reese et al., 2009; Mirouh et al.,
2019). In addition, the asteroseismic inference from $\delta$ Sct light curves
has lagged behind many other types of pulsating variables until recently
(Bedding et al., 2020), making precise stellar parameters for $\delta$ Sct
stars difficult to obtain. Now, however, some $\delta$ Sct stars have ultra-
precise parameters derived from asteroseismology (Murphy et al., 2021),
opening them up as a powerful diagnostic as exoplanet host stars.

With the advent of extremely high-precision space-based telescopes such as
Kepler, TESS and the upcoming PLATO mission (Koch et al., 2010; Borucki et
al., 2010; Ricker et al., 2014; Rauer et al., 2014), a large influx of light
curves for $\delta$ Sct stars are already available. It is estimated that
approximately half of the stars located within the classical instability strip
are pulsating above the $\sim$10 ppm detection threshold provided by Kepler
(Murphy et al., 2019).

The photometric amplitude of the stellar pulsations in $\delta$ Sct variables
may be on the order of 10 to 100,000 ppm. In comparison, the decrease in
stellar flux from a transit of an Earth-like planet is on the order of 100
ppm. Transits whose amplitudes are bigger than the pulsation amplitudes are
relatively easy to spot. Difficulties arise when the pulsational variability
exceeds that of the transits, where it can be said that such transits are
‘hidden’ in the pulsations. To find these hidden transit signals, the light
curve should be processed in a manner that removes the pulsation signal
(referred to as ‘cleaning’ in this paper) but preserves the transit. This
method was used by Zieba et al. (2019) to find transiting exocomets around the
$\delta$ Sct star $\beta$ Pictoris. We are fortunate in the fact that most
transits have clearly non-sinusoidal shapes. On the other hand, pulsations in
$\delta$ Sct variables behave highly sinusoidally. This lets them be easily
expressed as the sum of sine waves that can be subtracted (or divided out)
from the light curves, allowing for transit searches in the residuals. In
frequency space, the transits appear as peaks at integer multiples of the
orbital frequency as a result of their rectangular shape.

Sowicka et al. (2017) conducted such a search for transits around pulsating
stars in _Kepler_ by first modeling the stellar pulsations as sinusoids and
then subtracting this model. This search was limited to stars with short-
cadence observations and temperatures $6000\text{K}<T<8500~{}\text{K}$, and
found two candidates, around KIC 5613330 and KIC 8197761; the latter of which
was confirmed by RV follow-up to be an eclipsing binary.

In this paper, we seek to extend their search and characterize the occurrence
rate of exoplanets around the $\delta$ Sct variables in the entire _Kepler_
long-cadence field. Taking a similar approach to Sowicka et al. (2017), we
design a pipeline that iteratively subtracts the $\delta$ Sct pulsations to
facilitate a transit search, and introduce a simple method for distinguishing
between pulsations and transits in the low-frequency regime. We then search
for transit events in the cleaned light curves through a combination of manual
inspection of Box Least Squares (BLS) periodograms and a semi-automated
transit search with Transit Least Squares (TLS) in Section 3. We identify 3
previously unknown candidate transit events, and fit their planetary
parameters with a transit model in Section 4. We then search the light curves
of the candidate and confirmed planets in the Kepler objects of interest table
for $\delta$ Sct oscillations, completing our sample of $\delta$ Sct host
stars. Finally in Section 5, we exploit the timing variations of the stellar
pulsations over the orbit to provide a robust upper limit on the planetary
mass, and consequently, an upper limit on the radial velocity precision
required to rule them out as candidates or confirm their planetary status. We
make available our code and pulsation subtracted light
curves111https://github.com/danhey/dsct-exoplanet.

##  2 Observational data

Figure 1: HR diagram of the $\delta$ Sct sample. We make no distinction for
hybrid pulsators – those which oscillate simultaneously in p- and g-modes. The
evolutionary tracks were calculated in MESA v10108 (Paxton et al., 2011) with
$X=0.71$ and $Z=0.01$. See section 4.2 of Murphy et al. (2019) for a full
description of the tracks.

We chose our sample to be the known $\delta$ Sct variables in the _Kepler_
field for three reasons. Firstly, most $\delta$ Sct stars in the _Kepler_
field have been identified. Secondly, their masses, temperatures, and other
fundamental parameters have been well-characterized over the past several
years (Murphy et al., 2019; Berger et al., 2020; Guzik, 2021). Finally, the
4-year time-span of _Kepler_ data allows pulsation timing variation studies,
whose sensitivity scales with the SNR of the pulsations (see, for example,
Murphy et al., 2014). Despite our focus only on the _Kepler_ $\delta$ Sct
stars, we note that all of the principles in this paper are readily applicable
to other space based photometric surveys, including TESS.

We compiled a catalog of _Kepler_ $\delta$ Sct stars by combining the $\delta$
Sct catalog of Murphy et al. (2019) with those of Balona (2018) and Bradley et
al. (2015). We excluded the slowly pulsating B-stars and perhaps mythical
‘Maia’ variables (White et al., 2017) cataloged by Balona (2018). We also
chose not to include the rapidly oscillating Ap stars, since only few are
known and they have been intensely scrutinized already (Holdsworth et al.,
2018; Hey et al., 2019). After merging the catalogs, we were left with 2354
stars with $\delta$ Sct pulsations, some of which are hybrid pulsators, which
pulsate in both high-frequency p-modes and low-frequency gravity (g) modes
simultaneously, typically known as $\delta$ Sct / $\gamma$ Doradus hybrids
(Grigahcène et al., 2010; Uytterhoeven et al., 2011).

In order to search for transits, we used the light curves from the 4-year
nominal Kepler mission. Kepler was a space-based telescope that collected
precise photometric data for around 150,000 stars simultaneously in a 115
${\deg}^{2}$ field of view. The core science goal of the Kepler mission was to
detect Earth-like planets in the habitable zone (Borucki et al., 2010; Koch et
al., 2010). A secondary goal of this mission was asteroseismology (Gilliland
et al., 2011): with a photometric precision verging on the $\mu$mag level,
Kepler provided unprecedented insight into stellar pulsations.

The Kepler data are available in two observing modes: long cadence (LC) and
short cadence (SC). The LC data are integrated over 29.4-min observations,
whereas the SC data are integrated over 1-min observations, with up to 512
stars being allocated to SC at any given time on a priority basis. Both sets
of data were stored on-board the spacecraft and regularly down-linked to Earth
every 32 d, along with a periodic roll to re-orient the spacecraft’s solar
panels, introducing semi-regular gaps in the data. The Kepler data are thus
organized into quarters, which we downloaded from the Mikulski Archive for
Space Telescopes (MAST) repository. Light curves have both simple aperture
photometry (SAP) and pre-search data conditioning simple aperture photometry
(PDCSAP; Twicken et al. 2010; Smith et al. 2012). We chose to use PDCSAP v9.3,
which applies co-trending basis vectors to the SAP flux to remove long-term
and systematic trends, which is effective for both pulsations and transits.

###  2.1 Stellar properties

We took temperatures and luminosities for our sample from Berger et al.
(2020). For stars not in this catalogue, we calculated the luminosity
following the same method which we briefly outline now. Reddening was obtained
from the 3D maps provided by Green et al. (2018), and distances were
calculated using the Bayesian approach described by Bailer-Jones et al. (2018)
with Gaia eDR3 parallaxes. We did not apply an offset in the parallax, since
Murphy et al. (2019) found it results in underestimated luminosity for stars
in this temperature range. We used the Gaia G-band magnitudes, and bolometric
corrections from the MIST isochrones (Dotter, 2016). We show the location of
the stars on the HR diagram in Fig. 1.

##  3 The pipeline

Figure 2: Flowchart of the main functions of the pipeline.

We designed a pipeline to search for signals hidden by the pulsations of the
$\delta$ Sct stars. The pipeline has multiple components which perform the
following operations in order: light curve corrections, iterative fitting and
subtraction of pulsation modes, and a transit search. We describe each aspect
of the pipeline in detail below, and provide a flowchart of its main
operations in Fig. 2.

###  3.1 Light curve corrections

In the first phase of the pipeline, we processed the individual _Kepler_
quarters to remove slow variations for each star. To do this, we performed a
3-sigma clipping of the light curves and subtracted a 1D spline fit of 2nd
degree made using the Scipy package (Virtanen et al., 2019). The purpose of
the spline fit was to remove systematic variations which were unaffected by
the PDCSAP pipeline, in particular, variations that result in extremely non-
sinusoidal low-frequency signals in the final light curve. We chose a $2^{\rm
nd}$ order spline so any potential transits would not be removed. We then
stitched each quarters into a final combined light curve.

###  3.2 Removing pulsations

Figure 3: Example of light curve cleaning for the known planet hosting
$\delta$ Sct variable KIC 5202905. Left panel: the original flux (red) and
after cleaning away the pulsations (blue). Right panel: The amplitude spectrum
of the original flux. Over-plotted are the fitted frequencies – the opacity of
the points are proportional to their amplitude. Note that the planetary signal
is almost undetectable at this scale.

The cleaning routine is an iterative procedure that is widely used for
pulsating stars (also known as prewhitening, Lenz & Breger 2004). The method
fits and subtracts signals in the time domain of the form

$y(t)=A\cos{(\omega t+\phi)},$ (1)

where $A$, $\omega$, and $\phi$ are the amplitude, angular frequency, and
phase of the oscillation mode respectively. The routine selects the frequency
peak in the amplitude spectrum of highest amplitude at each step, calculates
an initial estimate of the amplitude and phase, and then fits Eq. 1 to the
light curve. The resulting fit is subtracted from the total flux for
subsequent iterations. The iteration is continued until either one of two
conditions is satisfied: there is no signal remaining above a signal-to-noise
ratio (SNR) of 5, or more than 100 signals have been removed. We calculated
the SNR by dividing the amplitude spectrum by an estimate of the background
noise, calculated through a moving median filter. The pipeline only cleans
between the frequencies of 1 to 47 d-1. The upper limit of 47 d-1 was chosen
as it is approximately $f_{s}-1$, where $f_{s}$ is the sampling frequency of
the _Kepler_ LC data. Murphy et al. (2019) found that 17.9% of $\delta$ Sct
stars have their strongest pulsation frequency above the Nyquist limit
($0.5f_{s}$), and so we consider our chosen limit to be acceptable. Even if
the pipeline were to clean an aliased signal, the true signal would also be
removed in the process. We note that calculation of the amplitude spectrum is
the most computationally intensive part of the pipeline, so our frequency
choice is a trade-off between accuracy and speed. We discuss low-frequency
signals in the next section.

All of the parameters of Eq. 1 can be well determined and used as starting
points before fitting. Prior to fitting to Eq. 1, we calculated an initial
estimate of the frequency, amplitude, and phase. The frequency of maximum
amplitude was calculated from a three-point parabolic interpolation routine
applied to the periodogram. The amplitude was then calculated from the
periodogram at this single frequency estimate. Finally, the phase was obtained
from the real ($\Re$) and imaginary ($\Im$) components of the Discrete Fourier
Transform ($z$) at the given frequency,

$\phi=\tan{\Big{(}\frac{\Im{z}}{\Re{z}}\Big{)}}^{-1}.$ (2)

To fit Eq. 1 we used a non-linear least squares algorithm as implemented in
Scipy (Virtanen et al., 2019). We calculated the gradient function of Eq. 1 as
a faster alternative to numerical estimates required by the solver,

$\displaystyle\nabla y=\left(\begin{array}[]{c}-At\sin{(\omega t+\phi)}\\\
\cos{(\omega t+\phi)}\\\ -A\sin{(\omega t+\phi)}\end{array}\right).$ (6)

We note that in most cases, actual optimization of the parameters is
unnecessary. Indeed, the initial estimate of the parameters is usually
sufficiently close to the true value to the point that optimization completes
within a few iterations, and the optimized value is functionally identical to
the initial guess. Despite this, the optimization is cheap to perform (<1 s
per pulsation mode) and thus included for the sake of completeness.

The iterative aspect of the cleaning is key to the light curve processing. It
is not feasible to select, fit, and subtract all oscillations lying above a
given SNR simultaneously from the light curve, because most $\delta$ Sct stars
exhibit amplitude modulation (Bowman et al., 2016). Amplitude modulation
causes the mode amplitude to vary over the the length of the _Kepler_ data. As
a result, it sometimes takes several or more iterations before a single
oscillation mode is subtracted. We show pre-whitening results for the known
$\delta$ Sct planet host, KIC 5202905 in Fig. 3. Note that the cleaning halts
when it reaches the low-frequency regime, where the transit appears.

###  3.3 Dealing with low-frequency variability

A major difficulty in removing pulsations from $\delta$ Sct light curves is
that they commonly exhibit low-frequency variations, caused either by spot-
based rotational modulation or low frequency gravity (g) modes (Breger, 2011;
Van Reeth et al., 2015, 2018). Attempting to clean at low frequencies requires
the user to make a choice between whether the observed signal is intrinsic to
the star or instead caused by a transit. Here, we devise a simple method for
differentiating between transits and intrinsic stellar variability through the
use of the Bayesian Information Criterion (BIC; Schwarz, 1978; Neath &
Cavanaugh, 2012). The BIC is a type of model selection that calculates a
quantitative value that can be used to determine which model best represents
the data:

${\rm BIC}=k\ln{n}-2\ln{\hat{L}},$ (7)

where $k$ is the number of parameters in the model, $n$ is the number of
samples, and $\hat{L}$ is the likelihood of the model calculated at the
maximum a posteriori (MAP). That is, the value of the likelihood at the
optimized model parameter values.

To implement the BIC in our search, we modelled intrinsic variability
(pulsation or rotation) following Eq. 1 and modelled transits using a Box
Least Squares (BLS) model. For each signal below 1 d-1 with a SNR greater than
5, we followed the iterative cleaning procedure prescribed above. However, at
each step of cleaning, the transit model was also fitted to the observed
signal and the BIC was calculated. If the BIC favored the pulsation model then
the signal was removed and the process repeated. On the other hand, if instead
the BIC favored the transit model, then the procedure was halted and the
pipeline moved onto the next star.

To test the efficiency of the BIC as a discriminator between transits and
intrinsic variability, we ran it on 1000 randomly selected light curves from
our _Kepler_ sample. For each light curve, we injected a transit calculated
using the batman Python package (Kreidberg, 2015), with a random orbital
period between 0.5 and 50 d, and planetary radius from 0.01 to 0.5 R⊙, both
drawn from uniform distributions. For each signal to be cleaned, we calculated
the BIC for both a transit and pulsation model fit. We recorded the true
positive, true negative, false positive, and false negative rate. For example,
if the algorithm identified the signal as a transit at the correct orbital
period (within a tolerance of 0.001 d), it counted one true positive.
Similarly, if the algorithm identified the signal as a transit and it was
actually a pulsation, we counted one false positive.

Figure 4: Confusion matrix of injection testing using the BIC to differentiate
between transits and intrinsic variability in a light curve. The BIC has been
normalized to 1.

We show the results of our injection testing as a confusion matrix in Fig. 4.
We note that the algorithm is heavily biased towards classifying variability
as intrinsic – because the cleaning will continue if a pulsation is observed,
whereas the cleaning stops if a transit is detected. As a result, there are
many more instances of pulsation true positives than transits in the
simulation. In utilizing the BIC to clean only pulsations, we seek to maximize
our recall score at the cost of precision, so we do not accidentally mis-
classify a transit as a pulsation. We see from our results that our recall is
effectively maximized: the number of transits accidentally classified as
pulsations is around 7%, although 23% of pulsations were mis-classified as
transits.

###  3.4 Transit search

Once the pulsations and intrinsic variability were removed, we then searched
each light curve for transiting planets. For each quarter of _Kepler_ data, we
removed variability on timescales significantly longer than transit durations
with the Savitsky-Golay filter implemented in the lightkurve package
(Barentsen et al., 2019). We filtered using a second-order polynomial and a
window length of 201 cadences (4.1 days), and we treated independently any
light curve segments separated by a gap larger than 11 cadences (5.5 hours),
typically the result of monthly data downlinks. We then re-stitched the light
curve into a single data set.

We searched for transiting planets with the transitleastsquares (TLS)
algorithm of Hippke & Heller (2019). TLS broadly follows the approach of the
Box Least Squares (BLS) algorithm (Kovács et al., 2002), updated to search for
periodic signals with realistic transit shapes and appropriate limb-darkening
rather than a simple inverse tophat. We restricted our search to periods
between 0.5 and 50.0 days; 77% of all _Kepler_ objects of interest fall in
that period range. Given the large radii of the stars in our sample, we expect
planets with periods shorter than 0.5 days to be quickly tidally engulfed
(e.g. Li et al., 2014; Patra et al., 2017) and longer period transiting
planets to be below the threshold for detectability. We applied an
oversampling factor of 0.2, which results in a search over 10,655 periods over
this range. We also performed a complementary search using the Box Least
Squares periodogram, with the orbital period restricted instead to between 0.5
and 100 days.

For each system, we recorded the signal detection efficiency as a function of
period, the binned and un-binned phase-folded flux at the time of the most
significant signal and offset by a phase of 0.1, and summary statistics on the
best-fitting orbital period, transit duration, and transit depth. We then
visually inspected each of these outputs for all systems to identify
candidates.

To quantify the efficiency of our pulsation removal and transit detection
pipeline, we performed a series of basic injection-and-recovery tests. We
generated a series of synthetic transiting planets using the batman package
(Kreidberg, 2015) by varying their radii from 0.1 to 5 Jup at a fixed orbital
period of 10 days. These were then injected into the original PDCSAP light
curves. For simplicity, we set the other transit parameters to fixed values.
We assumed circular orbits ($e=0$), and a radius for the host star of
$1.7~{}R_{\odot}$, a typical value for an A/F-type star.

After running the injected light curves through our pipeline, we recovered
100% of planets larger than around 0.5 RJup, around 80% with 0.36 RJup, and
50% at 0.25 RJup. We only detected the smallest radius (1 RJup) planet in one
of our systems. These tests give us confidence that our method is sensitive
to, at the least, the 0.5 RJup planets.

##  4 Results

KIC | Cadence | Transit period | Transit $T_{0}$ | Transit duration | Radius | a/R∗ | $\frac{\rho_{\rm implied}}{\rho_{\rm calculated}}$
---|---|---|---|---|---|---|---
|  | (d) | (d) | (d) | (R⊕) |  |
New candidates |  |  |  |  |  |  |
7767699 | LC | $1.12828^{+0.00003}_{-0.00003}$ | $121.66^{+0.02}_{-0.02}$ | $0.16^{+0.02}_{-0.02}$ | $0.56^{+0.09}_{-0.07}$ | $0.98^{+0.2}_{-0.2}$ | $0.53$
8249829 | LC | $1.0127^{+0.0001}_{-0.0001}$ | $1472.679^{+0.006}_{-0.009}$ | $0.090^{+0.009}_{-0.008}$ | $2.8^{+0.4}_{-0.3}$ | $1.6^{+0.3}_{-0.5}$ | $2.6$
9895543 | LC | $1.21889^{+0.00002}_{-0.00002}$ | $121.42^{+0.02}_{-0.01}$ | $0.15^{+0.02}_{-0.02}$ | $1.0^{+0.1}_{-0.1}$ | $1.1^{+0.2}_{-0.3}$ | $0.81$
KOI planet candidates |  |  |  |  |  |  |
with $\delta$ Sct pulsations |  |  |  |  |  |  |
3964109 | SC | $21.41624^{+0.00007}_{-0.00007}$ | $176.250^{+0.003}_{-0.003}$ | $0.295^{+0.004}_{-0.004}$ | $2.1^{+0.4}_{-0.2}$ | $23.4^{+0.6}_{-0.6}$ | $0.86$
9111849 | LC | $63.0732^{+0.0003}_{-0.0003}$ | $191.253^{+0.003}_{-0.004}$ | $0.225^{+0.005}_{-0.005}$ | $17.8^{+4.5}_{-8.9}$ | $62.8^{+25.2}_{-25.2}$ | $3.77$
9845898 | LC | $45.1262^{+0.0005}_{-0.0005}$ | $165.421^{+0.009}_{-0.01}$ | $0.28^{+0.01}_{-0.01}$ | $2.3^{+0.8}_{-0.3}$ | $46.2^{+17.0}_{-17.0}$ | $0.96$
5202905b | LC | $22.8245^{+0.0002}_{-0.0002}$ | $144.468^{+0.008}_{-0.009}$ | $0.28^{+0.01}_{-0.01}$ | $3.4^{+0.6}_{-0.7}$ | $25.8^{+4.9}_{-4.9}$ | $0.94$
5202905c | LC | $14.8445^{+0.0002}_{-0.0002}$ | $143.57^{+0.01}_{-0.01}$ | $0.25^{+0.02}_{-0.02}$ | $2.5^{+0.4}_{-0.5}$ | $18.7^{+7.2}_{-7.2}$ | $0.84$
6116172b | LC | $69.64679$ | $138.28306$ | $0.04^{+0.02}_{-0.02}$ | $2121^{+1290}_{-467}$ | $799^{+232}_{-232}$ | $4968$
6116172c | LC | $111.6555^{+0.0007}_{-0.0007}$ | $148.894^{+0.005}_{-0.005}$ | $0.04^{+0.02}_{-0.02}$ | $2121^{+1290}_{-467}$ | $799^{+232}_{-232}$ | $4968$
6116172d | LC | $86.13202$ | $164.22370$ | $0.04^{+0.02}_{-0.02}$ | $2121^{+1290}_{-467}$ | $799^{+232}_{-232}$ | $4968$
11013201b | SC | $13.11896^{+0.00001}_{-0.00001}$ | $261.5394^{+0.0003}_{-0.000}$ | $0.1794^{+0.0005}_{-0.0005}$ | $8.3^{+0.5}_{-2.5}$ | $23.69^{+0.04}_{-0.04}$ | $2.51$
11013201c | SC | $7.82187^{+0.00002}_{-0.00002}$ | $123.499^{+0.003}_{-0.003}$ | $0.160^{+0.003}_{-0.003}$ | $2.8^{+0.2}_{-0.8}$ | $15.7^{+1}_{-1}$ | $2.04$
6032730 | SC | $96.509^{+0.001}_{-0.001}$ | $223.808^{+0.008}_{-0.008}$ | $0.02^{+0.01}_{-0.01}$ | $860^{+222}_{-274}$ | $639^{+283}_{-283}$ | $1514$
5617259 | LC | $1.04813$ | $132.35280$ | $0.23628$ | $4.5^{+0.4}_{-2.4}$ | $1.53664$ | $0.42$
9775385 | SC | $45.163^{+0.005}_{-0.005}$ | $159.9^{+0.1}_{-0.1}$ | $0.30^{+0.09}_{-0.09}$ | $2.4^{+1.2}_{-0.4}$ | $48.6^{+18.6}_{-18.6}$ | $1.97$
6670742 | SC | $72.282^{+0.001}_{-0.001}$ | $156.75^{+0.02}_{-0.02}$ | $0.10^{+0.03}_{-0.03}$ | $13.1^{+2.0}_{-4.6}$ | $224.0^{+111.0}_{-111.0}$ | $211$
9289704 | LC | $34.136^{+0.002}_{-0.002}$ | $157.80^{+0.05}_{-0.05}$ | $0.38^{+0.08}_{-0.08}$ | $14.0^{+4.7}_{-8.8}$ | $29.1^{+10.0}_{-10.0}$ | $0.68$
3965201 | LC | $44.114^{+0.002}_{-0.002}$ | $164.83^{+0.03}_{-0.02}$ | $0.16^{+0.04}_{-0.04}$ | $4.9^{+0.9}_{-1.6}$ | $86.3^{+3.3}_{-3.3}$ | $56$
11180361 | SC | $0.53306$ | $171.22744$ | $0.07759$ | $133.6^{+26.0}_{-78.0}$ | $1.45285$ | $2.13$
Table 1: Properties of the systems investigated in this paper. IDs in italics
mean that the system was also found in our prewhitening transit search, and
IDs in bold mean the system is confirmed. For the new candidates,
uncertainties were obtained from the MCMC analysis performed in Section 4. The
uncertainties for the KOI planet candidates were taken from the KOI catalog.
Some systems in the KOI did not have uncertainties and thus they are left
blank. $b$, $c$, and $d$ designations in the ID denote identification for
multiple systems.

We identified 32 possible transiting exoplanet candidates from our cleaned
light curves. We cross-matched this sample against the KOI catalogue and found
that 13 are listed as False Positives, two are Confirmed, two are Candidates,
and the remaining 16 are not in the catalogue.

For some of our candidate transits, the dips did not appear as a typical
transit-like profile (with a U-shaped transit due to limb-darkening). However,
as transiting systems exist whose geometry does not generate flat minima in
the light curves, we decided to retain them in our sample. A number of effects
can change the transit shape, e.g. gravity darkening of a rapidly rotating,
oblate host star can cause asymmetric transit shapes (Barnes, 2009). We
caution that some of these events may not be real transits, but could rather
be due to imperfectly removed pulsations mimicking a transit-like light-curve
shape. As such, we do not claim that the candidates here show true transits,
but rather that they are worthy of more in-depth study.

Figure 5: Contamination for KIC 5565497, a known False Positive from the
Kepler Objects of Interest catalog. Top: the target pixel file with the
locations of the center of pipeline aperture and transit source marked. The
transit is several arcseconds away from the center of the pipeline aperture
and is likely to originate from a nearby eclipsing binary. Bottom: Light
curves created on the pipeline (blue) and transit (red). The pipeline aperture
light curve is constant.

We checked each candidate for obvious signs of blending and nonphysical
scenarios, using the contaminante software
222https://github.com/christinahedges/contaminante, which models the source of
the transit signal in relation to the center of the aperture pipeline in the
target pixel file (TPF). Stars with deviations exceeding one arcsecond between
the center of transit and aperture on the TPF are likely to be visual blends
caused by a background eclipsing binary. We show an example of an obvious
background eclipsing binary system, KIC 5565497, in Fig. 5. We note that for
15 of our stars, we were not able to isolate the transit on the TPF. This was
due to the fact that most of the transits are hidden behind the pulsations,
and so can not be accurately modeled until the pulsations are removed, which
was not done on a TPF aperture.

We also checked whether the density of the star implied from the transit
agreed with independent calculations from our sample. From Seager & Mallén-
Ornelas (2003); Sandford & Kipping (2017), it is known that the stellar
density ($\rho_{*}$) can be measured from the transit period ($P$) and
duration ($T$),

$\rho_{*}=\frac{3P}{\pi^{2}GT^{3}},$ (8)

where $G$ is the gravitational constant. We do not exclude candidates from our
sample if the implied stellar density is significantly different from our
calculations, because those calculations assume a perfect sphere, which is not
the case for the typically oblate $\delta$ Sct variables. Likewise, there
remains a possibility of the transit occurring on another nearby star in a
binary system, where the $\delta$ Sct pulsations are contaminating the light
curve. Regardless, we calculate the implied stellar density for all of our
stars and used them to inform our vetting process.

Figure 6: Transit fits for the 3 candidates in our sample. We plot the flux in
black points, the binned flux in red points, and plot the median and standard
distribution of the posterior light curve fit as a blue line.

For each candidate that was not listed as a KOI, we fitted a transit model to
the light curve using the exoplanet Python package, written on top of PyMC3
(Foreman-Mackey et al., 2021). This model consisted of both a standard transit
model and a Gaussian Process noise model with a simple harmonic oscillator
kernel. For the orbit, we parametrized over duration and assumed a circular
orbit, after preliminary models indicated that the eccentricity is poorly
constrained.

###  4.1 KOI $\delta$ Sct variables

In addition to our candidate sample of known $\delta$ Sct stars, we also
searched for $\delta$ Sct oscillations in all KOIs with a temperature between
6000 and 10,000 K, roughly corresponding to the classical instability strip
for $\delta$ Sct pulsators. We considered both ‘confirmed’ and ‘candidate’
planet designations. $\delta$ Sct stars were then identified by calculating
the skewness of the amplitude spectrum above 10 d-1, following Murphy et al.
(2019) and Bedding et al. (2020). Stars with a skewness less than unity are
typically non-pulsating and were removed. The remaining stars were visually
inspected to identify $\delta$ Sct pulsations. We note that this sample does
not include the single planet discovered around a $\delta$ Sct variable by
pulsation timing (Murphy et al., 2016), because the KOI catalog only lists
transiting planets. We summarize the key results of the fitting and KOI
$\delta$ Sct variables found in Table 1. We report the observing cadence, the
transit fit parameters; the orbital period, time of transit ($T_{0}$), the
duration of the transit, the fitted radius of the planet, the ratio of the
semi-major axis to the stellar radius (a/R∗), and the ratio of implied to
calculated stellar densities from Eq. 8. We use the KOI values for stars that
were already known.

In our search for $\delta$ Sct variables in the KOI, we found 13 stars that
show both transits and $\delta$ Sct pulsations out of a total 1171 objects of
interest which we list in Table 1. Ideally, all of these systems would have
appeared in our cleaning transit search. However, only four systems from the
KOI were found in our transit search. We attribute this to two causes: our
$\delta$ Sct sample is incomplete, and the transit search algorithm is not
100% effective. The largest contributor to the $\delta$ Sct sample was based
on a 6,500 - 10,000 K temperature cut (Murphy et al., 2019), whereas the KOI
search was performed between 6,000 - 10,000 K. This excluded three of the KOI
sample from our $\delta$ Sct sample: KIC 3964109, KIC 9111849, and KIC
11013201. Additionally, our search was only performed on the Kepler LC data.
From Table 1, seven of the 13 stars were observed in LC mode. Of those seven,
six were in our $\delta$ Sct planet search sample, and we recovered four
transit events. The two systems that we did not identify in our search were
KIC 5617529 and KIC 6116172. Although KIC 6116172 is a candidate for a 3
planet system, it is not surprising that it did not appear in our search
because strongest transit signal in the light curve is at an orbital period of
111 days, which exceeds the limits of our search (1 to 100 days). The
remaining sample in the KOI were all observed in SC mode, which our $\delta$
Sct sample did not use.

###  4.2 Candidates

From our 16 new candidates from the transit search, only three transit models
converged, of which only KIC 8249289 shows the typical flat-bottom minimum
expected of a transit. The other two show an extremely long duration and
either a shallow transit, or a deep V-shaped eclipse, implying that the
transit geometry is grazing and/or it is a background EB (Fig. 6). We are
unable to confirm or rule out these candidates from their transit shapes
alone, and we caution that it is likely they are all false positives. This is
supported by the fact that the binned flux outside the transit appears to vary
sinusoidally, which is indicative of an incorrectly removed pulsation or data
processing error. We inspected the amplitude spectrum of these three
candidates and found no significant pulsation structure around the orbital
frequency. In saying that however, the shallow, long duration transits in KIC
9895543 and KIC 7767699 are not unusual: the candidate planetary system KIC
6116172 from the KOI also shows a similar transit profile. For all three
candidates, although they lie close to the boundary of their host star
(a/R${}_{*}\sim$1), their implied stellar density from the transit duration is
somewhat close to the calculated value.

As a final test of the candidates, we performed ephemeris matching by cross-
matching our candidates against the entire KOI catalog (Coughlin et al.,
2014). We checked the period and time of transit ($T_{0}$), and found no
matching objects within a period tolerance of 0.01 days and $T_{0}$ tolerance
of 1 day. We note however that the radii of the candidate planets are much
lower than our expected limits of recovery from the injection tests.

###  4.3 False positives

KIC | Reason
---|---
4380834 | a/R∗ << 1
5724523 | a/R∗ << 1
9471419 | a/R∗ << 1
9655419 | Did not converge
8912730 | Did not converge
7779942 | Did not converge
7618364 | Did not converge
5702637 | Did not converge
5565497 | Did not converge
5302643 | Did not converge
4952341 | Did not converge
4035667 | Did not converge
3444098 | Did not converge
2167444 | Did not converge
11607193 | Did not converge
Table 2: Table of false positives identified in the search and their reason
for exclusion.

Our transit search yielded 28 false positives. As mentioned, 13 of them are
previously known false positives from the KOI catalog, and the remaining 15
were either found to be contaminated, or the transit fit failed to converge.
We summarize the new false positives and their reason for exclusion in Table
2. Of these false positives, KIC 4380834, KIC 5724523 and KIC 9471419 have a
nonphysical ratio between their semi-major axis and radius of their host star
(a/R∗ << 1). Their values imply an orbital position within the envelope of
their $\delta$ Sct host, and thus, were downgraded to false positives.

##  5 Pulsation timing variations

Hot stars, which include $\delta$ Sct stars, are not amenable to radial
velocity surveys for planetary sized companions because of their rapid
rotation. However, their highly coherent pulsations allow for the study of
timing variations in a manner similar to eclipse timing variations and pulsar
timing (Murphy et al., 2014; Murphy & Shibahashi, 2015). In a multiple system,
the pulsating stars’ orbit around the barycenter leads to a change in path
length for starlight traveling to Earth. If at least one star is pulsating,
the path length changes manifest as variations in the light arrival time of
the pulsations. The minimum detectable mass of a companion found through
pulsation timing relies on the signal-to-noise ratio (SNR) of the pulsation,
and its coherency over the orbital phase. These values are independent of the
rotation of the host star. Previously, it has been shown that pulsation timing
can be applied to _Kepler_ objects of interest to recover a signal (Balona,
2014). However, information about the system can still be gained even when no
signal is observed. That is, the upper limit of the mass of the companion can
be found.

For an exoplanet of sufficient mass, mutual gravitation between the planet and
its pulsating host star leads to variations in the arrival times of the
stellar pulsations. These variations manifest as time-dependent phase changes
in the pulsations modes. Orbital phase changes are distinguished from
frequency variations intrinsic to the star by the fact that they must affect
all pulsation modes equally.

The sensitivity of pulsation timing depends on the quality of the celestial
‘clock’ used (Compton et al., 2016). The milli-second level precision of
pulsars has been used to great success for identifying exoplanets through
their light travel time variations (Wolszczan & Frail, 1992; Sigurdsson et
al., 2003; Suleymanova & Rodin, 2014; Starovoit & Rodin, 2017). On the other
end of the scale, applications of pulsation timing towards variable stars
oscillating at lower frequencies, such as the A/F-type $\delta$ Scuti stars,
has been highly successful in identifying binary systems (Murphy et al., 2018,
2020). The sensitivity of pulsation timing is proportional to the oscillation
frequency, and, as a result, only one planet has successfully been detected
using pulsation timing with $\delta$ Sct stars (Murphy et al., 2016). We
recently introduced a new approach which simultaneously fits the phase
variations across the orbit of a star using a forward model (Hey et al.,
2020). This method was shown to be more sensitive and robust than previous
approaches for identifying planets with pulsation timing.

The orbital period obtained from a transiting exoplanet can be used to
constrain a model of pulsation timing variations. We can use the pulsation
timing variations as a proxy for radial velocities of exoplanets transiting
$\delta$ Sct stars. Even if no timing variation is observed in the pulsations,
the absence of signal can be used to place an upper limit on the mass of the
companion. We performed this analysis on all 13 KOIs with $\delta$ Scuti
pulsations, and our three new candidates to constrain the planetary mass
function. To the best of our knowledge, this is the first attempt to vet
transiting exoplanet candidates from timing of their host star oscillations.

Prior | Description
---|---
Pulsations |
$\omega_{j}$: $\mathcal{N}\sim(P,5)$ | Angular frequency of oscillations
$A_{j}$: $\mathcal{N}\sim(P,5)$ | Amplitude of oscillation modes
$\phi_{j}$: $\mathcal{N}\sim(P,5)$ | Phase of oscillation modes
Orbit model |
$P_{\rm orb}$: $\mathcal{N}\sim(P^{*},P_{e}^{*})$ | Orbital period (days)
$\rm{a_{j}}\sin{i}/c$: $\mathcal{F}$ | Projected semi-major axis (s)
$\varpi$: $\mathcal{U}\sim(0,2\pi)$ | Ascending node and periapsis angle
$\phi_{p}$: $\mathcal{U}\sim(0,2\pi)$ | Phase of periapsis passage
Table 3: Priors used in the model. Note that $\omega_{j}$, $A_{j}$ and
$\phi_{j}$ are defined as a vector of length equal to the number of
oscillation modes used in the model. A star on the value indicates that it has
been obtained from transit information.

We briefly describe the theory of pulsation timing below. For a full
explanation, see Murphy et al. (2014); Hey et al. (2020). The observed
relative flux $y(t)$ of a pulsating star with $J$ modes undergoing mutual
gravitation with a companion around a common barycenter varies according to

$y(t)=\sum_{j=1}^{J}A_{j}\cos(\omega_{j}[t-\tau]+\phi_{j}),$ (9)

where $A_{j}$ is the amplitude of mode $j$, $\omega_{j}=2\pi\nu_{j}$ is its
angular frequency, and $\phi_{j}$ is its phase. $\tau$ acts as an additional
phase term, and represents the time delay – the projected light arrival time
of the pulsations from the star – and is described by the same orbital
elements as radial velocities:

$\tau=-\frac{a\sin{i}}{c}\frac{1-e^{2}}{1+e\cos{f}}\sin{(f+\varpi)},$ (10)

in units of seconds, corresponding to a change in path length measured in
light-seconds. We follow the conventions that $a_{1}\sin i$ denotes the
projected semi-major axis of the primary, $e$ is the eccentricity, $\varpi$ is
the angle between the ascending node and the periapsis, and $c$ is the speed
of light. These conventions are chosen so that when the star is on the far
side of the barycenter with respect to Earth, the time delay will be at its
(positive) maximum.

In previous papers, the time-delay has been modeled for binary systems where
the orbital period was not known a priori (Murphy et al., 2020; Hey et al.,
2020). In those cases, the orbital period was obtained by either subdividing
the light curve into equal segments or by ‘brute-forcing’ a model over a grid
of orbital periods to observe where the model likelihood peaks. For the stars
in this paper, however, the orbital period is extremely well constrained from
the observed transit signal. This allows us to fix the pulsation timing model
at the exact orbital period and fit the remaining parameters.

Figure 7: Results of pulsation timing on the stars in our sample. The dashed black line indicates the expected sensitivity for a typical _Kepler_ $\delta$ Sct (Hey et al., 2020). The gray shaded region denotes the area of brown dwarf masses. KIC ID | KOI | a${}_{1}\sin{i}/c$ | Mass limit | $K_{1,\rm{min}}$
---|---|---|---|---
|  | (s) | ($M_{\rm Jup}$) | (km/s)
Candidates |  |  |  |
7767699 |  | $30$ | $4300$ | $580$
8249829 |  | $4.7$ | $670$ | $100$
9895543 |  | $6.9$ | $920$ | $120$
Known |  |  |  |
3964109 | K00393.01 | $150$ | $2200$ | $150$
9111849 | K02042.01 | $65$ | $790$ | $23$
9845898 | K02423.01 | $33$ | $310$ | $16$
5202905b | K01932.01 | $2.8$ | $48$ | $2.7$
5202905c | K01932.02 | $4.7$ | $110$ | $6.9$
6116172b | K06142.01 | $6.0$ | $44$ | $1.5$
6116172c | K06142.02 | $5.7$ | $35$ | $1.1$
6116172d | K06142.03 | $6.0$ | $44$ | $1.5$
11013201b | K00972.01 | $1.2$ | $37$ | $2.0$
11013201c | K00972.02 | $4.4$ | $190$ | $12$
6032730 | K06141.01 | $4.4$ | $33$ | $1.0$
5617259 | K07733.01 | $12$ | $1900$ | $250$
9775385 | K04462.01 | $10$ | $120$ | $5.0$
6670742 | K06753.01 | $14$ | $140$ | $4.3$
9289704 | K02222.01 | $6.0$ | $120$ | $3.9$
3965201 | K07546.01 | $14$ | $190$ | $6.9$
11180361 | K00971.01 | $320$ | $88000$ | $13000$
Table 4: Results of the pulsation timing model. Note that the values are
reported as the 95th percentile of the posterior distribution. A bold ID
indicates the planet is confirmed. For reference, 1 au/c$\sim$500 s.

We fitted each light curve with the model in Eq. 9, using a Hamiltonian Markov
Chain Monte Carlo No-U-Turn (NUTS) sampler implemented in PyMC3 (Salvatier et
al., 2016). The prior on $a\sin{i}$ was chosen to be a bounded flat prior,
ensuring that its value could not go negative. We fixed the eccentricity to be
0 for simplicity. The key aspect of this fit was that the orbital period was
fixed to its value determined from transit analysis. The remaining priors are
described in Table 3.

We ran the model for 2000 tuning steps and 2000 draw steps over 2 chains
simultaneously, resulting in a posterior size of 4000 points. We summarize the
results of the pulsation timing model for each system in Table 4, and plot the
maximum possible companion masses in Fig 7. We report the mass function ($f$),
calculated from the value of a$\sin{i}$:

$f(m_{1},m_{2}\sin{i})=\frac{4\pi^{2}c^{3}}{GP_{\rm
orb}^{2}}\Big{(}a\sin{i}\Big{)}^{3},$ (11)

where $c$ and $G$ are the speed of light and gravitational constant,
respectively. The RV semi-amplitude ($K_{1}$) follows from the mass function,

$K_{1}=(2\pi G)^{1/3}\sqrt{1-e^{2}}\Big{[}\frac{f(m_{1},m_{2}\sin{i})}{P_{\rm
orb}}\Big{]}^{1/3},$ (12)

which provides an estimate of the RV precision necessary to observe the
planetary companion, if it exists. We note that the posterior distribution of
nearly all of the stars for the a$\sin{i}$ and related mass quantities is
highly asymmetric. This is a result of the nature of the sampling: although
the distribution for most of the systems is centered around 0, the prior on
$a\sin{i}$ ensures that the mass cannot go negative. Thus, the constraint on
the mass is largely provided by how narrow the distribution is. We therefore
report the 95th percentile of each posterior distributions, which is the value
that encompass 2$\sigma$ of the posterior distribution.

For our sample, the pulsation timing model sets an upper limit on the
companion mass limit that ranges from 33 to 90,000 MJup. Most of these mass
limits are too large to be useful because of the short orbital periods, which
strongly limits the sensitivity of pulsation timing. However, most of the
longer period KOI candidates are very well constrained. Hey et al. (2020)
found that a typical $\delta$ Sct variable in the _Kepler_ LC data had a
pulsation timing precision of around 2 s, corresponding to the black dashed
line in Fig. 7. Stars that fall above this line are either of low SNR, or
potentially have stellar mass companions that would cause a non-zero signal.
Stars below this line are of higher SNR, and are thus provide better limits on
the mass of their companion. For the best case, KIC 6032370, the 95th
percentile of the mass limit is less than 33 $M_{\rm Jup}$, implying a
necessary RV precision of at least 1.0 km/s. Obtaining such precise RVs for a
$\delta$ Sct variable would be highly challenging, although significant work
has gone into ameliorating the effects of rotation and pulsations on RV
extraction in A/F stars (Galland et al., 2005, 2006b; Lagrange et al., 2009;
Desort et al., 2007).

A downside to this method is that it cannot decisively rule out candidates as
false positives. This is because some companion mass limits lie well above the
typical $\delta$ Sct sensitivity line, even though they are confirmed planets.
The reason for this is that the $\delta$ Sct oscillations are either not
coherent enough, or the SNR of the pulsations is too low (Compton et al.,
2016). Despite this, it is still useful in providing limits on possible future
surveys.

##  6 Discussion

Figure 8: HR diagram of the A/F stars (blue) with the confirmed (orange
crosses) and candidate (red stars) overlaid. The black line marks the zero age
main sequence for Fe/H = 0.1 dex. The top panel shows the combined (confirmed
& candidate) occurrence rate as a function of temperature.

Fig. 8 shows the occurrence rate of transiting planets among the hot A/F stars
(6500 K to 10000 K) by using the catalog of A/F stars from Murphy et al.
(2019), and planetary candidates from the KOI with temperatures and
luminosities provided from Berger et al. (2020).

We calculated the occurrence rate as a function of temperature in 500 K bins
by dividing the combined candidate and confirmed planetary systems by the
number of A/F stars in each temperature bin. With such a small sample size of
transiting planets above 7000 K, we cannot reliably provide any conclusions on
the occurrence rate. Indeed, for stars between 7500 and 8500 K there are few
confirmed planets, and only nine candidate planets between 8000 and 8500 K.
While we cannot draw conclusions from such a small sample size, we suggest
that the dearth of planetary candidates could be due to the fact that the
$\delta$ Sct occurrence rate peaks at around 8000 K (Murphy et al., 2019). If
so, there may still be a significant number of transits in the _Kepler_ data
hidden by the pulsations that our pipeline was not successful in identifying.

Note that we can only comment on the occurrence rate for transiting planets.
The KOI list does not contain candidate planets which have been detected by
other methods, and thus, they have been left out. The most obvious example of
this is the single planet discovered by pulsation timing around a $\delta$ Sct
variable: KIC 7917485 (Murphy et al., 2016).

If we assume that all of our candidates in Section 4 are false positives, then
we can estimate an upper limit for the occurrence rate ($\epsilon$) based on
our transiting pipeline. The number of candidates $n$ from $N$ observations
with a true occurrence rate $\epsilon$ is distributed according to the
binomial distribution

$B(n|N,\epsilon)=\frac{N!}{n!(N-n)!}\epsilon^{n}(1-\epsilon)^{N-n},$ (13)

$N$=2354 is the number of $\delta$ Sct variables in our sample, and $n$=0 is
the number of transiting systems. By using Bayes’ theorem, the posterior for
$\epsilon$ is given by the beta distribution:

$\epsilon=B(n+1,N+1).$ (14)

Evaluating this distribution at the 95th percentile gives an upper limit on
the occurrence rate of 0.13% for our pipeline, which was tested to have a 100%
recovery rate on planets larger than 0.5 RJup with orbital periods of 10 days,
falling to 50% at 0.25 RJup (see Sec. 3.4).

A detailed estimate of the number of transiting planets around A/F stars
expected in TESS is beyond the scope of this paper. However, we can make a
rough estimate based on the number of $\delta$ Sct variables in Kepler (2354)
and the number of late A/F-type stars between 6,500 and 10,000 K (12,135).
Dividing these numbers, we estimate the $\delta$ Sct occurrence rate to be
19.3%. TESS has observed 45,232 unique stars in the same temperature range up
to Sector 30. Thus, we expect TESS to already have light curves for
approximately 9000 $\delta$ Sct stars.

We used a Monte-Carlo approach to propagate uncertainties and estimate the
expected number of planet-hosting $\delta$ Sct stars by randomly drawing
50,000 samples from our posterior for $\epsilon$ (Eq. 14). We then evaluated
distribution for $n$ for each sample from $B(n|N,\epsilon)$, and obtained an
upper limit (95th percentile) of 12 planets. That is, given the values above
and the efficiency of our pipeline, we predict that at the 95th percentile
there are 12 detectable planets above 0.5 RJup in the current TESS 2-minute
cadence data. At the 50th percentile, there are 2 expected detectable planets.

Figure 9: Frequency (top) and amplitude (bottom) distributions for all
_Kepler_ $\delta$ Sct stars in our sample. The dashed black lines in the
frequency distribution are the modeled components of the simple Gaussian
Mixture Model, showing the two frequency regimes present in $\delta$ Sct
stars: high frequency p-modes and low-frequency g-modes. The gray region
denotes the typical range of orbital frequencies. In the amplitude
distribution, the median is marked in red. The top x-axis on both plots is the
non-log version of the x-axis.

Another outcome of our cleaning routine is the frequency, amplitudes, and
phases of almost every $\delta$ Sct pulsation down to 4 SNR in the _Kepler_
sample. This results in 84,352 values across all the stars in our sample,
which represents the most complete set of pulsation data for the $\delta$ Sct
stars ever compiled. In the context of exoplanet analysis, this allows us to
explore the parameter space for transits. We show in Fig. 9 the complete
frequency and amplitude distribution of all stars in our sample. The frequency
distribution is clearly split into two distinct distributions. The low-
frequency region is dominated by g-modes, with some contamination from
rotational variability. The high-frequency region is the $\delta$ Sct p-modes.

We modeled both frequency distributions as a simple Gaussian Mixture Model
(GMM) assuming two mixed distributions. Overlaid on Fig. 9 is the frequency
range of planetary transits, which overlaps with the peak of g-mode
pulsations. This is why it is useful to have a metric to distinguish low-
frequency intrinsic signals from transit signals. In our case, the BIC worked
remarkably well.

The distribution of amplitudes peaks at a median value of 166 ppm. Although
most of the high amplitudes in this distribution are due to p-mode pulsations
outside the frequency range of transit signals, the transit will nevertheless
be “hidden” beneath such a high noise floor. We advise that all such high-
frequency signals be removed prior to searching for transits in the $\delta$
Sct variables.

Figure 10: Folded light curves of KIC 9289704 for the cleaned light curves
created in this work (orange), and those made by the Kepler validation
pipeline (blue). The dashed lines show the binned flux of the transit.

Finally, we performed a quick comparison of our cleaned light curves against
the automated Kepler planet search pipeline (Jenkins et al., 2017; Twicken et
al., 2018). We selected one of our low-SNR candidates that is listed as a KOI
and found by our pipeline, KIC 9289704, and calculated the combined
differential photometric precision (CDPP) for our cleaned light curve, and the
de-trended Kepler pipeline light curve used in the planet validation reports.
The CDPP is a metric that estimates the amount of remaining scatter after all
long term trends have been removed. We calculated the CDPP using Lightkurve,
at a fixed transit duration of 6.5 hours, and found that our pipeline
performed somewhat better with a CDPP of 163.9 ppm, compared to the Kepler
CDPP of 213.1 ppm (Fig. 10). In the Kepler pipeline, the light curves are
corrected by process termed ‘whitening’, whereby a wavelet-based matched
filter is used to correct over a range of band passes. This has the effect of
removing pulsations but does so in a less targeted way than our approach,
hence the difference in performance.

##  7 Conclusion

In this paper, we searched for transit events among the $\delta$ Sct variables
by subtracting their pulsations through an automated routine. We identified 3
possible new candidates, but warn that they are likely false positives, with
shallow transits that will be difficult to confirm photometrically without
Kepler. We also identified 13 additional Kepler Objects of Interest which show
$\delta$ Sct pulsations, and used these pulsations to constrain the mass of
the planetary companion.

Despite the dearth of planetary candidates found around the A/F stars, we note
that the pipeline we presented is applicable to all space-based photometric
missions. Both TESS and PLATO are expected to yield a massive amount of
$\delta$ Sct variables across the sky. Because of this, techniques for dealing
with coherent pulsations in light curves will be fundamental for searching for
planets around the hot variable stars.

We thank the anonymous referee for their careful review, which greatly
improved the quality of this manuscript. DRH gratefully acknowledges the
support of the AGRTP scholarship. We also acknowledge support from the
Australian Research Council (DP210103119) and from the Danish National
Research Foundation (Grant DNRF106) through its funding for the Stellar
Astrophysics Center (SAC). We acknowledge the traditional owners of the lands
on which the University of Sydney is located, the Gadigal people of the Eora
Nation, and we pay our respect to the knowledge embedded forever within the
Aboriginal Custodianship of Country. BJSP would like to acknowledge the
traditional owners of the land on which the University of Queensland is
situated, the Turrbal and Jagera people. We pay respects to their Ancestors
and descendants, who continue cultural and spiritual connections to Country.
This paper includes data collected by the _Kepler_ mission and obtained from
the MAST data archive at the Space Telescope Science Institute (STScI).
Funding for the _Kepler_ mission is provided by the NASA Science Mission
Directorate. STScI is operated by the Association of Universities for Research
in Astronomy, Inc., under NASA contract NAS 5–26555. This research has made
use of the NASA Exoplanet Archive, which is operated by the California
Institute of Technology, under contract with the National Aeronautics and
Space Administration under the Exoplanet Exploration Program.

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