# Following up TESS Single Transits With Archival Photometry and Radial
Velocities

Xinyu Yao Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015, USA  Joshua Pepper Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015, USA  B. Scott Gaudi Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA  Paul A. Dalba NSF Astronomy & Astrophysics Postdoctoral Fellow  Department of Earth & Planetary Sciences, University of California Riverside, 900 University Ave, Riverside, CA 92521, USA  Jennifer A. Burt Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena CA 91109, USA  Robert A. Wittenmyer University of Southern Queensland, Centre for Astrophysics, West Street, Toowoomba, QLD 4350, Australia  Diana Dragomir Department of Physics and Astronomy, University of New Mexico, 1919 Lomas Blvd NE, Albuquerque, NM 87131, USA  Joseph E. Rodriguez Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA  Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Steven Villanueva, Jr. Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Daniel J. Stevens Eberly Fellow  Center for Exoplanets and Habitable Worlds, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA  Department of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey Lab, University Park, PA 16802, USA  Keivan G. Stassun Vanderbilt University, Department of Physics & Astronomy, 6301 Stevenson Center Lane, Nashville, TN 37235, USA  David J. James ASTRAVEO LLC, PO Box 1668, MA 01931

###### Abstract

NASA’s Transiting Exoplanet Survey Satellite (TESS) mission is expected to
discover hundreds of planets via single transits first identified in their
light curves. Determining the orbital period of these single transit
candidates typically requires a significant amount of follow-up work to
observe a second transit or measure a radial velocity orbit. In Yao et al.
(2019), we developed simulations that demonstrated the ability to use archival
photometric data in combination with TESS to “precover” the orbital period for
these candidates with a precision of several minutes, assuming circular
orbits. In this work, we incorporate updated models for TESS single transits,
allowing for eccentric orbits, along with an updated methodology to improve
the reliability of the results. Additionally, we explore how radial velocity
(RV) observations can be used to follow up single transit events, using
strategies distinct from those employed when the orbital period is known. We
find that the use of an estimated period based on a circular orbit to schedule
reconnaissance RV observations can efficiently distinguish eclipsing binaries
from planets. For candidates that pass reconnaissance RV observations, we
simulate RV monitoring campaigns that enable one to obtain an approximate
orbital solution. We find this method can regularly determine the orbital
periods for planets more massive than $0.5M_{\rm J}$ with orbital periods as
long as 100 days.

planets and satellites: detection — planets and satellites: general — methods:
data analysis

##  1 Introduction

Follow-up observations of transiting exoplanets generally require precise
ephemerides. Any attempt to conduct transmission or emission spectroscopy,
whether from the ground or from space, requires the ability to accurately
predict future transits so as not to waste valuable telescope time or miss
part of the event. When candidate transiting planets have large uncertainties
in their ephemerides, it becomes difficult to schedule and thus successfully
perform follow-up observations. That is most often the case for long-period
transit candidates, which we define here as candidates with orbital periods
comparable to or greater than the time baseline of their originating transit
survey. This situation was vividly demonstrated by Benneke et al. (2017), who
found that the initially calculated ephemeris of the planet K2-18b ($P=33$
days with two transits shown in K2 data) was off by 2 hours before they
recovered the correct ephemeris using the Spitzer telescope. More recently,
Ikwut-Ukwa et al. (2020) combined observations from Kepler and TESS for known
K2 planets to reduce the uncertainty on future transit times from hours to
minutes through 2030. Ephemeris errors of order hours would preclude the
possibility of future observations from facilities such as the James Webb
Space Telescope (JWST).

Reliable ephemerides are also crucial to further dynamical studies of
transiting planets. Wang et al. (2015) found that half of the ten long period
exoplanets (periods between 430 days and 670 days) discovered by Kepler show
transit timing variations (TTVs) ranging from $\sim$2 to 40 hours. For planet
discoveries in which the initial ephemerides are poorly constrained,
additional transit observations might be needed to fix the ephemerides to
permit later TTV analysis (Dalba & Muirhead, 2016; Dalba & Tamburo, 2019).
Obtaining such ephemerides from archival data saves valuable observing time,
which can be helpful even in the cases of shorter-period planets that TESS is
likely to detect.

The Transiting Exoplanet Survey Satellite (TESS) is designed to detect
transiting planets orbiting bright stars across the whole sky. Launched in
April 2018, TESS has observed 26 sectors and has discovered $\sim$ 2000 planet
candidates so far. Among those, 51 have been confirmed and published. However,
90% of the TESS planet discoveries have orbital periods shorter than 20
days111https://exoplanetarchive.ipac.caltech.edu/ due to the short observing
time for most of the sky ($\sim 27$ days). Cooke et al. (2018) and Villanueva
et al. (2019) estimated that TESS will detect hundreds of planets with long
orbital periods via single transits in their TESS light curves.

In Yao et al. (2019) (henceforth Y19), we investigated the ability to recover
the ephemerides of TESS single transit candidates using archival data from the
Kilodegree Extremely Little Telescope (KELT) ground-based transit survey
(Pepper et al., 2007, 2012). The process of using archival data to detect a
signal originally revealed in later observations is sometimes called
“precovery”. In that work, we inserted simulated transit signals into KELT
light curves, and explored the recoverability of the signals when combined
with the information from TESS observations. We found that a significant
subset of large planets in long orbits that show single transits in TESS could
be detected in KELT light curves, enabling precise measurements of their
ephemerides. This type of approach was successfully carried out by Gill et al.
(2020c) to recover the ephemeris of a single-eclipse TESS eclipsing binary
using archival photometry from the WASP survey (Pollacco et al., 2006).

There are many ways to consider how to best follow up and confirm single-
transit candidates. These approaches are different from those used to follow
up transit candidates where the orbital period is known, since in those cases,
photometric follow-up can be scheduled at specific times to catch future
transits, and radial velocity (RV) observations can be timed to properly
sample the orbital phase. In one approach to follow up single transits, Cooke
& Pollacco (2020) explored the use of photometric versus spectroscopic follow-
up observations to confirm single transits. That analysis considered the use
of three specific instruments: photometric observations using the Next
Generation Transit survey (NGTS; Wheatley et al., 2018), and radial velocity
observations using the HARPS (Mayor et al., 2003) and CORALIE (Queloz et al.,
2000) spectrographs. To compare cases, Cooke & Pollacco (2020) considered the
observing time required for a given instrument to achieve a detection of the
planet past a given signal-to-noise ratio (SNR) threshold. In §5 below, we
compare our approach to that work.

In this paper, we improve upon the simulations in Y19 by incorporating more
realistic transit models and orbital configurations. We also explore the use
of RV observations of single-transit candidates to both eliminate certain
types of false positives, and to confirm their planetary nature by measuring
their orbits. We do not compare single-transit follow-up strategies to those
used for multi-transit candidates, since the unavailability of a known period
means that the efficiency or expense of the efforts cannot be directly
compared. The paper is structured as follows: §2 updates the earlier analysis
by deriving recovery rates of TESS single transit candidates with KELT
photometry using realistic eccentricities for the simulated sample, and
applying a more sophisticated calculation for the recovery rate. §3 discusses
the use of RV observations to distinguish planetary systems from high-mass-
ratio eclipsing binaries. §4 presents simulations of RV observations to
confirm planet candidates by constraining their orbital periods. We explore
the results and implications of the result in §5, comparing this approach to
other techniques, and in §6 we summarize our findings.

##  2 Updates to the Recovery Analysis

In Y19, we explored transit recovery of TESS single transits using KELT
photometry to pre-cover the signals. We simulated TESS single transits with
known periods ranging from 13.5 to 300 days in circular and centrally
transiting (i.e. equatorial) orbits, and the transit depth was assigned
randomly from 3 mmag to 20 mmag in log space. Then we inserted the periodic
transit signal into detrended KELT photometry, using a subset of 130,000 KELT
light curves with RMS scatter below 30 mmag, out of all 5.8 million KELT light
curves. The selection of only low-noise KELT light curves effectively
eliminates cases with stellar variability larger than the KELT photometric
scatter. We then used the Box-fitting Least Squares (BLS) algorithm (Kovács et
al., 2002) to try to recover the signal. We use the version of BLS with a
fixed transit duration and a fixed $T_{C}$, since the parameters $T_{C}$ and
duration for a given transit signal will be known with high fidelity from the
TESS observations. The period was free to vary and we searched from 13.5 days
to 300 days, with a frequency resolution of 300,000 (the number of trial
frequencies scanned, evenly spaced in frequency). The recovered period was
identified as the period corresponding to the strongest peak in the BLS
periodogram. As the number of the light curves for the simulations is large,
no visual inspection was conducted to check the recovered period. For a
successful recovery of an inserted transit signal, we require that the percent
difference between the recovered period and the inserted period be within
0.01%. In that approach, we calculated the recovery rate for the KELT
photometry for a range of transit durations and transit depths. That is
essentially an injection/recovery test of long-period transits with the KELT
photometry under the assumption that the transit time and duration are known
before searching for the signal in the KELT data. Since planets with orbital
periods longer than 10 days actually have a broad distribution of orbital
eccentricities (Marcy & Butler, 2000; Winn & Fabrycky, 2015), the duration and
window coverage of transits is affected by the orbital eccentricity (Barnes,
2007; Burke, 2008; Kane et al., 2012; Kipping, 2014). Transiting planets with
eccentric orbits will not have uniformly distributed arguments of periastron
$\omega$; in eccentric orbits, $\omega$ is more likely to be close to $\pi/2$
(as shown in Figure 4 from Burke, 2008) and the planet is more likely to
transit, which means the typical transit duration is shorter, for a fixed
orbital period. Therefore, compared with circular orbits, planets in eccentric
orbits will on average have fewer data points during the transit, which causes
the SNR of the transit signals to decrease, and thus recovery rate for planets
with more realistic (eccentric) orbits will be lower than cases with circular
orbits. Therefore the recovery results from Y19 represent an overestimate of
the recovery rates.

We address that issue by repeating the Y19 simulations but now consider
eccentric orbits. To determine the orbital eccentricities in the updated
simulations, we adopt the beta distribution with parameters $\alpha=0.867$ and
$\beta=3.03$ from Kipping (2013), and use the algorithm ECCSAMPLES (Kipping,
2014) to generate the orbital eccentricity and argument of periastron for each
planet. Other parameter distributions such as orbital period, planetary
radius, and orbital inclination are kept the same as in the circular orbital
case. Using the same criteria from Y19 to calculate the fraction of
successfully recovered orbital periods (a period precision of better than
0.01%), the average recovery rate declines as expected, and we find 5% fewer
recovered planets, where the overall recovery rate across all ranges of
orbital period and planet radius drops from 33% to 28%.

We now explore ways to improve the utility of the recovery process. We
continue to operate under the assumption that the transit signal in the TESS
light curve has a high enough signal-to-noise ratio for the transit duration
and $T_{\rm C}$ to be calculated precisely, but without enough signal-to-noise
ratio to strongly constrain the eccentricity just from the light curve.

Along with updating the Y19 analysis by incorporating eccentric orbits, we
utilize the transit duration as observed by TESS in an additional way. In Y19
we required the recovered transit signal to have a similar duration as in the
TESS light curve, but we did not constrain the range of possible periods to
search based on the transit duration. Here, we consider the fact that the
orbital period of a transiting planet can be estimated from the observed
parameters of single transit assuming a circular orbit with a central transit
(Seager & Mallén-Ornelas, 2003; Yee & Gaudi, 2008; Winn, 2010). We do not
assume that the orbits are necessarily circular, but rather make an educated
guess that the orbit is not extremely eccentric, which allows us to more
efficiently and reliably search a more limited range of possible periods, as
described below.

An alternate approach to our assumption of central transits is to use the
transit shape as measured by TESS to estimate the impact parameter based on
the transit shape in the TESS light curve. However, that method relies on high
SNR of the TESS detection, along with assumptions for the stellar limb-
darkening. Since we want to apply this method to as many of the TESS single
transits as possible, we instead take the alternate approach of assuming
central transits throughout this analysis.

Equation (7) in Yee & Gaudi (2008) expresses the relation between the orbital
period and transit duration for a transiting planet. The way we incorporate
this constraint is by calculating the orbital period from the transit duration
under the assumption of circular and centrally transiting orbits ($P_{\rm
cal}$) (Winn, 2010),

$P_{cal}=365\;\mathrm{d}\left(\frac{T_{dur}}{13\;\mathrm{hr}}\right)^{3}\left(\frac{\rho_{\star}}{\rho_{\odot}}\right).$
(1)

and requiring that the recovered orbital period from the BLS search ($P_{\rm
BLS}$) be within the range of $P_{\rm cal}$ described below. Even though
$P_{\rm cal}$ is based on the assumption of a circular orbit, we can still use
it to constrain the search for eccentric transits. The transit duration in Eq.
1 is the full-width half-maximum of the transit in hours.

Figure 1 shows the distribution of the ratio $P_{\rm cal}/P_{\rm BLS}$, along
with the recovery rates. Since the BLS search is performed across period space
for all light curves, we expect some spurious signals at all period ranges. By
requiring $P_{\rm BLS}$ to be sufficiently close to $P_{\rm cal}$, we can
eliminate many spurious cases from the analysis and improve the reliability of
the surviving signals. We thus implemented an additional cut in the analysis
by requiring that $P_{\rm cal}/P_{\rm BLS}$ to be between 0.25 and 1.96, which
effectively removes cases where the expected recovery rate is below 20%. We
selected that particular cutoff to reflect a subjective judgement regarding
the balance between improving the overall recovery rate, and retaining a large
fraction of the total sample. We then calculated the overall recovery rate for
the remaining simulated light curves.

Using this constraint, the overall recovery rate improves by $\sim$12%, from
28% to 40%, as a result of excluding cases where the period calculated from
BLS is likely inconsistent with the transit duration. In this case, that
restriction excluded $\sim$41% of the original set of simulated light curves,
finding the BLS-derived period to be unlikely to be consistent with the
transit duration. This recovery rate is also higher than the overall recovery
rate found in Y19, even though we now account for eccentric orbits. The
improvement in the recovery rate from 28% to 40% does not mean that we can
detect more long-period planets from TESS, but that the reliability of the
results from searching the KELT data is more trustworthy.

Figure 1: Distributions of the ratios between $P_{\rm cal}$ and $P_{\rm BLS}$
for eccentric orbits (blue) and the associated recovery rate as a function of
the period ratio (orange). The green horizontal dotted line marks the average
recovery rate in Y19 (restricted to circular orbits). The green horizontal
dashed line marks the 20% recovery rate. The vertical dashed grey lines
indicate the period ratio corresponding to the 20% recovery rate.

To show the result in a more useful form for follow-up observers, we adopt the
signal-to-pink noise ratio (SPN) as a signal strength criterion to estimate
the confidence of the recovery result. The SPN ratio is a variant of the S/N
computed using a “pink” total noise that includes both uncorrelated (“white”)
and correlated (“red”) noise sources. More details of the definition of SPN
can be found in Hartman & Bakos (2016), and details of how we utilize this
statistic are in section 4.2 of Y19. In short, we calculate the empirical SPN
threshold where a given percentage of all test cases are successfully
recovered, and use that as the ”recovery confidence”. For instance, in a given
range of transit depth and duration, if we find that 90% of all test signals
are recovered in the KELT light curves, then that SPN value is our 90%
confidence threshold. Similarly to Figure 13 in Y19, Figure 2 shows the SPN
thresholds for 10%, 50% and 90% recovery confidence with the fraction of KELT
light curves that are above those thresholds across transit depth versus
transit duration, using the same parameter ranges as in Y19. In that figure,
we show the results separately for the KELT-South and KELT-North data sets,
since the KELT-North dataset generally includes more epochs spread over a
longer time baseline as compared to the KELT-South dataset. For example,
consider the center bin in the upper panel. The lower right row within that
bin refers to the SPN required for a transit to be recovered 90% of the time.
We find that SPN value to be 7.3, so that if a transit signal with an SPN
value greater than 7.3 is seen, an observer can have 90% confidence the signal
is real. The percentage in that row indicates that of the KELT light curves
with inserted transit durations and depth in that corresponding range, 26%
yielded SPN values greater than 7.3.

Figure 2: The fractions of KELT-North (top) and KELT-South (bottom) light
curves that pass SPN thresholds in transit depth / transit duration bins, and
the corresponding SPN values in parentheses. In each box are three percentage
values and corresponding SPN values. The percentages reflect the fraction of
KELT light curves in that bin of depth/duration space such that if the SPN
value is greater than the indicated value, there is a given likelihood that
the recovered period is correct. Those likelihoods are 10%, 50%, or 90%, from
upper left to lower right in each bin. The color bar indicates the fractional
change in the fraction of KELT light curves that pass the 50% confidence
threshold between this analysis and the results from Y19, (discussion in
text). The items marked ‘nan’ represent cases where none of the inserted
signals were recovered in the KELT light curves at the corresponding
confidence level for the depth and duration ranges for that bin.

The color of the cells in Figure 2 indicate whether the updated analysis
yields better (blue) or worse (red) recovery rates compared to Figure 13 in
Y19. That is, the colors indicate the differential improvement (or decline) in
recovery rates compared to the assumption of a circular orbit. The figure
conveys two sets of information - the absolute recovery rates, using the
methodology described above, as well as the relative increase or decrease in
recovery rates across parameters space compared to Y19.

The colors in the figure reflect the impact of the two key changes we have
made. The use of eccentric orbits decreases the recovery rate throughout the
analysis. That is because for the same transit duration, the real orbital
periods for eccentric transiting planets are generally longer than the ones
with circular orbits. Therefore the SNR of the transit signal in the light
curves is lower (the converse of the effect mentioned above) when we are
observing multiple transits. On the other hand, the use of the additional
information coming from the transit duration using $P_{\rm cal}$ improves the
reliability of the recovery (and thus the overall recovery rates) due to
constraining the period search range. Figure 2 shows that for transit
durations shorter than 8 hours, the former effect dominates, while at longer
transit durations, the latter effect dominates.

##  3 Distinguishing Eclipsing Binaries From Planets

Although in this analysis we have inserted transit-like signals into the KELT
light curves as a phenomenological feature, agnostic as to the physical cause
of the signal, we now explore not just how to detect those photometric
signals, but also how to distinguish their physical cause.

One feature of the mass-radius relationship for stars and planets is that the
onset of electron degeneracy in the core leads to a flat mass-radius curve
from the massive planet regime to the small star regime (Zapolsky & Salpeter,
1969; Chabrier & Baraffe, 2000; Lynden-Bell & O’Dwyer, 2001; Fortney et al.,
2007). That is, compact objects with masses between Jovian planets and late-
type stars all have radii of roughly 1 $R_{\rm J}$, despite an almost two
orders of magnitude range in masses. Among our simulation samples, $\sim$65%
of planetary candidates have radii larger than $1R_{\rm J}$. That means a
large number of transit signals in our simulations may be caused by eclipsing
binaries instead of planets. They can include a larger star being eclipsed by
a later-type dwarf star, yielding a primary eclipse depth comparable to that
of a transiting giant planet, or an eclipsing binary with a large primary
transit that is diluted by blending with a nearby bright star. In this section
we will concentrate on ways to identify the first of these scenarios. We will
ignore transit-like signals caused by blended eclipsing binaries. Those types
of false-positive scenarios can be addressed through careful analysis of the
TESS pixel data by looking for centroid shifts (Bryson et al., 2013, Guerrero,
et al. submitted), or via photometric monitoring with higher angular
resolution or other spectroscopic techniques (see Collins et al. (2018) for a
discussion on vetting different false positive types).

One tool for distinguishing transiting planets from eclipsing binaries is the
use of radial velocity (RV) observations. Here we explore how to conduct RV
observations in the case where the ephemeris of the transit signal is unknown
or poorly constrained, as in the case of single transits. This process often
involves two stages of follow-up observations. The first stage, often referred
to as reconnaissance spectroscopy, involves two spectroscopic observations.
These observations are taken at the predicted orbit’s quadrature phases, and
can identify SB1s, and visual inspection of the spectra can identify SB2s.
Once those scenarios are ruled out, further RV observations are obtained over
the full phase of the orbit to characterize the orbital elements, e.g., the RV
semiamplitude, argument of periastron, and eccentricity.

In the case of an unknown orbital period, a single spectroscopic observation
can still identify SB2-type cases, but without knowing the predicted
quadrature times, it is not obvious how to perform reconnaissance spectroscopy
so as to rule out EBs. However, if the system has a large semi-amplitude, two
random observations could show a large offset indicating the presence of an
EB. There has been extensive work conducting both RV and photometric follow-up
observations of single-transit candidates from the NGTS team, recovering the
ephemerides of two EBs that showed single transits in TESS photometry (Lendl
et al., 2020; Gill et al., 2020a), along with the discovery of a long-period
planet from what was initially thought to be a single transit in TESS data,
which was later revealed to contain an additional transit initially obscured
by scattered light in the photometry (Dalba et al., 2020; Gill et al., 2020b).
Here we investigate a comprehensive approach to RV follow-up.

The dataset used in Y19 and in §2 consists of KELT light curves in which we
inserted a set of simulated transit signals. In this section, we do not use
any light curves, but we apply the properties of the simulated planetary
systems from that analysis, namely the orbital periods, radii of eclipsing
objects, eccentricities, and argument of periastron, and the masses and radii
of the associated stars, with the stellar parameters taken from the TESS Input
Catalog (TIC-8).

We first introduce EB cases to the simulated signals. Note that the signals
were drawn from a range of empirical properties (depth, duration, period,
etc.), without reference to physical properties (companion mass, radius,
semimajor axis, etc.). We selected $\sim$48,000 targets from the KELT-North
sample of stars described in §2 that have radii of the transiting body that
are larger than $1R_{\rm J}$, as inferred from the simulated transit depths
and the estimated stellar radii from the TIC-8. This subset of the full light
curve simulation includes the candidates for which there is ambiguity about
the physical nature of the eclipsing body. In the process of assigning masses,
we now consider two cases: if the transiting objects are all planets, the mass
($M_{p}$) was randomly assigned between $0.2M_{\rm J}$ and $3M_{\rm J}$ in log
space; and if the transiting objects are stars, the mass ($\rm M_{\sec}$) was
calculated based on the mass-radius relation in the stellar region derived
from Chen & Kipping (2017). We are essentially assuming that objects with
masses between $3M_{\rm J}$ and $80M_{\rm J}$ are rarer than planets or stars,
i.e. that there exists a “Brown-dwarf desert” (e.g., Grether & Lineweaver,
2006). The semi-amplitude ($K$) of the radial velocity curve for planets was
calculated using the above parameters:

$K=28.4\;\mathrm{m/s}\left(\frac{P}{1\;\mathrm{yr}}\right)^{-1/3}\left(\frac{M_{\rm
p}}{M_{\rm J}}\right)\left(\frac{M_{\star}}{\rm
M_{\odot}}\right)^{-2/3}\left({1-e^{2}}\right)^{-1/2}.$ (2)

For the case of binary stars, we replace $M_{\rm p}$ with $M_{\sec}$ and
$M_{\star}$ with $(M_{\star}+M_{\sec})$ in Equation 2. Note that, since we
know these systems are seen nearly edge-on because they exhibit transits or
eclipses, Equation 2 assumes $\sin{i}$ $\simeq$ 1\. We therefore ignore the
$\sin{i}$ dependence on $K$ in Equation 2. We used the python package RadVel
(Fulton et al., 2018) to synthesize RV curves based on those parameters.

While the observational strategies of RV follow-up can vary significantly, we
simulated a plausible follow-up approach that might be undertaken by a mid-
size RV survey facility such as TRES (Szentgyorgyi & Furész, 2007), MINERVA
(Swift et al., 2015), MINERVA-Australis (Addison et al., 2019), the APF (Vogt
et al., 2014), or a similar facility. We assume that the time interval between
the TESS observations of the single transit and the first RV observation is
three months, which is consistent with the TESS data release procedure, and
the time required to identify candidates and begin scheduling RV observations.
We also tested lengths of time intervals of 1 month and 12 months,
respectively, and found very little effect on the results.

The RV precision of an observation depends on the telescope, the instrument
and various stellar properties such as brightness, effective temperature, and
rotational velocity, among other factors. Although we explored the possibility
of simulating the RV precision for each observation based on the stellar
properties and assumed noise model for a given instrument, we ultimately found
such an effort to be unfeasible. Even for a specific telescope and noise
model, there is a spread in RV precision for stars of a given brightness, with
many ways that the stellar properties can influence the RVs, such as the
effects of stellar rotation, spot coverage, and chromospheric activity.
Therefore, we decided to adopt a uniform RV precision of 20 m/s in the
simulation, assuming the stellar rotation velocities are lower than 10 km/s.
These assumptions were based on the statistics of real observational data from
the CHIRON spectrograph (Tokovinin et al., 2013).

Since the true orbital period is not precisely known from TESS single transit
light curves particularly when assuming eccentric orbits, we modeled the
approach of an observer making an educated guess about the quadrature times
based on $P_{\rm cal}$. We then determined the RV values at those times
according to the model RV, and then added offsets to the calculated RV using
values drawn from a Gaussian distribution with a width of 20 m/s. We then
approximate the RV semi-amplitude as half the difference in the simulated RV
measurements at the estimated times of quadrature. This makes the inherent
assumption that the orbit is approximately circular, and thus the quadrature
times are approximately known, and that the semiamplitude is just one half the
difference between the measurements at quadrature (ignoring measurement
errors).

We do not consider whether the calculated quadrature time is during local
night, but since the potential orbital periods are known to be much longer
than 24 hours, we assume that observations will take place within 12 hours of
the calculated quadrature, and that issue does not have a major impact on
these results. Another potential issue with this approach is that we assume
that all planet candidates included are observable at some point in the night
through the campaign. If targets are randomly distributed in right ascension,
then some targets will inevitably be unobservable for significant lengths of
time. But with a significant number of total candidates, the observers can
select a subset that should be observable through the whole campaign.

In the upper panel of Figure 3, we show the distribution of estimated $\rm K$
for the planet and star samples, respectively. The lower panel indicates the
fraction of the samples at a given estimated $\rm K$ in which the candidates
are planets. If the difference between the two measured RVs is smaller than 50
m/s, the transit signal has a less than 10% probability of being a stellar
companion, and more RV follow-up is merited to determine the full dynamical
orbit. If the RV difference is larger than 200 m/s, it is likely to be an EB
with greater than 90% probability. For RV differences between those ranges,
additional observations are required to identify potential false positives.

Figure 3: The upper panel shows the distributions of half the difference
between two RV measurements at estimated quadrature times for the case of
planetary companions (orange) and a portion of the distribution for stellar
companions (blue). The difference between the RV observations is a rough proxy
for the semiamplitude. The lower panel indicates the fraction of the samples
at a given calculated semi-amplitude in which the candidates are planets.

##  4 Radial Velocity Detection of Planets

RV observations to dynamically detect planets are generally conducted in one
of two ways. If the orbital period of a system is known from multiple
transits, RV observations can be obtained across a full orbital phase, to
measure the orbital reflex motion of the star due to the planet (e.g. Burt et
al., 2018; Medina et al., 2018). If the orbital period is not known, as in a
blind RV search, one star or a set of stars can be monitored by (semi)regular
RV observations until orbital motion is seen in a phase-folded search of the
RV data. In these situations, RV monitoring, which can detect the
gravitational reflex motion that a planet induces on its host star throughout
the entire orbit, can be used to determine the orbital parameters.

In this section we conduct a simulation of what such an RV campaign might look
like, and we examine the distribution of stellar and planet parameters for the
systems for which that campaign could successfully detect the orbit. It should
be noted that RV confirmation of a single-transit candidate must operate quite
differently than for a multiple-transit candidate. Since the photometry
provides no direct measure of the orbital period, the RV observations must
effectively operate as a blind RV search, as in the second case noted above.
The approach described here aims to provide a more efficient way to conduct
that sort of search.

To create the simulation, we randomly selected 10% of the stars from the
sample in §2 that have $T_{\rm eff}$ between 4000K and 7400K (i.e., FGK type
stars) which are best suited for RV follow-up. This results in a sample of
$\sim$5600 stars in the KELT-North set and $\sim$5400 stars in the KELT-South
set. Note again that the data set involved in this analysis consists of the
simulated transit signals, not the KELT light curves themselves. We start with
the empirical signal properties, and translate those into associated physical
properties of the transiting companions, as we did in §3. In this case, if the
transiting objects have radii larger than $1R_{\rm J}$, the mass was randomly
assigned between $0.2M_{\rm J}$ and $3M_{\rm J}$ in log space; if planets have
radii smaller than $1R_{\rm J}$, the mass was calculated based on the mass-
radius relation in the Neptunian region derived from Chen & Kipping (2017).

The RV observations are simulated over a 3-month time span. That duration was
chosen as a plausible length of time for a dedicated RV campaign of this type.
A natural consequence of this is that systems with true periods longer than 3
months will be incompletely sampled, which we discuss below. The parameters
$P$, $e$, $\omega$ and $T_{\rm C}$ used to generate the RV curves are the same
as those from §2, in which we have included eccentric orbits, and we again
assume an RV precision of 20 m/s. We randomly selected 7 days each month in
which one observation of the target star is obtained each night, for a total
of 21 observations per star over that time frame. We assume that the
stipulation of 7 RV observations per month can be met even in the presence of
weather, which we account for by the random placement of the observations
within each month. We assigned RV values according to the theoretical RV
curve, and then added Gaussian noise with a distribution width of 20 m/s to
the RV points.

It should be noted that this approach does not account for dynamic scheduling
of upcoming RV observations during a campaign based on an evaluation of the RV
results up to that point. That topic has been explored extensively before
(e.g. Kane et al., 2007; Ford, 2008; Loredo et al., 2011), but not in the
specific case of a singly-transiting planet where a date of conjunction and
the planet size are known, but not the orbital period. In a related analysis,
Cabona et al. (2020) examined the efficacy of different scheduling strategies
when conducting follow-up RV observations of small TESS targets using the
ESPRESSO instrument (Pepe et al., 2014) on the VLT. However, that work
considers only transit candidates with known periods, and so addresses a
different scientific question than considered here.

There are various software tools that perform RV fitting, such as RadVel
(Fulton et al., 2018) and the Joker (Price-Whelan et al., 2017) and EXOFASTv2
(Eastman et al., 2019). In the next step, we conducted a maximum likelihood
Keplerian model fit as implemented in the RadVel package for the synthesized
RV data for each target. In the fitting process, the transit time $T_{\rm C}$
was fixed to the simulated value as would be measured from the TESS single
transit, and we set boundaries from 13.5 days to 300 days for the orbital
period. Other parameters ($K$, $e\sin\omega$, $e\cos\omega$) were free to
vary. For the initial guess of the orbital period, we used $P_{\rm L-S}$ from
a Lomb-Scargle period search of the simulated RV data. The initial guess of
$K$ was calculated based on the assumption of a circular and centrally
transiting orbit with $P_{\rm L-S}$, known stellar mass $M_{\star}$, and
$M_{\rm p}$ calculated based on the $M_{p}-R_{p}$ relation from Chen & Kipping
(2017) as described above.

There are a number of ways to consider the precision on the period of an
exoplanet derived from RV observations. For the purposes of confirming a
planet, a fractional period precision of tens of percent may be sufficient.
For conducting intensive transit follow-up however, such as atmospheric
characterization, an absolute precision of better than 30 minutes would
typically be required. For the analysis here, we consider an RV-fitted period
to be correct if the fractional difference between the RV-fitted period
$P_{\rm RV,fit}$ and $P_{\rm real}$ is smaller than 5%. While that precision
would not be sufficient for atmospheric observations, or even for photometric
ephemeris confirmation in some cases, it would be more than sufficient to
identify good candidates for long-period transiting planets from the TESS
sample of single-transit candidates. At that point, an observer can conduct
additional RV observations beyond the campaign envisioned here, with higher
cadence, or with another facility with greater RV precision to bring the
measured period from the new RV observations to a level sufficient for
photometric ephemeris confirmation to obtain the higher absolute precision on
the transit time needed for intensive transit observations. For similar
reasons, we consider an RV-fitted $K$ to be correct if the fractional
difference between $K_{\rm fit}$ and $K_{\rm real}$ is smaller than 50%. Our
goals at this stage are to obtain approximate orbital solutions and to
differentiate likely planets from false positives, rather than obtaining a
complete system solution at the end of the campaign.

It is the case that while the simulated RV campaign lasts for only 90 days, we
search for orbital periods out to 300 days. We made this choice for two
reasons. First, it is possible to obtain an RV detection of the orbital signal
with only partial phase coverage, although the fractional period precision of
the resulting fit is typically quite poor. Since a real campaign, regardless
of its duration, will end up observing some systems with long orbital periods,
we wanted to test the ability to extract such signals. In most cases of
periods much longer than the campaign duration, only a linear trend will be
identified. We discuss such cases in more detail below.

Figure 4 shows the distributions of the real and fitted parameters $P_{\rm
real}$, $P_{\rm RV,fit}$, $K_{\rm real}$, $K_{\rm fit}$ and $e_{\rm real}$.
There is clearly a significant population along the diagonal in each panel
which represents the correctly fitted samples, along with clusters running
along the diagonals in the period plots which indicate cases off by factors of
2 or 1/2. As expected, systems with high eccentricity and small semi-
amplitudes tend to be recovered least well. Systems with long orbital periods
are also poorly recovered, although that is partly due to the fact that the
90-day span of the RV observations is shorter than the orbital period for some
of the systems. By comparing the fitted parameters with real values, we found
the median absolute deviation (MAD) of ($P_{\rm RV,fit}$ versus $P_{\rm
real}$) is 7.2 days and the MAD of ($K_{\rm fit}$ versus $K_{\rm real}$) is
8.2 m/s. Although these intermediate results provide a rough estimate of the
ephemerides of the candidates, these ephemerides are not sufficiently precise
to schedule follow up observations during transits. We improve upon these
results below by identifying the more reliable fitted periods.

Figure 5 shows two examples that were successfully fitted and two examples
where the fitting failed. We indicate the locations of these four examples on
the plots in Figure 4. The two successful fits (cases A and B) show good
agreement between the true periods and the fitted periods for both shorter and
longer orbital periods. Example C shows a case where the true semiamplitude is
smaller than the scatter in the RV points, and example D shows a case where
the true period is significantly longer than the time baseline of the RV data.

Figure 4: Scatter plots for the simulated samples, comparing the real orbital
periods and semi-amplitudes with their fitted values, colored by different
real parameter values. We indicate the real and fitted values for the 4
examples in Figure 5 with filled black points. In the upper plots, the
populations that run along side the diagonal represent cases where the fitted
period is off by a factor of 2 or 1/2.

Figure 5: Two successfully fitted RV samples (top panel) and two failed fitted
samples (bottom panel). All these four examples are identified in Figure 4 and
Figure 6.

We compute the reduced chi square ($\chi_{\rm dof}^{2}$) between the best-fit
model and the simulated data. By inspecting the fitting results, we found
nearly all samples have $\chi_{\rm dof}^{2}$<2, which indicates that
$\chi_{\rm dof}^{2}$ is not an efficient tool to measure the confidence of the
fitted RV period. That situation arises because the amplitudes of the RV
signals in our sample are often not that much larger than the per-point errors
that we are modeling. As $\chi_{\rm dof}^{2}$ is not an efficient tool to
measure the confidence of the fitted RV period, we instead conduct a False
Alarm Probability (FAP) test of the RV fits. For each star in the simulation,
we ran 100 iterations of the above analysis after randomizing the RV
observations and the observing times. We selected the smallest reduced chi
square $\chi_{\rm dof{FAP<1\%}}^{2}$ to indicate the 1% FAP value for that
star. Figure 6 shows the distribution of period recovery accuracy compared to
the ratio between $\chi_{\rm dof}^{2}$ and $\chi_{\rm dof{FAP<1\%}}^{2}$,
which we refer to as the 1% FAP ratio. The plots in that figure include
horizontal lines indicating the 5% accuracy threshold on period recovery, and
vertical lines at the FAP ratio of unity indicating whether the $\chi_{\rm
dof}^{2}$ fit was larger or smaller than the 1% FAP threshold. The region to
the right of the FAP threshold indicates systems where the RV fit is not very
reliable, and tends to include long-period and low-amplitude systems. The
region to the left of the FAP threshold indicates reliable results, and we
find that 85% of those results yield orbital periods that are within 5% of the
true period. Of the 15% cases that are considered reliable from the FAP cut
but do not match the orbital period (the upper left region in the first
panel), 82% of those have true periods longer than the observing campaign,
indicating that we see a broad trend without the leverage to accurately
measure the period. We conclude that the 1% FAP ratio can be used as an
efficient tool to measure the reliability of the fit. Similar to Figure 4,
systems with a relatively short period and large semi-amplitude are most
likely to be correctly fitted, as expected.

Figure 6: Scatter plots of the percentage difference between the fitted period
$P_{\rm RV,fit}$ and $P_{\rm real}$ versus the 1% FAP ratio $\chi_{\rm
dof}^{2}$/$\chi_{\rm dof{FAP<1\%}}^{2}$. The color bars represent the real
value of period, eccentricity, and semi-amplitude.

Figure 7 provides a different way to view the utility of the 1% FAP ratio. The
figure shows the distribution of the 1% FAP ratios for all test cases, with
one set of values for the period fit, and another for the semi-amplitude fit,
separated according to passing the fit threshold. At a 1% FAP ratio around
unity, half of the cases were correctly fitted, and at a 1% FAP ratio around
0.3, nearly all samples are correctly fit. These distributions provide
confidence metrics for the fitted period and semi-amplitude.

As discussed above, the precision of a fitted period has different
implications depending on the goals of an investigation. While a given fitted
period might have a low fractional uncertainty, if an observer wants to
establish the true transit ephemeris, they will often want to obtain a fitted
period with a specific absolute uncertainty, corresponding to the time span
over which the second transit is expected at a confidence of, say, 1$\sigma$
or 68%, so as to schedule follow-up photometry to measure the transit. Figure
8 shows another version of Figure 7, by using a fixed absolute uncertainty on
the period of 8 hours rather than a fixed fractional uncertainty on the
period. This version provides a more useful reference for observers to plan
photometric follow-up observations, assuming an 8-hr night for observing.
Specifically, if an observer wants to obtain follow-up photometry for targets
with an RV-fitted period with precision smaller than 8 hours, with at least
50% confidence, then they should select targets with a 1% FAP ratio smaller
than 0.3 (see §5 below for details on implementing this process.)

Figure 7: Distribution of the 1% FAP ratio ($\chi_{\rm dof}^{2}$/$\chi_{\rm
dof{FAP<1\%}}^{2}$) for tested RV fits, colored by whether the fitted period
was close to the correct period based on a period precision of 5%, with the
grey color indicating the overlap of the two. The dot-dashed histogram
represents the cases where the semi-amplitude was near the real value. The
lower panel indicates the fraction of the tested RV fittings at a given bin in
1% FAP ratio in which the period and semi-amplitude were correctly fitted. At
a 1% FAP ratio of about 1, half of the test samples have well-fitted periods
and semi-amplitudes.

Figure 8: Distribution of the 1% FAP ratio ($\chi_{\rm dof}^{2}$/$\chi_{\rm
dof{FAP<1\%}}^{2}$) for tested RV fits, colored by whether the fitted period
was close to the correct period based on an 8-hour absolute accuracy. The dot-
dashed histogram represents the cases where the semi-amplitude was within 50%
of the correct value.

Figure 9 reproduces Figure 4, showing the distributions of the real and fitted
parameters, but this time only for the samples with 1% FAP ratio smaller than
1. In this case, the median absolute deviation (MAD) of the difference between
$P_{\rm RV,fit}$ and $P_{\rm real}$ is 0.3 days, and the MAD of the offset
between $K_{\rm fit}$ and $K_{\rm real}$ is 6.5 m/s. There remain some
clusters of fits that are off by a factor of 2 or 1/2. This result shows the
clear improvement in the accuracy of the RV results after excluding poor-
quality fits. We find that the systems most amenable to successful recovery
are those with an orbital period shorter than 100 days and planetary mass
greater than half a Jupiter mass (see Figure 10).

We found that orbital periods as long as $\sim$180 days can be determined to a
fractional accuracy of 10-20%, in which case around half of the orbital phase
was observed, and the RV data usually show some linear trends. These systems
usually contain a planet with a high mass yielding a large semi-amplitude.
Detection of such systems in a campaign as mapped out here would prompt
observers to take additional RV observations to refine the period.

Figure 9: Scatter plots for the simulated samples, similar to Figure 4. In
this figure, only cases where the 1% FAP ratio is smaller than 1 are
displayed.

Figure 10: Distribution of the planet masses for the successfully and
unsuccessfully recovered simulated systems.

##  5 Discussion

In this work, we have updated the simulation process introduced in Y19 by
incorporating orbits with realistic eccentricity distributions, and improved
the recovery rates by using $P_{\rm cal}$ to refine the searched periods. The
results from this work use more realistic descriptions of planetary orbits,
and should be more useful for the observers to schedule photometric follow-up.
If an observer has a particular single transit TESS candidate, they can
calculate $P_{\rm cal}$, use the KELT light curve to compute $P_{\rm BLS}$ and
the associated SPN, and check the confidence of $P_{\rm BLS}$ from Figure 2.
They can use that information to prioritize and schedule photometric
observations during the predicted transit window for that candidate to confirm
the ephemeris. A similar procedure can be used with any archival photometry,
so long as a sensitivity analysis along the lines of Y19 and 2 is conducted to
determine the SPN threshold associated with a given confidence level for a
particular range of transit depth and duration.

That analysis can still be improved. We assume central transits in our
simulations, but real transits, with a range of impact parameters, will be
shorter for the same stellar properties and orbital period, leading to
additional errors in the value of $P_{\rm cal}$. However, the overall size of
that effect is small (Seager & Mallén-Ornelas, 2003). It is the case that one
could estimate the impact parameter and other orbital properties from the TESS
light curve, but we have assumed here that while the TESS light curve is
likely to have high enough photometric precision to obtain fairly reliable
measurements of the transit depth and duration, it should not be assumed that
the transit shape, along with ingress/egress times, can always be measured to
high precision. In cases with extremely high photometric precision, the
transit shape itself can be used to identify or discount false positive
scenarios, but that typically can only be done for very bright stars, such as
the case of HR 858, which has multiple planets orbiting a $T=5.9$ star
(Vanderburg et al., 2019). Another issue we did not consider is whether the
presence of additional planets or stellar variability due to rotational
modulation from spots/plages in the system could create RV signatures that
would interfere with the ability to recover the orbit of the transiting
planet.

We also explored in §3 how RV observations can be used for initial vetting of
single-transit candidates, by simulating the use of two RV observations tied
to the predicted quadrature times based on simplified orbital assumptions. We
showed that for most system configurations, such observations should be
readily able to distinguish stellar binaries from planetary systems.

We then examined an example campaign of RV observations to confirm the single-
transit candidates by measuring the orbital motion of the stellar host due to
the planet. We posited a particular observing campaign that is within the
capability of multiple current facilities. For a TESS single transit, even
after an initial pair of RV observations discounts the presence of an
unblended stellar binary, there is a broad range of possible periods (Figure
1) even after accounting for the transit duration and stellar mass. Our
simulations show that RV campaigns of the type we considered can successfully
measure the orbital motion due to the planet in $\sim$ 30% of cases we
simulated. The results of such an RV campaign, making use of modest telescope
and spectroscopic facilities, can constrain orbital period to within 5% with
85% confidence and the mass of the companion to within 50% with 88%
confidence. That information can then be used to plan a photometric campaign
over one or several nights to catch another transit and precisely determine
the ephemeris for future atmospheric study, or conduct additional, targeted RV
observations, potentially with higher RV precision to further constrain the
period before conducting photometric observations. A flow chart in Figure 11
provides a schematic illustration of this RV follow-up process we have
outlined. While this approach requires a number of steps to refine the precise
orbital period, it will typically be more feasible for most observers than
blind photometric follow-up of a candidate, as in the case of NGTS-11b
described below.

Figure 11: A flow chart illustrating the process to determine the ephemerides
of TESS single transits described in this paper.

The achievable precision of an actual RV follow-up campaign depends on many
factors, such as weather, observational windows, the number of RV observations
that can be acquired, the properties of the target stars (e.g., magnitude,
rotational velocity, effective temperature), the resolution of the
spectrograph ($\frac{\lambda}{\Delta\lambda}$), the exposure time, etc.
Therefore, we find it impractical to incorporate all these factors into a
generalized simulation, but these results can be used as a set of guidelines
for real-world follow-up efforts. In future work, we intend to explore how the
fitting of the RV signals can be improved with other software tools such as
EXOFASTv2 (Eastman et al., 2019) or Juliet (Espinoza et al., 2019), as opposed
to RadVel.

The TESS Single Transit Planet Candidates (TSTPC) working group has identified
a sample of more than 100 single transit candidates in TESS FFI data during
Sectors 1 through 13 (Villanueva et al in prep). Assuming circular orbits to
calculate the orbital period, and estimating the planetary mass using the
calculated planetary radius and the mass-radius relation from Chen & Kipping
(2017) we find that about 70% of the TSTPC-identified candidates would be
amenable to confirmation using these techniques.

Cooke & Pollacco (2020) have approached follow-up observations of TESS single
transits in a different way, as mentioned in section 1. That work asks a
narrower question than we deal with here. They investigate how one should
conduct follow-up observations when deciding between a blind photometric
search or RV search, for two (or three) specific observing facilities. They
calculate the effectiveness of using photometry or spectroscopy for exoplanet
follow-up as a function of $\rm R_{\star}$, $\rm R_{p}$ and $\rm P$. They
assume circular orbits for the transit candidates, and expect full, dedicated
access the the observing facilities, which in that case consist of the NGTS
photometric telescopes, and the CORALIE and HARPS spectrographs. For
photometric follow-up observations, they model regular, nightly observations,
in a mode exemplified by the successful confirmation of the ephemeris of the
planet NGTS-11b. In that case, Gill et al. (2020b) describe how the follow-up
procedure dedicated 79 nights of photometric monitoring in a blind search for
the detection of a second transit of a TESS single-transit candidate. Such an
approach is complementary to the process we describe in this paper and in Y19.
When archival photometry is available, we can restrict the likely ephemerides
to certain time ranges or to exact values, but if such photometry is not
available, the approach used by Cooke & Pollacco (2020) provides an alternate
path, when sufficient resources are available. The RV follow-up approach they
describe accounts for a more exact calculation of the RV precision as a
function of target magnitude for NGTS, HARPS, and CORALIE and the associated
SNR of the RV detection, and the assumption of circular orbits rather than a
range of eccentricities represents a different set of assumptions than we use.
We believe that the approach described in this work should be useful for
observers working with a range of follow-up observing facilities and various
levels of access, while the Cooke & Pollacco (2020) analysis is most
appropriate for those with dedicated access to high-performance instruments.

##  6 Summary

TESS will discover hundreds of planet candidates with long orbital periods via
single transits in their TESS light curves during the prime mission (Cooke et
al., 2018; Villanueva et al., 2019). Y19 demonstrated that KELT data can
recover the ephemerides of some single-transit candidates to a fractional
precision of 0.01%. In this work, we have incorporated more realistic models
for TESS single transits with eccentric orbits instead of circular orbits,
which is common for long-period exoplanets. We also improved the reliability
of precovery using transit duration constraints.

In addition to improving the precovery simulations, we explored the use of RV
follow-up observations in the case of single transit events. We found that the
use of the estimated period based on a circular orbit to schedule
reconnaissance RV observations around quadrature can efficiently distinguish
EBs from planets. For candidates that pass reconnaissance RV observations, we
simulated RV monitoring campaigns to obtain an orbital solution. We found that
the use of a $\chi_{\rm dof}^{2}$ fit to a Keplerian model, combined with an
FAP analysis, is sufficient to obtain an approximate orbital solution for
planets more massive than $0.5M_{\rm J}$ with orbital periods as long as 100
days, providing sufficient constraints for additional detailed orbital
refinement and photometric ephemeris determination.

JP and XY acknowledge support from NASA grant 80NSSC19K0387 under the TESS
Guest Investigator program G011229. PD acknowledges support from a National
Science Foundation (NSF) Astronomy & Astrophysics Postdoctoral Fellowship
under award AST-1903811. Part of this research was carried out at the Jet
Propulsion Laboratory, California Institute of Technology, under a contract
with the National Aeronautics and Space Administration (NASA). KGS
acknowledges partial support from NASA grant 17-XRP17 2-0024. BSG was
supported by a Thomas Jefferson Chair for Space Exploration endowment at the
Ohio State University. DJS is supported as an Eberly Research Fellow by the
Eberly College of Science at the Pennsylvania State University. The Center for
Exoplanets and Habitable Worlds is supported by the Pennsylvania State
University, the Eberly College of Science, and the Pennsylvania Space Grant
Consortium. KELT was partially supported by NSF CAREER Grant AST-1056524.
Funding for the TESS mission is provided by the NASA Explorer Program. This
work has made use of NASA’s Astrophysics Data System. This research made use
of Astropy,222http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018). We acknowledge
support for the KELT project through the Vanderbilt Initiative in Data-
intensive Astrophysics, Ohio State University, and Lehigh University.

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