#  Kepler and the Behemoth: Three Mini-Neptunes in a 40 Million Year Old
Association

L. G. Bouma 51 Pegasi b Fellow  Cahill Center for Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA  R. Kerr Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA  J. L. Curtis Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA  H. Isaacson Astronomy Department, University of California, Berkeley, CA 94720, USA  L. A. Hillenbrand  Cahill Center for Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA  A. W. Howard Cahill Center for Astrophysics, California Institute of Technology, Pasadena, CA 91125, USA  A. L. Kraus Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA  A. Bieryla Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  D. W. Latham Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  E. A. Petigura Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA  D. Huber Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

(Received 2022 May 2; Revised 2022 Sep 14; Accepted 2022 Sep 17)

###### Abstract

Stellar positions and velocities from Gaia are yielding a new view of open
cluster dispersal. Here we present an analysis of a group of stars spanning
Cepheus ($l=100^{\circ}$) to Hercules ($l=40^{\circ}$), hereafter the Cep-Her
complex. The group includes four Kepler Objects of Interest: Kepler-1643 b
($R_{\rm p}=2.32\pm 0.13\,R_{\oplus}$, $P=5.3\ {\rm days}$), KOI-7368 b
($R_{\rm p}=2.22\pm 0.12\,R_{\oplus}$, $P=6.8\ {\rm days}$), KOI-7913 Ab
($R_{\rm p}=2.34\pm 0.18\,R_{\oplus}$, $P=24.2\ {\rm days}$), and Kepler-1627
Ab ($R_{\rm p}=3.85\pm 0.11\,R_{\oplus}$, $P=7.2\ {\rm days}$). The latter
Neptune-sized planet is in part of the Cep-Her complex called the $\delta$ Lyr
cluster (Bouma et al., 2022). Here we focus on the former three systems, which
are in other regions of the association. Based on kinematic evidence from
Gaia, stellar rotation periods from TESS, and spectroscopy, these three
objects are also $\approx$40 million years (Myr) old. More specifically, we
find that Kepler-1643 is $46^{+9}_{-7}$ Myr old, based on its membership in a
dense sub-cluster of the complex called RSG-5. KOI-7368 and KOI-7913 are
$36^{+10}_{-8}$ Myr old, and are in a diffuse region that we call CH-2. Based
on the transit shapes and high resolution imaging, all three objects are most
likely planets, with false positive probabilities of $6\times 10^{-9}$,
$4\times 10^{-3}$, and $1\times 10^{-4}$ for Kepler-1643, KOI-7368, and
KOI-7913 respectively. These planets demonstrate that mini-Neptunes with sizes
of $\approx$2 Earth radii exist at ages of 40 million years.

exoplanet evolution (491), open star clusters (1160), stellar ages (1581)

††journal: Astronomical Journal††software:  astropy (Astropy Collaboration et
al., 2018), astroquery (Ginsburg et al., 2018), exoplanet (Foreman-Mackey et
al., 2020), and its dependencies (Agol et al., 2020; Kipping, 2013; Luger et
al., 2019; Theano Development Team, 2016), PyMC3 (Salvatier et al., 2016),
tesscut (Brasseur et al., 2019), unpopular (Hattori et al., 2021), VESPA
(Morton, 2012, 2015), ††facilities:  Astrometry: Gaia. Imaging: Second
Generation Digitized Sky Survey. Keck:II (NIRC2). Spectroscopy:
Tillinghast:1.5m (TRES). Keck:I (HIRES). Photometry: Kepler, TESS, ZTF.

##  1 Introduction

The discovery and characterization of planets younger than a billion years is
a major frontier in current exoplanet research. The reason is that the
properties of young planets provide benchmarks for studies of planetary
evolution. For instance, young planets can inform our understanding of when
hot Jupiters arrive on their close-in orbits (Dawson & Johnson, 2018), how the
sizes of planets with massive gaseous envelopes evolve (Rizzuto et al., 2020),
the timescales for close-in multiplanet systems to fall out of resonance
(Izidoro et al., 2017; Arevalo et al., 2022; Goldberg & Batygin, 2022), and
whether and how mass-loss explains the radius valley (Lopez et al., 2012; Owen
& Wu, 2013; Fulton et al., 2017; Ginzburg et al., 2018; Lee & Connors, 2021).

The discovery of a young planet requires two claims to be true: the planet
must exist, and its age must be secured. Spaced-based photometry from K2 and
TESS has yielded a number of exemplars for which the planetary evidence comes
from transits, and the age is based on either cluster membership (Mann et al.,
2017; David et al., 2019; Newton et al., 2019; Bouma et al., 2020; Nardiello
et al., 2020) or else on correlates of youth such as stellar rotation,
photospheric lithium content, x-ray activity, and emission line strength (Zhou
et al., 2021; Hedges et al., 2021).

In this work, we leverage recent analyses of the Gaia data, which have greatly
expanded our knowledge of stellar groups (e.g., Cantat-Gaudin et al., 2018;
Kounkel & Covey, 2019; Kerr et al., 2021). So far, these analyses have mostly
leveraged 3D stellar positions and 2D on-sky tangential velocities. One
important result has been the discovery of diffuse streams and tidal tails
comparable in stellar mass to the previously known cores of nearby open
clusters (Meingast et al., 2019, 2021; Gagné et al., 2021). Even though these
streams are spread over tens to hundreds of parsecs, their velocity
dispersions can remain coherent at the $\sim$1 km s-1 level. Internal dynamics
and projection effects can also drive them to be much more kinematically
diffuse: in the Hyades, stars in the tidal tails are expected to span up to
$\pm 40{\rm\,km\,s^{-1}}$ in velocity relative to the cluster center
(Jerabkova et al., 2021). The stars in such diffuse regions can be verified to
be the same age as the core cluster members through analyses of color–absolute
magnitude diagrams (Kounkel & Covey, 2019), stellar rotation periods (Curtis
et al., 2019; Bouma et al., 2021), and chemical abundances (Hawkins et al.,
2020; Arancibia-Silva et al., 2020). While there are implications for our
understanding of star formation and cluster evolution (Dinnbier & Kroupa,
2020), a separate consequence is that we now know the ages of many more stars,
including previously known planet hosts.

The prime Kepler mission (Borucki et al., 2010) found most of the currently
known transiting exoplanets, and it was conducted before Gaia. It is therefore
sensible to revisit the Kepler field, given our new constraints on the stellar
ages.

Here, we expand on our earlier study of a $38^{+7}_{-6}$ Myr old Neptune-sized
planet in the Kepler field (Kepler-1627 Ab; Bouma et al. 2022). This planet’s
age was derived based on its host star’s membership in the $\delta$ Lyr
cluster. While our analysis of the cluster focused on the immediate vicinity
of Kepler-1627 in order to have a reasonable scope, it became clear that the
$\delta$ Lyr cluster seems to also be part of a much larger group of similarly
aged stars. This association, which is at an average distance of 330 pc from
the Sun, spans Cepheus to Hercules (galactic longitudes, $l$, between 40∘ and
100∘), at galactic latitudes between 0∘ and 20∘. An assessment of its
membership, substructure, and age distribution will be provided as part of the
1 kpc expansion of the SPYGLASS project (R. Kerr et al. in prep), where it is
given the name Cep-Her, after the endpoint constellations.

Our focus is on the intersection of the Cep-Her complex with the Kepler field.
Cross-matching the stars thought to be in Cep-Her against known Kepler Objects
of Interest (KOIs; Thompson et al. 2018) yielded four candidate cluster
members: Kepler-1627, Kepler-1643, KOI-7368, and KOI-7913. Given our earlier
analysis of Kepler-1627, we focus here on the latter three objects. After
analyzing the relevant properties of Cep-Her (Section 2), we derive the
stellar properties (Section 3) and validate the planetary nature of each
system using a combination of the Kepler photometry and high-resolution
imaging (Section 4). We conclude with a discussion of mini-Neptune size
evolution, and point out possible directions for future work (Section 5).

##  2 The Cep-Her Complex

Figure 1:  Positions and velocities of candidate members of the Cep-Her
complex. Top row: On-sky positions in galactic coordinates. Black points are
stars for which group membership is more secure than for gray points.
Kepler-1627 is in the outskirts of the $\delta$ Lyr cluster (Bouma et al.,
2022), which is centered at $(l,b)\approx(66^{\circ},12^{\circ})$. The Kepler
footprint is shown in gray. Middle row: Galactic positions. The Sun is at
$(X,Y,Z)=(0,0,20.8)$ pc; lines of constant heliocentric distance are shown
between 250 and 400 pc, spaced by 50 pc. Bottom row: Galactic tangential
velocities (left) and galactic longitudinal velocity versus galactic longitude
(right). The gray band in the lower-right shows the $\pm$1-$\sigma$ projection
of the Solar velocity with respect to the local standard of rest (Schönrich et
al., 2010). There is a strong spatial and kinematic overlap between
Kepler-1643 and RSG-5 (magenta; smaller circles). The local population of
candidate young stars around KOI-7368 and KOI-7913 is more diffuse – we call
this region “CH-2” (lime-green; larger circles). The selection method for
these groups is described in Section 2.2.

###  2.1 Previous Related Work

Our focus is on a region of the Galaxy 200 to 500 pc from the Sun, above the
galactic plane, and spanning galactic longitudes of $40^{\circ}$ to
$100^{\circ}$. Two rich clusters in this region are the $\delta$ Lyr cluster
(Stephenson, 1959) and RSG-5 (Röser et al., 2016). Each of these clusters was
known before Gaia. Their reported ages are between 30 and 60 Myr. Early
empirical evidence that these two clusters could be part of a large and more
diffuse population was apparent in the Gaia-based photometric analysis of pre-
main-sequence stars by Zari et al. (2018, compare their Figures 11 and 13 to
our Figure 1). Further kinematic connections and complexity were highlighted
by Kounkel & Covey (2019), who included these previously known groups in the
larger structures dubbed “Theia 73” and “Theia 96”111See their visualization
online at http://mkounkel.com/mw3d/mw2d.html (accessed 15 March 2022).
Important caveats, particularly for extended groups $\gtrsim 100$ Myr old,
were presented by Zucker et al. (2022).. The connection made by Kounkel &
Covey (2019) between the previously known open clusters and the other groups
in the region was made as part of an unsupervised clustering analysis of the
Gaia DR2 positions and on-sky velocities with a subsequent manual “stitching”
step. Their results support the idea that there is an overdensity of 30 to 60
Myr old stars in this region of the Galaxy. Kerr et al. (2021), in a volume-
limited analysis of the Gaia DR2 point-source catalog out to one third of a
kiloparsec, identified three of the nearest sub-populations of Cep-Her, dubbed
“Cepheus-Cygnus”, “Lyra”, and “Cerberus”. Kerr et al. (2021) reported ages for
each of these subgroups between 30 and 35 Myr.

###  2.2 Member Selection

The possibility that the $\delta$ Lyr cluster, RSG-5, and the sub-populations
identified by Kerr et al. (2021) share a common origin has yet to be fully
substantiated, but preliminary clustering results from the 1 kpc SPYGLASS
analysis (R. Kerr et al. in prep) suggest the presence of contiguous stellar
populations connecting each of these groups in both space and velocity
coordinates. In other words, the stars appear to be comoving, though with a
continuous gradient in velocity as a function of position. The lower panels of
Figure 1 show this in detail, where $v_{b}$ is the distance-corrected proper
motion in the direction of increasing galactic latitude, and
$v_{l^{*}}=v_{l}\cos b$ is the distance-corrected proper motion in the
direction of increasing galactic longitude after accounting for the local
tangent plane correction. Some, but not all, of the gradient in the
$v_{l^{*}}$ vs. $l$ plane can be understood through a projection effect
stemming from the Sun’s motion with respect to the local standard of rest (see
also Figure 11 by Zari et al. 2018). In this work, our primary interest in
this region of sky stems from the fact that a portion of it was observed by
Kepler (Figure 1, top panel). To further explore this sub-population, we
select candidate Cep-Her members through four steps, the first three being
identical to those described in Section 3 of Kerr et al. (2021). We briefly
summarize them here.

The first step is to select stars that are photometrically distinct from the
field star population based on Gaia EDR3 magnitudes $\\{G,G_{\rm RP},G_{\rm
BP}\\}$, parallaxes and auxiliary reddening estimates (Lallement et al.,
2019). This step yielded 1,097 stars with high-quality photometry and
astrometry. These stars are either pre-main-sequence K and M dwarfs due to
their long contraction timescales, or massive stars near the zero-age main
sequence due to their rapid evolutionary timescales.

The second step is to perform an unsupervised HDBSCAN clustering on the
photometrically selected population (Campello et al., 2015; McInnes et al.,
2017). The parameters we use in the clustering are
$\\{X,Y,Z,cv_{b},cv_{l^{*}}\\}$, where $c$ is the size-velocity corrective
factor, which is taken as $c=6\,{\rm pc/km\,s}^{-1}$ to ensure that the
spatial and velocity scales have identical standard deviations. Positions are
computed assuming the astropy v4.0 coordinate standard (Astropy Collaboration
et al., 2018). As input parameters to HDBSCAN, we set the minimum $\epsilon$
threshold past which clusters cannot be fragmented as $25$ pc in physical
space, and $25/c$ km s-1 in velocity. The minimum cluster size $N$ is set to
10, as is $k$, the parameter used to define the “core distance” density
metric. Core distance is the distance to the $k^{\rm th}$ nearest star, and
therefore $k$ acts as a smoothing parameter, where a larger value reduces the
influence of local overdensities smaller than the scale that interests us.

The unsupervised clustering in this case yielded 8 distinct subgroups. These
groups are then used as the “seed” populations, in which the stellar members
each have their own individually-assigned distances to their tenth-nearest
photometrically-young neighbor. Using those distances, we search the entire
Gaia EDR3 point source catalog for stars that fall within each star’s 10th
nearest-neighbor distance. This third step yields stars that are spatially and
kinematically close to the photometrically young stars, but which cannot be
identified as young based on their positions in the color–absolute magnitude
diagram.

The outcome of the analysis up to the point of the third step is shown in
Figure 1. To enable a selection cut that filters out field-star contaminants,
we also compute a weight metric, $D$, defined such that the group member with
the smallest core distance has $D=1$, the group member with the greatest core
distance has $D=0$, and the weight $D$ scales linearly between the two
extremes. After applying a set of quality cuts on the astrometry and
photometry222$\varpi/\sigma_{\varpi}>5$; $G/\sigma_{G}>50$; $G_{\rm
RP}/\sigma_{G_{RP}}>20$; $G_{\rm BP}/\sigma_{G_{BP}}>20$ this procedure yields
a distribution of weights $D$ that is well described by a log-normal
distribution with $\log_{10}\mathcal{N}(-1.55,0.61)$. To visualize the
results, in Figure 1 we show 12,436 objects with $D>0.02$ as gray points, and
4,763 objects with $D>0.10$ as black points. These thresholds were selected
visually based on the apparent purity with which they yielded pre-main-
sequence stars on a color–absolute magnitude diagram. The $\delta$ Lyr cluster
is visible at $(l,b)=(68^{\circ},15^{\circ}$) and
$(v_{l^{*}},v_{b})=(-4.5,-4){\rm\,km\,s^{-1}}$. RSG-5 is visible at
$(l,b)=(83^{\circ},6^{\circ})$,
$(v_{l^{*}},v_{b})=(5.5,-3.5){\rm\,km\,s^{-1}}$. Most of the other
subclusters, including in Cep-Cyg ($l,b=90^{\circ},7^{\circ}$) and Cerberus
($l,b=48^{\circ},18^{\circ}$) are too small or dispersed to have previously
been analyzed in great detail.

Our fourth and final step was to cross-match the candidate Cep-Her member list
against all known Kepler Objects of Interest. We used the Cumulative KOI table
from the NASA Exoplanet Archive from 27 March 2022, and also compared against
the q1_q17_dr25 table (Thompson et al., 2018). From the candidate members with
weights exceeding 0.02, this yielded 11 known false positives, 6 confirmed
planets, and 8 candidate planets (see Appendix A). To determine whether these
objects were potentially consistent with being i) planets, and ii) $\lesssim
10^{8}$ years old, we inspected the Kepler data validation reports and
Robovetter classifications. Youth was assessed based on the presence of
rotational modulation at the expected period and amplitude for stars at least
as young as the Pleiades (e.g., Rebull et al., 2020). Planetary status was
assessed through the Robovetter flags, and by requiring non-grazing transits
with ${\rm S/N}>10$. Four objects passed both cuts: Kepler-1627, Kepler-1643,
KOI-7368, and KOI-7913.

Figure 1 shows the positions of these KOIs along various projections.
Kepler-1643 is near the core RSG-5 population both spatially and
kinematically. KOI-7368 and KOI-7913 are in a diffuse region $\approx$40 pc
above RSG-5 in $Z$ and $\approx$100 pc closer to the Sun in $Y$. In tangential
galactic velocity space, there is some kinematic overlap between the region
containing the latter two KOIs and the main RSG-5 group.

We define two sets of stars in the local vicinity of our objects of interest.
For candidate RSG-5 members near Kepler-1643, we require:

$\displaystyle X/{\rm pc}$ $\displaystyle\in[45,75]$ $\displaystyle Y/{\rm
pc}$ $\displaystyle\in[320,350]$ $\displaystyle Z/{\rm pc}$
$\displaystyle\in[40,80]$ $\displaystyle v_{b}/{\rm km\,s^{-1}}$
$\displaystyle\in[-4,-2.5]$ $\displaystyle v_{l^{*}}/{\rm km\,s^{-1}}$
$\displaystyle\in[3.5,6],$

though RSG-5 does have a greater spatial extent toward smaller $X$ (Figure 1,
middle panels). For the diffuse stars near KOI-7368 and KOI-7913, we require

$\displaystyle X/{\rm pc}$ $\displaystyle\in[20,70]$ $\displaystyle Y/{\rm
pc}$ $\displaystyle\in[230,270]$ $\displaystyle Z/{\rm pc}$
$\displaystyle\in[75,105]$ $\displaystyle v_{b}/{\rm km\,s^{-1}}$
$\displaystyle\in[-3.5,-1.5]$ $\displaystyle v_{l^{*}}/{\rm km\,s^{-1}}$
$\displaystyle\in[2,6]$

and we call this latter set of stars “CH-2”, using the preliminary Cep-Her
(CH) subgroup identifier from R. Kerr et al. (in prep). These cuts yielded 173
candidate RSG-5 members, and 37 candidate CH-2 members. These stars are listed
in Appendix A, as is the set of Cep-Her candidates that was observed by
Kepler.

###  2.3 The Cluster’s Age

(a)

(b)

Figure 2:  Age-diagnostic diagrams from the stellar groups near Kepler-1643,
KOI-7368, and KOI-7913. Top row: Color–absolute magnitude diagram of candidate
Cep-Her members, plotted over candidate members of the $\delta$ Lyr cluster
($\approx 38$ Myr; Bouma et al. 2022) and the Gaia EDR3 Catalog of Nearby
Stars (gray background). The left and right columns shows stars in RSG-5 and
CH-2, respectively. The range of colors is truncated to emphasize the pre-
main-sequence; approximate spectral types are shown on the upper axes. Stars
that fall far below the cluster sequences are field interlopers. Bottom row:
TESS and ZTF-derived stellar rotation periods, with the Pleiades ($\approx
112$ Myr) and Praesepe ($\approx 650$ Myr) shown for reference (Rebull et al.,
2016; Douglas et al., 2017). The detection efficiency for reliable rotation
periods falls off beyond $(G_{\rm BP}-G_{\rm RP})_{0}\gtrsim 2.6$.

####  2.3.1 Color–Absolute Magnitude Diagram

Color–absolute magnitude diagrams (CAMDs) of the candidate RSG-5 and CH-2
members are shown in the upper row of Figure 2. The stars from the $\delta$
Lyr cluster are from Bouma et al. (2022), and the field stars are from the
Gaia EDR3 Catalog of Nearby Stars (Gaia Collaboration et al., 2021b). To make
these diagrams, we imposed the data filtering criteria from Gaia Collaboration
et al. (2018, Appendix B), which include binaries while omitting artifacts
from for instance low photometric signal to noise, or a small number of
visibility periods. We then corrected for extinction using the Lallement et
al. (2018) dust maps and the extinction coefficients $k_{X}\equiv A_{X}/A_{0}$
from Gaia Collaboration et al. (2018), assuming that $A_{0}=3.1E(B-V)$. This
yielded a mean and standard deviation for the reddening of $E(B-V)=0.036\pm
0.002$ for RSG-5, and $E(B-V)=0.017\pm 0.001$ for CH-2. By way of comparison,
in Bouma et al. (2022) the same query for the $\delta$ Lyr cluster yielded
$E(B-V)=0.032\pm 0.006$. Finally, for the plots we set the color axis to best
visualize the region of maximal age information content: the pre-main-
sequence.

The CAMDs show that for RSG-5, all but one of the candidate members are on a
tight pre-main-sequence locus. Quantitatively, 88/89 stars with $(G_{\rm
BP}-G_{\rm RP})_{0}\geq 1.5$ are consistent with being on the pre-main-
sequence. This implies a false positive rate of a few percent, at most. In
comparison, our reference sample (the $\delta$ Lyr candidates) has a false
positive rate of $\approx$12%, based on the number of stars that
photometrically overlap with the field population. For CH-2, our membership
selection gives 27 objects in the color range displayed, and 23 of them appear
to be consistent with being on the pre-main-sequence. This would imply a false
positive rate in CH-2 of $\approx$15%.

Figure 2 also shows that most RSG-5 and CH-2 members overlap with the $\delta$
Lyr cluster on the CAMD, and that the groups are therefore roughly the same
age. To quantify this, we use the method introduced by Gagné et al. (2020,
their Section 6.3). The idea is to fit the pre-main-sequence loci of a set of
reference clusters, and to then model the locus of the target cluster as a
linear combination of these reference cluster loci. For our reference
clusters, we used UCL, IC 2602, and the Pleiades, with the memberships
reported by Damiani et al. (2019) and Cantat-Gaudin et al. (2018)
respectively. We adopted ages of 16 Myr for UCL (Pecaut & Mamajek, 2016), 38
Myr for IC 2602 (David & Hillenbrand, 2015; Randich et al., 2018) and 112 Myr
for the Pleiades (Dahm, 2015). These assumptions and the subsequent processing
steps taken to exclude field stars and binaries were identical to those
described in Bouma et al. (2022). The mean and uncertainty of the resulting
age posterior are $46^{+9}_{-7}$ Myr for RSG-5, and $36^{+10}_{-8}$ Myr for
CH-2. For comparison, this procedure yields an age for the $\delta$ Lyr
cluster of $38^{+6}_{-5}$ Myr. The older isochronal age of RSG-5 is consistent
with its location relative to the $\delta$ Lyr cluster in the upper left panel
of Figure 2. Generally speaking, this method is expected to be accurate
provided that the metallicities of IC 2602 and the Cep-Her groups (RSG-5,
CH-2, and the $\delta$ Lyr cluster) are roughly identical. The spectroscopic
metallicities that we find in Section 3 suggest that this is indeed the case.
While in reality stellar populations do not evolve linearly in the dimensions
of absolute magnitude versus color, in our case the Cep-Her loci are nearly
indistinguishable from IC 2602 (e.g., Figure 3 of Bouma et al. 2022).
Systematic errors incurred in the age from the non-linear evolution are
therefore likely much smaller than the $\approx$10 Myr systematic uncertainty
in the absolute reference age for IC 2602 itself (David & Hillenbrand, 2015;
Randich et al., 2018).

####  2.3.2 Stellar Rotation Periods

An independent way to assess the age of the candidate cluster members is to
measure their stellar rotation periods. This approach can be achieved using
surveys such as TESS (Ricker et al., 2015) and the Zwicky Transient Facility
(ZTF, Bellm et al. 2019); it leverages a storied tradition of measuring
rotation periods of stars in benchmark open clusters (see e.g., Skumanich,
1972; Curtis et al., 2020). The TESS data in our case are especially useful,
since they provide 3 to 5 lunar months of photometry for all of our candidate
CH-2 and RSG-5 members.

We selected stars suitable for gyrochronology by requiring $(G_{\rm BP}-G_{\rm
RP})_{0}\geq 0.6$ to focus on FGKM stars that experience magnetic braking. For
TESS, we also restricted our sample to $G<16$, to ensure the stars are bright
enough to extract usable light curves from the full-frame images. The
magnitude cut corresponds to $(G_{\rm BP}-G_{\rm RP})_{0}<2.6$ ($\sim$M3V) at
the relevant distances. These cuts gave 19 stars in CH-2 and 42 stars in
RSG-5. We extracted light curves from the TESS images using the unpopular
package (Hattori et al., 2021), and regressed them against systematics with
its causal pixel model. We measured rotation periods using Lomb-Scargle
periodograms and visually vetted the results using an interactive program that
allows us to switch between TESS Cycles, select particular sectors, flag stars
with multiple periods, and correct half-period harmonics. For ZTF, we used the
same color cut to focus on FGKM stars, but restricted the sample to $13<G<18$
to avoid the saturation limit on the bright end and ensure sufficient
photometric precision at the faint end. We followed the procedure outlined in
Curtis et al. (2020): we downloaded $8^{\prime}\times 8^{\prime}$ image
cutouts, ran aperture photometry for the target and neighboring stars
identified with Gaia, and used them to define a systematics correction to
refine the target light curves.

The lower panels of Figure 2 show the results. In RSG-5, 39/42 stars have
rotation periods at least as fast as the Pleiades (93%). This numerator omits
the two stars with periods $>$$12$ days visible in the lower-left panel of
Figure 2. The age interpretation for these latter stars, particularly the
$\approx$M2.5 dwarf, is not obvious. Rebull et al. (2018) for instance have
found numerous M-dwarfs with 10-12 day rotation periods at ages of USco
($\sim$$8$ Myr), and some still exist at the age of LCC ($\sim$$16$ Myr;
Rebull et al. 2022). Regardless, since only one field star outlier seems to be
present on the RSG-5 CAMD, the fact that we do not detect rotation periods for
$\approx$7% of stars should perhaps be taken as an indication for the fraction
of stars for which rotation periods might not be detectable, due to e.g.,
pole-on stars having lower amplitude starspot modulation. Field star
contamination is another possible contributor.

For CH-2, 13/19 stars have rotation periods that are obviously faster than
their counterparts in the Pleiades. 4 stars, not included in the preceding
numerator, are M-dwarfs with rotation periods between 10 and 12.5 days. As
previously noted, the age interpretation for these M-dwarfs is ambiguous. If
none are cluster members, the rotation period detection fraction is 68%; if
all are members, it is 89%.

This sets an upper bound on the contamination fraction in our candidate CH-2
members at about one in three. Combined with the roughly one in six
contaminant rate implied by the earlier CAMD analysis, this suggests that the
sample of candidate CH-2 members is more polluted by field stars than the
RSG-5 sample.

It is challenging to convert these stellar rotation periods to a precise age
estimate, since on the pre-main-sequence the stars are spinning up due to
thermal contraction rather than down due to magnetized braking. Regardless,
the rotation period distributions of both CH-2 and RSG-5 seem consistent with
other 30 Myr to 50 Myr clusters (e.g., IC 2602 and IC 2391; Douglas et al.
2021). They also seem consistent with the false positive rates estimated from
the color–absolute magnitude diagrams.

##  3 The Stars

Table 1: Selected system parameters of Kepler-1643, KOI-7368, and KOI-7913.

Parameter | Value | Uncertainty | Comment
---|---|---|---
_Kepler-1643_
Stellar parameters: |  |  |
Gaia $G$ [mag] | $13.836$ | $\pm 0.003$ | A
$T_{\rm eff}$ [K] | $4916$ | $\pm 110$ | B
$\log g_{\star}$ [cgs] | $4.502$ | $\pm 0.035$ | C
$R_{\star}$ [R⊙] | $0.855$ | $\pm 0.044$ | C
$M_{\star}$ [M⊙] | $0.845$ | $\pm 0.025$ | C
$\rho_{\star}$ [g cm-3] | $1.910$ | $\pm 0.271$ | C
$P_{\rm rot}$ [days] | $5.106$ | $\pm 0.044$ | D
Li EW [mÅ] | $130$ |  $+6$, $-5$ | E
Transit parameters: |  |  |
$P$ [days] | $5.3426258$ | $\pm 0.0000101$ | D
$R_{\rm p}/R_{\star}$ | $0.025$ | $\pm 0.001$ | D
$b$ | $0.58$ | $\pm 0.05$ | D
$R_{\rm p}$ [R⊕] | $2.32$ | $\pm 0.14$ | D
$t_{14}$ [hours] | $2.41$ | $\pm 0.06$ | D
_KOI-7368_
Stellar parameters: |  |  |
Gaia $G$ [mag] | $12.831$ | $\pm 0.004$ | A
$T_{\rm eff}$ [K] | $5241$ | $\pm 100$ | F
$\log g_{\star}$ [cgs] | $4.499$ | $\pm 0.030$ | C
$R_{\star}$ [R⊙] | $0.876$ | $\pm 0.035$ | C
$M_{\star}$ [M⊙] | $0.879$ | $\pm 0.018$ | C
$\rho_{\star}$ [g cm-3] | $1.840$ | $\pm 0.225$ | C
$P_{\rm rot}$ [days] | $2.606$ | $\pm 0.038$ | D
Li EW [mÅ] | $236$ |  $+16$, $-14$ | E
Transit parameters: |  |  |
$P$ [days] | $6.8430341$ | $\pm 0.0000125$ | D
$R_{\rm p}/R_{\star}$ | $0.023$ | $\pm 0.01$ | D
$b$ | $0.50$ | $\pm 0.06$ | D
$R_{\rm p}$ [R⊕] | $2.22$ | $\pm 0.12$ | D
$t_{14}$ [hours] | $2.79$ | $\pm 0.07$ | D
_KOI-7913_
Stellar parameters: |  |  |
Gaia $G$ [mag] | $14.200$ | $\pm 0.003$ | A
$T_{\rm eff,A}$ [K] | $4324$ | $\pm 70$ | B
$T_{\rm eff,B}$ [K] | $4038$ | $\pm 70$ | B
$\log g_{\star,A}$ [cgs] | $4.523$ | $\pm 0.043$ | C
$R_{\star,A}$ [R⊙] | $0.790$ | $\pm 0.049$ | C
$M_{\star,A}$ [M⊙] | $0.760$ | $\pm 0.025$ | C
$\rho_{\star,A}$ [g cm-3] | $2.172$ | $\pm 0.379$ | C
$P_{\rm rot,A}$ [days] | $3.387$ | $\pm 0.016$ | D
$P_{\rm rot,B}$ [days] | $2.642$ | $\pm 0.067$ | D
(Li EW)A [mÅ] | $65$ |  $+8$, $-6$ | E
(Li EW)B [mÅ] | $42$ |  $+12$, $-19$ | E
$\Delta G_{\rm AB}$ [mag] | $0.51$ | $\pm 0.01$ | G
Apparent sep. [au] | $959.4$ | $\pm 1.9$ | G
Transit parameters†: |  |  |
$P$ [days] | $24.278571$ | $\pm 0.000263$ | D
$R_{\rm p}/R_{\star}$ | $0.027$ | $\pm 0.001$ | D
$b$ | $0.30$ | $\pm 0.15$ | D
$R_{\rm p}$ [R⊕] | $2.34$ | $\pm 0.18$ | D
$t_{14}$ [hours] | $4.40$ | $0.21$ | D

Note. — †The planet orbits KOI-7913 A (Section 4.3). (A) Gaia Collaboration et
al. (2021a). (B) HIRES SpecMatch-Emp (Yee et al., 2017). (C) Cluster isochrone
(Choi et al., 2016; Bressan et al., 2012). (D) Kepler light curve. (E)
HIRES/TRES (Bouma et al., 2021). (F) TRES SPC (Buchhave et al., 2012; Bieryla
et al., 2021). (G) Magnitude difference and apparent physical separation
between primary and secondary; from Gaia EDR3. (H) HIRES SpecMatch-Synth
(Petigura et al., 2017).

Many of the salient properties of the Kepler objects of interest in Cep-Her
can be gleaned from Figure 2. The stars span spectral types of G8V
(Kepler-1627) to K6V (KOI-7913 A). The secondary in the KOI-7913 system has
spectral type $\approx$K8V. And since a star with Solar mass and metallicity
arrives at the zero-age main sequence at $t\approx 40$ Myr (Choi et al.,
2016), these stars are all in the late stages of their pre-main-sequence
contraction.

The adopted stellar parameters are listed in Table 1. The stellar surface
gravity, radius, mass, and density are found by interpolating against the MIST
isochrones in reddening-corrected absolute $G$-band magnitude as a function of
$(G_{\rm BP}-G_{\rm RP})_{0}$ color (Choi et al., 2016). The statistical
uncertainties from this technique mostly originate from the parallax
uncertainties; the systematic uncertainties are taken to be the absolute
difference between the PARSEC (Bressan et al., 2012) and MIST isochrones.
Reported uncertainties are a quadrature sum of the statistical and systematic
components.

To verify these parameters, determine the stellar effective temperatures, and
to analyze youth proxies such as the Li 6708 Å doublet and H$\alpha$, we
acquired high resolution optical spectra. We also acquired high resolution
imaging for each system, to constrain the existence of visual companions,
including possible bound binaries. We give the system-by-system details in
Sections 3.1 through 3.3, and summarize their implications for the youth of
the stars in Section 3.4.

###  3.1 Kepler 1643

##### Spectra

For Kepler-1643, we acquired two iodine-free spectra from Keck/HIRES on the
nights of 2020 Aug 16 and 2021 Oct 25. The acquisition and analysis followed
the usual techniques of the California Planet Survey (Howard et al., 2010). We
derived the stellar parameters ($T_{\rm eff},\log g,R_{\star}$) using
SpecMatch-Emp (Yee et al., 2017), which yielded values in $<$$1$-$\sigma$
agreement with those from the cluster-isochrone method. This approach also
yielded $[{\rm Fe/H}]=0.13\pm 0.09$. Using the broadened synthetic
templates333The broadening is calculated using the joint rotational and
macroturbulent broadening kernel from Hirano et al. (2011), assuming that the
macroturbulent velocity scales with effective temperature similar to the
prescription from Doyle et al. (2014). The latter assumption could be a source
of systematic uncertainty in our equatorial velocity measurements, since the
macroturbulent velocity could be systematically higher (or lower) on the pre-
main-sequence than it is for more slowly rotating field stars. from SpecMatch-
Synth (Petigura et al., 2017), we found $v\sin i=9.3\pm
1.0\,{\rm\,km\,s^{-1}}$. The systemic radial velocity at the two epochs was
$-9.1\pm 1.9\,{\rm\,km\,s^{-1}}$ and $-7.8\pm 1.2\,{\rm\,km\,s^{-1}}$
respectively, and was calculated following the methods of Chubak et al.
(2012). To infer the equivalent width of the Li I 6708 Å doublet, we followed
the procedure described by Bouma et al. (2021). In brief, this involved
calculating the line width by numerically integrating a single best-fit
Gaussian over a local window, and estimating the uncertainties through a Monte
Carlo procedure in which the continuum normalization was allowed to vary
through a bootstrap approach based on the local scatter in the spectra. For
Kepler-1643, this yielded a strong detection: ${\rm EW}_{\rm
Li}=130^{+6}_{-5}$ mÅ, with values consistent at $<$$1$-$\sigma$ between the
two epochs. The quoted value does not correct for the Fe I blend at 6707.44 Å.
Given the purported age and effective temperature of the star, the lithium
equivalent width is somewhat low. We discuss this in greater depth in Section
3.4.

##### High-Resolution Imaging

We acquired adaptive optics imaging of Kepler-1643 on the night of 2019 June
28 using the NIRC2 imager on Keck-II. Using the narrow camera (FOV = 10.2″),
we obtained 4 images in the $K^{\prime}$ filter ($\lambda=2.12\,\mu$m) with a
total exposure time of 320 s. The images did not show any additional visual
companions. We analyzed the data following Kraus et al. (2016), and determined
the detection limits by analyzing the residuals after subtracting an empirical
PSF template. This procedure yielded contrast limits of $\Delta
K^{\prime}=4.1$ mag at $\rho=150$ mas, $\Delta K^{\prime}=5.8$ mag at
$\rho=300$ mas, and $\Delta K^{\prime}=8.3$ mag at $\rho>1000$ mas.

###  3.2 KOI-7368

##### Spectra

For KOI-7368, we acquired a spectrum on 2015 June 1 using the echelle
spectrograph (TRES; Fűrész et al. 2008) mounted at the Tillinghast 1.5m at the
Fred Lawrence Whipple Observatory. The Stellar Parameter Classification
pipeline for TRES has been described by Bieryla et al. (2021). It is based on
the synthetic template library constructed by Buchhave et al. (2012). The
resulting stellar parameters ($T_{\rm eff},\log g,R_{\star}$) agreed with
those from the cluster-isochrone method within $1$-$\sigma$. Auxiliary
spectroscopic parameters included the metallicity $[{\rm Fe/H}]=-0.02\pm
0.08$, the equatorial velocity $v\sin i=20.2\pm 1.0\,{\rm\,km\,s^{-1}}$, and
the systemic velocity ${\rm RV}_{\rm sys}=-10.9\pm 0.2\,{\rm\,km\,s^{-1}}$.
The Li 6708Å EW measurement procedure yielded ${\rm EW}_{\rm
Li}=236^{+16}_{-14}$ mÅ.

##### High-Resolution Imaging

We acquired adaptive optics imaging of KOI-7368 on the night of 2019 June 12,
again using NIRC2. The observational configuration and reduction were
identical as for Kepler-1643. No companions were detected, and the analysis of
the image residuals yielded contrast limits of $\Delta K^{\prime}=5.2$ mag at
$\rho=150$ mas, $\Delta K^{\prime}=6.7$ mag at $\rho=300$ mas, and $\Delta
K^{\prime}=8.7$ mag at $\rho>1000$ mas.

###  3.3 KOI-7913

##### Binarity

KOI-7913 is a binary. The north-west primary is $\approx$0.5 magnitudes
brighter than the south-east secondary in optical passbands. The two stars are
separated in Gaia EDR3 by $3\farcs 5$ on-sky, and have parallaxes consistent
within $1$-$\sigma$ (with an average $\varpi=3.66\pm 0.01$ mas). The apparent
on-sky separation is $959\pm 2$ au. The Gaia EDR3 proper motions are also very
similar. Since the two stars were resolved in the Kepler Input Catalog and are
roughly one Kepler pixel apart, an accurate crowding metric has already been
applied in the NASA Ames data products to correct the mean flux level (Morris
et al., 2017). This is important for deriving accurate transit depths.

##### Spectra

We acquired Keck/HIRES spectra for KOI-7913 A on the night of 2021 Nov 13, and
KOI-7913 B on the night of 2021 Oct 26. The SpecMatch-Emp machinery yielded
$T_{\rm eff,A}=4324\pm 70\,{\rm K}$, and $T_{\rm eff,B}=4038\pm 70\,{\rm K}$.
These temperatures as well as the other spectroscopic parameters agreed with
those from the cluster isochrone method within 1-$\sigma$. For the primary, we
also found $[{\rm Fe/H}]=-0.06\pm 0.09$, $v\sin i=13.3\pm
1.0\,{\rm\,km\,s^{-1}}$, and ${\rm RV}_{\rm sys}=-17.8\pm
1.1\,{\rm\,km\,s^{-1}}$. For the secondary, these same parameters were $[{\rm
Fe/H}]=-0.01\pm 0.09$, $v\sin i=10.7\pm 1.0\,{\rm\,km\,s^{-1}}$, and ${\rm
RV}_{\rm sys}=-18.8\pm 1.1\,{\rm\,km\,s^{-1}}$. The primary showed lithium in
absorption with ${\rm EW}_{\rm Li}=65^{+8}_{-6}$ mÅ, while the secondary had a
marginal detection of ${\rm EW}_{\rm Li}=42^{+12}_{-19}$ mÅ. . Both components
displayed H$\alpha$ in emission. Given the spectral types of the stars, these
observations are consistent with a $\approx$40 Myr age for KOI-7913 (see
Section 3.4).

##### High-Resolution Imaging

We acquired adaptive optics imaging of KOI-7913 on the night of 2020 Aug 27
using the NIRC2 imager. The observational configuration and reduction were
identical as before. The images showed KOI-7913 A, KOI-7913 B, and an
additional faint neighbor $\approx$$0\farcs 99$ due East of KOI-7913 B.
Applying the PSF-fitting routines from Kraus et al. (2016), the tertiary
object has a separation $\rho=4397\pm 3$ mas from the primary, at a position
angle $231.17^{\circ}\pm 0.02^{\circ}$, with $\Delta K^{\prime}=6.97\pm 0.04$.
While it is too faint to affect the interpretation of the transit signal, it
would be amusing if this faint neighbor were comoving and therefore part of
the system, because it would have a mass between 10 and 15 MJup at an assumed
age of 40 Myr. Additional imaging epochs will tell.

###  3.4 Spectroscopic Youth Indicators

Figure 3:  Spectroscopic youth diagnostics for Kepler-1627, KOI-7368,
Kepler-1643, and KOI-7913 AB.  The spectra are shown in the observed frame,
and the stars are sorted left-to-right in order of decreasing effective
temperature.

Figure 3 shows key portions of the HIRES and TRES spectra for the Kepler
objects in Cep-Her. Lithium absorption is obvious at 6708Å in all stars except
KOI-7913 B. H$\alpha$ is in emission for both components of KOI-7913, and in
absorption for the hotter stars. Here, we compare these observations against
benchmark open clusters in order to assess their implications for the stellar
ages.

####  3.4.1 Lithium

(a)  (b)

(c)

Figure 4:  Lithium 6708Å and H$\alpha$ equivalent widths for the objects of
interest compared to young open clusters and field stars.  Positive equivalent
width means absorption; negative equivalent width means emission. Top &
middle: The field stars are KOIs from Berger et al. (2018). The “40-50 Myr”
reference stars (left) are from IC 2602 (Randich et al., 2001) and Tuc-Hor
(Kraus et al., 2014). The “112 Myr” stars are from the Pleiades (Soderblom et
al., 1993; Jones et al., 1996; Bouvier et al., 2018). The statistical
uncertainties on the equivalent widths are shown, or else are smaller than the
markers. Bottom: The H$\alpha$ comparison is against Tuc-Hor ($\approx$$40$
Myr; Kraus et al., 2014) and the Pleiades (Fang et al., 2018).

Figure 4 compares the measured lithium equivalent widths of the Kepler objects
against a few reference populations. We selected reference studies from the
literature only when upper limits were explicitly reported. KOI-7368 and
KOI-7913 A have secure lithium detections, while for KOI-7913 B the detection
is marginal (${\rm EW}_{\rm Li}=42^{+12}_{-19}$ mÅ). For all three stars, as
well as for Kepler-1627 A, the observed lithium equivalent width is consistent
with the stellar effective temperatures and a $\approx 40$ Myr age.

Kepler-1643, in RSG-5, is conspicuously below the 40-50 Myr sequence in the
top panel of Figure 4, though above the field stars (${\rm EW}_{\rm
Li}=130^{+6}_{-5}$ mÅ).

Quantitatively, there are 14 reference stars within $\pm$150 K of Kepler-1643.
The mean and standard deviation of their lithium EWs is $255\pm 31$ mÅ, which
implies that Kepler-1643 is $4.0$-$\sigma$ discrepant from expectations.

The middle panel shows a comparison against the Pleiades, where Kepler-1643 is
more consistent with the observed dispersion in lithium.

One explanation for the low Li equivalent width in Kepler-1643 relative to the
comparison stars could be that it is a field interloper; another could be that
RSG-5 is much older than 50 Myr. We do not favor either explanation. RSG-5
cannot be much older than 50 Myr based on its proximity to the $\delta$ Lyr
cluster and IC 2602 in the CAMD, and because it is below the Pleiades in the
rotation versus color diagram (Figure 2). Kepler-1643 also seems highly
unlikely to be a field interloper, because we demonstrated a few-percent false
positive probability in our spatio-kinematic selection of RSG-5 members, and
there is a similar independent chance ($\approx$$1\%$) of a field K2V star
having a rotation period below the Pleiades (McQuillan et al., 2014). This
yields a puzzle: how could a star have spatial, kinematic, and rotational
evidence consistent with being in a $\approx$50 Myr cluster, but a low lithium
content?

Our preferred explanation for Kepler-1643’s meager lithium content is that the
reference samples of IC 2602 and Tuc-Hor stars may not fully explore all
possible lithium equivalent widths at this age. This would be somewhat
surprising since over a dozen stars have already been analyzed in the relevant
effective temperature range. However, considering the top panels of Figure 4,
it is also remarkable that in 50 million years, stars between $4500$ K and
$5200$ K go from having a tight lithium sequence to one with a dispersion
$\approx$$10\times$ greater. The existence of the Li dispersion in Pleiades-
age K-dwarfs has been known for decades; it has also been known that the stars
with the largest lithium abundances are also the most rapidly rotating (Butler
et al., 1987; Soderblom et al., 1993). More recent analyses of this
correlation have been reviewed by Bouvier (2020). The conclusion of that work
was that the origin of the rotation-lithium correlation likely lies within
pre-main-sequence stellar physics. If so, one would expect the IC 2602 and
Tuc-Hor K-dwarfs to show a larger intrinsic lithium dispersion. A recent
analysis of the $\approx$40 Myr NGC 2547 by Binks et al. (2022) suggests that
this may be the case, though that study only had $\approx$10 stars in the
relevant effective temperature range. An alternative explanation could be that
the overall metallicity of Cep-Her is different from Tuc-Hor and IC 2602, but
this seems unlikely given the near-solar metallicities we have measured for
the Kepler Objects of Interest. Broadly, these considerations suggest that
Cep-Her is a worthy object for further spectroscopic analyses of lithium near
the zero-age main sequence.

####  3.4.2 H$\alpha$

As shown in Figure 3, H$\alpha$ is in emission for both components of
KOI-7913, and in absorption for the hotter stars. Additionally, the emission
appears double-peaked for both of the KOI-7913 components. An important note
is that KOI-7913 A and KOI-7913 B were spatially resolved from each other
during data acquisition. Performing a cross-correlation between each of the
stars and the nearest matches in the Keck/HIRES template library, we also
found that the CCFs for both components of KOI-7913 showed no indications of
double-lined binarity (Kolbl et al., 2015).

Balmer line emission, particularly in H$\alpha$, is expected for low-mass
stars of this age. Kraus et al. (2014) for instance, in their survey of Tuc-
Hor ($\approx$40 Myr), observed that all cluster members with spectral types
$>$${\rm K4.5V}$ had H$\alpha$ in emission. This is consistent with our
observations: KOI-7913 shows H$\alpha$ in emission for both components, and in
absorption for all of our other Kepler objects (Figure 3, lower panel). The
double-peaked nature of the emission, though not always present, is also
common for active stars. Proxima Centauri, for instance, has double-peaked
H$\alpha$ emission (Collins et al., 2017). Given that we have ruled out
spectroscopic binarity, the most likely explanation is self-absorption:
photons near the center of the line see a greater optical depth from higher
layers of the chromosphere, while photons on the wings are too far from the
rest wavelength to excite electrons and be re-absorbed in the upper layers.
The exact details of when a star’s atmosphere reaches the conditions for such
self-absorption require non-local thermal equilibrium models of the
chromosphere (Short & Doyle, 1998; Fuhrmeister et al., 2005).

##  4 The Planets

(a)

(b)

(c)

Figure 5:  Raw and processed light curves for the Kepler Objects of Interest
in Cep-Her. Top: 50 day light curve segment from the 3.9 years of Kepler data.
The ordinate shows the PDCSAP median-subtracted flux in units of parts-per-
thousand ($\times 10^{-3}$). The dominant signal is from starspots; planetary
transit times are indicated with vertical dashed lines, but the individual
transits are not visible at this scale. Bottom: Phase-folded transits of
Kepler-1643, KOI-7913, KOI-7368, and Kepler-1627 with stellar variability
removed. The maximum a posteriori model is shown with the gray line, and the
residual after subtracting the transit model is vertically displaced. Windows
over 10 hours are shown. Gray points are individual flux measurements; black
points are binned to 20 minute intervals, and have a representative 1-$\sigma$
error bar in the center-right of each panel.

###  4.1 Kepler Data

The Kepler space telescope observed Kepler-1643, KOI-7913, and KOI-7368 at a
30-minute cadence between May 2009 and April 2013. For all three systems
quarters 1 through 17 were observed with minimal data gaps. The top panel of
Figure 5 shows a 50-day slice of the PDCSAP light curves for the three new
Cep-Her candidates, along with Kepler-1627. In PDCSAP, non-astrophysical
variability is removed through a cotrending approach that uses a set of basis
vectors derived by applying singular value decomposition to a set of
systematics-dominated light curves (Smith et al., 2017). In our analysis, we
used the PDCSAP light curves with the default optimal aperture (Smith et al.,
2016). Cadences with non-zero quality flags were omitted. In all cases, the
stars are dominated by spot-induced modulation with peak-to-peak variability
between 2% and 10%. These signals are much larger than the transits, which
have depth $\approx$0.1%. To quantify the stellar rotation periods, we
calculated the Lomb-Scargle periodogram for each Kepler quarter independently.
The resulting means and standard deviations are in Table 1.

###  4.2 Transit and Stellar Variability Model

Our goals in fitting the Kepler light curves are twofold. First, we want to
derive accurate planetary sizes and orbital properties. Second, we want to
remove the spot-induced variability signal to enable a statistical assessment
of the probability that the transit signals are planetary.

We fitted the data as follows. Given the transit ephemeris from Thompson et
al. (2018), we first trimmed the light curve to a local window around each
transit that spanned three transit durations before and after each transit
midpoint. The out-of-transit points in each local window were then fitted with
a fourth-order polynomial, which was divided out from the light curve. The
resulting flattened transits were then fitted with a transit model that
assumed quadratic limb darkening. The model therefore included 8 free
parameters for the transit ($\\{P,t_{0},\log R_{\rm
p}/R_{\star},b,u_{1},u_{2},R_{\star},\log g\\}$), 2 free parameters for the
light curve normalization and a white noise jitter ($\\{\langle
f\rangle,\sigma_{f}\\}$), and 5 fixed parameters for each transit.

We fitted the data using exoplanet (Foreman-Mackey et al., 2020). We assumed a
Gaussian likelihood, and sampled using PyMC3’s No-U-Turn Sampler (Hoffman &
Gelman, 2014), after having initialized to the the maximum a posteriori (MAP)
model. We used the Gelman & Rubin (1992) statistic, $\hat{R}$, as our
convergence diagnostic. The resulting fits are shown in the lower panels of
Figure 5, and the important derived parameters are in Table 1. The set of full
parameters and their priors are given in Appendix B.

A potential drawback of our approach is that to remove the starspot-induced
variability, we fixed 5 parameters per transit to their MAP values. An
alternative could be to fit the planetary transits simultaneously with the
starspot-induced variability using a quasiperiodic Gaussian process (GP). We
explored this approach, but ultimately prefer our model for its simplicity,
and for the benefit that the white noise jitter never trades off with any
parameter equivalent to a damping timescale for the coherence of the GP. It is
also computationally efficient, and it captures the planetary parameters about
which we care the most.

###  4.3 Planet Validation

In the future, it may be possible to obtain independent evidence for the
planetary nature of the Cep-Her planets, for instance by observing
spectroscopic transits. For now, it is of interest whether the transit signals
might be astrophysical false positives, or whether they are statistically more
likely to be planetary. We adopt the Bayesian framework implemented in VESPA
to assess the relevant probabilities (Morton, 2012, 2015). Briefly summarized,
the priors in VESPA assume the binary star occurrence rate from Raghavan et
al. (2010), direction-specific star counts from Girardi et al. (2005), and
planet occurrence rates as described by Morton (2012, Section 3.4). The
likelihoods are then evaluated by forward-modeling a synthetic population of
eclipsing bodies for each astrophysical model class, in which each population
member has a known trapezoidal eclipse depth, total duration, and ingress
duration. These summary statistics are then compared against the actual
photometric data to evaluate the probabilities of false positive scenarios
such as foreground eclipsing binaries, hierarchical eclipsing binaries, and
background eclipsing binaries.

##### Kepler-1643

Kepler-1643 b (KOI-6186.01) was already validated as a transiting planet by
Morton et al. (2016), who found a probability for any of the aforementioned
false positive scenarios of 9$\times$10-6. Repeating the calculation with our
own stellar-variability correction and the new NIRC2 imaging constraints, we
find ${\rm FPP}=6\times 10^{-9}$. Figure 5 shows the justification: the
transit is flat and has a high S/N ($\approx$$47$). The shape is therefore
nearly impossible to reproduce with eclipsing binary models.

Intriguingly, Kepler-1643 failed one of the data validation centroid shift
tests (see the q1_q17_dr25_koi data release): the angular distance between the
target star’s KIC catalog position and the position of the transiting source
was measured as $1\farcs 0$ at 4.4-$\sigma$. The reports show however that two
outlying quarters (2 and 6) drive the offset — the centroid locations from the
other Kepler quarters are consistent at $\lesssim 0\farcs 4$ (3-$\sigma$).
Bryson et al. (2013) showed that for typical field star KOIs without centroid
offsets, the mean offset distribution peaks at 0.3′′ (their Figure 23). By
comparison, stars with centroid offsets that can be localized to nearby stars
have a distribution that peaks at 7′′ (their Figure 32). The stellar
variability in Kepler-1643 complicates the centroid-based vetting tests,
because the shifts measured by these tests are determined from the in- and
out-of-transit flux-weighted centroids. For stars with significant spot-
induced variability there is no static baseline in either the in- or out-of-
transit phases, and so the centroid location may shift depending on the
rotation phase combined with the local scene. Based on these considerations,
the centroid-level diagnostics for Kepler-1643 appear to be consistent with
the transit signal being localized to the target star.

##### KOI-7368

KOI-7368.01 is listed on the NASA Exoplanet Archive as a “candidate” planet.
Morton et al. (2016) did not compute a false positive probability for the
system because their default trapezoidal fitting routine failed, presumably
due to the spot-induced variability. Our fitting approach rectifies this
point, and our new NIRC2 images revealed no new stellar companions. Performing
the relevant calculation, we find ${\rm FPP}=4\times 10^{-3}$. Though not as
convincing as Kepler-1643, this clears the threshold probability of 1 in 100
suggested by Morton et al. (2016) for calling a planet statistically
validated. The S/N of the transit is $\approx$$32$, which indicates that it is
unlikely to be caused by systematic noise in the light curve (see Figure 5).
The positional probability444Columns pp_host_rel_prob and pp_host_prob_score
on the KOI Positional Probabilities table at the NASA Exoplanet Archive
(Akeson et al., 2013). calculated by Bryson & Morton (2017) also indicates
that the transit signal shares its position with the target star.

It bears mentioning that KOI-7368 shows a centroid shift in the
q1_q17_dr25_koi validation reports, similar to Kepler-1643. For KOI-7368, the
reported offset is smaller, and less formally significant ($0\farcs 2$;
3.0-$\sigma$). Again, the data validation reports show that the shift is
caused by a few outlying quarters (4, 5, 8, and 12). Since the remaining
quarters show consistent scatter in their centroid locations, these outlying
quarters are likely also caused by the stellar variability, because their
directions are inconsistent across different quarters. Our NIRC2 imaging
independently shows that there are no known neighboring sources that could
cause an offset of the observed amplitude, as is also the case for
Kepler-1643.

##### KOI-7913

KOI-7913.01 is also currently listed on the NASA Exoplanet Archive as a
“candidate” planet. The Morton et al. (2016) analysis was of Q1-Q17 KOIs from
DR24, and therefore spanned KOI-1.01 to KOI-7620.01 (omitting KOI-7913.01).
However the results of the subsequent DR25 analysis by Morton et al. are
listed at the NASA Exoplanet Archive. The relevant table gives a probability
for the system being an astrophysical false positive of $1.4\times 10^{-4}$,
with the most likely false positive scenario being a blended eclipsing binary.
Repeating the calculation with our new detrending and NIRC2 contrast curves,
we find a similar result: ${\rm FPP}=1.3\times 10^{-4}$. Though the transit
has the lowest S/N of any of the objects discussed ($\approx$$14$), its low
FPP can be understood through its flat-bottomed shape, combined with its long
transit duration relative to most eclipsing binary models (Figure 5). The
positional probability calculation performed by Bryson & Morton (2017) yielded
a near-unity probability that the transit event is at the same location as the
host star, and so the cumulative evidence suggests that KOI-7913 Ab is indeed
a statistically validated planet. Its disposition has however previously
fluctuated from “false positive” to “candidate” (see Appendix C). The most
likely explanation is the presence of KOI-7913 B, which is located $\approx
0.9$ Kepler pixels away from Kepler-7913 A. While the $\approx$1.5 pixel FWHM
of the Kepler pixel response function implies that there is blending between
the two stars, the target-pixel level data for KOI-7913 B reveals an entirely
different stellar rotation period (Table 1), and no hint of the transit
signal. This implies that KOI-7913 B cannot host the planet.

##  5 Discussion & Conclusion

Figure 6:  Radii, orbital periods, and ages of transiting exoplanets. Planets
younger than a gigayear with ages more precise than a factor of three are
emphasized. The Cep-Her planets are Kepler-1643 b ($\square$), KOI-7368 b
($\triangledown$), KOI-7913 Ab (X), and Kepler-1627 Ab (+). Interesting trends
in the population of planets younger than $10^{8}$ years old include i) their
large sizes and ii) the lack of hot Jupiters. The new objects of interest in
Cep-Her have normal mini-Neptune sizes between 2 and 3 $R_{\oplus}$, which is
a novelty given their ages. Parameters are from the NASA Exoplanet Archive
(2022).

###  5.1 Normal-Sized Mini-Neptunes Exist at 40 Myr

The most significant novelty about the planets in Kepler-1643, KOI-7368, and
KOI-7913 is that their sizes (2.2 to 2.3 $R_{\oplus}$) are normal relative to
the known population of mini-Neptunes from Kepler. At field star ages, mini-
Neptune sizes span 1.8 $R_{\oplus}$ to 3.6 $R_{\oplus}$, with the most common
size being $\approx 2.4\,R_{\oplus}$ (Fulton et al., 2017). The known planets
younger than $10^{8}$ years are almost all larger, with sizes between 4 and
10$\,R_{\oplus}$ (Mann et al., 2016; David et al., 2016; Benatti et al., 2019;
David et al., 2019; Newton et al., 2019; Rizzuto et al., 2020; Bouma et al.,
2020; Mann et al., 2022). Figure 6 explores this by showing the sizes, orbital
periods, and ages of the known transiting planets, emphasizing planets with
precise ages. The smallest previously known planets comparable to the new Cep-
Her mini-Neptunes are AU Mic c ($3.0\pm 0.2\,R_{\oplus}$, see Martioli et al.
2021 and Gilbert et al. 2022), Kepler-1627 Ab ($3.8\pm 0.2\,R_{\oplus}$; Bouma
et al. 2022), and AU Mic d ($4.2\pm 0.2\,R_{\oplus}$; Plavchan et al. 2020).

The theoretical expectation is that mini-Neptunes with sizes of 2 to 3
$R_{\oplus}$ should be common at ages of $10^{7}$ to $10^{8}$ years. This
expectation is tied to inferences about the initial distributions of planetary
core mass, core composition, and atmospheric mass fraction (Owen & Wu, 2017).
The Kelvin-Helmholtz cooling timescale, which is tied to the entropy of the
planetary interior shortly after disk dispersal, also plays a significant role
(Owen, 2020). As an example, Rogers & Owen (2021) predicted that given a core
mass distribution peaked at $\approx$$4$ $M_{\oplus}$, an ice-poor rock/iron
core composition, and a typical H/He mass fraction of $\approx$4%, there
should be a single local maximum in planet occurrence rates at 2 to 3
$R_{\oplus}$, at times between 10 and 100 Myr. In other words, Rogers & Owen
(2021) predict the existence of a “radius mountain” at these early times,
rather than a “radius valley”. The models advanced by Gupta & Schlichting
(2020) and Lee & Connors (2021) agree that this local maximum should exist;
their differences lie in the mechanism for producing the radius valley, and in
whether a population of rocky planets is predicted to exist at the time of
disk dispersal.

Systems such as K2-25, V1298 Tau, HIP-67522, TOI-837, and TOI-1227 have sizes
that are anomalously large relative to the predicted peak in planet occurrence
at 2 to 3 $R_{\oplus}$. However, their large sizes can be accommodated by
invoking any of i) larger core masses, ii) more volatile-rich compositions,
iii) larger initial atmospheric mass fractions, or iv) longer thermal cooling
times. Secure mass measurements would help constrain this parameter space, but
the $\sim$1 km s-1 spot-induced radial velocity semi-amplitudes make measuring
the Doppler orbits very difficult (Cale et al., 2021; Zicher et al., 2022;
Klein et al., 2022). Regardless, the new Kepler-1643, KOI-7368, and KOI-7913
systems do demonstrate that at least some planets at 40 Myr have sizes that
are consistent with theoretical expectations for mini-Neptunes. While
selection effects imposed by spot-induced photometric variability are a likely
explanation for why planets this small have not previously been identified
(e.g., Zhou et al., 2021), future work should quantify this bias more
carefully, in order to enable empirical studies of how the planetary size
distribution changes at early times.

###  5.2 Is CH-2 a Coeval Population?

RSG-5, and Kepler-1643’s membership inside it, meet typical expectations for a
star claimed to be in an open cluster. RSG-5 is an obvious overdensity
relative to the local field, and our membership selection easily yielded a
clean pre-main-sequence locus (Figure 2). CH-2, and KOI-7913 and KOI-7368’s
membership inside it, do not meet these expectations in as obvious a manner.
This is because the CH-2 association is diffuse.

To quantify the density difference between CH-2 and RSG-5, we can compare the
spatial and velocity volumes searched for each group. For RSG-5, we drew 173
candidate members from a $30\,{\rm pc}\times 30\,{\rm pc}\times 40\,{\rm pc}$
rectangular prism, given a $1.5{\rm{\rm\,km\,s^{-1}}}\times
2.5{\rm{\rm\,km\,s^{-1}}}$ rectangle in apparent galactic velocity. For CH-2,
our 37 candidate members came from a rectangular prism of dimension $50\,{\rm
pc}\times 40\,{\rm pc}\times 30\,{\rm pc}$, and a rectangular box of
$2{\rm{\rm\,km\,s^{-1}}}\times 4{\rm{\rm\,km\,s^{-1}}}$. If we define the
searched volume in units of ${\rm pc}^{3}\,{\rm km^{2}}\,{\rm s^{-2}}$, then
the volume ratio of CH-2 to RSG-5 is 3.5 to 1. The ratio of number densities
(candidate members per unit searched volume) in RSG-5 relative to CH-2 is 16
to 1.

Given its low density, is CH-2 truly a star cluster? For this discussion, we
adopt the definition that a star cluster is a group of at least 12 stars that
was physically associated at its time of formation. The “12” is set to
distinguish star clusters from high-order multiples (see Krumholz et al.,
2019). We explicitly do not require a “star cluster” to be gravitationally
bound: dissolved clusters as well as their tidal tails are included in our
adopted definition of “clusters”. We similarly do not require a threshold
number of stars per unit spatial volume. The latter point acknowledges that an
important factor in cluster identification is also coherence in velocity
space. For instance, the Psc-Eri stream, which has a shape that can be
approximated as a 600 parsec-long cylinder with a radius of 30 parsecs, has a
number density roughly a factor of three times lower than even CH-2 (Röser &
Schilbach, 2020). However its existence is discernible because of the
$\lesssim 2.5$ km s-1 scatter in its cylindrical velocities. Perhaps once
stellar rotation periods and chemical abundances reach the same level of
ubiquity as stellar proper motions, they might enable further refinement in
our ability to discover stars that formed as part of the same event.

From a data-driven perspective, demonstrating that a group of stars was
physically associated at its time of formation is challenging. While some
young groups show kinematic evidence for expansion (Kuhn et al., 2019), many,
including Sco-Cen, do not (Wright & Mamajek, 2018). This complicates the
feasibility of deriving kinematic ages through traceback, as well as through
the expansion itself (see Crundall et al., 2019). A more minimal approach is
that suggested by Tofflemire et al. (2021): search for coeval, phase-space
neighbors, measure their ages, and determine if they share a common age. This
approach can demonstrate whether a star is currently associated with a set of
coeval stars, though it falls short of determining what the association looked
like in the past. Our analysis of CH-2 meets the latter standard for
demonstrating the existence of a $\approx$40 Myr stellar association.

It would be a worthy exercise to perform a similar search for coeval phase-
space neighbors on the entire dataset of known exoplanet hosts. For the time
being, we can offer the anecdotal point that in our experience, most stars do
not have dozens of 40 Myr neighbors within a local volume of a few  km s-1
and tens of parsecs.

###  5.3 Future work

##### Cep-Her

Our analysis to date has focused only on portions of Cep-Her that were
observed by Kepler: RSG-5, CH-2, and the $\delta$ Lyr cluster. In Bouma et al.
(2022) as well as this work, we have shown that these groups share similar
ages, and have kinematic correlations that suggest a common origin. With that
said, the membership and kinematics of the other Cep-Her groups shown in
Figure 1 deserve independent attention. An important aspect of the remaining
work will be to acquire radial velocities for a larger subset of the stars,
and to determine whether the traceback approach could be applicable. Wide-
field spectroscopic surveys such as LAMOST (Zhao et al., 2012) or SDSS-V
(Kollmeier et al., 2017) could enable such analyses for the brightest members,
while also providing sensitivity to the Li 6708 Å line. The Gaia DR3 RVS
spectra (released during review of this manuscript) could contain similar
velocity information down to spectral types of $\approx$K5V ($G_{\rm
RVS}\lesssim 14$), and perhaps also enable analyses of the calcium infrared
triplet as a youth indicator. The combination of more complete kinematics and
youth indicators would help in definitively unraveling the formation history
of the complex.

A number of worthy photometric projects also seem possible given the new
understanding of Cep-Her. One is asteroseismology of the $\delta$ Sct stars,
using either TESS or Kepler data (Bedding et al., 2020). For cases in which
the modes are resolved, this might yield age or metallicity estimates for the
subgroups independent of other methods. Other projects could include a more
comprehensive analysis of the stellar rotation periods, searches of the Kepler
light curves for exocomets (Zieba et al., 2019), and searches for missed
planets around the most rapid rotators.

##### Exoplanet demographics at early times

Our main motivation for finding new young planets is to help benchmark models
for planetary evolution. However demographic analyses of the known planets
between $10^{7}$ and $10^{9}$ years have so far been rather limited.
Approximately 40 such planets are now known (Figure 2). About half come from
K2, a quarter from TESS, and now a quarter from Kepler.

Given the current state of the field, a few reflections regarding experimental
design of a demographic survey focused on planetary evolution over the first
gigayear might be useful. The first is that such a project requires a set of
target stars with known ages. A promising way to compile relevant stars could
be to combine automated spatio-kinematic clustering from Gaia with rotation
periods measured using TESS (see the appendices of Bouma et al., 2022). The
second consideration is that all the known young planets smaller than $3$
$R_{\oplus}$ come from either K2 or Kepler. Demographic inferences based on
TESS are therefore limited to planetary sizes $\gtrsim 4$ $R_{\oplus}$, for
planets close-in to their host stars. It would be worthwhile to compare the
occurrence rates of both types of planets with those from the main Kepler
sample. One specific question that seems within reach would be to clarify
whether enough young stars have been searched for the dearth of young hot
Jupiters to be significant. Since the hot Jupiter occurrence rate is strongly
dependent on stellar mass and metallicity (Petigura et al., 2018, 2022),
particular care would be needed to select a sample of well-studied FGK dwarfs
for the measurement, likely using stars in Sco OB2, Cep-Her, and Orion. For
demographic studies focused on how mini-Neptune sizes evolve, the combined K2
and Kepler dataset would be the better primary source.

###  5.4 Summary

We have shown that Kepler-1643 b, KOI-7368 b, and KOI-7913 Ab are 40 to 50
million years old, and that each system is most likely planetary. The evidence
for the planetary interpretation comes from an application of VESPA to the
Kepler data, alongside new imaging from NIRC2. The validity of the VESPA
framework rests on the premise that non-astrophysical false positives can be
rejected. This seems to be the case for all three objects, even though
Kepler-1643 and KOI-7368 both show weak centroid offsets in specific quarters.
For both systems, the observed shifts are consistent with being caused by
starspot-induced variability in specific quarters spuriously moving the
stellar center-of-light. Independently, our imaging rules out companion stars
with the brightnesses and positions that would be needed to explain the
reported shifts. All three objects are therefore most likely planets.

Each system has multiple indicators of youth that support the reported ages.
For Kepler-1643, the strongest youth indicator is its physical and kinematic
association with RSG-5. Based on the color–absolute magnitude diagram, we are
able to select members of this cluster with a false positive rate of a few
percent (Figure 2). Kepler-1643 is one such member. While the stellar rotation
period period agrees with this assessment, the star’s lithium equivalent width
is marginally low, which might motivate future exploration of lithium
depletion across FGKM stars in RSG-5 (see Section 3.4).

The spatio-kinematic argument for the youth of KOI-7368 and KOI-7913 is weaker
because they are in an association of stars, CH-2, that is more diffuse. For
KOI-7913, stronger indicators of its age come from its binarity. Both stellar
components in KOI-7913 have isochronal ages consistent with $40$ Myr. Both
components also show H$\alpha$ in emission, which for the transit-hosting
$\approx$K6V primary is a strong indicator that the star is $\lesssim$100 Myr
old. KOI-7368 is more massive, and its Li 6708 Å measurement and stellar
rotation period provide independent verification of the star’s youth.

The astrophysical implication of these considerations is that planets
$\approx$2 Earth radii in size exist at ages of 40 million years. It will be
interesting to continue the push down to smaller planetary sizes at comparable
ages – the planetary detections we have presented are well above the average
detection significance for Kepler planets. There may still be room at the
bottom.

L.G.B. is supported by the Heising-Simons Foundation 51 Pegasi b Fellowship
and the NASA TESS GI Program (80NSSC21K0335 and 80NSSC22K0298). R.K. is
supported by the Heising-Simons Foundation. J.L.C. is supported by NSF
AST-2009840 and the NASA TESS GI Program (80NSSC22K0299). D.H. is supported by
the Alfred P. Sloan Foundation and NASA (80NSSC19K0597).

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##  Appendix A Candidate Cep-Her Members

##### Table 2

contains 338 candidate Cep-Her members with weights $D>0.02$ observed by
Kepler. The complete catalog of candidate Cep-Her members will be provided by
R. Kerr et al. in prep. using Gaia DR3; Table 2 is from an early version of
that analysis based on Gaia EDR3. Note that more restrictive weight cuts
should be imposed if one wishes to remove the majority of field star
interlopers. Table 2 was created by cross-matching candidate Cep-Her members
(selected using Gaia EDR3; Section 2.2) against a Kepler to Gaia DR2 cross-
match (the gaia-kepler.fun crossmatch database created by Megan Bedell). The
kic_dr2_ang_dist column is from the latter table. The EDR3 to DR2 match was
performed using the gaiaedr3.dr2_neighbourhood table, and the closest proper
motion and epoch-corrected angular distance neighbor was taken as the single
best match. The edr3_dr2_mag_diff column gives some indication of the
reliability of this EDR3 to DR2 conversion, as there are a few cases between
Gaia DR2 and EDR3 where partially resolved binaries became fully resolved.

##### Candidate matches between Cep-Her and the Kepler Objects of Interest:

The full list of candidate matches between Cep-Her and the Kepler Objects of
Interest is as follows – the objects are listed in order of descending
weights, $D$. Objects designated as confirmed planets included Kepler-1627,
Kepler-1643, Kepler-1331, Kepler-1062, and Kepler-1933. Objects designated as
candidate planets included KOI-5264, KOI-8007, KOI-7572, KOI-7375, KOI-7368,
KOI-7638, KOI-5632, and KOI-7913. Objects designated known false positive
planet candidates included KOI-6437, KOI-5988, KOI-7871, KOI-7655, KOI-5024,
KOI-61, KOI-4336, KOI-6812, KOI-3399, and KOI-6277. Finally, Kepler-1902
(KOI-3090) has one confirmed planet (KOI-3090.02), and one false positive
(KOI-3090.01). Of these objects, only Kepler-1627, Kepler-1643, KOI-7368, and
KOI-7913 met our requirements for potentially both i) having real planets, and
ii) being $\lesssim 10^{8}$ years old, based on the presence of rotational
modulation at the expected period and amplitude. Of the 14 confirmed and
candidate planets, 6 failed the first filter, and 7 independently failed the
second. One object was ambiguous: Kepler-1933. This system has a confirmed
$\approx$$1.4\,R_{\oplus}$ planet, a stellar rotation period of 6.5 days, and
an effective temperature of $\approx$$5750\,{\rm K}$. This places it near the
upper envelope of the rotation period vs. color distribution for the Pleiades,
making it unlikely to be $\approx$40 Myr old. Nonetheless, we acquired a
reconnaissance HIRES spectrum, and it yielded ${\rm EW}_{\rm Li}=93\pm 5$ mÅ.
Combined with the rotation period, this suggests an age for Kepler-1933
between 100 and 300 Myr. Based on these indicators, the system is unlikely to
be part of Cep-Her, but could merit further study.

##### Table 3

contains spatial, kinematic, astrometric, and rotation period information for
the 173 candidate RSG-5 members and 37 candidate CH-2 members described in
Section 2.2. These are the data used to make the lower panels of Figure 2; as
with Table 2, these are from a preliminary version of the SPYGLASS 1 kpc
expansion (R. Kerr et al. in prep). We adopted the ZTF period over the TESS
period in three cases: (1) Gaia EDR3 2081755809272821248: the top ZTF Lomb-
Scargle peak gave 6.61 days, while our default pipeline favored a TESS peak of
13.34 days; manual inspection of the light curve favors the former; (2) Gaia
EDR3 2081737529891330560: we found 3.06 days with TESS and 6.64 days with ZTF;
we suspect that TESS captured the 1/2-period harmonic and adopt the
approximately double value from ZTF; (3) 2134851775526125696: for this star,
we measured 1.91 days with TESS from Cycle 2, but noted that the signal
appeared to be missing in Cycle 4; ZTF found a strong signal at 12.23 days and
we adopt this as the star’s period. In the remaining overlap cases, we adopted
the average between TESS and ZTF as the final period. For these overlap stars,
the median absolute deviation is 0.01 days, showing remarkable consistency
between the surveys. For three stars, we failed to detect a period in TESS but
recovered one from ZTF; in all cases the periods appear to be 13–16 days.
These stars were: (1) Gaia EDR3 2129930258400157440, for which TESS showed a
flat light curve while ZTF yielded a 15.3-day period; (2) Gaia EDR3
2082376861542398336, LS found a 7.6-day period which we rejected during visual
validation; we found 15.4 days with ZTF, and we suspect that the weak/rejected
signal form TESS might have been a 1/2 period harmonic; (3) Gaia EDR3
2082397099429013120, similar to the previous case, we rejected a 6.7-day
signal from TESS and recovered a 12.8-day period with ZTF.

##  Appendix B Table of Transit Fit Parameters

Table 4 gives the full set of fitted and derived parameters from the model
described in Section 4.2. Priors and convergence statistics are also listed.

##  Appendix C Disposition History of KOI-7913

The disposition of KOI-7913.01 has been debated: in q1_q17_dr25_koi the source
was flagged as a false positive, with the comment “cent_kic_pos—halo_ghost”.
This comment and disposition were removed in the q1_q17_dr25_sup_koi data
release, which renamed the planet a “candidate”. In this note, we discuss the
interpretation of these flags (which do not apply to the system, according to
the latest analysis). We also discuss how the relative on-sky positions of
KOI-7913 A and KOI-7913 B affect the interpretation of the Kepler data.

As described by Thompson et al. (2018), the “cent_kic_pos” flag is an
indication that the measured source centroid is offset from its expected
location in the Kepler Input Catalog. The final Kepler data validation
reports, generated 2016 Jan 30, do not show this to be the case for KOI-7913.
Moreover, the statistical significance of any centroid offset is lower than
for KOI-7368 and Kepler-1643 (which both show centroid offsets that are likely
explained by the stellar variability).

What of the “halo_ghost” flag? This test measures the transit strength for the
pixels inside the aperture, and compares it to that measured in the ring of
pixels around said aperture (the “halo”). One usually expects the transit
signal to be strongest in the central aperture, rather than the halo. Two
types of false positive scenarios can change this and trigger the flag: the
first is when optical ghosts from bright eclipsing binaries reflect off the
CCD, and contaminate the target star. The second is when the PRF of nearby
stars directly overlaps with the PRF of the target star (see Thompson et al.
2018, Section A.5.2). The most obvious explanation for KOI-7913 is the latter
case, given that KOI-7913 B is $\approx 0.9$ Kepler pixels away from
Kepler-7913 A and so it usually part of the “halo”. Due to the on-sky
orientation of KOI-7913 A and KOI-7913 B, the default “optimal aperture”
selected in quarters 3, 7, 11, and 15 in fact included both stars, while for
the remaining quarters KOI-7913 B was excluded from the optimal aperture but
was included as part of the halo (see pages 35 through 71 of the data
validation reports.)

Given the orientation of the stars and the $\approx$1.5 pixel FWHM of the
Kepler pixel response function, some blending between the two stars is
present. The pointing geometries from quarters 3, 7, 11, and 15 however did
not affect the observed transit depths, which is an indication that the
crowding metric applied in the data products accurately correct the mean flux
level (Morris et al., 2017). Analysis of the target-pixel data that was
separately acquired for KOI-7913 B also reveals a different stellar rotation
period, and no hint of the transit signal.

Table 2: Candidate Cep-Her members observed by Kepler

Parameter | Example Value | Description
---|---|---
dr2_source_id | 2073765172933035008 | Gaia DR2 source identifier.
dr3_source_id | 2073765172933035008 | Gaia (E)DR3 source identifier.
kepid | 5641711 | KIC identifier.
ra | 297.40986 | Gaia EDR3 right ascension [deg].
dec | 40.89719 | Gaia EDR3 declination [deg].
weight | 0.041 | Strength of connectivity to other candidate cluster members.
v_l | -0.51 | Longitudinal galactic velocity, $v_{l^{*}}$ [km s-1].
v_b | -8.23 | Latitudinal galactic velocity [km s-1].
x_pc | -8035.4 | Galactocentric $X$ position coordinate [pc].
y_pc | 331.4 | Galactocentric $Y$ position coordinate [pc].
z_pc | 65.3 | Galactocentric $Z$ position coordinate [pc].
kic_dr2_ang_dist | 0.298 | Separation between KIC and Gaia DR2 positions [arcsec].
edr3_dr2_mag_diff | 0.002 |  $G$-band difference between EDR3 and DR2 source match [mag].

Note. — Table 2 is published in its entirety in a machine-readable format. One
entry is shown for guidance regarding form and content.

Table 3: Rotation periods and kinematics for candidate RSG-5 and CH-2 members.

Parameter | Example Value | Description
---|---|---
dr3_source_id | 2127562009133684480 | Gaia (E)DR3 source identifier.
ra | 291.02306 | Gaia EDR3 right ascension [deg].
dec | 46.43843 | Gaia EDR3 declination [deg].
parallax | 3.7099 | Gaia EDR3 parallax [milliarcsec].
ruwe | 0.981 | Gaia EDR3 renormalized unit weight error.
weight | 0.087 | Strength of connectivity to other candidate cluster members.
v_l | 2.78 | Longitudinal galactic velocity, $v_{l^{*}}$ [km s-1].
v_b | -2.87 | Latitudinal galactic velocity [km s-1].
x_pc | -8068.5 | Galactocentric $X$ position coordinate [pc].
y_pc | 256.0 | Galactocentric $Y$ position coordinate [pc].
z_pc | 86.3 | Galactocentric $Z$ position coordinate [pc].
(BP-RP)0 | -0.115 | Gaia $G_{\mathrm{BP}}$-$G_{\mathrm{RP}}$ color, minus $E$($G_{\mathrm{BP}}$-$G_{\mathrm{RP}}$).
(M_G)0 | 0.442 | Absolute $G$-band magnitude, corrected for extinction.
cluster | CH-2 | RSG-5 or CH-2.
Prot_Adopted | NaN | Adopted rotation period if available, else NaN [days].
Prot_TESS | NaN | TESS rotation period if available, else NaN [days].
Prot_ZTF | NaN | ZTF rotation period if available, else NaN [days].
Prot_Confused | NaN | Boolean flag; true when stars are photometrically blended.

Note. — Table 3 is published in its entirety in a machine-readable format. One
entry is shown for guidance regarding form and content.

Table 4:  Priors and posteriors for the transit models with local polynomials
removed.

Param. | Unit | Prior | Median | Mean | Std. Dev. | 3% HDI | 97% HDI | ESS | $\hat{R}-1$
---|---|---|---|---|---|---|---|---|---
_Kepler-1643_
$P$ | d | $\mathcal{N}(5.34264;0.01000)$ | 5.3426257 | 5.3426258 | 0.0000101 | 5.3426071 | 5.3426454 | 7884 | 1.1e-03
$t_{0}^{(1)}$ | d | $\mathcal{N}(134.38;0.02)$ | 134.3820 | 134.3820 | 0.0011 | 134.3799 | 134.3841 | 7390 | 3.7e-04
$\log R_{\rm p}/R_{\star}$ | – | $\mathcal{U}(-6.215;0.000)$ | -3.688 | -3.689 | 0.021 | -3.728 | -3.653 | 4449 | -7.8e-05
$b$ | – | $\mathcal{U}(0;1+R_{\mathrm{p}}/R_{\star})$ | 0.583 | 0.578 | 0.051 | 0.485 | 0.673 | 4705 | 1.9e-04
$u_{1}$ | – | Kipping (2013) | 0.26 | 0.29 | 0.21 | 0.00 | 0.68 | 5324 | 7.9e-04
$u_{2}$ | – | Kipping (2013) | 0.32 | 0.31 | 0.32 | -0.26 | 0.88 | 4908 | 8.4e-04
$R_{\star}$ | $R_{\odot}$ | $\mathcal{N}(0.855;0.044)$ | 0.851 | 0.851 | 0.045 | 0.766 | 0.933 | 7473 | 7.2e-04
$\log g$ | cgs | $\mathcal{N}(4.502;0.035)$ | 4.507 | 4.507 | 0.035 | 4.442 | 4.576 | 6530 | -1.4e-04
$\log\sigma_{f}$ | – | $\mathcal{N}(\log\langle\sigma_{f}\rangle;2.000)$ | -8.520 | -8.520 | 0.019 | -8.556 | -8.486 | 7966 | 2.1e-04
$\langle f\rangle$ | – | $\mathcal{N}(1.000;0.100)$ | 1.000 | 1.000 | 0.000 | 1.000 | 1.000 | 7488 | 3.2e-04
$R_{\rm p}/R_{\star}$ | – | – | 0.025 | 0.025 | 0.001 | 0.024 | 0.026 | 4449 | -7.8e-05
$\rho_{\star}$ | g cm-3 | – | 1.94 | 1.95 | 0.19 | 1.60 | 2.31 | 6081 | 9.4e-05
$R_{\rm p}$ | $R_{\mathrm{Jup}}$ | – | 0.207 | 0.207 | 0.012 | 0.184 | 0.231 | 6326 | 2.5e-04
$R_{\rm p}$ | $R_{\mathrm{Earth}}$ | – | 2.32 | 2.32 | 0.13 | 2.06 | 2.59 | 6326 | 2.5e-04
$a/R_{\star}$ | – | – | 14.31 | 14.32 | 0.47 | 13.49 | 15.23 | 6081 | 8.2e-05
$\cos i$ | – | – | 0.041 | 0.040 | 0.005 | 0.032 | 0.049 | 4929 | 2.4e-04
$T_{14}$ | hr | – | 2.41 | 2.41 | 0.06 | 2.30 | 2.53 | 4774 | 5.3e-04
$T_{13}$ | hr | – | 2.23 | 2.23 | 0.07 | 2.11 | 2.36 | 4561 | 6.2e-04
_KOI-7368_
$P$ | d | $\mathcal{N}(6.84294;0.01000)$ | 6.8430344 | 6.8430341 | 0.0000125 | 6.8430107 | 6.8430574 | 10045 | 6.5e-05
$t_{0}^{(1)}$ | d | $\mathcal{N}(137.06;0.02)$ | 137.0463 | 137.0463 | 0.0014 | 137.0437 | 137.0489 | 10303 | 9.2e-05
$\log R_{\rm p}/R_{\star}$ | – | $\mathcal{U}(-4.605;0.000)$ | -3.760 | -3.763 | 0.031 | -3.819 | -3.708 | 4043 | 6.3e-04
$b$ | – | $\mathcal{U}(0;1+R_{\mathrm{p}}/R_{\star})$ | 0.508 | 0.500 | 0.064 | 0.380 | 0.612 | 4434 | 3.5e-04
$u_{1}$ | – | Kipping (2013) | 0.98 | 0.95 | 0.27 | 0.43 | 1.42 | 5809 | -5.6e-05
$u_{2}$ | – | Kipping (2013) | -0.19 | -0.16 | 0.31 | -0.66 | 0.42 | 4387 | 2.6e-04
$R_{\star}$ | $R_{\odot}$ | $\mathcal{N}(0.876;0.035)$ | 0.874 | 0.874 | 0.036 | 0.804 | 0.938 | 9902 | 7.3e-04
$\log g$ | cgs | $\mathcal{N}(4.499;0.030)$ | 4.503 | 4.502 | 0.030 | 4.445 | 4.557 | 7527 | 2.7e-05
$\log\sigma_{f}$ | – | $\mathcal{N}(\log\langle\sigma_{f}\rangle;2.000)$ | -8.314 | -8.314 | 0.012 | -8.337 | -8.292 | 10636 | 1.3e-03
$\langle f\rangle$ | – | $\mathcal{N}(1.000;0.100)$ | 1.000 | 1.000 | 0.000 | 1.000 | 1.000 | 9742 | -2.9e-04
$R_{\rm p}/R_{\star}$ | – | – | 0.023 | 0.023 | 0.001 | 0.022 | 0.025 | 4043 | 6.3e-04
$\rho_{\star}$ | g cm-3 | – | 1.87 | 1.88 | 0.15 | 1.59 | 2.16 | 6829 | 3.4e-04
$R_{\rm p}$ | $R_{\mathrm{Jup}}$ | – | 0.198 | 0.198 | 0.011 | 0.177 | 0.218 | 5676 | 2.8e-04
$R_{\rm p}$ | $R_{\mathrm{Earth}}$ | – | 2.22 | 2.22 | 0.12 | 1.98 | 2.44 | 5676 | 2.8e-04
$a/R_{\star}$ | – | – | 16.67 | 16.68 | 0.45 | 15.86 | 17.54 | 6829 | 3.3e-04
$\cos i$ | – | – | 0.030 | 0.030 | 0.004 | 0.022 | 0.038 | 4518 | 5.4e-04
$T_{14}$ | hr | – | 2.79 | 2.79 | 0.07 | 2.65 | 2.93 | 4845 | 5.0e-04
$T_{13}$ | hr | – | 2.62 | 2.62 | 0.08 | 2.47 | 2.78 | 4575 | 3.1e-04
_KOI-7913_
$P$ | d | $\mathcal{N}(24.27838;0.01000)$ | 24.278553 | 24.278571 | 0.000263 | 24.278112 | 24.279085 | 4413 | 1.5e-03
$t_{0}^{(1)}$ | d | $\mathcal{N}(154.51;0.05)$ | 154.5121 | 154.5124 | 0.0063 | 154.4998 | 154.5237 | 5612 | 6.0e-04
$\log R_{\rm p}/R_{\star}$ | – | $\mathcal{U}(-5.298;0.000)$ | -3.599 | -3.602 | 0.046 | -3.689 | -3.519 | 4290 | 5.6e-04
$b$ | – | $\mathcal{U}(0;1+R_{\mathrm{p}}/R_{\star})$ | 0.312 | 0.298 | 0.153 | 0.005 | 0.523 | 2373 | 1.8e-03
$u_{1}$ | – | Kipping (2013) | 0.27 | 0.34 | 0.28 | 0.00 | 0.86 | 4491 | -6.1e-05
$u_{2}$ | – | Kipping (2013) | 0.21 | 0.23 | 0.32 | -0.31 | 0.86 | 5935 | 7.0e-04
$R_{\star}$ | $R_{\odot}$ | $\mathcal{N}(0.790;0.049)$ | 0.788 | 0.788 | 0.049 | 0.699 | 0.881 | 6847 | 2.8e-04
$\log g$ | cgs | $\mathcal{N}(4.523;0.043)$ | 4.526 | 4.527 | 0.042 | 4.450 | 4.606 | 5714 | 6.6e-04
$\log\sigma_{f}$ | – | $\mathcal{N}(\log\langle\sigma_{f}\rangle;2.000)$ | -7.197 | -7.197 | 0.019 | -7.230 | -7.161 | 6976 | 1.4e-04
$\langle f\rangle$ | – | $\mathcal{N}(1.000;0.100)$ | 1.000 | 1.000 | 0.000 | 1.000 | 1.000 | 6998 | 2.8e-04
$R_{\rm p}/R_{\star}$ | – | – | 0.027 | 0.027 | 0.001 | 0.025 | 0.030 | 4290 | 5.6e-04
$\rho_{\star}$ | g cm-3 | – | 2.20 | 2.21 | 0.25 | 1.78 | 2.70 | 5357 | 5.6e-04
$R_{\rm p}$ | $R_{\mathrm{Jup}}$ | – | 0.209 | 0.209 | 0.016 | 0.179 | 0.238 | 4882 | 1.3e-03
$R_{\rm p}$ | $R_{\mathrm{Earth}}$ | – | 2.34 | 2.34 | 0.18 | 2.01 | 2.67 | 4882 | 1.3e-03
$a/R_{\star}$ | – | – | 40.92 | 40.95 | 1.54 | 38.14 | 43.84 | 5357 | 6.6e-04
$\cos i$ | – | – | 0.008 | 0.007 | 0.004 | 0.000 | 0.013 | 2344 | 1.9e-03
$T_{14}$ | hr | – | 4.39 | 4.40 | 0.21 | 3.98 | 4.76 | 3952 | 5.6e-04
$T_{13}$ | hr | – | 4.13 | 4.13 | 0.22 | 3.72 | 4.55 | 3632 | 7.6e-04

Note. — ESS refers to the number of effective samples. $\hat{R}$ is the
Gelman-Rubin convergence diagnostic. Logarithms in this table are base-$e$.
$\mathcal{U}$ denotes a uniform distribution, and $\mathcal{N}$ a normal
distribution. (1) The ephemeris is in units of BJKD (BJDTDB-2454833).