# Early Results from the HUMDRUM Survey: A Small, Earth-mass Planet Orbits
TOI-1450A

Madison Brady Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA  Jacob L. Bean Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA  Andreas Seifahrt Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA  David Kasper Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA  Rafael Luque Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA  Guðmundur Stefánsson Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08540, USA  NASA Sagan Fellow  Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands  Julian Stürmer Landessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, D-69117 Heidelberg, Germany  David Charbonneau Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138, USA  Karen A. Collins Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  John P. Doty  Noqsi Aerospace Ltd., 15 Blanchard Avenue, Billerica, MA 01821, USA  Zahra Essack Department of Physics and Astronomy, The University of New Mexico, 210 Yale Blvd NE, Albuquerque, NM 87106, USA  Akihiko Fukui Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan  Instituto de Astrofisica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain  Ferran Grau Horta Observatori de Ca l’Ou, Carrer de dalt 18, Sant Martí Sesgueioles 08282, Barcelona, Spain  Christina Hedges  NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA  Coel Hellier Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK  Jon M. Jenkins NASA Ames Research Center, Moffett Field, CA 94035, USA  Norio Narita Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan  Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Instituto de Astrofisica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain  Samuel N. Quinn Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Avi Shporer Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Richard P. Schwarz Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Sara Seager Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  Keivan G. Stassun Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA  Department of Physics, Fisk University, Nashville, TN 37208, USA  Stephanie Striegel SETI Institute, Mountain View, CA 94043 USA/NASA Ames Research Center, Moffett Field, CA 94035 USA  Cristilyn N. Watkins Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Joshua N. Winn Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Roberto Zambelli  Società Astronomica Lunae, Castelnuovo Magra, Italy

###### Abstract

M dwarf stars provide us with an ideal opportunity to study nearby small
planets. The HUMDRUM (HUnting for M Dwarf Rocky planets Using MAROON-X) survey
uses the MAROON-X spectrograph, which is ideally suited to studying these
stars, to measure precise masses of a volume-limited ($<\,30$ pc) sample of
transiting M dwarf planets. TOI-1450 is a nearby (22.5 pc) binary system
containing a M3 dwarf with a roughly 3000 K companion. Its primary star,
TOI-1450A, was identified by TESS to have a 2.04d transit signal, and is
included in the HUMDRUM sample. In this paper, we present MAROON-X radial
velocities which confirm the planetary nature of this signal and measure its
mass at a nearly 10% precision. The 2.04d planet, TOI-1450Ab, has
$R_{b}\,=\,1.13\,\pm\,0.04\,R_{\oplus}$ and
$M_{b}\,=\,1.26\,\pm\,0.13\,M_{\oplus}$. It is the second-lowest-mass
transiting planet with a high-precision RV mass measurement. With this mass
and radius, the planet’s mean density is compatible with an Earth-like
composition. Given its short orbital period and slightly sub-Earth density, it
may be amenable to JWST follow-up to test whether the planet has retained an
atmosphere despite extreme heating from the nearby star. We also discover a
non-transiting planet in the system with a period of 5.07 days and a
$M\mathrm{sin}i_{c}\,=\,1.53\,\pm\,0.18\,M_{\oplus}$. We also find a 2.01d
signal present in the systems’s TESS photometry that likely corresponds to the
rotation period of TOI-1450A’s binary companion, TOI-1450B. TOI-1450A,
meanwhile, appears to have a rotation period of approximately 40 days, which
is in-line with our expectations for a mid-M dwarf.

Extrasolar rocky planets (511), M dwarf stars (982)

††software: AstroImageJ (Collins et al., 2017), Astropy (Astropy Collaboration
et al., 2013, 2018, 2022), astroquery (Ginsburg et al., 2019), batman
(Kreidberg, 2015), dynesty (Speagle, 2020), emcee (Foreman-Mackey et al.,
2013), exoplanet (Foreman-Mackey et al., 2020), juliet (Espinoza et al.,
2019), lightkurve (Lightkurve Collaboration et al., 2018), Numpy (Harris et
al., 2020), PyAstronomy (Czesla et al., 2019), PyMC3 (Salvatier et al., 2016),
Scipy (Virtanen et al., 2020), spectres (Carnall, 2017), TAPIR(Jensen, 2013),
theano (Theano Development Team, 2016), tpfplotter (Aller et al., 2020)

=50

##  1 Introduction

M dwarfs are ideal targets for the discovery of exoplanets, as many detection
methods have signal amplitudes inversely proportional to the size of the host
star. Atmospheric characterization surveys, using instruments like JWST, are
similarly well-suited to small stars. This fact, combined with the extreme
abundance of M dwarfs in the universe, means that we expect to find a large
number of small, rocky planets around M dwarfs, and that the compositions,
atmospheres, and formation histories of these planets should be relatively
easy to study compared to similar planets around Sun-like stars.

Predictions of the single-transit detection capabilities of JWST found that
the majority of rocky planets with detectable atmospheric molecules orbit
stars within 10 pc and with effective temperatures $T_{\mathrm{eff}}\,<\,3400$
K (Wunderlich et al., 2019). This highlights the importance of precisely
measuring the properties of M dwarf planets in our solar neighborhood. While
longer campaigns would allow us to target warmer and more distant stars, these
targets would be far more resource-intensive.

While it is relatively easy to study M dwarf planets, they may differ
substantially from planets around hotter stars. Firstly, M dwarfs tend to have
much longer active periods than other types of stars, and possess more
frequent and more energetic flare activity during these periods (see, e.g.,
Hawley et al., 2000, for more details). These events could contribute to the
vaporization of any surface oceans, resulting in substantial water loss as the
water dissociates in the atmosphere and the hydrogen escapes to space (Luger &
Barnes, 2015). Vaporized oceans could contribute to a strong abiotic O2
signature, which could be detected with JWST (Fauchez et al., 2020) and tell
us whether or not a planet’s surface is dessicated. However, these signatures
may be difficult to detect, as the high stellar activity and flare rate of M
dwarfs could contribute to rapid planet atmosphere escape compared to planets
at equivalent instellations around more luminous stars (Tilley et al., 2019).
As escape results in thinner atmospheres with higher mean molecular weights, M
dwarf planetary atmospheres may be difficult to characterize with instruments
like JWST.

Additionally, due to the dramatic lowering of an M dwarf’s luminosity as it
moves onto the main sequence (Baraffe et al., 2015), planets’ instellations
change drastically over their lifetimes. An in-situ planet around a M dwarf
likely received a much larger amount of radiation in the past than it does
today. As the relative location of the snow line during planet formation
dictates what sorts of planetesimals are incorporated into a forming planet,
this may mean that planets that are currently in the HZ (habitable zone)
formed when the snowline was far away (Mulders et al., 2015). This would
result in M dwarf planets having volatile-poor compositions with higher
densities than planets at similar instellations around other types of stars.
This has important implications with respect to the expected yields of
transiting atmosphere surveys, as well as the generalizability of their
results.

This situation is further complicated by the sample of M dwarf planets studied
in Luque & Pallé (2022), which found that low-radius M dwarf planets can
largely be split into two populations, and interpreted those populations as
rocky planets (like Earth) and planets that are 50% water by mass. It is
unlikely for these water-rich planets to form so close to the host star, so
these strange targets may be planets that formed beyond the ice line (where
the rock:volatile ratio is about 1:1, see Thiabaud et al., 2014; Marboeuf et
al., 2014) and migrated inwards towards their host star (which has been shown
to be possible in the population synthesis models by Burn et al., 2021).
However, Rogers et al. (2023) have shown that it is possible for a similar
population partitioning to be caused by a divide between rocky cores and
planets with massive hydrogen atmospheres, which is the more familiar
explanation for the well-known exoplanet radius gap (Fulton et al., 2017).
Understanding which of these scenarios is more likely could have a dramatic
impact on our understanding of M dwarf planet formation and habitability.

To evaluate these hypotheses, we need to measure the compositions of a large,
relatively unbiased sample of M dwarf planets. As planets of different
compositions can have very similar masses and radii (see, e.g., the mass-
radius models included in Zeng et al., 2019), it is vital to collect high-
precision masses and radii for nearby M dwarf planets with wide-field and/or
ground-based instruments. This will allow us to properly prioritize targets
for follow-up with expensive single-target instruments like JWST. Precise mass
measurements will also be helpful for atmospheric retrievals with JWST, as
Batalha et al. (2019) found that a 20% planetary mass precision is necessary
to perform precise planetary atmospheric retrievals. With these mass
measurements, we will be able to understand whether or not M dwarf planets
tend to differ systematically than planets around FGK stars.

While many nearby M dwarfs have transiting planets with 5% precision radii
measured with wide-field photometric surveys like Kepler (Borucki et al.,
2010) and TESS (Ricker et al., 2015), there is no accompanying radial velocity
(RV) survey that seeks to provide every single one of these planets with
precise masses. Thus, we are conducting HUMDRUM (HUnting for M Dwarf Rocky
planets Using MAROON-X), a volume-limited survey of nearby ($d\,<\,30\,pc$) M
dwarf ($T_{\mathrm{eff}}$ < 4000 K) planets with transits identified via TESS
with MAROON-X. By selecting targets that TESS has identified, the biases of
TESS can be estimated (using simulations such as Sullivan et al., 2015; Brady
& Bean, 2022, or with more advanced methods) to inform the biases of this
survey. The completeness of this survey will be discussed in greater detail in
following papers.

In this paper, we present RV measurements of one of the targets in the 30 pc
sample, TOI-1450A, and confirm the planetary nature of its 2.04d transit
signal. We are also able to measure the $\approx\,40$d rotation period of the
primary star, as well as identify a 2d signal in the TESS photometry that may
be due to the rotation of TOI-1450B. The planet’s short orbital period places
it well within the conservative limits for tidal locking from Barnes (2017),
meaning that the planet is likely tidally locked over Gyr timescales. As the
primary star is relatively cool ($T_{\mathrm{eff}}\,\approx\,3400$ K) and
nearby, it is relatively easy to observe with JWST. Its planet is likely rocky
and contributes to the growing sample of small nearby rocky planets that are
prime targets for follow-up with JWST.

In Section 2, we describe the TOI-1450 system. In Section 3, we describe the
observations and data used in this paper. We characterize the stars in Section
4 and the planets in Section 5. We discuss the fit system parameters in
Section 6 and provide a summary of our conclusions in Section 7.

##  2 The TOI-1450 System

TOI-1450 is a multiple star system, consisting of an A and B component
separated by a projected 76 AU on-sky (Mugrauer & Michel, 2020). At a distance
of 22.44 pc (Gaia Collaboration, 2020), the target falls in the 30 pc HUMDRUM
sample. As described by the TESS catalog (Stassun et al., 2019), TOI-1450A is
a $R_{\star}\,=\,0.474\,R_{\odot}$ M dwarf with an effective temperature of
$T_{\mathrm{eff}}\,=\,3407\,\pm\,157$ K. TOI-1450B is described by Mugrauer &
Michel (2020) as an M dwarf with a temperature of around 3000 K. Relations
from Giovinazzi & Blake (2022) indicate that, based on its Gaia RP magnitude,
TOI-1450B has a mass of around $M_{*}\,=\,0.14\,\pm\,0.01\,M_{\odot}$.
TOI-1450A and TOI-1450B are separated by 3.4 arcseconds and are thus easily
resolved by MAROON-X (Seifahrt et al., 2018), but not by TESS. The two stars
are close enough on-sky that TOI-1450B is not identified separately in the
2MASS catalogue (Cutri et al., 2003; Skrutskie et al., 2006), meaning that the
observed $JHK$ magnitudes of the system are likely influenced by blending.
Thus, it is important to consider the influence of TOI-1450B when evaluating
any photometric signals. The stellar parameters of TOI-1450A are provided in
Table 1.

Property | Value | Reference
---|---|---
RA | 19:07:24.83 | Gaia Collaboration (2020)
Declination | +59:05:09.36 | Gaia Collaboration (2020)
Spectral Type | M3.0V | Lépine et al. (2013)
$J$ Mag | 8.457 | Cutri et al. (2003)
$H$ Mag | 7.820 | Cutri et al. (2003)
$K$ Mag | 7.565 | Cutri et al. (2003)
M${}_{\star,~{}K_{s}}$ ($M_{\odot}$) | 0.471 $\pm$ 0.020 | Stassun et al. (2019)
R${}_{\star,~{}K_{s}}$ ($R_{\odot}$) | 0.474 $\pm$ 0.014 | Stassun et al. (2019)
L⋆ ($L_{\odot}$) | 0.0027 $\pm$ 0.007 | Stassun et al. (2019)
M${}_{\star,~{}\mathrm{SED}}$ ($M_{\odot}$) | 0.480 $\pm$ 0.024 | This work
R${}_{\star,~{}\mathrm{SED}}$ ($R_{\odot}$) | 0.483 $\pm$ 0.025 | This work
Distance (pc) | 22.443 $\pm$ 0.045 | Gaia Collaboration (2020)
Prot (d) | 30-40 | This work
Teff (K) | 3437 $\pm$ 86 | This work
[Fe/H] | -0.12 $\pm$ 0.17 | This work
log $g$ | 4.76 $\pm$ 0.04 | This work
Table 1: Properties of TOI-1450A, the host star.

##  3 Observations

###  3.1 MAROON-X Radial Velocities

MAROON-X (Seifahrt et al., 2018, 2022) is a high-precision echelle
spectrograph mounted on the 8.1-m telescope Gemini-North. It has a wavelength
coverage in the visible and near infrared encompassing 500 — 920 nm, selected
due to the expected density of M dwarf spectral lines in this regime. The
instrument has two CCDs, one “blue” (500 — 670 nm) and the other “red” (650 —
920 nm), which are both exposed simultaneously whenever a target is observed.
These two channels are treated as separate instruments for the purposes of
data analysis, as they are in different wavelengths and thus capture different
stellar signals. The data is reduced with a Python3 pipeline developed using
tools that were originally developed for the CRIRES instrument (Bean et al.,
2010) and the RVs are calculated using a version of serval (Zechmeister et
al., 2020) updated to work with MAROON-X data. RVs are found by stacking the
individual spectra to form a co-added template, which is then compared to the
individual observations using least-squares fitting to find the individual
exposure radial velocities. Telluric lines from Earth’s atmosphere are fully
masked to prevent them from skewing our analysis. MAROON-X observations are
calibrated with etalon spectra, which are observed simultaneously with the
science targets. The etalon spectra correct for velocity drifts of the
spectrograph but are themselves prone to long-term drifts due to aging of the
Zerodur spacer in the etalon optics. The long-term drifts as well as
additional offsets between runs were calibrated out using the spectra of a
ThAr lamp.

We observed TOI-1450A as a part of the HUMDRUM sample, observing the system
118 times between the months of April 2021 and July 2023. As the planet has an
orbital period very close to 2 days, we attempted to observe it multiple times
a night to improve our phase coverage. The data are shown in Figure 1. Most
exposures, observed simultaneously in the red and blue channels, were ten
minutes long, allowing for peak per-pixel SNRs of approximately 80 in the blue
channel and 200 in the red channel. The higher signal in the red channel is
expected due to the cool nature of the star. These peak SNRs translate to a
per-measurement observed RV error of roughly 1.0 m/s in the blue channel and
0.6 m/s in the red.

Figure 1: The MAROON-X RV observations of TOI-1450 with calibrated offsets
between runs applied when available (and mean-subtractions otherwise). The
red-channel data are plotted in red, while the blue-channel data are plotted
in blue.

Over the course of our observations, MAROON-X was operated in campaign mode on
Gemini-North, sharing a port with several other instruments. Thus, the target
was observed over the course of eight runs, discrete periods in which the
instrument was connected to the telescope. There are frequently small (on the
order of a few m/s or less) RV offsets between these runs. These offsets are
the result of both a constant, linear drift in our etalon calibrations over
time (due to the aging of the ThAr lamp) in addition to some smaller, random
instrumental shifts that are on the order of 1-2 m/s and occur between each
run, likely as the result of connecting and disconnecting the fiber optic feed
of the instrument from the telescope. Major instrumental failures or
interruptions, such as the major cooler system shutdown on the instrument
between April and May 2021, also imparted additional offsets. These offsets
were calibrated by the examination of RV standard stars, such as HD 88230, GJ
908, HD 3651, and HD 32147, during each MAROON-X run. Offsets between each
MAROON-X run were then calculated using the measured RVs of these stars. These
offsets (listed in Table 9) typically had errors on the order of 0.5 m/s,
which we included as priors when fitting the RV data. We also included the
effects of an observed linear RV drift observed in all MAROON-X targets in
August 2021, calibrated using several standard stars. As there were only three
observations in August, any error on this drift calibration is unlikely to
meaningfully affect our results.

We also observed TOI-1450B with MAROON-X twice in July 2022, in an effort to
characterize the star. As TOI-1450B is much dimmer than TOI-1450A, we
performed 30 minute exposures when observing this target. The resulting
spectra had per-pixel SNRs of around 20 in the blue channel and 70 in the red
channel. This data is insufficient to do a full characterization of the star’s
orbit but give us some insight into its spectral characteristics.

Our RVs are included in Table 3.1.

[centered tabular= |c|c|c|c|, table head=Time (BTJD) Channel RV (m/s) RV Error
, late after line=
, late after last line =
, ] TOI1450_RVs_truncated.csvt=,͡ channel=, rv=, erv= \-

Table 2: The first several MAROON-X RVs collected on TOI-1450A. The times are
in BTJD (BJD - 2,457,000). The rest of the RVs, as well as the values of the
activity indicators, are available online.

###  3.2 TESS Photometry

The Transiting Exoplanet Survey Satellite (TESS , Ricker et al., 2015) is a
wide-field all-sky photometric survey which observes large chunks of the sky
in 27-day sectors.

The transiting signal around TOI 1450 was originally identified after the SPOC
conducted a transit search of Sector 14 with an adaptive, noise-compensating
matched filter (Jenkins, 2002; Jenkins et al., 2010, 2020) identifying a
threshold-crossing event (TCE) for which an initial limb-darkened transit
model was fitted (Li et al., 2019) and a suite of diagnostic tests were
conducted to help determine the planetary nature of the signal (Twicken et
al., 2018). The transit signature was also detected in a search of Full Frame
Image (FFI) data by the Quick Look Pipeline (QLP) at MIT (Huang et al., 2020a,
b). The TESS Science Office (TSO) reviewed the vetting information and issued
an alert on 14 November 2019 (Guerrero et al., 2021). The signal was
repeatedly recovered as additional observations were made and the transit
signature passed all the diagnostic tests presented in the Data Validation
reports. According to the difference image centroiding tests, the host star is
located within 5.95$\pm$ 3.48 arcseconds of the source of the transit signal.

TOI-1450 has been observed in 27 sectors (14-17, 19-26, 40-41, 47, 49-60), and
the data encompass dates from July 2019 through January 2023. This is
substantially more data than is available for most other TESS targets, which
typically have only a few sectors’ worth of data available. The data are
available at a two-minute cadence, much faster than the thirty-minute cadence
full frame images (FFIs). Starting in sector 40, as a part of the TESS
extended mission, the data are available with a 20-second cadence, allowing
for further refinements in our understanding of the transit shape. The data
from sectors 49-54 overlap with MAROON-X observations. All image data were
reduced and analyzed by the TESS Science Processing Operations Center at NASA
Ames Research Center (Jenkins et al., 2016).

Given the large pixels of TESS (21 arcseconds), we checked the target field to
estimate how much the transit signal was diluted by light from other stars and
to ensure that we targeted the correct star for follow-up. The presence of
nearby bright stars can result in smaller planetary transits and thus an
underestimate of the planet’s radius.

Investigating the crowding is fairly simple using the code tpfplotter (Aller
et al., 2020), which plots the location of the TESS target, the TESS aperture
mask, and the locations of any nearby stars found via the GAIA EDR3 (Gaia
Collaboration, 2020). Figure 2 shows the stars that have a chance of causing
flux contamination in the Sector 14 data. It is obvious that most of the stars
that fall within the TOI-1450 aperture are substantially dimmer than
TOI-1450A, with the brightest other star (TOI-1450B) having a $\Delta
m_{G}\,=\,3.78$.

Figure 2: The crowding around TOI-1450. The sky, as seen by TESS in Sector 14,
is shown, with the specific aperture mask used marked in red. The star studied
in this paper is marked with a white X, while nearby stars (identified via
GAIA) are shown as red circles, with their size representative of their $G$
magnitude relative to TOI-1450A. Only stars with a $\Delta m_{G}\,\leq\,6$ are
shown, as the brightest stars contribute maximally to the light curve flux
contamination.

We can determine the precise amount of dilution from the B star in the TESS
light curve by estimating its $T$ magnitude $m_{T}$. Using the relationship
between the Gaia colors and $m_{T}$ from Stassun et al. (2019), we find that
TOI-1450B has $m_{T}\,=\,13.70$. This means that $\Delta m_{T}\,=\,3.67$,
corresponding to a dilution of around 3.3%. Even when also considering the
other stars in the aperture, the dilution is less than 5%. If the transit is
occurring around the B star, the dilution is $>\,90\%$, meaning that the
transit appears to be much shallower than it actually is. However, our
MAROON-X RVs confirm that the 2d transiting planet is orbiting around the
primary star (see Section 5.2). Thus, the corrections we need to make to the
light curve due to dilution are minimal, and are already performed by the
Presearch Data Conditioning SAP (PDCSAP) algorithm (for more details on how
crowding is dealt with by the PDCSAP algorithm, see Stumpe et al., 2012; Smith
et al., 2012; Stumpe et al., 2014).

###  3.3 ASAS Photometry

The TOI-1450 system has also been observed with ASAS-SN (Shappee et al., 2014;
Kochanek et al., 2017), an all-sky photometric survey. The photometry, as of
11/14/2023, is shown in Figure 3. Given ASAS-SN’s precise calibrations and
long time baseline, it is a better photometer for examining long-term signals
in the TOI-1450 light curve than TESS. However, as ASAS-SN images typically
have cadences on the order of days (while the transits have one-hour
durations), they are not useful for detecting the planetary transits. ASAS-SN
images have larger pixels (8 arcseconds) than the separation between TOI-1450A
and TOI-1450B, meaning that the observed light curves are the result of the
blended light of the two targets. Given how much dimmer TOI-1450B is than
TOI-1450A, however, we can expect TOI-1450A to be the dominant source of flux
in the ASAS-SN curve.

Figure 3: Top: ASAS-SN $V$ and $g$ band photometry of TOI-1450, normalized to
unit mean flux. Bottom: 1d-binned WASP photometry of TOI-1450.

TOI-1450 has flux measurements in $V$ and $g$ (filters centered on 551 and 480
nm, respectively) available in the ASAS-SN public database111https://asas-
sn.osu.edu/. We downloaded flux measurements from the online database,
processing them using the publicly available image subtraction photometry
pipeline, which performs photometry on coadded, image subtracted data. As
these light curves use coadded data, they contain fewer individual data points
but are less noisy than the pure aperture photometry light curves. There were
309 total points in the $V$ band and 572 total measurements in the $g$ band.

The $V$ band data are older, encompassing times from November 2012 to October
2018, with an median observation rate of once per every three days. The $g$
data are the result of an overall instrument overhaul and are both more recent
(October 2017 to the present day) and have a more rapid cadence (with a median
cadence of one observation every two days).

###  3.4 LCOGT Photometry

The TESS pixel scale is $\sim 21\arcsec$ pixel-1 and photometric apertures
typically extend out to roughly 1 arcminute, generally causing multiple stars
to blend in the TESS photometric aperture. To attempt to determine the true
source of the TESS detection, we acquired ground-based time-series follow-up
photometry of the field around TOI-1450A as part of the TESS Follow-up
Observing Program (TFOP; Collins, 2019)222https://tess.mit.edu/followup.

We observed two full transit windows of TOI-1450.01 on UTC 2022 July 6 and UTC
2022 August 17 in Pan-STARRS $z$-short band from the the Las Cumbres
Observatory Global Telescope (LCOGT; Brown et al., 2013) 1 m network nodes at
Teide Observatory on the island of Tenerife (TEID) and McDonald Observatory
near Fort Davis, Texas, United States (McD), respectively. We also observed
one full transit window on UTC 2022 September 03 simultaneously in Sloan
$g^{\prime}$, $r^{\prime}$, $i^{\prime}$, and Pan-STARRS $z$-short from the
LCOGT 2 m Faulkes Telescope North at Haleakala Observatory on Maui, Hawai’i.
The Faulkes Telescope North is equipped with the MuSCAT3 multi-band imager
(Narita et al., 2020). The images were calibrated by the standard LCOGT BANZAI
pipeline (McCully et al., 2018) and differential photometric data were
extracted using AstroImageJ (Collins et al., 2017). We used circular
photometric apertures centered on TOI-1450A with radius $5\farcs 8$ for the 1
m observations and $4\farcs 9$ for the MuSCAT3 observation to extract the on-
target differential photometry. The nearest known neighbor in the Gaia DR3
catalog (Gaia DR3 2155540255030594688; TOI-1450B) is $\sim 3\farcs 3$ north of
TOI-1450A and is blended in the apertures. The light curves are included in
the joint modeling described in Section 5.3. Using smaller $\sim 3\arcsec$
photometric apertures, we tentatively detected the $\sim 0.5$ ppt transit in
TOI-1450A, but with higher scatter and systematics in the light curves. We
also ruled out nearby eclipsing binaries (NEBs) within 2.5$\arcmin$ of
TOI-1450A, except for the $\sim 3\farcs 2$ neighbor TOI-1450B, which is
heavily contaminated by the $\sim 30$ times brighter target star.

###  3.5 WASP Photometry

TOI-1450 was also observed by the WASP project at the Observatorio del Roque
de los Muchachos (Pollacco et al., 2006) between the years of 2008 and 2010.
The photometry is shown in Figure 3. The WASP transit-search operated arrays
of 8 cameras using 200mm, f/1.8 lenses with a broadband filter spanning
400–700 nm, backed by 2048x2048 CCDs giving a plate scale of
$13.7\arcsec$/pixel (Pollacco et al., 2006). The large pixel size means that
the WASP data also features blending between TOI-1450A and TOI-1450B on a
single pixel. TOI-1450 was observed over spans of $\sim$ 120 days in each
year. Observations on every clear night, with a typical 15-min cadence,
accumulated 27 600 photometric data points.

##  4 Analysis

###  4.1 Stellar Parameter Fits

####  4.1.1 Direct Comparison to PHOENIX Spectra

As we currently lack an interferometic measurement of the star’s radius, the
parameters of TOI-1450 are those from the TESS catalog (Stassun et al., 2019),
which takes its parameters from Muirhead et al. (2018), which makes use of the
relations from Mann et al. (2015) and Mann et al. (2019) to estimate the
stellar characteristics.

We can provide an independent measurement of the stellar parameters by
comparing the MAROON-X spectra directly to stellar atmospheric models,
allowing us to find the $T_{\mathrm{eff}}$, log $g$, and [Fe/H] of TOI-1450
directly. We thus produced a code, with a similar methodology as that from
Passegger et al. (2018), which estimates the stellar parameters by comparing
the gathered MAROON-X spectrum to a set of interpolated, broadened PHOENIX
(Husser et al., 2013) models. We consider both rotational broadening and the
derived instrument broadening order-by-order of MAROON-X, which can be
inferred directly from the etalon calibration files.

Instead of fitting a model to the entire MAROON-X spectrum, we have decided to
primarily focus on regions that have shown to be sensitive to the parameters
of interest, inspired by those used by CARMENES (Passegger et al., 2018),
which is an instrument with very similar wavelength coverage to MAROON-X.
Avoiding a full spectral comparison both makes our algorithm run substantially
faster and allows us to focus primarily on the regions that are known to be
the most indicative and/or modelled accurately. We have removed wavelength
regions that are not typically observed by MAROON-X and have avoided including
regions where the model spectra typically are very poor fits to the observed
data. Table 3 indicates the precise wavelengths of these regions, as well as
the molecular lines these regions typically encompass. We note that we have
removed the K I region from our analysis, as we found that the PHOENIX models
were poor fits to this region in our models.

Line | $\lambda_{\mathrm{min}}$ | $\lambda_{\mathrm{max}}$
---|---|---
TiO | 7055.0 | 7074.0
TiO | 7085.0 | 7114.0
TiO, Ca I | 7125.0 | 7170.0
Ti I | 8414.3 | 8415.0
Ti I | 8428.2 | 8429.5
Ti I | 8436.8 | 8438.7
Ti I, Fe I | 8470.2 | 8471.1
Fe I | 8516.0 | 8517.0
Fe I, Ti I | 8676.5 | 8678.2
Ti I | 8684.5 | 8685.8
Mg I | 8808.7 | 8809.7
Fe I | 8826.0 | 8827.5
Table 3: The rest-frame wavelengths of the regions considered in our spectral
analysis, inspired by (Passegger et al., 2018) and slightly modified to
include only regions where the models don’t show strong systematic deviations
from the data.

The PHOENIX models from Husser et al. (2013) have varying temperatures,
surface gravities, metallicities, and $\alpha$ element abundances. For the
sake of our fits, we included models with $4\,<\,$log $g\,<6\,$,
$2300\,<\,T_{\mathrm{eff}}\,<\,5000$ K, and $-1.0\,<\,[Fe/H]\,<\,1.0$. We
allow the $\alpha$ abundances of the models to remain fixed at the solar
values, as PHOENIX models with nonzero $\alpha$ values are not available for
stars under 3500 K. These limits are guided both by the the types of stars in
our sample (M dwarfs) and the actual PHOENIX models available for download. We
have found that all of the stars in the HUMDRUM sample fall well within these
limits (according to the TESS parameter estimates).

We also applied rotational and instrumental broadening to these models before
performing any fits. We first used the rotational convolution kernel from Gray
(2008), and then applied a kernel that represents the instrumental broadening
of MAROON-X (estimated from the instrument’s etalon calibrations). After this,
we resampled the models according to the MAROON-X wavelengths using the
spectres python module (Carnall, 2017). We allowed the rotation to vary
between $0\,<\,v$sin$i\,<\,5$ km/s.

After performing broadening on each of these model spectra, we used the scipy
module LinearNDInterpolator to interpolate between the model spectra, with
grid spacing $\Delta T_{\mathrm{eff}}\,=\,100$ K, $\Delta$log $g$ = 0.5,
$\Delta$[Fe/H] = 0.5 dex, and $v$sin$i$ = 0.5 m/s. The grid spacings for the
first three parameters are guided by the available PHOENIX models, while the
grid spacing of the stellar rotational velocities was user-specified and
selected to be fine enough to account for the somewhat nonlinear relationship
between stellar rotation and line broadening but coarse enough to allow for a
reasonable computation time.

We also fit an additional error term $\sigma$, added in quadrature to the
observed spectral flux errors, in order to get a sense of the scale of the
systematic differences between the models and the data. As we normalized both
the data and models such that the flux continuum was at a value of 1, this
error value is expected to be between 0 and 1. We then performed MCMC fits of
our hightest-SNR observed spectra to the interpolated model spectra using the
emcee package, with free parameters of $T_{\mathrm{eff}}$, [Fe/H], $v$sin$i$,
and $\sigma$. Our interpolations are performed in their entirety before the
MCMC run is initiated instead of having the broadening recalculated at every
step for the sake of reducing computation time. We did not independently fit
log $g$ because, much like Passegger et al. (2018), we found that there
existed strong degeneracies between the fit surface gravity and the other
parameters. We thus avoided fitting the log $g$ directly, and instead
estimated log $g$ at each MCMC step. To do so, we used the estimated
metallicity and the $K_{s}$ magnitude from the 2MASS survey (Cutri et al.,
2003) to calculate the stellar radius using the relation from Mann et al.
(2015) and the stellar mass using the relation from Mann et al. (2019). Mann
et al. (2015) quotes a 2.7% error on the estimated radius, while Mann et al.
(2019) quotes a roughly 3% error, which overall result in our log $g$
estimates having an additional error on the order of $\delta$log
$g\,=\,0.025\,-\,0.030$. This systematic error will be added in quadrature to
the fit error when listing the log $g$ value.

In general, we found typical $\sigma$ values that were on the order of ten to
one hundred times larger than the spectral flux errors. This indicates that
the deviations between the PHOENIX models and are data are dominated by
systematic differences between the models and data and not merely data errors.
The fit parameter errors of this method are likely an underestimate, as there
are significant discrepancies with the underlying models.

Passegger et al. (2022) found that, even when analyzing similar spectra,
different methods for stellar parameter determination can report differences
in $T_{\mathrm{eff}}$ on the order of $\geq$ 100 K and [Fe/H] on the order of
$\approx 0.1\,$— 0.3. Thus, instead of calibrating our code by looking at the
results of other stellar parametric determinations, we estimate the errors on
our parameters by comparing them directly to data. To estimate the accuracy of
our algorithm, we compared our code’s results to the known temperatures of
nearby M dwarfs who have had these parameters estimated directly using
interferometry. We use these to check our algorithm’s accuracy, as their
parameters were determined in a manner independent of the PHOENIX models.
Table 4 lists the stars we use as references, along with their radii and
temperatures. Observations of M dwarfs in binary systems with well-
characterized primary stars may be a useful way to more accurately calibrate
this relationship.

Name | Radius (R⊙) | Teff (K) | Ref. | Fit Teff (K) | Fit [Fe/H]
---|---|---|---|---|---
GJ 15A | 0.387 $\pm$ 0.002 | 3563 $\pm$ 11 | a | 3566${}^{+13}_{-10}$ | -0.36${}^{+0.04}_{-0.03}$
GJ 273 | 0.32 $\pm$ 0.005 | 3253 $\pm$ 39 | b | 3342${}^{+17}_{-20}$ | -0.30${}^{+0.04}_{-0.03}$
GJ 380 | 0.642 $\pm$ 0.005 | 4081 $\pm$ 15 | a | 4021${}^{+17}_{-19}$ | -0.23${}^{+0.03}_{-0.02}$
GJ 406 | 0.159 $\pm$ 0.006 | 2657 $\pm$ 20 | b | 3209${}^{+13}_{-19}$ | 0.14$\pm 0.04$
GJ 411 | 0.387 $\pm$ 0.004 | 3547 $\pm$ 40 | a | 3519${}^{+20}_{-74}$ | -0.35${}^{+0.04}_{-0.15}$
GJ 447 | 0.196 $\pm$ 0.01 | 3264 $\pm$ 24 | b | 3289${}^{+10}_{-11}$ | -0.25$\pm 0.03$
GJ 486 | 0.339 $\pm$ 0.015 | 3291 $\pm$ 75 | c | 3387${}^{+9}_{-19}$ | -0.13${}^{+0.02}_{-0.03}$
GJ 699 | 0.185 $\pm$ 0.001 | 3221 $\pm$ 32 | a | 3250${}^{+9}_{-8}$ | -0.48${}^{+0.03}_{-0.02}$
GJ 725A | 0.356 $\pm$ 0.004 | 3407 $\pm$ 15 | a | 3341${}^{+9}_{-8}$ | -0.49${}^{+0.02}_{-0.01}$
GJ 725B | 0.323 $\pm$ 0.006 | 3104 $\pm$ 28 | a | 3304$\pm 7$ | -0.46$\pm 0.03$
GJ 729 | 0.205 $\pm$ 0.006 | 3162 $\pm$ 30 | b | 3316${}^{+9}_{-30}$ | -0.39${}^{+0.03}_{-0.04}$
GJ 820A | 0.665 $\pm$ 0.005 | 4548 $\pm$ 64 | a | 4356${}^{+8}_{-11}$ | -0.42$\pm 0.02$
GJ 820B | 0.61 $\pm$ 0.018 | 3954 $\pm$ 28 | a | 4028${}^{+6}_{-4}$ | -0.32$\pm 0.01$
Table 4: Stars observed by MAROON-X with direct interferometric radius and
temperature measurements, as well as the temperature and metallicity estimates
from our code. We note that errors associated with our fit values are purely
taking the fit errors into account, and don’t consider systematics. They are
thus likely systematics. References: a) Boyajian et al. (2012), b) Rabus et
al. (2019), c) Caballero et al. (2022).

Figure 4 compares the temperatures of our calibrators derived via
interferometry with those derived using our method. For each calibrator, we
utilized the highest-SNR MAROON-X spectrum that we had available. It is
obvious that our code is less accurate at temperatures above 4200 K and at
temperatures at or below around 3200 K. At these temperatures, the code’s
sensitivity to temperature drops and the estimated temperatures appear to
approach a constant value. The lines we selected for analysis do not appear to
be useful for measuring the properties of stars outside of this temperature
range, which makes sense because our list was inspired by Passegger et al.
(2018)’s studies of M dwarfs. While the code underestimates the temperatures
for stars with temperatures above 4200 K, such stars are not included in the
HUMDRUM sample and are thus not a problem. TOI-1450A (with an estimated
temperature around 3400 K in the TESS catalog) is unlikely to fall outside of
our range of sensitivity, but TOI-1450B, which is expected to have
$T_{\mathrm{eff}}\,\approx\,3000$ K, is likely too cool to obtain accurate
results.

Overall, omitting Gl 406 and Gl 820A (both of which fall substantially outside
of the sensitive temperature range) from our calibrator sample, we find a
typical RMS deviation between our fits and the interferometric temperatures of
about $\sigma_{T}\,=\,82$ K. This indicates the necessity of adding an
additional error term in quadrature to the model-estimated $T_{\mathrm{eff}}$
(which is used to estimate log $g$) on top of the statistical fit error and
the systematic errors. As we are using similar spectral regions as Passegger
et al. (2018), we quote a fit metallicity error of
$\sigma_{\mathrm{Fe/H}}\,=\,0.16$. Unfortunately, this number may not be
accurate, but we do not have a sample of M dwarfs with independently-verified
metallicities to confirm our measurements.

Figure 4: The interferometric temperatures of the calibrator stars compared to
the MAROON-X estimations.

With our code, we find that TOI-1450A has a
$T_{\mathrm{eff}}\,=\,3433\,\pm\,87$ K, log $g$ = $4.76\,\pm\,0.04$, and
[Fe/H]$\,=\,-0.12\,\pm\,0.17$. Its metallicity is similar to that of the Sun,
and may be slightly subsolar, while its temperature is exactly what we would
expect for a mid-M dwarf.

Unfortunately, TOI-1450B likely falls in the regime in which our temperature
measurements are unreliable. An effort to measure its temperature yields
$T_{\mathrm{eff}}\,=\,3136\,\pm\,82$ K and its metallicity is measured as
[Fe/H]$\,=\,0.18\,\pm\,0.17$. Given the fact that the line list chosen seems
to be unable to accurately estimate stellar parameters for stars with
temperatures below about 3200 K, these values are likely inaccurate, though it
is difficult to more precisely quantify these errors without more calibrator
stars. We thus do not claim a measurement of the TOI-1450B stellar parameters.

####  4.1.2 SED Fitting

As an independent determination of the basic stellar parameters, we performed
an analysis of the broadband spectral energy distribution (SED) of the star
together with the Gaia DR3 parallax (with no systematic offset applied; see,
e.g., Stassun & Torres, 2021), in order to determine an empirical measurement
of the stellar radius, following the procedures described in Stassun & Torres
(2016); Stassun et al. (2017, 2018). We pulled the the $JHK_{S}$ magnitudes
from 2MASS, the W1–W4 magnitudes from WISE, the $z$ magnitude from PAN-STARRS,
and the $GG_{\rm BP}G_{\rm RP}$ magnitudes from Gaia. We also utilized the
absolute flux-calibrated spectrophotometry from Gaia. Together, the available
data spans the full stellar SED over the wavelength range 0.4–20 $\mu$m (see
Figure 5).

Figure 5: SED fit (upper black curve) to the observed photometry of TOI-1450A
(red symbols). The absolute flux-calibrated Gaia spectrum is shown in gray and
shown in detail in the inset plot. The SED of the faint companion star,
TOI-1450B, is shown as the lower black curve atop its broadband photometric
measurements (red symbols).

We performed a fit using PHOENIX stellar atmosphere models (Husser et al.,
2013), with the effective temperature ($T_{\rm eff}$) and metallicity ([Fe/H])
adopted from the analysis above. The extinction $A_{V}$ was fixed at zero due
to the close proximity of the system. The resulting fit (Figure 5) has a
reduced $\chi^{2}$ of 1.4. Integrating the model SED gives the bolometric flux
at Earth, $F_{\rm bol}=1.991\pm 0.046\times 10^{-9}$ erg s-1 cm-2. Taking the
$F_{\rm bol}$ and $T_{\rm eff}$ together with the Gaia parallax, gives the
stellar radius, $R_{\star}=0.500\pm 0.026$ R⊙. In addition, we can estimate
the stellar mass from the empirical $M_{K}$ relations of Mann et al. (2019),
giving $M_{\star}=0.496\pm 0.025$ M⊙.

Next, we sought to account for the small flux contribution from TOI-1450B. To
do this, we pulled the available broadband photometry from catalogs in which
the companion is separately resolved, namely Gaia and PAN-STARRS. Next, we
estimated the companion star’s $T_{\rm eff}$ by interpolating the relative
Gaia $G$ magnitudes in the empirical tables provided in Pecaut & Mamajek
(2013), and found that a PHOENIX model atmosphere of that $T_{\rm eff}$ (and
adopting the same [Fe/H] as for TOI-1450A) indeed provides a good fit to the
photometry (see Figure 5). Finally, we computed from this SED model the
companion star’s contribution to $F_{\rm bol}$ overall and to the $K_{S}$
magnitude specifically, then recalculated the radius and mass for TOI-1450A,
giving $R_{\star}=0.483\pm 0.025$ $R_{\odot}$ and $M_{\star}=0.480\pm 0.024$
$M_{\odot}$. These values agree quite closely with the values quoted by the
TESS Input Catalog (TIC, Stassun et al., 2019; STScI, 2018).

###  4.2 Stellar Rotation Period

####  4.2.1 TESS Photometry

The TESS photometry for this system is abundant, encompassing 27 sectors. We
examined the TESS Single Aperture Photometry (SAP) for signs of stellar
rotation, as long-term trends can sometimes be attenuated or removed by
Presearch Data Conditioning when it identifies and removes instrumental
effects. However, the SAP photometry may still contain meaningful instrumental
systematics, so we will need to check to make sure our results are stellar in
origin.

The 120s cadence SAP photometric flux collected by TESS (and read in using
lightkurve, Lightkurve Collaboration et al., 2018) is shown in Figure 6. Each
sector has been mean-normalized to emphasize the variability and periodicity
of the data as opposed to the slight photometric offsets between sectors. We
examined the 120s data (which has a longer time baseline than the 20s data)
for any evidence of rotational signals using the
PyAstronomy333https://github.com/sczesla/PyAstronomy (Czesla et al., 2019)
implementation of the generalized Lomb-Scargle (GLS) periodogram (Zechmeister
& Kürster, 2009). The resulting periodogram is shown in the middle panel of
Figure 6.

Figure 6: Top: The 120s TESS SAP fluxes for the TOI-1450 system for Sectors
14-60. Each sector’s data is mean-normalized. Middle: GLS periodogram of the
TESS 120s data. Dashed vertical lines are drawn on the periodogram
corresponding with the peaks at 40 days, 27 days (the length of a single TESS
sector), and 2 days. Bottom: TESS photometry of sectors 57-60 the TOI-1450
system. The different sectors are colored differently in order to draw
attention to where potential breaks in the baseline are expected. Sinusoidal
variations on 40d and 2d timescales are obvious in these sectors.

The strongest peak in the GLS periodogram corresponds to a signal of around 40
days, which is visually obvious even when examining the raw data by-eye (see
the bottom panel of Figure 6). There is also a signal at around 27 days, which
is likely merely related to the length of a TESS sector and is probably
present due to systematics in the light curve. Finally, there is a third
signal corresponding to 2.01 days, with a amplitude varying between
0.02%—0.08%. This signal is very close to the observed planet transit period
($P\,=\,2.04$ days), but appears to be distinct from the signal of the
transiting planet, as it has a slightly different period (the 2.01 day peak in
the periodogram is very narrow and does not encompass the transiting planet’s
period) and there is obvious sinusoidal variation on this timescale in the
photometric data. Thus, this 2.01d signal may be related to a star’s rotation.

We thus notice two apparent rotation signals in the photometry, one at 40 days
and one at 2.01 days. These could be representative of the rotation of the A
and B stars, which are blended in the TESS pixel. While the primary is thirty
times brighter than the secondary, the secondary is younger and thus may
possess a proportionally stronger time-varying rotational signal. Thus, we
cannot immediately determine which signal comes from which star based on the
TESS data alone. In general, we would expect that a shorter-period signal
belongs to the smaller star (see, e.g., the distributions of rotation periods
with spectral type from Popinchalk et al., 2021). Taking the 97% dilution from
the A star into account, if the 2d signal does come from the B star, it
actually corresponds to a 1—3% amplitude signal in the TESS flux band. As a
comparison, the rotation of TRAPPIST-1 (another late M dwarf) manifests in its
TESS photometry as a a roughly 0.6% signal. The slightly stronger amplitude of
TOI-1450B may be the result of the star’s shorter rotation period (and thus
higher activity levels). Therefore, it would be logical for the 2d signal to
come from TOI-1450B, as its amplitude is reasonable compared to other short-
period late M dwarfs.

Additionally, a 40 day rotation signal would align with expectations from
Popinchalk et al. (2021) for a field M3 star. However, the exact period of the
40 day rotation signal is somewhat uncertain, as TESS has problems with
detecting signals with periods longer than the length of the 27 day TESS
sector (Canto Martins et al., 2020). In Section 4.2.3, we study the Doppler
broadening of the MAROON-X spectrum to resolve which rotation period is
associated with which star, finding that the smaller star is likely the faster
rotator, in agreement with theoretical expectations.

It is obvious that the system has flared multiple times during the TESS
observation window, with one flare energetic enough to produce 15% of the
combined system’s luminosity. As lower-mass stars tend to flare more
frequently (Davenport et al., 2019) and these two stars are likely the same
age, the majority of these flares are likely the result of TOI-1450B and not
the exoplanet host star. There is no clear photometric evidence that the star
was flaring during any of our RV observations, so we were unable to confirm
whether or not the flares come from TOI-1450A or TOI-1450B spectroscopically.

####  4.2.2 ASAS-SN and WASP Photometry

We studied both the ASAS-SN and WASP photometry in order to gain further
insight on the potential 40d rotation period present in the TESS photometry.
Both instruments had pixel sizes large enough to result in blending between
TOI-1450A and TOI-1450B. The system appears stable, as the photometry, shown
in Figure 3, shows no evidence of of long-term trends on year-length
timescales. To search for signs of rotation, we produced GLS periodograms for
each instrument, examining periods between 0.5—500 days. Overall, we expected
both ASAS-SN and WASP to be more reliable than TESS over long periods of time.

Figure 7 shows the periodograms for three ASAS-SN cameras in two wavelength
bands. In general, we typically expect to see stronger activity-induced
variability in the bluer $g$ band, and the $g$ band also has more statistical
power due to the larger number of observations. There are signals at around 1
day and 172 days that likely correspond with the nightly observing cadence and
half of one year. Additionally, both the ba and bA cameras find a high-FAP
($>\,10\%$) signal at $P\,\approx\,43$d, but the signal is not present in the
bq data. This signal coincides with the 40d signal observed by TESS,
strengthening the hypothesis that one of the stars in the TOI-1450 system has
a 40d rotation period. The data are noticeably missing the strong 2.01d signal
noticed in the TESS photometry. However, this can be explained by the fact
that the typical flux error for the ASAS-SN data ($\approx\,0.18\%$) is much
larger than the amplitude of the rotational signal observed in the TESS data
($\approx\,0.03\%$).

Figure 7: Lomb-Scargle periodograms of the ASAS-SN photometry. The red dashed
horizontal line corresponds to a 1% false alarm probability, and the golden
vertical line indicates a 43d period.

Figure 8 shows the periodogram of the WASP photometry. Overall, we notice GLS
periodogram peaks with a $<\,1$% FAP in between $40\,$—$\,47$ days in all
three datasets, but the 40d peak is by far the strongest in the 2010 data.
Both the 2008 and 2009 datasets feature stronger signals at 20d, at half the
proposed rotation period. The 2009 data feature a strong peak at around 60
days, but this signal is suspect given the fact that it is approximately half
the length of the observing season. Overall, the data are consistent with the
$\approx$ 40 day rotation period found by other instruments.

Figure 8: GLS periodograms of the WASP data. The red dashed horizontal line
corresponds to a 1% false alarm probability, and the golden vertical span
indicates periods from 40-47 days. The signal at 1d is likely related to the
nightly sampling and is thus unlikely to be physical.

####  4.2.3 Identifying the Fast Rotator by Estimating v sin i

We can directly measure the projected rotational velocity $v\,$sin$\,i$ of
TOI-1450A and TOI-1450B by examining the line broadening present in the
MAROON-X spectra. We can then check to see if this value is consistent with
the observed 2d periodicity in the TESS data. However, as MAROON-X has a
resolution of roughly 85,000 (Seifahrt et al., 2018), this method will be
imprecise for spectra with $v\,$sin$\,i\,\lesssim\,2$
$\mathrm{km}\,\mathrm{s}^{-1}$.

To estimate $v\,$sin$\,i$, we used the cross correlation (CCF) comparison
method described in Gray (2005) and used with MAROON-X data in previous works
(see Section 3.3 in Brady et al., 2023, for a more comprehensive description).
With this method, we compared the spectra of TOI-1450A and TOI-1450B to other
spectra taken by MAROON-X of known slow rotators with similar spectral types,
which allowed us to estimate the rotational velocities of TOI-1450A and
TOI-1450B.

As TOI-1450A is a M3 star, we compared it to Luyten’s Star, a M3.5V star
(Hawley et al., 1996) with a rotation period of $115.9\,\pm\,19.4$ days
(Suárez Mascareño et al., 2015). Using this star as a template, we find that
the $v\,$sin$\,i$ of TOI-1450A is $1.8\,\pm\,0.6$
$\mathrm{km}\,\mathrm{s}^{-1}$, which is below the $v$sin$i\,<\,2$ km/s lower
limit of recordable rotational velocities given the MAROON-X resolution limits
(see Brady et al., 2023, for more details). It is thus possible that the
$v$sin$i$ of TOI-1450A is below 2 km/s, making the accuracy of this
measurement suspect. Given the radius of TOI-1450A (0.474 $R_{\odot}$), a two-
day rotation period would correspond to a measured rotational velocity of
about 12 km/s. This would be easily detectable by MAROON-X if the planet
system had a low obliquity. The fact that the observed rotational velocity is
below the measurement threshold of MAROON-X indicates that either the observed
signal does not correspond to the rotational signal of TOI-1450A, or that the
sin$\,i$ of TOI-1450A is extremely low ($i\,<\,10\arcdeg$). The chances of the
system randomly being oriented at such a low inclination is fairly small
($<\,20\%$), but not impossible.

We also used this method to estimate the rotational velocity of TOI-1450B and
compared it to the observed photometric signal. We used a spectrum of
Teegarden’s Star as a template, as it is the latest-type slow-rotating M dwarf
observed by MAROON-X (with a rotation period of $99.6\,\pm\,1.4$ days Terrien
et al., 2022). With it, we found that TOI-1450B has a rotation velocity of
roughly $3.0\,\pm\,0.7$ $\mathrm{km}\,\mathrm{s}^{-1}$. Given TOI-1450B’s mass
of about 0.14 $M_{\odot}$, we would expect it to have a radius of around 0.15
— 0.18 $R_{\odot}$ (comparing the mass to those of other late M dwarfs, such
as those in Parsons et al., 2018), which would correspond to an equatorial
velocity of around 3.8 - 4.5 $\mathrm{km}\,\mathrm{s}^{-1}$ if it had a two-
day rotational period and we were observing it equator-on. We cannot eliminate
the possibility that TOI-1450B has a rotation period of approximately 4d and
the observed 2d signal in the photometry is due to persistent features on
opposite hemispheres, though this possibility is slightly more discrepant with
our measured rotation $v$sin$i$. Thus, the observed signal could easily be
explained by the rotation of TOI-1450B, especially if it is somewhat inclined
relative to the transiting planet. Given its rapid rotation speed, it is
unlikely to be the source of the 40d rotation signal.

From this analysis, we conclude that the short-period rotation signal in the
photometry is likely to come from TOI-1450B, which makes sense given the fact
that it is much smaller than TOI-1450A and is thus likely to rotate more
rapidly. This means that the 40d signal is likely the product of TOI-1450A’s
rotation.

####  4.2.4 Activity

We can also estimate the rotation period of the star by studying the
spectroscopic activity indicators. serval measures several activity indicators
of the input spectra, including the chromatic index, differential line width,
$H\alpha$ index, the infrared Ca II triplet indices, and the NaI doublet
indices. We can examine periodograms of these activity indicators to get a
sense of the stellar rotation period. This examination is crucial to
distinguishing genuine planetary signals from those that are merely stellar in
nature, as we typically do not expect to see any planetary signals in the
indicators.

The actual values of the activity indicators are shown in Figure 9. There are
obvious offsets in the measured dLWs between each MAROON-X run. This is a
known issue and is believed to be due to small instrumental profile shifts as
a result of the instrument front-end unit being unplugged from Gemini between
each observation run. The fact that the offsets are not the same in the red
and blue channels between April and May 2021 could explain the fact that our
RV offsets between runs are chromatic at that time. There also appear to be
long-period trends in the activity data that correspond to signals with
$P\,=\,1$ year, which are likely related to the barycentric motion of the
Earth causing telluric lines to influence our activity measurements.

Figure 9: Time series of the MAROON-X activity indicators.

There is an obvious outlier in the May 2021 blue-channel data in both the dLW
measurements and in the Na D indices. A similar outlier of a smaller magnitude
is also present in the red-channel dLW data at this time. However, there is no
obvious outlier in the corresponding RV measurement. An examination of the
spectrum in question reveals strong emission in the wings of the sodium
doublet at time of observation. However, these emissive features are the
result of sky emission of sodium and are masked out by serval when estimating
RV. They are also not included in serval’s calculation of the actual Na D
index, and thus cannot explain the observed large index shift. However, there
are weaker telluric lines that cross the Na D line center during that
exposure, and it is possible that they are responsible for the anomalous
measurement. However, as the shift is also recorded in the measured chromatic
index, it is possible that this observation was actually taken during some
active event, in which chromospheric Na D emission spiked. We also note a
small outlier in the H $\alpha$ index as recorded by both the red and blue
channels in April 2021, though an examination of the involved spectra does not
show any obvious flare or telluric activity. As these observations may be the
result of flares or failures by serval to accurately estimate the line
indices, we removed them from our analysis for the purposes of creating the
GLS periodograms.

The GLS periodograms of the activity indices are shown in Figure 10. In this
figure, we removed the two observations that correlated with potential flare
events, and also normalized the dLW data by subtracting the mean dLW value
from each run. We see that most of the activity indicators used by serval have
signals at the FAP $<$ 1% level at around 40 days, with the exact periods
varying between about 39 — 47 days. While these periods are longer than the
length of a typical ($\approx$30 day) MAROON-X run, they are in agreement with
the longer rotation period estimate from the TESS , ASAS-SN, and WASP data.

Figure 10: GLS periodograms of the signals present in the MAROON-X activity
indicators, with outliers due to flares removed and the dLW data normalized to
the mean within each run. The dashed gold line indicates the location of a 39d
period. A red dashed line corresponding to a false alarm probability of 1% is
also shown.

A closer examination of the lines studied showed that the centers of three of
our known activity indicator lines (the redder sodium doublet line and the
bluest two Ca II lines) were contaminated by atmospheric tellurics during the
majority of our observations, making their measured activity signals dubious.
As measured trends in line indices between the two sodium doublet lines agree
despite the fact that one aligns with a known telluric region and the other
does not, it appears that the telluric contamination in the line center is
slight. This telluric contamination does appear to be more of an issue with
regards to the Ca II lines, however, as the three datasets are less obviously
correlated. To be conservative, if we ignore all activity indicators with
known telluric crossings, we would only use the indices of H $\alpha$, the
bluest sodium doublet line, and the reddest Ca II line. All of these
indicators have observed periodogram peaks around 39 days, however, so
dropping the other lines from our analysis does not affect our results. It may
be interesting in the future to attempt to fit out these lines (using a
package like molecfit, see Smette et al., 2015; Kausch et al., 2015) for the
purposes of determining telluric-corrected activity indicators, but is
unnecessary in this case due to the relatively unambigous nature of the signal
in the other indices.

The activity indicators give us a clear sign that the rotation period of the A
star is around 40 days. This supports our theory that TOI-1450A is not the
2.01d rotator in the system, and that TOI-1450A has a rotation period of 40
days, which is in-line with what is expected for a field M3 dwarf (Popinchalk
et al., 2021).

If TOI-1450A has a rotation period of 40 days and TOI-1450B has a rotation
period of 2 days, we can attempt to estimate the age of the system using
gyrochronology. Using the relations from (Engle & Guinan, 2023) for old
M2.5-6.5 dwarfs, we recovered an age of $3.4\,\pm\,0.5$ Gyr for TOI-1450A.
However, this relationship does not account for temperature. We thus studied
the gyrochrone of the 4 Gyr-old M67 cluster from Dungee et al. (2022), and
find that TOI-1450A is a slightly faster rotator than the 3400 K star observed
in M67. This agreed with the sub-4 Gyr age estimate, indicating that TOI-1450A
is a middle-aged star. Unfortunately, it is much more difficult to provide a
gyrochronological estimate of TOI-1450B’s age, as Pass et al. (2022) shows
that there is significant variability in the spindown rates of fully-
convective M dwarfs. While the 2 d rotation rate is not inconsistent with a
$\approx\,3$ Gyr system age, acquiring a more precise system age from
TOI-1450B is not possible. However, the rotation rates of the two stars are
not inconsistent with each other, suggesting an overall system age of a few
Gyr.

##  5 Planet Fitting

###  5.1 Transit Photometry

We performed a fit to the transit photometry of the TOI-1450 system using
juliet (Espinoza et al., 2019), which makes use of transit modeling from
batman (Kreidberg, 2015) and dynesty (Speagle, 2020) to perform nested
sampling. In our analysis of the TESS data, we considered the 2-minute cadence
data for light curves taken prior to the TESS extended mission, and the
20-second data for those taken afterwards (using the highest-cadence data
available for each sector). We used the PDCSAP light curves in our analysis.
For both the 120s and 20s data, we performed 5$\sigma$ clipping to reduce the
influence of outliers on our analysis of the transits.

We searched the TESS data for planet signals using the transitleastsquares
algorithm from Hippke & Heller (2019) to estimate the signal period and
$t_{0}$. transitleastsquares searches for signs of planetary transits using a
realistic model of transit ingress, egress, and stellar limb-darkening.
transitleastsquares found the 2.01d rotation signal at a high confidence, but
when the period search range is limited to exclude this signal, it recovered
the suspected planet at $P\,=\,2.04392$ days and $t_{0}\,=\,2458685.34341$
BJD, which agrees with the values quoted online by the TESS team.

Before fitting the transits of the 2.04d planet, we performed light curve
detrending. As discussed in Section 4.2.1, there are nonplanetary signals in
the data due to stellar rotation. One of these signals, the rotation of
TOI-1450B, has a period similar to the period of the suspected transiting
planet. Due to the large amount of photometric data available for the TOI-1450
system, we did not simultaneously perform gaussian process (GP) detrending and
transit fitting of the data. Instead, we detrended the TESS data before
fitting the transits. We detrended the data by fitting a GP (using an
implementation of the approximate Matern kernel from celerite Foreman-Mackey
et al., 2017) to the 120s PDCSAP TESS data from the system (the shorter-
cadence 20s data is unnecessary for studying a 2d sinusoidal rotation signal),
assuming an uninformative prior on the amplitude and a normal prior on the
period centered around the suspected 2d rotation period. We also masked out
the transits (assuming the parameters from transitleastsquares) when fitting
for the GP trend. The fit trend captured the 2d variation signal in the
photometry while preserving shorter-period variations (such as transits).

We then proceeded to fit the planet transits. First, we detrended the full
TESS dataset, including both 120s data and 20s data when it was available. To
do this, we divided each TESS data point by the GP model. We did not perform
the same extensive detrending for the ground-based transit data, as we do not
have enough data for the ground-based instruments to estimate the GP amplitude
of each instrument. Once our detrending was complete, we identified TESS data
points that were more than 45 minutes pre- or post-transit and omitted them
from our transit analysis. This step was primarily performed as the extremely
large amount of out-of-transit TESS data dramatically increased our
computation time. A phase-folding analysis of the transit data suggested a
transit duration of 70 minutes or less, so this masking was unlikely to remove
any in-transit data.

We calculated the planet transit parameters using juliet, modeling the transit
light curve of a planet with quadratic limb-darkening. The priors used for the
fit are listed in Table 5. We fit the period $P$ and transit time $t_{0}$
using narrow priors informed by our transitleastsquares analysis, as the
individual transits are very shallow and thus difficult to recover with
uninformative priors.

We fixed the eccentricity at $e\,=\,0$. This is because it is difficult to
constrain the eccentricity from transit data alone unless a secondary transit
is detected, TTVs are detected, or the stellar density is highly constrained
(see, e.g., Van Eylen & Albrecht, 2015). We will revisit the validity of this
assumption when analyzing the RV data in later sections.

The priors for the limb-darkening coefficients, $R_{p}/R_{\star}$ ratio, and
impact parameter $b$ are set by the sampling methods used by juliet. To
calculate the planet radius, we used the sampling technique described in
Espinoza (2018), which parameterizes $R_{p}/R_{\star}$ and $b$ using the
uniformly distributed variables $r_{1}$ and $r_{2}$. The limb-darkening
coefficients were sampled according to Kipping (2013) using the uniformly
distributed variables $q_{1}$ and $q_{2}$. The stellar density $\rho_{\star}$
has priors based on the values for the mass and radius of TOI-1450A listed in
Table 1. As TOI-1450A is a member of the TESS Cool Dwarf list, these values
come from the observationally-derived relations from Mann et al. (2015) and
Mann et al. (2019) and are thus not likely to suffer from systematic modelling
errors.

Our final analysis included the TESS data, the LCO-Sinistro data, and the LCO-
MuSCAT3 data. We also fit parameters to the mean flux and jitter of each
instrument to capture any variations between instrument flux baselines. We
considered the 120s and 20s data to be two separate datasets for the sake of
fitting jitter, mean, and limb-darkening terms.

Fit parameters |
---|---
$P$ | N(2.04392, 0.0001)
$t_{0}$ (BJD) | N(2458685.34341, 0.005)
$r_{1}$ | U(0, 1)
$r_{2}$ | U(0, 1)
$\rho_{\star}$ (g cm-3) | N(6.23, 0.62)
$\mu$ | U(0, 1)
ln$\sigma_{\mathrm{ppm}}$ | U(ln(1), ln(10000))
$q_{1}$ | U(0, 1)
$q_{2}$ | U(0, 1)
Table 5: The priors used in the transit fit as well as the resulting fits. The
$\mu$, ln$\sigma$, $q_{1}$, and $q_{2}$ for the different instruments (the
120s TESS data, the 20s TESS data, the four separate LCO-MuSCAT3 colors and
the two separate LCO-Sinistro transit observations) are being fit separately,
but all share the same priors so we omit listing them individually in this
table.

The resulting parameters for the transit fit are shown in Tables 6 and 7. The
observed transit signal, in combination with the stellar radius from the TIC
(Mann et al., 2015, using), is consistent with a
$R_{p}\,=\,1.13\,\pm\,0.04R_{\oplus}$ planet (similar in size to Earth) with
an elevated impact parameter ($b\,=\,0.74\,\pm\,0.02$). Our fits are primarily
dependent upon the TESS data, as the ground-based data didn’t have enough
statistical power to constrain the planet parameters. The transit data alone
did not constrain the stellar density, with our fits merely recovering the
prior. This is likely due to a combination of the short transit length and the
shallow transit signal.

Fit parameters |
---|---
$P$ (d) | $2.0439276\pm 0.0000010$
$t_{0}$ (BJD) | $2458685.34223\pm 0.00042$
$r_{1}$ | $0.8256\pm 0.0143$
$r_{2}$ | $0.0219\pm 0.0006$
$\rho_{\star}$ (g cm-3) | $6.32\pm 0.58$
Derived parameters |
$a/R_{\star}$ | $11.18\pm 0.35$
$R_{p}$ ($R_{\oplus}$) | $1.130\pm 0.044$
$b$ | $0.738\pm 0.022$
$I$ (degrees) | $86.213\pm 0.227$
$T_{\mathrm{dur}}$ (h) | $0.989\pm 0.013$
Table 6: The planet parameters from the juliet transit fit to the detrended TESS data. Instrument |  $\mu$ (ppm) |  $\sigma$ (ppm) | $q_{1}$ | $q_{2}$ | $u_{1}$ | $u_{2}$
---|---|---|---|---|---|---
TESS 120s | $-3\pm 23$ | $6^{+17}_{-4}$ | $0.29^{+0.26}_{-0.18}$ | $0.32^{+0.37}_{-0.23}$ | $0.32\pm 0.27$ | $0.16\pm 0.31$
TESS 20s | $10\pm 18$ | $865^{+25}_{-25}$ | $0.32^{+0.19}_{-0.14}$ | $0.54^{+0.30}_{-0.34}$ | $0.57\pm 0.31$ | $-0.04\pm 0.33$
SinistroJul | $31\pm 96$ | $1141^{+88}_{-83}$ | $0.57^{+0.29}_{-0.36}$ | $0.51^{+0.33}_{-0.34}$ | $0.66\pm 0.48$ | $-0.01\pm 0.42$
SinistroAug | $6\pm 86$ | $802^{+84}_{-83}$ | $0.54^{+0.31}_{-0.34}$ | $0.53^{+0.32}_{-0.34}$ | $0.66\pm 0.47$ | $-0.04\pm 0.41$
MuSCAT3g | $-4\pm 61$ | $569^{+73}_{-72}$ | $0.56^{+0.30}_{-0.34}$ | $0.50^{+0.33}_{-0.32}$ | $0.64\pm 0.46$ | $0.01\pm 0.41$
MuSCAT3r | $39\pm 66$ | $1718^{+56}_{-52}$ | $0.59^{+0.28}_{-0.34}$ | $0.51^{+0.33}_{-0.34}$ | $0.67\pm 0.48$ | $-0.01\pm 0.42$
MuSCAT3i | $25\pm 71$ | $2248^{+55}_{-56}$ | $0.44^{+0.36}_{-0.30}$ | $0.44^{+0.35}_{-0.30}$ | $0.49\pm 0.42$ | $0.06\pm 0.37$
MuSCAT3z | $26\pm 56$ | $2093^{+46}_{-45}$ | $0.45^{+0.34}_{-0.31}$ | $0.45^{+0.36}_{-0.31}$ | $0.51\pm 0.42$ | $0.05\pm 0.38$
Table 7: The instrument parameters from juliet transit fit to the detrended
TESS data.

We also tested whether or not it was feasible to detect TTVs in the TOI-1450
data using exoplanet (Foreman-Mackey et al., 2020). We chose to use exoplanet
(which uses MCMC sampling) instead of juliet (which uses nested sampling) at
this step due to the large dimensionality of the problem. We first generated a
light curve with batman with noise taken directly from the out-of-transit
light curve from TESS for TOI-1450 and added 15 minute TTVs, which are in line
with some of the timing discrepancies observed by the ground-based
observations. Overall, we found that we could not accurately recover a 15
minute TTV signal from the simulated data, as the individual transits were too
shallow compared to the noise to derive precise transit times. Thus, we did
not attempt to perform a more complete TTV analysis of the TESS data for this
system.

###  5.2 Radial Velocity

We fit the RV data for TOI-1450A using juliet. The framework is very similar
to what we used in Section 5.1, but with different variables to account for
the different observation methodologies. The fit planet parameters include
$K$, the planetary RV semi-amplitude, the period $P$, the time of conjunction
$t_{0}$, the eccentricity $e$, and the argument of periastron $\omega$. We
parameterized $e$ and $\omega$ as $\sqrt{e}$cos$\omega$ and
$\sqrt{e}$sin$\omega$, and allowed these values to vary between -1 and 1. The
$t_{0}$ and $P$ priors were taken from the transit fits in Table 6. We also
included some additional fit parameters to reflect instrumental effects. As
the red and blue channel cover different wavelength ranges, we allowed them to
have separate fit mean and jitter terms. We have priors on the RV offsets
between several of the individual MAROON-X data runs (described in Section
3.1, see Table 9 for the numerical values), so these priors were included on
the fit means. There were several runs in which offset calibration was
impossible due to insufficient or ambiguous calibration data. When no offset
priors were available, we set broad uninformative priors on the individual
offsets.

####  5.2.1 Model Selection

Figure 11: GLS periodograms of the RV data of TOI-1450 with various signals
subtracted. Each panel has a note describing which signals have been
subtracted from the data to produce the residual periodogram. The bottom panel
shows the window function. A red dashed line corresponding to a FAP of 0.1% is
shown on each plot. In addition, vertical dotted lines corresponding to the
periods of the fit planets are also included. The red line is the 2.04d
signal, the gold is the $\approx$3d signal, the green is the $\approx$5d, the
grey is the $\approx$15d signal, and the purple is the 20d signal.

To properly characterize the transiting planet, we must determine if there are
any additional planets in the system and their characteristics. We performed a
simple period search of our RV data by progressively fitting circular
Keplerian orbits to GLS periodogram peaks using juliet, searching for peaks
between 0.5 — 50 days. Figure 11 shows some of these periodograms, as well as
our window function. We ignored $P\approx 1$d signals given the strong peak at
that period in our window function.

First, we performed a simple zero-planet fit including the offset priors to
get a sense of the signals present in the data. There was a peak at
$P\,=\,2.04$d with a FAP $<$ 0.1%, corresponding to the period of the known
transiting planet. The presence of this significant signal supports our
statement that the transiting planet orbits this star and not TOI-1450B. While
this peak was not the dominant one in the periodogram, we were certain of its
period and phase from the transit data and thus fit it out first, using priors
on $P$ and $t_{0}$ from our analysis in Section 5.1).

After fitting out the transiting planet, the periodogram had several very low-
FAP ($<\,0.1\%$) peaks remaining, one of which was at 20.2 days (likely
corresponding to one half of the rotation period) and the other at 14.9 days.
After attempting to fit it with a keplerian, we found that the 14.9d signal
had a significantly higher amplitude in the blue channel of MAROON-X
($1.90\,\pm\,0.22$ m/s) than it did in the red channel ($1.01\,\pm\,0.18$
m/s). This indicates that it may be heavily influenced by either stellar
activity or some kind of instrumental systematic. While it does not obviously
have a timescale related to the stellar rotation period, it could potentially
be an alias of the 20 d signal related to the 57d peak in the window function
(see the bottom panel of Figure 11). The 20d signal is also chromatic
($1.64\,\pm\,0.18$ m/s in the blue channel and $0.95\,\pm\,0.21$ m/s in the
red channel), supporting the hypothesis that it is related to stellar
rotation. However, this analysis is complicated by the fact that most MAROON-X
runs are 1-4 weeks long, making it hard to disentangle the influence of a
long-period activity signal on any long-period planets. We thus hesitate to
claim this signal as a planet despite its high significance.

As we are confident that the star has a rotation period of around 40d, we next
performed a fit including the transiting planet and a keplerian at 20.2 days
(as a keplerian is an imperfect way to model an activity signal, we modeled it
differently later on). After this fit, the significance of the 14.9d signal
was reduced and the highest residual peak was at 5.06d. This 5.06d signal may
correspond to a second planet in the system. Fitting out the 5.06d signal
(using a uniform prior from 4 — 5.5d) resulted in a forest of low-FAP peaks in
the residual periodogram around 3.3 days. However, this final $\approx$ 3d
signal was difficult to fit, with its period strongly depending on our
treatment of the rotation signal. It is possible that there is a planet around
this period in the system (a planet at 3.18d or 3.39d would be in a
$p,\,q\,=\,2,\,3$ or $p,\,q\,=\,1,\,2$ three-body Laplace resonance with the
other two planets, respectively), but the signal is currently too low to
confirm its presence.

To understand the influence of aliasing on our inferred model, we performed an
analysis of the TOI-1450A system using the $\ell_{1}$
periodogram444https://github.com/nathanchara/l1periodogram (Hara et al.,
2017). The $\ell_{1}$ periodogram is similar to a Lomb-Scargle periodogram in
that it searches for signals in unevenly sampled data, but is specifically
designed to reduce the number of observed peaks due to aliasing. We performed
a $\ell_{1}$ periodogram analysis of all of the MAROON-X RV data, with the
same offsets applied to the RVs as in Table 9 and mean-subtractions when no
calibrated offsets were available.

For the purposes of noise modeling, considered a grid of models with different
white noise ($\sigma_{w}\,=\,$0 — 1.5 m/s, $\Delta\sigma_{w}\,=\,0.25$ m/s),
red noise ($\sigma_{r}\,=\,$0 — 1.5 m/s, $\Delta\sigma_{r}\,=\,0.25$ m/s),
calibration noise ($\sigma_{c}\,=\,$0 — 1.5 m/s, $\Delta\sigma_{c}\,=\,0.25$
m/s), and red noise timescales ($\tau\,=\,$1, 2, 4, 10, 20, 40, 120). For each
noise model, we computed the cross-validation score using the same methods as
described in Hara et al. (2020). In general, for each noise model, we selected
signals with log10FAP$\,<\,-0.5$. Then, the data were split into a randomly-
selected training and test set (with a 60/40 split). A sinusoidal model with
the selected frequencies was fit on the training set, and the likelihood of
this model was evaluated on the test set. This process was repeated 400 times
for each noise model, and we quote the median of the likelihoods for each
noise model as the cross validation score.

Figure 12 shows the resulting FAPs and periods of the selected signals for the
top ten highest-ranking models (representative of about 0.5% of the models
evaluated). We note the presence of both the transiting planet signal and the
5.06d planet signal at FAPs that are less than 0.1%, indicating that these
planets are present in the data. We also see signals at 15d and 20d that
correspond with the chromatic signals we have observed in our previous
analysis that are unlikely to be planetary in origin. We also see a low-FAP
signal at around 7.5d, which corresponds with half of the 15d signal and is
thus also unlikely to be planetary in origin. Interestingly, we do see the
presence of a signal at $P\,=\,3.44$d in this analysis, but the FAP is greater
than 1% and thus we hesitate to claim it as a planet. More data may be
necessary to confirm or deny the presence of this signal. Several other high-
FAP signals were also identified in the $\ell_{1}$ periodogram, but we do not
claim planetary detections at these periods for similar reasons.

Figure 12: A $\ell_{1}$ periodogram of the TOI-1450 RV data. Vertical dotted
lines corresponding to various periods. The red line is a 2.044d signal, the
gold a 3.44d signal, the green a 5.06d signal, the grey a 14.9d signal, and
the purple a 20.1d signal. The horizontal dashed line corresponds to a 1% FAP.

In summary, we identified the presence of signals corresponding to the known
transiting planet ($P\,=\,2.04$d), as well as signals at approximately 5, 15,
20, and 3 days, though we suspect that the 15d, 20d, and 3d signals may be
related to stellar activity and/or systematics (or at least contaminated
heavily by these effects). To investigate the significance of these signals,
we performed nested sampling fits with normal priors centered around the
identified periodogram peaks, and found the log evidence of each model with
juliet.

We also included models in which we used Gaussian Processes (GPs) to model the
stellar activity using a quasi-periodic kernel (see, e.g., Aigrain et al.,
2012; Stock et al., 2023). The quasi-periodic kernel is frequently used to
model the effect of stellar activity on RVs and photometry, as it is able to
reproduce the periodic signature of stellar rotation. The kernel is

$\kappa(t-t^{\prime})=\sigma_{i}^{2}\,\mathrm{exp}\Bigg{(}-\frac{\alpha(t-t^{\prime})^{2}}{2}-\Gamma\mathrm{sin}^{2}\bigg{(}\frac{\pi(t-t^{\prime})}{P_{\mathrm{rot}}}\bigg{)}\Bigg{)},$
(1)

where $\alpha$ is the inverse square of the correlation timescale (which
usually corresponds with the spot evolution timescale), $P_{\mathrm{rot}}$ is
the rotation period of the star, $\Gamma$ describes the relative strength of
the periodic to the aperiodic portion of the modeled signal, and $\sigma$ is a
wavelength-dependent parameter that describes the amplitude of the signal. The
first term of the equation describes the aperiodic portion of the signal and
the second term describes the periodic portion. We fitted $\sigma$ separately
for the red and blue channels of MAROON-X but assumed all of the other GP
parameters were instrument-independent.

We set our GP priors by following the suggestions of Stock et al. (2023). We
allowed $P_{\mathrm{rot}}$ to have a normal prior of $40\pm 3$ days, which
both encompasses the range of periods suggested by both the activity
indicators and photometry. We set $\alpha$ such that the correlation timescale
falls between 40 days and the length of our data collection, as it has been
shown observationally that M dwarfs typically have spot evolution timescales
longer than one rotation period (see, e.g., Robertson et al., 2020, and the
references therein). We have allowed $\Gamma$ to have a log-uniform prior from
0.01 to 10 following the suggestions of Stock et al. (2023), as very high
values of $\Gamma$ can result in GP overfitting. We have placed an upper limit
of 4 m/s on $\sigma$, reflecting the degree of scatter observed in the RVs.
The priors for our GPs are listed in Table 9 in Section 5.3.

The models examined, with their associated transiting planet masses and
relative Bayesian evidences, are shown in Table 8. The model with the highest
Bayesian evidence is the model that the data supports the most heavily, though
we note that differences in ln$Z$ on the order of 1 — 2.5 only indicate weak
to moderate evidence for a model, as discussed in Trotta (2008).

Model |  $M_{\mathrm{b}}$sin$i_{\mathrm{b}}$ ($M_{\oplus}$) | dln$Z$ | Prot
---|---|---|---
- |  | -78.1 |
2d | $1.24\pm 0.19$ | -60.3 |
2d, 5d | $1.30\pm 0.16$ | -43.0 |
2d, 15d | $1.18\pm 0.15$ | -27.8 |
2d, 20d | $1.01\pm 0.19$ | -41.1 |
2d, 3d, 5d | $1.29\pm 0.13$ | -40.6 |
2d, 5d, 15d | $1.34\pm 0.09$ | -25.3 |
2d, 5d, 20d | $1.16\pm 0.15$ | -16.4 |
2d, 3d, 5d, 15d | $1.27\pm 0.09$ | -37.6 |
2d, 3d, 5d, 20d | $1.25\pm 0.13$ | -11.5 |
2d, 3d, 5d, 15d, 20d | $1.27\pm 0.16$ | -7.6 |
GP |  | -37.5 | $39.9^{+1.7}_{-1.4}$
2d + GP | $1.08\pm 0.13$ | -13.5 | $38.0^{+1.4}_{-1.4}$
2d, 5d + GP | $1.25\pm 0.13$ |  | $40.0^{+1.7}_{-1.4}$
2d, 15d + GP | $1.05\pm 0.13$ | -13.5 | $38.0^{+1.7}_{-1.6}$
2d, 20d + GP | $1.15\pm 0.12$ | -7.0 | $29.9^{+2.4}_{-1.4}$
2d, 3d, 5d + GP | $1.26\pm 0.10$ | 13.8 | $38.9^{+0.5}_{-0.5}$
2d, 5d, 15d + GP | $1.23\pm 0.12$ | 8.5 | $39.9^{+1.8}_{-1.5}$
2d, 5d, 20d + GP | $1.24\pm 0.13$ | 10.0 | $39.5^{+1.9}_{-1.8}$
Table 8: List of all of the models used to fit the data with juliet, with the
associated log evidences and mass of the transiting planet signal. The log
evidences are listed in terms of the difference between the final chosen
model’s evidence and the individual models’ evidences, so lower numbers
indicate more preferred models. The final chosen model is bolded. The typical
error on ln$Z$ is 0.4 — 0.8, and is omitted from the table to save space.

The four models with the highest evidence are a two-planet model featuring the
transiting planet and the 5d planet, as well as three three-planet models with
the transiting planet, 5d planet, and a 3d, 15d, or 20d planet in addition.
However, we are hesitant to select any of these three-planet models, as we
have found in our previous analysis that the 15d and 20d signals in our data
are highly chromatic and the 3d signal has a broad period prior and cannot be
recovered by the $\ell_{1}$ periodogram with a <1% FAP. A closer look at our
3d planet fit shows that the observed $K$ of the 3d planet is degenerate with
the GP $\sigma$, meaning that the fit signal may be spurious and dependent
upon how we chose to fit the stellar rotation. While it is possible that there
is a 3d planet in our data, we are hesitant to claim its presence given our
data. If it does exist, it has a $K\,\leq\,0.6$ m s-1 and would thus likely
need far more data to characterize accurately.

We also studied the influence of allowing the eccentricites of the planets to
vary on our selected 2-planet model. We found that, when allowing
$\sqrt{e}$cos$\omega$ and $\sqrt{e}$sin$\omega$ to vary uniformly between -1
and 1 with our selected model, there was an increase in the BIC ($\Delta$BIC =
7.1), but the fit planet eccentricities were low (with broad posterior
distributions) and the planet masses were consistent (within 0.2$\sigma$) with
our circular-orbit model. We also found that allowing for nonzero
eccentricities in our model without the fit GP resulted in essentially no
change ($\Delta$BIC = 0.8). The fact that the planets only appear to have
eccentric orbits with our GPs in place indicates that the small eccentricities
observed may be the result of cross-talk between our keplerian and stellar
activity model and may not be physical. We thus conclude that we do not have
sufficient evidence to conclude the planets have nonzero eccentricities. This
aligns with the expectation that these short-period planets would likely have
circular orbits, as the 2d planet has a tidal circularization timescale of
$<\,2$ Myr and the 5d planet (assuming a rocky planet with a a nearly edge-on
orbit and a larger radius than the 2d planet) has a circularization timescale
of $<\,100$ Myr. Given that this system is likely several Gyr old, we expect
the orbits to be near-circular.

Thus, we select the two-planet model as our final model, as the models with
higher evidences are likely the result of overly flexible GP fitting. We note
that all of the models with higher BIC values have fit transiting planet
masses within 1$\sigma$ of our chosen model, so the transiting planet mass is
not sensitive to our choice of model. Overall, the data support the presence
of a transiting planet, as well as a planet with a 5.1d period. There are
longer-period signals present in the data, but it is difficult to disentangle
aliases of potential rotational signals from long-period planets, as they seem
to encompass similar ranges of periods. There also appears to be tentative
evidence for a planet with a roughly 3d period in the data (which could be in
period commensurability with the two known planets), but we currently lack the
statistical evidence to confirm it.

###  5.3 Joint Transit and Radial Velocity Models

In this section, we perform a joint fit of the RV and transit data using
juliet to determine our final planet parameters.

Before performing a joint fit to the RV and transit data, we checked the data
with transitleastsquares to see if there were any transit signals consistent
with the 5d signal. A closer examination of the TESS light curves yields no
obvious signals of a transiting planet at the newly-identified planet’s period
and $t_{0}$. This is not surprising given the high impact parameter of
TOI-1450Ab. If TOI-1450Ab and TOI-1450Ac were coplanar, TOI-1450Ac would have
an impact parameter $b_{c}\,>\,1$ and thus not transit.

As we do not find any evidence of any additional transiting planets in the
TESS data, we used the RV data alone to characterize the non-transiting planet
in the system. For the priors on the RV fit, we adopted the model from Section
5.2.1. This model included the transiting planet, a 5d planet, and a quasi-
periodic GP to model the $\approx$40d rotational signal. The orbital
eccentricities were fixed at 0. We use the same priors for the transit model
that we used in Section 5.1, with the same data and detrending techniques. Our
priors for the joint fit are listed in Table 9.

Fit parameters |
---|---
$P_{b}$ (d) | N(2.04392, 0.0001)
$P_{c}$ (d) | U(4, 5.5)
$t_{0,b}$ (BJD) | N(2458685.3434, 0.005)
$t_{0,c}$ (BJD) | U(2459321, 2459326.5)
$K_{b}$ (m s-1) | U(0, 3)
$K_{c}$ (m s-1) | U(0, 3)
$e_{bc}$ | 0 (fixed)
$\omega_{bc}$ | 90 (fixed)
$\rho_{\star}$ (g cm-3) | N(6.4, 1)
$\mu_{\mathrm{Apr21,R}}$ (m s-1) | U(-10, 10)
$\mu_{\mathrm{Apr21,B}}$(m s-1) | U(-10, 10)
$\mu_{\mathrm{May21,R}}-\mu_{\mathrm{Apr21,R}}$(m s-1) | N(2.5, 1)
$\mu_{\mathrm{May21,B}}-\mu_{\mathrm{Apr21,B}}$(m s-1) | N(2.5, 1)
$\mu_{\mathrm{Aug21,R}}$(m s-1) | U(-10, 10)
$\mu_{\mathrm{Aug21,B}}$(m s-1) | U(-10, 10)
$\mu_{\mathrm{Nov21,R}}-\mu_{\mathrm{Aug21,R}}$(m s-1) | N(2.5, 1)
$\mu_{\mathrm{Nov21,B}}-\mu_{\mathrm{Aug21,B}}$(m s-1) | N(1.5, 1)
$\mu_{\mathrm{Mar22,R}}$(m s-1) | U(-10, 10)
$\mu_{\mathrm{Mar22,B}}$(m s-1) | U(-10, 10)
$\mu_{\mathrm{May22,R}}-\mu_{\mathrm{Mar22,R}}$(m s-1) | N(1.5, 1)
$\mu_{\mathrm{May22,B}}-\mu_{\mathrm{Mar22,B}}$(m s-1) | N(-1.5, 1)
$\mu_{\mathrm{Jul/Aug22,R}-\mu_{\mathrm{Mar22,R}}}$(m s-1) | N(3.7, 1.12)
$\mu_{\mathrm{Jul/Aug22,B}}-\mu_{\mathrm{Mar22,B}}$(m s-1) | N(0.5, 1.41)
$\mu_{\mathrm{Jul23,R}}$(m s-1) | U(-10, 10)
$\mu_{\mathrm{Jul23,B}}$(m s-1) | U(-10, 10)
ln$\sigma_{\mathrm{all}}$(m s-1) | U(ln(0.001), ln(5))
ln$\sigma_{GP,~{}red}$ | U(ln(0.01, ln(4))
ln$\sigma_{GP,~{}blue}$ | U(ln(0.01, ln(4))
ln$\alpha_{GP}$ | U(ln($1.5\times 10^{-6}$), ln($6.25\times 10^{-4}$))
ln$\Gamma_{GP}$ | U(ln(0.01), ln(10))
$P_{\mathrm{rot}}$ | N(40, 3)
Table 9: The priors used in the RV fit. The joint fit model included both
these priors and the transit priors from Table 5. The ln$\sigma$ for the
different MAROON-X runs and channels are being fit separately, but all share
the same priors so we omit listing them individually in this table.

The numerical results of our joint fits are listed in Tables 10 and 11, and
the resulting plots are shown in Figures 13, 14, and 15. We found that, using
the TIC-derived stellar parameters, TOI-1450Ab has a mass of $1.26\,\pm\,0.13$
$M_{\oplus}$ and a radius of $1.13\,\pm\,0.04$ $R_{\oplus}$. This corresponds
to a density of $4.8\,\pm\,0.7$ g cm-3. If we instead use the SED-derived
stellar radius and mass, we recover a planet mass of
$1.27\,\pm\,0.13\,M_{\oplus}$ and radius of $1.16\pm 0.07\,R_{\oplus}$,
leading to a density of $4.6\,\pm\,1.0$ g cm-3. These values are all within
1$\sigma$ of the planet parameters using the TIC-derived stellar parameters.
This demonstrates that our measured planet mass is somewhat robust to changes
in how the stellar parameters are calculated. For the purposes of this paper,
we will use the planet parameters derived using the TIC stellar parameters
given their higher precision and empirically-calculated nature.

Our extensive dataset allows us to measure the mass of the transiting planet
with a 10% precision despite its very small size and the complicating
influence of the additional planet, activity signal, and near-integer day
orbital period. We are confident in our precise mass measurement, as we found
that our derived planet mass was insensitive to our model choice or method for
determining stellar mass.

Fit parameters | b | c
---|---|---
$P$ (d) | $2.0439274\pm 0.0000010$ | $5.0688\pm 0.0019$
$t_{0}$ (BJD) | $2458685.34221\pm 0.00042$ | $2459321.76\pm 0.16$
$r_{1}$ | $0.8243\pm 0.0147$ | -
$r_{2}$ | $0.0219\pm 0.0005$ | -
$\rho_{\star}$ (g cm-3) | $6.42\pm 0.63$ |
$K$ (m s-1) | $1.05\pm 0.10$ | $0.94\pm 0.11$
Derived parameters |  |
$a/R_{\star}$ | $11.23\pm 0.37$ | -
$R_{p}$ ($R_{\oplus}$) | $1.130\pm 0.043$ | -
$b$ | $0.737\pm 0.022$ | -
$I$ (degrees) | $86.245\pm 0.237$ | -
$T_{\mathrm{dur}}$ (h) | $0.988\pm 0.013$ | -
$T_{\mathrm{eq}}$ (K) | $722\pm 35$ | $533\pm 26$
$M$sin$i$ ($M_{\oplus}$) | $1.256\pm 0.128$ | $1.527\pm 0.181$
$M$ ($M_{\oplus}$) | $1.258\pm 0.128$ | -
Table 10: The planet parameters from the juliet fit to both the detrended TESS data and the MAROON-X RV data. Transit parameters |  |  |  |  |  |
---|---|---|---|---|---|---
Instrument |  $\mu$ (ppm) |  $\sigma$ (ppm) | $q_{1}$ | $q_{2}$ | $u_{1}$ | $u_{2}$
TESS 120s | $-2\pm 22$ | $15^{+21}_{-10}$ | $0.30^{+0.24}_{-0.17}$ | $0.34^{+0.40}_{-0.24}$ | $0.33\pm 0.29$ | $0.15\pm 0.32$
TESS 20s | $10\pm 18$ | $863^{+26}_{-26}$ | $0.32^{+0.17}_{-0.15}$ | $0.52^{+0.30}_{-0.30}$ | $0.54\pm 0.30$ | $-0.03\pm 0.30$
SinistroJul | $26\pm 105$ | $1140^{+77}_{-95}$ | $0.46^{+0.35}_{-0.30}$ | $0.60^{+0.28}_{-0.36}$ | $0.69\pm 0.46$ | $-0.09\pm 0.39$
SinistroAug | $2\pm 81$ | $830^{+73}_{-89}$ | $0.73^{+0.17}_{-0.29}$ | $0.68^{+0.23}_{-0.31}$ | $1.09\pm 0.45$ | $-0.28\pm 0.40$
MuSCAT3g | $-3\pm 61$ | $557^{+79}_{-73}$ | $0.54^{+0.31}_{-0.34}$ | $0.58^{+0.25}_{-0.35}$ | $0.75\pm 0.45$ | $-0.09\pm 0.38$
MuSCAT3r | $30\pm 68$ | $1725^{+51}_{-60}$ | $0.49^{+0.31}_{-0.27}$ | $0.41^{+0.37}_{-0.27}$ | $0.48\pm 0.42$ | $0.10\pm 0.39$
MuSCAT3i | $24\pm 74$ | $2243^{+61}_{-62}$ | $0.40^{+0.38}_{-0.27}$ | $0.52^{+0.32}_{-0.36}$ | $0.55\pm 0.42$ | $-0.02\pm 0.38$
MuSCAT3z | $27\pm 58$ | $2098^{+47}_{-42}$ | $0.40^{+0.34}_{-0.27}$ | $0.43^{+0.36}_{-0.29}$ | $0.46\pm 0.41$ | $0.07\pm 0.37$
RV parameters |  |  |  |  |  |
Parameter | Red | Blue |  |  |  |
$\mu_{\mathrm{Apr21}}$ (m s-1) | $-7.12\pm 0.80$ | $-7.36\pm 0.99$ |  |  |  |
$\mu_{\mathrm{May21}}-\mu_{\mathrm{Apr21,R}}$(m s-1) | $2.81\pm 0.75$ | $2.71\pm 0.87$ |  |  |  |
$\mu_{\mathrm{Aug21}}$(m s-1) | $-4.99\pm 0.79$ | $-2.81\pm 1.08$ |  |  |  |
$\mu_{\mathrm{Nov21}}-\mu_{\mathrm{Aug21,R}}$(m s-1) | $2.44\pm 0.77$ | $1.52\pm 0.84$ |  |  |  |
$\mu_{\mathrm{Mar22}}$(m s-1) | $1.45\pm 0.57$ | $1.17\pm 0.71$ |  |  |  |
$\mu_{\mathrm{May22}}-\mu_{\mathrm{Mar22,R}}$(m s-1) | $1.64\pm 0.77$ | $-0.87\pm 0.92$ |  |  |  |
$\mu_{\mathrm{Jul/Aug22}-\mu_{\mathrm{Mar22,R}}}$(m s-1) | $3.31\pm 0.69$ | $0.67\pm 0.87$ |  |  |  |
$\mu_{\mathrm{Jul23}}$(m s-1) | $6.84\pm 0.79$ | $4.70\pm 1.06$ |  |  |  |
$\sigma_{\mathrm{Apr21}}$(m s-1) | $0.79^{+0.80}_{-0.72}$ | $0.03^{+0.34}_{-0.02}$ |  |  |  |
$\sigma_{\mathrm{May21}}$(m s-1) | $0.46^{+0.13}_{-0.22}$ | $0.51^{+0.45}_{-0.50}$ |  |  |  |
$\sigma_{\mathrm{Aug21}}$(m s-1) | $0.03^{+0.31}_{-0.03}$ | $0.20^{+1.13}_{-0.19}$ |  |  |  |
$\sigma_{\mathrm{Nov21}}$(m s-1) | $0.02^{+0.17}_{-0.02}$ | $0.04^{+0.26}_{-0.04}$ |  |  |  |
$\sigma_{\mathrm{Mar22}}$(m s-1) | $0.70^{+0.39}_{-0.64}$ | $0.05^{+0.30}_{-0.05}$ |  |  |  |
$\sigma_{\mathrm{May22}}$(m s-1) | $0.03^{+0.26}_{-0.03}$ | $0.37^{+1.13}_{-0.36}$ |  |  |  |
$\sigma_{\mathrm{Jul/Aug22}}$(m s-1) | $0.01^{+0.18}_{-0.01}$ | $0.06^{+0.49}_{-0.05}$ |  |  |  |
$\sigma_{\mathrm{Jul23}}$(m s-1) | $0.05^{+0.22}_{-0.05}$ | $0.02^{+0.12}_{-0.01}$ |  |  |  |
GP parameters |  |  |  |  |  |
$\sigma_{\mathrm{GP,red}}$ | $1.40\pm 0.22$ |  |  |  |  |
$\sigma_{\mathrm{GP,blue}}$ | $1.85\pm 0.30$ |  |  |  |  |
$\alpha_{\mathrm{GP}}$ | $0.000025^{+0.000196}_{-0.000022}$ |  |  |  |  |
$\Gamma_{\mathrm{GP}}$ | $8.70^{+0.89}_{1.40}$ |  |  |  |  |
$P_{\mathrm{rot}}$ | $40.1^{+1.7}_{-1.5}$ |  |  |  |  |
Table 11: The instrument parameters from the juliet fit to both the detrended
TESS data and the MAROON-X RV data. Figure 13: Top: The offset-subtracted RV
data for TOI-1450A. The MAROON-X red and blue channels are colored
accordingly. Middle: The RV data with both the offsets and best-fit GPs
subtracted. The best fitting 2-planet Keplerian model is shown in gray.
Bottom: The data residuals (subtracting out the offsets, GPs, and best-fit
Keplerians). Figure 14: The phase-folded RVs for TOI-1450Ab and TOI-1450Ac.
The offsets, activity, and other planets are subtracted out in each panel to
focus on the relevant signal. The MAROON-X red and blue channels are colored
accordingly, and binned data is shown in black. The best-fit model is shown in
gray. Figure 15: The phase-folded transit data for TOI-1450Ab, with our best-
fit model shown in red. Binned data are shown as black points. Each panel
highlights a different dataset.

##  6 Discussion

###  6.1 Dynamics

We must check to see if the reported planet system is stable before claiming a
detection, especially given the fact that there is some ambiguity in the model
selection. An analysis that shows that the system is dynamically stable on
long timescales would support the two-planet model selected in Section 5.2.1.
We can do so quickly with an analytical method inspired by Lissauer et al.
(2011), which makes use of the mutual Hill sphere radius:

$R_{H_{i,o}}=\bigg{(}\frac{M_{i}+M_{o}}{3M_{\star}}\bigg{)}^{1/3}\frac{a_{i}+a_{o}}{2}\
$ (2)

where $i$ and $o$ are adjacent planet indices, with $i$ being an inner planet
and $o$ being an outer planet. The dynamical orbital separation $\Delta$ is
defined as

$\Delta=\frac{a_{o}-a_{i}}{R_{H_{i,o}}}$ (3)

If we assume that the planets are roughly coplanar and that
sin$i\,\approx\,1$, $\Delta_{cb}$ is approximately equal to 33. As shown in
Gladman (1993), a pair of small planets on circular orbits with
$\Delta\,>\,2\sqrt{3}$ are stable against close approaches for very long
periods of time. This indicates that this system would be stable if there are
no other planets in the system. should be stable.

If a planet is present at 3.18d or 3.39d (and thus in three-body resonance
with the other two planets) $\Delta\,>9$ even if the 3d planet had a mass of
$\geq\,10\,M_{\oplus}$. If such a planet was present, it would also satisfy
the stability boundary for three-planet systems discussed in Lissauer et al.
(2011), which would be fulfilled if

$\Delta_{cb}+\Delta_{dc}\,>\,18.$ (4)

If this planet was present, the planets would be relatively evenly-spaced,
aligning with the “peas-in-a-pod" pattern of orbital spacing noted in the
Kepler transiting planet sample by Weiss et al. (2018). With period ratios 1.5
— 1.7, the system would fall on the boundary between correlated and
uncorrelated period ratios described in Jiang et al. (2020), which concluded
that evenly spaced period ratios are likely the result of mean motion
resonances (MMRs). If the TOI-1450A planetary system is the result of disk-
driven inward planet migration, we may expect to see the presence of this
additional planet.

Regarding the secondary star, we note that Ballantyne et al. (2021) states
that, if the binary stars TOI-1450A and TOI-1450B have a circular orbit, any
planet orbiting the primary with $a\,\leq\,28.6$ AU is stable with respect to
the binary system. As all of the planets discussed here have a measured $a$
well within this limit, it’s clear that this system should be stable with
regards to the companion star.

###  6.2 Transiting Planet Composition

With our joint fit, we find that the transiting planet has a mass of
$M_{p}\,=\,1.26\,\pm\,0.13M_{\oplus}$ and a radius of
$R_{p}\,=\,1.13\,\pm\,0.04R_{\oplus}$. Given our knowledge of the mass and
radius of planet, we can estimate its internal composition using the
ManipulatePlanet tool (Zeng & Sasselov, 2013; Zeng et al., 2016).
ManipulatePlanet solves for the planet’s structure based on an extrapolation
based on Earth’s seismic density profile. It estimates the planet’s iron,
water, and rock contents, though these values are somewhat degenerate with the
planet’s central pressure, which is an unknown value. We note that this code
assumes an atmosphere is not present, which means that it may underestimate
the iron content of the planet.

The resulting ternary diagram from ManipulatePlanet is shown in Figure 16. It
appears that this planet may have a core mass fraction that is lower than that
of the Earth ($32.5\,\pm\,0.3\%$, from Wang et al., 2018), though mass and
radius measurements are not precise enough to allow for a better composition
estimate. More precision on the mass and radius measurements will be necessary
to prove whether or not this planet is less dense than the Earth. TOI-1450Ab’s
low density could hint at the presence of an atmosphere, but could also just
be a sign of a small core or an enhanced bulk volatile composition, which is
possible if the planet formed at larger orbital separations and migrated
inward (see, e.g., the results of the simulations from Burn et al., 2021).
Even if it does possess an atmosphere, however, its high equilibrium
temperature rules out the possibility of the planet being habitable, even when
adopting the most optimistic HZ limits from Kopparapu et al. (2016).

Figure 16: Ternary diagram of the composition of TOI-1450Ab generated using
ManipulatePlanet (Zeng & Sasselov, 2013; Zeng et al., 2016). The solid black
line indicates the planet composition for a variety of different central
pressure values, while the dotted black lines indicate the range of values
given the mass and radius errors.

Figure 17 shows where TOI-1450Ab sits on a mass-radius plot compared to other
exoplanets downloaded from TEPCat (Southworth, 2011) as of May 2, 2024.
Composition models from Zeng et al. (2019) are overplotted for comparison.
TOI-1450Ab appears to have a lower density than what we would expect from an
Earthlike planet, indicating that it either lacks a core or has some
meaningful volatile component. If we adopt the stellar paramters from SED
fitting, the planet has an even lower density. However, a higher mass
precision (and better stellar parameter characterization) will be necessary to
make any confident conclusions about the planet’s composition. We also note
that TOI-1450Ab is one of the lowest-mass planets with a sub-20% mass error,
with the only smaller (and more precise) planet masses being the TTV-derived
masses of the TRAPPIST-1 system and the RV-derived mass of the ultra-short-
period planet GJ 367b (Goffo et al., 2023). TOI-1450Ab is the second lowest-
mass planet with a precise RV mass.

Figure 17: The mass and radius of TOI-1450Ab (black star) compared to both
interior composition models from Zeng et al. (2019) (colored dashed lines) and
other exoplanets with mass and radius errors $<\,20\%$ from TEPCat (gray
points). Earth and Venus are also included for reference.

We can only estimate the minimum mass of TOI-1450Ac. If it is coplanar with
TOI-1450Ab, with $i\,\approx\,86^{o}$, it has a mass of about 1.5 Earth
masses, making it only slightly larger than the Earth. It could be very
similar in composition to TOI-1450Ab, though it is impossible to estimate its
composition without a radius measurement.

###  6.3 Suitability for Atmospheric Characterization

Our measured mass precision for TOI-1450Ab is approximately 10%. As the
measured error is less than 20%, the planet’s mass precision is now at the
level that the planet is amenable to high-precision atmospheric
characterization with JWST (Batalha et al., 2019). However, it is first
important to determine whether or not the planet is suitable for such a
measurement. Kempton et al. (2018) created two metrics for the purpose of
identifying transiting planets most suitable for atmospheric characterization.
These are primarily useful for prioritizing planets for follow-up studies with
JWST .

The transmission spectroscopy metric (TSM) gives a description of how suitable
the planet is for transmission spectroscopy. Assuming a low albedo, TOI-1450Ab
has a TSM of $14.3\,\pm\,2.1$. This value is slightly (but not significantly)
higher if we assume the SED-derived stellar characteristics, with a TSM of
$14.5\,\pm\,2.4$. Given the planet’s small radius, it is above the
$TSM\,=\,12$ threshold described by Kempton et al. (2018) for high-quality
atmospheric characterization targets, though only at the $\approx\,1\sigma$
level. As TOI-1450Ab has a very similar mass and radius to that of the Earth,
it at least superficially appears that this target should be prioritized for
follow-up atmospheric characterization efforts. However, given its low mass
and short orbital period, it is possible that any substantial atmosphere that
the planet may have had has already been lost.

We can estimate the photoevaporative mass loss rate of the planet’s atmosphere
given the following equation from Sanz-Forcada et al. (2011):

$\dot{M}=\frac{\pi R_{p}^{3}F_{\mathrm{XUV}}}{\mathrm{G}KM_{p}}$ (5)

where $F_{\mathrm{XUV}}$ is the XUV flux of the host star at the planet’s
orbit, G is the gravitational constant, and $K$ is a term that describes the
potential energy reduction due to tidal forces, using the formula from Erkaev
et al. (2007).

While there are no direct $F_{\mathrm{XUV}}$ measurements of the host star,
Freund et al. (2022) identified that a point X-ray source from the Second
ROSAT all-sky survey (2RXS Boller et al., 2016) seemed to roughly correspond
with the known right ascension and declination of TOI-1450A. However, the
X-ray flux of this source ($2.56*10^{-14}$ erg/s/cm2) is close to the
detection limit of the instrument and the resolution of the instrument is low
enough that this source likely also includes the light from TOI-1450B, so we
can consider this X-ray flux to be an upper limit. Given the star’s distance
from Gaia Collaboration (2020), this corresponds to an X-ray luminosity
$L_{\mathrm{X}}$ of $1.54*10^{27}$ erg/s. As there are no available EUV flux
measurements of the host star, we used the relation from Sanz-Forcada et al.
(2011) to estimate the EUV luminosity, finding
$L_{\mathrm{EUV}}\,\approx\,1.5*10^{28}$ erg/s, though we note that this
relation has a high degree of scatter. This results in a $F_{\mathrm{XUV}}$ at
the planet’s orbit of about $9.7*10^{3}$ erg/s/cm2. Plugging these numbers
into the above equation, we find that $\dot{M}\,\approx\,0.12$
$M_{\oplus}$/Gyr. This mass loss rate seems to be high enough that it would
indicate that the planet could lose any substantial atmosphere over the course
of its life, especially given that the host star likely had a higher XUV flux
earlier in its lifetime. However, we note that the X-ray flux is a likely
upper limit and the method used to estimate the EUV flux is imprecise, so it
is certainly possible that the actual mass loss rate is much slower. The
current data are insufficient to conclude whether or not TOI-1450Ab has lost
its atmosphere.

We can evaluate the likelihood of TOI-1450Ab of having an atmosphere by
comparing it to similar planets with JWST measurements. TRAPPIST-1b, which
superficially resembles TOI-1450Ab in that it is a roughly Earth-sized M dwarf
planet at a sub-3d orbital period (see, e.g., Agol et al., 2021), was recently
observed by JWST and found to have little to no atmosphere (Greene et al.,
2023). As TOI-1450Ab has an even higher equilibrium temperature than
TRAPPIST-1b, it seems possible that it was able to undergo more extreme
atmospheric loss. However, TRAPPIST-1 has a much later spectral type than
TOI-1450A, which could be influential on the planet’s atmosphere due to the
star’s enhanced activity levels and longer activity lifetime (Hawley et al.,
2000). TOI-1450Ab is a challenging target with MIRI/LRS as envisioned by
Kempton et al. (2018). However, Greene et al. (2023) and Zieba et al. (2023)
have been able to successfully detect the thermal emission of TRAPPIST-1b and
TRAPPIST-1c (which have lower emission spectroscopy metrics) at 15 microns,
demonstrating that these challenging targets can still be characterized with
the at longer wavelengths.

##  7 Summary and Conclusions

TOI-1450 is a binary star system with a mid-M primary and a late-M secondary
component. The two stars are too close to separate on an individual TESS
pixel, complicating efforts at follow-up given the secondary’s strong rotation
signal. However, by performing a follow-up RV survey of TOI-1450A with
MAROON-X, we were able to confirm the presence of a 2.044d planet signal that
transits the primary star at the correct period and phase, and found that it
is consistent with a $1.13\,\pm\,0.04$ $R_{\oplus}$ planet with a mass of
$1.26\,\pm\,0.13$ $M_{\oplus}$. This planet has a density consistent with
Earth (given the models from Zeng et al., 2019), but could also possess some
volatiles or even an atmospheric envelope given its slightly sub-Earth
density. Given its mass and radius, TOI-1450Ab is above the cutoff for
suitability for atmospheric transmission spectrum characterization with JWST ,
which may provide us with an interesting opportunity to study the atmosphere
of a hot, low-density M dwarf planet. However, given the planet’s short
orbital period, it is unclear whether or not TOI-1450Ab would be able to
retain an atmosphere over long periods of time, necessitating further modeling
efforts. We also discovered the presence of an additional non-transiting
planet in the system with a 5.07 day period and and a $M_{p}$sin$i_{p}$
similar to TOI-1450Ab.

In addition, while the photometric data contains a sinusoidal 2d signal that
is likely a rotation signal, we were able to use a combination of
spectroscopic activity indicators and chromatic RV signals to support the
hypothesis that TOI-1450B is the rapid rotator in the system. Additionally,
spectroscopic activity indicators and undetrended TESS photometry support a
roughly 40d rotation period for TOI 1450A- a long enough rotation period that
the star’s rotation is unlikely to strongly influence our results.

This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant No. DGE 1746045. GS acknowledges
support provided by NASA through the NASA Hubble Fellowship grant HST-
HF2-51519.001-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
for NASA, under contract NAS5-26555. This work is partly supported by JSPS
KAKENHI Grant Number JP18H05439 and JST CREST Grant Number JPMJCR1761. This
research has also made use of NASA’s Astrophysics Data System Bibliographic
Services.

The University of Chicago group acknowledges funding for the MAROON-X project
from the David and Lucile Packard Foundation, the Heising-Simons Foundation,
the Gordon and Betty Moore Foundation, the Gemini Observatory, the NSF (award
number 2108465), and NASA (grant number 80NSSC22K0117). The Gemini
observations are associated with programs GN-21A-Q-120, GN-21B-Q-216,
GN-22A-Q-120, GN-22A-Q-218, GN-22B-Q-214, and GN-23A-Q-120 (PI: Bean).

Resources supporting this work were provided by the NASA High-End Computing
Capability (HECC) Program through the NASA Advanced Supercomputing (NAS)
Division at Ames Research Center for the production of the SPOC data products.

This work makes use of observations from the LCOGT network. Part of the LCOGT
telescope time was granted by NOIRLab through the Mid-Scale Innovations
Program (MSIP). MSIP is funded by NSF.

This research has made use of the Exoplanet Follow-up Observation Program
(ExoFOP; DOI: 10.26134/ExoFOP5) website, which is operated by the California
Institute of Technology, under contract with the National Aeronautics and
Space Administration under the Exoplanet Exploration Program.

Funding for the TESS mission is provided by NASA’s Science Mission
Directorate. KAC and CNW acknowledge support from the TESS mission via
subaward s3449 from MIT. We acknowledge the use of public TESS data from
pipelines at the TESS Science Office and at the TESS Science Processing
Operations Center.

This paper is based on observations made with the MuSCAT3 instrument,
developed by the Astrobiology Center and under financial supports by JSPS
KAKENHI (JP18H05439) and JST PRESTO (JPMJPR1775), at Faulkes Telescope North
on Maui, HI, operated by the Las Cumbres Observatory.

Some of the data presented in this paper were obtained from the Mikulski
Archive for Space Telescopes (MAST) at the Space Telescope Science Institute.
The specific observations analyzed can be accessed via MAST Team (2021).

Exoplanet Archive, LCOGT

## References

  * Agol et al. (2021) Agol, E., Dorn, C., Grimm, S. L., et al. 2021, The Planetary Science Journal, 2, 1, doi: 10.3847/PSJ/abd022
  * Aigrain et al. (2012) Aigrain, S., Pont, F., & Zucker, S. 2012, MNRAS, 419, 3147, doi: 10.1111/j.1365-2966.2011.19960.x
  * Aller et al. (2020) Aller, A., Lillo-Box, J., Jones, D., Miranda, L. F., & Barceló Forteza, S. 2020, A&A, 635, A128, doi: 10.1051/0004-6361/201937118
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74
  * Ballantyne et al. (2021) Ballantyne, H. A., Espaas, T., Norgrove, B. Z., et al. 2021, MNRAS, 507, 4507, doi: 10.1093/mnras/stab2324
  * Baraffe et al. (2015) Baraffe, I., Homeier, D., Allard, F., & Chabrier, G. 2015, A&A, 577, A42, doi: 10.1051/0004-6361/201425481
  * Barnes (2017) Barnes, R. 2017, Celestial Mechanics and Dynamical Astronomy, 129, 509, doi: 10.1007/s10569-017-9783-7
  * Batalha et al. (2019) Batalha, N. E., Lewis, T., Fortney, J. J., et al. 2019, ApJ, 885, L25, doi: 10.3847/2041-8213/ab4909
  * Bean et al. (2010) Bean, J. L., Seifahrt, A., Hartman, H., et al. 2010, ApJ, 713, 410, doi: 10.1088/0004-637X/713/1/410
  * Boller et al. (2016) Boller, T., Freyberg, M. J., Trümper, J., et al. 2016, A&A, 588, A103, doi: 10.1051/0004-6361/201525648
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977, doi: 10.1126/science.1185402
  * Boyajian et al. (2012) Boyajian, T. S., von Braun, K., van Belle, G., et al. 2012, ApJ, 757, 112, doi: 10.1088/0004-637X/757/2/112
  * Brady et al. (2023) Brady, M., Bean, J. L., Seifahrt, A., et al. 2023, AJ, 165, 129, doi: 10.3847/1538-3881/acb5f7
  * Brady & Bean (2022) Brady, M. T., & Bean, J. L. 2022, AJ, 163, 255, doi: 10.3847/1538-3881/ac64a0
  * Brown et al. (2013) Brown, T. M., Baliber, N., Bianco, F. B., et al. 2013, PASP, 125, 1031, doi: 10.1086/673168
  * Burn et al. (2021) Burn, R., Schlecker, M., Mordasini, C., et al. 2021, A&A, 656, A72, doi: 10.1051/0004-6361/202140390
  * Caballero et al. (2022) Caballero, J. A., Gonzalez-Alvarez, E., Brady, M., et al. 2022, arXiv e-prints, arXiv:2206.09990.  https://arxiv.org/abs/2206.09990
  * Canto Martins et al. (2020) Canto Martins, B. L., Gomes, R. L., Messias, Y. S., et al. 2020, ApJS, 250, 20, doi: 10.3847/1538-4365/aba73f
  * Carnall (2017) Carnall, A. C. 2017, arXiv e-prints, arXiv:1705.05165.  https://arxiv.org/abs/1705.05165
  * Collins (2019) Collins, K. 2019, in American Astronomical Society Meeting Abstracts, Vol. 233, American Astronomical Society Meeting Abstracts #233, 140.05
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., Stassun, K. G., & Hessman, F. V. 2017, AJ, 153, 77, doi: 10.3847/1538-3881/153/2/77
  * Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data Catalog, II/246
  * Czesla et al. (2019) Czesla, S., Schröter, S., Schneider, C. P., et al. 2019, PyA: Python astronomy-related packages.  http://ascl.net/1906.010
  * Davenport et al. (2019) Davenport, J. R. A., Covey, K. R., Clarke, R. W., et al. 2019, ApJ, 871, 241, doi: 10.3847/1538-4357/aafb76
  * Dungee et al. (2022) Dungee, R., van Saders, J., Gaidos, E., et al. 2022, ApJ, 938, 118, doi: 10.3847/1538-4357/ac90be
  * Engle & Guinan (2023) Engle, S. G., & Guinan, E. F. 2023, ApJ, 954, L50, doi: 10.3847/2041-8213/acf472
  * Erkaev et al. (2007) Erkaev, N. V., Kulikov, Y. N., Lammer, H., et al. 2007, A&A, 472, 329, doi: 10.1051/0004-6361:20066929
  * Espinoza (2018) Espinoza, N. 2018, Research Notes of the American Astronomical Society, 2, 209, doi: 10.3847/2515-5172/aaef38
  * Espinoza et al. (2019) Espinoza, N., Kossakowski, D., & Brahm, R. 2019, MNRAS, 490, 2262, doi: 10.1093/mnras/stz2688
  * Fauchez et al. (2020) Fauchez, T. J., Villanueva, G. L., Schwieterman, E. W., et al. 2020, Nature Astronomy, 4, 372, doi: 10.1038/s41550-019-0977-7
  * Foreman-Mackey et al. (2017) Foreman-Mackey, D., Agol, E., Angus, R., & Ambikasaran, S. 2017, ArXiv.  https://arxiv.org/abs/1703.09710
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Foreman-Mackey et al. (2020) Foreman-Mackey, D., Luger, R., Czekala, I., et al. 2020, exoplanet-dev/exoplanet v0.4.0, doi: 10.5281/zenodo.1998447
  * Freund et al. (2022) Freund, S., Czesla, S., Robrade, J., Schneider, P. C., & Schmitt, J. H. M. M. 2022, A&A, 664, A105, doi: 10.1051/0004-6361/202142573
  * Fulton et al. (2017) Fulton, B. J., Petigura, E. A., Howard, A. W., et al. 2017, AJ, 154, 109, doi: 10.3847/1538-3881/aa80eb
  * Gaia Collaboration (2020) Gaia Collaboration. 2020, VizieR Online Data Catalog, I/350
  * Ginsburg et al. (2019) Ginsburg, A., Sipőcz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98, doi: 10.3847/1538-3881/aafc33
  * Giovinazzi & Blake (2022) Giovinazzi, M. R., & Blake, C. H. 2022, arXiv e-prints, arXiv:2208.12112.  https://arxiv.org/abs/2208.12112
  * Gladman (1993) Gladman, B. 1993, Icarus, 106, 247, doi: https://doi.org/10.1006/icar.1993.1169
  * Goffo et al. (2023) Goffo, E., Gandolfi, D., Egger, J. A., et al. 2023, ApJ, 955, L3, doi: 10.3847/2041-8213/ace0c7
  * Gray (2005) Gray, D. F. 2005, The Observation and Analysis of Stellar Photospheres (Cambridge University Press)
  * Gray (2008) —. 2008, The Observation and Analysis of Stellar Photospheres (Cambridge University Press)
  * Greene et al. (2023) Greene, T. P., Bell, T. J., Ducrot, E., et al. 2023, arXiv e-prints, arXiv:2303.14849, doi: 10.48550/arXiv.2303.14849
  * Guerrero et al. (2021) Guerrero, N. M., Seager, S., Huang, C. X., et al. 2021, ApJS, 254, 39, doi: 10.3847/1538-4365/abefe1
  * Hara et al. (2017) Hara, N. C., Boué, G., Laskar, J., & Correia, A. C. M. 2017, MNRAS, 464, 1220, doi: 10.1093/mnras/stw2261
  * Hara et al. (2020) Hara, N. C., Bouchy, F., Stalport, M., et al. 2020, A&A, 636, L6, doi: 10.1051/0004-6361/201937254
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
  * Hawley et al. (2000) Hawley, S., Reid, I. N., & Gizis, J. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 212, From Giant Planets to Cool Stars, ed. C. A. Griffith & M. S. Marley, 252
  * Hawley et al. (1996) Hawley, S. L., Gizis, J. E., & Reid, I. N. 1996, AJ, 112, 2799, doi: 10.1086/118222
  * Hippke & Heller (2019) Hippke, M., & Heller, R. 2019, A&A, 623, A39, doi: 10.1051/0004-6361/201834672
  * Huang et al. (2020a) Huang, C. X., Vanderburg, A., Pál, A., et al. 2020a, Research Notes of the American Astronomical Society, 4, 204, doi: 10.3847/2515-5172/abca2e
  * Huang et al. (2020b) —. 2020b, Research Notes of the American Astronomical Society, 4, 206, doi: 10.3847/2515-5172/abca2d
  * Husser et al. (2013) Husser, T. O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6, doi: 10.1051/0004-6361/201219058
  * Jenkins (2002) Jenkins, J. M. 2002, ApJ, 575, 493, doi: 10.1086/341136
  * Jenkins et al. (2020) Jenkins, J. M., Tenenbaum, P., Seader, S., et al. 2020, Kepler Data Processing Handbook: Transiting Planet Search, Kepler Science Document KSCI-19081-003
  * Jenkins et al. (2010) Jenkins, J. M., Chandrasekaran, H., McCauliff, S. D., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7740, Software and Cyberinfrastructure for Astronomy, ed. N. M. Radziwill & A. Bridger, 77400D, doi: 10.1117/12.856764
  * Jenkins et al. (2016) Jenkins, J. M., Twicken, J. D., McCauliff, S., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, ed. G. Chiozzi & J. C. Guzman, 99133E, doi: 10.1117/12.2233418
  * Jensen (2013) Jensen, E. 2013, Tapir: A web interface for transit/eclipse observability, Astrophysics Source Code Library.  http://ascl.net/1306.007
  * Jiang et al. (2020) Jiang, C.-F., Xie, J.-W., & Zhou, J.-L. 2020, AJ, 160, 180, doi: 10.3847/1538-3881/abb01b
  * Kausch et al. (2015) Kausch, W., Noll, S., Smette, A., et al. 2015, A&A, 576, A78, doi: 10.1051/0004-6361/201423909
  * Kempton et al. (2018) Kempton, E. M. R., Bean, J. L., Louie, D. R., et al. 2018, PASP, 130, 114401, doi: 10.1088/1538-3873/aadf6f
  * Kipping (2013) Kipping, D. M. 2013, MNRAS, 435, 2152, doi: 10.1093/mnras/stt1435
  * Kochanek et al. (2017) Kochanek, C. S., Shappee, B. J., Stanek, K. Z., et al. 2017, PASP, 129, 104502, doi: 10.1088/1538-3873/aa80d9
  * Kopparapu et al. (2016) Kopparapu, R. k., Wolf, E. T., Haqq-Misra, J., et al. 2016, ApJ, 819, 84, doi: 10.3847/0004-637X/819/1/84
  * Kreidberg (2015) Kreidberg, L. 2015, Publications of the Astronomical Society of the Pacific, 127, 1161, doi: 10.1086/683602
  * Lépine et al. (2013) Lépine, S., Hilton, E. J., Mann, A. W., et al. 2013, AJ, 145, 102, doi: 10.1088/0004-6256/145/4/102
  * Li et al. (2019) Li, J., Tenenbaum, P., Twicken, J. D., et al. 2019, PASP, 131, 024506, doi: 10.1088/1538-3873/aaf44d
  * Lightkurve Collaboration et al. (2018) Lightkurve Collaboration, Cardoso, J. V. d. M., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python, Astrophysics Source Code Library.  http://ascl.net/1812.013
  * Lissauer et al. (2011) Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8, doi: 10.1088/0067-0049/197/1/8
  * Luger & Barnes (2015) Luger, R., & Barnes, R. 2015, Astrobiology, 15, 119, doi: 10.1089/ast.2014.1231
  * Luque & Pallé (2022) Luque, R., & Pallé, E. 2022, Science, 377, 1211, doi: 10.1126/science.abl7164
  * Mann et al. (2015) Mann, A. W., Feiden, G. A., Gaidos, E., Boyajian, T., & von Braun, K. 2015, ApJ, 804, 64, doi: 10.1088/0004-637X/804/1/64
  * Mann et al. (2019) Mann, A. W., Dupuy, T., Kraus, A. L., et al. 2019, ApJ, 871, 63, doi: 10.3847/1538-4357/aaf3bc
  * Marboeuf et al. (2014) Marboeuf, U., Thiabaud, A., Alibert, Y., Cabral, N., & Benz, W. 2014, A&A, 570, A36, doi: 10.1051/0004-6361/201423431
  * MAST Team (2021) MAST Team. 2021, TESS Light Curves - All Sectors, STScI/MAST, doi: 10.17909/T9-NMC8-F686
  * McCully et al. (2018) McCully, C., Volgenau, N. H., Harbeck, D.-R., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10707, Proc. SPIE, 107070K, doi: 10.1117/12.2314340
  * Mugrauer & Michel (2020) Mugrauer, M., & Michel, K.-U. 2020, Astronomische Nachrichten, 341, 996, doi: 10.1002/asna.202013825
  * Muirhead et al. (2018) Muirhead, P. S., Dressing, C. D., Mann, A. W., et al. 2018, AJ, 155, 180, doi: 10.3847/1538-3881/aab710
  * Mulders et al. (2015) Mulders, G. D., Ciesla, F. J., Min, M., & Pascucci, I. 2015, ApJ, 807, 9, doi: 10.1088/0004-637X/807/1/9
  * Narita et al. (2020) Narita, N., Fukui, A., Yamamuro, T., et al. 2020, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 11447, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, 114475K, doi: 10.1117/12.2559947
  * Parsons et al. (2018) Parsons, S. G., Gänsicke, B. T., Marsh, T. R., et al. 2018, MNRAS, 481, 1083, doi: 10.1093/mnras/sty2345
  * Pass et al. (2022) Pass, E. K., Charbonneau, D., Irwin, J. M., & Winters, J. G. 2022, ApJ, 936, 109, doi: 10.3847/1538-4357/ac7da8
  * Passegger et al. (2018) Passegger, V. M., Reiners, A., Jeffers, S. V., et al. 2018, A&A, 615, A6, doi: 10.1051/0004-6361/201732312
  * Passegger et al. (2022) Passegger, V. M., Bello-García, A., Ordieres-Meré, J., et al. 2022, A&A, 658, A194, doi: 10.1051/0004-6361/202141920
  * Pecaut & Mamajek (2013) Pecaut, M. J., & Mamajek, E. E. 2013, ApJS, 208, 9, doi: 10.1088/0067-0049/208/1/9
  * Pollacco et al. (2006) Pollacco, D. L., Skillen, I., Collier Cameron, A., et al. 2006, PASP, 118, 1407, doi: 10.1086/508556
  * Popinchalk et al. (2021) Popinchalk, M., Faherty, J. K., Kiman, R., et al. 2021, ApJ, 916, 77, doi: 10.3847/1538-4357/ac0444
  * Rabus et al. (2019) Rabus, M., Lachaume, R., Jordán, A., et al. 2019, MNRAS, 484, 2674, doi: 10.1093/mnras/sty3430
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003, doi: 10.1117/1.JATIS.1.1.014003
  * Robertson et al. (2020) Robertson, P., Stefansson, G., Mahadevan, S., et al. 2020, ApJ, 897, 125, doi: 10.3847/1538-4357/ab989f
  * Rogers et al. (2023) Rogers, J. G., Schlichting, H. E., & Owen, J. E. 2023, ApJ, 947, L19, doi: 10.3847/2041-8213/acc86f
  * Salvatier et al. (2016) Salvatier, J., Wiecki, T. V., & Fonnesbeck, C. 2016, PeerJ Computer Science, 2, e55
  * Sanz-Forcada et al. (2011) Sanz-Forcada, J., Micela, G., Ribas, I., et al. 2011, A&A, 532, A6, doi: 10.1051/0004-6361/201116594
  * Seifahrt et al. (2018) Seifahrt, A., Stürmer, J., Bean, J. L., & Schwab, C. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10702, Ground-based and Airborne Instrumentation for Astronomy VII, ed. C. J. Evans, L. Simard, & H. Takami, 107026D, doi: 10.1117/12.2312936
  * Seifahrt et al. (2022) Seifahrt, A., Bean, J. L., Kasper, D., et al. 2022, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 12184, Ground-based and Airborne Instrumentation for Astronomy IX, ed. C. J. Evans, J. J. Bryant, & K. Motohara, 121841G, doi: 10.1117/12.2629428
  * Shappee et al. (2014) Shappee, B. J., Prieto, J. L., Grupe, D., et al. 2014, ApJ, 788, 48, doi: 10.1088/0004-637X/788/1/48
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163, doi: 10.1086/498708
  * Smette et al. (2015) Smette, A., Sana, H., Noll, S., et al. 2015, A&A, 576, A77, doi: 10.1051/0004-6361/201423932
  * Smith et al. (2012) Smith, J. C., Stumpe, M. C., Van Cleve, J. E., et al. 2012, PASP, 124, 1000, doi: 10.1086/667697
  * Southworth (2011) Southworth, J. 2011, MNRAS, 417, 2166, doi: 10.1111/j.1365-2966.2011.19399.x
  * Speagle (2020) Speagle, J. S. 2020, MNRAS, 493, 3132, doi: 10.1093/mnras/staa278
  * Stassun et al. (2017) Stassun, K. G., Collins, K. A., & Gaudi, B. S. 2017, AJ, 153, 136, doi: 10.3847/1538-3881/aa5df3