# TOI-1231 b: A Temperate, Neptune-Sized Planet Transiting the Nearby M3 Dwarf
NLTT 24399

Jennifer A. Burt Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA  Diana Dragomir
Department of Physics and Astronomy, University of New Mexico, 1919 Lomas Blvd
NE, Albuquerque, NM 87131, USA  Paul Mollière  Max-Planck-Institut für
Astronomie, Königstuhl 17, 69117, Heidelberg, Germany  Allison Youngblood
Laboratory for Atmospheric and Space Physics, 1234 Innovation Drive, Boulder,
CO, 80303, USA  Antonio García Muñoz AIM, CEA, CNRS, Université Paris-Saclay,
Université de Paris, Gif-sur-Yvette, France.  John McCann  Department of
Physics, University of California, Santa Barbara, CA 93106, USA  Laura
Kreidberg Max-Planck-Institut für Astronomie, Königstuhl 17, 69117,
Heidelberg, Germany  Chelsea X. Huang Juan Carlos Torres Fellow  Department of
Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Karen A. Collins Center for
Astrophysics ${\rm\mid}$ Harvard & Smithsonian, 60 Garden Street, Cambridge,
MA 02138, USA  Jason D. Eastman Center for Astrophysics ${\rm\mid}$ Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Lyu Abe  Université
Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de
l’Observatoire, CS 34229, 06304 Nice cedex 4, France  Jose M. Almenara  Univ.
Grenoble Alpes, CNRS, IPAG, 38000, Grenoble, France  Ian J. M. Crossfield
Physics & Astronomy Department, University of Kansas, Lawrence, KS, USA  Carl
Ziegler  Dunlap Institute for Astronomy and Astrophysics, University of
Toronto, 50 St. George Street, Toronto, Ontario M5S 3H4, Canada  Joseph E.
Rodriguez Center for Astrophysics ${\rm\mid}$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Department of Physics and Astronomy,
Michigan State University, East Lansing, MI, 48824 USA  Eric E. Mamajek Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Keivan G. Stassun Department of Physics and
Astronomy, Vanderbilt University, Nashville, TN 37235, USA  Department of
Physics, Fisk University, Nashville, TN 37208, USA  Samuel P. Halverson Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, CA 91109, USA  Steven Jr. Villanueva Pappalardo Fellow
Department of Physics and Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  R. Paul
Butler Earth & Planets Laboratory, Carnegie Institution for Science, 5241
Broad Branch Road, NW, Washington, DC 20015, USA  Sharon Xuesong Wang The
Observatories of the Carnegie Institution for Science, 813 Santa Barbara
Street, Pasadena, CA 91101, USA  Department of Astronomy, Tsinghua University,
Beijing 100084, People’s Republic of China  Richard P. Schwarz Patashnick
Voorheesville Observatory, Voorheesville, NY 12186, USA  George R. Ricker
Department of Physics and Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Roland
Vanderspek Department of Physics and Kavli Institute for Astrophysics and
Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  David W. Latham Center for Astrophysics ${\rm\mid}$ Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  S. 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  Joshua
N. Winn Department of Astrophysical Sciences, Princeton University, 4 Ivy
Lane, Princeton, NJ 08544, USA  Jon M. Jenkins NASA Ames Research Center,
Moffett Field, CA 94035, USA  Abdelkrim Agabi  Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de
l’Observatoire, CS 34229, 06304 Nice cedex 4, France  Xavier Bonfils Univ.
Grenoble Alpes, CNRS, IPAG, 38000, Grenoble, France  David Ciardi  NASA
Exoplanet Science Institute, Infrared Processing & Analysis Center, Jet
Propulsion Laboratory, California Institute of Technology, Pasadena CA 91125,
USA  Marion Cointepas  Univ. Grenoble Alpes, CNRS, IPAG, 38000, Grenoble,
France  Observatoire de l’Université de Genéve, Chemin des Maillettes 51, 1290
Versoix, Switzerland  Jeffrey D. Crane The Observatories of the Carnegie
Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA
Nicolas Crouzet  European Space Agency (ESA), European Space Research and
Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
Georgina Dransfield School of Physics & Astronomy, University of Birmingham,
Edgbaston, Birmingham, B15 2TT, UK  Fabo Feng Tsung-Dao Lee Institute,
Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, People’s
Republic of China  Department of Astronomy, School of Physics and Astronomy,
Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, People’s
Republic of China  Elise Furlan  NASA Exoplanet Science Institute, Infrared
Processing & Analysis Center, Jet Propulsion Laboratory, California Institute
of Technology, Pasadena CA 91125, USA  Tristan Guillot Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de
l’Observatoire, CS 34229, 06304 Nice cedex 4, France  Arvind F. Gupta
Department of Astronomy & Astrophysics, 525 Davey Laboratory, The Pennsylvania
State University, University Park, PA, 16802, USA  Center for Exoplanets and
Habitable Worlds, 525 Davey Laboratory, The Pennsylvania State University,
University Park, PA, 16802, USA  Steve B. Howell NASA Ames Research Center,
Moffett Field, CA 94035, USA  Eric L. N. Jensen Dept. of Physics & Astronomy,
Swarthmore College, Swarthmore PA 19081, USA  Nicholas Law  Department of
Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel
Hill, NC 27599-3255, USA  Andrew W. Mann Department of Physics and Astronomy,
The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255,
USA  Wenceslas Marie-Sainte  Concordia Station, IPEV/PNRA, Antarctica  Rachel
A. Matson U.S. Naval Observatory, Washington, DC 20392, USA  Elisabeth C.
Matthews  Observatoire de l’Université de Genève, Chemin des Maillettes 51,
1290 Versoix, Switzerland  Djamel Mékarnia Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de
l’Observatoire, CS 34229, 06304 Nice cedex 4, France  Joshua Pepper Department
of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015,
USA  Nic Scott  NASA Ames Research Center, Moffett Field, CA 94035, USA
Stephen A. Shectman  The Observatories of the Carnegie Institution for
Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA  Joshua E.
Schlieder Exoplanets and Stellar Astrophysics Laboratory, Code 667, NASA
Goddard Space Flight Center, Greenbelt, MD, 20771, USA  François-Xavier
Schmider  Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4,
France  Daniel J. Stevens Eberly Research Fellow  Johanna K. Teske  Earth &
Planets Laboratory, Carnegie Institution for Science, 5241 Broad Branch Road,
NW, Washington, DC 20015, USA  Amaury H.M.J. Triaud  School of Physics &
Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK  David
Charbonneau Center for Astrophysics ${\rm\mid}$ Harvard & Smithsonian, 60
Garden Street, Cambridge, MA 02138, USA  Zachory K. Berta-Thompson University
of Colorado Boulder, Boulder, CO 80309, USA  Christopher J. Burke Department
of Physics and Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Tansu Daylan
Kavli Fellow  Department of Physics and Kavli Institute for Astrophysics and
Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139,
USA  Thomas Barclay NASA Goddard Space Flight Center, 8800 Greenbelt Road,
Greenbelt, MD 20771, USA  University of Maryland, Baltimore County, 1000
Hilltop Circle, Baltimore, MD 21250, USA  Bill Wohler SETI Institute, Mountain
View, CA 94043, USA  NASA Ames Research Center, Moffett Field, CA 94035, USA
C. E. Brasseur Space Telescope Science Institute

###### Abstract

We report the discovery of a transiting, temperate, Neptune-sized exoplanet
orbiting the nearby ($d$ = 27.5 pc), M3V star TOI-1231 (catalog ) (NLTT 24399
(catalog ), L 248-27 (catalog ), 2MASS J10265947-5228099 (catalog )). The
planet was detected using photometric data from the Transiting Exoplanet
Survey Satellite and followed up with observations from the Las Cumbres
Observatory and the Antarctica Search for Transiting ExoPlanets program.
Combining the photometric data sets, we find that the newly discovered planet
has a radius of 3.65${}^{+0.16}_{-0.15}$ $R_{\earth}$, and an orbital period
of 24.246 days. Radial velocity measurements obtained with the Planet Finder
Spectrograph on the Magellan Clay telescope confirm the existence of the
planet and lead to a mass measurement of 15.5$\pm$3.3 $M_{\earth}$. With an
equilibrium temperature of just 330K TOI-1231 b is one of the coolest small
planets accessible for atmospheric studies thus far, and its host star’s
bright NIR brightness (J=8.88, Ks=8.07) make it an exciting target for HST and
JWST. Future atmospheric observations would enable the first comparative
planetology efforts in the 250-350 K temperature regime via comparisons with
K2-18 b. Furthermore, TOI-1231’s high systemic radial velocity (70.5 k$\rm
m\,s^{-1}$) may allow for the detection of low-velocity hydrogen atoms
escaping the planet by Doppler shifting the H I Ly-alpha stellar emission away
from the geocoronal and ISM absorption features.

Exoplanets (498), Planetary system formation (1257), Radial velocity (1332),
Transit photometry (1709)

††facilities: TESS, Magellan:Clay (Planet Finder Spectrograph), Gemini-South
(Zorro), SOAR, LCOGT, ASTEP††software: AstroImageJ (Collins et al., 2017),
TAPIR (Jensen, 2013), EXOFASTv2 (Eastman et al., 2019), petitCODE (Mollière et
al., 2017), PyMultiNest (Buchner, 2014), MultiNest (Feroz et al., 2009),
Astropy (Astropy Collaboration et al., 2013)

##  1 Introduction

The observing strategy adopted by NASA’s Transiting Exoplanet Survey Satellite
(_TESS_ , Ricker et al., 2014), wherein each hemisphere is divided into 13
sectors each of which is surveyed for roughly 28 days, is producing the most
comprehensive all-sky search for transiting planets. This approach has already
proven its capability to detect both large and small planets (Wang et al.,
2019; Rodriguez et al., 2019; Dragomir et al., 2019; Luque et al., 2019; Burt
et al., 2020) around stars ranging from Sun-like (Huang et al., 2018) down to
low-mass M dwarf stars (Vanderspek et al., 2019).

Although it enables the detection of exoplanets across the sky, TESS’s survey
strategy also produces significant observational biases based on orbital
period. Exoplanets must transit their host stars at least twice within _TESS_
’s observing span in order to be detected with the correct period by the
Science Processing Operations Center (SPOC) pipeline, which searches the
2-minute cadence TESS data obtained for pre-selected target stars (Jenkins et
al., 2016a). Because 74% of _TESS_ ’ total sky coverage is only observed for
$\leq$28 days, the majority of _TESS_ exoplanets detected by the SPOC are
expected to have periods less than 14 days. Simulations of the _TESS_
exoplanet yield presented in Sullivan et al. (2015) find mean and median
orbital periods of 13.48 and 8.19 days, respectively, among all detected
planets. Similarly, Barclay et al. (2018) found mean and median periods of
10.23 and 7.03 days among the full set of detected planets, and those values
drop to just 7.42 and 5.89 days when considering the stars only observed
during a single _TESS_ sector. And additional simulations from Jiang et al.
(2019) show that for stars observed only in a single sector the expected mean
value of the most frequently detected orbital period is 5.01 days, with a most
detected range of 2.12 to 11.82 days. Even when considering the Ecliptic
poles, where _TESS_ carries out 351 days of observing coverage during its
primary mission, the expected mean orbital period is still only 10.93 days
with a most detected range from 3.35 to 35.65 days.

Of the 1994 _TESS_ objects of interest (TOIs) identified as planet candidates
using Sectors 1-26 of the primary mission, only $\sim$14% have periods longer
than 14 days111https://tev.mit.edu/data/collection/193/. But these longer
period (and thereby cooler) candidates are some of the most intriguing targets
for atmospheric characterization. This is especially true for Neptune sized
planets whose lower temperatures could spark several marked changes in the
expected atmospheric chemistry: disequilibrium due to rain out is relevant,
water clouds may form, and ammonia is the dominant carrier of nitrogen (see,
e.g. Morley et al., 2014). Thus these cooler TOIs merit additional attention
and focused follow up efforts both to confirm their planetary nature and to
obtain the precise mass measurements necessary for correct interpretation of
future transmission spectroscopy observations (Batalha et al., 2019).

Here we report the discovery of a Neptune-sized planet transiting TOI-1231
(catalog ) (NLTT 24399, L 248-27, TIC 447061717, 2MASS J10265947-5228099), a
$V$ = 12.3 mag M3V star (Gaidos et al., 2014) in the Vela constellation at $d$
= 27.493 $\pm$ 0.0123 pc ($\varpi$ = 36.3726 $\pm$ 0.0163 mas; GaiaEDR3 van
Leeuwen et al., 2021). This paper is organized as follows. In Section 2 we
characterize the host star using details from published catalogs and new data
obtained once the _TESS_ planet candidate was identified. In Section 3 we
describe the initial discovery of TOI-1231 b and the follow up data obtained
in an effort to characterize the planet. In Section 4 we outline the procedure
used to perform a joint fit to the host star and planet, and then in Section 5
we conclude with a discussion of the planet’s promising potential for future
atmospheric characterization.

##  2 Stellar Data & Characterization

###  2.1 Background

TOI-1231 was first reported as a high proper motion star (0$\farcs$35 yr-1) by
Luyten (1957, as LTT 3840 and L 248-27), and later in the Revised NLTT catalog
(Luyten, 1979, as NLTT 24399). Over the past decade, the star has appeared in
several surveys of high proper motion 2MASS and WISE stars and nearby M dwarfs
(e.g. Lépine & Gaidos, 2011; Frith et al., 2013; Kirkpatrick et al., 2014;
Schneider et al., 2016). The only previous spectral characterization of
TOI-1231 was by Gaidos et al. (2014) in the CONCH-SHELL survey, who reported a
spectral type of M3V. Previous estimates of the basic stellar parameters were
reported by Gaidos et al. (2014); Muirhead et al. (2018); Stassun et al.
(2019).

Table 1: Astrometry & Photometry for TOI-1231 Parameter | Value | Source
---|---|---
Designations | TIC 447061717 | Stassun et al. (2019)
| NLTT 24399 | Luyten (1979)
RA (ICRS, J2000) | 10:26:59.34 | Gaia EDR3
Dec (ICRS, J2000) | -52 28 04.16 | Gaia EDR3
$\mu$ RA (mas yr-1) | -89.394 $\pm$ 0.019 | Gaia EDR3
$\mu$ Dec (mas yr-1) | 361.546 $\pm$ 0.015 | Gaia EDR3
Parallax (mas) | 36.3726 $\pm$ 0.0163 | Gaia EDR3a
Distance (pc) | 27.4932 $\pm$ 0.0123 | Gaia EDR3a
$v_{R}$ (km s-1) | 70.48 $\pm$ 0.54 | Gaia DR2
SpT | M3V | Gaidos et al. (2014)
$B$ | 13.739 $\pm$ 0.028 | APASS/DR10
$V$ | 12.322 $\pm$ 0.023 | APASS/DR10
$g^{\prime}$ | 12.997 $\pm$ 0.044 | APASS/DR10
$r^{\prime}$ | 11.806 $\pm$ 0.070 | APASS/DR10
$i^{\prime}$ | 10.754 $\pm$ 0.175 | APASS/DR10
TESS | 10.2565 $\pm$ 0.007 | TIC8
$G$ | 11.3612 $\pm$ 0.0009 | Gaia EDR3
$G_{BP}$ | 12.5440 $\pm$ 0.0022 | Gaia EDR3
$G_{RP}$ | 10.2735 $\pm$ 0.0015 | Gaia EDR3
$J$ | 8.876 $\pm$ 0.027 | 2MASS
$H$ | 8.285 $\pm$ 0.038 | 2MASS
$K_{s}$ | 8.069 $\pm$ 0.026 | 2MASS
$W_{1}$ | 7.922 $\pm$ 0.024 | WISE
$W_{2}$ | 7.826 $\pm$ 0.020 | WISE
$W_{3}$ | 7.732 $\pm$ 0.017 | WISE
$W_{4}$ | 7.515 $\pm$ 0.091 | WISE
U (km s-1) | -20.25 $\pm$ 0.24 | This work
V (km s-1) | -73.37 $\pm$ 0.36 | This work
W (km s-1) | 39.45 $\pm$ 0.33 | This work
Stot (km s-1) | 85.73 $\pm$ 0.35 | This work
a: Correction of -17 $\mu$as applied to the Gaia parallax following
the prescription in Lindegren et al. (2020)

###  2.2 Astrometry & Photometry

Stellar astrometry and visible and infrared photometry are compiled in Table
1. The positions, proper motions, parallax, and Gaia photometry are from Gaia
EDR3 (van Leeuwen et al., 2021), while the radial velocity is from GaiaDR2
(Gaia Collaboration et al., 2018). We convert the astrometry to Galactic
velocities following (ESA, 1997)222$U$ towards Galactic center, $V$ in
direction of Galactic spin, and $W$ towards North Galactic Pole (ESA, 1997)..
Photometry is reported from APASS Data Release 10 (Henden et al.,
2016)333https://www.aavso.org/apass, the TESS Input Catalog (TIC8), 2MASS
(Cutri et al., 2003), and WISE (Cutri et al., 2012).

###  2.3 Spectral Energy Distribution

We analyzed TOI-1231’s broadband spectral energy distribution (SED) alongside
its Gaia EDR3 parallax to determine an empirical measurement of the star’s
radius, following Stassun & Torres (2016); Stassun et al. (2017, 2018).
Together, the available photometry described here and listed in Table 1 cover
the full stellar SED from 0.4–22 $\mu$m (see Figure 1). We exclude the
APASS/DR10 data from the SED fit in favor of the Gaia EDR3 passbands, which
cover the same wavelength range and have smaller errors, but note that we
adopt an error floor of 0.03 mag on the photometry because the systematics on
the absolute flux calibration between photometric systems is 2-3%.

We performed the fit using NextGen stellar atmosphere models, placing a prior
on the star’s surface gravity ($\log g$) from the TESS Input Catalog (TIC-8).
We set the stellar effective temperature equal to the result from the Cool
Dwarf Catalog ($T_{\rm eff}$ = 3557 $\pm$ 82 K, Muirhead et al., 2018) and the
extinction to zero due to the star’s proximity (within the Local Bubble) which
is consistent with the reddening value of $E_{B-V}=0.001\pm 0.015$ from
Lallement et al. (2018)444STILISM: https://stilism.obspm.fr/. The resulting
SED fit is quite good (Figure 1, top) with a reduced $\chi^{2}$ of 1.9. The
best fit stellar metallicity is [Fe/H] = 0.0 $\pm$ 0.3. Integrating the model
SED results in the bolometric flux at Earth being $F_{\rm bol}=(1.345\pm
0.047)\times 10^{-9}$ erg s-1 cm-2 ($m_{bol}$ = 10.682 $\pm$ 0.032 on IAU 2015
bolometric magnitude scale)555Besides this analysis, we also estimate $m_{\rm
bol}$ via other methods. Applying the M dwarf bolometric magnitude relations
of Casagrande et al. (2008) with the 2MASS $JHK_{s}$ photometry, we estimate
$m_{\rm bol}$ = 10.74 $\pm$ 0.02. Using the $V-J$ vs. $BC_{Ks}$ relation from
Mann et al. (2015) applied to the 2MASS $K_{s}$ magnitude we estimate $m_{\rm
bol}$ = 10.79. Fitting BT-Settl-CIFIST synthetic spectra to the photometry in
Table 1 using the VOSA SED Analyzer (Bayo et al., 2008) yielded $F_{\rm
bol}=(1.363\pm 0.116)\times 10^{-9}$ erg s-1 cm-2 ($m_{\rm bol}$ = 10.666
$\pm$ 0.095) for $T_{\rm eff}$ = 3600 K, $\log g$ = 4.5, [M/H] = 0. These
values provide independent checks, but we adopt the $F_{bol}$ and $m_{\rm
bol}$ in the text.. Taking the $F_{\rm bol}$ and $T_{\rm eff}$ together with
the Gaia EDR3 parallax gives the stellar radius as $R_{\star}$ = 0.466 $\pm$
0.021 $R_{\sun}$ and the stellar mass as $M_{\star}$ = 0.485 $\pm$ 0.024
$M_{\sun}$ based on the empirical $M_{Ks}$ vs. mass relations from Mann et al.
(2019)666For $M_{Ks}$ = 5.861 $\pm$ 0.026 and assuming [Fe/H] = 0..

Figure 1:  Top: Best fit SED for TOI-1231. The red symbols are the observed
photometric measurements and the horizontal lines show the effective width of
each filter. The blue symbols depict the model flux values from the best-fit
NextGen atmosphere model, which is shown in black. Middle: SOAR contrast curve
and image (inset) for I-band speckle observations of TOI-1231. Bottom: Zorro
contrast curves and images (inset) for 562 nm and 832 nm speckle observations
of TOI-1231. No visual companions are detected in the field of view of either
SOAR or Zorro.

####  2.3.1 Speckle Observations

High-angular resolution imaging is needed to search for nearby sources that
can contaminate the TESS photometry, resulting in a diluted transit and an
underestimated planetary radius, and also to search for faint stars that might
be responsible for the transit signal. We searched for nearby sources to
TOI-1231 with SOAR speckle imaging (Tokovinin et al., 2018) on UT 12 December
2019 in I-band, a similar near-IR bandpass as used by _TESS_. Further details
of observations from the SOAR _TESS_ survey are available in Ziegler et al.
(2020). We detected no nearby stars within 3″ of TOI-1231 within the 5$\sigma$
detection sensitivity limits of the observation, which are plotted along with
the speckle auto-correlation function in the middle panel of Figure 1. Using
the measured detection sensitivity from the SOAR observation and the estimated
$T_{\rm eff}$ of the target star along with main-sequence stellar SEDs (Kraus
& Hillenbrand, 2012), we can effectively rule out a main-sequence companion
within angular resolutions between 0.2″ to 3.0″ or projected physical
separations of 5.5 au to 83 au.

Speckle interferometric images of TOI-1231 were also obtained on UT 13 March
2020 using the Zorro777https://www.gemini.edu/sciops/instruments/alopeke-
zorro/ instrument mounted on Gemini-South. Zorro observes simultaneously in
two bands (832$\pm$40 nm and 562$\pm$54 nm) obtaining diffraction limited
images with inner working angles of 0.026 and 0.017 arcseconds, respectively.
The TOI-1231 data set consisted of 5 minutes of total integration time taken
as sets of 1000 $\times$ 0.06 sec images. All the images were combined using
Fourier analysis techniques, examined for stellar companions, and used to
produce reconstructed speckle images (see Howell et al., 2011). The speckle
imaging results reveal TOI-1231 to be a single star to contrast limits of
$\sim$5 to 8 magnitudes, eliminating the possibility of any main sequence
companions to TOI-1231 ($d$ = 27.6 pc) within the spatial limits of 3 to 33 au
(Figure 1, bottom).

###  2.4 Stellar Kinematics and Population

Using the astrometry and radial velocity data from Gaia Collaboration et al.
(2018), we calculate the heliocentric Galactic velocity for TOI-1231 to be
($U,V,W$) = (-20.25, -73.37, 39.45 $\pm$ 0.24, 0.36, 0.33) km s-1, with total
velocity $S_{tot}$ = 85.73 $\pm$ 0.35 km s-1. Compared to the Local Standard
of Rest (LSR) of Schönrich et al. (2010), we estimate velocities of ($U,V,W$)
= (-10.2, -62.4, 46.5) km s-1, with $S_{LSR}$ = 78.4 km s-1. We use the BANYAN
$\Sigma$ (Bayesian Analysis for Nearby Young AssociatioNs $\Sigma$; Gagné et
al., 2018) tool to estimate membership probabilities to nearby young
associations within 150 pc, however the probabilities are $\ll$0.1% for any of
the known nearby stellar groups (all with ages $<$1 Gyr), and the star is
classified as “field”. Following Bensby et al. (2014), we use the Galactic
velocity to estimate kinematic membership probabilities to the Milky Way’s
principal populations, using a 4-population model for the thin disk, thick
disk, halo, and the Hercules stream888The Hercules stream stars contain a mix
of $\alpha$-enhanced old stars and younger less $\alpha$-enhanced around the
solar [Fe/H] (e.g. Bensby et al., 2014), likely from the inner part of the
Galaxy and kinematically heated by the Galactic bar (Dehnen, 2000) and halo,
although the exact type of resonant interaction responsible for the stream is
still controversial (Monari et al., 2019).. We estimate kinematic membership
probabilities of $P$(thin) = 16.7%, $P$(thick) = 64.5%, $P$(Hercules) = 18.7%,
and $P$(halo) = 0.06%. However, the star’s LSR velocity places it among the
Hercules stream member in Fig. 29 of Bensby et al. (2014). Mackereth & Bovy
(2018) calculated parameters of the star’s Galactic orbit using Stäckel
approximation with the Gaia Collaboration et al. (2018) astrometry, and find
an eccentricity of 0.332, $z_{max}$ = 0.905 kpc, perigalacticon of $r_{peri}$
= 4.02 kpc and apogalacticon of $r_{ap}$ = 8.03 kpc999$R_{O}$ = 8 kpc is
assumed., i.e. we are catching the star near its apogalacticon.

We also searched for companions of TOI-1231 in the Gaia Collaboration et al.
(2018) catalog via the 50 pc sample of Torres et al. (2019). Given the mass of
0.46 $M_{\sun}$, we estimate the tidal radius for TOI-1231 (where bound
companions would likely be found) to be 1.04 pc (Mamajek et al., 2013), which
corresponds to a projected radius of $\sim$2∘.2. Querying the Torres et al.
(2019) catalog for stars with proper motions and parallaxes within 20% of that
of the star within 2 tidal radii (4∘.4) yielded no candidate companion.
Therefore, we conclude TOI-1231 to be a single star.

###  2.5 Metallicity

The $V-K_{s}$ vs. absolute magnitude position of an M dwarf can be used to
infer a photometric metallicity estimate (e.g. Johnson & Apps, 2009).
TOI-1231’s combination of $V-K_{s}$ color (4.25) and absolute magnitude
($M_{Ks}$ = 5.86) are consistent with it being 0.17 mag brighter than the
locus for nearby M dwarfs, which Schlaufman & Laughlin (2010) estimate
represents an isometallicity trend of [Fe/H] = -0.14. Using the calibrations
of Johnson & Apps (2009) and Schlaufman & Laughlin (2010), this offset is
consistent with a predicted metallicity of [Fe/H] = +0.05 and -0.03,
respectively, i.e., approximately solar.

Gaia DR2 (Gaia Collaboration et al., 2018) has an estimate of [Fe/H] = -1.5,
but for an unrealistic giant-like surface gravity of $\log g$ = 3.0 and hot
$T_{\rm eff}$ of 4000 K. Taken at face value, the Gaia DR2 metallicity would
predict that the star’s $V-K_{s}$ vs. $M_{Ks}$ position should be more than a
magnitude below the main sequence (extrapolating the metallicity vs. $\Delta
M_{Ks}$ relations of Schlaufman & Laughlin, 2010), well below where it is
observed ($\sim$0.2 mag above the MS).

Anders et al. (2019) uses the _StarHorse_ code to fit $G_{B},\ G,\ G_{R},\ J,\
H,\ Ks,\ W1,\ \rm{and}\ W2$ photometry to solve for a photometric metallicity
estimate consistent with [Fe/H] = 0.095${}^{+0.028}_{-0.061}$.

Taking the mean of our independent photometric metallicity estimates and that
of Anders et al. (2019), we adopt a metallicity of [Fe/H] = +0.05 $\pm$ 0.08.

##  3 Exoplanet Detection & Follow Up

###  3.1 _TESS_ Time Series Photometry

TOI-1231 was selected for transit detection observations by _TESS_ from two
input lists. It was included in the exoplanet candidate target list (CTL) that
accompanied version 8 of the TESS Input Catalog (TIC; Stassun et al., 2019),
and also in the Cool Dwarf Catalog (Muirhead et al., 2018). Its CTL observing
priority was 0.00734, placing it among the top 3% of targets selected for
transit detection by the mission, due to its brightness and small estimated
stellar radius (see sections 3.1 and 3.3 of Stassun et al. (2019) for details
on the prioritization process). It was also selected for observations by TESS
Guest Investigator proposal GO11180 (C. Dressing). TOI-1231 was observed by
_TESS_ from UT 28 February through 26 March 2019 as part of the Sector 9
campaign and again from UT 26 March through 22 April 2019 as part of Sector
10. The star fell on Camera 3 in both sectors, but shifted from CCD 1 in
Sector 9 to CCD 2 in Sector 10.

The SPOC data for TOI-1231 can be accessed at the the Mikulski Archive for
Space Telescopes (MAST) website101010https://mast.stsci.edu, and includes both
the simple aperture photometry (SAP) flux measurements (Twicken et al., 2010;
Morris et al., 2017) and the presearch data conditioned simple aperture
photometry (PDCSAP) flux measurements (Smith et al., 2012; Stumpe et al.,
2012, 2014). These data products differ in that the instrumental variations
present in the SAP measurements are removed from the PDCSAP data. The main
variations are due to thermal effects and strong scattered light present at
the start of each orbit, which impact the systematic error removal in PDC (see
_TESS_ data release notes111111https://archive.stsci.edu/tess/tess_drn.html
DRN16 and DRN17). We therefore use the quality flags provided by SPOC to mask
out unreliable segments of the time series before executing the global fitting
process in Section 4. We further detrend the _TESS_ data by separating the
individual spacecraft orbits (two in each sector) and fitting each orbit’s
flux measurements with a low order spline in order to mitigate residual trends
in the photometry (Figure 2).

Figure 2: TESS light curves: Simple Aperture Photometry (_SAP, top_),
presearch data conditioned SAP (_PDCSAP, middle_), and orbit-by-orbit
detrended PDCSAP (_bottom_). Lighter points depict the _TESS_ two-minute
cadence flux measurements, darker points are the same data binned into 30
minute intervals. The two transit events can be clearly seen as brief flux
dips in the middle and bottom panels.

####  3.1.1 TESS Transit Detection

TOI-1231 b transited once in each of sectors 9 and 10. It was first identified
as a planet candidate five months prior to becoming a TOI, by the TESS Single
Transit Planet Candidate Working Group (TSTPC WG). The TSTPC WG focuses on
searching light curves produced by the MIT Quick Look Pipeline for single
transit events, and validating and/or confirming those that are true planets,
with the aim of increasing the yield of intermediate-to-long-period planets
found by _TESS_ (Villanueva et al. 2019, Villanueva et al. in prep.).

The two transits of TOI-1231 b were later also detected by both the MIT Quick
Look Pipeline (QLP), which searches for evidence of planet candidates in the
_TESS_ 30 minute cadence Full Frame Images, and the SPOC pipeline, which
analyzes the 2-minute cadence data that _TESS_ obtains for pre-selected target
stars (Jenkins et al., 2016b). The _TESS_ transits, one of which occurs in
Sector 9 and the other in Sector 10, have a measured depth of 6453 ppm, a
duration of 3.26 hours, and a measured period of 24.246 days.

While the depth and flat bottomed shape of the TOI-1231 transits were
suggestive of the transit signal being planetary in nature, there are a
variety of false positives that can mimic this combination. The main source of
false positives in the TESS Objects of Interest (TOIs) are eclipsing binaries,
either as two transiting stars on grazing orbits or in the case of a
background blend which reduces the amplitude of a foreground eclipsing binary
signal, causing it to be fallaciously small (e.g. Cameron, 2012). The _TESS_
vetting process is designed to guard against these false positives, and so we
inspected the star’s Data Validation Report (DVR, Twicken et al., 2018; Li et
al., 2019), which is based upon the SPOC two minute cadence data. The multi-
sector DVR shows no signs of secondary eclipses, odd/even transit depth
inconsistencies, nor correlations between the depth of the transit and the
size of the aperture used to extract the light curve, any of which would
indicate that the transit signal originated from by a nearby eclipsing binary.
The DVR also showed that the location of the transit source is consistent with
the position of the target star. Upon passing these vetting checks, the
transit signal was assigned the identifier TOI-1231.01 and announced on the
MIT _TESS_ data alerts website121212http://tess.mit.edu/alerts (Guerrero et
al., 2021).

###  3.2 Ground-based Time-Series Photometry

We acquired ground-based time-series follow-up photometry of TOI-1231 during
the times of transit predicted by the _TESS_ data. We used the TESS Transit
Finder, which is a customized version of the Tapir software package (Jensen,
2013), to schedule our transit observations.

####  3.2.1 LCO 1m Observations

Two partial transits of TOI-1231 b were observed using the Las Cumbres
Observatory Global Telescope (LCOGT) 1m network (Brown et al., 2013) in the
Pan-STARSS $z$ band on UTC 2020 January 16 by the LCOGT node at Cerro Tololo
Inter-American Observatory and May 6 2020 by the LCOGT node at Siding Spring
Observatory (Figure 5, second panel). The telescopes are equipped with
$4096\times 4096$ LCO SINISTRO cameras having an image scale of 0$\farcs$389
pixel-1 resulting in a $26\arcmin\times 26\arcmin$ field of view. The images
were calibrated by the standard LCOGT BANZAI pipeline and the photometric data
were extracted using the AstroImageJ (AIJ) software package (Collins et al.,
2017). Circular apertures with radius 12 pixels (4$\farcs$7) were used to
extract the differential photometry.

####  3.2.2 ASTEP 0.4m Observations

We observed two full transits of TOI-1231 with the Antarctica Search for
Transiting ExoPlanets (ASTEP) program on the East Antarctic plateau (Guillot
et al., 2015; Mékarnia et al., 2016). The $0.4$m telescope is equipped with an
FLI Proline science camera with a KAF-16801E, $4096\times 4096$ front-
illuminated CCD. The camera has an image scale of 0$\farcs$93 pixel-1
resulting in a $1^{\circ}\times 1^{\circ}$ corrected field of view. The focal
instrument dichroic plate splits the beam into a blue wavelength channel for
guiding, and a non-filtered red science channel roughly matching an Rc
transmission curve. The telescope is automated or remotely operated when
needed. Due to the extremely low data transmission rate at the Concordia
Station, the data are processed on-site using an automated IDL-based pipeline,
and the result is reported via email and then transferred to Europe on a
server in Roma, Italy. The raw light curves of about 1,000 stars are then
available for deeper analysis. These data files contain each star’s flux
computed through various fixed circular apertures radii, so that optimal
lightcurves can be extracted (Figure 5, third panel). For TOI-1231 an 11
pixels (10$\farcs$3) radius aperture was found to give the best results.

The observations took place on UTC 2020 May 6 and August 11. Weather was good
to acceptable, and air temperatures ranged between $-50^{\circ}$C and
$-70^{\circ}$C. Two full transits, including the ingress and egress, were
detected. Two other transits of TOI-1231 b were also detected on May 30 and
June 26, but were partial or affected by technical issues, are generally of
lower signal-to-noise and are thus not included in the present analysis. In
each case, the ingress and egress occurred at the predicted times (with an
uncertainty of a few minutes or less), indicating that any transit time
variation must be small.

####  3.2.3 ExTrA 0.6m Observations

The ExTrA facility (Bonfils et al., 2015), located at La Silla observatory,
consists of a near-infrared (0.85 to 1.55 $\mu$m) multi-object spectrograph
fed by three 60-cm telescopes. At the focal plane of each telescope five fiber
positioners pick the light from the target and four comparison stars. We
observed two full transits of TOI-1231 b on UTC 2020 March 18 and 2021 January
27 (Figure 3). The first night we observed with three telescopes using the
fibers with 8″ apertures. The second night we observed with two telescopes and
4″ aperture fibers. Both nights we used the high resolution mode of the
spectrograph ($R\sim$200) and 60-seconds exposures. We also observed 2MASS
J10283882-5234151, 2MASS J10285562-5220140, 2MASS J10273532-5206553, and 2MASS
J10271569-5239330, with J-magnitude (Skrutskie et al., 2006) and Teff (Gaia
Collaboration et al., 2018) similar to TOI-1231, for use as comparison stars.
The resulting ExTrA data were analysed using custom data reduction software.

Figure 3: Light curves obtained by the ExTrA facility on nights UTC 2020 March
18 (left) and 2021 January 27 (right). Top panels show the Raw ExTrA
photometry of each telescope (labeled 1, 2, and 3), displaced vertically for
clarity and a black curve depicting the median of the posterior of the
modeling. Bottom panels show the light curves after correction with the
maximum a posteriori model, with individual observations as light points, 5
minute bins as dark circles, and the maximum a posteriori transit model as a
black line.

The ExTrA light curves are affected by systematic effects that are currently
under investigation. To account for them, we modeled the transits observed by
ExTrA with juliet (Espinoza et al., 2019; Kreidberg, 2015; Speagle, 2020), and
included the quasi-periodic kernel Gaussian Process implemented in celerite
(Foreman-Mackey et al., 2017), with different kernel hyperparameters for each
ExTrA telescope and for each night. We used a prior for the stellar density of
$\rho_{\star}=6.48\pm 0.30~{}\rm g\,cm^{-3}$ (Stassun et al., 2019), and non-
informative priors for the rest of the parameters. The posterior provides the
timings of the observed transits: $2458927.5716\pm 0.0013$ BJDTDB, and
$2459242.7657\pm 0.0013$ BJDTDB, a planet to star radius ratio of
$R_{P}/R_{*}=0.0660_{-0.0086}^{+0.0072}$, and an impact parameter of
$b=0.201_{-0.12}^{+0.091}$. These transit light curves are not used in the
global EXOFATv2 fit described in Section 4 in order to avoid biasing the final
results due to the systematics present in the ExTrA data. However, the timing
of the second transit is used as a prior for the time of conjunction ($T_{C}$)
to better constrain the period of the planet.

###  3.3 Time Series Radial Velocities

Shortly after the discovery of the planet candidate by the TSTPC WG, we began
radial velocity (RV) follow up efforts using the Planet Finder Spectrograph
(PFS) on Las Campanas Observatory’s 6.5m Magellan Clay telescope (Crane et
al., 2006, 2008, 2010). PFS is an iodine cell-based precision RV spectrograph
with an average resolution of $R\simeq$ 130,000. RV values are measured by
placing a cell of gaseous I2, which has been scanned at a resolution of 1
million using the NIST FTS spectrometer (Nave, 2017), in the converging beam
of the telescope. This cell imprints the 5000-6200Å region of the incoming
stellar spectra with a dense forest of I2 lines that act as a wavelength
calibrator and provide a proxy for the point spread function (PSF) of the
spectrometer (Marcy & Butler, 1992).

The spectra are split into 2Å chunks, each of which is analyzed using the
spectral synthesis technique described in Butler et al. (1996), which
deconvolves the stellar spectrum from the I2 absorption lines and produces an
independent measure of the wavelength, instrument PSF, and Doppler shift. The
final Doppler velocity from a given observation is the weighted mean of the
velocities of all the individual chunks ($\sim$800 for PFS). The final
internal uncertainty of each velocity is the standard deviation of all 800
chunk velocities about that mean.

A total of 28 PFS radial observations were obtained from May 2019 to February
2020, binned into 14 velocity measurements, with a mean internal uncertainty
of 1.22 $\rm m\,s^{-1}$(Table 2). A Generalized Lomb-Scargle (GLS) periodogram
of the PFS RV data shows a significant peak at 24 days (Figure 4), which
matches the orbital period determined from the _TESS_ data.

Figure 4:  Top: Generalized Lomb-Scargle periodogram of the radial velocity measurements from PFS. There is a clear peak at period of TOI-1231 b (orange dotted line). Bottom: GLS of the radial velocity residuals, after the signal for planet b has been removed. No significant peaks remain in the data to suggest the presence of additional planets and/or stellar rotational signals.  Table 2: Binned RV data of TOI-1231 Date [BJDTDB] | RV [$\rm m\,s^{-1}$] |  $\sigma_{RV}$ [$\rm m\,s^{-1}$]
---|---|---
2458618.501473 | -10.13 | 1.19
2458625.551234 | 0.32 | 1.44
2458627.574495 | 3.73 | 1.17
2458677.491563 | 7.16 | 1.27
2458679.483330 | 5.96 | 1.22
2458685.500395 | 2.12 | 1.44
2458827.826089 | 3.29 | 1.06
2458828.842260 | -0.34 | 1.08
2458833.804469 | -9.11 | 1.27
2458836.814827 | -4.43 | 1.01
2458883.799738 | -4.85 | 1.21
2458885.807135 | -0.57 | 1.24
2458886.798816 | -2.90 | 1.31
2458890.781648 | 1.10 | 1.17

##  4 System Parameters from EXOFASTv2

To fully characterize the TOI-1231 system, we used the EXOFASTv2 software
package (Eastman et al., 2013, 2019) to perform a simultaneous fit to the
star’s broadband photometry, the _TESS_ , LCO, and ASTEP time series
photometry, and the PFS radial velocity measurements. We applied Gaussian
priors to the parallax and V-band extinction of the star using the results of
Gaia EDR3 ($\varpi=36.3726\pm 0.0163$, corrected using the Lindegren et al.
(2020) prescription) and Lallement et al. (2018) (Av=$0.003\pm 0.0465$),
respectively. Gaussion priors were also placed on the TOI-1231’s mass
($M_{\star}$= 0.461 $\pm$ 0.018 $M_{\sun}$, calculated using the prescription
in Mann et al. 2019), effective temperature (Teff=3562$\pm$101 K, taken from
Gaidos et al. 2014) and metallicity ([Fe/H] = 0.05 $\pm$ 0.08 (see S2.5 for
details). We disabled EXOFASTv2’s MIST isochrone fitting option, which is less
reliable for low mass stars (Eastman et al., 2019). Finally, we placed a prior
on the planet’s time of conjunction derived from the ExTrA photometry (3.2.3).
We did not place any constraints on the planet’s period or orbital
eccentricity.

EXOFASTv2’s SED fitting methodology differs from the approach used in the SED-
only fit that helped verify TOI-1231’s suitability for PRV follow up in
Section 2.3. In place of the NextGen atmospheric models, EXOFASTv2 instead
uses pre-computed bolometric corrections in a grid of $\log g$, $T_{\rm eff}$,
[Fe/H], and V-band extinction
131313http://waps.cfa.harvard.edu/MIST/model_grids.html##bolometric. This grid
is based on the ATLAS/SYNTHE stellar atmospheres (Kurucz, 2005) and the
detailed shapes of the broadband photometric filters.

We note that neither the raw TESS time series photometry nor the PFS RV
measurements exhibit the type of sinusoidal variations that we would expect to
see if the star was subject to rotation-based activity due to active regions
such as star spots or plages crossing the visible hemisphere (Saar & Donahue,
1997; Robertson et al., 2020). This lack of rotational modulation suggests an
inactive star, a claim further supported by the lack of emission or any
detectable temporal changes in the core of the H-$\alpha$ line (see, e.g.,
Reiners et al., 2012; Robertson et al., 2013). We therefore do not include any
additional activity-based terms or detrending efforts when fitting either the
photometric or radial velocity data.

Figure 5: Results of the EXOFASTv2 joint fit to the TESS photometry, ground-
based LCO and ASTEP, and the PFS radial velocities. The top three panels show
the phase folded photometry data from TESS (two full transits), LCO (two
partial transits), and ASTEP (two full transits). All three photometry panels
depict the unbinned data as light points, the five minute binned data as dark
circles, and the best fit model as a solid red line. The bottom panel shows
the phase folded RV measurements from PFS in cyan and the best fit model in
red. The RV residuals after the best fit model has been subtracted from the
data are displayed underneath the fit. Error bars in both RV panels are the
quadrature sum of the PFS internal uncertainties and the RV jitter estimate
from the EXOFASTv2 fit.

The median EXOFASTv2 parameters for the TOI-1231 system are shown in Table 3
and the best fits to the TESS, LCO, and ASTEP photometry and the PFS radial
velocity data are shown in Figure 5. The scatter in a star’s radial velocity
measurements includes any unmodeled instrumental effects or stellar
variability. To address this, we include a ‘jitter’ term in the RV fit which
is used to encompass uncorrelated signals in the star’s own variability or
PFS’s systematics that occur on timescales shorter than the observational
baseline. This value is added in quadrature to the internal uncertainties
reported in the PFS data set to produce the RV error bars seen in Figure 5.
The best fit orbital eccentricity ($e$ = 0.087${}^{+0.012}_{-0.061}$) should
not be regarded as statistically significant as it does not meet the criteria
of being at least 2.45$\sigma$ from 0 that is necessary to avoid falling
subject to the Lucy-Sweeney bias (Lucy & Sweeney, 1971). Even though the best
fit eccentricity is consistent with a circular orbit, we do not enforce a zero
eccentricity fit because even a small amount of non-modeled eccentricity can
bias the resulting orbital parameters and underestimate the uncertainties in
many covariant parameters.

The mass of TOI-1231 b is measured to be 15.5$\pm$3.3 $M_{\earth}$, which,
when combined with the measured planet radius of 3.65${}^{+0.16}_{-0.15}$
$R_{\earth}$, results in a bulk density of 1.74${}^{+0.47}_{-0.42}$ g cm-3
making the planet slightly denser than Neptune ($\rho_{Nep}=1.638$ g cm-3).

##  5 Discussion

Barclay et al. (2018) predict that _TESS_ will find just one non-rocky
Neptune-sized or smaller planet in the habitable zone of a M dwarf brighter
than $J$ = 10. While the final number may be slightly higher, TOI-1231 b is
the first confirmed _TESS_ planet to meet these criteria. One method for
comparing planets’ potential for atmospheric characterization via transmission
spectroscopy is the Transmission Spectroscopy Metric (TSM, Kempton et al.,
2018). A planet’s TSM is proportional to its transmission spectroscopy signal-
to-noise and is based on the strength of its expected spectral features
(derived from its radius and scale height and the radius of the host star) and
the host star’s apparent J-band magnitude. With a transmission spectroscopy
metric (TSM) of 99 $\pm$ 25, TOI-1231 b planet ranks among the highest TSM
Neptunes of any temperature in the pre-TESS era (Figure 6, Guo et al., 2020).

Figure 6: Transmission spectroscopy metric (TSM) values for small planets with
Teq <1000K compared to their radii and host star’s J magnitude, data are taken
from Guo et al. (2020). The four planets that have already had their
atmospheres characterized by HST are shown as solid circles and labeled.
TOI-1231 b (green outline) offers another promising target in the cool, small
planet regime and would, for the first time, enable comparisons in the Teq =
250-350 K range when contrasted with K2-18 b.

TOI-1231 b is one of the coolest planets accessible for atmospheric studies
with Teq = 330 K141414We note that Teq is calculated using Eqn. 1 of Hansen &
Barman (2007), which assumes no albedo and perfect redistribution. As such the
quoted statistical error is likely severely underestimated relative to the
systemic errors inherent in this assumption.. Until recently it appeared that
cooler planets had smaller spectral features, perhaps due to the increasing
number of condensates that can form at lower temperatures (Crossfield &
Kreidberg, 2017). However, new observations of water features in the
habitable-zone planet K2-18 b break this trend (Tsiaras et al., 2019; Benneke
et al., 2019b). The K2-18 b water feature is very intriguing: it is suggestive
of a qualitative change in atmospheric properties near the habitable zone.
Perhaps condensates rain out (analogous to the L/T transition in brown
dwarfs), and/or photochemical haze production is less efficient (Saumon &
Marley, 2008; Morley et al., 2013). However, K2-18 b is the only planet below
350 K with a measured transmission spectrum. TOI-1231 b provides an intriguing
addition to the atmospheric characterization sample in this temperature range
to determine whether K2-18 b is representative or an outlier. Recently, four
HST transit observations were awarded to measure the near-infrared
transmission spectrum of TOI-1231 b with the Wide Field Camera 3 (WFC3)
instrument (GO 16181; PI L. Kreidberg).

###  5.1 Simulated Atmospheric Retrievals

In order to estimate how well the atmospheric properties could be extracted
with HST, we used the open-source petitRADTRANS package (Mollière et al.,
2019) to derive transmission spectra of TOI-1231 b, based on a simple
atmospheric model. The atmosphere probed by the observations was assumed to be
isothermal, at the equilibrium temperature derived for the planet in this
work. Next, equilibrium chemistry was used to calculate the absorber
abundances in the atmosphere, obtained with the chemistry model that is part
of petitCODE (Mollière et al., 2017). We assumed two different compositional
setups, 3 and 100 $\times$ solar (Jupiter and Neptune-like, respectively), at
a solar C/O. In addition, we introduced a gray cloud deck and modeled its
effect on the spectrum when placing it between 100 and $10^{-6}$ bar, in 1 dex
steps. The model with the highest cloud pressure was assumed to be our cloud-
free model, because the atmosphere will become optically thick at lower
pressures. We included the gas opacities of the following line absorbers: H2O,
CH4, CO, CO2, Na, and K. In addition to the gray cloud, continuum opacity
sources arising from H2, He, CO, H2O, CH4, and CO2 Rayleigh scattering, as
well as H2-H2 and H2-He collision-induced absorption, were included. We refer
the reader to Mollière et al. (2019) for the references used for the
opacities.

Figure 7: Synthetic HST observations and retrieval of the clear, 100 $\times$
solar enrichment case. The 16-18 and 2-98 percentile envelopes of the
retrieved flux distribution are shown for the retrievals with the full model
(green envelopes) and the model neglecting CH4 or H2O (black or orange
envelopes, respectively).

We generated mock observations for all cases described above and retrieved
them with petitRADTRANS, using the PyMultiNest package (Buchner et al., 2014).
The latter uses the nested sampling implementation MultiNest (Feroz et al.,
2009). The synthetic HST WCF3 observations were created assuming a wavelength
range of 1.12 to 1.65 $\mu$m, with 12 points spaced equidistantly in
wavelength space. We estimated the uncertainties on the spectroscopic transit
depths using the Pandexo_HST tool151515https://exoctk.stsci.edu/. Assuming
four HST transit observations, we expect uncertainties of 18 ppm on the
transit in each spectral channel. This corresponds to $\sim 0.6$ times the
transit signal of the planet’s scale height, when assuming a 100 $\times$
solar composition. For these retrievals we placed special emphasis on the
detectability of H2O and CH4, for which we implemented the method described in
Benneke & Seager (2013): the abundances of all metal absorbers were retrieved
freely, assuming vertically constant abundance profiles. The abundance of H2
and He was found by requiring that the mass fractions of all species (metals +
H2 and He) add up to unity, with an abundance ratio of 3:1 between H2 and He.
Three retrievals were run for every synthetic observation. (i): nominal model,
retrieving the abundances of all metal absorbers, as well as the cloud deck
pressure. (ii): same as (i), but neglecting the CH4 opacity and CH4 abundance
as a free parameter. (iii): same as (i), but neglecting the H2O opacity and
H2O abundance as a free parameter.

For every synthetic observation we then derived three evidences for models
(i), (ii), and (iii), using nested sampling retrievals. The Bayes factor $B$,
which is the ratio of these evidences, then allows us to assess how strongly
models including CH4 or H2O are preferred (Bayes factor of models (i) and (ii)
or models (i) and (iii), respectively). We used a boundary value of $B>3$ to
express strong preference for a given model, following Kass & Raftery (1995).

We show the synthetic HST observation of the clear, 100 $\times$ solar case in
Figure 7. Because the 100 $\times$ solar case has a larger mean molecular
weight, it is the more challenging of the two enrichment cases. In addition to
the synthetic observation, the 16-18 and 2-98 percentile envelopes of the
retrieved transit depth distribution are shown for the retrievals with models
(i), (ii), and (iii), that is the full model and the model neglecting the CH4
or H2O opacity, respectively.

For these cases, we find very strong preference for including CH4 in the model
($B=10^{8}$) and no preference for including H2O (H2O and CH4 have roughly
equal abundances in the input model but the CH$4$ opacity is larger than the
H2O opacity at all wavelengths). Clouds are a possibility for such planets
(Crossfield & Kreidberg, 2017) and we find that from $P_{\rm cloud}<1$ mbar on
it becomes challenging to detect atmospheric features at all, and the three
models (i), (ii) and (iii) become indistinguishable. In summary, we conclude
that TOI-1231 b is an excellent target for atmospheric characterization. With
a few transit observations, it will be possible to detect spectral features in
an atmosphere similar to that of K2-18 b (Benneke et al., 2019b; Tsiaras et
al., 2019), enabling the first comparative planetology in the temperature
range $250-350$ Kelvin.

###  5.2 Prospects for _JWST_ Observations

With just one transit observed with _JWST_ ’s NIRISS, for a clear, solar
composition atmosphere, we expect to detect TOI-1231 b’s spectrum (dominated
by water at NIRISS wavelengths) with 90$\sigma$ significance (Kempton et al.,
2018).

We also investigate prospects for a cloudy atmosphere, using PLATON (Zhang et
al., 2019) to generate a solar composition model spectrum with a cloud deck
pressure of 10 mbar (similar to GJ 3470 b; Benneke et al. 2019a). We then used
PandEXO (Batalha et al., 2017) to simulate a transmission spectrum for such an
atmosphere. We find that even in this scenario, one transit with the NIRISS
instrument would be sufficient to detect water absorption with 7.5$\sigma$
significance. For reference, this is 2$\sigma$ higher than the detection
significance obtained with six HST WFC3 transits for GJ 3470b (Benneke et al.,
2019a), a planet with a similar size but much lower density (and also orbiting
a 0.5 $R_{\sun}$ star).

###  5.3 Probing Atmospheric Escape

Given TOI-1231 b’s low gravitational potential and expected XUV instellation,
we consider the likelihood that atmospheric escape is occurring and traceable
with H I Lyman $\alpha$ (Ly$\alpha$; 1216 Å) and the meta-stable He I line
(10830 Å). TOI-1231 b’s bulk density is similar to that of GJ 436 b
(1.80$\pm$0.29 g cm-3; Maciejewski et al. 2014), a planet well known for its
vigorously escaping atmosphere (Kulow et al., 2014; Ehrenreich et al., 2015).
TOI-1231’s fundamental stellar properties are very similar to GJ 436’s, and
our PFS spectra indicate that TOI-1231’s log10 R${}^{\prime}_{HK}$ = -5.06 is
nearly equivalent to GJ 436’s (-5.09; Boro Saikia et al. 2018), indicating the
same level of magnetic activity. Since R${}^{\prime}_{HK}$ is known to
correlate well with UV emission (Youngblood et al., 2017), we assume GJ 436’s
synthetic X-ray and UV spectrum from Peacock et al. (2019) as a proxy for
TOI-1231’s. The integrated flux from 100-912 Å at 0.13 au is 172 erg cm-2 s-1,
but could be as low as $\sim$53 erg cm-2 s-1 according to the estimates from
the MUSCLES Treasury Survey (France et al., 2016; Youngblood et al., 2016;
Loyd et al., 2016) based on Linsky et al. (2014). In the energy-limited
approximation (Salz et al., 2016), the mass loss rate scales inversely with
the planet’s bulk density and inversely with the square of the orbital
distance. Thus, under this approximation we expect a mass loss rate about 14
times lower for TOI-1231 b than for GJ 436 b.

Figure 8: Results from the simulated upper atmosphere of TOI-1231 b. Top
panel: the number density of H2 (blue), H and H+ (red), He and He+ (brown),
and H${}_{3}^{+}$ (green) are shown as a function of distance (r/Rp) from the
planet’s 1-$\mu$m pressure level, that we approximate as the planet’s optical
radius (Rp). Neutrals are shown as solid lines and ions are shown as dotted
lines. Bottom panel: The temperature (103 K; blue line) and outward bulk
velocity (km s-1; green line) are shown as a function of distance from the
planet’s 1-$\mu$m pressure level.

To better understand the properties of the escaping atmosphere of TOI-1231 b,
we constructed a 1D model of this planet’s upper atmosphere, which solves the
hydrodynamics equations for an escaping atmosphere and considers
photochemistry at pressures $\lesssim$1 $\mu$bar, and radial distances from
the planet center $r/R_{P}$ = 1–10. We assume the planet’s bulk composition is
dominated by H2 and He, and set volume mixing ratios at the 1 $\mu$bar
boundary of 0.9 for H2 and 0.1 for He. The model is one-dimensional and
spherically-symmetric, appropriate to the sub-stellar line. See García Muñoz
et al. (2020) and references within for more details about the method. The
model simulations shown in Figure 8 are specific to the stellar spectrum from
Peacock et al. (2019) and assume supersonic conditions at r/R${}_{p}\sim$ 10\.
We take the stellar spectrum in its original
format161616http://archive.stsci.edu/hlsp/hazmat, correcting only for orbital
distance. A one-dimensional approach is expected to give a good representation
of the flow within a few planetary radii from the surface, but cannot capture
the shape of the flow after its interaction with the stellar wind. In any
case, our predictions provide helpful insight to understand the prevalent
forms of hydrogen, and the range of temperatures and velocities expected in
the vicinity of the planet.

Figure 8 shows that H2 remains abundant up to very high altitudes, and that
the transition from H to H+ also occurs at high altitude, a condition
favorable for Ly$\alpha$ transit spectroscopy. Unlike for typical hot
Jupiters, H${}_{3}^{+}$ remains relatively abundant over an extended column
and contributes to cooling of the atmosphere and to reducing the mass loss
rate. The model predicts that the planet is losing 2.3$\times$109 g s-1
(integrated over a solid angle 4$\pi$).

We use the H I profile predicted by the model to estimate the Ly$\alpha$
transit depth attributable to hydrogen escaping the planet and before
interacting with the stellar wind. This ‘cold’ component is typically hidden
by ISM absorption and has so far remained undetected. The Ly$\alpha$
absorption reported in other systems including GJ 436 b (e.g., Ehrenreich et
al. 2015) is attributed to a ‘hot’ component that results from charge exchange
of the hydrogen atoms from the planet and the stellar wind protons
(Khodachenko et al., 2019).

Figure 9: The flux densities of TOI-1231 at Earth without (blue) and with
(orange) ISM attenuation and with ISM and in-transit planetary absorption
(green). TOI-1231’s intrinsic Ly$\alpha$ profile is scaled from GJ 436’s and
shifted to +70.5 km s-1, and the ISM attenuation is conservatively assumed to
be log N(HI) = 18.6, centered at -4 km s-1 (Redfield & Linsky, 2008). The
planetary absorption during transit is centered at 0 km s-1 in the stellar
rest frame. The dotted and dash-dotted grey lines represent the ISM and in-
transit planetary absorption, respectively (right axis). The transmission
fraction ($e^{-\tau}$) is unity where all photons are transmitted and zero
where all photons are absorbed.

Despite the smaller expected escape rate with respect to other planets,
TOI-1231 b’s exosphere may be observable during a H I Ly$\alpha$ (1215.67 Å)
transit. The host star’s radial velocity (+70.5 km s-1) Doppler shifts the
entire system partially out of the bulk of ISM’s H I attenuation region,
allowing access to the core of the Ly$\alpha$ line and therefore to the ‘cold’
component.

In order to assess the utility of a Ly$\alpha$ transit with HST for studying
this planet, we estimate the profiles of the stellar Ly$\alpha$ emission, ISM
attenuation, and planetary absorption (Figure 9). We use the reconstructed
Ly$\alpha$ profile of GJ 436 (Youngblood et al., 2016) rescaled to match
TOI-1231’s distance. For the ISM, we assume a velocity centroid for the H I
absorbers of -4.3 km s-1 based on a kinematic model for the local ISM
(Redfield & Linsky, 2008) and a conservative H I column density log10 N(HI) =
18.6 based on measured column densities of nearby sight lines (Wood et al.,
2005; Youngblood et al., 2016; Waalkes et al., 2019). Using STScI’s online
exposure time calculator for STIS with the G140M, 1222 central wavelength, and
52″$\times$0.2″ slit, we find that the expected planetary absorption around
the Ly$\alpha$ line core (the ‘cold’ component) could be detected at high
confidence in a single transit.

We have not modeled the expected transit signature for metastable He I, but
note the potential for exploring this planet’s upper atmosphere with He I
transits. However, as noted above TOI-1231’s XUV spectrum, which is
responsible for both driving escape and populating the metastable He I state,
is likely very similar to GJ 436’s, and GJ 436 b’s vigorously escaping
atmosphere was not detected in He I (Nortmann et al., 2018). TOI-1231 b’s
larger orbit (a = 0.1266 AU) further lowers its likelihood to be a promising
He I target, as the furthest planet with a confirmed He I detection thus far
is WASP 107 b which has a semi-major axis of only 0.055 au and orbits a more
favorable (to He I excitation) K6V star (Spake et al., 2018). See Oklopčić &
Hirata (2018) and Oklopčić (2019) for more details on the prospects tracing
atmospheric escape with metastable He I.

##  6 Conclusions

We reported the _TESS_ discovery and confirmation (using several ground-based
facilities) of TOI 1231 b, a temperate, Neptune-sized planet orbiting a nearby
(27.6 pc) M dwarf star. The mass and radius of TOI-1231 b were measured to be
15.5$\pm$3.3 $M_{\earth}$ and 3.65${}^{+0.16}_{-0.15}$ $R_{\earth}$,
respectively. By virtue of its volatile rich atmosphere, long transit duration
and small host star, TOI 1231 b appears to be one of the most promising small
exoplanets for transmission spectroscopy with HST and _JWST_ detected by the
_TESS_ mission thus far. It represents a rare and valuable addition to the
current sample of just one other low-density Neptune-sized or smaller planet
with an equilibrium temperature in the 250-350 K range and a transmission
spectrum (K2-18 b). Moreover, its high systemic radial velocity makes it a
particularly attractive target for atmospheric escape observations via the H I
Lyman $\alpha$ , and possibly the meta-stable He I line.

This planet also serves as excellent motivation for follow up efforts focused
on _TESS_ single transit events (Villanueva et al., 2019), which is how
TOI-1231 b would have presented itself if the host star was only observed for
a single TESS sector.

Table 3: Median values and 68% confidence interval for EXOFASTv2 results on
TOI-1231.
Notes from Eastman et al. (2019): The optimal conjunction time ($T_{0}$) is
the time of conjunction that minimizes the covariance with the planet’s period
and therefore has the smallest uncertainty. The equilibrium temperature of the
planet ($T_{eq}$) is calculated using Equation 1 of Hansen & Barman (2007) and
assumes no albedo and perfect heat redistribution. The tidal circularization
timescale ($\tau_{\rm circ}$) is calculated using Equation 3 from Adams &
Laughlin (2006) and assumes Q = 106. The 3.6$\mu$m and 4.6$\mu$m secondary
occultation depths use a black-body approximation of the stellar flux,
$F_{\star}$, at Teff and of the planetary flux, $F_{p}$, at Teq and are
calculated using
$\delta_{S,\lambda}=\frac{(R_{p}/R_{\star})^{2}}{(R_{p}/R_{\star})^{2}+(F_{\star}/F_{p})}$.

Parameter | Units | Values
---|---|---
EXOFASTv2 Gaussian priors: |  |  |
$M_{*}$ | Stellar mass ($M_{\sun}$) | $0.485\pm 0.024$ |  |
$T_{\rm eff}$ | Effective Temperature (K) | $3562\pm 101$ |  |
$[{\rm Fe/H}]$ | Metallicity (dex) | $0.05\pm 0.08$ |  |
$\varpi$ | Parallax (mas) | $36.3726\pm 0.0163$ |  |
$A_{V}$ | V-band extinction (mag) | $0.0031\pm 0.0465$ |  |
Stellar Parameters: |  |  |
$M_{*}$ | Mass ($M_{\sun}$) | $0.485\pm 0.024$ |  |
$R_{*}$ | Radius ($R_{\sun}$) | $0.476^{+0.015}_{-0.014}$ |  |
$R_{*,SED}$ | Radius1 ($R_{\sun}$) | $0.4766\pm 0.0084$ |  |
$L_{*}$ | Luminosity ($L_{\sun}$) | $0.0326\pm 0.0010$ |  |
$F_{Bol}$ | Bolometric Flux (10-9 erg s-1 cm-2) | $1.381^{+0.043}_{-0.044}$ |  |
$\rho_{*}$ | Density (g cm-3) | $6.31^{+0.68}_{-0.64}$ |  |
$\log{g}$ | Surface gravity (log(cm s-2)) | $4.767^{+0.033}_{-0.035}$ |  |
$T_{\rm eff}$ | Effective Temperature (K) | $3553^{+51}_{-52}$ |  |
$T_{\rm eff,SED}$ | Effective Temperature1 (K) | $3553\pm 31$ |  |
$[{\rm Fe/H}]$ | Metallicity (dex) | $0.041^{+0.069}_{-0.063}$ |  |
$A_{V}$ | V-band extinction (mag) | $0.030^{+0.033}_{-0.021}$ |  |
$\sigma_{SED}$ | SED photometry error scaling | $2.09^{+0.52}_{-0.37}$ |  |
$\varpi$ | Parallax (mas) | $36.373\pm 0.016$ |  |
$d$ | Distance (pc) | $27.493\pm 0.012$ |  |
Planetary Parameters: | b |  |
$P$ | Period (days) | $24.245586^{+0.000064}_{-0.000066}$ |  |
$R_{P}$ | Radius ($\,R_{\rm E}$) | $3.65^{+0.16}_{-0.15}$ |  |
$M_{P}$ | Mass ($\,M_{\rm E}$) | $15.4\pm 3.3$ |  |
$T_{C}$ | Time of conjunction ($\rm{BJD_{TDB}}$) | $2458563.88838^{+0.00057}_{-0.00058}$ |  |
$T_{T}$ | Time of minimum projected separation ($\rm{BJD_{TDB}}$) | $2458563.88839^{+0.00057}_{-0.00058}$ |  |
$T_{0}$ | Optimal conjunction Time ($\rm{BJD_{TDB}}$) | $2458685.11630^{+0.00048}_{-0.00049}$ |  |
$a$ | Semi-major axis (AU) | $0.1288^{+0.0021}_{-0.0022}$ |  |
$i$ | Inclination (Degrees) | $89.73\pm 0.18$ |  |
$e$ | Eccentricity | $0.087^{+0.12}_{-0.061}$ |  |
$\omega_{*}$ | Argument of Periastron (Degrees) | $176^{+81}_{-90}$ |  |
$T_{eq}$ | Equilibrium temperature (K) | $329.6^{+3.8}_{-3.7}$ |  |
$\tau_{\rm circ}$ | Tidal circularization timescale (Gyr) | $20600^{+9500}_{-8600}$ |  |
$K$ | RV semi-amplitude ($\rm m\,s^{-1}$) | $5.6\pm 1.2$ |  |
$R_{P}/R_{*}$ | Radius of planet in stellar radii | $0.0701^{+0.0019}_{-0.0017}$ |  |
$a/R_{*}$ | Semi-major axis in stellar radii | $58.1\pm 2.0$ |  |
$\delta$ | Transit depth (fraction) | $0.00492^{+0.00027}_{-0.00024}$ |  |
$Depth$ | Flux decrement at mid transit | $0.00492^{+0.00027}_{-0.00024}$ |  |
$\tau$ | Ingress/egress transit duration (days) | $0.00946^{+0.0018}_{-0.00066}$ |  |
$T_{14}$ | Total transit duration (days) | $0.1350^{+0.0017}_{-0.0014}$ |  |
$T_{FWHM}$ | FWHM transit duration (days) | $0.1251\pm 0.0013$ |  |
$b$ | Transit Impact parameter | $0.27^{+0.19}_{-0.18}$ |  |
$b_{S}$ | Eclipse impact parameter | $0.27^{+0.14}_{-0.18}$ |  |
$\tau_{S}$ | Ingress/egress eclipse duration (days) | $0.00961^{+0.00085}_{-0.00077}$ |  |
$T_{S,14}$ | Total eclipse duration (days) | $0.136^{+0.012}_{-0.013}$ |  |
$T_{S,FWHM}$ | FWHM eclipse duration (days) | $0.126^{+0.012}_{-0.013}$ |  |
$\delta_{S,2.5\mu m}$ | Blackbody eclipse depth at 2.5$\mu$m (ppm) | $0.000522^{+0.00012}_{-0.000097}$ |  |
$\delta_{S,5.0\mu m}$ | Blackbody eclipse depth at 5.0$\mu$m (ppm) | $0.99^{+0.13}_{-0.11}$ |  |
$\delta_{S,7.5\mu m}$ | Blackbody eclipse depth at 7.5$\mu$m (ppm) | $10.49^{+1.0}_{-0.86}$ |  |
$\rho_{P}$ | Density (g cm-3) | $1.74^{+0.47}_{-0.42}$ |  |
$logg_{P}$ | Surface gravity | $3.054^{+0.094}_{-0.11}$ |  |
$\Theta$ | Safronov Number | $0.079\pm 0.017$ |  |
$\langle F\rangle$ | Incident Flux (109 erg s-1 cm-2) | $0.00263^{+0.00013}_{-0.00014}$ |  |
$T_{P}$ | Time of Periastron ($\rm{BJD_{TDB}}$) | $2458543.7^{+4.8}_{-6.3}$ |  |
$T_{S}$ | Time of eclipse ($\rm{BJD_{TDB}}$) | $2458575.45^{+0.95}_{-2.3}$ |  |
$T_{A}$ | Time of Ascending Node ($\rm{BJD_{TDB}}$) | $2458557.57^{+0.57}_{-1.4}$ |  |
$T_{D}$ | Time of Descending Node ($\rm{BJD_{TDB}}$) | $2458569.62^{+0.63}_{-1.1}$ |  |
$V_{c}/V_{e}$ |  | $0.993^{+0.062}_{-0.048}$ |  |
$e\cos{\omega_{*}}$ |  | $-0.036^{+0.061}_{-0.15}$ |  |
$e\sin{\omega_{*}}$ |  | $0.002^{+0.046}_{-0.066}$ |  |
$M_{P}\sin i$ | Minimum mass ($\,M_{\rm E}$) | $15.4\pm 3.3$ |  |
$M_{P}/M_{*}$ | Mass ratio | $0.000096^{+0.000021}_{-0.000020}$ |  |
$d/R_{*}$ | Separation at mid transit | $57.5^{+4.3}_{-4.4}$ |  |
$P_{T}$ | A priori non-grazing transit prob | $0.0162^{+0.0014}_{-0.0011}$ |  |
$P_{T,G}$ | A priori transit prob | $0.0186^{+0.0016}_{-0.0013}$ |  |
$P_{S}$ | A priori non-grazing eclipse prob | $0.01607^{+0.0016}_{-0.00069}$ |  |
$P_{S,G}$ | A priori eclipse prob | $0.01849^{+0.0019}_{-0.00080}$ |  |
Wavelength Parameters: | R | z’ | TESS
$u_{1}$ | linear limb-darkening coeff | $0.31^{+0.33}_{-0.22}$ | $0.20^{+0.25}_{-0.15}$ | $0.31\pm 0.19$
$u_{2}$ | quadratic limb-darkening coeff | $0.36^{+0.33}_{-0.44}$ | $0.22^{+0.34}_{-0.27}$ | $0.07^{+0.29}_{-0.23}$
$A_{D}$ | Dilution from neighboring stars | – | – | $-0.001^{+0.046}_{-0.049}$
Telescope Parameters: | PFS velocities |  |
$\gamma_{\rm rel}$ | Relative RV Offset (m/s) | $0.11\pm 0.91$ |  |
$\sigma_{J}$ | RV Jitter (m/s) | $3.03^{+1.0}_{-0.72}$ |  |
$\sigma_{J}^{2}$ | RV Jitter Variance | $9.2^{+7.2}_{-3.8}$ |  |

11footnotetext: This value ignores the systematic error and is for reference
only

This paper includes data collected by the _TESS_ mission. Funding for the
_TESS_ mission is provided by NASA’s Science Mission directorate. We
acknowledge the use of public _TESS_ Alert data from pipelines at the _TESS_
Science Office and the _TESS_ Science Operations Center. This paper includes
data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas
Observatory, Chile. This work makes use of observations from the LCOGT network
and from the ASTEP telescope. ASTEP benefited from the support of the French
and Italian polar agencies IPEV and PNRA in the framework of the Concordia
station program. This work has made use of data from the European Space Agency
(ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. Some of the observations in
the paper made use of the High-Resolution Imaging instrument Zorro. Zorro was
funded by the NASA Exoplanet Exploration Program and built at the NASA Ames
Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett
Quigley. Zorro was mounted on the Gemini South telescope of the international
Gemini Observatory, a program of NSF’s OIR Lab, which is managed by the
Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation on behalf of the
Gemini partnership: the National Science Foundation (United States), National
Research Council (Canada), Agencia Nacional de Investigación y Desarrollo
(Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina),
Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and
Korea Astronomy and Space Science Institute (Republic of Korea). Resources
supporting this work were provided by the NASA High-End Computing (HEC)
Program through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center for the production of the SPOC data products. Some of the data
presented in this paper were obtained from the Mikulski Archive for Space
Telescopes (MAST). Support for MAST for non-HST data is provided by the NASA
Office of Space Science via grant NNX13AC07G and by other grants and
contracts. This research has made use of the NASA Exoplanet Archive, which is
operated by the California Institute of Technology, under contract with the
National Aeronautics and Space Administration under the Exoplanet Exploration
Program. This research has made use of NASA’s Astrophysics Data System. This
research has also made use of the Exoplanet Follow-up Observation Program
website, which is operated by the California Institute of Technology, under
contract with the National Aeronautics and Space Administration under the
Exoplanet Exploration Program. This research made use of Astropy, a community-
developed core Python package for Astronomy (Astropy Collaboration et al.,
2013). Part of this research was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National
Aeronautics and Space Administration (NASA). D. D. acknowledges support from
the TESS Guest Investigator Program grant 80NSSC19K1727 and NASA Exoplanet
Research Program grant 18-2XRP18_2-0136. TD acknowledges support from MIT’s
Kavli Institute as a Kavli postdoctoral fellow. E.M. acknowledges support from
NASA award 17-K2G06-0030. P.M. acknowledges support from the European Research
Council under the European Union’s Horizon 2020 research and innovation
program under grant agreement No. 832428. T.G., A.A., L.A., D.M., F.-X.S.
acknowledge support from Idex UCAJEDI (ANR-15-IDEX-01). D.J.S. acknowledges
funding support from the Eberly Research Fellowship from The Pennsylvania
State University Eberly College of Science. The Center for Exoplanets and
Habitable Worlds is supported by the Pennsylvania State University, the Eberly
College of Science, and the Pennsylvania Space Grant Consortium. This research
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement n∘
803193/BEBOP), and from the Science and Technology Facilities Council (STFC;
grant n∘ ST/S00193X/1). This publication makes use of VOSA, developed under
the Spanish Virtual Observatory project supported by the Spanish MINECO
through grant AyA2017-84089. VOSA has been partially updated by using funding
from the European Union’s Horizon 2020 Research and Innovation Programme,
under Grant Agreement nº 776403 (EXOPLANETS-A).

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# Investigating the Two-Dimensional Generalized XY Model using Tensor Networks

Vamika Longia  Anosh Joseph  Department of Physical Sciences, Indian Institute
of Science Education and Research - Mohali, Knowledge City, Sector 81, SAS
Nagar, Punjab 140306, India  National Institute for Theoretical and
Computational Sciences, School of Physics and Mandelstam Institute for
Theoretical Physics, University of the Witwatersrand, Johannesburg, Wits 2050,
South Africa  Abhishek Samlodia  Department of Physics, Syracuse University,
Syracuse, New York 13244, United States

###### Abstract

The critical behavior of the two-dimensional XY model has been explored in the
literature using various methods. They include the high-temperature expansion
(HTE) method, Monte Carlo (MC) approach, strong coupling expansion method, and
tensor network (TN) methods. This model undergoes a Berezinskii-Kosterlitz-
Thouless (BKT) type of phase transition. This model can be modified by adding
spin-nematic interaction terms with a period to give rise to the generalized
XY model. The modified model contains excitations of integer and half-integer
vortices. These vortices govern the critical behavior of the theory and
produce rich physics. With the help of tensor networks, we investigate the
transition behavior between the integer vortex binding and half-integer vortex
binding phases of the model and how this transition line merges into two BKT
transition lines.

##  1 The Model

The model we study, the two-dimensional generalized XY model on a square
lattice, has the following Hamiltonian

$H=-\left(J\sum_{\langle
ij\rangle}\cos(\theta_{i}-\theta_{j})\;\;+\;\;J_{1}\sum_{\langle
ij\rangle}\cos(q(\theta_{i}-\theta_{j}))\;\;+\;\;h\sum_{i}\cos\theta_{i}\right),$
(1)

where $\theta_{i}\in[0,2\pi]$ is the spin orientation at the lattice site $i$,
$\cos(q(\theta_{i}-\theta_{j}))$, with $q>1$, is the nematic term in the
model, $J>0$ and $J_{1}>0$ are the coupling constants between the nearest-
neighbor spins, $h$ is the applied external magnetic field which breaks the
global $O(2)$ symmetry, $\langle ij\rangle$ denotes the summation over the
nearest-neighbor sites $i$ and $j$ for the square lattice.

For $q=2$, $J=\Delta$, $J_{1}=(1-\Delta)$, and $h=0$ the Hamiltonian takes the
form [1]

$H=-\left(\sum_{\langle
ij\rangle}\Delta\cos(\theta_{i}-\theta_{j})\;\;+\;\;\sum_{\langle
ij\rangle}(1-\Delta)\cos(2(\theta_{i}-\theta_{j}))\right).$ (2)

The usual XY spin model with integer vortex excitations can be obtained from
Eq. (2) by setting $\Delta=1$. By setting $\Delta=0$ in Eq. (2) we can study a
purely spin-nematic model possessing symmetry under the
$\theta_{i}\to\theta_{i}+\pi$ transformation.

We study the model in the regime, $0\leq\Delta\leq 1$, where the system
undergoes two phase transitions: Ising-like, which exhibits continuous phase
transition, and BKT class, which exhibits infinite order phase transition,
with a multi-critical point. The three phases are the integer vortex pair
phase, the half-integer vortex pair phase, and a disordered phase [2]. After
that, we study the model in the presence of non-zero $h$.

##  2 Tensor Network Construction

###  2.1 The Partition Function

The partition function is given by

$Z=\prod_{i}\int\frac{d\theta_{i}}{2\pi}\prod_{\langle
ij\rangle}e^{\beta[\Delta\cos(\theta_{i}-\theta_{j})+(1-\Delta)\cos(2(\theta_{i}-\theta_{j}))]}\prod_{i}e^{\beta
h\cos(\theta_{i})}.$ (3)

Using the character expansion of the Boltzmann factor to construct the tensor
network representation where the phase variable changes into number indices on
the links [3]. This transformation, in terms of modified Bessel functions of
the first kind, $I_{n}(x)$ is

$e^{x\cos\theta}=\sum_{n=-\infty}^{\infty}I_{n}(x)e^{in\theta}.$ (4)

Using the above expansion, the partition function takes the form

$Z=\prod_{s}\int\frac{d\theta_{s}}{2\pi}\prod_{l\in
L}\sum_{n_{l}}a_{n_{l}}(\beta,\Delta)e^{in_{l}(\theta_{s_{i}}-\theta_{s_{j}})}\times\sum_{p_{l}}I_{p_{l}}(\beta
h)e^{ip_{l}\theta_{s_{i}}}.$ (5)

In the above $l$ runs over all the links, $s$ labels all the lattice sites,
and

$a_{n}(\beta,\Delta)=\sum_{m=-\infty}^{\infty}I_{n-2m}(\beta\Delta)I_{m}(\beta(1-\Delta)).$
(6)

After integration over the $\theta$ variables, we obtain the tensor form of
the partition function

$Z={\rm tTr}\left(\prod_{s}T_{n_{1},n_{2}}^{n_{3},n_{4}}(s)\right),$ (7)

with

$T_{n_{1},n_{2},n_{3},n_{4}}(s)=\left(\prod_{i=1}^{4}a_{n_{i}}(\beta,\Delta)\right)^{\frac{1}{2}}~{}I_{n_{1}+n_{2}-n_{3}-n_{4}}(\beta
h),$ (8)

and ${\rm tTr}$ indicates the tensor trace.

The indices $(n_{1}$, $n_{2}$, $n_{3}$, $n_{4})$ $\in$ $(-\infty$, $\infty)$
denote the four legs of the tensor. For computational purposes, the size of
each leg is truncated down to a preferred bond dimension $D$. Similarly, the
index $m$ in Eq. (6) is truncated to a fixed number large enough for
convergent results. Furthermore, for $h=0$ the second term in Eq. (8) becomes
$\delta_{n_{1},n_{2}}^{n_{3},n_{4}}$ [4].

###  2.2 Observables

We use the following set of observables to find the transition temperature in
the model

$F=-\frac{1}{V}\frac{\ln
Z}{\beta},\;\;C_{{}_{V}}=\frac{\partial^{2}F}{\partial\beta^{2}},\;\;M=-\frac{\partial
F}{\partial h}=\frac{1}{\beta}\frac{\partial\ln Z}{\partial
h},\;\;\chi=\frac{\partial M}{\partial
h}\;=\;\frac{1}{\beta}\frac{\partial^{2}\ln Z}{\partial h^{2}},$ (9)

where, $F$ is the free-energy density, $C_{{}_{V}}$ is the specific heat, $M$
is the magnetization, and $\chi$ is the magnetic susceptibility.

Using the partition function given in Eq. (7), the tensor formulation of $M$
can be expressed as [4]

$M={\rm
tTr}\left(\sqrt{\prod_{i=1}^{4}a_{n_{i}}(\beta,\Delta)}\times\frac{I_{n_{1}+n_{2}-n_{3}-n_{4}-1}(\beta
h)+I_{n_{1}+n_{2}-n_{3}-n_{4}+1}(\beta h)}{2}\right).$ (10)

##  3 Numerical Simulations

We use the Higher Order Tensor Renormalization Group (HOTRG) method for the
coarse-graining step of the tensor, as described in [5, 6]. This step is
repeated for a fixed number of iterations to converge the simulations. Then,
in the end, we compute the final partition function and other observables,
such as free-energy density and magnetization. We first study the case where
the external magnetic field is set to zero and find the critical temperature
by locating the specific heat peak for a fixed $\Delta$. Since, there are both
types of phase transitions of which the BKT being infinite-order has the
continuous derivatives of $\log Z$, the specific heat is not enough to
precisely locate the critical points. Hence, we compute the peak of magnetic
susceptibility with respect to temperature for various values of $h$ to obtain
a much more accurate transition.

##  4 Summary

We have reported on how to investigate the phase diagram of the two-
dimensional generalized XY model for $q=2$ in $\cos(q(\theta_{i}-\theta_{j}))$
using the HOTRG method. We can use the specific heat and magnetic
susceptibility to locate the critical points of the transitions belonging to
different types - Ising for the first and BKT for the second.

We use a GPU-accelerated [7] version of the HOTRG method to reduce the
simulation time considerably. Our initial results are consistent with the
conventional Monte Carlo studies [8] but with greater precision. We will soon
report on our detailed simulation results elsewhere.

### Acknowledgements

We thank Raghav Jha for ongoing discussions and collaboration on this work.
Numerical calculations were carried out at IISER Mohali and the PARAM SMRITI
supercomputing facility at the National Agri-Food Biotechnology Institute -
Center of Innovative and Applied Bioprocessing (NABI-CIAB). We acknowledge the
National Supercomputing Mission (NSM) for providing computing resources of
PARAM SMRITI at NABI, Mohali, which is implemented by C-DAC and supported by
the Ministry of Electronics and Information Technology (MeitY) and Department
of Science and Technology (DST), Government of India. Part of the simulations
were carried out at Condor, Syracuse University Cluster as well.

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