TOI-3757 b: A Low-density Gas Giant Orbiting a Solar-metallicity M Dwarf

TOI-3757 (TIC-445751830, Two Micron All Sky Survey (2MASS) J06040089+5501126, Gaia Early Data Release (EDR) 3 996878131494639488, UCAC4 726-038940) is an M0 dwarf observed by TESS in Sector 19 in Camera 2 from 2019 November 27 to 2019 December 24 at ∼30 minute cadence (Figure 1). The planet candidate was identified using the Quick Look Pipeline (QLP) algorithm developed by Huang et al. (2020), under the "faint-star search" (Kunimoto et al. 2021) with a period of ∼3.43 days.

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We extract the photometry from the TESS full-frame images using eleanor (Feinstein et al. 2019), which uses the TESScut ³⁰ service to obtain a cutout of 31 × 31 pixels from the calibrated full-frame images centered on TOI-3757. The TESS light curve is derived from the CORR_FLUX values, in which eleanor uses linear regression with pixel position, measured background, and time to remove signals correlated with these parameters. The final aperture is a 2 × 1 rectangle centered on TOI-3757 and was selected by minimizing the combined differential photometric precision (CDPP) after the data were binned in 1 hour timescales. The CDPP is formally the rms of the photometric noise on transit timescales and was originally defined for Kepler (Jenkins et al. 2010). We obtain a CDPP of 2730 ppm for the TESS photometry.

2.2.1. Red Buttes Observatory

We observed TOI-3757 on the night of 2021 December 21 using the NN-Explore Exoplanet Stellar Speckle Imager (NESSI) on the WIYN 3.5 m telescope at Kitt Peak National Observatory to search for faint background stars and nearby stellar companions. A 9 minute sequence of 40 ms diffraction-limited exposures was collected using the Sloan ${r}^{{\prime} }$ filter on NESSI. The speckle images were reconstructed following the procedures described in Howell et al. (2011). No stellar sources were detected down to a magnitude limit of ${\rm{\Delta }}{r}^{{\prime} }$ = 4.0 at separations >02, as shown in Figure 2.

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2.4.1. HPF

2.4.2. NEID

We also observed TOI-3757 using NEID, a new ultra-precise (Halverson et al. 2016), environmentally stabilized (Robertson et al. 2019) spectrograph at the WIYN 3.5 m telescope at Kitt Peak National Observatory. NEID is a high-resolution spectrograph (R ∼ 110, 000) with an extended red wavelength coverage (380–930 nm; Schwab et al. 2016); it has a fiber-feed system similar to HPF (Kanodia et al. 2018) with three fibers—science, sky, and simultaneous calibration. For these observations we use the NEID high-resolution (HR) mode ³² with resolution R ∼ 110, 000. Between 2021 November 1 and 2022 January 10, we obtained 12 visits on TOI-3757 with NEID, of which we exclude the visit from 2021 November 2 with a S/N of 2.3 at 850 nm due to patchy clouds during the observation. Of the visits included for analysis, each consisted of an 1800 s exposure with a median S/N per 1D extracted pixel of 9.3 at 850 nm.

The NEID data were reduced using the NEID data reduction pipeline ³³ (DRP), and the Level-2 1D extracted spectra were retrieved from the NEID archive. ³⁴ We used a modified version of the SERVAL template-matching algorithm to obtain the NEID RVs (Stefansson et al. 2022), similar to that for HPF. The NEID RVs presented here were calculated using orders spanning 4560–8960 Å (order indices 40 to 104), across the central 7000 pixels which masks out the low S/N spectra outside the free spectral range. We also mask out the telluric and sky emission lines, which were identified following an identical procedure to our HPF processing (Section 2.4.1), and obtain the barycentric velocities using Kanodia & Wright (2018). The final NEID RVs are listed in Table 1.

The early spectral type (M0) of TOI-3757 leads to higher RV information in the optical than in the NIR (Bouchy et al. 2001). Coupled with negligible rotational broadening (v sin i < 2 km s⁻¹), this leads to NEID RVs (R ∼ 110, 000) that are much more precise than HPF (R ∼ 55, 000; Table 1, Figure 3).

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We use the HPF-SpecMatch ³⁵ package (Stefansson et al. 2020) to empirically determine stellar parameters from HPF spectra using the template-matching method based on Yee et al.(2017).

The HPF-SpecMatch algorithm employs a two-step process, which uses the χ² metric twice to find the library stars that best fit the target spectrum. The HPF spectral library contains 166 stars and spans the following parameter space: 2700 K < T_e < 6000 K, $4.3\lt \mathrm{log}{g}_{\star }\lt 5.3$, and −0.5 < [Fe/H] < 0.5.

In the first step, each stellar library spectrum is compared with the target spectrum using the χ² metric and ranked from best to worst-fitting library star to the target spectrum. In the second step, only the top five best-fit ranked library stars are used. The χ² metric is applied to assign scaling constants to each of the five best-fit library stars and create a composite spectrum that fits the target spectrum even more closely. These scaling constants are also used to determine a weighted average for the precise parameter estimates of effective temperature T_eff, surface gravity (log g), and metallicity ([Fe/H]) of the target star.

The errors we adopt for the spectroscopic parameters are determined via a leave-one-out cross validation, broadly following the methodology in Stefansson et al. (2020). In this process, a library star of interest is removed from the rest of the stellar library pool. Then, the HPF-SpecMatch algorithm is run to estimate its stellar parameters independently of its true parameter. The difference between these calculated parameters and the true parameter values are noted. The spectral matching is performed on HPF order index 5 (8534–8645 Å) for TOI-3757 because this order has negligible telluric contamination. The resolution limit of HPF places a constraint of $v\sin i\lt 2\,\mathrm{km}\,{{\rm{s}}}^{-1}$ for TOI-3757. Table 2 presents the derived spectroscopic parameters with their uncertainties.

Table 2. Summary of Stellar Parameters for TOI-3757

Parameter	Description	Value	Reference
Main identifiers:
TOI	TESS Object of Interest	3757	TESS mission
TIC	TESS Input Catalogue	445751830	Stassun
2MASS	⋯	J06040089+5501126	2MASS
Gaia EDR3	⋯	996878131494639488	Gaia EDR3
Equatorial Coordinates, Proper Motion, and Spectral Type:
αJ2016	R.A. (RA)	06:04:00.87	Gaia EDR3
δJ2016	decl. (Dec)	55:01:11.90	Gaia EDR3
μα	Proper motion (RA, mas/yr)	−9.03 ± 0.02	Gaia EDR3
μδ	Proper motion (Dec, mas/yr)	−43.13 ± 0.02	Gaia EDR3
d	Distance in pc	177.4 ± 0.7	Bailer-Jones
${A}_{V,{\max }}$	Maximum visual extinction	0.26	Green
Optical and Near-infrared Magnitudes:
B	Johnson B mag	16.2 ± 0.2	APASS
V	Johnson V mag	14.8 ± 0.1	APASS
${g}^{{\prime} }$	Sloan ${g}^{{\prime} }$ mag	15.5 ± 0.1	APASS
${r}^{{\prime} }$	Sloan ${r}^{{\prime} }$ mag	14.2 ± 0.1	APASS
${i}^{{\prime} }$	Sloan ${i}^{{\prime} }$ mag	13.5 ± 0.1	APASS
J	J mag	12.00 ± 0.03	2MASS
H	H mag	11.31 ± 0.03	2MASS
Ks	Ks mag	11.15 ± 0.02	2MASS
W1	WISE1 mag	11.06 ± 0.02	WISE
W2	WISE2 mag	11.10 ± 0.02	WISE
W3	WISE3 mag	11.0 ± 0.1	WISE
Spectroscopic Parameters a :
Teff	Effective temperature in K	3913 ± 56	This work
[Fe/H]	Metallicity in dex	0.0 ± 0.20	This work
$\mathrm{log}(g)$	Surface gravity in cgs units	4.68 ± 0.04	This work
Model-dependent Stellar SED and Isochrone fit Parameters b :
M*	Mass in M⊙	0.64 ± 0.02	This work
R*	Radius in R⊙	0.62 ± 0.01	This work
L*	Luminosity in L⊙	0.087 ± 0.003	This work
ρ*	Density in g cm−3	3.7 ± 0.2	This work
Age	Age in Gyrs	7.1 ± 4.5	This work
Av	Visual extinction in mag	${0.067}_{-0.047}^{+0.078}$	This work
Other Stellar Parameters:
$v\sin {i}_{* }$	Rotational velocity in km s−1	<2 km s−1	This work
Δ RV	"Absolute" radial velocity in km s−1	21.86 ± 0.04	This work
U, V, W	Galactic velocities in km s−1	−36.12±0.07, −17.81±0.10, −15.82±0.09	This work
U, V, W c	Galactic velocities (LSR) in km s−1	−25.02±0.85, −5.57±0.69, −8.56±0.61	This work

Note. References are: Stassun (Stassun et al. 2018), 2MASS (Cutri et al. 2003), Gaia EDR3 (Collaboration et al. 2021), Bailer-Jones (Bailer-Jones et al. 2018), Green (Green et al. 2019), American Association of Variable Star Observers Photometric All Sky Survey (APASS; Henden et al. 2018), Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010).

^a Derived using the HPF spectral matching algorithm from Stefansson et al. (2020). ^b EXOFASTv2 derived values using MIST isochrones with the Gaia parallax and spectroscopic parameters in a) as priors. ^c The barycentric UVW velocities are converted into local standard of rest (LSR) velocities using the constants from Schönrich et al. (2010).

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We derive model-dependent stellar parameters by modeling the spectral energy distribution (SED) for TOI-3757 using the EXOFASTv2 analysis package (Eastman et al. 2013). The SED fit uses the precomputed bolometric corrections ³⁶ in $\mathrm{log}{g}_{\star }$, T_eff, [Fe/H], and A_V from the Modules for Experiments in Stellar Astrophysics Isochrones and Stellar Tracks (MIST) model grids (Dotter 2016; Choi et al. 2016).

We place Gaussian priors on the (i) broadband photometry listed in Table 2, (ii) the spectroscopic stellar parameters derived with HPF-SpecMatch, and (iii) the geometric distance calculated from Bailer-Jones et al. (2021). We set the upper limit of the visual extinction to the estimate from Green et al. (2019) calculated at the distance determined by Bailer-Jones et al. (2021). The extinction from Green et al. (2019) is converted to a visual magnitude extinction using the R_v = 3.1 reddening law from Fitzpatrick (1999). Table 2 contains the stellar priors and derived stellar parameters with their uncertainties.

We use the systemic velocity derived from HPF and proper motion from GAIA EDR3 to calculate the UVW velocities in the barycentric frame using GALPY (Bovy 2015). ³⁷ We also provide the UVW velocities in the local standard of rest using the offsets from Schönrich et al. (2010). Using the BANYAN tool (Gagné et al. 2018), we classify TOI-3757 as a field star in the thin disk with very high probability (>99%; Bensby et al. 2014).

We run a generalized Lomb–Scargle (GLS) periodogram (Lomb 1976; Scargle 1982; Zechmeister & Kürster 2009) on the TESS photometry (after masking the transits on TOI-3757 b) using its astropy implementation and find a significant peak (<0.1% False Alarm Probability) at ∼7 days (Figure 4). This is consistent with the results from our Gaussian process (GP) stellar rotation kernel ³⁸ applied to the TESS photometry, which suggest a stellar rotation period of ${6.9}_{-0.7}^{+0.5}$ days. This kernel consists of two simple harmonic oscillator terms—one at the rotation period, with the second one at half the period.

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On further inspection using eleanor we find a similar significant peak at ∼7 days for the photometry in the majority of the adjoining pixels in a 6 × 6 grid centered on the centroid for TOI-3757. This suggests that the periodic signal seen in the eleanor reduction of the TESS FFI photometry is not astrophysical in origin, and TOI-3757 does not have significant rotational modulation. This is further corroborated by the lack of a detectable rotational broadening signal in the HPF spectra, using which we can place a limit of v sin i < 2 km s⁻¹ on the host star. ³⁹

We also access publicly available data from the Zwicky Transient Facility (ZTF; Masci et al. 2019) and ASAS-SN (Kochanek et al. 2017) for this target to perform a GLS periodogram analysis, and do not detect any rotation signal present in the photometry. The photometry spans ∼800 days for ZTF and ∼1000 days for ASAS-SN. We also analyze the Ca infrared triplet from the HPF spectra, and Hα from the NEID spectra, but do not find any periodic signals in the time series.

3.5.1. Unresolved Stellar Companions

3.5.2. Resolved Stellar Companions

Figure 5 presents a comparison of the region contained in the 11 × 11 pixel footprint from Sector 19 using a Palomar Observatory Sky Survey (POSS-1; Harrington 1952; Minkowski & Abell 1963) image from 1951 and a ZTF (Masci et al. 2019) image from 2019. The POSS-1 plate images were taken with Eastman 103a-E spectroscopic plates in conjunction with a No. 160 red plexiglass filter with a bandpass between 6000 and 6700 Å, and have a limiting magnitude of R ∼ 19 (Harrington 1952). TOI-3757 has low proper motion, and the change in coordinates between the two epochs is negligible. There are no bright targets with Δ G_RP < 3 present in the 2 × 1 TESS aperture. There are a few targets with Δ G_RP < 4 that may dilute the TESS transit. We use the ground-based photometry to estimate the dilution term for the TESS photometry. Additionally, using the NESSI observations we are able to rule out stellar sources with a ${\rm{\Delta }}{r}^{{\prime} }\lt 4.0$ at separations >02.

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We perform a joint fit of the photometry (TESS + ground-based sources) and the RVs (HPF + NEID) using the Python package exoplanet, which builds upon PyMC3, the Hamiltonian Monte Carlo (HMC) package (Salvatier et al. 2016). The exoplanet package uses starry (Luger et al. 2019; Agol et al. 2020) to model the planetary transits, using the analytical transit models from Mandel & Agol (2002), and a quadratic limb-darkening law. These limb-darkening priors are implemented in exoplanet using the reparameterization from Kipping (2013) for uninformative sampling. We fit each phased transit (Figure 6) with separate limb-darkening coefficients. Our likelihood function for the TESS photometry includes the GP kernel to model the stellar rotation signal. To account for the long-cadence photometry from TESS, exoplanet (Foreman-Mackey et al. 2021a) oversamples the time series while evaluating the model.

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We model the RVs using a standard Keplerian model, where we let the eccentricity float. We include a separate RV offset for each instrument (HPF and NEID), along with a common linear RV trend to account for long-term drifts. For the joint modeling of the photometry + RVs, we also include a simple white-noise model in the form of a jitter term that is added in quadrature to the measurement errors from each data set.

Using scipy.optimize, we find the initial maximum a posteriori (MAP) parameter estimates, which uses the default Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (Broyden 1970; Fletcher 1970; Goldfarb 1970; Shanno 1970). These parameter estimates are then used as the initial conditions for parameter estimation using "No U-Turn Sampling" (NUTS; Hoffman & Gelman 2014), implemented for the HMC sampler PyMC3, where we check for convergence using the Gelman–Rubin statistic ($\hat{{\rm{R}}}\leqslant 1.1;$ Ford 2006).

The final derived planet parameters from the joint fit are included in Table 3, with the phased RVs shown in Figure 7. We obtain a ∼ 10σ mass for TOI-3757 b of 85.3${}_{-8.7}^{+8.8}$ M_⊕, and a radius of 12.0${}_{-0.5}^{+0.4}$ R_⊕.

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Table 3. Derived Parameters for the TOI-3757 System

Parameter	Units	Value a
Orbital Parameters:
Orbital Period...	P (days) ...	3.438753±0.000004
Eccentricity...	e ...	0.14±0.06
Argument of Periastron...	ω (degrees) ...	130±23
Semi-amplitude Velocity...	K (m s−1)...	49.24±5.07
Systemic Velocity b ...	γHPF (m s−1)...	−37.91${}_{-7.85}^{+7.80}$
	γNEID (m s−1)...	17.58±4.15
RV trend...	dv/dt (m s−1 yr−1)	−0.20${}_{-4.94}^{+4.96}$
RV jitter...	σHPF (m s−1)...	${17.71}_{-9.49}^{+10.01}$
	σNEID (m s−1)...	${5.82}_{-3.99}^{+5.66}$
Transit Parameters:
Transit Midpoint ...	TC (BJDTDB)...	${2458838.77148}_{-0.00061}^{+0.00062}$
Scaled Radius...	Rp /R* ...	0.1769${}_{-0.0065}^{+0.0056}$
Scaled Semimajor Axis...	a/R* ...	${13.26}_{-0.25}^{+0.26}$
Orbital Inclination...	i (degrees)...	${86.76}_{-0.20}^{+0.23}$
Transit Duration...	T14 (days)...	${0.0800}_{-0.0030}^{+0.0038}$
Photometric Jitter c ...	σTESS (ppm)...	${2569}_{-59}^{+62}$
	σRBO20211117 (ppm)...	${2953}_{-283}^{+332}$
Limb Darkening d ...	u1,TESS, u2,TESS ...	${0.39}_{-0.28}^{+0.41}$, ${0.10}_{-0.33}^{+0.38}$
	u1,RBO20211117, u2,RBO20211117 ...	${0.59}_{-0.40}^{+0.43}$, ${0.07}_{-0.41}^{+0.43}$
Dilution e ...	DTESS ...	0.994±0.053
Planetary Parameters:
Mass...	Mp (M⊕)...	85.3${}_{-8.7}^{+8.8}$
Radius...	Rp (R⊕)...	12.0${}_{-0.5}^{+0.4}$
Density...	ρp (g cm−3)...	${0.27}_{-0.04}^{+0.05}$
Semimajor Axis...	a (au) ...	0.03845±0.00043
Average Incident Flux f ...	〈F〉 ( 105 W/m2)...	0.75±0.05
Planetary Insolation	S (S⊕)...	55.4±3.8
Equilibrium Temperature g ...	Teq (K)...	759±13

Note.

^a The reported values refer to the 16%–50%–84% percentiles of the posteriors. ^b In addition to the absolute RV from Table 2. ^c Jitter (per observation) added in quadrature to photometric instrument error. ^d Where u₁ + u₂ < 1, and u₁ > 0 according to Kipping (2013). ^e Dilution due to the presence of background stars in the TESS aperture, not accounted for in the eleanor flux. ^f We use a solar flux constant = 1360.8 W m⁻² to convert insolation to incident flux. ^g We assume the planet to be a blackbody with zero albedo and perfect energy redistribution to estimate the equilibrium temperature.

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From the joint RV + photometry fit we obtain an eccentricity of 0.14 ± 0.06, which is consistent with the eccentricity estimate derived from the RBO photometry using the photoeccentric effect (Dawson & Johnson 2012). Here, we compare the stellar density obtained from isochrones and SED fitting in Section 3, to that from the measured transit duration, which is affected by the eccentricity of this orbit. Using this comparison, we measure the eccentricity of the orbit from a joint fit of the TESS and RBO photometry to be ${0.12}_{-0.09}^{+0.31}$. This is consistent (to within 1σ) with the eccentricity derived from the joint RV + photometry fit (Table 3).

The low bulk density of this planet makes it a promising candidate to detect mass loss via atmosphere escape. We observed TOI-3757 b during its transit with HPF (described in Section 2.4.1), to constrain the absorption in the He 10830 Å triplet due to the planetary exosphere. Not only does HET's fixed-altitude design drastically reduces the number of transits observable with HPF, but also the stellar barycentric velocity shifts the helium triplet feature against the bright hydroxyl sky emission lines. Considering these restrictions, we observed the best available transit of TOI-3757 b with HPF in 2021 December.

The individual spectra were sky subtracted using the simultaneous sky spectra obtained via an adjacent sky fiber (Kanodia et al. 2018). Conservatively, we inflate the error bars in the pixels that are corrected for sky emission lines to avoid potential systematics. Telluric correction was not performed on this spectra, as the wavelength region of interest for He 10830 Å detection was well outside the telluric absorption lines. We considered all spectra taken 1.9 hr (one full transit duration) away from transit midpoints in the 2021 November and December for the out-of-transit spectra. The individual out-of-transit spectra were then combined by weighted averaging to obtain a high signal-to-noise ratio template spectrum of the star. Individual in-transit spectra were then divided by this template to obtain a set of transmission (or ratio) spectra. These ratio spectra were then aligned to the planet's rest frame (based on the orbit model) and combined by weighted average to obtain the final transmission spectrum shown in Figure 8. No statistically significant He 10830 Å absorption signal was detected.

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For calculating an upper limit, we used a Gaussian absorption line model with an FWHM width of 0.89 Å. This is the typical width of the reported helium detections in the literature across different planets. ⁴⁰ For a circular orbit, we obtain an upper limit on the maximum depth of the line as 6.9% (with 90% confidence; see Figure 8). If we consider an eccentricity of 0.14 and argument of periastron to be 130 degrees, the net radial velocity of the planet during the transit is redshifted by 11.2 km s⁻¹ in comparison to the circular orbit. This places the expected planetary signal on top of a bright hyroxyl sky emission line. Therefore, an eccentric orbit ephemeris precludes us from placing any meaningful upper limit. With this caveat, while our upper limit shows the signal is weaker than some of the strongest known signal in exoplanets (Allart et al. 2019; Kirk et al. 2020), we encourage future spectroscopic and photometric observations to place tighter limits on the presence of He 10830 Å absorption in TOI-3757.

TOI-3757 b is shown in Figure 9 with respect to other M dwarf giant planets (R_p > 4 R_⊕) with mass measurements at 3σ or higher. The data are taken from the NASA Exoplanet Archive (Akeson et al. 2013) and include recent M dwarf transiting planets discovered by TESS. While TOI-3757 b is Jovian in size (∼1.05 R_J ), its mass is about one-quarter that of Jupiter (∼0.26 M_J ); due to this TOI-3757 b has the lowest density (ρ ∼ 0.27 g cm⁻³) hitherto of M dwarf planets with precise mass and radius measurements (>3σ).

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We examine different hypotheses to explain the low density (and puffy nature) of TOI-3757 b:

• 1.  
  Stellar insolation—The inflated radii of hot Jupiters, ⁴¹ i.e., the so-called radius anomaly, is primarily explained by the high stellar insolation flux incident on these planets with typical equilibrium temperatures >1000 K (Thorngren & Fortney 2018). Here, a fraction of the absorbed energy from the host star is transported deep into the planetary atmosphere, which then through various mechanisms can cause its inflation. ⁴² However as seen in Figure 10(a), the insolation flux incident on TOI-3757 b is not anomalously higher compared to similar M dwarf gas giants. Therefore, we consider it unlikely that the inflated nature of TOI-3757 b is due to stellar insolation.
• 2.  
  Low metallicity—As indicated in Figure 10(b), TOI-3757 b has the lowest metallicity of all known gas giants around M dwarfs. It is important to note the caveat here that the M dwarf gas giants form a heterogeneous sample relying on different techniques for metallicity determination, ⁴³ and hence this must be interpreted cautiously. While we pursue a detailed discussion on the role of stellar metallicity on the formation of gas giants around M dwarfs in an upcoming manuscript (S. Kanodia et al. in preparation), we discuss it briefly here. Assuming that the progenitor protoplanetary disk had the same metallicity as the host star, the disk for TOI-3757 is ∼0.3 dex poorer in metallicity compared to the median metallicity of the other transiting gas giants around M dwarfs (Figure 10(b)), albeit with the caveat that our metallicity estimate for the host star TOI-3757 has a 1σ error of 0.2 dex. Gan et al. (2022) discuss the metallicity trends seen in the transiting M dwarf Jovian sample, which have a median metallicity of ∼0.3 dex, when the sample is updated to include the Jovian planets that have been confirmed since. This directly influences the formation of the planet in two ways—(i) There will be ∼2x lesser material in the disk than those of comparable gas giants with higher metallicities (10^+0.3 ∼ 2); (ii) Yasui et al. (2009) suggest that low-metallicity protoplanetary disks have shorter lifetimes and disperse faster (than comparable high-metallicity disks). This would further exacerbate the problem of slow formation timescales for gas giants around M dwarfs, relative to the disk lifetime (Laughlin et al. 2004). The initiation of runaway gas accretion under the core-accretion theory requires the formation of a rocky core of mass ∼10M_⊕ (Pollack et al. 1996) and an envelope comparable in mass (${M}_{\mathrm{core}}\sim {M}_{\mathrm{env}}$), in a timely manner, before the disk dissipates. Under the hypothesis that the low stellar metallicity is driving the density of TOI-3757 b, due to the two reasons mentioned above, we postulate that for such planets the process of runaway gas accretion did not initiate in a timely enough manner. This would involve a core, which though massive enough for runaway accretion, did not form quickly enough to accrete substantial gas before the disk dissipated. This would explain why TOI-3757 b could not form a planet closer to Jupiter in mass, and is a failed Jupiter.
• 3.  
  Excess internal heat—We explore the possibility that TOI-3757 b's radius is inflated by internal heating (as opposed to external insolation). Multiple studies (e.g., Bodenheimer et al. 2001; Burrows et al. 2007; Lopez & Fortney 2014; Millholland 2019) show that a planet's radius is influenced by its internal heat—hotter interiors lead to "puffier" radii and thus lower densities compared to cooler interior planets of the same composition. One driver of a hotter interior is simply set by a planet's age; younger planets naturally possess hotter interiors (Marley et al. 2007). The lack of a detectable stellar rotation period and SED fit indicate that TOI-3757 is an older star, and therefore we expect its core to have naturally cooled. If TOI-3757 b's inflated radius is due to a hotter interior, it must be heated by some other mechanism. We investigate tidal heating created by orbital eccentricity as a method for inflating the radius of TOI-3757 b. While our eccentric orbit determination for TOI-3757 b is only at the ∼2σ level with an eccentricity of 0.14 ± 0.06, we use the framework presented in Leconte et al. (2010) and calculate a tidal-to-irradiation luminosity ratio of ∼0.05 (assuming an eccentricity of 0.14, zero obliquity, and a reduced tidal quality factor—Q'—of 10⁵). Millholland et al. (2020) find that a tidal-to-irradiation luminosity ratio of >10⁻⁵ is required for potential radius inflation due to eccentricity driven tides. With a tidal luminosity ratio of 5%, TOI-3757 b could be experiencing significant tidal heating from its eccentric orbit. From this power we calculate an interior temperature of 500 K for TOI-3757 b. We also determine that any eccentricity >0.001 would lead to some tidal heating in TOI-3757 b and provide interior temperatures >200 K. Therefore, it is possible that tides are responsible for at least a part of TOI-3757 b's radius inflation. Quantifying the full impact of tidal heating on the radius inflation however requires a detailed investigation into the interior structure of TOI-3757 b and a better estimate of eccentricity.
• 4.  
  Planetary rings—Recent work by Piro & Vissapragada (2020) investigate whether rings could explain the apparent inflated radii of low-density superpuffs. For rings to create the appearance of inflated radii due to deeper than expected transits, the rings would need to be at an oblique angle to the planet's orbital plane. If a planet becomes tidally locked to its star however, the ring system must remain in the orbital plane of the planet (no tilt). In this case, we would only observe the edge on portion of the ring during transit and have a negligible impact on the overall transit depth. Using Equation (11) from Piro & Vissapragada (2020), we calculate a synchronous timescale of 5.3 Myr for TOI-3757 b. If TOI-3757 b did possess rings, they would lie in its orbital/rotation plane and not explain the planet's inflated radii. That said, we are viewing this system from a slightly inclined angle, which may tilt the rings into our vantage point. We thus further investigate whether a ring system could remain stable given TOI-3757 b's proximity to its star. Ohta et al. (2009) note that the Poynting–Robertson drag timescale is extremely short for close-in planets. Using Equation (17) from Ohta et al. (2009), we calculate a drag timescale between 0.1 and 5 Myr for TOI-3757 b dependent on the assumed ring particle densities and sizes. Unless TOI-3757 b also possesses some outside method for stabilizing its ring (such as shepherding moons similar to Saturn), it cannot dynamically maintain a significant ring system.

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Due to the electron degeneracy pressure, Jovian-sized objects can span ∼100x in mass, ranging from 0.3 M_J (Saturn mass) to ∼80 M_J (brown dwarfs and very-low-mass stars). Therefore, 2D mass–radius relationships are unable to accurately model Jovian-sized gas giants. The low density of TOI-3757 b also makes it an outlier for mass–radius relationships such as Wolfgang et al. (2016), Chen & Kipping (2017), Ning et al. (2018), and Kanodia et al. (2019). Attempts have been made to include the orbital period (or insolation) as a third dimension (Ma & Ghosh 2021) in these models to capture the radius anomaly due to host stellar insolation (Miller & Fortney 2011; Thorngren & Fortney 2018). However, as shown above, stellar insolation is not the driving force behind the low density of TOI-3757 b, suggesting the need for higher (>3) dimensional analysis including stellar metallicity or eccentricity (among others). We also use the giant planet models from Fortney et al. (2007) to predict the rocky core mass for TOI-3757 b. While we do not have a precise age estimate, the models assuming a system 1 Gyr old, predict a core mass of ∼6 M_⊕. The models for 10 Gyr (the oldest planetary population modeled by Fortney et al. 2007) do not accommodate objects as low-density as TOI-3757 b; however a planet as large as TOI-3757 b would need to have a mass of ∼140 M_⊕ (which is about 1.6x the mass of TOI-3757 b) to have a core just ∼1 M_⊕ in mass. We obtain the same result with the planetary models from Baraffe et al. (2008), which are also unable to accommodate the low density of TOI-3757 b.

TOI-3757 b spans multiple unique regions of the parameter space, in terms of its low bulk density and low host-star metallicity (Table 4). Understanding how TOI-3757 b formed and evolved could provide the context for explaining the low density of this unusual planet. Atmospheric characterization with the Hubble Space Telescope (HST), JWST, and future instruments could provide the necessary information to test our hypotheses regarding TOI-3757 b: is its low density a result of its formational history, or is TOI-3757 b's radius inflated from present day tidal heating?

Table 4. List of Giant Planets with (R_p > 4 R_⊕) Around M Dwarfs, with Precise Masses (>3σ), and Spectroscopic Stellar Metallicity Estimates, as Shown in Figure 10

Host Star	Pl. Radius	Pl. Mass	Pl. Density	Orbital Period	Teff	Metallicity	Reference
	R⊕	M⊕	g cm−3	days	K	dex
GJ 436	${4.19}_{-0.11}^{+0.11}$	${23.1}_{-0.8}^{+0.8}$	${1.73}_{-0.148}^{+0.148}$	2.64388	${3479}_{-60}^{+60}$	${0.1}_{-0.2}^{+0.2}$	Turner et al. (2016)
LP 714-47	${4.7}_{-0.3}^{+0.3}$	${30.8}_{-1.5}^{+1.5}$	${1.636}_{-0.323}^{+0.323}$	4.05204	${3950}_{-51}^{+51}$	${0.41}_{-0.16}^{+0.16}$	Dreizler et al. (2020)
TOI-1728	${5.04}_{-0.16}^{+0.16}$	${26.82}_{-5.44}^{+5.13}$	${1.155}_{-0.247}^{+0.247}$	3.492	${3985}_{-30}^{+30}$	${0.25}_{-0.1}^{+0.1}$	Kanodia et al. (2020)
TOI-674	${5.25}_{-0.17}^{+0.17}$	${23.6}_{-3.3}^{+3.3}$	${0.899}_{-0.153}^{+0.153}$	1.97714	${3514}_{-57}^{+57}$	${0.11}_{-0.07}^{+0.07}$	Murgas et al. (2021)
TOI-532	${5.79}_{-0.19}^{+0.18}$	${60.38}_{-8.77}^{+9.1}$	${1.715}_{-0.304}^{+0.304}$	2.32665	${3957}_{-69}^{+69}$	${0.38}_{-0.04}^{+0.04}$	Kanodia et al. (2021)
TOI-3629	${8.29}_{-0.22}^{+0.22}$	${82}_{-6}^{+6}$	${0.8}_{-0.1}^{+0.1}$	3.93655	${3870}_{-90}^{+90}$	${0.5}_{-0.1}^{+0.1}$	Cañas et al. (2022b)
HATS-75	${9.91}_{-0.15}^{+0.15}$	${156.05}_{-12.4}^{+12.4}$	${0.884}_{-0.08}^{+0.08}$	2.78866	${3790}_{-5}^{+5}$	${0.52}_{-0.03}^{+0.05}$	Jordán et al. (2021)
Kepler-45	${10.76}_{-1.23}^{+1.23}$	${160.49}_{-28.6}^{+28.6}$	${0.71}_{-0.275}^{+0.275}$	2.45524	${3820}_{-90}^{+90}$	${0.28}_{-0.14}^{+0.14}$	Johnson et al. (2012)
HATS-6	${11.19}_{-0.21}^{+0.21}$	${101.0}_{-22.0}^{+22.0}$	${0.397}_{-0.089}^{+0.089}$	3.32527	${3724}_{-18}^{+18}$	${0.2}_{-0.09}^{+0.09}$	Hartman et al. (2015)
TOI-3714	${11.32}_{-0.34}^{+0.34}$	${222.0}_{-10.0}^{+10.0}$	${0.85}_{-0.08}^{+0.08}$	2.15485	${3660}_{-90}^{+90}$	${0.1}_{-0.1}^{+0.1}$	Cañas et al. (2022b)
HATS-74 A	${11.57}_{-0.24}^{+0.24}$	${464.03}_{-44.5}^{+44.5}$	${1.653}_{-0.188}^{+0.188}$	1.73186	${3776}_{-9}^{+9}$	${0.51}_{-0.02}^{+0.03}$	Jordán et al. (2021)
TOI-3757	${12.02}_{-0.49}^{+0.44}$	${85.25}_{-8.67}^{+8.75}$	${0.271}_{-0.041}^{+0.041}$	3.43875	${3913}_{-56}^{+56}$	${0.0}_{-0.2}^{+0.2}$	This work
TOI-1899	${12.89}_{-0.56}^{+0.45}$	${209.77}_{-22.25}^{+22.25}$	${0.54}_{-0.08}^{+0.08}$	29.02	${3841}_{-45}^{+54}$	${0.31}_{-0.12}^{+0.11}$	Cañas et al. (2020), Lin et al. (in prep.)

Note. Data are taken from the NASA Exoplanet Archive (Akeson et al. 2013). We exclude au mic b, NGTS-1, and TOI-530 from this table due to lack of spectroscopic metallicity estimates.

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Despite the relatively faint host star (J mag ∼ 12), the ∼3% transit depth and large scale height of ∼500 km for TOI-3757 b lend it a large transmission spectroscopy measurement (TSM; Kempton et al. 2018) of 190. As shown in Figure 11, it has one of the highest TSMs of any gas giant (R_p > 6 R_⊕) with an equilibrium temperature <1000 K. Beyond its detectability, TOI-3757's mass (and scale height) is constrained to >9σ precision allowing us to resolutely analyze its spectrum and removing the planet's mass as a confounding factor during atmospheric retrievals (Batalha et al. 2019).

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With one JWST/NIRSpec-Prism transit of TOI-3757 b, we should easily retrieve an abundance for water, methane, and carbon dioxide in its atmosphere. This combination of molecules provides a constraint on both the overall atmospheric metallicity and the C/O ratio (Moses et al. 2013; Heng 2017); measurements that inform where in the disk this planet formed (Öberg et al. 2011; Figure 12). We also show in Figure 12 that we should be able to observe molecular absorption features at wavelengths >2 μm even with an aerosol layer at 0.1 mbars. While degeneracies between aerosols and atmospheric compositions occur, we demonstrate that TOI-3757 b is a promising target for future JWST observations even if it has high-altitude aerosols present.

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Aerosols (condensation clouds or photochemically created hazes) are ubiquitous across all exoplanet atmospheres; flattening or muting features in some while having little apparent effect in the atmosphere of others. By studying Titan in our own solar system, we know that UV flux (even a minimal amount) drives the production of photochemical tholin hazes from the photodissociation of methane (e.g., Hörst 2017). Even though TOI-3757 is an inactive M dwarf, it is likely still producing enough UV flux to create photochemical hazes in TOI-3757 b's atmosphere. We would therefore hypothesize that TOI-3757 b has a hazy atmosphere. However, both Crossfield & Kreidberg (2017) and Dymont et al. (2021) found no statistically significant relationship between the UV flux of the star and the water amplitude feature observed by HST/WFC3. Alternatively, Dymont et al. (2021) do detect a tentative correlation between the planet's density and the amplitude indicating that lower-density planets may also be more hazy. This trend is greatly influenced by the four superpuff spectra, all of which appear to be featureless (Libby-Roberts et al. 2021; Chachan et al. 2020; Alam et al. 2022). TOI-3757 b's density is similar to these superpuffs though it is >8× more massive.

TOI-3757 b is instead most similar to the low-density sub-Saturns HAT-P-18 b and HAT-P-19 b, both of which orbit early K dwarf stars (Hartman et al. 2011). Observed by WFC3, Tsiaras et al. (2018) retrieved a clear water abundance for HAT-P-18 b. However, when this spectrum was normalized by the scale height of the atmosphere in Dymont et al. (2021), the spectrum became nearly featureless. It is therefore unclear what we can expect for TOI-3757 b in regards to aerosol formation making it an interesting target for further aerosol studies.

As discussed in Fortney et al. (2020), we may also be able to probe the interior temperature of TOI-3757 b by searching for signs of disequilibrium chemistry in the planet's JWST spectrum. The model plotted in Figure 12 assumes chemical equilibrium; however, if TOI-3757 b possesses a hotter interior from tidal heating, methane will appear depleted compared to water, which remains unaffected. Moreover, with a strong vertical diffusion coefficient (K_zz >10⁴) in its atmosphere, ammonia would also be present in TOI-3757 b's JWST spectrum. If neither methane depletion or ammonia is observed, this would also place an upper limit on the interior temperature of TOI-3757 and thus a limit on the tidal heating as a source for radius inflation.

Given TOI-3757 b's low density and extended atmosphere, the planet is potentially experiencing some atmospheric mass loss (even though TOI-3757 is fairly quiescent). Verifying this mass loss is occurring and quantifying the rate would enable us to build a picture of the planet's atmosphere over time. The helium 10830 Å line provides an observable mass-loss tracer. As we mentioned in Section 4.2, we rule out >7% excess of helium absorption around the planet (assuming a circular orbit), a feature that may be due to a lack of mass loss or a lack of UV stellar flux pumping helium up to its metastable state. We note that helium was recently detected around the low-density sub-Saturn HAT-P-18 b with an excess of 0.46% (Paragas et al. 2021). Considering the similarities between the two planets, it is possible that TOI-3757 b maintains an exosphere beneath the precision we achieved with a single transit on HPF. TOI-3757 b would therefore be an interesting target for future helium observations that achieve a higher precision measurement.