TOI-6324b: An Earth-Mass Ultra-Short-Period Planet Transiting a Nearby M Dwarf

A spectrum of TOI-6324 (TIC 372207328; 2MASS J22032128+6729596; Gaia DR2 2220012421430629632) was obtained with the High Resolution Echelle Spectrometer (HIRES; Vogt et al., 2014) on the 10-m Keck I Telescope situated at Mauna a Wākea (Maunakea). The observation was taken on 2023-11-26UT at 900 s exposure time, with a peak SNR of $\sim$150. We employed the SpecMatch-Emp routine (Yee et al., 2017) to empirically derive spectroscopic parameters by cross-matching the observed HIRES spectrum with an extensive library of well-calibrated stellar spectra observed by the California Planet Search collaboration. This routine mitigates the significant systematic effects found in direct spectral modeling of low-mass stars, such as poor continuum identification and normalization, or incomplete molecular line lists, which are especially troublesome for cool stars. From SpecMatch-Emp we find T_eff = 3247 $\pm$ 70 K, consistent with the TESS Input Catalog estimate (TIC; Stassun et al., 2018; Muirhead et al., 2018). We also find [Fe/H] = -0.32 $\pm$ 0.09, consistent with the estimate from Dittmann et al. (2016), and an upper limit on $v\sin i$ of < 2 km s^-1.

We utilized the Python package isoclassify (Huber et al., 2017) to derive stellar parameters based on our derived spectroscopic stellar parameters and Gaia astrometry. Isoclassify directly computes the stellar radius ($R_{\star}$) using the Stefan-Boltzmann Law, based on the observed K-band magnitude, Gaia parallax, and T_eff. We adopted the MESA Isochrones and Stellar Tracks (MIST; Choi et al., 2016) and adhered to the default settings recommended by isoclassify. We compared our results with the empirical M dwarf mass-luminosity relation from Mann et al. (2019), and found that they are consistent within model uncertainties. The derived stellar parameters are given in the top section of Table 1.

To validate transiting exoplanets and assess possible contamination of bound or unbound companions on the derived planetary radii (Ciardi et al., 2015), we observed TOI-6324 using near-infrared adaptive optics (AO) imaging and optical speckle imaging. The AO observations were conducted on 2023-08-05UT using NIRC2 on Keck-II as part of the K2 and TESS follow-up and validation survey by Schlieder et al. (2021). We utilized a 3-point dither pattern. The narrow-angle mode provided a field of view of  10″ with a pixel scale of $\sim 0.0099442\arcsec$/pixel, using the narrow-band K_cont filter $(\lambda_{o}=2.2706;\Delta\lambda=0.0296~\mu$m).

Each frame had an integration time of 2.9 seconds, totaling 26.1 seconds on-source. Flat fields were taken on-sky, dark subtracted, and median averaged, and sky frames were created from the median average of dithered science frames. These were sky-subtracted and flat-fielded, resulting in a combined image with a resolution of 0.053″. Sensitivity was assessed by injecting simulated sources around the primary target at 20^∘ intervals, scaling brightness until detection at 5$\sigma$ significance (Furlan et al., 2017). We ruled out $\delta$mag $<$7 companions within 0.5″ (Fig. 1).

In addition to high contrast imaging, we investigated Gaia DR3 to search for stellar companions to TOI-6324. We identified no co-moving sources in Gaia DR3 within a radius of 10″. We further assessed the astrometry of TOI-6324 for evidence of a stellar companion. The Gaia DR3 Renormalised Unit Weight Error (RUWE) value represents the reduced $\chi^{2}$ of Gaia’s single-star astrometric solution and can be used to identify unresolved binaries (Belokurov et al., 2020) at separations of $\sim$0.1-1.0″ (Wood et al., 2021). For TOI-6324, RUWE = 1.165, well below the empirical threshold (RUWE = 1.4) at which binarity is likely.

It is notoriously difficult to determine the age of M dwarfs using the methods applied to more massive stars. Their asteroseismic oscillations are too small to be detected with current capabilities (Rodr´ıguez et al., 2016; Berdiñas et al., 2017), and their isochrone ages are poorly calibrated (Engle & Guinan, 2018). To assess the age of TOI-6324, we turned to gyrochronology (Barnes, 2003, 2007), which relates the rotation rate of a star to its age based on empirically-calibrated, mass-dependent spin-down rates (Skumanich, 1972; Barnes, 2007). However, we did not recover a robust rotation signal from a Lomb-Scargle periodogram (Lomb, 1976; Scargle, 1982) of TESS (Ricker et al., 2015) light curves (further described in Sec. 3) using a standard lightkurve routine (Lightkurve Collaboration et al., 2018). This may be attributed to the approximate upper limit of $P=13.7$ days imposed by the length of the TESS orbit. Older stars with rotation periods longer than 13.7 days are difficult to age-date using TESS photometry alone. Newton et al. (2016) also report a non-detection of the rotation period of TOI-6324, utilizing long-term photometric monitoring data from MEarth (Berta et al., 2012; Irwin et al., 2015). Furthermore, using BANYAN $\Sigma$ (Gagné et al., 2018), we found the Gaia DR3 kinematics of TOI-6324 are consistent with that of field stars. It is therefore unlikely that TOI-6324 is a member of any known nearby young moving groups. We are unable to derive a precise age estimate based on gyrochronology or by association, but it is likely that TOI-6324 is not a young ($\lesssim$100 Myr) star.

We calculated the probability that TOI-6324 is a member of the thick disk using the prescription of Bensby et al. (2014). TOI-6324 has a $TD/D=84$, i.e., it is likely a thick disk member based on its Galactic UVW velocity (U, V, W = 9.4$\pm$1.0, -65.7$\pm$2.1, -60.4$\pm 0.5$ km s^-1) after correcting for the Local Standard of Rest (LSR). This is consistent with the trend that USPs tend to occur around older stars (Schmidt et al., 2024). We adopted the LSR as U_⊙,V_⊙,W_⊙ = 10.0$\pm 1.0$, 11.0$\pm 2.0$, 7.0$\pm 0.5$ km s^-1 from Bland-Hawthorn & Gerhard (2016). Unfortunately, we do not have $[\alpha/Fe]$ data to back up the thick disk membership using the host star’s abundances, though we do find a non-detection of H$\alpha$ emission in the HIRES spectrum (Sec. 2.1).

TOI-6324 was observed by TESS during Sectors 16, 17, 18, 24, 58, 77, and 78 (September-November 2019, April-May 2020, November 2022, and April-May 2024). The data were processed by the TESS Science Processing Operations Center (SPOC at NASA Ames Research Center; Jenkins et al., 2016) and the MIT Quick-Look Pipeline (QLP; Huang et al., 2020). The transit signal of TOI-6324 b at 0.279 days was first alerted in a faint-star search in QLP, reported on the Exoplanet Follow-up Observing Program (ExoFOP ¹¹1https://exofop.ipac.caltech.edu/tess/target.php?id=372207328), and the SPOC pipeline detected the transit signature in every sector observed. We collected the 2-minute cadence SPOC light curves using the Python package lightkurve (Lightkurve Collaboration et al., 2018). We applied transit-depth bias correction to the Year 2 data (Sectors 16, 17, 18, and 24) which were affected by a sky background bias issue documented in the S27 Data Release Notes. We utilized the Presearch Data Conditioning Simple Aperture Photometry (PDC-SAP) light curves (Smith et al., 2012; Stumpe et al., 2012, 2014; Twicken et al., 2010) for transit modeling after removing data points with non-zero data quality flags.

To prepare the PDC-SAP light curve for transit modeling we first removed instrumental and long-term stellar systematics using the wōtan Python package (Hippke et al., 2019). We first masked the transit signal of TOI-6324 b based on the QLP transit parameters. We then used an iterative sigma-clipping spline fit (rspline) with a width of 0.5 days to detrend the transit-masked light curve.

Our transit model was constructed using the exoplanet package (Foreman-Mackey et al., 2021), a gradient-based probabilistic inference toolkit for modeling time-series astronomical data. We used the transit parameters reported by QLP (depth, duration, orbital period, and epoch) as the basis for uninformative priors (either uniform or broad normal distributions) to initialize our model. This allowed for a thorough and unconstrained search of the parameter space. We additionally imposed Gaussian stellar radius and density priors derived in this work (see Sec. 2 and Table 1). Limb darkening coefficients q₁ and q₂ were re-parameterized following the following the quadratic law formulation of Kipping (2013). We assumed a circular orbit ($e$ = 0) for model simplicity. Since the orbital period is extremely short, we expect this to be a valid assumption due to tidal circularization. The transit model further included the planet-to-star radius ratio $R_{\rm p}/R_{\star}$ derived from transit depth, and the impact parameter $b$. The orbital inclination $i$ was derived from $b$. We fit all transits assuming a constant period, since we did not identify any significant transit-timing variations.

A maximum a posteriori (MAP) transit model solution was determined from an initial non-linear optimzation using pyMC3 and set as the initial condition for sampling. We then explored the parameter space using No U-Turn Sampling (NUTS; Hoffman & Gelman, 2011) in a gradient-based Hamiltonian Monte Carlo (HMC; Neal, 2011; Betancourt, 2017) framework in pyCM3 to generate the transit parameter posteriors. We ran 2 chains for 10000 draws, with 5000 iterations to tune (15,000 steps total). We list the median values of the posterior distributions for the transit parameters in Table 1 along with the 68.3% confidence intervals. Fig. 2 shows the binned and phase-folded PDC-SAP light curve of TOI-6324, with a robust detection of TOI-6324 b, along with the best-fit transit model. We report a radius of $R_{\rm p}$ = 1.059 $\pm$ 0.041 R_⊕ for TOI-6324 based on the best-fit planet-to-star radius ratio $R_{\rm p}/R_{\star}$ = 0.0331 $\pm$ 0.0006 and our derived stellar radius of $R_{\star}$ = 0.293 $\pm$ 0.010 R_⊙(Table 1).

In addition to the SPOC light curves, we obtained TESS-Gaia light curves (TGLC; Han & Brandt, 2023), which are created from FFIs forward-modeled with the effective point-spread function (PSF) to remove contamination from nearby stars. Based on Gaia DR3 positions and magnitudes, these light curves achieve a very high-treatment account of dilution. We performed the identical analysis on the TGLC light curves, and find $R_{\rm p}$ = 1.075 $\pm$ 0.05 R_⊕  consistent with the result from the SPOC light curves within 1$\sigma$. Furthermore, TOI-6324 b was observed with the 200-in Hale Telescope at Palomar Observatory for transit validation (Gomez Barrientos et al., 2024). The radius measurement of Gomez Barrientos et al. (2024), using both ground-based and TESS SPOC light curves, is consistent with the measurement from this work.

No additional transiting planet candidates of TOI-6324 have been reported by QLP nor SPOC. To search for additional transit signals in the PDC-SAP light curves described in Sec. 3.2, we masked the transits of TOI-6324 b and performed a Box-Least-Square (BLS; Kovács et al., 2002) analysis using the lightcurve.to_periodogram function of lightkurve (Lightkurve Collaboration et al., 2018). We allowed for periods between 0.1 and 30 days, and transit durations between 0.1 and 2.15 hr. We phase-folded the light curve on the significant periodic signals from this search, and we did not identify any additional planetary transit signals.

TOI-6324 b is $\sim$20% from the Roche limit, making it likely to be tidally distorted (see Sec. 5.3). We searched the TESS light curves for evidence of phase curve variation and secondary eclipse signal from TOI-6324 b. The light curve was iteratively sigma-clipped and detrended in the same manner as in Sec. 3.2, but with a wider spline width of $2*P_{orb}$ or $\sim$13.4 hr, to ensure any phase curve signal was not removed in the detrending step. The light curve was then phase-folded and binned to $\sim$10 minute intervals (50 bins total). We used batman (Kreidberg, 2015) to construct a secondary eclipse model based on the transit parameters derived in Sec. 3.2 (Table 1), but with the limb darkening coefficients (u₁, u₂) set to 0. The eclipse time ($t_{sec}$) was fixed to a 0.5 phase offset from the primary transit time.

In addition to the secondary eclipse, we modeled the out-of-eclipse phase curve variation, initially characterized by a combination of the illumination effect from reflected stellar light ($A_{ill}$), an offset of the peak illumination effect ($\theta$), and ellipsoidal light variation ($A_{ELV}$). $A_{ill}$ was fixed to be the same as the secondary eclipse depth ($\delta_{sec}$). We found that the TESS data do not have sufficient SNR to robustly constrain $\theta$, and thus we set $\theta=0$ for model simplicity.

The ellipsoidal light variation $A_{ELV}$ can be due to both the tidal distortion of the star and the planet itself. We estimated that the stellar component is likely smaller than 1 ppm (e.g. Faigler et al., 2012), however the planetary distortion can be substantial given the extremely short orbital period of the planet. We compared two phase curve models: with and without the ellipsoidal light variation $A_{ELV}$. Combined with the secondary eclipse model, the best-fit solutions were determined using the Levenberg-Marquardt least-squares likelihood maximization method (lmfit; Newville et al., 2014). We employed a nested sampling routine using dynesty (Speagle, 2020; Koposov et al., 2024) to sample the posteriors and compare the Bayesian evidence ($\log(Z)$) of the two models, and found that the non-ELV model is preferred over the ELV-inclusive model ($\Delta\log(Z)=5$). We show in Fig. 3a the best-fit phase curve models with and without ELV. The posterior medians of the non-ELV and ELV models are consistent within $1\sigma$ uncertainties, and from the preferred non-ELV model we find $\delta_{sec}\equiv A_{ill}=42\pm 28$ ppm ($<2\sigma$). From the ELV model, we report a tentative ($\sim 1\sigma$) detection of $A_{ELV}=21\pm 17$ ppm, corresponding to a Bond Albedo of $A_{B}=0.36^{+0.28}_{-0.36}$ (Fig. 3b).

We obtained high-resolution spectra of TOI-6324 with the Keck Planet Finder (KPF; Gibson et al., 2016, PI: A. Howard) on Keck I. KPF is a high-resolution (R$\sim$98,000), optical (445-870 nm), fiber-fed echelle spectrometer newly commissioned in late 2022. The confirmation of TOI-6324 b in this work is among the first science results of KPF. KPF boasts an internal instrumental RV precision of $\sim$50 cm s^-1 and has demonstrated the capability to detect a 1.44 M_⊕ USP to greater than 5$\sigma$ precision (TOI-6255 b; Dai et al., 2024). We collected a total of 77 KPF spectra of TOI-6324 between 2023-06-30UT and 2024-01-01UT (see Table 2). Each exposure was 15 min and the average SNR achieved was 65 at 550 nm. We aimed to obtain multiple exposure per night in order to mitigate some of the instrumental drift. The spectra were reduced using the publicly-accessible KPF Data Reduction Pipeline (DRP)²²2https://github.com/Keck-DataReductionPipelines/KPF-Pipeline, the radial velocities were extracted with the template matching code serval (Zechmeister et al., 2018) after minor modifications for KPF (Dai et al., 2024). RVs from nights with only a single measurement, as well as nightly 3$\sigma$ outlier measurements, noted with a (*) in Table 2, were excluded, leaving 65 RV measurements taken over 23 nights for fitting.

To disentangle the planetary RV modulation from the significant instrumental noise in the RV variations of TOI-6324, we employed a Gaussian Process (GP) regression model. We used the package radvel (Fulton et al., 2018) for this analysis. The correlated noise is likely a result of stochastic instrumental drift. Stellar pulsations which normally affect observations in more massive stars have very low amplitudes in M dwarfs, and because they occur on much shorter timescales, are averaged out in each exposure (Kjeldsen & Bedding, 1995; Chaplin et al., 2019). In our analysis of the light curves, we did not identify any periodic variations that could be attributed to rotationally modulated stellar activity or pulsations. As such, we used a squared-exponential GP kernel in our model, rather than the widely-used quasi-periodic kernel (e.g., Grunblatt et al., 2015; Rajpaul et al., 2015; Brinkman et al., 2023), for model simplicity (Dai et al., 2019, 2021, 2024). This is included in the covariance matrix which is factored into the likelihood function. See Dai et al. (2024) for a detailed explanation of these expressions.

The model basis included the orbital elements of orbital period $P_{\rm orb}$, time of conjunction $T_{c}$, and RV semi-amplitude $K$, and the eccentricity $e$ and argument of pericenter $\omega$ were included jointly as $\sqrt{e}\cos\omega$ and $\sqrt{e}\sin\omega$. We imposed Gaussian priors on P_orb and $T_{c}$ derived from our transit analysis (see Sec. 3 and Table 1). We placed a hard lower bound for semi-amplitude of $K>0$, and we additionally imposed Jeffreys priors on the jitter term for KPF $\sigma_{\rm jit,KPF}$, as well as the GP hyperparameters of amplitude $h$ and length $l$. We set a lower bound of $l>1$ day to ensure the GP did not fit out the 6.7 hour planetary signal of TOI-6324 b. As in the transit model, we fixed the eccentricity to 0.

The standard radvel routine employs emcee (Foreman-Mackey et al., 2013) to explore the parameter space in a Markov Chain Monte Carlo (MCMC) framework. We ran 128 walkers for 10⁴ runs in 3 ensembles for parallelization, achieving a maximum Gelman-Rubin (Max G-R) statistic of 1.001. We show the best-fit GP model in panel (a) and the best-fit Keplerian solution for TOI-6324 b in panel (c) of Fig. 4. The median values of the posterior distributions for the orbital parameters and GP hyperparameters are presented in Table 1, respectively. We found that the RV semi-amplitude imposed by TOI-6324 b is robustly detected at 5$\sigma$ precision with $K=2.69\pm 0.51\;{\rm m\,s}^{-1}$. We did not detect any additional periodic signals or longer-term RV trends in this dataset for evidence of another RV planet.

To check if there was any over-fitting GP analysis, we additionally performed a fit to the RVs using the floating chunk offset (FCO) method (Hatzes et al., 2010, 2011; Hatzes, 2014, e.g.,). FCO is a common method for determining the RV semi-amplitudes of USPs (P $\lesssim$ 1 day), whose nightly RV variations are typically larger than individual RV measurement uncertainties (Deeg et al., 2023). By assigning nightly additive RV offsets to grouped observations, we focus entirely on intra-night RV variations and effectively remove any long-term (P $>$ 1 day) stellar and instrumental systematics.

As done for the GP fit, RVs from single-measurement nights and nightly 3$\sigma$ outliers were excluded, noted with a (*) in Table 2. We assigned an offset term for each of 23 nights of observation, $\gamma_{night}$. This model differs from the GP model in that we remove the squared-exponential kernel and use only the white noise component in the covariance function. We implicitly include nightly offsets in the measured RV values in the likelihood function. We again used emcee for sampling in an MCMC to explore the posteriors of the model, mirroring the procedure in our GP analysis. From the FCO method we detected TOI-6324 b at 2$\sigma$ precision, with a RV semi-amplitude of $K=2.19\pm 1.02\;{\rm m\,s}^{-1}$. This is consistent with the result of the GP model within 1$\sigma$ uncertainties. Panel (d) of Fig. 4 shows the Keplerian solution from the FCO model in phase with the GP model. Because the FCO model requires many more parameters (due to each $\gamma_{night}$) than the GP model, therefore introducing larger uncertainty, we chose to adopt the GP model result for further analyses.

With precise mass and radius measurements (both $<$ 20% uncertainties) of TOI-6324 b, we are able to approximate its interior composition. As mentioned, because USPs are so strongly irradiated by their host stars, it is suggested that any thick H/He envelopes should be stripped away (e.g., Sanchis-Ojeda et al., 2014; Lundkvist et al., 2016; Wang & Dai, 2018; Kreidberg et al., 2019; Zhang et al., 2024), allowing for direct constraint of their compositions. To compare with other confirmed USPs (queried from the NASA Exoplanet Archive³³3https://exoplanetarchive.ipac.caltech.edu) and Earth-sized planets, we plotted TOI-6324 b on a mass-radius diagram along with the model composition curves from Zeng et al. (2016), shown in Fig. 5. The light purple colored points represent all confirmed R${}_{\rm p}<2$ R_⊕ planets with well-constrained masses and radii ($<$ 20% uncertainties), while the dark purple points highlight the USPs among these with P $<$ 1 day. Similarly to other USPs, TOI-6324 b closely follows the Earth-like composition curve of 30% iron and 70% MgSiO₃ silicate rock.

To further characterize the interior composition of TOI-6324 b we modeled the iron mass fraction (Fe-MF), which describes the fraction of the planet’s mass that is comprised of the iron core. We modeled the core mass fraction (CMF) using Manipulate Planet⁴⁴4https://lweb.cfa.harvard.edu/~lzeng/manipulateplanet.cdf, a numerical tri-layer planetary interior model with an iron core, silicate rock mantle, and surface water layer (Zeng & Sasselov, 2013; Zeng et al., 2016). The model assumes an undifferentiated core and numerically solves the interior structure of a planet based on the Equation of State (EOS) extrapolated from Earth’s Seismic Density Profile (using the Preliminary Reference Earth Model (PREM; Dziewonski & Anderson, 1981)). We adjusted the model to remove the water layer, and from this bi-layer iron and rock model we found a Fe-MF (adopted as the CMF in this model) of 27$\pm 37$% for TOI-6324 b.

We additionally employed a more sophisticated model, exopie (Plotnykov & Valencia, 2024), based on superearth (Valencia et al., 2006; Plotnykov & Valencia, 2020), which computes an iron mass fraction Fe-MF, distinct from the general CMF. The Fe-MF calculation takes into account the iron that could be in the mantle and removes any light elements in the core, as well as different degrees of core differentiation. We assumed that there is a variable amount of Fe in the mantle and Si in the core (0-20%), but set the Mg/Si ratio in the mantle to be Earth-like ($\sim$0.9). Ni in the core was fixed to 10%. Using a Monte Carlo scheme, we found that TOI-6324 b has a CMF=32$\pm 19$% and Fe-MF=30$\pm 14$%. These results are in agreement with the simpler model of Zeng et al. (2016).

The extremely short (6.7 hr) orbit of TOI-6324 b gives rise to the possibility that it is substantially tidally distorted. Such an effect has been suggested for TOI-6255 b, a 1.079 R_⊕ USP on a 5.7 hr orbit (Dai et al., 2024). Tidal distortion would have a non-negligible effect on the volumetric radius $R_{\rm vol}$ and the effective transit radius of a planet, and thus must be accounted for when approximating the bulk planet composition.

Following the method of Dai et al. (2024) we assess the degree of deformation due to tidal stress first by comparing the orbital period to the Roche limit of an incompressible fluid (Rappaport et al., 2013). The Earth is not very well modeled as an incompressible fluid, so this formula is a very rough approximation:

To estimate the tidal distortion $\delta R_{\rm p}$ of TOI-6324 b, we used the Love number $h_{2}$ for a solid homogeneous planet (Love, 1944), which describes the radial tidal displacement of a planet’s surface. $h_{2}$ depends on the mean tensile strength of the planetary material. For Earth, $h_{2}$ has been estimated to be 0.6 to 0.9 (Lambeck, 1980). We adopted $h_{2}=1$ assuming a weaker material strength for USPs, as in Dai et al. (2024). This gives $\delta R_{\rm p}=0.07$ for TOI-6324 b. A more robust detection of ellipsoidal light variation than we could recover in the TESS phase curve (Sec. 3.5, Fig. 3a) would provide more conclusive evidence that TOI-6324 b is tidally deformed.

USPs such as TOI-6324 b are particularly excellent targets for phase curve and secondary eclipse observations with JWST. Several such systems have been selected or have already been successfully observed with JWST (e.g., 55 Cnc e, GJ 367 b, K2-141 b, LHS 3844 b, TOI-561 b, TOI-1685 b, TOI-2445 b, TOI-4481 b, and WASP-47 e). Precise study of their surfaces will revolutionize our understanding of planetary atmospheres and geology. USPs tend to orbit late-type stars (Dai et al., 2024), and thus their flux contrasts with their host stars are observationally favorable, on the order of 100 ppm in the infrared. We computed the Emission Spectroscopy Metric (ESM; Kempton et al., 2018) for TOI-6324 b to be 25. This is the highest ESM target among Earth-sized (0.8 R_⊕ $<$ R_p $<$ 1.2 R_⊕) planets. The TESS data alone are insufficient for robustly constraining the phase curve variation, though we do find preliminary evidence of several tens of ppm in the optical ($A_{ill}=42\pm 28$ ppm and $A_{ELV}=21\pm 17$ ppm; Sec. 3.5, Fig. 3a). For USPs, the planet-to-star flux ratio is expected to be much more significant in the mid-IR (Dai et al., 2024). In the MIRI/LRS bandpass (5-12 $\mu$m), the planet’s thermal emission increases to $\sim$280 ppm assuming a low Bond albedo. We aim to include more detailed modeling of the TESS phase curve variation in a future paper with comparison to JWST observations.

There are several scientific avenues for detailed characterization of TOI-6324 b with JWST MIRI/LRS (Kendrew et al., 2015) observations. Since USPs are tidally locked, they will have constant daysides and nightsides, resulting in a significant longitundal temperature contrasts across their surfaces. An observed offset in the phase curve would signify heat re-circulation over the surface of the planet due to the presence of an outgassed secondary atmosphere (e.g., Knutson et al., 2009; Demory et al., 2013; Showman et al., 2015; von Paris et al., 2016; Angelo & Hu, 2017). A lack of phase offset indicates a bare rock surface (e.g., Zhang et al., 2024; Weiner Mansfield et al., 2024). In this case, variations in the phase curve will trace the projected shape of the planet across its orbit and help to constrain the extent of tidal distortion and provide key insights into the tidal interactions USPs undergo with their host stars (see Sec. 5.3; Dai et al. 2024).

For a bare rock surface, the emission spectrum will constrain the dominant surface rock type of the planet (Hu et al., 2012; Kreidberg et al., 2019; Whittaker et al., 2022; Zhang et al., 2024). The approximate equilibrium temperature of TOI-6324 b is $T_{eq}=1216\pm 60$ K (Table 1), assuming a low albedo of 0.1. This is below the zero-pressure melting point of common silicates found on Earth (such as peridotite at 1390 K), and thus it is unlikely that TOI-6324 b has a pure lava surface, although the hotter dayside, which receives constant irradiation may be partially molten. Assuming zero albedo, the temperature at the substellar point would be approximately 1824 K, which is sufficient to melt silicates. Excessive tidal heating may also contribute to violent volcanism at the surface (Jackson et al., 2008; Hu et al., 2012), if the planet actually has a non-zero eccentricity, contrary to our assumption in Sec 3.2. Detection of basaltic or ultramafic surface mineralogy, based on the Si-O absorption feature from 8-10 $\mu$m, will constrain the ongoing geologic processes. Ultramafic rock is formed from high-temperature volcanism and has strong Si-O features, while basaltic rock is formed from Earth-like extrusive volcanism and typically exhibits weaker Si-O features. The presence of granitoid rock (the primary component of Earth’s continental crust) is unlikely at the extreme temperatures of TOI-6324 b, which are high enough to melt feldspar and quartz (the primary minerals in granite). Dedicated simulations of these observations and effects will be included in future works.