TESS discovery of a sub-Neptune orbiting a mid-M dwarf TOI-2136

TOI-2136 (TIC 336128819) was first observed by TESS on its Camera 1 with the two-minute cadence mode in Sector 26 during the primary mission from 9th June 2020 to 4th July 2020 and it was re-observed in Sector 40 between 24^th June 2021 and 23^th July 2021 during the Extended Mission. The left panel of Figure 1 shows the POSSI image of TOI-2136 taken in 1950. Based on the relatively large stellar proper motion ($\sim 180$ mas/yr), we rule out the possibility that the light from an unassociated distant eclipsing binary system with $V\lesssim 21$ mag caused the TESS detection. The other panels of Figure 1 show the target pixel files (TPFs) and the Simple Aperture Photometry (SAP) apertures used in each sector , plotted with tpfplotter (Aller et al., 2020). A nearby star (Gaia DR2 2096535788163295744, $T_{\rm mag}=13.23$) 33^′′ away is located at the edge of the aperture, which is expected to make only a slight contribution to the TESS signal. We summarize the host star properties in Table 1.

The TESS time-series data were initially processed by the Science Processing Operations Center (SPOC; Jenkins et al. 2016) pipeline. After correcting the instrumental and systematic effects as well as the light dilution with the Presearch Data Conditioning (PDC; Stumpe et al. 2012; Smith et al. 2012; Stumpe et al. 2014) module, transit signals were searched using the Transiting Planet Search (TPS; Jenkins, 2002; Jenkins et al., 2020) algorithm, which resulted in a periodic signal with an orbital period of 7.85 days and a duration of 1.61 hours. Validation tests were then conducted to confirm the transit signature (Twicken et al., 2018; Li et al., 2019), including locating the source of the transit signal to within $1-3^{\prime\prime}$ of the target star, and searching for additional transiting planet signatures in the residual light curve before TOI-2136 was finally alerted as a planet candidate in the TESS Object of Interest catalog (TOI-2136.01).

We retrieved the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) light curve from the Mikulski Archive for Space Telescopes¹¹1http://archive.stsci.edu/tess/(Twicken et al., 2010; Morris et al., 2020). We found a total of 16941 and 15319 useful measurements within the data from Sector 26 and Sector 40, respectively. We then performed our own transit search by utilizing the Transit Least Squares (TLS; Hippke & Heller 2019) algorithm, which is an advanced version of Box Least Square (BLS; Kovács et al. 2002). We confirmed the 7.85 days signal with a signal detection efficiency (SDE) of 34 but we did not find additional significant signals existing in the light curve. To detrend the TESS light curve and remove the systematic trends left in the PDCSAP light curve, we fit a Gaussian Process (GP) model with a Matérn-3/2 kernel using the celerite package (Foreman-Mackey et al., 2017), after masking out all in-transit data. We show the SAP, raw PDCSAP and detrended PDCSAP light curves in Figure 2.

Due to the large pixel scale of TESS (21^′′/ pixel, Ricker et al. 2015), the host star is likely to be blended with close stars in a single TESS pixel. Consequently, the transit signal of TOI-2136 detected in the space data could be caused by nearby eclipsing binaries. Even though the transit signal is on target, the depth might be biased to a smaller value because of light contamination. With all of the above in mind, we collected a series of ground-based observations of TOI-2136, as part of the TESS Follow-up Observing Program (TFOP²²2https://tess.mit.edu/followup), to validate the planetary nature and refine both the transit ephemeris and the radius measurement. We scheduled these photometric time-series by using the TESS Transit Finder (TTF) tool, which is a customized version of the Tapir software package (Jensen, 2013). We summarize the details in Table 2 and describe individual observations below. We show the raw and detrended ground-based light curves in Figure 3 (see Section 4.1.2).

High-resolution imaging is one of the standard follow-up observations made for exoplanet host stars. Spatially close companions, bound or line of sight, can create a false-positive transit signal and provide “third-light” flux leading to an underestimated planetary radius (Ciardi et al., 2015), incorrect planet and star properties (Furlan & Howell, 2017, 2020) and can cause non-detections of small planets residing with the same exoplanetary system (Lester et al., 2021). Additionally, the discovery of close, bound companion stars provides crucial information toward our understanding of exoplanetary formation, dynamics and evolution (Howell et al., 2021). Generally, Gaia is not capable to recover binaries with separations smaller than $0.7\arcsec$ (Ziegler et al., 2020). Thus, to search for close-in bound companions unresolved in Gaia, TESS or other ground-based follow-up observations, we obtained high-resolution imaging observations of TOI-2136.

We first estimate the absolute $K$ band magnitude from the 2MASS observed $m_{K}$ and the parallax from Gaia EDR3 (Gaia Collaboration et al., 2021), which yields $M_{K}=6.73\pm 0.02$ mag. Taking use of the polynomial relation between $R_{\ast}$ and $M_{K}$ derived by Mann et al. (2015), we obtain a stellar radius of $R_{\ast}=0.34\pm 0.01\ R_{\odot}$, assuming a typical uncertainty of 3% (see Table 1 in Mann et al. 2015). This is consistent with the estimation $R_{\ast}=0.34\pm 0.02\ R_{\odot}$ within $1\sigma$ using the angular diameter relation in Boyajian et al. (2014).

Based on the empirical relation between bolometric correction ${\rm BC}_{K}$ and stellar color $V-J$ found by Mann et al. (2015), we obtain a ${\rm BC}_{K}$ of $2.73\pm 0.21$ mag, leading to a bolometric magnitude $M_{\rm bol}=9.46\pm 0.22$ mag. We then calculate the bolometric luminosity to be $L_{\ast}=0.013\pm 0.003\ L_{\odot}$. We further estimate the stellar effective temperature of TOI-2136 using two different methods. Combined with the stellar radius and bolometric luminosity, we find $T_{\rm eff}=3324\pm 55$ K by utilizing the Stefan-Boltzmann law. Additionally, we also obtain $T_{\rm eff}$ following the empirical relation with stellar color $V-J$ and $J-H$ reported by Mann et al. (2015), and we find $T_{\rm eff}=3314\pm 104$ K. Both estimations agree well with the result $T_{\rm eff}=3267\pm 133$ K from Pecaut & Mamajek (2013).

We also evaluate that TOI-2136 has a mass of $M_{\ast}=0.33\pm 0.02\ M_{\odot}$ using Equation 2 in Mann et al. (2019) according to the $M_{\ast}$-$M_{K}$ relation. This is consistent with the value $M_{\ast}=0.35\pm 0.02\ M_{\odot}$ given by the $M_{K}$-Mass empirical relation of Benedict et al. (2016).

As an independent check, we performed an analysis of the broadband Spectral Energy Distribution (SED) of TOI-2136 using MIST stellar models (Dotter, 2016; Choi et al., 2016) along with the Gaia EDR3 parallax (Gaia Collaboration et al., 2021) in order to derive the stellar parameters of TOI-2136. We made use of the EXOFASTv2 package (Eastman et al., 2019) to conduct the SED fit. We used the MIST method (the favored method reported in Eastman et al. (2019), -MISTSEDFILE) that interpolates the 4D grid of $\log g$, $T_{\rm eff}$, [Fe/H], and an extinction grid from Conroy et al., (in prep) to determine the bolometric corrections in each of the observed band. We pulled the $JHK_{S}$ magnitudes from 2MASS (Cutri et al., 2003), the W1-W4 magnitudes from WISE (Wright et al., 2010), and three Gaia magnitudes $G,G_{\rm BP},G_{\rm RP}$ (Gaia Collab. et al., 2018). Together, the available photometry spans the full stellar SED over the wavelength range 0.5-22$\mu$m (see Figure 6). We applied an upper limit on the V-band extinction from the dust maps of Schlafly & Finkbeiner (2011) and a Gaussian prior on the [Fe/H] taken from the spectroscopic analysis. The EXOFASTv2 analysis ran until convergence when the Gelman-Rubin statistic (GR) and the number of independent chain draws (Tz) were less than 1.01 and greater than 1000, respectively. The full results of the SED fit are provided in Table 3, which are in excellent agreement with our previous estimation.

Taking all the results above into account, we adopt the weighted-mean values of effective temperature $T_{\rm eff}$, stellar radius $R_{\ast}$ and stellar mass $M_{\ast}$ with conservative uncertainties as listed in Table 1. Combining the derived stellar radius with mass, we find a mean stellar density of $\rm\rho_{\ast}=12.20\pm 2.53$ g cm^-3.

Finally, we also estimate the systemic radial velocity of TOI-2136 to be $-28.8\pm 6.0$ km/s by RV-correcting our SpeX spectrum using tellrv (Newton et al., 2014). To determine the stellar type, we further compare our SpeX spectrum with the IRTF library (Rayner et al., 2009) and find that TOI-2136 is consistent with a star of spectral type M4.5V (Figure 4). Lastly, following the procedure described in Gan et al. (2022), we obtain the metallicity of TOI-2136 based on the relations defined in Mann et al. (2013) for cool dwarfs with spectral types between K5 and M5. Our analysis yield metallicities of [Fe/H] = $0.03\pm 0.07$ and [M/H] = $-0.01\pm 0.08$.

Combined with the tangential velocity ($\mu_{\alpha}$, $\mu_{\delta}$) and the stellar parallax ($\varpi$) from Gaia EDR3 as well as the spectroscopically determined systemic RV from the SpeX spectrum, we calculate the three-dimensional space motion with respect to the LSR based on the methodology described in Johnson & Soderblom (1987). We obtain three-dimensional space velocities $U_{\rm LSR}=-25.15\pm 2.26$ km s^-1, $V_{\rm LSR}=-9.42\pm 5.27$ km s^-1, $W_{\rm LSR}=13.16\pm 1.75$ km s^-1, respectively. We further identify the Galactic population membership of TOI-2136 following the criterion first used in Bensby et al. (2003). We compute the relative probability $P_{\rm thick}/P_{\rm thin}=0.01$ of TOI-2136 to be in the thick and thin disks by taking use of the recent kinematic values from Bensby et al. (2014), indicating a thin-disk origin. Finally, we integrate the stellar orbit with the “MWPotential2014” Galactic potential using galpy (Bovy, 2015) following the procedure described in Gan et al. (2020), and we estimate that the maximal height $Z_{\rm max}$ of TOI-2136 above the Galactic plane is about $206$ pc. Therefore, we conclude that TOI-2136 belongs to the thin-disk population, which is also consistent with its solar-like metallicity.

Stellar activity, often manifesting as stellar rotation signals, is expected to affect the RV measurements and make it challenge to accurately determine the planet mass (Queloz et al., 2009; Howard et al., 2013; Pepe et al., 2013), especially when its timescale is close to the planet orbital period (Gan et al., 2021). In order to evaluate the effect of the stellar activity on the RVs, we first search for the periodic signals in the TESS PDCSAP light curve after masking the known in-transit data using the generalized Lomb-Scargle (GLS) periodogram (Zechmeister & Kürster, 2009), and we find no signs of stellar variation. Therefore, we do not present the periodograms here. However, we note that the PDCSAP photometry from TESS flattens variability on timescales greater than about 15 day and TESS is insensitive to long-term stellar rotational features due to its sector-by-sector observational strategy. Thus, we further examine the archival long-term photometric time-series data from ground-based surveys. We look into the rotational modulation of TOI-2136 in the publicly available light curve taken by the Zwicky Transient Facility (ZTF; Masci et al. 2019). A total of 1054 measurements were acquired in $r-$ band spanning 1112 days. After removing the observations flagged as bad-quality, we have 1011 measurements left with a standard deviation of 0.011 mag. We compute the GLS periodogram and find a clear peak at $75\pm 5$ days (see Figure 7). This is consistent with the estimation of $P_{\rm rot}\sim 82.97$ days derived by Newton et al. (2016) using the data from MEarth (Nutzman & Charbonneau, 2008). Additionally, the lack of significant flaring activity existing in the TESS light curves also suggests that the host star is quiet and inactive. We thus attribute this $75\pm 5$ days signal to the stellar rotation. Adopting the empirical relations from Engle & Guinan (2018), we estimate that TOI-2136 has an age of $4.6\pm 1.0$ Gyr, consistent with our thin-disk population conclusion.

We perform an RV-only fit using juliet, which employs the radvel package (Fulton et al., 2018) to model the Keplerian RV signals. Since the expected RVs caused by the planet perturbation is expected to be small, we choose to fix the orbital period $P_{b}$ and mid-transit epoch $T_{0,b}$ at the best-fit transit ephemeris derived from the TESS only fit to reduce introducing additional uncertainties. As the TESS photometric data do not show evidence for eccentricity, we assume a circular orbit and fix eccentricity $e$ at 0, and the argument of periastron $\omega$ at $90^{\circ}$. Moreover, we do not take the RV slope $\dot{\gamma}$ or the quadratic trend $\ddot{\gamma}$ into consideration and fix them at zero due to the short time span of our RV data. We include a simple jitter term $\sigma_{\rm RV}$ that is added in quadrature to the error bars of each data point to account for the white noise. We set uniform priors on both the RV semi-amplitude $K_{b}$ and the systemic velocity $\mu$ but a log-uniform prior on $\sigma_{RV}$. We obtain $K_{b}=4.1\pm 1.5$ m/s, which is consistent with the expected value $\sim 3.7$ m/s supposing a planet mass estimated using the mass-radius relation from Chen & Kipping (2017).

We next construct a Keplerian model with free $e$ and $\omega$ to look for the significance of the eccentricity in the RV data. We find that the circular orbit model is slightly preferred with a Bayesian evidence improvement of $\Delta\ln Z=\ln Z_{\rm Circular}-\ln Z_{\rm Eccentric}=1.2$, agreeing with our findings in the TESS photometric data (see Section 4.1.1).

Building on the results from the independent transit and RV fits, we finally carry out a joint-fit using juliet to simultaneously model all detrended space and ground light curves together with the SPIRou RVs to infer the properties of the TOI-2136 b. We place the same priors as we did in Section 4.1.1 except that we adopt Gaussian priors for the linear limb darkening coefficients of the ground-based light curves, centered at the estimates from the LDTK package (Husser et al., 2013; Parviainen & Aigrain, 2015) with a $1\sigma$ value of 0.1. As there is less contamination in the ground data, we fix all dilution factors $D_{i}$ to 1. For the RV part, we use the same priors as in Section 4.2. The phase-folded TESS and ground light curves along with the best-fit transit models are shown in Figures 8 and 9. The RV timeseries and the best-fit RV model are presented in Figure 10. The fitted RV semi-amplitude is $4.2\pm 1.4$ m/s, a detection close to a $3\sigma$ significance. Our joint-fit model reveals that the planet has a radius of $2.19\pm 0.17\ R_{\oplus}$ with a mass of $6.37\pm 2.45\ M_{\oplus}$. All priors and the median of the posterior distributions for each fitted parameter are summarized in Table LABEL:allpriors. We also run a separate joint-fit using EXOFASTv2 (Eastman et al., 2019), and we verify that similar results were obtained within $1\sigma$.

We search for the transit timing variations (TTVs) with all the photometric datasets (TESS, LCOGT, TRAPPIST-North, SPECULOOS-North) using EXOFASTv2. EXOFASTv2 uses the Differential Evolution Markov chain Monte Carlo method to derive the values and their uncertainties of the stellar and planetary parameters of the system. It fits a linear ephemeris to the transit times and adds a penalty for the deviation of the step’s linear ephemeris from the best-fit linear ephemeris of the transit times. For the TTV analysis of TOI-2136 b, we fix the stellar parameters to the values as in Table 1 and orbital parameters to the results obtained from the joint-fit performed. The results of the analysis showing the difference between the observed transit times and the calculated linear ephemeris from all the transits is presented in Figure 11. We find no evidence of a significant TTV signal in the current photometric data.

Since the mass constraint on the planet has a significance slightly below $3\sigma$, we make use of the TRICERATOPS package (Giacalone et al., 2021) to vet and statistically validate the planetary nature of TOI-2136 b. TRICERATOPS is a Bayesian tool that takes host and nearby stars into consideration and calculate the probabilities of different transit-producing scenarios. The output false positive probability (FPP) value quantifies the possibility that the transit signal is not due to a planet around the host star. We first apply TRICERATOPS to the TESS light curve along with the contrast curve obtained by the ‘Alopeke speckle imaging (832 nm). The resulting FPP value 0.014 is close to the normal FPP threshold of 0.015 (1.5%) to classify a validated planet (Giacalone et al., 2021). Wells et al. (2021) found that ground light curves sometimes put a better photometric constraint than the TESS data. We thus rerun the pipeline using the same contrast curve but the SPECULOOS-North/Artemis time-series, which yields a FPP value of $4\times 10^{-3}$. Therefore, we consider this TOI to be a validated planet.

We use the radius constraint on TOI-2136 b derived from the transit photometry and the measured mass from the SPIRou RV data to investigate the location of this planet in the mass-radius diagram. Figure 12 shows the mass and radius distribution of a sample of well-characterized planets with the precisions on both measurements better than $30\%$ taken from the TEPcat database (Southworth, 2011). The composition curves are retrieved from Zeng et al. (2016). The mass and radius of TOI-2136 b are compared to the two-layer internal structure models of (Zeng et al., 2016). As can be seen from Figure 12, TOI-2136 b appears to have a composition consistent with a pure water-ice world or a rocky planet with moderate atmosphere.

We further investigate the composition of TOI-2136 b using the Exoplanet Composition Interpolator⁵⁵5https://tools.emac.gsfc.nasa.gov/ECI/. The algorithm takes the planet evolution models proposed by Lopez & Fortney (2014) and interpolates between the grid of these pre-computed models to explore the interiors and compositions of the planets. Taking the planet mass, radius, insolation flux as well as stellar age as inputs, we find the rocky core and gaseous envelope of TOI-2136 b have mass fractions of $98.7^{+1.0}_{-1.5}\%$ and $1.3^{+1.5}_{-1.0}\%$, respectively.

The bimodality of radius distribution shown in small planets around FGK stars, which splits them into super-Earths and sub-Neptunes, is known as a transition between planets with and without extended gaseous envelopes (Fulton et al., 2017; Fulton & Petigura, 2018). Martinez et al. (2019) found that this transition radius is orbital period dependent, following a power law of $r_{\rm p,valley}\propto P^{-0.11}$. This finding approximately agrees with the prediction from the thermally driven atmospheric mass-loss scenarios including photoevaporation and core-powered envelope escape ($r_{\rm p,valley}\propto P^{-0.15}$; Lopez & Rice 2018). However, observational results from Cloutier & Menou (2020) suggested that the transition radius of small planets around low mass stars is likely in accordance with the gas poor formation model (Lee et al., 2014; Lee & Chiang, 2016), following $r_{\rm p,valley}\propto P^{0.11}$. For early type M dwarfs with mass around $0.64\ M_{\odot}$, Cloutier et al. (2020) found that the thermally driven atmospheric mass-loss scenario remains efficient at sculpting their close-in planets. Recent work from Luque et al. (2021), instead, tentatively reached a different conclusion. They proposed that the planetary radius valley for stars within a mass range between $0.54$ and $0.64\ M_{\odot}$ probably results from gas poor formation. The relative dominance of these two kinds of competing physical processes at the low stellar mass end remains unclear. Thus, populating the number of small planets with known bulk composition is crucial to solve the puzzles.

Figure 13 shows the orbital period and radius of planets with mass determination around M dwarfs ($M_{\ast}\lesssim 0.65\ M_{\odot}$). We can see that TESS has doubled the number of small planets with known density, making them important for further investigating the strength of the two aforementioned envelope escape physical processes. With a period of $P_{b}=7.85$ days and a radius of $R_{p}=2.19\pm 0.17\ R_{\oplus}$, TOI-2136 b is located slightly above the radius valley for low mass dwarfs predicted by the thermally driven atmospheric mass loss model (See Figure 13). Theoretical studies infer that TOI-2136 b should be predominantly gaseous. Indeed, our previous analysis shows that TOI-2136 b likely retains a H/He envelope with a small mass fraction. Given an estimated stellar age of $4.6\pm 1.0$ Gyr, TOI-2136 has, in principle, finished the photoevaporation stage, which has a timescale of hundreds of Myrs (Owen & Wu, 2013, 2017). However, it is possible that TOI-2136 is still undergoing the mass loss process following the core powered mechanism that has a Gyr timescale (Ginzburg et al., 2018).

Another interesting question is the behaviour of the radius valley as a function of stellar mass. Both competing physical processes predict a positive correlation between the center of radius valley and stellar mass, although different models show a difference in the slope at each mass bin (Lopez & Rice, 2018; Gupta & Schlichting, 2019; Wu, 2019). Consequently, comparing the theoretical predictions with the observational findings may rule out certain models. Cloutier & Menou (2020) obtained a similar positive trend using a sub-sample of Kepler and K2 planets. However, the sample size is small, especially for planets around mid-to-late M dwarfs, leading to a relatively large statistical uncertainty. Nevertheless, TOI-2136 b joins the small but growing sample of planets around mid-M dwarfs that may help understand evolution of the transition radius with stellar mass in the future.

Given the proximity, small size and brightness in the near infrared, TOI-2136 is a promising star for atmospheric studies of its planet. Following the criteria proposed in Kempton et al. (2018), we compute the Transmission Spectroscopy Metric (TSM) of TOI-2136 b to examine its potential opportunities for atmospheric characterization with the James Webb Space Telescope (JWST, Gardner et al. 2006). We derive a TSM of $65^{+20}_{-32}$ for TOI-2136 b. We compare the TSM factor of TOI-2136 b with other small planets ($R_{p}\leq 4\ R_{\oplus}$) harbored around low mass stars ($M_{\ast}\leq 0.65\ M_{\odot}$) with mass measurements from RVs or TTVs in Figure 14. Kempton et al. (2018) quantified $\rm TSM=90$ as a recommended threshold for planets with $1.5<R_{p}<10\ R_{\oplus}$ to be high-quality atmospheric characterization targets. Thus, TOI-2136 b is located close to the first rank of targets with a relatively low equilibrium temperature $T_{\rm eq}$. In addition, we also estimate the signal amplitude of TOI-2136 b in the transit transmission spectroscopy following the approach described in Gillon et al. (2016):

where $R_{p}$ and $R_{\ast}$ are the planet and its host star radius, $h_{\rm eff}$ is the effective atmospheric height. We calculate the signal amplitude under the typical case that $h_{\rm eff}/H=7$, where $H=kT/\mu g$ is the atmospheric scale height. We find a $\it S$ of $382\pm 196$ ppm, assuming a bond albedo of 0 and a mean molecular mass $\mu$ of 2.3 amu for sub-Neptunes (Demory et al., 2020). The large uncertainty mainly comes from the loose constraint on the planet mass. Schlawin et al. (2020) reported a noise floor level 10 ppm for JWST for NIRSpec ($\lambda=5.0-11\ \mu{\rm m}$). Supposing the lower limit on $\it S$ measurement of TOI-2136 b, it would be between 18.6 times and the higher limit, 57.8 times the 10 ppm uncertainty. Taking two aspects into consideration, we suggest that TOI-2136 b is an exciting target for further atmospheric researches. A number of studies on the diversity of sub-Neptunian atmospheres have already been made (e.g., Lavvas et al., 2019; Chouqar et al., 2020).

As noted above, the mean molecular mass $\mu$ is degenerated with the surface gravity of the planet (i.e., planet mass $M_{p}$). Thus, a well-measured planet mass is required to fully understand the compositions of the planet atmosphere. Otherwise, the accuracy and precision of the retrieved atmospheric parameters will be largely limited (Batalha et al., 2019). Since the current SPIRou RVs only provide a $2\sigma$ mass constraint and TOI-2136 is a quiet M dwarf without strong stellar activity, future subsequent spectroscopy observations are encouraged to determine the planet mass at the $3\sigma$ confidence level and look for other potential non-transiting planets. Due to the faintness of TOI-2136 (Vmag=14.3), it challenges most optical spectroscopy instruments on the ground. However, it is still accessible by NIR facilities like InfrRed Doppler spectrograph (IRD; Kotani et al. 2018) and Habitable-zone Planet Finder (HPF; Mahadevan et al. 2014) or red-optical spectrographs on large telescopes like MAROON-X (Seifahrt et al., 2018), which is dedicated to conducting RV measurements for mid-to-late M dwarfs.

Based on the results from the Kepler survey, Muirhead et al. (2015) found that $21^{+7}_{-5}\%$ of mid-M dwarf stars like TOI-2136 host compact multiple planets with periods all shorter than 10 days. This rate is not very different from that of early-type M dwarfs but much higher than solar-like stars. Therefore, we perform an injection-and-recovery test using MATRIX ToolKit⁶⁶6https://github.com/PlanetHunters/tkmatrix (Pozuelos et al., 2020; Demory et al., 2020) to explore the detection limits of the current TESS data and determine the type of planets we perhaps miss. We make use of all available PDC-SAP light curves of TOI-2136 after removing the known transits of TOI-2136 b. We explore a period-radius space of $1\sim 15\ {\rm days}$ and $0.5\sim 3.0\ R_{\oplus}$ with step sizes of 1 day and 0.25 $R_{\oplus}$. During the injection, we assume that the synthetic “planet” has an inclination of $i=90^{\circ}$ on a circular orbit, and randomly generate ten light curves with different $T_{0}$ for each grid. We thus examine a total of 1680 scenarios. For each light curve, we use a biweight filter with a window size of 0.5 day to remove the systematic trends. MATRIX ToolKit defines a successful recovery if the the detected period is within 5% of the injected period and the transit duration is within 1 hr when compared with the set value. Figure 15 depicts our test results. We find that: (1) most planets smaller than super Earths ($R_{p}\lesssim 1.5\ R_{\oplus}$) across the period range we searched are likely to be still buried in the light curve and remain undetectable (Brady & Bean, 2021); (2) planets that have $R_{p}\gtrsim 2.0\ R_{\oplus}$ with periods up to 15 days can be ruled out with a recovery rate $\geq 80\%$. Since the current TESS dataset is insensitive to planets with period larger than 15 days, here we note that future TESS observations to be done in Sector 53 and 54 between 13^th Jun. 2022 and 5^th Aug. 2022 during the Extended Mission would help better understand the architecture of this system.