Vetting of 384 TESS Objects of Interest with TRICERATOPS and Statistical Validation of 12 Planet Candidates

Initially, each star within 10 pixels of the target is considered a potential origin of the transit-like event. Because each star is contributing a different amount of flux to the aperture, the size that the transiting object must be to produce the observed transit depth is different for each star. Because the transiting object size is important for determining the probability of each scenario, the relative flux contributed by each star in the aperture is essential information.

We calculate the flux ratio contributed by each star using a method similar to that used in Stassun et al. (2018) to determine the contamination ratios reported for candidate target stars in the TIC. Specifically, we assume that the point-spread function (PSF) of each star takes the form of a circular 2D Gaussian where the area under each Gaussian (i.e., the total flux) is determined using the TESS magnitudes reported in the TIC. We estimate the standard deviation of the Gaussian using the TESS pixel response function (PRF) models on MAST.³⁵ Due to effects relating to the design of the TESS optics, the exact PRF for a star is dependent on the location on the CCD on which it is observed. These models allow one to estimate the PRF for a given star by providing the size and shape of the TESS PRF at 25 locations on each CCD. We fit each PRF model to a circular 2D Gaussian and record the best-fit standard deviation, finding that it typically ranges between 0.6 and 0.9 pixels. For simplicity, we adopt a standard deviation of 0.75 pixels for all stars, regardless of CCD location. For each star, we integrate the flux in the aperture and divide by the total flux contributed to the aperture by all stars to determine its flux ratio, X_s. For targets that are observed in multiple sectors, we assume that the flux ratio for a given star is the average of its flux ratios across each sector.

To ensure that our method provides reliable flux ratios, we compare in Figure 1 the target star flux ratios for 228 TOIs obtained using our method with those reported by the TESS Science Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016), which calculates flux ratios using the actual PRF models discussed above.³⁶ Both of these calculations are carried out with the aperture used by the TESS SPOC pipeline to extract the light curve of the target star. The figure shows good agreement between the two calculations, with a slightly better agreement for fainter stars.

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After flux ratios are determined, we eliminate stars that are too faint to be the source of the observed dimming event. If the observed transit depth is δ_obs, the relative transit depth for each star is simply δ_s = δ_obs/X_s. For stars that contribute relatively little flux to the aperture, it is possible for δ_s to exceed unity. We exclude these stars from further analysis.

After calculating the flux ratio for each star in the aperture, we determine the scenarios that can produce the observed transit-like event. Our procedure considers a total of 15 scenarios for the target star and an additional three scenarios for each nearby star with δ_s < 1. These scenarios are summarized in Table 1.

The 15 target star scenarios can be classified into three configurations. The first is the case where the target star has no unresolved stellar companion (where we define "companion" to encompass both bound and foreground/background stars). In this case, we consider the scenarios of a transiting planet with the reported orbital period around the target star (TP), an EB with the reported orbital period around the target star (EB), and an EB with twice the reported orbital period around the target star (EBx2P). The last of these scenarios is meant to capture the possibility that the observed transit is caused by eclipsing binary stars of roughly equal size, such that the primary and secondary eclipses are mistaken for the primary transit of a smaller object with half the orbital period. The second configuration is that in which the target star has an unresolved bound stellar companion. In this case, we consider the scenarios of a transiting planet around the target star with the reported orbital period (primary TP, or PTP), an eclipsing binary with the reported orbital period around the target star (primary EB, or PEB), an eclipsing binary with twice the reported orbital period around the target star (primary EBx2P, or PEBx2P), a transiting planet with the reported orbital period around the companion (secondary TP, or STP), an eclipsing binary around the companion (secondary EB, or SEB), and an eclipsing binary with twice the reported orbital period around the companion (secondary EBx2P, or SEBx2P). The third configuration is that in which there is an unresolved foreground or background star along the line of sight to the target star. In this case, we again consider the scenarios of a transiting planet with the reported orbital period around the target star (diluted TP, or DTP), an eclipsing binary with the reported orbital period around the target star (diluted EB, or DEB), an eclipsing binary with twice the reported orbital period around the target star (diluted EBx2P, or DEBx2P), a transiting planet with the reported orbital period around the companion (background TP, or BTP), an eclipsing binary with the reported orbital period around the companion (background EB, or BEB), and an eclipsing binary with twice the reported orbital period around the companion (background EBx2P, or BEBx2P).³⁷

For nearby stars with δ_s < 1, we also consider the scenarios of a transiting planet with the reported orbital period around that star (nearby TP, or NTP), an eclipsing binary with the reported orbital period around that star (nearby EB, or NEB), and an eclipsing binary with twice the reported orbital period around that star (nearby EBx2P, or NEBx2P). Each of these scenarios operates under the assumption that the nearby star has no unresolved stellar companion. These scenarios can also be omitted by the calculation if false positives originating from the respective nearby stars have been ruled out through supplementary follow-up.

Whenever possible, we use the stellar properties listed in the TIC in our calculations. However, for reasons that will be discussed, there are times in our procedure where we must estimate the properties (i.e., mass M_⋆, radius R_⋆, and effective temperature T_eff) of a star in order to determine the probability of the corresponding scenario. We do so using the empirical and semiempirical relations between stellar properties used to populate these fields in the TIC.

For stars with M_⋆ > 0.63M_⊙ (corresponding roughly to T_eff > 4000 K), we determine stellar properties using the results from Torres et al. (2010a). Using the same method discussed in Section 3 of Stassun et al. (2018), we draw spline curves through the distribution of points in M_⋆−T_eff and M_⋆−R_⋆ space. For stars with M_⋆ ≤ 0.63M_⊙, we repeat this process using a sample of stars from the specially curated TESS Cool Dwarf Catalog (Muirhead et al. 2018). We select nodal points using the sample such that they are continuous with the curves obtained for hotter stars.

The spline curves and the samples on which they are based are shown in Figure 2. The result of this process is a set of relations that allows us to estimate the R_⋆ and T_eff of a star given M_⋆.

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We employ a Bayesian framework in our procedure and thus make use of Bayes's theorem:

$$\begin{eqnarray}&&p({S}_{j}| D)\propto p({S}_{j})p(D| {S}_{j}),\end{eqnarray} \tag{ 1 }$$

where p(S_j∣D) is the posterior probability of the jth scenario S_j given the data D, p(S_j) is the prior probability of scenario S_j, and p(D∣S_j) is the marginal likelihood of the data D given the scenario S_j (sometimes also referred to as the global likelihood, or the Bayesian evidence). Because we work with a transit model characterized by the parameter vector θ_j, we express the marginal likelihood as the marginalization of the likelihood p(D∣θ_j, S_j) over θ_j:

$$\begin{eqnarray}&&p(D| {S}_{j})=\int p({\theta }_{j}| {S}_{j})p(D| {\theta }_{j},{S}_{j})d\theta ,\end{eqnarray} \tag{ 2 }$$

where p(θ_j∣S_j) is the prior distribution of the model parameters. We discuss how these quantities are calculated throughout the remainder of this section.

After calculating p(S_j∣D) for each scenario, we determine the relative probability of each scenario using the equation

$$\begin{eqnarray}&&{{ \mathcal P }}_{j}=\displaystyle \frac{p({S}_{j}| D)}{{\displaystyle \sum }_{j}p({S}_{j}| D)}.\end{eqnarray} \tag{ 3 }$$

From here, we define two quantities that are useful for vetting and validation purposes. First, the FPP is given by

$$\begin{eqnarray}&&\mathrm{FPP}=1-({{ \mathcal P }}_{\mathrm{TP}}+{{ \mathcal P }}_{\mathrm{PTP}}+{{ \mathcal P }}_{\mathrm{DTP}}).\end{eqnarray} \tag{ 4 }$$

This quantity represents the probability that the observed transit is due to something other than a transiting planet around the target star. Second, the nearby false-positive probability (NFPP) is given by

$$\begin{eqnarray}&&\mathrm{NFPP}=\sum ({{ \mathcal P }}_{\mathrm{NTP}}+{{ \mathcal P }}_{\mathrm{NEB}}+{{ \mathcal P }}_{\mathrm{NEBx}2{\rm{P}}})\end{eqnarray} \tag{ 5 }$$

(i.e., the sum of all scenarios involving nearby stars). This quantity represents the probability that the observed transit originates from a resolved nearby star rather than the target star.

2.4.1. Scenario Priors

The scenario prior represents the prior probability of a given scenario before the data are considered. The only scenario prior we employ in our calculation is the probability of a transiting planet or eclipsing binary having the P_orb applied to the model. For both transiting planets and eclipsing binaries, we assume that the probability distribution of P_orb takes the form of a broken power law in the range of 0.1–50 days. Using these probability distributions, we calculate the prior probability of an orbital period ${P}_{\mathrm{orb}}^{{\prime} }$ by integrating the probability distribution between ${P}_{\mathrm{orb}}^{{\prime} }-0.1$ and ${P}_{\mathrm{orb}}^{{\prime} }+0.1$:

$$\begin{eqnarray}&&p({P}_{\mathrm{orb}}^{{\prime} })={\int }_{{P}_{\mathrm{orb}}^{{\prime} }-0.1}^{{P}_{\mathrm{orb}}^{{\prime} }+0.1}p({P}_{\mathrm{orb}}){{dP}}_{\mathrm{orb}}.\end{eqnarray} \tag{ 6 }$$

For transiting planets we base the behavior of this distribution on studies of planet occurrence rates as a function of orbital period (e.g., Howard et al. 2012; Dong & Zhu 2013; Petigura et al. 2013; Dressing & Charbonneau 2015; Mulders et al. 2015a, 2018). We express p(P_orb) as a broken power law with a break at P_orb = 10 days and the form

$$\begin{eqnarray}p({P}_{\mathrm{orb}})\sim \left\{\begin{array}{ll}{P}_{\mathrm{orb}}^{1.5} & \quad 0.1\,\mathrm{days}\leqslant {P}_{\mathrm{orb}}\leqslant 10\,\mathrm{days}\\ {P}_{\mathrm{orb}}^{0.0} & \quad 10\,\mathrm{days}\lt {P}_{\mathrm{orb}}\leqslant 50\,\mathrm{days}\end{array}\right..\end{eqnarray} \tag{ 7 }$$

Note that while planet occurrence is typically expressed as a nonseparable function of both planet radius and P_orb, we treat the two variables as independent in our calculation procedure.

For eclipsing binaries we base the behavior of this distribution on the results of the Kepler Eclipsing Binary Catalog (Kirk et al. 2016), which contains the properties of thousands of objects that were classified as EBs based on their light-curve morphologies. After correcting the catalog for eclipsing binaries that were not detected owing to orbital misalignment, we find that p(P_orb) is best expressed as a broken power law with a break at P_orb = 0.3 days and the form

$$\begin{eqnarray}p({P}_{\mathrm{orb}})\sim \left\{\begin{array}{ll}{P}_{\mathrm{orb}}^{5.0} & \quad 0.1\,\mathrm{days}\leqslant {P}_{\mathrm{orb}}\leqslant 0.3\,\mathrm{days}\\ {P}_{\mathrm{orb}}^{0.5} & \quad 0.3\,\mathrm{days}\lt {P}_{\mathrm{orb}}\leqslant 50\,\mathrm{days}\end{array}\right..\end{eqnarray} \tag{ 8 }$$

It is common for validation procedures to also include priors that capture the overall planet occurrence and stellar multiplicity rate. Planet occurrence rate studies have revealed that the probability of an FGKM dwarf hosting a planet with P_orb < 50 days ranges from 10% to 100%, decreasing as a function of increasing host star mass (e.g., Fressin et al. 2013; Petigura et al. 2013; Dressing & Charbonneau 2015). Stellar multiplicity rate studies have determined that the probability of an FGKM dwarf hosting a stellar companion with P_orb < 50 days ranges from 1% to 10%, increasing as a function of increasing host mass (Moe & Di Stefano 2017). This implies that all scenarios involving transiting planets should have a prior probability 10–100× higher than those involving eclipsing binaries. At first, we included this prior in the algorithm. However, after testing the performance of our tool on known transiting planets and astrophysical false positives (see Section 4), we concluded that the prior gave transiting planet scenarios too much of an advantage. This advantage often caused an underestimation of FPP, which led the algorithm to classify astrophysical false positives as transiting planets. To avoid this apparent bias, we omit these priors from our calculation procedure.

2.4.2. Parameter Prior Distributions

Every scenario we test is associated with a vector θ_j of parameters that are needed for modeling the light curves of each scenario. The parameters that compose these vectors for each scenario are shown in Table 1. To reflect the fact that certain values of these parameters are more common than others, each is associated with a probability distribution. In this section, we define each of these parameters and their respective probability distributions. Examples of these distributions are shown in Figure 3 for a sample size of 10⁶.

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The parameter i represents the inclination of the orbit of a transiting planet or eclipsing binary. Assuming an isotropic distribution of orbits, the distribution of inclinations takes the form

$$\begin{eqnarray}&&p(i)\sim \sin i.\end{eqnarray} \tag{ 9 }$$

The parameter R_p represents the radius of a transiting planet. Because this distribution is known to be dependent on host star mass, we use different distributions for M dwarfs and FGK dwarfs. The two distributions differ in the prevalence of giant planets (R_p > 6R_⊕), which are known to be less common around M dwarfs than they are around their more massive counterparts by a factor of ∼10 (e.g., Dressing & Charbonneau 2013; Fressin et al. 2013; Petigura et al. 2013; Mulders et al. 2015b). We express these distributions as broken power laws with breaks at R_p = 3R_⊕ and R_p = 6R_⊕ and a range of R_p = 0.5R_⊕–20R_⊕ (e.g., Mulders et al. 2015b, 2018).³⁸ For M dwarfs the distribution takes the form

$$\begin{eqnarray}p({R}_{{\rm{p}}})\sim \left\{\begin{array}{ll}{R}_{{\rm{p}}}^{0.0} & \quad 0.5{R}_{\oplus }\leqslant {R}_{{\rm{p}}}\leqslant 3{R}_{\oplus }\\ {R}_{{\rm{p}}}^{-7.0} & \quad 3{R}_{\oplus }\lt {R}_{{\rm{p}}}\leqslant 6{R}_{\oplus }\\ {R}_{{\rm{p}}}^{-0.5} & \quad 6{R}_{\oplus }\lt {R}_{{\rm{p}}}\leqslant 20{R}_{\oplus }\end{array}\right.,\end{eqnarray} \tag{ 10 }$$

and for FGK dwarfs the distribution takes the form

$$\begin{eqnarray}p({R}_{{\rm{p}}})\sim \left\{\begin{array}{ll}{R}_{{\rm{p}}}^{0.0} & \quad 0.5{R}_{\oplus }\leqslant {R}_{{\rm{p}}}\leqslant 3{R}_{\oplus }\\ {R}_{{\rm{p}}}^{-4.0} & \quad 3{R}_{\oplus }\lt {R}_{{\rm{p}}}\leqslant 6{R}_{\oplus }\\ {R}_{{\rm{p}}}^{-0.5} & \quad 6{R}_{\oplus }\lt {R}_{{\rm{p}}}\leqslant 20{R}_{\oplus }\end{array}\right..\end{eqnarray} \tag{ 11 }$$

The parameter q_short represents the mass ratio between the host star and a short-period stellar companion (i.e., an eclipsing binary). To calculate this distribution, we extrapolate from the results of Moe & Di Stefano (2017) for Sun-like stars. In the study, q is parameterized as a broken power law with a break at q = 0.3 and a range of q = 0.1–1.0. In addition, the parameterization takes into account the excess of stellar "twins" (stellar companions with q > 0.95) with a term ${{ \mathcal F }}_{\mathrm{twin}}$ (defined as the fraction of stars with q > 0.3 that have q > 0.95) that boosts the prevalence of these stars in the probability distribution. For short-period stellar companions, the distribution takes the form

$$\begin{eqnarray}p({q}_{\mathrm{short}})\sim \left\{\begin{array}{ll}{q}_{\mathrm{short}}^{0.3} & \quad 0.1\leqslant q\leqslant 0.3\\ {q}_{\mathrm{short}}^{-5.0} & \quad 0.3\lt q\leqslant 1.0\end{array}\right.\end{eqnarray} \tag{ 12 }$$

with ${{ \mathcal F }}_{\mathrm{twin}}=0.3$.

The parameter q_long represents the mass ratio between the target star and a long-period stellar companion (i.e., an unresolved bound companion). Again, we utilize the parameterization and extrapolate results of Moe & Di Stefano (2017) for Sun-like stars. For long-period stellar companions, the distribution takes the form

$$\begin{eqnarray}p({q}_{\mathrm{long}})\sim \left\{\begin{array}{ll}{q}_{\mathrm{long}}^{0.3} & \quad 0.1\leqslant q\leqslant 0.3\\ {q}_{\mathrm{long}}^{-0.95} & \quad 0.3\lt q\leqslant 1.0\end{array}\right.\end{eqnarray} \tag{ 13 }$$

with ${{ \mathcal F }}_{\mathrm{twin}}=0.05$.

The parameter "simulated star" represents the properties of a star drawn from a population of stars simulated with TRILEGAL. To determine the properties of blended stars used in DTP, DEB, DEBx2P, BTP, BEB, and BEBx2P scenarios, we simulate a population of stars in a 0.1 deg² region of the sky centered at the target star. We then produce a distribution of possible foreground/background stars by removing all stars with TESS magnitudes brighter than the target and fainter than 21, which typically yields between 300 and 1000 stars. When simulating an instance of these scenarios, we draw a star directly from this distribution.

2.4.3. Marginal Likelihoods

Because the integral in Equation (2) is typically impossible to solve analytically, it is common to approximate the integral by sampling p(θ_j∣S_j). This is, in fact, what is done when calculating odds ratios between competing scenarios in the PASTIS and VESPA validation procedures. In this work, we calculate the marginal likelihood using arithmetic mean estimation (Kass & Raftery 1995). This method allows us to calculate the marginal likelihood using Monte Carlo sampling by approximating Equation (2) as

$$\begin{eqnarray}&&p(D| {S}_{j})\sim \displaystyle \frac{1}{N}\displaystyle \sum _{n=1}^{N}p(D| {\theta }_{j}^{(n)},{S}_{j}),\end{eqnarray} \tag{ 14 }$$

where ${\theta }_{j}^{(n)}$ is the nth sample from the parameter prior distribution and N is the total number of samples. This is typically regarded as the simplest estimator of the marginal likelihood, but it is often avoided because it can produce a large variance in p(D∣S_j) if N is not sufficiently high and is relatively inefficient when integrating over a large number of parameters. We take two approaches to combat these drawbacks: (1) we chose an N high enough to produce results that are consistent between consecutive calculations (which we determine to be N = 10⁶), and (2) we make simplifying assumptions in our transiting planet and eclipsing binary models that minimize the number of parameters we must marginalize over.

The first simplifying assumption we make is to assume that the M_⋆, R_⋆, and T_eff of each resolved star are known precisely. Unless the user provides these parameters, they are assumed to be equal to those listed in the TIC. In addition, any other stars added to our transit model that do not have estimates for these quantities (e.g., eclipsing binaries or unresolved companions) are assumed to be precisely characterized based on their M_⋆ (see Section 2.3). Because the transit models are sensitive to these parameters, this assumption saves us from having to marginalize over a distribution of target star properties.

The second simplifying assumption we make is to assume a fixed orbital period and zero eccentricity (e) in all scenarios considered, which significantly simplifies the orbital solution of the system. There is strong evidence that short-period planets are biased toward lower e (e.g., Kane et al. 2012; Kipping 2013; Shabram et al. 2016). According to the NASA Exoplanet Archive,³⁹ 84% of confirmed planets with P_orb < 30 days and reported eccentricities have e < 0.2. The same justification can be applied to short-period eclipsing binaries. Moe & Di Stefano (2017) showed that the e distribution of binary stars with P_orb < 10 days goes like e^−0.8. This implies that 72% of short-period eclipsing binaries have e < 0.2. Because a majority of TOIs will have P_orb < 30 days (due to the ∼27-day intervals in which sectors are observed and the general requirement for at least two transits be observed for a system to become a planet candidate), we believe that the assumption of circular orbits is justified in most cases. However, users of TRICERATOPS should be aware that this assumption becomes less valid as longer orbital periods are considered.

We calculate $p(D| {\theta }_{j}^{(n)},{S}_{j})$ as the product of two terms:

$$\begin{eqnarray}&&p(D| {\theta }_{j}^{(n)},{S}_{j})=p({D}_{\mathrm{tra}}| {\theta }_{j}^{(n)},{S}_{j})\times {w}^{(n)},\end{eqnarray} \tag{ 15 }$$

where the first term is the likelihood of the transit data and w⁽ⁿ⁾ is a weight that encapsulates our ability to rule out unresolved companions near the target star using high-resolution imaging follow-up. This weight is intended to decrease the likelihood of scenarios involving unresolved companions when stronger constraints on the existence such companions are applied.

The likelihood of the transit data is calculated using the equation

$$\begin{eqnarray}&&p({D}_{\mathrm{tra}}| {\theta }_{j}^{(n)},{S}_{j})\propto \prod \exp \left[-\frac{1}{2}{\left(\frac{{y}_{l}-f({t}_{l}| {\theta }_{j}^{(n)})}{\sigma }\right)}^{2}\right],\end{eqnarray} \tag{ 16 }$$

where y_l is the flux of the lth data point, $f({t}_{l}| {\theta }_{j}^{(n)})$ is the flux given by the model for the parameter vector ${\theta }_{j}^{(n)}$ at the time of the lth data point, and σ is the characteristic uncertainty of the flux.

For PTP, PEB, PEBx2P, STP, SEB, and SEBx2P scenarios we calculate w⁽ⁿ⁾ using Equation (23) of Moe & Di Stefano (2017). Equation (23) of Moe & Di Stefano (2017) provides the frequency of bound stellar companions as a function of primary mass and orbital period. We calculate this quantity for the nth sample of the parameter prior distribution using the following steps: (1) determine magnitude difference between the primary and secondary star using the mass of the target and the nth draw of q_long, (2) use the contrast curve obtained from high-resolution imaging to determine the angular separation beyond which the simulated secondary would have been detected, (3) convert this angular separation to an orbital period using the parallax of the target and the masses of the target and simulated secondary, and (4) use this orbital period and Equation (23) of Moe & Di Stefano (2017) to calculate the corresponding frequency of bound stellar companions. If no high-resolution imaging data are available to fold in, the angular separation used in step (2) is assumed to be 22 (Brown et al. 2018).

For DTP, DEB, DEBx2P, BTP, BEB, and BEBx2P scenarios we calculate w⁽ⁿ⁾ using the results of the TRILEGAL simulation discussed in Section 2.4.2. Specifically, we calculate this likelihood as the frequency of unresolved foreground and background stars aligned with the target star in the sky. This calculation is performed with the following steps: (1) determine the magnitude difference between the target star and the nth drawn foreground/background star, (2) use the contrast curve obtained from high-resolution imaging to determine the angular separation beyond which the simulated foreground/background star would have been detected, (3) use this separation and the total number of simulated stars to estimate the frequency of unresolved foreground/background stars near the target. As for the previous scenarios, if no high-resolution imaging data are available to fold in, the angular separation used in step 2 is assumed to be 22 (Brown et al. 2018).

We set the maximum value of w⁽ⁿ⁾ for each scenario to 1. We also set w⁽ⁿ⁾ = 1 for TP, EB, EBx2P, NTP, NEB, and NEBx2P scenarios, which do not involve unresolved companions.

2.4.4. Light-curve Modeling

We calculate Equation (16) by modeling light curves using a modified version of batman (Kreidberg 2015). Here we describe the steps that go into simulating the transits of each scenario.

The simplest scenario to model is the TP scenario, in which we assume that all of the flux originates from the host star. For this scenario, we use batman in its default form. For this scenario, as well as all other scenarios, we use quadratic limb-darkening coefficients chosen based on the T_eff and log g of the host star (Claret 2018).

For all scenarios involving eclipsing binaries, we must account for the fact that the flux is split between the host star and the short-period companion. Doing so requires an estimate for the flux contributed by the eclipsing binary, which we find by determining a relation between M_⋆ and TESS magnitude. We begin by querying the TIC for all stars located a distance between 99 and 101 pc away. We then draw a spline curve through the distribution of points in the TESS magnitude—M_⋆ plane, which is shown in Figure 4. This relation allows us to calculate the TESS band flux ratio between two stars given their masses and adjust the in-transit flux of the light curve accordingly.

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For scenarios involving unresolved companions, we again must account for the flux dilution from the additional star. For scenarios involving an unresolved bound companion (whose mass is determined by q_long), we use the spline relation shown in Figure 4 to determine the flux contribution of the star. For scenarios involving an unresolved foreground/background star, we use the TESS magnitude provided by TRILEGAL to determine the flux contribution of the star.

Lastly, we apply constraints to our transit models for all "EB" and "EBx2P" scenarios. For the former, we require q_short < 0.95 and for the expected secondary eclipse depth to be shallower than 1.5× the scatter of the TESS light-curve flux (otherwise the secondary eclipse would have been detected and identified as such). For the latter, we require q_short > 0.95. If the nth model light curve does not satisfy these conditions, we set the likelihood of the transit to zero.

We apply our algorithm on the previously confirmed TOI 465.01 (WASP-156b; Demangeon et al. 2018), a ∼6R_⊕ planet orbiting a K dwarf with a 3.84-day orbital period. The host star, which has a TESS magnitude of 10.73 and is located 122 pc away, was observed with a 2-minute cadence in sector 4.

We begin by searching for all other stars within 10 pixels of the target star. This is shown in Figure 5, where the location of each nearby star relative to the local TESS pixels is shown on the left and the corresponding TESS image is shown on the right. Next, we calculate the flux contribution of each star and determine which contribute enough flux to the aperture to produce a transit with the reported depth. In this case, the target star is the only star bright enough to host the signal. We therefore ignore NTP, NEB, and NEBx2P scenarios for the remainder of this analysis, which leaves 15 scenarios to be considered.

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Next, we determine the best-fit model parameters for each of the 15 scenarios considered. The results of this step are displayed in Figure 6 and Table 2. Figure 6 shows the best-fit transit models for each scenario compared to the extracted TESS light curve. Table 2 shows the best-fit values for several transit model parameters. We see in both of these that the best-fitting scenario is the TP scenario.

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Table 2.  Scenario Probabilities for TOI 465.01

Scenario	TIC ID	M⋆ (M⊙)	R⋆ (R⊙)	Porb (days)	i (deg)	Rp (R⊕)	REB (R⊙)	${{ \mathcal P }}_{j}$	${{ \mathcal P }}_{j}$ with AO
TP	270380593	0.81	0.85	3.84	87.3	6.27		0.39	0.61
EB	270380593	0.81	0.85	3.84	85.3		0.10	<0.01	<0.01
EBx2P	270380593	0.81	0.85	7.67	85.3		0.84	<0.01	<0.01
PTP	270380593	0.81	0.85	3.84	87.4	6.35		0.22	0.14
PEB	270380593	0.81	0.85	3.84	86.4		0.10	<0.01	<0.01
PEBx2P	270380593	0.81	0.85	7.67	85.4		0.83	<0.01	<0.01
STP	270380593	0.79	0.82	3.84	87.8	8.71		0.31	0.19
SEB	270380593	0.63	0.65	3.84	89.8		0.10	0.01	<0.01
SEBx2P	270380593	0.48	0.49	7.67	87.3		0.49	<0.01	<0.01
DTP	270380593	0.81	0.85	3.84	87.5	6.26		0.06	0.06
DEB	270380593	0.81	0.85	3.84	85.7		0.10	<0.01	<0.01
DEBx2P	270380593	0.81	0.85	7.67	85.3		0.83	<0.01	<0.01
BTP	270380593	0.55	0.48	3.84	89.3	19.36		<0.01	<0.01
BEB	270380593	0.81	0.75	3.84	89.7		0.19	<0.01	<0.01
BEBx2P	270380593	0.83	1.01	7.67	85.4		0.85	<0.01	<0.01
TICa	270380593	${0.81}_{-0.10}^{+0.10}$	${0.85}_{-0.06}^{+0.06}$
WASP-156bb	270380593	${0.84}_{-0.05}^{+0.05}$	${0.76}_{-0.03}^{+0.03}$		${89.1}_{-0.9}^{+0.6}$	${5.72}_{-0.22}^{+0.22}$

Notes.

^aHost star properties from version 8 of the TIC (Stassun et al. 2018). ^bBest-fit host star and planet properties from Demangeon et al. (2018).

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The final step in the procedure is to calculate the relative probability of each scenario using Equation (3). These probabilities are shown in the rightmost columns of Table 2. For this TOI, we find that FPP = 0.33 and NFPP = 0.0.

The above calculation was done assuming that unresolved companions near the target star can be ruled out beyond 22. However, if one is able to further constrain the separation beyond which an unresolved star could exist, this number can be decreased to that new separation. On 2019 July 10, we obtained adaptive optics (AO) assisted high-resolution images of this TOI with ShARCS/ShaneAO on the Shane 3 m telescope at Lick Observatory, shown in Figure 7. These images were reduced using the steps outlined in Hirsch et al. (2019) and Savel et al. (in preparation), to which we refer the reader for more information. With these observations, we produce a contrast curve (also shown in Figure 7) that can be folded into the TRICERATOPS analysis in order to further constrain the probabilities of scenarios involving unresolved companions.

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To show how this changes the results of our tool, we repeat the calculation with this constraint applied. The impact that this AO follow-up has on the probability of each scenario is shown in the rightmost column of Table 2, which now yields FPP = 0.19.

We also apply our algorithm on TOI 529.01, a candidate with a 1.67-day orbital period that has been ruled out as an NEB around the nearby star TIC 438490748 (see Section 5 for more details). The originally proposed host star is an M dwarf with a TESS magnitude of 14.14 and a distance of 63 pc away. This TOI was observed with a 2-minute cadence in sector 6.

We again begin by searching for all other stars within 10 pixels of the target star, as shown in Figure 8. After calculating the flux contribution due to each star, it is determined that two nearby stars, TIC 438490736 and TIC 438490748, contribute enough light to the aperture for them to host the observed transit. As a result, there are 21 scenarios to be considered for this TOI.

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Figure 9 and Table 3 show the best-fit transits and transit model parameters for these scenarios, respectively. According to these results, the most probable scenario is the NEBx2P scenario around the nearby star TIC 438490748. In fact, the preference for this scenario is so strong that this TOI has FPP > 0.99 and NFPP > 0.99.

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Table 3.  Scenario Probabilities for TOI 529.01

Scenario	TIC ID	M⋆ (M⊙)	R⋆ (R⊙)	Porb (days)	i (deg)	Rp (R⊕)	REB (R⊙)	${{ \mathcal P }}_{j}$
TP	438490744	0.21	0.24	1.67	89.9	6.89		<0.01
EB	438490744	0.21	0.24	1.67	86.6		0.10	<0.01
EBx2P	438490744	0.21	0.24	3.33	87.3		0.24	<0.01
PTP	438490744	0.21	0.24	1.67	90.0	8.61		<0.01
PEB	438490744	0.21	0.24	1.67	89.5		0.10	<0.01
PEBx2P	438490744	0.21	0.24	3.33	87.7		0.24	<0.01
STP	438490744	0.09	0.10	1.67	89.2	19.70		<0.01
SEB	438490744	0.18	0.22	1.67	89.7		0.10	<0.01
SEBx2P	438490744	0.48	0.24	3.33	87.7		0.24	<0.01
DTP	438490744	0.21	0.24	1.67	89.2	9.82		<0.01
DEB	438490744	0.21	0.24	1.67	89.5		0.10	<0.01
DEBx2P	438490744	0.21	0.24	3.33	87.7		0.24	<0.01
BTP	438490744	0.51	0.45	1.67	89.8	19.92		<0.01
BEB	438490744	1.05	1.42	1.67	89.6		1.05	<0.01
BEBx2P	438490744	0.93	1.67	3.33	84.4		0.97	<0.01
NTP	438490736	0.67	0.69	1.67	89.5	19.94		<0.01
NEB	438490736	0.67	0.69	1.67	88.1		0.56	<0.01
NEBx2P	438490736	0.67	0.69	3.33	89.5		0.69	<0.01
NTP	438490748	0.51	0.45	1.67	89.7	19.98		<0.01
NEB	438490748	1.12	1.75	1.67	89.8		0.76	0.06
NEBx2P	438490748	1.08	1.54	3.33	85.2		1.16	0.94
TICa	438490744	${0.21}_{-0.02}^{+0.02}$	${0.24}_{-0.01}^{+0.01}$

Note.

^aHost star properties from version 8 of the TIC (Stassun et al. 2018).

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We begin by running our code on TOIs identified in 2-minute cadence data collected by TESS. In the first 2 yr of the TESS mission, these observations were collected for ∼200,000 nearby dwarf stars across nearly the entire sky. These observations are processed by the TESS Science Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016), which identifies TCEs and generates data validation reports that contain information useful for further vetting. These stars are then subjected to manual vetting by the TESS Science Office to compile a set of TOIs that consist of the TCEs with the best chances of being actual planets.

We use publicly available information from the TESS Follow-up Observation Program (TFOP) website⁴⁰ and 2-minute cadence TESS light curves from MAST to obtain the phase-folded light curves and apertures that we input into TRICERATOPS for each TOI. Because a key function of our algorithm is the identification of TOIs that are false positives around nearby stars, we use light curves extracted using simple aperture photometry instead of those processed with the pre-data-conditioning step of the SPOC pipeline, which removes contamination and variability originating from nearby stars. In order to recreate the conditions under which one would use our tool on new TOIs, we only use data from the first sector in which each TOI is observed and restrict the analysis to TOIs with at least three transits.

In order to have a ground truth with which to compare the results of our algorithm, we restrict our sample of TOIs to those that have been designated as confirmed planets (CPs) and those that have been designated as false positives (FPs) by the TFOP. We also discard TOIs that have been designated FPs due to instrumental false alarms (which our tool does not test for); TOIs without estimates for M_⋆, R_⋆, and T_eff in the TIC; and TOIs for which we are unable to feasibly recover a transit with the purported orbital parameters. Lastly, we only include planets with best-fit planet radii R_p < 8R_⊕ under the TP scenario. This radius corresponds roughly to the minimum radius of a brown dwarf (e.g., Sorahana et al. 2013) and has been used as an upper limit in the size of objects that can be validated in past validation studies (e.g., Mayo et al. 2018), due to the fact that giant planets, brown dwarfs, and low-mass stars are typically indistinguishable based on radius alone. This leaves 68 TOIs in total, 28 of which are confirmed planets and 40 of which are false positives. The system properties of these TOIs are displayed in Figure 10.

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After generating light curves for these TOIs, we calculate the FPP and NFPP for each to determine the limits within which TRICERATOPS can be used reliably. First, we explore how our predictions depend on the signal-to-noise ratio (S/N) of the data. We define the S/N as

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}=\frac{{\delta }_{\mathrm{obs}}}{{\sigma }_{\mathrm{CDPP}}}\sqrt{{n}_{\mathrm{tra}}},\end{eqnarray} \tag{ 17 }$$

where δ_obs is the observed transit depth (i.e., not corrected for dilution from nearby stars), σ_CDPP is the combined differential photometric precision (CDPP; Christiansen et al. 2012) of the 2-minute cadence data, and n_tra is the number of observed transits. We calculate σ_CDPP by applying the estimate_cdpp method of lightkurve (Lightkurve Collaboration et al. 2018) over the duration of the transit. Because this quantity incorporates our confidence in the size of a transiting object and the overall density of data points in-transit, it should correlate with the ability of TRICERATOPS to characterize the shape of a given transit.

The results of this analysis are shown in Figure 11. For both CPs and FPs, TRICERATOPS generally has more accurate predictions when S/N is higher. Specifically, FPP alone does not appear to be a reliable predictor of TOI disposition when S/N < 15, where FPs are frequently assigned low values of FPP that would ideally be reserved for CPs.

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Second, we explore how our algorithm performs when NFPP is also considered. Figure 12 shows the distribution of the TOIs in NFPP–FPP space for S/N < 15 (on the left) and S/N > 15 (on the right). In the figure, we differentiate TOIs that are CPs, TOIs that have been ruled out as FPs around nearby stars (nearby false positives, or NFPs), and TOIs that have been ruled out as FPs originating from the immediate vicinity of the target star (target false positives, or TFPs). The most salient feature of this figure is the region defined by NFPP < 10⁻³ and FPP < 0.5 that contains nearly all of the CPs, contains none of the NFPs or TFPs, and is independent of S/N. We designate TOIs that exist within this region as likely planets.

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Another visible feature of Figure 12 is the pileup of CPs in the region defined by NFPP < 10⁻³ and FPP < 0.05. Because this region is representative of TOIs with the best chances of being bona fide planets, we use it as a guide in defining our criteria for validating planets. Typically, the standard for validating planets (e.g., with VESPA) is to achieve an FPP below 1%. We therefore define validated planets as TOIs with NFPP < 10⁻³ and FPP < 0.015 (or FPP ≤ 0.01, when rounding to the nearest percent).

As a cross-check of our definition of a validated planet, we calculate the FPP of the TOIs in Figure 12 using VESPA. We run VESPA using the coordinates, stellar photometry (TESS mag, Bmag, Vmag, Jmag, Hmag, and Kmag), T_eff, $\mathrm{log}g$, and parallax listed for each TOI in the TIC. We use the same transit data used in our TRICERATOPS runs and assume a maximum unresolved star separation of 22. The FPPs obtained with VESPA are compared to the FPPs obtained with TRICERATOPS in Figure 13. According to the figure, TOIs that score a low FPP with TRICERATOPS generally score a similar FPP with VESPA. When it comes to FPs, and NFPs in particular, TRICERATOPS typically assigns higher FPPs than VESPA does. This is a reflection of our calculation procedure, which considers each star that contributes flux to the target aperture as a potential source of the observed transit. One might also note that there are a few NFPs that are scored low FPPs with both tools. However, because of our condition that a TOI have NFPP < 10⁻³ to be classified as a validated planet or a likely planet, TRICERATOPS would not identify these candidates as planets. Conversely, because VESPA explicitly requires the assumption that no contaminating stars exist within a specified radius of the target star, it could classify these candidates as planets if all nearby stars are not ruled out as transit sources prior to the analysis. To avoid outcomes like this, VESPA requires a separate calculation of the probability that the transit originates from the target star prior to its FPP calculation (e.g., Morton et al. 2016).

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It is also worth noting that the calculation procedures between the tools are not identical. An important difference is that TRICERATOPS takes into account the STP scenario, which involves a planet transiting an unresolved bound companion, whereas VESPA does not. This false-positive scenario typically has a nonnegligible probability of being the ground truth and therefore inflates the FPP obtained with TRICERATOPS relative to that of VESPA. To test how this impacts the FPP comparison, we calculate the TRICERATOPS FPP for each TOI both using (left panel of Figure 13) and omitting (right panel of Figure 13) the STP scenario. We see that when this scenario is included, there are several TOIs that score a validation-worthy FPP with VESPA that do not with TRICERATOPS. However, when this scenario is omitted from the calculation, the two tools return more consistent results. This suggests that TRICERATOPS is more conservative when validating TOIs and will oftentimes rely on supplementary follow-up observations to achieve FPP ≤ 0.01.

One might expect our code to have a more difficult time distinguishing CPs from FPs when using data with a longer cadence, as they would yield fewer points with which to characterize the shape of the transit. To test this, we also run our code on 30-minute cadence light curves of the same TOIs. We use eleanor (Feinstein et al. 2019) to extract these light curves from TESS Full Frame Images (FFIs) within the same sectors and apertures used to obtain the 2-minute cadence light curves.⁴¹

In Figure 14, we show how S/N affects the new FPP calculations. As in the previous section, TRICERATOPS is able to correctly identify CPs and FPs more frequently when S/N is high, but the correlation is weaker overall. Specifically, the FPPs of CPs are less concentrated near zero here than those calculated with the 2-minute data.

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In Figure 15, we reproduce the NFPP versus FPP analysis from the previous section using the 30-minute cadence data. We again see that most CPs are contained within a region defined by NFPP < 10⁻³ and FPP < 0.5, with very few FPs also falling within this region. Specifically, the region contains 18 CPs and only 2 FPs. In addition, almost no CPs have an FPP > 0.7 (with the exception of one, which is mistaken for a nearby false positive), which implies that a high FPP is still indicative of actual FPs. We thus again designate TOIs with NFPP < 10⁻³ and FPP < 0.5 as likely planets. However, unlike the results obtained with the 2-minute cadence data, there does not appear to be a region of parameter space in which planets can be confidently validated. Nonetheless, TRICERATOPS results involving long-cadence TESS data are useful for vetting TOIs and prioritizing them for follow-up observations to further investigate the nature of the signal.

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TFOP SG1 confirms that the true host of the signal is TIC 260043722. Previous HAT-South data suggested that this TOI is an NEB, which was confirmed by PEST Observatory R_C-band observations with a depth of 200 ppt. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, one of which is TIC 260043722. The total NFPP calculated by TRICERATOPS is 0.0063. TIC 260043722 has an NFPP of 0.0059, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.0806 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 279740439. The signal was a nearby planet candidate (signal not on the original TOI, but still possibly planetary) based on observations from the TRAPPIST telescope that show a depth of 40 ppt in a custom I + z-band filter. Later observations with LCOGT (Brown et al. 2013) showed a V-band depth of 30 ppt on the nearby candidate; the wavelength-dependent eclipse depth indicates that it is an eclipsing binary. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, one of which is TIC 279740439. The total NFPP calculated by TRICERATOPS is 0.5377. TIC 279740439 has an NFPP of 0.2041, making it the second most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.6095 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 250386182. The TOI is an NEB, based on LCOGT observations in the PanSTARRS zs filter showing a depth of roughly 350 ppt. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, one of which is TIC 250386182. The total NFPP calculated by TRICERATOPS is 0.0311. TIC 250386182 has an NFPP of 0.0311, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP. However, the calculated FPP of 0.9703 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 219388775. The TOI is an NEB with depth of 130 ppt, based on LCOGT zs observations. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies one nearby source other than the target star bright enough to host the signal, which is TIC 219388775. The total NFPP calculated by TRICERATOPS is 0.0101. TIC 219388775 has an NFPP of 0.0101, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.2882 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 176778114. The TOI is an NEB with primary and secondary eclipse depths of ∼430 ppt and ∼300 ppt, respectively, in LCOGT r' observations. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, one of which is TIC 176778114. The total NFPP calculated by TRICERATOPS is 0.0650. TIC 176778114 has an NFPP of 0.0438, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.3405 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 20178112. The TOI is an NEB, based on PEST Observatory R_C observations that show a ∼55 ppt eclipse on TIC 20178112, which Gaia shows as two stars with magnitudes G = 14.2 and G = 15.9. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies three nearby sources other than the target star bright enough to host the signal, one of which is TIC 20178112. The total NFPP calculated by TRICERATOPS is 0.2332. TIC 20178112 has an NFPP of 0.1638, making it the most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.4065 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 427352247. The TOI is an NEB, based on LCOGT r' observations that show a 200 ppt, V-shaped eclipse. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies three nearby sources other than the target star bright enough to host the signal, one of which is TIC 427352247. The total NFPP calculated by TRICERATOPS is 0.5219. TIC 427352247 has an NFPP of 0.0693, making it the second most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.6498 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 108645800. The TOI is an NEB, based on LCOGT r' observations with a depth of at least 100 ppt, and confirmed by archival HAT-South data. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies four nearby sources other than the target star bright enough to host the signal, one of which is TIC 108645800. The total NFPP calculated by TRICERATOPS is 0.6361, but the NFPP around TIC 108645800 was not calculated owing to unknown stellar parameters. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.8299 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 760244235. The TOI is an NEB with depth of at least 200 ppt, based on LCOGT r' observations. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies 10 nearby sources other than the target star bright enough to host the signal, one of which is TIC 760244235. The total NFPP calculated by TRICERATOPS is 0.0507. TIC 760244235 has an NFPP of 0.0030, making it the sixth most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.1215 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 431999916. The TOI is an NEB with depth of at least 90 ppt, based on LCOGT i' observations. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies eight nearby sources other than the target star bright enough to host the signal, one of which is TIC 431999916. The total NFPP calculated by TRICERATOPS is 0.9230. TIC 431999916 has an NFPP of 0.1482, making it the second most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.9819 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 438490748. The TOI is an NEB with depth of ∼80 ppt, based on K2 and HAT-South data. TIC 438490748 (the source of the signal) is a pair of stars in Gaia, so the true depth may be deeper. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, one of which is TIC 438490748. The total NFPP calculated by TRICERATOPS is 0.9938. TIC 438490748 has an NFPP of 0.9938, making it the most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 1.0 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 302895984. The TOI is an NEB with a depth of 200 ppt in the I band from LCOGT observations. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies five nearby sources other than the target star bright enough to host the signal, one of which is TIC 302895984. The total NFPP calculated by TRICERATOPS is 0.7971. TIC 302895984 has an NFPP of 0.0509, making it the third most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.9477 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 53593470. The TOI is an NEB with a depth of ∼250 ppt in both g' and i' in LCOGT observations. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies five nearby sources other than the target star bright enough to host the signal, one of which is TIC 53593470. The total NFPP calculated by TRICERATOPS is 0.6436, but the NFPP around TIC 53593470 was not calculated owing to unknown stellar parameters. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.9580 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 59003118. This is K2-78b (EPIC 210400751; Crossfield et al. 2016), which was later shown to be an NEB (Cabrera et al. 2017). This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, one of which is TIC 59003118. The total NFPP calculated by TRICERATOPS is 0.0258. TIC 59003118 has an NFPP of 0.0050, making it the second most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.2854 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 830310300. The TOI is an NEB, based on observations from LCOGT and Mt. Kent Observatory in i' and from El Sauce Observatory in R_C. The depth is at least 500 ppt in i'. In this case, the SPOC centroid offset analysis failed to identify the presence of a background source at the 3σ level of significance. TRICERATOPS identifies nine nearby sources other than the target star bright enough to host the signal, one of which is TIC 830310300. The total NFPP calculated by TRICERATOPS is 0.8854. TIC 830310300 has an NFPP of 0.0124, making it the sixth most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.9687 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 146463868. The TOI is an NEB, based on LCOGT I_C-band observations with a depth of 300 ppt. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies three nearby sources other than the target star bright enough to host the signal, one of which is TIC 146463868. The total NFPP calculated by TRICERATOPS is 0.8640, but the NFPP around TIC 146463868 was not calculated owing to unknown stellar parameters. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.9887 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 432008934. The TOI is an NEB, based on the centroid offset from the SPOC S01–S09 vetting report. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, but neither is TIC 432008934. The total NFPP calculated by TRICERATOPS is 1e − 5. This NFPP is too low to classify the TOI as a likely NFP. However, the calculated FPP of 0.9996 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 54085149. The TOI is an NEB, based on LCOGT i' observations that show a depth of 400 ppt at two different epochs. In this case, the SPOC centroid offset analysis found a significant offset, but the offset did not point directly to the true host. TRICERATOPS identifies two nearby sources other than the target star bright enough to host the signal, one of which is TIC 54085149. The total NFPP calculated by TRICERATOPS is 0.0013. TIC 54085149 has an NFPP of 0.0008, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.2747 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 147660207. This candidate was retired from SG1 as a nearby planet candidate. Observations show the true source of the signal to be a ∼4 ppt event in the nearby star TIC 147660207, which is still an active planet candidate as of this writing. The event was seen in R_C from El Sauce Observatory and in i' from Mt. Kent and Hazelwood Observatories. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies nine nearby sources other than the target star bright enough to host the signal, one of which is TIC 147660207. The total NFPP calculated by TRICERATOPS is 0.1652. TIC 147660207 has an NFPP of 0.0868, making it the most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.6543 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the TOI is an NFP. The TOI is an NEB, based on large scatter in the image centroid from sector to sector in a very crowded field and a possible secondary eclipse. From the SPOC S01–S09 report, this is a clear NEB ∼35'' west. Although the exact source of the NEB is not clear from the SPOC centroid offset analysis, it is likely too faint, and thus the event is too deep to be planetary. TRICERATOPS identifies 141 nearby sources other than the target star bright enough to host the signal. The total NFPP calculated by TRICERATOPS is 0.1760. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.5955 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 373424060. The TOI is an NEB, based on LCOGT observations that show a depth of ∼200 ppt in the zs filter. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies 31 nearby sources other than the target star bright enough to host the signal, one of which is TIC 373424060. The total NFPP calculated by TRICERATOPS is 0.2268. TIC 373424060 has an NFPP of 0.0006, making it the 23rd most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.4377 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 271596416. The TOI is an NEB, based on LCOGT observations that show a depth of 70–100 ppt in zs and ∼30 ppt in i'. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies seven nearby sources other than the target star bright enough to host the signal, one of which is TIC 271596416. The total NFPP calculated by TRICERATOPS is 0.0259. TIC 271596416 has an NFPP of 0.0078, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.6551 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 1310226289. The TOI is an NEB, based on LCOGT observations that show a depth of at least 75 ppt in zs. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies four nearby sources other than the target star bright enough to host the signal, one of which is TIC 1310226289. The total NFPP calculated by TRICERATOPS is 0.0131. TIC 1310226289 has an NFPP of 0.0068, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.0645 is too high to classify the TOI as a validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 253985122. The TOI is an NEB, based on PEST Observatory R_C-band observations with a depth of ∼600 ppt. In this case, the SPOC centroid offset analysis failed to identify the presence of a background source at the 3σ level of significance. TRICERATOPS identifies nine nearby sources other than the target star bright enough to host the signal, one of which is TIC 253985122. The total NFPP calculated by TRICERATOPS is 0.0315. TIC 253985122 has an NFPP of 0.0037, making it the fourth most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.5030 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the TOI is an NEB. Stellar parameters from Gaia and TIC indicate R_* > 40R_⊙, but the orbital period of <1 day would place the companion's orbit inside the star if it were on target. The SPOC centroid offset suggests that the signal originates from a star to the south. TRICERATOPS identifies 108 nearby sources other than the target star bright enough to host the signal. The total NFPP calculated by TRICERATOPS is 0.0383. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.7727 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 274762865. The TOI is an NEB, based on archival MEarth-North (Nutzman & Charbonneau 2008; Irwin et al. 2015) observations that show no event on target and eclipses at the TESS ephemeris in a neighboring star. SPOC difference image analysis correctly identified this star as the true host. TRICERATOPS identifies six nearby sources other than the target star bright enough to host the signal, one of which is TIC 274762865. The total NFPP calculated by TRICERATOPS is 0.9869, but the NFPP around TIC 274762865 was not calculated owing to unknown stellar parameters. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.9981 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 267561450. The TOI is an NEB, based on observations by the University of Louisville Manner Telescope and MuSCAT2 at Teide Observatory in g', r', i', and z' that show a ∼200 ppt eclipse. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies 13 nearby sources other than the target star bright enough to host the signal, one of which is TIC 267561450. The total NFPP calculated by TRICERATOPS is 0.2818. TIC 267561450 has an NFPP of 0.0235, making it the fourth most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.7151 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 274662220. The TOI is an NEB, based on observations at the University of Louisville Manner Telescope that show a depth of 150 ppt in r'. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies 21 nearby sources other than the target star bright enough to host the signal, one of which is TIC 274662220. The total NFPP calculated by TRICERATOPS is 0.3031. TIC 274662220 has an NFPP of 0.0501, making it the second most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.6880 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 408203452. The TOI is an NEB, based on observations in a long-pass GG495 filter at the Steward Observatory Phillips 0.6 m Telescope on Mount Lemmon that show a 35 ppt eclipse. Observations at the University of Louisville Moore Observatory show a depth of 60 ppt in r'. This star was also correctly identified as the host of the signal by the SPOC centroid offset analysis. TRICERATOPS identifies 10 nearby sources other than the target star bright enough to host the signal, one of which is TIC 408203452. The total NFPP calculated by TRICERATOPS is 0.3258. TIC 408203452 has an NFPP of 0.1435, making it the most probable NFP host. This NFPP is high enough to classify the TOI as a likely NFP. In addition, the calculated FPP of 0.8512 is too high to classify the TOI as a likely planet or validated planet.

TFOP SG1 confirms that the true host of the signal is TIC 233681148. The TOI is an NEB, based on SPOC S14–S16 reports that show a centroid offset to the closest star SW. Single-pixel photometry on the TESS FFIs supports this conclusion. TRICERATOPS identifies one nearby source other than the target star bright enough to host the signal, which is TIC 233681148. The total NFPP calculated by TRICERATOPS is 0.0309. TIC 233681148 has an NFPP of 0.0309, making it the most probable NFP host. This NFPP is too low to classify the TOI as a likely NFP and too high to classify the TOI as a likely planet. In addition, the calculated FPP of 0.0947 is too high to classify the TOI as a validated planet.