An Estimate of the Yield of Single-transit Planetary Events from the Transiting Exoplanet Survey Satellite

The Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015), launched in the spring of 2018, will discover thousands of transiting exoplanets that exhibit two or more transits during the mission. TESS will have a number of advantages over previous transiting planet surveys, including ground-based surveys such as the Hungarian Automated Telescope (Bakos et al. 2004) survey, the Wide Angle Search for Planets (Pollacco et al. 2006) survey, and the Kilodegree Extremely Little Telescope (Siverd et al. 2012) survey, as well as space-based missions like Corot (Baglin 2003), Kepler (Borucki et al. 2010), and K2 (Howell et al. 2014). Ground-based surveys are essentially limited to planets whose transits have depths above ∼0.1%, but do so around bright stars that are amenable to follow-up observations from ground-based telescopes and radial velocity measurements. Kepler has exquisite photometric precision down to several tens of parts-per-million (ppm) (Koch et al. 2010), but the majority of the planets found by Kepler are orbiting stars that are too faint to be confirmed via radial velocity using the current generation of telescopes and instruments. Both the original Kepler campaign and the extended K2 mission campaigns have relatively long baselines of almost 4 yr and 80 days respectively, but both are also limited in their sky coverage.

By virtue of TESS's observing strategy and design, it will observe 85% of the entire sky, monitoring and discovering planets transiting bright stars, which are amenable to both photometric and radial velocity follow-up, as well as detailed characterization of their atmospheres via ground- and space-based telescopes. The tradeoff of achieving this nearly all-sky coverage is that 63% of the sky, or 74% of the mission's total sky coverage, will only be observed for 27 days (as compared to 80 days for each K2 campaign and nearly 4 yr for the primary Kepler campaign). In this regime of many millions of stars monitored for a relatively short amount of time, the number of single-transit planetary events found by TESS will be much larger than that expected or found by Kepler (Yee & Gaudi 2008; Wang et al. 2015; Foreman-Mackey et al. 2016; Osborn et al. 2016; LaCourse & Jacobs 2018). Single-transit events require significantly more resources to confirm than planets that exhibit two or more transits, but nevertheless they can be quite scientifically valuable.

With the planned survey strategy, and a requirement of at least two transits to confirm a planet, the majority of these planets will have periods of less than 10 days. The primary mission of the survey is to measure masses and radii of 50 terrestrial planets. This leaves open the opportunity to discover planets outside of this regime, including giant planets and planets on long orbits that transit only once. However, recovering, confirming and studying planets that transit only once pose difficulties. Some of these difficulties include the fact that their ephemerides are difficult to constrain for the purpose of scheduling future observations, and they are easily confused with false positives.

Previous studies have investigated single-transit events in Kepler (Yee & Gaudi 2008). Multiple simulations have been performed to estimate the (two or more transit) yield of TESS (Sullivan et al. 2015; Bouma et al. 2017; Ballard 2018; Barclay et al. 2018). In each of these simulations, only systems that exhibit two or more transits are reported, and to date there has not been an estimate of the expected yield of single transits in TESS, or their properties. Given the number of stars and observing strategy of TESS, we expect a 100-fold increase in the number of single-transit events in TESS relative to Kepler.

It is both worthwhile and possible to follow up these longer-period transiting planets, as they represent an opportunity to investigate a number of questions related to planet formation, such as the migration mechanism for hot Jupiters and the physical mechanisms that lead to inflated radii of close-in giant planets. Using the definition of habitable zones described by Kopparapu et al. (2013), transiting planets of main-sequence stars of spectral type earlier than roughly M5 (${T}_{\mathrm{eff}}\approx 2800$ K) will have the inner edge of the habitable zone at periods of ≈11 days. For the majority of the TESS survey, any habitable zone planets around M4 or earlier stars are expected to only display single transits.

The expected total number of planets detectable by TESS with exactly one or more transits is the integral over all periods and all planetary radii of the geometric probability of observing a transit around a star ℘_tr, the probability of observing the transit(s) during the finite baseline of observations ${\wp }_{B}$, and the planet occurrence rates f(P) with a Heaviside step function cut on the signal-to-noise ratio ${\rm{\Theta }}({\rm{\Delta }}{\rm{S}}/{\rm{N}})$, multiplied by the total number of stars observed by TESS N_⋆:

$$\begin{eqnarray}&&{N}_{\det }={N}_{\star }\int {\wp }_{\mathrm{tr}}{\wp }_{B}f(P){\rm{\Theta }}({\rm{\Delta }}{\rm{S}}/{\rm{N}}){dP},\end{eqnarray} \tag{ 1 }$$

where ${\rm{\Delta }}{\rm{S}}/{\rm{N}}={\rm{S}}/{\rm{N}}-{\rm{S}}/{{\rm{N}}}_{\min }$, where each term is for a fixed period and planetary radius. In reality, each of the terms [N_⋆, ℘_tr, ${\wp }_{B}$, f(P), ${\rm{\Theta }}({\rm{\Delta }}{\rm{S}}/{\rm{N}})]$ depend on more than just the period P and planet radius, but also depend on other variables such as the stellar mass, stellar radius, apparent stellar magnitude, and intrinsic stellar variability. All of these variables are considered in the final analysis.

We will also assume circular orbits and that ${r}_{p}\ll {R}_{\star }$. With this assumption, the geometric transit probability is then

$$\begin{eqnarray}&&{\wp }_{\mathrm{tr}}=\displaystyle \frac{{R}_{\star }}{a}\end{eqnarray} \tag{ 2 }$$

for non-grazing geometries, where ${R}_{\star }$ is the host star radius. It is possible to evaluate Equation (1) in units of the semimajor axis a, but it is more convenient to use the period P as this is the direct observable in both transit and radial velocity detections of exoplanets. We use Kepler's third law assuming that the planet's mass is much smaller than the stellar mass to convert semimajor axis to period

$$\begin{eqnarray}&&{P}^{2}={a}^{3}\displaystyle \frac{4{\pi }^{2}}{G({M}_{\star }+{m}_{p})}\approx {a}^{3}\displaystyle \frac{4{\pi }^{2}}{{{GM}}_{\star }},\end{eqnarray} \tag{ 3 }$$

where ${M}_{\star }$ is the stellar mass. The geometric transit probability then becomes a function of stellar mass, stellar radius, and orbital period:

$$\begin{eqnarray}&&{\wp }_{\mathrm{tr}}={\left(\displaystyle \frac{4{\pi }^{2}}{G}\right)}^{1/3}{R}_{\star }{M}_{\star }^{-1/3}{P}^{-2/3}.\end{eqnarray} \tag{ 4 }$$

The geometric probability decreases as ${\wp }_{\mathrm{tr}}\propto {P}^{-2/3}$ and leads to a decreased probability of observing a transit at long periods.

We also consider the probability of a transit occurring during the finite baseline of observation B of the TESS mission ${\wp }_{B}$. During this paper, we will evaluate cases where two or more transits occur, or exactly one transit occurs. In the case where two or more transits are observed with planets on periods shorter than $B/2$, the probability of observing two or more transits occurring during the observing baseline is unity. However, for planets on periods longer than $B/2$, the probability decreases until only one or no transits occur during the observing baseline. For a finite observing baseline B and ignoring the finite duration of the transits, the probability of exactly one ${\wp }_{B,1}$ or two or more ${\wp }_{B,2+}$ transits occurring is given by

$$\begin{eqnarray}\begin{array}{rcl}{\wp }_{B,1} & = & \displaystyle \frac{B}{P},P\geqslant B\\ & = & \displaystyle \frac{2P}{B}-1,\displaystyle \frac{B}{2}\leqslant P\leqslant B\\ {\wp }_{B,2+} & = & 2-\displaystyle \frac{2P}{B},\displaystyle \frac{B}{2}\leqslant P\leqslant B\\ & = & 1,P\leqslant \displaystyle \frac{B}{2}.\end{array}\end{eqnarray} \tag{ 5 }$$

These are analogous to Equations (3) and (4) from Yee & Gaudi (2008), but we include the case of two or more transits, instead of exactly two transits. Yee & Gaudi (2008) incorrectly chose the lower limit for the two-transit case to be $P/4$, instead of $P/3$; however, this only biases their yields for the two-transit cases and their single-transit yields should be unaffected. When investigating single-transit events with periods longer than the observing baseline, the total probability is

$$\begin{eqnarray}\begin{array}{rcl}{\wp }_{\mathrm{tot}} & \equiv & {\wp }_{\mathrm{tr}}{\wp }_{B}\\ & = & 0.026\left(\displaystyle \frac{{R}_{\star }}{{R}_{\odot }}\right){\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{-1/3}{\left(\displaystyle \frac{P}{27.4\mathrm{days}}\right)}^{-5/3}\left(\displaystyle \frac{B}{27.4\,\mathrm{days}}\right)\end{array}\end{eqnarray} \tag{ 6 }$$

and scales as ${\wp }_{\mathrm{tr}}{\wp }_{B}\propto {P}^{-5/3}$.

The product of these two terms can be seen in Figure 1, for representative host stars and a observing baseline of B = 27.4 days. To observe two transits, the probability is the geometric transit probability ${R}_{\star }/a$ until periods of $B/2$, while the probability of observing a single transit peaks at B, with the transition for observing one versus two transits occurring between $\tfrac{B}{2}\leqslant P\leqslant B$.

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For simplicity, we assume each star observed by TESS will have an observing baseline of an integer multiple of $N\,\times 27.4$ days where $1\leqslant N\leqslant 13$. This ignores the 16 hr downtime during each baseline, which results in a ∼2.4% reduction in observing time; the observing gaps between sectors, which may evolve during the mission; systematics that may increase near the beginning and ending of observations; and partial single transits. Our estimates should still be accurate at the order of magnitude level, but may be overestimated by ∼2.5%–5%. The amount of sky covered in each observing baseline is summarized in Table 1. The dominant baselines are 73.8% of the mission covered for 27.4 days, 17.8% for 54.8 days, and 3.5% for 82.2 days. There is an uptick at the ecliptic poles, which cover 2% of the mission for 356 days. Each remaining observing baseline covers less than 1% of the mission.

For the total number of stars observed by TESS, we use the TESS Candidate Target List-6 (CTL-6) provided online by Stassun et al. (2018). The catalog has ∼4 million sources, with estimated host star masses, radii, and TESS magnitudes, the host star's coordinates, and an estimate of the blended flux from nearby stars that is expected to dilute the depth of transits. The CTL also includes evolved stars, and although the CLT has been vetted to avoid them there are some that vetting may have missed in our sample. This method deviates from those used in simulations by Sullivan et al. (2015) and Bouma et al. (2017) as we do not use a Galactic model, but instead calculate our yields directly from the CTL-6. For each of the 4 million stars in the CTL-6, we use the estimated mass, radius, effective temperature, magnitude, and ecliptic latitude of the star. As the the longitude of the first sector was not yet known, we use the ecliptic latitude to assign the number of sectors in which the star will be observed by TESS and determine the total observing baseline. We do this by taking all stars above a given ecliptic latitude, such that the total area on sky is the same as described in Table 1. All observed stars are sorted by the CTL-6 priority, where the top 200,000 stars are classified as postage stamp (PS) stars, and the remaining stars are placed in the full frame image (FFI) sample. Barclay et al. (2018) have shown that using only the CTL priority may result in an overselection of target stars in the ecliptic poles relative to the true mission; however, we remain agnostic as to the final selection strategy of the targeted stars and default to CTL-6 priority over speculation as to what the final selected stars will be.

We use the planet occurrence rates of Fressin et al. (2013) for stars with ${T}_{\mathrm{eff}}\geqslant 4000\,{\rm{K}}$, and of Dressing & Charbonneau (2015) for stars with ${T}_{\mathrm{eff}}\lt 4000\,{\rm{K}}$. The planet occurrence rates are quoted in bins of planet radius and period. Rates are only complete to periods of ∼100 days, but we extrapolate these rates to periods of $\gt 1000$ days to explore the probability of finding planets at longer periods. The assumed total planet occurrence rates across all radii can be seen in Figure 1. We use a solid line to represent periods that are complete for all radius bins. The planet occurrence rates are not uniformly complete to the same maximum period, so we use dashed lines where a combination of measured occurrence rates, where available, and extrapolated rates are used. The extrapolated rate per radius bin is assumed to be constant in $\mathrm{log}P$ set equal to the longest-period bin for which an occurrence rate is available. For each period and radius bin of the respective occurrence rates, we draw a radius and period from a random uniform logarithmic distribution in that bin. For our analytic estimate we ignore the uncertainties on the planet occurrence rates, as they should not effect final yield at the order of magnitude level, although caution should be used when drawing conclusions from planet demographics with small number statistics. From the period of the planet, host star mass, and host star radius we calculate the geometric probability using Equation (4). From the ecliptic latitude of the star and the period of the planet, we can calculate the probability of the object being observed for one, two, or more transits from Equation (5). We note that, because the probability of observing a single transit drops precipitously with period ($\propto {P}^{-5/3}$), changing our assumed form for the extrapolation in period (within reason) is unlikely to change our results substantially.

We only refer to planets as detectable if the S/N is above 7.3. To calculate the S/N, we follow the formula used in Bouma et al. (2017):

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}=\sqrt{{N}_{\mathrm{tr}}}\displaystyle \frac{\delta D}{{\left(\tfrac{{\sigma }_{1\mathrm{hr}}^{2}}{T}+{\sigma }_{v}^{2}\right)}^{1/2}},\end{eqnarray} \tag{ 7 }$$

using the number of transits ${N}_{\mathrm{tr}}=1$ for single transits, the transit depth δ; the dilution from background stars and contamination D bounded from 0 to 1; the total noise per hour ${\sigma }_{1\mathrm{hr}}^{2}$ from CCD read noise, photon-counting noise, and zodiacal noise; a systematic 60 ppm hr${}^{1/2}$ instrumental noise floor; the transit duration T; and an intrinsic variability term ${\sigma }_{v}$.

We use the host star's T_eff to assign an intrinsic stellar variability based on Basri et al. (2013), following the procedures of Sullivan et al. (2015) and Bouma et al. (2017). Using the planet's period and radius, along with the host star's variability and magnitude, we calculate the S/N of each planet, in each period and radius bin, around every star in the sample. For those that have an S/N > 7.3, we define the planet as being detectable. We then sum the product of the geometric probability, probability of being observed, and planet occurrence rate, over all detectable planets around all stars. The results are shown in Figure 1. The total number of single-transit events is 1218, with 241 of the detectable planets being found in the PSs; 201/241 have periods >25 days, and 19/241 have periods >250 days. Finally, we recover an estimate of the total integrated number of planets detectable. As we integrate fractional probability over all stars, we only recover the total number of planets detectable, and not the total number of host stars. As such, we cannot make any quantitative statements on the expected multiplicity of the systems, but one could assume that each star hosts only one planet.

2.1. Demographics of Detectable Single Transits

We present the demographics of the detectable planets. In Figure 2, we show the distribution of detectable planets in host star magnitude, host star effective temperature, planet radius, and stellar insulation relative to the Earth. Of the 1218 expected single transits, 173 are around stars brighter than T = 10 with 74 around PS stars and 99 in the FFIs. In the PSs, the detectable planets are split equally, 118/124, among cool (T_eff < 4000 K) and warm (T_eff ≥ 4000 K) stars, but the FFIs favor the warm stars with a 93/883 split between the cool/warm stars. We also find that 196 sub-Neptunes with ${r}_{p}\lt 4\,{R}_{\oplus }$ will be detectable as single transits in the PSs, with an additional 230 detectable in the FFIs of CTL-6 stars. All of the planets detectable around cool stars have ${r}_{p}\lt 4\,{R}_{\oplus }$ as there are no planets above $4\,{R}_{\oplus }$ in the Dressing & Charbonneau (2015) occurrence rates. Clanton & Gaudi (2016) show that giant planets do indeed exist around cool stars at long periods, but are uncommon relative to small planet occurrence rates.

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Fifty-six planets will be detectable in the PSs with a stellar insulation within a factor of two of the Earth ($0.5\leqslant S/{S}_{\odot }\leqslant 2$) and another 73 in the FFIs. Using the conservative habitable zone defined by Kopparapu et al. (2013), we expect 34 habitable zone planets from the PSs, 29 are around cool stars and another 45 habitable zone planets from the FFIs with 32 coming from cool stars. If we limit the habitable zone planets to terrestrial planets ($R\leqslant 1.5\,{R}_{\oplus }$), then we expect only one planet to be detectable, which happens around a PS star with T_eff ≤ 4000 K, although we urge caution on the robustness of an expected yield of one. In a recent simulation of the TESS yield from Barclay et al. (2018), there were no planets found beyond ≈85 days that resulted in no planets being found in the habitable zone of FGK stars, where we find 5 in the PSs and 13 in the FFIs. It is worth noting that we differ from Barclay et al. (2018) not only in the number of habitable zone planets around stars earlier than M, but also in our target star selection criteria, extrapolation of planet occurrence rates, and in the definition of habitable zone.

We adopt the same S/N threshold S/N = 7.3 as used by both Sullivan et al. (2015) and Bouma et al. (2017) for multiple-transit events. Given the added uncertainty of single-transit events (e.g., false positives) we follow Barclay et al. (2018) and also look at the distribution of S/N for all of the detectable planets in Figure 3 for those wishing to adopt a more stringent S/N cut. We find that of the 241 PS detectable planets with S/N ≥ 7.3, 162 have S/N ≥ 10, and 14 have robust detectable planets at S/N ≥ 100. Among the 977 detectable planets around FFI stars with S/N ≥ 7.3, 695 have S/N ≥ 10, and 90 have S/N ≥ 100.

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An important aspect of identifying the single-transit candidates is the ability to predict the time of future transits to confirm their ephemerides and schedule future observations. One can relate the observed properties of the light curve, the velocity of the planet assuming a circular orbit and zero impact parameter (b = 0), and Kepler's third law (Equation (3)) to relate the period, stellar density, and the light-curve observable quantities (Seager & Mallén-Ornelas 2003; Yee & Gaudi 2008; Winn 2010). Beginning with the velocity of the planet,

$$\begin{eqnarray}&&{v}_{p}=\displaystyle \frac{2\pi a}{P}=\displaystyle \frac{2{R}_{\star }}{{T}_{0}},\end{eqnarray} \tag{ 8 }$$

where T₀ is the duration of the transit at b = 0. One can relate T₀ to the observable quantities T and τ, the measured duration of the transit, and ingress/egress time, with

$$\begin{eqnarray}&&{T}_{}={T}_{0}\sqrt{1-{b}^{2}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\tau =\displaystyle \frac{\sqrt{\delta }{T}_{0}}{\sqrt{1-{b}^{2}}},\end{eqnarray} \tag{ 10 }$$

where $\sqrt{\delta }$ is the transit depth in non-limb-darkened stars, and must be corrected for limb darkening to arrive at

$$\begin{eqnarray}&&P=\displaystyle \frac{G{\pi }^{2}}{3}{\rho }_{* }{\left(\displaystyle \frac{{T}_{}\tau }{\sqrt{\delta }}\right)}^{3/2}.\end{eqnarray} \tag{ 11 }$$

This leads to a degeneracy between the period and the host star density. Seager & Mallén-Ornelas (2003) showed that it is possible to estimate the period when the mass and radius of the host star are known. Yee & Gaudi (2008) show that the fractional uncertainty in the period $\left({\sigma }_{P}/P\right)$ will come from the fractional uncertainty in the density $\left({\sigma }_{\rho }/\rho \right)$ and the fractional uncertainty on the period due to the TESS photometry ${({\sigma }_{P}/P)}_{\mathrm{TESS}}$ added in quadrature:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{P}}{P}\right)}^{2}={\left(\displaystyle \frac{{\sigma }_{\rho }}{\rho }\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{P}}{P}\right)}_{\mathrm{TESS}}^{2}.\end{eqnarray} \tag{ 12 }$$

3.1. Uncertainty on the Period Due to the Stellar Density

3.2. Uncertainty on the Period Due to the Photometry

The fractional uncertainty in the period due to the photometry ${({\sigma }_{P}/P)}_{\mathrm{TESS}}$ is dominated by the ability to measure the ingress/egress time τ and the total S/N of the single transit. From Equation (9) in Yee & Gaudi (2008), we get that the fractional uncertainty in the period due to the photometric precision is

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\sigma }_{P}}{P}\right)}_{\mathrm{TESS}}^{2}\approx \displaystyle \frac{9}{4}{\left(\displaystyle \frac{{\sigma }_{\tau }}{\tau }\right)}^{2}\approx \displaystyle \frac{1}{{Q}^{2}}\left(\displaystyle \frac{27{T}_{}}{2\tau }\right),\end{eqnarray} \tag{ 13 }$$

and can be related to Q, the approximate total S/N of the transit, and the ratio of the transit duration T to the ingress/egress time τ, assuming $\tau \ll {T}_{}.$ A detailed investigation into the details of the uncertainties in the observables can be found in Carter et al. (2008) and the Appendix of Yee & Gaudi (2008).

In Figure 4, we show the fractional uncertainty expected from single transits based on their photometry. 146/1218 planets will have a fractional uncertainty on the period to better than 10%, with 16 coming from the PSs and 130 of those coming from the FFIs. It is worth noting that even in the event of a 1% constraint on the period from photometry, the uncertainty on the density will likely dominate and limit the constraint on the inferred period. The planets with the tightest constraints on the periods from the photometry are typically giant planets around warm stars in the FFIs, which have typically higher S/N than the smaller planets around cool stars. This can be seen by comparing the high S/N planets in Figure 3 to the planets in Figure 4 with the tightest precision. Another 72 planets will have a fractional uncertainty on the period of 10%–15%, with 5 around PS stars and 72 coming from the FFIs. For cases where the ingress/egress time is shorter than the exposure time, we can only place an upper limit on the ingress/egress time, and therefore lose the ability to constrain the period. This happens with 373 planets, all identified in the 30 minute cadence FFIs. The remaining 627 planets all have fractional uncertainties of greater than 15%; where the approximation in Equation (13) breaks down, we lose the ability to place a constraint on the period, and follow-up becomes difficult.

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3.3. Uncertainty on the Period Due to Eccentricity

Up until now, we have assumed circular orbits for all of the estimates. In reality, a number of these planets will likely have noncircular orbits. This will change the duration of the transit and will lead to an incorrect estimation of the period of the planet under the assumption of a circular orbit. Yee & Gaudi (2008) show that the maximum and minimum deviation from the true period under this assumption is given as

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{\rm{\Delta }}P}{P}\right)}_{\min /\max }={\left(\displaystyle \frac{1+e}{1-e}\right)}^{\pm 3/2}.\end{eqnarray} \tag{ 14 }$$

Barnes (2007) has shown that for planets with eccentric orbits, their transit probability increases by a factor of $1/(1-{e}^{2})$ for a given eccentricity. This is the result of two competing effects. Planets on eccentric orbits spend more time during their orbit at longer separations, but pass closer to their host stars at periastron, increasing the geometric odds of creating a transit. This effect is strongest at periastron, and leads to planets transiting closer to their host stars. This also corresponds to an increase in the velocity of the planet, and a decrease in transit duration on average. If you assume that same transit was caused by a planet on a circular orbit, you would systematically underestimate the orbital period.

We show these combined effects in Figure 5 where the minimum and maximum deviation from the true period under the assumption of a circular orbit of a planet for a given eccentricity are shown as the upper and lower dashed lines, corresponding to the maximum and minimum deviation for a planet observed at periastron and apastron, respectively. We also calculate the transit probability weighted deviation (the black line) and the mean deviation from a random draw of planets on elliptical orbits (black data points). We find that the mean underestimation is small, of order ${(1-{e}^{2})}^{-1/4}$, compared to the maximum or minimum deviation possible.

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Assuming the median eccentricity of e = 0.17 from the general Beta distribution in Kipping (2013) to describe the global population of RV planets, we get a maximum and minimum range of true periods from 0.59 to 1.69 times that of the assumed circular period, or a ≈50% uncertainty in the period for planets at e = 0.17. This would imply that for noncircular systems, the $e\ne 0$ uncertainties will limit over our ability to place a constraint using both the density and the photometry. However, many of these systems will be in multiple-planet systems (Ballard 2018). Zhu et al. (2018) showed that Kepler planet systems become dynamically cooler as the number of planets in the system increases. This follows the results from Xie et al. (2016) that found the mean eccentricity of Kepler multi-planet systems of e = 0.04 to be much lower than that of the single-planet systems, e = 0.3. For an eccentricity of e = 0.04, the range of possible periods relative to circular is only 0.89–0.12, which is in line with the expected uncertainty from the density and photometry. It is also worth noting that although eccentric planets will be more difficult to place constraints on their periods, they have an increased transit probability, and would thus increase the yield of single-transit systems detectable by TESS.

4.1. Recovery with Additional Photometry or Precovery in Archival Data

After estimating the timing of a future transit, we also need to consider which of the single-transit candidates will be observable from a typical ground-based facility. An additional resource is to look for signals present in existing data sets given a known depth and approximate period. We present the distribution of the undiluted transit depths of single transits in Figure 6. 197/241 planets will be detectable at $\delta \geqslant 0.1 \%$ in the PSs, with 894/976 of planets detectable around stars in the FFIs. Of these, 40 planets around PS stars will have deep $\delta \geqslant 1 \%$ transit depths and 253 planets around stars in the FFIs.

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4.2. Expected Radial Velocity Signal

To estimate the expected radial velocity semiamplitude K, we first assign a mass to each planet based on its radius. For planets with radii $\lt 4.0\,{R}_{\oplus }$ we use the planetary mass–radius relations from Weiss & Marcy (2014) and for planets with radii $\geqslant 4.0\,{R}_{\oplus }$ we use the planetary mass–radius relations from Mordasini et al. (2012). After assigning a planetary mass m_p, we use the following equation to assign the radial velocity semiamplitude K:

$$\begin{eqnarray}&&K=\displaystyle \frac{{m}_{p}}{{({m}_{p}+{M}_{\star })}^{2/3}}{\left(\displaystyle \frac{P}{2\pi G}\right)}^{-1/3}.\end{eqnarray} \tag{ 15 }$$

The distribution of expected RV signals from the single transits is shown in Figure 7. The majority of single-transit planets, 1195/1218, will have RV semiamplitudes detectable by modern RV instruments $K\geqslant 1$ m s⁻¹. 556/1218 will have $K\geqslant 10$ m s⁻¹, and 20 will have $K\geqslant 100$ m s⁻¹.

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As no one has published yields from single transits from TESS simulations, we present the expected yield of the TESS mission proper (i.e., planets detectable with two or more transits) for our analysis as a way to compare and scale our results to previous studies. We perform the same analysis described in Section 2 to provide an updated estimate of the yield of the primary TESS mission while only considering the number of planets detectable with two or more transits at an S/N ≥ 7.3. We find 2114 planets detectable in the PSs and another 5130 in the FFIs around stars in the CTL-6. These can be seen in Figure 8. Again, these numbers are incomplete fainter than $T\gt 12$ and are lower limits for the FFIs. 255/2114 have periods $\gt 25$ days, and only 2/2114 have periods $\gt 250$ days in the PSs, while and 211/5130 have periods $\gt 25$ days and $\lt 1/5130$ have periods $\gt 250$ days in the FFIs. We also show the yield from single transits as the dashed lines in Figure 8 for the PSs (black) and FFIs (gray). We find that within PSs the single transits match the TESS mission yield beyond ≈25 days and dominate relative to the expected yield in the FFIs. Beyond ≈250 days nearly all detectable planets come from the single transits in both the PSs and FFIs.

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We find that our predicted yield from the PSs of 2114 represents a 20%–25% increase over the yield of both the Sullivan et al. (2015; 1734) and Bouma et al. (2017; 1670) estimates using a galactic model and a 70% increase over the more realistic Barclay et al. (2018; 1250) simulation. The Sullivan et al. (2015) yield predicts >20,000 planets detectable in the FFIs, where we only expect 5130 detectable planets in the FFIs, noting that we are incomplete and this is only a lower limit. Bouma et al. (2017) provide a lower limit of 3342 for the FFIs, which is consistent with our estimate. Barclay et al. (2018) found 1250 planets in the PSs, with another 3200 planets in the FFIs using the CTL-6 and TIC-6, with another 10,000 planets around faint stars not included in the CTL. Although our input catalog is most similar to the Barclay et al. (2018) simulations than the others, Barclay et al. (2018) included a much more robust selection of the target stars, distribution of target stars per sector (i.e., fewer stars in the polar fields), and did not extrapolate the planet occurrence rates to longer periods. We find that in general we overestimate the number of planets in the PSs relative to other simulations, but are more consistent for the FFIs where the differences in target star selection are less important. It is worth noting that we have extrapolated our planet occurrence rates to much longer periods than all of the above simulations, and we have a more simplistic target star selection criteria, which is the likely source of the increased yields. Given that each group has used a different approach to input catalog, target start selection, and planet occurrence rate extrapolation, it is no surprise that the community disagrees at the factor of 2 level.

We were able to obtain rough numbers for the estimated number of single-transit events from individual trials of other simulations via (P. Sullivan 2016, private communication; L. Bouman 2017, private communication). Again, we find 977 single transits in the FFIs and 241 in the PSs in our work. From one trial from the Sullivan et al. (2015) simulation, we estimate $\approx 1300$ single transits in the FFIs and ≈100 single transits in the PSs (P. Sullivan 2018, private communication). From one trial from the Bouma et al. (2017) simulation, we estimate ≈850 single transits in the FFIs and ≈150 single transits in the PSs (L. Bouman 2018, private communication). Although unpublished, we find that the total number of single-transit events from each simulation, including our work, varies from 1000 to 1400 with each group disagreeing on the relative fraction found in the PSs versus the FFIs.

To understand the uncertainties in our analytic method, we additionally perform a random draw of single-transit events with discrete planets instead of an integrated probability of observing a planet. Our methodology follows the same as in Section 2, with a few notable differences.

We use the CTL-6 in the same manner as in Section 2 to determine, for each star with a TICID, if the star will be observed and the observing baseline B, i.e., how long the star will be observed by TESS. We also use the same method in Section 2 to determine which stars are observed as target stars and those that will fall in the FFIs.

Instead of using the probability of a star hosting a planet in a given period and radius bin, we draw from a Poisson distribution including the uncertainty in the bin itself from Fressin et al. (2013) and Dressing & Charbonneau (2015). For each planet drawn in a system, we assign a random period uniform in $\mathrm{log}P$ and a random radius uniform in $\mathrm{log}{R}_{p}$ from within their respective bins. We make no assumptions about the stability of theses systems and use the periods and radii drawn. We maintain the assumption that all planets are on circular orbits, and we assume all planets planets are coplanar by assigning a random inclination uniform in $\cos i$. Planets are detectable when their impact parameter $b\leqslant 1$, where the impact parameter for e = 0 is defined in Winn (2010):

$$\begin{eqnarray}&&b=\displaystyle \frac{a}{{R}_{\star }}\cos i.\end{eqnarray} \tag{ 16 }$$

The semimajor axis a and planetary radius R_⋆ are taken from Equation (3) and the CTL-6.

To determine the number of transits N_tr that are observed for a planet, we divide the period by the observing baseline to obtain the minimum number of transits observed. We then select a random phase to determine if an additional transit is observed.

We assign stellar noise, calculate the transit duration T, and the signal to noise S/N the same as in Section 2. Again, we use the same S/N ≥ 7.3 threshold for planets to be detectable. For detectable planets we also use the same methodology used above to assign planet mass, stellar insulation, habitability, and RV semiamplitude.

We repeat this exercise 100 times to create 100 trials of single-transit events.

6.1. Results of the Random Draw of Single-transit Events

We find ${1115}_{-31}^{+39}$ detectable single-transit events from our random draws, where we quote the median value from all 100 trials along with the 16th and 84th percentiles as the uncertainties. We find that ${224}_{-13}^{+18}$ and ${891}_{-28}^{+34}$ detectable single transits will come from the target stars and FFIs, respectively. These numbers are lower, but are in rough agreement with our analytic yields of 1218, 241, and 977, respectively. In general, we find that the results and demographics from the random draw are comparable to our analytic yields, although they are systematically lower and are shown in Table 2.

Table 2.  Comparison of 100 Random Draws of Single Transits to Analytic Estimates

Parameter	Analytic Total	Analytic PS	Analytic FFI	Random Draw Total	Random Draw PS	Random Draw FFI
Total	1218	241	977	${1115}_{-31}^{+39}$	${224}_{-13}^{+18}$	${891}_{-28}^{+34}$
$P\gt 25$ days	⋯	201	⋯	${966}_{-29}^{+36}$	${186}_{-12}^{+17}$	${780}_{-26}^{+32}$
$P\gt 250$ days	⋯	19	⋯	${58}_{-7}^{+9}$	${15}_{-3}^{+5}$	${43}_{-6}^{+7}$
$T\lt 10$	173	74	99	${164}_{-13}^{+15}$	${68}_{-8}^{+11}$	${96}_{-10}^{+10}$
${T}_{\mathrm{eff}}\lt 4000\,{\rm{K}}$	211	118	93	${198}_{-16}^{+17}$	${111}_{-14}^{+12}$	${87}_{-9}^{+12}$
R < 4 R⊕	426	196	230	${478}_{-23}^{+26}$	${195}_{-14}^{+16}$	${283}_{-18}^{+20}$
Habitable Zone	79	34	45	${74}_{-8}^{+11}$	${32}_{-6}^{+8}$	${42}_{-5}^{+8}$
S/N ≥ 10	857	162	695	${783}_{-28}^{+30}$	${156}_{-13}^{+12}$	${627}_{-25}^{+27}$
S/N ≥ 100	104	14	90	${72}_{-8}^{+9}$	${10}_{-3}^{+3}$	${62}_{-8}^{+8}$
δ ≥ 0.1%	1091	197	894	${983}_{-31}^{+33}$	${179}_{-12}^{+16}$	${804}_{-28}^{+28}$
δ ≥ 1%	293	40	253	${219}_{-15}^{+16}$	${34}_{-5}^{+7}$	${185}_{-14}^{+14}$
$K\geqslant 1$ m s−1	1195	⋯	⋯	${1099}_{-32}^{+37}$	${215}_{-15}^{+17}$	${884}_{-28}^{+32}$
$K\geqslant 10$ m s−1	556	⋯	⋯	${459}_{-19}^{+23}$	${20}_{-5}^{+4}$	${439}_{-19}^{+23}$
$K\geqslant 100$ m s−1	20	⋯	⋯	${10}_{-3}^{+2}$	${0}_{-0}^{+1}$	${10}_{-3}^{+2}$
$K\geqslant 1$ m s−1, ${T}_{\mathrm{mag}}\leqslant 12$	⋯	⋯	⋯	${817}_{-28}^{+31}$	${148}_{-13}^{+13}$	${669}_{-24}^{+28}$

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We also plot the distribution of single transits in period–radius space as shown in Figure 9. The relative frequency of detections from all 100 random draws are shown in the background gray scale. Trial 23 is overplotted with the FFI detectable planets in squares and detectable planets observed in PSs as larger circles. The stars are color-coded according to those detected around warm ${T}_{\mathrm{eff}}\geqslant 4000$ and cool ${T}_{\mathrm{eff}}\lt 4000$ stars as we have done throughout this paper. The planets' distribution in both period and radius space matches those found in Figures 1 and 2.

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We also plot the distribution of single transits in magnitude–RV space as shown in Figure 10. The color-coding and symbols are the same as in Figure 9. Although many planets will be detectable with RV semiamplitudes $K\geqslant 1$ m s⁻¹, many will be around targets too faint for current RV instruments. When we restrict that criteria to detectable planets with RV semiamplitudes $\geqslant 1$ m s⁻¹ and ${T}_{\mathrm{mag}}\leqslant 12$, there are ${817}_{-28}^{+31}$ planets detectable with ${148}_{-13}^{+13}$ coming from the PSs and ${669}_{-24}^{+28}$ from the FFIs. These are dominated by planets around warm stars, and we estimate that only ${1}_{-1}^{+1}$ planet will meet these conditions from a cool star in the FFIs, but ${39}_{-5}^{+7}$ will be found around cool stars in the PSs.

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Given that nearly all of the 1218 single-transit events detectable in TESS will have either photometric (90%) or radial velocity (98%) signals measurable from current ground-based observatories, there will be more planets detected than could possibly be followed up. Follow-up observers should coordinate to prioritize which planets will be targeted for follow-up observations, either by their ability to constrain the period, signal-to-noise ratio, or scientific merit. With 98% of planets detectable in RV, that the RV measurements are required to determine the mass and planetary nature of the planets, and that RV measurements can help constrain the period, RV resources should be immediately allocated toward confirming single-transit events. RV will be crucial to the single-transit detections in single-planet systems with poorly constrained eccentricities. Additionally, for those with photometric signals, searches in archival data and planned observations around the predicted next transit can be used to determine the period and to constrain the timing of future transits.

The number of single-transit planets from TESS is expected to be an order of magnitude greater than those found in Kepler, with 241 single-transit planets detectable in the PSs and another 977 detectable planets from the FFIs around stars brighter than T = 12. Single transits require greater follow-up resources than the typical TESS planet, and there will be more single-transit planet signals than follow-up resources will be able to observe or confirm. This is despite the fact that 90% and 98% of all such planets detectable will have photometric and RV signals, respectively, that will be observable from current ground-based observatories.

It is possible to predict future transits of single transits by placing constraints on the light-curve observables and on the density of the stellar host. The uncertainties from the density of the host star will be ≈10% in many cases; however, only 10% of the planets (146) will have photometry sufficient to provide constraints on the period to better than a 10% due to uncertainties on the photometry assuming circular orbits. The uncertainty due to eccentric orbits will make constraining the true period difficult, but multi-planet systems represent the best systems to place constraints on both the stellar density and eccentricity.

Our single-transit yields predict a 80% increase in the number of planets detectable beyond 25 days compared to the TESS mission and a factor of 12 increase in the yield for planets beyond 250 days. This includes 79 habitable zone planets, although only ∼1 may be a terrestrial planet. This opportunity to substantially augment the yield of the TESS mission should not be overlooked. However, given that there will be more opportunities to recover long-period planets by these single-transit events than there will be resources available, we recommend community collaboration to make the most of these opportunities.

We would like to thank Chelsea X. Huang for her feedback, helpful advice, and for sharing her yields with our group. We would like to thank Tom Barclay, Luke Bouma, and Peter Sullivan for sharing their yields with our group. We would like to thank Daniel Stevens for his insight and feedback. We would also like to thank our anonymous referee for helpful feedback and comments.

Work by S.V. is supported by the David G. Price Fellowship for Astronomical Instrumentation and by the National Science Foundation Graduate Research Fellowship under grant No. DGE-1343012. D.D. acknowledges support provided by NASA through Hubble Fellowship grant HST-HF2-51372.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. Work by B.S.G. is supported by National Science Foundation CAREER Grant AST-1056524.