THE TRANSITING EXOPLANET SURVEY SATELLITE: SIMULATIONS OF PLANET DETECTIONS AND ASTROPHYSICAL FALSE POSITIVES

TESS will observe from a 13.7 days elliptical orbit around the Earth. Over 2 years, it will observe the sky using 26 pointings. Two spacecraft orbits (27.4 days) are devoted to each pointing. Because the cameras are fixed to the spacecraft, the spacecraft must re-orient for every pointing. The pointings are spaced equally in ecliptic longitude, and they are positioned such that the top camera is centered on the ecliptic pole and the bottom camera reaches down to an ecliptic latitude of ${6}^{\circ }$. Figure 1 shows the hemispherical coverage resulting from this arrangement.

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The spectral response of the TESS cameras is limited at its red end by the quantum efficiency of the CCDs. TESS employs the MIT Lincoln Laboratory CCID-80 detector, a back-illuminated CCD with a depletion depth of 100 μm. This relatively deep depletion allows for sensitivity to wavelengths slightly longer than 1000 nm.

At its blue end, the spectral response is limited by a longpass filter with a cut-on wavelength of 600 nm. Figure 2 shows the the complete spectral response, defined as the product of the quantum efficiency and filter transmission curves.

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It is convenient to define a TESS magnitude T normalized such that Vega has T = 0. We calculate the T = 0 photon flux by multiplying the template A0V spectrum provided by Pickles (1998) by the TESS spectral response curve and then integrating over wavelength. We assume Vega has a flux density of ${F}_{\lambda }=3.44\times {10}^{-9}$ erg s⁻¹ cm⁻² Å⁻¹ at λ = 5556 Å (Hayes 1985). We find that T = 0 corresponds to a flux of $4.03\times {10}^{-6}$ erg s⁻¹ cm⁻², and a photon flux of $1.514\times {10}^{6}$ ph s⁻¹ cm⁻².

By repeating the calculation for different template spectra from the Pickles (1998) library, we obtain the photon fluxes for stars of other spectral types. These are shown in Table 1. To facilitate comparisons with the standard Johnson-Cousins I_C band (which is nearly centered within the T-band), Table 1 also provides synthetic ${I}_{C}-T$ colors. We note that the ${I}_{C}-T$ color for an A0V star is $+0.035$, which is equal to the apparent I_C magnitude defined for Vega.

Table 1.  Fluxes in the ${\text{}}{TESS}$ Bandpass and ${I}_{C}-T$ Colors

Spectral Typea	${T}_{\mathrm{eff}}$	IC = 0 Photon Fluxb	${I}_{C}-T$
	(K)	(106 ph s−1 cm−2)	(mmag)
M9V	2450	2.38	306
M5V	3000	1.43	−191
M4V	3200	1.40	−202
M3V	3400	1.38	−201
M1V	3700	1.39	−174
K5V	4100	1.41	−132
K3V	4500	1.43	−101
K1V	5000	1.45	−80.0
G2V	5777	1.45	−69.5
F5V	6500	1.48	−40.0
F0V	7200	1.48	−34.1
A0V	9700	1.56	35.0

Notes.

^aThe mapping between ${T}_{\mathrm{eff}}$ and spectral type is based on data compiled by E. Mamajek^k. ^bThe photon flux at T = 0 is $1.514\times {10}^{6}$ ph s⁻¹ cm⁻².

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The most important stellar characteristics that affect planet detectability are apparent magnitude and stellar radius. Here we provide a simple calculation for the limiting apparent magnitude (as a function of stellar radius) that permits TESS to detect planets smaller than Neptune (${R}_{{\rm{p}}}\lt 4{R}_{\oplus }$). This gives an overview of TESS's planet detection capabilities and establishes the necessary depth of our more detailed simulations of the population of nearby stars.

We assume the noise in the photometric observations to be the quadrature sum of read noise and the photon-counting noise from the target star and the zodiacal background (see Section 6.4 for the more comprehensive noise model). We require a signal-to-noise ratio (S/N) of 7.3 for detection (see Section 6.6 for the rationale). We assume that the total integration time during transits is 6 hr, which may represent two or more transits of shorter duration. Using these assumptions, Figure 3 shows the limiting apparent magnitude as a function of stellar radius at which transiting planets of various sizes can be detected.

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To gauge the necessary depth of the detailed simulations, we consider the detection of small planets around two types of stars represented in Figure 3, a Sun-like star and an M dwarf with ${T}_{\mathrm{eff}}=3200$ K. These two choices span the range of spectral types that TESS will prioritize; stars just larger than the Sun give transit depths that are too shallow, and dwarf stars just cooler than 3200 K are too faint in the TESS bandpass.

For the Sun-like star, a 4 ${R}_{\oplus }$ planet produces a transit depth of 0.13%. The limiting magnitude for transits to be detectable is T = 13.9. This also corresponds to ${K}_{s}=13.2$ and a maximum distance of 1.4 kpc, assuming no extinction.

For the M dwarf with ${T}_{\mathrm{eff}}=3200$ K, we assume ${R}_{\star }=0.155\;{R}_{\odot }$, based on the Dartmouth Stellar Evolution Database (Dotter et al. 2008) for solar metallicity and an age of 1 Gyr. Since Dressing & Charbonneau (2015) found that M dwarfs very rarely have close-in planets larger than 3 ${R}_{\oplus }$, we consider a planet of this size rather than 4 ${R}_{\oplus }$. At 3 ${R}_{\oplus }$, the transit depth is 3.1% and the limiting apparent magnitude for detection is T = 17.3. This corresponds to ${K}_{s}=14.9$ and a maximum distance of 190 pc, assuming no extinction.

A similar calculation can be carried out for EB stars. Some TESS target stars will turn out to be EBs, and others will be blended with faint binaries in the background. The maximum eclipse depth for an EB is approximately 50%, which occurs when two identical stars undergo a total eclipse. Assuming the period is 1 day, and that TESS observes the system for 27.4 days, the limiting apparent magnitude for detection of the eclipse signals is $T\lt 21$, corresponding to many kiloparsecs.

To summarize, TESS is sensitive to small planets around Sun-like stars within ≲1 kpc. For M dwarfs, the search distance is ≲200 pc. EBs can be detected across the Milky Way. These considerations set the required depth of our simulations of the stellar population, which must also take into account the structure of the galaxy and extinction.

The TRILEGAL simulation is accessed through a web-based interface.⁹ We use the default input parameters for the simulation (Table 2); the post facto adjustments that we make to dwarf properties, binarity, and the disk LF are discussed below. The runtime of a TRILEGAL query is limited to 10 minutes, so we build an all-sky catalog by performing repeated queries over regions with small solid angles.

Table 2.  TRILEGAL Input Settings

Parameter	Value
Galactic radius of Sun	8.70 kpc
Galactic height of Sun	24.2 pc
IMF (log-normal, Chabrier 2001)
Characteristic mass	0.1 ${M}_{\odot }$
Dispersion	0.627 ${M}_{\odot }$
Thin Disk
Scale height (sech2)	94.69 pc
Scale radius (exponential)	2.913 kpc
Surface density at Sun	55.4 ${M}_{\odot }$ pc−2
Thick Disk
Scale height (sech2)	800 pc
Scale radius (exponential)	2.394 kpc
Density at Sun	${10}^{-3}{M}_{\odot }$ pc−3
Halo (${R}^{1/4}$ Oblate Spheriod)
Major axis	2.699 kpc
Oblateness	0.583
Density at Sun	${10}^{-4}{M}_{\odot }$ pc−3
Bulge (Triaxial, Vanhollebeke et al. 2009)
Scale length	2.5 kpc
truncation length	95 pc
Bar: $y/x$ aspect ratio	0.68
Bar-Sun angle	15°
$z/x$ ratio	0.31
Central Density	406 ${M}_{\odot }$ pc−3
Disk Extinction
Scale height (exponential)	110 pc
Scale radius (exponential)	100 kpc
Extinction at Sun (${{dA}}_{V}/{dR}$)	0.15 mag kpc−1
${A}_{V}(z=\infty )$	0.0378 mag
Randomization (rms)	10%

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We divide the sky into 3072 equal-area tiles using the HEALPix scheme (Górski et al. 2005). Each tile subtends a solid angle of 13.4 deg². For the 164 tiles closest to the galactic disk and bulge, the stellar surface density is too large for the necessary TRILEGAL computations to complete within the runtime limit. The high background level and high incidence of EBs will also make these areas difficult to search for transiting planets, so we simply omit these tiles from consideration. This leaves 2908 tiles covering 95% of the sky.

For each of the 2908 sightlines through the centers of tiles, we make three queries to TRILEGAL:

• 1.  
  The "bright catalog" with ${K}_{s}\lt 15$ and a solid angle of 67. This is intended to include any star that could be searched for transiting planets; the magnitude limit of ${K}_{s}\lt 15$ is based on the considerations in Section 2.3. Using the K_s band to set the limiting magnitude is a convenient way to allow the catalog to have a fainter T magnitude limit for M stars than for FGK stars. The full solid angle of 13.4 deg² cannot be simulated due to the 10-minute maximum runtime of the simulation. Instead, we simulate a 6.7 deg² field and simply duplicate each star in the catalog. Once duplicated, we assign coordinates to each star randomly from a probability distribution that is spatially uniform across the entire tile. Across all of the tiles, this catalog contains 2.11 $\times {10}^{7}$ stars.
• 2.  
  The "intermediate catalog" with $T\lt 21$ and a solid angle of 0.134 deg². This is intended to include stars for which TESS would be able to detect a deep eclipse of a binary star. We use this catalog to assign blended background binaries to the target stars in the bright catalog and also to evaluate background fluxes. This deeper query is limited to a smaller solid angle (1/100th of the area of the tile) to limit computational time. The simulation then re-samples from these stars 100 times when assigning background stars to the target stars. We also restrict this catalog to ${K}_{s}\gt 15$ in the simulation to avoid double-counting stars from the bright catalog. Across all tiles, this catalog contains $1.19\times {10}^{9}$ stars.
• 3.  
  The "faint catalog" with $21\lt T\lt 27$ and a solid angle of 0.0134 deg². This is used only to calculate background fluxes due to unresolved background stars. The limiting magnitude is not critical because the surface brightness due to unresolved stars is dominated by stars at the brighter end rather than at the fainter end of the population of unresolved stars. Stars from this catalog are re-sampled 1000 times. Across all tiles, this catalog contains $7.39\times {10}^{9}$ stars.

Low-mass dwarf stars are of particular importance for TESS because they are abundant in the solar neighborhood and their small sizes facilitate the detection of small transiting planets. Although the TRILEGAL model is designed to provide simulated stellar populations with realistic distributions in spatial coordinates, mass, age, and metallicity, we noticed that the radii of low-mass stars for a given luminosity or ${T}_{\mathrm{eff}}$ in the TRILEGAL output were smaller than have been measured in recent observations or calculated in recent theoretical models.

Figure 4 illustrates the discrepancy. It compares the radius–magnitude relation employed by TRILEGAL with that of the more recent Dartmouth models (Dotter et al. 2008) as well as empirical data based on optical interferometry of field stars and analysis of EB stars. The interferometric radius measurements are from Boyajian et al. (2012). The measurements based on EBs are from the compilation of Andersen (1991) that has since been maintained by J. Southworth.¹⁰ We also include the systems tabulated by Winn et al. (2011a) in their study of Kepler-16. The published data specify ${T}_{\mathrm{eff}}$ rather than absolute I_C magnitude; in preparing Figure 4, we converted ${T}_{\mathrm{eff}}$ into absolute I_C using the temperature–magnitude data compiled by E. Mamajek¹¹ and Pecaut & Mamajek (2013).

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Figure 4 shows that the Dartmouth stellar-evolutionary models give better agreement with measured radii, especially those from interferometry. Therefore, to bring the key properties of the simulated stars into better agreement with the data, we replaced the TRILEGAL output for the apparent magnitudes and radii of low-mass stars (0.15–0.78 ${M}_{\odot }$) with the properties calculated with the Dartmouth models. To make these replacements, we use a trilateral interpolation in mass, age, and metallicity to determine the absolute magnitudes, ${T}_{\mathrm{eff}}$, and radii from the grid of Dartmouth models. For simplicity, we assume the helium abundance is solar for all stars. Furthermore, motivated by Fuhrmann (1998), we only select the grid points that adhere to the following one-to-one relation between [α/Fe] and [Fe/H]:

$$\begin{eqnarray}&&[\mathrm{Fe}/{\rm{H}}]\geqslant 0\ \Longleftrightarrow \ [\alpha /\mathrm{Fe}]=0.0\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&[\mathrm{Fe}/{\rm{H}}]=-0.5\ \Longleftrightarrow \ [\alpha /\mathrm{Fe}]\;=+0.2\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&[\mathrm{Fe}/{\rm{H}}]\leqslant -1.0\ \Longleftrightarrow \ [\alpha /\mathrm{Fe}]\;=+0.4.\end{eqnarray} \tag{ 3 }$$

In calculating the apparent magnitudes of the stars with properties overwritten from the Dartmouth models, we preserve the distance modulus from TRILEGAL and apply reddening corrections using the same extinction model that TRILEGAL uses. TRILEGAL reports the extinction A_V for each star, and for bands other than V, we use the ${A}_{\lambda }{/A}_{V}$ ratios from Cardelli et al. (1989).

Binary companions to the TESS target stars have three important impacts on the detection of transiting planets. First, whenever a "target star" is really a binary, there are potentially two stars that can be searched for transiting planets. The effective size of the search sample is thereby increased. However, there is a second effect that decreases the effective size of the search sample: if there is a transit around one star, the constant light from the unresolved companion diminishes the observed transit depth, making it more difficult to detect the transit. Even if the transit is still detectable, the radius of the planet may be underestimated due to the diminished (or "diluted") depth. The third effect is that a planet around one member of a close binary has a limited range of periods within which its orbit would be dynamically stable.

Furthermore, EBs that are blended with target stars, or that are bound to the target star in hierarchical triple or quadruple systems, can produce eclipses that resemble planetary transits. Because EBs produce larger signals than planetary transits, the population of EBs needs to be simulated down to fainter apparent magnitudes than the target stars.

To capture these effects in our simulations, we need a realistic description of stellar multiplicity. We are guided by the review of Duchêne & Kraus (2013). The multiplicity fraction (MF) is defined as the fraction of systems that have more than one star; it is the sum of the binary fraction (BF), triple fraction (TF), quadruple fraction (QF), and so on. Our simulations consider systems with up to 4 stars.

The MF has been observed to increase with the mass of the primary, which is reflected in our simulation. In our TRILEGAL queries, every star is originally a binary, and we decide randomly whether to keep the secondary based on the primary mass and the MF values in Table 3. Next, we turn a fraction of the remaining binaries into triple and quadruple systems according to the desired TF and QF. The MF, TF, and QF are adopted as follows.

• 1.  
  For primary stars of mass 0.1–0.6 ${M}_{\odot }$, we adopt the MF of 26% from Delfosse et al. (2004, p. 166). For systems with n = 3 or 4 components, the fraction of higher-order systems is taken to be ${3.9}^{2-n}$ from Duchêne & Kraus (2013).
• 2.  
  For stars of mass 0.8–1.4 ${M}_{\odot }$, we draw on the results of Raghavan et al. (2010). Primary masses of 0.8–1.0 ${M}_{\odot }$ have a MF of 41%, while primary masses of 1.0–1.4 ${M}_{\odot }$ have a MF of $50\%$. The fraction of higher-order systems is ${3.8}^{2-n}$ for both ranges (Duchêne & Kraus 2013).
• 3.  
  For stars of mass 0.6–0.8 ${M}_{\odot }$, we adopt an intermediate MF of 34%. The fraction of higher-order systems is ${3.7}^{2-n}$.
• 4.  
  For primaries more massive than 1.4 ${M}_{\odot }$, we use the results for A stars from Kouwenhoven et al. (2007), giving a MF of 75%. We assume that the fraction of higher-order systems is ${3.7}^{2-n}$.

Table 3.  Binary Properties as a Function of the Mass of the Primary

Mass (${M}_{\odot }$)	MF	$\bar{a}$ (AU)	$\sigma (\mathrm{log}P)$	γ	TF	QF
<0.1	0.22	4.5	0.5	4.0	n/a	n/a
0.1–0.6	0.26	5.3	1.3	0.4	0.067	0.017
0.6–0.8	0.34	20	2.0	0.35	0.089	0.023
0.8–1.0	0.41	45	2.3	0.3	0.11	0.030
1.0–1.4	0.50	45	2.3	0.3	0.14	0.037
>1.4	0.75	350	3.0	−0.5	0.20	0.055

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Next, we consider the properties of the binary systems. TRILEGAL originally creates binaries with a uniform distribution in the mass ratio between the secondary and the primary, q, between 0.1 and 1. However, a more realistic distribution in q is

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dq}}\propto {q}^{\gamma },\end{eqnarray} \tag{ 4 }$$

where the power-law index γ is allowed to vary with the primary mass, as specified in Table 3. When we select the binary systems to obtain the desired MF, we choose the systems to re-create this distribution in q over the range $0.1\lt q\lt 1.0$.

The period P is not specified by TRILEGAL, so we assign it from a log-normal distribution. Duchêne & Kraus (2013) parameterizes the distribution in terms of the mean semimajor axis ($\bar{a}$) and the standard deviation in $\mathrm{log}P$; both parameters vary with the primary mass as shown in Table 3. We convert from $\bar{a}$ to $\bar{P}$ with Kepler's third law.

The orbital inclination i is drawn randomly from a uniform distribution in $\mathrm{cos}i$. The orbital eccentricity e is drawn randomly from a uniform distribution, between zero and a maximum value

$$\begin{eqnarray}&&{e}_{\mathrm{max}}=\displaystyle \frac{1}{\pi }{\mathrm{tan}}^{-1}(2[\mathrm{log}P-1.5])+\displaystyle \frac{1}{2},\end{eqnarray} \tag{ 5 }$$

where P is specified in days, to provide a good fit to the range of eccentricities shown in Figure 14 of Raghavan et al. (2010). The argument of pericenter ω is drawn randomly from a uniform distribution between 0° and 360°.

For the systems that are designated as triples, we assign the properties using the approach originally suggested by Eggleton (2009). Although there is no physical reason why this method should work well, it has been found to reproduce the multiplicity properties of a sample of Hipparcos stars (Eggleton & Tokovinin 2008). First, we create a binary according to the prescriptions described above with a period P₀. Then, we split the primary or secondary star (chosen randomly) into a new pair of stars. The new pair of stars orbit their barycenter with a higher-order period ${P}_{\mathrm{HOP}}$ according to

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{\mathrm{HOP}}}{{P}_{0}}=0.2\times {10}^{-2u},\end{eqnarray} \tag{ 6 }$$

where u is uniformly distributed between 0 and 1. This procedure ensures that ${P}_{\mathrm{HOP}}$ is $\lt 1/5$ the orbital period of the original binary system, a rudimentary method for enforcing dynamical stability. The mass of a star is conserved when it is split, so the barycenter of the original binary remains the same, and the orbital period of the companion star about this barycenter is unchanged.

The original prescription given by Eggleton (2009) assigns P₀ from a distribution peaking at 10⁵ days and allows the new period to vary over 5 decades. Since our assumed distribution for $\mathrm{log}({P}_{0})$ peaks at a shorter period (for stars ≲1 ${M}_{\odot }$), we only allow the higher-order orbital period to vary over 2 decades in our implementation. In this way, we avoid generating unphysically short periods.

The total mass of a new pair of stars is set equal to that of the original star, and the mass ratio q is assigned in the following manner. The parent distribution of q is taken from the sample of triples presented in Figure 16 of Raghavan et al. (2010). We model this distribution by setting q = 1.0 for 23% of the pairs and drawing q from a normal distribution with $(\mu ,{\sigma }^{2})=(0.5,0.04)$ for the other 77% of the pairs. Finally, for each star in a higher-order pair, we calculate the absolute and apparent magnitudes, radius, and ${T}_{\mathrm{eff}}$ from the new stellar mass in combination with the age and metallicity inherited from the original star. We do so using the same interpolation onto the Dartmouth model grid described in Section 3.2.

For the systems that are turned into quadruples, we create a binary and then split both stars using the procedure described above. This results in two higher-order pairs that orbit one another with the original binary period P₀.

After modifying the TRILEGAL simulation to improve upon the properties of low-mass stars and assign multiple-star systems, we ensure that the LF is in agreement with observations. For this purpose, we rely on two independent J-band LFs reported in the literature. The first LF is from Cruz et al. (2007). It is based on volume-limited samples: a 20 pc sample for ${M}_{J}\gt 11$ and an 8 pc sample for ${M}_{J}\lt 11$ (Reid et al. 2003). Both samples use 2MASS photometry and are limited to $J\lesssim 16$. The second LF, from Bochanski et al. (2010), is based on data from the Sloan Digital Sky Survey for stars with $16\lt r\lt 22$. The resulting LF is reported for the range $5\lt {M}_{J}\lt 10$. Where the Cruz et al. (2007) and Bochanski et al. (2010) LFs overlap, we use the mean of the two LFs reported for single and primary stars (the brightest member of a multiple system). This results in the "empirical LF" to which the TRILEGAL LF is adjusted.

Next, we compute the LF of our TRILEGAL-based catalog by selecting all of the single and primary disk stars with distances within 30 pc. Then, we bin the stars according to M_J and compare the result to the empirical LF. For each M_J bin, we find the ratio of the TRILEGAL LF to the empirical LF. This ratio ranges from 0.5 to 11 across all of the magnitude bins.

We then return to each HEALPix tile individually, and we bin the stars by M_J. Using the ratio computed above for each M_J bin, we select stars at random for duplication or deletion to bring the simulated LF into agreement with the empirical LF. This process results in a net reduction of ≈30% in the total number of stars in the catalog and a shift in the LF peak toward brighter absolute magnitudes.

The left panel of Figure 5 shows the LF of the TRILEGAL simulation before and after this adjustment. The final LF is also quantified in Table 4. Each column of the table considers stellar multiplicity in a different fashion: "Singles and Primaries" counts single stars and the brightest member of a multiple system; "Systems" counts the combined flux of all stars in a system, regardless of whether it is single or multiple; and "Individual Stars" counts the primary and secondary members separately.

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As a sanity check, we make some further comparisons between our simulated LF and other published LFs. Figure 5 shows a comparison to the 10 pc RECONS sample (Henry et al. 2006), the Hipparcos catalog (Perryman et al. 1997 and van Leeuwen 2007), and the I_C-band LF of Zheng et al. (2004). The agreement with the Hipparcos sample is good up until V ≈ 8, where the Hipparcos sample becomes incomplete. The RECONS LF has a lower and blunter peak, and the Zheng et al. (2004) LF has a sharper and taller peak than the simulated LF, but are otherwise in reasonable agreement.

As another sanity check, we examine star counts as a function of limiting apparent magnitude in Figure 6. We compare the number of stars per unit magnitude per square degree in the simulated stellar population against star counts from the classic Bahcall & Soneira (1981) star-count model in the I_C band as well as actual star counts from the 2MASS point source catalog (Skrutskie et al. 2006) in the J-band. In all cases, multiple systems are counted as a single "star" with a magnitude equal to the total system magnitude. The agreement seems satisfactory; we note that the comparison with 2MASS becomes less reliable at faint magnitudes because of photometric uncertainties as well as extra-galactic objects in the 2MASS catalog.

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Intrinsic stellar variability is a potentially significant source of photometric noise for the brightest stars that TESS observes. To each star in the simulation, we assign a level of intrinsic photometric variability from a distribution corresponding to the spectral type. Our assignments are based on the variability of Kepler stars reported by Basri et al. (2013). For each star, they calculated the median differential variability (MDV) on a 3 hr timescale by binning the light curve into 3 hr segments and then calculating the median of the absolute differences between adjacent bins. Since each transit is a flux decrement between one segment of a light curve relative to a much longer timeseries, rather than two adjacent segments of equal length, the noise statistic relevant to transit detection is approximately $\sqrt{2}$ smaller than the MDV.

G. Basri et al. (2013) kindly provided the data from their Figures 7–10. Their sample is divided into four subsamples according to stellar ${T}_{\mathrm{eff}}$. We select 100 stars in each subsample with ${m}_{\mathrm{Kep}}\lt 11.5$ to minimize the contributions of instrumental noise from Kepler. Since red giants exhibiting pulsations can contaminate the subsample with ${T}_{\mathrm{eff}}\lt 4500$ K, particularly at brighter apparent magnitudes, we select stars with $12.5\lt {m}_{\mathrm{Kep}}\lt 13.1$ for these temperatures.

Figure 7 shows the resulting distributions of variability. Each star in our simulated population is assigned a variability index from a randomly chosen member of the 100 stars in the appropriate ${T}_{\mathrm{eff}}$ subsample. The variability of the ${T}_{\mathrm{eff}}\lt 4500$ K subsample is roughly 5 times greater than that of solar-type stars. However, M dwarfs are the faintest stars that TESS will observe, so instrumental noise and background will dominate the photometric error of these targets.

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Since the photometric variations associated with stellar variability exhibit strong correlations on short timescales, we assume that the level of noise due to intrinsic variability is independent of transit duration: we do not adjust it according to ${t}^{-1/2}$ as would be the case for white noise. However, we do assume that stellar variations are independent from one transit to the next, so the noise contribution from stellar variability scales with the number of transits as ${N}^{-1/2}$. In summary, the standard deviation in the relative flux due to stellar variability, after phase-folding all of the transits together, is taken to be

$$\begin{eqnarray}&&{\sigma }_{V}=\displaystyle \frac{\mathrm{MDV}(3\;\mathrm{hr})}{\sqrt{2}}{N}^{-1/2}.\end{eqnarray} \tag{ 7 }$$

The planet assignments are based on several recent studies of Kepler data. The Kepler sample has high completeness for the planetary periods ($P\lesssim 20$ days) and radii (${R}_{{\rm{p}}}\gtrsim {R}_{\oplus }$) that are most relevant to TESS.

For FGK stars, we adopt the planet occurrence rates from Fressin et al. (2013). For ${T}_{\mathrm{eff}}\lt 4000$ K, we adopt the occurrence rates from Dressing & Charbonneau (2015), who updated the results that were originally presented by Dressing & Charbonneau (2013). We note that Dressing & Charbonneau (2015) corrected their planet occurrence rates for astrophysical false positives by using the false-positive rates presented by Fressin et al. (2013) as a function of the apparent planet size.

In both cases, the published results are provided as a matrix of occurrence rates and uncertainties for bins of planetary radius and period. The incompleteness of the Kepler sample is considered for each bin. Because the bins are relatively coarse, we allow the radius and period of a given planet to vary randomly within the limits of each bin. Periods are assigned from a uniform distribution in $\mathrm{log}P$. (We omit planets for which the selected period would place the orbital distance within $2\;{R}_{\star }$, on the grounds that tidal forces would destroy any such planets.)

For the smallest radius bin examined by Fressin et al. (2013), we choose the planet radius from a uniform distribution between 0.8–1.25 ${R}_{\oplus }$. For the larger-radius bins, we choose the planet radius within each bin according to the distribution

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{{dR}}_{{\rm{p}}}}\propto {R}_{{\rm{p}}}^{-1.7}.\end{eqnarray} \tag{ 8 }$$

These intra-bin distributions were chosen ad hoc to provide a relatively smooth function in the radius–period plane. Likewise, when applying the occurrence rates from Dressing & Charbonneau (2015), for the smallest radius bin we choose the planet radius from a uniform distribution between 0.5–1.0 ${R}_{\oplus }$. For the bin extending from 1.0–1.5 ${R}_{\oplus }$, we chose the planet radius from a distribution with a power-law index of −1. For ${R}_{{\rm{p}}}\gt 1.5\;{R}_{\oplus }$ we use a power-law index of −1.7. The maximum planet size in the Fressin et al. (2013) matrix is 22 ${R}_{\oplus }$, and the maximum planet size in the Dressing & Charbonneau (2015) matrix is 4 ${R}_{\oplus }$. The final distributions are illustrated in Figure 8.

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We allow our simulation to assign more than one planet to a given star with independent probability. The only exceptions are (1) we require the periods of adjacent planetary orbits to have ratios of at least 1.2, and (2) planets around a star with a binary companion cannot have orbital periods that are within a factor of 5 of the binary orbital period. The result is that 53% of the transiting systems around FGK stars and 55% of those around M stars are multiple-planet systems. Figure 9 shows the resulting distribution of period ratios. The orbits of multi-planet systems are assumed to be perfectly coplanar, both for simplicity and from the evidence for low mutual inclinations in compact multi-planet systems (Figueira et al. 2012; Fabrycky et al. 2014).

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As a sanity check, we compare the proportion of planets in multi-transiting systems in our simulated stellar population to the proportion of multi-transiting Kepler candidates. In our simulation, 26.2% of planets around FGK stars and 33.6% of planets around M stars reside in multi-transiting systems. Out of the 4178 Kepler objects of interest, 41% are in multi-transiting systems.

For simplicity, we assume that all planetary orbits are circular. The orbital inclinations i are assigned randomly from a uniform distribution in $\mathrm{cos}i$. We identify the transiting systems as those with $| b| \lt 1$, where

$$\begin{eqnarray}&&b=\displaystyle \frac{a\mathrm{cos}i}{{R}_{\star }}\end{eqnarray} \tag{ 9 }$$

is the transit impact parameter.

We then calculate the properties of the planets and their transits and occultations. The transit duration Θ is given by Equations (18) and (19) of Winn (2011) in terms of the mean stellar density ${\rho }_{\star }$:

$$\begin{eqnarray}&&{\rm{\Theta }}=13\;\mathrm{hr}{\left(\displaystyle \frac{P}{365\;\mathrm{days}}\right)}^{1/3}{\left(\displaystyle \frac{{\rho }_{\star }}{{\rho }_{\odot }}\right)}^{-1/3}\sqrt{1-{b}^{2}}.\end{eqnarray} \tag{ 10 }$$

The depth of the transit ${\delta }_{1}$ is given by ${({R}_{{\rm{p}}}{/R}_{\star })}^{2}$. The depth of the occultation (secondary eclipse) is found by estimating the effective temperature of the planet (T_p) and then computing the photon flux ${{\rm{\Gamma }}}_{{\rm{p}}}$ within the TESS bandpass from a blackbody of radius R_p. The photon flux from the planet is then divided by the combined photon flux from the planet and the star:

$$\begin{eqnarray}&&{\delta }_{2}=\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{p}}}}{{{\rm{\Gamma }}}_{{\rm{p}}}+{{\rm{\Gamma }}}_{\star }}.\end{eqnarray} \tag{ 11 }$$

The equilibrium planetary temperature T_p is determined by assuming radiative equilibrium with an albedo of zero and isotropic radiation (from a recirculating atmosphere), giving

$$\begin{eqnarray}&&{T}_{{\rm{p}}}={T}_{\mathrm{eff}}\sqrt{\displaystyle \frac{{R}_{\star }}{2a}}.\end{eqnarray} \tag{ 12 }$$

We also keep track of the relative insolation of the planet $S{/S}_{\oplus }$, defined as

$$\begin{eqnarray}&&\displaystyle \frac{S}{{S}_{\oplus }}={\left(\displaystyle \frac{a}{1\;\mathrm{AU}}\right)}^{-2}{\left(\displaystyle \frac{{R}_{\star }}{{R}_{\odot }}\right)}^{2}{\left(\displaystyle \frac{{T}_{\mathrm{eff}}}{5777\;{\rm{K}}}\right)}^{4}.\end{eqnarray} \tag{ 13 }$$

We identify the EBs by computing the impact parameters b₁ and b₂ of the primary and secondary eclipses, respectively:

$$\begin{eqnarray}&&{b}_{\mathrm{1,2}}=\displaystyle \frac{a\mathrm{cos}i}{{R}_{\mathrm{1,2}}}\left(\displaystyle \frac{1-{e}^{2}}{1\pm e\mathrm{sin}\omega }\right)\end{eqnarray} \tag{ 14 }$$

(see Equations (7)–(8) of Winn 2011). Non-grazing primary eclipses are identified with the criterion

$$\begin{eqnarray}&&{b}_{1}{R}_{1}\lt {R}_{1}-{R}_{2},\end{eqnarray} \tag{ 15 }$$

while grazing primary eclipses have larger impact parameters:

$$\begin{eqnarray}&&{R}_{1}-{R}_{2}\lt {b}_{1}{R}_{1}\lt {R}_{1}+{R}_{2}.\end{eqnarray} \tag{ 16 }$$

The eclipse depth of non-grazing primary eclipses is given by

$$\begin{eqnarray}&&{\delta }_{1}={\left(\displaystyle \frac{{R}_{2}}{{R}_{1}}\right)}^{2}\displaystyle \frac{{{\rm{\Gamma }}}_{1}}{{{\rm{\Gamma }}}_{1}+{{\rm{\Gamma }}}_{2}}\end{eqnarray} \tag{ 17 }$$

where ${{\rm{\Gamma }}}_{1}$ and ${{\rm{\Gamma }}}_{2}$ are the photon fluxes from each star. In the event that ${R}_{2}\gt {R}_{1}$, the area ratio is set equal to unity; in that case, the primary undergoes a total eclipse. We neglect limb-darkening in these calculations for simplicity. Secondary eclipses are identified and quantified in a similar manner.

For grazing eclipses, the area ratio ${({R}_{2}/{R}_{1})}^{2}$ is replaced with the overlap area of two uniform disks with the appropriate separation of their centers, given by Equations (2.14–5) of Kopal (1979). The durations and timing of eclipses are calculated from Equations (14)–(16) of Winn (2011, pp. 55–77).

We discard EBs when the assigned parameters imply $a\lt {R}_{1}$ or $a\lt {R}_{2}$. We also exclude systems where a is less than the Roche limit a_R for either star, assuming they are tidally locked:

$$\begin{eqnarray}&&{a}_{{\rm{R}}\mathrm{1,2}}={R}_{\mathrm{2,1}}{\left(3\displaystyle \frac{{M}_{\mathrm{1,2}}}{{M}_{\mathrm{2,1}}}\right)}^{1/3}.\end{eqnarray} \tag{ 18 }$$

For primaries with ${I}_{C}\lt 12$, our simulated stellar population has 97461 EBs over the $95\%$ of the sky that is covered by the simulation. Another 21441 systems contain eclipsing pairs in a hierarchical system. As another sanity check, we compare the simulated density of eclipsing systems on the sky to the catalog of EBs in the Kepler field. We use Version 2 of the compilation¹² from Prša et al. (2011) and Slawson et al. (2011) to plot the density of EBs as a function of apparent system magnitude in Figure 10. Within the range of $0.5\lt P\lt 50$ days, this catalog contains 1.85 EBs deg⁻² with ${m}_{\mathrm{Kep}}\lt 12$. A 203 deg² subsample of our TRILEGAL catalog, taken from 15 HEALPix tiles and centered on galactic coordinates l = 76° and b = 134 for similarity to the Kepler field, contains 1.04 EBs deg⁻² with ${Kp}\lt 12$. This disparity suggests that our model of the eclipsing-binary population could have systematic errors of nearly 80%, at least for the relatively low galactic latitude of the Kepler field, where the TRILEGAL simulation loses accuracy, and the steep increase in the stellar surface density makes it difficult to accurately match the simulation results to the Kepler field.

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The synthetic images are constructed from the PRF, which describes the fraction of light from a star that is collected by a given pixel. It is calculated by numerically integrating the point-spread function (PSF) over the boundaries of pixels. The photometric aperture for a star is the collection of pixels over which the electron counts are summed to create the photometric signal; they are selected to maximize the photometric S/N of the target star. Throughout this study, we assume that the pixel values are simply summed without any weighting factors.

The TESS lens uses seven elements with two aspheres to deliver a tight PSF over a large focal plane and over a wide bandpass. Due to off-axis and chromatic aberrations, the TESS PSF must be described as a function of field angle and wavelength. We calculate the PSF at four field angles from the center (0°) to the corner (17°) of the field of view. Chromatic aberrations arise both from the refractive elements of the TESS camera and from the deep-depletion CCDs absorbing redder photons deeper in the silicon. We calculate the PSF for nine wavelengths, evenly spaced by 50 nm, between 625 and 1025 nm. These wavelengths are shown with dashed lines in Figure 2. These wavelengths also correspond to a set of bandpass filters that will be used in the laboratory to measure the performance of each flight TESS camera.

The TESS lens has been modeled with the Zemax ray-tracing software. We use the Zemax model to trace 250,000 simulated rays through the camera optics for each field angle and wavelength. The model is set to the predicted operating temperature of −75°C. Rays are propagated through the optics and then into the silicon of the CCD. A probabilistic model is used to determine the depth of travel in the silicon before the photons are converted to electrons. Finally, the diffusion of the electrons within the remaining depth of silicon is modeled to arrive at the PSF.

Pointing errors from the spacecraft will effectively enlarge the PSF because the 2 s exposures are summed into 2-minute stacks without compensating for these errors. The spacecraft manufacturer (Orbital Sciences) has provided a simulated time series of spacecraft pointing errors from a model of the spacecraft attitude control system. Using two minutes of this time series, we offset the PSF according to the pointing error and then stack the resulting time series of PSFs. The rms amplitude of the pointing error is ≈1'', which is small in comparison to the pixel size and the full width half-maximum of the PSF. Thus, the impact of pointing errors on short timescales turns out to be minor. Long-term drifts in the pointing of the cameras will also introduce photometric errors, but this effect is budgeted in the systematic error described in Section 6.4.2.

Limits in the manufacturing precision of TESS cameras will also increase the size of the PSF from its ideal value. In a Monte Carlo simulation drawing from the tolerances prescribed in the optical design, the the fraction of the flux captured by the brightest pixel in the PRF is reduced by ≲3% in 80% of cases. To capture this effect, we simply increase the size of the PSF by ≈3% to achieve the same reduction.

Even after considering jitter and manufacturing errors, the PSF is still under-sampled by the 15 μm pixels of the TESS CCDs. Therefore, we must recalculate the PRF for a given offset and orientation between the PSF and the pixel boundaries. We numerically integrate the PSF over a grid of 16 × 16 pixels to arrive at the PRF. We do so over a $10\times 10$ grid of sub-pixel centroid offsets and two different azimuthal orientations (0° and 45°) with respect to the pixel boundaries. For the corner PSF (at a field angle of 17°), only the 45° azimuth angle is considered.

We can also view the PRF in terms of the cumulative fraction of light collected by a given number of pixels. In Figure 13, we average over all of the centroid offsets and both azimuthal angles. For clarity, only three of the field angles and three values of ${T}_{\mathrm{eff}}$ are shown. There is little change in the PRF across the range of ${T}_{\mathrm{eff}}$, but the PRF degrades significantly at the corners of the field.

For each target star with eclipses or transits, we create a synthetic image in the following manner. First, we determine the appropriate PRF based on the star's color and location in the camera field. We calculate the field angle from its ecliptic coordinates and the direction in which the relevant TESS camera is pointed. We randomly assign an offset between the star and the nearest pixel center, and we randomly assign an azimuthal orientation of either 0° or 45°. We then look up the nine wavelength-dependent PRFs for the appropriate field angle, centroid offset, and azimuthal angle. The nine PRFs are summed with weights according to the stellar effective temperature.

The weight of a given PRF is proportional to the stellar photon flux integrated over the wavelengths that the PRF represents. Outside of the main simulation, we considered a Vega-normalized stellar template spectrum of each spectral type from the Pickles (1998) library. We multiplied each template spectrum by the spectral response function of the TESS camera, and we integrated the photon flux for each of the nine PRF bandpasses. Next, we fitted a polynomial function to the relationship between the stellar effective temperatures and the photon flux in each bandpass. During the simulation, the polynomial functions are used to quickly calculate the appropriate PRF weights as a function of stellar effective temperature.

Once the PRFs are summed, the result is a synthetic 16 × 16-pixel image of each target star. We only consider the central 8 × 8 pixels when determining the optimal photometric aperture; the left panel of Figure 12 shows an example.

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After synthesizing the image of each eclipsing or transiting target star, a separate 16 × 16 image is synthesized of all the relevant neighboring stars and companion stars. The neighboring stars are drawn from all three star catalogs described in Section 3. The stars are assumed to be uniformly distributed across each HEALPix tile, allowing us to randomly generate the distances between the target star and the neighboring stars. Stars from the target catalog are added to the synthesized image if they are within a radius of 6 pixels from the target star. Stars from the intermediate catalog are added if they are within 4 pixels, and stars from the faint catalog are added if they are within 2 pixels. The synthesized images are created in the same manner as described above: by weighting, shifting, and summing the PRFs associated with each star. The right panel of Figure 12 shows an example.

Synthetic images are also created for the EB systems drawn from the intermediate catalog, but a slightly different approach is taken. For each EB, we search for any target stars within 6 pixels. If any are found, the brightest is added to the list of target stars with apparent transits or eclipses. Separate synthetic images are created for the target star, the EB, and the non-eclipsing neighboring stars. Hierarchical binaries are treated in a similar fashion; the non-eclipsing component is treated as the target star, and a separate synthetic image is created for the eclipsing pair so that its apparent depth can be diluted. While this approach may appear to strongly depend upon the somewhat arbitrary magnitude limits adopted for the different catalogs, this is not really the case. Both the EBs from the target catalog and the BEBs from the intermediate catalog end up being diluted by neighboring stars drawn from all of the catalogs.

For each target star that is associated with an eclipse or transit (whether it is due to the target star itself or a blended eclipsing binary), we select the pixels that provide the optimal photometric aperture from the central 8 × 8 pixels of its synthetic image. Starting with the three brightest pixels in the PRF, we add pixels in order of decreasing brightness one at a time. At each step, we sum the flux of the pixels from the synthetic image of the target star and from the synthetic image of the neighboring stars. We also consider the read noise and zodiacal noise, which are discussed in Section 6.4. As the number of pixels in the photometric aperture increases, more photons are collected from the target star, and more noise is accumulated from the readout, sky background, and neighboring stars. The optimal photometric aperture maximizes the S/N of the target star even if the eclipse is produced by a blended binary. We assume that the data will be analyzed with prior knowledge of the locations of neighboring stars (but no prior knowledge of whether they eclipse).

Once the optimal aperture is determined, we calculate the dilution parameter D, which is the factor by which the true eclipse or transit depth is reduced by blending with other stars in the photometric aperture. Specifically, the dilution parameter is defined as the ratio of the total flux in the aperture from the neighboring stars (${{\rm{\Gamma }}}_{{\rm{N}}}$) and target star (${{\rm{\Gamma }}}_{{\rm{T}}}$) to the flux from the target star:

$$\begin{eqnarray}&&D=\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{N}}}+{{\rm{\Gamma }}}_{{\rm{T}}}}{{{\rm{\Gamma }}}_{{\rm{T}}}}.\end{eqnarray} \tag{ 19 }$$

For blended binaries and hierarchical systems, the denominator is replaced with the flux from the binary ${{\rm{\Gamma }}}_{{\rm{B}}}$, and the target star becomes a source of dilution:

$$\begin{eqnarray}&&D=\displaystyle \frac{{{\rm{\Gamma }}}_{{\rm{N}}}+{{\rm{\Gamma }}}_{{\rm{T}}}+{{\rm{\Gamma }}}_{{\rm{B}}}}{{{\rm{\Gamma }}}_{{\rm{B}}}}.\end{eqnarray} \tag{ 20 }$$

With this definition, D = 1 signifies an isolated system, and in general, $D\gt 1$. This parameter is later reported for all detected eclipses under the "Dil." column of Table 6.

Table 6.  Catalog of Simulated TESS Detections

α (°)	δ (°)	Rp	P (days)	$S/{S}_{\oplus }$	K (m s−1)	${R}_{\star }$ (${R}_{\odot }$)	${T}_{\mathrm{eff}}$ (K)	V	IC	J	Ks	DM	Dil.	${\mathrm{log}}_{10}({\sigma }_{V})$	S/N	Mult.
0.439	45.217	3.31	9.14	361.7	2.03	1.41	6531	8.47	7.97	7.63	7.41	5.00	1.00	−4.87	16.8	1
0.480	−66.204	2.19	14.20	2.1	3.11	0.32	3426	15.08	12.83	11.56	10.79	3.90	1.01	−4.22	12.3	3
0.646	42.939	1.74	4.96	235.0	1.66	0.95	5546	10.12	9.35	8.81	8.41	4.95	1.00	−4.64	7.5	0
0.924	−26.065	1.48	2.16	1240.1	1.95	1.12	5984	8.06	7.42	6.98	6.67	3.65	1.00	−4.50	8.4	0
1.314	−24.954	2.29	9.75	5.9	2.95	0.42	3622	14.19	12.15	10.99	10.19	4.10	1.00	−4.44	8.6	2
1.384	10.606	2.32	13.99	2.1	3.32	0.32	3425	15.04	12.79	11.52	10.74	3.85	1.07	−3.50	7.8	2
1.783	−71.931	3.29	8.42	4.5	5.23	0.34	3444	15.29	13.06	11.79	11.02	4.25	1.00	−4.39	26.4	2
1.789	−9.144	2.81	5.62	2.6	9.15	0.17	3228	15.05	12.52	11.13	10.36	1.85	2.28	−4.41	28.7	3
1.948	−16.995	17.15	1.34	10164.5	16.03	2.11	6668	7.51	7.05	6.72	6.53	5.00	1.00	−4.42	457.4	2
2.172	−15.533	4.80	17.14	341.2	2.10	2.11	6668	7.51	7.05	6.72	6.53	5.00	1.00	−4.68	13.1	2
4.071	9.507	1.97	11.45	1.5	4.09	0.22	3300	14.96	12.55	11.21	10.44	2.65	1.00	−4.50	17.7	2
4.634	−23.500	4.71	5.17	116.3	4.60	0.80	5000	9.52	8.54	7.85	7.32	3.35	1.00	−4.68	64.0	0
4.788	78.625	1.56	0.62	71.2	8.92	0.22	3283	15.94	13.50	12.14	11.38	3.50	1.00	−4.43	16.9	2
5.322	−55.554	2.85	18.16	2.4	3.01	0.41	3592	14.23	12.15	10.98	10.18	4.00	1.00	−4.46	15.5	2
5.704	50.726	2.24	2.75	308.3	2.82	0.80	5188	11.28	10.40	9.77	9.28	5.35	1.01	−4.67	10.6	2
5.951	−28.675	3.75	17.75	4.3	3.46	0.50	3844	11.73	9.94	8.89	8.08	2.55	1.00	−4.29	52.5	1
6.166	32.455	1.11	0.79	53.4	2.95	0.22	3304	14.81	12.41	11.08	10.32	2.65	1.00	−3.50	8.5	2
6.521	−4.048	1.53	7.51	4.8	2.78	0.32	3435	13.66	11.43	10.17	9.41	2.50	1.00	−3.49	8.3	2
6.662	−79.377	2.31	1.89	45.5	5.48	0.40	3551	14.42	12.30	11.10	10.31	4.00	1.00	−3.49	19.9	0
7.592	−79.924	7.72	3.92	320.7	7.41	0.91	5623	11.06	10.32	9.79	9.40	5.85	1.00	−4.54	96.5	1
8.071	−77.185	1.49	1.02	887.9	3.50	0.70	5030	8.00	7.05	6.40	5.85	1.60	1.00	−4.74	40.7	1
8.396	−53.966	2.38	5.59	6.8	4.55	0.31	3442	14.14	11.91	10.66	9.89	2.95	1.25	−3.60	20.3	3
8.919	67.419	2.94	29.94	1.3	2.61	0.42	3611	13.33	11.30	10.14	9.35	3.20	1.09	−3.69	18.1	1
9.843	−11.832	2.82	6.28	62.9	2.98	0.69	4819	11.69	10.60	9.88	9.26	4.90	1.00	−4.85	17.2	0
10.467	−40.605	1.71	15.14	1.3	2.83	0.26	3359	14.51	12.18	10.87	10.11	2.70	1.02	−3.49	9.1	1
10.551	−27.297	3.60	5.99	283.9	2.96	1.04	5970	9.85	9.21	8.76	8.44	5.25	1.00	−4.52	22.5	0
11.067	−52.752	2.40	17.18	0.9	4.35	0.22	3287	15.54	13.10	11.74	10.98	3.10	1.00	−4.32	17.8	1
11.145	29.347	2.43	8.66	13.3	2.80	0.53	3948	12.80	11.09	10.08	9.27	3.95	1.01	−4.46	13.8	2
11.145	−49.044	1.51	8.12	6.4	2.27	0.39	3557	13.67	11.56	10.37	9.58	3.25	1.03	−4.45	10.7	1
11.207	37.355	2.64	1.98	28.9	7.02	0.32	3437	14.67	12.43	11.18	10.40	3.55	1.02	−4.01	22.6	3
11.547	−44.670	5.26	39.43	56.6	1.90	1.47	6577	8.65	8.16	7.82	7.62	5.30	1.00	−4.01	14.5	0
11.909	−67.746	3.67	4.78	216.5	3.69	0.84	5598	11.69	10.94	10.40	10.00	6.25	1.01	−3.77	20.4	0
12.015	74.529	3.85	1.85	11.8	17.24	0.17	3225	16.89	14.36	12.95	12.18	3.75	1.10	−3.49	45.8	3
12.085	−51.662	11.96	40.02	16.1	4.97	0.99	5598	10.31	9.55	9.02	8.63	5.25	1.01	−4.74	169.9	0
12.261	−60.310	2.12	10.35	3.4	3.19	0.33	3467	13.88	11.68	10.44	9.67	2.90	1.00	−3.53	13.3	2
12.320	75.640	6.92	5.25	513.2	5.25	1.20	6295	10.12	9.57	9.18	8.91	6.10	1.54	−4.12	64.3	1
12.410	−10.212	2.97	6.84	2.2	8.51	0.18	3230	16.63	14.10	12.70	11.93	3.60	1.00	−4.39	9.0	0
12.640	−53.861	1.73	2.40	20.9	4.48	0.31	3442	14.14	11.91	10.66	9.89	2.95	1.00	−3.95	17.7	3
12.928	−13.886	12.98	6.03	8419.9	5.13	2.50	10593	6.03	6.14	6.15	6.21	5.55	1.00	−4.16	61.4	1
12.974	58.302	2.77	13.71	422.9	1.35	1.56	7603	9.02	8.77	8.57	8.48	6.45	1.06	−4.50	7.9	1
13.048	74.712	1.47	1.06	81.4	5.18	0.36	3490	15.02	12.84	11.60	10.82	4.25	1.30	−3.61	7.9	2
13.408	27.048	3.21	11.65	84.2	2.21	0.96	5689	10.53	9.81	9.30	8.93	5.50	1.01	−4.84	13.2	1
13.494	−57.211	1.98	19.54	2.3	2.08	0.42	3606	14.03	11.97	10.80	10.01	3.90	1.00	−4.53	8.0	3
13.690	−81.593	1.85	2.20	15.3	6.25	0.24	3324	14.95	12.57	11.24	10.48	2.85	1.15	−3.84	29.3	3
13.824	−20.209	1.05	15.96	0.6	1.16	0.16	3228	14.83	12.30	10.91	10.15	1.60	1.00	−3.96	8.3	1
14.214	79.814	1.06	0.50	659.2	1.50	0.55	3996	12.87	11.21	10.21	9.40	4.20	1.05	−4.10	7.9	3
14.313	32.413	2.29	5.80	7.6	4.23	0.34	3470	12.33	10.15	8.92	8.15	1.40	1.00	−4.54	46.1	2
14.807	−14.485	1.99	12.55	0.6	5.39	0.16	3027	14.98	12.76	11.10	10.32	1.45	1.00	−3.53	19.7	1
15.256	48.530	2.27	7.68	4.4	3.93	0.31	3435	14.47	12.23	10.98	10.21	3.25	1.21	−3.70	15.1	2
15.926	75.902	1.94	5.06	333.5	1.72	1.05	5888	8.68	8.02	7.56	7.23	4.05	1.00	−4.64	16.7	2

Note.This catalog is based on one realization of the Monte Carlo simulation. The detections are drawn from the $2\times {10}^{5}$ target stars that are observed with a 2-minute cadence. The larger sample of detections from stars that are only observed in full-frame images is not provided here. The dilution parameter "Dil." is defined in Equations (9) and (10).

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

Download table as:  DataTypeset image

The photometric noise model includes the photon-counting noise from all of the stars in the photometric aperture, photon-counting noise from zodiacal light (ZL), stellar variability, and instrumental noise. Stellar variability and background stars are randomly assigned from distributions, while the other noise terms are more deterministic in nature. Figure 14 shows the relative photometric noise as a function of apparent magnitude and also breaks down the contributions from the deterministic sources of noise. Each subsection below describes the noise terms in more detail.

6.4.1. Zodiacal Light

Although TESS avoids the telluric sky background by observing from space, it its still affected by the ZL and its associated photon-counting noise. Our model of the zodiacal flux is based on the spectrum measured by the Space Telescope Imaging Spectrograph on the Hubble Space Telescope.¹³ We multiply this ZL spectrum by the TESS spectral response function and integrate over wavelength. This gives the photon flux of

$$\begin{eqnarray}&&2.56\times {10}^{-3}{10}^{-0.4(V-22.8)}\;\mathrm{ph}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}\;{\mathrm{arcsec}}^{-2},\end{eqnarray} \tag{ 21 }$$

where V is the V-band surface brightness of the ZL in mag arcsec⁻². For TESS, the pixel scale is 211 and the effective collecting area is 69 cm². To model the spatial dependence of V, we fit the tabulated values of V as a function of helio-ecliptic coordinates,¹⁴ with a function

$$\begin{eqnarray}&&V={V}_{\mathrm{max}}-{\rm{\Delta }}V{\left(\displaystyle \frac{b-90^\circ }{90^\circ }\right)}^{2}\end{eqnarray} \tag{ 22 }$$

where b is the ecliptic latitude and ${V}_{\mathrm{max}}$ and ΔV are free parameters. Because TESS will generally be pointed in the anti-solar direction (near helio-ecliptic longitude l ≈ 180°), and because V depends more strongly on latitude than longitude in that region, we only fitted to the data with $l\geqslant 120^\circ$ and weighted the points in proportion to ${(l-90^\circ )}^{2}$. The least-squares best-fit has ${V}_{\mathrm{max}}=23.345$ mag and ΔV = 1.148 mag.

Based on these results, we find that the ZL collected in a 2 s image ranges from 95–270 ${e}^{-}$ pix⁻¹ depending on ecliptic latitude. The photon-counting noise associated with this signal varies from 10–16 ${e}^{-}$ pix⁻¹ rms, as mentioned in Section 2.

6.4.2. Instrumental Noise

6.4.3. Saturation

6.4.4. Cosmic Rays (CRs)

The S/N of transits or eclipses will depend critically on how long the star is observed. Figure 1 is a sky map showing the number of times that TESS will point at a given location as a function of ecliptic coordinates. As noted above, the simulations assign coordinates to each star through a uniform random distribution across the HEALPix tile to which it belongs. The star's ecliptic coordinates are then converted to x and y pixel coordinates for each TESS pointing. We tally the number of pointings for which the target falls within the field of view of a TESS camera. The total amount of observing time is calculated as the total duration of all consecutive pointings.

The duty cycle of observations must also be considered. At each orbital perigee, TESS interrupts observations in order to transmit data to Earth and perform other housekeeping operations. This takes approximately 0.6 days. We model this interruption in the simulation, so each 13.6 days spacecraft orbit actually results in 13.0 days of data.

The presence of the Earth or moon in the field of view of any camera will also prohibit observations. We do not model this effect since predicting their presence depends upon the specific launch date of TESS. However, our simulations do show that if observations are interrupted near TESS's orbital apogee in addition to its perigee, then the planet yields are approximately proportional to the duty cycle of observations.

The model for the detection process is highly simplified: we adopt a threshold for the S/N, and we declare a signal to be detected if the total S/N exceeds the threshold. In other words, the detection probability is modeled as a step function of the computed S/N. (The matched-filter technqiues of the TESS pipeline probably have a smoother profile, such as a standard error function Jenkins et al. 1996.) For transiting planets, all of the observed transits contribute to the total S/N. For EBs, we allow both the primary and secondary eclipses to contribute to the total S/N.

The choice of an appropriate S/N threshold was discussed in detail by Jenkins et al. (2002) in the context of the Kepler mission. Their criterion was that the threshold should be sufficiently high to prevent more than one "detection" from being a purely statistical fluke after analyzing all of the data from the entire mission. We adopt the same criterion here. Since the number of astrophysical false positives is at least several hundred (as discussed below), this criterion allows statistical false positives to be essentially ignored.

To determine the appropriate threshold, we use a separate Monte Carlo simulation of the transit search. We produce $2\times {10}^{5}$ light curves containing uncorrelated, Gaussian noise and analyze them for transits in a similar manner as will be done with real data. Then, we find the S/N threshold that results in approximately one statistical false positive. Each light curve consists of 38,880 points, representing two 27.4 days TESS pointings with 2 minute sampling. We chose a timeseries length of two pointings rather than one to account for the stars observed with overlapping pointings.

To search for transits, we scan through a grid of trial periods, times of transit, and transit durations. At each grid point, we identify the data points belonging to the candidate transit intervals. The S/N is computed as the mean of the in-transit data values divided by the uncertainty in the mean.

The grid of transit durations Θ starts with 28 minutes (14 samples) and each successive grid point is longer by 4 minutes (2 samples). The grid of periods p is the range of periods that are compatible with the transit duration. The periods are calculated by inverting Equation (10):

$$\begin{eqnarray}&&P=(365\;\mathrm{days}){\left(\displaystyle \frac{{\rm{\Theta }}}{78\;\mathrm{minutes}}\right)}^{3}\displaystyle \frac{{\rho }_{*}}{{\rho }_{\odot }}{\left(1-{b}^{2}\right)}^{-3/2}.\end{eqnarray} \tag{ 23 }$$

We allow p to vary over a sufficient range to include plausible stellar densities ${\rho }_{\star }/{\rho }_{\odot }$ from 0.5 to 100. The fractional step size in the period ${\rm{\Delta }}P/P$ is then 3ΔΘ/Θ, which has a minimum value of 0.43 for the shortest periods. We consider orbital periods ranging from 1.7 hr (which is below the period corresponding to Roche limit) to 27.4 days (half of the nominal observing interval). The transit phase is stepped from zero to the orbital period in increments of one-half the transit duration.

Figure 15 shows how the number of false-positive detections scales with the detection threshold. We find that an S/N of 7.1 produces approximately one statistical false positive within the library of $2\times {10}^{5}$ light curves. By coincidence, this is equal to the S/N threshold of 7.1 that was calculated for the Kepler mission by Jenkins et al. (2002). TESS searches twice as many stars as the 10⁵ considered in the Kepler study, and over a larger dynamic range in period; Kepler searches for planets with longer periods using longer intervals of data.

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To account for the expected reduction in S/N due to the CR rejection algorithm (see Section 6.4.4), we adopt a slightly higher threshold of 7.3 in this paper. In addition, we only consider a transit or eclipse to be detected if two or more events are observed. We also record the single events that exceed the S/N threshold, but we do not count them as "detections" in the tallies and the discussion that follows. The planets detected with a single transit generally have longer periods than the multiple-transit detections. It is also worth noting that TESS may detect some single transits from the population of planets with periods longer than a year, which we have not simulated at all, because our sources for planet occurrence rates do not extend to such long periods. The single-transit detections may represent interesting opportunities to study the properties of more distant planets. However, they will require additional ground-based follow-up observations to determine the orbital period and discriminate against astrophysical or statistical false positives.

From the 2.11 $\times {10}^{7}$ stars in the ${K}_{s}\lt 15$ catalog, we must select the $2\times {10}^{5}$ target stars for which pixel data will be saved and transmitted with 2 minute time sampling. In our simulation, the target stars are chosen according to the prospects for detecting the transits of small planets, which depend chiefly on stellar radius and apparent magnitude.

In the simulation, we have complete knowledge of the properties of each star, which makes it straightforward to determine whether a fiducial transiting planet with a given radius and period could be detected with TESS. We adopt an orbital period of 20 days; for each 27.4 days pointing that TESS spends observing a star, we assume that two transits are observed. The stellar radius and mass are used to calculate the transit duration with a 20 days period, thereby determining the total exposure time during transits. Then, we use the simplified noise model from Section 2.3 that considers the read noise and photon-counting noise of the star and ZL. We then check to see if the fiducial transiting planet would be detectable with a S/N exceeding of 7.3.

The number of stars meeting this detection criterion depends strongly on the radius of the fiducial planet. Starting from small values, we increase the radius until the number of stars for which the planet would be detectable is $2\times {10}^{5}$. This is achieved for ${R}_{{\rm{p}}}=2.25\;{R}_{\oplus }$. Through this procedure, the target star catalog is approximately complete for planets smaller than 2.25 ${R}_{\oplus }$ with orbital periods shorter than 20 days. There is a higher density of target stars assigned near the ecliptic poles due to the longer duration of TESS observations in those regions.

In selecting the target stars, we do not assume prior knowledge of whether a star is part of a multiple-star system. If it is, we assume that all components of the system fall within a single photometric aperture, and they are all observed at the 2-minute cadence.

Figure 16 illustrates the selection of the target stars on a Hertzsprung–Russell diagram. For clarity, we show a magnitude-limited subsample (${K}_{s}\lt 6$) of our "bright" catalog as well as a randomly selected subsample of the $2\times {10}^{5}$ target stars. Nearly all main-sequence stars with ${T}_{\mathrm{eff}}\lt 6000$ K are selected as target stars. Stars that are larger than the Sun are only included if they have a sufficiently bright apparent magnitude. White dwarfs could also be interesting targets for TESS, but we do not include them in our simulation because the occurrence rates of planets around white dwarfs is unknown.

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Figure 17 shows the distribution of target stars as a function of effective temperature, along with their apparent I_C magnitudes. The distribution in effective temperature of the target stars is bimodal, with a sharp peak near 3400 K and a broader peak near 5500 K.

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In reality, it will not be quite as straightforward to select the target stars for TESS. While proper-motion surveys (e.g., Lépine & Shara 2005) can readily distinguish red giants from dwarf stars, it is much more difficult to distinguish dwarfs from subgiant stars (Stassun et al. 2014). Ultimately, the selection of the TESS target stars may rely on parallaxes from the ongoing Gaia mission (Perryman et al. 2001). Errors in selecting the target stars might be mitigated by simply observing a larger number of stars at a 2-minute cadence. There is also the possibility of detecting transits in the full-frame images (FFIs), which is described below.

TESS will record and downlink a continuous sequence of FFIs with an effective integration time of 30 minutes or shorter. Transiting planets can still be detected with 30-minute sampling, but the longer integration time of the FFIs reduces the sensitivity to events with a short duration. Our simulation estimates the yield of transiting planets from the FFIs in the following fashion.

First, we identify all the transiting or eclipsing stars that are not among the pre-selected $2\times {10}^{5}$ target stars. We assign to each system a random phase between the beginning of a 30-minute window and the beginning of an eclipse. Next, we calculate the number of 30-minute data points that are required to cover the transit or eclipse duration. The data points at the beginning and end of the series are omitted if they do not increase the signal-to-noise. Finally, we compute the effective depth of the transit or eclipse by averaging over all of the 30-minute data points spanning the event. This step can reduce the depth because some of the data points include time outside of the transit or eclipse.

For transits with durations shorter than 1 hr, the 30-minute integration time of the FFIs causes the apparent transit duration to be lengthened and the apparent transit depth to become more shallow. However, the depths and durations of transits with longer durations are largely unaffected. The effects of time averaging on the uncertainties in transit parameters derived from light-curve fitting have been analyzed by Kipping (2010) and Price & Rogers (2014).

Our calculated detection threshold of 7.3 only ensures that no more than one statistical false positive is detected among the $2\times {10}^{5}$ target stars. Since many more stars can be searched for transits in the FFIs, the number of statistical false positives will be much greater than one if the same threshold is adopted.

Total number of detections—Based on five trials with the $2\times {10}^{5}$ target stars, we expect TESS to find 70 ± 9 planets smaller than 1.25 ${R}_{\oplus }$, 486 ± 22 planets in the range 1.25–2 ${R}_{\oplus }$, 1111 ± 122 planets in the range 2–4 ${R}_{\oplus }$, and 67 ± 8 planets larger than 4 ${R}_{\oplus }$. Table 6 presents the catalog of planets from one of these five trials. Figure 20 shows the distribution of detected planets plotted on the radius-period plane, in the same fashion that the input planet occurrence rates were plotted in Figure 8.

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The top panel of Figure 19 maps the simulated planet detections in ecliptic coordinates. Detections among the target stars (red points) are enhanced in the vicinity of the ecliptic poles because of the overlapping pointings they receive. Apart from that conspicuous feature, the detections are nearly uniformly distributed across the sky. The detections from stars that are only observed in the full-frame images (blue dots) show a strong enhancement near the galactic plane. This is due to the vast number of faint and distant stars around which giant planets can be detected.

Habitable-zone planets—Of the 556 planets smaller than 2 ${R}_{\oplus }$, a subset of 48 ± 7 have a relative insolation on the range $0.2\lt S/{S}_{\oplus }\lt 2$ and are therefore near the habitable zone. We also expect a smaller subset of 14 ± 4 to be within the more restricted zone defined by Kopparapu et al. (2013). This definition of the habitable zone extends approximately from $0.2\lt S/{S}_{\oplus }\lt 1$, with the exact bounds depending on stellar effective temperature. Figure 21 shows the distribution of $S/{S}_{\oplus }$ and ${T}_{\mathrm{eff}}$ for the simulated detections in the vicinity of the habitable zone. Because the sensitivity of TESS favors short periods, the potentially habitable planets must orbit low-mass, cool stars with ${T}_{\mathrm{eff}}\lesssim 4000$ K. Furthermore, the yield of such planets depends strongly upon the definition of the inner edge of the habitable zone, but much less so upon the outer edge.

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Small planets with measurable masses—The smallest planets will be of particular interest for mass measurement since there are presently very few small (and potentially rocky) planets with measured masses and sizes. Among the 70 simulated planets smaller than 1.25 ${R}_{\oplus }$, the median period is 2.1 days, and the median stellar effective temperature is 3450 K. The median I_C magnitude is 11.6.

Survey completeness—The degree of completeness of the TESS survey can be assessed by comparing the simulated planet detections against the total number of transiting planets on the sky (as discussed in Section 4.1). Plotted in Figure 22 are the cumulative numbers of transiting planets as a function of the limiting apparent magnitude of the host star. We make the comparison for short-period planets around Sun-like and smaller stars for planets of different sizes as well as small planets near the HZ. For planets with ${R}_{{\rm{p}}}\lt 2\;{R}_{\oplus }$, the completeness of the TESS survey is limited by instrumental noise. For planets with ${R}_{{\rm{p}}}\gt 4\;{R}_{\oplus }$, the completeness is limited by the maximum number of target stars ($2\times {10}^{5}$).

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Diluting flux—Whenever the photometric aperture contains flux from neighboring stars, the measured transit depth will be smaller than it would be if the star were observed in isolation. If this effect is not taken into account (by using observations with higher angular resolution), then the planet's radius will be underestimated. The source of the "diluting flux" can be a star that is gravitationally bound to the target star, or it can be one or more completely unrelated stars along the same line-of-sight. In our simulation, we find that 12% of detected planets suffer dilution by more than $\gt 21\%$, making them vulnerable to radius underestimation by $\gt 10\%$. For 6% of planets, the radii could be underestimated by $\gt 20\%$. We note that we do not consider cases of underestimated planet sizes to be "false positives," in contrast to Fressin et al. (2013). Those authors considered the detection of transits with significant dilution to be a false positive because they were concerned with determining the occurrence rates of planets as a function of planet radius.

A separate scenario in which the transit depth can be diluted is when the transiting planet is actually orbiting a background star rather than the target star. Simulating these background transiting planets is a more computationally challenging problem which we conducted separately from the main simulations. We generated planets around the background stars represented by in "faint" star catalog and simulated the detection of the transiting planets blended with target stars. We found this type of transit detection to be very rare. Of the $2\times {10}^{5}$ target stars, we find that only ∼1 planet transiting a background star will be detectable with TESS. In the 30-minute FFIs, approximately 70 such planets might be detected. The transit depths of these planets must be very deep to overcome the diluting flux of the brighter target star. In the simulations, the median radius of blended transiting planets is $17\;{R}_{\oplus }$. Our conclusion is in agreement with that of Fressin et al. (2013), who found that transits of background stars are a less important source of detections than transits of planets around gravitationally bound companion stars (see their Figure 10).

Single-transit detections—In a few notable cases, the S/N of a transit exceeds the threshold of $7.3$, but only a single transit is observed. We expect 110 such planets to be detected with one transit. These are not counted as detections in the tallies given above, but they are included in Figure 21 as gray points. These planets have longer periods and lower equilibrium temperatures than the rest of the TESS sample. There may even be additional single-transit detections from planets with orbital periods exceeding 1 year, which we have not modeled at all. Although the periods will not be well-constrained using TESS data alone, and the probability of a "detection" being a statistical fluke is higher, it may still be worthwhile to conduct follow-up observations of these stars. The single-transit detections have a median planet size of ∼3 ${R}_{\oplus }$, a median orbital period of ∼30 days, and a median insolation of 1.9 ${S}_{\oplus }$.

Among the $2\times {10}^{5}$ target stars, TESS detects 1103 ± 33 EB systems along with the transiting planets. The uncertainty in this figure is based only on the Poisson fluctuations; we acknowledge that the true uncertainty is likely to be significantly larger. Based on our comparison with the Kepler EB catalog (see Section 4.2), the uncertainty may be as large as 80% for relatively low galactic latitudes.

The false-positives can be divided into the following cases:

• 1.  
  EB: the target star is an EB with grazing eclipses. There are 250 ± 16 detections of EBs.
• 2.  
  Hierarchical EB (HEB): the target star is a triple or quadruple system in which one pair of stars is eclipsing. There are 410 ± 20 detections of HEBs.
• 3.  
  BEB: the target star is blended with a BEB. There are 443 ± 20 detections of BEBs.

These tallies are also illustrated in Figure 18. The bottom panel of Figure 19 shows a sky map of the astrophysical false positives in the same coordinate system as the top panel. The surface density of false positives is a much stronger function of galactic coordinates than the density of planet detections, for binary eclipses are deeper than planetary transits and can be detected out to greater distances. The period and depth distributions of the EB population is discussed in Section 8.6.

The members of a close binary exert strong tidal gravitational forces on one another, causing their photospheres to deform into ellipsoids. These deformations lead to ellipsoidal variations in the light curve. A model for these photometric variations was presented by Morris & Naftilan (1993). Mazeh (2008) gave a simple expression for the dominant component, which has a period equal to half of the orbital period, and a semi-amplitude

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}{{\rm{\Gamma }}}_{1}}{{{\rm{\Gamma }}}_{1}}=0.15\displaystyle \frac{(15+{u}_{1})(1+\tau )}{(3-{u}_{1})}q{\left(\displaystyle \frac{{R}_{1}}{a}\right)}^{2}{\mathrm{sin}}^{2}i,\end{eqnarray} \tag{ 24 }$$

where R₁ is the primary radius, a is the orbital distance, i is the orbital inclination, u₁ is the linear limb-darkening coefficient, τ is the gravity-darkening coefficient, and q is the mass ratio. To estimate the amplitude of this effect for our simulated TESS detections, we adopt an appropriate value of u₁ for each star using the tables of Claret et al. (2012, 2013), which come from the PHOENIX stellar models. For gravity darkening, we use a value of τ = 0.32 for all stars, which is thought to be appropriate for stars with convective envelopes (Lucy 1967).

The formal detection limits for ellipsoidal variations are quite low because the signal is present throughout the entire light curve rather than being confined to eclipses of a narrower duration. Since the period and phase are fixed from the observed eclipses, we model the detection of the ellipsoidal variations as a cross-correlation of the light curve with a cosine function of the appropriate period. If the fractional uncertainty in the flux of each data point is σ, and the total number of data points is N, then the S/N of ellipsoidal variations is

$$\begin{eqnarray}&&{{\rm{S}}/{\rm{N}}}_{\mathrm{EV}}=\displaystyle \frac{{\rm{\Delta }}{{\rm{\Gamma }}}_{1}}{{{\rm{\Gamma }}}_{1}}\displaystyle \frac{\sqrt{N}}{D\sigma \sqrt{2}}.\end{eqnarray} \tag{ 25 }$$

Here, D denotes the dilution of the target star in the photometric aperture, which is defined in Section 6.3. Due to this factor, ellipsoidal variations from BEBs are more difficult to detect since their eclipses are usually more diluted than EBs and HEBs. The factor of $\sqrt{2}$ arises from the rms value of a cosine function.

It seems likely that correlated noise will prevent the detection limit from averaging down to extremely low values as the duration of observations is extended. Somewhat arbitrarily, we require the semi-amplitude of the ellipsoidal variations to exceed 10 ppm, in addition to the criterion ${\rm{S}}/{{\rm{N}}}_{\mathrm{EV}}\gt 5$, to be counted as "detectable." We also require that the orbital period of the binary, which is twice the period of ellipsoidal variations, is shorter than one spacecraft orbit (13.6 days) out of concern that thermal or other variations of the satellite will induce systematic errors with a similar frequency. Under these detection constraints, shown in Figure 23, ellipsoidal variations are detected for 34% of the EBs in the simulation. The majority of these are grazing-eclipse binaries rather than HEBs or BEBs. The results are summarized in the second column of Table 5.

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Another key difference between EBs and transiting planets is that the secondary star in a binary is more luminous than a planetary companion. This distinction is somewhat blurred when comparing brown-dwarf and hot-Jupiter companions but is quite clear between ordinary stars and lower-mass planets. If the two stars in a binary have nearly the same surface brightness, then the depths of the primary and secondary eclipses will be indistinguishable. In this case, the system might appear to be a planet with an orbital period equal to half of the true orbital period of the binary. However, if the surface brightnesses of the stars differ and both eclipses are detected with a sufficiently high S/N, then the secondary eclipse can be distinguished from the primary eclipse and the system can be confidently classified as an EB.

To estimate the number of cases for which the primary and secondary eclipses are distinguishable, we identify the simulated systems for which signal-to-noise of the secondary eclipses, S/N₂, is $\gt 5$, and the S/N in the difference between the primary and secondary eclipse depths, ${\rm{S}}/{{\rm{N}}}_{1-2}$, is also $\gt 5$. The latter quantity is calculated as

$$\begin{eqnarray}&&{{\rm{S}}/{\rm{N}}}_{1-2}=\displaystyle \frac{{\delta }_{1}-{\delta }_{2}}{\sqrt{{\sigma }_{1}^{2}+{\sigma }_{2}^{2}}},\end{eqnarray} \tag{ 26 }$$

where ${\delta }_{\mathrm{1,2}}$ denote the depths of the eclipses and ${\sigma }_{\mathrm{1,2}}$ denote the noise in the relative flux over the observed duration of each eclipse. Figure 24 shows the detectability of secondary eclipses by plotting ${\rm{S}}/{{\rm{N}}}_{1-2}$ versus S/N₂. The secondary eclipse can be distinguished from the primary eclipse for the systems that lie in the upper-right quadrant of the plot.

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The results are also summarized in the third column of Table 5. A majority of the false positives have detectable secondary eclipses that are distinguishable in depth from the primary eclipses. The notable exceptions include the HEBs in which the eclipsing pair consists of equal-mass stars ($q\approx 1$). In such cases, ${\delta }_{1}\approx {\delta }_{2}$ and it is impossible to distinguish between primary and secondary eclipses. For the BEBs, the difficulty is that the eclipse depths are often strongly diluted and the secondary eclipses are not detectable. Most planets are too small and faint to produce detectable secondary eclipses in the TESS bandpass. In the simulations, the fraction of detected planets with detectable secondary eclipses is only 0.01%.

EBs can also be distinguished from transiting planets based on the more prolonged ingress and egress phases of stellar eclipses. As above, we adopt an S/N threshold of 5 for the ingress/egress phases to be detectable. The average "signal" during ingress and egress is half the maximum eclipse depth, and the "noise" is calculated for the combined durations of ingress and egress. In order to ensure that the ingress/egress can be temporally resolved, we require the duration of the ingress or egress to be more than twice as long as the duration of an individual data sample (2 minutes for the target stars and 30 minutes for the rest of the stars).

Since transiting planets generally have ingrees or egress phases lasting a few minutes, TESS will only be able to detect the ingress/egress for a small fraction (≈10%) of transiting planets observed with 2 minute sampling. Only large planets observed in the 30 minutes. FFIs would have resolvable ingress/egress. However, the the ingress/egress phases of EBs are more readily detectable.

We note that the detection of the ingress/egress alone does not classify a signal as an EB. One would next examine the period and shape of the eclipse signals to determine whether the radius of the eclipsing body is consistent with the observed depth.

Figure 25 illustrates the detection of ingress/egress for planets and false positives. The fourth column of Table 5 summarizes the results. Approximately 70% of the EB systems that TESS detects among the target stars might be classified as false positives by virtue of a lengthy ingress or egress duration. For stars that are only observed at a 30-minute cadence, this method is not as effective.

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Another diagnostic of false positives, particularly BEBs, is the centroid motion that accompanies the photometric variations. If there are detectable shifts in the centroid of the target star during transit or eclipse events, it is more likely that the target is a blended EB rather than a transiting planet or an eclipse of the target star itself. Transits or eclipses of the target star can still have significant centroid motion if another bright star is blended with the target.

With real data, one could interpret the amplitude and direction of the measured centroid shift using the known locations of neighboring stars in order to determine the most likely source of the photometric variations. This is a complicated process to simulate, so we simply investigate the issue of the detecting the centroid shift. As verified in our simulations, the systems with detectable centroid shifts are much more likely to be false positives than transiting planets.

We simulate the detectability of centroid shifts by calculating the two-dimensional centroid (center-of-light) of the target star, C_x and C_y, within the 8 × 8 synthetic images described in 6.2. We calculate the centroids both during and outside of the loss of light to find the magnitude and direction of the centroid shift. Next, we calculate the uncertainty in the centroid ${\sigma }_{{Cx}}$ and ${\sigma }_{{Cy}}$, which stems from the photometric noise of each pixel. If each pixel $(i,j)$ has coordinates $(x,y)$, and its photometric noise relative to the total flux is denoted by ${\sigma }_{i,j}$, then the noise propagates to the centroid measurement uncertainty through

$$\begin{eqnarray}&&{\sigma }_{{Cx}}^{2}=\displaystyle \sum _{i}{({x}_{i}-{C}_{x})}^{2}{\sigma }_{{ij}}^{2}\quad \mathrm{and}\quad {\sigma }_{{Cy}}^{2}=\displaystyle \sum _{j}{({y}_{j}-{C}_{y})}^{2}{\sigma }_{{ij}}^{2}.\end{eqnarray} \tag{ 27 }$$

In an analogous fashion to determining the optimal photometric aperture, we select the pixels that maximize the S/N of the centroid measurement. Finally, we project the x and y centroid uncertainties in the direction of the centroid shift. The S/N of the centroid measurement is the magnitude of the centroid shift divided by the centroid uncertainty projected in the direction of the centroid shift. We consider a centroid shift to be detectable if the signal-to-noise is 5 or greater.

In practice, the centroid measurement uncertainty could be much larger if the spacecraft jitter does not average down during the hour-long timescales of transits and eclipses. On the other hand, monotonic drifts in the spacecraft pointing during a transit or eclipse are less likely to impact the centroid measurement since the motion is common to all stars.

We find that centroid shifts can be detected for 69% of the BEBs and HEBs. These results are illustrated in Figure 26 and summarized in column 5 of Table 5. The BEBs have a higher fraction of detectable centroid shifts from the larger angular separations between the eclipsing system and the target star. Only 6% of planet transits produce a detectable centroid shift.

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As shown in Table 5, the simulations suggest that blended EBs are the type of false positive that is most difficult to identify based only on TESS data. Assuming that all of the false-positive tests described in the previous sections are applied, approximately 150 of the 1103 ± 33 false positives would fail to be identified. The large majority (78%) of these more stubborn cases are BEBs.

If archival images or catalogs do not reveal a system in the vicinity of a TESS target star that is consistent with any measurable centroid motion, then additional imaging is needed. An effective way to identify these BEBs is through ground-based imaging with higher angular resolution than the TESS cameras. A series of images spanning an eclipse could reveal which star (if any) is the true source of variations. Due to the large pixel scale of the TESS optics, it will not be difficult to improve upon the angular resolution with ground-based observations. Even modest contrast and a well-sampled PSF can resolve many ambiguous cases.

Figure 27 illustrates the requirements on angular resolution and contrast. For each BEB, we have plotted the angular separation and the J-band magnitude difference between the BEB and the target star. Natural-seeing images with 1'' resolution would be sufficient to resolve all of the simulated BEBs. In more difficult cases, adaptive optics might be necessary to enable high contrast.

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Figure 28 shows the photometric requirements to detect the planets as well as BEBs and other eclipsing systems for which the TESS photometry cannot distinguish whether the candidate is a false positive. We plot the eclipse depth against apparent system magnitude to indicate the photometric precision that is required of the facilities performing these observations.

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The false positives and transiting planets have significantly different distributions of orbital period, eclipse/transit depth, and galactic latitude. Therefore, the likelihood that a given source is a false positive can be estimated from the statistics of these distributions in addition to the characteristics described above that can be observed on a case-by-case basis.

Figure 29 shows the distributions of apparent period and apparent depth of the eclipses caused by transiting planets and false positives. Here, the "apparent period" is the period one would be likely to infer from the TESS photometry; if the secondary eclipse is detectable but not distinguishable from the primary eclipse, one would conclude that the period is half of the true orbital period. The "apparent depth" takes into account the dilution of an eclipse from background stars or, in the case of BEBs, the dilution from the target star.

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These populations are seen to be quite distinct. EB systems tend to have larger depths and shorter periods than planets. Simply by omitting sources which have eclipse/transit depths >5% or periods <0.5 days, approximately 83% of the false positives among the target stars would be discarded.

The galactic latitude b of the target also has a strong influence on the likelihood that a given source is a false positive. Figure 30 shows the fraction of detections that are due to planets, BEBs, and other false positives as a function of galactic latitude. Only the events with apparent depth <10% are included in this plot. For $| b| \lt$ ${10}^{\circ }$, the density of background stars is very high, and any observed eclipse is far more likely to be from a BEB than any other kind of eclipse. For $| b| \gt$ 20°, planets represent a majority over false positives. A weaker dependence on galactic latitude is seen for grazing-eclipse binaries and HEBs.

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The TESS planets should be attractive targets for radial-velocity observations because the host stars will be relatively bright and their orbital periods will be relatively short. Both of these factors facilitate precise Doppler spectroscopy. To evaluate the detectability of the Doppler signal we assign masses to the simulated planets using the empirical mass–radius relation provided by Weiss et al. (2013). For ${R}_{{\rm{p}}}\lt 1.5\;{R}_{\oplus }$, the planet mass M_p is calculated as

$$\begin{eqnarray}&&{M}_{{\rm{p}}}={M}_{\oplus }\left[0.440{\left(\displaystyle \frac{{R}_{{\rm{p}}}}{{R}_{\oplus }}\right)}^{3}+0.614{\left(\displaystyle \frac{{R}_{{\rm{p}}}}{{R}_{\oplus }}\right)}^{4}\right],\end{eqnarray} \tag{ 28 }$$

and for ${R}_{{\rm{p}}}\geqslant 1.5\;{R}_{\oplus }$, the mass is calculated as

$$\begin{eqnarray}&&{M}_{{\rm{p}}}=2.69\;{M}_{\oplus }{\left(\displaystyle \frac{{R}_{{\rm{p}}}}{{R}_{\oplus }}\right)}^{0.93}.\end{eqnarray} \tag{ 29 }$$

This simple one-to-one relationship between mass and radius is used here for convenience. In reality, there is probably a distribution of planet masses for a given planet radius (see, e.g., Rogers 2014).

From the masses calculated here, we then find the radial-velocity semiamplitude K, which is reported in Table 6. Figure 31 shows K values of each planet detected in one trial as a function of the apparent magnitude of the host star. Because of the short periods, even planets smaller than 2 ${R}_{\oplus }$ will produce a radial-velocity semiamplitude K close to 1 m s⁻¹, putting them within reach of current and upcoming spectrographs.

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The composition of planetary atmospheres can be probed with transit spectroscopy. Such measurements can be carried out with space-based or balloon-based facilities, or even from ground-based facilities if the resolution is high enough to separate telluric features from stellar and planetary features. The enhanced sensitivity of TESS to transiting planets near the ecliptic poles will provide numerous targets for observations inside or near the continuous viewing zone of the James Webb Space Telescope (JWST). The prospects for follow-up with JWST have been detailed in Deming et al. (2009) and elsewhere. More specialized space missions, including FINESSE (Deroo et al. 2012) and EChO (Tinetti et al. 2012), have also been proposed to perform transit spectroscopy.

Here, we use the simulation results to explore the relative difficulty of transit spectroscopy of the TESS planets independent from the facility that is used to observe them. We compute a figure-of-merit ${\delta }_{{\rm{H}}}$, which is the fractional loss-of-light from an annulus surrounding the planet (with radius R_p) and a thickness equal to the scale height, H:

$$\begin{eqnarray}&&{\delta }_{{\rm{H}}}=\displaystyle \frac{2{{HR}}_{{\rm{p}}}}{{R}_{\star }^{2}}.\end{eqnarray} \tag{ 30 }$$

The scale height is calculated from

$$\begin{eqnarray}&&H=\displaystyle \frac{{k}_{{\rm{B}}}{T}_{{\rm{p}}}{R}_{{\rm{p}}}^{2}}{{{GM}}_{{\rm{p}}}\mu {m}_{{\rm{p}}}},\end{eqnarray} \tag{ 31 }$$

where M_p is the planet mass and m_p is the proton mass. We calculate the temperature of the planet, T_p, assuming it is in radiative equilibrium with zero albedo and isotropic re-radiation (see Equation (12)). We assume a mean molecular weight μ of 2 amu, which corresponds to an atmosphere consisting purely of H₂. In any other case, the atmospheric transit depth ${\delta }_{{\rm{H}}}$ is reduced by a factor of μ/2. An Earth-like atmosphere would have μ = 29 amu, and a Venusian atmosphere would have μ = 44 amu.

Figure 32 shows ${\delta }_{{\rm{H}}}$ for all of the detected planets in the simulation as a function of the apparent magnitude of the host star. For a molecular species to be identifiable, one must observe transits with a sensitivity on the order of ${\delta }_{{\rm{H}}}$ both in and out of the absorption bands of that species. The detection of various species therefore depends on the depth of the absorption bands and the spectral resolution used to observe them. The presence of clouds and haze can reduce the observable thickness of the atmosphere.

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Next, we look specifically at the number of planets with a relative insolation $0.2\lt S{/S}_{\oplus }\lt 2$, placing them within or near the habitable zone. These planets are especially attractive targets for atmospheric spectroscopy because they may have atmospheres similar to that of the Earth, and may present "biomarkers" indicative of life. Such observations are most feasible for the planets with the brightest possible host stars. For that reason, we show in Figure 33 the cumulative distribution of apparent K_s magnitudes of stars hosting planets with $0.2\lt S/{S}_{\oplus }\lt 2$. With the statistical errors in the planet occurrence rates and Poisson fluctuations in the number of detected planets, between 2 and 7 planets with $0.2\lt S/{S}_{\oplus }\lt 2$ and ${SR}\lt 2\;{R}_{\oplus }$ have host stars brighter than ${K}_{s}=9$.

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Of particular interest for atmospheric spectroscopy with JWST are the planets that are located near the continuous-viewing zones of JWST, which will be centered on the ecliptic poles. A subset of 18 ± 5 planets with ${R}_{{\rm{p}}}\lt 2\;{R}_{\oplus }$ and $0.2\lt S/{S}_{\oplus }\lt 2$ are found within 15° of the ecliptic poles. The brightest stars hosting these planets have ${K}_{s}\;\approx$ 9.