EDEN: Sensitivity Analysis and Transiting Planet Detection Limits for Nearby Late Red Dwarfs

EDEN observations are currently conducted with eight unique telescopes at seven observatories in North America, Europe, and Asia. The telescopes are the Kuiper 1.55 m (Mount Bigelow, Arizona), Bok 2.3 m (Kitt Peak, Arizona), Vatican Advanced Technology Telescope 1.8 m (VATT; Mount Graham, Arizona), Phillips 0.6 m and Schulman 0.8 m (Mount Lemmon, Arizona), Calar Alto 1.23 m (Calar Alto, Spain), Cassini 1.52 m (Mount Orzale, Italy), and Lulin 1 m (Mount Lulin, Taiwan). Table 1 details the location, design, and CCD imager of each telescope. With the exception of the robotic Schulman and Phillips telescopes, each of them is manually controlled by an observer, who actively monitors weather conditions and instrument performance during the course of a night. While the telescope designs are varied, each of the telescopes has been carefully evaluated for photometric performance before its inclusion in EDEN and, when necessary, changes have been made in the telescope's operation and setup, which will be detailed in D. Apai et al. (2020, in preparation). Systematic differences between telescopes therefore have very minor effects on the final lightcurves and transit search. These differences can be compensated for during the data reduction and detrending steps, discussed in Sections 3 and 4.

Table 1.  EDEN Telescopes

Telescope	Location	Operation	Mount	CCD Imager	Det. Size	FOV	Px. Scale	Qe at 700 nm
Phillips 0.6 m	Mount Lemmon, Arizona	Robotic	EQ	SBIG STX (KAF-16803)	4096 × 4096	22' × 22'	035	40%
Schulman 0.8 m	Mount Lemmon, Arizona	Robotic	EQ	SBIG STX (KAF-16803)	4096 × 4096	22' × 22'	035	40%
Lulin 1.0 m	Mount Lulin, Taiwan	Classical	EQ	Sophia 2048B CCD	2048 × 2048	1308 × 1308	039	60%
Calar Alto 1.23 m	Calar Alto, Spain	Remote	EQ	DLR-MKIII camera with e2v CCD231-84-NIMO-BI-DD sensor	4k×4k	215 × 215	031	93%
Cassini 1.52 m	Mount Orzale, Italy	Classical	EQ	Bologna Faint Object Spectrograph and Camera	1300 × 1340	13' × 126	034	75%
Kuiper 1.55 m	Mount Bigelow, Arizona	Classical	EQ	Mont4K SN3088 (Weiner et al. 2018)	4096 × 4097	97 × 97	014	62%
VATT 1.8 m	Mount Graham, Arizona	Classical	Alt-Az	VATT4K STA0500A CCD	4064 × 4064	125 × 125	0188	80%
Bok 2.3 m	Kitt Peak, Arizona	Classical	EQ	90 Prime Focus Wide-Field Imager (Grant Williams et al. 2004)	4 × 4032 × 4096	116 × 116	04	80%

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The majority of the EDEN telescopes are not solely dedicated to EDEN, so observations are scheduled at each facility individually in blocks usually from two to 10 days per month, depending on availability. Observing science targets at these sites has been ongoing since 2018 June (following a six month long EDEN pilot program), with observations of the targets discussed in this paper occurring between 2018 June and 2019 February.

EDEN's primary focus is to search for potentially habitable planets within 15 pc (∼50 lt-yr). Correspondingly, for the EDEN Transit Survey, our target selection prioritizes M4 and later-spectral-type host stars, which offer favorable planet-to-star projected areal ratios, making broadly Earth-sized planets detectable in our data. We eliminate known close binary stars that may reduce the stability of putative planets and would complicate the interpretation of the lightcurve. We then prioritize sources that are too faint (I > 15 mag) to be efficiently searched by TESS or are outside TESS's sky coverage. In addition to these high-priority EDEN targets we also include separately targets of particular interest in our source catalog. Such targets may be exoplanet candidate host stars (from radial velocity (RV) or transit searches), for which EDEN data can prove valuable for candidate verification. Such follow-up targets (where prior knowledge about a planet's presence exists) will not be used in future exoplanet occurrence rate studies.

EDEN targets, including those discussed in this paper, are late-M dwarfs scattered throughout the Northern Hemisphere sky and thus must be observed one at a time. For planets orbiting within the habitable zone of these stars or closer (e.g., Kasting et al. 1993; Kopparapu et al. 2014), expected transit durations range from 0.5 to 3 hr at periods of roughly 0.5–10 days.

To maximize the probability of observing transits with these parameters and to take advantage of the longitudinal coverage of EDEN telescopes, we designed our observing strategy around two pillars. First, we observe each target for as long as possible on a given night. This typically means that on a clear night we observe a primary target for >6 hr, and then a secondary target for 2–3 hr when the primary is not observable. This also increases the chance of observing a full transit, which is easier to detrend and detect than fractional transits. Second, whenever possible, we schedule simultaneous observing campaigns in Arizona, Europe, and Taiwan to allow the potential for continuous 24 hr monitoring of one target for multiple days. On such longer, coordinated runs—given good weather at all sites—we can obtain roughly week-long continuous sequences, limited only by our allocated time on these facilities.

These pillars allow us to quickly get good phase coverage of a target for shorter-period planets. Practically, continuous observation has been difficult to fully exploit because of the rarity of getting good weather on three continents during the entire run. The number of nights dedicated to any target is based on the probability that we would have observed two transits of a planet with an orbital period of less than 10 days. As this probability increases, we deprioritize a given target so that more targets can be adequately sampled. While it is not practically possible to reach 100% detection probability for planets throughout the entire habitable zone (from inner to outer edge; Kopparapu et al. 2014), we aim to reach high sensitivity for transiting planets that orbit at the inner edge of the habitable zone (i.e., ∼50% successful detection of Earth-size transiting planets), which typically translates to some sensitivity (≳10%) throughout the habitable zone.

2.3.1. Observational Procedures

Although each of our telescopes has somewhat different capabilities and performance, we adopt the same observational procedures at each telescope to minimize systematic differences.

Filter. For each telescope we use a near-infrared (NIR) (or blue-blocking) filter, such as Harris-I or similar. This filter choice maximizes the collected photons from our targets, which are brightest in the NIR, while blocking unwanted sky background from the Moon and skyglow. Since spring 2019, the filter has been standardized at all telescopes to an uncoated GG 495¹⁶ glass long-pass filter (transparent at >500 nm). Redder filters such as I or z' have been occasionally used for bright targets if the sky background is very high, for example, during a full Moon. The z' is otherwise generally avoided because of the low quantum efficiency of most CCD detectors at those wavelengths and the greater presence of telluric absorption bands from water vapor (Bailer-Jones & Lamm 2003; Blake et al. 2008).

Exposure Time. The exposure time is chosen to balance competing signal-to-noise and cadence considerations. We never allow the peak target flux to go above ∼60% the detector's full well, where the detector begins to exhibit non-linear behavior. In a given period of time, such as a transit duration, the total Poisson-noise-driven signal-to-noise ratio follows the relationship

$$\begin{eqnarray*}&&{{\rm{S}}/{\rm{N}}}_{\mathrm{tot}}\propto \sqrt{\displaystyle \frac{R}{1+R}},\end{eqnarray*}$$

where R is the ratio of the exposure time to readout time (Howell & Tavackolimehr 2019). This relationship levels off at R ∼ 3.5, and we thus aim for an exposure time of ∼3.5× the readout time. For our telescopes with a diameter larger than 1 m and targets with magnitude I ∼ 14, this gives a cadence <60 s.

Focus. Previous work (e.g., Southworth et al. 2009) has shown that defocusing can result in more precise lightcurves as the point-spread function (PSF) is spread across more pixels. We aim for a slight-to-moderate defocus of 2''–3'', so that pixel-to-pixel variations are reduced, but the PSF maintains a Gaussian shape. Since defocusing also reduces the peak of the PSF, it has the additional benefit of allowing longer exposures.

We follow standard calibration procedures for flat, bias, and dark corrections to reduce systematic effects on our lightcurves. Detailed tests (complete re-reduction and analysis of selected data sets) show that the details of the basic calibration do not affect the resulting lightcurve precision significantly.

For our calibration procedure, before or after every night of observation, we collect ∼10 twilight flat-fields with exposure times chosen to maintain a sky flux approximately at 50% the detector's full well, the same as our desired peak target flux. In some cases of inclement weather during twilight, we may use dome flats, but these are not preferred since they have less uniform illumination. The minimum flat exposure time is always long enough so that the shutter time has <1% effect on the precision of the flat.

Generally, at least once per observing run, we collect a set of bias and dark frames. The dark current for our exposure times is nearly zero at all telescopes and is usually not subtracted. At some telescopes darks are not collected for this reason. There is no evidence for persistence on any of our detectors.

The first step in edenAP is to calibrate the raw science frames using the calibration frames discussed in Section 2.4. In the event that calibration frames are not available or are of poor quality, this step can be skipped with the rest of the pipeline remaining the same. To create master calibration frames, we collect all bias frames within one month of the observation, and all dark and flat frames within the observation run. Monitoring of flat fields has indicated that these stay mostly constant over the course of a run, with the exception of minor localized dust accumulation and chance occurrences such as insects getting trapped in the optical path. In cases where many hundreds of calibration frames are available in the above time periods, we narrow the period and only collect calibration frames within two to three days of the observation.

We then derive the astrometric solution for every science frame by using a local installation of the astrometry.net software package (Lang et al. 2010). While this solution provides accurate astrometric calibration for most frames, it can fail in case of partial cloud cover or poor seeing. If no astrometric solution can be found for a particular image, the solution from the preceding image is used, despite these data typically being very poor. The astrometric solution derived is used as a first guess for placing photometric apertures; however, we always refine the centroid using the photutils¹⁷ DAOStarFinder method (Bradley et al. 2019), based on the DAOFIND algorithm (Stetson 1987). Position refinement is key to getting sub-pixel centroid precision, especially for our high proper motion target stars.

Aperture photometry is performed using the photutils package (Bradley et al. 2019). For every star in the field of view, we measure the intensity in apertures ranging from 5 to 50 pixels in steps of 1 pixel. The aperture size that minimizes the rms scatter of the target star lightcurve is selected as the best aperture for all sources. The optimal size depends on detector and seeing, but typical sizes are roughly a few arcseconds. Sky background is calculated as the median of a 60 × 60 pixel sub-image around the star with other sources clipped. Photometry is saved into a Python pickle file with other important information for each star, such as centroid positions, stellar magnitudes, background, FWHM, airmass, etc., which can later be used for detrending steps and vetting transit-like signals.

The final step in edenAP is to detrend the target lightcurve on the basis of comparison star lightcurves. Trends are long- or short-term photometric variations in the lightcurve that decrease transit detection sensitivity, and can arise from instrumental, atmospheric, and stellar variability. We select the best comparison stars by first filtering out stars that are saturated, are too faint (several magnitudes dimmer than the target), or have too many failed photometric measurements. Next, we divide the flux-normalized target lightcurve by the normalized lightcurves of every comparison star, and rank them based on the average standard deviation in windows of 20 data points. The six with the lowest average deviation (i.e., those with the most similar data trends) are median-combined into a "super comparison" lightcurve, which the target lightcurve is then divided by. For crowded fields with many available comparison stars, it is conceivable that this selection method could weaken or remove transit signals. We believe this is highly unlikely, however, due to the improbability that comparison lightcurves would have the necessary shape to remove a transit, and because the duration of the window is shorter than any expected non-grazing transit. Nevertheless, we account for this in our sensitivity analysis (Section 6.2) by re-selecting comparison stars after injecting transits.

We visually inspect every lightcurve on a single EDEN target to ensure that lightcurve anomalies are recognized and managed correctly. We select high-quality data for further analysis without relying on automatic algorithms. To streamline this process, we have implemented an interactive data viewer that displays each lightcurve along with systematic trends, allowing the user to flag large sections of problematic data (e.g., stellar flares, passing clouds) for removal and points of interest (a transit-like feature) for further analysis. Excluding poor-quality data is exceedingly important because strong systematic trends can be fit as transits, and they can throw off the correct period determination if one transit of an otherwise detectable period happened to occur within it. Individual outlier data points are ignored in this step, but are efficiently removed by our automatic filtering in the next step.

After visual inspection, lightcurves undergo automated data cleaning and detrending. We fit a long-term trend with a median filter of two hours and 2σ-clip upper outlying data before dividing out the trend. We do not clip below the median because of the risk of clipping deep transits. Median filtering will reduce the depth of all transits slightly, though our use of a two-hour filter window minimizes this effect for transits with durations of less than one hour, which comprises most of our discovery space. An example of median detrending applied to a real EDEN lightcurve with an injected transit of ∼1% depth and TRAPPIST-1 b parameters is shown in Figure 1.

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While median detrending is a simple method, its effects are predictable and robust. Although the median filtering will not remove short-period, transit-like trends, it will not remove real transits either, if they are deeper than a few tenths of a percent (a danger of more complicated detrending techniques). Other trend-fitting methods with which we have experimented when performing transit injection tests include Savitzky–Golay (Savitzky & Golay 1964), biweight, and multivariate polynomials constructed from external parameters such as airmass, and centroid positions. Savitzky–Golay and biweight filtering have very similar results to median detrending, and while multivariate polynomials can outperform median filters, they are also more likely to accidentally remove a real transit feature. Despite their relative simplicity, median filters are reliable (Hippke et al. 2019).

To search for transits in our detrended lightcurves, we utilize the package transit least squares (TLS; Hippke & Heller 2019). The primary improvement over box least squares (BLS; Kovács et al. 2002) is that, rather than fitting a boxcar model to a time series, TLS fits a more realistic, fixed transit shape with limb-darkening included, but the same parameters as BLS otherwise. We optimize the TLS algorithm for our search by setting upper and lower limits on the stellar radius and mass to those for M dwarfs ($0.1\mbox{--}0.6{R}_{\odot }$, $0.08\mbox{--}0.5{M}_{\odot }$) and the maximum period to correspond to the approximate outer edge of the habitable zone (∼10 days). We rely on our previously described data-cleaning and detrending steps to remove bad data, and all lightcurves are weighed equally regardless of photometric precision. For each search we save a median-smoothed periodogram, as well as the phase folded model, transit parameters, false alarm probability (FAP), and signal detection efficiency (SDE) for the highest power period.

Most transit candidates identified by TLS are false positives—and often obvious ones. Currently, vetting is done manually, but it may be automated in the future. The first check of a candidate is inspection of the viability of the TLS output: are the transit parameters physical? Does the phase-folded lightcurve have obvious flares or systematic trends? What are the SDE and FAP values? If these are viable, the interactive data viewer is used to look at systematic trends during transit times, which usually reveal systematic noise sources that introduced the feature. We pursue follow-up observation to eliminate astrophysical false positives (such as eclipsing binaries) only after identifying a promising transit candidate not explainable by other means. We do not specifically set SDE or FAP values to eliminate transit candidates, and consider even those with poor statistics. However, we do perform an analysis in Section 6 of the SDE and FAP values that indicate a robust detection.

EIC-1 is an M8.5V dwarf located 5.7 pc away (Reid et al. 2003). It was discovered and identified as a nearby dwarf by Lépine et al. (2002) as part of the Digitized Sky Survey. Its brown dwarf status is currently unknown due to differing lines of evidence (Reiners & Basri 2009; Berdyugina et al. 2017; Saur et al. 2018). It is a known radio pulsator with a strong magnetic field and a rapid 2.84 hr rotation period (Berger et al. 2008; Hallinan et al. 2008, 2015; Berdyugina et al. 2017; Kuzmychov et al. 2017). A possible detection of auroral emission has recently been reported for this target (Hallinan et al. 2015).

EIC-1 has been the target of RV observations by CARMENES (Tal-Or et al. 2018) and Keck NIRSPEC (Tanner et al. 2012), some photometric monitoring by MEarth (Dittmann et al. 2016), a wide-orbiting companion search by Spitzer IRAC (Carson et al. 2011), and Subaru adaptive optics observations (Siegler et al. 2005), as well as numerous spectroscopic studies from UV to radio wavelengths. We are unaware of any companion candidates from these observations, but note that CARMENES identified it as "active RV-loud," potentially making the detection of habitable planets difficult by RV. EIC-1 was not observed by K2 and is scheduled to be observed by TESS in Sector 26 in 2020 June.

EIC-2 is an M8V dwarf located 14.7 pc away, identified by Kirkpatrick et al. (1995). It has a rotational period of 0.61 days (Irwin et al. 2011) and is a known flare star with a previously observed giant flare by XMM-Newton (Stelzer et al. 2006).

EIC-2 has been the target of RV observations by the Red-Optical Planet Survey (Barnes et al. 2014) and Keck NIRSPEC (Rodler et al. 2012; Tanner et al. 2012), which have 2σ sensitivity to $M\sin i\gt 3.0$ M_⊕ throughout the habitable zone. It has also had periodic observations by MEarth (Dittmann et al. 2016). It was not monitored by K2 or Spitzer and is not scheduled to be observed by TESS until after the primary mission due to its location near the ecliptic.

EIC-3 is an M8 dwarf located 15.6 pc away, identified by Lépine & Shara (2005). Compared to EIC-1 and EIC-2, it has been the target of relatively few observations. It has been observed as part of MEarth and the SDSS-III APOGEE Radial Velocity Survey (Deshpande et al. 2013). It was not observed by K2 or Spitzer and is scheduled to be observed by TESS in Sector 20 in 2020 January.

EIC-1, EIC-2, and EIC-3 were observed for 200–300 hr each from 2018 June to 2019 February, with 40–60 individual observations per target (see Table 3). The observations are highly clustered in time, with a few periods of continuous or nearly-continuous observations at different observatories lasting 24 hr or more.

Roughly 60%–80% of the cleaned, detrended data are of sufficient quality for a subsequent transit search; the rest is affected by bad weather conditions or technical issues. Durations for the individual high-quality lightcurves range between 2 and 10 hr, depending on target priority, observability, and weather. Some gaps less than 2 hr long exist within longer lightcurves because of passing clouds, temporary technical issues, or manual removal of flares or poor data sections. Cadences vary by a factor of  ∼2–3 depending on the telescope (with higher cadence for larger primary mirrors) and detector readout times. The average median unbinned precision for lightcurves on a target is ∼0.28%. Trends are variable, but most lightcurves have nearly linear or parabolic variations of 1%–3% over their duration, possibly attributable to changing airmass or ponting drift. A sample of detrended lightcurves for EIC-2 for each telescope is shown in Figure 2.

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Each target shows evidence for stellar activity, which is expected given their spectral type and previous observations described in Section 5. EIC-2 and EIC-3 have occasional flaring activity above 1%. Lightcurve segments with clearly identifiable flares were removed manually before the transit search. Less than five flares were removed for both targets, representing a negligible loss in time. EIC-1 exhibits regular variability with a 0.5%–1% amplitude, consistent with the rotational period of ∼3 hr (Berger et al. 2008). This variation can mimic transit-like signals, and thus reduces our transit detection sensitivity for the target.

To assess the transit detection capability of our observations, we implement a transit injection and recovery routine. We inject realistic transits into our raw target lightcurves using the analytic solutions of Mandel & Agol (2002) as implemented in batman (BAsic Transit Model cAlculatioN; Kreidberg 2015), re-select comparison stars with the same procedure described in Section 3.4, and attempt to recover the transit signals using our detrending and transit search pipeline. We also perform a limited visual transit recovery test to compare the sensitivity of the pipeline to a manual search by eye.

6.2.1. Manual Transit Search

6.2.2. Visual Transit Recovery Tests

6.2.3. Automated Transit Recovery Tests

6.2.4. Positive Identification of Transits

For us to consider a transit detected by TLS to be a true positive result, it must meet one of the following two criteria: (1) the best period is less than 0.5 hr different from the true period of the injected planet, or (2) at least one identified transit midpoint time is within 20 minutes of a real injected transit midpoint (i.e., a transit candidate is correctly identified, but the period is incorrect). All candidates that meet condition (1) naturally meet condition (2).

We make an additional distinction between true positives recovered by TLS and "successful recoveries," which we count in our sensitivity analysis. Successful recoveries are a subset of true positives that also pass a detection significance criterion. We make this distinction because it is possible in a real search to detect a shallow transit only to dismiss it due to low signal. We do not want to consider these cases as successful in our analysis. Therefore, we limit successful recoveries in this analysis to detections that exceed a minimum SDE (Hippke & Heller 2019), corresponding to a detection in a real search that would likely pass vetting and trigger follow-up observations. The SDE is the significance of a period relative to the average significance of all other periods.

We determine the minimum SDE for each target individually based on the global SDE distribution of false positives resulting from our injection recoveries. We set the minimum SDE required for a successful detection as the SDE that is greater than 95% of false positives (i.e., only 5% of false positives have a higher SDE). For our three targets, the minimum robust SDE value ranges for EIC-1, 2, and 3, are roughly 6, 7, and 11.

The true and false positive distributions are shown for EIC-1, EIC-2, and EIC-3 in Figure 3. The differences result from the unique structure of each target's set of lightcurves, which produce higher and lower significance false positives. One noticeable feature of these plots (especially for EIC-3) is that the false positive distribution does not continually increase for lower SDE values, but is instead centered at a specific SDE. This potentially counter-intuitive distribution is caused by both the structure of each target's set of lightcurves, as well as the range and step size of the injection grid. Each target has a dominant false positive signal that is returned when there is no transit injection. Our grid range includes two rows of sub-Earth-size planets that are extremely shallow in depth, and each injection in these rows will return nearly the same false positive SDE as if there were no injection. This leads to a build-up of a high fraction of false positives around the no-injection SDE value, which corresponds roughly to the maximum of the false positive distribution. The higher fraction of true positives at lower SDE values is due to the fact that there is a certain range of injection depths that will only be a marginally higher power than the no-injection false positive and thus will have a low SDE, but they will still be detected successfully at high rates.

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It is important to note that the SDE cutoff is not used to determine the significance of transit candidates in the real transit search and is only used in finding the significance of injection recoveries after concluding by other means (Section 6.2.1) that the data contain no real transits. Therefore, it likely provides a conservative sensitivity estimate. Finally, the SDE cutoff cannot be expected to fully capture the probability that a true positive candidate would be followed-up and confirmed, but rather is a best attempt at conservatively estimating the likelihood given subjective human involvement in deciding what is and what is not a convincing candidate. While it would be more desirable to build a completely automatic vetting algorithm, for our observations the algorithm would need to be prohibitively intelligent and complex, and could result in more missed planets.

6.2.5. Pipeline Sensitivity

We illustrate our transit detection sensitivity for EIC-1, EIC-2, and EIC-3 in Figures 4–6, respectively. The top plots show the efficiency of our pipeline in detecting transiting planets, while the bottom plots represent total detection probability for all planets, both transiting and non-transiting, based on our transit detection sensitivity and the geometric transit probability for planets as a function of semimajor axis $\left({P}_{\mathrm{tr}}=\tfrac{{R}_{* }}{a}\right)$. To calculate the overall sensitivity within a specific range of periods and radii, we simply average the detection sensitivity in that range. Mean sensitivities for select ranges are shown in Table 4.

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Table 4.  EDEN Sensitivity

	Transit Sensitivity (%)			Total Detectability (%)
ID	0.5–2 days	2–5 days	5+ days	0.5–2 days	2–5 days	5+ days
EIC-1	60 ± 10	35 ± 5	12 ± 1	4.0 ± 0.5	1.1 ± 0.25	0.18 ± 0.05
EIC-2	40 ± 10	22 ± 5	10 ± 1	2.5 ± 0.5	0.6 ± 0.25	0.17 ± 0.05
EIC-3	80 ± 10	40 ± 5	8 ± 1	5.3 ± 0.5	1.1 ± 0.25	0.13 ± 0.05

Note. Reported transit sensitivity and total detectability values are averages for planets between one and two Earth radii. Listed errors are the standard error of the mean.

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The sensitivity maps for EIC-1, EIC-2, and EIC-3 show that we have the potential to successfully detect transiting Earth-sized planets in the habitable zones of nearby, ultracool dwarfs. Furthermore, they show that in a few cases we can detect sub-Earth-sized planets on closer orbits provided two or more transits occur during high-quality observations. To compare these results with TESS, the estimated photometric precisions for EIC-1, EIC-2, and EIC-3 are 0.136%, 0.299%, and 0.343% respectively in one hour periods of observation (TESS Mag. 13.28, 14.35, and 14.52) (Stassun et al. 2018). These are very similar to the median achieved precisions of unbinned EDEN lightcurves typically at a one minute cadence (0.163%, 0.315%, and 0.380% respectively). Thus, with long-term targeted observations it is possible we could achieve better sensitivities than TESS for single targets, in cases where the benefit of our increased photometric precision can outweigh the benefit of TESS's continuous 28 day coverage.

The primary goal of our sensitivity analysis is setting planetary limits around the target stars that will be useful for future observations. These limits can potentially improve the efficiency of similar transit surveys, and in the case of any future RV companion candidates, help to constrain the inclination. The secondary goal is to help to identify strengths, weaknesses, and biases of our observations and routines. Using this information we can improve our future observations, data reduction, detrending, and search methods. That being stated, we believe our methods are nearly optimized, and only minor improvements can still be expected, which would not significantly change our sensitivity results.

The sensitivity maps in Figures 4–6 show two distinct gradients of decreasing sensitivity. As one would expect, these gradients are for smaller planets (<1R_⊕, i.e., lower transit signal-to-noise), and longer periods (>3 days, i.e., fewer observed transits). Both regions of low sensitivity have more true positives than are considered successful, since many detections will have low significance that may not be followed-up. It is possible that some of these true positives would be followed-up, therefore it is likely that the map is somewhat conservative. Furthermore, our manual injection and recovery by eye test estimated that our sensitivity to TRAPPIST-1 b analogs is ∼80%, while the average automated sensitivity is ∼30%. This provides additional evidence that the automated sensitivity is conservative, especially for longer-period planets where one transit can be successfully detected by eye. As a final point, on the right side of the bottom plot of Figures 4–6, where geometric probability is considered, the gradient for longer-period planets becomes steeper, reflecting the decreasing transit probability at greater distance from the host star.

One noticeable aspect of our sensitivity maps is higher noise than similar plots from space-based missions. The noise is due to four primary factors, including the limited grid size, random transit times, the relatively low number of planets injected, and the sporadic and discontinuous schedule of EDEN observations. Most single blocks with relatively high or low sensitivity are simply due to the random sample times. Unlike observations from Kepler, it is possible that by misfortune a short-period planet never transits during an observation. Some columns may also have lower or higher sensitivity compared to their surroundings depending on whether or not the period is close to a harmonic of the period of observations, and therefore are more or less sensitive to phase.

Our detection limits for inner, shorter-period planets can place significant constraints on the probability of outer, longer-period planets, where observational coverage is lacking, in light of the occurrence rates of small planets around M dwarfs (Mulders et al. 2015a). The strongest example of this is the TRAPPIST-1 system. TRAPPIST-1b and c were detected by ground-based observations that motivated space-based follow-up, which discovered longer-period planets. For our targets, the approximate probability to detect transiting planets analogous to TRAPPIST-1b and c with one or more transits is ∼50%. The lack of close-in transiting planets in the extensive data sets on our targets decreases the probability that there are transiting planets at longer periods, and suggests continued observation to increase sensitivity for them is not pragmatic, given the much larger volume of data needed.

The sample of planets around very cool stars is still small, since late-M dwarfs are too faint for wide-field transit surveys. In addition, higher stellar activity can further complicate the analyses of their lightcurves (e.g., Perger et al. 2017). EDEN has unique capabilities to target these stars and any planet our survey may detect will serve as a valuable addition to this small sample. The examples of TRAPPIST-1 (Gillon et al. 2017) and GJ 3512b (Morales et al. 2019) showed how individual discoveries can challenge our current understanding of planet formation and inform tests of competing formation theories. To assess such discoveries in terms of the actual underlying population of exoplanets, it is crucial to be aware of and able to quantify the relevant selection biases. With a well-defined target selection function, an automated detection pipeline, and the thorough sensitivity analysis presented here, we are prepared to accurately model the EDEN selection biases. Correcting for these biases enables detailed occurrence rate measurements and builds the foundation to study the demographics of late-M-dwarf planetary systems.

The inferred bias can also be applied to synthetic planets from a theoretical formation model. The resulting observable synthetic population enables statistical comparisons between theory and observations (e.g., Mordasini et al. 2009). Detailed forward models of well-characterized exoplanet surveys can directly test planet formation models and even optimize free parameters (Mulders et al. 2018, 2019). Such dedicated M-dwarf population syntheses are powerful tools to constrain planet formation in a parameter space different from that around solar-type stars. The predictive power of exoplanet surveys depends on the survey's sensitivity and the number of targets observed: as the number of targets observed by EDEN increases, the emerging planet statistics will increase in significance. Currently, we are surveying targets at an increasing rate.