© ESO, 2014

1. Introduction

The Kepler mission has provided more than 6000 planet candidates¹ (Kepler objects of interest, KOI) in its more than four years of almost continuous operation. The end of phase K1 operations of the mission (extrasolar planets search) is by contrast the starting point of a new phase, which is the systematic analysis of the immense database produced by the observatory. In particular, the validation of these planet candidates is the first step to obtain a large catalog of confirmed extrasolar planets that help us to understand the formation, properties, evolution and death of planetary systems. More than 200 Kepler planets have been confirmed so far, which still represents less than 5% of the total sample of candidates. Several techniques (such as radial velocity, light curve variations, or transit timing variations) have been used to that end. However, the large pixel size of the Kepler camera (around 4 × 4 arcsec), the broad point spread function (PSF) of the Kepler telescope (with a typical² full width at half maximum of FWHM ≈ 6.4 arcsec), and the size of the aperture (typically 6–10 arcsec) used to extract the photometry implies the need for obtaining high-resolution images prior to applying these (somehow expensive and time-consuming) confirmation techniques. Exhaustive statistical analysis has also provided hundreds of validated planets (e.g. Rowe et al. 2014; Lissauer et al. 2014).

High-resolution imaging observations have been previously carried out in other planetary samples (apart from Kepler candidates) with interesting results, such as the cases of WASP-2, TrEs-2 and TrEs-4 (see Daemgen et al. 2009), where the properties of the confirmed planets were revised due to the presence of close companions detected by the lucky imaging technique. To date, there are four extensive works on the Kepler sample that use different high-resolution techniques in different wavelength ranges: speckle imaging in the optical range (Howell et al. 2011), adaptive optics in the near-infrared (Adams et al. 2012, 2013), adaptive optics in the optical (Law et al. 2013), and lucky imaging in the optical range (our previous catalog presented in Lillo-Box et al. 2012).

These high-spatial resolution surveys are important for three main reasons: 1) ruling out the possibility of chance-aligned sources in specific configurations that could mimic a planetary transit (e.g. background eclipsing binaries); 2) improving the orbital and physical parameters of the transiting object by accounting for possible extra sources in the Kepler aperture; and 3) detecting possible bound companions forming S-type binary systems, where the planet orbits one of the components of the system and acts on the other as a gravitational perturber (see Kley 2010). These points are crucial in our understanding of planetary systems (formation and evolution) and are the first step of the confirmation of Kepler planets. Indeed, in some cases, these observations represent a key step in the statistical validation of very small planets (with a mass too low to be detected by current radial velocity instruments) as was the case of Kepler-37b (Barclay et al. 2013).

In the present work, we release a new sample of lucky imaging observations of Kepler candidates and provide the isolated sample of candidates observed in our previous release (Lillo-Box et al. 2012). These isolated Kepler objects of interest (hereafter KOIs) represent an excellent sample of candidates to be followed-up, given the very low probability of contamination of their Kepler light curves (Barrado et al. 2013). In Sect. 2, we describe the target sample, the observations and the reduction of the data. The sensitivity limits of the images are shown in Sect. 2.5. In Sect. 3 we provide an update of our previous survey and the sample of isolated KOIs in our entire lucky imaging dataset. Statistics on the number of detected companions are given in Sect. 3.2. The analysis of the high-resolution images in terms of quality and how they reduce the probability for a particular KOI to have blended eclipsing binaries is explained in Sect. 4.1. In the case of KOIs with detected close companions, we provide estimations in Sect. 4.2 on how the transit depth and the planetary radius are modified due to the presence of such additional sources. A useful and comprehensive comparison between the different high-resolution imaging surveys of Kepler candidates, using different techniques, is presented in Sect. 5, and conclusions are summarized in Sect. 6.

2. Observations and data reduction

2.1. Target selection

Among the different releases of Kepler planet host candidates (Borucki et al. 2011; Batalha et al. 2013; Burke et al. 2013), a sample of few hundred targets were selected to be observed with high spatial resolution imaging. The selection criteria were based on both the interests of the planets themselves and the observational limitation imposed by the instrument/telescope configuration. The latter restriction is given by the combination of the CAHA 2.2 m telescope and the AstraLux instrument, which provides detectability limits of m_SDSSi = 20–21 mag in total exposure times of around 2700 s. Since we wanted to detect possible companions that are at least 5.0 mag fainter at 1.0 arcsec (fainter visual companions would usually not affect the planet-star properties significantly), the faintest targets that we observed were of m_Kep< 18 mag. From a practical point, except in few exceptions, we avoided observing KOIs that are fainter than m_Kep = 16 mag to ensure the significance of our results.

In total, we observed 230 KOIs (101 KOIs in 2011, already reported in Lillo-Box et al. 2012, with detailed information about 44 objects with possible companions, 21 KOIs in 2012, and 108 KOIs in 2013) hosting 376 planet candidates. Unfortunately, after some of these KOIs were observed, some of their hosted planet candidates were rejected for different reasons (re-analysis of the light curve by Kepler team, radial velocity observations, etc.). As a consequence, a total of 56 KOIs among our targets (including all but one of the KOIs with m_Kep> 16) do not seem to host a planet candidate anymore. On the positive side, we have used these 56 demoted KOIs as sample control so that the same study was carried out for this sample. This leaves us with 174 planet host candidates in our sample (97 KOIs in 2011, 20 KOIs in 2012, and 57 KOIs in 2013) by hosting 313 planet candidates. Among these 174 KOIs, nine have all their candidates already confirmed³: KOI-0041 or Kepler-100, KOI-0069 or Kepler-93, KOI-0082 or Kepler-102, and KOI-1925 or Kepler-409 from Marcy et al. (2014), KOI-1529 or Kepler-59 (Steffen et al. 2012), KOI-0196 or Kepler-41 (Quintana et al. 2013), KOI-0351 or Kepler-90 (Cabrera et al. 2014), KOI-0245 or Kepler-37 (Barclay et al. 2013), KOI-0094 or Kepler-89 (Weiss et al. 2013), KOI-2133 or Kepler-91 (Lillo-Box et al. 2014), and KOI-0571 or Kepler-186 (Quintana et al. 2014).

2.2. Data acquisition and reduction

We applied the lucky imaging technique to the selected targets to achieve diffraction-limited resolution. We used the AstraLux North instrument located at the 2.2 m telescope at the Calar Alto Observatory (Almería, Spain). The targets were observed along three visibility windows of the Kepler field during 2011, 2012, and 2013. The results regarding the non-isolated KOIs of observations on 2011 were published in Lillo-Box et al. (2012). In the present work, we report the results concerning the isolated candidates observed in 2011 and the new results for the 2012–2013 observing runs.

We used exposure times for the single frames in the range 30–90 milliseconds (which is below the coherence time of the atmospheric turbulence) and set the number of frames accordingly to accomplish our depth requirement (typically 20 000–40 000 frames). In all cases, we used the full CCD array of the camera (24 × 24 arcsec). In the same line as in our previous work, this observing configuration ensures the aimed coverage both in contrast and angular separation from the main target. Table 1 lists the observing characteristics (date, individual exposure times, and number of frames) for each target.

Data cube images were reduced by using the online pipeline of the instrument (see Hormuth 2007), which performs basic reduction and selects the highest quality images. Then, it combines the best 1.0%, 2.5%, 5.0%, and 10% frames with the highest Strehl ratios (Strehl 1902). It calculates the shifts between the single frames, performs the stacking, and resamples the final image to have half the pixel size (i.e. around 0.023 arcsec/pixel). In this paper, we only use the 10% selection rate images (which we simply call AstraLux images). We chose this particular selection rate, since it provides the best compromise between a good angular resolution and the largest magnitude depth, according to our previous experience with the instrument and recommendations from Felix Hormuth (PI of the instrument).

2.3. Astrometric calibration

We acquired images of the M15 globular cluster in all three observing seasons to obtain the relative plate solution of the CCD. We used the more than 100 cross-matched sources with the Yanny et al. (1994) catalog to obtain the plate scale and position angle of the CCD. We compared the angular separations and position angles of more than 1000 randomly selected star pairs in the latter catalog (separations in arcsec) and in our own catalog (separations in pixel units). The derived pixel size and position angle of the CCD for each observing season are shown in Table 2. We obtained typical uncertainties of 0.20 mas/px (around 1% of relative error) for the pixel size.

2.4. Source detection and photometry

Sources were identified in each image by using our semi-automatic routine specifically designed for the instrument. The algorithm first detects possible sources in the image whose integrated flux over an aperture of 10 pixels is, at least 3 times greater than the corresponding flux of the sky in the image (measured as the median value of all pixels, assuming that most of the image is not covered by stars). Then, each source candidate is individually checked to fulfill specific criteria, such as having a PSF-like radial profile shape (to reject possible artifacts and cosmic rays) or having magnitudes in the range of 0–30 to reject possible remaining bad pixels. All images were then manually inspected to check the final detected sources.

We then applied aperture photometry to measure the relative magnitudes between objects in the same image. We used the aper routine in IDL to extract the flux contained within a specified aperture. This aperture is selected for each image by taking the close objects in the field to avoid contamination of close companions into account. Thus, for each image, we have the instrumental magnitudes for all sources and the magnitude differences with respect to the KOI (which we call Δm). In cases where a close companion (below 3 arcsec) was found, we obtained additional photometry in the z_AstraLux filter (equal to the SDSSz filter from the Sloan Digital all-Sky Survey) to characterize the secondary object.

Absolute calibration was then performed by using the KIC photometry of the KOI and the instrumental magnitudes and magnitude differences of the surrounding objects with respect to the KOI. First, KIC magnitudes were converted to SDSS magnitudes by using the photometric transformations presented by Pinsonneault et al. (2012) in their Eqs. (3) and (4). According to Brown et al. (2011), the KIC images have a full width at half-maximum of 2.5 arcsec. Hence, as stated by the authors, the KIC photometry is unable to resolve the components of close binary stars. According to this, we can consider that their PSF photometry cannot resolve visual companions closer than 2.5 arcsec, so the magnitudes of such KOIs account for the flux of all sources inside such radius. Thus, we can distinguish between two cases to calibrate our photometry: KOIs with and without companions closer than 2.5 arcsec.

When any close companion was detected, we derived the photometric zeropoint of our Astralux images using the i-mag provided in the KIC, neglecting atmospheric or instrumental effects.

In the case where a close companion was found, we assume that the KIC magnitude that is converted to the SDSS system (m_SDSS) is actually the sum of the fluxes coming from all sources inside 2.5 arcsec. If we call to the magnitude difference of the jth object inside 2.5 arcsec from the KOI, as detected by AstraLux, the calibrated magnitude of the KOIs must read (1)Having the calibrated magnitude of the KOI, we can obtain the absolute magnitudes of all companions in the image as , where C/ represents the values for the companion. This scheme was then applied to both filters i and z to obtain the absolute SDSS magnitudes of all objects detected in the images.

In Table 3, we provide the complete catalog of sources detected within 6 arcsec from the KOIs observed during 2012 and 2013. In Fig. 1, we show the location of all companions found within 6 arcsec for KOIs that are observed during our three observing seasons. The color-code in the figure shows the magnitude difference in the i filter. Figure 1 illustrates the high density of close visual companions and the need to obtain high resolution images of all candidates to better characterize the systems.

The identified isolated KOIs are studied in more detail in Sect. 4.1. These targets are thus suitable to proceed with radial velocity studies, since no objects have contaminated the Kepler light curve and cannot contaminate the radial velocity data within our sensitivity and detectability limits (presented in the next section).

2.5. Completeness, detectability, and sensitivity limits

In high-spatial resolution studies, it is crucial to determine the limitations of our images in terms of completeness, contrast, and how contrast depends on the separation to the main target. These three concepts completely describe the quality of the observation and, thus, should be individually calculated and reported for each image.

We refer to Lillo-Box et al. (2012) for a detailed explanation about the employed method to measure both the completeness and detectability magnitudes. In brief, we used three images with a total exposure time each of 200 s for the globular cluster M15 to compute these parameters. According to the detected sources in these images, we found the mean completeness value to be i_complete = 18.4 ± 0.3 mag and to reach detectability down to approximately i_detect = 21.7 mag at 5σ level. Since different exposure times were set for each image and selection rate, we have to extrapolate these values individually. In particular, we have scaled the completeness and detectability limits for each particular object and selection rate by adding the quantity − 2.5log (200 s /t_exp(s)) to the detectability and completeness magnitudes shown above. Here, t_exp is the effective exposure time of each image (i.e. the individual exposure time per frame multiplied by the number of selected and stacked frames). The resulting values are shown in Table 1 for each individual object.

The sensitivity limits (i.e. the faintest stars detectable in our images at each angular separation, which we also call sensitivity curve, C_sens) were also determined for each image once the KOI was identified. We artificially add a similar PSF compared to the one of the main target but scaled by a factor of Δm (i.e. F₂ = F_KOI10^{− 0.4Δm}) at different positions of the image. We used 20 angular separations between α = 0.1 and α = 3.0 arcsec and 20 relative magnitudes (Δm) between 0 and 10 mag. For every pair [d, Δm], we added ten artificial stars that are distributed at random angles (hence, 4000 artificial stars were included), and we run our detection algorithm used to detect the companion sources (see Sect. 2.4) to try detecting these artificially added companions. The sensitivity curve (C_sens) is then computed as the contour line in the [d, Δm] plane, which shows that at least 7 out of the 10 artificially added stars were detected with a 5σ minimum requirement for the detection. The sensitivity curve calculated for KOI-0082 is shown in Fig. 2 to illustrate the results and to compare other high-spatial resolution surveys to that also observed this target (a quantitative comparison of these studies for the coincident objects is performed in Sect. 5). The sensitivity limits at different angular separations for each KOI in this work are presented in Table 4 for the corresponding i and z filters.

3. Results

In what follows, we use the isolated designation to those KOIs that do not present companions closer than 6 arcsec from the KOI. This limit comes from the typical values of the Kepler PSF, and the typical apertures used to extract the photometry, which ranges between 6–10 arcsec. Note that stars beyond the 6 arcsec limit would have been detected by more conventional surveys.

3.1. Update on 2011 results

In Lillo-Box et al. (2012), we observed 98 KOIs and concluded that 57 were actually isolated, 27 presented at least one companion between 3–6 arcsec, and 17 presented at least one companion closer than 3 arcsec. The new (more accurate) astrometric calibration presented in this paper with the update of the list of false positives implies that these numbers have slightly changed. The updated results for the 2011 observing season provide 63 isolated KOIs (65.0%), 22 KOIs with companions at 3–6 arcsec (22.7%), and 15 KOIs with companions closer than 3 arcsec (15.5%) out of 97 KOIs. (We removed those that currently do not present any planet candidate, namely KOI-0644, KOI-0703, and KOI-0465, and added other two not included in the previous analysis, namely KOI-0490 and KOI-0408.)

3.2. Results from the new sample

In this work, we add information about another 77 KOIs with active planet candidates (nine of them with all their transiting candidates already confirmed, see Sect. 2.1). Among this new sample, we find that 54 KOIs are isolated (70.1% of the new sample), and 23 KOIs (29.9%) have at least one companion closer than 6 arcsec (inside the typical Kepler PSF). Among the non-isolated KOIs, we find that 12 KOIs (15.6%) show at least one companion between 3–6 arcsec, and 15 KOIs (19.4%) show at least one object below 3 arcsec (which means that 4 KOIs present companions in both ranges). These numbers are relatively similar and consistent to those obtained in the 2011 run.

According to the updated numbers from the 2011 observing season and to the addition of the new sample, we find that 111 are isolated (67.2%), 35 present at least one companion between 3–6 arcsec (20.1%), and 30 have companions closer than 3 arcsec (17.2%), among the 174 KOIs observed with remaining planet candidates,. All these values are summarized in Table 5. These results imply that we have a 67.2% probability that a KOI is isolated, regardless of the possible biases related to the target selection. However, there is a non-negligible 32.8% probability that a companion object inside the typical Kepler PSF exists, thus contaminating the Kepler light curve and modifying the derived properties of any planet candidate.

4. Analysis

4.1. Isolated planet host candidates

Once the sensitivity limits (i.e. the sensitivity curve) have been calculated for the images of the isolated KOIs (Sect. 2.5), the relevant information to be supplied by our high-resolution observations is how well we can be assured that no blended background or foreground sources can contaminate the detected transit signal.

Generally, our high-resolution images can play an important role in the rejection of two false positive scenarios. Other configurations, such as hierarchical triples or grazing eclipses cannot be ruled out by high-spatial resolution images, but their occurrence probabilities are extremely low. First and most critical, specific configurations of a blended unassociated eclipsing binary can reproduce the detected planetary transit of the candidate. According to Morton & Johnson (2011, hereafter MJ11), this case is of particular importance for shallow transits (with apparent depths below 10³ ppm), which should be the case of smaller planets. In Sect. 4.1.1, we deeply investigate and quantify how our high-resolution images can reduce the probability of such scenario being the responsible of the detected transit.

Secondly, even if the transiting object is actually eclipsing the target star, the mere presence of single blended sources not accounted for in the light curve analysis can importantly dilute the transit depth. As a consequence, the transiting object would seem smaller than it actually is. This scenario is discussed and quantified in Sect. 4.1.2.

4.1.1. Blended source confidenc: Rejection of background eclipsing binary scenarios

As stated in MJ11, the probability of having a blended eclipsing binary (P_BB) inside the PSF of the KOI can be split into two factors: the probability of having a blended source (P_BS) and the probability for that source to be an eclipsing binary with the appropriate configuration to produce the observed transit (P_appEB). While the second factor only depends on the galactic latitude and magnitude of the KOI (see Eq. (14) in MJ11), the first factor can be observationally constrained to some extent with high-resolution images.

To assess the contamination probability, we define a new parameter, the blended source confidence level (BSC), as the observational level of confidence that no blended sources are located within a given angular separation to the host candidate. The BSC is evaluated as the complementary probability of having a blended object that could mimic the observed transit.

i. Probability of having a blended source

For a given KOI, we can calculate the maximum contrast (with respect to the measured flux) that a hypothetical blended EB must have to mimic a transit of fractional depth δ (see Eq. (7) in MJ11). This equation reads (2)This value is valid for the Kepler filter. Since we are working with the i band of the SDSS, we have to compute the magnitude conversion. If we use the KIC (Kepler Input Catalog) magnitudes, we can easily see that the Kepler and i_SDSS magnitudes are linearly correlated. Since the vast majority of our isolated candidates lie in the range 13.0 <m_K< 16.0 (only 2 have m_K< 13.0), we have only used KOIs in this range to compute the linear coefficients. We obtain that The linear correlation goodness of this fit is R² = 0.98, which is acceptable enough for this work. Thus, we can estimate the constrast in the i_SDSS band as Δi_SDSS = 0.947·Δm_K, so that (3)For clarity, we refer to this maximum constrast in the i_SDSS band as Δm_max. In Fig. 3, we show the sensitivity curves of two KOIs and the Δm_max line that marks the limit to be reached by observations that minimize the probability of existence of a non-detected blended source capable of mimicking the transit signal.

Let us now, as a first step, assume one particular KOI with a magnitude m_i (in the SDSS photometric system) at galactic latitude b. The expected number of stars within an angular separation r from our KOI and with magnitudes in the range [ m_i,Δm_max ] is given by (4)where ρ(b,m,Δm) represents the number of stars per unit area (density of stars) and depends on the galactic latitude (b) of the target and the requested magnitude range ([ m,m + Δm ]). For small areas⁴, this value can be interpreted as the probability for this area that contains one chance-aligned star within this magnitude range. In that case, we can define the probability of an object having one companion source within a certain magnitude range as P_aligned = N(r,b,m,Δm). This equation clearly shows that the probability of a chance-aligned source decays with the square of the angular separation as we approach to the star. Contamination sources above 3.0 arcsec could have been easily detected by photometric observations as dedicated to the Kepler field (such as in Brown et al. 2011, or the UKIRT J-band survey observed and supplied by Phil Lucas⁵), or by photocenter centroid analysis of the Kepler images (see, for example, Batalha et al. 2010). Hence, in this work, we only take the 0.0–3.0 arcsec region into account.

We can now integrate P_aligned in the parameter space α = [ 0.0,3.0 ] arcsec and Δm = [ 0,Δm_max ] to compute the total a priori probability (P_BS,0) for a particular target of magnitude m_K that contains a chance-aligned source with magnitude m_i<m<m_i + Δm_max within 3 arcsec: (5)

ii. Calculating the ρ parameter

We have calculated the density of stars parameter (ρ) for each planet candidate in a similar way as MJ11. We used the online tool TRILEGAL⁶ to compute the number of expected stars with a limiting magnitude in a particular region of the sky. We used the default parameters for the bulge, halo, thin/thick disks, and the lognormal initial mass function from Chabrier (2001). As the Kepler field is relatively large and encompasses around 12 degrees in each direction (and almost 20 degrees in galactic latitude), we have found important differences in the stellar density from galactic latitudes that are close to the galactic disk to those farther from it. Due to the large number of targets in this paper and given that it is not possible to perform an automatic query in TRILEGAL (the user must proceed object by object), it is not possible to obtain individual populations for each target. Since the ρ parameter just depends on the galactic latitude, we obtained instead stellar populations for regions of 1 deg² centered at b = 6° to b = 22° in steps of 1° and a galactic longitude fixed to the center of the Kepler field (i.e. 76°), as seen in Fig. 4. We then simulate stars up to a magnitude limit of i_SDSS = 28 inside each region according to the galactic model.

For a particular KOI at a galactic latitude b_KOI with a magnitude m_i and a needed depth in magnitude of Δm_i, we determine ρ(b_j,m_i,Δm_i) (j subscript representing each galactic latitude for which we run the TRILEGAL simulations) at all galactic latitudes in the grid by just counting stars within the magnitude range [ m_i,m_i + Δm_i ] and dividing by the box area of 1 deg². Then we perform a low-order polynomial fit to the rho versus galactic latitude values and infer the corresponding ρ(b_KOI,m_i,Δm_i) by evaluating the fitted function at b_KOI. We found that a polynimial of order five fits the data sufficiently well for the purposes of this work (see the example in Fig. 5). By following this scheme, we can precisely estimate the density of stars in a concrete magnitude range at any position in the Kepler field.

iii. Observing constraints and the BSC parameter

In Sect. 4.1.1, we have described how we define the probability of having a blended source within some observational constraints (namely, the star position and the magnitude range of possible blended stars). Since we have calculated a sensitivity line for each observation, we can reduce the a priori probability (P_BS,0) by limiting this calculation to only that region where our image is not sensitive. In the angular separation versus magnitude difference plane, this non-sensitive zone is defined as the region between our sensitivity curve (C_sens) and the maximum magnitude difference of a possible blended eclipsing binary that could mimic our transit signal (Δm_max). In Fig. 3, this region has been shaded with diagonal lines. Hence, with the CAHA/AstraLux high-resolution observations, the probability is given by Eq. (5) but now integrates only over the diagonally shaded region: (6)where Δm₀(α) = max [ C_sens(α),Δm_max ] and the second term in the right hand side of the expression represents the contribution of the high-resolution image.

It is thus clear that the better and deeper our image (i.e. the closer C_sens is to Δm_max), the more we diminish the blended source probability and thus improve the planetary candidacy. We can now determine an observational value of P_BS and define the BSC parameter as BSC = 1 − P_BS, representing the confidence for this source by not having blended eclipsing binaries mimicking the planetary transit. We propose this parameter to be used in all high-resolution studies to compare the different surveys with the adaptive optics, speckle, or lucky-imaging techniques. In column six of Table 7, we show the updated values of the P_BS according to our AstraLux observations (hereafter called in %) for the isolated targets.

As an example, we obtain BSC = 99.5% in the upper panel of Fig. 3, given our high-resolution image of the planet host KOI-0094 and for its KOI-0094.01 planet candidate. In other words, there is a small probability (lesser than 0.5%) that we are missing a background source with the right magnitude to mimic the planetary transit (P_BS< 0.5%). It must be remembered that the P_BS probability must be multiplied by the probability of that source being an appropriate eclipsing binary (P_appEB). This value, as calculated with the correlation explained in Eq. (14) in MJ11, is shown for each planet in the eighth column of Table 7.

iv. Corrections to the BSC due to incompleteness

In the bottom panel of Fig. 3, we can see an example of an image providing poor quality information. It is the case of KOI-0473. In this case, we have to make a correction to the BSC value because the completeness limit of our observation (corresponding to i_complete, see Sect. 2.5) is above the sensitivity curve. Since the detectability limit (i_detect) is below the sensitivity curve (i.e. i_KOI + Δm_max<i_detect), we still could detect a percentage of sources with magnitudes above the completeness magnitude but not the all of them. Thus the contribution of the vertically shaded region (which we call incompleteness region, Incomp.Reg. in Eq. (7)) must be weighted by a function of the magnitude difference according to the decay in the probability detection. This function can be calculated by fitting the decay of the distribution of detected sources in the M15 images to an exponential function like f(Δm) = C + Ae^{− Δm/B}, where Δm = m − m_complete. We can appropriately set C = 0 and A = 1, so that f(Δm = 0.0) = 1.0. From the M15 data (see Sect. 2.5), we derived the value B = 0.667. Thus, in the cases where the completeness line (upper yellow dashed line in Fig. 3) lies above the sensitivity curve (C_sens, green line), the P_BS must be calculated as (7)where and r₀ represents the angular separation at which m_i + C_sens(α = r₀) = m_comp. The second term corresponds to the contribution of the high-resolution image in the magnitude range where it is complete (non-shaded region above the C_sens line in Fig. 3). The third term represents the weightned contribution of the high-resolution image according to our exponential incompleteness function derived above, and it is represented by the vertically shaded region in the bottom panel of Fig. 3.

The results for KOI-0473.01 are worse than for KOI-0094.01, since we obtain a BSC parameter of 84.9% here, indicating that the depth of the image is less than what was needed for this particular object.

We have calculated the parameter for all KOIs without companions closer than 3 arcsec. The results are presented in Col. 6 of Table 7. In this table, we also show the P_BS,0 value (Col. 5) and the corresponding improvement obtained with our high-resolution observations (Col. 7). These values allow us to compare the quality of different high-resolution imaging techniques.

4.1.2. Rejecting diluted single-star scenarios

The second scenario that our high-resolution images can rule out is the case in which the presence of single-blended stars can dilute the transit depth of the eclipsing object so that it could mimic a planetary eclipse. Let us start with the simple case of one single star blended in the Kepler aperture. The observed transit depth can be calculated as δ = (F_nt − F_t) /F_nt, where F_nt represents the measured flux when the object is not passing in front of the target star and F_t represents the flux when it is transiting the star. In the case of one single star contributing with a flux F₂ and the target star contributing with a flux F₁, we have F_nt = F₁ + F₂ and F_t = F₁(1 − ϵ), where ϵ is the actual fraction of the star covered by the transiting object and its value can be easily demonstrated to be equal to the true transit depth (i.e. ϵ = δ_true). By using this, we can correct the transit depth due to the presence of a blended source as (10)where Δm = m₂ − m₁ represents the magnitude difference between the blended (m₂) and the target star (m₁) in the Kepler band. At this point, as stated by Law et al. (2013), we can distinguish between two cases: A) the transited star is brighter than the blended star (Δm> 0); and B) the transited star is fainter than the blended star (Δm< 0). To get the true radius of the transiting object, case B requires some knowledge about the radius ratio between the two stars involved. This requires additional knowledge of both stars (which means more free parameters rather than just the magnitude difference), which is out of the scope of this theoretical analysis.

In case A, however, assuming that the transit depth is releated to the radius ratio between the transiting object and the parent star as δ = (R_p/R_⋆)², the true radius of the transiting object is given by (11)According to the most updated catalog of confirmed exoplanets⁷, the empirical maximum possible radius for a planet is . Thus, we can calculate the maximum magnitude difference that the blended source can have (i.e. how faint could it be) to dilute the transit depth, such that a non-planetary object (i.e. , regardless of its nature) appears as the true planet-size object: (12)This equation indicates that the presence of undetected blended objects with magnitudes can dilute the transit depth of a non-planetary object down to that of a planetary object. According to this expression, case A only applies to candidates with . In our sample, we only have six objects with (namely, KOI-0338.01, 1353.01, 1452.01, 2481.01, 3728.01, and 3765.01 ). For those cases, we can proceed exactly as we did for the blended eclipsing binary scenario to get the P_BS, but now we use as the maximum magnitude value to get the probability of the presence of a diluter source P_DS. The results show that this probability is dimminished from P_DS,0 = 10^-3 − 10^-2 to . Although the starting probabilities were already small, our high-resolution images showing no blended sources within our detection limits practically discard this possibility as a false positive scenario for these candidates.

We note that due to the mathematical shape of Eq. (12), we cannot perform this calculation for planet candidates with (36 in our sample). Also, as stated by Law et al. (2013), case B would only affect few planet candidates with observed radii close to the limit and present blended stars with very small radius. Candidates with a small calculated radius are not affected by this scenario, although the presence of blended sources can modify their properties as we show in Sect. 4.2.1.

For cases with more than one blended star, the P_DS probability is insignificant for case A, since the probability of having two or more undeteced sources within our sensitivity limits is far smaller.

4.2. Non-isolated KOIs

We have found close companions (below 3 arcsec) for 15 KOIs among the sample of targets in the 2012 and 2013 observing seasons (see Fig. 6). The mere presence of such objects affects both the KOI status as a planet candidate and (if confirmed by other techniques) the derived planet properties, such as planet radius or impact parameter. Thus, the light curves of these targets should be studied in more detail, taking this additional sources into account. In Table 3, we provide the list of KOIs with companions below 6 arcsec. Among them, 15 KOIs present close companions (below 3 arcsec) that should be added to the 17 KOIs with detected sources by Lillo-Box et al. (2012).

Among the 15 KOIs with close companions, we obtained photometry in the z_SDSS band for five of them. In these cases, we can compute the i − z color as (13)where the subscript /C denotes that the value corresponds to the companion source and Δi = i_/C − i_KOI and Δz = z_/C − z_KOI. By following the prescriptions described in Lillo-Box et al. (2012), we can estimate the spectral types of the objects by having the i − z color information. For those without the z magnitude (mostly KOIs with companions at 3–6 arcsec), we searched for photometric information in public catalogs by using the last version of the Virtual Observatory SED Analyzer (Bayo et al. 2008, 2013, and in prep.). In cases where more than five photometric points are found, this tool fits a SED model to obtain the effective temperature of the source. Table 3 (Col. 10) shows the results for the sources for which this study was possible.

4.2.1. Effect on the planet properties

The diluted light from the blended companion that is not accounted in the light curve analysis provides erroneous determinations of the properties of the transiting object. We note that Kepler light-curves are deblended from sources detected by either the UKIRT J-band survey or by the KIC photometric survey on the Kepler field. Among the 15 KOIs with detected companions that are closer than 3 arcsec by the present survey, two of them have all their companions detected by the UKIRT J-band survey (namely KOI-0650 and KOI-1452). Thus, we can remove this KOIs from the current study. In the other two cases (KOI-1546 and KOI-1812), only one out of the two detected companions within 3 arcsec have also been detected by the UKIRT J-band survey. Thus, for these two, we only take the non-detected source into account. Finally, there are another two cases (KOI-2324 and KOI-3886), where the detection of the companions in the UKIRT J-band survey and their correspondence to our detected companions remains unclear. In what follows, we proceed for these targets as if their companions were non-detected by the UKIRT J-band survey. In total, 13 KOIs hosting 24 planet candidates are affected by additional non-accounted fluxes of blended companions.

In Sect. 4.3 of Lillo-Box et al. (2012) and Sect. 4.1.2 of this paper, we provided theoretical estimations of how the planetary radius of KOIs with detected close companions change because of the detection of blended stars (assuming the brighter object as the host star). In this work, we proceed in the same manner to estimate the corrected planetary radius. The results for the 24 affected planets in the 2012 and 2013 observed targets are listed in Table 8. The estimations clearly show the increase in the planetary radius caused by the non-accounted flux of the blended star. In three cases (namely, KOI-1230.01, KOI-3649.01, and KOI-3886.01), the planet candidate has a new estimated radius that according to mass-radius relations by Chabrier & Baraffe (2000) would yield to a typical mass of the transiting object in the stellar regime. We also note that the largest extrasolar planets confirmed so far⁸ have a maximum radius below 2.2 R_j. Thus, we can flag these candidates as probable false positives.

We note that other orbital parameters, such as the semi-major axis or the impact parameter, are also affected by the presence of blended stars. However, the correction in these cases as well as a fine correction of the planetary radius involves careful re-analysis of the transit signals, which is out of the scope of this work.

4.2.2. Physical bond of blended companions

For those KOIs with observations in both i and z filters, we can further constrain whether the close companions are bound or not to the central object using the color information. We proceed in the same way as in Lillo-Box et al. (2012, Sect. 4.2.2) by testing whether the close companions lie in the same zero-age main-sequence (ZAMS) as the central objects by projecting them to the same distance (see details in the aforementioned paper). Figure 7 summarizes the results for the five KOIs with remaining planet candidates and i,z observations during the 2012–2013 observing seasons. Among them, the companion to KOI-3158 (called KOI-3158B) seems to have a consistent age (within the uncertainties) with the main object, since they lie in the same ZAMS. This result agrees with a common formation of the two components, thus being a possible bound binary system. If this were the case, the estimated projected distance between the two components⁹ would be 1100 ± 1000 AU for KOI-3158AB. According to our spectral type analysis based on the i − z measured color in Sect. 4.2, the compainon is a redder object in the range K5-M1 (assuming that it is a main-sequence star).

Although the orbital parameters must be revised to account for the detection of the blended object, we can conclude that KOI-3158 is a potential candidate to be a S-type planetary systems. Indeed, it has five planet candidates as detected by Kepler.

According to their position in the Hertzprung-Russell diagram of Fig. 7, the remaining close companions (KOI-0191B, KOI-3886B, KOI-2481B, and KOI-3444C) are probably background sources.

4.3. A control sample: demoted KOIs

As was mentioned in Sect. 2.1, 56 KOIs were rejected as candidates after our lucky-imaging observations were performed (and we call them demoted KOIs). We have taken profit of these observations to contrast the multiplicity results. Among the demoted KOIs, we have found that 38 (67.8%) are actually isolated; 5 (8.9%) present one source closer than 1.4 arcsec; 11 (19.6%) present at least one companion closer than 2.5 arcsec; 13 (23.2%) have at least one source within 3 arcsec; and another 13 KOIs (23.2%) present at least one object between 3−6 arcsec. These results are summarized in the last section of Table 6. Compared to the real KOIs sample, we can see that these values are similar. Indeed, we found the same rate of isolated targets in both samples. However, there is a slightly higher amount of close companions in this sample compared to the real KOIs.

Although the detected planets around these KOIs were rejected, the light curves of those presenting close sources should be re-analyzed by taking this into account. Depending on the causes that led the different works to demote these objects as candidates, some of them could change the properties of the transiting objects and, perhaps, return them back to the planet candidate status.

5. Comparison to other high spatial resolution techniques in the Kepler sample

There are three main techniques that provide high-resolution (diffraction limited) images from the ground: speckle imaging, adaptive optics, and lucky imaging. Regarding the high spatial resolution studies performed for the Kepler candidates, there are three main works that have provided exhaustive observations of the candidates appart from our survey. Howell et al. (2011, hereafter H11) published the first results of the speckle imaging observations for 156 KOIs, using the 3.5 m-telescope WIYN on Kitt Peak. Adams et al. (2012 , hereafter A12) and Adams et al. (2013) provided the adaptive optics multiplicity results in the near-infrared regime for a total of 102 KOIs using both the 6.5 m Multiple Mirror Telescope (MMT) and the Palomar Hale 5.1 m telescope. A posterior, shallow survey by Law et al. (2013) with Robo-AO (hereafter L13) provided adaptive optics observations for 715 Kepler candidates, using the robotic Palomar 1.5 m telescope (Cenko et al. 2006). The results of the current paper added to those of Lillo-Box et al. (2012) complete the set of high spatial resolution techniques by providing lucky imaging for 234 KOIs in the optical range.

The distribution of Kepler magnitudes is similar for all surveys, peaking L13 and the present work at slightly fainter magnitudes (m_kep ≈ 14) than H11 and A12 (m_kep ≈ 12). In Fig. 8, we show this distribution for the four studies.

However, since these works were published, some of their KOIs have been rejected as planet candidates due to several reasons. In particular, 24 KOIs from H11 (out of 156, 15%), 16 KOIs from A12 (out of 102, 16%), and 17 KOIs from L13 (out of 714, 2.3%) currently do not present any planet candidate. In our case, the percentage of non-planet KOIs is slightly higher (56 out of 230, 24%) because we observed several KOIs, which still had the non-dispositioned flag in the Kepler archive (meaning that they did not yet passed all requirements to be planet candidates), in 2013 observing season.

The speckle imaging study by H11 provides the highest resolution images (with detection limits at 0.05 arcsec) but in a very small field of view that only allows to detect companions at a limiting separation of 1.4 arcsec. They also provide a typical depth magnitude limitation of Δm = 4.0 mag. The large majority of the transits of planet candidates could be mimicked by blended stellar systems fainter than this magnitude difference (as we can see in the fourth column of Table 7). With these observing limitations, they found that four out of the 127 KOIs (3%) with remaining planet candidates do present a stellar companion closer than 1.4 arcsec.

The adaptive optics work by A12 seems to be more complete in magnitude depth and field coverage (more than 20 × 20 arcsec). We have used their updated Tables 2 and 4 to compute statistics that could be compared to the H11 and our own study. In particular, they find that among their 85 KOIs with remaining planet candidates, 37 are isolated (no companion closer than 6 arcsec, 44%), 12 KOIs (14%) present a stellar companion within 1.4 arcsec, 28 KOIs (33%) present a stellar companion within 3 arcsec, and 30 KOIs (35%) present at least one companion in the range 3–6 arcsec.

The recently published survey by Law et al. (2013) provides the largest catalog of AO observations of Kepler candidates. Their observations determine that 29 out of the 697 KOIs with remaining candidates (4.2%) present some companion within 1.4 arcsec, and 49 (7%) do present companions closer than 2.5 arcsec. Since this survey is limited to 2.5 arcsec of separation, we cannot include the remaining 648 in the isolated group.

All these numbers are summarized and compared to the lucky imaging results provided in this paper in Tables 6 and 9. Figure 9 also illustrates the coincident KOIs between the different surveys. In the next subsections, we compare these works to our survey summarizing coincident objects and BSC results. For the latter, please note that all four studies provide 5σ level sensitivity limits, so that direct comparisson of the BSC values can be done.

5.1. Comparison with Howell et al. (2011)

Among the 12 coincident objects between H11 and this work, none do present companions within 1.4 arcsec (the largest separation detectable by H11). Since sensitivity curves and photometric transformations of the filters used to the SDSS system are not provided by the authors for these targets, we cannot exactly compare how both studies have improved the BSC values. As a zero-order approximation, we can assume that the limiting magnitude presented in their Table 2 as Δ_max and calculated for an angular separation of 0.2 arcsec is constant over the 1.4 arcsec of spatial coverage and obtained in a similar filter¹⁰. We can then determine a zero-order BSC_H11 and thus compare to our values. The results (see Table 10) show that the speckle-imaging limiting magnitudes are smaller than the required magnitude differences for discarding possible background configurations that are able to mimic the planetary transit (that we have called Δm_max in this paper). This happens for all 32 planet candidates orbiting the 12 common KOIs. In all cases, our AstraLux observations are better (in terms of reducing the BSC parameter) than the H11 observations. The small contribution of the H11 study to reduce the probability of a blended eclipsing binary is mostly due to the reduced field of view, which avoids detection of 1.5−3.0 arcsec companions, where the probability of having a background source is maximum in the 0–3 arcsec range.

5.2. Comparison with Adams et al. (2012)

Only four out of the ten coincident objects with Adams et al. (2012) do present at least one companion below 6 arcsec.

KOI-0111 & KOI-0555. The companions to KOI-0111 and KOI-0555 detected by this work are not detected by A12. Since the magnitude differences in both cases are relatively small (Δ_i = 6.1 for KOI-0111 and Δ_i = 3.8 for KOI-0555), the non-detection by A12 could mean that these companions are bluer, but we would need photometry in different bands to provide conclusions about this result.

KOI-0372. On the contrary, we do not detect any of the faint companions to KOI-0372 with Δm_Kep> 10.0 due to their faintness. However, the maximum magnitude difference for a companion star that could mimic the planetary transit of the candidate KOI-0372.01 is Δm_max = 4.99, so that the detected companions by A12 do not affect the planetary candidacy of this object.

KOI-0115. The latter case affects KOI-0115 with three planet candidates for which Δm_max(.01) = 7.6, Δm_max(.02) = 8.8, and Δm_max(.03) = 11.1. The observations from A12 detected two companions with magnitude differences below those values. We do not detect the closest target at 2.43 arcsec and Δm_kep = 11.4 mag due to sensitivity restrictions in the present study. However, this companion has a magnitude difference that is higher than the maximum difference that would affect any of the three planets detected in this system. Hence, we could say that this is a negligible blended star for this system.

In the case of the six remaining KOIs with non-detected companions closer than 6 arcsec, we obtain smaller values of the blended source probability, given that our images are deeper for these particular objects. Table 10 summarizes these results compared to our values according to the updated sensitivity limits that are provided by A12 for each target in the Kepler band and transformed to the i_SDSS filter using our own transformation determined in Sect. 4.1.1. In this case, we can see that A12 reduces the probability of having a background source more than H11. The only handicap of this survey is that possible blue non-negligible objects could not be detected by this survey (as we have shown in the cases of KOI-0555 or KOI-0111), since Kepler observations are performed in the optical wavelengths and A12 observations are obtained in J and Ks bands.

5.3. Comparison with Law et al. (2013)

A total of 112 KOIs have counterparts in both L13 and the present study. Among this subsample, 13 KOIs have been detected to have companions within 2.5 arcsec (the largest separation that L13 can achieve). In four cases, both studies detect the companions (KOI-0401, KOI-0191, KOI-0628, and KOI-1375). In one case (KOI-0640), our survey does not detect the companion object at 0.44 arcsec with a contrast magnitude of 0.62 mag in the i band. We have examined the AstraLux image and concluded that the ambient conditions were poor for this particular night. This is also reflected in its sensitivity curve with poor quality. Finally, we have detected companions to the remaining eight KOIs (namely, KOI-0658, KOI-1452, KOI-0703, KOI-0704, KOI-0721, KOI-2481, KOI-0111, and KOI-1812) that were not detected by the L13 survey. In L13, the authors justify this non-detection compared to our previous study by arguing that the companions are fainter than their detections limits. However, all planet candidates in these eight planetary systems have calculated a Δm_max that is fainter than the calculated magnitude differences of the companion sources. Thus, the detected companions in our survey and those not detected by L13 can actually severely affect the candidacy of the planet candidates or, at least, their planet properties, which are, thus, non-negligible detections. It must be noted that we have detected companions in the range 2.5–6.0 arcsec for another 26 coincident KOIs that could also affect the derived properties of the planet candidate or even their candidacy and that they are not detected in L13 due to field of view restrictions.

Since no individual sensitivity limits are provided for each KOI in L13, we can use their quality definition for each KOI (low, medium, or high), use the correspondent sensitivity curve in the i_SDSS band provided in their paper to estimate the BSC, and compare it to the values found for our isolated KOIs. The results are presented and compared in Table 11. In general, our observations reduce the P_BS by a more significant amount.

5.4. General comparison of Kepler high-resolution imaging surveys

We can compare the results of the surveys by using the BSC parameter defined in previous sections. In particular, we can estimate how each of these high-resolution surveys have contributed to the validation of the planet candidates by measuring how it has diminished the probability of a KOI to have a blended non-detected source. We can calculate the BSC parameter for each observed target in each survey and compare the BSC value prior and after the imaging observations (as we have done in Sect. 4.1.1 for the KOIs observed in the present study). We can define the Improvement parameter as the relative difference between the prior BSC value and the new BSC value obtained with the high-resolution image (i.e. Improvement = (P_BS,0 − P_BS) /P_BS,0). By doing so, we can summarize the results by the histogram shown in Fig. 10. According to this, we can see that A12 obtained a similar distribution of improvements than our work. The only handicap of this survey lies is that the targets were observed in the near-infrared while Kepler observations are performed in the optical band. Thus, they could miss some bluer companions that affect the Kepler photometry. This was demonstrated in Sect. 5.2 with the cases of KOI-0111 and KOI-0555. On the contrary, we could be missing redder companions that are possibly bound (such KOI-0372B), which could have implications in the knowledge of the formation and evolution of the planetary system.

The H11 speckle imaging study does not reduce the probability of a blended scenario in more than 10% for the large majority of their observations. This is mostly due to the limited contrast magnitude and small field of view that they use.

In the case of L13, they present the largest sample of high-resolution images, which are also observed in the optical range. Their distribution of improvement for L13 is rather broad. With these observations, the 93% of the planet candidates hosted by their observed KOIs (1163 planet candidates in total) diminish the probability of a blended scenario by less than 50%. The remaining 7% of the planet candidates (87 in total) reduced this probability by more than 50%. However, since we calculated their P_BS by assuming the typical sensitivity curves that are provided by L13 for each target, according to their quality definition of the AO image (namely low, medium, and high), we warn that applying the particular sensitivity curves for each KOI could significantly modify these results.

Finally, our survey provides high-resolution observations for 230 Kepler host candidates (174 still active KOIs) in the optical range. Our results show improvements in the blended source probability above 50% for the 62% of the planet candidates studied (186 in total) and below 50% for the remaining 38% of the planet candidates (115 in total).

In Fig. 11, we show all the companions detected by the four surveys. Empirical sensitivity curves according to these detections are also plotted for each of the surveys.

6. Conclusions

In this work, we have delivered a second release of high-resolution observations of Kepler candidates with the AstraLux/CAHA instrument. In total, 230 KOIs were observed, and 174 currently kept at least one of their planets as candidates. Our complete multiplicity study shows that 111 KOIs (67.2%) do not present any visual companion closer than 6 arcsec; 35 KOIs (20.1%) do present at least one source between 3–6 arcsec; and 30 KOIs (17.2%) present close companions within 3 arcsec. Among the new sample of close companions, we have concluded that KOI-3158B could be physically associated to the planet host, thus being an S-type binary system with multiple planet candidates orbiting one of the components of the binary. These results clearly shows the need for obtaining high-resolution images of planet candidates prior to starting the confirmation process by other (more expensive and time-consuming) techniques, such as radial velocity.

We have analyzed the quality of the images by defining the BSC parameter (background source confidence) that provides the level of confidence by which one can assure that the KOI is isolated within some specific magnitude difference and angular separation. We calculated the BSC parameter for all targets without close companions below 3 arcsec, reducing this probability by more than 50% for the 62% of our targets. This implies that thanks to the AstraLux observations we have increased the level of confidence that the KOIs are not blended by an eclipsing binary or affected by additional sources contaminating the Kepler light curve.

We have performed a comparison to the other three main catalogs of high-resolution images published so far. With the adaptive optics study by Adams et al. (2012, 2013) our conclusions show that our work reduces the blended source probability by a high percentage for the majority of the observed targets. The only handicap of the former study is that it is performed in the near-infrared regime; thus, it misses possible hotter companions that could affect the Kepler photometry obtained in the optical range. This becomes clear in the cases of KOI-0111 and KOI-0555, where we detect close companions non-detected by them. By contrast, we could be missing redder physically bound companions (although they do not affect the planet candidacy since the optical Kepler light curve would not be significantly affected). The other adaptive optics survey provides a large set of observations performed by a robotic telescope (Robo-AO) by Law et al. (2013). They obtained high-resolution images for 714 KOIs in the optical regime. However, although their resolution is high enough, their field of view is too limited (only detecting sources at less than 2.5 arcsec) and their limiting contrast is too small to improve the isolated confidence by more than 50% for a significant percentage of their candidates. Hence, this survey must be combined with other observations to achieve useful conclusions. Finally, the speckle study by Howell et al. (2011) is too limited in both field coverage and magnitude difference, although they achieve very high angular resolutions. This implies that they do not improve the isolation confidence for their targets by more than 10% and is, thus, not conclusive for this purpose.

In this paper, we have also included the high-spatial resolution results for the Kepler-186 (KOI-0571) system, hosting the recently validated planet Kepler-186f, an Earth-like planet in the

habitable zone of its cool dwarf (Quintana et al. 2014). The authors obtained high-contrast images in the Ks band using AO with NIRC2 at the Keck-II telescope and optical speckle imaging with DSSI at WIYN telescope. The present work adds additional observational support to the conclusion that Kepler-186 is isolated with optical information that is farther than the 1.2 arcsec of the WIYN observations and in the wavelength regime of Kepler observations.

Our high-resolution survey of Kepler candidates with similar important surveys has proven the need for complementary observations of the Kepler candidates.

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Acknowledgments

This research has been funded by Spanish grants AYA 2010-21161-C02-02, AYA2012-38897-C02-01, and PRICIT-S2009/ESP-1496. J. Lillo-Box thanks the CSIC JAE-predoc programme for the Ph.D. fellowship. We also thank Calar Alto Observatory, both the open TAC and Spanish GTO panel, for allocating our observing runs, and the CAHA staff for their effort and passion in their work, being very helpful during our visitor and service AstraLux observations. This publication made use of VOSA, developed under the Spanish Virtual Observatory project supported from the Spanish MICINN through grant AyA2008-02156. We also thank the Kepler Team for providing valuable information about the targets to be observed. This paper made use of iPython (Pérez & Granger 2007), and the python libraries numpy, matplotlib, matplotlib_venn, and asciitable.

References

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