Ultra-short-period Planets in K2. III. Neighbors are Common with 13 New Multiplanet Systems and 10 Newly Validated Planets in Campaigns 0–8 and 10

Previous work has turned up numerous cases of USPs in multiplanet systems, often identified because the companion planet(s) had obvious transits in the photometry (e.g., Adams et al. 2017). It is also common, however, for multiple candidates at different periods to be reported individually in separate works, with no combined system analysis performed or even consistent candidate numbering (e.g., EPIC 206215704/K2-302; see Section 6.3.3). Motivated by these discrepancies, for the first time we have systematically searched every candidate USP in K2 Campaigns 0–8 and 10 for evidence of transiting companions.

We note that for completeness, a full census of USPs in multiplanet systems would require searching all known transiting planet candidates at any period for heretofore missed USPs, using the targeted short-period search methods described in this paper. Complementary programs to identify nontransiting, longer-period companions are also recommended, such as long-term RV monitoring, through which planets have been found around HD 3167 (Christiansen et al. 2017) and WASP-47 (Neveu-VanMalle et al. 2016). Both projects are, however, well beyond the scope of the present work.

To conduct our search, we first compiled values from the literature for all of the known companions to our 74 USP candidates. For each candidate that has already been identified in the literature, starting with the USP (which we designate p1, to avoid confusion with other designations in the literature), we folded the transit data at the known period and fit for the transit light curve (as in Section 2).

Overlapping transits are common in multiplanet systems, particularly for USPs with frequent transits. We found, however, that the impact of overlapping exterior transits on the parameters derived for the USP is minimal because the fraction of USP transits that are overlapped is typically small. Overlap is more important for the outer planets, particularly those with only a handful of transits. Thus, we subtracted the best-fit model for the USP from the light curve before searching for the next potential candidate planet. For systems with companions that have already been identified, we used the reported orbital periods to progressively search for, fit, and remove each of the companion transit signals. After exhausting the list of known companion periods, we searched for new periodic signals, both between known orbital periods (if there are already multiple candidates identified) and beyond the longest period identified (we formally searched out to half the length of the time series). For simplicity, we used the same EEBLS routine for longer-period planets as we used to search for USPs, which may affect recovery efficiencies, though with a median P = 4.5 days, most companions have short-enough transit durations that the 1 day smoothing window will not have a huge effect. We also dropped the initial S/N detection threshold for periodic signals and gave everything above 3σ a visual examination, because the S/N of a periodic signal can be difficult to calculate for a multiple-planet system with many overlapping periodic peaks. Some candidates that originally appeared at low S/N in the power spectra were actually detected at moderate S/N once we had fit for the S/N of the transit light curve with the other signals removed. We found seven companions with S/N = 3–7, and another six between S/N = 7–10. None of the low and marginal S/N candidates are listed as validated; see discussion in Section 5. In this manner, we identified 24 systems with 30 longer-period candidates, in addition to our 74 USP candidates. Thirteen of these systems are newly identified as hosting multiple candidates, and of these four are entirely new systems (both the USP and companion are new).

Both the USPs and their companions tend to be small (see Figure 1). Of the 24 candidate multiplanet systems we identified using EVEREST photometry, we documented only three cases where both the USP and the companion(s) would have been detectable using k2sff photometry: EPIC 206103150 (a.k.a. WASP-47), three transiting planets (Becker et al. 2015); EPIC 220383386 (a.k.a. HD 3167), two transiting planets (Vanderburg et al. 2016a) (as well as one nontransiting planet, Christiansen et al. 2017); and EPIC 220674823 (a.k.a. K2-106), two transiting planets (Adams et al. 2017).

We observed targets using two similar speckle imaging instruments at three different telescopes. The Differential Speckle Survey Instrument (DSSI; Horch et al. 2009) was deployed as a visiting instrument at the WIYN 3.5 m, Gemini North 8 m, and Gemini South 8 m telescopes. The NN-EXPLORE Exoplanet Stellar Speckle Imager (NESSI; Scott et al. 2018) was installed at WIYN in late 2016 to replace DSSI at that facility. Both DSSI and NESSI operate by splitting the incoming light into blue and red channels and recording simultaneous images of each target with an Andor iXon Ultra 888 EMCCD camera on each channel. For speckle imaging, we acquire a data set consisting of a small number (1–10) of 1000-frame data cubes on each target. The number of 1000-frame sets taken normally depends on target brightness and observing conditions. Each frame is 40 ms long at WIYN and 60 ms at the Gemini telescopes, exposure times intended to maximize S/N for the typical seeing at each site. Along with data sets taken of science targets, we observed nearby single stars to serve as point-source calibrators. For each data set, the fields of view are confined to 256 × 256 pixel subarray readouts of the full EMCCDs. These narrow fields are sufficient to cover the typically ∼2'' wide isoplanatic atmospheric patch centered on the target with a plate scale that critically samples the diffraction-limited resolution at each telescope. At Gemini, the field of view is 2.8 × 28 with a plate scale of 0011 pixel⁻¹ and at WIYN it is 4.6 × 46 with a plate scale of 0018 pixel⁻¹. Filters with bandpass widths of 40–54 nm are used in each channel. For these data, filters with central wavelengths at 692, 832, and 880 were used.

A standardized data reduction pipeline was applied to all of these speckle data. The basic reduction and production of high-level data products were previously detailed by Howell et al. (2011). The high-level data pipeline products consist of a reconstructed, diffraction-limited resolution image of the target and surrounding field in each filter, relative photometry and astrometry of any companion sources detected near the target, and a contrast curve that gives the detection limit for point sources as a function of angular separation from the target.

We searched for stellar companions to EPIC 201650711 with speckle imaging using the HRCam instrument on the 4.1 m Southern Astrophysical Research (SOAR) telescope (Tokovinin 2018) on 2019 March 18 UT. Observations were in Cousins I band, a similar visible bandpass to Kepler. The 5σ detection sensitivity and speckle autocorrelation functions from the observation are shown in Figure 2. A 1.2 mag fainter companion was detected in the SOAR observation at a separation of 179. This companion also appears in the Gaia DR2 catalog (GAIA DR2 source id 3812335125094701056, g' = 13.79, Δg' = 1.35) and has a comparable parallax and proper motion to the target star, indicating the two are likely associated; see Section 6.4.2 for further discussion of this system.

For high-resolution imaging in the near-infrared, we utilized two adaptive optics systems: the first was on the Hale 5 m telescope and the second was on the Keck II 10 m telescope.

The Palomar Observatory observations were made of EPIC 220554210 and EPIC 228801451 with the PHARO instrument (Hayward et al. 2001) behind the natural guide star AO system P3K (Dekany et al. 2013) in 2016 and 2017 (see Table 4). The observations were taken in a standard five-point quincunx dither pattern with steps of 5'' in the narrowband K_short filter (λ_o = 2.145). Each dither position was observed three times, offset in position from each other by 05 for a total of 15 frames, with an integration time of 30 s per frame for EPIC 220554210 and 3 s per frame for EPIC 228801451. PHARO has a pixel scale of 0025 per pixel for a total field of view of ∼25''.

The Keck Observatory observations of five USP candidates (Table 4) were made with the NIRC2 instrument on Keck II behind the natural guide star AO system (Wizinowich et al. 2000) in 2015 and 2016. The observations were made in the standard three-point dither pattern that is used with NIRC2 to avoid the left lower quadrant of the detector, which is typically noisier than the other three quadrants. The dither pattern step size was 3'' and was repeated twice, with each dither offset from the previous dither by 05. NIRC2 was used in the narrow-angle mode with a full field of view of ∼10'' and a pixel scale of 00099442 per pixel. The Keck observations were made in the K filter (λ_o = 2.196; Δλ = 0.336 μm).

The AO data were processed and analyzed with a custom set of IDL tools. The science frames were flat-fielded and sky-subtracted. The flat fields were generated from a median average of dark-subtracted flats taken on-sky. The flats were normalized such that the median value of the flats is unity. The sky frames were generated from the median average of the dithered science frames; each science image was then sky-subtracted and flat-fielded. The reduced science frames were combined into a single combined image using an intrapixel interpolation that conserves flux, shifts the individual dithered frames by the appropriate fractional pixels, and median-coadds the frames. The final resolution of the combined dithers was determined from the FWHM of the point-spread function with typical final resolutions of 01 and 005 for the Palomar and Keck observations, respectively.

To within the limits of the AO observations, no stellar companions were detected. The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around the primary target every 20° at separations of integer multiples of the central source's FWHM (Furlan et al. 2017; M. B. Lund et al. 2021). The brightness of each injected source was scaled until standard aperture photometry detected it with 5σ significance. The resulting brightness of the injected sources relative to the primary target set the contrast limits at that injection location. The final 5σ limit at each separation was determined from the average of all of the determined limits at that separation, and the uncertainty on the limit was set by the rms dispersion of the azimuthal slices at a given radial distance.

EPIC 210605073, from Campaign 4, was previously reported as a candidate by Adams et al. (2016). There are no spectroscopically derived stellar parameter values for T_eff, [Fe/H], and log(g) because it is too faint for follow-up observations (Kep = 17.887). This system is our largest candidate and the only UHJ. For its size, its period is short enough (P = 0.567 days) that its confirmation would be highly significant because tidal effects are quite strong on giant planets at this distance; the shortest-period confirmed UHJ, NGTS-10 b, has P = 0.767 days (Figure 1). For the sake of completeness, we attempted to validate the system despite the highly uncertain stellar parameters. We inferred a value of T_eff = 7918 K from the available stellar photometric magnitudes (g = 17.859, r = 17.896, i = 18.004, z = 18.125) and the corresponding temperatures in Table 1 from Pinsonneault et al. (2012); the number we use is the median of the three estimates for T_eff using the differences in each band, using 500 K for the error bar. We arbitrarily assigned [Fe/H] = 0.0 ± 0.5, and log(g) = 4.0 ± 0.5. Even with these favorable guesses, vespa returned a high FPP of 0.6, and this object remains an uncertain, though unlikely, candidate.

We do not find any compelling short-period desert candidates in this work. Just three candidates are within 1σ of the lower radius edge of the desert (R_p > 3 R_⊕): EPIC 201085153, EPIC 201089381, and EPIC 203533312.

EPIC 201085153 (R_p = 2.4 ± 1 R_⊕, P = 0.26 days, Kep = 17.3) is a candidate planet in a two-candidate system (see also Section 6.3.1), but its planetary radius is low enough and its radius error high enough that it likely falls below the lower edge of the short-period desert.

The only other plausible short-period desert candidates are EPIC 201089381 (R_p = 2.8 ± 0.5 R_⊕, P = 0.16 days, Kep = 14.3) and EPIC 203533312 (R_p = 3.5 ± 0.3 R_⊕, P = 0.176 days, Kep = 12.2), which are also the two shortest-period planets in our sample. These two candidates are so close to their stars that the validation software fails to converge, with vespa returning a "Roche Lobe" error, indicating that its orbital solutions are all necessarily unstable. Given that the location of the Roche limit depends very strongly on the stellar and planetary parameters, a closer examination of these systems is required to determine if these are legitimate planets that we have caught on the brink of disruption (Jackson et al. 2016), or (more likely) a pair of false positives.

EPIC 201609593, from Campaign 1, has two new candidates (P = 0.318 days and P = 1.39 days). Both were detected with marginal S/N = 9 and 4, respectively. The USP is among the smallest candidates in our sample (R_p = 0.5 R_⊕) and has one of the shortest periods (P = 7.6 hr), while the companion is the shortest-period companion to a USP candidate yet found. (No system has yet been found with two USPs.) Neither planet was validated due to their low S/N.

EPIC 220440058, from Campaign 8, has two new candidates (P = 0.627 days and P = 3.836 days). The USP was robustly detected with S/N = 17, while the companion has S/N = 7. No contrast curves are available, and both remain unvalidated as we do not apply a multiplicity boost due to the companion's marginal S/N.

EPIC 201085153, from Campaign 10, has two new candidates (P = 0.257 days and P = 2.26 days). The USP was detected with high S/N = 26, though the companion only has S/N = 4. We note that further follow-up of this system is difficult s it is a faint (Kep = 17.3, J = 15.8) M dwarf (T_eff = 3945 ± 386 K).

EPIC 201438507, from Campaign 10, has two new candidates (P = 0.327 days and P = 2.536 days). Although the USP was detected with S/N = 13, the companion has a very low S/N = 3.4. The USP has not been validated; no contrast curves are available, though the star is sufficiently bright for future observations (Kep = 14.067).

EPIC 212624936 (2 planets), from Campaign 6: Kruse et al. (2019) reported a long-period candidate at P = 11.81 days and S/N = 10, and we found a new USP candidate at P = 0.57 days and S/N = 12.9. Both planets have been validated for the first time in this work.

EPIC 201239401, from Campaign 1: the USP was reported by Vanderburg et al. (2016b) and has S/N = 24, and we have newly identified a companion at P = 6.784 days (low S/N =6). Due to the low S/N of the companion, we have not applied a multiplicity boost, and both remain candidates.

EPIC 205152172, from Campaign 2: the USP (Vanderburg et al. 2016b), with S/N = 28, has a newly identified companion at P = 3.225 days (S/N = 10). Both the USP and the companion have been validated after applying a multiplicity boost.

EPIC 206215704/K2-302, from Campaign 3: A USP with P = 0.623 days was first reported by Vanderburg et al. (2016b), while two candidate outer planets, but not the USP, were reported by Kruse et al. (2019). We have confirmed the validation of the planet at P = 2.25 days, assigned the planetary designation K2-302 b by Heller et al. (2019). However, we do not find the signal at 3.1 days, which is spurious and likely an alias (3.1 = 5 × 0.62 days, the USP period); instead, we find a different third planetary signal at P = 2.9 days. The planets have S/N = 14, 13, and 12, ordered by distance from the star. We have verified that three separate transit signals are present in the data and fit each candidate separately. Even after applying a multiplicity boost, the USP and outermost companions narrowly missed validation, with FPP = 0.017 and 0.011, respectively.

EPIC 206298289, from Campaign 3: the USP at P = 0.43 days (Vanderburg et al. 2016b) has a newly identified companion at P = 3.215 days, with S/N = 26 and 7.8, respectively. Due to the marginal S/N of the companion, we have not applied a multiplicity boost, and both remain candidates.

EPIC 210801536, from Campaign 4, has two candidates (P = 0.893 days and P = 2.536 days), with the USP previously reported by Kruse et al. (2019). Both candidates were detected at marginal S/N = 9.6 and 4.4, respectively. There are no contrast curves available to provide constraints on background stars. The host star appeared to be a moderately evolved G or K star, with MAST values of log g = 3.83 ± 0.76 and R_⋆ = 2.0 ± 1.7 R_Sun; however, the isochrones value for the stellar radius is ${0.85}_{-0.09}^{+0.28}$, which we have used. Follow-up observations to improve the stellar parameters and search for nearby background stars are recommended, as the host star has magnitudes Kep = 13.176 and J = 11.7.

EPIC 220256496/K2-221, from Campaign 8, has two candidates. The USP (P = 0.66 days) was reported by Mayo et al. (2018), and we have found a new candidate with P = 2.601 days. We validate the USP at S/N = 21, but the companion has low S/N = 8.7, and hence remains a candidate.

EPIC 220554210/K2-282, from Campaign 8, has three planets. Kruse et al. (2019) reported two planets, at P = 4.17 and P = 0.705 days (Kruse et al. 2019), and Heller et al. (2019) confirmed the USP as K2-282 b. We did not identify the USP in our search because the dominant signal is from the planet at P = 4.17 days. The larger planet did show up in our longer-period search out to P = 3 days, at an alias at P = 2.08 days or half its true period, but it was evidently not a USP, and this system was set aside as outside our survey parameters. After finding this system during a review of the literature, we easily recovered the two known planets and discovered a third signal at P = 27 days, which has three transits during 78 days of data. A plot of the transits of each of these three systems is shown in Figure 5. The planets have S/N = 21, 18, and 8.6 respectively. All three planets were validated independently, even without a multiplicity boost.

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EPIC 206024342/K2-299: Three candidates have been previously reported in this system (P = 0.91 days, P = 4.507 days, and P = 14.637 days; Vanderburg et al. 2016a; Kruse et al. 2019). The second planet (P = 4.5 days) was previously validated as K2-299 b by Kruse et al. (2019); we have validated the other two for the first time. By distance from the Sun, the candidates have S/N = 11.2, 11.8, and 15. We have also identified a fourth transit-like signal at P = 27.7 days, albeit with low S/N = 3.6. Although the fourth candidate received a low FPP from vespa, it remains a candidate due to its low S/N and long period, with only three transits detected.

Although the scope of this paper is confined to K2 Campaigns 0–8 and 10, two candidate planets were found in systems from Campaign 5 that were also observed in Campaign 18. We conditioned and searched the Campaign 18 data for just those two systems in the same manner as the rest of our data, recovering the known USP candidates at high S/N. In both cases, previous tentative detection of longer-period candidates was made more secure by the addition of the extra campaign of data. (Coincidentally, the companions both have periods near P = 11.6 days, though the transit midtimes do not line up.)

EPIC 211357309, from Campaign 5 and 18, has a known USP candidate at P = 0.46 days with S/N = 26 (Adams et al. 2016; Dressing et al. 2017). We also identified a new candidate, at P = 11.61 days. We combined data from Campaigns 5 and 18 to search for the best period for both candidates and found that the outer companion was detected during nine transits across both campaigns, at S/N = 8.8 (Figure 6). Neither candidate is validated because the S/N of the exterior candidate precludes a multiplanet boost.

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EPIC 211305568, also from Campaign 5 and 18, has two candidates identified by Dressing et al. (2017), who calculated high FPPs for both candidates. Our FPPs are similarly high, and due to the low S/N (4.6) of the outer companion, we did not apply a multiplanet boost to the USP (at S/N = 15). There is a nearby faint stellar companion (48, ΔJ = 2, from 2MASS via ExoFOP website), which we have not corrected for because its dilution factor in visible wavelengths is unknown. The transits of the outer planet are shown in Figure 7.

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EPIC 228721452/K2-223 is a multiplanet system (P_b = 0.505 days and P_c = 4.56 days) identified by Mayo et al. (2018) and Livingston et al. (2018). In our initial survey out to longer periods (P < 3 days), we clearly identified a signal at P = 2.28 days (an alias of half the true period of K2-223 c) with S/N = 15. However, we did not identify the USP (S/N = 10) until after reviewing the literature. We easily identified both planets and found fitted parameters are almost identical to those of Mayo et al. (2018) and Livingston et al. (2018). We have validated both planets.

EPIC 228801451, a.k.a. K2-229, from Campaign 10, has three candidates at P = 0.58 days, P = 8.33 days, and a single transit with P = 31 ± 1 days (Livingston et al. 2018; Santerne et al. 2018). Although we easily identified and validated the USP, neither of the longer-period planets appears in our data. Our light curve of the USP also contains points that appear, in unbinned plots, to be artificially close to zero, due to some unknown processing problem in the smoothing step. Both of these issues are likely due to the star being active and variable; the K2-229 system required special processing to confirm in Santerne et al. (2018).

EPIC 220250254, a.k.a. K2-210, from Campaign 8, was validated as K2-210 b (Mayo et al. 2018) with FPP = 2.31 × 10⁻⁴; our FPP was high enough (FPP = 0.036) that it remains a candidate, with a background eclipsing binary (BEB) the next most likely hypothesis. It is not clear where the discrepancy lies, and a higher-contrast image than the WIYN curve in Table 4 or other follow-up is recommended.

EPIC 201650711, from Campaign 1: We originally flagged EPIC 201650711 p1 as a false positive in a single-planet system with an FPP of 0.92. However, during the course of this work we reexamined the system after a second planet candidate was reported by Kruse et al. (2019) at P = 5.54 days (which did not note the presence of a USP candidate). We call the USP candidate "p1" and the longer-period candidate "p2" to avoid confusion, because both have been referred to as candidate 1 in the literature (Adams et al. 2016; Vanderburg et al. 2016b; Kruse et al. 2019). We have verified that both sets of transits are present in the data (with S/N = 18 and 9.4, respectively). As noted in Section 4.3.2, there is a close, likely bound GAIA companion star 179 away. No significant odd–even differences were detected in the transit depths for either p1 or p2 (odd–even σ = 0.01 and 0.06 respectively); no additional stellar companions were found with high-resolution imaging by SOAR (Figure 2).

Unfortunately, we have not been able to determine which star hosts the candidates (or indeed if they are in the same system); an analysis of the centroid positions of the stars using the Python package lightkurve did not turn up any significant shifts during transit, not unexpected given the close proximity of the stars (less than half a Kepler pixel). The properties of the fainter star are largely unknown apart from having the same GAIA DR2 proper motion (R.A. = 80.1 ± 0.3 mas yr⁻¹, decl. = −29.1 ± 0.3 mas yr⁻¹) and parallax (10.8 ± 0.2 mas) as the brighter star (all values within 1–2σ), indicating a likely bound companion.

For either host star scenario, we estimated the effect of light-curve dilution due to the other star on the calculated planetary radii. For transits around the brighter star (s1), the radius must be increased by ∼15%, to R_{p1, s1} = 0.36R_⊕ and R_p2,s1 = 0.45R_⊕. If the transits are around the fainter star (s2) instead, assuming the stellar properties are identical, both planets would be larger by a factor of 2, or R_p1,s2 = 0.64R_⊕ and R_p2,s2 = 0.85R_⊕. The radius values scale directly with the stellar radius value, so adjusting from the nominal radius of EPIC 201650711 (R_⋆ = 0.34 ± 0.05R_Sun, Dressing et al. 2019) for any reasonable main-sequence stellar radius would still yield planetary-sized values.

More work needs to be done to determine which star(s) host the planets and what the stellar properties are. The relative orbital periods of the two planets, at P = 0.26 days and P = 5.54 days, are qualitatively consistent with most other observed USPs in multiplanet systems, as is the USP being the smaller planet (Section 8.2.4). Assuming that the transit is around the central star, we reran the validation analysis including the SOAR contrast curve, which was not available when we first attempted to validate. The contrast curve halved the FPP for the USP (though it remains unvalidated), though the companion remains unvalidated due to its marginal S/N. We note that the planetary radius values in Table 2 have not been adjusted for dilution.

EPIC 212303338, from Campaign 6, has a USP candidate with P = 0.595 days (Mayo et al. 2018) at S/N = 9. It also has a very close stellar companion (01) in high-resolution images taken with WIYN in 2016, with Δmag = 2.45 at 692 nm (g') and Δmag = 1.99 at 880 nm (r') (Matson et al. 2018). We first consider how much dilution the companion star introduces to the light curve. The candidate planet is small enough (R_p = 0.46 R_⊕) and the stellar magnitudes similar enough that most scenarios are plausibly planetary. We consider two scenarios: (1) The candidate orbits the brighter star. In this case, the true radius should be increased from the value reported in Table 2 by 5%–7%, to 0.5 R_⊕. (2) The candidate orbits the fainter star. In this case, the radius should increase by a factor of about 3, to 1.5 R_⊕, if we were to assume that the fainter star had the same radius as the bright star. The radius of the fainter star is also unknown; but if it is on the main sequence (R_⋆ ≈ 0.2–2 R_⊙), the estimated radius will change by an additional factor of 0.25–2.5, using the brighter star radius of 0.8R_⊙; even in the maximal case, that would be R_p ≤ 4R_⊕. We note that EPIC 212303338 p1 has a relatively high validation FPP (0.67), dominated by BEB scenarios. We leave the final disposition of this candidate as undetermined. We also do not attempt to correct for dilution in the radius value reported in Table 2 due to the uncertainty of which star the signal is around.

EPIC 228732031, a.k.a. K2-131, from Campaign 10, has a confirmed USP at P = 0.37 days (Dai et al. 2017; Livingston et al. 2018; Mayo et al. 2018). We were unable to validate this planet using the EVEREST light curve, and closer inspection revealed high-frequency noise that appeared as quasi-sinusoidal oscillations, likely negatively affecting vespa's score. Noting that our EVEREST best-fit period was 7.8 s longer than the published period (though only by 1.6σ), we redid our analysis using the exact orbital period of Mayo et al. (2018), which reduced, although did not entirely eliminate, the high-frequency noise and also improved, but did not validate, the validation score (EVEREST FPP = 0.45). When we instead used the k2sff light curve, in which no oscillations are visible, we easily revalidated the system with FPP = 2.6 ∗ 10⁻⁴, demonstrating that the issue is with the photometry and not our validation framework. For consistency, the parameters for fits to the light curves derived from EVEREST photometry are the ones reported in the table Table 2.

This was the only instance we noted in EVEREST photometry of such high-frequency noise, though it is possible that other candidates with similar noise were rejected. Understanding the source of the noise would help understand whether this is a significant issue for completeness, but we leave that for future work.

Validation with vespa produced 17 USPs that are more likely than not to be false positives, with FPPs between 50% and 100%. Listed in decreasing order of FPP, the suspect USP candidates (above 90%) are in the following systems: EPIC 228739208*, EPIC 205002291*, EPIC 214539781*, EPIC 219752140, EPIC 214189767; and from 50% to 90%: EPIC 201461352, EPIC 212303338, EPIC 211336288*, EPIC 201231940*, EPIC 201609593, EPIC 210605073, EPIC 210931967*, EPIC 201469738, EPIC 211305568, EPIC 212443973, EPIC 206151047, and EPIC 201239401*. Three of these candidates (EPIC 201239401, 201609593, and 211305568) have additional candidates in their system but do not receive a multiplanet boost due to the low S/N of either the USP or the candidate. An additional three candidates in multicandidate systems had individual FPPs > 50%, but which fell below 5% after applying a multiplicity boost: EPIC 211305568 p2, EPIC 201239401 p2*, and EPIC 220440058 p2*.

Objects labeled with an asterisk (*) are candidates whose signals were only identified in our search of the EVEREST pipeline and not the k2sff pipeline, which may indicate that the signals are more likely to be spurious. This applies to 3 of the 5 candidates with FPP above 90% and 4 of 12 single candidates with FPP from 50% to 90%, as well as two potential companions in multiplanet systems.

One additional candidate we failed to validate was EPIC 201637175 a.k.a. K2-22 b, a disintegrating planet confirmed by RV mass measurements elsewhere (Sanchis-Ojeda et al. 2015a). We do not include this in our statistics as a true false positive, as we did not attempt to correct for the well-known light-curve variability when fitting.

High-resolution imaging data are available for just one system with an elevated FPP (EPIC 212303338, FPP = 0.74, which has a known stellar companion and is discussed in Section 6.4.2). For those targets on the elevated FPP list for which follow-up observations are possible, they are recommended because imaging contrast curves can make a huge difference in validation results, as noted previously for EPIC 201650711 where the FPP dropped in half once imaging was available. A more dramatic case was EPIC 213715787 a.k.a. K2-147 b, where we estimated the FPP both with and without the Subaru AO imaging constraints and found that including them dropped the FPP from 0.6 to 3 × 10⁻⁵.

Mayo et al. (2018) completed a similar survey of K2 Campaigns 0–10 using their k2sff pipeline (Vanderburg & Johnson 2014). That work searched for candidates at all orbital periods and found 14 of our 74 USP candidates. A combination of the improved photometric sensitivity of the EVEREST pipeline over k2sff (as discussed in Sections 2 and 7.2) and our smoothing and detection algorithms, which have been tailored to detect short periodic signals, probably accounts for why they found only one-fifth of the candidates that we did. Our candidates also skew smaller: 43% of our USP sample has R_p < 1 R_⊕ but only 17% of theirs did (3/17).

They also reported three candidates that we do not have listed as planets, all of which we have identified as false positives in Table 6: EPIC 206169375, which has a close stellar companion and large odd/even differences visible in the light curve, and EPIC 212645891 and EPIC 229039390, two eclipsing binary stars with true periods at twice that identified by Mayo et al. (2018).

Table 6. Rejected Planet Candidates

EPIC	Reported P (days)	Real P (days)	Notes
201577017	0.4775252	0.955	EB
201606542	0.44437	0.8887	Spectroscopic binary (SB2, double peak)
202092559	0.6687	1.3374	EB
202094740	0.6897	1.3794	Spectroscopic binary (SB2, single peak)
202139294	0.5217136	1.0434	EB
202971774	0.2358684	0.47171	EB
203518244	0.84	2.1495	EB
203633064	0.7099504	1.4199	EB
204880394	0.2338108	0.71007	EB
205377483	0.3938505	0.78763	EB
206169375	0.36745	⋯	Close AO companion + high odd/even difference
206315178	0.3176828	0.63532	EB
210961508	0.349935	0.769	EB
211158357	0.2990737	0.5981	EB
211613886	0.9587591	1.9175	EB
211685045	0.769057	⋯	Variable star
211843564	0.4520313	0.90399	EB
211995325	0.279	⋯	Pipeline depth mismatch
212066407	0.8218241	1.6436	EB
212150006	0.89838	⋯	Pipeline depth mismatch, close AO companion (05)
212270970	0.7165461	1.4331	EB
212362217	0.6962935	1.3928	EB
212432685	0.531684	1.0634	EB
212645891	0.32815384	0.65626	EB
213123443	0.546127	1.0923	EB
214984368	0.2633809	0.5268	EB
215125108	0.738067	1.4761	EB
215353525	0.9085915	1.8172	EB
215381481	0.533393	1.0668	EB
218195416	0.4951253	0.9903	EB
218952024	0.56007	1.1201	Spectroscopic binary (SB2, double peak)
219266595	0.944284	1.8886	EB
219420915	0.5150196	1.02988	EB
220198551	0.7988453	1.59758	EB
220201207	0.4378443	⋯	Variable star
220282718	0.5551606	1.1103	EB
220263105	1.44473	⋯	Variable star
220316844	0.6719977	1.34404	EB
220569187	0.6045	⋯	Pipeline depth mismatch
229039390	0.25605	0.5033	EB, close AO comp. (06)

Note. EB = eclipsing binary from visual inspection of EVEREST photometry. Spectroscopic binary = binary found through follow-up spectroscopy, Section 4.1. Close AO companion = nearby star likely contaminating the system, from https://exofop.ipac.caltech.edu/. Pipeline depth mismatch = EVEREST and k2sff depths differ by >50%, indicating different blending fractions.

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

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Livingston et al. (2018) analyzed only Campaign 10 of K2 using their own photometric pipeline and found six USP candidates: EPIC 201110617/K2-156, EPIC 201595106 (which we validate), EPIC 228721452 (two planets: P = 0.5 and P = 4.5 days), EPIC 228732031/K2-131 b, EPIC 228801451/K2-229 (two planets: P = 0.58 and P = 8.3 days), and EPIC 228836835 (which we validate). We identified five of the USP candidates and discussed the one we missed, EPIC 228721452, in Section 6.3.6. We found an additional nine candidates that were not present in Livingston et al. (2018), including EPIC 201438507 (two candidates), EPIC 228739208, EPIC 228814754, EPIC 228877886, EPIC 201427007 (which we validate), EPIC 201130233/K2-157, EPIC 228813918/K2-137, EPIC 201085153 (two candidates), and EPIC 201089381.

Kruse et al. (2019) used the earlier EVEREST 1.0 photometry to announce 818 planet candidates in C0–8 at all orbital periods (their Table 7). By design, their survey made conservative cuts (10σ) on odd–even depth differences, allowing likely eclipsing binaries to pass through. By comparing our set of candidates to theirs, we can offer a real-world assessment of the reliability of results from the EVEREST 1.0 catalog, particularly at the shortest periods where EVEREST 1.0 performed worst (Kruse et al. 2019).

Removing our candidates from Campaign 10, we found 58 USP candidates in C0–8 (excluding EPIC 220554210, which we originally missed), of which Kruse et al. (2019) detected 24 (33%). Again, our sample skews smaller: 32% of our sample had R_p < 1R_⊕ but only 4% of theirs did. They also reported 31 USP candidates that do not appear in our list (Table 2). We examined the folded light curves for all of these candidates and found that almost all of the signals are either due to eclipsing binary blends (28) or to stellar variability (2), often at twice the period listed in Kruse et al. (2019). These objects are included in our table of rejected candidates in Table 6. Meanwhile, none of our candidates appear in their list of false positives or eclipsing binaries (their Tables 6 and 9). Just one of their candidates that we originally missed is genuine (EPIC 220554210) as discussed in Section 6.3.3.

Because the P < 1 day sample of Kruse et al. (2019) had so many false positives, we briefly examined some of their longer-period candidates (1 ≤ P ≤ 3 days) to estimate what fraction might not be planets. For this paper, we did not vet any candidates with P > 1 day (aside from companions to USPs), but our initial search went out to P < 3 days. This was to guard against missing objects where the period had been aliased out of our search region, as discussed in Section 2. Our initial diagnostic plots include the light curve folded to the nominal period, P, and 2 × P. These plots are useful for quickly identifying eclipsing binary stars, which may exhibit different transit depths between odd and even transits, patterns of out-of-transit variability (often due to ellipsoidal variations in binary star systems), and other features that can be obscured when the data are folded to shorter periods than the true orbital period for the system. An example is shown in Figure 8, where a plausible transit of EPIC 205377483 at the originally identified period (P = 0.39 days) is shown to have unequal transit depths, an EB signature, when folded at twice the nominal period. By examining plots like this for the Kruse et al. (2019) sample of 69 candidates with 1 ≤ P < 2 days, we found 9 clear eclipsing binaries and 20 with some other kind of periodic signal, likely stellar variability—2 more candidates we identified at somewhat different periods than they did. Repeating this exercise for candidates with 2 ≤ P ≤ 3 days, we found that 15 of their 68 candidates in this range were likely nonplanets (11 stellar noise/variability, 4 EBs) and 3 were candidates at different periods (including EPIC 206215704, discussed in Section 6.3.3). We thus find that the short-period planet candidates reported in Kruse et al. (2019) (P < 3 days) include a significant fraction of variable and eclipsing binary stars mixed among the viable planetary candidates, with the percentage of false positives decreasing with longer orbital periods (47% with P ≤ 1 day, 42% with 1 ≤ P < 2 days, and 22% with 2 ≤ P ≤ 3 days). Additional false positives are also expected after further vetting and follow-up observations, as in this work.

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We first examined the distance of USPs in multiplanet systems from their stars. We found that the planetary semimajor axis follows a trend of increasing distance from the star with stellar radius so that USPs around smaller stars are closer to their stars. This trend holds for both the USPs and for their nearest companions, although the USPs follow a shallower trend. The ratio of semimajor axes, however, a₂/a₁, has a minimum value of 2.4 that holds across all stellar radii, as shown in Figure 10.

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We next examined how closely USPs in multiplanet systems are spaced to each other. We compared our sample to the work of Weiss et al. (2018), which examined 909 planets in 355 multiplanet systems from the California Kepler Survey (CKS). We followed their formulations to estimate the planetary masses from the radius values and used that to calculate the Hill radius, which is defined following Gladman (1993) as

$$\begin{eqnarray}&&{R}_{{\rm{H}}}={\left(\displaystyle \frac{{m}_{j}+{m}_{j+1}}{3{M}_{\star }}\right)}^{1/3}\displaystyle \frac{({a}_{j}+{a}_{j+1})}{2},\end{eqnarray} \tag{ 1 }$$

where planets of masses m_j and m_j+1 orbit a star of mass M_⋆ at semimajor axes a_j and a_j+1. The mutual separation between two planets, in units of mutual Hill radii, is

$$\begin{eqnarray}&&{\rm{\Delta }}{R}_{{\rm{H}}}=({a}_{j+1}-{a}_{j}){R}_{{\rm{H}}}.\end{eqnarray} \tag{ 2 }$$

We found a similar average separation to Weiss et al. (2018), with a median ΔR_H = 18 for USP systems versus 22 for CKS sample, but a higher fraction of USP systems are more closely spaced, with 22% of the USP sample less than 10 Hill radii apart, compared to 7% in the CKS sample (Figure 11). The minimum separation found is almost identical (ΔR_H = 4.5 for USPs and ΔR_H = 4.9 for the CKS sample). We note that Chambers et al. (1996) found separations of less than 10 Hill radii to be always unstable for systems with three or more planets. We also found that the USP sample had a larger fraction of planet pairs with very large separations (>50 Hill radii): 19% of USPs versus 8% in the CKS sample; in our case these are all small, Mercury-sized planets with very low estimated masses and hence very small Hill radii. What our sample is missing, compared to the CKS sample, is planet pairs at intermediate separations. Whether this is a selection bias effect (Kepler targets were observed for much longer than K2 targets, making longer-period detections possible) or an artifact of different formation histories is left for future research.

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The inclination distribution of 24 systems with one USP and one or more transiting companions are shown in Figure 12. We find that USPs have smaller inclinations than their companions, which is consistent with their geometrical advantage because more distant companions need to be more tightly aligned to be observed to transit (see Section 8.3). We calculated the mutual inclination as the difference between the two fitted inclinations and calculated the joint error using the maximal values of the asymmetrical errors (corresponding to the lower error bar, after requiring that i ≤ 90° from the fitted values). We also find that the mutual inclinations of systems with USPs are quite low (mean Δi = 12 ± 12), with few cases significantly different from zero, and none with Δi > 8°. The mutual inclinations presented here are much lower than those in the sample reported by Dai et al. (2018), which comprises a different set of objects with little overlap, as discussed in Section 8.4. It is possible that our sample is more strongly biased against objects with higher inclination dispersion, as we include many small candidates at moderate to low S/N and would not easily detect grazes, which can result in even shallower transit depths. We also note that many of our objects have higher errors than the difference in inclinations.

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The distribution of periods and radii for all 74 USPs is shown in Figure 4. There is no difference in the observed period distribution of USPs with and without companions, which is essentially flat between 0.2 and 1 day, with a median of 0.58 days. For companion planets, the median period is 4.5 days for the second planet in the system and 18.6 days for the third or fourth planet, with a maximum of 31 days for transiting companions (WASP-47 c, at 588 days, is one of only two companions known that does not transit).

Little evidence is seen for current mean-motion resonances involving USPs. We examined period ratios between candidate planets in multiplanet systems. If the ratio between the periods is within ±0.1 of an integer or half-integer ratio (i.e., 0.0 ± 0.1 or 0.5 ± 0.1), it was considered near resonance. This was intended as a first look because actual resonance or past resonance membership would need to be determined with dynamical modeling. For period ratios up to a factor of 10, just four USPs meet that criteria, which is fewer than if the period ratios were randomly distributed (40% of numbers end in those digits, which would be 0.4 × 30 combinations = 12 planets). None of the USPs are very near low-order resonances, with the closest being (1) EPIC 206024342/K2-299, where the period ratio of the USP to the second-closest planet is within 5% of a 5:1 ratio; (2) EPIC 220554210/K2-282, where the period ratio of the USP to the second closest planet is within 9% of a 6:1 ratio; (3) EPIC 201239401, where the period ratio of the USP to the second closest planet is within 1% of a 15:2 ratio; and (4) EPIC 228721452/K2-223, where the period ratio of the USP to the second closest planet is within 2% of a 9:1 ratio.

Some additional near-resonances can be found among the outer companions, though with four out of nine combinations flagged it is consistent with random chance. (4) In EPIC 212157262/K2-187, the second and third planets are within 1% of a 5:2 ratio. Two pairs are within 10% of a 2:1 ratio: (5) EPIC 211562654/K2-183, second and third planets, and (6) EPIC 212157262/K2-187, third and fourth planets. (7) EPIC 220554210/K2-282 has the second and third planets within 3% of a 13:2 ratio.

Only 6out of 24 systems have any members near a resonance, as broadly construed here. Clearly, if dynamical processes required mean-motion resonances to emplace the USPs, they have long since decoupled from the resonances in all of the systems we examined. A more detailed dynamical study would be required to determine if any of the near-resonant period ratios actually correspond to potential current or past resonances.

We did not confirm any UHJs (R_p ⪆ 10R_⊕) and found just one UHJ candidate with moderately high FPP; however, by optimizing our search to discover small planets it is possible that some of the deepest transits might have been sigma-clipped, so we do not consider our UHJ count to be complete. Medium to large planets (R > 3R_⊕) also appear to be rare among planetary companions to USPs, with just one known among 24 systems (WASP-47 b). We also did not identify any compelling candidates in the short-period desert (∼3–10 R_⊕), as discussed in Section 6.1.2.

Excluding the lone UHJ candidate, EPIC 210605073, and focusing on the smaller population, we find a few correlations of note. First, USPs with companions are marginally more likely to be larger (median: 1.36 R_⊕, sample N = 24) than those without (median: 1.07 R_⊕, sample N = 50), with KS test p-value = 0.02. This is unlikely to be a detection effect because there may, if anything, be a slight bias toward finding a smaller USP in a system with a larger known companion than to find one on its own.

Second, we do not find evidence that the intrinsic population of USPs must have smaller radius values than their companions. Although it is true that the observed median radius of all USPs (1.14 R_⊕, N = 74) is different from that of their known companions (second planets: median = 1.5 R_⊕, N = 24, p-value = 0.04; third/fourth planets: median = 2.6 R_⊕, N = 11, p-value = 3 × 10⁻⁴), this difference is entirely consistent with different detection efficiencies based on stacking shallow transits. At the median orbital period for a USP, a second planet, and a third planet (at 0.58 days, 4.5 days, and 22.6 days respectively), a K2 campaign of 80 days would yield a maximum of 138, 18, and 4 transits (ignoring gaps). Assuming that stacking yielded improvements proportional to the square root of the number of transits, the USP would be detectable at 2.8 and 5.9 times shallower depths than the second and third planets; because the radius is proportional to the square root of the depth, the detected companions would have larger radius values by a factor of 1.7 for second planets and 2.4 for third/fourth, very similar to the observed differences in median values. We do not account here for any of the other detection biases that are also at play, including inclination effects (making more distant and smaller planets less likely to transit) as well as sampling effects (making USPs with the shortest periods harder to detect, because each transit has only 1–3 points per transit with K2's 30 minute observing cadence).

The first paper to provide a mechanism for the origins of USPs involving high-eccentricity migration was Petrovich et al. (2019). In particular, they found that USP planets should have spin–orbit angles and inclinations relative to outer planets in the range of 10°–50°. They also predicted that most USPs should have companions that are either more distant than P > 20 days or with high mutual inclination (i > 20°). As seen in Figure 12, we see very low mutual inclinations in multicandidate systems (albeit with large error bars). We also see many companions at shorter orbital periods, although our sample is far from complete and transit detectability decreases rapidly for more distant companions.

Pu & Lai (2019) also presented a model that requires the presence of additional companions to emplace planets on short-period orbits. In this scenario, the future USP starts out as the innermost one in a chain of planets, with a modest orbital eccentricity (∼0.1), and encounters resonances that increase the mutual inclination with respect to the exterior planets. This theory predicts that (1) planets exterior to the USP must orbit close enough that secular interactions can maintain a nonzero eccentricity for the USP for a long enough time that tidal interactions can emplace the USP; (2) the USPs should be less massive than the exterior planets; and (3) the USP may have an orbit that is significantly misaligned with the rest of the planetary system.

Our results have broad agreement with Pu & Lai (2019) on point 1, qualified agreement on 2, and qualified disagreement on point 3:

• 1.  
  Planets must orbit close enough: we have found that a third of our sample of K2 USPs has a transiting companion within P < 31 days, and our results are consistent with most of the apparently singleton systems hosting nearby planets that do not transit (Section 8.3). The separations between planets are small both physically (Figure 10) and dynamically (Figure 11), with an average mutual Hill radius of ΔR_H = 18; nearly a quarter have ΔR_H < 10.
• 2.  
  The observed sample of USPs have smaller radii than their companions, but this is consistent with detection biases toward finding small planets with more frequent transits (Section 8.2.4). A further caveat is that smaller radii may not correspond to smaller masses. Measured masses are unavailable for more than a handful of planets.
• 3.  
  While we observe that USP inclinations span a larger range than those of the companions (Figure 12), it is not clear how significant this finding is, due to high errors on most of our inclinations. All of our systems are statistically consistent with zero mutual inclination, in contrast with Dai et al. (2018), one of the works that Pu & Lai (2019) drew on, who found an inclination dispersion of 67 (using a sample that extends to longer periods than our own, although their USPs also have a higher range of mutual inclinations). Panel (c) of Figure 12 reproduces their Figure 3 for our sample, which has a mean Δi = 12 ± 12. One possible cause for the discrepancy could be sample size and/or quality; their sample comprises few K2 candidates and has little overlap with ours. Many of our objects are shallow and/or have low to moderate S/N, meaning that a planet in a grazing orbit (i.e., significantly different from i = 90°) might be more likely to be missed. We also do not account for the possibility of oppositely aligned orbits; a pair of planets at 88° and 89° will have the same Δi in Figure 12 as a pair at 92 and 89°. Because, however, most of our nominal values hew closely to 90°, this is unlikely to account for the full difference. A more in-depth analysis of the inclinations of USPs from K2 is recommended before drawing conclusions on this point.

The model of Millholland & Spalding (2020) is obliquity-driven tidal migration, where it is the planetary obliquity (the angle between the planet's orbit and its rotational spin) that drives migration of the innermost planet with at least one close companion, using the quick passage through Cassini states (configurations in which precession of the planet's spin axis matches precession of its orbital plane), which can result in substantial tidal dissipation. Five testable predictions are made by that work; we broadly agree with points (1) and (5), disagree with (3) and (4b), and leave the corroboration of points (2) and (4a) for future work.

• 1.  
  USPs should have close companions (P ≲ 10 days), which may or may not transit. This agrees well with Figure 4, which finds the closest companion has a median P = 4.5 days. About a quarter (6 of 24) of the closest companions have 10 < P₂ < 15 days, still broadly compatible with this theory.
• 2.  
  No trends are expected for eccentricity, whereas the Pu & Lai (2019) eccentricity-driven tidal migration model would expect some remnant eccentricity in the companion planets. Unfortunately, we cannot test this with our current fits because we assumed zero eccentricity for all planets to avoid fitting extra parameters to light curves that are often near the S/N limit. Some of the companion light curves are of sufficient quality that they could be examined for nonzero eccentricity.
• 3.  
  USPs should have nonzero mutual inclinations. In particular, (a) they should have modest misalignments with their planetary companions, in agreement with Dai et al. (2018) but not our results in Figure 12, and (b) they should be aligned with the stellar spin (which is mostly unknown). Measuring the stellar obliquity of the USPs in multiplanet systems is an excellent goal to constrain this theory.
• 4.  
  In this model, the USP rapidly breaks out of the Cassini states. This implies that (a) few USPs should be currently undergoing rapid tidal decay; long-term monitoring of the transit timing of these systems would be needed to address this question. Another prediction is that (b) planets will pile up near 1 day, with occurrence rates decreasing at shorter periods, as found in the earlier work of Sanchis-Ojeda et al. (2014). As noted in Section 8.2.3 we instead observed an even spread in the orbital period for both USPs in known multiplanet systems and USPs that are apparently single.
• 5.  
  Finally, USPs are predicted to be more common around smaller host stars as in Sanchis-Ojeda et al. (2014). Although we have not debiased our discoveries to produce a detection rate by stellar type, we found that only 4 of 24 multiplanet host stars have R_* > 1R_⊙.