KEPLER-445, KEPLER-446 AND THE OCCURRENCE OF COMPACT MULTIPLES ORBITING MID-M DWARF STARS

The Kepler-42 exoplanetary system is rather remarkable: it consists of three sub-Earth-sized planets all orbiting and transiting a mid-M dwarf host star with periods of less than two days (Muirhead et al. 2012), a so-called "compact multiple." Consider that Kepler-42 is a metal-poor star, with a measured [M/H] of −0.27 in Muirhead et al. (2012). Following the calculation of Schlaufman & Laughlin (2010), if we assume the planets formed from material in a protoplanetary disk with a total disk mass equal to 1% of the current host star mass, and that the disk had identical metal abundance to the star today, that leads to a disk metal content totaling 4.1 Earth masses. Assuming the three planets have primarily rocky compositions, and combining the predicted planetary mass–radius relationships of Fortney et al. (2007) with the measured planetary radii in Muirhead et al. (2012), the total mass of the three planets is calculated to be 0.71 Earth masses. Under these admittedly simplistic assumptions, nearly 20% of Kepler-42's disk metals went into the formation of these three rocky planets. The same calculation for the Sun results in only 5% of disk metals contributing to rocky planets (Earth, Venus, Mars, and Mercury), with significantly more contributing to the cores of the solar system's gas-giant planets.

The preference for metals to contribute to rocky planets rather than gas-giant cores would be strong evidence for the planet-formation scenario suggested by Laughlin et al. (2004), in which gas-giant-core embryos form in the protoplanetary disks around M dwarf stars; however, the gas in the disk dissipates before those embryos grow large enough to accrete and are cut-off as terrestrial planets. The scenario is already supported by the relative scarcity of gas-giant exoplanets found to orbit M dwarf stars. Using radial velocity observations, Johnson et al. (2010) found a statistical decrease in giant planet occurrence with decreasing host star mass, including M dwarfs in the radial velocity sample. However, Gaidos & Mann (2014) do not find strong support for a statistical deficiency of gas-giant planets orbiting M dwarfs, though they cannot statistically rule out a deficiency. Regardless, the presence of failed embryos in some consistent proportion to the amount of available metals in the protoplanetary disk would provide support for the cut-off accretion scenario.

It is important to determine whether the planetary system orbiting Kepler-42 is an outlier or a common type of system around mid-M dwarf stars, and whether metallicity affects the sizes of failed embryos. The occurrence of compact multiple systems orbiting mid-M dwarf stars has not been thoroughly analyzed. Recent investigations aimed at understanding the planet occurrence statistics from Kepler, such as Howard et al. (2012); Petigura et al. (2013), and Silburt et al. (2014) do not consider mid-M dwarf stars. Dressing & Charbonneau (2013), Gaidos et al. (2014), and Morton & Swift (2014) investigated the occurrence of planets around specifically M dwarf stars and found a propensity for Earth-sized planets, the latter finding roughly two planets per M dwarf star with periods of less than 150 days. However, those investigations did not address the occurrence of compact multiples, and many Kepler mid-M dwarf stars were not included in that study, including Kepler-42. Swift et al. (2013) investigated the occurrence of compact multiples orbiting Kepler M dwarf stars using Kepler-32 as a representative of the Kepler M dwarf stars. They found that Kepler planet detections around M dwarf stars could be recreated reasonably well assuming all planetary systems are clones of the Kepler-32 system, indicating a high occurrence of compact multiples.

In this paper we confirm and characterize two new compact multiple systems with mid-M-dwarf host stars: Kepler-445¹² and Kepler-446,¹³ initially discovered by the Kepler exoplanet search pipeline to host two and three short-period planets, respectively. Kepler-445c, Kepler-445b, Kepler-446b, and Kepler-446d were first reported as planet candidates by Burke et al. (2014) as KOI-2704.01, KOI-2704.02, KOI-2842.01, and KOI-2842.02, respectively, and Kepler-446c was reported as a threshold-crossing event by Tenenbaum et al. (2013), and later added to the NASA Exoplanet Archive (NEA) as KOI-2842.03. With regard to the host stars, Kepler-445 was identified as a mid-M dwarf star in an optical spectroscopic survey of late-type Kepler Objects of Interest (KOIs) by Mann et al. (2013a), and both Kepler-445 and Kepler-446 were identified as mid-M dwarf stars in a separate infrared spectroscopic survey of late-type KOIs by Muirhead et al. (2014). We searched the light curves for additional planets, and found an additional planet orbiting Kepler-445 not detected by the Kepler pipeline, which we refer to as Kepler-445b. Refining the parameters for the host stars and refitting the planet transit light curves, we find that Kepler-445c is similar to GJ 1214b: both are likely mini-Neptunes orbiting metal-rich mid-M dwarf stars, and that Kepler-446 is similar to Kepler-42 in multiple ways: the low metallicity of the host star, the multiplicity, sizes, and orbital periods of the orbiting planets, and the planets' initial mischaracterization.

We present the observations, data and analysis of these systems in Section 2. In Section 3 we calculate the false-positive probabilities (FPPs) for the planets orbiting Kepler-445 and Kepler-446, confirming the planetary nature of the transits. In Section 4 we combine Kepler-445, Kepler-446, and Kepler-42 with the full sample of mid-M dwarf stars observed by Kepler with similar precision from quarters 1 to 16 to estimate the occurrence of compact multiple systems around mid-M dwarf stars, which we define as two or more planets orbiting with periods of less than 10 days. Finally, in Section 5, we discuss the implications of these systems for planet formation scenarios and accretion of disk metals by protoplanets orbiting mid-M dwarf stars.

2.1. High-contrast Imaging

High-contrast imaging of Kepler planet candidates serves two purposes: with high-contrast images, the level of contamination by unresolved objects within the Kepler aperture can be determined, and in the event that there are no contaminating objects within the aperture, high-contrast imaging provides constraints on false-positive scenarios that could mimic the detected transit signal (e.g., Adams et al. 2012; Lillo-Box et al. 2012, 2014; Law et al. 2014; Dressing et al. 2014).

We observed Kepler-445 and Kepler-446 with the Keck II Telescope at Mauna Kea Observatories using the facility near-infrared adaptive-optics (AO) imager NIRC2, operated in K' band. We observed the stars using both conventional imaging and using non-redundant aperture masks, which can be placed near an image of the telescope pupil within the NIRC2 cryostat. Non-redundant aperture masks enable high-contrast imaging by measuring individual spatial frequencies corresponding to each baseline produced by a pair of apertures (Haniff et al. 1987; Nakajima et al. 1989; Tuthill et al. 1999; Monnier et al. 2004; Lloyd et al. 2006; Martinache et al. 2009; Kraus & Ireland 2012; Ireland 2013). By making the apertures non-redundant, high fringe-contrast, and phase information is not lost to incoherent addition of fringes from redundant baselines, enabling contrast performance better than the full-dish diffraction limit of the telescope. Phases from combinations of three individual baselines are combined into closure phases, which probe azimuthal asymmetries in the image while remaining robust against phase errors from instrument distortions or uncorrected atmospheric fluctuations.

We observed Kepler-445 on UT 2013 July 17 and Kepler-446 on UT 2014 July 29. For each target, we acquired three, conventional K'-band AO images, each with 20 s integrations, each dithered across the detector, using multiple-correlated sampling with a Fowler depth of 16 (Fowler & Gatley 1990). The non-redundant aperture masking images were acquired in the same manner as Kraus & Ireland (2012): we used a nine-hole mask, with 1.5 m equivalent apertures creating baselines with separations of 1.5–9.2 m. We acquired six aperture-masked images in K band, each with 20 s integrations and using multiple-correlated sampling.

We reduced the data using the methods described in Kraus et al. (2008), as well as a kernel-phase technique with the POISE calibration algorithm (Ireland 2013), that included subsampling of the Keck pupil sub-apertures. The kernel-phase technique yielded marginally superior contrast limits, and those limits are reported here.

For Kepler-445, we detected a faint object at a separation of 3908 ± 0004, with a position angle of 27962 ± 005 east of north, and a magnitude difference of ΔK' = 7.60 ± 0.07 mag. With such a large contrast, the object's contribution to the Kepler light curve is negligible and it is unlikely the object showing the transit signals. Nevertheless, we consider the possibility in our false-positive analysis in Section 3. Aside from this detected object, we detect no other objects near Kepler-445, with 6σ contrast limits shown in Figure 1. Interestingly, this is inconsistent with the Kepler Input Catalog (KIC; Batalha et al. 2010; Brown et al. 2011) and the NOMAD Catalog (Zacharias et al. 2005), both of which record a similar visible-brightness star ∼22 away: KIC 9730159 and NOMAD 1364-0341824, respectively. Considering the infrared AO imaging, and the seeing-limited visible-wavelength imaging we report in the following section which also lacks this nearby object, we believe that the NOMAD catalog and the KIC may have inadvertently recorded Kepler-445 twice.

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2.2. Host-star Magnitudes

2.3. Moderate-resolution Spectra and Stellar Parameters

Neither Kepler-445 nor Kepler-446 currently has a published astrometric parallax measurement. With astrometric parallaxes, relatively accurate mass–luminosity relations can be used to determine stellar masses based on their absolute infrared magnitudes (Henry & McCarthy 1993; Delfosse et al. 2000). The stellar masses can be combined with empirical or theoretical mass–radius relationships to determine stellar radius (e.g., Torres et al. 2010). Or, in some cases, stellar parameters can be determined from the transit light curve itself, either by measuring photometric signatures from asteroseismic pulsations (e.g., Bedding et al. 2011; Huber et al. 2013), or by using an exoplanet transit light curve to infer the host star's density assuming a low eccentricity or circular orbit (e.g., Seager & Mallén-Ornelas 2003; Carter et al. 2011), or a combination of both techniques (e.g., Ballard et al. 2014).

In the case of Kepler-445 and Kepler-446, asteroseismic signatures are very difficult to measure because, being M dwarf stars, they are dense, with mean densities over 10 gm cm⁻³, where the asteroseismic signals are low and the stellar oscillation frequencies are high. Also, the signal-to-noise of the Kepler light curves is simply not high enough to provide strong constraints on the stellar density from transit light curve fitting. Without astrometric parallaxes, asteroseismic or transit light curve constraints, we must rely on colors and spectroscopy to determine the stellar parameters. In the case of M dwarf stars burning hydrogen on the main-sequence, spectroscopy probes primarily effective temperature, T_eff, and metallicity, [Fe/H], or [M/H]. Typically, surface gravity, log(g), is another physical parameter probed by spectroscopy. In the case of M dwarf stars on the main sequence, log(g) is predicted to be a strict function of T_eff and metallicity. With T_eff and metallicity alone, the stellar mass, radius, and bolometric luminosity can be determined using either predictions from stellar evolutionary models (e.g., Dotter et al. 2008), or empirical relations (e.g., Ségransan et al. 2003; Boyajian et al. 2012).

Moderate-resolution infrared spectra (R ∼ 2700, 1.5–2.5 μm) for Kepler-445 and Kepler-446 were obtained by Muirhead et al. (2014), who determined stellar parameters based on the K-band indices of Rojas-Ayala et al. (2012), then interpolated those values on a new set of models based on the Dartmouth Stellar Evolution Database (Dotter et al. 2008), calculated by G. Feiden. A visible low-resolution spectrum for Kepler-445 (R ∼ 900, 3200–9200 Å) was obtain by Mann et al. (2013a), who used a new calibration to measure effective temperature. They then used the mass–radius–temperature relations of Boyajian et al. (2012) for their sample, but were unable to determine the stellar mass and radius of Kepler-445 as it was too cool for these relations. Kepler-446 was not discovered by the Kepler pipeline at the time of their study.

In this paper, we combine the techniques from Mann et al. (2013a) and Muirhead et al. (2014) to produce the best possible physical parameters of the stars. We choose to use the Mann et al. (2013a) visible-light spectroscopic effective temperature calibration, since it is calibrated using truly empirical measurements of nearby M dwarf stars from optical-long baseline interferometry (Boyajian et al. 2012). We use the K-band spectroscopic metallicity measurements from Muirhead et al. (2014), as those used empirical calibrations by Rojas-Ayala et al. (2012), who verified the metallicities by comparison to space motions of nearby M dwarf stars. Finally, we interpolate those values onto the new suite of Dartmouth evolutionary models as was done in Muirhead et al. (2014), but with a small correction based on the empirical measurements of Barnard's Star (similar to Muirhead et al. 2012).

To do this, we used primarily the archival spectroscopy described above, but acquired a new visible spectrum of Kepler-446. Our additional spectrum of Kepler-446 was acquired with the SNIFS instrument on the University of Hawaii 88 inch Telescope at Mauna Kea Observatory. We followed identical observing and reduction procedures described in Mann et al. (2013a).

To stitch the visible and near-infrared spectra together for each star, we combine the r magnitudes measured in the previous section with the H and Ks magnitudes from 2MASS. We acquired the filter transmission curves for each band (r, H, and K) and computed a synthetic spectral magnitude using the transmission curve and the measured spectrum, keeping in mind that 2MASS used a modified Vega system, and our measured SDSS magnitudes use the AB system (Oke & Gunn 1983). We calculated the difference between the synthetic spectral magnitude and the measured magnitude from each respective survey, and used that difference to renormalize and combine the visible and near-infrared spectra.

Figure 2 plots the visible and infrared spectra of Kepler-445 and Kepler-446, with spectra of Barnard's Star from Mann et al. (2013a) and Kepler-42 from Muirhead et al. (2014) for comparison, subject to an arbitrarily total normalization such that they overlap in H band. Clearly, the spectra are very similar: are all identifiably M4 dwarf stars. We did not correct for interstellar reddening because all stars are likely well within 150 pc of the sun (see Section 2). None of the stars show Hα in emission, indicating low quiescent activity and that all of the stars are likely old (t > 5 Gyr; West et al. 2008). However, we note that Barnard's Star does show occasional Hα flaring (e.g., Paulson et al. 2006), and that age–activity relationships are a statistical tool for estimating the age of an ensemble of stars, and are subject to large uncertainties when applied to any specific star.

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Table 2. Stellar Parameters^a

Mid-M Dwarf Star	Teff	[Fe/H]	[M/H]	M⋆	R⋆	Distance
Kepler-445	3157 ± 60 K(a)	+0.27 ± 0.13(c)	+0.19 ± 0.12(c)	0.18 ± 0.04 M☉	0.21 ± 0.03 R☉	∼ 90 pc
Kepler-446	3359 ± 60 K	−0.30 ± 0.12(c)	−0.21 ± 0.12(c)	0.22 ± 0.05 M☉	0.24 ± 0.04 R☉	∼ 120 pc
Kepler-42	3241 ± 57 K(a)	−0.48 ± 0.12(d)	−0.33 ± 0.12(d)	0.15 ± 0.03 M☉(d)	0.18 ± 0.02 R☉(d)	∼ 40 pc(d)
Barnard's Star	3238 ± 11 K(a)	−0.39 ± 0.17(b)	−0.27 ± 0.12(b)	0.159 ± 0.013 M☉(a)	0.1867 ± 0.0012 R☉(e)	1.824 ± 0.005 pc(f)

Note. ^aValues without a mark are from this work, values with a mark are from the following: ^(a)Mann et al. (2013a), ^(b)Rojas-Ayala et al. (2012), ^(c)Muirhead et al. (2014), ^(d)Muirhead et al. (2012), ^(e)Boyajian et al. (2012), and ^(f)Hipparcos astrometric parallax from van Leeuwen (2007).

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To determine any offsets to apply to the new Dartmouth models, we interpolated the empirical effective temperature and metallicity measurements of Barnard's Star onto the new Dartmouth models to determine the predicted stellar mass and radius. The predicted mass and radius are marginally different from the empirical mass (determined using mass–luminosity relationships of Delfosse et al. 2000) and the empirical radius (measured using optical long-baseline interferometry; Lane et al. 2001; Boyajian et al. 2012), with offsets of −0.004 M_☉ and 0.002 R_☉, both of which are well below the typical uncertainty in those quantities. Nevertheless we still applied those corrections to the interpolated values.

Our results for the stellar parameters appear in Table 2. In addition to calculating the parameters for Kepler-445 and Kepler-446, we also recalculated the parameters for Kepler-42 using identical methods, and find agreement with Muirhead et al. (2012), who used a similar technique, but applied a correction to the original Dartmouth models (Dotter et al. 2008), rather than the new set used in Muirhead et al. (2014) and this study. We estimated the distance to the stars by inverting the mass–luminosity relations of Delfosse et al. (2000) to determine the stars absolute K-band magnitudes, and compared that to the measured K-band magnitude from 2MASS. All three stars are relatively nearby with distances less than 150 pc.

The stellar parameter determinations for Kepler-445 and Kepler-446 presented here revise values published in the literature. Compared to Muirhead et al. (2014), the masses and radii presented here are slightly larger owing to the use of the Mann et al. (2013a) effective temperature calibration rather than the K-band technique. The most recent evaluation of stellar parameters for Kepler targets was compiled by Huber et al. (2014), who assign both Kepler-445 and Kepler-446 smaller and larger radii (respectively) than presented here. We ascribe this to erroneous metallicity determinations in that catalog, which lists Kepler-445 as a metal-poor star ([Fe/H] = −0.380) and Kepler-446 as a metal-rich star ([Fe/H] = +0.30), the inverse of our measurements. The source for the Kepler-446 parameters in Huber et al. (2014) is Dressing & Charbonneau (2013), who used photometry to estimate stellar parameters. As Dressing & Charbonneau (2013) admit, photometry alone places poor constraints on M dwarf metallicity, and Kepler-446's erroneously large radius in both Dressing & Charbonneau (2013) and Huber et al. (2013) is a direct consequence of the imprecise metallicity measurement.

2.4. High-resolution Spectra and Space Motions

We also acquired higher-resolution spectra in order to measure the absolute radial velocities of the stars and their galactic space motions. We observed Kepler-445 and Kepler-446 with the Echellette Spectrograph and Imager (ESI) on the Keck-II Telescope (Sheinis et al. 2002) on UT 2014 July 25. We operated ESI in echellette mode using a slit width of 05, achieving a resolving power of ∼8000 from 4000 to 10000 Å, spread across ten cross-dispersed orders. For Kepler-446 we acquired a single 1000 s exposure, and for Kepler-445 we co-added three 900 s exposures. We also observed GJ 687 and GJ 905 as M dwarf radial velocity standard stars with published radial velocities and uncertainties in Nidever et al. (2002) and Deshpande et al. (2012), respectively. We reduced the data using the publicly available ESIRedux pipeline,¹⁴ using dome flats for flat-fielding and arc lamps for wavelength calibration (Prochaska et al. 2003; Bochanski et al. 2009).

We then cross-correlated the radial velocity standard spectra with Kepler-445 and Kepler-446 to measure their absolute radial velocities. We used the FWHM of the peak in the cross-correlation function as our measurement uncertainty, combined in quadrature with the archival measurement errors for the radial velocity standard stars. We found agreement within our uncertainties when using GJ 687 or GJ 905 separately as standards, and we report the mean of those measurements here. For Kepler-445 we measured an absolute radial velocity of −61  ±  1 km s⁻¹, and for Kepler-446 we measured −118  ±  1 km s⁻¹.

When combined with archival proper motion measurements and our distance estimates from the previous section, we can estimate the stars' space motions. The most recent proper motion measurements available for Kepler-445 and Kepler-446 are from the PPMXL catalog (Roeser et al. 2010). They report a proper motion of μ_α = 42.2 μ_δ = 132.7 mas yr⁻¹ for Kepler-445 and μ_α = −13.2 μ_δ = −30.6 mas yr⁻¹ for Kepler-446. Both proper motion measurements are relatively large and are consistent with M dwarf stars within 150 pc.

Combining proper motion, radial velocity, and distance estimates, we measure the galactic space velocity motions of Kepler-445 to be U = 59, V = −39, W = 9 km s⁻¹, and for Kepler-446 we measure U = 8, V = −98, W = −31 km s⁻¹. We corrected for the galactic solar motion using the values of Coşkunoǧlu et al. (2011):U = −8.5, V = 13.38, and W = 6.49 km s⁻¹. Figure 3 plots the space motions for Kepler-445, Kepler-446, Kepler-42 and Barnard's Star with nearby dwarf stars in a Toomre diagram with thin disk, thick disk and halo boundaries from Fuhrmann (2004). Kepler-42 and Kepler-446 are located in the thick disk regime, consistent with their low metallicities, whereas Kepler-445 is located in the thin disk regime, consistent with its high metallicity. We note that the distances to Kepler-42, Kepler-445, and Kepler-446 and their corresponding tangential motions are highly uncertain, as is the true solar motion through the galaxy. Therefore, we estimate the uncertainty in their galactic space motions to be ∼10 km s⁻¹.

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2.5. Kepler Photometry

Kepler-445 was observed by Kepler in quarters 6, 8, and 9 in long-cadence mode as part of guest observer program GO20031, a search for microlensing by observing high-proper motion stars (Di Stefano 2010). It was later observed in quarters 12–14, and 16 in long-cadence mode and in quarter 17 in short-cadence mode as an exoplanet search target as part of the primary Kepler Mission. However, quarter 17 was cut short due to a malfunction in the spacecraft ending the primary mission, so we did not include it in our analysis.

Kepler-446 was observed by Kepler in quarter 7 in long-cadence mode as part of guest observer program GO20001, a search for eclipsing binary systems in a sample of M dwarf stars (Harrison 2010). It was again observed in quarters 12–16 in long-cadence mode and in quarter 17 in short-cadence mode as an exoplanet search target as part of the primary Kepler Mission, though we do not include the quarter 17 data for the reason stated above.

2.5.1. Known Transiting Planets

At the time this manuscript was submitted, the NEA (Akeson et al. 2013) listed two planets orbiting Kepler-445 (KOI-2704), with orbital periods (P) of 4.871229  ±  1.1e-05 and 2.984151  ±  1.1e-05 days, for Kepler-445c and Kepler-445b, respectively (KOI-2704.01 and KOI-2704.02), with planetary radii (R_P) of 2.9 and 1.96 R_⊕, respectively. The NEA listed three planets orbiting Kepler-446 (KOI-2842) with orbital periods of 1.5654088  ±  3.3e-06, 5.148921  ±  2.2e-05, 3.03617925  ±  5.49e-06 days for Kepler-446b, Kepler-446d, and Kepler-446c respectively (KOI-2842.01, KOI-2842.02, and KOI-2842.03), with planetary radii (R_P) of 25  ±  15, 26  ±  15, and 2.393  ±  0.582 R_⊕. For Kepler-446, the NEA listed notably large impact parameters (b) of 1.17  ±  0.92, 1.19  ±  0.99, and 0.90  ±  0.177, for Kepler-446b, Kepler-446d, and Kepler-446c, respectively.

The planetary radii listed in the NEA as initially downloaded were determined using either an Markov chain Monte Carlo (MCMC) fitting routine, as in the case of Kepler-445c, Kepler-445b, Kepler-446b, and Kepler-446d (Burke et al. 2014), or by the Kepler pipeline when initially searching for threshold-crossing events, as in the case of Kepler-446c (Tenenbaum et al. 2014), with the stellar parameters from Huber et al. (2014). For Kepler-446b and Kepler-446d We attribute the large radii and uncertainties in the NEA to correlations between the transit impact parameter and planet-to-star radius ratio. The parameters are typically not strongly correlated; however in the case of Kepler-446 the transit durations (T) are relatively short, lasting only about an hour each, and the long-cadence integrations times are ∼30 minutes. This results in the light curve being significantly smoothed and appearing to be V-shaped, where a truly flat-bottomed, low-impact parameter transit curve is indistinguishable from a grazing eclipse.

We note that when this manuscript was accepted, the parameters listed in the NEA had changed, listing more reasonable values for the planet radii for Kepler-446, but still significantly larger than the values we calculate in Section 2.5.3.

2.5.2. Independent Planet Search

The Kepler-445 and Kepler-446 transiting planet candidates found by the Kepler pipeline did not utilize all quarters of data available currently. Therefore, we conducted our own search for additional transiting planets on the complete, available Kepler data set for Kepler-445, Kepler-446, and Kepler-42. We downloaded the Kepler light curves for Kepler-445 and Kepler-446 from the NASA's Mikulski Archive for Space Telescopes (MAST). The light curve data files contain flux values measured using simple aperture photometry on the Kepler pixels using pre-defined apertures (SAP_FLUX), and flux values that have been detrended using custom tools to remove slowly varying instrumental fluctuations in the Kepler photometric response as well as fluctuations due to stellar rotation and intrinsic variability: the "pre-search data conditioning simple aperture photometry flux" (PDCSAP_FLUX; Smith et al. 2012). We choose to use the PDCSAP_FLUX for the independent planet search and for fitting the transit events.

For each star, we fitted the PDCSAP_FLUX light curve with a cubic basis spline (cubic B-spline) with breakpoints located 1.5 days apart, and divided the light curve by the B-spline fit to remove stellar and instrumental variability. We excluded outlier data points (both astrophysical and otherwise) from the B-spline fit by iteratively fitting the B-spline, locating points falling more than 3σ away from the fit, and re-calculating the B-spline while excluding the outliers. We repeated this process until convergence, typically five iterations. We then searched for transits by calculating a box least squared (BLS) periodogram (Kovács et al. 2002) for each star. We evaluated the BLS periodogram over periods ranging from 0.15 days (or 3.6 hr) to the total duration of the Kepler observations. We evaluated the BLS power spectrum at roughly 10⁵–10⁶ discrete periods, depending on the total time baseline of Kepler observations, and we spaced the trial periods to ensure that the BLS signal of a short duration transit around a mid-M dwarf would not be smeared out by coarse period spacing. After we calculated the BLS power spectrum, we subtracted away a noise floor from the power spectrum and estimated its typical scatter by calculating the median absolute deviation and dividing by 0.67 to convert to an equivalent standard deviation. We considered any peak with a BLS signal-to-noise ratio (S/N) of greater than nine to be significant. Upon detecting a significant signal, we masked out the signal in question and re-calculated the periodogram.

In all three systems, we recovered the planets discovered by the Kepler pipeline at high significance. For Kepler-42 and Kepler-446, after removing the three signals detected by the Kepler pipeline, there remained no significant peaks in the BLS, but for Kepler-445, after removing the two known planet candidates, there remained a series of significant peaks spaced like harmonics of a transiting planet signal, shown in Figure 4. The highest peak in the BLS spectrum indicated a candidate period of 8.15275 days. Hereafter, we refer to this new planet candidate as Kepler-445d, orbiting near a 5:3 resonance with Kepler-445c, and we confirm all of the planet in Section 3. We also detected no significant photocenter shift in the centroid of the Kepler image of Kepler-445 during the Kepler-445d transit events. Taking the difference between the centroid of the image in and out of transit, we calculate a photocenter shift of 000054  ±  000033.

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2.5.3. Transit Fits

Given the large uncertainties on the physical radii of the planets orbiting Kepler-446, and the new detection of Kepler-445d, we chose to fit the transit light curves for Kepler-445 and Kepler-446 independently. We fitted transit light curves to the data using a modified version of the Transit Analysis Package, or TAP, an IDL software package developed by Gazak et al. (2012). TAP employs a MCMC algorithm within a Bayesian framework for determining transit parameters. In order to reduce the degeneracy between impact parameter and planet-to-star radius ratio, we modified TAP to impose a strict prior on the stellar densities for Kepler-445 and Kepler-446. As Seager & Mallén-Ornelas (2003) showed, ingress/egress duration and full transit duration can be combined in such a way so as to determine the density of the host star, assuming knowledge of the planet's orbital period, orbital eccentricity, and longitude of periastron. Likewise, knowledge of the host star density, orbital eccentricity, and longitude of periastron can be combined to constrain the relationship between ingress/egress duration and transit duration. In the case of the planets orbiting Kepler-445 and Kepler-446, ingress/egress duration is difficult to measure due to the V-shaped nature of the light curves as discussed above, and imposing such a constraint is a powerful way to fit accurate transit parameters. Indeed, Muirhead et al. (2012) used this technique to fit the transit light curves for the Kepler-42 system, although they fixed the stellar density rather than applying a prior.

We assumed a stellar density prior based on the measured masses and radii for Kepler-445 and Kepler-446 from spectroscopy described in the previous section. Due to the use of evolutionary models, the mass and radius uncertainties are nearly 100% covariant for the stars, so we only use the uncertainty in mass when calculating the density prior. We assumed a Gaussian prior with a mean of 26.51 gm cm⁻³ and a standard deviation of 6.54 gm cm⁻³ for Kepler-445, and a mean of 22.73 gm cm⁻³ and a standard deviation of 6.06 gm cm⁻³ for Kepler-446.

We also assumed that all of the planets orbiting Kepler-445 and Kepler-446 have low eccentricity (e), and fix the value to zero in our fit. Following Wu & Goldreich (2002, Equation (1)), and assuming reasonable planet densities with tidal $Q_P^{\prime }$ values of 100–10,000, all of the planets orbiting Kepler-445 and Kepler-446 have circularization timescales of less than 1 Gyr. Given the lack of quiescent Hα or u-band emission in either Kepler-445 or Kepler-446, and Kepler-446's high galactic space motion, we are confident that the stars are older than 1 Gyr and likely older than 5 Gyr. We note, however, that planets can sustain eccentricities over long timescales via spin–orbit coupling, and that in multiple-planet systems non-zero eccentricity can be sustained via orbital resonances. The assumption of zero eccentricity also negates the importance of the longitude of periastron parameter for the transit fit or for the imposed prior on stellar density.

For limb darkening, we fit a linear (u1) and quadratic (u2) coefficient to the transits. For each system, Kepler-445 and Kepler-446, we tied the limb-darkening parameters across each of the transiting planet fits. We used Gaussian priors for the limb-darkening coefficients, based on the expected coefficients from Claret & Bloemen (2011) using our measured stellar parameters and their uncertainties. For Kepler-445 we used a Gaussian prior with a mean linear limb-darkening parameter of 0.50 with standard deviation of 0.17, and a quadratic limb-darkening parameter of 0.35 with a standard deviation of 0.13. For Kepler-446, we used a Gaussian prior with a mean linear limb-darkening parameter of 0.42 with standard deviation of 0.12, and a quadratic limb-darkening parameter of 0.35 with a standard deviation of 0.11.

Finally, we chose to fix the orbital periods and transit epochs (t₀) of the planets to the values listed in the NEA for all but Kepler-445d, for which we fit a separate transit model to the Kepler light curve using a Levenberg–Marquardt minimization routine. We used the best fit period and transit epoch for the TAP fit. Allowing the periods of the planets to vary within TAP had no significant effect on the resulting transit parameters. Our final transit parameters for the six planets are listed in Tables 3 and 4, and we show the phase-folded Kepler light curves and fitted transit curves in Figure 5. We combine the stellar parameters with the transit parameters to determine the physical parameters for the planets, which are also listed in Table 3, and Figure 6 illustrates the planets with Kepler-42 and the Galilean moons of Jupiter for comparison. We also calculate the incident flux on the planets as a fraction of the solar flux incident on the Earth's upper atmosphere (S₀ = 1360 W m⁻²). The values indicate that all of the planets are likely too hot to be located within their host stars' habitable zones, using habitable-zone limits calculated by Kopparapu (2013). However, with an S/S₀ of 2.24, one could argue that Kepler-445d is near the habitable zone.

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Table 3. Transit Parameters for Kepler-445

Parameter	Kepler-445b	Kepler-445c	Kepler-445d
P (days)a	2.984151 ± 0.000011	4.871229 ± 0.000011	8.15275 ± 0.00040
t0 − 2454833 (BJD)a	133.1194 ± 0.0033	133.6408 ± 0.0019	3.7512226 ± 0.05
RP/R⋆	0.0676 ± 0.0018	0.1075 ± 0.0014	0.0533 ± 0.0029
b	0.001$^{+0.203}_{-0.001}$	0.000$^{+0.101}_{-0.000}$	0.011$^{+0.470}_{-0.010}$
T (hours)	1.0304 ± 0.0180	1.2287 ± 0.0154	1.3860 ± 0.0888
μ1b	0.141 ± 0.080	0.141 ± 0.080	0.141 ± 0.080
μ2b	0.263 ± 0.071	0.263 ± 0.071	0.263 ± 0.071
ea	0	0	0
Inc (degrees)c	89.74$^{+0.18}_{-0.28}$	89.91$^{+0.07}_{-0.10}$	89.61$^{+0.27}_{-0.25}$
a/R⋆c	21.94 ± 0.30	30.21 ± 0.38	42.95 ± 0.58
RP (R⊕)c	1.58 ± 0.23	2.51 ± 0.36	1.25 ± 0.19
S/S0d	8.57 ± 0.60	4.52 ± 0.31	2.24 ± 0.17
MP (M⊕)e	4–6	8–9	3–4
K (m s−1)e	6–8	9–10	2–4

Notes. ^aHeld fixed in fitting procedure. Periods and ephemerides for Kepler-445c and Kepler-445b are from the NASA Exoplanet Archive (Akeson et al. 2013). ^bLimb-darkening coefficients were tied between planets, and subject a prior based on Claret & Bloemen (2011). ^cCalculated from parameterization. ^dS₀ = 1360 W m⁻², the solar flux incident on Earth's upper atmosphere. ^eCoarsely estimated using the empirically measured planet mass–radius relationships of Marcy et al. (2014).

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Table 4. Transit Parameters for Kepler-446

Parameter	Kepler-446b	Kepler-446c	Kepler-446d
P (days)a	1.565409 ± 0.0000033	3.036179 ± 0.0000055	5.148921 ± 0.000022
t0 − 2454833 (BJD)a	132.9135 ± 0.0019	134.069573 ± 0.000945	133.3196 ± 0.0042
RP/R⋆	0.0574 ± 0.0026	0.0424 ± 0.0018	0.0519 ± 0.0022
b	0.601$^{+0.096}_{-0.088}$	0.025$^{+0.529}_{-0.025}$	0.705$^{+0.057}_{-0.066}$
T (hours)	0.6456 ± 0.0456	0.9432 ± 0.0504	0.8832 ± 0.0480
μ1b	0.447 ± 0.059	0.446 ± 0.059	0.446 ± 0.059
μ2b	0.353 ± 0.065	0.353 ± 0.065	0.353 ± 0.065
ea	0	0	0
Inc (degrees)c	87.42$^{+0.62}_{-0.37}$	88.97$^{+0.57}_{-0.46}$	88.72$^{+0.17}_{-0.19}$
a/R⋆c	14.20 ± 0.94	22.40 ± 1.36	31.60 ± 2.08
RP (R⊕)c	1.50 ± 0.25	1.11 ± 0.18	1.35 ± 0.22
S/S0d	26.22 ± 4.70	10.54 ± 2.03	5.30 ± 0.83
MP (M⊕)e	4–5	2–4	3–5
K (m s−1)e	6–8	2–5	3–5

Notes. ^aHeld fixed in fitting procedure. Periods and ephemerides are from the NASA Exoplanet Archive (Akeson et al. 2013). ^bLimb-darkening coefficients were tied between planets, and subject a prior based on Claret & Bloemen (2011). ^cCalculated from parameterization. ^dS₀ = 1360 W m⁻², the solar flux incident on Earth's upper atmosphere. ^eCoarsely estimated using the empirically measured planet mass–radius relationships of Marcy et al. (2014).

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We also include coarse estimates for the planet masses and expected semi-amplitude radial velocity signatures in Table 3, using the recent empirically measured planet mass–radius relations of Marcy et al. (2014). All of the planets have anticipated radial velocity semi-amplitudes of over 1 m s⁻¹. However, being mid-M dwarf stars, the stars are relatively faint for current-generation visible-light precision-radial-velocity spectrometers (e.g., Bottom et al. 2013). In the future, the stars may be compelling targets for next-generation, infrared precision-radial-velocity instruments (Mahadevan et al. 2012; Halverson et al. 2014; Quirrenbach et al. 2012; Barrick et al. 2012; Artigau et al. 2012; Thibault et al. 2012; Micheau et al. 2012; Parès et al. 2012; Crepp et al. 2014; Ge et al. 2014). Such measurements would prove useful for precisely measuring the planet masses and for placing constraints on their atmospheres (e.g., Miller-Ricci Kempton et al. 2012) and interior structures (e.g., Rogers & Seager 2010; Valencia et al. 2013), as well as measuring any non-zero eccentricity (e.g., Anglada-Escudé et al. 2013).

As with the vast majority of Kepler planet candidates, the transit signals detected in the light curves of Kepler-445 and Kepler-446 are not amenable to dynamical confirmation as bona fide planets, either by radial velocity or transit timing variation measurements. Confirming their planetary nature thus requires probabilistic validation; that is, demonstrating that the probability for them to be caused by a blended stellar eclipsing binary, or any other astrophysical false-positive scenario, is very low.

While broad arguments have demonstrated that only a small number of Kepler planet candidates are expected to turn out to be astrophysical false positives (Morton & Johnson 2011; Fressin et al. 2013), any specific planet candidates of particular interest should be individually investigated more thoroughly to ensure their planetary nature. To this end, we have employed the method described in detail by Morton (2012) to calculate FPPs for all six of the candidates discussed in this paper. This method compares the shapes of the observed light curves to the shapes of simulated planet and astrophysical false-positive scenarios in order to determine the relative likelihoods of the signals to be caused by the different scenarios.

The basic assumption of this analysis is that the observed transit signal is roughly spatially coincident with the target star—that is, that the flux decrement in the aperture of the target star is not due, for example, to a different star several arcseconds separated from the target star. An important part of the Kepler vetting pipeline is looking for such offsets (Bryson et al. 2013)—only candidates that do not show significant offsets get promoted to "planet candidate" status. The results of these tests are summarized in the NEA tables in the column titled "PRF Δθ_MQ" and associated uncertainty, which indicates the difference in position between the Kepler pixel response function (PRF) fitted to the in- and out-of-transit data. For Kepler-446b, c, and d and Kepler-445b and d, all these offsets are significantly smaller than 1''. For Kepler-445d, which is not in the Kepler catalogs, we perform our own in- and out-of-transit centroid analysis, finding no noticeable shift. While we note that this analysis is not as detailed as the Bryson et al. (2013) PRF analysis, it certainly indicates that there is no large offset of the candidate signal.

While the pixel-level analysis of these candidates localizes them to be at least within 1'' of the target stars' locations, limiting the possible on-sky area in which could exist a potential false-positive blend, the constraints from the AO and aperture masking sensitivity curves presented in Section 2.1 further restrict the parameter space of possible false positives. Incorporating all these constraints into the Morton (2012) analysis, we obtain FPPs for all candidates of <10⁻⁴, except for Kepler-445d and Kepler-446d, for which we obtain FPPs of approximately 1 in 500 and 1 in 200, respectively. For both Kepler-445d and Kepler-446d the most probable false-positive scenario is an eclipsing binary within a hierarchical triple stellar system.

We note, however, that these FPP calculations do not account for the fact that the candidates are observed to be in multiple-transiting systems. As it is now known that many multiple planet systems have low mutual inclinations (Fabrycky et al. 2012; Fang & Margot 2012), the presence of one transiting planet means that additional planets have a significantly higher probability to transit than if their orbital inclinations were randomly distributed. Accounting for this effect gives a "multiplicity boost" to the probability of the planet scenario of about a factor of ∼5–10 (depending on the exact assumptions), thus lowering the FPPs of both Kepler-445d and Kepler-446d to below 0.1%—all six of the planet candidates discussed in this paper are probabilistically validated.

However, there is another scenario, while not strictly an astrophysical false positive, that we cannot completely rule out. That is, the AO and aperture masking observations presented here do not completely exclude the presence of stellar companions around Kepler-445 and Kepler-446. This can be understood by noting that the inner working angle of 20 mas achieved by the aperture masking observations corresponds to a projected distance of 2 AU at 100 pc, and that a snapshot of a binary star system will generally not be at maximum separation, due to orbital position and inclination. Using a Monte Carlo simulation of potential stellar companions distributed according to the stellar binary period and eccentricity distribution of Raghavan et al. (2010; with randomized orbital elements), we find that about 25% of potential stellar companions would be undetected by these high-resolution imaging observations. However, a close stellar companion would be detrimental to the existence of the observed planets, so we place another constraint requiring putative stellar companions to have periods longer than 200 days; this removes another ∼15% of potential stellar companions. Thus, starting with an assumption that about 40% of M dwarf stars are in stellar binary systems (Fischer & Marcy 1992; Clark et al. 2012; Duchêne & Kraus 2013), then our observations allow about a 4% chance for either of Kepler-445 or Kepler-446 to be binary systems.

As discussed in Muirhead et al. (2012), there would be two potential consequences of the presence of such a stellar companion in either of these systems. First, the planet radii would be larger than we estimate due to the unaccounted-for dilution (by a maximum of $\sqrt{2}$ if the planets are around the primary star, and by a larger factor if they are around a fainter secondary star). Second, there is a possibility that the transiting planets could be distributed around both stars in the system, rather than all being around the same, single star. Such a "split-multi" configuration, while predicted to be rare by Fabrycky (2011), does undoubtedly exist in the archetype KOI-284/Kepler-132 system (Lissauer et al. 2014), a visual binary with two of its three candidates having orbital periods of 6.18 and 6.42 days. However, given the low probability for either of these two systems to be binaries, and the even lower probability that the planets would be split between the two components, we neglect this scenario for the remainder of this paper, although we do acknowledge its possibility at the ∼1% probability level.

With three confirmed compact multiple systems orbiting mid-M dwarf stars—Kepler-42, Kepler-445, and Kepler-446—we can estimate the occurrence of compact multiples like these around mid-M dwarf stars in general: defined here to be a nearly coplanar system of two or more planets all orbiting with periods of less than 10 days. First, we downloaded and combined the lists of all targets observed by Kepler for at least one quarter from quarters 1–16 from MAST. ¹⁵ The files include measurements of the combined differential photometric precision (CDPP; Christiansen et al. 2012) for each target for each quarter, calculated over 3, 6, and 12 hr durations.

To calculate the occurrence of compact multiples orbiting mid-M dwarf stars, we use two approaches. The first follows the methodology of Howard et al. (2012), who calculated planet occurrence as a function of planet radius and orbital period using the Kepler planet candidate discoveries. We considered the number of mid-M dwarf stars around which Kepler and the Kepler pipeline could have found compact multiples like the ones orbiting Kepler-42, Kepler-445 or Kepler-446. We exclude Kepler-445d from our calculation, as that was not discovered by the Kepler pipeline. We isolated mid-M dwarf stars from the Kepler targets applying two color cuts using magnitudes from the KIC: r − J > 3.2 to isolate red objects, and J − K < 0.0555 (r − J) + 0.7622 to remove evolved stars (giant and highly reddened sub-giant stars). At first we considered using J − H to remove giant stars; however, we found that several giant stars identified in Mann et al. (2012) were not excluded using J − H, but were excluded using J − K, so we chose the latter. We also excluded targets that were only observed in the very first quarter of Kepler (quarter 0), which was only 30 days and has lower intrinsic photometric precision than subsequent quarters, and targets that were added in quarter 17, since that has not yet been searched by the Kepler pipeline and was cut short due to a spacecraft malfunction. The color cuts resulted in 509 mid-M dwarf stars with at least one full quarter of Kepler observations. Figure 7 illustrates the color–color cuts, and the position of Kepler-42, Kepler-445, and Kepler-446.

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Of the 509 mid-M dwarf stars, Kepler-42, Kepler-445, and Kepler-446 are the only stars with multi-planet-candidate systems listed on the NEA at the time of this study. In addition to these three stars, KOI-2862, KOI-4290, KOI-3855, KOI-3138, and KOI-5327 also host single transiting planet candidates and fall within the defined color cuts. We do not consider the single-transiting-planet hosts in this occurrence calculation.

We then calculated the number of mid-M dwarf stars with the S/N necessary to discover each of the planets orbiting Kepler-42, Kepler-445, or Kepler-446. To calculate the S/N for an arbitrary transiting planet orbiting a particular Kepler target, we used Equation (A1) from Fressin et al. (2013), repeated here with a slight modification:

$$\begin{equation} {\rm S/N = {\delta \over CDPP_{{\rm eff}}}}\sqrt{t_{\rm obs} \over P}, \end{equation} \tag{ 1 }$$

where δ is the transit depth, CDPP_eff is the combined differential photometric precision over the duration of the transit, t_obs is the total Kepler observing time used to search for the planet by the Kepler pipeline, and P is the planet orbital period. To determine CDPP_eff, we first averaged the 3, 6, and 12 CDPP values for each mid-M dwarf target over all quarters, then fit a function to the CDPP versus time with the form ${\rm CDPP}_{{\rm eff}} = {\rm CDPP}_0/\sqrt{t_{\rm dur}}$, where t_dur is the duration of a transit. We then calculated the signal-to-noise that would be achieved on each of the planets orbiting Kepler-42, Kepler-445 (excluding Kepler-445d), and Kepler-446, if they were orbiting every single mid-M dwarf observed by Kepler with the same transit durations.

We adopt a value of 10 as the minimum signal-to-noise required to detect a planet, following Howard et al. (2012). This is a reasonable value, considering the signal-to-noise needed to detect the Kepler-42, Kepler-445, and Kepler-446 transit signals and give them a planet-candidate disposition. For the three planets orbiting Kepler-42, they were detected by the Kepler pipeline using only the first quarter of data (Borucki et al. 2011). For Kepler-445 and Kepler-446, we combine the "disposition provenance" listed on the NEA with the number of observed quarters within the provenance boundaries to determine the number of quarters required to detect their orbiting planets. Following this, Kepler-445c and Kepler-445b were detected by the Kepler pipeline using two quarters of data (6 and 8). The three planets orbiting Kepler-446 were detected by the Kepler pipeline using 1 quarter of data (7) for 0.01 and 0.02, and two quarters (7 and 12) for 0.03. Table 5 indicates the resulting S/N values for the eight planets achieved when they were detected, all of which are greater than 10. We also list the number of Kepler mid-M dwarf stars for which the S/N of the respective planet would be higher than 10, if it were transiting all of the Kepler mid-M dwarf stars. Lastly, we list the individual transit probabilities (R_⋆/a).

When calculating the number of mid-M dwarf targets for which Kepler-42, Kepler-445 or Kepler-446-like systems could be detected, we make two assumptions. First, we assume the mid-M dwarf targets have similar radii to Kepler-42, Kepler-445, and Kepler-446, all roughly 0.20 R_☉. The true stellar radii of the mid-M dwarf targets affects the signal-to-noise of the Kepler-42, Kepler-445, or Kepler-446-like systems if there were transiting other stars. The assumption of similar radii across all the mid-M dwarf targets is a reasonable one considering that Kepler-42, Kepler-445, and Kepler-446 themselves have similar radii and nearly subtend the full color–color space defined in the mid-M dwarf target cut (see Figure 7). Even so, without spectra for the full sample of mid-M dwarf stars we must make this assumption. Second, we assume that the impact parameter has a negligible impact on the signal-to-noise of the planet transits if they were transiting other mid-M dwarf targets. In fact, an especially large impact parameter will reduce the signal-to-noise of a transit event since the duration will be shorter and the transit could even be grazing.

To calculate the occurrence of compact multiples, we use Equation (2) from Howard et al. (2012), who calculated the frequency of planets for a specific range of orbital periods and planet sizes, but modified for compact multiple systems:

$$\begin{equation} f_{\rm CM} = \sum _j^{n_{{\rm sys}}} { 1 / p_j \over n_{{\rm sys},j} }, \end{equation} \tag{ 2 }$$

whereas Howard et al. (2012) calculated the number of planets per star for specific ranges of orbital period and planet radius, we, on the other hand, want to calculate the fraction of mid-M dwarfs that host compact multiple systems, which we call f_CM. Instead of indexing each planet detection in the summation, instead we index each system: j refers to Kepler-42, Kepler-445, and Kepler-446, successively, p_j refers to the transit probability of the whole compact multiple (corresponding to the transit probability of the outermost system), and n_{sys, j} is the number of stars around which the system could be detected. Equation (2) implicitly defines a compact multiple as a system as being like Kepler-42, Kepler-445, and Kepler-446: two or more planets all orbiting with periods of less than 10 days.

Also implicit in this formalism is the assumption that compact multiple systems are co-planar: if the outermost planet is detected, the inner planets would also be detected, presuming their signal-to-noise is high enough. If, in fact, compact multiple systems are not coplanar, then f_CM represents a lower limit to their frequency around mid-M dwarf stars, as the single transiting-planet-candidate hosts mentioned above could harbor many more interior planets that do not transit.

We calculated the uncertainty in f_CM following the same procedure as described in Howard et al. (2012): we calculated the binomial probability distribution of drawing compact multiple planets from n_pl/f stars, and use the closest values to 1σ above and below f_CM of the binomial distribution as the uncertainties (∼15% and ∼85% of the cumulative binomial distribution).

Performing this calculation we arrive at a compact-multiple occurrence of 23$^{+13}_{-7}$%. The uncertainties correspond to drawing 5 and 2 compact multiple systems from 14 stars, based on the binomial distribution. We note that we include mid-M dwarf stars of all metallicities observed by Kepler in this calculation, which roughly matches the metallicity distribution of the solar neighborhood (Mann et al. 2013b), and that Kepler-445 has significantly super-solar metallicity ([Fe/H] = +0.25) and Kepler-446 and Kepler-42 both have sub-solar metallicity ([Fe/H] < −0.30). We conclude that roughly one-fifth to one quarter of mid-M dwarf stars host compact multiple systems, for the full spectrum of metallicities in the solar neighborhood.

4.1. A Monte Carlo Approach

As a consistency check, we also performed the occurrence calculation using a Monte Carlo approach, taking into account the effect of transit duration on S/N, as well as the probability of a Kepler-pipeline detection for a given S/N. We assigned each of the Kepler mid-M dwarf targets with at least one-quarter of data a random inclination, then calculated the S/N of the transit signals if the Kepler-42, Kepler-445, or Kepler-446 planets were transiting at the random inclination. We then used the detectability ramp described in Fressin et al. (2013) to randomly determine whether each system would be detected by the Kepler pipeline. This process results in the number of Kepler-42-, Kepler-445-, and Kepler-446-like systems that should have been detected if they were orbiting all mid-M dwarf stars. Since one Kepler-42, one Kepler-445, and one Kepler-446 system was detected, we use the following summation to determine f_CM:

$$\begin{equation} f_{\rm CM} = { 1 \over n_{Kepler\hbox{-}42} } + { 1 \over n_{\rm {Kepler}\hbox{-}445} } + { 1 \over n_{\rm {Kepler}\hbox{-}446} }, \end{equation} \tag{ 3 }$$

where n refers to the number of systems that would have been detected if they orbited all mid-M dwarf stars. We repeated the simulation 10,000 times, and the resulting distribution of f_CM is shown in Figure 8. The distribution has a median of 22%, a mode 21%, and asymmetric uncertainties of ${}^{+7}_{-5}$%, corresponding to the equivalent of ±1σ, which is consistent with the value determined by the first method.

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We note that planets too small to be detected are not included in this occurrence calculation. When including planets smaller than the Kepler-42 planets within the definition of "compact multiples," this result becomes a lower limit on the occurrence of compact multiples orbiting mid-M dwarfs.

4.2. Comparison to Sun-like and Early-M Dwarf Stars

Interestingly, Kepler-445c is somewhat similar in nature to GJ 1214b: both are super-Earths or mini-Neptunes orbiting metal-rich mid-M dwarf stars. Kepler-445c orbits with a longer period: 4.87 versus 1.58 days (Carter et al. 2011), and is subject to much lower stellar irradiation. It would be illuminating to study the atmosphere of this planet via transit transmission spectroscopy, as has been done for GJ 1214b (Bean et al. 2010; Croll et al. 2011; Désert et al. 2011; Berta et al. 2012; Fraine et al. 2013; Kreidberg et al. 2014), if not for the star's faintness: with a Ks-band magnitude of 12.61, transit-transmission spectroscopy will be challenging.

Both the Kepler-446 and Kepler-42 systems present cautionary tales for future transit surveys such as NASA's Transit Exoplanet Survey Satellite (TESS; Ricker 2014). The Kepler pipeline assigned unphysically large planet-to-star radius ratios with significant uncertainties for the planets in both systems. In the future, systems like these may have accurate astrometric parallax measurements from ESA's Gaia Mission. Combining the Gaia absolute magnitudes with mass–radius–luminosity relations would enable accurate stellar density priors to be employed during the pipeline transit fitting algorithm. We suggest such a procedure, as TESS is expected to observe significantly more M dwarf stars than Kepler.

Following the same calculation in the introduction performed on Kepler-42, but for Kepler-445 and Kepler-446, we calculate that their "1% of M_⋆" protoplanetary disks consisted of 16.8 and 8.0 M_⊕ of metals, respectively. In the case of Kepler-446, where the three planets have radii less than or equal to 1.5 R_⊕, we speculate that the three planets have primarily rocky compositions based on the rocky/non-rocky threshold calculated by Rogers (2014); however, we note that the planet radii are very near the rocky/non-rocky threshold. Assuming all are rocky, we calculate that the three planets constitute 6.5 M_⊕ of metals, meaning that ∼80% of the Kepler-446's disk metals went into these three planets. We repeat that this calculation relies on simplistic assumptions, but could indicate that rocky planet formation is efficient around low-mass stars with compact multiple exoplanets.

The same calculation for Kepler-445 is trickier. At 2.51 R_⊕, Kepler-445c is very likely to have a thick gaseous envelope. However, at 1.58 and 1.25 R_⊕, Kepler-445b and Kepler-445d could be rocky, with masses of 3.9 and 1.6 M_⊕, when interpolating onto the Fortney et al. (2007) relations for primarily rocky composition. That leads to 30% of the metal content of Kepler-445's protoplanetary disk going into its orbiting planets, not accounting for the core of Kepler-445c, which is presumably more massive than Kepler-445b in order to have accreted gas during the protoplanetary disk phase. Assuming Kepler-445c has a rocky core as massive as Kepler-445b, this leads to 56% of the metal content in these three planets alone.

These results can be interpreted as one or a combination of the following: (1) rocky planet formation is highly efficient around mid-M dwarf stars with compact multiple planetary systems, (2) disk mass fractions around M dwarf stars are significantly higher than 1% of the host star mass, and/or (3) the planets presented in this work are not in fact rocky, with significant mass in the form of hydrogen and helium. As stated earlier, next-generation near-infrared precise-radial-velocity spectrometers may be able to confirm the rocky nature of these planets. If, in fact, rocky planet formation is efficient for compact multiple systems, we would not expect to find significantly more planets beyond the orbits of Kepler-42 d, Kepler-445c, or Kepler-446c, simply because there is not enough rocky material in the protoplanetary disk to create significantly more planets. Therefore, we anticipate a dearth of outer, long-period planets with radii ⩾ 0.8 R_⊕ orbiting mid-M dwarf stars with compact multiple systems.

The planet–metallicity correlation, wherein higher-metallicity stars are more likely to host gaseous planets, has been shown to be true for Sun-like stars as well as early-M dwarf stars (Santos et al. 2001; Fischer & Valenti 2005; Johnson et al. 2010; Rojas-Ayala et al. 2010). However, it has not been thoroughly investigated for mid-M dwarf stars due to the challenges in finding planets around them. It is therefore intriguing that Kepler-445 ([Fe/H] = +0.27 ± 0.13; Muirhead et al. 2014) and GJ 1214 ([Fe/H] = +0.20; Rojas-Ayala et al. 2012) are the only mid-M dwarf stars with transiting planets larger than 2R_⊕, and both stars are metal-rich compared to the solar neighborhood ([Fe/H] = −0.05; Gaidos & Mann 2014). Based on this alone, we speculate that the planet–metallicity correlation may in fact extend down to mid-M dwarf stars for large planets (>2 R_⊕).

We thank the anonymous referee for the thoughtful report and useful suggestions. We thank Andrew West for the suggestions regarding measuring accurate SDSS magnitudes. A.V. is supported by the National Science Foundation Graduate Research Fellowship under grant No. DGE1144152. This paper includes data collected by the Kepler Mission. Funding for the Kepler Mission is provided by the NASA Science Mission Directorate. The Kepler data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX13AC07G and by other grants and contracts.

This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. This research has made use of the Exoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org (Wright et al. 2011).

These results made use of the Discovery Channel Telescope at Lowell Observatory, supported by Discovery Communications, Inc., Boston University, the University of Maryland, the University of Toledo, and Northern Arizona University. The Large Monolithic Imager was funded by the National Science Foundation via grant AST-1005313.

Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community.

SNIFS on the UH 2.2 m telescope is part of the Nearby Supernova Factory project, a scientific collaboration among the Centre de Recherche Astronomique de Lyon, Institut de Physique Nuclaire de Lyon, Laboratoire de Physique Nuclaire et des Hautes Energies, Lawrence Berkeley National Laboratory, Yale University, University of Bonn, Max Planck Institute for Astrophysics, Tsinghua Center for Astrophysics, and the Centre de Physique des Particules de Marseille.

The infrared spectrum of Barnard's Star was acquired using the SpeX instrument on the Infrared Telescope Facility, which is operated by the University of Hawaii under Cooperative agreement No. NNX-08AE38A with the National Aeronautics and Space Administration, Science Mission Directorate, Planetary Astronomy Program.

These results made use of the Sloan Digital Sky Survey (SDSS) III. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III Web site is http://www.sdss3.org/. SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University. The SDSS data were accessed using the VizieR catalog access tool, CDS, Strasbourg, France (Ochsenbein et al. 2000).

Facilities: DCT (LMI) - , Hale (TSPEC) - Palomar Observatory's 5.1m Hale Telescope, IRTF (SpeX) - Infrared Telescope Facility, Keck:II (NIRC2, ESI) - KECK II Telescope, Kepler - The Kepler Mission, UH:2.2m (SNIFS) - University of Hawaii 2.2 meter Telescope