INFLUENCE OF STELLAR MULTIPLICITY ON PLANET FORMATION. I. EVIDENCE OF SUPPRESSED PLANET FORMATION DUE TO STELLAR COMPANIONS WITHIN 20 AU AND VALIDATION OF FOUR PLANETS FROM THE KEPLER MULTIPLE PLANET CANDIDATES

Our sample consists of bright host stars with multi-planet transiting systems from Kepler. Bright stars provide a higher signal-to-noise ratio for photometric measurements, which helps to better determine stellar and orbital properties. Follow-up observations will be relatively easier for brighter stars. Multi-planet systems have lower false-positive rates; they are more likely to be bona fide exoplanet systems. The presence of a second planet transit signal increases the likelihood of a bona fide planet by a factor of at least 25, and additional planet transit signals around one star provide an even larger boost of this likelihood (Lissauer et al. 2012). Out of 5779 Kepler Objects of Interest⁷ (KOIs; Borucki et al. 2010; Batalha et al. 2013), we selected all the systems with a Kepler magnitude (K_P) brighter than 13.5 mag and with at least two planet candidates (not with false-positive disposition). Most of these are FGK stars with 3700 K ⩽T_eff  ⩽ 7500 K and log g higher than 4.0. The sample includes 343 planet candidates in 138 systems. For this sample, we have refined their stellar and orbital solutions, searched for stellar companions, and calculated the planet confidence levels. These analyses allow us to compare the stellar multiplicity rate with field stars and infer the planet occurrence rate for single- and multiple-star systems.

Stellar and orbital parameters from the KOI table are obtained independently from stellar evolution modeling and transit light-curve analysis (Batalha et al. 2013). Occasionally the calculations of some parameters are not consistent between stellar evolution modeling and transit light-curve analysis. For example, a/R_S, the ratio of the planet orbital semimajor axis and the stellar radius, is sometimes different from these two analysis techniques. a/R_S can be directly obtained from the transit light-curve analysis based on a model described in Mandel & Agol (2002). We note that in their paper the notation d/R_S, the ratio of star–planet separation during transit and the stellar radius, is used and only circular orbits are considered. We used a/R_S instead of d/R_S and considered eccentric orbits in this paper. Alternatively, a/R_S can be calculated from Kepler's third law when the stellar mass and radius are known from a stellar evolution model and the orbital period is known from transit observations. Figure 1 (left panel) shows the comparison of a/R_S from these two calculations based on KOI values. The values for a/R_S are not always consistent with each other and can be off by a factor of three to four, which is much larger than the predicted error bar. An algorithm that iterates between stellar evolution modeling and transit light-curve analysis is needed to alleviate this discrepancy.

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We therefore developed an iterative algorithm that uses a/R_S to link the results from the stellar evolution modeling and transit light-curve analysis. A similar approach can also be found in Torres et al. (2012) and Dawson & Johnson (2012). For stellar parameters, we used the Yonsei–Yale interpreter (Demarque et al. 2004) to calculate stellar mass, radius, and luminosity based on the observation results reported in the KOI table, such as effective temperature (T_eff), surface gravity (log g), and metallicity ([Fe/H]). In a Monte Carlo simulation, we generated a large sample of inputs (including T_eff, log g, [Fe/H], α-element abundance, and age) for the Yonsei–Yale interpreter. The values for T_eff, log g, and [Fe/H] follow a Gaussian distribution centered at the reported value from the KOI table and with a standard deviation equal to the reported error bar. For α-element abundance and age, we assumed uniform distribution with a range of [0.0 dex, 0.2 dex] and [0.08 Gyr, 15 Gyr], respectively. We estimated the median values and the 68% credible interval for stellar mass, radius, and luminosity based on their distributions from the outputs of the interpreter. With the orbital period, which is usually the most precisely determined parameter in a transit observation, the value and uncertainty of a/R_S are calculated based on stellar evolution modeling.

Once the distribution of a/R_S is obtained from the stellar evolution modeling, it can be used as a prior for the estimation of orbital parameters in the transit light-curve analysis. Instead of searching for a/R_S values based purely on the light-curve fitting, we considered only a/R_S values that are consistent with the prior distribution. In the light-curve fitting process, we used the model described in Mandel & Agol (2002). We adopted a boot-strapping algorithm to estimate the uncertainties of orbital parameters, in which we ran a larger number of trials in analyzing light curves that are perturbed based on the reported photometric measurement uncertainty. To avoid the dependence of results on the initial guess, we also perturbed the initial guess based on distributions of orbital parameters from previous runs. Initial guesses within 5σ dispersion were tried in order to avoid the sensitivity of the initial guess, and to explore a large parameter space. Orbital parameters are determined based on the distribution of the fitting results. The best-fit values and their 68% credible intervals are reported. A distribution of a/R_S will be given at the end of the transit light-curve analysis. The a/R_S distribution will be fed into the Yonsei–Yale interpreter as a sampler. Only the outputs of the interpreter will be selected for a/R_S distribution for the next round of iteration.

Figure 1 (right panel) shows two examples of an a/R_S distribution from stellar evolution modeling (blue dashed line) and transit light-curve analysis (green dotted line) and the converged a/R_S distribution from the iterative algorithm (red solid line). For KOI 94.02, because the stellar parameters (such as T_eff, log g, and [Fe/H]) are determined by the Spectroscopy Made Easy analysis (Valenti & Fischer 2005), the uncertainties for these parameters are smaller. Therefore, the constraint on a/R_S from the stellar evolution modeling is tighter than that from the transit light-curve analysis. Thus, the converged a/R_S distribution is mainly determined by the a/R_S distribution from the stellar evolution modeling. Conversely, for KOI 289.02, since T_eff, log g, and [Fe/H] are determined by J − K color, the constraint on a/R_S from the stellar evolution modeling is weak, and the converged distribution of a/R_S is mainly determined by the transiting light-curve analysis.

The iterative algorithm unifies the stellar evolution modeling and the transit light-curve analysis, resulting in a consistent set of stellar and orbital parameters. The results of stellar and orbital parameters for the systems in our sample are presented in Tables 2 and 3. Figure 2 compares the results of the iterative algorithm with the values from the KOI table. The iterative results generally agree well with the KOI values, as the data points are mostly located along the 1:1 line within the error bars. However, for cases when independent stellar evolution modeling and the transit light-curve analysis show discrepancies, the iterative algorithm will give a different result from the KOI value because it considers measurement errors of both methods and thus gives a more consistent result. As a general trend, we found that stellar radii from KOI values are underestimated, which is in agreement with previous studies (Brown et al. 2011; Verner et al. 2011; Plavchan et al. 2012). Everett et al. (2013) suggested that 87% of the KOIs' radii need an upward correction and 26% of the KOIs' radii are underestimated by more than a factor of 1.35. In our case, we find that 86% have radii larger than KOI values and 17% are at least 1.35 larger than KOI values. Figure 3 shows the distribution of revised planet radii for the sample of bright Kepler multiple-planet candidates. The sample is dominated by smaller planet candidates. There are 233 planet candidates with radii smaller than 2.5 earth radii, 68 with radii between 2.5 and 5.0 earth radii, and 42 larger than 5.0 earth radii.

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Table 2. Stellar Parameters

KOI†	KOI							Derived Values
	KIC†	α†	δ†	$K_P^{\dagger }$	$T_{\rm eff}^{\dagger }$	log g†	[Fe/H]†	Teff	log g	[Fe/H]	M*	R*	L*	ρ*
		(deg)	(deg)	(mag)	(K)	(cgs)	(dex)	(K)	(cgs)	(dex)	(M☉)	(R☉)	(L☉)	(g cm−3)
K00005	8554498	289.739716	44.647419	11.66	5861	4.19	0.12	$5865^{+60}_{-58}$	$4.17^{+0.07}_{-0.08}$	$0.11^{+0.49}_{-0.47}$	$1.13^{+0.09}_{-0.11}$	$1.44^{+0.16}_{-0.15}$	$2.19^{+0.79}_{-0.60}$	0.53 ± 0.18
K00041	6521045	291.385986	41.990269	11.20	5909	4.30	−0.12	$5902^{+53}_{-74}$	$3.62^{+0.20}_{-0.12}$	$-0.13^{+0.43}_{-0.38}$	$1.23^{+0.21}_{-0.13}$	$2.86^{+0.53}_{-0.65}$	$4.73^{+3.63}_{-2.22}$	0.07 ± 0.05
K00070	6850504	287.697998	42.338718	12.50	5443	4.45	−0.15	$5444^{+44}_{-43}$	$4.43^{+0.07}_{-0.07}$	$-0.17^{+0.52}_{-0.47}$	$0.88^{+0.08}_{-0.12}$	$0.94^{+0.09}_{-0.08}$	$0.66^{+0.26}_{-0.19}$	1.49 ± 0.45
K00072	11904151	285.679382	50.241299	10.96	5627	4.39	−0.81	$5627^{+61}_{-53}$	$4.35^{+0.06}_{-0.07}$	$-0.80^{+0.48}_{-0.51}$	$0.79^{+0.11}_{-0.07}$	$0.98^{+0.09}_{-0.08}$	$0.67^{+0.28}_{-0.16}$	1.16 ± 0.32
K00082	10187017	281.482727	47.208031	11.49	4908	4.61	0.20	$4909^{+41}_{-41}$	$4.58^{+0.08}_{-0.07}$	$0.18^{+0.42}_{-0.47}$	$0.81^{+0.04}_{-0.09}$	$0.76^{+0.07}_{-0.07}$	$0.28^{+0.05}_{-0.06}$	2.61 ± 0.75
K00085	5866724	288.688690	41.151180	11.02	6172	4.36	−0.05	$6170^{+47}_{-48}$	$4.33^{+0.08}_{-0.07}$	$-0.03^{+0.48}_{-0.48}$	$1.18^{+0.12}_{-0.15}$	$1.22^{+0.13}_{-0.11}$	$1.76^{+0.72}_{-0.46}$	0.92 ± 0.30
K00089	8056665	299.821167	43.814270	11.64	7713	3.90	0.06	$7715^{+194}_{-200}$	$3.88^{+0.18}_{-0.29}$	$0.06^{+0.51}_{-0.48}$	$1.96^{+0.14}_{-0.20}$	$2.68^{+1.06}_{-0.62}$	$24.99^{+11.94}_{-9.49}$	0.14 ± 0.13
K00094	6462863	297.333069	41.891121	12.21	6184	4.18	−0.76	$6182^{+60}_{-52}$	$4.17^{+0.04}_{-0.05}$	$-0.74^{+0.48}_{-0.49}$	$1.05^{+0.18}_{-0.09}$	$1.40^{+0.14}_{-0.10}$	$2.19^{+1.06}_{-0.56}$	0.53 ± 0.15
K00102	8456679	299.704742	44.435749	12.57	5838	4.42	−0.36	$5836^{+52}_{-51}$	$4.41^{+0.06}_{-0.08}$	$-0.33^{+0.49}_{-0.52}$	$0.99^{+0.13}_{-0.13}$	$1.03^{+0.11}_{-0.09}$	$0.92^{+0.31}_{-0.27}$	1.28 ± 0.41
K00108	4914423	288.984558	40.064529	12.29	5975	4.33	−0.10	$5973^{+46}_{-41}$	$4.30^{+0.06}_{-0.07}$	$-0.06^{+0.43}_{-0.51}$	$1.11^{+0.12}_{-0.16}$	$1.22^{+0.12}_{-0.11}$	$1.52^{+0.88}_{-0.45}$	0.86 ± 0.27

Notes. Columns marked with a † are values from NASA Exoplanet Archive. Stars marked with a ^†† have missing T_eff. Their T_eff values are estimated based on g − r and g − i values (Pinsonneault et al. 2012); the adopted T_eff is the average of the results. There are three exceptions: values of T_eff for K03158 and K0368 are adopted from Ammons et al. (2006), and T_eff for K04021 is based on the infrared flux method in Pinsonneault et al. (2012).

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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Table 3. Orbital Parameters

KOI	KIC	T0	Period	Impact	RPL/R*	e	ω	RPL	Inclination	a/R*	a	TPL	Depth	Dur	Sig	p1	p2
		(BKJD)	(days)	Parameter			(radian)	(R⊕)	(deg)		(AU)	(K)	(ppm)	(hr)	(σ)
K00005.01	8554498	132.9733	$4.78033^{+0.00001}_{-0.00001}$	$0.93^{+0.01}_{-0.01}$	$0.045^{+0.001}_{-0.002}$	$0.00^{+0.02}_{-0.00}$	$0.00^{+0.13}_{-0.00}$	8.07 ± 0.64	$82.72^{+0.61}_{-0.46}$	$7.6^{+0.5}_{-0.4}$	0.059 ± 0.002	1434 ± 45	983 ± 3	2.3	5.1	0.932	0.997
K00005.02	8554498	133.3670	$7.05186^{+0.00000}_{-0.00000}$	$0.92^{+0.16}_{-0.17}$	$0.003^{+0.002}_{-0.001}$	$0.00^{+0.05}_{-0.00}$	$0.04^{+0.21}_{-0.04}$	0.51 ± 0.26	$85.30^{+0.72}_{-0.60}$	$11.2^{+1.0}_{-1.0}$	0.075 ± 0.002	1159 ± 69	12 ± 4	1.9	0.4	0.015	0.278
K00041.01	6521045	122.9506	$12.81574^{+0.00004}_{-0.00005}$	$0.57^{+0.20}_{-0.55}$	$0.014^{+0.001}_{-0.001}$	$0.00^{+0.30}_{-0.00}$	$0.00^{+0.88}_{-0.00}$	3.08 ± 0.74	$87.33^{+2.57}_{-2.06}$	$12.0^{+2.5}_{-2.4}$	0.114 ± 0.005	1120 ± 130	219 ± 5	6.7	1.5	0.615	0.988
K00041.02	6521045	133.1758	$6.88710^{+0.00016}_{-0.00015}$	$0.90^{+0.02}_{-0.09}$	$0.009^{+0.001}_{-0.001}$	$0.11^{+0.19}_{-0.11}$	$0.01^{+1.11}_{-0.01}$	3.13 ± 0.58	$81.11^{+1.65}_{-1.50}$	$5.4^{+1.3}_{-0.5}$	0.076 ± 0.003	1628 ± 119	72 ± 4	4.4	0.3	0.067	0.783
K00041.03	6521045	153.9839	$35.33314^{+0.00109}_{-0.00204}$	$0.93^{+0.10}_{-0.14}$	$0.012^{+0.000}_{-0.003}$	$0.00^{+0.22}_{-0.00}$	$0.00^{+3.09}_{-0.00}$	3.78 ± 0.88	$86.88^{+0.83}_{-0.55}$	$17.0^{+4.8}_{-2.4}$	0.226 ± 0.010	947 ± 106	96 ± 10	6.3	1.0	0.077	0.807
K00070.01	6850504	138.6075	$10.85409^{+0.00002}_{-0.00003}$	$0.31^{+0.26}_{-0.31}$	$0.028^{+0.002}_{-0.001}$	$0.00^{+0.41}_{-0.00}$	$0.16^{+3.20}_{-0.16}$	2.94 ± 0.26	$89.13^{+0.87}_{-0.78}$	$20.4^{+1.1}_{-1.2}$	0.092 ± 0.003	803 ± 32	1060 ± 9	4.0	0.5	0.959	0.999
K00070.02	6850504	134.5009	$3.69612^{+0.00002}_{-0.00002}$	$0.36^{+0.17}_{-0.36}$	$0.017^{+0.001}_{-0.001}$	$0.00^{+0.27}_{-0.00}$	$0.00^{+0.34}_{-0.00}$	1.77 ± 0.18	$87.96^{+2.04}_{-1.23}$	$10.1^{+0.7}_{-0.7}$	0.045 ± 0.002	1140 ± 53	400 ± 5	2.7	1.7	0.821	0.996
K00070.03	6850504	164.7276	$77.61153^{+0.00017}_{-0.00022}$	$0.32^{+0.21}_{-0.32}$	$0.026^{+0.001}_{-0.000}$	$0.00^{+0.07}_{-0.00}$	$0.00^{+0.23}_{-0.00}$	2.68 ± 0.27	$89.76^{+0.24}_{-0.18}$	$77.4^{+6.0}_{-6.4}$	0.341 ± 0.012	409 ± 20	834 ± 28	7.4	0.5	0.549	0.984
K00070.04	6850504	135.9338	$6.09850^{+0.00018}_{-0.00022}$	$0.07^{+0.59}_{-0.07}$	$0.008^{+0.000}_{-0.002}$	$0.52^{+0.19}_{-0.50}$	$3.16^{+1.61}_{-2.67}$	0.82 ± 0.13	$89.70^{+0.30}_{-2.46}$	$14.3^{+1.1}_{-1.2}$	0.063 ± 0.002	951 ± 54	74 ± 7	2.8	0.3	0.037	0.657
K00070.05	6850504	135.2142	$19.57709^{+0.00077}_{-0.00066}$	$0.34^{+0.44}_{-0.31}$	$0.009^{+0.002}_{-0.001}$	$0.01^{+0.37}_{-0.01}$	$0.00^{+1.02}_{-0.00}$	0.89 ± 0.16	$89.37^{+0.58}_{-0.83}$	$31.0^{+2.4}_{-2.5}$	0.136 ± 0.005	646 ± 35	104 ± 13	4.5	1.5	0.023	0.538

Notes. The last three columns are significance of pixel centroid offset between in- and out-of-transit, planet confidence before and after considering the planetary likelihood boost in Lissauer et al. (2012). Planet candidates with significant pixel centroid offset (>3σ) and planet candidates with higher than 0.997 planet confidence are marked in bold.

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The UKIRT data archive in Edinburgh provides positions and magnitudes for J-band sources within the Kepler field. Archive tools generate high-resolution cut-out images on the fly and allow source cross matches around user-specified lists of coordinates. The current UKIRT data set covers 99.5% of the field and was observed and supplied by Phil Lucas. The images have a typical spatial resolution of 08–09. They are therefore useful for separating blended stellar pairs and spatially resolving external galaxies.

We used UKIRT images to calculate brightness contrast curves and to detect stellar companions around planet candidate host stars. For each star in our sample, we downloaded a UKIRT image from the WFCAM Science Archive.⁸ We calculated the median and the standard deviation of brightness (with outlier rejection) for many concentric annuli centering at the planet host star. The brightness ratio of 5σ above the median and the central star is defined as contrast. A contrast curve plots contrast as a function of the radii of concentric annuli, or the distance to the central star in arcseconds. The contrast curve defines a brightness limit above which a visual companion can be detected. If there was a source with a brightness of at least 5σ brighter than the median brightness, then we recorded a detection of a visual companion to the central star.

Within a 20'' radius centered on the central star, we detected 177 visual companions around 99 planet host stars. The other 39 stars in our sample have no visual companions down to the contrast limit. Table 4 presents the results of visual companions detected with the UKIRT images. The average contrast curve and its 3σ variation are plotted in Figure 4. The detections are also plotted in Figure 4 as asterisks, which agree well with the average contrast curve. In this sample of 138 stars, we do not find any stars with a visual companion within 2''. We note that KOI 119 and KOI 2311 have elongated point-spread functions; however, the possible stellar companions failed the 5σ detection criteria. About 7% (10/138) have a visual companion within 6''. The numbers are lower than those numbers from Adams et al. (2012) by a factor of nine. This result implies the incompleteness of visual companion searches using the UKIRT archival images, and the strength of AO observations in spatial resolution and contrast limit. However, the UKIRT images provide a contrast curve for each star and will be used to constrain parameter space in planet confidence estimation in Section 2.4. There are 177, 82, 32, and 5 stars with companion star separations less than 20'', 15'', 10'', and 5'', respectively. These numbers are roughly proportional to the sky area around the central star, implying that they are likely to be background stars rather than physically bound companions.

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The likelihood of a planet candidate being a bona fide exoplanet can be estimated by calculating the probability of false positives and the probability of a true planet transiting event. The BLENDER technique has been used to validate Kepler planet candidates (e.g., Torres et al. 2011; Fressin et al. 2011). A similar approach was later developed for planet candidate validation (Wang et al. 2013; Barclay et al. 2013a). In order to perform planet confidence calculation, several observational constraints should be taken into consideration to exclude unlikely regions in a ΔK_P-separation parameter space. ΔK_P is the differential magnitude of a possible contaminating object, and separation is its projected distance from the central star. These observational constraints include pixel centroid offset, transit depth, and UKIRT contrast curve.

The Kepler light-curve files contain information on both the position of the star at a given time (flux-weighted centroids, called MOM_CENTR1 and MOM_CENTR2) and the predicted position of the star based on the position of reference stars (called POS_CORR1 and POS_CORR2). By subtracting one from the other, we can find the centroid position of a star. We measured the centroid position during the prospective transit, and immediately before and after transit to search for centroid shifts. If the transit occurs on a star other than the target star, the centroid position should move away from that star by an amount proportional to the transit depth (Bryson et al. 2013). The magnitude of the shift can be used to calculate how far away this source is from the target. Even if no obvious shift is detected, we can still calculate a confusion region around the star. In our planet confidence calculations, we used the 3σ confusion radius as the outer limit on the separation between the target and a false-positive source.

The KOI table gives results of pixel centroid offset measurement. We used values from the flux-weighted method because this method is similar to our independent analysis. Out of 343 planet candidates in our sample, there are 240 with measured pixel centroid offset values from the KOI table, and 34 of them show significant (above 3σ) pixel centroid offsets. In comparison, for these 240 planet candidates, our independent pixel centroid offset analysis found 20 candidates with significant offsets, 14 of them overlapped with systems characterized by significant offsets in the KOI table. The difference between our independent analysis and the measured offsets from the KOI table may be attributed to a different window size and de-trending functional form selected for analysis. For the polynomial functional form that we used for de-trending the centroid measurements, the optimal polynomial order depends on the variability of the measurement results and the window sizes for both in- and out-of-transit measurement. However, we chose a second-order polynomial function and fixed window sizes to serve the purpose of a streamline data reduction. These choices are empirical and may be responsible for the differences from the values from the KOI table. For the 103 planet candidates without measured pixel centroid offset values from the KOI table, we found 7 with significant offsets. In the column of pixel centroid offset significance in Table 3, we marked values larger than 3σ in bold text.

Transit depth places a lower brightness limit to a possible contaminating object. The contrast curve of each star is calculated as described in Section 2.3. Once the observational constraints are put on the parameter space, the planet confidence will be calculated based on a galactic stellar population model (Robin et al. 2003). The details of planet confidence calculations can be found in Wang et al. (2013) and Barclay et al. (2013a). Table 3 gives the planet confidences for each of the KOIs in our sample. Two sets of planet confidences are given. The p1 column in Table 3 contains the planet confidences before considering the existence of other candidates in the same system. The p2 column contains the augmented planet confidences for a multi-planet system (Lissauer et al. 2012; Barclay et al. 2013a).

There are 14 planet candidates with p2 higher than 0.997, a value high enough to promote a planet candidate to a validated planet. KOI 70.01 (Fressin et al. 2012), KOI 72.01 (Batalha et al. 2011), KOI 85.01 (Chaplin et al. 2013), KOI 94.01 (Weiss et al. 2013), KOI 244.01, KOI 244.02 (Steffen et al. 2012), KOI 245.01 (Barclay et al. 2013b), and KOI 246.01 (Gilliland et al. 2013) are planet candidates that were previously validated or confirmed. For these KOIs, we note that all but KOI 70.01 have pixel centroid offsets larger than 3σ. KOI 72, KOI 85, KOI 244, KOI 245, and KOI 246 are bright stars in the Kepler field that cause pixel saturation, resulting in an imprecise pixel centroid offset measurement. KOI 94 is in a crowded field with four visual companions detected within 20''. We therefore caution that the flux-weighted pixel centroid offset measurement may not be precise for stars brighter than K_P = 11, or for stars in a crowded field. There are 10 stars (7.2%) in the sample that are brighter than 11th Kepler magnitude and 9 stars (6.5%) with more than three detected stellar companions within 22''. In addition to previously confirmed KOIs, we find six other KOIs with higher than 0.997 planet confidence: KOI 5.01, KOI 82.01, KOI 115.01, KOI 282.01, KOI 1781.01, and KOI 1781.02. However, KOI 5.01 and KOI 1781.01 have a significant pixel centroid offset, and their K_P values are 11.6 mag and 12.2 mag, respectively, which raises concerns about their planet nature. Therefore, our planet confidence calculation results in four newly validated KOIs: KOI 82.01, KOI 115.01, KOI 282.01, and KOI 1781.02. The validation plots for them are shown in Figures 5–8. We note that KOI 115, KOI 282, and KOI 1781 all have one visual stellar companion within 85. Although the possibility of the transit signal originating from the companion is excluded, flux contamination may affect the determination of the planet radius. However, given the relatively large magnitude difference between the companion and the center star (ΔK_P ⩾ 3.0), the major contribution to the uncertainty of the planet radius is the stellar radius estimation rather than the flux contamination. We note that the planet confidence considers only scenarios in which a planet is physically associated with the primary. There is a chance that it orbits an undetected physical companion. This chance is likely low but is not accounted for.

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There are 23 KOIs in our sample with more than two epochs of RV measurement results. They are available through the Kepler Community Follow-up Observing Program (CFOP). The purpose of the Web site is to facilitate collaboration on follow-up observing projects of KOIs and optimize the use of available facilities. Most of the RV measurements were conducted using the Keck HIRES or SAO TRES spectrographs. Table 5 summarizes the RV data for 23 KOIs. The median measurement precision is 7.3 m s⁻¹, and the median observation baseline is 447.8 days. Among 23 KOIs, one non-transiting gas giant planet (Kepler-68d) was revealed by the RV technique (Gilliland et al. 2013), and two (KOI 5 and KOI 148) exhibit a long-term RV trend. KOI 148 (Kepler-48; Steffen et al. 2013; Xie 2013) shows an RV slope of ∼80 m $\rm {s}^{-1}\ \,\rm {yr}^{-1}$, but this slope is caused by a ∼2 Jupiter-mass planet with an orbital period of ∼1000 days (H. Isaacson 2013, private communication). KOI 5 shows an RV slope of ∼100 m $\rm {s}^{-1}\ \rm {yr}^{-1}$. If the mass of this companion is in the stellar regime, it must be more than 9.5 AU away from the central star in an edge-on orbit with respect to an observer. An edge-on orbit for a potential stellar companion is a probable configuration for stars with multiple transiting planets (see detailed discussions in Section 3.1). We therefore flag KOI 5 as a possible multiple stellar system. However, close-in (a ⩽ ∼10 AU) stellar companions are excluded.

We studied the completeness of Doppler measurements based on the reported RV precision and observation cadence. We defined a parameter space of orbital separation and companion inclination. For each point in the parameter space, we generated a set of simulated RV signals at the reported observation epochs, but with random mass and argument of periastron (ω). The mass ratio follows a distribution of stellar companion mass ratios from Duquennoy & Mayor (1991) with a median of 0.23 and a standard deviation of 0.46. The distribution of ω is uniform between 0 and 2π. Eccentricity is assumed to be 0.4 for all simulations. This value roughly corresponds to the median eccentricity determined in studies of stellar companions around solar-type stars (Duquennoy & Mayor 1991; Raghavan et al. 2010). If the simulated RV set has an rms scatter three times larger than the reported RV measurement error, then a stellar companion in a certain range of parameter space can be detected. One hundred simulations were run at each point in a grid of orbital separation and inclination space. The fraction of detectable trials is reported in Figure 9. Completeness decreases with increasing separation and decreasing inclination. This information will be used in assessing the probability of multiple stars in calculations of the stellar multiplicity rate and the planet occurrence rate in Section 3.

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A companion star can affect the stability of a planetary orbit. Given the semimajor axis of the outmost planet's orbit (a_p), we have a critical (lower limit) semimajor axis of the companion (Holman & Wiegert 1999),

$$\begin{eqnarray} a_{\rm c1} &=& a_{\rm p}/\left(0.464-0.380\mu -0.631e_{\rm B}+0.586\mu e_{\rm B}\right. \nonumber\\ &&\left. + 0.150e_{\rm B}^{2}-0.198\mu e_{\rm B}^{2}\right), \end{eqnarray} \tag{ 1 }$$

where e_B is the orbital eccentricity of the stellar companion and μ = m_B/(m_A + m_B) is the ratio between the mass of the companion and the total mass of the stellar system. Note that a_c1 is only an empirical estimate based on the test particle simulation. Considering the mutual perturbation of planets, the real lower limit should be larger than a_c1. In any case, a_c1 can be treated as a lower limit.

Besides stability, a stellar companion can affect the coplanarity of a multiple-planet system if the companion is on an inclined orbit with respect to the planets' orbits. This occurs when the planet's orbital node precession is dominated by stellar–planet interaction rather than the planets' mutual interaction. For a two-planet system, the node precession timescale of the outer planet due to the stellar companion t_Bp can be estimated as twice the Kozai timescale (Kiseleva et al. 1998), i.e.,

$$\begin{eqnarray} &&t_{\rm Bp}\sim \frac{4}{3\pi }\frac{P_{\rm B}^{2}}{P_{\rm 2}}\left(1-e_{\rm B}^{2}\right)^{3/2}\frac{m_{\rm A}+m_{\rm B}}{m_{\rm B}}, \end{eqnarray} \tag{ 2 }$$

where P_B and P₂ are the orbital period of the stellar companion and the outer planet, respectively. The precession timescale due to planet–planet interaction can be estimated using the second-order Laplace–Lagrange secular theory (Murray & Dermott 1999), i.e.,

$$\begin{eqnarray} &&t_{\rm pp}\sim \frac{4P_{\rm 2}}{b_{3/2}^{(1)}(\alpha)} \frac{m_{\rm A}}{\alpha m_{1}+\alpha^{1/2} m_{2}}, \end{eqnarray} \tag{ 3 }$$

where m₁ and m₂ are the masses of the two planets, α = a₁/a₂ is the ratio of their semimajor axes, and $b_{3/2}^{(1)}(\alpha)$ is the Laplace coefficient. Equating the above two timescales for precession due to stellar–planet interaction and planet–planet interaction, we obtain a critical orbital period for the companion, i.e.,

$$\begin{eqnarray} P_{\rm B} &=& P_{\rm 2}\left(\frac{3\pi }{b_{3/2}^{(1)}(\alpha)}\right)^{1/2}\left(\frac{m_{\rm A}}{\alpha m_{1}+\alpha^{1/2} m_{2}}\right)^{1/2}\nonumber\\ &&\times\left(\frac{m_{\rm B}}{ m_{\rm A}+ m_{\rm B}}\right)^{1/2}\left(1-e_{\rm B}^{2}\right)^{-3/4}, \end{eqnarray} \tag{ 4 }$$

corresponding to a critical semimajor axis, i.e.,

$$\begin{eqnarray} &&a_{\rm c2} = a_{\rm 2}\left(\frac{3\pi }{b_{3/2}^{(1)}(\alpha)}\right)^{1/3}\left(\frac{m_{\rm B}}{\alpha m_{1}+\alpha^{1/2} m_{2}}\right)^{1/3}\left(1-e_{\rm B}^{2}\right)^{-1/2}.\quad\quad \end{eqnarray} \tag{ 5 }$$

For a system with more than two planets, we calculate Equation (5) for each pair of planets and adopt the minimum value as the a_c2 of the system. Below this critical semimajor axis, planetary coplanarity would be significantly affected by the companion's perturbation. Generally, a_c2 > a_c1 for a typical planet system, and thus coplanarity puts a stronger constraint on the companion's orbit.

Our analysis gives a qualitative estimate of how planetary stability and coplanarity would be affected by a stellar companion. To quantitatively measure such effects, we perform the following N-body simulation. For each multiple KOI system, we add a test stellar companion and use the N-body simulation package MERCURY (Chambers & Migliorini 1997) to integrate the orbits of the system up to a timescale of 10t_Bp. In each integration, we monitor the relative orbital inclinations of all the planets and calculate the fraction of time during which their relative inclinations are all less than 5°. This time fraction is then treated as the probability⁹ (P_DA) at which the proposed stellar companion cannot be ruled out. Here, the proposed stellar companion is drawn from a range of semimajor axes of 0.5a_c2 < a_B < 2a_c2 and a range of inclinations of 0 < i_B < 90°. The companion's mass and orbital eccentricity are set to m_B = 0.1 M_☉ and e_B = 0.1, to get a conservative estimate of P_DA. The relative inclination cutoff at 5° is chosen because Kepler planets are believed to be highly coplanar with a relative inclination within 1°–2° (Lissauer et al. 2011; Fang & Margot 2012; Tremaine & Dong 2012; Fabrycky et al. 2012; Figueira et al. 2012; Johansen et al. 2012). In all simulations, the planet mass is set to a nominal value, m_p = (R_p/R_⊕)^2.06 M_⊕ (Lissauer et al. 2011), and the planets are started with circular and coplanar orbits. As an example, Figure 10 plots the result for KOI 275.

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The Doppler technique and the dynamical stability analysis are two complementary methods, with the former sensitive to edge-on companions and the latter sensitive to face-on companions. The difference between these two methods is that the Doppler technique sets constraints from observation while the constraints from the dynamical analysis come from numerical simulations.

At each point in the a–i parameter space for a possible stellar companion in a planetary system, two constraining numbers are given. One is from the Doppler analysis, and the other one is from the dynamical analysis. For example, the completeness of the Doppler technique at [a, i] = [10 AU, 50°] is 60%, and the dynamical analysis predicts that 10% of simulations do not meet the coplanarity condition. The probability of an undetected companion is therefore 40% × 90% × p(a, i), where 40% is the incompleteness of Doppler observations, 90% is the probability of coplanar configuration from the dynamical analysis, and p(a, i) is the probability of a stellar companion at a given a and i. We note that the results from Doppler observation and the dynamical analysis may be correlated. For example, the 10% rejected orbits in the previous example may be partially detected by Doppler observations. The correlation will result in an underestimation of the probability of an undetected companion. However, the correlation should be small because they are sensitive to different phase spaces and the results are thus nearly orthogonal. Most face-on companions missed by Doppler observations will be excluded by the dynamical analysis, and vice versa. Highly eccentric orbits that are occasionally missed by Doppler observations due to limited phase coverage can also be excluded by the dynamical analysis.

We treat the probability p(a, i) as a product of p(a) and p(i). We used the Gaussian distribution of stellar companion periods of solar-type stars (Raghavan et al. 2010) and converted periods to separations to describe p(a). For p(i), the distribution of inclinations for stellar companions, we adopted the result from Hale (1994). All systems in our sample are multi-planet systems, and there is evidence of alignment of stellar spin and planet orbital planes for multi-planet systems (Hirano et al. 2012; Albrecht et al. 2013). Hale (1994) suggested that the spin–orbit alignment is likely for binaries with separations less than ∼15 AU and the spin–orbit angle becomes random when separations exceed ∼30 AU. Therefore, if there is a stellar companion within 15 AU around a star in our sample, its orbital plane should align approximately with the orbital plane of the planets. This alignment probability becomes smaller and randomized as the separation grows larger than 30 AU. We used different functions of p(i) for different separations. For separations less than 15 AU, we adopted a Gaussian form with a median of 80° and a standard deviation of 5°; for 15 AU ⩽ a ⩽ 30 AU, we used a combination of the Gaussian form and a uniform distribution between 0° and 90° with equal probability; for separations larger than 30 AU, we used a uniform distribution between 0° and 90° to describe p(i). These functions at different separations agree well with the observations in Hale (1994). The probability of a star being in a multiple stellar system is then calculated by integrating over the a–i phase space.

Figure 11 shows the average sensitivity contours after combining the Doppler observations and the dynamical analysis. When compared to Figure 9, the Doppler observation incompleteness is complemented by the dynamical analysis at low inclinations. There are a total of 23 systems with Doppler observations among 138 systems in our sample.

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Figure 12 shows the comparison of the stellar multiplicity rate for field stars and planet host stars as a function of separation. We performed calculations for two samples. The first sample contained stars with both Doppler measurements and dynamical analysis (RV sample, N = 23), and the second sample included all stars (N = 138). The dashed line is the field star multiplicity rate (Raghavan et al. 2010). The blue region is the calculated stellar multiplicity rate, i.e., N_M/N, for the RV sample, where the vertical height of the region at a given separation indicates the 1σ error bar. The error bar of N_M is estimated based on Poisson statistics. The square root of the closest integer to N_M is used to estimate the error of N_M unless the closest integer is 0, in which case we used 1 for the error of N_M. At a = 20.8 AU, the blue hatched area crosses over the dashed line, suggesting that the stellar multiplicity rates for field stars and for the RV sample become indistinguishable beyond ∼20 AU. The lower stellar multiplicity rate for planet host stars suggests that the presence of a close-in stellar companion may have a suppressive effect on planet formation and evolution. From the completeness contours in Figure 11, the completeness is ∼50% around 20 AU. At this completeness level, the majority of stellar companions within 20 AU would be detected or excluded due to orbit stability concern, leading to a conclusion of suppressive influence of a close-in stellar companion within 20 AU.

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The red hatched area is the stellar multiplicity rate for the entire sample of multi-planet host stars. The error bar is smaller due to a larger sample. The crossover with the dashed line takes place at 85 AU. However, the significant incompleteness beyond 20 AU (see Figure 11) prevents us from determining whether the crossover is due to an indistinguishable stellar multiplicity rate between planet host stars and field stars, or merely an incomplete survey. If it is the former case, then the large sample helps to reduce the statistical error and push the effective separation of a stellar companion from 20 AU to 85 AU. If it is the latter case, a longer RV measurement baseline or AO imaging would detect more stellar companions, pushing the red hatched area upward and the effective separation closer than 85 AU. Nonetheless, the suppressive influence of a close-in stellar companion is shown. It is the effective separation that is currently lacking a satisfactory constraint due to statistical error and survey incompleteness.

Section 3 provides a method of calculating the planet occurrence rate for single and multiple stars, respectively. The result depends on f, the overall planet occurrence rate. Since f is still an issue of debate (Cumming et al. 2008; Howard et al. 2010, 2012; Mayor et al. 2011; Catanzarite & Shao 2011; Youdin 2011; Traub 2012; Mann et al. 2012; Gaidos 2013; Swift et al. 2013; Kopparapu 2013; Bonfils et al. 2013; Fressin et al. 2013; Dressing & Charbonneau 2013; Parker & Quanz 2013; Petigura et al. 2013a, 2013b), we will leave it in the equation as a variable to be determined. Planet occurrence rate calculations need to cover a wide range of separations to avoid omitting any possible companions. The distribution of separations for stellar companions approaches zero when a gets to more than 10⁵ AU; therefore, we truncate the a range at 10⁵ AU. We get the ratio of f_S and f_M very close to 1, indicating that a planet is equally probable to appear around a single star or around a multiple-star system. This result is not surprising due to the incompleteness of Doppler observations and the dynamical analysis. There is a large unexplored phase space as the calculation covers a wider separation range. The prior information, e.g., the stellar multiplicity rate for the field star, outweighs the observations and analysis, naturally leading to a conclusion purely based on priors of the calculation.

We could potentially reduce the separation range and therefore reduce the impact of priors. However, this would also reduce the value of this analysis. Follow-up observations that address the unexplored phase space are the only way to reach a sensible planet occurrence rate estimation for both single and multiple stars. It requires a longer time baseline of Doppler observations for detection of close-in companions and imaging techniques with higher spatial resolutions (e.g., AO, speckle imaging, and Hubble Space Telescope snapshot program¹⁰) to probe for companions at wide orbits with deeper contrast.

We selected a total of 138 multi-planet candidate systems from the Kepler mission and studied the influence of a nearby stellar companion on planet formation. For these systems, we used archival data from Kepler to characterize their stellar and orbital properties. We developed an iterative algorithm that combines the Kepler photometric data with a stellar evolution interpolator. This algorithm uses a/R_S as a bridge to obtain a self-consistent stellar and orbital solution. We used this algorithm to refine stellar and orbital solutions for the 138 systems in our sample. The results are reported in Tables 2 and 3. The majority of the planet candidates are in compact systems: 90% (310/343) of them have semimajor axes less than 0.28 AU. We used the archival data from UKIRT to study the stellar environment of these planet host stars and found 177 visual companions within 20'', but we argued that these are likely to be background stars according to a statistical test.

We calculated the planet confidence for these multi-planet candidate systems (Table 3). The calculation involves pixel centroid offset analysis, transit depth analysis, and contrast curve calculations. A total of 14 planet candidates have a planet confidence higher than 0.997, indicating a high likelihood of being a bona fide planet. Eight of them are previously confirmed or validated planets. The other six planet candidates are validated for the first time, but two of them have a significant pixel centroid offset. Thus, KOI 82.01, KOI 115.01, KOI 282.01, and KOI 1781.02 become four newly validated planets from Kepler data. Future AO observations will further reduce the parameter space for possible contaminating stellar companions and increase the planet confidences for other unvalidated planet candidates.

The majority of previous work on stellar multiplicity made use of imaging techniques, such as AO and Lucky Imaging, and searched for stellar companions at wide separations. In comparison, we used a unique combination of the Doppler technique and dynamical stability analysis to evaluate stellar companions out to ∼100 AU. This approach demonstrates the influence of a close-in stellar companion on planet formation. We presented a statistical method for calculating the stellar multiplicity rate and the planet occurrence rate for both single stars and multiple stars. We found that the stellar multiplicity within 20.8 AU for planet host stars is significantly lower than that of field stars, indicating a strong suppressive effect of a stellar companion on planet occurrence. The influence of a stellar companion at larger separations is uncertain because of statistical error and survey sensitivity. Doppler observations for the 23 KOIs in our sample are not optimal for a search of stellar companions at larger separations. Future follow-up Doppler observations with a longer time baseline will help set further constraints on stellar companions.

In our dynamical analysis, we used several conservative assumptions. First, we assumed a moderate eccentricity of 0.1 in the analysis. If this is replaced by 0.5, roughly the median of eccentricity for binaries (Raghavan et al. 2010), a larger fraction of orbital configurations will become non-coplanar for the same a and i. The dynamical analysis thus becomes even more effective with a higher eccentricity assumption. Second, we assumed a mutual inclination limit to be 5°, which is a weaker constraint compared to the highly coplanar orbits found by Kepler with a mutual inclination of less than 2° (Lissauer et al. 2011; Fang & Margot 2012; Tremaine & Dong 2012; Fabrycky et al. 2012; Figueira et al. 2012; Johansen et al. 2012). A decreasing limit of mutual inclination will result in more rejections in dynamical analysis and thus increase the constraint on undetected stellar companions. However, we must note some limitations of the dynamical analysis. For example, we do see several cases in which a possible planet candidate is found in a region predicted to be unstable by the dynamical analysis, e.g., KOI 191 and 204 (Lissauer et al. 2011) and ν Octantis (Goździewski et al. 2013).

The stellar multiplicity rate for field stars in the solar neighborhood has been calculated by Raghavan et al. (2010) for solar-type stars within 25 pc of the Sun. The majority of the stars in our sample are within 550 pc of the Sun based on the magnitude cut at 13.5 Kepler mag and the assumption that most of them are main-sequence stars. The caveat of extrapolating the measurement to a larger volume must be noted. If the stellar multiplicity rate of the Kepler field is different from the solar neighborhood, the ratio (0.54/0.46) in Section 3 should be replaced with a different (but unknown) value. In addition, we compared the stellar multiplicity rate for field stars and planet host stars, but we do not know the fraction of field stars hosting planets. If all field stars have planets, then the comparison provides information about the difference of stellar companions in terms of separation distribution and the stellar multiplicity rate, because we would have compared two groups of planet-hosting stars in two environments. One is the solar neighborhood, and the other is the Kepler field. However, if (1) not all field stars have a planet and (2) the statistics of multiple stars (multiplicity, separation distribution, etc.) is similar for the nearby solar-type stars and the stars in our sample, then the difference in Figure 12 indeed suggests the impact of a close-in stellar companion on planet occurrence. In this case, the field stars are a sample contaminated by planet host stars. If a difference is seen when compared to a sample of planet host stars, then the difference would have been more distinct when comparing a planet host sample and a non-planet host sample. The latter is difficult to obtain because of current detection precision limitation. However, a planet mass or radius limit can be set in investigations of a certain type of planet, e.g., comparing the stellar multiplicity rate for the giant planet host stars and stars without a gas giant. For future studies, the combination of the Doppler technique, AO observation, and dynamical stability analysis will contribute an almost complete survey for stellar companions around planet host stars and will ultimately reveal how stellar multiplicity influences planet formation.