1. Preliminary draft. To be submitted to ApJ. Architecture and Dynamics of Kepler’s Candidate Multiple Transiting Planet Systems Jack J. Lissauer1 , Darin Ragozzine2 , Daniel C. Fabrycky3,4 , Natalie M. Batalha5 , William J.arXiv:1102.0543v1 [astro-ph.EP] 2 Feb 2011 Borucki1 , Stephen T. Bryson1 , Douglas A. Caldwell6 , Edward W. Dunham13 , Eric B. Ford7 , Jonathan J. Fortney3 , Thomas N. Gautier III8 , Matthew J. Holman2 , Jon M. Jenkins1,6 , David G. Koch1 , David W. Latham3 , Geoﬀrey W. Marcy9 , Robert Morehead7 , Jason Rowe6 , Elisa V. Quintana6 , Dimitar Sasselov2 , Avi Shporer10,11 , Jason H. Steﬀen12 Jack.Lissauer@nasa.gov ABSTRACT Borucki et al. (2011a) [ApJ submitted, hereafter B11] report on characteristics of over 1200 candidate transiting planets orbiting nearly 1000 Kepler spacecraft target stars detected in the ﬁrst four months of spacecraft data. Included among these tar- gets are 115 targets with two transiting planet candidates, 45 targets with three, 8 with four, and one each with ﬁve and six sets of transit signatures. We character- ize herein the dynamical properties of these candidate multi-planet systems. We ﬁnd that virtually all systems are stable, as tested by numerical integration assuming a mass-radius relationship. The distribution of observed period ratios is also clustered 1 NASA Ames Research Center, Moﬀett Field, CA, 94035, USA 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 3 Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA 95064, USA 4 Hubble Fellow 5 Department of Physics and Astronomy, San Jose State University, San Jose, CA 95192, USA 6 SETI Institute/NASA Ames Research Center, Moﬀett Field, CA, 94035, USA 7 University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611, USA 8 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA 9 Astronomy Department, UC Berkeley, Berkeley, CA 94720 10 Las Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102, Santa Barbara, CA 93117, USA 11 Department of Physics, Broida Hall, University of California, Santa Barbara, CA 93106, USA 12 Fermilab Center for Particle Astrophysics, P.O. Box 500, MS 127, Batavia, IL 60510 13 Lowell Observatory, 1400 W. Mars Hill Road, Flagstaﬀ, AZ 86001, USA
2. –2– just outside resonances, particularly the 2:1 resonance. Neither of these characteristics would emerge if the systems were signiﬁcantly contaminated with false positives, and these combined with other considerations strongly suggest that the majority of these multi-candidate systems are true planetary systems. Using the observed multiplicity frequencies (e.g., the ratio of three-candidate systems to two-candidate systems), we ﬁnd that a single population that matches the higher multiplicities underpredicts the number of singly-transiting systems. We provide constraints on the true multiplicity and mutual inclination distribution of the multi-candidate systems, which includes a population of systems with multiple small planets with low ( 5◦ ) inclinations. In all, multi-transiting systems from Kepler provide a large and rich dataset which can be used to powerfully test theoretical predictions of the formation and evolution of plan- etary systems. [This is a preliminary draft; some numbers may change slightly in the submitted version.] Subject headings: celestial mechanics – planetary systems 1. Introduction Kepler is a 0.95 m aperture space telescope using transit photometry to determine the fre-quency and characteristics of planets and planetary systems (Borucki et al. 2010; Koch et al. 2010;Jenkins et al. 2010; Caldwell et al. 2010). Besides searching for small planets in the habitable zone,Kepler ’s ultra-precise long-duration photometry is also ideal for detecting systems with multipletransiting planets. Borucki et al. (2011b) presented, for the ﬁrst time, 5 Kepler systems withmultiple transiting planet candidates, and these candidate systems were analyzed by Steﬀen et al.(2010). A system of three planets (Kepler-9=KOI-377, Holman et al. 2010; Torres et al. 2011) wasfound, in which a resonant eﬀect on the transit times allowed Kepler data to conﬁrm two of theplanets and characterize the dynamics of the system. A compact system of six transiting planets(Kepler-11, Lissauer et al. 2011), ﬁve of which were conﬁrmed by their mutual gravitational inter-actions exhibited through transit timing variations (TTVs), has also been reported. These systemswere highly anticipated because of their unique value in understanding the formation and evolutionof planetary systems Koch et al. (1996); Agol et al. (2005); Holman & Murray (2005); Fabrycky(2009); Holman (2010); Ragozzine & Holman (2010). A catalog of all candidate planets evident in the ﬁrst four and a half months of Kepler datais presented in B11. They also summarize the observational data and follow-up programs beingconducted to conﬁrm and characterize planetary candidates. Each object in this catalog is assigneda vetting ﬂag (reliability score), with 1 signifying a conﬁrmed or validated planet (>98% conﬁdence),2 being a strong candidate (> 80% conﬁdence), 3 being a somewhat weaker candidate (> 60%conﬁdence), and 4 signifying a candidate that has not received ground-based follow-up (this samplealso has an expected reliability of >60%). For most of the remainder of the paper, we refer to all
3. –3–of these objects as “planets”, although 99% remain unvalidated and thus should more properly betermed planet candidates. Ragozzine & Holman (2010) and Lissauer et al. (2011) point out thatsystems with multiple planet candidates are less likely to be due to astrophysical false positivesthan stars with single candidates, although the possibility that one of the candidate signals is dueto, e.g., a background eclipsing binary, is still non-negligible. Here we only examine the candidatesthat are known to be in multiple systems; many of the apparently single systems are likely to haveadditional planets that escape detection by being too small, too long-period, or too highly inclinedwith respect to the line of sight. Ford et al. (2011, in preparation) discuss candidates exhibitingTTVs that may be caused by non-transiting planets. B11 list ﬁve dozen planetary candidates withradii Rp > 15 R⊕ (Earth radii), but none of these are in multiple candidate systems. Lathamet al. (2011, in preparation) compare the radius distribution of planets in multi-planet systemswith those of planets which don’t have observed transiting companions. We examine herein thedynamical aspects of Kepler’s multi-planet systems. Ragozzine & Holman (2010) discussed in detail the value of systems with multiple transitingplanets. Besides their higher likelihood of being true planetary systems, multi-transiting systemsprovide numerous other insights that are diﬃcult or impossible to gain from single-transiting sys-tems. These systems harness the power of radius measurements from transit photometry in combi-nation with the illuminating properties of multi-planet orbital architecture. Observable interactionsbetween planets in these systems, seen in transit timing variations (TTVs), are much easier to char-acterize, and our knowledge of both stellar and planetary parameters is improved. The true mutualinclinations between planets in these systems, while not directly observable from the light curvesthemselves, are much easier to obtain than in other systems, both individually and statistically (seeSection 6 and Lissauer et al. 2011). The distributions of planets in these systems, both in termsof physical and orbital parameters, are invaluable for comparative planetology and provide uniqueand invaluable insights into the processes of planet formation and evolution. It is thus very exciting that B11 report 115 doubles, 45 triples, 8 quads, 1 quintuple and 1sextuple transiting system. While very few of these systems contain conﬁrmed planets, it is verylikely that this set of multi-transiting candidates contains more than twice as many as planets asthose known in the pre-Kepler era. We begin by summarizing the characteristics of Kepler’s multi-planet systems in Section 2. Thereliability of the data set is discussed in Section 3. Sections 4 – 6 present our primary statisticalresults: Section 4 presents an analysis of the long-term dynamical stability of candidate planetarysystems for nominal estimates of planetary masses and unobserved orbital parameters. Evidence oforbital commensurabilities in among planet candidates is analyzed in Section 5. The characteristicdistributions of numbers of planets per system and mutual inclinations of planetary orbits arediscussed in Section 6. We compare the Kepler-11 planetary system (Lissauer et al. 2011) to otherKepler candidates in Section 7. We conclude with a discussion for the implications of these dataand summarize the most important results presented herein.
4. –4– 2. Characteristics of Kepler Multi-Planet Systems The light curves derived from Kepler data can be used to measure the radii and orbital periodsof planets. Periods are measured to high accuracy. The ratios of planetary radii to stellar radiiare also measured quite precisely, except for low signal to noise transits (e.g., planets much smallerthan their stars, those for which few transits are observed, and planets orbiting faint or variablestars). Besides errors in estimated stellar radii, in some cases the planet’s radius is underestimatedbecause a signiﬁcant fraction of the ﬂux in the target aperture is provided by a star other than theone being transited by the planet. This extra light can often be determined and corrected for byanalyzing displacements of the centroid of light from the target between in-transit and out-of-transitobservations (Bryson et al. 2010), high-resolution imaging and/or spectroscopic studies conductedfrom the ground , although dilution by faint physically bound stars is diﬃcult to detect (Torres et al.2011). Spectroscopic studies can and in the future will be used to estimate the stellar radii, therebyallowing for improved estimates of the planetary radii, and also the stellar masses. We list the key observed properties of Kepler two-planet systems in Table 1, three-planetsystems in Table 2, four-planet systems in Table 3, and those systems with more than four planetsin Table 4. Planet indexes given in these tables signify orbit order, with 1 being the planet with theshortest period. These do not always correspond to the post-decimal point portion (.01, .02, etc.)of the KOI number designation of these candidates given in B11, for which the numbers signify theorder in which the candidates were identiﬁed. These tables are laid out diﬀerently from one anotherbecause of the diﬀerence in parameters that are important for systems with diﬀering numbers ofplanets. In addition to directly observed properties, these tables contain results of the dynamicalanalyses discussed below. We plot the period and radius of each active Kepler planetary candidate observed duringthe ﬁrst four months of spacecraft operations, with special emphasis on multi-planet systems,in Figure 1. The ratio(s) of orbital periods is a key factor in planetary system dynamics; thedistribution of these period ratios is plotted in Figure 3. Figure 4 shows a cumulative distributionof the period ratio. To check whether the observed transiting planets are randomly distributed among host starswe compared the observed distribution of planets per system to a statistically random Poissondistribution, assuming there are 1229 planets orbiting some of the 150,000 stars Kepler monitors,i.e., a Poisson distribution with a mean of 1229/150,000≈ 0.008. Such a Poisson distribution givesan expectation of 1219 single planet systems, 5.0 doubles, 0.014 triples, 2.8 × 10−5 systems withfour planets and above. For single planet systems there are fewer observed systems than expectedfrom a random distribution, but, for systems with two and more planets there are far more systemsdetected by Kepler than predicted randomly. This shows that transiting planets tend to “come inpacks” or cluster around stars, meaning that once a planet is detected around a given star, that staris more likely to be orbited by an additional planet than a star with no detected planets. Some ofthe observed clustering is attributable to SNR considerations, with stellar brightness, size, intrinsic
5. –5–variability, multiplicity, crowdedness of ﬁeld, and position on the Kepler detectors all contributing.Quantifying the eﬀects of SNR variations is beyond the scope of this paper, but they are clearly fartoo small to account for the bulk of the observed clustering. Given the bias of the transit methodto detect planets on edge-on orbits, a possible interpretation of the clustering of transiting planetsis that planetary systems (or at least some fraction of them) tend to be coplanar, like the SolarSystem. It is diﬃcult to estimate the corresponding distribution for planets detected through radialvelocity (RV) surveys, since we do not know how many stars were spectroscopically surveyed forplanets for which none were found. However, a search through the Exoplanet Orbit Database1(Wright et al. 2010) shows that 17% of the planetary systems detected by RV surveys are multi-planet systems (have at least two planets), compared to 21% for Kepler. Including a linear RVtrend as evidence for an additional planetary companion, Wright et al. (2009) show that at least28% of known RV planetary systems are multiple. Therefore, the clustering observed for transitingplanets detected by Kepler is likely to occur also for RV planets, and maybe even at a higherfraction, which is expected given the bias of the transit detection method. 3. Reliability of the Sample Very few of the planetary candidates presented herein have been validated as true exoplan-ets. As discussed in B11, the overwhelming majority of false positives are expected to come fromeclipsing binary stars. Some of these will be grazing eclipses of the target star, others eclipses offainter stars (physically associated to the target or background/faint foreground objects) within thetarget aperture. Similarly, transits of fainter stars by large planets may also mimic small planetstransiting the target star. The vast majority of all active Kepler planet candidates were identiﬁedusing automated programs that do not discriminate between single candidates and systems of mul-tiple planets. The 1% of single and multiple candidate systems that were identiﬁed in subjectivevisual searches are too small to aﬀect our results in any statistically signiﬁcant manner, and eventhe signs of such changes are unclear. The fact that planetary candidates are clustered around targets in the sense that there are farmore targets with more than one candidate than would be expected for randomly distributed can-didates (section 2) suggests that the reliability of the multi-candidate systems is likely to be higherthan that of the single candidate targets. We know from radial velocity observations that planetsfrequently come in multiple systems (Wright et al. 2009), whereas astrophysical false positives arenearly random. Nonetheless, the presence of multiple candidates increases our conﬁdence that one or both areplanets is real since the probability of two false positive signals is the product of the probabilities 1 exoplanet.org, as of January 2011.
6. –6–of two relatively rare cases, signiﬁcantly lowering the overall probability than none of the signalsis due to planets Ragozzine & Holman (2010). (Triple star systems would be dynamically unstablefor most of the orbital period ratios observed. Their dynamics would also give rise to very largeeclipse timing variations — e.g., Carter et al. 2011 — which are generally not seen.) Additionally,the concentration of candidate pairings with period ratio near 1.5 and 2 (ﬁg. 3) suggest that thesesubsamples likely have an even larger fraction of true planets, since such concentrations would notbe seen for random eclipsing binaries. However, at present these qualitative factors have not beenquantiﬁed. Strong TTV signals are no more common for planets in observed multi-transiting candidatesthan for single candidates (Ford et al. 2011, in preparation), suggesting that large TTVs areoften the result of stellar systems masquerading as planetary systems (Fabrycky et al. 2011, inpreparation, Steﬀen et al., 2011, in preparation). Nonetheless, it is possible that weaker TTVscorrelate with multi-transiting systems, but these require more data and very careful study toreveal (e.g., those in the Kepler-11 planetary system, Lissauer et al. 2011). The ratio of durations given the observed periods of candidates in multi-transiting systemscan be used to potentially identify false positives (generated by a blend of two objects eclipsingtwo stars of diﬀerent densities) or as a probe of the eccentricity and inclination distributions. Ananalysis of the corrected duration ratio as in (?) shows that essentially all pairs of candidate havethe expected duration ratio if they were both orbiting the same star. The consistency of durationratios is not a strong constraint, but the observed systems also pass this test. 4. Long-term Stability of Planetary Systems Kepler measures planetary sizes, whereas masses are the key parameter for dynamical studies.We convert sizes to masses using the following simple formula: 2.06 Mp = Rp (1)where the planet’s mass, Mp , is expressed in Earth masses and its radius, Rp , in units of Earth’smean radius. The power-law of equation 1 is ﬁt to Earth and Saturn; it slightly overestimates themass of Uranus (17.2 M⊕ vs. 14.5 M⊕ ) and slightly underestimates the mass of Neptune (16.2 M⊕vs. 17.1 M⊕ ). (Note that Uranus is larger, but Neptune is more massive, so ﬁtting both with amonotonic mass-radius relation is not possible.) A convenient metric for the dynamical proximity of planets is the separation of their orbitalsemi-major axes, ai+1 − ai , measured in units of their mutual Hill sphere radius mi + mi+1 1/3 (ai + ai+1 ) RHi,i+1 = , (2) 3M⋆ 2where the index i = 1, N −1, with N being the number of planets in the system under consideration,mi and ai are the planetary masses and semi-major axes, and M⋆ is the central star’s mass.
7. –7– The dynamics of two-planet systems are a special case of the 3-body problem that is amenableto analytic treatment and simple numerically-derived scaling formulas. For instance, a pair ofplanets initially on coplanar circular orbits with orbital separation √ ∆ ≡ (ao − ai )/RH > 2 3 ≈ 3.46 (3)cannot develop crossing orbits later, and are thus called “Hill stable” (Gladman 1993). Orbitalseparations of all planet pairs in two planet systems for stellar masses given in B11 and planetarymasses given by Equation 1 are listed in Table 1. We see that they all obey this criterion. The actual dynamical stability of these systems also depends on their eccentricity and mutualinclination (Veras & Armitage 2004), neither of which can be measured well from the transit dataalone. Circular coplanar orbits have the lowest angular momentum deﬁcit (AMD), and thus arethe most stable, except for protective resonances that produce libration of angles and prevent closeapproaches; a second exception exists for retrograde orbits (Smith & Lissauer 2009), but these areimplausible on cosmogonic grounds (Lissauer & Slartibartfast 2008). Hence, the Kepler data canonly be used to suggest that a pair of planets may be potentially unstable, particularly if theirsemi-major axes are closer than the resonance overlap limit (Wisdom 1980; Malhotra 1998): aouter − ainner Mp 2/7 2 < 1.5 . (4) ainner + aouter M⋆(This is derived only for one massive planet; the other is massless.) Conversely, if it is assumedthat both candidates are true planets, limits on the eccentricities and mutual inclinations can beinferred from the requirement that the planetary systems are presumably stable for the age of thesystem, which is generally of order 1011 times longer than the orbital timescale for the short-periodplanets that represent the bulk of Kepler ’s candidates. However, we may turn the observationalproblem around: the ensemble of duration measurements can be used to address the eccentricitydistribution (Ford et al. 2008); since multiplanet systems have eccentricities constrained by stabilityrequirements, they can be used as a check to those results (Moorhead et al. 2011, in preparation). Systems with more than two planets have additional dynamically complexity, but are veryunlikely to be stable unless all neighboring pairs of planets satisfy equation 3. Nominal separationsbetween neighboring planet pairs in three planet systems are listed in Table 2, the separations in fourplanet systems are listed in Table 3, and those for the 5 and 6 planet systems in Table 4. We ﬁnd thatin these cases, equation 3 is satisﬁed by quite a wide factor. From suites of numerical integrationsSmith & Lissauer (2009), dynamical survival of systems with more than three comparably-spacedplanets for at least 1010 orbits has been shown only if the relative spacing between orbital semi-major axes exceeds a critical number (∆crit 9) of mutual Hill-spheres. For each adjacent setof 3 planets in our sample, we plot the ∆ separations of both the inner and the outer pairs inFigure 9. We ﬁnd that in some cases ∆ < ∆crit , but the other separation is quite a bit bigger than∆crit . In our simulations of a population of systems in section 6, we impose the stability boundary∆max + ∆min > 18, which accounts for this possibility, in addition to requiring that inequality 3 issatisﬁed.
8. –8– We investigated long-term stability of all 55 systems with three or more planets using the hybridintegrator within the Mercury package (Chambers 1999), run on the supercomputer Pleiades atUniversity of California, Santa Cruz. We set the switchover at 3 Hill radii, but in practice weaborted simulations that violated this limit, so for the bulk of the simulation the Wisdom-Holmann-body mapping (Wisdom & Holman 1991) was used, with a time step of 0.05 times as large as theorbital period of the innermost planet. The simplest implementation (Nobili & Roxburgh 1986) ofgeneral relativistic precession was used, an additional potential UGR = −3(GM⋆ /cr)2 , where G isNewton’s constant, c is the speed of light, and r is the instantaneous distance from the star. Moresophisticated treatments of general relativity (Saha & Tremaine 1994) are not yet required, due tothe uncertainties of the masses of the planets and stars whose dynamics are being modeled. Weneglected precession due to tides or rotational ﬂattening, which are only signiﬁcant compared togeneral relativity for Jupiter-size planets in few-day orbits (Ragozzine & Wolf 2009). The stellarmass was derived from the R⋆ and log g estimates from color photometry, as given by B11. Weassumed initially circular and coplanar orbits that matched the observed periods and phases, andchose the nominal masses of equation 1. For the triples and quadruples, we ran these integrations for 4 × 109 orbital periods of theinnermost planet, and found the nominal system to be stable for this span in nearly all cases; thethree exceptions are described below. An integration of the ﬁve-planet system KOI-500 has run2 × 109 days (the inner planet’s period is 0.99 day), suggesting stability for astronomically rele-vant timescales is possible. However, the short orbital periods and high planet number means thisconclusion needs to be revisited in future work, especially once masses are estimated observation-ally. (We consider the near-resonances of KOI-500 in section 5.5.) The most populous transitingplanetary system discovered so far is the six-planet system Kepler-11. In the discovery paper,Lissauer et al. (2011) reported a circular model, with transit time variations yielding an estimateof the masses of the ﬁve inner planets. They also proposed two more models with planets on mod-erate eccentricities, and slightly diﬀerent masses, which also ﬁt the transit timing signals. Now,both those eccentric solutions have become unstable, at 610 Myr for the all-eccentric ﬁt, and at427 Myr for an integration that was a very slight oﬀset from the b/c-eccentric ﬁt. The all-circularﬁt remains stable at 640 Myr, so we conclude that the ﬁts of that paper are physically plausible,but that introducing a small amount of eccentricity can compromise stability. Future work shouldaddress how high the eccentricities could be, while requiring stability for several Gyr. Only three systems became unstable at the nominal masses, KOI-284, KOI-191, and KOI-730;the ﬁrst is described next, the latter two are described in section 5. KOI-284 has a pair of candidateswith a period ratio 1.0383. Both planets would need to have masses about that of the Earth orsmaller for the system to satisfy equation 3 and thus be Hill stable. But the vetting ﬂags are 3for both these candidates — meaning that they are not good candidates anyway — and the datadisplay signiﬁcant correlated noise which may be responsible for these detections. We expect oneor both of these candidates are not planets, or if they are, they are not orbiting the same star. In our investigation, we also were able to correct a poorly ﬁt radius of 3.2 RJup (accompanied
9. –9–by a grazing impact parameter in the original model, trying to account for a “V” shape) for KOI-1426, which resulted in too large a nominal mass, driving the system unstable. It is remarkablethat stability considerations allowed us to identify a poorly conditioned light curve ﬁt. Once thecorrection was made to 1.2 RJup , the nominal mass allowed the system to be stable. That so many of the systems passed basic stability measurements suggests that there are nota large number of planets that with substantially higher densities than given by the simple formulaof equation (1). If these candidates are assumed to be orbiting the same stars, stability constraintscould potentially be used to place upper limits on the masses and densities. 5. Resonances 5.1. Resonance Abundance Analysis The ratio of periods of multi-transiting candidates in the same system is signiﬁcantly diﬀerentfrom ratios obtained by the ratio of randomly selected periods from all 408 multi-transiting can-didates. Multiply-transiting planets provide better data for the measurement of resonances thanRV planetary systems for two reasons. First, periods are measured to very high accuracy, evenfor systems with low SNR, so period ratios can be conﬁdently determined without the harmonicambiguities that aﬀect RV detections (Anglada-Escud´ et al. 2010; Dawson & Fabrycky 2010). Sec- eond, resonances signiﬁcantly enhance the signal of transit timing variations (Agol et al. 2005;Holman & Murray 2005; Veras & Ford 2010), which can also be measured accurately, either todemonstrate resonance occupation (as will be possible with the Kepler-9 system with future data;Holman et al. 2010 or exclude it with conﬁdence. Planets can be librating in a resonance even when they have apparent periods that are notperfectly commensurate (Ragozzine & Holman 2010). Nevertheless, the period ratios of small tran-siting resonant planets will generally be within a few percent of commensurability, unless masses oreccentricities are relatively high. In the cumulative distribution of period ratios (Figure 3), we seethat few planets appear to be directly in the resonance (with the important exception of KOI-730). In the solar system, there is a known excess of satellite pairs near mean-motion resonance(MMR) (Goldreich 1965) and theoretical models of planet formation and migration suggest thatthis should be the case (e.g., Terquem & Papaloizou 2007). We can ask whether such an excess canbe seen in the sample of multiply transiting Kepler planetary systems. To do this, we divide theperiod ratios of the multiply transiting systems into bins which surround the ﬁrst and second-order(j:j − 1 and j:j − 2) MMRs. The boundaries between these “neighborhoods” are chosen at theintermediate, third-order MMRs. Thus, all candidate systems with period ratios less than 4:1 arein some neighborhood. For example, the neighborhood of the 3:1 MMR runs between period ratiosof 5:2 and 4:1, for the 2:1 it runs from 7:4 to 5:2. With this method the ﬁrst-order MMRs haveneighborhoods that are essentially twice as large as those for second-order MMRs.
10. – 10 – We deﬁne a statistic, ξ, that is a measure of the diﬀerence between an observed period ratioand the MMR in its neighborhood. In order to treat each neighborhood uniformly, they are scaledsuch that the ξ statistic runs from −1 to 1. For ﬁrst-order MMRs, ξ is given by 1 1 ξ1 ≡ 3 − Round (5) P −1 P−1where P is the observed period ratio (always greater than unity) and “Round” is the standardrounding function that returns the integer nearest to its argument. Similarly, for second-orderMMRs, ξ is given by 2 2 ξ2 ≡ 3 − Round . (6) P −1 P−1Table 5 gives the number of period ratios taken from tables 1–4 that are found in each neighborhood.The ξ statistic as a function of the period ratio, along with the data, is shown in Figure 6. We calculate ξ for each planetary period ratio and stack the results for all of the resonancesin each MMR order. The resulting distribution in ξ is compared to a randomly-generated sampledrawn from a uniform distribution in log P , a uniform distribution in P , and from a sample con-structed by taking the ratios of a random set of the observed periods in the multiple candidatesystems. This third sample has a distribution that is very similar to the logarithmic distribution.Figure 7 shows a histogram of the number of systems as a function of ξ for both the ﬁrst andsecond-order MMRs. We see that the results for the second-order MMRs have some qualitativesimilarities to the ﬁrst-order MMR, though the peaks and valleys are less prominent. In our testsbelow we also consider the case where all ﬁrst and second-order MMRs are stacked together. To test whether the observed distributions in ξ are consistent with those from our varioustest samples, we took the absolute value of ξ for each collection of neighborhoods and used theKolmogorov-Smirnov test (KS test) to determine the probability that the observed sample is drawnfrom the same distribution as our test sample. The results of these tests are shown in Table 6.Figure 8 shows the probability density function (PDF) for the combined distribution of ﬁrst andsecond-order MMRs and the PDF for the logarithmically distributed sample. A few notable results of these tests include: 1) the combined ﬁrst and second-order neighbor-hoods are distinctly inconsistent with any of the trial period distributions, 2) the neighborhoodssurrounding the ﬁrst-order MMRs are also very unlikely to originate from the test distributions, 3)there is signiﬁcant evidence that the neighborhood surrounding the 2:1 MMR alone is inconsistentwith the test distributions, and 4) there is a hint that second-order resonances are not be consistentwith the test distributions, particularly the distribution that is uniform in P . Additional data arenecessary to produce higher signiﬁcance results for the second-order resonances. Given the distinct diﬀerences between our test distributions and the observed distribution,a careful look at the histograms shown in Figure 7 shows that the most common location for apair of planets to reside is slightly exterior to the MMR (the planets are farther apart than theresonance location) regardless of whether the resonance is ﬁrst or second-order. Also, a slight
11. – 11 –majority (just over 60%) of the period ratios have ξ statistics between -1/2 and +1/2 while allof the test distributions have roughly 50% between these two values in ξ (note that ξ = ±1/2corresponds to sixth-order resonances in ﬁrst-order neighborhoods and twelfth-order resonances insecond-order neighborhoods—none of which are likely to be strong in these planet pairs thoughthey may play some minor role). There are very few examples of systems that lie slightly closer toeach other than the ﬁrst-order resonances and for the second-order resonances only the 3:1 MMRhas systems just interior to it. Finally, there is a hint that the planets near the 2:1 MMR have arange of orbital periods, while those near the 3:2 appear to have shorter periods and smaller radiion average. While additional data are necessary to claim these statements with high conﬁdence, theobservations are nonetheless interesting and merit some investigation to determine the mechanismsthat might produce such distributions. A preference for resonant period ratios cannot be exhibited by systems without direct dynam-ical interaction. Thus, the preference for periods near resonances in our dataset is statisticallysigniﬁcant evidence that many of the systems are actually systems of three or more planets. (Starswith such close orbits are not dynamically stable, thus dynamical interaction between objects withperiod ratios 5 indicates low masses due to planets.) Assessing the probability of dynamicalproximity to resonance for the purpose of eliminating the false positive hypothesis in any partic-ular system requires a speciﬁc investigation; here, we can only say that statistically many of thenear-resonant systems appear to be planetary in nature. The excess of pairs of planets wider than nominal resonances was predicted by (Terquem & Papaloizou2007), who pointed out that tidal interactions with the star will, under some circumstances, breakresonances and will more strongly aﬀect the inner planets. Interestingly KOI-730 seems to havemaintained stability and resonance occupation despite diﬀerential tidal evolution. 5.2. Frequency of Resonant Systems Doppler planet searches suggest that roughly one third of multiple planet systems well-characterizedby Doppler observations are near a low-order period commensurability with one sixth near the 2:1MMR (Wright et al. 2011). However, an approximate degeneracy with a single planet on an eccen-tric orbit makes it diﬃcult for Doppler observations to detect a low-mass planet in an interior 2:1MMR (Giuppone et al. 2009; Anglada-Escud´ et al. 2010). Among Doppler-discovered systems, eonly one quarter of the pairs near the 2:1 MMR have an inner planet signiﬁcantly less massivethan the outer planet. The two exceptions (GJ 876 b&c, µ Ara e&b) beneﬁtted from an unusuallylarge (≥ 100) number of Doppler observations. Thus, it is possible that the true rate of planetarysystems near the 2:1 MMR is signiﬁcantly greater than one sixth. Kepler observations ﬁnd that at least ∼ 16% of multiple transiting planet candidate systemscontain at least one pair of transiting planets close to a 2:1 period commensurability (1.83 ≤ Pout /Pin ≤2.18). Even using our narrow range of period ratios, the true rate of systems near a 2:1 period
12. – 12 –commensurability could be signiﬁcantly greater than ∼ 16%, since not all planets will transit. Ifwe assume that planets near the 2:1 MMR are in the low inclination regime, then the outer planetshould transit ≃ 63% of the time, implying that the true rate of detectable planets near the 2:1MMR (if both had a favorable inclination) is at least ∼ 25%. If a signiﬁcant fraction of thesesystems are not in the low inclination regime, then the true rate of pairs of planets near the 2:1MMR would be even larger. This mildly suggests that radial velocity surveys could be missing some2:1 pairs due to observational degeneracy. To consider this possibility further requires investigatingthe expected mass ratios of planets near the 2:1. Based on the B11 catalog, neighboring transiting planet candidates tend to have similar radii,and in the majority of cases, the outer planet is slightly larger. This is consistent with the typicaloutcome of planet formation models that predict more distant planets can more easily grow tolarger sizes. The distribution of planetary radii ratios (Rp,out /Rp,in ) for neighboring pairs of transitingplanet candidates near a 2:1 period commensurability is concentrated between 0.9 and 1.25 witha tail extending to larger ratios (see Fig. 10, solid curve). For reasonable assumptions of a mass-radius ratio (Eqn. 1), this implies that most neighboring transiting planet candidate systems havemasses within 20% of each other. Fortunately, such pairs are not signiﬁcantly aﬀected by the radialvelocity observational degeneracy with a single eccentric planet. However, the tail of ∼ 20% ofsystems with a radii ratio exceeding 1.7 are more problematic, as the Doppler signatures of woulddiﬀer from that of a single eccentric planet by less than 20% of the velocity amplitude of the outerplanet. Based on the planet radii ratio distribution from Kepler, we estimate that the abundance ofneighboring planets near the 2:1 MMR could be ∼ 20% greater than suggested by Doppler surveysdue to the diﬃculty in distinguishing the Doppler signature of such systems from a single eccentricplanet. A more precise determination of the frequency of resonant or near-resonant systems basedon Kepler and/or Doppler surveys is left for future work. 5.3. KOI-730: A Multi-Resonant Candidate System While few nearly exact mean motion resonances are evident in the sample of Kepler planetarycandidates, one system stands out as exceptional: the periods of the four candidates in KOI-730 satisfy the ratio 6:4:4:3 to ∼ 1 part in 1000 or better. This system is the ﬁrst to showevidence for extrasolar “Trojan” or co-orbital planets, which have been suggested to be theoreticallypossible (Laughlin & Chambers 2002; Go´dziewski & Konacki 2006; Smith & Lissauer 2010). In zKepler data, the two co-orbital candidates began separated by ∼118◦ , with the trailing candidatereducing the gap at the rate of ∼1◦ per month. The numerical integration (section 4) matchingthe periods and phases in B11, circular orbits, and nominal masses went unstable at 25 Myr,precipitated by a close encounter between the coorbital planets.
13. – 13 – This system will be diﬃcult to study because of the faintness of the target star (Kepler magni-tude 15.34) and its location within the ﬁeld which implies that it cannot be observed by Kepler dur-ing winter quarters (Q4, Q8, ...). Nonetheless, the remarkably commensurate period ratios of thesefour candidates give us strong conﬁdence that they all will eventually be conﬁrmed as planets. Thedynamics of this class of planetary systems are analyzed by Fabrycky et al. (2011; in preparation),whose results further strengthen our conﬁdence in these candidates. 5.4. KOI-191: A System Protected by Resonance? Two of the planets in the KOI-191 system have a period ratio of 1.258. One of those planetshas a ∼ 1RJup size. Its mass would need to be 17M⊙ to satisfy equation 3. Alternatively, thetwo planets could be locked in 5:4 mean-motion resonance that prevents close approaches even iforbits cross. Neptune and Pluto are locked in 3:2 mean-motion resonance of this type. Relativelyfast precession, on the order of 100 orbital periods, would be needed as well, as the observed periodratio is signiﬁcantly diﬀerent from the nominal resonance location. 5.5. KOI-500: (Near-)Resonant Five-Candidate System KOI-500 is a 5-candidate system with periods 0.986779 d, 3.072166 d, 4.645353 d, 7.053478d, and 9.521696 d. Neighboring pairs of the outer four of these planets are all near two-bodymean motion resonances, but unless there is an unexpectedly large amount of apse precession,they do not appear to be locked in librating two-body resonances. Perhaps this is a consequenceof diverging tidal evolution, as predicted by Papaloizou & Terquem (2010). The three adjacentperiod spacings between these four planets contribute to the statistics of period ratios just wide ofresonance (negative ξ). Moreover, the spacing of these systems from exact resonance, Poﬀset = 1/(1/Pout − 1/Pin ) (7)is equal to ∼ 190 days in all cases. The combination of mean motions 2n2 − 5n3 + 3n4 ≈ 1.6 × 10−5and 2n3 − 6n4 + 4n5 ≈ 1.3 × 10−5 are so small that we suspect two 3-body Laplace-like resonancesand/or a 4-planet resonance may be active and controlling the dynamics of this system. 6. Coplanarity of Planetary Systems The orbits of the planets in our Solar System lie close to the same plane. The distributionof planetary orbits around the invariable plane of the Solar System has a Rayleigh distributionwidth of σi ≃ 1.5◦ ; if Mercury is excluded, then σi ≃ 1◦ . The strong level of co-planarity has beenrecognized as an important constraint on models of planet formation for over 250 years (Kant 1755;Laplace 1796) and our current understanding is that growth within a dissipative protoplanetary
14. – 14 –disk generally yields circular orbits and low relative inclinations (e.g., Safronov 1969; Lissauer1993). Some mechanisms for explaining the higher eccentricity distribution of large exoplanetsare likely to increase inclinations as well, such as planet-planet scattering (e.g., Chatterjee et al.2008; Juri´ & Tremaine 2008). Whatever the mechanism, it is clear that measuring the inclination cdistribution of typical planetary systems will create an important test for planet formation theories. The interplay between eﬀects due to the protoplanetary disk, distant perturbers, and planet-planet scattering will leave a unique ﬁngerprint in the coupled eccentricity-inclination distributionof exoplanetary systems and the exoplanet inclination distribution is thus of immense value. Yetneither transit observations, nor the radial velocity technique can directly measure the true mutualinclination between planets observed in multiple systems, except in unusually fortuitous circum-stances. This continues to be true in multi-transiting systems: even though the inclinations tothe line of sight of all transiting planets must be small, the orbits could be rotated around theline of sight and mutually inclined to each other. Indirect constraints, however, can be obtainedfor systems with multiple transiting planets that make them the best probe of mutual inclinationsaround main sequence stars. For example, the lack of transit duration variations of the planets ofKepler-9 (Holman et al. 2010) and Kepler-11 (Lissauer et al. 2011) already place interesting lim-its on mutual inclinations to be 10◦ , while the inclinations to the line-of-sight of the Kepler-10planets (Batalha et al. 2011, in prep.) require a mutual inclination 5◦ . Furthermore, given the large number of multi-candidate systems identiﬁed in B11, we canapproach the question of the inclination distribution of the population statistically. The moremutually inclined a given pair of planets is, the smaller the probability that multiple planetstransit (Ragozzine & Holman 2010), and thus the smaller the fraction of multi-planet systems thatwe would expect to see. Therefore, an investigation of coplanarity requires also addressing thenumber of planets expected per planetary system. Of course, the question of planetary multiplicityis also inherently interesting to investigate. Here we attempt to provide a self-consistent rigorousestimate of the multiplicity and coplanarity of typical systems using a statistical approach. 6.1. Simulated Population Model 6.1.1. Model Background The number of observed transiting planets per star is clearly aﬀected by multiple observationalbiases, most notably that the probability of transiting decreases with increasing orbital period.With the current observations, we have little to no insight on the non-transiting planets in thesesystems, though eventually investigations of transit timing and duration variations (or the lackthereof) will be able to place additional limits on such planets (Steﬀen et al. 2010, Ford et al. inpreparation). If we knew about the presence of every planet, transiting or not, then the problemof determining the typical inclination would be much easier.
15. – 15 – A useful way to address the problem of non-detections is to create a forward model of transitingplanet detections based on a minimal number of assumptions and a relatively small number oftunable parameters and to compare the output with the properties of the observed systems. Themodel described here attempts use Kepler ’s observed frequency of planet multiplicity to determinetypical values for the true multiplicity of planetary systems as well as the mutual inclination betweenplanets in the same system. After choosing a distribution for the number of planets assigned to arandom Kepler star (characterized by a parameter Np ) and their relative inclinations (characterizedby σi ), the model computes how many of these planets would be detected, requiring a geometricalalignment that leads to transit and a signal-to-noise ratio (SNR) large enough to be detected. Thisprocess is done for over 100000 planetary systems and the results compared to frequency of observedmulti-transiting systems. Though imperfections in the model and observations will only provide anapproximate answer, Kepler enables for the ﬁrst serious attempt at using observations to measurethe value of Np and σi for a large number of systems. To isolate the true multiplicity and inclination distributions, other aspects of the simulatedsystems (such as radius and period distributions) must be consistent with the true underlyingdistribution. This requires some interpretation of the observed distribution and the identiﬁcationof potential biases. At the end of Kepler ’s mission, a simulated population that attempts to matchall the observations could be employed; here, we hold the period and radius distributions ﬁxed andthey are pre-selected in a way that the output of the simulation will closely match the observeddistribution. A good example of this process is the choice of the period distribution. The diﬀerence betweenthe observed and true underlying period distribution is almost entirely determined by the geometricbias of observing planets in transit that scales as 1/a ∝ P −2/3 . As discussed in B11 and conﬁrmed byour investigations, after correcting for the geometric bias and discarding very short periods (below3 days) and large planets (above 6 Earth radii), the remaining distribution is very well describedby a uniform distribution in log(P ). Hence, our simulator uses a period distribution uniform inlog(P ) for a pre-speciﬁed range of periods: 3-125 days. An improvement could be placed on thisprocess that takes into account the observed relationship (generally expected from planet formationtheories), that larger period planets have larger radii (Figure 10), but this covariance is not includedin the simulated planetary systems discussed below. Estimating the true radius distribution of Kepler planets is more complicated. A simple his-togram of planet radii shows that above ∼2 Earth radii, the observed population follows approxi-mately a 1/r 2 distribution (B10, B11). Below ∼1.5-2 Earth radii, the number of planets decreasessigniﬁcantly, which is at least partly due to small planets escaping detection due to low SNR. The1/r 2 distribution is very sensitive to the minimum allowed radius; to avoid this, we draw radii fromthe distribution of observed planetary radii around stars with Kepler magnitude greater than 14and through-Q5 SNR (from Table 1 in B11) greater than 16. A histogram of this distribution, andthe full distribution of all detected planets, is shown in Figure 11. In particular, our simulatedplanets have radii drawn from this distribution between the limits of 1.5 and 6 Earth radii; except
16. – 16 –at the smallest radii, this nearly approximates the 1/r 2 distribution observed in B11. Finally, an inspection of the B11 SNR ratios (which are calculated from Quarters 1-5) suggestthat completeness eﬀects begin to dominate below a SNR of ∼16, corresponding to a SNR of 9.2from Quarters 1-2. We require that our simulated planets match this SNR in order to be detectable.We did not consider the FOP vetting ﬂag in our choice of systems as a large fraction of the multipleshave not yet been investigated in detail. Applying these same cuts to Kepler ’s observed multi-candidate systems from B11 results in437, 63, 17, 1, and 1 systems with 1, 2, 3, 4, and 5 transiting planets, respectively. It is possiblethat some small fraction of the observed planets could be false positives and our simulation doesnot attempt to account for this. Note that the sixth planet of Kepler-11, planet g, had a transitthat fell into the data gap between Quarters 1 and 2, so this system counts as a quintuple forcomparison to the simulated population (a minor diﬀerence that does not aﬀect our results). 6.1.2. Model Description Under the constraints described above, we want to create simulated planetary populations thatwill be compared to the Kepler observations. In particular, for each simulated population, we willcompared the number of systems with i detectable transiting planets for 1 ≤ i ≤ 5 to the observeddistribution from Kepler . The model starts by taking the full stellar population from the 161836 Q2 target stars in the“EX”oplanet program (the list of which is available from MAST), with the assumed masses andradii from the Kepler Input Catalog (Latham et al. 2005, Brown et al. in preparation). Afterrandomly choosing a star, the model then assigns a number of planets, Np . Three distributionswere considered for the number of planets: a “uniform” distribution, a Poisson distribution, and anexponential distribution. In the exponential distribution, the probability of having i + 1 planets isa factor of α smaller than the probability of having i planets, with α a free parameter. This modelcreated a very poor ﬁt to the data (particularly the triples to doubles ratio) for any value of α orσi , so we do not consider it further. The “uniform” distribution parameter Np,U is the mean number of planets per system, whichis a ﬁxed number of planets per star if Np,U is an integer or a random mix of two ﬁxed values ifNp,U is not an integer. For example, if Np,U = 3.25, 75% of systems would be assigned 3 planetsand 25% would be assigned 4 planets. The Poisson distribution parameter Np,λ assigns the numberof planets per system based on a Poisson distribution with width Np,λ . The frequency of stars withplanetary systems is considered separately (see below), so we require the number of planets to benon-zero; for Np,λ 1, the distribution diﬀers somewhat from a true Poisson distribution. Once the each star is hosting a speciﬁc non-zero number of planets, the orbital and physicalproperties of these planets are assigned. As discussed above, the period of each planet is selected
17. – 17 –by drawing from a uniform distribution in log(P ) with a minimum of 3 and a maximum of 125days. Although the period ratios in multiple systems deviate from a random distribution due toclustering near resonance (Section 5), this is a minor eﬀect compared to the overall geometricbias. Each planetary radius is assigned based on the Kepler observed radii independently of orbitalperiod, as described above. As the observed multiples are almost entirely stable on long timescales (Section 4), there is adesire to impose a proximity constraint that rejects simulated multiples that are too closely packed.This will require assigned a mass to each planet as well. We follow the mass-radius relation usedabove (eq. 1), i.e., M = R2.06 with masses and radii in units of Earth masses and radii. The mutualHill separation between each pair of planets, ∆ can then be calculated as described in equation 3. In practice, the systems are built one planet at a time until all the planets assigned to this star(based on Np ) are selected. The ﬁrst planet is assigned its period, radius, and corresponding mass.Starting with the second planet, a new period, radius, and mass are assigned and ∆ calculated. Ifany two planets have ∆ < 3.46 or if ∆inner + ∆outer < 18 for any three consecutive planets, then theplanet is rejected on stability grounds and the process begins again until all the planets have beenassigned. Very rarely, over the wide parameter space we are considering, large numbers of planets( 10) are assigned which will not ﬁt within the period range and stability criteria; in this case, asmany planets as can ﬁt are given to the star. Inclinations with respect to a reference plane are drawn with a Rayleigh distribution (appro-priate for this kind of geometric problem) of width σi and random nodal longitudes are assigned.In general, we assume that eccentricities are zero, and our investigations that considered non-zeroeccentricities indicated that, statistically speaking, the observed multiplicities did not change sig-niﬁcantly, though we did not consider the possibility that eccentricities and inclinations may becorrelated. At this point, the full orbital and physical characteristics of the planetary systems have beendetermined and we can begin to assess observability. First, these planetary systems are then rotatedby a uniform random rotation matrix in a manner equivalent to choosing a random point on a spherefor the direction of the normal to the reference plane. In the new random orientation, the impactparameter of each planet is calculated. If the planets are transiting, the SNR of a single transit iscalculated assuming a box-shaped transit with depth of (Rp /R∗ )2 and duration calculated from theequation in (Kipping 2010). A random epoch is assigned and the number of transits in a 127-dayperiod (corresponding to the duration of Q1-Q2) is calculated. To account for the duty cycle of92%, each of these transits can be independently and randomly lost with 8% probability. Thetotal signal is then calculated as the signal for a single transit times the square root of the numberof observed transits. This is compared to the estimated noise over the course of the duration ofthe transit, calculated from the TMCDPP values for this star from Quarter 2 (see Kepler DataRelease Notes). As described above, we require that our simulated planets have SNR of 9.2 inthe simulated Quarters 1 and 2 in order to be detected. We ﬁnd that approximately 30-50% of
18. – 18 –simulated transiting planets are “missed” due to insuﬃcient SNR. The output of the model is the number of stars with N detectable transiting planets, for all N .The next step is a statistical comparison between a variety of simulated populations (with diﬀerentchoices for Np and σi ) with the observed distribution from Kepler . An assessment of whethera particular simulation is rejectable is done using a G-test (similar to the categorical χ2 test) bycalculating the G-statistic: G= 2Oi ln(Oi /Ei ) (8) iwhere Oi is the number of systems with i planets observed by Kepler and Ei is the expected numberof such systems from the simulated population, which is scaled so that the total values of O andE are the same. The values Oi and Ei always refer to the number of systems with i observed andexpected detectable transiting planets. Note that this statistic gives the most weight to categorieswith large numbers of objects and no weight to the categories where there are no observed orexpected systems (i.e. i > 5). The G-statistic grows as the two populations become more diﬀerent.Though the value of the G-statistic follows the χ2 distribution when O and E are large enough, itis more robust to compare the G value to a suite of G-statistics from a Monte Carlo simulation.Using the expected frequencies Ei , 100 random Monte Carlo populations are created with the trueunderlying distribution and their G-statistics are calculated. The signiﬁcance of the observed valueof G is calculated by comparing it to the Monte Carlo generated G-statistics. For example, if theobserved value of G is 10 and this is larger than 95% of the Monte Carlo values of G, then thehypothesis that the population observed by Kepler is the same as the simulated population can berejected with 95% conﬁdence. 6.2. Results and Discussion The simulated population assumes that all 161836 Q2 target stars have planetary systems.B11 report that the expected number of planets per star based on the Kepler observations is about0.4, though this describes planets of all types (unlike the limited size range we are considering)and converting this to the fraction of stars with planetary systems requires dividing by the typicalmultiplicity. Our investigation could address the frequency of planetary systems by adding a newfree parameter fp and ﬁtting to Oi with 0 ≤ i ≤ 5. However, this ﬁt would be equivalent to: ﬁttinga scaled population of systems that all contained planets to Oi , i > 0; calculating the scaled numberof systems that had planets but did not transit, E0 ; and setting E0 = (1 − fp )O0 . That is, we canreturn to the question of the frequency of planetary systems after we ﬁnd simulated populationsthat match the observed frequencies of systems where at least one transiting planet is detected. The ﬁrst result is that there is no simulated population that adequately ﬁts the observeddistribution Oi for 1 ≤ i ≤ 5. The only population that adequately describes the singly-transitingsystems has a uniform distribution of two planets per star (Np,U = 2) and an inclination width ofσi = 4◦ . While the G-statistic shows that this solution has high conﬁdence, clearly any population
19. – 19 –that has only 2 planets per star cannot explain the large number of triples and higher-multiplicitiesseen in the Kepler observations. The “good ﬁt” here is biased by the larger weight that the G-statistic gives to categories with more members and that categories with no observations haveno weight. However, this model is useful to point out as one that correctly reproduces the ratioof doubles to singles by requiring a typical inclination dispersion of σi = 4◦ , corresponding to aRayleigh distribution with a mean of ¯ ≡ σi π/2 ≃ 5◦ . i All other simulated populations produce relatively fewer singly-transiting systems than areobserved. The diﬃculty can be seen qualitatively by comparing the ratio of doubly-transitingsystems to singly-transiting systems (O2 /O1 = 14.5%) to the ratio of triply-transiting systems todoubly-transiting systems (O3 /O2 = 25.8%). These two ratios are quite diﬀerent, yet the proba-bility of an additional planet transiting depends only on the semi-major axis ratio and the mutualinclination (Ragozzine & Holman 2010). We have also found that the probability that a planetpassed our detection threshold is more-or-less independent of multiplicity, i.e., the detectability canbe estimated on a planet-by-planet basis. (In detail, systems with high multiplicities tend to haveplanets with longer periods which lowers the SNR somewhat.) The simulated populations haveboth the geometric and detectability probabilities decreasing at an approximately constant rate asadditional planets are added, leading to ratios Ei+1 /Ei that are nearly constant. Unlike in the single homogeneous simulated populations, the observed ratios are not similarand Kepler seems to be seeing an “excess” of singly-transiting systems. Note that this cannot beattributed to the fact that “hot Jupiters” appear to be singletons (Latham et al. 2011, in prep),since we are considering objects that are Neptune size or smaller. While it is possible that theinability to create a simulated population that matches the observations of multiples by Kepler isdue to a shortcoming in the model, a theoretically motivated alternative is that the true planetpopulation that has a bimodal distribution of characteristic inclination, with some have smallinclinations and typically more planets than the other group. This is strongly suggested for distantplanets (e.g., Levison et al. 1998) and may be present in the close-in population we are modelinghere due to a distinction between systems that have had large dynamical scatterings in the pastversus those systems which have remained relatively calm. As we are only attempting to match 5 multiplicity frequencies Oi , we cannot justify adding afull second population to our ﬁts since this would require 4 highly degenerate parameters. Instead,we suppose that there may be a excess population of singly-transiting systems, and the simulatedpopulations are compared again to the observations, but are now ﬁt only to the observed populationwith two or more transiting planets (i.e., 2 ≤ i ≤ 5) using the same G-statistic method describedabove. Most of the constraint on this population is from the ratio of O3 /O2 . Note that a secondplanetary population with a large inclination dispersion would also contribute slightly to the numberof doubles and possibly triples, but that we are ignoring this contribution. The best ﬁt (not rejectable at the 93% level) in this case is from a population of systems withNp,U = 3.25 (i.e., 75% of systems with 3 planets and 25% of systems with 4 planets) and with a
20. – 20 –Rayleigh inclination width of σi = 2◦ , implying a mean inclination of ¯ ≃ 2.5◦ . This population iproduced Ei = (62.1, 17.5, 1.4, 0) for i from 2-5, which is an excellent match to the observed valuesOi = (63, 17, 1, 1). (Recall that the expected numbers are scaled so that the total number of planetsis the same as the observations.) When scaled to the observed values, this population producedonly E1 = 130.1 singly-transiting systems, suggesting that over half of the observed singles cannotbe explained in a simple model that matches the higher multiplicities. Note that these ﬁts have only 1 degree of freedom (4 categories - 1 from scaling - 2 parameters),so it is not a surprise that a wide variety of models provide moderately good ﬁts to the data. Figure12 gives a contour plot of the acceptability of simulated populations as a function of Np,U and σi . Asexpected, increasing the number of planets per system requires the inclination dispersion to increasein order to provide a good match to the Kepler data, which can be demonstrated quantitatively.These ﬁts also cannot explain the large number of singly-transiting systems: the better the ﬁt toOi , 2 ≤ i ≤ 5, the lower the expected number of singly-transiting systems. Some of the simulated populations where the number of planets is taken from a Poisson dis-tribution also give moderately good ﬁts. The best is rejectable at the 70% level and has Np,λ = 2.5with σi = 4◦ . This simulation can (rarely) produce systems with more than 4 detectable transitingplanets, Ei = (65.3, 14.3, 1.9, 0.4) for i from 2-5, but underproduces the number of triply-transitingsystems. A contour plot showing these the acceptability levels as a function of Np,λ and σi is shownin Figure 13. This ﬁgure also demonstrates the trend that more planets require a larger inclinationdispersion to match the Kepler observations. It is expected that not every star has planets in this size and period range that would havebeen detected by Kepler ; the Sun is an example of such a star. The population of doubles that bestmatched the observations for 1 ≤ i ≤ 5 required E0 =9699 stars, i.e., ∼6% of the Kepler sample. Thebest-ﬁt population for 2 ≤ i ≤ 5 described above (Np , U = 3.25, σi = 2◦ ) required only E0 = 3225stars (∼2%) to have, on average, 3.25 planets (within the period and radius limits) to reproduce theobservations seen by Kepler . The Poisson distributed simulated population (Np , λ = 2.5, σi = 4◦required 4925 stars (∼3%). (Other simulated populations that were adequate ﬁts as shown inFigures 12 and 13 had similar values of E0 .) In these latter cases, an additional set of stars withplanets are needed to account for the additional singly-transiting systems. Nevertheless, thesenumbers appear quite small and fall about a factor of 2 short of the completeness calculationsreported in B11, even after accounting for the fact that we are considering a diﬀerent size rangeand that the number of stars with planetary systems is the average number of planets with stars(calculated by B11) divided by the typical multiplicity. The source of this discrepancy is unclear. In multiple systems with signiﬁcant width in the inclination distribution, a substantial fractionof observers will see at least one planet in transit, even if the periods are somewhat large. Oursimulations suggest that only a small fraction (∼5%) of the ∼162000 stars observed by Kepler havelarge numbers of planets with periods less than 125 days and radii between 1.5 and 6 Earth radii.As our completeness SNR cutoﬀ is just an approximation to the actual fraction of planetary signals
21. – 21 –that Kepler will detect among all actual planetary transits, the true fraction of stars with planetswill be diﬀerent from this estimate, with a more detailed assessment of completeness to be done bythe Kepler Mission in the future. For example, the large errors and, in some cases, biases in theKIC will propagate into our simulations to some degree, though the ratio of observed multiplicitieswhich drive the ﬁts above are not as strongly aﬀected as the estimate of the number of stars withplanetary systems. With these caveats, Kepler is providing incredibly powerful insight into the formation of plan-etary systems. A single population model does not match the observed distribution, suggesting asuperposition of two populations. A small fraction of stars have multiple (3 or more) super-Earth orNeptune size planets with periods less than 125 days. It appears that many of the doubly-transitingsystems and less than half of the singly-transiting systems are actually systems with 3-4 planets,with non-transiting planets potentially observable through TTVs (Ford et al. 2011, in preparation).Systems with large values of ∆ in Tables 1 and 2 are good candidates for systems where additionalnon-transiting planets may be missing in between. We have also provided the ﬁrst estimate of the inclination dispersion that has been given for alarge sample of observed planetary systems. To match the numbers of higher multiples observed byKepler requires a population with relatively low inclination dispersion; mean inclinations greaterthan 5◦ are poor ﬁts. It does not appear that all of the multiples observed by Kepler come froma population of purely coplanar systems. Systems with multiple super-Earth and Neptune sizeplanets with periods less than 125 days at low relative inclinations suggest interactions with aprotoplanetary disk that induce migration while damping inclinations. There are good prospects for improving our understanding of the planetary populations ob-served by Kepler . Besides continued observations which will increase the detectability of planetsand ﬁnd new candidates in multiple systems, a more detailed population simulator could be de-veloped. For example, though the errors in the impact parameters of Kepler planets are usuallyrelatively large, they do contain information on relative inclinations. This can be used both in astatistical population sense as well as for individual systems. In some systems, the detection of or upper limits to Transit Duration Variations (TDVs) willconstrain the unobserved component of the relative inclinations of planets. Potential mutual eventsin Kepler multi-planet systems will be discussed and studied in more detail by Ragozzine et al.(2011, in preparation). Measurements of the Rossiter-McLaughlin eﬀect for planets within multi-planet systems to measure true mutual inclination would constrain orbital inclinations to star’sequator, though this is observationally challenging for most Kepler stars (Ragozzine & Holman2010). Each measurement of the true mutual inclination in individual systems helps ﬁll in thepicture for the typical inclination distribution of planetary systems.
22. – 22 – 7. How Rare are Planetary Systems Similar to Kepler-11 The Kepler-11 (= KOI-157) planetary system, with six transiting planets, is clearly an outlierto the number/frequency distribution for detected transiting planets. Moreover, the period ratioof the two inner planets is only 1.264, which is the 4th lowest ratio among the 238 neighboringpairs of transiting Kepler exoplanets (Figure 3). The smallest period ratio is 1.001, between thecandidate co-orbital planets in KOI-730 (Section 5.3). The second smallest period ratio, 1.038, isbetween two weak (vetting ﬂag 3) candidates in the KOI-284 system; this system would be unstablefor any reasonable planetary masses if indeed both of these candidates are actually planets in orbitabout the same star (section 4). The third smallest period ratio, 1.258, is for a pair of candidates inthe KOI-191 system that stability considerations strongly suggest are locked in a 5:4 mean motionresonance that prevents close approaches (section 5.4). Thus, Kepler-11b and Kepler-11c may wellhave the smallest period ratio of any non-resonant pair of planets in the entire candidate list.Additionally, the ﬁve inner planets of Kepler-11 travel on orbits that lie quite close to one another,both in terms of period ratio and absolute distance. Kepler-11 clearly does not possess a “typical”planetary system, but are systems of this type quite rare, or simply somewhat scarce? The two candidate most analogous to Kepler-11 are KOI-500 and KOI-707. With 5 planetcandidates, KOI-500 is second in abundance to Kepler-11 and four of these candidates are close toone another in terms of period ratio (Table 4). The target star is signiﬁcantly smaller than theSun (∼0.74 R⊙ ; B11), and the orbital periods of the candidates are shorter, so if conﬁrmed theplanets in this system would be most closely spaced in terms of physical distance between orbitsof any several planetary systems known. Moreover, four of the planets orbiting KOI-500 appearto be locked in three-planet resonances (Section 5.5), whereas no analogous situation is observedin Kepler-11. Unfortunately, KOI-500 is quite faint, so it will be more diﬃcult to study than isKepler-11. The four candidates in KOI-707 have periods within the range of the periods of the 5inner planets observed in the Kepler-11 system. Periods ratios of neighboring candidates in thefour planet KOI-117 are all less than two, but nonetheless signiﬁcantly larger than in Kepler-11and KOI-707. None of the six other four candidate systems is nearly as closely spaced. Furthermore, the better-ﬁtting population simulations described in 6 rarely produced 5-planetsystems, suggesting that Kepler-11 is not just a natural extension of Kepler multiple systems.Although its solitary nature makes determining a lower bound on planetary systems like Kepler-11an ill-posed question (Kepler could have just gotten lucky), we can estimate the number of Kepler-11 systems that would not be seen in transit probabilitistically. If all the Kepler-11 planets arecoplanar (which is slightly inconsistent with the observed impact parameters), then the probabilityof seeing all six planets transit is the same as seeing Kepler-11g transit, which is 1.2%, suggestingthat there are dozens of systems similar to Kepler-11, unless we were observing from a particularlyfavorable position.
23. – 23 – 8. Conclusions Analysis of the ﬁrst four months of Kepler data reveals 170 targets with more than one tran-siting planet candidate (B11). While the vast majority of these candidates have yet to be validatedas true planets, we expect that the ﬁdelity of this subsample of Kepler planet candidates to be high(Section ), and the small fraction of false positives not to aﬀect the robustness of our statisticalresults. Our major conclusions are: 1. The large number of multiple planet systems observed by Kepler show that ﬂat multi-planet systems are common in short-period orbits around other stars. This result holds for planets in the size range of ∼ 1.5 – 6 R⊕ , but not for giant planets. Not enough data are yet available to assess its viability for Earth-size and smaller worlds, nor for planets with orbital periods longer than a few months. The abundance and distribution of multiply transiting systems implies that the mean number of planets per star with at least one planet of this type is ∼3-4. Thus, only ∼3% of stars have planets within these in size-period limits. 2. Almost all candidate systems survived long-term dynamical integrations that assume circular, planar orbits and a mass-radius relationship derived from planets within our Solar System 1. This bolsters the evidence for the ﬁdelity of the sample. 3. Mean-motion orbital resonances and near resonances are favored over random statistical prob- ability, but only a small fraction of multiple candidates are in or very near such resonances. In particular, resonances seem to be less common among nearly coplanar planets with short periods and radii a few times that of Earth than among the typically more massive and dis- tant planets detected by RV (Wright et al. 2011). Nonetheless, we note that a few systems are likely to have particularly interesting resonances conﬁgurations (Section 5). 4. Taken together, the properties of the observed systems suggest that the majority are free from false positives. It is unlikely that all of the candidates in multiple system are false positives, but there may be systems where one of the candidates is not a transiting planet around the same star as the other candidate(s). Those very near mean-motion resonances are most likely to be true planetary systems. 5. No single population of multiple planets with a single inclination dispersion can describe all the Kepler transiting systems. It appears that Kepler systems come from two populations which produce an excess in singly-transiting systems. Among systems with two or more transiting planets, the inclination dispersion appears to have a mean between 1-5◦ , suggesting relatively low mutual inclinations similar to the Solar System. Many more Kepler targets are multi- planet systems where additional non-transiting planets are present and these may eventually show TTVs.
24. – 24 – The rich population of multi-transiting systems discovered by Kepler and reported by B11have immense value both as individuals and as a population for improving our understanding ofthe formation and evolution of planetary systems. The Kepler spacecraft is scheduled to continue toreturn data on these multi-planet system for the remainder of its mission, and the longer temporalbaseline aﬀorded by these data will allow for the discovery of more planets and more accuratemeasurements of the planets and their interactions. Kepler was competitively selected as the tenth Discovery mission. Funding for this mission isprovided by NASA’s Science Mission Directorate. The authors thank the many people who gaveso generously of their time to make the Kepler mission a success. D. C. F. acknowledge NASAsupport through Hubble Fellowship grants #HF-51272.01-A and #HF-51267.01-A, respectively,awarded by STScI, operated by AURA under contract NAS 5-26555. We thank Avi Loeb, JoshCarter, and others for valuable discussions. REFERENCESAgol, E., Steﬀen, J., Sari, R., & Clarkson, W. 2005, MNRAS, 359, 567Anglada-Escud´, G., L´pez-Morales, M., & Chambers, J. E. 2010, ApJ, 709, 168 e oBorucki, W. J., Kepler, & Team. 2011a, ApJ—. 2011b, ApJ, in pressBorucki, W. J., Koch, D., Basri, G., Batalha, N., Brown, T., Caldwell, D., Caldwell, J., Christensen- Dalsgaard, J., Cochran, W. D., DeVore, E., Dunham, E. W., Dupree, A. K., Gautier, T. N., Geary, J. C., Gilliland, R., Gould, A., Howell, S. B., Jenkins, J. M., Kondo, Y., Latham, D. W., Marcy, G. W., Meibom, S., Kjeldsen, H., Lissauer, J. J., Monet, D. G., Morrison, D., Sasselov, D., Tarter, J., Boss, A., Brownlee, D., Owen, T., Buzasi, D., Charbonneau, D., Doyle, L., Fortney, J., Ford, E. B., Holman, M. J., Seager, S., Steﬀen, J. H., Welsh, W. F., Rowe, J., Anderson, H., Buchhave, L., Ciardi, D., Walkowicz, L., Sherry, W., Horch, E., Isaacson, H., Everett, M. E., Fischer, D., Torres, G., Johnson, J. A., Endl, M., MacQueen, P., Bryson, S. T., Dotson, J., Haas, M., Kolodziejczak, J., Van Cleve, J., Chandrasekaran, H., Twicken, J. D., Quintana, E. V., Clarke, B. D., Allen, C., Li, J., Wu, H., Tenenbaum, P., Verner, E., Bruhweiler, F., Barnes, J., & Prsa, A. 2010, Science, 327, 977Bryson, S. T., Tenenbaum, P., Jenkins, J. M., Chandrasekaran, H., Klaus, T., Caldwell, D. A., Gilliland, R. L., Haas, M. R., Dotson, J. L., Koch, D. G., & Borucki, W. J. 2010, ApJ, 713, L97Caldwell, D. A., Kolodziejczak, J. J., Van Cleve, J. E., Jenkins, J. M., Gazis, P. R., Argabright, V. S., Bachtell, E. E., Dunham, E. W., Geary, J. C., Gilliland, R. L., Chandrasekaran, H.,
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34. – 34 –Table 6: KS test p-values for diﬀerent resonance sets and for test distributions . Resonance set log(P ) P Random 2:1 only 0.0084 0.0042 0.0044 j:j − 1 0.00093 0.00048 0.00067 j:j − 2 0.089 0.019 0.054 j:j − 1 and j:j − 2 0.00023 9.8 × 10−6 8.9 × 10−5 1.0 10 1 RJupiter Rp (RJupiter) Rp (REarth) 1 RNeptune 0.1 1 1 REarth 1 10 100 Period (days)Fig. 1.— Planet period vs. radius for all planetary candidates listed in B11. Those planets that arethe only candidate for their given star are represented by black dots, those in two-planet systemsas blue circles (open for the inner planets, ﬁlled for the outer ones), those in three-planet systemsas red triangles (open for the inner planets, ﬁlled for the middle ones, ﬁlled with black borders forthe outer ones), those in four planet systems as purple squares (inner and outer members ﬁlledwith black borders, second members open, third ﬁlled), the ﬁve candidates of KOI 500 as orangepentagons and the six planets orbiting KOI-157 (Kepler-11) as green hexagons. It is immediatelyapparent that there is a paucity of giant planets in multi-planet systems; this diﬀerence in the sizedistributions is quantiﬁed and discussed by Latham et al. (2011, in preparation).
35. – 35 – 1200 40 1000 30 20 800 # Systems 10 600 0 0 1 2 3 4 5 6 400 200 0 0 1 2 3 4 5 6 # Planets per SystemFig. 2.— Comparison of the number of planetary systems of given multiplicity observed by Kepler(candidates with Rp < 15 R⊕ ), shown by the shaded histogram, with a Poisson distribution of thesame number of objects and the same mean value (thick green line and ﬁlled circles). The columnfor stars with no detected planets goes far beyond the scale of the ﬁgure. Also shown (by thedashed red line and ﬁlled circles) are the numbers of exoplanet systems detected by radial velocitysurveys (data obtained from exoplanets.org ¡http://exoplanets.org¿ on 2011 Jan 28) scaled to thesame total number of objects as Kepler candidates. The inset in the upper right zooms in to revealthe structure of the curves for systems with three or more planets.
36. – 36 – 700 All adjacent pairings All pairings 600 All 2−candidate pairings All adjacent 3−candidate pairings All 3−candidate pairings Inner 3−candidate pairing Outer 3−candidate pairing 500 All adjacent 4−5−6−candidate pairings All 4−5−6 candidate pairings 400 N 300 200 100 0 1 1.5 2 2.5 3 3.5 4 4.5 5 Period RatioFig. 3.— Cumulative number of pairs of Kepler planets orbiting the same star with period ratio lessthan the value speciﬁed. Black dashed curve shows all pairings, solid black curve shows pairings ofneighboring planets; solid red curve shows all pairings in two candidate systems; solid blue curveshows all pairings in three-planet systems, solid light blue curve shows inner pair of planets inthree-planet systems, the solid orange curve shows outer pair of planets in three-planet systems;dashed purple curve shows all pairings in the 4, 5 and 6 planet systems, solid purple curve showsneighboring pairings in the 4, 5 and 6 planet system.
37. – 37 – 1 0.9 Kepler adjacent pairings RV adjacent pairings 0.8 0.7 0.6 N 0.5 0.4 0.3 0.2 0.1 0 1 1.5 2 2.5 3 3.5 4 4.5 5 Period RatioFig. 4.— Cumulative fraction of neighboring planet pairs for Kepler candidate multiplanet sys-tems with period ratio exceeding the value speciﬁed (solid curve). The cumulative fraction forneighboring pairs in multiplanet systems detected via radial velocity is also shown (dashed curve),and includes data from exoplanets.org as of 29 January 2011.
38. – 38 – 1 Kepler adjacent pairings RV adjacent pairings 0.9 Slope of Cumulative Period Ratio 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1.5 2 2.5 3 3.5 4 4.5 5 Period RatioFig. 5.— ”Slope” of the cumulative fraction of Kepler neighboring planet pairs (solid curve) andmultiplanet systems detected via radial velocity (dashed curve) with period ratio exceeding thevalue speciﬁed. The slopes were computed by taking the diﬀerence in period ratio between pointswith N diﬀering by 4 and dividing by 4. The spikes show excess planets piling up near periodcommensurabilities.
39. – 39 – 1.0 0.5 0.0 Ξ 0.5 1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Period RatioFig. 6.— The ξ statistic as a function of period ratio for the ﬁrst order (solid, blue) and secondorder (dashed, red) MMRs. Also shown are the observed period ratios in the Kepler data.
40. – 40 – 2.0 1.5 PDF 1.0 0.5 0.0 1.0 0.5 0.0 0.5 1.0 Ξ 2.0 1.5 PDF 1.0 0.5 0.0 1.0 0.5 0.0 0.5 1.0 ΞFig. 7.— Probability density of the systems as a function of ξ for both ﬁrst (top) and second(bottom) order MMRs. The most common value for ξ in both samples lies just outside the corre-sponding MMR while one of the least common values (especially for the j : j − 1 MMRs) lies justinside the MMR. Note that the neighborhood of the 3:1 MMR has some planets just interior tothe resonance while none of the other second-order resonances do (see Figure 6. The probabilitydensity for the logarithmically distributed test sample is also shown for reference and appears as amonotonically decreasing function in both plots (blue).
41. – 41 – 1.0 0.8 0.6 CDF 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ΞFig. 8.— Cumulative distribution function (CDF) of the absolute value of the ξ statistic for thecombined ﬁrst and second-order resonances (blue). Also shown is the CDF for the logarithmicallydistributed test sample (red).
42. – 42 –Fig. 9.— Larger and smaller of the orbital separations, expressed in mutual Hill radii, for adjacent3-planet sub-systems of the 3-, 4-, 5-, and 6-planet systems of our sample. The symbols are the sameas in ﬁgure 1, except we also include the spacings in the Solar System as black dots. Gray region: √stability boundary for 2-planet systems, ∆ > 2 3 (eq. 3). Dashed line: stability boundary usedfor the simulated population of multiplanet systems: ∆inner + ∆outer > 18; some of our integrationssurvived long-term integrations despite transgressing that boundary. The unstable nominal systemsKOI-191, KOI-248, and KOI-730 all lie within the gray region, and all the long-term survivors areoutside of it.
43. – 43 –Fig. 10.— Cumulative distribution of the ratio of planetary radii (Rp,out /Rp,in ) for neighboringpairs of transiting planet candidates. The distribution of radii ratios for all neighboring pairs(dotted line) is consistent with that of neighboring pairs of planet candidates which are near a 2:1MMR (solid line). For neighboring pairs near the 2:3 (dotted-dashed) and 1:3 (dashed) MMRs,there may be a decreased frequency of planets with extreme ratios. Neighboring planets tend tohave similar radii, and in the majority of cases, the outer planet is slightly larger.
44. – 44 –Fig. 11.— Radius distribution of all candidates and for candidates around stars brighter than 14(Kepler magnitudes) and with SNR greater than 16. This is the radius distribution used in thesimulated populations of Section 6.
45. – 45 –Fig. 12.— Contour plot of the probability that a simulated population matches the Kepler observedpopulation of systems with 2 or more transiting planets with radii between 1.5 and 6 Earth radii,periods less than 125 days, and planets with SNR of 9.2 or greater (see text). The x-axis showsthe average number of planets assigned to all stars in the “uniform” model and the y-axis showsthe Rayleigh width parameter σi for the inclinations in degrees. A plus sign marks the best-ﬁtmodel (which cannot be rejected at a 93% conﬁdence level) described in the text. Note the cleartrend that the simulated population must have an increased inclination dispersion if the numberof planets per star is increased in order to match the Kepler observations. Populations with meaninclinations greater than ¯ ≡ σi π/2 ≃ 5◦ are poor ﬁts, as are completely coplanar conﬁgurations. i
46. – 46 –Fig. 13.— Same as ﬁgure 12 but where the number of planets is drawn from a Poisson distribution(without allowing zero planets) with width Np,λ reported on the x-axis. These populations areworse ﬁts to the observed numbers than the “uniform” model, but show the same trends.
47. 1 0.9 0.8 0.7 0.6N 0.5 0.4 All adjacent pairings All pairings 0.3 All 2−candidate pairings All adjacent 3−candidate pairings All 3−candidate pairings 0.2 Inner 3−candidate pairing Outer 3−candidate pairing All adjacent 4−5−6−candidate pairings 0.1 All 4−5−6 candidate pairings 0 1 1.5 2 2.5 3 3.5 4 4.5 5 Period Ratio
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