Architecture and dynamics of kepler's candidate multiple transiting planet systems

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

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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 multiple
transiting planets. Borucki et al. (2011b) presented, for the first time, 5 Kepler systems with
multiple 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) was
found, in which a resonant effect on the transit times allowed Kepler data to confirm two of the
planets 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 systems
were highly anticipated because of their unique value in understanding the formation and evolution
of 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 data
is presented in B11. They also summarize the observational data and follow-up programs being
conducted to confirm and characterize planetary candidates. Each object in this catalog is assigned
a 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 sample
also has an expected reliability of >60%). For most of the remainder of the paper, we refer to all

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of these objects as “planets”, although 99% remain unvalidated and thus should more properly be
termed planet candidates. Ragozzine & Holman (2010) and Lissauer et al. (2011) point out that
systems with multiple planet candidates are less likely to be due to astrophysical false positives
than stars with single candidates, although the possibility that one of the candidate signals is due
to, e.g., a background eclipsing binary, is still non-negligible. Here we only examine the candidates
that are known to be in multiple systems; many of the apparently single systems are likely to have
additional planets that escape detection by being too small, too long-period, or too highly inclined
with respect to the line of sight. Ford et al. (2011, in preparation) discuss candidates exhibiting
TTVs that may be caused by non-transiting planets. B11 list five dozen planetary candidates with
radii Rp > 15 R⊕ (Earth radii), but none of these are in multiple candidate systems. Latham
et al. (2011, in preparation) compare the radius distribution of planets in multi-planet systems
with those of planets which don’t have observed transiting companions. We examine herein the
dynamical aspects of Kepler’s multi-planet systems.

Ragozzine & Holman (2010) discussed in detail the value of systems with multiple transiting
planets. Besides their higher likelihood of being true planetary systems, multi-transiting systems
provide 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 interactions
between 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 mutual
inclinations between planets in these systems, while not directly observable from the light curves
themselves, are much easier to obtain than in other systems, both individually and statistically (see
Section 6 and Lissauer et al. 2011). The distributions of planets in these systems, both in terms
of physical and orbital parameters, are invaluable for comparative planetology and provide unique
and 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 1
sextuple transiting system. While very few of these systems contain confirmed planets, it is very
likely that this set of multi-transiting candidates contains more than twice as many as planets as
those known in the pre-Kepler era.

We begin by summarizing the characteristics of Kepler’s multi-planet systems in Section 2. The
reliability of the data set is discussed in Section 3. Sections 4 – 6 present our primary statistical
results: Section 4 presents an analysis of the long-term dynamical stability of candidate planetary
systems for nominal estimates of planetary masses and unobserved orbital parameters. Evidence of
orbital commensurabilities in among planet candidates is analyzed in Section 5. The characteristic
distributions of numbers of planets per system and mutual inclinations of planetary orbits are
discussed in Section 6. We compare the Kepler-11 planetary system (Lissauer et al. 2011) to other
Kepler candidates in Section 7. We conclude with a discussion for the implications of these data
and summarize the most important results presented herein.

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2. Characteristics of Kepler Multi-Planet Systems

The light curves derived from Kepler data can be used to measure the radii and orbital periods
of planets. Periods are measured to high accuracy. The ratios of planetary radii to stellar radii
are also measured quite precisely, except for low signal to noise transits (e.g., planets much smaller
than their stars, those for which few transits are observed, and planets orbiting faint or variable
stars). Besides errors in estimated stellar radii, in some cases the planet’s radius is underestimated
because a significant fraction of the flux in the target aperture is provided by a star other than the
one being transited by the planet. This extra light can often be determined and corrected for by
analyzing displacements of the centroid of light from the target between in-transit and out-of-transit
observations (Bryson et al. 2010), high-resolution imaging and/or spectroscopic studies conducted
from 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, thereby
allowing 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-planet
systems in Table 2, four-planet systems in Table 3, and those systems with more than four planets
in Table 4. Planet indexes given in these tables signify orbit order, with 1 being the planet with the
shortest 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 the
order in which the candidates were identified. These tables are laid out differently from one another
because of the difference in parameters that are important for systems with differing numbers of
planets. In addition to directly observed properties, these tables contain results of the dynamical
analyses discussed below.

We plot the period and radius of each active Kepler planetary candidate observed during
the 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; the
distribution of these period ratios is plotted in Figure 3. Figure 4 shows a cumulative distribution
of the period ratio.

To check whether the observed transiting planets are randomly distributed among host stars
we compared the observed distribution of planets per system to a statistically random Poisson
distribution, 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 gives
an expectation of 1219 single planet systems, 5.0 doubles, 0.014 triples, 2.8 × 10−5 systems with
four planets and above. For single planet systems there are fewer observed systems than expected
from a random distribution, but, for systems with two and more planets there are far more systems
detected by Kepler than predicted randomly. This shows that transiting planets tend to “come in
packs” or cluster around stars, meaning that once a planet is detected around a given star, that star
is more likely to be orbited by an additional planet than a star with no detected planets. Some of
the observed clustering is attributable to SNR considerations, with stellar brightness, size, intrinsic

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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 far
too small to account for the bulk of the observed clustering. Given the bias of the transit method
to detect planets on edge-on orbits, a possible interpretation of the clustering of transiting planets
is that planetary systems (or at least some fraction of them) tend to be coplanar, like the Solar
System.

It is difficult to estimate the corresponding distribution for planets detected through radial
velocity (RV) surveys, since we do not know how many stars were spectroscopically surveyed for
planets 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 RV
trend as evidence for an additional planetary companion, Wright et al. (2009) show that at least
28% of known RV planetary systems are multiple. Therefore, the clustering observed for transiting
planets detected by Kepler is likely to occur also for RV planets, and maybe even at a higher
fraction, 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 from
eclipsing binary stars. Some of these will be grazing eclipses of the target star, others eclipses of
fainter stars (physically associated to the target or background/faint foreground objects) within the
target aperture. Similarly, transits of fainter stars by large planets may also mimic small planets
transiting the target star. The vast majority of all active Kepler planet candidates were identified
using 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 subjective
visual searches are too small to affect our results in any statistically significant manner, and even
the signs of such changes are unclear.

The fact that planetary candidates are clustered around targets in the sense that there are far
more 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 higher
than that of the single candidate targets. We know from radial velocity observations that planets
frequently come in multiple systems (Wright et al. 2009), whereas astrophysical false positives are
nearly random.

Nonetheless, the presence of multiple candidates increases our confidence that one or both are
planets is real since the probability of two false positive signals is the product of the probabilities

1
exoplanet.org, as of January 2011.

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of two relatively rare cases, significantly lowering the overall probability than none of the signals
is due to planets Ragozzine & Holman (2010). (Triple star systems would be dynamically unstable
for most of the orbital period ratios observed. Their dynamics would also give rise to very large
eclipse 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 these
subsamples likely have an even larger fraction of true planets, since such concentrations would not
be seen for random eclipsing binaries. However, at present these qualitative factors have not been
quantified.

Strong TTV signals are no more common for planets in observed multi-transiting candidates
than for single candidates (Ford et al. 2011, in preparation), suggesting that large TTVs are
often the result of stellar systems masquerading as planetary systems (Fabrycky et al. 2011, in
preparation, Steffen et al., 2011, in preparation). Nonetheless, it is possible that weaker TTVs
correlate with multi-transiting systems, but these require more data and very careful study to
reveal (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 systems
can be used to potentially identify false positives (generated by a blend of two objects eclipsing
two stars of different densities) or as a probe of the eccentricity and inclination distributions. An
analysis of the corrected duration ratio as in (?) shows that essentially all pairs of candidate have
the expected duration ratio if they were both orbiting the same star. The consistency of duration
ratios 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’s
mean radius. The power-law of equation 1 is fit to Earth and Saturn; it slightly overestimates the
mass 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 a
monotonic mass-radius relation is not possible.)

A convenient metric for the dynamical proximity of planets is the separation of their orbital
semi-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⋆ 2
where 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.

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The dynamics of two-planet systems are a special case of the 3-body problem that is amenable
to analytic treatment and simple numerically-derived scaling formulas. For instance, a pair of
planets 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). Orbital
separations of all planet pairs in two planet systems for stellar masses given in B11 and planetary
masses 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 mutual
inclination (Veras & Armitage 2004), neither of which can be measured well from the transit data
alone. Circular coplanar orbits have the lowest angular momentum deficit (AMD), and thus are
the most stable, except for protective resonances that produce libration of angles and prevent close
approaches; a second exception exists for retrograde orbits (Smith & Lissauer 2009), but these are
implausible on cosmogonic grounds (Lissauer & Slartibartfast 2008). Hence, the Kepler data can
only be used to suggest that a pair of planets may be potentially unstable, particularly if their
semi-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 assumed
that both candidates are true planets, limits on the eccentricities and mutual inclinations can be
inferred from the requirement that the planetary systems are presumably stable for the age of the
system, which is generally of order 1011 times longer than the orbital timescale for the short-period
planets that represent the bulk of Kepler ’s candidates. However, we may turn the observational
problem around: the ensemble of duration measurements can be used to address the eccentricity
distribution (Ford et al. 2008); since multiplanet systems have eccentricities constrained by stability
requirements, 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 very
unlikely to be stable unless all neighboring pairs of planets satisfy equation 3. Nominal separations
between neighboring planet pairs in three planet systems are listed in Table 2, the separations in four
planet systems are listed in Table 3, and those for the 5 and 6 planet systems in Table 4. We find that
in these cases, equation 3 is satisfied by quite a wide factor. From suites of numerical integrations
Smith & Lissauer (2009), dynamical survival of systems with more than three comparably-spaced
planets 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 set
of 3 planets in our sample, we plot the ∆ separations of both the inner and the outer pairs in
Figure 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 is
satisfied.

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We investigated long-term stability of all 55 systems with three or more planets using the hybrid
integrator within the Mercury package (Chambers 1999), run on the supercomputer Pleiades at
University of California, Santa Cruz. We set the switchover at 3 Hill radii, but in practice we
aborted simulations that violated this limit, so for the bulk of the simulation the Wisdom-Holman
n-body mapping (Wisdom & Holman 1991) was used, with a time step of 0.05 times as large as the
orbital period of the innermost planet. The simplest implementation (Nobili & Roxburgh 1986) of
general relativistic precession was used, an additional potential UGR = −3(GM⋆ /cr)2 , where G is
Newton’s constant, c is the speed of light, and r is the instantaneous distance from the star. More
sophisticated treatments of general relativity (Saha & Tremaine 1994) are not yet required, due to
the uncertainties of the masses of the planets and stars whose dynamics are being modeled. We
neglected precession due to tides or rotational flattening, which are only significant compared to
general relativity for Jupiter-size planets in few-day orbits (Ragozzine & Wolf 2009). The stellar
mass was derived from the R⋆ and log g estimates from color photometry, as given by B11. We
assumed initially circular and coplanar orbits that matched the observed periods and phases, and
chose the nominal masses of equation 1.

For the triples and quadruples, we ran these integrations for 4 × 109 orbital periods of the
innermost planet, and found the nominal system to be stable for this span in nearly all cases; the
three exceptions are described below. An integration of the five-planet system KOI-500 has run
2 × 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 this
conclusion 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 transiting
planetary 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 estimate
of 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 at
427 Myr for an integration that was a very slight offset from the b/c-eccentric fit. The all-circular
fit 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 should
address 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 candidates
with a period ratio 1.0383. Both planets would need to have masses about that of the Earth or
smaller for the system to satisfy equation 3 and thus be Hill stable. But the vetting flags are 3
for both these candidates — meaning that they are not good candidates anyway — and the data
display significant correlated noise which may be responsible for these detections. We expect one
or 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

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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 remarkable
that stability considerations allowed us to identify a poorly conditioned light curve fit. Once the
correction 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 not
a large number of planets that with substantially higher densities than given by the simple formula
of equation (1). If these candidates are assumed to be orbiting the same stars, stability constraints
could 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 different
from 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 than
RV planetary systems for two reasons. First, periods are measured to very high accuracy, even
for systems with low SNR, so period ratios can be confidently determined without the harmonic
ambiguities that affect RV detections (Anglada-Escud´ et al. 2010; Dawson & Fabrycky 2010). Sec-
e
ond, 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 to
demonstrate 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 not
perfectly 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 or
eccentricities are relatively high. In the cumulative distribution of period ratios (Figure 3), we see
that 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 that
this should be the case (e.g., Terquem & Papaloizou 2007). We can ask whether such an excess can
be seen in the sample of multiply transiting Kepler planetary systems. To do this, we divide the
period 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 the
intermediate, third-order MMRs. Thus, all candidate systems with period ratios less than 4:1 are
in some neighborhood. For example, the neighborhood of the 3:1 MMR runs between period ratios
of 5:2 and 4:1, for the 2:1 it runs from 7:4 to 5:2. With this method the first-order MMRs have
neighborhoods that are essentially twice as large as those for second-order MMRs.

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We define a statistic, ξ, that is a measure of the difference between an observed period ratio
and the MMR in its neighborhood. In order to treat each neighborhood uniformly, they are scaled
such 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−1

where P is the observed period ratio (always greater than unity) and “Round” is the standard
rounding function that returns the integer nearest to its argument. Similarly, for second-order
MMRs, ξ is given by
2 2
ξ2 ≡ 3 − Round . (6)
P −1 P−1
Table 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 resonances
in each MMR order. The resulting distribution in ξ is compared to a randomly-generated sample
drawn 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 candidate
systems. 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 and
second-order MMRs. We see that the results for the second-order MMRs have some qualitative
similarities to the first-order MMR, though the peaks and valleys are less prominent. In our tests
below 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 various
test samples, we took the absolute value of ξ for each collection of neighborhoods and used the
Kolmogorov-Smirnov test (KS test) to determine the probability that the observed sample is drawn
from 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 and
second-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 neighborhoods
surrounding 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 inconsistent
with the test distributions, and 4) there is a hint that second-order resonances are not be consistent
with the test distributions, particularly the distribution that is uniform in P . Additional data are
necessary 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 a
pair of planets to reside is slightly exterior to the MMR (the planets are farther apart than the
resonance location) regardless of whether the resonance is first or second-order. Also, a slight

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majority (just over 60%) of the period ratios have ξ statistics between -1/2 and +1/2 while all
of the test distributions have roughly 50% between these two values in ξ (note that ξ = ±1/2
corresponds to sixth-order resonances in first-order neighborhoods and twelfth-order resonances in
second-order neighborhoods—none of which are likely to be strong in these planet pairs though
they may play some minor role). There are very few examples of systems that lie slightly closer to
each other than the first-order resonances and for the second-order resonances only the 3:1 MMR
has systems just interior to it. Finally, there is a hint that the planets near the 2:1 MMR have a
range of orbital periods, while those near the 3:2 appear to have shorter periods and smaller radii
on average. While additional data are necessary to claim these statements with high confidence, the
observations are nonetheless interesting and merit some investigation to determine the mechanisms
that 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 statistically
significant evidence that many of the systems are actually systems of three or more planets. (Stars
with such close orbits are not dynamically stable, thus dynamical interaction between objects with
period ratios 5 indicates low masses due to planets.) Assessing the probability of dynamical
proximity 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 the
near-resonant systems appear to be planetary in nature.

The excess of pairs of planets wider than nominal resonances was predicted by (Terquem & Papaloizou
2007), who pointed out that tidal interactions with the star will, under some circumstances, break
resonances and will more strongly affect the inner planets. Interestingly KOI-730 seems to have
maintained 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-characterized
by Doppler observations are near a low-order period commensurability with one sixth near the 2:1
MMR (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 2:1
MMR (Giuppone et al. 2009; Anglada-Escud´ et al. 2010). Among Doppler-discovered systems,
e
only one quarter of the pairs near the 2:1 MMR have an inner planet significantly less massive
than the outer planet. The two exceptions (GJ 876 b&c, µ Ara e&b) benefitted from an unusually
large (≥ 100) number of Doppler observations. Thus, it is possible that the true rate of planetary
systems near the 2:1 MMR is significantly greater than one sixth.

Kepler observations find that at least ∼ 16% of multiple transiting planet candidate systems
contain 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 –

commensurability could be significantly greater than ∼ 16%, since not all planets will transit. If
we assume that planets near the 2:1 MMR are in the low inclination regime, then the outer planet
should transit ≃ 63% of the time, implying that the true rate of detectable planets near the 2:1
MMR (if both had a favorable inclination) is at least ∼ 25%. If a significant fraction of these
systems are not in the low inclination regime, then the true rate of pairs of planets near the 2:1
MMR would be even larger. This mildly suggests that radial velocity surveys could be missing some
2:1 pairs due to observational degeneracy. To consider this possibility further requires investigating
the 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 typical
outcome of planet formation models that predict more distant planets can more easily grow to
larger sizes.

The distribution of planetary radii ratios (Rp,out /Rp,in ) for neighboring pairs of transiting
planet candidates near a 2:1 period commensurability is concentrated between 0.9 and 1.25 with
a 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 have
masses within 20% of each other. Fortunately, such pairs are not significantly affected by the radial
velocity observational degeneracy with a single eccentric planet. However, the tail of ∼ 20% of
systems with a radii ratio exceeding 1.7 are more problematic, as the Doppler signatures of would
differ from that of a single eccentric planet by less than 20% of the velocity amplitude of the outer
planet.

Based on the planet radii ratio distribution from Kepler, we estimate that the abundance of
neighboring planets near the 2:1 MMR could be ∼ 20% greater than suggested by Doppler surveys
due to the difficulty in distinguishing the Doppler signature of such systems from a single eccentric
planet. A more precise determination of the frequency of resonant or near-resonant systems based
on 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 planetary
candidates, 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 show
evidence for extrasolar “Trojan” or co-orbital planets, which have been suggested to be theoretically
possible (Laughlin & Chambers 2002; Go´dziewski & Konacki 2006; Smith & Lissauer 2010). In
z
Kepler data, the two co-orbital candidates began separated by ∼118◦ , with the trailing candidate
reducing the gap at the rate of ∼1◦ per month. The numerical integration (section 4) matching
the 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 –

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 these
four candidates give us strong confidence that they all will eventually be confirmed as planets. The
dynamics 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 planets
has a ∼ 1RJup size. Its mass would need to be 17M⊙ to satisfy equation 3. Alternatively, the
two planets could be locked in 5:4 mean-motion resonance that prevents close approaches even if
orbits cross. Neptune and Pluto are locked in 3:2 mean-motion resonance of this type. Relatively
fast precession, on the order of 100 orbital periods, would be needed as well, as the observed period
ratio 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, 7.053478
d, and 9.521696 d. Neighboring pairs of the outer four of these planets are all near two-body
mean 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 consequence
of diverging tidal evolution, as predicted by Papaloizou & Terquem (2010). The three adjacent
period spacings between these four planets contribute to the statistics of period ratios just wide of
resonance (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 × 10−5
and 2n3 − 6n4 + 4n5 ≈ 1.3 × 10−5 are so small that we suspect two 3-body Laplace-like resonances
and/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 distribution
of planetary orbits around the invariable plane of the Solar System has a Rayleigh distribution
width of σi ≃ 1.5◦ ; if Mercury is excluded, then σi ≃ 1◦ . The strong level of co-planarity has been
recognized 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 –

disk generally yields circular orbits and low relative inclinations (e.g., Safronov 1969; Lissauer
1993). Some mechanisms for explaining the higher eccentricity distribution of large exoplanets
are 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
c
distribution 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 distribution
of exoplanetary systems and the exoplanet inclination distribution is thus of immense value. Yet
neither transit observations, nor the radial velocity technique can directly measure the true mutual
inclination 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 to
the line of sight of all transiting planets must be small, the orbits could be rotated around the
line of sight and mutually inclined to each other. Indirect constraints, however, can be obtained
for systems with multiple transiting planets that make them the best probe of mutual inclinations
around main sequence stars. For example, the lack of transit duration variations of the planets of
Kepler-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-10
planets (Batalha et al. 2011, in prep.) require a mutual inclination 5◦ .

Furthermore, given the large number of multi-candidate systems identified in B11, we can
approach the question of the inclination distribution of the population statistically. The more
mutually inclined a given pair of planets is, the smaller the probability that multiple planets
transit (Ragozzine & Holman 2010), and thus the smaller the fraction of multi-planet systems that
we would expect to see. Therefore, an investigation of coplanarity requires also addressing the
number of planets expected per planetary system. Of course, the question of planetary multiplicity
is also inherently interesting to investigate. Here we attempt to provide a self-consistent rigorous
estimate 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 observational
biases, 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 these
systems, though eventually investigations of transit timing and duration variations (or the lack
thereof) will be able to place additional limits on such planets (Steffen et al. 2010, Ford et al. in
preparation). If we knew about the presence of every planet, transiting or not, then the problem
of determining the typical inclination would be much easier.

– 15 –

A useful way to address the problem of non-detections is to create a forward model of transiting
planet detections based on a minimal number of assumptions and a relatively small number of
tunable parameters and to compare the output with the properties of the observed systems. The
model described here attempts use Kepler ’s observed frequency of planet multiplicity to determine
typical values for the true multiplicity of planetary systems as well as the mutual inclination between
planets in the same system. After choosing a distribution for the number of planets assigned to a
random Kepler star (characterized by a parameter Np ) and their relative inclinations (characterized
by σi ), the model computes how many of these planets would be detected, requiring a geometrical
alignment that leads to transit and a signal-to-noise ratio (SNR) large enough to be detected. This
process is done for over 100000 planetary systems and the results compared to frequency of observed
multi-transiting systems. Though imperfections in the model and observations will only provide an
approximate answer, Kepler enables for the first serious attempt at using observations to measure
the value of Np and σi for a large number of systems.

To isolate the true multiplicity and inclination distributions, other aspects of the simulated
systems (such as radius and period distributions) must be consistent with the true underlying
distribution. This requires some interpretation of the observed distribution and the identification
of potential biases. At the end of Kepler ’s mission, a simulated population that attempts to match
all the observations could be employed; here, we hold the period and radius distributions fixed and
they are pre-selected in a way that the output of the simulation will closely match the observed
distribution.

A good example of this process is the choice of the period distribution. The difference between
the observed and true underlying period distribution is almost entirely determined by the geometric
bias of observing planets in transit that scales as 1/a ∝ P −2/3 . As discussed in B11 and confirmed by
our investigations, after correcting for the geometric bias and discarding very short periods (below
3 days) and large planets (above 6 Earth radii), the remaining distribution is very well described
by a uniform distribution in log(P ). Hence, our simulator uses a period distribution uniform in
log(P ) for a pre-specified range of periods: 3-125 days. An improvement could be placed on this
process that takes into account the observed relationship (generally expected from planet formation
theories), that larger period planets have larger radii (Figure 10), but this covariance is not included
in 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 decreases
significantly, which is at least partly due to small planets escaping detection due to low SNR. The
1/r 2 distribution is very sensitive to the minimum allowed radius; to avoid this, we draw radii from
the distribution of observed planetary radii around stars with Kepler magnitude greater than 14
and through-Q5 SNR (from Table 1 in B11) greater than 16. A histogram of this distribution, and
the full distribution of all detected planets, is shown in Figure 11. In particular, our simulated
planets have radii drawn from this distribution between the limits of 1.5 and 6 Earth radii; except

– 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) suggest
that completeness effects begin to dominate below a SNR of ∼16, corresponding to a SNR of 9.2
from 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 multiples
have not yet been investigated in detail.

Applying these same cuts to Kepler ’s observed multi-candidate systems from B11 results in
437, 63, 17, 1, and 1 systems with 1, 2, 3, 4, and 5 transiting planets, respectively. It is possible
that some small fraction of the observed planets could be false positives and our simulation does
not attempt to account for this. Note that the sixth planet of Kepler-11, planet g, had a transit
that fell into the data gap between Quarters 1 and 2, so this system counts as a quintuple for
comparison 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 that
will be compared to the Kepler observations. In particular, for each simulated population, we will
compared the number of systems with i detectable transiting planets for 1 ≤ i ≤ 5 to the observed
distribution 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 and
radii from the Kepler Input Catalog (Latham et al. 2005, Brown et al. in preparation). After
randomly choosing a star, the model then assigns a number of planets, Np . Three distributions
were considered for the number of planets: a “uniform” distribution, a Poisson distribution, and an
exponential distribution. In the exponential distribution, the probability of having i + 1 planets is
a factor of α smaller than the probability of having i planets, with α a free parameter. This model
created a very poor fit to the data (particularly the triples to doubles ratio) for any value of α or
σi , so we do not consider it further.

The “uniform” distribution parameter Np,U is the mean number of planets per system, which
is a fixed number of planets per star if Np,U is an integer or a random mix of two fixed values if
Np,U is not an integer. For example, if Np,U = 3.25, 75% of systems would be assigned 3 planets
and 25% would be assigned 4 planets. The Poisson distribution parameter Np,λ assigns the number
of planets per system based on a Poisson distribution with width Np,λ . The frequency of stars with
planetary systems is considered separately (see below), so we require the number of planets to be
non-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 physical
properties of these planets are assigned. As discussed above, the period of each planet is selected

– 17 –

by drawing from a uniform distribution in log(P ) with a minimum of 3 and a maximum of 125
days. Although the period ratios in multiple systems deviate from a random distribution due to
clustering near resonance (Section 5), this is a minor effect compared to the overall geometric
bias. Each planetary radius is assigned based on the Kepler observed radii independently of orbital
period, as described above.

As the observed multiples are almost entirely stable on long timescales (Section 4), there is a
desire 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 used
above (eq. 1), i.e., M = R2.06 with masses and radii in units of Earth masses and radii. The mutual
Hill 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. If
any two planets have ∆ < 3.46 or if ∆inner + ∆outer < 18 for any three consecutive planets, then the
planet is rejected on stability grounds and the process begins again until all the planets have been
assigned. 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, as
many 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-zero
eccentricities indicated that, statistically speaking, the observed multiplicities did not change sig-
nificantly, though we did not consider the possibility that eccentricities and inclinations may be
correlated.

At this point, the full orbital and physical characteristics of the planetary systems have been
determined and we can begin to assess observability. First, these planetary systems are then rotated
by a uniform random rotation matrix in a manner equivalent to choosing a random point on a sphere
for the direction of the normal to the reference plane. In the new random orientation, the impact
parameter of each planet is calculated. If the planets are transiting, the SNR of a single transit is
calculated assuming a box-shaped transit with depth of (Rp /R∗ )2 and duration calculated from the
equation in (Kipping 2010). A random epoch is assigned and the number of transits in a 127-day
period (corresponding to the duration of Q1-Q2) is calculated. To account for the duty cycle of
92%, each of these transits can be independently and randomly lost with 8% probability. The
total signal is then calculated as the signal for a single transit times the square root of the number
of observed transits. This is compared to the estimated noise over the course of the duration of
the transit, calculated from the TMCDPP values for this star from Quarter 2 (see Kepler Data
Release Notes). As described above, we require that our simulated planets have SNR of 9.2 in
the simulated Quarters 1 and 2 in order to be detected. We find that approximately 30-50% of

– 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 different
choices for Np and σi ) with the observed distribution from Kepler . An assessment of whether
a particular simulation is rejectable is done using a G-test (similar to the categorical χ2 test) by
calculating the G-statistic:
G= 2Oi ln(Oi /Ei ) (8)
i

where Oi is the number of systems with i planets observed by Kepler and Ei is the expected number
of such systems from the simulated population, which is scaled so that the total values of O and
E are the same. The values Oi and Ei always refer to the number of systems with i observed and
expected detectable transiting planets. Note that this statistic gives the most weight to categories
with large numbers of objects and no weight to the categories where there are no observed or
expected 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, it
is 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 true
underlying distribution and their G-statistics are calculated. The significance of the observed value
of G is calculated by comparing it to the Monte Carlo generated G-statistics. For example, if the
observed value of G is 10 and this is larger than 95% of the Monte Carlo values of G, then the
hypothesis that the population observed by Kepler is the same as the simulated population can be
rejected 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 about
0.4, 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 typical
multiplicity. Our investigation could address the frequency of planetary systems by adding a new
free parameter fp and fitting to Oi with 0 ≤ i ≤ 5. However, this fit would be equivalent to: fitting
a scaled population of systems that all contained planets to Oi , i > 0; calculating the scaled number
of systems that had planets but did not transit, E0 ; and setting E0 = (1 − fp )O0 . That is, we can
return to the question of the frequency of planetary systems after we find simulated populations
that 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 observed
distribution Oi for 1 ≤ i ≤ 5. The only population that adequately describes the singly-transiting
systems has a uniform distribution of two planets per star (Np,U = 2) and an inclination width of
σi = 4◦ . While the G-statistic shows that this solution has high confidence, clearly any population

– 19 –

that has only 2 planets per star cannot explain the large number of triples and higher-multiplicities
seen 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 have
no weight. However, this model is useful to point out as one that correctly reproduces the ratio
of doubles to singles by requiring a typical inclination dispersion of σi = 4◦ , corresponding to a
Rayleigh distribution with a mean of ¯ ≡ σi π/2 ≃ 5◦ .
i

All other simulated populations produce relatively fewer singly-transiting systems than are
observed. The difficulty can be seen qualitatively by comparing the ratio of doubly-transiting
systems to singly-transiting systems (O2 /O1 = 14.5%) to the ratio of triply-transiting systems to
doubly-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 mutual
inclination (Ragozzine & Holman 2010). We have also found that the probability that a planet
passed our detection threshold is more-or-less independent of multiplicity, i.e., the detectability can
be estimated on a planet-by-planet basis. (In detail, systems with high multiplicities tend to have
planets with longer periods which lowers the SNR somewhat.) The simulated populations have
both the geometric and detectability probabilities decreasing at an approximately constant rate as
additional 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 similar
and Kepler seems to be seeing an “excess” of singly-transiting systems. Note that this cannot be
attributed 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 the
inability to create a simulated population that matches the observations of multiples by Kepler is
due to a shortcoming in the model, a theoretically motivated alternative is that the true planet
population that has a bimodal distribution of characteristic inclination, with some have small
inclinations and typically more planets than the other group. This is strongly suggested for distant
planets (e.g., Levison et al. 1998) and may be present in the close-in population we are modeling
here due to a distinction between systems that have had large dynamical scatterings in the past
versus those systems which have remained relatively calm.

As we are only attempting to match 5 multiplicity frequencies Oi , we cannot justify adding a
full 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 simulated
populations are compared again to the observations, but are now fit only to the observed population
with two or more transiting planets (i.e., 2 ≤ i ≤ 5) using the same G-statistic method described
above. Most of the constraint on this population is from the ratio of O3 /O2 . Note that a second
planetary population with a large inclination dispersion would also contribute slightly to the number
of 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 with
Np,U = 3.25 (i.e., 75% of systems with 3 planets and 25% of systems with 4 planets) and with a

– 20 –

Rayleigh inclination width of σi = 2◦ , implying a mean inclination of ¯ ≃ 2.5◦ . This population
i
produced Ei = (62.1, 17.5, 1.4, 0) for i from 2-5, which is an excellent match to the observed values
Oi = (63, 17, 1, 1). (Recall that the expected numbers are scaled so that the total number of planets
is the same as the observations.) When scaled to the observed values, this population produced
only E1 = 130.1 singly-transiting systems, suggesting that over half of the observed singles cannot
be 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. Figure
12 gives a contour plot of the acceptability of simulated populations as a function of Np,U and σi . As
expected, increasing the number of planets per system requires the inclination dispersion to increase
in 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 to
Oi , 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 = 2.5
with σi = 4◦ . This simulation can (rarely) produce systems with more than 4 detectable transiting
planets, Ei = (65.3, 14.3, 1.9, 0.4) for i from 2-5, but underproduces the number of triply-transiting
systems. A contour plot showing these the acceptability levels as a function of Np,λ and σi is shown
in Figure 13. This figure also demonstrates the trend that more planets require a larger inclination
dispersion to match the Kepler observations.

It is expected that not every star has planets in this size and period range that would have
been detected by Kepler ; the Sun is an example of such a star. The population of doubles that best
matched the observations for 1 ≤ i ≤ 5 required E0 =9699 stars, i.e., ∼6% of the Kepler sample. The
best-fit population for 2 ≤ i ≤ 5 described above (Np , U = 3.25, σi = 2◦ ) required only E0 = 3225
stars (∼2%) to have, on average, 3.25 planets (within the period and radius limits) to reproduce the
observations 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 in
Figures 12 and 13 had similar values of E0 .) In these latter cases, an additional set of stars with
planets are needed to account for the additional singly-transiting systems. Nevertheless, these
numbers appear quite small and fall about a factor of 2 short of the completeness calculations
reported in B11, even after accounting for the fact that we are considering a different size range
and 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 fraction
of observers will see at least one planet in transit, even if the periods are somewhat large. Our
simulations suggest that only a small fraction (∼5%) of the ∼162000 stars observed by Kepler have
large 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 –

that Kepler will detect among all actual planetary transits, the true fraction of stars with planets
will be different from this estimate, with a more detailed assessment of completeness to be done by
the Kepler Mission in the future. For example, the large errors and, in some cases, biases in the
KIC will propagate into our simulations to some degree, though the ratio of observed multiplicities
which drive the fits above are not as strongly affected as the estimate of the number of stars with
planetary 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 a
superposition of two populations. A small fraction of stars have multiple (3 or more) super-Earth or
Neptune size planets with periods less than 125 days. It appears that many of the doubly-transiting
systems 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 additional
non-transiting planets may be missing in between.

We have also provided the first estimate of the inclination dispersion that has been given for a
large sample of observed planetary systems. To match the numbers of higher multiples observed by
Kepler requires a population with relatively low inclination dispersion; mean inclinations greater
than 5◦ are poor fits. It does not appear that all of the multiples observed by Kepler come from
a population of purely coplanar systems. Systems with multiple super-Earth and Neptune size
planets with periods less than 125 days at low relative inclinations suggest interactions with a
protoplanetary 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 planets
and 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 usually
relatively large, they do contain information on relative inclinations. This can be used both in a
statistical population sense as well as for individual systems.

In some systems, the detection of or upper limits to Transit Duration Variations (TDVs) will
constrain the unobserved component of the relative inclinations of planets. Potential mutual events
in 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’s
equator, though this is observationally challenging for most Kepler stars (Ragozzine & Holman
2010). Each measurement of the true mutual inclination in individual systems helps fill in the
picture for the typical inclination distribution of planetary systems.

– 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 outlier
to the number/frequency distribution for detected transiting planets. Moreover, the period ratio
of the two inner planets is only 1.264, which is the 4th lowest ratio among the 238 neighboring
pairs of transiting Kepler exoplanets (Figure 3). The smallest period ratio is 1.001, between the
candidate co-orbital planets in KOI-730 (Section 5.3). The second smallest period ratio, 1.038, is
between two weak (vetting flag 3) candidates in the KOI-284 system; this system would be unstable
for any reasonable planetary masses if indeed both of these candidates are actually planets in orbit
about the same star (section 4). The third smallest period ratio, 1.258, is for a pair of candidates in
the KOI-191 system that stability considerations strongly suggest are locked in a 5:4 mean motion
resonance that prevents close approaches (section 5.4). Thus, Kepler-11b and Kepler-11c may well
have 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 planet
candidates, KOI-500 is second in abundance to Kepler-11 and four of these candidates are close to
one another in terms of period ratio (Table 4). The target star is significantly smaller than the
Sun (∼0.74 R⊙ ; B11), and the orbital periods of the candidates are shorter, so if confirmed the
planets in this system would be most closely spaced in terms of physical distance between orbits
of any several planetary systems known. Moreover, four of the planets orbiting KOI-500 appear
to be locked in three-planet resonances (Section 5.5), whereas no analogous situation is observed
in Kepler-11. Unfortunately, KOI-500 is quite faint, so it will be more difficult to study than is
Kepler-11. The four candidates in KOI-707 have periods within the range of the periods of the 5
inner planets observed in the Kepler-11 system. Periods ratios of neighboring candidates in the
four planet KOI-117 are all less than two, but nonetheless significantly larger than in Kepler-11
and 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-planet
systems, 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-11
an 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 are
coplanar (which is slightly inconsistent with the observed impact parameters), then the probability
of seeing all six planets transit is the same as seeing Kepler-11g transit, which is 1.2%, suggesting
that there are dozens of systems similar to Kepler-11, unless we were observing from a particularly
favorable position.

– 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 validated
as 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 statistical
results.

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 –

The rich population of multi-transiting systems discovered by Kepler and reported by B11
have immense value both as individuals and as a population for improving our understanding of
the formation and evolution of planetary systems. The Kepler spacecraft is scheduled to continue to
return data on these multi-planet system for the remainder of its mission, and the longer temporal
baseline afforded by these data will allow for the discovery of more planets and more accurate
measurements of the planets and their interactions.

Kepler was competitively selected as the tenth Discovery mission. Funding for this mission is
provided by NASA’s Science Mission Directorate. The authors thank the many people who gave
so generously of their time to make the Kepler mission a success. D. C. F. acknowledge NASA
support 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, Josh
Carter, and others for valuable discussions.

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– 28 –

Table 1. Characteristics of Systems with Two Transiting Planets

KOI # Rp,1 (R⊕ ) Rp,2 (R⊕ ) P2 /P1 ∆

72 1.30 2.29 54.083733 90.8
82 6.84 3.70 1.565732 7.6
89 4.36 5.46 1.269541 4.8
112 3.68 1.71 13.770977 55.7
115 3.37 2.20 1.316598 7.4
116 4.73 4.81 3.230779 21.6
123 2.26 2.50 3.274199 34.1
124 2.33 2.83 2.499341 25.5
139 5.66 1.18 67.267271 54.4
150 3.41 3.69 3.398061 26.3
153 3.17 3.07 1.877382 14.2
209 7.55 4.86 2.702205 14.8
220 2.62 0.67 1.703136 17.3
222 2.06 1.68 2.026806 20.5
223 2.75 2.40 12.906148 55.8
232 3.58 1.55 2.161937 20.1
244 4.53 2.65 2.038993 15.5
260 1.19 2.19 9.554264 70.3
270 0.90 1.03 2.676454 49.1
271 1.99 1.82 1.654454 17.5
279 4.90 2.09 1.846201 13.4
282 2.77 0.86 3.252652 36.0
291 1.45 1.05 3.876548 57.0
313 3.10 2.17 2.220841 21.0
314 1.95 1.57 1.675518 14.8
339 1.47 1.08 3.240242 50.6
341 3.32 2.26 1.525757 11.0
343 2.19 1.64 2.352438 28.6
386 3.40 2.90 2.462733 21.7
401 6.24 6.61 5.480100 23.3
416 2.95 2.82 4.847001 35.8
431 3.59 3.50 2.485534 19.6
433 5.78 13.40 81.440719 28.7
440 2.80 2.24 3.198296 30.0
442 1.86 1.43 7.815905 67.9
446 2.31 1.69 1.708833 15.5
448 2.33 3.79 4.301984 28.2
456 3.12 1.66 3.179075 31.6
459 3.69 1.28 2.810266 26.6
464 7.08 2.66 10.908382 33.2
474 2.34 2.32 2.648401 28.7
475 2.36 2.63 1.871903 17.0
490 2.29 2.25 1.685884 15.1
497 2.49 1.72 2.981091 33.8
508 3.76 3.52 2.101382 16.3
509 2.67 2.86 2.750973 25.6

– 29 –

Table 1—Continued

KOI # Rp,1 (R⊕ ) Rp,2 (R⊕ ) P2 /P1 ∆

510 2.67 2.67 2.172874 20.6
518 2.37 1.91 3.146856 32.3
523 7.31 2.72 1.340780 4.9
534 2.05 1.38 2.339334 28.7
543 1.89 1.53 1.371064 11.1
551 2.12 1.79 2.045843 23.5
555 1.51 2.27 23.366026 77.3
564 2.35 4.99 6.073030 34.0
573 3.15 2.10 2.908328 28.3
584 1.58 1.51 2.138058 28.5
590 2.10 2.22 4.451198 44.3
597 2.60 1.40 8.272799 59.7
612 3.51 3.62 2.286744 18.1
638 4.76 4.15 2.838628 19.5
645 2.64 2.48 2.796998 28.9
657 1.59 1.90 4.001270 42.5
658 1.53 2.22 1.698143 18.1
663 1.88 1.75 7.369380 52.0
672 3.98 4.48 2.595003 18.8
676 4.48 2.94 3.249810 20.4
691 2.88 1.29 1.828544 18.8
693 1.80 1.72 1.837696 22.2
700 3.05 1.93 3.297027 32.2
708 2.19 1.71 2.262657 27.3
736 2.64 1.96 2.789104 24.7
738 3.30 2.86 1.285871 6.2
749 1.98 1.35 1.357411 11.0
752 2.69 3.39 5.734874 39.3
775 2.11 2.47 2.079976 17.9
787 2.87 2.19 1.284008 7.0
800 2.73 2.52 2.659942 26.3
837 1.83 1.45 1.919049 22.3
841 4.00 4.91 2.042803 13.0
842 2.78 3.14 2.835637 22.9
853 2.87 2.12 1.767103 15.2
869 3.19 4.33 4.845276 30.2
870 3.62 3.45 1.519922 8.9
877 2.50 2.32 2.021803 17.0
881 2.54 3.92 10.793324 43.8
896 3.85 2.81 2.574418 20.6
904 2.11 2.96 12.635882 50.2
936 3.46 2.04 10.601824 40.6
938 2.89 1.28 9.512348 57.5
945 2.04 2.64 1.575035 13.6
954 2.35 2.45 4.550137 40.9
1015 2.36 1.63 2.305817 27.5

– 30 –

Table 1—Continued

KOI # Rp,1 (R⊕ ) Rp,2 (R⊕ ) P2 /P1 ∆

1060 1.54 1.23 2.545144 39.7
1089 9.64 5.69 7.094116 22.6
1102 3.22 0.94 1.513919 12.3
1113 2.65 2.81 3.217321 30.7
1151 1.17 1.20 1.420286 16.6
1163 1.90 1.77 2.729492 33.8
1198 2.04 1.53 1.561840 16.0
1203 2.51 2.44 2.256556 23.0
1215 2.21 2.12 1.905318 20.8
1221 5.03 5.31 1.694236 10.0
1236 2.78 1.67 5.807534 49.5
1241 10.43 6.99 2.039957 9.3
1278 1.57 2.39 3.575571 40.0
1301 1.86 2.32 2.954666 32.3
1307 2.96 2.81 2.204782 20.1
1360 2.73 2.29 2.520266 24.0
1364 2.86 2.74 2.952861 27.1
1396 2.53 1.86 1.790301 17.8
1475 1.79 2.18 5.910858 43.3
1486 8.45 2.38 8.433622 27.9
1589 2.23 2.28 1.476360 12.0
1590 2.75 1.90 10.943201 55.8
1596 2.26 3.45 17.785655 54.0

– 31 –

Table 2. Characteristics of Systems with Three Transiting Planets

KOI # Rp,1 (R⊕ ) Rp,2 (R⊕ ) Pmiddle/Pinner ∆inner Rp,3 (R⊕ ) Pouter /Pmiddle ∆outer

85 3.24 1.67 2.719393 28.2 2.01 1.387583 9.2
94 12.64 4.01 2.143302 9.0 6.89 4.052306 14.9
111 2.46 2.25 2.071194 21.5 2.51 4.373353 40.8
137 6.04 8.56 2.180367 14.4 2.32 1.944495 8.9
148 2.12 2.97 2.024651 18.7 2.04 4.434225 37.6
152 4.92 2.54 2.032345 20.4 2.41 1.900850 13.5
156 1.90 1.64 1.549841 14.0 2.77 1.464437 10.0
168 3.69 1.86 1.395701 11.9 2.00 1.511764 10.9
248 2.86 2.49 2.795806 23.1 1.99 1.515099 9.1
250 3.63 3.61 3.465802 25.5 1.28 1.404620 6.1
284 2.53 1.96 1.038334 1.3 1.87 2.807616 31.5
351 8.50 6.02 3.522872 23.1 1.95 1.575869 6.2
377 5.73 6.20 12.089890 40.6 1.04 2.020508 10.7
398 8.61 3.49 2.417105 21.8 1.92 12.403143 29.2
408 3.63 2.87 1.701564 12.5 2.56 2.454333 23.6
481 2.47 1.65 4.922978 45.4 2.97 4.478249 36.0
520 3.06 1.96 2.348524 22.6 2.75 2.018139 17.2
528 3.14 3.43 2.146104 18.2 3.19 4.703530 33.6
567 2.89 2.29 1.899657 17.5 2.23 1.429507 10.9
571 1.70 1.97 1.869767 17.7 1.81 1.836067 16.7
623 2.04 1.74 1.848406 21.8 1.80 1.514823 15.0
665 2.28 1.19 1.905525 33.9 0.84 1.910436 23.9
701 2.22 1.52 3.178366 35.2 1.73 6.737918 52.2
711 2.74 1.33 12.349935 64.8 2.63 2.785831 26.8
718 1.59 1.77 4.953555 54.9 1.49 2.108971 27.9
723 2.81 2.87 2.562570 22.2 3.17 2.783488 23.9
733 2.25 1.93 1.891175 20.2 1.50 1.915474 19.3
756 3.65 2.60 1.610874 14.8 1.76 2.683265 23.3
757 4.84 3.29 2.569790 18.5 2.28 2.563540 17.3
806 8.99 12.05 2.068502 8.3 3.10 2.373352 8.7
812 2.45 2.19 6.005814 39.0 2.12 2.302249 20.8
829 2.63 1.93 1.912284 19.4 2.72 2.067646 19.5
864 2.17 1.79 2.265275 25.5 1.82 2.052797 24.3
884 3.00 2.74 2.829393 27.8 1.24 2.169267 18.1
898 3.03 2.41 1.889897 13.8 2.54 2.056092 15.3
899 1.69 1.28 2.151439 22.7 1.72 2.160301 20.8
907 3.52 3.41 3.446956 30.7 2.07 1.824674 13.7
921 2.30 2.68 2.717125 31.1 1.48 1.762252 15.2
934 3.16 2.02 2.130198 20.2 2.41 1.510392 12.6
935 3.59 3.16 2.043775 16.5 2.47 2.055839 18.8
941 5.39 3.43 2.762270 18.3 6.61 3.747567 19.1
961 3.94 14.44 2.677706 8.5 10.72 1.536625 4.6
1306 2.14 2.20 1.930557 20.5 1.81 1.705578 17.6
1422 3.06 2.92 1.613089 10.5 2.02 3.398094 23.4
1426 3.29 7.74 1.926989 10.3 12.30 2.002740 7.5

– 33 –

Table 3. Characteristics of Systems with Four Transiting Planets

KOI # Rp,1 (R⊕ ) Rp,2 (R⊕ ) P2 /P1 ∆12 Rp,3 (R⊕ ) P3 /P2 ∆23 Rp,4 (R⊕ ) P4 /P3 ∆34

70 2.27 1.60 1.649977 22.6 1.97 1.779783 20.8 0.60 7.150242 53.5
117 2.39 1.32 1.541373 19.4 1.29 1.623476 25.1 0.68 1.853594 22.3
191 11.56 2.81 3.412824 35.8 1.39 6.350795 20.2 1.49 1.258207 2.9
707 3.36 2.64 1.652824 13.4 2.48 1.459577 9.8 2.19 1.290927 7.3
730 3.09 2.29 1.333812 9.4 2.45 1.001034 0.0 1.83 1.499447 10.7
834 4.87 1.85 2.943873 44.2 1.39 2.149832 28.5 1.39 1.787503 12.5
880 4.90 5.76 2.476866 25.6 2.76 4.480164 28.7 2.01 1.948715 11.1
952 2.27 2.32 2.037706 19.5 2.44 1.483153 9.2 1.14 2.602734 21.3

Table 4. Characteristics of Systems with Five or Six Transiting Planets

property KOI-500 KOI-157 (nominal Kepler-11) Kepler-11 (Lissauer et al. 2011)

Rp,1 (R⊕ ) 2.75 3.03 1.97±0.19
Rp,2 (R⊕ ) 2.83 3.50 3.15±0.30
P2 /P1 3.113327 1.264079 1.26410
∆12 40.5 6.8 7.0
Rp,3 (R⊕ ) 1.47 4.21 3.43±0.32
P3 /P2 1.512077 1.741808 1.74182
∆23 12.7 13.1 15.9
Rp,4 (R⊕ ) 2.06 2.22 4.52±0.43
P4 /P3 1.518394 1.410300 1.41031
∆34 10.4 7.3 10.9
Rp,5 (R⊕ ) 1.23 3.24 2.61±0.25
P5 /P4 1.349929 1.459232 1.45921
∆45 6.8 8.8 13.3
Rp,6 (R⊕ ) — 1.70 3.66±0.35
P6 /P5 — 2.535456 2.53547
∆56 — 24.1 —

Table 5: Counts of systems found in each resonance neighborhood.
MMR # Systems
2:1 90
3:2 24
4:3 7
5:4 3
3:1 80
5:3 15
7:5 5
9:7 3

– 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 are
the only candidate for their given star are represented by black dots, those in two-planet systems
as blue circles (open for the inner planets, filled for the outer ones), those in three-planet systems
as red triangles (open for the inner planets, filled for the middle ones, filled with black borders for
the outer ones), those in four planet systems as purple squares (inner and outer members filled
with black borders, second members open, third filled), the five candidates of KOI 500 as orange
pentagons and the six planets orbiting KOI-157 (Kepler-11) as green hexagons. It is immediately
apparent that there is a paucity of giant planets in multi-planet systems; this difference in the size
distributions is quantified and discussed by Latham et al. (2011, in preparation).

– 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 System

Fig. 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 the
same number of objects and the same mean value (thick green line and filled circles). The column
for stars with no detected planets goes far beyond the scale of the figure. Also shown (by the
dashed red line and filled circles) are the numbers of exoplanet systems detected by radial velocity
surveys (data obtained from exoplanets.org ¡http://exoplanets.org¿ on 2011 Jan 28) scaled to the
same total number of objects as Kepler candidates. The inset in the upper right zooms in to reveal
the structure of the curves for systems with three or more planets.

– 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 Ratio

Fig. 3.— Cumulative number of pairs of Kepler planets orbiting the same star with period ratio less
than the value speciﬁed. Black dashed curve shows all pairings, solid black curve shows pairings of
neighboring planets; solid red curve shows all pairings in two candidate systems; solid blue curve
shows all pairings in three-planet systems, solid light blue curve shows inner pair of planets in
three-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 shows
neighboring pairings in the 4, 5 and 6 planet system.

– 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 Ratio

Fig. 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 for
neighboring 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 –

1
Kepler 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 Ratio

Fig. 5.— ”Slope” of the cumulative fraction of Kepler neighboring planet pairs (solid curve) and
multiplanet systems detected via radial velocity (dashed curve) with period ratio exceeding the
value specified. The slopes were computed by taking the difference in period ratio between points
with N differing by 4 and dividing by 4. The spikes show excess planets piling up near period
commensurabilities.

– 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 Ratio

Fig. 6.— The ξ statistic as a function of period ratio for the ﬁrst order (solid, blue) and second
order (dashed, red) MMRs. Also shown are the observed period ratios in the Kepler data.

– 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
Ξ

Fig. 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 just
inside the MMR. Note that the neighborhood of the 3:1 MMR has some planets just interior to
the resonance while none of the other second-order resonances do (see Figure 6. The probability
density for the logarithmically distributed test sample is also shown for reference and appears as a
monotonically decreasing function in both plots (blue).

– 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
Ξ

Fig. 8.— Cumulative distribution function (CDF) of the absolute value of the ξ statistic for the
combined ﬁrst and second-order resonances (blue). Also shown is the CDF for the logarithmically
distributed test sample (red).

– 42 –

Fig. 9.— Larger and smaller of the orbital separations, expressed in mutual Hill radii, for adjacent
3-planet sub-systems of the 3-, 4-, 5-, and 6-planet systems of our sample. The symbols are the same
as 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 used
for the simulated population of multiplanet systems: ∆inner + ∆outer > 18; some of our integrations
survived long-term integrations despite transgressing that boundary. The unstable nominal systems
KOI-191, KOI-248, and KOI-730 all lie within the gray region, and all the long-term survivors are
outside of it.

– 43 –

Fig. 10.— Cumulative distribution of the ratio of planetary radii (Rp,out /Rp,in ) for neighboring
pairs 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 2:1
MMR (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 to
have similar radii, and in the majority of cases, the outer planet is slightly larger.

– 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 the
simulated populations of Section 6.

– 45 –

Fig. 12.— Contour plot of the probability that a simulated population matches the Kepler observed
population 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 shows
the average number of planets assigned to all stars in the “uniform” model and the y-axis shows
the Rayleigh width parameter σi for the inclinations in degrees. A plus sign marks the best-fit
model (which cannot be rejected at a 93% confidence level) described in the text. Note the clear
trend that the simulated population must have an increased inclination dispersion if the number
of planets per star is increased in order to match the Kepler observations. Populations with mean
inclinations greater than ¯ ≡ σi π/2 ≃ 5◦ are poor fits, as are completely coplanar configurations.
i

– 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 are
worse ﬁts to the observed numbers than the “uniform” model, but show the same trends.

1

0.9

0.8

0.7

0.6
N

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

500

450 RV adjacent pairings

400

350

300
N

250

200

150

100

50

0
1 1.5 2 2.5 3 3.5 4 4.5 5
Period Ratio

500
450
Slope of Cumulative Period Ratio

400

350

300

250

200

150

100

50

0
1 1.5 2 2.5 3 3.5 4 4.5 5
Period Ratio

Architecture and dynamics of kepler's candidate multiple transiting planet systems

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