#  Almost All of Kepler’s Multiple Planet Candidates are Planets

Jack J. Lissauer11affiliation: NASA Ames Research Center, Moffett Field, CA
94035, USA , Geoffrey W. Marcy22affiliation: Astronomy Department, University
of California, Berkeley, CA 94720, USA , Jason F. Rowe11affiliation: NASA Ames
Research Center, Moffett Field, CA 94035, USA 33affiliation: SETI
Institute/NASA Ames Research Center, Moffett Field, CA 94035, USA , Stephen T.
Bryson11affiliation: NASA Ames Research Center, Moffett Field, CA 94035, USA ,
Elisabeth Adams44affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138, USA , Lars A. Buchhave55affiliation: Niels
Bohr Institute, University of Copenhagen, DK-2100, Copenhagen, Denmark
66affiliation: Centre for Star and Planet Formation, Natural History Museum of
Denmark, University of Copenhagen, DK-1350 Copenhagen, Denmark , David R.
Ciardi77affiliation: Exoplanet Science Institute/Caltech, Pasadena, CA 91125,
USA , William D. Cochran88affiliation: Department of Astronomy, University of
Texas, Austin, TX 78712, USA , Daniel C. Fabrycky99affiliation: Department of
Astronomy & Astrophysics, University of California, Santa Cruz, CA 95064, USA
1010affiliationmark:  , Eric B. Ford1212affiliation: University of Florida,
211 Bryant Space Science Center, Gainesville, FL 32611, USA , Francois
Fressin44affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA , John Geary44affiliation: Harvard-
Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138,
USA , Ronald L. Gilliland1313affiliation: Space Telescope Science Institute,
Baltimore, MD 21218, USA , Matthew J. Holman44affiliation: Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA , Steve B.
Howell11affiliation: NASA Ames Research Center, Moffett Field, CA 94035, USA ,
Jon M. Jenkins11affiliation: NASA Ames Research Center, Moffett Field, CA
94035, USA 33affiliation: SETI Institute/NASA Ames Research Center, Moffett
Field, CA 94035, USA , Karen Kinemuchi11affiliation: NASA Ames Research
Center, Moffett Field, CA 94035, USA 1414affiliation: Bay Area Environmental
Institute, CA , David G. Koch11affiliation: NASA Ames Research Center, Moffett
Field, CA 94035, USA , Robert C. Morehead1212affiliation: University of
Florida, 211 Bryant Space Science Center, Gainesville, FL 32611, USA , Darin
Ragozzine44affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA , Shawn E. Seader11affiliation: NASA Ames
Research Center, Moffett Field, CA 94035, USA 33affiliation: SETI
Institute/NASA Ames Research Center, Moffett Field, CA 94035, USA , Peter G.
Tanenbaum11affiliation: NASA Ames Research Center, Moffett Field, CA 94035,
USA 33affiliation: SETI Institute/NASA Ames Research Center, Moffett Field, CA
94035, USA , Guillermo Torres44affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA , Joseph D.
Twicken11affiliation: NASA Ames Research Center, Moffett Field, CA 94035, USA
33affiliation: SETI Institute/NASA Ames Research Center, Moffett Field, CA
94035, USA<EMAIL_ADDRESS>
###### Abstract

We present a statistical analysis that demonstrates that the overwhelming
majority of Kepler candidate multiple transiting systems (multis) indeed
represent true, physically-associated transiting planets. Binary stars provide
the primary source of false positives among Kepler planet candidates, implying
that false positives should be nearly randomly-distributed among Kepler
targets. In contrast, true transiting planets would appear clustered around a
smaller number of Kepler targets if detectable planets tend to come in systems
and/or if the orbital planes of planets encircling the same star are
correlated. There are more than one hundred times as many Kepler planet
candidates in multi-candidate systems as would be predicted from a random
distribution of candidates, implying that the vast majority are true planets.
Most of these multis are multiple planet systems orbiting the Kepler target
star, but there are likely cases where (a) the planetary system orbits a
fainter star, and the planets are thus significantly larger than has been
estimated, or (b) the planets orbit different stars within a binary/multiple
star system. We use the low overall false positive rate among Kepler multis,
together with analysis of Kepler spacecraft and ground-based data, to validate
the closely-packed Kepler-33 planetary system, which orbits a star that has
evolved somewhat off of the main sequence. Kepler-33 hosts five transiting
planets with periods ranging from 5.67 to 41 days.

###### Subject headings:

planetary systems; stars: individual (Kepler-33/KOI-707/KIC 9458613); methods:
statistical

††slugcomment: ApJ in press1111affiliationtext: Hubble Fellow

##  1\.  Introduction

Roughly one-third of Kepler’s planet candidates announced by Borucki et al.
(2011) are associated with targets that have more than one candidate planet.
False positives (FPs) plague ground-based transit searches, but the exquisite
quality of Kepler photometry, combined with the ability to measure small
deviations in center of light during transits (Jenkins et al., 2010; Batalha
et al., 2010), have been used to cleanse the sample prior to presentation in
Borucki et al. (2011). Accounting for candidates on each one’s individual
merit, Morton & Johnson (2011) estimated the fidelity of Kepler’s planet
candidates (fraction of the candidates expected to be actual planets) to be
above $90\%$. Yet the fidelity of multiple planet candidates is likely to be
higher than that for singles (Latham et al., 2011; Lissauer et al., 2011a). We
show herein that the vast majority of Kepler’s multiple planet candidates are
true multiple planet systems.

The majority of Kepler planet discoveries announced to date have been
confirmed dynamically, using radial velocity variations of the target star or
departures of observed transit times from a linear ephemeris (transit timing
variations, TTVs). However, three of the first sixteen Kepler planets do not
have dynamical evidence supporting the discovery, but rather these planets
have been “validated” by showing that the probability that the observed signal
is produced by a planet around the host star is at least 100 times as large as
that of a false positive (Torres et al., 2011; Fressin et al., 2011; Lissauer
et al., 2011b). For these three planets, the validation process was based on
the results of the BLENDER code, which performs detailed comparisons between
the observed Kepler lightcurve and the lightcurves predicted for an ensemble
of theoretically conceivable FPs to reject FP models that are not consistent
with the observations. BLENDER then sums the a priori probability for the
allowed false positives and compares that to the a priori probability for the
planet model. BLENDER validates a planet when the resulting odds ratio very
strongly favors the planet model. Although all three of these planets orbit
stars that also have dynamically-confirmed planets, planetary multiplicity was
not used quantitatively to aid in their validation.

Morton & Johnson (2011) do a more cursory analysis, but one that is far
easier, enabling them to perform it for all of the candidate planets
(“candidates”, henceforth) that are listed in the Borucki et al. (2011) paper.
The study by Morton & Johnson (2011) assumes a flat 20% planet occurrence rate
as a prior for their Bayesian analysis (i.e., the average number of planets
per star is assumed to be 0.2), and they do not consider planets orbiting
background stars to be FPs. Morton & Johnson (2011) use the depth and period
of the transit, together with the magnitude, stellar properties and galactic
latitude of the target star (henceforth “target”) to provide an upper bound on
the FP prior; for high SNR transits, this upper bound on the FP prior can be
much higher than the one computed using detailed lightcurve matching by
BLENDER to rule out many FP scenarios (Torres et al., 2011; Fressin et al.,
2011). Indeed, Morton & Johnson (2011) do not examine Kepler lightcurves, but
rather assume for their calculations that various false-positive vetting
procedures using Kepler photometry have been done, such as elimination of
candidates with V-shaped lightcurves or where the centroid of light received
on the Kepler focal plane is in a different position during a transit than it
is outside of transit, and that the planet candidates show no evidence for
being FPs. However, these cuts were not applied to all of the candidates
listed in Borucki et al. (2011). Some candidates were not vetted as
extensively as assumed by Morton & Johnson (2011), and other candidates did
not pass one of the cleanness tests but still remain viable, if somewhat more
suspect, planetary candidates (e.g., grazing planetary transits, as well as
grazing stellar eclipses, produce V-shaped lightcurves), and thus remain on
Borucki et al. (2011)’s candidate list.

We develop herein a quantitative method to estimate the true planet fraction
of the overall sample of Kepler candidate multiple planets that is based upon
the observation that there are far more multiples than would be expected for a
random distribution of candidates among Kepler targets (Lissauer et al.,
2011a). This “overabundance” of multiples is illustrated by the fact that
fewer than 1% of targets have a candidate on Borucki et al. (2011)’s candidate
list, whereas more than 15% of targets with at least one candidate have
multiple candidates, and more than 25% of targets with at least two candidates
have a third candidate. Such a result is expected if planets often appear
within systems of planets, especially if such systems tend to be flat in the
distribution of inclinations. High multiplicity, low inclination, planetary
systems are consistent with Kepler observations (Lissauer et al., 2011a).

We describe an approach to validation of specific multiple planet system
candidates that is intermediate between those of Morton & Johnson (2011) and
BLENDER, and we apply these techniques to validate the very flat and
dynamically-packed KOI-707 system111Targets with planetary candidates are
given an integer KOI (Kepler Object of Interest) number, and individual
candidates are denoted by the target KOI number followed by a decimal point
and a two digit number specifying the order in which they were identified. For
example, KOI-707.02 is the second planetary candidate to have been identified
by transit-like signatures in the lightcurve of the target star KOI-707.. In
this approach, applied herein exclusively to KOI-707, we examine the target
and candidates individually, supplementing the Kepler Input Catalog (KIC,
Brown et al., 2011) with spectroscopic observations of the target and high-
resolution imaging, including Keck guider camera exposures and speckle imaging
(Howell et al., 2011) and/or adaptive optics (Troy et al., 2000; Hayward et
al., 2001), to search for other stars in the target aperture. We check the
lightcurve for a shape consistent with a transit. We verify that the centroid
motion is nil or consistent with a transit around the target star and not a
transit/ eclipse of any other stars observed nearby. But we do not perform
detailed matching of observed lightcurves to FP scenarios. We then use
multiplicity to effectively increase the planet priors and allow for
validation in a Bayesian sense. Note that for candidate giant ($\sim$ 1
Jupiter radius) planets, one would need to consider the possibility of the
transit signal being produced by a late M star partially eclipsing the target
star (perhaps with dilution); for smaller candidates, the possibility of
transits of fainter stars by larger planets must be taken into account.

The probability of a candidate in a multiple system being a false positive is
contrasted to that of a single candidate in Section 2. Other factors involved
in validating transiting planet candidates are discussed in Section 3. We
apply our method to the KOI-707 candidate multi-planet system in Section 4. We
conclude in Section 5 with a summary of our results and a discussion of their
implications.

##  2\.  Statistical Validation of Multi-Planet Candidates

Kepler has found far more multiple candidate systems than would be the case if
candidates were randomly distributed among target stars (Lissauer et al.,
2011a). False positives are expected to be nearly randomly distributed among
Kepler targets, whereas true transiting planets could be clustered if planets
whose sizes and periods are adequate for transits to be detected often come in
multiples, as is the case for planets detected by radial velocity variations
(Wright et al., 2009), and/or if planetary systems tend to be flat, so
geometry leads to higher transit probabilities for other planets if one planet
is seen to transit (Ragozzine & Holman, 2010). We quantify the excess of
multiples and its implications for the overall fidelity of the sample of
multiple candidates in this section.

We use the following notation and input parameters: The number of planet
candidates is $n_{c}=1199$, the number of targets with two or more candidates,
i.e., the number of candidate multiple planet systems, is $n_{m}=170$, and the
number of targets from which the sample is drawn is $n_{t}$ = 160,171. These
numerical values are taken from Lissauer et al. (2011a), who removed from the
1235 candidates listed in Borucki et al. (2011) those objects that were
estimated to have radii greater than twice that of Jupiter, those for which
only one transit had been observed, and those identified as FPs by Howard et
al. (2011). We denote the number of actual planets among the candidates as
$n_{p}$, the number of false positives as $n_{f}$, and the fidelity of the
sample (fraction of candidates that are planets) as $P$, so
$n_{c}=n_{p}+n_{f}$ and $P=n_{p}/n_{c}$.

For our calculations in Equations (1–9), we make the following two
assumptions:

i) FPs are randomly distributed among the targets;

ii) there is no correlation between the probability of a target to host one or
more detectable planets and to display FPs.

In contrast, we do not assume that planets are randomly distributed among the
targets. The previous verifications of the two-planet Kepler-10 system
(Batalha et al., 2011; Fressin et al., 2011), the three-planet Kepler-9
(Holman et al., 2010; Torres et al., 2011) and Kepler-18 systems (Cochran et
al., 2011), and especially the six-planet Kepler-11 system (Lissauer et al.,
2011b), provide strong evidence of the non-random distribution of transiting
planets among Kepler target stars.

The above two assumptions, together with observed values of $n_{c}$, $n_{m}$
and $n_{t}$ and an assumed lower bound on $P$, allow us to estimate lower
bounds on the fidelity of candidates in various classes of candidate multi-
planet systems using simple algebra. For example, assumption (1) implies that
the expected number of targets with $k$ false positives, $E(k)$, is given by a
Poisson distribution of $n_{t}$ members whose mean is given by the average
number of false positives per target, $\lambda=(1-P)n_{c}/n_{t}$. This
expectation value is given by the following formula:

$\displaystyle E(k)=\frac{\lambda^{k}e^{-\lambda}}{k!}n_{t}$ $\displaystyle=$
$\displaystyle\frac{((1-P)\frac{n_{c}}{n_{t}})^{k}e^{-(1-P)\frac{n_{c}}{n_{t}}}}{k!}n_{t}$
$\displaystyle\approx$
$\displaystyle\frac{((1-P)\frac{n_{c}}{n_{t}})^{k}}{k!}n_{t},$

where the approximation in Equation (1) is valid for $\lambda\ll 1$.

We next compute the expected number of Kepler targets with multiple candidates
at least one of which is a FP. In some cases, we make approximations that
yield conservative estimates of the expected numbers of true planets (i.e.,
overestimate the expected number of FPs). Our results, presented in Equations
(2–7), are given first as general formulae, then as a function of the fraction
of candidates that are true planets, $P$, for the observed values of 160,171
targets, 1199 candidates, and 170 multiple candidate systems stated above, and
finally these value if $P$ = 0.5 or 0.9222We choose to present numerical
results for $P=0.9$ because we consider it to be a reasonable estimate of the
actual planet fidelity rate, and $P=0.5$ to show that even for an overall FP
rate far higher than studies suggest, the expected number of FPs among the
multis is remarkably low.. These final numbers are to be compared to the
observation that there are 115 candidate 2-planet systems, with a total of 230
planet candidates, and 55 candidate systems of 3 or more transiting planets,
with a total of 178 planet candidates.

Equation (1) yields estimates of the number of targets with two or three false
positives:

$\displaystyle 2~{}{\rm FPs:}~{}\frac{((1-P)n_{c})^{2}}{2n_{t}}$
$\displaystyle=$ $\displaystyle 4.49(1-P)^{2}$ $\displaystyle=$ $\displaystyle
1.12~{}{\rm or}~{}0.045;$ $\displaystyle 3~{}{\rm
FPs:}~{}\frac{((1-P)n_{c})^{3}}{6n_{t}^{2}}$ $\displaystyle=$ $\displaystyle
0.011(1-P)^{3}$ $\displaystyle=$ $\displaystyle 0.0014~{}{\rm or}~{}1.1\times
10^{-5}.$

The number of targets expected to have four or more false positives is very
small and can be neglected for our purposes. Equation (1) also yields the
expected number of targets with a single FP under the same assumptions, 597.3
or 119.8.

The assumed lack of correlation between the propensity of a target to have
false positives and true planets implies that the probability that a given
target hosts both a planet and one or more false positive is equal to the
product of these individual probabilities. The probability of a target having
a detected transiting planet is given by $Pn_{c}/n_{t}$, and the probability
of it showing a false positive is given by Equation (1). Thus, the estimated
numbers of targets with at least one planet as well as one or more false
positives are given by:

$\displaystyle 1~{}{\rm planet}~{}+1~{}{\rm FP:}$
$\displaystyle\frac{n_{c}P}{n_{t}}\times\frac{n_{c}(1-P)}{n_{t}}\times n_{t}$
$\displaystyle=$ $\displaystyle P(1-P)\frac{n_{c}^{2}}{n_{t}}$
$\displaystyle=$ $\displaystyle 8.98P(1-P)=2.25~{}{\rm or}~{}0.81;$
$\displaystyle 1~{}{\rm planet}~{}+2~{}{\rm FPs:}$ $\displaystyle
P(1-P)^{2}\frac{n_{c}^{3}}{2n_{t}^{2}}$ $\displaystyle=$ $\displaystyle
0.034P(1-P)^{2}$ $\displaystyle=$ $\displaystyle 0.0042~{}{\rm or}~{}3\times
10^{-4}.$

The number of targets expected to have both a planet and three or more false
positives is very small and can be neglected for our purposes. Somewhat lower
estimates of the number of multi- candidate systems with one planet and at
least one FP than given by Equations (4) and (5) can be derived by noting that
the observed multiplicity implies fewer (than $n_{c}$) targets with a nonzero
number of planet candidates.

To derive an estimate of the expected number of targets with multiple true
planets as well as at least one FP, replace the term in Equations (4) and (5)
that represents the probability of a given target having a true planet,
$Pn_{c}/n_{t}$, with a term representing the probability of a given target
having a multi-planet system, $n_{m}/n_{t}$. (Assuming all multis are multi-
planet systems to estimate of this term is conservative in the sense that it
may lead to an overestimate in the number of false positives in multi-
candidate systems.)

$\displaystyle{\rm 2~{}or~{}more~{}planets}~{}+1~{}{\rm FP:}$
$\displaystyle\frac{n_{m}}{n_{t}}\times\frac{n_{c}(1-P)}{n_{t}}\times n_{t}$
$\displaystyle=$ $\displaystyle(1-P)\frac{n_{m}n_{c}}{n_{t}}=1.27(1-P)$
$\displaystyle=$ $\displaystyle 0.64~{}{\rm or}~{}0.13;$

$\displaystyle{\rm 2~{}or~{}more~{}planets}~{}+2~{}{\rm FPs:}$
$\displaystyle\frac{n_{m}}{n_{t}}\times(1-P)^{2}\times\frac{n_{c}^{2}}{2n_{t}}$
$\displaystyle=$ $\displaystyle 0.0048(1-P)^{2}$ $\displaystyle=$
$\displaystyle 0.0012~{}{\rm or}~{}5\times 10^{-5}.$

The number of targets expected to have both multiple planets and three or more
false positives is extremely small and can be neglected for our purposes.

###  2.1. Numerical Estimates for Kepler Candidates

Adding up the above estimates of FPs of different types yields expected totals
of 4.5 or 0.9 FPs (for assumed FP rates of 50% and 10%, respectively) in
doubles (compared to a total of 230 candidates) and 0.65 or 0.13 FPs in
systems with three or more candidates (compared to a total of 178 candidates).
These numbers suggest that validation of doubles at the 99% level by this
method alone is marginal. Triples and higher multiplicity candidates are more
strongly validatable, enough so that the one dynamically-unstable candidate
triple system noted by Lissauer et al. (2011a), KOI-284, which has a pair of
candidates with a period ratio $1.0383$, should be viewed as somewhat
surprising (but see Section 2.4).

It is of interest to compare the (low) false positive rates for multis
estimated in the previous paragraph with the presumed FP rates for singles
upon which their calculations are based. For 50% FPs in singles, our estimates
above correspond to 2% FPs in doubles (compare the first numbers at the end of
equations 2 and 4 with the 230 candidates in Kepler candidate 2-planet
systems) and 0.4% FPs for targets with three or more candidates (compare the
first numbers at the end equations 3 and 5–7 with the 178 candidates in Kepler
candidate 3 or more planet systems); for 10% FPs in singles, we estimate 0.4%
FPs in doubles (compare the final numbers at the end of equations 2 and 4 with
the 230 candidates in Kepler candidate 2-planet systems) and 0.1% for targets
with three or more candidates (compare the final numbers at the end equations
3 and 5–7 with the 178 candidates in Kepler candidate 3 or more planet
systems). These values are consistent with a “multiplicity boost” in the
Bayesian prior for planets of $\sim$ 25 for a system with one additional
candidate and of $\sim$ 100 for a system with more than one additional
candidate. The probability that a target hosts a planet candidate, 1/150, can
be compared to the probability of a target that hosts one candidate hosts
another, 1/6, and that an additional candidate has been found for higher
multiplicity systems, 1/3. These latter values, together with the assumption
that FPs and true planets are uncorrelated, lead to an estimate of the
multiplicity boost for candidate doubles of 30, slightly larger than the value
of 25 computed above, and to a somewhat reduced estimate of $\sim$ 50 for
candidates in systems of higher multiplicity.

The multiplicity boost introduced in the previous paragraph can be used to
quantify the increased probability that a set of transit-like signatures
represent a real transiting planet if one or more additional set of transit-
like signatures are detected for the same Kepler target; i.e., to compute an
estimate the probability of a particular candidate in a multiple candidate
system being a true planet from an estimate of that candidate being a planet
that does not account for multiplicity. We denote the probability of
planethood of the candidate in question computed without factoring in
multiplicity by $P_{1}$ and the probability of it being a planet accounting
for multiplicity by $P_{2}$ if it is in a two candidate system and by $P_{3+}$
if it is a member of a three or more candidate system. In our formulation,
multiplicity effectively increases the Bayesian estimate of the planet prior
but leaves the FP prior unchanged. Thus the probability of the planet
interpretation accounting for multiplicity is related to that neglecting
multiplicity according to the following approximate formulae:

$\displaystyle P_{2}$ $\displaystyle\approx$
$\displaystyle\frac{25P_{1}}{25P_{1}+(1-P_{1})}$ $\displaystyle~{}($
$\displaystyle\approx 1-\frac{1-P_{1}}{25}~{}{\rm for}~{}1-P_{1}\ll 1,~{}{\rm
and}~{}\approx 25P_{1}~{}{\rm for}~{}P_{1}\ll 1)$ $\displaystyle P_{3+}$
$\displaystyle\approx$ $\displaystyle\frac{50P_{1}}{50P_{1}+(1-P_{1})}$
$\displaystyle($ $\displaystyle\approx 1-\frac{1-P_{1}}{50}~{}{\rm
for}~{}1-P_{1}\ll 1,~{}{\rm and}~{}\approx 50P_{1}~{}{\rm for}~{}P_{1}\ll 1),$

where in each case we use the smaller value computed in the previous
paragraph. Rough estimates for the planet probabilities of individual Kepler
candidate multis can be obtained by applying Expressions (8) and (9) to the
estimates of planet probability by Morton & Johnson (2011); the Morton &
Johnson (2011) planet probabilities are given by $P_{1}$ in the above formulae
since their numbers do not account for multiplicity. Note that this
multiplicity boost incorporates self-consistently both the coplanarity boost
discussed in Cochran et al. (2011) and the systems boost (that planets tend to
come in systems) without need to disentangle these effects.

The above calculations of the multiplicity boost do not assume that any of the
candidates of the target under consideration have been confirmed or validated
as planets. Does validating one or more candidate as a true planet(s)
substantially increase the probability that other candidates of the same
target are also planets? For realistic estimates of FP rates (e.g., 10%), the
equations above show that most FPs in multis are likely to be combinations of
1 FP with one or more true planets, so validating one candidate does not
substantially increase confidence in the others from a statistical viewpoint.
Nonetheless, validating one candidate does eliminate some of the non-
statistical issues raised in the following three subsections. In particular,
if that validated candidate is shown to not only be a planet but also to orbit
the Kepler target star, then the probability that the other candidates are
physically-associated with the target increases.

###  2.2. Caveats

To examine the validity of the three assumptions stated above, we divide FPs
into two classes, (I) chance-alignment blends such as background eclipsing
binaries (BGEBs) and (II) physical triple stars (PTs), where grazing eclipses
of the target star by a stellar companion are considered together with PTs for
this purpose. We note that it is a matter of choice how to count cases where
the transit signal is produced by a planet transiting a star other than the
Kepler target. Larger planets transiting stars that are not physically
associated with the star providing most of the target’s flux can be considered
to fall within the chance-alignment blends/BGEB class of FPs (as they must be
for Kepler’s statistical census of planets, because background stars are not
in the denominator of the ratio taken to calculate the fraction of stars with
planets, so their planets must not be included in the numerator), or
alternatively not be regarded as FPs when assessing individual candidates and
the reliability of the sample. Planets transiting physical companions to the
star providing most of the flux can either be viewed as members of the PT
class of FPs or not classified as FPs; how these planets are categorized for
Kepler’s statistical census of planets is also a matter of choice.

Assumptions (i) and (ii) lead to the following caveats:

(i): The expected number of BGEBs (chance-alignment blends) that can cause FPs
varies with the magnitude and galactic latitude of the target star, whereas
those of PTs do not (except to the extent that the distribution of stellar
types among Kepler targets depends somewhat on magnitude). The variations in
expected BGEBs act to increase the expected total number of systems with 2 or
more FPs compared to the value given by the above formulae. When looking at
individual candidate systems, the FP probability of those around faint targets
and at low galactic latitude is larger, whereas the planetary interpretation
becomes more likely for brighter targets and those at higher galactic
latitude. However, the distributions in galactic latitude of the planet
candidates and of the multiple candidate systems both track that of all Kepler
targets, whereas the distribution of expected BGEB FPs is quite different
(Figure 1); this suggests that the fraction of candidates (both singles and
multis) that are produced by BGEBs is small.

Figure 1.— Distributions in galactic latitude of all 163,986 Kepler Quarter 2
exoplanet targets (solid turquoise curve), the 961 targets with planet
candidates meeting our criteria (solid dark red curve) and the 170 multiple
candidate systems (solid black curve). The dashed curve shows the distribution
of expected BGEB FPs, which we compute by weighting each target by the sky
density of stars with magnitudes between 15.0 and 20.0 according to the
Besancon model of the galaxy (Robin et al. 2003, 2004). All curves are
normalized to give a total of unity. Error bars represent the statistical
uncertainty (1$\sigma$ binomial variation) of each of the planet candidate
curves at quartile points in their distributions. The basic shape of the three
solid curves is governed by the orientation of the Kepler field on the sky,
where the amount of area covered increases from a point at galactic latitude
$5^{\circ}$ to a region spanning $16^{\circ}$ in longitude at $13^{\circ}$ in
galactic latitude and then decreases to a point at $21^{\circ}$ latitude; the
second most important factor is a weighting towards the galactic plane because
of the higher spatial density of target stars in that portion of the field.
The dashed curve is more heavily weighted towards the galactic plane because,
to first approximation, it is weighted quadratically in the spatial density of
stars (more precisely, the product of the spatial densities of target stars
and background stars), whereas the other curves are only weighted linearly.
The spatial density of background stars drops by a factor of 17 from the
lowest latitude Kepler observes to the highest latitude. The similarity of
curves for targets, planets and multis, and the difference of these from the
curve estimating BGEB FPs, suggests that the fraction of candidates that are
produced by BGEBs (as well as that resulting from planets transiting
background stars) is small.

The statistical analysis that we presented above does not discriminate between
multiple candidates resulting from multiple planets orbiting a background star
and a single candidate resulting from a single planet orbiting a background
star. However, the correlations and lack thereof between the curves shown in
Figure 1 also imply that the fraction of KOIs caused by single or multiple
planets transiting a background star is small. In some cases, it is especially
unlikely that a multiple candidate system is produced by several planets
orbiting a background star, either because its largest member would be greater
than planetary in size, or because the requirement of dynamical stability
would imply implausibly small densities for the larger planets orbiting the
fainter star. Additionally, as the fraction of giant planets among the multis
is smaller than it is for the singles (Latham et al., 2011), the probability
of a multi being a system of larger planets orbiting a fainter star is less
than that of a single candidate orbiting a star fainter than the target.

A second class of multiple FPs can be produced from a single physical system,
either a triple star with two sets of eclipses or a background star with
multiple transiting planets. Triple star systems are unstable for small period
ratios and produce large eclipse timing variations for moderate period ratios,
so if large timing variations are not detected, we do not need to consider
triple star system FPs (nor grazing eclipsing binary stars with a transiting
planet) unless period ratios are large.

Quadruple star systems consisting of two pairs of eclipsing binaries (e.g.,
Graczyk et al., 2011) could produce a small excess probability of a pair of
FPs. Such systems, as well as line of sight pairs of physically-unrelated EBs,
also might be more likely to mimic planetary-depth transits because of mutual
dilution of the fraction of light in the Kepler aperture.

We note also that FPs are inherently more likely among both small (Earth-size
and smaller) and large (Jupiter-size and larger) candidates than among
intermediate (Neptune-size) candidates, independent of multiplicity. Most
small candidates produce low-amplitude transits that reduce our ability to
remove false positives from the candidate list on the basis of the shape of
the lightcurve or from differences in the centroid of the location of the dip
relative to that of the target. Small candidates, including multis, are more
likely to be planets orbiting stars fainter than the target simply because the
Neptune-size planets that could produce such signals are far more common
according to Kepler observations than are giant planets (Borucki et al.,
2011), and this difference in abundance is even larger for the multis than for
the singles (Latham et al., 2011). At the other extreme of the planet size
distribution, giant candidates appear to be more contaminated by stellar false
positives (Demory & Seager, 2011) of the type which plague ground-based
surveys than do Neptune-size candidates.

(ii): Planets and FPs of both types are more easily detectable for quieter
targets; planets and PTs are more detectable for brighter targets, but BGEBs
that are bright enough to mimic observable transits of the target star are
more likely for fainter targets. A grazing eclipsing binary with a
circumstellar or circumbinary transiting planet analogous to Kepler-16 (Doyle
et al., 2011) would violate our assumption of a lack of correlation between
planets and FPs, but such a configuration requires a large period ratio for
stability and an even larger period ratio for the planet to have modest TTVs.

The assumption of no correlation between planets and FPs would be violated if
the orientations of orbital planes of PTs with one star having a planet are
not random. Alignments of planetary orbital planes for binary stars for which
both stars possess planets also violate this assumption, although members of
this class are true planets that are all physically associated (albeit less
directly) with the target star and thus would for many purposes not be
classified as FPs.

The numbers quoted in Equations (2–9) do not incorporate the caveats noted in
this subsection. Correlations between the propensity of a target to have one
or more FPs and to host one or more identified true planets (ii) are likely to
be at most a factor of a few. False positives of the BGEB class are likely to
be concentrated among faint target stars that are located at low galactic
latitude (Morton & Johnson, 2011); this concentration increases the expected
number of targets with two (or more) FPs, but also points to a substantial
portion of Kepler’s target list whose members have a priori FP likelihood that
is lower than or comparable to those given by the numbers above.

###  2.3. Period Ratios: Resonances & Stability

The distribution of candidate period ratios exhibits spikes near strong mean
motion resonances, which implies that candidates with such period ratios are
more likely to be true planets than those candidates not in or very near such
resonances (Lissauer et al., 2011a). This clustering can be used to increase
confidence in candidates with periods near resonances; such a “near-resonance
boost” can be quantified using the measured values of the normalized distance
from resonance computed from the distribution of planet period ratios in
Lissauer et al. (2011a). Candidate resonant planets can, however, arise from
period aliases in analysis of light curves; such aliases are not of the form
of the astrophysical FPs considered in this paper, and we note that special
care to search for them is an important aspect of validating candidate
resonant planets (see the discussion of KOI-730 and the Note Added in Proof in
Lissauer et al. (2011a)).

Resonant period ratios increase confidence in various candidates, and as
mentioned at the beginning of Section 2.1, one of the candidate systems,
KOI-284, is highly suspect because two of the candidates have periods that
differ from one another by $<4\%$, which would rapidly lead to a dynamical
instability on a very short time scale (but see the discussion of this system
in Section 2.4). Do non-resonant periods reduce the confidence in individual
multis and do stable systems produce an increase in our confidence? The
answers to both parts of this question are yes, but in most cases these
effects are minor. While resonances are overpopulated, a substantial majority
of multis are not in nor near such resonances (Lissauer et al., 2011a; Wright
et al., 2011). And while drawing planetary periods randomly from the observed
ensemble of transiting planet periods gives many more nominally unstable
systems than are observed, the fraction of unstable configurations from such a
random draw is still small. Nonetheless, in high-multiplicity densely-packed
systems, wherein most configurations with the same number of planets randomly
spread over the same range in period would likely be unstable, the boost in
confidence from stability can be significant. Note that special period ratio
boosts and stability boosts are in addition to the standard multiplicity
boost, i.e., they can be applied together as two multiplicative factors in the
planet prior for appropriate systems.

###  2.4. KOI-284 & Multis Composed of Planets Transiting Differing Stars in a
Binary

A very rough estimate of the expected number of multiple candidate systems
that orbit different members of a binary/multiple star system (rather than a
single system of planets orbiting the same star) can be made by assuming that
half of Kepler’s targets are binaries, that each star within a binary is as
likely to host both single and multiple transiting planets as are single
stars, and that the probability of one star within a binary hosting transiting
planets is uncorrelated with that of its binary companion. These assumptions
imply $n_{c}$ planets spread around $3n_{t}/2$ stars, and thus
$(2/3)(n_{c}/n_{t})$ planets per star and $2n_{c}^{2}/(9n_{t})\approx 2$ cases
in which multis are composed of planets transiting differing members of a
stellar binary. Positive correlations between the propensity of stars in a
binary to have transiting planets (resulting from the abundance of planets
and/or alignment of orbital planes) would increase this number, but the fact
that a significant majority of binaries are composed of stars of substantially
differing surface brightness combined with the paucity of giant planets
transiting low luminosity stars should reduce the number of multis comprised
of two planetary systems because of the difficulty of detecting planets around
secondary stars.

As mentioned above, KOI-284, with three planetary candidates having periods of
6.18, 6.42, 18.0 days and nominal sizes of $\sim$ 2 R⊕ (Earth radii), is the
only one of 170 multi-candidate system identified in Borucki et al. (2011)
that would be clearly dynamically unstable if all of the planets orbited the
same star (Lissauer et al., 2011a)333The other multi that was unstable
according to the integrations reported in Lissauer et al. (2011a) was the 4
candidate system KOI-191. This instability resulted from the strong
perturbations of giant planet candidate KOI-191.01 on the nearby outermost
candidate KOI-191.04. As the period ratio of these two candidates was 1.258,
Lissauer et al. (2011a) speculated that the system was librating in a 5:4 mean
motion resonance and that this resonance protected the planets from close
approaches. Subsequent data analysis shows that the period of KOI-191.04 was
underestimated by a factor of two. The updated value of the period of this
candidate is 38.6516 $\pm$ 0.0012 days. The revised period places the planets
much farther apart, and the system is stable with the updated parameters..
Analysis of a McDonald observatory spectrum of this bright (Kepler magnitude
$Kp=11.8$) target yields a temperature of $\sim$ 6250 K and log$g=4.5$, which
suggest a F7V star that is twice as luminous as the Sun. Speckle images of
KOI-284 obtained on 24 June 2010 revealed two stars differing in brightness by
less than one magnitude and separated from one another by slightly less than
1′′ (Howell et al., 2011). Spectroscopic observations of each individual star
obtained at Keck in 2011 show the two stars to have nearly identical spectra
and have a difference in radial velocity of 0.94 $\pm$ 0.1 km/s. The nearly
identical velocities are consistent with their being gravitationally bound.
Their separation of $0.9^{\prime\prime}$ at a distance of roughly 500 pc
implies a projected separation of 450 AU; the relative orbital velocities for
two 1 $M_{\odot}$ stars on a circular orbit with semimajor axis $a$ = 450 AU
is 2 km/s. These observations suggest that the three candidates may well all
be transiting planets in the same stellar system, with one of the stars
hosting one of the six day period planets and the other two planets orbiting
the other star in the binary. Thus, this system would not call into question
the statistical arguments presented in Section 2.1. A detailed analysis of
both Kepler and ground-based data to test this hypothesis and elucidate
particulars is currently underway (Bryson et al., 2012).

##  3\.  Transit Characteristics: Planets vs. False Positives

The numbers derived in Section 2 suggest that the sample of multi-planet
candidates identified by Borucki et al. (2011) is likely to include only a
small fraction of false positives, of order 1%. But because of the
uncertainties associated with the caveats discussed in Section 2, as well as
any factors that we may have omitted, we do not consider it appropriate to
view all of Kepler’s multi-planet candidates to be validated planets at the
99% level. Nonetheless, it is possible to identify candidates with various
favorable characteristics as detailed below, and to examine the available data
for systems individually for signs of potential inconsistencies with the
planet hypothesis. See Haswell (2011) for a more detailed discussion of
transit characteristics.

The shape and duration of a candidate transit can be used to distinguish
events produced by planets from false positives produced by known
astrophysical sources. The BLENDER code (Torres et al., 2011) compares the
observed lightcurve to synthetic lightcurves of both a planet transiting the
target star and a vast array of possible astrophysical false positives
involving binary stars and larger planets transiting fainter stars. If the
planet hypothesis does not provide an acceptable fit to the data, the
candidate is not viable; candidates for which the planet fit is marginal are
called into question, and such candidates are not considered further herein.
Astrophysical false positives are considered credible if they provide an
acceptable fit (3$\sigma$) to the data. Then the a priori likelihoods of the
planet hypothesis and that of a false positive are compared, and a probability
that the apparent transit is caused by a blend is computed. The process can be
very involved, especially for candidates with orbital periods long enough that
astrophysical false positives could plausibly be produced by objects with
eccentric (not tidally-damped) orbits (Fressin et al., 2011; Lissauer et al.,
2011b).

True planetary transits of the star that is the dominant source of light in
the aperture are wavelength-independent (neglecting the small effects of
stellar limb-darkening and contamination by flux from other stars in the
aperture), whereas false positives are not (except in the unlikely case that
the effective temperatures of the contributing stars are similar). By
providing infrared time series spanning times of transit, the Warm Spitzer
Mission can assist in the validation of many transiting planet systems.
Several candidates in multiple systems have been observed, and they all
exhibit achromatic transit depths, consistent with planetary signals (Desert
et al., 2012).

The durations of transit-like signatures can be used to assess the probability
of the planetary interpretation, especially when more than one planet
candidate is present for a given target. Transit and eclipse durations depend
on several factors, including the masses and radii of the two objects, the
impact parameter, $b$, which is defined as the minimum projected separation
between the center of the smaller body (planet in the case of a planetary
transit) and that of the larger body (star) measured in units of the radius of
the larger body, and the period and eccentricity of the orbit. Central
transits (those for which the center of the transiting body passes over the
center of the transited body) by a planet on a circular orbit have durations
that vary as the cube root of the orbital period (Kepler, 1619) and inversely
to the cube root of the stellar density (Seager & Mallén-Ornelas, 2003); these
relationships hold for FPs as long as the bright star is substantially larger
and more massive than the other body. When the system’s mass is dominated by
the primary but the radius of the secondary cannot be ignored, then the
duration, $T_{\rm dur}$, (1st contact – 4th contact, which corresponds to the
values reported in Borucki et al. (2011) and in Table 3 of this paper), of a
central transit for a circular orbit satisfies the following proportionality
relationship:

$T_{\rm dur}\propto\frac{R_{\star}+R_{p}}{R_{\star}}P_{\rm orb}^{1/3}.$ (10)

If the transit duration is measured over the time interval when the center of
the planet covers the disk of the star, the first factor on the right-hand
side of Expression (10) is no longer relevant and the proportionality is
simply to $P_{\rm orb}^{1/3}$. Note that the orbital period, $P_{\rm orb}$, is
very accurately measured by Kepler, as is the ratio of the planet’s radius,
$R_{p}$, to that of the star, $R_{\star}$, provided dilution by nearby stars
is small.

Planetary transits with an impact parameter smaller than half the star’s
radius have a duration that exceeds $3^{1/2}/2\approx 86.67\%$ that of a
central transit. Such transits account for almost half of all transiting
planets, and over half of the transits observed to a specified SNR threshold.
The fraction of transits with durations greater than half that of a central
transit is $3^{1/2}/2$; again specifying a SNR threshold increases the
observed fraction. Orbital eccentricities also influence the durations of
transits and FPs; in most cases, durations are reduced because speeds are
highest near periastron (Kepler, 1609), where geometric considerations imply
that transit probabilities are larger.

##  4\. Validation of the 5-planet system Kepler-33

There are two primary purposes of this work: demonstrating the very high
credibility of the ensemble of Kepler multi-planet candidates and providing a
framework to validate particular systems of candidates as planets. We now turn
to a specific system that is both intrinsically interesting because it is the
closest analog to Kepler-11 observed to date (Lissauer et al., 2011a) and has
characteristics very favorable for validation by multiplicity.

KOI-707 (KIC 9458613) is a $Kp=14$ star for which four planet candidates,
ranging in period from 13 to 41 days, were announced by Borucki et al. (2011);
a fifth candidate planet, with shorter orbital period and smaller size, has
subsequently been identified. The lightcurve of this target is shown in Figure
2.

Figure 2.— Kepler photometric time series data for KOI-707 (KIC 9458613;
Kepler-33) in 29.426 minute intervals (standard Kepler long cadence
observations). Calibrated data from the spacecraft with each quarter
normalized to its median are shown in the top panel. The bottom panel displays
the lightcurve after detrending with a polynomial filter (Rowe et al., 2010).
The entire 8 Quarters of Kepler data analyzed herein are displayed. Note the
difference in vertical scales between the two panels. The midpoint times of
the five sets of periodic transits (including those that were not observed
because they occurred during data gaps) are indicated by triangles of
differing colors. Close-ups of folded lightcurves near transit times of each
of the planets are shown in Figures 7 and 8.

We performed an analysis of pixel-level Kepler spacecraft data that shows that
the transit signals come from a location on the sky that is coincident with
that of the target star within small uncertainty, and we obtained high-
resolution images of the target that did not show any nearby stars capable of
producing the transit signals; these studies are presented in Section 4.1. Our
analyses of the transit lightcurves and spectra taken of the target star are
presented in Section 4.2. In Section 4.3, we review the characteristics of
this system that allow for its validation by multiplicity. We discuss the
properties of the Kepler-33 planetary system in Section 4.4.

###  4.1. Location of Transit Signature on the Plane of the Sky

A seeing-limited image taken at Lick shows a star $7^{\prime\prime}$ to the NW
that is 4.5 magnitudes fainter than KOI-707, but no stars closer to the
target. On 12 June 2011, KOI-707 was imaged in the 880 nm filter
($\Delta\lambda$ = 50 nm) of our speckle camera at the WIYN 3.5-m telescope
located on Kitt Peak. This star showed no companions within a 2.8 $\times$ 2.8
arcsec region down to 3.1 magnitudes fainter than the target star; stars
within $0.05^{\prime\prime}$ of the target star would not have been detected.
Details of this speckle imaging data collection and reduction precesses are
fully described in Howell et al. (2011).

We check to see whether the transit signal is due to a source other than
KOI-707 using the difference image technique described in Torres et al.
(2011). This method fits the measured Kepler pixel response function (PRF) to
a difference image formed from the average in-transit and average out-of-
transit pixel images in order to calculate the position of the difference
image centroid. This difference image centroid position is compared with the
position of the PRF-fit centroid of the average out-of-transit image, as well
as to the predicted deviations in centroid locations due to scene crowding.
The results for all 5 planet candidates are presented in Table 1 and shown in
Figure 3.

Figure 3.— Centroid fitting results are shown for all 5 planet candidates of
KOI-707. The origin is the center of light received outside of transit times.
The red asterisk is the expected centroid of light during transit assuming the
transit is on the target star; it is displaced from the origin because some of
the light entering the aperture comes from known background stars whose
brightness would not be affected by transits of the target star. (The numbers
following the red asterisk are the KIC number and Kepler magnitude of
KOI-707.) The fitted centroids of the transit signatures observed in Quarters
1 through 8 are shown as the thin black crosses, where the arms of these
crosses show the uncertainty in RA and Dec. Some quarters as missing for some
candidates due to failure of the PRF fit. The robust average centroid across
quarters is shown by the thick black cross, with the black solid circle giving
the 3$\sigma$ uncertainty in the offset distance. The robust average of each
modeled quarterly offsets is shown as a thick gray cross, with the gray dashed
circle showing the average model 3$\sigma$ offset distance uncertainty. When
the circle showing the observed 3$\sigma$ uncertainty in the difference image
has significant intersection with the dashed circle showing the modeled
3$\sigma$ uncertainty in the position of the target star, the data are
consistent with the observed transit signal being on KOI-707.

The uncertainty in the quarterly centroid locations displayed in Figure 3 is
based upon propagating pixel-level uncertainty and does not include possible
PRF fit bias. Sources of PRF fit bias include scene crowding, because the fit
is of a single PRF that assumes a single star, and PRF error. The centroid
offset is the difference between the positions of the centroids of the
difference image and out-of-transit images, so common biases such as PRF error
should approximately cancel. There is, however, a quarter-dependent residual
bias. Statistical analysis across many uncrowded Kepler targets indicates that
the residual bias due to PRF error is zero-mean, so we average the quarterly
centroid positions. Specifically, we compute a robust $\chi^{2}$ minimizing
best-fit position to the quarterly position data. The uncertainties of the
average are estimated by taking the uncertainties of the fit and performing a
bootstrap analysis. This average for each planet candidate is shown on Figure
3.

Bias due to crowding (stars close enough to the target that they contribute
light observed in the aperture), however, does not cancel because, to the
extent that variations in other field stars are not correlated with transits,
field stars do not contribute to the difference image. This introduces a
systematic bias component that remains in the average over quarters. To
investigate the possibility that the observed offsets are due to crowding
bias, we model the local scene using stars from the KIC and the measured PRF,
induced the transit on KOI-707 with the ephemeris of each planet candidate,
and performed the same PRF fit analysis on the model difference and out-of-
transit images as that performed on the pixel data. If the resulting modeled
average offset matches the observed average offset, then the observed offset
can be explained by crowding bias.

The locations of the transit signatures (difference image centroids) of each
of KOI-707’s five planet candidates lie within $1^{\prime\prime}$ of the
predicted position, and given the uncertainties listed in Table 1, none of the
signatures are significantly offset and all are fully consistent with transits
on the target star. Furthermore, the lack of near neighbor stars seen in the
speckle image rules out some of the possible false positive scenarios that
could produce a centroid shift of the magnitude suggested by the marginally
significant offset observed.

###  4.2. Lightcurves, Spectra, Derived Properties

We obtained a spectrum of the target star KOI-707/Kepler-33 using the HiRES
spectrometer at Keck I 10-m telescope. Classification of this spectrum using
SME (Spectroscopy Made Easy, (Valenti & Piskunov, 1996; Valenti & Fischer,
2005)) yields the stellar parameter values shown in Table 2 (the final two
columns in this table show results from constrained fits described at the end
of this subsection). The mean stellar density based on spectroscopic
information was estimated by matching the determined distributions of $T_{\rm
eff},\log g$ and [Fe/H] to the Yonsei-Yale evolution tracks. This allowed the
determination of stellar mass and radius and mean stellar density posterior
distribution based on spectroscopy. The spectrum of KOI-707 is consistent with
two sets of stellar evolution tracks (Figure 4), leading to double-peaked
probability distributions and asymmetric uncertainties in several stellar
parameters. We plot the probability distributions of the fundamental
parameters of stellar mass and age in Figure 5 and those of the stellar
density, which is a key input to transit fitting, in Figure 6, and we include
in Table 2 various measures of uncertainty beyond the simple standard
deviation for the values of various derived stellar parameters that are either
critical to the validation and properties of KOI-707 and/or distributed highly
asymmetrically. Note that both solutions are somewhat above the main sequence,
implying an evolved star.

Figure 4.— Plotted are the Yonsei-Yale evolutionary tracks for 1.1 M⊙ (red),
1.2 M⊙ (green), 1.3 M⊙ (blue), 1.4 M⊙ (cyan) and 1.5 M⊙ (magenta) stars with
[Fe/H] = 0.15. Ages, in Gyr, are marked along the tracks. Note the uneven
rates of evolution along these tracks. The gray dotted tracks are spaced by
0.01 M⊙ in stellar mass and 0.02 Gyr in time to illustrate the distribution of
field stars on the $\log g-T_{\rm eff}$ plane; the apparent
gaps/discontinuities in these tracks are caused by rapid changes at various
stages of stellar evolution. The three boxes mark the 1, 2 and 3 $\sigma$
confidence regions of the spectroscopic SME analysis of the spectrum of
KOI-707 taken at Keck. There are degeneracies in determining a unique stellar
mass and radius for specified values of $T_{\rm eff}$ and $\log g$. For
example, a 4.4 Gyr, 1.3 M⊙ model and a 5.5 Gyr, 1.2 M⊙ model have nearly
identical $T_{\rm eff}$ and $\log g$. The low-mass/old peaks in the
probability distribution functions shown in Figure 5 result from the pile-up
of stellar evolution tracks as core H is exhausted in the transition regime
from radiative cores of 1.1 M⊙ and smaller stars to convective cores in 1.2 M⊙
and larger stars. The high-mass/young peaks result from the overlap between
tracks for the early shell H burning phase of stars with convective cores with
tracks for the late core H burning phase of stars that are $\sim
0.12~{}M_{\odot}$ more massive.  Figure 5.—  The relative probability
distribution of estimates for the mass and age of the star Kepler-33
(KOI-707), computed using the SME analysis of the star’s spectrum and a prior
based on the frequency distribution of stars, as described in the text. Both
curves are double-peaked because two solutions, both recently evolved off the
main sequence, are consistent with the data. Integration under the curves
implies that the younger, more massive, solution is a bit more probable than a
somewhat older and fainter star.  Figure 6.—  The relative probability
distribution of estimates for the density of the star Kepler-33 (KOI-707). The
top curve was computed using an MCMC analysis of the transit photometry data,
whereas the results shown in the bottom panel incorporated the SME analysis of
the star’s spectrum and a prior based on the frequency distribution of stars.
The spectral solution is highly asymmetric because two groups of stars are
consistent with the data (see Figure 4). Note the difference in horizontal
scales between the two panels.

As an independent check of the values of the SME parameters, we also derived
values by matching the spectrum to synthetic spectra (Torres et al., 2002;
Buchhave et al., 2010), and in addition we employ a new fitting scheme
(Stellar Parameter Classification, SPC) that is currently under development
and being readied for publication, allowing us to extract precise stellar
parameters from the spectra. The SPC analysis yielded the following
parameters: $T_{\rm eff}=5849\pm 50$ K, $\log g=4.07\pm 0.10$, [m/H]=$0.12\pm
0.08,v\sin i=3.4\pm 0.5$ km/s, RV = $13.779\pm 0.020$ km/s. The SME and SPC
parameters are fully consistent, and the nominal value of the key parameter
$\log g$ from SPC is slightly ($<1\sigma$) larger than the SME value. In
contrast, the derived stellar radius from SME is $1.82\pm 0.16~{}R_{\odot}$,
substantially larger (more than $3\sigma$ formal error, see Table 2) than the
radius estimate of 1.29 R⊙ based upon the KIC that was used by used by Borucki
et al. (2011).

We measured the radial velocity of KOI-707 relative to the barycenter of our
Solar System. The method involves measuring the Doppler shift of the stellar
spectrum obtained with the Keck I telescope and HiRES spectrometer on BJD =
2455782.9441 relative to the solar spectrum by a $\chi^{2}$ fit of the two
spectra. To set the zero point of the wavelength scale accurately, we measure
the displacement of telluric lines from the Earth’s atmosphere, serving as the
“iodine cell”. The resulting radial velocities are accurate to 0.1 km/s, based
on measurements of 2000 stars (Chubak & Marcy, 2012) and comparison with IAU
standard stars. For KOI-707, the radial velocity is $v_{\star}$ = 14.09 km/s,
indicating motion of the star away from the Solar System at a speed typical of
the velocity dispersion of stars in the Galactic disk. Orbital periods of the
planets quoted in Table 3 are as perceived from the barycenter of our Solar
System. As the Kepler-33 system is receding from us at $4.7\times 10^{-5}c$,
where $c$ represents the speed of light, the actual orbital periods in the
rest frame of the Kepler-33 system are 0.999953 times as large as the
tabulated values.

We used Q1-Q8 long-cadence Kepler aperture photometry (i.e., only pixel-level
corrections such as bias and dark smear are applied to the data and there is
no co-detrending). At this level of data reduction, there are still
instrumental artifacts on timescales similar to the planetary orbits. To
detrend the data (remove long-period variations), a second-order polynomial
was fitted to segments of sequential Kepler photometry data outside of
transits (i.e., in-transit observations were given zero weight), and then this
polynomial was subtracted from the data. A segment is defined as a length of
data that is uninterrupted for 5 or more consecutive long cadences ($\sim$2.5
hours). Each segment is then combined to produce a final time series that is
then normalized by the median.

We adopt a simple model to fit the data. We fit for the mean stellar density
($\rho_{\star}$), and each planet’s orbital period ($P_{\rm orb}$), transit
epoch ($T_{0}$), scaled planetary radius ($R_{p}/R_{\star}$) and impact
parameter ($b$). The planets are assumed to be on circular orbits; planet-
planet interactions are implicitly accounted for by allowing the center of
transit time to vary in order to best-fit transit shape (minimization of
scatter on residuals). The transit was described with the analytic formulae of
Mandel & Agol (2002), and we adopted a non-linear limb darkening law with
coefficients fixed based on the Claret & Bloemen (2011) tabulation for the
Kepler band with stellar parameters from SME analysis. A best fit model was
computed using a Levenberg-Marquardt algorithm to minimize the $\chi^{2}$
statistic and initial uncertainties were estimated via the construction of a
co-variance matrix.

The best fit model and co-variances were used to seed a hybrid-MCMC (Markov
Chain Monte-Carlo) algorithm to determine posterior distributions of our model
parameters. The model is considered hybrid, as we randomly use a Gibbs-sampler
and a buffer of previous chain parameters to produce proposals to jump to a
new location in the parameter space. The addition of the buffer allows for a
calculation of vectorized jumps that allow for efficient sampling of highly
correlated parameter space.

A direct comparison of the mean stellar density determined by the circular
transit-model (0.236 $\pm$ 0.080) and the mean stellar density determined from
spectroscopy (0.301 $\pm$ 0.067) differ by less than 1$\sigma$. This shows
that we can produce a self-consistent circular model in which all five planets
orbit the same star and produce stellar parameters that agree with
spectroscopy.

Close-ups of the lightcurves of each of the planets near planetary transit are
shown in Figure 7. Transit durations of all five KOIs vary in proportion to
the $P_{\rm orb}^{1/3}$ to within 5%, and to within 3% when planet sizes are
factored in as specified in Expression (10). The remarkable similarity of
these transit durations when appropriately scaled to orbital periods is
apparent in Figure 8, which shows the same data as Figure 7 but with the time
coordinate scaled by $P_{\rm orb}^{1/3}$. Such similarity would be the
consequence of an ensemble of planets transiting the same star on low
eccentricity orbits with similar impact parameters. The impact parameters
given in Table 3 are indeed quite similar for all of the planets. But the
inclinations implied to produce these parameters require a very fortuitous
arrangement in which tilt to the line of sight is just enough larger in the
inner planets than the outer ones. In contrast, if the system is viewed nearly
edge on, i.e., $i\approx 90^{\circ}$, then both a wider range of impact
parameters would be allowed (because transit durations are less sensitive to
the precise value of $b$ when $b\ll 1$) and similar values of $b$ imply a very
flat system rather than a fortuitous ensemble of tilts.

Figure 7.— Detrended Kepler flux from Kepler-33 shown phased at the period of
each transit signal and zoomed to a 30-hour region around mid-transit. Black
dots represent individual Kepler long cadence observations. The folded light
curves corresponding to the model fits are shown in colors that correspond to
these used in Figure 2. In each panel, the best-fit model for the other 4
planets was removed before plotting. Each panel has an identical vertical
scale, to show the relative depths, and identical horizontal scale, to show
the relative durations. Figure 8.— The folded lightcurves near the transit of
each planet in the Kepler-33 system, as shown in Figure 7, but in this case
the time coordinate in each panel has been scaled to $P_{\rm orb}^{1/3}$ of
the planet in question. Vertical lines mark the beginning and end times for
each transit. For planets on circular orbits with the same impact parameter,
durations normalized in this manner should be the same for all planets. The
remarkable similarity in normalized durations of these planets is very strong
evidence that they all orbit the same star.

We are thus motivated to perform a second set of fits constrained by the
assumptions of circular orbits for all planets and $b=0$ for the planet with
the longest normalized (according to Expression 10) transit duration,
KOI-707.01. Two such fits were performed, one for the low mass stellar
solution (where we imposed the constraint $M_{\star}<1.21~{}M_{\odot}$) and
the other for the high mass ($M_{\star}>1.21~{}M_{\odot}$) solution. We show
in the far right columns of Tables 2 and 3 both the high and low stellar mass
“Flat” fits; no uncertainties are quoted for these fits because the $b=0$
assumption constrains results so tightly that computed uncertainties are
unrealistically low.

The fitting procedure that we used to compute the planetary parameters listed
in Table 3 did not account for the small (1 – 2%) contribution of the second
star in the aperture to the flux measured by Kepler. This dilution implies
that the planet radii quoted are overestimated by 0.5 – 1%, a difference far
smaller than the uncertainties in their actual values.

###  4.3. Validation of the Kepler-33 Planetary System

The five planetary candidates of KOI-707 are extremely likely to represent a
true 5 planet system orbiting the target star. There are several lines of
evidence that support this conclusion. First, the analysis presented in
Section 2 implies that the overwhelming majority of Kepler multiple-planet
candidates are true planets and bound within the same physical system.

Second, the planetary system is already closely-packed dynamically for the
sizes of planets around an undiluted target star (see Section 4.4). Were the
planetary system to orbit another star that is not seen because it is
significantly fainter than the target, the larger planets required to match
the observations would necessitate implausibly low densities for the planetary
system to be stable.

Third, the spatial location analysis presented in Section 4.1, combined with
the galactic latitude distribution of targets and planet candidates shown in
Figure 1 and the relatively high galactic latitude of KOI-707 (15∘), together
imply that it is unlikely that the signals are due to a source not physically
associated with the target star.

Fourth, the long durations of the transits relative to the planets’ orbital
periods imply that they are produced by light being blocked from a star whose
density is significantly less than those of cool dwarfs, and/or they are
produced by apocentric transits of objects with substantial orbital
eccentricity. Substantial orbital eccentricities would make the system
unstable if the planets orbited the same star. It is highly implausible that
two stars in an unresolved stellar binary would be so similarly coeval to have
nearly equal densities characteristic of a somewhat evolved star.

Fifth, the remarkable similarity of the transit durations when appropriately
scaled to orbital periods (Expression (10) and Figure 8) can be understood as
the consequence of an ensemble of planets transiting the same star with
similar impact parameters, but would otherwise be an unlikely coincidence.

The above characteristics of KOI-707.01–05, combined with the overall high
validity of Kepler multis, imply that these candidates have a well above 99%
probability of composing a true multi-planet system, hence we consider these
candidates to be validated planets and name the system Kepler-33.

###  4.4. Characteristics of the Kepler-33 Planetary System

Numerical integrations to test stability of the system were performed using
masses of the planets from the nominal mass-radius relationship based upon
planets within our Solar System presented in Lissauer et al. (2011a). This
relationship is given by:

$M_{p}=\Big{(}\frac{R_{p}}{R_{\oplus}}\Big{)}^{2.06}M_{\oplus},$ (11)

where R⊕ and M⊕ are the radius and mass of the Earth, respectively. In
previous work, Lissauer et al. (2011a) integrated the KOI-707 system using
parameters based on the data presented by Borucki et al. (2011) and assuming
planar initially circular orbits; the model system survived intact for the
entire $3\times 10^{8}$ years simulated. All of our integrations also assume
that the system is planar. For all planets having zero initial eccentricity,
all three models of the system listed in Tables 2 and 3 survived intact for
the entire $10^{7}$ years simulated.

Of the three Kepler-33 planetary system models that we integrated, the one
based directly on the spectroscopic stellar parameters had the largest
planetary sizes and thus largest estimated planetary masses, with the
fractional differences in estimated planetary masses from one model to another
being larger than those for stellar mass. Thus, the spectroscopic parameters
yielded the largest interplanetary perturbations, and presumably the largest
TTV signatures. We examined distributions of TTV signals over 2 year periods
during the first 2 Myr of this simulation. These theoretical TTVs for the
inner planet were smaller than 30 seconds, and those for Kepler-33c less than
90 seconds, both values below current measurement capabilities. The outer
three planets exhibited far larger TTVs, 9 – 18 minutes for Kepler-33d, 23 –
38 minutes for Kepler-33e and 23 – 35 minutes for Kepler-33f, roughly
consistent with variations detected. While this model used relatively high
mass estimates, the integration began with circular orbits, which produce
smaller TTVs, so the results do not yield a preference between the three
models of planetary sizes listed in Table 3. A detailed analysis of planetary
TTVs is beyond the scope of this paper, but we note that it may well soon be
possible to derive estimates of the masses of some or all of the outer three
planets using observed TTVs.

The properties of the 5 planet Kepler-33 system are analogous to those of the
6 planet Kepler-11 system (Lissauer et al., 2011b). Both systems include 4
planets with orbital periods between 13 and 47 days that are comparable in
radius to, or somewhat smaller in size than, Neptune, as well as a smaller
planet closer to the star. Both systems are thus very closely-packed in a
dynamical sense. The Kepler-11 system is slightly more dynamically close-
packed than is the Kepler-33 system, both in the sense that the ratios of the
orbital periods of neighboring planets are a bit larger for Kepler-33’s
planets (Lissauer et al., 2011a) and as evidenced from smaller TTVs detected
in the Kepler-33 system.

In some aspects, the Kepler-33 system can be viewed as a less extreme cousin
of Kepler-11, with 5 known transiting planets rather than 6 and spacing that
not quite as close, yet dynamically much tighter than most systems (Lissauer
et al., 2011a). However, the new system’s inner planet, Kepler-33b, is smaller
and closer to its star than is its counterpart, Kepler-11b. The high density
of Kepler-11b relative to its brethren, coupled with its smaller size,
indicates a composition richer in heavy elements (Lissauer et al., 2011b).
While the mass and thus the density of Kepler-33b is not known, it is unlikely
that a planet this small ($1.5-1.8~{}R_{\oplus}$) and close (0.07 AU) to its
bright star (3 L⊙) could have retained a substantial H/He atmosphere. This
lends support to the suggestion by Lissauer et al. (2011b) that planets such
as Kepler-11b may have once had atmospheres of light gases that subsequently
escaped as a result of stellar irradiation. The slightly shorter normalized
duration of the small inner planet Kepler-33b suggests that this planet has
somewhat larger inclination and/or eccentricity than do the outer four
planets. We note that Mercury, the innermost and smallest planet in our Solar
System, also has the largest eccentricity and inclination of the 8 planets
known to orbit our Sun.

Stability of the Kepler-33 planetary system excludes large eccentricities of
the four outer planets, although interplanetary perturbations must cause
eccentricities to be non-zero. The remarkable similarity in the normalized
transit durations of the five planets orbiting Kepler-33 would be very
unlikely if the eccentricities and inclinations of planetary orbits are
significant (Figure 9). Moreover, the consequent similarity in computed impact
parameters despite the planets’ differing orbital distances requires that
either the system is viewed near edge-on, with impact parameters $b\ll 1$, or
a very fortuitous coincidence producing an anticorrelation between
inclinations and orbital periods. We view such a coincidence as unlikely, and
therefore prefer the planetary parameters closer to those listed in final two
columns of Tables 2 and 3 (the “Flat” fits) than to those listed in the second
column of Tables 2 and 3.

Figure 9.— The probability of observing all five planets in the Kepler-33
system to have period-normalized transit durations within 3.0% of one another
given that all five planets are seen to transit the host star is shown as a
function of mean mutual inclination. Points were derived using Monte Carlo
simulations that assumed parameters from the spectroscopic fit given in the
second column of Tables 2 and 3. Results with the flat fit parameters (not
shown) are nearly identical. Mutual inclinations for each realization are
drawn from a Rayleigh distribution with respect to random reference plane
(i.e., random observer). The black dots are for the $e=0$ case. The blue
triangles, yellow diamonds and red squares show results from simulations in
which the eccentricity assigned to each planet was drawn form a Rayleigh
distribution with mean value $<e>~{}=0.01$, $<e>~{}=0.03$ and $<e>~{}=0.05$,
respectively. The addition of even a small range of eccentricities greatly
reduces probability of highly correlated transit durations.

##  5\. Conclusions

The calculations presented in Section 2 show that the vast majority of
Kepler’s multi-planet candidates are indeed planets. The essence of our
argument is that fewer than 10-2 of Kepler targets have a planet candidate
listed in Borucki et al. (2011). If 10% or fewer of these candidates are FPs,
then fewer than 10-3 of Kepler targets have a false positive, and if FPs are
not correlated with other planet candidates, then fewer than 10-5 of Kepler’s
$\sim$ 160,000 targets, i.e., a total of at most 2 targets, are expected to
have a false positive as well as another transit-like signature that makes
them a candidate multi-planet system. This number can be compared to 170
candidate multi-planet systems with a total of 408 planet candidates announced
by Borucki et al. (2011). Accurate estimates of the false positive rate in
this population require detailed analysis of both Kepler and ground-based data
to account for the caveats expressed in Section 2.2. Nonetheless, we expect
that $\gtrsim 400$ of the 408 planet candidates in multis that were announced
by Borucki et al. (2011) are indeed exoplanets, although a small fraction of
these planets orbit stars other than their nominal Kepler target star. Note
that the smallest planet candidates are most prone to false positives, because
low SNR limits the restrictiveness in parameter space of tests of lightcurve
shape and location of the centroid of the transit signal on the sky plane, and
the abundance of astrophysical signals that are capable of producing lower
amplitude events is greater. Also note that stellar parameters, and thus
planetary sizes, are prone to large uncertainties for candidates that have no
high-SNR spectroscopic observations from the ground.

From a Bayesian statistical perspective, multiplicity effectively allows us to
increase the planet prior, and thus enables the validation of candidates as
being true planets at a specified high probability level without going to the
extreme lengths often required to provide a very small upper bound on the FP
prior. We use these arguments along with a detailed analysis of Kepler data to
validate the Kepler-11 analog KOI-707 system, which we name Kepler-33.

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, chief among them Bill Borucki, who has devoted decades to
developing and implementing Kepler. Useful comments were provided by Natalie
Batalha, Ruth Murray-Clay, Dimitar Sasselov, Jason Steffen, and especially by
Tim Brown. Kevin Zahnle and Mark Marley provided constructive comments on the
manuscript. D. C. F. acknowledges NASA support through Hubble Fellowship grant
#HF-51272.01-A, awarded by STScI, operated by AURA under contract NAS 5-26555.

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Table 1Centroid offsets for KOI-707 based on PRF fits. Planet | Transit Offset from KOI-707 | Transit Offset from Prediction
---|---|---
| (arcsec) | (Offset/$\sigma$) | (arcsec) | (Offset/$\sigma$)
.01 | $0.17\pm 0.29$ | $0.59$ | $0.39\pm 0.26$ | $1.50$
.02 | $0.30\pm 0.22$ | $1.35$ | $0.53\pm 0.22$ | $2.38$
.03 | $0.25\pm 0.39$ | $0.63$ | $0.65\pm 0.40$ | $1.64$
.04 | $0.37\pm 0.93$ | $0.39$ | $0.50\pm 0.67$ | $0.75$
.05 | $1.06\pm 0.42$ | $2.53$ | $0.71\pm 0.48$ | $1.47$
Table 2 Characteristics of the Star Kepler-33  Parameter | Median | $\sigma$ | $+1\sigma$ | $-1\sigma$ | $+3\sigma$ | $-3\sigma$ | Flat, $M_{\star}>1.2~{}M_{\odot}$ | Flat, $M_{\star}<1.2~{}M_{\odot}$ |
---|---|---|---|---|---|---|---|---|---
Spectroscopy |  |  |  |  |  |  |  |  |
$M_{\star}$ (M⊙) | 1.291 | 0.082 | 0.063 | -0.121 | 0.212 | -0.200 | 1.271 | 1.164 |
$R_{\star}$ (R⊙) | 1.82 | 0.16 | 0.14 | -0.18 | 0.49 | -0.42 | 1.663 | 1.615 |
$Z$ | 0.0250 | 0.0019 |  |  |  |  | 0.0248 | 0.0241 |
$T_{\rm eff}$ (K) | 5904 | 47 |  |  |  |  | 5899 | 5880 |
log($L_{\star}$/L⊙) | 0.556 | 0.080 |  |  |  |  | 0.476 | 0.446 |
Age (Gyr) | 4.27 | 0.87 | 0.74 | -1.03 | 2.68 | -1.65 | 4.31 | 5.92 |
log$g$ (cm2/s) | 4.027 | 0.056 |  |  |  |  | 4.0994 | 4.087 |
$v_{\star}$ (km/s) | 14.09 | 0.1 |  |  |  |  |  |  |
$\rho_{\star}$ (g/cm3) | 0.300 | 0.066 | 0.049 | -0.079 | 0.230 | -0.147 | 0.3883 | 0.3885 |
Transit Model |  |  |  |  |  |  |  |  |
$\rho_{\star}$ (g/cm3) | 0.288 | 0.081 | 0.091 | -0.066 | 0.112 | -0.215 | 0.03897 | 0.3897 |
Table 3 Characteristics of the Kepler-33 Planets  Parameter | Median | $\sigma$ | $+1\sigma$ | $-1\sigma$ | $+3\sigma$ | $-3\sigma$ | Flat, $M_{\star}>1.2~{}M_{\odot}$ | Flat, $M_{\star}<1.2~{}M_{\odot}$
---|---|---|---|---|---|---|---|---
Kepler-33b = KOI-707.05 |  |  |  |  |  |  |  |
$T_{0}$ (BJD-2454900) | 64.8981 | 0.0075 |  |  |  |  | 64.8973 | 64.8973
$P_{\rm orb}$ (days) | 5.66793 | 0.00012 |  |  |  |  | 5.667939 | 5.667939
$b$ | 0.50 | 0.16 | 0.16 | -0.20 | 0.34 | -0.41 | 0.298 | 0.298
$R_{p}/R_{\star}$ | 0.00877 | 0.00046 |  |  |  |  | 0.00853 | 0.00853
$R_{p}$ (R⊕) | 1.74 | 0.18 |  |  |  |  | 1.549 | 1.504
$i$ (degrees) | 86.39 | 1.17 | 0.96 | -1.62 | 2.90 | -3.16 | 88.03 | 88.03
$a/R_{\star}$ | 7.87 | 0.87 |  |  |  |  | 8.714 | 8.714
$a$ (AU) | 0.0677 | 0.0014 |  |  |  |  | 0.06740 | 0.06544
Trdepth (ppm) | 86.8 | 6.8 |  |  |  |  | 87.7 | 87.7
Trdur (hr) | 4.88 | 0.16 | 0.16 | -0.15 | 0.45 | -0.54 | 4.80 | 4.80
Kepler-33c = KOI-707.04 |  |  |  |  |  |  |  |
$T_{0}$ (BJD-2454900) | 76.6764 | 0.0042 |  |  |  |  | 76.6777 | 76.6777
$P_{\rm orb}$ (days) | 13.17562 | 0.00014 |  |  |  |  | 13.17563 | 13.17563
$b$ | 0.44 | 0.17 | 0.17 | -0.21 | 0.38 | -0.38 | 0.144 | 0.144
$R_{p}/R_{\star}$ | 0.01602 | 0.00057 |  |  |  |  | 0.01560 | 0.01560
$R_{p}$ (R⊕) | 3.20 | 0.30 |  |  |  |  | 2.833 | 2.751
$i$ (degrees) | 88.19 | 0.72 | 0.58 | -1.06 | 1.55 | -1.92 | 89.46 | 89.46
$a/R_{\star}$ | 13.8 | 1.5 |  |  |  |  | 15.292 | 15.292
$a$ (AU) | 0.1189 | 0.0025 |  |  |  |  | 0.11287 | 0.11484
Trdepth (ppm) | 297.7 | 9.1 |  |  |  |  | 299.8 | 299.8
Trdur (hr) | 6.700 | 0.104 | 0.096 | -0.107 | 0.325 | -0.295 | 6.616 | 6.616
Kepler-33d = KOI-707.01 |  |  |  |  |  |  |  |
$T_{0}$ (BJD-2454900) | 122.6342 | 0.0018 |  |  |  |  | 122.6341 | 122.6341
$P_{\rm orb}$ (days) | 21.77596 | 0.00011 |  |  |  |  | 21.775984 | 21.7759
$b$ | 0.44 | 0.17 | 0.14 | -0.23 | 0.38 | -0.40 | 0 (assumed) | 0 (assumed)
$R_{p}/R_{\star}$ | 0.02667 | 0.00087 |  |  |  |  | 0.02589 | 0.02589
$R_{p}$ (R⊕) | 5.35 | 0.49 |  |  |  |  | 4.700 | 4.564
$i$ (degrees) | 88.71 | 0.51 | 0.61 | -0.48 | 1.08 | -1.37 | 90 (assumed) | 90 (assumed)
$a/R_{\star}$ | 19.3 | 2.1 |  |  |  |  | 21.38 | 21.38
$a$ (AU) | 0.1662 | 0.0035 |  |  |  |  | 0.16533 | 0.16053
Trdepth (ppm) | 831.4 | 10.5 |  |  |  |  | 831 | 831
Trdur (hr) | 8.011 | 0.097 | 0.065 | -0.103 | 0.345 | -0.192 | 7.987 | 7.987
Kepler-33e = KOI-707.03 |  |  |  |  |  |  |  |
$T_{0}$ (BJD-2454900) | 68.8715 | 0.0048 |  |  |  |  | 68.8720 | 68.8720
$P_{\rm orb}$ (days) | 31.78440 | 0.00039 |  |  |  |  | 31.78438 | 31.78438
$b$ | 0.47 | 0.16 | 0.14 | -0.21 | 0.38 | -0.37 | 0.204 | 0.204
$R_{p}/R_{\star}$ | 0.02011 | 0.00072 |  |  |  |  | 0.01955 | 0.01955
$R_{p}$ (R⊕) | 4.02 | 0.38 |  |  |  |  | 3.550 | 3.446
$i$ (degrees) | 88.94 | 0.37 | 0.45 | -0.35 | 0.84 | -1.07 | 89.576 | 89.576
$a/R_{\star}$ | 24.9 | 2.8 |  |  |  |  | 27.51 | 27.51
$a$ (AU) | 0.2138 | 0.0045 |  |  |  |  | 0.2127 | 0.20656
Trdepth (ppm) | 467 | 12 |  |  |  |  | 469 | 469
Trdur (hr) | 8.90 | 0.13 | 0.12 | -0.13 | 0.40 | -0.38 | 8.819 | 8.819
Kepler-33f = KOI-707.02 |  |  |  |  |  |  |  |
$T_{0}$ (BJD-2454900) | 105.5763 | 0.0040 |  |  |  |  | 105.5757 | 105.5757
$P_{\rm orb}$ (days) | 41.02902 | 0.00042 |  |  |  |  | 41.02903 | 41.02903
$b$ | 0.43 | 0.17 | 0.15 | -0.23 | 0.39 | -0.42 | 0.130 | 0.130
$R_{p}/R_{\star}$ | 0.02227 | 0.00076 |  |  |  |  | 0.02171 | 0.02171
$R_{p}$ (R⊕) | 4.46 | 0.41 |  |  |  |  | 3.941 | 3.827
$i$ (degrees) | 89.17 | 0.34 | 0.33 | -0.42 | 0.68 | -0.98 | 89.772 | 89.772
$a/R_{\star}$ | 29.5 | 3.3 |  |  |  |  | 32.61 | 32.61
$a$ (AU) | 0.2535 | 0.0054 |  |  |  |  | 0.2522 | 0.2449
Trdepth (ppm) | 579 | 12 |  |  |  |  | 581 | 581
Trdur (hr) | 9.87 | 0.13 | 0.11 | -0.15 | 0.43 | -0.36 | 9.732 | 9.732